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For "perimeter" you could think: how long would a piece of string be to go around the edge of the circle (or square, or whatever shape).
For area, you could think how many brush strokes you would need to paint inside the circle (or square, etc).
To help you remember π, have a look at this
We also have a page on perimeters and areas here
Is any of that a help?
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The PI Section Line block implements a single-phase transmission line with parameters lumped in PI sections.
For a transmission line, the resistance, inductance, and capacitance are uniformly distributed along the line. An approximate model of the distributed parameter line is obtained by cascading several
identical PI sections, as shown in the following figure.
Unlike the Distributed Parameter Line block, which has an infinite number of states, the PI section linear model has a finite number of states that permit you to compute a linear state-space model.
The number of sections to be used depends on the frequency range to be represented.
An approximation of the maximum frequency range represented by the PI line model is given by the following equation:
N Number of PI sections
v Propagation speed (km/s) = ; l in H/km, c in F/km
ltot Line length (km)
For example, for a 100 km aerial line having a propagation speed of 300,000 km/s, the maximum frequency range represented with a single PI section is approximately 375 Hz. For studying interactions
between a power system and a control system, this simple model could be sufficient. However for switching surge studies involving high-frequency transients in the kHz range, much shorter PI sections
should be used. In fact, you can obtain the most accurate results by using a distributed parameters line model.
│ Note The Powergui block provides a graphical tool for the calculation of the resistance, inductance, and capacitance per unit length based on the line geometry and the conductor │
│ characteristics. │
Hyperbolic Correction of RLC Elements
Let us assume the following line parameters:
r Resistance per unit length (Ω/km)
l Inductance per unit length (H/km)
c Capacitance per unit length (F/km)
f Frequency (Hz)
lsec Line section length = ltot / N (km)
For short line sections (approximately lsec <50 km) the RLC elements for each line section are simply given by:
However, for long line sections, the RLC elements given by the above equations must be corrected in order to get an exact line model at a specified frequency. The RLC elements are then computed using
hyperbolic functions as explained below.
Per unit length series impedance at frequency f is
Per unit length shunt admittance at frequency f is
Characteristic impedance is
Propagation constant is
Hyperbolic corrections result in RLC values slightly different from the non corrected values. R and L are decreased while C is increased. These corrections become more important as line section
length is increasing. For example, let us consider a 735 kV line with the following positive-sequence and zero-sequence parameters (these are the default parameters of the Three-Phase PI Section Line
block and Distributed Parameter Line block):
Positive sequence r = 0.01273 Ω/km
l = 0.9337×10^−3 H/km
c = 12.74×10^−9 F/km
Zero sequence r = 0.3864 Ω/km
l = 4.1264×10^−3 H/km
c = 7.751×10^−9 F/km
For a 350 km line section, noncorrected RLC positive-sequence values are:
Hyperbolic correction at 60 Hz yields:
For these particular parameters and long line section (350 km), corrections for positive-sequence RLC elements are relatively important (respectively −6.8%, −3.4%, and + 1.8%). For zero-sequence
parameters, you can verify that even higher RLC corrections must be applied (respectively −18%, −8.5%, and +4.9%).
The PI Section Line block always uses the hyperbolic correction, regardless of the line section length.
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The digit at the 1st decimal place shows the number of Tenths. In the number above, there are 3 tenths.
The 2nd decimal place shows the number of Hundredths. In that number, there are 8 hundredths. And so on.
The first digit to the left of the decimal point -- the 2 -- shows the number of Ones.
As with whole numbers, each digit has its place value:
2.3875. . .
Ones, tenths, hundredths, thousandths, ten-thousandths, . . .
As with the whole numbers, each place is ten times the place to its right.
One is ten tenths.
One-tenth is ten hundredths.
One-hundredth is ten thousandths.
One-thousandth is ten ten-thousandths.
And so on.
Example 1. In this number
a) there are how many ones?
Answer. 4. The ones place is the first digit to the left of the decimal point.
b) How many hundredths?
Answer. 6. Hundredths (ordinal) is a decimal unit. It falls to the right of the decimal point.
c) How many hundreds?
Answer. 5. Hundred (cardinal) is a whole unit. It falls to the left of the decimal point.
Example 2. Expanded form. Just as the numeral for every whole number stands for a sum (Lesson 2), so does the numeral for every decimal. Here is the expanded form of
534.267 = 5 Hundreds + 3 Tens + 4 Ones + 2 Tenths + 6 Hundredths +
7 Thousandths.
At this point, please "turn" the page and do some Problems.
Continue on to Section 2.
Introduction | Home | Table of Contents
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2014 Lawrence Spector
Questions or comments?
E-mail: themathpage@nyc.rr.com
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Determining Inductance/Capacitance from S-Parameters
Newbie level 1
Join Date
Jun 2007
0 / 0
s-parameter inductance
I'm trying to determine the inductance and capacitance of chip inductors/capacitors at 1-2 GHz from the S-Parameters (S11 & S21) measured on an VNA. Does anyone know what equations/methods I
should use to get these values. Thank you for your help!
Advanced Member level 1
Join Date
May 2004
Heart of Europe
58 / 58
capacitance from s parameters
1. Find an appropriate model including the most important parasitics
2. Find some good starting component values for the I/C and their parasitics
3. Optimize the component value and parasitics (preferrably automatically) to the S-parameters match the model as good as possible.
A good starting point for inductors would be the Spice models of the Coilcraft inductors. You should be able to find something similar for the caps, but usually a series RLC circuit works
1 members found this post helpful.
Advanced Member level 3
Join Date
Aug 2005
228 / 228
s parameter inductance
Hello asubraman,
Import or read your S-pareameter file to any simulator (AWR MWO or Sonnet EM)
then plot the Y parameters Then use the following equation to get the Capacitance & Indutance values
Y11 = Im(Y(1,1))
C1=1.0E12 / ( 2 * _PI * _FREQ * ( 1 / Y11 ) )
L1=1.0E9 * ( 1 / Y11 ) / ( 2 * _PI * _FREQ )
L1: Effective Inductance (in nH) of a series RL network
C1: Effective Capacitance (in pF) of a series RC network
_FREQ: is project frequencies in GHz
Note: Sonnet has direct measurement option for the above...
1 members found this post helpful.
Junior Member level 2
Join Date
Jan 2010
1 / 1
Re: Determining Inductance/Capacitance from S-Parameters
dear manju,
thank you so much for the above post, it really helped me. can u tell me whether in this case, the resistance would be Re(Z(1 1)) or Re(1/Y(1 1))?
Advanced Member level 4
Join Date
Aug 2008
Hyderabad, Andhra Pradesh, India
243 / 243
Determining Inductance/Capacitance from S-Parameters
both should be equal by theory.
Advanced Member level 3
Join Date
Aug 2005
228 / 228
Re: Determining Inductance/Capacitance from S-Parameters
I hope this helps you...see the attached document...
2 members found this post helpful.
Junior Member level 2
Join Date
Jan 2010
1 / 1
Determining Inductance/Capacitance from S-Parameters
Dear Manju,
In the calculation of L1, the formula you have used.... is it 1/imag(Y(1 1)) or should it be imag(1/Y(1 1))? Please reply soon... Thanks so much!
Advanced Member level 3
Join Date
Aug 2005
228 / 228
Re: Determining Inductance/Capacitance from S-Parameters
Hello acekas,
it is imag(1/Y11)...
Also Note that on your query Resistance calculation;
If you are looking for the equivalent series resistance then
Z11 is only equal to 1/Y11 for a one-port (or if you short circuit port 2 device).
When you have a two-port, Y and Z are matrices and you cannot just take the reciprocal of each element in the matrix.
so whatever kspalla told was theoretically not true...
I hope this helps you...!!!
1 members found this post helpful.
Junior Member level 2
Join Date
Jan 2010
1 / 1
Determining Inductance/Capacitance from S-Parameters
Thanks Manju, yeah, that helped!
Regarding what you said about short circuiting port 2, can you please tell me how I can do that on cadence?
Further, isn't the expression imag(1/Y(1 1))... valid only for circuits with the second port short-circuited? If not, I would appreciate it tremendously if you could throw some light on how
we are arriving at the expression!
Thank you once again!
10. 3rd August 2013, 07:17 #10
Full Member level 3
Join Date
Feb 2012
Tamilnadu, India
22 / 22
Re: Determining Inductance/Capacitance from S-Parameters
Thanks Manju, yeah, that helped!
Regarding what you said about short circuiting port 2, can you please tell me how I can do that on cadence?
Further, isn't the expression imag(1/Y(1 1))... valid only for circuits with the second port short-circuited? If not, I would appreciate it tremendously if you could throw some light on how
we are arriving at the expression!
Thank you once again!
can i do it in cadence?
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Solving applications: Systems of three Equations
October 8th 2010, 09:01 PM #1
Oct 2010
Solving applications: Systems of three Equations
I have this word problem that i am having trouble on i am usually good with word problems but my instructor has completely caught me off guard with this one:
The problem is:
The sum of three numbers is 85. The second is 7 more than the first. The third is 2 more than four times the second. Find the numbers.
So far what i have is:
Let x = the first number
let y = the second number
let z = the third number
This is where i am stuck. I have read the chapter multiple times and have yet to come up with the correct answer. Am i writing the correct equations for the problem given? If so, what step should
i take about solving this? Please help thanks in advanced.
I have this word problem that i am having trouble on i am usually good with word problems but my instructor has completely caught me off guard with this one:
The problem is:
The sum of three numbers is 85. The second is 7 more than the first. The third is 2 more than four times the second. Find the numbers.
So far what i have is:
Let x = the first number
let y = the second number
let z = the third number
y=2+x The second number is 7 more than the first!!
This is where i am stuck. I have read the chapter multiple times and have yet to come up with the correct answer. Am i writing the correct equations for the problem given? If so, what step should
i take about solving this? Please help thanks in advanced.
The sum of three numbers is 85 .
So x+y+z=85.....(I)
The second is 7 more than the first
so, y = x+7....(II)
The third is 2 more than four times the second
s0, z = 2+4y....(III)
substitute (II) and (III) in (I) and find x,y, and z
So, x+y+z=85
or, x+(x+7)+(2+4y)=85
or, x+(x+7)+(2+4(x+7))=85
okay got it thanks for your help 8,15,62 this just made life like a million times easier
October 8th 2010, 09:07 PM #2
October 8th 2010, 09:25 PM #3
Oct 2010
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CRpuzzles Logic Problem Solution - The Good Sport
Logic Puzzle # 154 Logic Problems Help
Logic Problem Solution:
The Good Sport
From the introduction, no two donations Buzz made to sports teams were for the same amount of money, and the six donations totaled $1500. By clue 1, Buzz gave the basketball team twice as much as
the Knights, to whom he gave $50 more than to the Devils (clue 8); and by clue 6, Buzz gave the Hawks twice as much as the soccer team, to which he donated $50 more than to the Rovers (3). Between
the two clues, then, either all six teams are named or there is some commonality. If all six donations are listed, then either the Devils (1, 8) or the Rovers (6, 3) would have gotten the least, $50
(5). If the Devils had received the $50, by clues 8 and 1, the Knights would have received $100 and the basketball team $200. Then the three teams in clues 3 and 6 would have received $1500-$350, or
$1150. Adding the amounts in the two clues with the Rovers getting X, the soccer club would have gotten X+$50 and the Hawks 2X+$100, a total of 4X+$150. So, solving, 4X+$150 = $1150, or X would
equal $250. But the Hawks then would have received $600 from Buzz, contradicting clue 7. If the Rovers had received the $50 from Buzz, then the Devils would have had to get more than $50 and the
basketball team would have received the largest donation--no (9). Therefore, there must be some overlap between the two sets of teams in clues 1, 8 and 6, 3. The Hawks aren't the basketball team and
the Knights aren't the soccer team, since then the Devils and Rovers would have received the same sum, contrary to the introduction. If the Devils were the soccer team, then combining the four
clues, given that the basketball team didn't get the biggest amount (9) and that the Rovers therefore got the least, $50, the Devils would have received $100, the Knights $150, the Hawks $200, and
the basketball team $300, a total of $800. But the sixth team would have received $700, contradicting clue 7. So, the Rovers must be the basketball team. If the Devils weren't the $50 donees, then
the five teams amomng the four clues would have received $1450 as follows: the Devils X, the Knights X+$50, the basketball Rovers 2X+$100, the soccer team 2X+$150, and the Hawks 4X+$300. Solving, 10
X+$600 = $1450, or X = $85. But the Hawks then would have gotten $640--no (7). So, combining the clues, Buzz gave the Devils $50, the Knights $100, the basketball Rovers $200, the soccer squad $250,
and the Hawks $500, a total of $1100, with the sixth team then getting $400. By clue 4, the soccer team is the Lions, and the Knights play baseball. By elimination, the sixth team that got $400 is
the Comets. By clue 7, the Devils play hockey. The Hawks play football and the Comets softball (2). In sum, Buzz's deductible sports contributions in 2001 were as follows:
• $500 to the Hawks football team
• $400 to the Comets softball team
• $250 to the Lions soccer team
• $200 to the Rovers basketball team
• $100 to the Knights baseball team
• $50 to the Devils hockey team
CRpuzzles.com. Copyright © 2000-2007 by Calvin J. Hamilton & Randall L. Whipkey. All rights reserved. Privacy Statement.
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Determine if a matric is diagonalizable and diagonlize it
Okay, you have eigenvector [1, -1] corresponding to eigenvalue -3 and eigenvector [4, 1] corresponding to eigevalue 1. Those are correct.
You then put them together to form matrix "P" (you have it labeled [itex]\vec{V}[/itex] which is incorrect- this is a matrix, not a vector.)
[tex]\begin{bmatrix}1 & 4 \\ -1 & 1\end{bmatrix}[/tex] and declare that it this is incorrect. Why? You need to understand that there are, in fact, an infinite number of different matrices, P, so
that, for this matrix A, [itex]P^{-1}AP[/itex] is diagonal. The matrix which you say is incorrect is perfectly correct. Using it as P will give you the diagonal matrix
[tex]\begin{bmatrix}-3 & 0 \\ 0 & 1\end{bmatrix}[/tex]
Using the matrix that you say is correct,
[tex]\begin{bmatrix}4 & 1 \\ 1 & -1\end{bmatrix}[/tex]
has the two columns (eigenvectors) reversed and so gives
[tex]\begin{bmatrix}1 & 0 \\ 0 & -3\end{bmatrix}[/tex]
which is also a diagonal matrix, just the eigenvalues in different places.
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Properties of Recurrent Equations for the Full-Availability Group with BPP Traffic
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 547909, 17 pages
Research Article
Properties of Recurrent Equations for the Full-Availability Group with BPP Traffic
Communication and Computer Networks, Faculty of Electronics and Telecommunications, Poznan University of Technology, ul. Polanka 3, 60-965 Poznan, Poland
Received 27 April 2011; Accepted 1 August 2011
Academic Editor: Yun-Gang Liu
Copyright © 2012 Mariusz Głąbowski et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The paper proposes a formal derivation of recurrent equations describing the occupancy distribution in the full-availability group with multirate Binomial-Poisson-Pascal (BPP) traffic. The paper
presents an effective algorithm for determining the occupancy distribution on the basis of derived recurrent equations and for the determination of the blocking probability as well as the loss
probability of calls of particular classes of traffic offered to the system. A proof of the convergence of the iterative process of estimating the average number of busy traffic sources of particular
classes is also given in the paper.
1. Introduction
Dimensioning and optimization of integrated networks, that is, Integrated Services Digital Networks (ISDN) and Broadband ISDN (B-ISDN) as well as wireless multiservice networks (e.g., UMTS), have
recently developed an interest in multirate models [1–5]. These models are discrete models in which it is assumed that the resources required by calls of particular traffic classes are expressed as
the multiple of the so-called Basic Bandwidth Units (BBUs). The BBU is defined as the greatest common divisor of the resources demanded by all call streams offered to the system [6, 7].
Multirate systems can be analysed on the basis of statistical equilibrium equations resulting from the multidimensional Markov process that describe the service process in the considered systems [8–
13]. Such an approach, however, is not effective because of the quickly increasing—along with the system's capacity—number of states in which a multidimensional Markov process occurring within the
system can take place [14]. Consequently, for an analysis of multirate systems, there are used methods based on the convolution algorithm [11, 15] and the recurrent methods in which the
multidimensional service process—occurring in the considered systems—is approximated by one-dimensional Markov chain [16–21]. The convolution methods allow us to determine exactly the occupancy
distribution in the so-called full-availability systems servicing traffic streams with arbitrary distributions (i.e., systems with state-independent admission process and with both state-independent
and state-dependent arrival processes). In the case of the systems with state-dependent admission process (i.e., the system in which the admission of a new call is conditioned not only by the
sufficient number of free BBUs but also by the structure of the system and the introduced admission policy) the convolution methods lead to elaboration of approximate methods with quite high
computational complexity [22, 23].
Nowadays, in the analysis and optimization of multirate systems, the recurrent algorithms are usually used. This group of algorithms is based on the approximation of the multidimensional service
process in the considered system by the one-dimensional Markov chain. Such approach leads to a determination of the occupancy distribution in systems with state-independent admission process and
state-independent arrival process (in teletraffic engineering such system is called the full-availability group with Erlang traffic streams) on the basis of simple Kaufman-Roberts recurrence [24, 25]
and its modifications [16–19, 26, 27]. One of them, the so-called Delbrouck recurrence [18], allows us to determine the occupancy distribution in the system with state-independent admission process
(the full-availability group) and BPP traffic streams. The research on the full-availability group model, started by Delbrouck, was subsequently continued, for example, in [12, 28–30].
Because of the simplicity of the Kaufman-Roberts equation, in many works the attempts of its modification in order to analyse the systems with BPP traffic were undertaken. In [13] the modified form
of the Kaufman-Roberts equation that makes the value of offered traffic dependent on the number of active sources was presented. In [31] the approximation of the number of active sources with their
mean values in relation to the total value of occupied resources in particular states of the system was proposed. In [32], on the basis of the method proposed in [31], the Kaufman-Roberts equation
was generalized for systems with BPP traffic and state-dependent call admission process. The accuracy of the method for modelling systems with multirate BPP traffic—further on called the Multiple
Iteration Method-BPP (MIM-BPP)—proposed in [32] was verified in simulations for systems with both state-independent and state-dependent call admission process. In publications issued so far, no
attempt to formally prove the correctness of the MIM-BPP assumptions was taken up.
The aim of this paper is to formally prove that the MIM-BPP algorithm [32], considered earlier as an approximate algorithm, is exact. To this purpose we derive recurrent equations describing the
occupancy distribution in the full-availability group with multirate BPP traffic. We are going to demonstrate at the same time that the number of calls of particular Engset and Pascal classes
appearing in equations that determine the occupancy distribution is exactly determined with their average values. Additionally, we intend to prove the convergence of the iterative process of
estimating the average number of busy traffic sources of particular classes.
The paper is organized as follows. Section 2 presents an analysis of the call admission and the call arrival process in the full-availability group with BPP traffic at the micro- and macrostate
level. In Section 3 an iterative method for estimating the average number of busy traffic sources of particular classes is presented, and its convergence is proved. The paper ends with a summary
contained in Section 4.
2. Full-Availability Group with BPP Traffic
2.1. Basic Assumptions
Let us consider a model of the full-availability group with the capacity of BBUs (Figure 1). The group is offered traffic streams of three types: Erlang streams (Poisson distribution of call streams)
from the set , Engset streams (binomial distribution of call streams) from the set , and Pascal streams (negative binomial distribution of call stream) from the set . In the paper it has been adopted
that the letter “” denotes any class of Erlang traffic, letter “” any class of Engset traffic, and letter “” any class of Pascal traffic, whereas the letter “” any traffic class. (In relation to the
ITU-T recommendations [11], all types of discussed traffic are defined collectively by the term BPP traffic. Thus, we use the term BPP when we talk about all traffic types cumulatively, whereas when
we consider single traffic streams, then, because our study is focused on systems with limited capacity only, we use the terms Erlang, Engset, and Pascal streams.) The number of BBUs demanded by
calls of class is denoted by .
The call arrival rate for Erlang traffic of class is equal to . The parameter determines the call intensity for the Engset traffic stream of class , whereas the parameter determines the call
intensity for Pascal traffic stream of class . The arrival rates and depend on the number of and of currently serviced calls of class and . In the case for Engset stream, the arrival rate of class
stream decreases with the number of serviced traffic sources: where is the number of Engset traffic sources of class , while is the arrival rate of calls generated by a single free source of class .
In the case of Pascal stream of class , the arrival rate increases with the number of serviced sources: where is the number of Pascal traffic sources of classes , while is the arrival rate of calls
generated by a single free source of class .
The total intensity of Erlang traffic of class offered to the group amounts to whereas the intensity of Engset traffic and Pascal traffic of class and , respectively, offered by one free source, is
equal to In (2.3) and (2.4) the parameter is the average service intensity with the exponential distribution.
2.2. The Multidimensional Erlang-Engset-Pascal Model at the Microstate Level
Let us consider now a fragment of the multidimensional Markov process in the full-availability group with the capacity of BBUs presented in Figure 2. The group is offered traffic streams of three
types: Erlang, Engset, and Pascal. Each microstate of the process is defined by the number of serviced calls of each of the classes of offered traffic, where denotes the number of serviced calls of
the Poisson stream of class (Erlang traffic), denotes the number of serviced calls of the binomial stream of class (Engset traffic), whereas determines the number of serviced calls of the negative
binomial stream of class (Pascal traffic). To simplify the description, the microstate probability will be denoted by the symbol .
The multidimensional service process in the Erlang-Engset-Pascal model is a reversible process. In concordance with Kolmogorov reversibility test considering any cycle for the microstates shown in
Figure 2, we always obtain equality in the intensity of transitions (streams) in both directions. The property of reversibility implies the local equilibrium equations between any of the two
neighbouring states of the process. Such equations for the Erlang stream of class , the Engset stream of class , and Pascal stream of class can be written in the following way (Figure 2): Since the
call streams offered to the group are independent, we can add up, for the microstate , all equations of type (2.5) for the Erlang streams, equations of type (2.6) for the Engset streams, and
equations of type (2.7) for the Pascal streams. Additionally, taking into consideration traffic intensity (see (2.3) and (2.4)), we get
2.3. The Full-Availability Group with BPP Traffic at the Macrostate Level
It is convenient to consider the multidimensional process occurring in the considered system at the level of the so-called macrostates. Each macrostate determines the number of busy BBUs in the
considered group, regardless of the number of serviced calls of particular classes. Therefore, each of the microstates is associated with such a macrostate in which the number of busy BBUs is
decreased by BBUs, necessary to set up a connection of class , that is, with such a macrostate in which the number of busy BBUs equals . The following equation is then fulfilled: where determines the
number of all traffic classes offered to the system, that is, .
The macrostate probability defines then the occupancy probability of BBUs of the group and can be expressed as the aggregation of the probabilities of appropriate microstates: where is a set of all
such subsets that fulfil the following equation: The definition of the macrostate (2.11) makes it possible to convert (2.8) into the following form: Adding on both sides all microstates that belong
to the set , we get Following the application of the definition of macrostate probability, expressed by (2.10), we are in a position to convert (2.13) as follows: where , if , and the value ensues
from the normative condition .
In (2.14) the sums determine the value of the average number , of calls of class and in occupancy states (macrostates) and , respectively. In order to determine the relationship between the number of
serviced calls of particular traffic classes and the macrostate (for which the average values and are determined), in the subsequent part of the paper we have adopted the following notations: Taking
into consideration (2.15) and (2.16), we can rewrite (2.14) in the following way:
In (2.18) the value of Engset traffic of class and Pascal traffic of class depends on the occupancy state of the system. Let us introduce the following notation for the offered traffic intensity in
appropriate occupancy states of the group: Formula (2.18) can be now finally rewritten to the following form:
3. Modelling the Full-Availability Group
3.1. Average Number of Serviced Calls of Class in State
In order to determine the average number of calls serviced in particular states of the system, let us consider a fragment of the one-dimensional Markov chain presented in Figure 3 and corresponding
to the recurrent determination of the occupancy distribution in the full-availability group on the basis of (2.22). The diagram presented in Figure 3 shows the service process in the group with two
call streams (, BBU, BBUs).
Let us notice that each state of the Markov process in the full-availability group (Figure 3) fulfils the following equilibrium equation: where is the average number of calls of a given class being
serviced in state . From (3.1) it results that the sum of all service streams outgoing from state towards lower states is equal to : On the basis of (2.22) and (3.2), Formula (3.1) can be rewritten
in the following form: Equation (3.3) is a balance equation between the total stream of calls outgoing from state and the total service stream coming in to state . This equation is fulfilled only
when the local equilibrium equations for streams of particular traffic classes are fulfilled: On the basis of (3.4), the average number of calls of class in state of the group may be finally
expressed in the following way:
3.2. MIM-BPP Method
Let us notice that, in order to determine the parameter , it is necessary to determine first the occupancy distribution . Simultaneously, in order to determine the occupancy distribution , it is also
necessary to determine the value . This means that (2.22) and (3.5) form a set of confounding equations that can be solved with the help of iterative methods [32]. Let denote the occupancy
distribution determined in step , and let denote the average number of serviced calls of class , determined in step . In order to determine the initial value of the parameter , it is assumed,
according to [32], that the traffic intensities of Engset and Pascal classes do not depend on the state of the system and are equal to the traffic intensity offered by all free Engset sources of
class and Pascal sources of class , respectively: , . When we have the initial values of offered traffic, in the subsequent steps, we are in a position to determine the occupancy distribution, taking
into account the dependence of the arrival process on the state of the system. The iteration process finishes when the assumed accuracy is obtained.
On the basis of the reasoning presented above, in [32] the MIM-BPP method for a determination of the occupancy distribution, blocking probability, and the loss probability in the full-availability
group with BPP traffic is proposed. The MIM-BPP method can be presented in the form of the following algorithm.
Algorithm 3.1 (MIM-BPP method). Consider the following steps.(1)Determination of the value of Erlang traffic of class on the basis of (2.3).(2)Setting the iteration step: .(3)Determination of initial
values of the number of Engset serviced calls of class and the number of Pascal serviced calls of class : (4)Increase in each iteration step: .(5)Determination of the value of Engset traffic of class
and Pascal traffic of class on the basis of (2.20) and (2.21): (6)Determination of the state probabilities on the basis of (2.22): (7)Determination of the average number of serviced calls and on the
basis of (3.5): (8)Repetition of steps (3)–(6) until predefined accuracy of the iterative process is achieved: (9)Determination of the blocking probability for calls of class and the loss probability
for Erlang calls of class , for Engset calls of class , and for Pascal calls of class ,
3.3. Convergence of the Iterative Process of Estimation of the Average Number of Serviced Engset Calls
In this section we prove that the process for a determination of the average number of serviced traffic sources proposed in the MIM-BPP method is, in the case of multiservice Engset sources, a
convergent process. Thus, the following theorem needs to be proved.
Theorem 3.2. The sequence of the average number of serviced class Engset calls in the system with BPP traffic, where is convergent.
Proof. In order to prove Theorem 3.2, we are going to show first that each succeeding element of sequence (3.12), starting from the first one, could be represented by finite series: Since , then on
the basis of (3.12) for Now, using (3.15), we can determine the value for on the basis of (3.12): Rearranging (3.16), we can present it in the following way: Proceeding in an analogical way for , we
obtain Generalizing, the value of succeeding element of sequence in step can be expressed by (3.14). Now, setting the limit to infinity (), we have
Regardless of the iteration step, for every , the probability that system is in a state is equal to 0 (i.e., ). Thus, we can rewrite (3.19) in the following way: A series appearing on the right side
of (3.20) is finite; therefore, there exists a finite limit of sequence , which was to be proved.
3.4. Convergence of the Iterative Process of Estimation of the Average Number of Serviced Pascal Calls
Let us demonstrate now that the process of a determination of the average number of serviced traffic sources proposed in the MIM-BPP method is a convergent process also in the case of multiservice
Pascal sources. The following theorem will be then proved.
Theorem 3.3. The sequence of the average number of serviced class Pascal sources in the system with BPP traffic, where is convergent.
Proof. Proceeding in the analogical way as we did in the case of sequence (3.12), we can prove that the elements of sequence can be expressed by the following expression:
Therefore, in order to show that sequence is convergent, we only need to prove that for the series is convergent.
Consider the elements of series (3.24): The elements of series are positive, which means that we can use the ratio test (d'Alembert criterium) for convergence to prove that series is convergent (if
in series with positive terms beginning from certain place (this means for all ), then the ratio of arbitrary term to previous term is permanently less than number less than 1; i.e, if for all , then
series is convergent [33]). The ratio of two consecutive elements of sequence is equal to For numerator and denominator of (3.26) converge to 0. Note also that the numerator converges to 0 faster
than the denominator. Hence, is equal to 0, that is, is permanently less than 1. Therefore, by virtue of the ratio test (d'Alembert criterium) for convergence series (3.24) is convergent. Thus,
sequence (3.21) is convergent as well.
3.5. Advantages and Possible Applications of MIM-BPP Method
The presented iterative algorithm for systems with state-independent admission process (i.e., the full-availability group) makes it possible to determine exactly the occupancy distribution and the
blocking and loss probabilities in systems that service Erlang (Poisson distribution of call streams), Engset (binomial distribution of call streams), and Pascal traffic streams (negative binomial
distribution of call stream). The call stream of the types investigated in the paper are typical streams to be considered in traffic theory. They are used for modelling at the call level, where any
occupancy of resources of the system, for example, effected by a telephone conversation or by a packet stream with characteristics defined at the packet level, can be treated as a call [11]. In the
case of the Integrated Services Digital Networks, resource occupancies were in the main related to voice transmission, whereas nowadays a call is understood to be a packet stream to which appropriate
equivalent bandwidth is assigned [34–36], and then the demanded resources, as well as the capacity of the system, are discretized [7]. In the case of wired systems, the most important is the Poisson
stream and the consequent Erlang traffic stream. This stream assumes stable intensity of generating calls, independent of the number of calls that are already being serviced. In the case of wireless
systems, it was soon noticed that, because of the limited number of subscribers serviced within a given area, the application of the Erlang model for certain traffic classes could lead to erroneous
estimation of the occupancy distribution. Hence, for certain traffic classes, the application of the Engset model was proposed, initially for single-service (single-rate) systems and then for
multiservice (multirate) systems [3, 4]. In general, the Engset distribution is used to model systems with noticeable limitation of the number of users. Currently, the main practical scope for the
usage of the Pascal distribution is a simplified modelling of systems with overflow traffic [11]. The presented algorithm makes it then possible to determine traffic characteristics for all three
call (traffic) streams considered in traffic theory.
The application of the notion of the basic bandwidth unit (BBU) used in the notation of the presented method makes it possible to obtain high universality for the method. BBU is determined as the
highest common divisor of all demands that are offered to the system. Depending on a system under consideration, the basic bandwidth unit can be expressed in bits per second or as the percentage of
the occupancy of the radio interface (the so-called interference load) [4, 37]. In the presented method for modelling multirate systems with BPP traffic streams, both required resources and the
capacity of the system are expressed as the multiplicity of BBU. The method can be thus applied to model both wired broadband integrated services networks as well as wireless networks (UMTS/WCDMA
networks in particular).
The algorithm worked out for modelling systems with BPP traffic can be treated as an extension to the Kaufman-Roberts model [24, 25] that has been worked out for systems with Poisson traffic streams
only. Both the algorithm proposed by Kaufman-Roberts and the algorithm presented in the paper are exact algorithms. Having an exact formula as a base, the algorithm can be extended—analogously as in
the case of the Kaufman-Roberts formula for systems with Erlang traffic—into systems with state-dependent call admission process and BPP traffic. In the case of communication system, state dependence
in the call admission process results mainly from the introduction of the control policy in allocating resources for calls of individual traffic classes (reservation mechanism [32], threshold
mechanism [38]) or a particular structure of the system (e.g., a limited-availability group [5]). An extension of the scope in which the presented algorithm can be applied, including systems with
state-dependent call admission process, entails only the introduction of the additional transition coefficient [32], without further changes, depending on the considered system. It should be stressed
that such a universality cannot be achieved by the convolution algorithm also worked out for systems with state-independent call admission process only.
3.6. Numerical Examples
The paper introduces a formula that makes it possible to determine exactly the occupancy distribution in systems with state-independent call admission process. It is then demonstrated that the
algorithm for a determination of the average number of serviced traffic sources of particular classes used in the MIM-BPP method is convergent.
In order to present the convergence of the MIM-BPP method (the number of required iterations), in Table 1 the results of relative errors of the number of busy class 3 sources in the full-availability
group with the capacity equal to 80BBUs are contained (with the instance of calls of class 1 and 2, the number of required iterations is lower than in the case of the presented results for class 3).
The results are presented depending on the average value of traffic offered to a single bandwidth unit of the group: . The group was offered three traffic classes, that is, Erlang traffic class:
BBU, Engset traffic class: BBUs, , and Pascal traffic class: BBUs, . The results presented in Table 1 indicate that the proposed iterative method converges very quickly.
In this section we limit ourselves to just presenting the results of the convergence of the presented algorithm for one selected system. A comparison of the analytical results for the blocking/loss
probability with the results of the simulation is presented in earlier works, for example, [4, 32], in which it was still assumed that the presented analytical method was an approximate method.
4. Conclusion
In the paper recurrent equations describing—at the macrostate level—the service process in the full-availability group with multirate BPP traffic were derived. The derived equations made it possible
to formulate an exact iterative algorithm for determining the occupancy distribution, blocking probability, and loss probability of calls of particular classes offered to the system. The convergence
of the proposed process of estimating the average number of busy sources of Engset and Pascal traffic was proved.
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Voice Recognition Using MATLAB
Summary: The following module describes the process behind implementing a voice recognition algorithm in MATLAB. The algorithm utilizes the Discrete Fourier Transform in order to compare the
frequency spectra of two voices. Chebyshev’s Inequality is then used to determine (with reasonable certainty) whether two voices came from the same person. All material in this module is the result
of a course project for a Partial Differential Equations course (Math 480) held at California State University Northridge during the Fall 2009 semester. The project was carried out under the guidance
of professors – Carol Shubin and Gloria Melara.
Voice Recognition M-Files
Initial Problem
A human can easily recognize a familiar voice however, getting a computer to distinguish a particular voice among others is a more difficult task. Immediately, several problems arise when trying to
write a voice recognition algorithm. The majority of these difficulties are due to the fact that it is almost impossible to say a word exactly the same way on two different occasions. Some factors
that continuously change in human speech are how fast the word is spoken, emphasizing different parts of the word, etc… Furthermore, suppose that a word could in fact be said the same way on
different occasions, then we would still be left with another major dilemma. Namely, in order to analyze two sound files in time domain, the recordings would have to be aligned just right so that
both recordings would begin at precisely the same moment.
How to Compare Recordings
Frequency Domain
Given the difficulties mentioned in the above paragraph, it became quite evident that any voice analysis in time domain would be extremely impractical. Instead, an analysis of the frequency spectra
in a voice (which remains predominately unchanged as speech is slightly varied) turned out to be a more viable option. Converting all recordings into frequency domain (by applying the Discrete
Fourier Transform) greatly simplified the process of comparing two recordings. That being said, working in frequency domain also provided a new set of issues that required attention.
Finding a Norm
Due to nature of human speech, all data pertaining to frequencies above 600Hz can safely be discarded. Therefore, once a recording is converted into frequencey domain, it could then be simply
regarded as a vector in 600-dimensional Euclidean space. At this point, a comparison between two vectors could easily be carried out by normalizing the vectors (giving them length 1) then computing
the norm of the difference betweeen the two (of course, the difference between two vectors in R^600 is performed by subtracting componentwise). Unfortunately, exactly which norm to use is not
immediately clear. After carefully comparing and contrasting the use of the Taxicab, Euclidean, and Maximum norms, it became clear that the Euclidean norm most accurately measured the closeness
between different frequency spectra. Once the norm function was chosen, all that remained was to decide exactly how small the norm of the difference of two vectors had to be in order to determine
that both recordings originated from the same person.
Chebyshev's Inequality
Recall that Chebyshev's Inequality states that in particular, at least 3/4 of all measurements from the same population fall within 2 standard deviations of the mean. Hence, in response to the
problem posed at the end of the previous paragraph, the following solution can be formulated:
By requiring that the norm of the difference fall within 2 standard deviations of the normal average voice, we are then ensured that at least 3/4 of the time, the algorithm would recognize a voice
Algorithm Instructions
All files pertaining to the algorithm are located within the zip-file VoiceRecognition.zip which can be downloaded by simply pressing the link. The following is a short synopsis regarding the proper
execution of the software.
Short Description
As mentioned before, all files pertaining to the project can be accessed using the link: Voice Recognition. As soon as the file is opened, the following folders will be accessable:
David's Recordings
Matlab Files
The contents of these folders will now be discussed in more detail. The folder Matlab Files contains 10 audio recordings of David Roberts saying his name 'David'. Moreover, the folder contains the
two m-files project.m and voicerec.m.
Project.m is the voice recognition algorithm that accomplishes the goals of the class project. The script file project.m can be executed by typing 'project' in the command window. Please make sure
that the directory in Matlab is set to the directory that contains project.m and the 10 audio recordings g1.wav through g10.wav. Once project.m is ran in Matlab, it will then request that you "Enter
the name that must be recognized". Since the recordings in that folder are of David Roberts, then type in 'David'. Next, the program will inform you that you have 2 seconds to say the name 'David'.
After recording, Matlab will playback the sample and give you the option to try again or to proceed if satisfied. A plot is then generated depicting how the normalized frequency spectra in your voice
(top window) compares to the average normal vector of David's Voice (bottom window). See the figure below for an example. At this point, the algorithm makes a comparison and displays in the command
window 'YOU ARE NOT DAVID!!!!' if you do not fall within 2 standard deviations of the normal average voice. If you do happen to fall within 2 standard deviations, then the command window displays
'HELLO DAVID!!!'.
Figure 1: Example of a Frequency Spectra Comparison.
The second m-file in that folder is voicerec.m. This script file is executed by typing 'voicerec' in the command window. Running voicerec.m will prompt the user to record their name 10 times. The
recordings are then saved as g1.wav through g10.wav in the directory. Therefore, the ten new recording will in fact replace the recordings of David Roberts. Doing this results in the conversion of
project.m into a voice recognition algorithm for the user's voice (as oppose to the voice of David Roberts). In this case, the user's name should be entered as the voice to be recognized (instead of
'David') when running project.m. Lastly, since voicerec.m replaces g1.wav through g10.wav in the directory, back-up copies of David Roberts' voice are conviently stored in the folder David's
1 comments:
krishna said...
hello !I need a speech recognition using neural network in matlab........any one having the code plz mail me to krishnasatish2007@gmail.com
Labels: Voice Recognition Using MATLAB
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Calculating DB size
03-22-2001, 01:03 AM
Can anyone help me in calculating the exact size of the Db. What are the files and Memory area has to considered in this calculation
Thanks in advance
03-22-2001, 02:05 AM
database sizing
Hi Felix,
We can't go for an exact calculation of an database. Eventhough if we do so , the size of the database is not static.
Calculating the Size of the database mainly depends on the transaction rate and volume. Make a chart of transaction rate and volume against number of users. And determine the avergage size of a
Initialize the storage settings for tablespace and table so that a single row can be accomodated in a block and a transaction can be accomodated in a rollback segment.
Take care in sizing redo log file. After setting the parameters check for the performance of the database and tune accordingly. Size SGA so that number of hits increases.
03-22-2001, 12:04 PM
If you're referring to allocated size, you can sum the sizes of all the datafiles at the OS level ("du -sk" in Unix). If you're referring to utilized size, that is indeed dynamic. A snapshot
valid for a particular time could be taken by "select sum(bytes)/(1024*1024) MB from dba_segments".
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average velocity from t=0 and t-3
August 2nd 2011, 09:40 PM #1
Junior Member
Apr 2011
average velocity from t=0 and t-3
position of particle is given s=t^3 -12t^2 +45t+10
Do subsitibe t=0 and t=3 into s=t^3 -12t^2 +45t+10 or into its derivative?
s=t^3 -12t^2 +45t+10
is s' =3t^2 -24t +45 = V(velocity)?
Re: average velocity from t=0 and t-3
August 2nd 2011, 09:42 PM #2
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Who Stole Second Base?
Date: 20 Feb 1995 15:42:57 -0500
From: Anonymous
Subject: The Stolen Base
Help!! (settle for a hint)
The umpire was convinced that either Archie, Buster, Cal, or Dusty
had stolen (really stolen) second base. Each player, in turn, made a
statement, but only one of the four statements was true.
Archie said, "I didn't take it."
Buster said, "Archie is lying."
Cal said, "Buster is lying."
Dusty said, "Buster took it."
Who told the truth?
Date: 20 Feb 1995 19:20:04 -0500
From: Dr. Ken
Subject: Re: The Stolen Base
Hello there!
