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Homework Help
Posted by Jessica Fuentes on Thursday, December 9, 2010 at 9:01pm.
Does this look right?
Mike runs twice as far as Jim, but Phillip runs three times as far as Jim. If the three of them run a total of 24 miles, how far does each man run?
2m + j = 24
3p + j = 24
I am so wrong aren't I? Can someone help me, please??
• algebra - bobpursley, Thursday, December 9, 2010 at 9:04pm
You are so wrong.
• algebra - Jessica Fuentes, Thursday, December 9, 2010 at 9:05pm
lol, I figured. i am so confused I hate word problems, ok can i post another one and see if I can get it??
• algebra - Jessica Fuentes, Thursday, December 9, 2010 at 9:17pm
TastyBake bakery sells three times as many chocolate donuts as plain glazed donuts, but only half as many as jelly-filled donuts. If they sold 40 dozen donuts, how many of each type did they
x = chocolate
y = plain
z = jelly
y = 3x
z = 1/2x
x + y + z = 40
How was that???
• algebra - Jessica Fuentes, Thursday, December 9, 2010 at 9:20pm
TastyBake bakery sells three times as many chocolate donuts as plain glazed donuts, but only half as many as jelly-filled donuts. If they sold 40 dozen donuts, how many of each type did they
x = chocolate
y = plain
z = jelly
y = 3x
z = 1/2x
x + y + z = 40
How was that???
• algebra - bobpursley, Thursday, December 9, 2010 at 9:26pm
J=1/2 P
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Monte Carlo Simulation
March 6th 2009, 02:21 AM #1
Mar 2009
I am currently working for a financial analyst using a modelling program which uses Monte Carlo simulations (Latin Hypercube Sampling method) to predict the annual cost of a set of potential
events. I don’t have much formal education or training in advanced probability/statistics so I need some help.
Background –
The model is set up like so:
□ Enter the data set of potential events
□ Specify a set of ranges of potential cost per event (for example, level 1 might be between $0 and $100k per event, level 2 between $100k and $1m, level 3 between $1m and $5m, etc.) – this is
the impact I.
□ Specify a set of ranges of potential number of events per unit time, generally one year (for example, level 1 between 0 and 0.5 events per year, level 2 between 0.5 and 1 events per year,
level 3 between 1 and 4 e.p.y., etc.) – this is the frequency F.
□ Select a particular range for both these categories for each event, for example the first event could have Impact at level 2 and Frequency at level 3 [This would give you a broad estimate of
2.5 occurrences per year costing $550,000 each time leaving a total expected cost of $1.375m per year, assuming likelihood of taking a particular i or f within a range is normally distributed
around the midpoint of that range]
□ The cost per event and events per year are each assigned a distribution curve. In this case, the Impact uses a Gumbel distribution while the Frequency uses a Poisson distribution. This is
used to randomly generate values within the specified range in the MC simulation.
Running the simulation –
As well as a sample size (50,000 simulations as this is enough to guarantee convergence using the LHS method), I specify a confidence level for each simulation. The best way I can describe this
confidence level is as the probability that events of this severity won’t be seen in a given year. For example running simulations at a CL of 0.95, or 95%, will give the costs of these events for
the worst year in twenty; using a CL of 99.93%, or 0.9993, will give the costs of these events for the seventh worst year in ten thousand and so on.
My question is, what confidence level do I use to simulate the ‘average year’? I believe it to be somewhere in the