I think I'll try giving you a hint first. When you are told that only one
of the four statements is true, you can then try each statement as the true
statement to see whether it works. For example, let's assume that the last
statement is true; this implies that the other three statements are false, so
let's see if that gives us a consistent set of statements. To do this,
let's write out the four statements, but with only the last statement
intact, and the for the other three let's use the negative of what they said.
So we get:
Archie took it.
Archie is telling the truth (i.e. Archie didn't take it)
Buster is telling the truth (i.e. Archie is telling the truth)
Buster took it.
Is this a consistent set of statements? I don't think so. The most obvious
contradiction is that it claims Archie and Buster both took the base. So
Dusty can't be the one telling the truth.
Try the same thing with the other three guys, and see which gives you a
consistent set of statements. Hopefully there will be only one that works,
and that will tell you which one is telling the truth (although not
necessarily who took the base!).
-Ken "Dr." Math
Is it possible for Archie to be telling the truth while Buster, Cal, and
Dusty are lying? Go through all of their statements. If Archie is telling
the truth, then we know: 1) he didn't take it; 2) Buster is lying (Buster
says Archie is lying, so that means that the truth is that Archie isn't
lying, so we are okay so far); 3) Cal is lying (Cal says Buster is lying,
so that means that since Cal is lying, Buster really is NOT lying, so
Buster is telling the truth). This contradicts the given conditions
of the problem, though, so we know that Archie isn't telling the truth.
Now you consider the 2 other cases!
If you need any more help, feel free to write back!
--Sydney, "dr.math"
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Contribution Dynamics in Defined Contribution Pension Plans During the Great Recession of
by Irena Dushi, Howard M. Iams, and Christopher R. Tamborini
Social Security Bulletin, Vol. 73 No. 2, 2013
Text description for Chart 1.
Distribution of contribution amounts in 2007 and their percentage change during the crisis period (2007–2009)
Chart 1 is comprised of two separate charts. The first one shows the frequency distribution of 2007 contribution amounts as bar charts overlaid by kernel density function. The x-axis denotes the
dollar amount from zero to the maximum amount of contributions, with each bin width of $500. The y-axis shows the percentage of the sample within each bin. The distribution of contributions is skewed
to the left, indicating that the median contribution amount is smaller than the mean. The second chart shows the frequency distribution of the percentage change in contributions in 2009 compared with
that in 2007 as bar charts overlaid by kernel density function. Thus, the x-axis shows the percentage point change, which ranges from -100 percent to +100 percent. Here each bin has a width of 5
percent. The median percentage change is -2.2 percent, and for about 37 percent of the sample the percentage change in contributions fell within plus or minus 15 percent. Around 16 percent of the
sample stopped contributing (that is, their contributions from 2007 through 2009 changed by -100 percent), and about 9 percent of the sample started contributing over the period (that is, their
contributions changed by +100 percent).
Text description for Chart 2.
Distribution of contribution amounts in 2005 and their percentage change during the precrisis period (2005–2007)
Chart 2 is also comprised of two separate charts. The first one shows the frequency distribution of 2005 contribution amounts as bar charts overlaid by kernel density function. The x-axis denotes the
dollar amount from zero to the maximum amount of contributions, with each bin width of $500. The y-axis shows the percentage of the sample within each bin. The distribution of contributions is skewed
to the left, indicating that the median contribution amount is smaller than the mean. The second chart shows the frequency distribution of the percentage change in contributions in 2007 compared with
that in 2005 as bar charts overlaid by kernel density function. Thus, the x-axis the percentage point change, which ranges from -100 percent to +100 percent. Here each bin has a width of 5 percent.
The median percentage change is -1.9 percent, and for about 44 percent of the sample the percentage change in contributions fell within plus or minus 15 percent. Around 13 percent of the sample
stopped contributing (that is, their contributions from 2005 to 2007 changed by -100 percent) and about 12 percent of the sample started contributing over the period (that is, their contributions
changed by +100 percent).
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A 53.7-kg person, running horizontally with a velocity of +4.99 m/s, jumps onto a 11.9-kg sled...
Get your Question Solved Now!!
A 53.7-kg person, running horizontally with a velocity of +4.99 m/s, jumps onto a 11.9-kg sled...
Introduction: I need help with part b
More Details: A 53.7-kg person, running horizontally with a velocity of +4.99 m/s, jumps onto a 11.9-kg sled that is initially at rest. (a) Ignoring the effects of friction
during the collision, find the velocity of the sled and person as they move away. (b) The sled and person coast 30.0 m on level snow before coming to rest. What is
the coefficient of kinetic friction between the sled and the snow?
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C++ Complex Numbers
The complex class provides a useful way to deal with complex numbers. It is defined as a basic template class in the std namespace in the <complex> header file.
template<typename Num>
struct complex;
The type parameter Num is the type that is used for each of the real and imaginary components of the complex number. It must be one of: float, double, or long double.
Constructors create complex numbers from the real and imaginary parts or other complex numbers
Operators arithmetic of complex numbers, equality of complex numbers, IO of complex numbers
abs absolute value of a complex number
arg argument (angle) of a complex number
conj conjugate of a complex number
cos cosine of a complex number
cosh hyperbolic cosine of a complex number
exp exponential of a complex number
imag imaginary part of a complex number
log natural logarithm of a complex number
log10 base-10 logarithm of a complex number
norm square of the absolute value of a complex number
polar constructs a complex number from polar coordinates
pow raising a complex number to a power, raising to a complex number power, or both
real real part of a complex number
sin sine of a complex number
sinh hyperbolic sine of a complex number
sqrt square root of a complex number
tan tangent of a complex number
tanh hyperbolic tangent of a complex number
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level sets of multivariate polynomials
up vote 8 down vote favorite
Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $P$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube
$\int_Q f(x) dx = 0.$
For $\alpha>0$ consider the sublevel sets of $P$,
$$E_\alpha= \{x\in Q: |p(x)|\leq \alpha\}$$
There are several known estimates for the Lebesgue measure of this set which in some sense or another are uniform over some classes of polynomials. For example, we have that
$$|E_\alpha| \lesssim \min(pd,n) \frac{ \alpha^{1/d} }{ \|p\|_{ L^p(Q) }^{1/d} } $$
This particular estimate is due to Carbery and Wright and can be found here.
I'm interested in studying the (induced Lebesgue) measure of the boundary of this set
$$|\partial E_\alpha|=|\{x\in Q: |P(x)|=\alpha\}| $$
Consider first the easy case of dimension $n=1$. Then the set $E_\alpha$ is a finite union of closed intervals and the question is trivial. It is obvious that in this case there are at most $O(d)$
intervals so the $0$-dimensional measure of the boundary is $O(d)$.
Now in many variables things will be much more complicated. For example can we say that the set $E_\alpha$ has $O(d)$ connected components? Is there an estimate for the measure of the boundary $\
partial E_\alpha $ in terms of $\alpha$, $d$ and $n$, assuming (say) that $\|p\| _ {L^1(Q)}=1$ ?
This question comes up naturally if one tries to study an oscillatory integral with phase $p$ and apply integration by parts (i.e Gauss theorem) imitating the one dimensional method of proving the
van der Corput lemma (for example).
add comment
3 Answers
active oldest votes
There is a trivial estimate for the measure of the boundary based on the observation that $|p|^2$ is still a polynomial, so the corresponding surface intersects any line at most $2d$
times. Averaging over directions, we get $O(d\sqrt[] n)$ regardless of $\alpha$. Now, the question is what exactly you want: a dimension-independent bound, a bound that is small for
up vote 10 large $\alpha$, or anything else. It may really help to try it from the other end: figure out what exactly you need and we'll try to figure out whether it is true or not. Otherwise, you
down vote may get plenty of answers with trivial and not so trivial estimates for everything, none of which will fit your real needs.
Thank you for the answer. I am a bit confused. So you consider all the lines that meet the set $E_ \alpha$. If you average over all the lines that meet the set you will get the
surface measure of the boundary (which is what we want). So we need to compute this in another way as well. I'm missing this other way. To answer your question now, I need an estimate
of the measure of the boundary that stays bounded in $n$ when $$d<<n$$. So we can suppose that $d$ is fixed and $n\rightarrow \infty$. I'm not sure if this is even possible of course.
You should think that $\alpha \sim d$ – ioannis.parissis Nov 9 '09 at 0:45
This other way is just noting that coordinate lines (they are enough to average over) are restricted to the cube (in other words, the area of the projection of your set to each
coordinate hyperplane is bounded by $1$). As to the bound independent of $n$ in the cube, it holds when $d=1$ (sections of the cube by hyperplanes have area at most $\sqrt[]2$) but
it'll take me some time (possibly, infinite) to figure out if it holds for other $d$ as well. That the cube is unit is crucial here, of course. Seems like a really nice question
(unless there is some trivial counterexample)! – fedja Nov 9 '09 at 1:30
1 A small update: apparently, even the question about dimension-free estimates for the Gaussian perimeters of algebraic surfaces of fixed degree is open (your question is not easier
because we can easily simulate independent Gaussians by long sums of coordinates). Even for $d=2$, the best that is known is that the Gaussian perimeters of all balls are uniformly
bounded and the Gaussian perimeters of all origin-symmetric quadratic surfaces are uniformly bounded. Fascinating! It might make a nice polymath project... – fedja Nov 9 '09 at 4:01
Let me rephrase your argument the way i understand it better. By crofton's formula for reasonable surfaced $S\subset \mathbb R ^n$,$ |S|=\int_{\mathcal L_S}n _{S}(l) d\nu(l)$ where $\
mathcal {L_S}$ is the space of lines that intersect $S$, $\nu$ is the motion invariant measure on the set of lines and $n_S(l)$ is the number of intersections of $l$ with $S$. –
ioannis.parissis Nov 9 '09 at 15:46
In our case this number is at most $2d$,and we can make the integral bigger by considering all lines that intersect the unit cube. We end up with $|S|\lesssim d n$, $n$ coming from
the measure of lines that intersect $Q$. We have a $\sqrt{n}$ discrepancy here! – ioannis.parissis Nov 9 '09 at 15:46
show 1 more comment
I don't know how to estimate the measure of the level sets, but I can answer the question about the number of connected components to an extent. Let $P_f(x) = (x-1)(x-2)\cdots(x-f)$, and
define $p(x_1, \ldots, x_n) = P_f(x_1)^2 + \cdots + P_f(x_n)^2$. Then $p$ has degree $2f$, and the level sets $p = \varepsilon$ for small $\varepsilon$ have $f^n = (d/2)^n$ connected
components (each $x_i$ must be close to one of the roots of $P_f$).
up vote 4
down vote Theorem 11.5.3 of Bochnak, Coste, & Roy, Real Algebraic Geometry, says the sum of the Betti numbers, hence the number of connected components, of an algebraic set in $R^n$ defined by
degree ≤d polynomials is at most $d(2d-1)^{n-1}$. So for fixed n, the maximum number of connected components of the level sets of a degree d polynomial in n variables is $\Theta(d^n)$.
Thank you for the answer. I forgot to add hypothesis that the polynomial $p$ is assumed to have mean value zero on the unit cube. The second part of your answer still applies of course.
I will edit my question to add this hypothesis. – ioannis.parissis Nov 8 '09 at 21:30
I think the example can be adapted; take (p-ɛ)*L where L is some linear function whose zero set does not meet the grid; you should be able to find some hypercube on which the mean value
is zero, and rescale the variables to make it the unit hypercube. Then consider the zero set of the resulting polynomial. – Reid Barton Nov 8 '09 at 21:39
add comment
An encouraging update: Daniel Kane posted his proof for the Gaussian case on arXiv yesterday. The proof is both very simple and brilliant. I won't be surprised if his technique can
up vote 2 down be modified to cover the cube case.
Thanks for the link. Reading the article now. – ioannis.parissis Dec 16 '09 at 13:42
add comment
Not the answer you're looking for? Browse other questions tagged fourier-analysis ca.analysis-and-odes ag.algebraic-geometry algebraic-surfaces open-problem or ask your own question.
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Coder's Revolution
Mar 27
General, Mathmatics
So I was at a friend's house Sunday night playing a game when this odd fact came up in conversation:
If you were to fold a piece of paper in half 42 times, it would reach the moon.
Several of those around the table scoffed at this, exclaiming that a single sheet of paper was simply too thin to have its thickness reach any substantial amount after only a few dozen folds. I
pointed out it was entirely possible seeing as how doubling the thickness with each fold would lead to an exponential increase in thickness that would increase slowly at first before quickly getting
larger. My friends were clearly imagining a linear increase in thickness.
I also knew that it is pretty much impossible to fold a single sheet of paper more than about 8 times -- though Myth Busters once folded a giant sheet the size of a football field 10 times. The
resulting thickness (after hitting it with a bulldozer) was almost a foot tall, though there was quite a bit of air mixed in with the 1,024 sheets. The formula for finding out how many of something
you'll have after doubling it N number of times is as follows where O is the original number (or size in our case).
o * 2^(n)
A standard sheet of paper is about 0.1 mm so 42 folds would give us this:
0.1 * 2^(42) = 439,804,651,110 mm
That's 440 billion millimeters, or 439,804 kilometers. The moon on average is 384,400 kilometers from Earth according to Google. I'd say this checks out.
To help visualize the data, I created a quick spreadsheet and graph that tracks the thickness of the paper for each fold.
# Folds Thickness (mm)
0 0.10
1 0.20
2 0.40
3 0.80
4 1.60
5 3.20
6 6.40
7 12.80
8 25.60
9 51.20
10 102.40
11 204.80
12 409.60
13 819.20
14 1,638.40
15 3,276.80
16 6,553.60
17 13,107.20
18 26,214.40
19 52,428.80
20 104,857.60
21 209,715.20
22 419,430.40
23 838,860.80
24 1,677,721.6
25 3,355,443.2
26 6,710,886.4
27 13,421,773
28 26,843,546
29 53,687,091
30 107,374,182
31 214,748,365
32 429,496,730
33 858,993,459
34 1,717,986,918
35 3,435,973,837
36 6,871,947,674
37 13,743,895,347
38 27,487,790,694
39 54,975,581,389
40 109,951,162,778
41 219,902,325,555
42 439,804,651,110
And to graph that out in kilometers looks like this:
In a recent pissing match between ColdFusion and PHP, Jared Rypka-Hauer was demonstrating the performance of a function that generated prime numbers. The discussion really wasn't about the BEST prime
generator as much as it was about how much ColdFusion can kick PHP's puny butt all over town. Never the less, I piped up in the comments to ask Jared to compare a prime number generator that I wrote
a while back based on the Sieve of Eratosthene. After Jared asked some good questions about how my code worked I figured it was time I stopped high-jacking the comments of the PHP pooper train. I
decided to spin off a new post to highlight some significant performance gains I was able to produce.
Here's another UDF I was tinkering with last week. I wanted to be able to count all of the numbers that divided evenly into a given integer. I couldn't find a ColdFusion implementation, so after
getting some advice from Stack Overflow I created my own.
Ahh... the quintessential math problem-- finding prime numbers. Last week while tinkering with a math challenge I needed to find all of the primes up to a given number. There was a version on
cflib.org, but I thought I could do it in less code, so I dug in myself.
Feb 01
Generating Primes Revisited: My Modifications To The Sieve of Eratosthenes
ColdFusion, Java, Mathmatics, Performance
Aug 14
ColdFusion UDF: Calculate A Number's Divisors
ColdFusion, Mathmatics, Performance
Aug 13
Calculate Prime Numbers: Sieve of Eratosthenes
ColdFusion, Mathmatics, Performance
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question about math module notation
Paul Rubin http
Fri Jul 27 02:29:26 CEST 2007
Dan Bishop <danb_83 at yahoo.com> writes:
> > I was surprised to find that gives an exact (integer, not
> > floating-point) answer. Still, I think it's better to say 2**64
> > which also works for (e.g.) 2**10000 where math.pow(2,10000)
> > raises an exception.
> It *is* binary floating point. Powers of 2 are exactly
> representable. Of course, it doesn't give an exact answer in general.
> >>> int(math.pow(10, 23))
> 99999999999999991611392L
Oh yikes, good point. math.pow(2,64) is really not appropriate for
what the OP wanted, I'd say. If you want integer exponentiation, then
write it that way. Don't do a floating point exponentiation that just
happens to not lose precision for the specific example.
>>> int(math.pow(3,50)) # wrong
>>> 3**50 # right
More information about the Python-list mailing list
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Eighth Grade Hands-On Activities
• Describe and draw vertical, supplementary and complementary angles. Verify the relationship between the angles by measuring.
• Using graph paper, apply transformations to geometric figures such as:
□ rotate or turn
□ reflect or flip horizontally and vertically
□ translate or slide
□ dilate or scale
Find applications of transformations such as tiling, fabric design or art.
• Describe and construct solid figures including prisms, pyramids, cylinders and cones.
• Verify the Pythagoreum Theorum by measuring. Apply the Pythagoreum Theorum to find the missing length of a side of a right triangle when the other two sides are given.
• Analyze games of chance and board games using probability and make predictions about the possible outcomes.
• Use information derived from line, bar, circle and picture graphs and histograms to make comparisons, predictions and inferences.
• Organize and describe data using a matrix.
• Graph a two variable linear equation on the coordinate plane, using a table of ordered pairs.
• Describe and represent relations using tables and graphs.
• Create and solve problems using proportions, formulas and functions.
© Copyright 2006 J. Banfill. All Rights Reserved.Legal Notice
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188 helpers are online right now
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
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Noninvadability implies noncoexistence for a class of cancellative systems
Jan M. Swart (Institute of Information Theory and Automation of the ASCR (UTIA))
There exist a number of results proving that for certain classes of interacting particle systems in population genetics, mutual invadability of types implies coexistence. In this paper we prove a
sort of converse statement for a class of one-dimensional cancellative systems that are used to model balancing selection. We say that a model exhibits strong interface tightness if started from a
configuration where to the left of the origin all sites are of one type and to the right of the origin all sites are of the other type, the configuration as seen from the interface has an invariant
law in which the number of sites where both types meet has finite expectation. We prove that this implies noncoexistence, i.e., all invariant laws of the process are concentrated on the constant
configurations. The proof is based on special relations between dual and interface models that hold for a large class of one-dimensional cancellative systems and that are proved here for the first
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-12
Publication Date: May 23, 2013
DOI: 10.1214/ECP.v18-2471
This work is licensed under a
Creative Commons Attribution 3.0 License
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Problem 2
ABCD is a two-by-two square. E is the midpoint of AD. Find the area of CDEF.
[Problem submitted by Juergen Pahl, LACC Professor of Mathematics.]
Solution for Problem 2:
[] and [], therefore the area of triangle ABE is 1 and
[] and [],
which means that ABE and BCF are similar triangles.
Area of triangle BCF is then []
Area of CDEF []
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Animated Lorenz strange attractor
delay r
Use your fingers or mouse to control the model (hold shift key or use mouse wheel to zoom it).
3D trajectory of motion of a point (to the left) and graphics { x(t), y(t), z(t) } (to the right) are ploted. Equations of the motion are
x' = a(y - x),
y' = -xz + rx - y,
z' = xy - bz.
where a = 3, b = 1, r = 26.5. The point starts at t = 0, x(0) = 1, y(0) = 0, z(0) = 0. 3d color is changed from red to violet in accordance with the HSB palette.
The ball oscillates around one of two centers then randomly goes to another. For small r (e.g. r = 10) there are simple attractors only. See Chaotic maps.
Resizable demos animated Lorenz and Rossler band.
WebGL Demos updated 30 May 2010
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Length Contraction
Accel-1D Length Contraction Plots
When I began to think about teaching special relativity for my first modern physics course, I could usually guess which observer in a given problem would see a ruler shrink relative to his own. But I
wondered, couldn't we then call the phenomenon "length expansion" from the vantage point of the frame in which it is shrunk? For the answer, continue reading...
In my desire to visualize this more concretely, I decided to illustrate the process with an x-ct diagram. Even though the sound byte "length contraction" was familiar to me, and even though my
intuition usually told me how to solve length contraction problems, I didn't really understand the concept until this was done. As Jonathon Swift's Gulliver might have said, "When I could hold the
fruit in my hand and taste it, the inadequacy of my prior wisdom became apparent." This was one of my first object lessons in the sloppiness of our own language for dealing with relativistic
spacetime effects.
In special relativity terms, "length contraction" problems involve THREE events - which we might refer to as left measure event common to both frames, and the right measure at rest and right measure
moving events which take place simultaneously on the other end of the ruler from the viewpoint of one observer or another. As I looked at the diagrams I realized that in length contraction problems,
one and only one inertial frame, namely the "uncontracted one", was accorded and deserved special status. This special frame, inherent to the statement of the problem, is the frame in which the ruler
is traveling. All other inertial frames see the rulers in this priviledged frame as contracted. Thus when you look at any moving ruler, or equivalently when any yardstick moves, it's length is
contracted in the direction of motion. Moving a ruler never causes its length to increase or "expand"!
To pose a specific problem, imagine a large ruler, one light-year in length, and an inertial frame which measures at one of its "instants" the length of this ruler while traveling at a constant
coordinate velocity of one third of a lightyear per inertial year (i.e. v = c/3) to the right.
To draw the x-ct plot below, the four steps are:
The rest frame of the ruler is chosen as the one with orthogonal axes.
Imagine that the tick marks on the green grid denote times (vertical) of one year, and distances (horizontal) of one light-year. (This would work as well with seconds & light-seconds, respectively,
if we wanted the ruler to be shorter.)
The red lines denote the world line (path through time) of the left and right ends of the ruler. The largest red box denotes the left side measurement event (i.e. point in time and space when a frame
yardstick is used to measure our ruler). The other two boxes denote the position of the other end of the ruler at the instant of the measurement, from the point of view of the frames in which the
ruler is at rest (medium square), and in which it is moving (smallest square).
The blue grid shows lines of constant maptime t (about 18 degrees up from the horizontal right) and constant position (about 18 degrees rightward from the vertical) for an inertial observer moving to
the right at one third of a lightyear per mapyear (i.e. v=c/3). The measurement occurs at "time zero" for each observer. Note that the point where the world-line of the right edge of the ruler
intersects the blue (rightward moving observer) time-zero line (denoted by the smallest red square) is inside of the dotted blue axis crossing -- hence the measured length of the ruler is less its
resting length of one light-year. Since gamma for the rightward frame is 1/Sqrt[1-(v/c)^2] = 1/Sqrt[1-1/9] = Sqrt[9/8] = 1.06, the length measured in the blue frame for the ruler is only 1/1.06 =
0.94 lightyears.
As mentioned on our x-ct plotting page, the same information could be obtained for a plot which uses orthogonal axes for the rightward moving frame. Then of course the ruler is moving to the left. As
in other examples elsewhere on these pages, the picture looks quite different even though the information it provides is the same!
Cite/Link: http://newton.umsl.edu/~run/contract.html
This release dated 06 Feb 1996 (Copyright by Phil Fraundorf 1988-1995)
This is part of our Map-Based Motion at Any Speed Project.
At UM-StLouis see also: cme, infophys, physics&astronomy, programs, stei-lab, & wuzzlers.
Other Theory pages: derivations, slow-example, fast-example&twins, x-tv Plots, 4-vectors, rap. Mindquilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter
linked subset of pages" since 4/7/2005: .
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What the heck is "?" ???
What the heck is "?" ???
I'm going over my Discrete Structures assignment and an operator or something is used that I've never seen in the book or the notes. The questions is- Describe the following sets using the
explicit method, with
U = {x|x E N ^ 1<x<10}
and then it gives sets A B C D and E explicitly, the questions im stuck on is - (C ? E) ? (A -(D ? B)) and B is complemented and D ? B is also complemented... what is that question mark, and what
do I do with it??
I don't remember much from my Discrete Math class, but I found this:
A group of three people, each of which is a liar (lies all the time) or is a truar (tells the truth all the time) are talking. B says that C and D are the same type (both liars or both truars).
Someone then asks D, ``Are B and C the same type?'' What does D answer?
Here's our solution. Let b stand for "B" is a truar", and similarly for c and d. The statement ``C and D are the same type'' is formalized as c == d. Since B said it, we have b == (c == d). D's
answer can be formalized as b ? c, where ? is either == if D answers ``yes'' or /== if D answers ``no'', and we have to determine which it is. Since D answers, we write this as d == (b ? c).
We manipulate d == b ? c under the truth of b == c == d. In the manipulation,we omit all parentheses because == is associative and == and /== are mutually associative.
d == b ? c --Fact
= < Fact b == c == d >
b == c == b ? c
By inference rule Equanimity, the last line is a theorem. But the last line is a theorem only if ? is ==. Therefore, we conclude that D answered ``yes''.
actually the GA just emailed me back saying that there aren't any ? in the assignment and then I figured out that because im using OOo it read the word file wrong, too bad he emailed me back
after the assignment was due.
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Greek alphabet
Mathematics requires a large number of symbols to stand for abstract objects, such as numbers, sets, functions, and spaces, so the use of Greek letters was introduced long ago to provide a collection
of useful symbols to supplement the usual Roman letters.
To us these symbols may seem quite foreign, and they are difficult to become familiar with. However, at the time they were introduced most scholars had been taught at least some Latin and Greek
during their education, so the letters did not seem nearly so strange to them as they do to us. Since then, each new generation of mathematicians has just gotten used to using them.
The table below lists all of the letters in the Greek alphabet, upper-case and lower-case, with their names and pronunciations. The lower-case letters are used more often than the upper-case letters,
but the latter are used often enough. (The upper-case letters in gray are not used as Greek symbols per se, because they are generally indistinguishable from their Roman-letter equivalents.) The
lower-case letters are most often used for variables, such as angles and complex numbers, and for functions and formulas, while the upper-case letters more commonly stand for sets and spaces, and
sometimes for repeated arithmetic operations such as adding and multiplying (see Sigma and Pi). In any particular textbook or paper, the way in which these symbols should be interpreted should
generally be clear from the context and definitions.
The pronunciations provided are not necessarily the “correct” ones, but reflect the most common pronunciations in use in English speaking countries.
The information and character table on this page are also available as a download and as an inexpensive poster.
CAP lower NAME (pronunciation) Description.
A \(\large\alpha\) ALPHA (AL-fuh) First letter of the Greek alphabet.
B \(\large\beta\) BETA (BAY-tuh)
\(\large\Gamma \(\large\gamma\) GAMMA (GAM-uh)
\(\large\Delta \(\large\delta\) DELTA (DEL-tuh)
E \(\large\epsilon,\ EPSILON (EP-sil-on) A stylized form of the lower case epsilon, \(\in\), is used as the “set membership” symbol.
Z \(\large\zeta\) ZETA (ZAY-tuh)
H \(\large\eta\) ETA (AY-tuh)
\(\large\Theta \(\large\theta\) THETA (THAY-tuh)
I \(\large\iota\) IOTA (eye-OH-tuh)
K \(\large\kappa\) KAPPA (KAP-uh)
\(\large\ \(\large\lambda\) LAMBDA (LAM-duh)
M \(\large\mu\) MU (MYOO)
N \(\large\nu\) NU (NOO)
\(\large\Xi\) \(\large\xi\) XI (KS-EYE)
O \(\large\omicron\) OMICRON (OM-i-KRON)
\(\large\Pi\) \(\large\pi\) PI (PIE) The lower-case \(\pi\) is universally used to represent that number which is the ratio of the circumference of a circle to its diameter. The
upper-case \(\Pi\) is used as the “product” symbol.
P \(\large\rho\) RHO (ROW)
\(\large\Sigma \(\large\sigma,\varsigma SIGMA (SIG-muh) The capital \(\Sigma\) is used as the “summation” symbol.
\) \)
T \(\large\tau\) TAU (TAU)
\(\large\ \(\large\upsilon\) UPSILON (OOP-si-lon)
\(\large\Phi\) \(\large\phi,\varphi\) PHI (FEE)
X \(\large\chi\) CHI (K-EYE)
\(\large\Psi\) \(\large\psi\) PSI (SIGH)
\(\large\Omega \(\large\omega\) OMEGA (oh-MAY-guh) Last letter of the Greek alphabet. The lower-case \(\omega\) denotes the smallest infinite ordinal in set-theory, isomorphic to the set of
\) natural numbers.
• [MLA] “Greek alphabet.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 2 Mar 2013. Web. 2 Mar 2013. <http://platonicrealms.com/>
• [APA] Greek alphabet (2 Mar 2013). Retrieved 2 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/Greek-alphabet/
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Matches for: Author/Editor=(Massey_David_B)
Memoirs of the American Mathematical Society
2003; 268 pp; softcover
Volume: 163
ISBN-10: 0-8218-3280-8
ISBN-13: 978-0-8218-3280-6
List Price: US$80
Individual Members: US$48
Institutional Members: US$64
Order Code: MEMO/163/778
The Milnor number is a powerful invariant of an isolated, complex, affine hypersurface singularity. It provides data about the local, ambient, topological-type of the hypersurface, and the constancy
of the Milnor number throughout a family implies that Thom's \(a_f\) condition holds and that the local, ambient, topological-type is constant in the family. Much of the usefulness of the Milnor
number is due to the fact that it can be effectively calculated in an algebraic manner.
The Lê cycles and numbers are a generalization of the Milnor number to the setting of complex, affine hypersurface singularities, where the singular set is allowed to be of arbitrary dimension. As
with the Milnor number, the Lê numbers provide data about the local, ambient, topological-type of the hypersurface, and the constancy of the Lê numbers throughout a family implies that Thom's \(a_f\)
condition holds and that the Milnor fibrations are constant throughout the family. Again, much of the usefulness of the Lê numbers is due to the fact that they can be effectively calculated in an
algebraic manner.
In this work, we generalize the Lê cycles and numbers to the case of hypersurfaces inside arbitrary analytic spaces. We define the Lê-Vogel cycles and numbers, and prove that the Lê-Vogel numbers
control Thom's \(a_f\) condition. We also prove a relationship between the Euler characteristic of the Milnor fibre and the Lê-Vogel numbers. Moreover, we give examples which show that the Lê-Vogel
numbers are effectively calculable.
In order to define the Lê-Vogel cycles and numbers, we require, and include, a great deal of background material on Vogel cycles, analytic intersection theory, and the derived category. Also, to
serve as a model case for the Lê-Vogel cycles, we recall our earlier work on the Lê cycles of an affine hypersurface singularity.
Graduate students and research mathematicians interested in several complex variables and analytic spaces.
• Overview
Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles
• Introduction
• Gap sheaves
• Gap cycles and Vogel cycles
• The Lê-Iomdine-Vogel formulas
• Summary of Part I
Part II. Lê Cycles and Hypersurface Singularities
• Introduction
• Definitions and basic properties
• Elementary examples
• A handle decomposition of the Milnor fibre
• Generalized Lê-Iomdine formulas
• Lê numbers and hyperplane arrangements
• Thom's \(a_f\) condition
• Aligned singularities
• Suspending singularities
• Constancy of the Milnor fibrations
• Another characterization of the Lê cycles
Part III. Isolated Critical Points of Functions on Singular Spaces
• Introduction
• Critical avatars
• The relative polar curve
• The link between the algebraic and topological points of view
• The special case of perverse sheaves
• Thom's \(a_f\) condition
• Continuous families of constructible complexes
Part IV. Non-Isolated Critical Points of Functions on Singular Spaces
• Introduction
• Lê-Vogel cycles
• Lê-Iomdine formulas and Thom's condition
• Lê-Vogel cycles and the Euler characteristic
• Appendix A. Analytic cycles and intersections
• Appendix B. The derived category
• Appendix C. Privileged neighborhoods and lifting Milnor fibrations
• References
• Index
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This Article
Bibliographic References
Add to:
Optimal Power Management of Residential Customers in the Smart Grid
Sept. 2012 (vol. 23 no. 9)
pp. 1593-1606
ASCII Text x
Yuanxiong Guo, Miao Pan, Yuguang Fang, "Optimal Power Management of Residential Customers in the Smart Grid," IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 9, pp. 1593-1606,
Sept., 2012.
BibTex x
@article{ 10.1109/TPDS.2012.25,
author = {Yuanxiong Guo and Miao Pan and Yuguang Fang},
title = {Optimal Power Management of Residential Customers in the Smart Grid},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {23},
number = {9},
issn = {1045-9219},
year = {2012},
pages = {1593-1606},
doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2012.25},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - Optimal Power Management of Residential Customers in the Smart Grid
IS - 9
SN - 1045-9219
EPD - 1593-1606
A1 - Yuanxiong Guo,
A1 - Miao Pan,
A1 - Yuguang Fang,
PY - 2012
KW - Batteries
KW - Electricity
KW - Renewable energy resources
KW - Smart grids
KW - Real time systems
KW - Pricing
KW - renewable energy generation
KW - Batteries
KW - Electricity
KW - Renewable energy resources
KW - Smart grids
KW - Real time systems
KW - Pricing
KW - real-time pricing.
KW - Smart grid
KW - optimal power management
KW - Lyapunov optimization
KW - energy storage
VL - 23
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
Recently intensive efforts have been made on the transformation of the world's largest physical system, the power grid, into a “smart grid” by incorporating extensive information and
communication infrastructures. Key features in such a “smart grid” include high penetration of renewable and distributed energy sources, large-scale energy storage, market-based online
electricity pricing, and widespread demand response programs. From the perspective of residential customers, we can investigate how to minimize the expected electricity cost with real-time
electricity pricing, which is the focus of this paper. By jointly considering energy storage, local distributed generation such as photovoltaic (PV) modules or small wind turbines, and inelastic or
elastic energy demands, we mathematically formulate this problem as a stochastic optimization problem and approximately solve it by using the Lyapunov optimization approach. From the theoretical
analysis, we have also found a good tradeoff between cost saving and storage capacity. A salient feature of our proposed approach is that it can operate without any future knowledge on the related
stochastic models (e.g., the distribution) and is easy to implement in real time. We have also evaluated our proposed solution with practical data sets and validated its effectiveness.
[1] A Century of Innovation: Twenty Engineering Achievements that Transformed Our Lives, US Nat'l Academy of Eng. Joseph Henry Press, 2003.
[2] A. Ipakchi and F. Albuyeh, "Grid of the Future," IEEE Power and Energy Magazine, vol. 7, no. 2, pp. 52-62, Mar./Apr. 2009.
[3] The Smart Grid: an Introduction. US Dept. of Energy (DOE), 2008.
[4] V. Vittal, "The Impact of Renewable Resources on the Performance and Reliability of the Electricity Grid," US Nat'l Academy of Eng., http://www.nae.edu/Publications/Bridge/TheElectricityGrid
18587.aspx, 2010.
[5] P. Ribeiro, B. Johnson, M. Crow, A. Arsoy, and Y. Liu, "Energy Storage Systems for Advanced Power Applications," Proc. IEEE, vol. 89, no. 12, pp. 1744 -1756, Dec. 2001.
[6] B. Roberts and C. Sandberg, "The Role of Energy Storage in Development of Smart Grids," Proc. IEEE, vol. 99, no. 6, pp. 1139-1144, June 2011.
[7] T. Markel and A. Simpson, "Plug-in Hybrid Electric Vehicle Energy Storage System Design," Proc. Advanced Automotive Battery Conf., 2006.
[8] M. He, S. Murugesan, and J. Zhang, "Multiple Timescale Dispatch and Scheduling for Stochastic Reliability in Smart Grids with Wind Generation Integration," Proc. IEEE INFOCOM, pp. 461-465, Apr.
[9] R. Anderson, A. Boulanger, W. Powell, and W. Scott, "Adaptive Stochastic Control for the Smart Grid," Proc. IEEE, vol. 99, no. 6, pp. 1098-1115, June 2011.
[10] "How a PV System Works," Florida Solar Energy Center, http://www.fsec.ucf.edu/en/consumer/solar_electricity/ basicshow_ pv_system_works.htm , 2012.
[11] A.-H. Mohsenian-Rad and A. Leon-Garci, "Energy-Information Transmission Tradeoff in Green Cloud Computing," Proc. IEEE GlobeCom '10, Mar. 2010.
[12] M.J. Neely, Stochastic Network Optimization with Application to Communication and Queueing Systems. Morgan Claypool, 2010.
[13] M. Neely, A. Tehrani, and A. Dimakis, "Efficient Algorithms for Renewable Energy Allocation to Delay Tolerant Consumers," Proc. IEEE First Int'l Conf. Smart Grid Comm. (SmartGridComm), Oct.
[14] A. Papavasiliou and S. Oren, "Supplying Renewable Energy to Deferrable Loads: Algorithms and Economic Analysis," Proc. IEEE Power and Energy Soc. General Meeting, July 2010.
[15] T.T. Kim and H.V. Poor, "Scheduling Power Consumption with Price Uncertainty," IEEE Trans. Smart Grid, vol. 2, no. 3, pp. 519-527, Sept. 2011.
[16] A. Kansal, J. Hsu, S. Zahedi, and M.B. Srivastava, "Power Management in Energy Harvesting Sensor Networks," ACM Trans. Embedded Computing Systems, vol. 6, p. 32, Sept. 2007.
[17] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, "Resource Management for Fading Wireless Channels with Energy Harvesting Nodes," Proc. IEEE INFOCOM, pp. 456-460, 2011.
[18] S. Chen, P. Sinha, N. Shroff, and C. Joo, "Finite-Horizon Energy Allocation and Routing Scheme in Rechargeable Sensor Networks," Proc. IEEE INFOCOM, pp. 2273-2281, Apr. 2011.
[19] M. Gatzianas, L. Georgiadis, and L. Tassiulas, "Control of Wireless Networks with Rechargeable Batteries," IEEE Trans. Wireless Comm., vol. 9, no. 2, pp. 581-593, Feb. 2010.
[20] L. Huang and M.J. Neely, "Utility Optimal Scheduling in Energy Harvesting Networks," Proc. ACM MobiHoc, May 2011.
[21] M. Marwali, H. Ma, S. Shahidehpour, and K. Abdul-Rahman, "Short Term Generation Scheduling in Photovoltaic-Utility Grid with Battery Storage," IEEE Trans. Power Systems, vol. 13, no. 3,
pp. 1057-1062, Aug. 1998.
[22] R.-H. Liang and J.-H. Liao, "A Fuzzy-Optimization Approach for Generation Scheduling with Wind and Solar Energy Systems," IEEE Trans. Power Systems, vol. 22, no. 4, pp. 1665-1674, Nov. 2007.
[23] R. Urgaonkar, B. Urgaonkary, M.J. Neely, and A. Sivasubramaniam, "Optimal Power Cost Management Using Stored Energy in Data Centers," Proc. ACM Int'l Conf. Measurement and Modeling of Computer
Systems (SIGMETRICS '11), June 2011.
[24] Y. Guo, Z. Ding, Y. Fang, and D. Wu, "Cutting Down Electricity Cost in Internet Data Centers by Using Energy Storage," Proc. IEEE GLOBECOM, 2011.
[25] D. Linden and T.B. Reddy, Handbook of Batteries. McGraw Hill Handbooks, 2002.
[26] A. Qureshi, R. Weber, H. Balakrishnan, J. Guttag, and B. Maggs, "Cutting the Electric Bill for Internet-scale Systems," Proc. ACM SIGCOMM, Aug. 2009.
[27] L. Georgiadis, M.J. Neely, and L. Tassiulas, Resource Allocation and Cross-Layer Control in Wireless Networks, vol. 1. Now Publishers Inc., 2006.
[28] D.P. Bertsekas, Dynamic Programming and Optimal Control, second ed. Athena Scientific, 2000.
[29] California ISO Open Access Same-Time Information System (OASIS), http:/oasis.caiso.com/, 2012.
[30] NREL: Measurement and Instrumentation Data Center, http://www.nrel.govmidc/, 2012.
Index Terms:
Batteries,Electricity,Renewable energy resources,Smart grids,Real time systems,Pricing,renewable energy generation,Batteries,Electricity,Renewable energy resources,Smart grids,Real time
systems,Pricing,real-time pricing.,Smart grid,optimal power management,Lyapunov optimization,energy storage
Yuanxiong Guo, Miao Pan, Yuguang Fang, "Optimal Power Management of Residential Customers in the Smart Grid," IEEE Transactions on Parallel and Distributed Systems, vol. 23, no. 9, pp. 1593-1606,
Sept. 2012, doi:10.1109/TPDS.2012.25
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A limit theorem on subinterval
Linda Green
A limit theorem on subintervals of interrenewal times
Adobe Acrobat PDF
Consider a renewal process {X[n], n ≥ 1} for which there is defined an associated sequence of independent and identically distributed random variables {B[n], n ≥ 1} such that B[n] is the length of a
subinterval of X[n]. We show that when attention is restricted only to B-intervals, the asymptotic joint distribution of the residual life and total life of a B-interval is that of a renewal process
generated by {B[n], n ≥ 1}.
Source: Operations Research
Exact Citation:
Green, Linda. "A limit theorem on subintervals of interrenewal times." Operations Research 30, no. 1 (1982): 210-216.
Volume: 30
Number: 1
Pages: 210-216
Date: 1982
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CMS Summer 2003 Meeting
Computational and Analytical Techniques in Modern Applications / Techniques numériques et analytiques dans les applications modernes
(Org: Peter Minev)
Around 1967, Kraichnan, Leith, and Batchelor (KLB) independently proposed the dual cascade theory, which is thought to describe turbulence in unbounded two-dimensional fluids. In a bounded
domain, however, the upscale energy cascade they discussed will be halted at the lowest wavenumber (corresponding to the domain size). An upper bound on the ratio of the total enstrophy to
total energy derived by Tran and Shepherd [Physica D, 2002] establishes that the energy must be dissipated at scales larger than the forcing scale. This result is based on the assumption that
the square root of the ratio of mean enstrophy to mean energy injection is spectrally localized to the forcing region. We investigate the conjecture that turbulence driven by a spectrally
localized temporally white-noise random forcing satisfies this assumption. We also provide numerical evidence that energetic reflections at the lower spectral boundary may eventually lead to
a large-scale k^-3 energy spectrum, in agreement with the large-scale k^-3 spectra observed in the atmosphere by Lilly and Peterson [Tellus 35A, 379 (1983)]. A spectral constraint derived by
Tran and Bowman [Physica D, 2003] establishes that the two inertial-range exponents must sum to -8. A large-scale k^-3 spectrum resulting from reflections at the lower spectral boundary would
then explain the small-scale k^-5 spectrum frequently observed in numerical simulations of the enstrophy range. We propose that combined supergrid and subgrid models based on Kolmogorov's
hypothesis of self-similar energy (or enstrophy) transfer could be used to mimic the behaviour of an unbounded fluid in a doubly periodic domain, thereby allowing one to address the validity
of the classical KLB theory for unbounded fluids.
The lecture begins with a short review of physical background of composite materials and porous media like periodicity or randomness (from static and kinetic mechanical problems of composite
materials, heat and mass transfer in porous media, wave propagation in heterogeous media, etc.), and then introduces several recent mathematical results on the asymptotic homogenization
methods. In particular, I would like to advance our methods and numerical results for the above physical problems. Finally, some open problems are also presented in corresponding sections.
New efficient algorithms are formulated and analyzed for the solution of a class of linear partial integro-differential equations of parabolic type in the unit square. In these methods,
orthogonal spline collocation (OSC) with C^1 piecewise polynomials of degree ³ 3 is used for the spatial discretization. For the time stepping, alternating direction implicit (ADI) methods
based on the backward Euler method, the Crank Nicolson method and the second order BDF scheme are considered. Such methods reduce the multidimensional problem to sets of independent
one-dimensional problems in which the OSC matrices are easily determined since, unlike the finite element Galerkin case, no integrals must be evaluated or approximated. The methods are shown
to be of first or second order accuracy in time and of optimal order accuracy in the L^2, H^1 and H^2 norms in space. From the analysis, a new optimal order H^2 estimate is obtained for an
ADI OSC Crank Nicolson method for the heat equation in two space variables. ADI OSC methods are also examined for the solution of a class of evolution equations with a positive type memory
term, and an optimal order L^2 estimate is derived for each method.
This is joint work with Amiya Pani and Bernard Bialecki.
There are numerous challenges faced by scientists when developing algorithm libraries for scientific applications. Issues of functionality, stability, performance, and flexibility depending
on the specific application areas are at the focus of researchers working in both theoretical and applied areas. This talk discusses some of the approaches to development and implementation
of two- and three-dimensional data structures for applications in molecular biology, mechanical engineering and GIS. The talk concentrates on issues of algorithm efficiency, precision of the
result and numerical stability. Challenges in the development of the ECL (Exact Computational Library) for performing exact computations in the fixed precision floating-point arithmetic are
discussed. The ECLibrary is based on the interval point arithmetic, iterative approximation methods, reduction technique and algorithms for performing complex transformations on
floating-point numbers. The ESAE algorithm (Exact Sign of Algebraic Expression) will be described. Some implementation issues will be also discussed.
The use of standard numerical discretization techniques, such as explicit RK methods, to integrate non-linear differential equations often leads to scheme-dependent instabilities and/or
convergence to spurious solutions when certain step-sizes or parameter values are used in their simulations. This paper presents some nonstandard finite-difference methods that are, in
general, free of the aforementioned drawbacks. These schemes are designed in such a way that they preserve the important features/properties of the continous model they approximate.
The convection diffusion equations, which describe many realistic procedures in many problems of science and technology; eg., fluid mechanics, heat and mass transfer, groundwater modelling,
petroleum reservoir simulation and environmental protection, are very important and difficult in numerical simulation. The standard finite difference methods or finite element methods will
introduce severe nonphysical oscillations into the numerical solutions since the corresponding discrete schemes are unstable for the problems. Because of satisfying both the stability and the
conservation of mass, the methods of finite-volume-type with upwinding techniques have obtained high successes in the numerical simulation of the convection diffusion problems. However, the
standard upwinding technique treating convection terms usually derives lower-order accuracy schemes for the problems. In this talk, we will present the nonstandard high-order upwinding finite
covolume methods for the convection diffusion problems. The conservation law of mass and the unconditional stability are analyzed, the high-order error estimates are obtained for the methods.
Numerical experiments are given to demonstrate the performance of the schemes.
We will discuss an immersed finite element method for axial symmetric three dimensional nonlinear interface problems. The basis functions in this method are piece-wise linear polynomials
satisfying the jump conditions approximately (or even exactly in many situations). In addition, the mesh in this method does not have to be aligned with the interface because the interface is
allowed to pass through the elements. Therefore, structured Cartesian meshes can be used in this method to facilitate efficient numerical solutions. We will show that this method has the
usual second and first order convergence rates in L^2 and H^1 norms, respectively. Numerical examples for a nonlinear interface problem arising from ion optics modelling in composite
structures will be provided to illustrate features of this method.
In this talk we will survey on recent results on nonconforming finite elements on quadrilaterals (http://www.nasc.snu.ac.kr/). These elements, originally developed for elliptic problems, are
applied to solving Stokes, elasticity, Helmholtz, and Maxwell's equations. Error estimates and selected numerical results will be shown.
Although interest in the biomechanics of the brain goes back over centuries, mathematical models of hydrocephalus and other brain abnormalities are still in their infancy and a much more
recent phenomenon. This is rather surprising, since hydrocephalus is still an endemic condition in the pediatric population. Treatment has dramatically improved over the last 30 years thanks
to the introduction of CSF-shunts. This too, however, is not without problems and the shunt failure rate at 2 years post shunt insertion is 50common causes of shunt failure is due to shunt
obstruction. Common sense suggests that the optimal shunt location would be in the ventricular region that remains largest after ventricular decompression drainage. In this talk, we will
report on some recent progress towards the solution of this problem.
An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. We combine a multiquadric integral
formulation with a previously employed coordinate stretching technique. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This
indicator is used to adaptively add data centres and collocation points. Our method resolves extremely thin layers accurately with fairly few basis functions. We demonstrate the effectiveness
and the robustness of our new method on a number of examples.
Joint work with Leevan Ling, SFU.
We consider spatially discrete FitzHugh-Nagumo equations as a model for ionic conductances that generate the action potential of nerve fibers in motor nerves of vertebrates. Existence results
for front and pulse solutions are discussed. Numerical techniques are considered to approximate solutions to mixed type functional differential equations obtained when considering traveling
fronts and pulses of these equations. Extensions to coupled systems of spatially discrete FitzHugh-Nagumo equations corresponding to bundles of nerve fibers will be discussed.
A new gradient recovery technique (PPR) is introduced to construct a posteriori error estimator. The recovery operator is polynomial preserving and insensitive to mesh distortion. Some
theoretical results are provided regarding the recovery operator. In addition, some numerical results are provided in comparison with the Zienkiewicz-Zhu error estimator based on the
superconvergent patch recovery (SPR).
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│Weight│ Topic │
│ │Study designs, measurement types, and which statistics to use with each. You may be given descriptions of studies and asked to describe their design and the kind of statistic you would use │
│50% │to analyze their results with. You may be asked to give an example of a particular kind of study design, including a research model and description of variables. You may be given an excerpt │
│ │of data from a study and asked what kind of statistic you should use. Also important is an understanding of extraneous/confounding variables and how to deal with them, and a conceptual │
│ │understanding of mediating and moderating variables, proximal vs. distal outcome variables, and independent vs. dependent variables (relative to a research model). │
│10% │Descriptive statistics. You should be able to determine the mean, median, mode, variance, and standard deviation for a data sample (when appropriate), and be able to construct a frequency │
│ │distribution for it and be able to state whether it is unimodal or bimodal, symmetric or positively or negatively skewed. │
│ │Definitions and concepts. You should be able to describe the following: Empirical vs. analytic research; scientific explanations; the scientific method; qualitative vs. quantative research │
│ │methods (and examples of each); primary vs. secondary research literature; meta-analysis; the three ethical principles of human subjects research (from lecture); sampling; generalization of │
│10% │study results; ethnographic research; the typical use of different types of studies during the software development process; internal vs. external validity of a study; reliability vs. │
│ │validity of a measure; system usability; coding manual (for behavioral/observational measures); inter-rater reliability; retrospective vs. prospective study; demographic measures; between │
│ │subjects design; distribution of individuals vs. distribution of means; levels/treatments/conditions in a study. │
│ │Hypothesis testing. You should be able to describe the basic logic of hypothesis testing, research vs. null hypotheses (and how they relate to the populations/population parameters in a │
│10% │study and whether the test is one vs. two-tailed), and significance levels. If provided output from R for Chi-squared goodness-of-fit, Chi-squared test for independence, Pearson correlation │
│ │or t-test of independent means you should be able to state what the results mean in English (relative to the study hypotheses) and reformat the results into publication format. │
│5% │Sampling and generalization. You should be able to describe simple random, systematic, and stratified sampling, and what constitutes a representative vs. biased sample and how this affects │
│ │the generalizeability of your study. │
│5% │You should be able to construct a scatter plot from sample data and be able to estimate (using the "eyeball" method) the Pearson correlation coefficient given a scatter plot. │
│ │Measures. You should be able to describe the steps you would take in validating a measure, including test-retest reliability, internal consistency, and face, content, criterion, and │
│5% │construct (convergent and discriminant) validity, or identify when these are mentioned in the description of a measure. You will not be asked to calculate any of the figures, but you may be │
│ │asked to interpret them. │
│5% │Survey measures. You should be able to describe and provide examples of: open-ended questions; restricted/closed-ended questions; rating scales (including Likert and semantic differential); │
│ │and composite measures. If provided with a filled out survey index (composite measure) you should be able to compute the composite score. │
│Weight│ Topic │
│ │Study designs, measurement types, and which statistics to use with each. You may be given descriptions of studies and asked to describe their design and the kind of statistic you would use │
│ │to analyze their results with. You may be asked to give an example of a particular kind of study design, including a research model and description of variables. You may be given an excerpt │
│ 50% │of data from a study and asked what kind of statistic you should use. Also important is an understanding of extraneous/confounding variables and how to deal with them, and a conceptual │
│ │understanding of mediating and moderating variables, proximal vs. distal outcome variables, and independent vs. dependent variables (relative to a research model). Includes descriptive │
│ │studies, correlational studies, demonstration studies, and two-group, multi-group and multi-factor between-subjects and two-group within-subjects experiments. Includes descriptive │
│ │statistics, chi-square goodness of fit, Pearson correlation, t-test for independent means, t-test for dependent means, one-way and multi-factor ANOVAs. │
│ │Data screening. You should be able to describe how to handle study data once it is acquired including: screening for outliers, floor or ceiling effects or other violations of analysis method│
│ 10% │assumptions; knowing how to apply data transformations and when they are appropriate; checking for potential confounds (between-subjects) or order effects (within-subjects); and how to do │
│ │subgroup analysis. │
│ │Definitions and concepts. You should be able to describe the following: Empirical vs. analytic research; scientific explanations; the scientific method; qualitative vs. quantative research │
│ 10% │methods (and examples of each); primary vs. secondary research literature; meta-analysis; the three ethical principles of human subjects research (from lecture); internal vs. external │
│ │validity of a study; reliability vs. validity of a measure; retrospective vs. prospective study; demographic measures; between subjects design vs. within-subjects design; levels/treatments/ │
│ │conditions in a study; longitudinal vs. cross-sectional experimental design; single-subject vs. group design; covariate; quasi-independent variable. │
│ │Hypothesis testing. You should be able to describe the basic logic of hypothesis testing, research vs. null hypotheses (and how they relate to the populations/population parameters in a │
│ 10% │study and whether the test is one vs. two-tailed), and significance levels. If provided output from R for Chi-squared goodness-of-fit, Chi-square test for independence, Pearson correlation, │
│ │t-test of independent means, t-test of dependent means or one-way or multi-factor ANOVAs you should be able to state what the results mean in English (relative to the study hypotheses) and │
│ │reformat the results into publication format. │
│ 5% │Descriptive statistics. You should be able to determine the mean, median, mode, variance, and standard deviation for a data sample (when appropriate), and be able to construct a frequency │
│ │distribution for it and be able to state whether it is unimodal or bimodal, symmetric or positively or negatively skewed. │
│ 5% │Study documentation. You should know the major parts and structure of a study proposal and report/publication. For example, you might be asked where your hypotheses should be stated, or what│
│ │parts of a sample proposal are missing. │
│ 5% │Power & significance. Be able to describe the relationship between power, effect size, and Type I and Type II errors, and the major study parameters that affect these. You should be able to │
│ │conduct a power analysis, given sufficient information, and be able to compute effect size for two-group between-subjects and within-subjects experiments. │
│ │Measure Validation. You should be able to describe the steps you would take in validating a measure, including test-retest reliability, internal consistency, and face, content, criterion, │
│ 5% │and construct (convergent and discriminant) validity, or identify when these are mentioned in the description of a measure. You will not be asked to calculate any of the figures, but you may│
│ │be asked to interpret them. │
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East Brunswick Algebra Tutor
Find an East Brunswick Algebra Tutor
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6 Subjects: including algebra 1, algebra 2, geometry, prealgebra
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36 Subjects: including algebra 2, algebra 1, chemistry, English
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18 Subjects: including algebra 2, writing, algebra 1, elementary (k-6th)
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4 Subjects: including algebra 1, geometry, ESL/ESOL, prealgebra
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analysis of circuits consisting of passive elements, and coursework incorporates techniques such as total imp...
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An Alternative to Simpson?
July 16th 2012, 07:03 AM #1
Feb 2010
An Alternative to Simpson?
Will this function integrate, or am I stuck with Simpsons rule? (It's 35 years since I did my calculus, and I'm getting a bit rusty at it.)
Re: An Alternative to Simpson?
This is what Wolfram gets...
integral[1/(a*x^4 + b*x + c)] - Wolfram|Alpha
Re: An Alternative to Simpson?
Thanks, Prove it, I needn't feel quite so embarrassed about not being able to solve that lot for myself. I think Simpson will be just fine!
July 16th 2012, 07:24 AM #2
July 18th 2012, 06:36 AM #3
Feb 2010
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Physics Forums - View Single Post - Diff. Geometry Question
I talk too much, and this post has gone 'way overboard. I won't be responding in this thread again, unless some very specific question (with a short answer!) comes up.
Hi Sal,
I am very ignorant of analysis, and so I would love to hear what it is
all about
"Analysis", at least where I came from, was multivariate calculus
"done over again, right". But I suspect the term may cover quite a
bit more than that depending on where you are and who's using it.
Real (as opposed to complex) analysis is calculus in R
with everything proven rigorously as you go; epsilonics are the order
of the day. A few things are a little hard to grasp at first, such as
the definition of continuity which may seem to have no apparent
relation to what "continuity" means in calculus-1 (a function is
continuous if and only if the inverse image of any open set is open).
Typically one doesn't go nearly as far into techniques for solving
particular integrals and such in an analysis class, as one does in
basic calculus; the issue is more with definitions than with solutions
to numeric problems.
(Some schools kind of merge calculus and analysis. I think Harvard
may do it that way.)
I was certainly not aware that
this is where tensors are from.
In my experience, almost nobody who has not actually learned tensor
calculus has a clue what a tensor is
Now, regarding tensors, are they as scary as many people say? Or is
this a percieved image with no realism to it?
You are driving too far ahead on the road. There are a few things you
should be aware of.
1) You should get through calculus, linear algebra, and all the other
things which come up in a good two-semester post-calculus algebra
course before you worry much about tensors. (A half-semester of
linear algebra is just an introduction, please note.)
2) Very few people who start out double-majoring in math and physics
actually get two degrees, one in math, and one in physics. Don't
bet the (psychological) farm on getting a double degree; odds are it
won't work out that way.
The reason (2) is true is mostly because it's a wide world and it's
hard to see what the possibilities are from the vantage point of a
high school classroom. Math and physics stand out, everybody knows
about them, they're really "techie", they have a certain caché, and a
lot of people think it would be really "cool" to be in both fields.
Maybe they seem like "ritzier" fields than some pedestrian old
engineering discipline, for instance. And so lots of folks enter
university thinking they're going to be math/physics majors all the
But when you get to university you find, first, that there are an
awful lot of fields with interesting work, and with extremely
challenging work -- theoretical ("pure") math and physics have
monopoly on being difficult to master or having cool,
sophisticated mathematical underpinnings!
And, second, you learn a fact about the world: A degree in
theoretical math or theoretical physics is excellend preparation for
... well, teaching other people to do it, at the university level, if
you happen to be brilliant and if you find a tenure slot somewhere. If
you can't find a tenure track post at a university, you can always,
um, paint houses or something. To put it bluntly, it's nice work
... if you can get it.
Mathwonk states in his paper on algebraic geometry (that I briefly
skimmed through) that this subject is related to the study of geometry
of polynomials, or as he now likes to think, the study of geometry of
rings (or was it groups?). This sounds interesting, but as with a
large chunk of higher mathematics, I am again very ignorant of the
subject. I am aware that groups and rings are important concepts in
abstract algebra, but that is all.
Mathwonk can be seriously obscure when he puts his mind to it, IMHO.
I will have a closer look at solid state physics, as it is an area of
theoretical physics that I have not looked at. I think, at my age, it
would be better for me to see the "big picture", perhaps, something
that I haven't been doing.
Solid state physics is seriously cool, very important, deep,
difficult, complex ... and not what I would call theoretical physics,
at least not in the same sense that, say, astrophysics is
"theoretical". In fact, I would describe it as "Applied" with a
capital A. As I said, it's solid state physics which makes your
computer go. If you don't know what I mean when I say that, then you
definitely have something to learn about solid state physics, and
about electrical engineering, as well.
Entering an applied field is not "death", and what's more there's more
demand for people who work on things which have applications than for
people who don't.
I am guessing, because higher-level geometry and analysis are "pure"
branches of mathematics, they are on a completely different level,
than say, partial diff. eqn's?
The term "level" is undefined in this context, so I can't really
answer that.
Is it not Fast Fourier Transforms (or FFTs) that govern the rendering
of computer graphics from the graphics pipeline?
Is it correct that
Fourier transforms are essentially methods to solve differential
No. You are thinking of Laplace transforms; they are not at all the
same thing.
Fourier transforms move a function from configuration space (where it
has, like, values and stuff) into phase space (where you get to see
the frequency spectrum).
They are used in quantum mechanics, where each quantum state could be
said to be one component of the Fourier transform, and they're used in
signal processing, because the transform of an acoustic signal is
exactly its frequency spectrum, and the transform of an image is its
spactial frequency spectrum. But AFAIK polygons get blasted into the
frame buffer in your graphics card with no use of Fourier transforms.
It was these types of questions that I liked solving, and as a
sidenote, geometry is my favourite branch of mathematics so far.
OK time for the bucket of ice water. Sorry, I apologize for this in
advance. I'll start with a little background, and I expect it'll
rapidly become obvious what I'm leading up to.
When I was an undergrad I took a semester of organic chem, and I
really loved the "carbohydrate game", as they called it in that
course, which was looking at how carbohydrate molecules could be
mushed around from one form to another. And when we finished that
unit of the course, I asked about further information or opportunities
to study that sort of thing, because it was far and away the coolest
bit of chemistry I'd ever encountered ... and they told me "There
isn't any more. That was all of it. That field is dead, it's been
mined out, nobody's studying it any more."
I took a semester of point set topology, and I thought it was just the
coolest thing ever. The textbook (Munkres) was more fun to read than
a dime store novel, and the "mental pictures" it all produced were
fabulous. Talka about "visualizeable" -- it practically
that word! At the end of the course, I asked about the
next level
of point set topology -- what was the follow-on
course? For surely, I wanted to take it! ... and they told me,
"There isn't any more. That was all of it. That field is dead, it's
been mined out, nobody's studying it any more." The "followon" to
point set topology was algebraic topology, which bears about as much
resemblance to point set topology as Mandarin Chinese bears to
Now, geometry.... Let me put it this way. I majored in math at a
good school, and in all my college years, I never took a geometry
course, in the sense that you mean it; what's more I'm not sure any
were offered at my school (aside from the differential/algebraic
sort). In fact, the only geometry class I've ever taken in my life
was a high school class called "coordinate geometry", and it didn't
bear much resemblance to the plane or solid geometry classes of
yesteryear. I started to read a bit about plane and solid geometry
last year, just for fun, from a book (published by Dover) in which the
author talks about the stuff "they never taught you in school"
because, for the most part ... geometry isn't even taught any more.
Calculus has almost entirely superseded it.
On the bright side, there are a lot of other fields in math which are
just as visual, and just as pleasing. The downside is that, for the
most part, you must work harder to visualize things when you're not
talking about the actual geometry of the universe we (appear to) live
Take note that these courses are generally taught by people who have a
really solid mathematics grounding (ie a degree in it, most commonly),
as opposed to what I've heard about the American AP Calculus courses
(I have heard rumours that they are often taught by incompetent
teachers. Is this true?).
I don't know; I only attended one high school, after all. The math
faculty in my high school was variable, some better than others;
however, the factulty were generally a whole lot better at college.
Also, a few questions regarding analysis, before I finish writing this
post. How intuitive is this field?
Very, at least to start with.
Does it throw you in the deep end?
I don't know what that means.
Since you've probably never encountered much in the way of serious
proofs before, and you've almost certainly never been called upon to
prove much of anything, it's likely to be difficult to start with, but
an introductory analysis class is geared to people who are all going
through the same struggle, so it's not really a problem.
The first analysis test I recall taking had ten questions, and each
one was just a statement, no questionmark, no "please do...", no
nothing. Just a statement. And for each one, we were expected to
either prove it or find a counterexample.
And how elegant is the field?
That's a rather poorly defined term, isn't it?
The "Topics in Advanced Mathematics" course covers various topics such
as geometry, set theory, number theory, history of mathematics and a
few others. If I'm not mistaken, you are able to choose the area of
the list of topics provided that you wish to study. The course itself
only requires that you have studied a second-year mathematics
unit. Perhaps if I can choose, I wold be wise to choose to study
algebraic and differential geometry?
It would be wise to choose something based on what you've learned at
the time you pick. It's too early to try to make a decision like
that; wait until you've gotten through calculus and algebra!
Differential equations and linear algebra
are covered in the same course
Quelle bizarre!
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Posts by
Total # Posts: 609
Intro to Computers
your brother just lost his job and wants to build an online profile with a social media site geared at possible employers and HR recruiters. He asks you if you have a suggestion as to where to go.
what do you tell him? A. WordPress B. LinkedIn C. Facebook d. Flickr
Linear Algebra
Fnd the left null space of matrix A = [5 -3 1] [-2 4 -6] [11 -8 5]
Linear Algebra
Use Gauss Jordan Elimination to write the solution of the system of equations: x1+4x2+2x3=17 3x1+x2-5x3=7 2x1-3x2-7x3=-10
Thanks we thought 8x8 but not sure
Ms. Fisher is using 64 carpet tiles to make a reading area in her classroom. Each tile is square that measures 1 foot by 1 foot. What is.the length and width of the rectangular area she can make with
the smallest possible perimeter?
Bronsted reaction HCO3-+HF<--> acid?+base?
Complete the Bronsted lowry reactions HPO42-+H+<--> HPO42-+OH-<-->
Chemistry 2
Calculate Kp for the reaction C(s)+CO2(g)<-->2CO(g) kp=? C(s)+2H2O(g)<--->CO2(g)+2H2 kp1=3.47 H2(g)+CO2(g)<--->H2O(g)+CO(g) kp2=0.735
sam and sue have only nickels and dimes. sue has $.35. sam has the same number of dimes as sue has nickels, and he has half as many nickels as sue has dimes. how many nickels does sue have?
Sorry I mean [(2x-y)^2]dx
What is the integral of [(2x-y)^3]dx?
Linear Algebra
Use the existence and uniqueness theorem to determine where solutions will exist and be unique for the differential equation: dy/dx=((2x-y)^2)/(x^2y+2xy^2)
The lawn is rectangular shaped with a length of 10 yards and a width of 40 yards. A bag of grass seeds covers 50 square yards at a cost of $ 4.99 per bag. What is the total area of the new lawn? How
many bags of grass seeds will they need to buy? What is the total cost to reno...
the lawn rectangular shaped with a lenght of 10 yards and a width of 40 yards. A bag of seed covers 50 square yards at a cost of $4.99 per bag. What is the total area of the new lawn?
Given below is the sequence of the mRNA ready for Translation" 5' - AUG CUA UAC CUC CUU UAU CUG UGA - 3' a) what is the first codon technically known as and which amino acid does it code for? b) how
many amino acid residues will make the polypeptide chain correspo...
American Government
American Government
Miranda v. Arizona (1966) A. Example of voting rights B. Example of racial - discrimination C. Example of criminal defendant case D. Example of a states rights case I think the answer is D. Please
Determine whether the statement is true/ false 1. Traveling 10 miles in 2 hours is same as traveling 4 miles in one hour 2. If two triangles are similar, the length of their sides are the same 3. x/
12 = 4/24. Then x=2 Multiple Choice. Choose the best answer for each question 4...
The parking cost at a garage is c=1.3(h-1) , where h is the number of hours you park and C is the cost in dollars. When is the cost of parking $10
Question 5. 5. The Tenth Amendment balances out national authority by (Points : 1) stating that states have first priority in those areas specifically not given to Congress. asserting that states
rights trump national authority. limiting state function to only those matt...
Social Studies
During the western expansion, what animals, plants did the settlers encounter on the California trail?
If GMAT scores for applicants at Oxnard Graduate School of Business are N(500, 50) then the top 5 percent of the applicants would have a score of at least (choose the nearest integer):
The yield curve is typically _____________, meaning that the annualized interest is higher for debt securities with longer terms to maturity. A. Upward B. vertical C. Downward D. Horizontal
Personal Finance
Aaron wants to put $200.00 per month into an IRA account at 15% for four years. What is he solving for using his financial calculator? A. Present Value B. Future Value C. Interest Rate D. Payment
You plan to saving 6,400 per year, beginning immediately. Making 4 deposits in an account that pays 5.7% interest. How much will there be in 4 years from today?
mat 117
How do you factor the difference of the two squares? How do you factor the perfect square trinomial? How do you factor the same and difference of two cubes? Which of these tree makes the most sense
to you?
show me step by step how to solve t hese problems. 5 7/8 +(3 1/3 - 1 1/2) i cannot get this down please help
Compute SEdiff Mean1=23 Mean2=18 SD1 =4 SD2 =6 N1 =9 N2 =6
if the perimeter of a rectangle is 3 times the length, plus twice the width. what is the equation to find the perimeter of the rectangle? I think it's P= 3 x l(length)+2. Is this right?
Healthcare Finance
2 A firm that owns the stock of another corporation does not have to pay taxes on the entire amount of dividends received. In general, only 30 percent of the dividends received by one corporation
from another are taxable. The reason for this tax law feature is to mitigate the ...
Healthcare Finance
Jane Smith currently holds tax-exempt bonds of Good Samaritan Healthcare that pay 7 percent interest. She is in the 40 percent tax bracket. Her broker wants her to buy some Beverly Enterprises
taxable bonds that will be issued next week. With all else the same, what rate must ...
Healthcare Finance
2 A firm that owns the stock of another corporation does not have to pay taxes on the entire amount of dividends received. In general, only 30 percent of the dividends received by one corporation
from another are taxable. The reason for this tax law feature is to mitigate the ...
if a swimmer swims 85.4 yards in five minutes, how many meters will he swim in 70.0 seconds?
Did I cite the start my source correctely according to APA format. The source did not have a date or an author listed. WebMD (n.d.) Attention Deficit Hyperactivity Disorder: What Is ADHD?
Is this a scholarly website it is called WebMD? I am writing a paper on Attention Deficit Hyperactivity Disorder: What Is ADHD?
business math
if I deposited $625 in a retirement account per month and it had a return of 3.84% a year compounded monthly. How much will I have in my account in 15 years?
Paul Warren is a self-employed mechanic. Last year he had total gross earnings of $44,260. What are Paul s quarterly social security payments due to the IRS?
Universal Exporting has three warehouse employees: John Abner earns $422 per week, Anne Clark earns $510 per week, and Todd Corbin earns $695 per week. The company s SUTA tax rate is 5.4%, and the
FUTA rate is 6.2% minus the SUTA. As usual, these taxes are paid on the fir...
Ransford Alda is a self-employed security consultant with estimated annual earnings of $90,000. His social security tax rate is 12.4%, Medicare is 2.9%, and his federal income tax rate 14%. How much
estimated tax must Ransford send to the IRS each quarter?
Paul Warren is a self-employed mechanic. Last year he had total gross earnings of $44,260. What are Paul s quarterly social security payments due to the IRS?
Fran Mallory is married, claims five withholding allowances, and earns $3,500 (gross) per month. In addition to Federal income tax (FIT), social security, and Medicare withholding, Fran pays 2.1%
state income tax, and ½% for state disability insurance (both based on her...
Mitch Anderson is a security guard. He earns $7.45 per hour for regular time up to 40 hours, time-and-a-half for overtime, and double time for the midnight shift. If Mitch worked 56 hours last week,
including 4 hours on the midnight shift, how much were his gross earnings?
soc 120
Virtue ethics regards which of these as virtues?
Epicure Market prepares fresh gourmet entrees each day. On Wednesday, 80 baked chicken dinners were made at a cost of $3.50 each. A 10% spoilage rate is anticipated. At what price should the dinners
be sold to achieve a 60% markup based on selling price? (price of the dinner i...
this is the answer I got at first but it was wrong. And I did this problem several times.
Epicure Market prepares fresh gourmet entrees each day. On Wednesday, 80 baked chicken dinners were made at a cost of $3.50 each. A 10% spoilage rate is anticipated. At what price should the dinners
be sold to achieve a 60% markup based on selling price? price of the dinner is...
Office Market is having a sale on printer paper. If you buy 2 reams for $8.99 each, you will get a third ream free. Calculate the markdown percent. (Round the percent to the nearest tenth.)
You are the buyer for The Shoe Outlet. You are looking for a line of men s shoes to retail for $79.95. If your objective is a 55% markup based on selling price, what is the most that you can pay for
the shoes and still get the desired markup?
sabina had $200 she spent 3/5 on clothes and 3/4 of the reminder on books. she saved the rest. what fraction of her money did she spend on books. how much did she spend on books and clothes
altogether. what fracion of her money didnshe save
Critically discuss 5 ways in which in which xenophobia imapact on the community?
How can you use a right angle to determine the classification of another angle?
Jill said that an angle is made of two rays. Is she correct? Explain
computer can do 1000 operations in 4.25 X 10 to the 6th power, how many operations can this computer do in an hour?
pre calculus
If you are driving down a 10% grade, you will always be 10% below where you would have been if you could have driven on flat ground from where you started. Suppose you descend to where you are 20
feet below your starting point. What concept does the grade relate to...
Question 10. 10. The principal powers of Congress include all except (Points : 1) the power of the purse. the power to declare war. implied powers. the power to hire generals.
pre calculus
I meant to say *4y instead of 4x
pre calculus
Write the equation of the circle in standard form. Find the center, radius, intercepts, and graph the circle. x^2+y^2-12x-4x+36=0 How on earth do I do this?
I am a 3 digit number My ones place is one more than my hundred My hundreds place is 3 greater then 6 My tens place is half of my hundred.
Finite Math
How to solve this problem using this equation P(T)=Poekt how long must $5700 be in bank at 8% compounded yearly to become $10,550.30 where P0 is the amount initially placed in savings, t is the time
in yaers and P(t) is the balance after time
You have an invoice for a major expenditure due in 5 years for $82,000 and the annual interest rate of 5%. How much do you need now to pay for this expenditure?
not sure where to begin
Look around you and think of what things make your life what it is. The home you live in, the neighborhood, the availability of services (fast food ... which is really a service, IMO, because someone
is cooking for you and cleaning up after you ... then look around and think o...
Math Perimeter
write as a decimal seven and eighty-seven hundreths
Understanding Perimeter
When finding the perimeter of a figure on a grid, why do you not count the spaces inside the grid?
Understanding Perimeter
A park has the shape of a trapezoid. Two of the sides are 25 meters long. The other two sides are 40 meters and 20 meters long. What is the perimeter of the park?
Math Perimeter
Draw a figure with 22 units
Math Perimeter
Draw a figure with the given perimeter, 10 units.
The average height for Chinese women is 62.5 inches with a standard deviation of 4.6 inches. What is the probability that a randomly selected woman in the United States would be ≥69 inches tall?
fundamental accounting principle II
How do I explain to a customer about the $6000. loss reported on the sale of its investment of common stock that the customer has 40% interest. When the 2010 income statement reported earning form
all investments were $126,000. On January 2011 the company sold the investment s...
Proofread find two missing commas in compound sentences. Let's say it is ten o clock in the morning. Your room is a mess! Your clothes are on the floor and games are everywhere. It won t take long to
clean but you can t do it. There isn t anything easie...
Estimated product. 81,000
How much would an initial investment of $500 be worth in 90 days if it earned 3% daily?
slope of the function f (x) = 4/3x -7 to the slope of the function that fits the data in the table.
the length of a rectangle is x +19 and the width is 6. the length of a scond rectangle is 6x +2 and width is 4. What is the expression equal to the area of the first rectangle?
The trailer measures nine feet by forty feet. how do I rewrite this to make it right
A calculator was dropped at an equation of d= 0.5*9.8t^2 + 5t How long would it take for the calculator to hit the ground? HINT: SET THE EQUATION TO ZERO.
college english
Talk to the Hand: The result when communication pathways are blocked
A 1.500m long uniform beam of mass 30.00 kg is supported by a wire as shown in the figure. The beam makes an angle of 10.00 degrees with the horizontal and the wire makes and angle of 30.00 degrees
with the beam. A 50.00 kg mass, m, is attached to the end of the beam. What is ...
. For each of the following, assume that the two samples are selected from populations with equal means and calculate how much difference should be expected, on average, between the two sample means.
a. Each sample has n _ 5 scores with s2 _ 38 for the first sample and s2 _ 42...
Calclte the number of moles of ethane needed to produce 10.0 grams of water according to the following reaction.
A number between 10 and 20. It rounds to 20. One of this number's digits is 5 greater than the other. What number is it?
What would happen to the profit maximizing level of output if the market price suddenly rose to $54 per case? Explain why the output level changes.
According to American Airlines, Flight 215 from Orlando to Los Angeles is on time 90% of the time. Suppose 150 flights are randomly selected. Find the probability that few than 125 flights are on
A sample of n _ 11 scores has a mean of M _ 4. One person with a score of X _ 16 is added to the sample. What is the value for the new sample mean?
4,195.43 1,408 625.00 + 239.00 i got 6,466.43 am i right
For this part of the assignment, we will focus on the demand curve. Draw the demand curve for the A-Phone. Explain how the graph, price, and quantity demanded will change if the following occurs:
There is an overall increase in income. It is discovered that there...
Foreign languages
How do you write in the Fiji language? Hello dear, how was your day?
Which of the following is NOT a technique for emphasizing material in a resume? A. Presenting it in vertical lists B. Including it as part of a lengthy paragraph C. Offsetting it with white space D.
Positioning it as the top of a page
If a particle moves with velocity V PA = (2.0 m/s) + (4.0 m/s) relative to frame A, what is the velocity V AP of the frame relative to the particle? a. -(2.0 m/s) + (4.0 m/s) b. -(2.0 m/s) - (4.0 m/
s) c. (2.0 m/s) + (4.0 m/s)
Did I choose the correct answer #3 In the mid- to late 1880s, the United States experienced a surge in immigrants from <China and Japan> 1)Japan and Cuba. 2)China, Japan, and Mexico. 3)China and
Japan < I chose this one< 4)China and Canada.
Sensory was intact to light touch, proprioception, and pinprick.
With vectors a = 5.5 and b = 4.2, what are (a) the z component of a x b , (b) the z component of a x c , and (c) the z component of b x c ? The angle between a and b is 90 degrees.
Albert Ellis depicts the relationship between irrational belief and negative emotions through use of: A. A contingency contract B. dialectical theory C. The abc model D. Observational learning
Thank you!
Did I answer this question correctly? wi If all of a company s Filipino employees sit together for lunch, sociologists would say they have formed their own _________. (c) ingroup A). diversified
group B). outgroup C). ingroup ( I chose this one, ingroup ) D). ethic group
If the concept is truck, the prototype might be: A. A ford pick up B. bulldozer C. An eighteen-wheeler D. Personnel carrier
It's not the 3rd option, because the stem stores water Roots do not contatin stomata I beleive it would be either of the first two; my guess would be the 2nd
According to Erikson the first psychosocial crisis or conflict in life involves the:
History/ Social Studies
How far was the US policy towards Europe between 1945 - 1952 based on self interest? What were the policies?
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finite and infinite sets...denumerable
November 22nd 2009, 01:55 PM #12
November 22nd 2009, 06:56 AM #11
November 21st 2009, 03:03 PM #10
November 21st 2009, 03:01 PM #9
November 21st 2009, 02:55 PM #8
November 21st 2009, 02:40 PM #7
November 19th 2009, 09:49 PM #6
November 19th 2009, 09:12 PM #5
November 19th 2009, 09:08 PM #4
November 19th 2009, 05:14 PM #3
November 19th 2009, 04:44 PM #2
November 19th 2009, 03:22 PM #1
I forgot to change the index to $\scriptstyle k$. The reason why it is $n\ge 2$ is because its if $n=1$ and wasn't worth the weriting. Lastly, doing the countable case requires your nice little
pictures which: A) I don't like pictures, B) I can't draw pictures, C) was uneccessary here. Also, the starred section above is a little misleading. Most books use countable as countably infinite
and "at most countable" for what you said.
I don't like the way it's stated. Imho there are some wrong things.
It suffices for n to be $\geq 1$. Then I don't understand why you put $1\le i \le n$, since i is just an index in the union, and we don't need any condition over it (it's given in the notation of
the union).
Then more generally, it is sufficient to put a countable union, not necessarily finite.
Note : E is countable means that there exists an injection from E to $\mathbb{N}$
Lemma 1 : $\mathbb{N}^2$ is countable.
Proof : The easiest way is to consider the attached picture.
Then you denote... :
point 1: (0,0)
point 2: (1,0)
point 3: (1,1)
point 4: (0,1)
point 5: (0,2)
point 6: (1,2)
This is a bijection from Nē to N.
Lemma 2 : A countable union of countable sets is countable.
Proof : Suppose we have a sequence $(E_n)_{n\geq 0}$ of countable sets.
Then $\forall n \in\mathbb{N}, \exists \varphi_n ~:~ E_n \to \mathbb{N}$, an injective mapping.
Let $E=\bigcup_{n\geq 0} E_n$
And for any $x\in E$, there exists $n\in\mathbb{N},x\in E_n$. So define $N(x)=\min \{n ~:~ x\in E_n\}$
Now consider $\phi ~:~ E \to \mathbb{N}^2$, where $\phi(x)=(N(x),\varphi_{N(x)}(x))$ which is an injection (easy to prove)
And for finishing it, (injection o bijection) is an injection.
Here is a list of things that you are assumed to know (if you doubt/don't know these then say so):
- $\mathbb{R}=\mathbb{R}\backslash\mathbb{Q} \cup \mathbb{Q}$
- A countable set can be mapped bijectively to the natural numbers {1,2,3...}
- $\mathbb{Z}$ is countable (take the mapping that sends negative integers to odd naturals and positive integers to even naturals)
- The composition of two bijections is a bijection (so a countable set is one who is bijective to an other countable set)
Suppose that R\Q is countable. Then as Q is countable we can bijectively map Q to N using a function (say f). Same is true for R\Q, say g maps R\Q to N bijectively. Notice that R and R\Q are
disjoint, hence if we define $h:\mathbb{R}\rightarrow \mathbb{Z}$ by
$<br /> h(x)=\begin{cases} f(x) & \mbox{if} \quad x\in \mathbb{Q} \\ -g(x) & \mbox{if} \quad x\in \mathbb{R}\backslash\mathbb{Q} \end{cases}<br />$
then we see that that h is a bijection. Hence R is countable (by the fact that Z is countable).
Has your background prepared you for this question?
Do you understand that the union of two countable set is countable?
Do you understand that $\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})=\mat hbb{R}?$
So what if $\mathbb{R}\setminus\mathbb{Q}$ is countable?
Thanks for the help, but your proof is rather complicated for me, maybe could you write a more simple proof so I can understand it?
He was using proof by contradiction, and showing that if we assumed $\mathbb{R}\backslash\mathbb{Q}$ was countable, we could reach a false conclusion. So therefore $\mathbb{R}\backslash\mathbb{Q}
$ must be uncountable.
What?? I don't even understand what you mean. It sounds like you are restating the question. Was there something unsatisfactory with my proof?
Hi, What I wrote before that that a union of countable sets is countable , but in this case we have Q u (R\Q)= R were its is countable U non countable = Non Countable, how do I show this law to
be be correct given we know Q is countable and R is not countable?
Problem: Given that $\mathbb{Q}\cong\mathbb{N}$ and $\mathbb{R}$ is not, prove that $\mathbb{R}-\mathbb{Q}$ is not as well.
Lemma: If $n\ge2\wedge n\in\mathbb{N}$ and $E_i\cong\mathbb{N}\quad 1\le i\le n$ then $\bigcup_{i=1}^{n}E_i\cong\mathbb{N}$.
Proof: It suffices to prove this for the case when $A\cap B=\varnothing$(why?). Since $A,B\cong\mathbb{N}$ there exists $f,f'$ such that $f:A\mapsto\mathbb{N}$ and $f':B\mapsto\mathbb{N}$
bijectively. Define a new mapping $\tilde{f}:A\cup B\mapsto\mathbb{Z}-{0}$ by $\tilde{f}(x)=\begin{cases} f(x) & \mbox{if} \quad x\in A \\ -f'(x) & \mbox{if} \quad x\in B \end{cases}$. Clearly
this mapping is bijective, therefore $A\cup B\cong\mathbb{Z}-{0}$. But it can easily be shown that $\mathbb{N}\cong\mathbb{Z}\cong\mathbb{Z}-{0}$. And since $\cong$ is an equivalence relation it
follows that $A\cup B\cong\mathbb{N}$. If $A\cap Be\varnothing$. The lemma follows by induction. $\blacksquare$
So assume that $\mathbb{R}-\mathbb{Q}$ was countable, then by the above lemma $\mathbb{Q}\cup\left(\mathbb{R}-\mathbb{Q}\right)=\mathbb{R}$ is countable. Contradiction.
Find injective map from the union of two countable sets to the naturals.
Given Q is denumerable, such that R is not denumerable. Show now that R\Q is not denumerable.
I don't like the way it's stated. Imho there are some wrong things.
It suffices for n to be $\geq 1$. Then I don't understand why you put $1\le i \le n$, since i is just an index in the union, and we don't need any condition over it (it's given in the notation of the
Then more generally, it is sufficient to put a countable union, not necessarily finite.
Note : E is countable means that there exists an injection from E to $\mathbb{N}$$\color{red}\star$
Lemma 1 : $\mathbb{N}^2$ is countable.
Proof : The easiest way is to consider the attached picture.
Then you denote... :
point 1: (0,0)
point 2: (1,0)
point 3: (1,1)
point 4: (0,1)
point 5: (0,2)
point 6: (1,2)
This is a bijection from Nē to N.
Lemma 2 : A countable union of countable sets is countable.
Proof : Suppose we have a sequence $(E_n)_{n\geq 0}$ of countable sets.
Then $\forall n \in\mathbb{N}, \exists \varphi_n ~:~ E_n \to \mathbb{N}$, an injective mapping.
Let $E=\bigcup_{n\geq 0} E_n$
And for any $x\in E$, there exists $n\in\mathbb{N},x\in E_n$. So define $N(x)=\min \{n ~:~ x\in E_n\}$
Now consider $\phi ~:~ E \to \mathbb{N}^2$, where $\phi(x)=(N(x),\varphi_{N(x)}(x))$ which is an injection (easy to prove)
And for finishing it, (injection o bijection) is an injection.
obvious if
That is certainly not been my experience in years of reviewing textbooks.
It is true that many texts make that distinction between finite and denumerable sets.
Then say a countable set is either.
Amen again. I have never liked that zigzag proof.
Here is a way that really teaches students the structure of that problem.
Let $\mathbb{N}=\{0,1,2,\cdots\}$ then define $\Phi :\mathbb{N} \times \mathbb{N} \mapsto \mathbb{Z}^ +$ as $\Phi (m,n) = 2^m (2n + 1)$.
In proving that $\Phi$ is a bijection a great many concepts are learned.
Who cares if you don't like sketches ????
You think I do too ? To be honest, my friend did this sketch, and had to explain it 3 times before I understood.
You're just a bunch of selfish people, who don't think that sometimes people reply to those who ask questions, and that maybe the ones who ask questions may understand better this way !!!!!!!!!!
And that's precisely the problem with you, M. Drexel28, who dare say that he doesn't like sketches, and (presumably) suppose it shouldn't be used in order to prove or to explain something.
But what about your Greek letters, that everybody would be such in an ease to manipulate ? What about that isomorphic sign that, of course, everybody would recognize at first sight and understand
? What about your so-perfect proof that everybody would understand where A and B come from ? And what about that excellent example of you saying that in most books, countable means being in
bijection with $\mathbb{N}$, with your so-long experience of analysis books ? For your information, I precised at the beginning of my message what countable would stand for in the rest of the
And why the heck would you say that the sketch is unnecessary here ? Do you think you can just memorize the function Plato gave and present it one year after reading it ?
The aim of my message was to give a way to prove a general thing, that's what "more generally" means.
I forgot to change the index to $\scriptstyle k$
And so, where does k intervene ? I wonder !!
Just things among others.
And, in response to your PM, I won't change my way, if I see something I don't like, or where there are problems (and there were !), I will say it. I won't shut up in order to be pleasant to M.
November 22nd 2009, 02:09 PM #13
November 22nd 2009, 02:48 PM #14
November 22nd 2009, 03:35 PM #15
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Post a reply
I input a,b,c,d use odds ratio calculator
Cases with positive outcome
Number in 1st group: a=25
Number in 2nd group: b=15
Cases with negative outcome
Number in 1st group: c=18
Number in 2nd outcome: d=30
and calculate
Odds ratio 2.8889
95 % CI 1.2184 to 6.8497
z statistic 2.408
P = 0.0160
I know odds ratio is Odds ratio = (a/c) / (b/d)
my question is
what z statistic and p mean ?
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Kurt Gödel
Birthplace: Brno, Brno, Jihomoravský kraj, Czech Republic
Death: Died in Princeton, Mercer, New Jersey, United States
Managed by: Martin Eriksen
Last Updated:
About Kurt Gödel
Kurt Gödel (1906-1978) mathematician and logician, was one of the most influential thinkers of the twentieth century andl probably the most strikingly original and important logician of the twentieth
century. He proved the incompleteness of axioms for arithmetic (his most famous result), as well as the relative consistency of the axiom of choice and continuum hypothesis with the other axioms of
set theory.
Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen.
In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty.
His 1940 book, better known by its short title, The Consistency of the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in
1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously
delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for
this problem.
Kurt Gödel's father was Rudolf Gödel whose family were from Vienna. Rudolf did not take his academic studies far as a young man, but had done well for himself becoming managing director and part
owner of a major textile firm in Brünn. Kurt's mother, Marianne Handschuh, was from the Rhineland and the daughter of Gustav Handschuh who was also involved with textiles in Brünn. Rudolf was 14
years older than Marianne who, unlike Rudolf, had a literary education and had undertaken part of her school studies in France. Rudolf and Marianne Gödel had two children, both boys. The elder they
named Rudolf after his father, and the younger was Kurt.
Gödel's work was the surprising culmination of a long search for foundations. Throughout the nineteenth century, mathematicians had tried to establish the foundations of calculus. First Cauchy gave
the modern definition of limits; later Weierstrass and Dedekind gave rigorous definitions of the real numbers. By the end of the century, the foundations of calculus rested on integers and their
arithmetic. This left the problem of putting the integers themselves on a sound logical basis, which Frege appeared to solve by defining the positive integers in terms of sets. But it soon became
clear that naive use of sets could lead to contradictions (such as the set of all sets that aren't members of themselves). Set theory itself would have to be axiomatized. In their massive 3-volume
Principia Mathematica, Russell and Whitehead built the foundations of mathematics on a set of axioms for set theory; they needed hundreds of preliminary results before proving that 1 + 1 = 2.
There remained the problem of analyzing the axioms of set theory. Mathematicians hoped that their axioms could be proved consistent (free from contradictions) and complete (strong enough to provide
proofs of all true statements). Gödel showed these hopes were overly naive. He proved that any consistent formal system strong enough to axiomatize arithmetic must be incomplete; that is, there are
statements that are true but not provable. Also, one can't hope to prove the consistency of such a system using the axioms themselves. The basic idea of Gödel's proof, indirect self-reference, is
strikingly simple, but tricky to grasp. A book-long explanation for the general reader is offered in Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.
Gödel was born in Brno, which was then part of the Austria-Hungary. In 1924 he matriculated at the University of Vienna. He became interested in logic and was influenced by Hahn, who was to be his
thesis advisor. From 1926-28 he participated in the Vienna Circle that was to become associated with Rudolf Carnap and logical positivism (though Gödel disagreed with most of Carnap's views). He
completed his dissertation (on the completeness of first-order logic) in 1929. The next year he had already proved his incompleteness theorem, and it was published in 1931. (It is ironic that Gödel's
first two major results were a completeness theorem and an incompleteness theorem. The two are not contradictory, but together they do show that no first-order axiomatization can capture all the
truths of arithmetic). Gödel submitted his incompleteness paper to the University of Vienna as his Habilitationsschrift (probationary essay), and in 1933 he was confirmed as a Privatdozent: this was
not a salaried position, but a certificate that gave him the right to lecture and collect fees from students. He taught his first course in the summer of 1933, and that fall he began a year-long
appointment at the newly formed Institute for Advanced Study (IAS) in Princeton, New Jersey.
Upon his return to Austria the next year, Gödel had the first of several breakdowns; he spent several months in a sanatorium recovering from depression. In 1935 he proved the (relative) consistency
of the axiom of choice with the other axioms of set theory. ("Relative" in this case means that if the axioms other than the axiom of choice are consistent, then so are these axioms together with the
axiom of choice. As noted above, one can't hope to prove the consistency of the axioms from themselves.) A second visit to the IAS was cut short by a relapse of depression, and Gödel remained
incapacitated until spring 1937. Later that year he proved the consistency of the generalized continuum hypothesis with the axioms of set theory, and he lectured on his set-theoretic results at the
IAS in 1938-39. By now Austria had been incorporated into Hitler's Germany, and when he returned home he faced liability for military service. Though he was not Jewish, Gödel's academic associations
put him in a precarious position. After protracted negotiations he received a U. S. visa late in 1939: in the early months of the Second World War he and his wife travelled to the U. S. via the
Soviet Union and Japan. He was given a one-year appointment to the IAS upon his arrival in Princeton; this was renewed yearly until 1946, when he was appointed a permanent member.
In 1942 Gödel attempted to prove that the axiom of choice and continuum hypothesis are independent of (not implied by) the axioms of set theory. He did not succeed, and the problem remained open
until 1963. (In that year, Paul Cohen proved that the axiom of choice is independent of the axioms of set theory, and that the continuum hypothesis is independent of both.) Gödel did little original
work in logic after this, though he did publish a remarkable paper in 1949 on general relativity: he discovered a universe consistent Einstein's equations in which there were "closed timelike
lines"--in such a universe, one could visit one's own past!
Gödel struck most people as eccentric. His political views were often surprising: for instance, while he condemned Truman for fomenting war hysteria and creating the climate for McCarthyism, he was a
great admirer of Eisenhower. While studying for his U. S. citizenship examination in 1948, he became convinced he had found an inconsistency in the Constitution. (Fortunately, this did not disrupt
Gödel's citizenship interview, as the judge brushed aside the point when Gödel tried to bring it up.) Gödel became increasingly reclusive in his later years. He was always somewhat prone to paranoia,
was distrustful of doctors, and tended to feed himself poorly. When his wife was incapacitated with illness, these factors combined to cause his death from self-starvation.
Birth of Kurt
1906 April 28, 1906
Brno, Brno, Jihomoravský kraj, Czech Republic
January 14, 1978 Death of Kurt
Age 71 Princeton, Mercer, New Jersey, United States
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Survey of Analysis Methods Part I
Posted March 2, 2010by
White Paper
Practical marketing research deals with two major problems: identifying key drivers and developing segments. In this two-part series TRC looks at key driver analysis and segmentation.
Key driver analysis is a broad term used to cover a variety of analytical techniques. It always involves at least one dependent or criterion variable and one or (typically) multiple independent or
predictor variables whose effect on the dependent variable needs to be understood. The dependent variable is usually a measure on which the manager is trying to improve the organization’s
performance. Examples include overall satisfaction, loyalty, value and likelihood to recommend.
When conducting a key driver analysis, there is a very important question that needs to be considered: Is the objective of the analysis explanation or prediction?
Answering this question before starting the analysis is very useful because it not only helps in choosing the analytical method to be used but also, to some extent, the choice of variables. When the
objective of the analysis is explanation, we try to identify a group of independent variables that can explain variations in the dependent variable and that are actionable. For example, overall
satisfaction with a firm can be explained by attribute satisfaction scores. By improving the performance on those attributes identified as key drivers, overall satisfaction can be improved. If the
predictors used are not actionable, then the purpose of the analysis is defeated.
In the case of prediction, we try to identify variables that can best predict an outcome. This is different from explanation because the independent variables here do not have to be actionable, since
we are not trying to change the dependent variable. As long as the independent variables can be measured, predictions can be made. For example, in the financial services industry, it is important to
be able to predict (rather than change) the creditworthiness of a prospective customer from the customer’s profile.
Beyond the issue of explanation versus prediction, there are two other questions that help in the choice of analytical technique to be used:
1. Is there one, or more than one, dependent variable?
2. Is the relationship being modeled linear or non-linear?
In the remainder of this article we will discuss analytical methods that would be appropriate if one or both of these questions is answered in the affirmative.
Single dependent variable
Key driver analyses often use a single dependent variable and the most commonly used method is multiple regression analysis. A single scaled dependent variable is explained using multiple independent
variables. Typically, the scale for the dependent variable ranges from five points to 10 points and is usually an overall measure such as satisfaction or likelihood to recommend.
The independent variables are some measures of attribute satisfaction usually measured on the same scale as the dependent variable, but not necessarily. There are two main parts to the output that
are of interest to the manager: the overall fit of the model and the relative importance.
The overall fit of the model is often expressed as R2 or the total variance in the dependent variable that can be explained by the independent variables in the model. R2 values range from 0 to 1,
with higher values indicating better fit. For attitudinal research, values in the range of 0.4-0.6 are often considered to be good. Relative importance of the independent variables is expressed in
the form of coefficients or beta weights. A weight of 0.4 associated with a variable means that a unit change in that variable can lead to a 0.4 unit change in the dependent variable. Thus, beta
weights are used to identify the variables that have the most impact on the dependent variable.
While regression models are quite robust and have been used for many years they do have some drawbacks. The biggest (and perhaps most common) is the problem of multicollinearity. This is a condition
where the independent variables have very high correlations among them and hence their impact on the dependent variable is distorted. Different approaches can be taken to address this problem.
A data reduction technique such as factor analysis can be used to create factors out of the variables that are highly correlated. Then the factor scores (which are uncorrelated with each other) can
be used as independent variables in the regression analysis. Of course, this would make interpretation of the coefficients harder than when individual variables are used. Another method of combating
multicollinearity is to identify and eliminate redundant variables before running the regression. But this can be an arbitrary solution that may lead to the elimination of important variables. Other
solutions such as ridge regression have also been used. But, if in fact the independent variables truly are related to each other, then suppressing the relationship would be a distortion of reality.
In this situation other methods, such as structural equation modeling, that use multiple dependent variables may be more helpful and will be discussed later in this article.
Categorical values
What if the dependent variable to be used is not scaled, but categorical? This situation arises frequently in loyalty research and examples include classifications such as customer/non-customer and
active/inactive/non-customer. Using regression analysis would not be appropriate because of the scaling of the dependent variable. Instead, a classification method such as linear discriminant
analysis (or its equivalent, logistic regression) is required. This method can identify the key drivers and also provide the means to classify data not used in the analysis into the appropriate
Key driver analyses with categorical dependent variables are often used for both explanation and prediction. An example of the former is when a health care organization is trying to determine the
reasons behind its customers dis-enrolling from the health plan. Once these reasons are identified, the company can take steps to address the problems and reduce dis-enrollment.
An example of the latter is when a bank is trying to predict to whom it should offer the new type of account it is introducing. Rather than trying to change the characteristics of the consumers, it
seeks to identify consumers with the right combination of characteristics that would indicate profitability.
Multiple dependent variables
As mentioned above, one problem with multiple regression models is that relationships between independent variables cannot be incorporated. It is possible to overcome this by running a series of
regression models. For example, if respondents answer multiple modules in a questionnaire relating to customer service, pricing etc., individual models can be run for each module. Following this an
overall model that uses the dependent variables from each model as independents can be run. However, this process can be both cumbersome and statistically inefficient.
A better approach would be to use structural equation modeling techniques such as LISREL or EQS. In these methods, a single model can be specified with as many variables and relationships as desired
and all the importance weights can be calculated at once. This can be done for both scaled and binary variables.
By specifying the links between the independent variables, their inherent relationships are acknowledged and thus the problem of multicollinearity is eliminated. But the drawback in this case is that
the nature of the relationships needs to be known up front. If this theoretical knowledge is absent, then these methods are not capable of identifying the relationships between the variables.
All of the methods discussed so far have been traditionally used as linear methods. Linearity implies that each independent variable has a linear (or straight-line) relationship with the dependent
variable. But what if the relationship between the independent and dependent variables is non-linear? Research has shown that in many situations, linear models provide reasonable approximations of
non-linear relationships and thus tend to be used since they are easier to understand. There are situations however, where the level of non-linearity or the predictive accuracy required is so high
that non-linear models may need to be used.
The simplest extensions to linear models use products (or interactions) of independent variables. When two independent variables are multiplied and the product is used as an independent variable in
the model, its relationship with the dependent variable is no longer linear. Similarly, other non-linear effects can be obtained by squaring a variable (multiplying it with itself), cubing it or
raising it to higher powers. Such models are referred to as polynomial regression models and they have useful properties. For example, squaring a variable can help model a U-shaped relationship such
as the one between a fruit juice’s tartness rating and the overall taste rating. Other variations such as logarithmic (or exponential) transformations can also be used if there is a curved
relationship between the dependent and independent variables.
The methods described above are not strictly considered to be non-linear methods. In real non-linear models the relationship between the dependent and independent variables is much more complex. It
is usually in a product form and linearity cannot be achieved by transforming the variables. Further, the user needs to specify the nature of the non-linear relationship to be modeled. This can be a
very important drawback, especially when there are many independent variables. The relationship between the dependent and independent variables can be very complicated, making it extremely hard to
specify the type of non-linear model required. A recent development in non-linear models that can help in this regard is the multivariate adaptive regression splines (MARS) approach that can model
non-linear relationships automatically with minimal input from the user.
Non-linear models are particularly useful if prediction rather than explanation is the objective. The reason for this is that the coefficients from a non-linear regression are much harder to
interpret than those from a linear regression. The more complicated the model, the harder the coefficients can be to interpret. This is not really a problem for prediction because the issue is only
whether an observation’s value can be predicted, not so much how the prediction can be accomplished. Hence, if explanation is the objective, it is better to use linear models as much as possible.
Artificial intelligence
The title of artificial intelligence covers several topic areas including artificial neural networks, genetic algorithms, fuzzy logic and expert systems. In this article we will discuss artificial
neural networks as they have recently emerged as useful tools in the area of marketing research. Although they have been used for many years in other disciplines, marketing research is only now
beginning to realize the potential of these tools. Artificial neural networks were originally conceived as tools that could mathematically emulate the decision-making processes of the human brain.
Their algorithm is set up in such a way that they “learn” the relationships in the data by looking at one (or a group of) observation(s) at a time.
Neural networks can model arbitrarily complex relationships in the data. This means that the user really doesn’t need to know the precise nature of the relationships in the data. If a network of a
reasonable size is used as a starting point, it can learn the relationships on its own. Often, the challenge is to stop the network from learning the data too well as this could lead to a problem
known as overfitting. If this happens, then the model would fit the data on which it is trained extremely well, but would fit new (or test) data poorly.
While complex relationships can be modeled with neural networks, obtaining coefficients or importance weights from them is not straightforward. For this reason, neural networks are much more useful
for prediction rather than explanation.
There are many types of neural networks, but the most commonly used distinction is between supervised and unsupervised networks. We will look at supervised networks here and at unsupervised networks
in the next article. Supervised neural networks are similar to regression/classification type models in that they have dependent and independent variables. Back-propagating networks are probably the
most common supervised learning networks. Typically they contain an input layer, output layer and hidden layer.
The input and output layers correspond to the independent and dependent variables in traditional analysis. The hidden layer allows us to model non-linearities. In a back-propagating network the input
observations are multiplied by random weights and compared to the output. The error or difference in the output is sent back over the network to adjust the weights appropriately. Repeating this
process continuously leads to an optimal solution. A holdout (or test) dataset is used to see how well the network can predict observations it has not seen before.
Recent advances
Several recent advances have been made in key driver methodology. The first of these relates to regression analysis and is called hierarchical Bayes regression. Consider an example where consumers
provide attribute and overall ratings for different companies in the marketplace. Different consumers may rate different companies based on their familiarity with the companies. An overall
market-level model can be obtained by combining all of the ratings and running a single regression model across everybody. But if we one could run a separate model for each consumer and then combine
all of that information, the resulting coefficients would be much more accurate than what we get from a regular regression analysis. This is what hierarchical Bayes regression does and is hence able
to produce more accurate information. Of course, this type of analysis can be used only in situations where respondents provide multiple responses.
For classification problems, there have been a series of recent advances such as stacking, bagging and boosting. In stacking, a variety of different analytical techniques are used to obtain
classification information and then the final results are based on the most frequent classification of data points into groups in each of those methods. Bagging is a procedure where the same
technique is used on many samples drawn from the same data and the final classifications are made based on the frequencies observed in each sample. Finally, boosting is a method of giving higher
weights to observations that are mis-classified and repeating the analysis several times. The final classifications are based on a weighted combination of the results from the various iterations.
Variety of tools
This article has touched upon both traditional methods and recent developments in key driver methodology that may be of interest to marketing research professionals. The particular method to be used
often hinges on the primary objective — explanation or prediction. Once this determination is made, there are a variety of tools that can be used that include linear and non-linear methods, as well
as those that employ multiple dependent variables.
This content was provided by TRC. Visit their website at www.trchome.com.
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The force pointing out involved in a buss-turn.
hi jegerjon! welcome to pf!
centripetal force
is not a separate force
it is only an alternative name for the radially inward component of tension or
or other force or forces on a body
in the case of the bus, it is the friction force, which of course is irrelevant to tipping, since it has no
about the tipping axis
the easiest way to solve this is to use the
rotating frame
of the bus …
in that frame, there are two relevant forces, mg vertically downward, and the
centrifugal force
/r horizontally outward
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[R] Vars package - specification of VAR
herrdittmann at yahoo.co.uk herrdittmann at yahoo.co.uk
Sun Dec 7 17:20:39 CET 2008
Hi useRs,
Been estimating a VAR with two variables, using VAR() of the package "vars".
Perhaps I am missing something, but how can I include the present time t variables, i.e. for the set of equations to be:
x(t) = a1*y(t) + a2*y(t-1) + a3*x(t-1) + ...
Y(t) = a1*x(t) + a2*x(t-1) + a3*y(t-1) + ...
The types available in function VAR() allow for seasonal dummies, time trends and constant term.
But the terms
always seem to be excluded by default, thus only lagged variables enter the right side.
How can I specify VAR() such that a1*y(t) and a1*x(t) are included?
Or would I have to estimate with lm() instead?
Many thanks in advance,
More information about the R-help mailing list
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Examples of theorems arising from many authors' work
up vote 3 down vote favorite
(Old question: How much of mathematics is or could be done by the 'geniuses'?)
A lot of important theorems or even theories end up being named after or otherwise attributed to one person or a small group of people. This is often fair, but taken as an overall trend it can give
the (hopefully false!) impression that the most important mathematics is being done by a small minority of mathematicians. What I'm wondering is, how much does the opposite phenomenon occur, where
it's very clear that a result is a large team effort and that no small group of authors deserves the lion's share of the credit? The most obvious example I can think of is the Classification of
Finite Simple Groups, but I'm sure there are others from different areas of maths.
5 This question seems too discussion-y to me. I think it is more appropiate for other fora. – Alberto García-Raboso Sep 28 '10 at 15:55
7 Exemplar for "subjective and argumentative" ;). Perhaps every mathematician should do it's best and not ponder what would happen without him ;). – Martin Brandenburg Sep 28 '10 at 15:56
2 On reflection, I agree. Perhaps I should instead have asked something more concrete, such as what are some examples of important theorems that can't be boiled down to the work of one or two big
names. The classification of finite simple groups would be an obvious example here - it seems unlikely that it would ever have been proved in a world with very few mathematicians, even if they
were all brilliant. – Colin Reid Sep 28 '10 at 16:53
Now that the question has been completely rewritten, perhaps someone should start a meta thread to attract attention for unclosing. – JBL Sep 28 '10 at 18:28
@JBL: done: tea.mathoverflow.net/discussion/687/… – Tony Huynh Sep 28 '10 at 23:14
add comment
3 Answers
active oldest votes
Aside from the Geometrization Theorem/Poincare conjecture, probably the deepest theorem in low-dimensional topology in the last 10 years is the classification of hyperbolic structures on
3-manifolds with finitely generated fundamental group. Aside from the topological type, it turns out they they are classified by certain invariants "at infinity" (either Riemann surfaces at
up vote infinity or so-called "ending laminations"). The proof of this uses work of an enormous number of people : Agol, Alhfors, Bers, Brock, Calegari, Canary, Gabai, Namazi, Kleineidam, Kra,
6 down Lecuire, Marden, Maskit, Masur, Minsky, Mostow, Ohshika, Prasad, Rees, Souto, Sullivan, Thurston, and probably people I'm forgetting.
add comment
Most theorems in Mathematics build upon the work of many people. Maybe a good recent example is the proof of Serre's conjecture by Khare and Winterberger. They contributed a decisive,
up vote 3 fundamental piece of the puzzle, but it depends on the work of e.g. Serre, Tate, Ribet, Mazur, Taylor, Wiles, Kisin, Dieulefait and many others.
down vote
add comment
One possible answer (and hopefully the source of many future answers) is Tim Gowers' polymath experiment. In short, the idea of the project is to test if massively collaborative
up vote 2 down mathematics over the internet is possible.
add comment
Not the answer you're looking for? Browse other questions tagged soft-question or ask your own question.
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Math Forum Discussions
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Back to the basics: Designing your embedded algorithms with high level graphical tools
Algorithms are the heart and soul of any control or monitoring application - without "intelligence" integrated into the design it's simply a “dumb” node on the network. Adding signal processing and
mathematics to embedded and distributed hardware adds significant value. Signal processing also can add new functionality to devices, therefore, the ability and freedom to easily design and iterate
on algorithms is critical to innovation.
Engineers consistently need processing closer and closer to the actual plant or unit under test to help speed performance and decentralize the processing from a central node. While there are many
hardware platforms such as PLCs, industrial PCs, custom controllers, and other off-the-shelf devices such as performance automation controllers (PACs), without being able to program “intelligence”
onto each distributed node, you simply have a bunch of “dumb” nodes on the network that are slaves to a central server.
Providing the intelligence on each distributed node arms the system against crashes if a component of the network goes down and increases processing capabilities since each node runs algorithms
independently instead of relying on one central server.
Signal processing and mathematic algorithms that can be programmed onto distributed devices add significant value to distributed and embedded systems. That said, the ability and freedom to design and
iterate on algorithms is critical to innovation. But how are these algorithms being developed? More importantly, how should they be developed?
Often, algorithms are designed in floating point by experts of a particular field (I will refer to him as the domain expert later) that understand the mathematics of an algorithm very well, but may
not understand the implications of deploying that algorithm to an embedded device. Deployed devices may be floating point PowerPCs or even x86 processors, but they are more often fixed-point MPUs,
DSPs or FPGAs.
Additionally, it is often times not the algorithm developer who is actually deploying the code onto the device, it is an embedded system designer who understands the silicon very well, but may not
have a complete understanding of the algorithm.
This communication gap can create inefficiencies and even flawed products if not addressed appropriately. We will discuss how a common interface to perform interactive algorithm design decreases
inefficiencies and provides a single programming platform to design, prototype and deploy your intelligent devices.
Rapid Innovation through Software Iterations
As stated earlier, the interactions between the domain expert designing the algorithm and the embedded system designer needs to be improved because the communication gap between these two leads to
large inefficiencies. To solve these problems, both groups need to use a better tool and ideally, the same tool.
Using a sufficiently sophisticated graphical programming approach provides intuitive models of computation for designing a variety of algorithms, as well as, implementing that algorithm onto a
As an example, let’s look at the process of designing a low pass filter, similar to the one shown in Figure 1 below, to see how the combination of graphical programming in conjunction with
interactive tools can empower a domain expert to easily implement an optimized, high-performance filter, without ever having to deal with some of the numerical complexities of the final fixed point
Figure 1: Frequency response of a low-pass filter
In this example, a Labview graphical toolchain will allow the user to design the algorithm based on the following steps:
1) Configure the filter specification and visually analyze the filter performance characteristics.
2) Automatically parameterize the filter coefficients for fixed or floating point implementation.
3) If fixed point is needed, generate a model for fixed point implementation of the filter and analyze the characteristics of the parameterized filter in the time and frequency domain.
4) Simulate and compare implementation model to original specification on a graphical interface.
5) Generate efficient and optimized code for implementation for a fixed or floating-point platform.
Following these five steps, the domain expert is guided all the way from a theoretical specification for the filter to the implementation of the filter on a target platform. The tool assumes the
burden of scaling appropriately across the floating and fixed point domains. This way the embedded systems designer can focus on the desired filter behavior instead of low-level optimizations for the
filter implementation. Since the end-code for the filter is generated at the lowest level in the tool’s graphical language, the engineer has the complete freedom to ‘tweak’ and further optimize or
modify the filter.
Figure 2: Configuring the low-pass filter and analyzing frequency response
Figure 2 above shows an example of using the LabVIEW Digital Filter Design Toolkit for configuring and implementing a lowpass digital filter. To characterize the filter, we set the filter parameters
using a LabVIEW function (or VI) that allows configuration through a dialog window.
Using the interactive Configure Classical Filter Design VI, we can select various filter characteristics such as sampling frequency, passband edge frequency, passband ripple, stopband edge frequency,
and stopband attenuation. Additionally, we can design this using various design windowing methods including but not limited to Equi-Ripple FIR, elliptical, or Butterworth.
Here you can visualize the filter performance through its frequency response and Root-locus plot to verify the expected behavior of your floating point filter. Once you configure the filter, you can
run the VI and observe the magnitude response through a frequency sweep for the filter sampling rate.
After the filter is designed, we need to generate a fixed point filter model to implement on a fixed point platform, such as an FPGA. Here we will add a function to automatically convert this filter
to a fixed point model.
Figure 3: Converting floating point coefficients to fixed point and comparing fixed and floating point models in the frequency domain.
Within the “DFD FXP Modeling for CodeGen.vi”, shown in Figure 3, above, we can set the integer word length (iwl) for the model inputs and outputs and the filter coefficients. When we run this VI, we
can simulate and analyze the fixed-point filter for discrepancies in the frequency domain when compared to the floating point filter and calculate overflows, underflows and zeros. We can use this
interactive structure to find the smallest iwl while still maintaining the integrity of the filter in the frequency domain.
The next step is to simulate and analyze the quantized fixed point filter to the floating point implementation in the time domain [Figure 4, below ]. Here we can further modify the integer word
length for the coefficients until the two filters behave similarly.
Figure 4: Comparing the fixed point and floating point filter coefficients in the time domain
The last step for developing this filter is to generate the code to be implemented within the appropriate target, either generating LabVIEW or C code for implementation on various embedded platforms,
as shown below, from the graphical description.
Figure 5: LabVIEW block diagram for a low pass filter
In this example, we can generate LabVIEW code for implementation on a FPGA, as shown in Figure 6 below.
Figure 6: LabVIEW code for a low pass filter
As you can see, implementing algorithms for such functions as digital filters is much more of an art than a science. With this in mind, using an interactive and iterative approach is necessary so
that you can quickly and efficiently move from design, to simulation, to implementation. Additionally, this approach encourages experimentation with various design parameters and can lead to further
Mike Trimborn is LabVIEW FPGA Product Manager at National Instruments, Inc.
|
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help with defining a relation ~ on Z_p
October 16th 2012, 06:58 PM #1
Sep 2012
help with defining a relation ~ on Z_p
The question is as follows:
Define a relation ~ on Z[p] where p is an odd prime, as follows: [a][p] ~ [b][p] if 2^ia = b mod p for some i in N. show that ~ is an equivalence relation on Z[p]
I know that to show an equivalence relation you have to show the reflexive, symmetric and transitive properties hold.
reflexive: [a][p] ~ [a][p] 2^ia = a mod p this is true since any multiple of a consequence class is equal to itself
symmetric: [a][p] ~ [b][p] , [b][p] ~ [a][p] I know that I need to show that 2^ia = b mod p and 2^ib = a mod p but I am unsure if the logic is sound.
2^ia = b mod p then 2^ia = pq + b for some integer q
2^ia - b = pq so p|(2^ia - b) however p is an odd prime so it can not divide 2^i so p|(a - b) and p|(b - a) then p|(2^ib - a)
then 2^ib - a = pq so 2^ib = a mod p
Transitive: [a][p] ~ [b][p] [b][p] ~ [c][p] [a][p] ~ [c][p] 2^ia = b mod p and 2^ib = c mod p then 2^ia = c mod p
2^ia = pq + b and 2^ib = pr + c for integers q, r
then 2^ia + 2^ib = pq + pr + c+ b
2^i(a + b) - (c + b) = p(q + r)
so p divides the LHS but as in the above argument p does not divide 2^i so p|[(a + b) - (c + b)]
so p|(a - c) then p|(2^ia - c) so 2^ia - c = p(q + r) so 2^ia = c mod p
have I done enough to prove the relation and is my arguments logically sound?
Re: help with defining a relation ~ on Z_p
What you should observe here is that $2^0a = a \mod p$. That's what shows that a ~ a.
That actually doesn't follow.
What you need to do is show that starting with $2^ia = b \mod p$,
you can show that there's some j such that $2^jb = a \mod p$. What could j be?
Working backwards, if both those hold, then: $2^ia = b \mod p$, and $2^jb = a \mod p$,
so $2^{i+j}a = 2^jb = a\mod p$, so $2^{i+j}aa^{-1} = aa^{-1} = 1 \mod p$, so $2^{i+j} = 1\mod p.$
So find n such that $2^n = 1\mod p,$ and then $2^ia = b \mod p$ will imply $2^{n-i}b = a \mod p$.
For the transitive: Those aren't "the same i".
From a~b and b~c you should conclude that:
$\text{"For some } i \in \mathbb{N}, \ 2^ia = b \mod p, \text{ and, for some } j \in \mathbb{N}, \ 2^jb = c \mod p \text{ ."}$
After that consider $2^{i+j}a = ? \mod p$.
Last edited by johnsomeone; October 16th 2012 at 08:00 PM.
Re: help with defining a relation ~ on Z_p
thank you very much for the help
October 16th 2012, 07:52 PM #2
Super Member
Sep 2012
Washington DC USA
October 17th 2012, 02:32 AM #3
Sep 2012
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Test if a Number is Prime using a Formula in Microsoft Excel
Prime Numbers
A prime number is a natural number which has exactly two distinct natural number divisors: One and itself.
The number one is not a prime number.
The formula below will test if a number is prime. It will test all whole numbers not exceeding 268,435,455.
=IF(C9=2,"Prime",IF(AND(MOD(C9,ROW(INDIRECT("2:"&ROUNDUP(SQRT(C9),0))))<>0),"Prime","Not Prime"))
This is an array formula. Therefore, you must press CTRL SHIFT ENTER after entering the formula and whenever you edit it later.
You must enter the test number in cell C9.
The formula relies on Excel's MOD function to calculate the remainder on every whole number divisor from 2 to
the square root of the test number. If there is always a remainder then the number is prime.
Excel's ability to calculate the remainder fails when the test number exceeds 268,435,455.
You can download a sample spreadsheet: Primality Test.
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[Numpy-discussion] subclassing matrix
Basilisk96 basilisk96@gmail....
Thu Jan 10 00:08:03 CST 2008
Hello folks,
In the course of a project that involved heavy use of geometry and
linear algebra, I found it useful to create a Vector subclass of
numpy.matrix (represented as a column vector in my case).
I'd like to hear comments about my use of this "class promotion"
statement in __new__:
ret.__class__ = cls
It seems to me that it is hackish to just change an instance's class
on the fly, so perhaps someone could clue me in on a better practice.
Here is my reason for doing this:
Many applications of this code involve operations between instances of
numpy.matrix and instances of Vector, such as applying a linear-
operator matrix on a vector. If I omit that "class promotion"
statement, then the results of such operations cannot be instantiated
as Vector types:
>>> from vector import Vector
>>> import numpy
>>> u = Vector('1 2 3')
>>> A = numpy.matrix('2 0 0; 0 2 0; 0 0 2')
>>> p = Vector(A * u)
>>> p.__class__
<class 'numpy.core.defmatrix.matrix'>
This is undesirable because the calculation result loses the custom
Vector methods and attributes that I want to use. However, if I use
that "class promotion" statement, the p.__class__ lookup returns what
I want:
>>> p.__class__
<class 'vector.Vector'>
Is there a better way to achieve that?
Here is the partial subclass code:
#---------- vector.py
import numpy as _N
import math as _M
#default tolerance for equality tests
TOL_EQ = 1e-6
#default format for pretty-printing Vector instances
FMT_VECTOR_DEFAULT = "%+.5f"
class Vector(_N.matrix):
2D/3D vector class that supports numpy matrix operations and more.
u = Vector([1,2,3])
v = Vector('3 4 5')
w = Vector([1, 2])
def __new__(cls, data="0. 0. 0.", dtype=_N.float64):
Subclass instance constructor.
If data is not specified, a zero Vector is constructed.
The constructor always returns a Vector instance.
The instance gets a customizable Format attribute, which
controls the printing precision.
ret = super(Vector, cls).__new__(cls, data, dtype=dtype)
#promote the instance to cls type.
ret.__class__ = cls
assert ret.size in (2, 3), 'Vector must have either two or
three components'
if ret.shape[0] == 1:
ret = ret.T
assert ret.shape == (ret.shape[0], 1), 'could not express
Vector as a Mx1 matrix'
if ret.shape[0] == 2:
ret = _N.vstack((ret, 0.))
ret.Format = FMT_VECTOR_DEFAULT
return ret
def __str__(self):
fmt = getattr(self, "Format", FMT_VECTOR_DEFAULT)
fmt = ', '.join([fmt]*3)
return ''.join(["(", fmt, ")"]) % (self.X, self.Y, self.Z)
def __repr__(self):
fmt = ', '.join(['%s']*3)
return ''.join(["%s([", fmt, "])"]) %
(self.__class__.__name__, self.X, self.Y, self.Z)
#### the remaining methods are Vector-specific math operations,
including the X,Y,Z properties...
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128 or 96 bit integer types?
Paul Rubin http
Sat Jul 28 09:28:12 CEST 2007
"mensanator at aol.com" <mensanator at aol.com> writes:
> has 146 digits. And that's just the begining. The above
> actually represents a polynomial with 264 terms, the
> exponents of which range from 0 to 492. One of those
> polynomials can have over 50000 decimal digits when
> solved.
You should use gmpy rather than python longs if you're dealing with
numbers of that size. Python multiplication uses a straightforward
O(n**2) algorithm where n is the number of digits. This is the best
way for up to a few hundred or maybe a few thousand digits. After
that, it's better to use more complicated FFT-based algorithms which
are O(n log n) despite their increased constant overhead. Gmpy does this.
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[R] hex format
Earl F. Glynn efg at stowers-institute.org
Thu Apr 7 16:52:04 CEST 2005
"Prof Brian Ripley" <ripley at stats.ox.ac.uk> wrote in message
news:Pine.LNX.4.61.0504071442200.25401 at gannet.stats...
> On Thu, 7 Apr 2005, Steve Vejcik wrote:
> > Has anyone used hex notation within R to represents integers?
> Short answer: yes.
> > as.numeric("0x1AF0")
> [1] 6896
> (which BTW is system-dependent, but one person used it as you asked).
I see this works fine with R 2.0.0 on a Linux platform, but doesn't work at
all under R 2.0.1 on Windows.
> as.numeric("0x1AF0")
[1] NA
Warning message:
NAs introduced by coercion
Seems to me the conversion from hex to decimal should be system independent
(and makes working with colors much more convenient). Why isn't this system
independent now?
The "prefix" on hex numbers is somewhat language dependent ("0x" or $)
perhaps but I didn't think this conversion should be system dependent.
I don't remember where I got this, but this hex2dec works under both Linux
and Windows (and doesn't need the "0x" prefix).
hex2dec <- function(hexadecimal)
hexdigits <- c(0:9, LETTERS[1:6])
hexadecimal <- toupper(hexadecimal) # treat upper/lower case the same
decimal <- rep(0, length(hexadecimal))
for (i in 1:length(hexadecimal))
digits <- match(strsplit(hexadecimal[i],"")[[1]], hexdigits) - 1
decimal[i] <- sum(digits * 16^((length(digits)-1):0))
> hex2dec(c("1AF0", "FFFF"))
[1] 6896 65535
"FFFF" can be interpreted as 65535 as unsigned and -1 as signed on the same
system depending on context. This isn't system dependent, but rather
context dependent.
I suggest "as.numeric" should perform the unsigned conversion on all
systems. What am I missing?
Earl F. Glynn
Scientific Programmer
Bioinformatics Department
Stowers Institute for Medical Research
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Good news, soundtrack aficionados! Pi and Moon composer Clint Mansell will be scoring The Source Code, Duncan Jones' upcoming flick about a soldier (Jake Gyllenhaal) whose consciousness is
transferred into the body of a commuter who's witnessed a train bombing. »
This creepy-cool trailer for EBBËTO's 25-minute-long film Analog is an agoraphobe's worst nightmare -a machine keeps a man alive during a deep space journey. Unfortunately, the machine begins to
reassess its programming, and a lot of symbolic oddness precipitates. »
Today (3/14) is Pi Day, a hallowed celebration in which radius and circumference lovers everywhere convene to eat circular pastries and challenge each other to decimal-reciting contests. Score your
festivities with the awesomely underrated soundtrack to Darren Aronofsky's Pi. »
Quantum computers, which would rely on quantum mechanical concepts like superposition and entanglement to perform operations of unimaginable complexity, remain a pipe dream. But physicists have
nevertheless come up with an algorithm that only quantum computers could use. »
If you saw Darren Aronofsky's frenetic, disturbing flick Pi, you know that its hero, a supergenius who invents a super algorithm, meets a rather terrible end. Though he wants to use his algorithm for
the forces of good, he's pursued by evil corporate schemers who want to use it to predict the stock market. Eventually… »
A new Spanish film features four rival scientists struggling to solve logic puzzles before the walls of the room they're trapped in squish them into jelly. Fermat's Room combines elements of Pi
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Must-see movies are futuristic classics that shouldn't be missed. Of course, not every must-see is perfect. That's why we've rated them 1-5 on the patented "crunchy goodness" scale. Written by
Sherilyn Connelly.
Title: Pi
Date: 1998
Vitals: A mathematician is obsessed with finding a pattern of numbers which (according … »
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chain rule
chain rule, in calculus, basic method for differentiating a composite function. If f(x) and g(x) are two functions, the composite function f(g(x)) is calculated for a value of x by first evaluating g
(x) and then evaluating the function f at this value of g(x), thus “chaining” the results together; for instance, if f(x) = sin x and g(x) = x^2, then f(g(x)) = sin x^2, while g(f(x)) = (sin x)^2.
The chain rule states that the derivative D of a composite function is given by a product, as D(f(g(x))) = Df(g(x)) ∙ Dg(x). In other words, the first factor on the right, Df(g(x)), indicates that
the derivative of f(x) is first found as usual, and then x, wherever it occurs, is replaced by the function g(x). In the example of sin x^2, the rule gives the result D(sin x^2) = Dsin(x^2) ∙ D(x^2)
= (cos x^2) ∙ 2x.
In the German mathematician Gottfried Wilhelm Leibniz’s notation, which uses d/dx in place of D and thus allows differentiation with respect to different variables to be made explicit, the chain rule
takes the more memorable “symbolic cancellation” form:d(f(g(x)))/dx = df/dg ∙ dg/dx.
The chain rule has been known since Isaac Newton and Leibniz first discovered the calculus at the end of the 17th century. The rule facilitates calculations that involve finding the derivatives of
complex expressions, such as those found in many physics applications.
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Ana B.
All of Ana’s current tutoring subjects are listed at the left. You can read more about Ana’s qualifications in specific subjects below.
ACT Math
I used to work for a company in Houston which specialized in ACT and SAT tutoring. We were taught special tricks and methods to improve students' ACT and SAT scores. I focus on both explaining the
methods to train for this particular test as well as to teach the mathematics behind the problems to explain the fundamental concepts necessary to not only achieve a good grade on the test, but also
succeed in future math courses.
Description of course: The ACT math consists of basic geometry, algebra, arithmetic and trigonometry. This test differs from the SAT in that you are not penalized for guessing and there are 60
problems to do in 60 minutes, but they are generally more straight forward and slightly easier.
ACT Science
I used to work for a company in Houston which specialized in ACT and SAT tutoring. We were taught special tricks and methods to improve students' ACT and SAT scores. I focus on both explaining the
methods to train for this particular test as well as to teach the mathematics behind the problems to explain the fundamental concepts necessary to not only achieve a good grade on the test, but also
succeed in future math courses.
Description: This test covers material from biology, chemistry and physics, but mainly the test focuses on data analysis. There will be 7 passages of which 6 will contain data given on a variety of
charts and graphs. At the end of each passage there will be a series of 5-7 questions most of which can be answered by analyzing the graphs and charts. There is always one analytical question which
is a bit more challenging at the end of each passage.
Algebra 1
When it comes to math, some students prefer learning by pictures and graphs and others like a more "algebraic" approach. I tend to explain concepts in multiple ways and let the student guide me by
seeing what they understand better. I adapt my teaching methods to my students to customize a plan that will work for them. Every person learns differently. There are several methods I generally
apply that have proved successful with most or even all of my students. I explain in a way that clicks for them. Description: Algebra 1 consists mainly of understanding the concept of a variable and
learning to solve linear equations with a variety of methods.
Algebra 2
When it comes to math, some students prefer learning by pictures and graphs and others like a more "algebraic" approach. I tend to explain concepts in multiple ways and let the student guide me by
seeing what they understand better. I adapt my teaching methods to my students to customize a plan that will work for them. Every person learns differently. There are several methods I generally
apply that have proved successful with most or even all of my students. I explain in a way that clicks for them. Description: The main goal is to understand the concept of a function and to be able
to work with a variety of different types of functions. You will learn to graph, solve and identify key elements of different types of functions. In certain cases, you will learn some formulas that
will help you solve or graph these functions.
Calculus is my favorite course to teach! I have been teaching calculus for about 6 years and have a LOT of success stories! I focus a lot on theory as this is the most important aspect of calculus
and it's the most fundamental tool to solve any of the problems. I also do a lot of practice problems and make problems up that tend to show the various way in which a concept can be applied. I try
to tie it as much as possible to something of interest for each particular student and do my absolute best to keep things from getting boring. I think that these tutoring sessions go by the fastest
and students tend to (rather reluctantly) enjoy themselves. As much as they'll hate to admit it, they have fun!
The goal of the course is to teach a variety of concepts which further analyze a certain family of functions. Some of these concepts are limits, derivatives and integrals. In the second part of the
course, you learn about sequences and series and the concept of convergence and divergence of each. You also learn more integration techniques and become acquainted with the concept of double
Discrete Math
I took an honors discrete math at the University of Texas at Austin and I got a very high A in it. I later graded for this same class and I also graded for a different discrete class. I also took and
graded a Number Theory class (which is very similar) and I have taken many abstract Algebra courses that are also very similar.
Discrete math is basically an introduction to proofs and basic logic all of which is extremely abundant throughout the major in mathematics.
This course mainly focuses on Euclidean Geometry which is a very nice and visual course. A lot of students struggle with this course because it is the first time where they are asked to write proofs
of theorems. At first, this is a daunting and intimidating task for most, but with time and guidance it gets a lot easier. I have had several students recently that have needed help with geometry so
I feel very comfortable with this subject and am very capable of explaining the concepts, proofs and ideas in many different ways. This course can be deceivingly fun and interesting, so hopefully I
can help make it less scary and more positive.
I used to work for a company in Houston which specialized in standardized test tutoring. We were taught special tricks and methods to improve students' scores, mainly in the SAT and ACT, but the GRE
is not very different. I focus on both explaining the methods to train for this particular test as well as to teach the mathematics behind the problems to explain the fundamental concepts necessary
to not only achieve a good grade on the test, but also succeed in future math courses.
Description of course: The GRE math consists of arithmetic, algebra, geometry, and data analysis. The computer based test is the most common one and can be taken throughout the year. It is timed, and
you cannot go back on questions once you have submitted them. It is an exam that you must prepare for in order to succeed, but it is not conceptually terribly difficult. The key is practice,
practice, practice!
Linear Algebra
Linear Algebra was one of the first proof based courses I ever took, and was one of the reasons I fell in love with math. At first it is an in depth review of what you learn about Matrices in high
school, so you'll learn matrix multiplication, row reduction techniques, determinants, and some properties of matrices, for example matrices don't necessarily commute under multiplication. Later you
will learn about vector spaces, linear transformations, eigen values and eigen vectors, changing bases and maybe touch on some complex numbers. It's an in depth course where you focus a lot on why
mathematical theorems are true and spend some time learning formal proofs. It's a very important class to understand well if you're a math major and it can be a bit challenging, but also it's very
interesting and can be a lot of fun!
I am a math major and I have a Master's degree in Mathematics.
Pre-calculus and trig are generally very challenging courses for students. Unfortunately, a lot of times you can actually get pretty good grades in these courses without understanding too much of the
material and relying heavily on memorization. This leads to the major issues we see in calculus every year, because a good pre-calculus foundation is crucial to succeed in calculus. I specialize in
explaining concepts and teaching students where the information comes from, allowing them to derive it all themselves if they forget what to do during an exam when they're nervous, or simply after
several months of not using it. I also do encourage some degree of memorization, but I am a firm believer in minimizing the usage of that part of your brain as much as possible since it is
notoriously the least reliable, and the most overused.
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Empowering students to own their learning solves maths problems
Empowerment isn’t just a buzz-word. It makes a difference to learning. Take a look at this problem: The question is this: what fraction of the shape is shaded?
The level of maths required is well within KS3. Area of a square? Simple. Area of a circle? Harder, but straight forward. But, ask students to do this faced with the blank diagram and many of
those who appear to know the basics will flounder. Where to begin? We’re not given any information about size – how can it be done?
But, throw in the labels and, hey presto, it all becomes clear. (πr2/(2r)2) giving π/4. The fact that this is independent of r can now be seen.
So, the main thing holding students back is the idea of labelling the diagram. What is this about? Our investigations involving students at KEGS and other partner schools suggest that for some it
is about permission. Students often ask ‘am I allowed to label the diagram?’. They don’t feel that this is permitted until given the go-ahead. Even then, the confidence to say ‘let’s call the
radius r’ is not common. Obviously there is some conceptual work to do to be sure that each side is 2r but, without the first step of labelling, this is impossible. Where there is higher level of
risk-taking, a greater emphasis on ‘wrong answers’ and problem-solving work (puzzles, games, investigations etc), we’ve found students are more likely to be successful in tackling this question.
In the problem below, this is even more difficult. The basic ideas are solid KS4 territory.- area of a semi circle and square are easy, and a simple application of Pythagorus’ theorem is needed.
However, in schools we’ve visited, few students in a typical top-set GCSE class had the natural confidence to add the radius r or even to do the simpler thing of giving the square an algebraic size –
say a or x.
The first step in developing the skills required for this kind of problem-solving is to create a culture where students confidently assume authority over the problem. This is my problem; my
diagram. I can write all over it if I want to. I can call things what I like and add lines if it helps me…. Without the simple empowerment stage, the maths itself can’t happen. So this is the
place to start! Try these problems out with your classes, with and without labels and see what happens.
(If you are still stuck, leave a comment and I’ll tell you the answer.)
This post presents a very similar argument: http://fawnnguyen.com/2013/05/07/20130506.aspx
Thanks to Clare Benton for the link.
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Related rates--find time two vehicles are closest
June 23rd 2011, 11:57 AM #1
Jun 2011
Related rates--find time two vehicles are closest
A truck is 30 km due west of an SUV and is travelling east at a constant speed of 70 km/h. Meanwhile, the SUV is going south at 90 km/h. When will the truck and the SUV be closest to each other?
I set up a triangle, defined dx/dt as -70t, dy/dt as 90t, and used Pythagorean theorem:
z^2 = x^2 + y^2
I then took the derivative of this.
I'm not sure where to go from here?
Last edited by jemray; June 23rd 2011 at 03:09 PM.
Re: Related rates--find time two vehicles are closest
You need to express the distance z (or its square) as a function of t only, take the derivative with respect to t and equate it to zero.
Thank you, I understand now. I don't know why I found that so confusing.?
June 23rd 2011, 02:33 PM #2
MHF Contributor
Oct 2009
June 23rd 2011, 03:06 PM #3
Jun 2011
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use cylindrical shells method to find the volume of the solid generated when the region enclosed by the given curves... - Homework Help - eNotes.com
use cylindrical shells method to find the volume of the solid generated when the region enclosed by the given curves is revolved about the y-axis
`y=2cos(9x^2)` , `x=0 ` , `x=(sqrt(pi))/6` , `y=0`
The formula for calculating the volume of a solid rotating about `y`-axis is
`V=2piint_a^bxf(x)dx,` `0leqa<b`
In your case `f(x)=2cos(9x^2)` `a=0,b=sqrt(pi)/6`
hence we have
`V=2piint_0^(sqrt pi/6)x cdot 2cos(9x^2)dx=|(t=9x^2),(dt=18xdx),(a_t=0),(b_t=pi/4)|=`
` ` ` ` `(2pi)/9int_0^(pi/4)costdt=(2pi)/9sint |_0^(pi/4)=(2pi)/9(sqrt2/2-0)=(sqrt2pi)/9`
` `
Hence the volume of your rotating solid is `(sqrt2 pi)/9`
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Aligned to:
• 1.OA.3. Apply properties of operations as strategies to add and subtract.
• 1.OA.6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
• 1.OA.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
• 2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
• 2.NBT.5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
• 2.NBT.6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
• 3.OA.1.Interpret products of whole numbers.
• 3.OA.2. Interpret whole-number quotients of whole numbers.
• 3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
• 3.OA.5. Apply properties of operations as strategies to multiply and divide.
• 3.OA.7.Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one–digit numbers.
• 4.OA.4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100
is a multiple of a given one–digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
• 3.NBT.2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
• 3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
• 4.OA.1. Interpret a multiplication equation as a comparison.
• 4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
• 4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
• 4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
• 5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.
• 5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
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What do violations of Bell's inequalities tell us about nature?
Not in the definition of locality! (Yes, such a thing does come up in the derivation of the inequality, though.)
I don't think he actually gave a definition of "locality". The way I interpreted what he was doing was describing a class of models, and then proving that no model in that class could reproduce the
predictions of quantum mechanics. If he gave an explicit definition of what "local" means, I didn't see one.
I am highly skeptical of this. First of all, the claim that was made here was that Bell's definition of locality is inapplicable if the space of λs is unmeasureable.
Maybe it would help the discussion if you wrote down what you consider Bell's definition of "local". What I have seen is this:
• Assume in an EPR-type experiment (assume the spin-1/2 version for definiteness) involving Alice and Bob that there is a deterministic function [itex]A(\hat{a}, \hat{b}, \lambda)[/itex] giving
Alice's result (+1 or -1) as a function of Alice's choice of detector orientation, [itex]\hat{a}[/itex], Bob's choice of detector orientation, [itex]\hat{b}[/itex], and some unknown parameter
[itex]\lambda[/itex] shared by the two particles by virtue of their having been produced as a twin-pair. Similarly, assume a deterministic function [itex]B(\hat{a}, \hat{b}, \lambda)[/itex]
giving Bob's result.
• Then, in terms of such a model, we can call the model "local", if [itex]A(\hat{a}, \hat{b}, \lambda)[/itex] does not depend on [itex]\hat{b}[/itex], and [itex]B(\hat{a}, \hat{b}, \lambda)[/itex]
does not depend on [itex]\hat{a}[/itex]. In other words, Alice's result is [itex]A(\hat{a}, \lambda)[/itex] and Bob's result is [itex]B(\hat{b}, \lambda)[/itex].
• Theorem, there are no such functions [itex]A(\hat{a}, \lambda)[/itex] and [itex]B(\hat{b}, \lambda)[/itex].
The proof of the theorem assumes that the unknown hidden variable [itex]\lambda[/itex] is measurable; in particular, that it makes sense to talk about things such as "the probability that [itex]\
lambda[/itex] lies in some range such that [itex]A(\hat{a},\lambda) = B(\hat{a},\lambda)[/itex]" for various choices of [itex]\hat{a}[/itex] and [itex]\hat{b}[/itex]. Pitowky showed that if you don't
assume measurability of [itex]\lambda[/itex], then the EPR correlations can be explained in terms of a non-measurable function [itex]F(\hat{r})[/itex] where [itex]\hat{r}[/itex] is a unit vector (or
alternatively, a point on the unit sphere), with the properties that:
(This is from memory, so I might be screwing these up):
• [itex]F(\hat{r})[/itex] is always either +1 or -1.
• [itex]\langle F \rangle = \frac{1}{2}[/itex]: The expectation value, over all possible values of [itex]\hat{r}[/itex], of [itex]F(\hat{r})[/itex] is 0.
• If [itex]\hat{r_1}[/itex] is held fixed, and [itex]\hat{r_2}[/itex] is randomly chosen so that the angle between [itex]\hat{r_1}[/itex] and [itex]\hat{r_2}[/itex] is [itex]\theta[/itex], then the
probability that [itex]F(\hat{r_1}) = F(\hat{r_2})[/itex] is [itex]cos^2(\dfrac{\theta}{2})[/itex]
Mathematically, you can prove that such functions exist (with the notion of "probability" in the above being flat lebesque measure on the set of possibilities). Pitowksy called it a "spin-1/2
function".But it's not a very natural function, and is not likely to be physically relevant.
But anyway, were it true, then wouldn't it follow that it was impossible to meaningfully assert that Pitowsky's model is local?
It's explicitly local: When a twin pair is created, a hidden variable, [itex]F[/itex] is generated. Then when Alice later measures the spin along axis [itex]\hat{a}[/itex], she deterministically gets
the result [itex]F(\hat{a})[/itex]. When Bob measures the spin of the other particle, he deterministically gets [itex]-F(\hat{b})[/itex]
I'd even be willing to bet real money that this isn't right -- that is, that there's no genuine example of a local theory sharing QM's predictions here. If it were true, it would indeed be big news,
since it would refute Bell's theorem!
Not in any serious way. Physicists routinely assume things like measurability and continuity, etc., in their theories, and whatever results they prove don't actually hold without these assumptions,
which are seldom made explicit.
In a brief Google search, I didn't see Pitowsky's original paper, but his spin-1/2 models are discussed here:
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On Categories of Fuzzy Petri Nets
Advances in Fuzzy Systems
Volume 2011 (2011), Article ID 812040, 5 pages
Research Article
On Categories of Fuzzy Petri Nets
^1Department of Mathematics and Center for Interdisciplinary Mathematical Sciences, Faculty of Science, Banaras Hindu University, Varanasi 221 005, India
^2Department of Mathematics, Indian School of Mines, Dhanbad 826004, India
Received 16 December 2010; Accepted 20 April 2011
Academic Editor: Uzay Kaymak
Copyright © 2011 Arun K. Srivastava and S. P. Tiwari. 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 introduce the concepts of fuzzy Petri nets and marked fuzzy Petri nets along with their appropriate morphisms, which leads to two categories of such Petri nets. Some aspects of the internal
structures of these categories are then explored, for example, their reflectiveness/coreflectiveness and symmetrical monoidal closed structure.
1. Introduction
Petri nets are a well-known model of concurrent systems [1]. A number of authors have been led to the study of the various categories of Petri nets and their appropriate morphisms [2–4], in the
belief that the categorical study provides a tool to compare different models of Petri nets. At the same time, fuzzy Petri nets have also been studied by many authors in different ways [5, 6]. In
this paper, taking a cue from [3], we introduce another concept of fuzzy Petri nets (although we model our definition on the lines of [2]). This, along with their appropriate morphisms, results in a
category of fuzzy Petri nets (and also of marked fuzzy Petri nets). The structure of these categories is then studied on the lines similar to those in [2], showing that one of the categories is
symmetric monoidal closed.
2. Category of Fuzzy Petri Nets
For categorical concepts used here, [7] may be referred. We begin by collecting some basic definitions.
A Petri net is a bipartite graph, consisting of two kinds of nodes, namely, places and transitions, where arcs are either from a place to a transition or vice versa [3]. Graphically, the places are
represented by circles, transitions by rectangles, and the arcs by arrows. We define a fuzzy Petri net as follows.
Definition 1. A fuzzy Petri net (in short, fPn) is 4-tuple , where and are sets, called the set of places and set of transitions, respectively, and , are functions, called the incidence functions.
We may interpret and defined above as follows.
For (resp., ) gives, the grade with which place is related to transition (resp., transition is related to place ). Thus, and describe some kind of fuzzy arcs between places and transitions.
Definition 2. An fPn-morphism from an fPn to fPn is a pair of functions, and such that the following two diagrams: (1) “hold”, by which it is meant that for all and .
Remark 3. Fuzzy Petri nets and fPn-morphisms form a category, denoted as FPN (the identity morphisms and the composition for this category are obvious to guess).
Definition 4. The product of two fPn's and is the fPn , where and are given by for all and .
Proposition 5. Let , be the two projections and let , be the two injections. Then , are fPn-morphisms.
Proof. To prove that is an fPn-morphism, we need to show that the following two diagrams hold. (3) The above diagrams hold because for all and . Similarly, one can prove that is also an fPn-morphism.
Proposition 6. The product of fPn's and is the categorical product of and in FPN.
Proof. Let be an fPn together with fPn-morphisms . We show that there exists a unique fPn-morphism such that , or equivalently that and . For this purpose, we choose the following and . Let be the
map given by and . We show that the diagrams (5) hold, that is, for all , and , that is, , if and , if , for all as well as , if and , if , for all . But as , are fPn-morphisms, the above
inequalities hold, whereby the diagrams (5) hold. Thus, is an fPn-morphism. Also, the definitions of and are such that we obviously have and .
To prove the uniqueness of , let there exist another fPn-morphism such that , that is, and . We then have , and , whereby and . Thus, , proving the uniqueness of . Hence the product is a categorical
Definition 7. The coproduct of two fPn's and is the fPn , where , and are given by for all and .
Similar to Propositions 5 and 6, the following two propositions can also be proved.
Proposition 8. Let , be the two projections and , be the two injections. Then , are fPn-morphisms.
Proposition 9. The coproduct of fPn's and is the categorical coproduct of and in FPN.
Definition 10. Given two fPn's and , we define two new fPn's and as follows (for sets and , shall denote the set of all functions from to ): (1), where are defined as(2), where are defined as
Proposition 11. The category FPN is a symmetric monoidal closed category (with the constructions in (1) and (2) above, respectively, giving the associated tensor product and hom-object).
Proof. For convenience, the notation is used to denote the fPn . We give a sketch of the proof of closedness of FPN. For this, the two functors are denoted, respectively, as and , which map any
FPN-morphism , respectively, to FPN-morphisms, , and , such that for all, and , for all .
It turns out that is a right adjoint to ; the associated unit of the adjunction, , is given for each , by , where and , are such that , and , with and , for all , and for all .
To establish the universality of , we need to produce, for any given and FPN-morphism , a unique FPN-morphism , such that the following diagram commutes.
We only describe , leaving out the verification of the commutativity of the above diagram and the uniqueness of . is given by and , such that and , with , and .
3. Marked Fuzzy Petri Nets
A marked Petri net is a Petri net together with a function, called marking defined from the set of places to the set of natural numbers [2]. Marking at a particular place gives the number of tokens
at that particular place. In this section, we introduce a concept of marked fuzzy Petri nets and thereby a category of marked fuzzy Petri nets.
Definition 12. An fPn , together with a function (called a fuzzy marking of ), is called a marked fuzzy Petri net (in short, an mfPn) and is denoted as .
Here, marking at a particular place may be interpreted as the degree of confidence to which a token can reside at that place.
Definition 13. Given an mfPn , a transition is said to fire at (or is enabled at ), if , for all .
In an mfPn , for fixed induces a function such that for , gives the degree of confidence to which a transition can fire at marking . Thus, a transition at a marking of mfPn can fire if the degree of
confidence to which it fires does not exceed the degree of confidence to which a token can reside at places.
After firing at the fuzzy marking , we get a new fuzzy marking of , given by , for all . We say that fires at to yield and denote this by . Also, is then said to be directly reachable from through
the transition .
Similar to [8], the marked fuzzy Petri net models of negation, disjunction, and conjunction of fuzzy proposition, can also be given. We illustrate these by following examples.
Example 14. Consider the following graphical representation of an mfPn, which gives the truth value of the negation of a fuzzy proposition. For this, take an mfPn with and . Given a fuzzy
proposition, the initial marking is so chosen that is the truth value of the fuzzy proposition and . Also, is so chosen that (so that the transition can fire) and we also take to be . After the
firing of the transition , at marking , the marking at is given by , the truth value of the negation of the fuzzy proposition.
Example 15. Similar to Example 14, consider the following graphical representation of an mfPn, which gives the disjunction of truth values of two fuzzy propositions. For this, take an mfPn, with and
. Given two fuzzy propositions, the initial marking is so chosen that and are the truth values of the given fuzzy propositions and . Also, and are so chosen that and (so that the transition can fire)
and we also take to be . After the firing of the transition , at marking , the marking at is given by , the truth value of the disjunction of the fuzzy propositions.
(Analogous to Example 15, one can design mfPn, which gives the conjunction of the truth values of two fuzzy propositions.)
4. Category of Marked Fuzzy Petri Net
In this section, a category of marked fuzzy Petri net, inspired from [2], is introduced.
Definition 16. For mfPn's and , a function , is said to be -ok if , for all .
Remark 17. MFPN shall denote the category of all mfPn's, with mfPn-morphisms being the fPn-morphisms such that is -ok.
Proposition 18. Let and be two mfPn's and let be an mfPn-morphism. Then for is enabled at , if is enabled at .
Proof. As is an mfPn-morphism, and , for all . Also, as is enabled at , we have , for all , whence for all. But , whereby, , for all. Thus, is enabled at .
Proposition 19. Let and be two mfPn's and be an mfPn-morphism. Then for is -ok, if .
Proof. From the above proposition, it is clear that . Also, as is an mfPn-morphism, , and , for all . Consequently, for all , whereby, . Thus, for all . Hence is -ok, for .
Definition 20. The product of two mfPn's and is the mfPn , where is the product of fPn's and and is given by for all .
Proposition 21. The FPN-morphisms , given in Proposition 5 are MFPN-morphisms from to .
Proof. Since and , and are -ok and -ok, respectively. Hence , are MFPN-morphisms.
Using Propositions 5 and 21, the next proposition is evident.
Proposition 22. The product of mfPn's is the categorical product in MFPN.
Definition 23. The coproduct of two mfPn's and is the mfPn , where is the coproduct of fPn's and and is given by , for all .
Similar to Propositions 21 and 22, the following two propositions can also be proved.
Proposition 24. The FPN-morphisms , given in Proposition 8 are MFPN-morphisms from to .
Proposition 25. The coproduct of mfPn's is the categorical coproduct in MFPN.
5. Relationship between FPN and MFPN
There is an obvious functor , given by and .
We omit the easy verification of the following observations.
Proposition 26. There are full and faithful functors , which, on objects, are respectively, given by and , where 0 and 1, are respectively, the 0-valued and the 1-valued constant functions, and which
leave the morphisms unchanged.
It is easy to prove the following.
Proposition 27. The functor (resp., ) is left adjoint (resp., right adjoint) to the functor .
Thus, we have the following.
Proposition 28. The category FPN is isomorphic to a full reflective subcategory, and also to a full coreflective subcategory, of MFPN.
6. Conclusion
We note that nothing has been said about the symmetric monoidal closed structure of the category MFPN of marked fuzzy Petri nets. An obvious attempt to make MFPN symmetric monoidal closed would
appear to be as follows. Given mfPn's and , the fPn's and (cf. Definition 10) can be made mfPn's by taking their respective fuzzy markings to be and , defined as and , for all , for all. However, for
each fixed mfPn , the resulting functors , do not turn out to be adjoint. As an attempt to repair the above situation, may be redefined as , so that the functor does, now, turn out to be right
adjoint to the (modified) functor . However, the symmetry of in this modified setup is now lost (this situation is similar to the one noted in [2]). So there may be a different symmetric monoidal
closed structure on MFPN which we have not been able to find presently.
1. W. Reisig, Petri Nets: An Introduction, EATCS Monographs on Theoretical Computer Science, Springer, Berlin, Germany, 1985.
2. C. Brown, D. Gurr, and V. de Paiva, “A linear specification language for Petri nets,” Tech. Rep. DAIMI PB-363, Aarhus University, Aarhus, Denmark, 1991.
3. J. Meseguer and U. Montanari, “Petri nets are monoids,” Information and Computation, vol. 88, no. 2, pp. 105–155, 1990.
4. G. Winskel, “Categories for model of concurrency,” in Lecture Notes in Computer Science, No. 197, pp. 246–267, Springer, Berling, Germany, 1994.
5. C. G. Looney, “Fuzzy Petri nets for rule-based decision making,” IEEE Transactions on Systems, Man and Cybernetics, vol. 18, no. 1, pp. 178–183, 1988. View at Publisher · View at Google Scholar
6. H. Scarpelli and F. G. Gomide, “Fuzzy reasoning and fuzzy Petri nets in manufacturing systems modelling,” Journal of Intelligent and Fuzzy Systems, vol. 1, pp. 225–241, 1993.
7. S. Maclane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, Berlin, Germany, 1971.
8. H. E. Virtanen, “A study in fuzzy Petri nets and the relationship to fuzzy logic programming, Reports on Computer Science and Mathematics,” Åbo Akademi A, no. 162, 1995.
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Useful mathematical tools
3.7 Useful mathematical tools
3.7.1 Curvature of space with a conical singularity
Consider a space [195] in the two-dimensional case by using topological arguments. These arguments were generalized to higher dimensions in [7 ]. One way to extract the singular contribution is to
use some regularization procedure, replacing the singular space [111 ]. In the limit [111 ]:
These formulas can be used to define the integral expressions [111 ]
The terms proportional to 56) – (59) are defined on the regular space 57) – (59) are something like a square of the [111 ].
Topological Euler number.
The topological Euler number of a Suppose that [111 ]A special case is when [111 ]An interesting consequence of this formula is worth mentioning. Since the introduction of a conical singularity can
be considered as the limit of certain smooth deformation, under which the topological number does not change, one has [111 ]A simple check shows that Eq. (63) gives the correct result for the Euler
number of the sphere 63): 63) the known identity 63) is valid for spaces with continuous abelian isometry and it may be violated for an orbifold with conical singularities.
Lovelock gravitational action.
The general Lovelock gravitational action is introduced on a d-dimensional Riemannian manifold as the following polynomial [166]where 60) and is thus topological. In other dimensions the action (64)
is not topological, although it has some nice properties, which make it interesting. In particular, the field equations, which follow from Eq. (64), are quadratic in derivatives even though the
action itself is polynomial in curvature. On a conical manifold [111 ]where the first term is the action computed at the regular points. As in the case of the topological Euler number, all terms
quadratic in 65). The surface term in Eq. (65) takes the form of the Lovelock action on the singular surface and 65) allows us to compute the entropy in the Lovelock gravity by applying the replica
formula. In [145] this entropy was derived in the Hamiltonian approach, whereas arguments based on the dimensional continuation of the Euler characteristics have been used for its derivation in [7].
3.7.2 The heat kernel expansion on a space with a conical singularity
The useful tool to compute the effective action on a space with a conical singularity is the heat kernel method already discussed in Section 2.8. In Section 2.9 we have shown how, in flat space,
using the Sommerfeld formula (22), to compute the contribution to the heat kernel due to the singular surface
where the coefficients in the expansion decompose into bulk (regular) and surface (singular) parts The regular coefficients are the same as for a smooth space. The first few coefficients are The
coefficients due to the singular surface The form of the regular coefficients (69) in the heat kernel expansion has been well studied in physics and mathematics literature (for a review see [219 ]).
The surface coefficient 70) was calculated by the mathematicians McKean and Singer [174] (see also [42]). In physics literature this term has appeared in the work of Dowker [69]. (In the context of
cosmic strings one has focused more on the Green’s function rather on the heat kernel [3, 100].) The coefficient [101 ] although in some special cases it was known before in works of Donnelly [64, 65
It should be noted that due to the fact that the surface 70) in the heat kernel expansion.
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Help with bouncing ball
November 25th 2006, 10:00 PM
Help with bouncing ball
Hi, anyone out there, my problem is pure logic that I obviously don't have at the mo.........
A ball bounces everytime at 2/5 of the height of its bounce. If we let it fall 75 cm, calculate the height it reaches on the 2nd bounce.
Please someone help me make sense of this:eek:
November 25th 2006, 11:08 PM
Hello, Asnera,
The ball starts at a height of 75 cm. The height after the first bounce is (2/5)*75 cm = 30 cm. The height after the seconf bounce is (2/5)*30 cm = 12 cm.
In general: Let n be the number of bounces and h the height with respect of n then:
$h(n)=75cm \cdot \left(\frac{2}{5}\right)^n$
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FOM: physical computability
JoeShipman@aol.com JoeShipman at aol.com
Wed Aug 23 18:42:29 EDT 2000
In a message dated 8/23/00 2:37:32 PM Eastern Daylight Time,
mlerma at math.northwestern.edu writes:
<< That is not clear to me. For instance, a measurement process performed
on a quantum system usually yields random (hence non computable) results,
but that does not refute the Church-Turing thesis ("every algorithm can be
carried out by a Turing machine"), since the (non causal, non deterministic)
time evolution of a quantum system during a measurement process could
hardly qualify as "algorithmic". I do not know what other people's intuition
for "algorithm" is, but I would not call "algorithmic" a process that
does not always yield the same result for identical inputs. >>
In the situation I described, the noncomputable sequence is not noncomputable
because it is random, it is noncomputable because it is a mathematically
definable but non-recursive sequence. The randomness only enters the picture
in making the probability that the output is the defined noncomputable
sequence be some real number arbitrarily close to 1 rather than equal to 1.
The version of the Church-Turing thesis I am using here is different from
yours. You are rendering it as "every algorithm can be carried out by a
Turing machine", while I am interested in the thesis "every function we can
calculate by an effective procedure is calculable by a Turing machine". The
difference is that "algorithm" connotes some sort of mental step-by-step
process, while "effective procedure" means ANYTHING we can do to obtain the
sequence that works reliably. Here "reliably" only need mean "with
probability arbitrarily close to 1", not with certainty.
Your version of the Church-Turing thesis is a weak one which just says that
Turing machines capture the intuitive notion of algorithm. It is a statement
about PSYCHOLOGY. My version is a stronger one which says that Turing
machines can accomplish anything that can be accomplished by an "effective
procedure". It is a statement about PHYSICS. A physical theory which
entailed that a certain experimental procedure would produce a definable
noncomputable bit sequence would refute my strong version of Church's thesis;
I can't see how your weak version could be falsified.
-- Joe Shipman
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2010.18: Computing Matrix Functions
2010.18: Nicholas J. Higham and Awad H. Al-Mohy (2010) Computing Matrix Functions. Acta Numerica, 19. 159 -208. ISSN 0962-4929
This is the latest version of this eprint.
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DOI: 10.1017/S0962492910000036
The need to evaluate a function $f(A)\in\mathbb{C}^{n \times n}$ of a matrix $A\in\mathbb{C}^{n \times n}$ arises in a wide and growing number of applications, ranging from the numerical solution of
differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a
description of two recent applications. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or
rational functions, and matrix iterations. Computation of the Fr\'echet derivative, which is important for condition number estimation, is also treated, along with the problem of computing $f(A)b$
without computing $f(A)$. A summary of available software completes the survey.
Item Type: Article
Additional Information:
Uncontrolled Keywords: matrix $p$th root, primary matrix function, nonprimary matrix function, Markov chain, transition matrix, matrix exponential, Schur-Parlett method, CICADA
Subjects: MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number: 2010.18
Deposited By: Nick Higham
Deposited On: 18 May 2010
Available Versions of this Item
• Computing Matrix Functions (deposited 18 May 2010) [Currently Displayed]
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Solomon Friedberg
Department of Mathematics
Boston College
Chestnut Hill, MA 02467-3806
(617) 552-3002
Department Fax: (617) 552-3789
Email: friedber@bc.edu
Curriculum Vitae (in pdf format) (updated December 18, 2013)
Ph.D. University of Chicago, 1982
M.S. University of Chicago, 1979
B.A. Summa cum Laude, University of California, San Diego, 1978
James P. McIntyre Professor of Mathematics and Chair of the Department of Mathematics. I have been Chair of the department since 2007. During my period of service, the Math Department has written
a self-study and had an external program review, started a Ph.D. program, instituted a new B.S. degree, hired superb young scholars and teachers into tenure-track and postdoctoral positions,
revised its undergraduate offerings significantly, started an annual Alumni Newsletter and the BC Math Alumni Network, organized a Distinguished Lecturer series and the BC-MIT Number Theory
Seminar, built new ties to the Lynch School of Education, hosted an American Mathematical Society sectional meeting, signed a Memorandum of Understanding with the Mathematical Sciences Center and
Department of Mathematical Sciences at Tsinghua University to encourage cooperation and the exchange of scholars, and carried out a planning process to determine next steps as we seek to become
one of the top departments in the country in both research and teaching. Here is a November 2012 news report on the department's progress.
During the 2013-2014 academic year, I will be on sabbatical leave. Prof. Robert Meyerhoff will serve as Acting Chair.
The websites for some of my previous courses may be found here. For more information about which mathematics course to take, please see the Mathematics Department's Advisement Website.
Research Areas:
Automorphic forms, number theory, and representation theory. Selected publications (including preprints). A good part of my work has concerned the study of families of L-functions by means of
analytic methods involving Dirichlet series in several complex variables. For example, my 1989 paper with Bump and Hoffstein used these to prove a first-order-vanishing theorem for GL(2)
L-functions under quadratic twists, which has applications to arithmetic. The study of such series has proved unexpectedly rich. I and my collaborators now refer to this area as the study of
Multiple Dirichlet Series (though it might be more accurate to tack on "Automorphic" in front). Multiple Dirichlet series, which are related to the theory of automorphic forms on metaplectic
covers of reductive groups, are not Euler products (in contrast to Langlands L-functions), but rather twisted Euler products---the interplay between the contributions from different primes is
governed by n-th order residue symbols. In many cases they have meromorphic continuation and a finite group of functional equations that is generated by reflections. In the last few years, I and
my collaborators Profs. Ben Brubaker (Minnesota) and Daniel Bump (Stanford) have established surprising links to combinatorial representation theory, quantum groups and statistical mechanics.
Multiple Dirichlet series may also be attached to other classes of mathematical objects, such as affine Weyl groups. If their continuation to a suitable region can be proven, it would lead to
striking advances.
I am one of the organizers for the BC-MIT Number Theory Seminar. The year 2013-2014 will be the sixth year of this joint seminar series.
Other Activities:
I founded the Boston College Mathematics Case Studies Project to develop new training materials--Case Studies--for use in TA-development programs for mathematics graduate students. Though the
project ended over 10 years ago, the materials are still in use, and I continue to give workshops and talks on them. More recently, a group of Chilean mathematicians has carried out a project to
improve the pedagogical skills of future Chilean high school teachers using, in part, case studies. I have made 3 trips to Chile in support of their efforts. A volume by Cristián Reyes of the
Centro de Modelmiento Matemático containing Spanish-language cases for secondary teachers appeared in 2011. Further work in this direction is now being planned.
I am also involved in pre-collegiate mathematics education in other ways. I am a member of the Board of Directors of
Math for America Boston
, and was an advisor to the Massachusetts Department of Elementary and Secondary Education concerning the Massachusetts mathematics framework and concerning its response to the Common Core
(2009-2010). Additional service includes: Massachusetts Board of Elementary and Secondary Education's
Math-Science Advisory Council
Focus on Mathematics Phase II
Advisory Board (2009-2011),
Community for Advancing Discovery Research in Education
Advisory Board (2011-2012), AMS representative on a JPBM committee to explore a Partnership for Mathematical Sciences in America (2009); member of the Advisory Board for the American Mathematical
Society's Working Group on Preparation for Technical Careers (2007-2009); member of the Steering Committee for the Commonwealth of Massachusetts's Mathematics and Science Partnerships Program
(2004-2007); member of the Arithmetic Test Online Math Content Board (2006). I was also on a team of mathematicians and math educators who developed essays concerning middle school and high
school mathematics (2008-2009). Here are some
from the project. And I served as an (unpaid) consultant in the writing of the Massachusetts Board of Education's
Guidelines for the Mathematical Preparation of Elementary Teachers (July 2007)
I have written three op-eds concerned with math education:
□ an Op-Ed concerning the implementation of the Common Core which appeared in the Los Angeles Times on December 11, 2013. (This op-ed was reprinted in The Anniston Star, Chattanooga Times Free
Press, The Columbian, Glen Falls Post-Star, The Gulf Today (United Arab Emirates), Honolulu Star-Advertiser, Long Island Newsday, The State, The Times-Tribune, The Virgin Islands Daily News,
and several on-line sites.)
□ an Op-Ed concerning the need to invest in math and science education which appeared in the Boston Globe on May 21, 2009. (This op-ed was reprinted under the title "Addressing the Crisis in
Math and Science" by Business West, June 8, 2009.)
□ an Op-Ed concerning the math education of future elementary school teachers which appeared in the Boston Herald on April 23, 2007.
I have been an editor of the CBMS book series Issues in Mathematics Education from 2006 on.
Locally, I have served as a mentor and as a content-advisor for preservice teachers at BC, as the BC Teachers for a New Era (TNE) point person for the mathematics department, and as Chair of the
TNE Coordinating Council (2011-2013). I hope to involve more math students in K-12 education, and more math faculty in interacting with pre- and in- service K-12 teachers. Please contact me about
this if you are interested. From 2009-2010 on I have also co-organized a monthly Seminar in Mathematics Education jointly with Prof. Lillie Albert of the Department of Teacher Education, and from
2012 on, also jointly with Prof. C.K. Cheung of the Mathematics department. This seminar series has been sponsored by TNE.
Last, I am co-PI on a 6-year NSF-funded project (2013-2019) "Exemplary Mathematics Educators for High-need Schools," joint with Profs. Albert and Cheung. Together with colleagues at BU, the EDC
and Math for America-Boston, we are working with both beginning and master math teachers for high-needs schools and developing infrastructure to support excellence in math instruction in
Boston-area schools. For more information about the Noyce Teaching Fellowships leading to an M.S.T., please see here.
Honors and Awards:
Phi Beta Kappa, University of California, San Diego, 1978
McCormick Graduate Fellowship, University of Chicago, 1978-81
NSF Postdoctoral Research Fellowship, 1982-84
NATO Postdoctoral Fellowship in Science, 1985-86
Indo-American (Fullbright) Fellowship, 1987-88
Sloan Fellowship, 1989-92
Distinguished Visiting Professor of Mathematics, Brown University, Spring 2002
MAA Northeastern Section Award for Distinguished College or University Teaching, 2009
McIntyre Chair, Boston College, appointed 2013
Fellow of the American Mathematical Society, Class of 2014
Distinguished Ordway Visitor, University of Minnesota, 2014
Here are a few photos from various trips and conferences.
Recent and Planned External Lectures (and occasional other travel or events):
□ Automorphic Forms: Advances and Applications, CIRM, Luminy, May 25-29, 2015.
□ Math Education project in Chile, January 5-9, 2015. (Tentative.)
□ Conference on Mathematicians and Math Education in the Americas, Guanajuato, Mexico, October 20-24, 2014. (Tentative.)
□ Analytic Number Theory and its Applications: A Conference in Honor of Jeff Hoffstein, Perrotis College, Thessaloniki, Greece, July 14-18, 2014.
□ Second US-EU Building Bridges Automorphic Forms Summer School and Workshop, Bristol, U.K., June 30-July 11, 2014. Mini-course instructor, "The Langlands Program" (co-instructor with Prof. Jim
Cogdell, Ohio State University); will also lecture in the workshop.
□ SUMO Colloquium/Speaker Series, Stanford University, April 2014.
□ Workshop on The roles of mathematics departments and mathematicians in the mathematical preparation of teachers, MSRI, March 26-28, 2014. (Workshop co-organizer.)
□ Northeastern University, Representation Theory Seminar, March 21, 2014.
□ Distinguished Ordway Visitor, University of Minnesota, February 24 to March 7, 2014. (Includes Colloquium lecture on February 27, 2014 and Lie Theory seminar lecture on February 28, 2014.)
□ Mathematicians and School Mathematics Education: A Pan-American Workshop, Banff International Research Station, January 26-31, 2014. (Workshop co-organizer.)
□ Workshop on Whittaker Functions: Number Theory, Geometry and Physics, Banff International Research Station, October 13-18, 2013. (Workshop co-organizer.)
□ Brown University, Algebra and Number Theory Seminar, September 30, 2013.
□ Stark's Conjecture and Related Topics (honoring Harold Stark on his retirement), UCSD, September 2013.
□ POSTECH Summer School in Number Theory, Pohang Math Institute, Republic of Korea, July 5-11, 2013, 5 lectures.
□ 2013 NCTS Special Day on Automorphic Forms, National Center for Theoretical Sciences, Hsinchu, Taiwan, July 1, 2013.
□ The Many Ways of IBL, University of Chicago, June 17-21, 2013 (attending but not presenting).
□ Preparing Graduate Students to Teach Undergraduate Mathematics: A Working Conference, Harvard University, June 12-13, 2013.
□ Initiative on a Common Core Mathematics Licensure Examination for Entry into Teaching, University of Michigan, May 31-June 1, 2013 (attending but not presenting).
□ Commencement Address to Boston College Arts & Sciences Masters and PhD Recipients, May 20, 2013.
□ Rutgers University, Number Theory Seminar and Teaching Math Seminar (2 separate talks), April 3, 2013.
□ ICERM, January 28, 29; February 4, 5, 6, 19, 26, March 12, April 9, 23: Lectures, tutorials, panels related to the special semester there.
□ AMS-MAA Joint Math Meetings, The Training and Professional Development of Teaching Assistants (sponsored by the AMS-MAA Joint Committee on TAs and Part-Time Instructors), Panel Member, San
Diego, CA, January 9-12, 2013. Slides available here.
□ Number Theory Seminar, Texas A&M University, December 12, 2012.
□ Georgetown University, Department of Mathematics and Statistics External Review Committee, December 5-7, 2012.
□ Palmetto Number Theory Series XVIII, Plenary Speaker, Wake Forest University, Winston-Salem, NC, September 2012.
□ Automorphic Forms, Representations and Combinatorics: A Conference in Honor of Daniel Bump, Stanford University, August 13-16, 2012, speaker and conference co-organizer.
□ Tsinghua University Mathematical Sciences Center, Colloquium, Beijing, China, July 27, 2012.
□ Workshop on Lie Group Representations and Automorphic Forms (3 lectures), Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, China, July 23-27, 2012.
□ Emory University, REU Program in Number Theory, June 25, 2012.
□ 2012 KIAS-POSTECH Number Theory Workshop (2 lectures), Pohang University of Science and Technology (POSTECH), Pohang, Republic of Korea, June 10-14, 2012.
□ Caltech Number Theory Seminar, March 8, 2012.
□ Modular Forms, Mock Theta Functions, and Applications, University of Cologne, Cologne, Germany, 27 February -1 March 2012.
□ Matemáticos en la Educación Matemática Escolar: En la búsqueda de impacto en nuestra realidad educacional (Mathematicians and School Mathematics Education: Looking for impact in transforming
our educational reality), Santiago, Chile, January 11-13, 2012.
□ Universidad de Chile, Public Lecture, January 10, 2012.
□ TORA I (TORA=Texas-Oklahoma Representations and Automorphic forms), plenary lecture, Denton, TX, September 17-18, 2011.
□ Undergraduate Math Research Colloquium, University of North Texas, Denton, TX, September 16, 2011.
□ Number Theory Xi'an 2011 - Workshop on Number Theory (4 lectures), Northwest University, Xi'an, China, June 2011.
□ ETH, Zürich, May 2011 (2 lectures).
□ Ohio State University, Number Theory Seminar, May 9, 2011.
□ Yale University, Number Theory/Algebraic Geometry Seminar, March 29, 2011.
□ Workshop on Knowledge of Mathematics for Teaching at the Secondary Level (workshop participant and advisory board member for the related research project), Institute for Mathematics and
Education, University of Arizona, March 23-26, 2011.
□ University of Rochester, Colloquium & Number Theory Seminar, March 16-17, 2011.
□ Brigham Young University, "Focus on Math" Colloquium Lecture & Number Theory Seminar, January 11, 2011.
□ First joint meeting of the AMS and the Sociedad de Matemática de Chile (SOMACHI), Session on Automorphic Forms and Dirichlet Series, Pucón, Chile, December 15-18, 2010. (Session
□ AMS Meeting, Session on Automorphic Forms and Number Theory, Los Angeles, October 9-10, 2010.
□ University of Maine, Colloquium in the Math Department & Lecture at the Center for Science and Mathematics Education Research, October 4, 2010.
□ NCTS International Conference on Automorphic Forms and Related Topics, National Center for Theoretical Sciences, Hsinchu, Taiwan, July 7-9, 2010; lecture for graduate students July 12, 2010.
□ Workshop on Whittaker Functions, Crystal Bases, and Quantum Groups, Banff International Research Station, June 6-11, 2010.
□ BC Faculty/Staff Nicaragua Immersion Trip, May 28-June 4, 2010. (Did not lecture; university-associated trip.)
□ University of Wisconsin, Number Theory Seminar, April 29, 2010.
□ College of the Holy Cross, Colloquium on the occasion of the Pi Mu Epsilon Induction Ceremony, April 14, 2010.
□ 24th Automorphic Forms Workshop, University of Hawaii at Manoa, March 22-26, 2010.
□ Columbia-CUNY-NYU Number Theory Seminar, February 18, 2010.
External lectures, 2001-2009
Ph.D. Students:
Ozlem Imamoglu, 1992, UCSC, Theta functions and Kubota homomorphisms for the symplectic group over the Gaussian integers.
Thomas Goetze, 1995, UCSC, On a cubic Shimura integral for a rank two symplectic group.
Nancy Allen, 1996, UCSC, On the spectra of certain graphs arising from finite fields.
Ji Li, 2005, Boston University, Determination of a GL(2) cuspform by twists of critical L-values.
Ting-Fang Lee, 2013, National Tsing Hua University, Taiwan, On arithmetic over function fields (co-advisor with Prof. Jing Yu).
I greatly enjoy having doctoral students, and welcome graduate students interested in writing a Ph.D. dissertation in automorphic forms or related areas of number theory or representation theory.
Please apply to our doctoral program if you are interested in working with me.
Math Department Home Page
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Undergraduate Alumni
*A recipient of the O.L. Hughes award.
Year 2013
Andrew Bowles
BS in Mathematics
Andrew Breckenridge
BSE in Mathematics Education
Annamarie Cantrell
BSE in Mathematics Education
Harold Crow
BS Mathematics
Thomas Crisco
BSE in Mathematics Education
Jacob De La Paz
BSE in Mathematics Education
Haley Laffoon**
BSE in Mathematics Education
Kristianna Leonard
BSE in Mathematics Education
Joel Lookadoo
BSE Mathematics Education
April Martin
BSE in Mathematics Education
Katrina McAfee
BS in Mathematics
Rebecca Moye
BS in Mathematics
Lindsey Parker
BS in Mathematics
Jessica Reznicek
BSE in Mathematics Education
Paige Sanderson
BSE in Mathematics Education
Brooke Tunstall
BS in Mathematics
Katelynn Whisnant
BSE in Mathematics Education
Year 2012
Derek, Anderson
BS in Mathematics
Merri, Beall
BS in Mathematics
Carlene, Crumby
BSE in Mathematics
Blake, Driscoll
BSE in Mathematics
Jeremy, Elsinger
BSE in Mathematics
Laura, Fisher
BSE in Mathematics
Hannah, Fitzmaurice
BS in Mathematics
Alex, Looney
BS in Mathematics
Vinh, Lu
BS in Mathematics
Andrea', Mack
BSE in Mathematics
Matthew, Mason
BS in Mathematics
Aaron, McMoran
BS in Mathematics
Brandon, McVay
BS in Mathematics
Lauren, Messer
BSE in Mathematics
James, Morgan
BSE in Mathematics
Zachary, Parham*
BS in Mathematics
Vanessa, Paz
BS in Mathematics
Joshua, Schoolcraft
BS in Mathematics
Rebecca, Smith
BS in Mathematics
Zachary, Stalling*
BS in Mathematics
Rachel, Tyler
BSE in Mathematics
Taylor, Vance
BS in Mathematics
Brandy. Walsh
BS in Mathematics
Year 2011
Choate, Odes
BSE in Mathematics
Counts, Chris
BSE in Mathematics
LeMaster, Sarah
BSE in Mathematics
Lunsford, Tamara
BSE in Mathematics
McGhehey, Kali
BS in Mathematics
Ng, Bo Sing
BS in Mathematics
Pruss, Philip
BSE in Mathematics
Steinbarger, Tabitha
BS in Mathematics
Stovall, Thomas
BS in Mathematics
Tate, Timothy
BS in Mathematics
Trainor, Kristen
BS in Mathematics
Watts, Lindsey
BSE in Mathematics
White, Megan
BSE in Mathematics
Year 2010
Bradley Baker
BSE in Mathematics
Lucille Busch
BS in Mathematics
Jing Voon Chen*
BS in Mathematics
Kristy Clark
BSE in Mathematics
Derek Damon
BS in Mathematics
Ezechiel Degny
BS in Mathematics
Kryshtene Henderson
BS in Mathematics
David Hieronymus
BS in Mathematics
Alexzandria Hook
BSE in Mathematics
Carlos Merino
BS in Mathematics
Andrew Muse
BS in Mathematics
Austin Muse
BS in Mathematics
Sean Patterson
BS in Mathematics
Mark Senia
BS in Mathematics
Brady Sharp
BS in Mathematics
Jordan Short
BSE in Mathematics
Jon Sumners
BS in Mathematics
Mary Whitney
BSE in Mathematics
Year 2009
Matthew Adams
BS in Mathematics
Benjamin Bell
BS in Mathematics
Kellie Black
BSE in Mathematics
Kevin Casey
BSE in Mathematics
Natalie Cox
BSE in Mathematics
Courtney Davis*
BS in Mathematics
Jessica Elliott
BSE in Mathematics
Peggy Enfinger
BS in Mathematics
James Fitch
BS in Mathematics
Newton Foster
BA in Mathematics
Lori Kagebein
BSE in Mathematics
Rachel Layman
BS in Mathematics
Rebecca Nichols
BS in Mathematics
Janna Palmer
BS in Mathematics
Darron Pitchford
BS in Mathematics
Brittany Russell
BSE in Mathematics
Eric Sellers
BS in Mathematics
Lori Smith
BSE in Mathematics
Luis Suazo
BS in Mathematics
Jonathan Taylor
BS in Mathematics
Christopher Willette
BS in Mathematics
Scott Wray
BS in Mathematics
Year 2008
Jonathan Bise
BS in Mathematics
Barrett Buck
BSE in Mathematics
Jillian Bullock
BSE in Mathematics
David Ekrut
BS in Mathematics
Dawn Gabbard
BSE in Mathematics
Amanda Green
BSE in Mathematics
Gage Harris
BS in Mathematics
Erikka Johnson
BS in Mathematics
Kara Kamruddin
BS in Mathematics
Cedean Kematic
BS in Mathematics
Kimberly King
BS in Mathematics
Jennifer Lachowsky
BSE in Mathematics
Ashley Matthews
BSE in Mathematics
Christine Minkler
BS in Mathematics
Jason Morris
BS & BSE in Mathematics
Triston Murray
BS in Mathematics
Nicholas Nelson
BS in Mathematics
Benjamin Perea
BS in Mathematis
Catherine Reyes
BSE in Mathematics
Anthony Rhodes
BS in Mathematics
Benjamin Rodery
BSE in Mathematics
Michael Schelkopf
BS in Mathematics
Patrick Sichmeller
BSE in Mathematics
Brittany Williams
BSE in Mathematics
Tabitha Wirges
BS in Mathematics
Matthew Yielding
BSE in Mathematics
Year 2007
Brittany Culberson
BS in Mathematics
Robert Burgess
BS in Mathematics
Samuel Elsinger
BS in Mathematics
Amber Fason
BS in Mathematics
Christa Fisher
BS in Mathematics
Ethan Hereth*
BS in Mathematics
Emily Johnson
BSE in Mathematics
Jonathan Johnson
BS in Mathematics
Kimberly Layman
BS in Mathematics
Elisha Lowry
BSE in Mathematics
Latasha Mullins
BSE in Mathematics
Hong Khank Phan
BS in Mathematics
Katie Reynolds
BS in Mathematics
Everett Lee Robbins
BS in Mathematics
Dennis Show II
BSE in Mathematics
Amber Strain
BA in Mathematics
Joel Terwilliger
BA in Mathematics
Emily Thomas
BSE in Mathematics
Joseph Walker
BS in Mathematics
Justin Warner
BSE in Mathematics
Heather Willis
BSE in Mathematics
William Womack
BS in Mathematics
Year 2006
Tara Beck
BS in Mathematics
Nicole Becnel
BS in Mathematics
Bradley Burnette
BS in Mathematics
Cara Cates
BS in Mathematics
Paige Covington
BS in Mathematics
Holly Easley
BS in Mathematics
Altankhuyag Gongor
BS in Mathematics
Emily Hodo
BS in Mathematics
Beau Jones
BS in Mathematics
Cailin Noble
BS in Mathematics
Steven Stock
BS in Mathematics
Jason Torrence
BS in Mathematics
David Watts*
BS in Mathematics
John Wharton
BS in Mathematics
Tonia Young
BS in Mathematics
Sydney Marks
BA in Mathematics
Mechelle Belanger
BSE in Mathematics
Mary Foster
BSE in Mathematics
Jessica Frye
BSE in Mathematics
Raymond Girdler
BSE in Mathematics
Bethany Knight
BSE in Mathematics
Katie Lilly
BSE in Mathematics
Britni Pulliam
BSE in Mathematics
Year 2005
Yousuf Abbasi
BS in Mathematics
Timothy Bennett
BS in Mathematics
Anna Bowden
BS in Mathematics
Andrew Brown
BS in Mathematics
Sheila Conyers
BS in Mathematics
Matthew Franklin
BS in Mathematics
Joel Harris
BS in Mathematics
Joshua Knight
BS in Mathematics
William Ledbetter
BS in Mathematics
Michael McVay
BS in Mathematics
Katy Ray
BS in Mathematics
Jeremy Roberts
BS in Mathematics
Justin Talley
BS in Mathematics
Genai Walker
BS in Mathematics
Angela West
BS in Mathematics
Curtis Wilder
BS in Mathematics
Kathryn Bergeron
BSE in Mathematics
Rachel Blackwell
BSE in Mathematics
Sarah Edwards*
BSE in Mathematics
Brandy Harvey
BSE in Mathematics
Kaci Palmer
BSE in Mathematics
Melissa Riley
BSE in Mathematics
Heather Trusty
BSE in Mathematics
Wilson Whitney
BSE in Mathematics
Year 2004
Brittany Benefield
BS in Mathematics
Teresa Barnett
BS in Mathematics
Aaron Cates
BS in Mathematics
Christa Chennault
BS in Mathematics
Matthew Dalke
BS in Mathematics
Ashley Dumas
BS in Mathematics
Jenny Gardner
BS in Mathematics
Leslie Gomes
BS in Mathematics
Abel Henson
BS in Mathematics
Philip Hern
BS in Mathematics
Ruth Miller
BS in Mathematics
Matthew Pace
BS in Mathematics
Windy Richardson*
BS in Mathematics
Angela Spicer
BS in Mathematics
Scott Sullivan
BS in Mathematics
Joshua Thacker
BS in Mathematics
Charity Washam
BS in Mathematics
Bonnie Wolf
BS in Mathematics
Hope Allen
BSE in Mathematics
Lillie Griffin
BSE in Mathematics
Danielle Severs
BSE in Mathematics
Year 2003
Brooke Boatman
BS in Mathematics
Gayla Nooner
BS in Mathematics
Elizabeth Nichols
BS in Mathematics
Matthew Flora
BS in Mathematics
Glenn Gamble
BS in Mathematics
Kurt Stein
BS in Mathematics
Joy Tenpenny
BS in Mathematics
Year 2002
Loi Clampit-Booher
BS in Mathematics
Sarah Jacobs
BS in Mathematics
Heather May
BS in Mathematics
Martha Wolfe
BSE in Mathematics
Year 2001
Austin Lovenstein
BS in Mathematics
Krista Cunningham*
BSE in Mathematics
Year 2000
Carlie Brown
BS in Mathematics
Ashley Smith
BS in Mathematics
Angela Whisenhunt
BS in Mathematics
Catherine Corless
BS in Mathematics
Tamekia Brown
BSE in Mathematics
John Rathbun
BSE in Mathematics
Debra Taylor
BSE in Mathematics
Amy Wheat
BSE in Mathematics
Jaime Brigance Walker
BSE in Mathematics
Jimmy Fetterly
BSE in Mathematics
Dana Goodwin*
BS in Mathematics
Year 1999
Lisa Monroe
BS in Mathematics
Amy Rumfelt Slack
BS in Mathematics
Year 1998
Jackie Williams
BSE in Mathematics
Received the Dorothy Long Award
Year 1997
Tara Jackson
BS in Mathematics
Bridget Warner Johnson
BS in Mathematics
Greg Lunsford
BS in Mathematics
Tommy Pyle*
BS in Mathematics
Dale Ward
BS in Mathematics
Year 1996
Dustin Jones
BS in Mathematics
Jennifer Mills
BSE in Mathematics
Trevor Seifert
BS in Mathematics
Kohnie Tingley
BS in Mathematics
Kerry Ballany*
BS in Mathematics
William Joyner
BS in Mathematics
Lori Neal
BS in Mathematics
Monica Jett
BS in Mathematics
Year 1995
Leela Holliman
BS in Mathematics
Betty Peterson
BS in Mathematics
Amy Calhoun
BS in Mathematics
Erin Jumper
BS in Mathematics
Anita Cegers
BSE in Mathematics
Marshall White
BS in Mathematics
Year 1994
Becky McMoran
BS in Mathematics
Paul Griep*
BSE in Mathematics
Felix Maul
BSE in Mathematics
Year 1993
Scott Schluterman
BS in Mathematics
Year 1992
Nancy Murphy
BS in Mathematics
Year 1991
Kennan Shelton*
BS in Mathematics
Dan Turner
BS in Mathematics
Jennifer James
BS in Mathematics
Anna Mercer Waters
BSE in Mathematics
Year 1990
Jeffrey Howard
BS in Mathematics
Warren Readnour
BS in Mathematics
Betty Stevens Owen*
BS in Mathematics
Year 1988
Pamela White
BSE in Mathematics
Year 1987
Robert Kingan
BS in Mathematics
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Re: FW: st: Regression with multiple age groups
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Re: FW: st: Regression with multiple age groups
From David Greenberg <dg4@nyu.edu>
To statalist@hsphsun2.harvard.edu, shirleysy@hotmail.co.uk
Subject Re: FW: st: Regression with multiple age groups
Date Wed, 25 Apr 2012 13:18:08 -0400
IF you have 21 years of annual observations, this is very few
observations for ARIMA modeling. To identify a model with confidence
you would want many more observations than that. A model for count
data, such as Poisson or negative binomial regression might be your
best option. However, you will only be able to enter a limited number
of predictors given the small number of observations available to you.
David Greenberg, Sociology Department, New York University
On Wed, Apr 25, 2012 at 11:21 AM, Shirley Sy <shirleysy@hotmail.co.uk> wrote:
> Hi David,
> Thanks for your reply!
> I took the data from the Office of National Statistics website for the years 1980 to 2000. My independent variable is the divorce rate (which I calculated myself using the total number of divorces in a given year divided by the total number of marriages in the same year) and my explanatory variables are: husband's age at divorce, wife's age at divorce, husband's previous marital status, wife's previous marital status, combination of husband's and wife's previous marital status (i.e. first marriage for both, one party previously divorced, both previously divorced), duration of marriage (under 2yrs, 2-5, 6-9, 10-14, 15-19, 20-24, 25-29, 30+, not stated), average number of children per couple, female unemployment rate and male unemployment rate.
> I was planning to do OLS and have not considered poisson or negative binomial as of yet. Unfortunately I was given this project to do without any supervision and absolute minimal help and I was only taught the very basics of Stata a year ago so I didn't intend on doing other models with the fear of doing it completely wrong. Would an ARIMA model be appropriate for this data? Shirley
> ----------------------------------------
>> Date: Wed, 25 Apr 2012 09:09:33 -0400
>> Subject: Re: st: Regression with multiple age groups
>> From: dchoaglin@gmail.com
>> To: statalist@hsphsun2.harvard.edu
>> Dear Shirley,
>> Others will agree that you need to tell the list more about the data
>> and the analysis that you intend to do, before we can make useful
>> suggestions.
>> It would be appropriate to handle the age categories, at least
>> initially, by using a separate dummy variable for each category except
>> the first (which will be fitted by the intercept term in your model).
>> You can then plot the coefficients against the midpoint of the
>> category and consider whether to revise the model.
>> You said that you have the age category, separately, for the husband
>> and the wife. Thus, you would use two separate sets of dummy
>> variables, one for the husband's age and the other for the wife's age.
>> You may want to explore the possibility of interactions between the
>> two ages.
>> Please say more about the counts that you listed for 1985 and 1986 (I
>> assume that those are two of the 20 years). They appear to be the
>> frequency distribution of the numbers of divorces by one of the sets
>> of age categories. If you are projecting divorce rates, and those
>> counts represent the number of divorces in one year, what is the
>> denominator for the rate? Do you have the denominator for each age
>> category (and even the combination of husband's age category and
>> wife's age category) or only for the year as a whole?
>> Do the data come from a survey? If so, you will need to take the
>> sampling design and the weights into account.
>> What type of regression are you planning to use? Ordinary regression
>> will probably not be appropriate for rates. You should consider
>> Poisson regression (and perhaps negative binomial regression)?
>> So far, I can envision a model that contains a time pattern (initially
>> based on a dummy variable for each year after the first), effects for
>> husband's age, and effects for wife's age (and perhaps some form of
>> husband-wife interaction). Do you have covariates that you are
>> planning to include?
>> I look forward to seeing more information.
>> David Hoaglin
>> On Wed, Apr 25, 2012 at 12:36 AM, Shirley Sy <shirleysy@hotmail.co.uk> wrote:
>> > Dear Statalisters,
>> > I am a complete beginner at Stata so my question is very basic but I am having trouble finding an answer on the web. I am doing a time series regression project forecasting divorce rates. My data spans 20 years and for both husband and wife the 'age at divorce' variable is split into groups i.e it looks something like this:
>> >
>> > Year under 20 20 to 29 30 to 39 40 to 49 50 to 59 60plus not stated1985 458 1154 78 52 3 2 3
>> > 1986 221 956 50 59 9 5 0
>> > How would I run a regression with the total numbers in each age group? Would I use dummy variables? I understand how I would do it if I had individual ages but since this is a time series model and I have the total number in each age group, I am finding it slightly more complicated.
>> >
>> > ThanksShirley
>> *
>> * For searches and help try:
>> * http://www.stata.com/help.cgi?search
>> * http://www.stata.com/support/statalist/faq
>> * http://www.ats.ucla.edu/stat/stata/
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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10 Interesting Math Facts
Math facts is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by
mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Let’s see the 10 interesting Math facts.
1. π=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 …
2. The other interesting Math facts that a sphere has two sides. However, there are one-sided surfaces.
3. There are shapes of constant width other than the circle. One can even drill square holes.
4. The other interesting Math facts, there are just five regular polyhedra.
5. In a group of 23 people, at least two have the same birthday with the probability greater than 1/2.
6. The Math facts that everything you can do with a ruler and a compass you can do with the compass alone.
7. Among all shapes with the same perimeter a circle has the largest area.
8. One of the interesting Math facts that there are curves that fill a plane without holes.
9. Much as with people, there are irrational, perfect, complex numbers.
10. Last but not the least about Math facts, Mathematics studies stability, projections and values, values are often absolute but may also be extreme, local or global.
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Express f(t) in terms of unit step functions, then find the Laplace
July 25th 2010, 06:24 AM #1
Junior Member
Jul 2010
Express f(t) in terms of unit step functions, then find the Laplace
I've been trying to do this problem forever. I'm not sure how to type this so I took a screenshot of the problem:
I have no idea how to do this. From what I could gather in my notes, I started off like this:
For the unit step functions of f(t):
t^2 - t^2U(t-1)
(t^2 - 1)U(t-1) - (t^2 - 1)U(t-2)
(t^2 - 1)U(t-2)
These may not be right... and even if they are right, can someone briefly explain why? I still don't have a thorough understanding of this.
it's like this
$t^2[u(t+11)-u(t-2)]$ or it's $(t^2-1)[u(t-1)-u(t-2)]$ don't know if u wrote there $( t^2)$ for t goes from -11 to 2 or its $(t^2-1)$ for t goes from 1 to 2
$(t^2 - 2)[u(t-2)]$
Step function $u(t)$ is defined : $1$ for $t\ge0$ , and $0$ for $t<0$
but when u need to make let's say square signal with amplitude 1 from 0 to 1 u'll do it like this... u put step function and then add one more but negative one that begins at 1
and for Laplace , just put it in integral and solve it
Laplace transform (bilateral):
$\displaystyle X(S) = \int _{-\infty} ^{+\infty} x(t) e^{-st} \,dt$
unilateral, just goes from $0-$ not from $-\infty$ ....
P.S. if needed i can write u detailed explanation
here are few signals with step functions, hope it will help u understand how to present signals by using step functions
signal $x(t)=u(t)$
signal $x(t)=(u(t)-u(t-5))$
signal $x(t)=t(u(t)-u(t-1))$
signal $x(t)=(u(t+1)-u(t-1))-2(u(t-1)+u(t-4))$
to see signals bigger just click on them
and here u are a few basic Laplace transformations
$\displaystyle\begin{array}[b]{||c||c||c||}\hline<br /> Signal&Laplace\; transformation&Region\; of\; convergence\\\hline\hline<br /> \delta(t)&1&ROC: \forall S\\\hline\hline<br /> u(t)&\frac{1}
{S}&ROC: Re(S)>0\\\hline\hline<br /> -u(-t)&\frac{1}{S}&ROC: Re(S)<0\\\hline\hline<br /> tu(t)&\frac{1}{S^2}&ROC: Re(S)>0\\\hline\hline<br /> e^{-at}u(t)&\frac{1}{S+a}&ROC: Re(S)>-Re(a)\\\hline\
hline<br /> te^{-at}u(t)&\frac{1}{(S+a)^2}&ROC: Re(S)>-Re(a)\\\hline\hline<br /> \cos(\omega_0 t)u(t)&\frac{S}{S^2+\omega_0 ^2}&ROC: Re(S)>0\\\hline\hline<br /> \sin(\omega_0 t)u(t)&\frac{\
omega_0}{S^2+\omega_0 ^2}&ROC: Re(S)>0\\\hline<br /> <br /> \end{array}$
well that's not all of "basic" but if u need more.. just say
Last edited by yeKciM; July 25th 2010 at 04:04 PM. Reason: added few signals
In the first line, I don't get why you put the original u(t) in there. Also, just to clarify, the 2nd part of f(t) is t^2 - 1, 1<t<2, there is no 11 in there (its hard to tell since the function
and boundaries are written so close together).
With the laplace, we never learned it using integrals, we were given the transformations of common ones. for Laplaces of things with U, its supposed to be in the form of f(t-a)U(t-a), the laplace
of which would be e^-as * L{f(t)}. But these aren't in that form =\
let's see
in the first line $[u(t)-u(t-1)]$, u must do it like that. that first signal of yours (or function) is $x(t) = t^2 \;$, for $t\;$, goes from $0$ to $1$. well if u look at definition of step
function that I wrote in the last post u'll see that step function is infinity function but with values 0 for $t<0$ and $1$ for $t\ge0$.
now, let's first look at simple function (not that first of yours, but simple one) let's look first and second one that i posted with picture $x(t)=u(t)$ and $x(t)=[u(t)-u(t-5)]$ meaning that
your question why do i put original $u(t)$ (or i didn't get u right) is because your signal is bounded from 0 to infinity for the first and from 0 to 5 in second one.... Now do u concur that in
fact that $x(t)=[u(t)-u(t-1)]$ (part of your first function) we use step functions to bound that signal from 0 to 1... and for every another value out from that interval it's 0 $t^2$ and U will
get your first signal (function) looking like this :
signal (function) $x(t)=t^2[u(t)-u(t-1)]$
i hope that I understand your question with that "original u(t) there"
now for Laplace, it's not much harder (if not easier) to do Laplace transformation with integral, (we where strictly forbid to use Laplace tables )... Juts put your let's say step function in the
integral and u'll see strait away that it's 1
just try using integral, and maybe u should put some work that is correctly done by your teachers and we'll see where is misunderstanding
P.S. please check that Laplace that U wrote there....
Last edited by yeKciM; July 26th 2010 at 10:13 AM.
the f(t) will be:
t^2U(t) - t^2U(t-1) + (t^2 - 1)U(t-1) - (t^2 - 1)U(t-2) + (t^-2)U(t-2)
which simplifies to:
t^2U(t) - U(t-1) - U(t-2)
^can you please double check if that is correct?
Assuming thats correct, then the laplace of it would be L{t^2U(t)} - e^-s - e^-2s, right?
Is there some way (an identity or something) to find the first laplace L{t^2 * U(t)}? and did I do the 2nd two terms correctly?
yes it's correct
your signal represented with step functions is :
$\displaystyle t^2[u(t)-u(t-1)]+(t^2-1)[u(t-1)-u(t-2)]+(t^2-2)[u(t-2)] = t^2u(t) -u(t-1) -u(t-2)$
and for your Laplace transformation, i truly don't know how do anyone can teach u to use Laplace without using Laplace integral... Your solution is hmmmm it's missing S
when u do Laplace transformation on signals :
$\displaystyle u(t-1)$ it will become $\displaystyle \int _{-\infty} ^{+\infty}u(t-1) e^{-st} \,dt=\int _1 ^{+\infty} e^{-st} \,dt = \frac {1}{Se^S}$
$\displaystyle u(t-2)$ it will become $\displaystyle \int _{-\infty} ^{+\infty}u(t-2) e^{-st} \,dt= \int _2 ^{+\infty} e^{-st} \,dt = \frac {1}{Se^{2S}}$
and first one u can put in integral and solve it with maybe one substitution ... i can do it but, latter (I have some important work to finish)
I'll try to do it as quickly as i can
here u go and first one
$\displaystyle t^2u(t)$ it will become $\displaystyle \int _{-\infty} ^{+\infty}t^2u(t) e^{-st} \,dt=\int _0 ^{+\infty} t^2e^{-st} \,dt = \frac {2}{S^3}$
hope i helped
P.S. one question
Last edited by yeKciM; July 26th 2010 at 11:49 AM. Reason: added first one
July 25th 2010, 07:41 AM #2
July 25th 2010, 06:14 PM #3
Junior Member
Jul 2010
July 25th 2010, 11:17 PM #4
July 26th 2010, 10:37 AM #5
Junior Member
Jul 2010
July 26th 2010, 11:04 AM #6
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Here's the question you clicked on:
An object is dropped from a height of 1700 ft above the ground. The function h=-16t^2+1700 gives the object’s height h in feet during free fall at t seconds. a. When will the object be at 1000ft
above the ground? b. When will the object be 940ft above the ground? c. What are a reasonable domain and range for the function h?
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both part a and b are asking you to find the time t for a given h. for a, solve the following equations for t: -16t^2 + 1700 = 1000
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t=(close to) 1.654 is what I got a.
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I got t = 6.614 -16t^2 +1700 = 1000 -16t^2 = -700 t^2 = 43.75 then take square root of both sides
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1000=-16t^2+1700 -700=-16t^2 _/-700=_/-16t^2 26.458/16=16t/16 t=1.654 is what I did
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You need to divide by 16 first, before you take the square root.
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Okay got a. and b. now I just need c.:)
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Domain = what values of time would be used here? t > 0, since negative time not reasonable. And you probably need to find when the ball hits the ground, since after that time, the ball would be
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1700=16t^2 _/-106.25=t^2 t=10.3078 Is what I got for how long it takes to hit the ground.
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Cool. So the a reasonable domain is from 0 to 10.3078.
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Range: what values of height seem reasonable? h > 0, because negative height would mean that the ball was underground. And ... since the ball was "dropped" from a height of 1700 (vs being "thrown
up" from a height) the maximum height would be 1700. So ... a reasonable range would be from 0 to 1700.|dw:1352328021424:dw|
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Thank you that helped a lot! I have one more question to post can you help me with it?
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sure, I'll look for you.
Your question is ready. Sign up for free to start getting answers.
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Rad Geek People
Shared Article from Slate Magazine
Zeno’s Paradox Is a Trick—But a Very Interesting Trick
The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. “Achilles and the Tortoise” is the easiest to understand, but it’s …
David Plotz @ slate.com
O.K., so, briefly: If you think that the point of Zeno’s Paradoxes of motion is to prove that the arrow never will reach its target, or that Achilles never does pass the tortoise, &c. — then I think
that you are mistaken about the point of raising the paradox in the first place. Of course, it’s hard to be confident about the motives of dead philosophers who have no surviving books. But what we
do know is that Zeno was a student of Parmenides; and Plato tells us that his books were written to defend Parmenides’s doctrines, by negative means,^[1] showing that the views of his opponents led
to contradictions.
So the most charitable understanding of Zeno’s aims is not that he’s trying to show you that Achilles can never catch the tortoise. Of course he does; just watch them race and you’ll see it happen.
His point is to ask, given that Achilles passes the tortoise, well, how is that possible? And, for good or for ill, to argue from the paradox that you can only make sense of Achilles passing the
tortoise if you reject presentism, and accept eternalist and Parmenidean conclusions about the nature of time and being.
Maybe he’s right about that, and maybe he’s wrong. (I’m inclined to think he’s wrong.) But note that if your solution is to try and settle the issue by introducing a lot of mathematical notation and
conceptual apparatus from modern calculus — for example infinitesimal limit processes, convergent and divergent series, etc. — as is done in the Slate article here, and as is probably the
overwhelmingly most common first response to Zeno’s paradoxes by mathematically-trained writers — then probably you are doing a better job than any pre-classical Greek philosopher could do in
elaborating the precise nature of the problem.^[2] But you’re not obviously refuting Zeno’s claims in any way, at least not yet. At the most you’re kicking the can down the road, and really you’re
sort of strengthening Zeno’s own position. After all, naive formulations of mathematical notation are more or less always going to involve you in all kinds of specifically eternalist language, for
example about moments in past and future time actually “existing,” instantiating the value of functions, etc. You cannot normally take the limit of ΔS(t) over values of t that don’t exist (no longer
exist, do not yet exist).^[3]
Or perhaps you can. But if you can, then doing so, and explaining what you’re doing when you do it, will take some very non-naive reinterpretation of ordinary mathematical language — and some nice
metaphysics, too, to justify your reinterpretation. In any case the solution is going to have to be deeply philosophical, not just a matter of applying a technical innovation in maths.
1. ^[1] In the Parmenides: “I see, Parmenides, said Socrates, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another
way You affirm unity, he denies plurality. Yes, Socrates, said Zeno… . The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of
him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack
I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one. ↩”
2. ^[2] Since the 19th century, we’ve done a lot to really nicely rigorize the mathematics of infinites and infinitesimals, in ways that sometimes anticipated by but never fully available to ancient
mathematicians. ↩
3. ^[3] If anything, this is even more true of late-modern mathematics than it was of classical mathematics. Contemporary mathematics constantly helps itself to a lot of the language of existence,
actuality, etc., for mathematical objects, in areas where Euclid and other classical mathematicians were typically much more circumspect about making “existence” claims for mathematical objects
that hadn’t yet been constructed. ↩
Happy Tyrannicide Day (observed)
Happy Tyrannicide Day (observed)! To-day, March 15^th, commemorates the assassination of two notorious tyrants. On the Ides of March in 2014 CE, we mark the 2,057 anniversary — give or take the
relevant calendar adjustments — of the death of Gaius Julius Caesar, ruthless usurper, war-monger, slaver and military dictator, who rose to power in the midst of Rome’s most violent civil wars, who
boasted of butchering and enslaving two million Gauls, who set fire to Alexandria, who battered and broke through every remaining restraint that Roman politics and civil society had against
unilateral military and executive power. Driving his enemies before him in triumphs, having himself proclaimed Father of His Country, dictator perpetuo, censor, supreme pontiff, imperator, the King
of Rome in all but name, taking unilateral command of all political power in Rome and having his images placed among the statues of the kings of old and even the gods themselves, he met his fate at
the hands of a group of republican conspirators. Led by Marcus Junius Brutus and Gaius Cassius Longinus, calling themselves the “Liberators”, on March 15, 44 BCE they surrounded Caesar and ended his
reign of terror by stabbing him to death on the floor of the Senate.
By a coincidence of fate, March 13^th, only two days before, also marks the anniversary (the 133^rd this year) of the assassination of Alexander II Nikolaevitch Romanov, the self-styled Imperator,
Caesar and Autocrat of All the Russias. A group of Narodnik conspirators, acting in self-defense against ongoing repression and violence that they faced at the hands of the autocratic state, put an
end to the Czar’s reign by throwing grenades underneath his carriage on March 13^th, 1881 CE, in an act of propaganda by the deed.
In honor of the coinciding events, the Ministry of Culture in this secessionist republic of one, together with fellow republics and federations of the free world, is happy to proclaim the 15^th of
March Tyrannicide Day (observed), a commemoration of the death of two tyrants at the hands of their enraged equals, people rising up to defend themselves even against the violence and oppression
exercised by men wrapped in the bloody cloak of the State, with the sword of the Law and in the name of their fraudulent claims to higher authority. It’s a two-for-one historical holiday, kind of
like President’s Day, except cooler: instead of another dull theo-nationalist hymn on the miraculous birth of two of the canonized saints of the United States federal government, we have instead one
day on which we can honor the memory, and note the cultural celebrations, of men and women who defied tyrants’ arbitrary claims to an unchecked power that they had neither the wisdom, the virtue, nor
the right to wield against their fellow creatures.
My favorite collectible coin. This silver denarius was actually minted and circulated in Macedonia by M. Junius Brutus after he and his fellow conspirators stabbed Caesar to death. The obverse
features Brutus’s head in profile. The thing in the middle, above EID MAR (Ides of March) and flanked by the two daggers, is a Liberty Cap, traditionally given to emancipated slaves on the day of
their freedom.
It is worth remembering in these days that the State has always tried to pass off attacks against its own commanding and military forces (Czars, Kings, soldiers in the field, etc.) as acts of
“terrorism.” That is, in fact, what almost every so-called act of “terrorism” attributed to 19^th century anarchists happened to be: direct attacks on the commanders of the State’s repressive forces.
The linguistic bait-and-switch is a way of trying to get moral sympathy on the cheap, in which the combat deaths of trained fighters and commanders are fraudulently passed off, by a professionalized
armed faction sanctimoniously playing the victim, as if they were just so many innocent bystanders killed out of the blue. Tyrannicide Day is a day to expose this for the cynical lie that it is.
There are in fact lots of good reasons to set aside tyrannicide as a political tactic — after all, these two famous cases each ended a tyrant but not the tyrannical regime; Alexander II was replaced
by the even more brutal Alexander III, and Julius Caesar was replaced by his former running-dogs, one of whom would emerge from the carnage that followed as Imperator Gaius Julius Son-of-God Caesar
Octavianus Augustus, beginning the long Imperial nightmare in earnest. But it’s important to recognize that these are strategic failures, not moral ones; what should be celebrated on the Ides of
March is not the tyrannicide as a strategy, but rather tyrannicide as a moral fact. Putting a diadem on your head and wrapping yourself in the blood-dyed robes of the State confers neither the
virtue, the knowledge, nor the right to rule over anyone, anywhere, for even one second, any more than you had naked and alone. Tyranny is nothing more and nothing less than organized crime executed
with a pompous sense of entitlement and a specious justification; the right to self-defense applies every bit as much against the person of some self-proclaimed “sovereign” as it does against any
other two-bit punk who might attack you on the street.
Every victory for human liberation in history — whether against the crowned heads of Europe, the cannibal-empires of modern Fascism and Bolshevism, or the age-old self-perpetuating oligarchies of
race and sex — has had these moral insights at its core: the moral right to deal with the princes and potentates of the world as nothing more and nothing less than fellow human beings, to address
them as such, to challenge them as such, and — if necessary — to resist them as such.
How did you celebrate Tyrannicide Day? (Personally, I toasted the event at home, watched the Season 1 finale of Rome, posted some special-occasion cultural artifacts to Facebook, and re-read
Plutarch’s Life of Brutus from a nice little Loeb edition that I picked up from Jackson Street Books in Athens, Georgia.) And you? Done anything online or off for this festive season? Give a
shout-out in the comments.
Thus always to tyrants. And many happy returns!
Beware the State. Celebrate the Ides of March!
Philosophical Tastes
This is a note from quite a while back, over at Kelly Dean Jolley’s common-place blog, which I stashed to chew on later, and which I’m chewing on a bit now. Here’s Jolley:
I’ve been thinking again about Wittgensteinian reminders, and, while I was doing so, I ran across the following from Henry James.
There are two kinds of taste, the taste for emotions of surprise and the taste for emotions of recognition.
It strikes me that much of the power of Wittgenstein’s work in PI is only available to those who have the taste for emotions of recognition. In fact, I wonder if the juxtaposition of PI 127^[1]
and 128^[2] is not itself a juxtaposition of the two tastes: in 127 Wittgenstein engages the taste for emotions of recognition and in 128 he denies the taste for emotions of surprise.
—Kelly Dean Jolley, Reminders and a Kind of Taste
Quantum Est In Rebus Inane (March 20, 2012)
1. ^[1] [Philosophical Investigations § 127: “The work of the philosopher consists in marshalling recollections for a particular purpose.” — CJ.] ↩
2. ^[2] [Philosophical Investigations § 128: “If someone were to advance theses in philosophy, it would never be possible to debate them, because everyone would agree to them.” — CJ] ↩
Wartime Logic
Suppose that you have — somehow or another — conclusively proven that there is just no way to have a modern war without bombing cities and massacreing innocent people.^[1] That leaves you with a hard
incompatibility claim between moralism and militarism — so if you go around morally condemning military tactics (like the atomic bombing of Hiroshima and Nagasaki, say, or the firebombing of Tokyo)
because they killed innocent people, then you’d end up having to condemn any modern war at all as immoral, no matter who fought it or how it was fought.
Many people, when they reach this point in the argument, want to shove it at you as if the incompatibility made for an obvious reductio ad absurdum of any kind of moralism about military tactics —
“Oh, well, if it’s always immoral to bomb cities then you couldn’t have any wars. That’s why it must not always be immoral to bomb cities.” I honestly don’t know why so few of the people who give
this argument ever even seem to have imagined that their conversation partner might take the incompatibility as an obvious reductio ad absurdum of any kind of militarism — “Oh, well, if it’s always
immoral to kill innocent people, you can’t bomb cities, and if you can’t bomb cities, you can’t have any wars. And that’s precisely why you shouldn’t have any wars.”
• Anthony Gregory @ Facebook (August 7, 2013) provided the inspiration for this post.
• GT 2006-09-01: One man’s reductio makes a similar logical point.
• GT 2003-09-30: Why there are no arguments for terrorism makes an even stronger claim.
1. ^[1] Actually, I think this has been more or less conclusively proven. And that’s precisely why you shouldn’t have any wars. ↩
Show me an axiomatic approach to ethics, ideology or anything else in the marketplace of ideas, and I’ll show you a recipe designed to produce a specific result. Besides, everyone since Gödel’s
proof knows formal systems degenerate into mental masturbation at some point.^[1]
Groundbreaking developments in the history of mathematics and logic: In 1931 Kurt Gödel published “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”^[2] in the
journal Monatshefte für Mathematik. The paper is famous among logicians and mathematicians for the two “Incompleteness Theorems” it contains,^[3] logically demonstrating that no formal system rich
enough to express truths of ordinary arithmetic can be both consistent and deductively complete while having a finite number of axioms.
The paper is famous among almost everyone else for containing a multi-page Rorschach inkblot, allowing a projection test in which the reader-subject can discern an easy dismissive response to
whichever deductive argument they happen to like the least; or, if they prefer, to the exercise of deductive logic as a whole.
1. ^[1] Lorraine Lee, Re: Julian Assange, the Left-Anarch. Comments at Social Memory Complex (21 April 2013). This is actually not even remotely what either of Gödel’s two major Incompleteness
Theorem proofs says. —CJ. ↩
2. ^[2] A PDF blob of the article in its original German is available online thanks to Wilhelm K. Essler. An English translation of most of the paper is also available online thanks to Martin
Hirzel. ↩
3. ^[3] Theorem VI and Theorem XI in the paper, specifically. ↩
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Mplus Discussion >> MLR acronym?
Alyson Zalta posted on Friday, April 13, 2007 - 5:29 pm
I'm reporting SEM results that were calculated with the MLR estimator. While this may seem trivial, I haven't been able to find what the MLR acronym stands for. I'd guess that it stands for "Maximum
Likelihood Robust", but I want to ensure that I cite it properly. If you could please let me know, I'd appreciate it.
Linda K. Muthen posted on Saturday, April 14, 2007 - 7:52 am
I don't think MLR is an acronym. It is an Mplus option for maximum likelihood estimation with robust standard errors.
Francis Huang posted on Monday, March 23, 2009 - 4:05 pm
I am running a two level MLSEM. I have slightly nonnormal continuous data and from what I understand, using a Satorra-Bentler x2 with robust standard errors should be used. Mplus has this under the
MLM estimator.
However, in a two level analysis, MLM is not available, but MLR is. In the manual, MLR also provides robust standard errors.
My question is: how is MLR related to MLM (in short-- how do I write this up aside from saying that I used a maximum likelihood estimator with robust standard errors)?
Linda K. Muthen posted on Monday, March 23, 2009 - 4:22 pm
MLM maximum likelihood parameter estimates with standard errors and a mean-adjusted chi-square test statistic that are robust to non-normality. The MLM chi-square test statistic is also referred to
as the Satorra-Bentler chi-square.
MLR maximum likelihood parameter estimates with standard errors and a chi-square test statistic (when applicable) that are robust to non-normality and non-independence of observations when used
with TYPE=COMPLEX. The MLR standard errors are computed using a sandwich estimator. The MLR chi-square test statistic is asymptotically equivalent to the Yuan-Bentler T2* test statistic.
See the Yuan and Bentler paper referenced in the user's guide. MLR is an extension of MLM that can include missing data.
Francis Huang posted on Tuesday, March 24, 2009 - 3:10 pm
Thanks for the info.
I have a follow up question. I am using MPLUS 5.2 and it displays the two-tailed p value-- how is it possible that in the unstandardized output-- it is nonsignificant (p>.05) and then in the
standardized results, it is significant (p<.05)?
I am modeling achievement (ACHW and ACHB) defined by reading and math at two levels (student and school level) and I am using the presence of basic facilities at the school level as a predictor
(i.e., presence of electricity, 1=yes, 0=no).
ELECTRIC 1.438 0.864 1.664 0.096
STDY Standardization
ELECTRIC 1.185 0.585 2.028 0.043
ELECTRIC 0.488 0.241 2.027 0.043
Thank you.
Bengt O. Muthen posted on Tuesday, March 24, 2009 - 7:19 pm
The unstandardized and standardized values have different sampling distributions and can give somewhat different z values.
Francis Huang posted on Tuesday, March 24, 2009 - 8:01 pm
If that is the case, which one should be 'trusted' and interpreted?
Bengt O. Muthen posted on Wednesday, March 25, 2009 - 8:36 am
I would go with the tests for the unstandardized coefficients, but I haven't seen this studied. It could be a good methods research project, simulating data to see for which type of coefficient the z
tests behave best at different sample sizes.
Paul A.Tiffin posted on Thursday, April 08, 2010 - 12:35 pm
I was just wondering, if you use mlr as the estimator method on a regression or path analysis is it still helpful to center explanatory variables?
Bengt O. Muthen posted on Thursday, April 08, 2010 - 2:11 pm
I don't see that the MLR choice and centering choice are related.
Paul A.Tiffin posted on Friday, April 09, 2010 - 12:22 am
Thanks for your quick reply.
Wayne deRuiter posted on Thursday, July 22, 2010 - 7:31 pm
If you use the estimator MLR without using the Type=Complex option, can you still get standard errors that are robust to non-normality and non-indepenence of observations?
Linda K. Muthen posted on Friday, July 23, 2010 - 8:36 am
No, without TYPE=COMPLEX MLR is robust only to non-normality.
Alexander Kapeller posted on Sunday, April 10, 2011 - 9:20 am
is the type=complex option required in the case of missing data (mcar or mar) or not.
Linda K. Muthen posted on Sunday, April 10, 2011 - 10:10 am
All missing data estimation using maximum likelihood assumes MAR.
Till posted on Tuesday, September 13, 2011 - 11:47 am
Dear Mrs. or Mr. Muthιn,
I'm running a latent growth curve analysis. This is the Input:
Variable: names= g1 e1 n1 g2 e2 n2 g3 e3 n3 l01 l02 l03 l04 l05 l06 l07 l08 l09;
model: i s | l01@0 l02 l03 l04 l05@-1 l06 l07 l08 l09;
F1 by n1 n2 n3;
F2 by e1 e2 e3;
F3 by g1 g2 g3;
i s on F1 F2 F3;
output: samp standardized tech4;
I would like to use the MLR estimator because the mardia coefficient shows me that I can't assume multivariate normal distribution for my data. Is the use of the MLR Estimator appropriate here or do
I have to use the normal ML?
Thank you in advance
Munajat Munajat posted on Monday, September 26, 2011 - 5:30 pm
Dear Dr. Bengt and Dr. Linda
In my model, I have 41 variables. 4 of them have kurtosis values > 3 (3.6, 3.6, 5.6 and 6.8). Do I need to run my model using MLM or MLV estimators? What is the rule of thumb to use the MLM/MLV
instead of ML? What is the difference between MLM and MLV?
Linda K. Muthen posted on Monday, September 26, 2011 - 6:20 pm
There are three estimators that are robust to non-normality. Following are brief descriptions. Only MLR is available with missing data. This is what I would recommend.
MLM maximum likelihood parameter estimates with standard errors and a mean-adjusted chi-square test statistic that are robust to non-normality. The MLM chi-square test statistic is also referred
to as the Satorra-Bentler chi-square.
MLMV maximum likelihood parameter estimates with standard errors and a mean- and variance-adjusted chi-square test statistic that are robust to non-normality
MLR maximum likelihood parameter estimates with standard errors and a chi-square test statistic (when applicable) that are robust to non-normality and non-independence of observations when used
with TYPE=COMPLEX. The MLR standard errors are computed using a sandwich estimator. The MLR chi-square test statistic is asymptotically equivalent to the Yuan-Bentler T2* test statistic.
Back to top
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The Purchasing Power of Money
§ 1 (TO CHAPTER III, § 2)
"Arrays" of k's and r's
Let k be the ratio of deposits to money in circulation M'/M which, on the average, the public prefers to keep; k will then be derivable from the like ratios for the different persons and business
firms in the community in the successive moments of the year, and we may, therefore, form an array on the analogy of previous arrays, of the form:—
PERSONS 1 2 AVERAGE
1 [1]k[1] [2]k[1] k[1]
2 [1]k[2] [2]k[2] k[2]
— — — —
— — — —
Average [1]k [2]k k
Each letter outside the array is a weighted arithmetical average either of the row to its left or of the column above it. k (in the lower right corner) also is both of these as well as the weighted
arithmetical average of all the elements inside the lines (the weights being in all cases the amounts of money in circulation, which are the denominators of the ratios represented in the arrays). The
same proportions hold true if "harmonic" be substituted for "arithmetic" (provided the weights be changed from the denominators to the numerators of the ratios, viz. the deposits). These theorems can
be easily proved analogously to those in § 7 of the Appendix to Chapter II, remembering that k =M'/M.
Similarly, we may let r stand for the average ratio, for the year, of the reserves of all banks (m) to their deposits (M'). This ratio (r, or m/M') is resolvable into an array expressing the ratios
for different banks at different moments, viz.:—
PERSONS 1 2 AVERAGE
1 [1]r[1] [2]r[1] r[1]
2 [1]r[2] [2]r[2] r[2]
— — — —
— — — —
Average [1]r [2]r r
Here each element outside the lines is a weighted arithmetic (or harmonic) average of the terms in the row to its left or the column above it, while r is both of these as well as a weighted
arithmetic (or harmonic) average of all the terms inside, the weights being (for the arithmetic average) the deposits in each case or (for the harmonic average) the money in each case. The total
currency of the community is m + M + M', although only M + M' is actually in circulation.
§ 2 (TO CHAPTER III, § 4)
Algebraic Demonstration of Equation of Exchange Including Deposit Currency
The money expended for goods by individual 1 at moment 1 is [1]e[1] and his check expenditure is [1]e'[1]. His total expenditure for goods by money and checks is, therefore, [1]e[1] + [1]e'[1] = [1]p
[1] [1]q[1] + [1]p'[1][1]q'[1] +....
By adding together all such equations for all persons in the community and all moments of the year, we obtain the equation
E + E' = SpQ
which becomes
MV + M'V' = SpQ
since, by definition, V = E/M and V' = E'/M'.
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Suggestion about generators of MathML code
From: <juanrgonzaleza@canonicalscience.com> Date: Thu, 27 Jul 2006 04:00:04 -0700 (PDT) Message-ID: <3207.217.124.69.241.1153998004.squirrel@webmail.canonicalscience.com> To: <www-math@w3.org>
MathML was developed to facilitate the transfer and re-use of mathematical
content between applications.
However, since each tool generates|understands completely different code
we cannot develop a generic parser receiving content MathML code from
several authors using different software.
The problem increase with feedback. Take authors J send us fragment of
content MathML next is checked or simbolically evaluated and output
returned to J. Depending of the MathML software J is using, she|he will be
able to open the content or no.
I could develop an one-to-one tool comparison and generate specific XSLT
templates for each MathML fragment. This is highly expensive and time
consuming but the main problem is that do not work with multiauthored
docs, containing content MathML fragments from different tools. Since each
fragment is not identified, templates cannot be applied because rules are
tool specific. for instance in some cases we wait eliminate <mi> in others
we wait introduce extra <mi>, etc.
Even the simple examples introduced in
1) a + b
2) sin π
3) -5
4) ∫ sin ω dω
5) 3/4
6) sqrt(x)/(y^2 -1)
7) -x
8) ∫_a^b ω dω
9) x >> 0
10) <p>My favourite Greek letter is β</p>
11) x_i = 5
12) {}^7log x
13) (x+3)^2
14) a/b; a=3, b=4
15) 123/456
are parsed with great difficulty. Take the first example, i would wait
something like
transforming it to Scheme (+ a b) for symbolic evaluation, but oops i got
error because the file i received from HERMES generated the s-expr
(+ <mi>a</mi> <mi>b</mi>)
All of above examples are extracted from current sites on MathML. Next is
of interest to the Center
16) (&partial;ρ / &partial;t) = L ρ + ε(&rho - ρ_0)
The RG-1 can work in the research of suitable relativistic expressions for
the L superoperator (Sch-KG, Dirac, or R-QFT propagators are plain wrong
since compact support for wavefunctions is not maintained -the current
research tendency is to obtain propagators are cuadratic in momenta-).
RG-2 can work in the research of generalizations of Abe kernels for the
Zubarev term ε(&rho - ρ_0) [*].
If RG-1 and RG-2 are using different MathML software the communication can
be difficult or even impossible.
Use an attribute informating of the profile and other informing of the
generator. For instance
<math generator="HERMES" profile="normal">
and i would apply special XSLT to the nodes generating the desired (+ a b).
<math generator="ConText" profile="TeX-annotation">
and i would apply special XSLT to the nodes introducing 'lacking' <mi> and
No MathML, GIFs.
Juan R.
Center for CANONICAL |SCIENCE)
Received on Thursday, 27 July 2006 11:04:48 GMT
This archive was generated by hypermail 2.2.0+W3C-0.50 : Saturday, 20 February 2010 06:12:59 GMT
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Introduction to Fortran
CHAPTER 7
Introduction to Fortran
The Fortran Programming Language
The Fortran programming language was one of the first (if not the first) “high level” languages developed
for computers. It is referred to as a high level language to contrast it with machine language or assembly
language which communicate directly with the computer’s processor with very primitive instructions. Since
all that a computer can really understand are these primitive machine language instructions, a Fortran
program must be translated into machine language by a special program called a Fortran compiler before it
can be executed. Since the processors in various computers are not all the same, their machine languages are
not all the same. For a variety of reasons, not all Fortran compilers are the same. For example, more recent
Fortran compilers allow operations not allowed by earlier versions. In this chapter, we will only describe
features that one can expect to have available with whatever compiler one may have available.
Fortran was initially developed almost exclusively for performing numeric computations (Fortran is an
acronym for “Formula Translation”), and a host of other languages (Pascal, Ada, Cobol, C, etc.) have been
developed that are more suited to nonnumerical operations such as searching databases for information.
Fortran has managed to adapt itself to the changing nature of computing and has survived, despite repeated
predictions of its death. It is still the major language of science and is heavily used in statistical computing.
The most standard version of Fortran is referred to as Fortran 77 since it is based on a standard
established in 1977. A new standard was developed in 1990 that incorporates some of the useful ideas from
other languages but we will restrict ourselves to Fortran 77.
The Basic Elements of Fortran
A Fortran program consists of a series of lines of code. These lines are executed in the order that they
appear unless there are statements that cause execution to jump from one place in the series of lines to
another. At the end of this chapter, we have listed the code of a complete program and we will refer to this
program as we go along. Learning to write Fortran programs is usually done by imitating programs written
by others. This chapter cannot possibly provide all of the details of Fortran, but rather to point out those
features that are particularly important in statistical computing.
1. Statements begin in column 7 of a line and end before column 72 (if a statement will not fit in columns
7–72, it can be continued onto the next line (or lines) as long as a character is placed in column 6 of
each continuation line). Columns 1–5 of a line can contain “a statement number” so that the statement
on that line can be referenced by other statements. Lines that begin with the letter c in column one
are interpreted as comments and are not executed. It is important to include enough comments in any
program in any language so that you (or somebody else) can later understand what the lines of code
are intended to do.
2. A file containing a Fortran program begins with a main program which may be followed by subprograms.
62 Fortran chap 7
Subprograms come in two kinds; functions and subroutines. The main program and the subprograms
it calls can be in different files which can be compiled separately and then linked together to form an
executable file.
4. A main program begins with a series of statements declaring the names of the variables that are going
to be used by the program and what type of variables they are (see “data types” below). Note that
variable names must begin with letters of the alphabet, contain letters and the digits 0 through 9, and
(to be safe as different compilers allow different numbers of characters) contain 6 characters or fewer.
The program ends with the lines stop and end.
5. Between the declarations and the stop and end there are a wide variety of operations that can be
performed, but usually a program consists of 1) reading some information from somewhere (for example,
from a file or from the keyboard from the person using the program 2) performing some numerical or
graphical task, and 3) placing the results of the task somewhere so that the user of the program can use
them (perhaps on the user’s screen or into a file).
6. Often there is some subtask that needs to be performed several times in a program. For example, it
may be necessary to form the plot of one vector versus another. Rather than have the same set of
lines of code in several different places in the main program, one can form them into what is called a
subprogram. A subprogram begins and ends with special lines of code that tell the compiler that it is
in fact a subprogram and not part of the main program. Another use of subprograms is in modular
programming where one breaks a complicated task into a series of tasks that are more managable and
writing a subprogram for each one.
Data Types
Fortran performs numerical and logical operations on variables of a few types, including:
1. Integers, that is, numbers that have no decimal part. There are two kinds of integers; long integers
(also called four-byte integers or integer*4’s, they range from roughly −231 to 231 and use four bytes of
memory) and short integers (also called two-byte integers or integer*2’s, they range from roughly 2−15
to 215 and use two bytes of memory). One declares variables to be short or long integers by statements
of the form
integer*2 n,m,k
integer*2 n,m,k
integer*4 n,m,k
2. Floating point numbers, that is, numbers that can contain decimal parts. There are two kinds of
floating point numbers; double precision (also called real*8 reals, they range from roughly 10−300 to
10300 , take eight bytes of memory, and are accurate to roughly 13 or 14 decimal places) and single
precision (also called real*4 reals, they range from roughly 10−38 to 1038 , take four bytes of memory,
and are accurate to roughly six or seven decimal places). Single and double precision variables are
declared by lines of the form
real*4 x,y
chap 7 Fortran 63
real*8 x,y
3. Complex numbers, that is variables that can take on complex values. Actually, complex variables
can be thought of as a pair of real*4 or real*8 floating point numbers. Single and double precision
complex variables are declared by lines of the form
complex*8 x,y
complex*16 x,y
4. Logical variables, that is, variables that can only take on the “values” .TRUE. or .FALSE..
5. Character variables, that is, variables that contain characters such as letters of the alphabet or
numerical digits. A character variable can be as long as desired. To declare variables called st1 and
st2 to be of length 10 and 20 respectively, one uses the statement
character st1*10,st2*20
6. Arrays of variables. In addition to scalar variables, one can declare a variable to be a vector (one
dimensional array) or matrix (two dimensional array), or a higher dimensional array. To declare a single
precision 100 by 3 matrix called x, one would have the statement
real*4 x(100,3)
while the statement
character st1(100)*10,st2(200)*20
would declare character arrays of length 100 and 200 called st1 and st2 where each of the 100 elements of
st1 can contain 10 characters, while each of the 200 elements of st2 can contain as many as 20 characters.
Note that Fortran can only perform operations on individual elements of arrays, that is, it has no built-in
ability to perform operations (such as addition) on all the elements of arrays. Most computer systems have
libraries of subprograms that perform operations on entire arrays.
Undeclared Names
It is very good programming pactice to declare every variable that a program uses (most compilers have
a way to warn a programmer of any variables that have not been declared). However, one should be aware of
the fact that any variable whose name begins with one of the letters i, j, k, l, m, or n, is considered to be an
integer (usually an integer*4) unless otherwise declared explicitly, while any variable whose name begins
with one of the other letters of the alphabet is taken to be a real (usually real*4). Often it is useful to
follow these conventions in the naming of variables (although they should also be explicitly declared) so that
someone reading the code will know what type the variables are without having to look at the declarations.
64 Fortran chap 7
Assignments and Arithmetic Operations
An important part of a Fortran program is assigning values to variables and performing arithmetic
operations on variables. There are five arithmetic operations, namely addition (+), subtraction (-), multi-
plication (*), division (/), and exponentiation (**). If more than one operation is contained in a statement,
then exponentiation is done first, then multiplication and division (if there are more than one of these they
are done from left to right), and finally addition and subtraction (again if there are more than one of these,
they are done from left to right). Expressions within parentheses are done first.
When all operands of an arithmetic expression are of the same type, then the final type of the value
of the expression will be of that type. When they are of different types, the value of the expression will
be that of the last binary operation performed where the value of a binary operation of data of different
types takes on the type of the higher of the two types, where types are ranked from lowest to highest by
integer*2, integer*4, real*4, real*8, complex*8, and complex*16. One must be careful of expressions
such as x*(i/j) or x**(i/j) when i and j are integers since the result of the division is truncated and not
Once an expression has been evaluated, it is converted to the type of the variable on the left hand side
of the expression.
Intrinsic Functions
Fortran has a wide variety of functions built into it. These functions are called intrinsic functions.
Examples include trigonometric and logarithm functions. The table below contains a list of the functions
that a Fortran programmer can expect to have available when writing code. Note that these functions need
not be declared before use. Also, starting with Fortran 77 the concept of generic functions was developed.
For example, in previous versions of Fortran there were four functions for finding the absolute value of a
number depending on its type. Now there is only the single function abs which will return a value having
the same type as its input type. The old functions iabs, abs, dabs, and cabs are still available as well. The
prefixes i, a, d, c are used in these functions for integer, real, double precision, and comples, respectively.
Intrinsic Functions1
functions Definition
Numerical functions:
sdc=COS(sdc), sdc=SIN(sdc), sd=TAN(sd) Trigonometric functions
sd=ACOS(sd), sd=ASIN(sd), sd=ATAN(sd) Inverse trig functions2
sd=COSH(sd), sd=SINH(sd), sd=TANH(sd) Hyperbolic trigonometric functions
sdc=LOG(sdc), sd=LOG10(sd) Log functions
sdc=SQRT(sdc) Square root
isds=ABS(isdc) Absolute value
sdc=EXP(sdc) Exponentiation
isd=MOD(isd1,isd2) Remainder3
isd=SIGN(isd1,isd2) Sign function4
isd=MAX(isd1,isd2,...), isd=MIN(isd1,isd2,...) Maximum and minimum
s=AMAX0(i1,i2,...), s=AMIN0(i1,i2,...) Integer input, single output
i=MAX1(s1,s2,...), i=MIN1(s1,s2,...) Single input, integer output
isd=DIM(isd1,isd2) Positive difference: MAX(isd1-isd2,0)
d=DPROD(s1,s2) Double precision product of arguments
Type conversion:
i=INT(isdc), s=REAL(isdc), d=DBLE(isdc)
c=CMPLX(isd) Input is real part, imaginary part is 0
c=CMPLX(c) Identity
chap 7 Fortran 65
c=CMPLX(isd1,isd2) First argument is real part, 2nd is imaginary part
Complex numbers:
s=REAL(c), s=AIMAG(c) Real and Imaginary parts
c=CONJG(c), s=CABS(c) Complex conjugate and modulus
Rounding and truncating5:
sd=AINT(sd), sd=ANINT(sd) Truncation and rounding
i=NINT(sd) Rounding to nearest integer
Operations with characters:
C=CHAR(i) Character having input ASCII code
i=ICHAR(C) ASCII code of input character
i=INDEX(str1,str2) Compare strings6
i=LEN(str) Number of characters in the string
L=LGE(str1,str2), L=LGT(str1,str2), Lexicographical ordering7
L=LLE(str1,str2), L=LLT(str1,str2)
The table lists the input and output type of the function, for example, the notation isds=ABS(isdc)
means that for integer, single, double, and complex input, the output is integer, single, double, and single
respectively (note that CABS(c) is single). For characters, C represents a single character while str represents
a string of characters. Logical variables are denotd by L.
The function ATAN2 has two arguments x1 and x2 and is defined to be ?????
The remaindering function applied to arguments x1 and x2 is defined to be x1 − INT(x1 /x2 ) ∗ x2 .
The sign function having two arguments x1 and x2 is |x1 | if x2 ≥ 0 and −|x1 | if x2 < 0.
Truncation means, for example, REAL(INT(s)). Rounding means, for example, REAL(INT(s+.5) if s≥ 0 or
REAL(INT(s-0.5)) if s< 0.
If str2 is contained in str1 then the result is the position of the first character of str2 in str1; otherwise
the result is 0.
For example, LGE(str1,str2) is true if the two strings are the same or if str1 follows str2 in lexicographical
order (comparing ASCII codes character by character).
Input and Output
Perhaps the most difficult part of Fortran programming are the operations of inputting information into
the program and outputting information from the program to a place where it can be used later. These
operations are performed using the read and write statements respectively. There are four basic parts of
these statements:
1. A description of where the information is coming from or going to,
2. A description of the format that the information is in if it is being read or a description of the format
that is desired for the information if it is being written,
3. The statement number of the statement to be executed next if something goes wrong during the reading
or writing,
4. The names of the variables that are to receive the information or the names of the variables whose
values are to be written.
It is almost impossible to describe all of the variations on these four basic ideas. Most Fortran pro-
grammers learn them by imitating what other programmers have done. An example which contains all four
elements is provided by the statements
read(2,10,err=20) ((x(i,j),j=1,10),i=1,5)
10 format(10x,6f10.0)
66 Fortran chap 7
The read statement says to read 50 numbers from the file having “logical unit number” 2 according to the
format statement having statement number 10, and assign these numbers to the upper left hand corner of
the matrix x (upper five rows and 10 columns). The err=20 means that if something goes wrong during
the reading process (such as if a nonnumeric character were mistakenly entered in the file) then the next
statement to be executed by the program is the one having statement number 20. The 6f10.0 in the format
statement says that the 50 numbers should be read six per line, with the first of the six in columns 11–20,
the next in 21-30, and so on, and the the numbers can be anywhere in these columns as long as there is a
decimal point (note that the 10x says to skip the first 10 columns on each line—this makes it posible to have
extraneous information in the first 10 columns (such as a date if one were reading information collected over
time and wanted the date in the file but didn’t want to input it into the program).
Opening a File for Reading or Writing
Before reading from or writing to a file, one must first alert the computer’s operating system what the
name of the file is using the open statement. One also uses the open statement to attach a logical unit
number to the file (such as the 2 in the write statement above) and then uses this number in future read
statements. Note that one can have several files open at the same time, each with its own logical unit
Looping and Branching
An important part of Fortran are the statements that allow one to perform loops, that is executing sets
of lines of code repeatedly. The most famous example is a “do loop”, which is of the form
do 10 i=i1,i2,i3
10 continue
where i1, i2, and i3 are integers (or integer expressions). The statements between the do and the continue
are all executed with the variable i having value i1, then they are done again with i having value i1+i3,
and so on, until the value of i is outside of the range i1 to i2. Note that do loops can be nested, and that
the do loop counter (the variable i in the example above) can not be changed during the loop.
Another example of a loop is the so-called “implied loop” which is of the form
5 continue
if(i.eq.n) go to 10
go to 5
10 continue
There are a number of branching constructs available. The go to used above is an unconditional branch.
The line
if(i.eq.3) go to 10
is an example of a conditional branch. In addition to .eq. we have .ne., .le., .lt., .ge., and .gt.. The
use of go to’s is discouraged as they make reading the code of a program difficult. Other, more readable
branches are available such as
if(i.eq.1) then
chap 7 Fortran 67
elseif(i.eq.2) then
Subroutines and Functions
There are two kinds of subprograms in Fortran; subroutines and functions. We have already seen
examples of functions in the intrinsic functions above, for example the statement x=sqrt(8.0*atan(1.0))
uses two intrinsic functions; atan and sqrt. Functions return single values “through their names” and any
number of values (scalars or arrays) through their argument list. A subroutine on the other hand returns
nothing through its name, can return any number of values through its argument list, and is invoked by a
statement of the form
call subname(x,n,m,y,z)
To illustrate writing functions and subroutines, consider the following code:
double precision function inprod(x,y,n)
dimension x(n),y(n)
do 10 i=1,n
10 inprod=inprod+dble(x(i))*dble(y(i))
subroutine mtmult(a,b,n,ndim,c)
dimension a(ndim,n),b(ndim,n),c(ndim,n)
do 10 i=1,n
do 10 j=1,n
do 20 k=1,n
20 c1=c1+a(i,k)*b(k,j)
10 c(i,j)=c1
Thus functions and subroutines both end with the statements return and end, and begin with a line that
gives their name (note how functions and subroutines name themselves differently and that functions declare
their own type) and specifies a list of arguments, that is variables that are to be used in the function or
There are a few simple rules that one must use in writing and calling subroutines:
1. Subprograms can call other subprograms but a subprogram can never call itself either directly or in-
directly (by calling another subprogram which in turn calls the original subprogram). Some modern
compilers allow this recursive calling, but most still do not so we will not discuss it.
2. A subroutine begins with a statement beginning with the word subroutine followed by a space, followed
by the name of the subroutine, followed (optionally) by a list of variable names in parentheses and
68 Fortran chap 7
separated by commas. These variables are referred to as the arguments of the subroutine. The variables
in the calling program are not available to the subprogram unless they are passed to it through the
argument list (one can also use a common statement but we ignore that for now). We consider arguments
in more detail later in the section discussing arrays in Fortran.
3. One of the most common errors in Fortran occurs if there is a mismatch between the type of a variable
in a main program and the type expected by the subroutine. For example, if a subroutine expects a
single precision variable and is passed a double precision number, then the number it actually receives
will not be what is expected and in many cases this will cause a problem (note that the compiler does
not check for “type mismatches.”)
Compiling and Linking
The first program considered later in this chapter (which is stored in the file regswp.f) is self-contained
except that it uses a subroutine called tcdf which is in another file called tcdf.f. To compile and link
regswp.f and tcdf.f and create an executable file called regswp, one does the following (the -c switch on
the first line says to only compile tcdf.f while the -o switch on the second line says to name the executable
file regswp).
f77 -c tcdf.f
f77 -o regswp regswp.f tcdf.o
Arrays in Fortran
Arrays are an often poorly understood part of Fortran, particularly in relation to how they are used in
subprograms. Unfortunately, arrays and subprograms are often covered at the end of introductory Fortran
courses, if at all. We will consider one dimensional arrays (vectors), two dimensional arrays (matrices having
row and column indices), and three dimensional arrays (often visualized as a vector of matrices, the first two
indices being the row and column numbers and the third index being the matrix number). The extension to
higher dimensional arrays should be obvious.
Arrays are Statically Allocated
With a few notable exceptions, Fortran compilers force the programmer to specify the size of an array
in a main program before it is compiled and linked. This is done with a dimension statement (such as
dimension a(10,10,4), which allocates a three dimensional array a of four 10 by 10 matrices). Thus the
memory for an array is said to be “statically allocated.” Many programming languages allow “dynamic
memory allocation,” that is, the program can ask the user to specify the size of a problem and then allocate
exactly the right amount of space for the arrays that are involved. In contrast, the Fortran programmer
must anticipate the maximum size that a user might want and do the dimension accordingly.
Arrays are Stored in Memory in “Column Order”
Most features of a language such as Fortran do not require the programmer to keep in mind how the
program interacts with a computer’s memory. However, the ordering of the elements of an array is a notable
exception to this rule. The elements are stored in consecutive locations in memory in what is called “column
order.” For a vector this just means that consecutive elements are stored in consecutive locations in memory,
but how the ordering is done for higher dimensional arrays is not obvious. For a matrix, the basic rule is
that the elements of the first column are stored in order followed by the elements of the second column, and
so on. Thus code such as
dimension a(4,4)
data a/11,21,31,41,12,22,32,42,13,23,33,43,14,24,34,44/
chap 7 Fortran 69
do 10 i=1,4
10 write(*,20) (a(i,j),j=1,4)
20 format(1x,4f4.0)
leads to
11. 12. 13. 14.
21. 22. 23. 24.
31. 32. 33. 34.
41. 42. 43. 44.
that is, the first four elements in the data statement are placed into the first column of a, the next four in
the second column, and so on.
For a three dimensional array, we have rows, columns, and matrix number. If we had dimensioned a
above by dimension a(2,2,4), then the four matrices would be
21 41 22 42 23 43 24 44,
that is, the first four elements go into the first matrix (with the first two in the first column and the next
two in the second), the next four in the second matrix, and so on.
Perhaps the best way to visualize this is to think of all arrays as vectors, with the dimensions specifying
how that vector gets reshaped into the array. The important point is that in the assignment of elements into
the array so that “the first index varies most rapidly.” Thus if we added the line
write(*,20) a
to the code above, we would get
11. 21. 31. 41.
12. 22. 32. 42.
13. 23. 33. 43.
14. 24. 34. 44.
Dimensioning Arrays in a Subprogram
An important part of programming is to create subprograms (subroutines or functions) to perform often
needed calculations (such as adding or multiplying matrices). In Fortran, variables defined in one “module”
(main program or subprogram) are unknown to any other module unless the programmer does something
special to “pass” the values from one module to the other. The standard method to do this in Fortran is
to have an argument in the subprogram list and in the call to the subprogram for the variable of interest.
This is called “passing a variable” to the subprogram. It is still necessary to dimension any passed array in
the subroutine. Further, it is posible to use variables to dimension arrays in a subprogram (note that such
a variable must also be in the argument list).
A simple example of this is to write a subroutine which writes out the elements of an n by n matrix
subroutine matprt(b,n)
dimension b(n,n)
do 10 i=1,n
10 write(*,20) (b(i,j),j=1,n)
20 format(1x,6f12.5)
70 Fortran chap 7
Note that the variables in such a subprogram are called “dummy variables” and need not have the same
names as the corresponding variables in the calling subprogram.
In keeping with what we said above, our subroutine will take the first n2 elements being passed from
the array in the calling routine, form the matrix b by column and then write out the rows of the result. But
what are the first n2 elements of the array in the calling routine? In the following main program,
dimension a(4,4)
data a/11,21,31,41,12,22,32,42,13,23,33,43,14,24,34,44/
call matprt(a,3)
the first 9 elements of a get passed to matprt and thus the output of the program will be
11.00000 41.00000 32.00000
21.00000 12.00000 42.00000
31.00000 22.00000 13.00000
which is probably not at all what the programmer intended (presumably they wanted the upper left hand 3
by 3 portion of the matrix displayed).
The “ndim problem”
This last example illustrates a problem which causes the vast majority of hard-to-find Fortran bugs.
What’s worse is that many times the programmer has no idea that there is a problem (in the example we
knew what the “correct” values should have been). One can only hope that the incorrect passing leads to a
big enough problem (such as division by zero) so that the subtle passing problem gets diagnosed.
I refer to this problem as the “ndim problem” since the solution is to include a third argument in the
subroutine (often called ndim) corresponding to the actual row dimension of the array in the calling routine
(in this case it would be 4). In our example, ndim is the natural name for this argument as it is the actual
dimension corresponding to the argument n in the calling routine. Thus the subroutine would be modified
to be
subroutine matprt(b,n,ndim)
dimension b(ndim,n)
do 10 i=1,n
10 write(*,20) (b(i,j),j=1,n)
20 format(1x,6f12.5)
and the call would be
call matprt(a,3,4)
which would give the intended output.
A Bonus
A very powerful (but often dangerous) feature of Fortran is that one can pass an array having one
number of indices (vector, matrix, three dimensional, etc.) to a subroutine which is expecting an array
having a different number. This allows us to do something such as
chap 7 Fortran 71
dimension x(8,2)
data x/11,21,31,41,12,22,32,42,13,23,33,43,14,24,34,44/
call mean(x(1,2),8,xbar)
write(*,10) xbar
10 format(1x,’xbar = ’,f12.5)
subroutine mean(x,n,xbar)
dimension x(n)
do 10 i=1,n
10 xbar=xbar+x(i)
which would find the sample mean of the second column of the matrix x. This example also illustrates
another important fact about array passing. We’ve seen that a subroutine takes consecutive elements being
passed from the calling routine, shapes them into an array according to the dimension in the subprogram,
and then operates on the array. In the line call mean(x(1,2),8,xbar), we see that the argument in the
call is actually only “pointing” to the first element in x to be passed (note that if the argument had just
been x, the pointer would be at the first element in x, that is, x(1,1)). This is useful in something such as
dimension x(100),xbar(100)
do 10 i=1,n
10 x(i)=i
do 20 i=1,n
20 call mean(x(n-i+1),i,xbar(i))
which finds the mean of the last i x’s for i = 1, . . . , 100.
A final example of this is given by writing a subroutine to calculate X T X for an n × m matrix X:
subroutine xprx(x,n,m,ndim,mdim,xtx)
dimension x(ndim,m),xtx(mdim,mdim)
double precision dbprod
do 10 i=1,m
do 10 j=1,i
10 xtx(j,i)=xtx(i,j)
double precision function dbprod(x,y,n)
dimension x(1),y(1)
do 10 i=1,n
10 dbprod=dbprod+dble(x(i))*dble(y(i))
There are several things to note here:
1. We need both an ndim and an mdim in this case.
72 Fortran chap 7
2. The (i, j)th element of X T X is the inner product of its ith column with its jth column so we are using
the inner product function dbprod and just pointing to the beginning of the two columns of interest in
the call to dbprod.
3. Since X T X is symmetric, xprx only calculates the the elements in the lower triangle and puts each
element into the corresponding place in the upper triangle as well.
4. The variables x and y have been “dimensioned with a 1” in the function dbprod. This is consistent
with what we have said above about the role of dimensions. This sort of thing is sometimes very useful
in avoiding extra arguments. For example, if we were writing a subroutine to find the zeros of the
polynomial g(z) = j=0 αj z j we would probably pass α0 , α1 , . . . , αp in the vector alpha of length p + 1
(since there is no zero index in standard Fortran) and to avoid passing the value of p + 1, we can just
dimension alpha with a 1.
An Example
We consider a program to do basic multiple linear regression which illustrates reading and writing and
how to use subroutines.
Review of Regression Analysis
A very common situation in statistics is to have a set of n “objects” with measurements on a set of m+1
“variables” for each object and we wish to explain the behavior of one of the variables y (the “dependent”
variable) in terms of the other m variables X1 , . . . , Xm (the “independent” or “regressor” variables) according
yi = β0 + β1 Xi1 + · · · + βm Xim + i , i = 1, . . . , n,
where yi and Xi1 , . . . , Xim are the measurements for the ith object, β0 , β1 , . . . , βm are a set of unknown
parameters, and i is called the error for the ith object. The ’s are assumed to be statistically independent
for the different objects, have expectation 0, all have the same variance σ 2 , and are included in the model
to explain the fact that not all objects whose X values are the same will have the same y value.
The aims of the regression analysis are twofold: 1) to determine which of the X’s are important in
explaining the behavior of y, and 2) to develop a prediction formula that can be used to find reasonable
values of y for a new object for which we only know the X’s.
An example of such a situation is the Hald data set included at the end of this report, wherein n = 13,
m = 4, y is the amount of heat generated during the hardening of cement, and the X’s are the amount of
four different chemicals used in the production of the cement. In the data file, y is listed first followed by
the four X’s.
If we rewrite the multiple linear regression model above in matrix form as
y = Xβ + ,
where y is the n × 1 vector of the y values, X is the n × (m + 1) matrix of the X values with a column of 1’s
appended at the beginning, β is the (m + 1) × 1 vector of coefficients, and is the n × 1 vector of ’s, then
it is easy to write down the basic results of a regression analysis. First, the best linear unbiased estimators
β of the β’s and an unbiased estimator s2 of σ2 are given by
β = (XT X)−1 XT y,
ˆ s2 = , RSS = eT e, e = y − y, y = Xβ ,
ˆ ˆ βˆ
where the vectors y and e are called the vectors of fitted values and residuals, and RSS is the residual sum
of squares.
If we further assume that the ’s are normally distributed, then the hypothesis that βj is zero while the
other β’s are not is rejected at the α significance level if |tj | > tα/2,n−m−1 where tα/2,n−m−1 is the upper
α/2 critical point of a t distribution having n − m − 1 degrees of freedom and
tj = .
s2 (XT X)−1
chap 7 Fortran 73
Note that the denominator of tj is called the standard error of βj . Typically, instead of just reporting
whether the hypothesis is rejected or not, one determines the p-value of βj , that is, the probability that a t
with n − m − 1 degrees of freedom is outside the interval ±tj , in which case the null hypothesis is rejected
if pj < α.
A measure of how much of the variability in y is explained by its linear dependence on the X’s is
R2 = 1 − e
where s2 and s2 are the sample variances of the e’s and the y’s, respectively.
e y
A Multiple Linear Regression Program Using SWEEP
In this section we give as an example a regression program which
1. Asks the user for a file name containing a regression data set to analyze and a file name to receive the
results of the analysis. The program assumes that the file starts with a line containing a label to be
displayed on the output and a line with the number of observations, n, and the number of independent
variables, m, in the regression (not counting an intercept term). Each of the remaining n lines in
the input file is assumed to contain the value of the dependent variable followed by the values of the
independent variables. An example input file is given below which contains the so-called Hald Data
which has 13 observations and four independent variables:
Hald Data (From Draper and Smith, pg. 296, 304, 630)
78.5 7 26 6 60
74.3 1 29 15 52
104.3 11 56 8 20
87.6 11 31 8 47
95.9 7 52 6 33
109.2 11 55 9 22
102.7 3 71 17 6
72.5 1 31 22 44
93.1 2 54 18 22
115.9 21 47 4 26
83.8 1 40 23 34
113.3 11 66 9 12
109.4 10 68 8 12
2. Reads the data into the vector y and the matrix X (putting a column of 1’s into the beginning of X so
that it is n × (m + 1) after reading the data), forms the augmented cross-product matrix
XT X XT y
A= ,
yT X yy
and then gets the regression information by using the matrix SWEEP algorithm.
3. The least squares estimates, their standard errors, t-statistics, and corresponding p-values are then
written out, as well as the value of R2 . The result for the Hald Data are:
Hald Data (From Draper and Smith, pg. 296, 304, 630)
Regression results, R-Squared = 0.98237562, d.f. = 8
i beta(i) se(i) t(i) pval(i)
0 62.405369 70.070959 0.890602 0.399134
1 1.551103 0.744770 2.082660 0.070822
2 0.510168 0.723788 0.704858 0.500901
74 Fortran chap 7
3 0.101909 0.754709 0.135031 0.895923
4 -0.144061 0.709052 -0.203174 0.844071
The Program
1 c -------------------------------------------
2 c REGSWP.F: Regression program using SWEEP
3 c
4 c Uses functions inprod and smean and subroutines
5 c tcdf, tpdf, betai, and gama
6 c -------------------------------------------
8 c ---------------------------------------------
9 c Declare variables and external functions:
10 c ---------------------------------------------
12 character*60 fnin,fnout,label
14 real*8 x(1000,20),y(1000),a(21,21),beta(21),se(21),t(21),pval(21)
16 real*8 cdf,pdf,inprod,rss,s2,ybar,ssy,rsq
18 real*8 smean
20 logical fnexist
22 data ndim/21/
24 c ------------
25 c Get data:
26 c ------------
28 write(*,10)
29 10 format(’ Enter file name containing the regression data: ’$)
31 read(*,20) fnin
32 20 format(a60)
34 inquire(file=fnin,exist=fnexist)
36 if(.not.fnexist) go to 100
38 write(*,30)
39 30 format(’ Enter the file name to get the output: ’$)
40 read(*,20) fnout
42 open(2,file=fnout,status=’new’)
43 open(1,file=fnin,status=’old’)
45 c Read first two lines of input file:
47 read(1,20,err=110) label
48 read(1,*,err=120) n,np
50 c Read data, putting X’s in columns 2,...,m+1 and 1’s in 1st column:
52 m=np+1
53 do 40 i=1,n
54 x(i,1)=1.0
55 40 read(1,*,err=130) y(i),(x(i,j),j=2,m)
57 c -----------------------------------------
58 c Form the augmented crossproduct matrix:
59 c -----------------------------------------
61 mp1=m+1
62 do 60 i=1,m
63 a(i,mp1)=inprod(x(1,i),y,n)
64 a(mp1,i)=a(i,mp1)
65 do 50 j=1,i
66 a(i,j)=inprod(x(1,i),x(1,j),n)
67 50 a(j,i)=a(i,j)
68 60 continue
69 a(mp1,mp1)=inprod(y,y,n)
71 c ----------------------------------------------------------
72 c Sweep it on first m diagonals and check for singularity:
73 c ----------------------------------------------------------
chap 7 Fortran 75
75 call sweep(a,ndim,mp1,1,m,ier)
77 if(ier.eq.1) go to 90
80 c -----------------------------------------------------------------
81 c Now get LSE’s, RSS, R squared, se’s, t-statistics, and p-values:
82 c -----------------------------------------------------------------
84 rss=a(mp1,mp1)
85 s2=rss/(n-m)
87 ybar=smean(y,n)
89 do 70 i=1,n
90 70 ssy=ssy+(y(i)-ybar)**2
92 rsq=1.-(rss/ssy)
94 do 80 i=1,m
95 beta(i)=a(i,mp1)
96 se(i)=sqrt(s2*a(i,i))
97 t(i)=beta(i)/se(i)
98 call tcdf(dabs(dble(t(i))),n-m,cdf,pdf)
99 80 pval(i)=2.*(1.-cdf)
101 c --------------------
102 c write out results:
103 c --------------------
105 write(2,81) label
106 81 format(1x,a60)
108 write(2,82) rsq,n-m
109 82 format(/,’ Regression results, R-Squared = ’,f10.8,
110 1 ’, d.f. = ’,i3)
112 write(2,83)
113 83 format(/,10x,’i’,5x,’beta(i)’,7x,’se(i)’,8x,’t(i)’,5x,’pval(i)’,/
114 1 6x,53(1h-))
116 do 84 i=1,m
117 84 write(2,85) i-1,beta(i),se(i),t(i),pval(i)
118 85 format(6x,i5,4f12.6)
119 go to 99
121 c -----------------
122 c Error handling:
123 c -----------------
125 90 write(*,*) ’ Regression matrix singular’
126 go to 99
128 100 write(*,*) ’ File doesn’’t exist’
129 go to 99
131 110 write(*,*) ’ Error reading label’
132 go to 99
134 120 write(*,*) ’ Error reading n and m’
135 go to 99
137 130 write(*,135) i
138 135 format(’ Error reading row ’,i5)
140 99 continue
141 stop
142 end
143 subroutine sweep(a,mdim,m,k1,k2,ier)
144 c ---------------------------------------------------------------
145 c
146 c Subroutine to sweep the mxm matrix a on its k1 st thru k2 th
147 c diagonals. ier is 1 if a is singular. mdim is row dimension
148 c of a in calling routine.
149 c
150 c ---------------------------------------------------------------
152 double precision a(mdim,1)
153 double precision d
76 Fortran chap 7
155 ier=1
157 do 50 k=k1,k2
159 if(abs(a(k,k)).lt.1.e-20) return
161 d=1./a(k,k)
162 a(k,k)=1.
163 do 10 i=1,m
164 a(k,i)=d*a(k,i)
165 10 if(i.ne.k) a(i,k)=-a(i,k)*d
167 do 20 i=1,m
168 do 20 j=1,m
169 20 if((i.ne.k).and.(j.ne.k)) a(i,j)=a(i,j)+a(i,k)*a(k,j)/d
171 50 continue
173 ier=0
175 return
176 end
178 double precision function inprod(x,y,n)
179 double precision x(n),y(n)
181 inprod=0.0d0
182 do 10 i=1,n
183 10 inprod=inprod+x(i)*y(i)
185 return
186 end
188 double precision function smean(x,n)
189 double precision x(n)
191 smean=0.0d0
192 do 10 i=1,n
193 10 smean=smean+x(i)
194 smean=smean/n
196 return
197 end
The External Statement and Numerical Integration
The Common and Named Common Statements and Newton-Raphson
See the chapter on “Iterative Methods for Parameter Estimation” for examples of using the common
and named common statements.
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Help!: Consider two functions f: X ->Y and g: Y->X where X and Y are two finite sets.
March 25th 2009, 07:26 PM #1
Junior Member
Feb 2009
I need some help with these questions from my textbook that I know these questions will be on an upcoming quiz and I am having difficulty with.
Consider two functions f: X ->Y and g: Y->X where X and Y are two finite sets.
g○f = 1x
Argue by either providing a detailed proof or counter example.
a) Does necessarily f○g = 1y?
b) Is f, g necessarily one-one?
c) Is f, g necessarily onto?
Can someone help me out here?
March 25th 2009, 10:09 PM #2
Junior Member
Feb 2009
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decidability of group homomorphism existence
up vote 5 down vote favorite
Fix a finitely-presented group $G$ with distinguished non-identity element $g$. For any finitely-presented group $H$ with element $h$, is it decidable whether there is a homomorphism $h: G \
rightarrow H$ such that $h(g) = h\ ?$
If we know $G$ is cyclic, the question is undecidable by reduction from the Word Problem. But what if we don't know anything about $G$? What if we know $g$ has finite order in $G$?
3 Maybe I'm missing something but if the problem is undecidable when G is cyclic, how could it not also be undecidable when we know less about G? – Reid Barton Oct 18 '09 at 21:22
I think the original poster means finite cyclic instead of cyclic. My understanding of this is that if we have two words x and y in H, and we want to know if they are equal, we just set h = xy^
{-1} and see if every generator of a finite cyclic group maps to it. If so, then it has to be the identity element because it has order n for all n. Well, I haven't shown that this question is
undecideable for a given finite cyclic group (just that we can't do it for all of them for any given h), so maybe there's a better way to see this. – Steven Sam Oct 20 '09 at 23:36
add comment
4 Answers
active oldest votes
The problem is decidable if and only if there exists a homomorphism from $G$ to the infinite cyclic group $\mathbb Z$ taking $g$ to 1 (the generator of $\mathbb Z$). Clearly, if such a
homomorphism exists, the answer to your question is "yes" for every $H,h$. Suppose that there is no such homomorphism. Consider the signature (group operations, nullary operation) of pairs $
(H,h)$ where $H$ is a group, $h\in H$. Let $X$ be the set of generators and $R$ be the set of defining relations of $G$, suppose $g$ is represented by a word $w$ in $X$. Then you are asking,
up vote whether a formula $\theta=\exists X (\& R \& (w=h))$ is true for the pair $(H,h)$. But the negation $\neg\theta$ is a Markov property. Indeed, there exists a pair, say, $({\mathbb Z},1)$
6 down which satisfies $\neg\theta$ because there is no homomorphism $G\to \mathbb Z$ that takes $g$ to $1$, and there exists a pair, say, $(G,g)$ which cannot be embedded into any pair $(G',g)$
vote which satisfies $\neg\theta$. The proof that Markov properties for pairs $(H,h)$ are undecidable is the same as the proof of the Adian-Rabin theorem for groups .
add comment
As people have already observed, if G is infinite cyclic and g is a generator, then the answer is always "yes". Steven Sam's comment shows that there can't be a uniform algorithm that
works for all cyclic groups Z/n. In fact, the problem is undecidable for any given n. For instance:-
For finite cyclic G of order n with generator g, the question can be rephrased as "Is it decidable whether the element h is of finite order dividing n?". Suppose H is torsion-free. If
this problem is decidable for some n then the word problem is solvable in H.
up vote 3
down vote But the word problem is not solvable in torsion-free groups. For instance, Collins and Miller constructed a sequence of presentations for torsion-free groups H_1, H_2,... with the
property that each H_i is torsion-free and it is undecidable which of the H_i are trivial. (More precisely, the set of all i such that H_i is trivial is recursively enumerable but not
add comment
If G is infinite cyclic and g generates G, then the answer is: "yes, there is such a map" (so in particular, it's not undecidable at all).
So I'm not quite sure what the original poster meant by saying that if G is cyclic, the problem is redicible to the word problem. Maybe, if someone sees how that argument would go (and what
the right hypothesis is), they could explain it.
Then it might be more clear whether this same argument can be applied if G is not cyclic.
up vote 1
down vote Charles is right, of course, to say that a map from G to H restricts to a map from the cyclic group generated by g, to H, but if you could determine that there was no appropriate map from G
to H, that wouldn't necessarily tell you that there was no appropriate map from < g> to H, so, on the face of it, it's possible that it could be decidable that there were no appropriate
maps from G to H, but not decidable whether or not there were appropriate maps from < g> to H. (Here, "appropriate" means "taking g to h".)
(Edited to correct html issue.)
add comment
I'm with Reid. If G,H are finitely presented, and g,h elements, and we want to know if there's a morphism f:G->H with f(g)=h, wouldn't that restrict to a morphism on the cyclic group
up vote 0 down generated by g, and thus be immediately undecidable?
add comment
Not the answer you're looking for? Browse other questions tagged gr.group-theory or ask your own question.
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Summary: Examples of dynamical systems
with innite invariant measure
Jean-Pierre Conze
(University of Rennes 1)
Abstract : We discuss two models of dynamical systems with an innite invariant mea-
In a rst part, we consider the non-dispersive billiard ow given by the movement of a ball
in the plane with identical rectangular obstacles which are Z2
-periodically distributed.
The behavior of this ow is quite dierent from the billiard ow with dispersive barriers.
Here the phase space can be decomposed into invariant subsets A corresponding to the
4 directions that can take the ow from one initial direction . For this model of billiards
in the plane, the problems are : recurrence (does the ball go back to any neighborhood of
its initial point ?) and ergodicity on the sets A for the innite measure associated with
the Lebesgue measure in the plane.
We will show that, for a very particular choice of the direction ( = /4), the billard is
recurrent (this was shown in 1980 by J. Hardy and J. Weber) and ergodic if the ratio of
the length and the width of the rectangle is irrational.
In a second part (work in collaboration with Nicolas Chevallier), we will discuss the recur-
rence for extensions of bi-dimensional rotations and give examples and counter-examples
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Mercury's precession
When applying GR to calculate Mercury's precession, the result is 43 arcseconds which coincides with the part of observed precession unexplained by newtonian theory . My question is: why the formula
from GR gives precisely this unexpained 43 arcseconds and not the total observed precession of 5600 arcseconds per century as if the calculation had implicit the rest of approximations? I guess it is
the way the GR derivation is set up but I'm curious abot how exactly.
Most of the observed precession (about 5557 arcsecs/cent) is simply due to precession of the equinox, and another 532 arcsecs/cent is due to the pulls exerted by the other planets. These effects
would be virtually identical for both Newtonian gravity and general relativity. The remaining 43 arcsecs/cent has no explanation within Newtonian gravity, but in general relativity this extra
precession is a natural feature of a single test particle orbiting in a spherical field, not related to the precession of the equinox or the pull of the other planets. When you set up the equations
to determine the magnitude of this effect, you omit the precession of the equinox and the pull of the other planets. That's why you get just the extra 43 arcsec/cent. You could do the calculation for
the whole effect, but it's much more complicated.
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I've found it's best to ask an accountant these types of questions. In this example, an accountant
might expect average cost to be calculated a certain way, and not worry so much about "penny
Daniel Kunkel wrote:
> Hi
> It's so wonderful to see an integrated accounting system for OFBiz.
> I've done some thinking about the algorithm used to calculate the
> average cost of an inventory item, and thought I'd share it again.
> Rather than trying to store an "average cost" for an item, I believe we
> would be much better off storing the "total investment" for each item.
> Most of the time, it won't make any difference, however I have seen that
> directly storing the average cost leads to all sorts of floating point
> resolution and residual adjustment issues, and complications when
> purchasing inventory at a varying price. Most notably that the total
> investments purchasing an item sometimes won't exactly equal the total
> cost of goods expensed after it is all expended.
>>>From http://lists.ofbiz.org/pipermail/dev/2006-March/010207.html:
> For example, if you bought a million pieces for $3,889,107,143, and sold
> them out of inventory one at a time for $3.89 without tracking/updating
> the residual average cost.. eventually you'd have to account for
> difference of more than $800,000!
> The solution:
> Keep a running "total investment" for each item. At
> the point that you utilize an item, calculate and subtract
> the cost of those items rounded to a penny for example. Other
> currencies will round to the degree necessary for their currency.
> The effect of doing this will be that the cost of goods will
> oscillate up and down, in the above example between 3.88, and 3.89
> in such a way that when you've sold your millionth item, the
> residual investment remaining is exactly 0.
> Thanks
> Daniel
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stlaurentv212010-10-05T16:19:00Z2010-10-05T16:19:00Z514878482Wentworth Institute of Technology7019995012Math 205 College Math I (Sect. 13/14) Fall 2010Course DescriptionAlgebra and trigonometry,
including algebraic fractions, systems of linear equations, quadratic equations, literal equations, word problems and their solutions, right triangles and vectors. Applications will be stressed.
Prerequisite: High School Algebra II .Professor: Dr. Naomi RidgeEmail: ridgen@wit.edu (preferred).AIM: naomiastroSkype: egdiranTelephone: 617-209-8122 (home: urgent questions only).Office Hours:Mon/
Tue/Wed 11am-noon in AMS Conference room,or by appointment.Class Meeting Times:Mondays 10-10:50am: Location TBDTuesdays 10-10:50am: Location TBDWednesdays 10-10:50am: Location TBDThursdays 10-11:50am
(“lab”): Annex Central 102Communication: I will send class announcements via email. Please ensure you provide me with an email address which you check daily.Required Text: Algebra & Trigonometry, 8th
Edition, Sullivan, with “MyMathLab” package.. Attempting to take this course without the textbook is not recommended!THE COLLEGE BOOKSTORE:Location:103 Ward Street Boston MA
02115Telephone:617-445-8814Learning ObjectivesAt the completion of this course, the student should be able to:Add, subtract multiply and divide real numbers, complex numbers, and polynomials with
real coefficients.Compute using the laws of exponents, including nth roots.Factor any polynomial of degree two or less and certain higher order polynomials.Recognize, state, and solve linear and
quadratic equations.Compute the distance and the midpoint between two points.Recognize and graph equations in two variables on a Cartesian plane.Recognize and graph (x − a)2 + (y − b)2 = c2 .Solve
problems involving proportionality.Solve 2 × 2 systems of linear equations by substitution and by elimination.Recognize, compute with, and convert between decimal degrees, degrees/minutes/seconds,
and radians.Compute sin θ, cos θ, tan θ, csc θ, sec θ, and cot θ for general angles θ.State and use both the law of sines and the law of cosines.Solve triangles, given sufficient data.Perform basic
arithmetic on vectors including adding, subtracting, and multiplication by a scalar.Class Participation and Attendance:You are required to attend all classes and lab sessions. Attendance will be
taken at the beginning of each class. Students arriving late for class will be marked as absent at the discretion of the instructor.Students missing more than 15% of scheduled classes (including
labs) may be withdrawn according to WIT policy. If you miss a class it is your responsibility to ensure that you learn the material covered, do the homework set and check on any announcements that
were made.Attendance bonus: Starting in week 2, if you never miss a single class then you will be awarded extra credit! 3 percentage points will be added to your final score if you don’t miss a
class; for example, if you are at every single class and say your final average is 93%, then your average will become 96%.The drop/add period for day students ends on Friday of the first week of
classes. Dropping and/or adding courses is done online. Courses dropped in this period are removed from the student's record. Courses to be added that require written permission, e.g. closed courses,
must be done using a Drop/Add form that is available in the Student Service Center. Non-attendance does not constitute dropping a course. If a student has registered for a course and subsequently
withdraws or receives a failing grade in its prerequisite, then the student must drop that course. In some cases, the student will be dropped from that course by the Registrar. However, it is the
student's responsibility to make sure that he or she meets the course prerequisites and to drop a course if the student has not successfully completed the prerequisite. The student must see his or
her academic advisor or academic department head for schedule revision and to discuss the impact of the failed or withdrawn course on the student's degree status.Homework:All homework and exam
problems will closely follow examples provided in the class. Homework will be set using the MyMathLab system (more information on this will be given in class).IMPORTANT: Your MyMathLab Course ID for
this class is: ridge83991Homework will generally be due twice per week, on Mondays and Thursdays. Classwork will be due at the end of the “lab” session in which it is set. I encourage a two-way
conversation during the class – if there is something from a homework that you do not follow, please let me know. The “lab” sessions in particular are a chance to spend more time working through
examples together.Make sure to seek help if you are struggling with any homework problems, either by email, phone or during office hours. Students are encouraged to work on homework problems
together, but should remember that you won't have a friend to help you in the exam!Credit will not be deducted if you seek help, but you will almost certainly lose points if you are struggling with a
topic and don't seek help! If you cannot do the homework problems, then you won't be able to do the exam problems either!Exams: Six exams will be given in this course: this includes the final exam
and the five in-class exams before the final exam (see class schedule). When calculating your final score, your lowest in-class exam score is dropped. The final exam is compulsory and that score
cannot be the one that is dropped.Calculators:Calculators, and in particular graphing calculators, are not required for Math 205. Calculators will not be allowed during quizzes or exams. They may be
used to help complete certain homeworks.Makeup Policy: If you miss an exam, then the makeup policy is that you must have a medical excuse (verifiable via a doctor’s note or a phone number of the
medical facility) or a special circumstance (funeral, etc.). Otherwise if you miss an exam, then that is a zero. Assessment and Grading:Grading of all in-class problems tests, and the final exam will
require you to show your working. If you use the correct methodology to a problem, but do not get the correct answer, you will still get most of the points for that problem. Therefore, make sure to
include every step of your working – a correct final answer will get zero points if you do not show how you got to it.Your final class grade will be calculated as follows:Cumulative homework average:
10% (missing problem sets count as a zero).Cumulative classwork (lab) average: 10% (missing labs count as a zero).Five in-class exams (Average will be calculated after dropping lowest score):
60%Final Exam: 17%Attendance bonus: 3%The following is the grading scale, as mandated by WIT policy:Letter GradePercent
RangesA96%-100%A-92%-95%B+88%-91%B84%-87%B-80%-83%C+76%-79%C72%-75%C-68%-71%D+64%-67%D60%-63%FBelow 60% Course ScheduleAlgebra ReviewR.2 Algebra EssentialsR.4 Polynomials R.5 Factoring PolynomialsR.7
Rational Expressions R.8 nth Roots; Rational ExponentsIn-class test 1Linear and Quadratic Equations1.1 Linear Equations,1.2 Quadratic Equations 1.3 Complex Numbers 1.7 Problem Solving In-class test
2Graphs2.1 Distance and Midpoint Formulas2.2 Graphs of Equations in Two Variables 2.3 Lines2.4 Circles2.5 Variation 12.1 Systems of Linear Equations (2×2 only) In-class test 3Right Angle
Trigonometry7.1 Angles and Their Measure, 7.2 Right Triangle Trigonometry7.3 Computing the Values of Trig. Functions of Acute Angles7.4 Trigonometric Functions of General AnglesIn-class test
4Analytical Trigonometry9.1 Applications Involving Right Triangles 9.2 The Law of Sines9.3 The Law of Cosines 10.4 VectorsIn-class test 5Academic Honesty Statement:Students at Wentworth are expected
to be honest and forthright in their academic endeavors. Academic dishonesty includes cheating, inventing false information or citations, plagiarism, tampering with computers, destroying other
peoples studio property, or academic misconduct (Academic Catalog). See your catalogue for a full explanation.Student Accountability Statement: Cheating on homework, a quiz, or an exam will result in
a zero for that activity. Repeated cheating will be reported to the Institute and will result in an F in the class. Plagiarism is cheating. If you do not understand what plagiarism is in the context
of a mathematics course, please come talk to me before you cross the line. To promote academic honesty, a random number of exams will be selected and photocopied before being returned to students.
All exams and tests are closed book unless otherwise specified, so please put all your books, laptops, cell phones, ipods, etc... out of sight during tests and exams. Furthermore, talking or looking
away from your paper during a test or exam is strictly prohibited. Leaving the classroom for any reason during a test isalso prohibited.Disability Services Statement:Any student who thinks s/he may
require a disability-related accommodation for this course should contact the Disability Services Office as soon as possible. Disability Services coordinates reasonable accommodations for students
with documented disabilities. They are located in Watson Hall 003 (the Counseling Center) and can be contacted at 617-989-4390 or counseling@wit.edu. Note that it is also your responsibility to
contact us privately as soon as possible to arrange for such accommodations in a timely manner. For more information on acceptable documentation and the Disability Services process, visit the
Disability Services website at http://www.wit.edu/disabilityservices.
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The Dimension of Matrices (Matrix Pencils) with Given Jordan (Kronecker) Canonical Forms
James W. Demmel and Alan Edelman
EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-92-706
September 1992
The set of n by n matrices with a given Jordan canonical form defines a subset of matrices in complex n^ 2 dimensional space. We analyze one classical approach and one new approach to count the
dimension of this set. The new approach is based upon and meant to give insight into the staircase algorithm for the computation of the Jordan Canonical Form as well as the occasional failures of
this algorithm. We extend both techniques to count the dimension of the more complicated set defined by the Kronecker canonical form of an arbitrary rectangular matrix pencil A -- lambda B.
BibTeX citation:
Author = {Demmel, James W. and Edelman, Alan},
Title = {The Dimension of Matrices (Matrix Pencils) with Given Jordan (Kronecker) Canonical Forms},
Institution = {EECS Department, University of California, Berkeley},
Year = {1992},
Month = {Sep},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6252.html},
Number = {UCB/CSD-92-706},
Abstract = {The set of <i>n</i> by <i>n</i> matrices with a given Jordan canonical form defines a subset of matrices in complex <i>n</i>^<i>2</i> dimensional space. We analyze one classical approach and one new approach to count the dimension of this set. The new approach is based upon and meant to give insight into the staircase algorithm for the computation of the Jordan Canonical Form as well as the occasional failures of this algorithm. We extend both techniques to count the dimension of the more complicated set defined by the Kronecker canonical form of an arbitrary rectangular matrix pencil <i>A</i> -- lambda<i>B</i>.}
EndNote citation:
%0 Report
%A Demmel, James W.
%A Edelman, Alan
%T The Dimension of Matrices (Matrix Pencils) with Given Jordan (Kronecker) Canonical Forms
%I EECS Department, University of California, Berkeley
%D 1992
%@ UCB/CSD-92-706
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6252.html
%F Demmel:CSD-92-706
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Topic: Happy Birthday Herman Rubin and thanks!
Replies: 3 Last Post: Nov 4, 2012 11:31 PM
Messages: [ Previous | Next ]
Re: Happy Birthday Herman Rubin and thanks!
Posted: Oct 29, 2012 5:44 AM
On Saturday, October 27, 2012 9:47:09 AM UTC-7, The poster formerly known as Colleyville Alan wrote:
> Happy Birthday, Herman. And thanks for recommending your late wife's book on set theory - I was able to find a copy and I am enjoying it very much.
> Alan
Hello . Set theory for mathematicians by Jean Rubin is indeed a very excellent book both for the system of natural logic and for the Kelly-Moore set theory .There is also another excellent book on
Kelly-Moore set theory by Donald Monk.
However both books are out of print. Perhaps Herman could give all of us a present by arranging for the book to be published in Dover so that it would be as inexpensive as possible for the benefit
of future generations .
And while I'm at it let me mention that the best book on set theory to appear lately did come out in Dover - Raymond Smullyan and Melvin fitting -set theory and the continuum hypothesis .
And yes of course Happy Birthday to Herman Rubin .smn
Date Subject Author
10/27/12 Happy Birthday Herman Rubin and thanks! The poster formerly known as Colleyville Alan
10/29/12 Re: Happy Birthday Herman Rubin and thanks! Stuart M Newberger
11/4/12 Re: Happy Birthday Herman Rubin and thanks! Alan Charbonneau
11/4/12 Re: Happy Birthday Herman Rubin and thanks! The poster formerly known as Colleyville Alan
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Sylow p-subgroups of locally finite groups
up vote 1 down vote favorite
Suppose $G$ is a locally finite group such that $G=\bigcup_{i=1}^\infty S_i$, where $S_i$ is a finite group and $S_i \triangleleft S_{i+1}$ for all $i \in \mathbb{N}$. Let $P$ be a Sylow (maximal
with respect to inclusion) $p$-subgroup of the group $G$. For each $i \in \mathbb{N}$ put $P_i=S_i \cap P$. Then $P_i$ is a Sylow $p$-subgroup of $S_i$ for all $i \in \mathbb{N}$. Or not?
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1 Answer
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Yes. Fix $i$. For $n\ge i$, let $L_n$ be a $p$-Sylow of $S_n$ containing $P_n$. Note that $S_i$ is subnormal in $S_n$. By the lemma below, $L_n$, the intersection $M_n=L_n\cap S_i$ is a
$p$-Sylow of $S_i$. Now there exists an infinite set $N$ of $n$ such that the Sylow $M=M_n$ does not depend on $n\in N$. Since $\langle P_n,M\rangle$ is a $p$-group for all $n$ and $P=\
bigcup P_n$, we deduce that $\langle P,M\rangle=\bigcup\langle P_n,M\rangle$ (increasing union) is a $p$-group. By maximality, we deduce that $M\subset P$, QED.
up vote 1 [Lemma: if $G$ is a finite group, $H$ a subnormal subgroup and $P$ a $p$-Sylow of $G$ then $P\cap H$ is a $p$-Sylow of $H$.
down vote
accepted Indeed, by an obvious induction we can reduce to the case when $N$ is normal, let $Q$ be a $p$-Sylow of $H$. Then $Q$ is contained in a $p$-Sylow of $G$, that is, a conjugate $gPg^{-1}
$. So $g^{-1}Qg\subset P\cap H$ (because $H$ is normal); by cardinality this is also a $p$-Sylow of $H$ and the lemma is proved.]
You seem to be assuming that $S_i\triangleleft S_j$ for all $j>i$, which does not automatically follow from the case $j=i+1$ as in the question. However, I suspect that the OP meant
to assume the stronger version. – Neil Strickland Jun 25 '13 at 21:49
@Neil Strickland. Yes, $S_i$ is normal olny in $S_{i+1}$. In other words, $S_i$ is subnormal in $S_j$ for $j>i$. But it's O.K., the Lemma is valid in that case too. And I proved
something similar too. But... – user35603 Jun 25 '13 at 22:40
But why $M$ does not depend on $n$? – user35603 Jun 25 '13 at 22:57
@user35603: As there are only finitely many possibilities for $M_n$ ($\le S_i$), at least one of them shows up infinitely times often. – j.p. Jun 26 '13 at 6:29
Thanks to all. I think you are right. – user35603 Jun 26 '13 at 11:24
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Right triangle PQR is to be constructed in the xy-plane so
Question Stats:
68%31% (02:28)based on 154 sessions
Official answer is "C" - 9900
This was a hard question for me.
Anyway, I took the following approach to solve this.
When y=6
We know -4<=x<=+5, there are 10 integral points.
The points are: {-4,6},{-3,6},{-2,6},{-1,6},{0,6},{1,6},{2,6},{3,6},{4,6},{5,6}
So, number of distinct line segments that can be formed are 10C2=45.
And these line segments PR as it should be parallel to x-axis.
With every line segment there can be 20 triangles:
Let's see how:
consider line segment with points: P={-4,6} and R={1,6}
We know there are 10 points above P that can serve us as Q. Q should always be
vertically above or below point P because P is the right angle.
So Q can be: {-4,7},{-4,8},{-4,9},{-4,10},{-4,11},{-4,12},{-4,13},{-4,14},{-4,15},{-4,16}
See for yourself that now: for just one segment PQ we have 10 triangles.
However, we can flip the points on the same line segment PR, so that P={1,6} and R={-4,6}
We know there are 10 points above P that can serve us as Q. Q should always be
vertically above or below point P because P is the right angle.
So in this case Q can be:
See for yourself that for just one segment PQ we have another 10 triangles.
So, a total of 20 distinct triangles for just one line segment.
thus: a total of 20 * 10C2 = 20 * 45 = 900 triangles for all line segments where PR
is on y=6.
Remember, we were talking only about PR that lies on y=6 line.
There are 11 such lines between y=6 and y=16 i.e. 16-6+1.
Thus; a total of 900 * 11 = 9900 triangles can be formed with given conditions.
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Numerical differentiation and regularization
- Mathem. of Computation , 1968
"... Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize h
being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
Cited by 39 (22 self)
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Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize h being a
regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra
integral equation is developed. 1.
- Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to
the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the
derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of
stable differentiation. 1.
, 1995
"... this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed
interval [a; b], we construct an associated functional, ..."
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this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a;
b], we construct an associated functional,
- in "48th IEEE Conference on Decision and Control , 2009
"... Résumé Ce rapport est consacré aux estimations des dérivées. Contrairement à la procédure de régularisation de Tikhonov, nous utilisons un cadre algébrique récent qui conduit enfin à une
projection dans la base de polynômes de Jacobi, afin d’estimer des dérivées des signaux bruités. Aucune informati ..."
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Résumé Ce rapport est consacré aux estimations des dérivées. Contrairement à la procédure de régularisation de Tikhonov, nous utilisons un cadre algébrique récent qui conduit enfin à une projection
dans la base de polynômes de Jacobi, afin d’estimer des dérivées des signaux bruités. Aucune information sur les propriétés statistiques du bruit est requise. Nous donnons quelques résultats
concernant le choix des paramètres dans cette méthode de manière à minimiser l’erreur due au bruit et les erreurs d’approximation. De plus, deux nouveaux estimateurs centraux fondés sur telles
techniques algébriques de la différenciation sont introduits. Une comparaison est faite entre ces estimations et quelques méthodes classiques de la différentiation numérique. inria-00439386, version
1- 7 Dec 2009 1
- J. COMP. APPL. MATH. , 2006
"... A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the
examples considered its solution is computed analytically. Some numerical results of its application are ..."
Cited by 2 (1 self)
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A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the
examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation
is numerically more efficient than some other methods, in particular, than variational regularization.
"... 3. NONPARAMETRIC RESGRESSION ESTIMATES 3.1 Defining the basic model ..."
, 2009
"... We consider the approximate solution of linear ill-posed inverse problems of high dimension with a simulation-based algorithm that approximates the solution within a low-dimensional subspace.
The algorithm uses Tikhonov regularization, regression, and low-dimensional linear algebra calculations and ..."
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We consider the approximate solution of linear ill-posed inverse problems of high dimension with a simulation-based algorithm that approximates the solution within a low-dimensional subspace. The
algorithm uses Tikhonov regularization, regression, and low-dimensional linear algebra calculations and storage. For sampling efficiency, we use variance reduction/importance sampling schemes,
specially tailored to the structure of inverse problems. We demonstrate the implementation of our algorithm in a series of practical large-scale examples arising from Fredholm integral equations of
the first kind. 1
, 905
"... Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by
geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude ..."
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Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical
optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complex-valued. The following mathematical principle is ultimately behind
the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an
alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs
the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise
beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand,
ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and
a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. 1.
, 2007
"... differentiation of experimental data: ..."
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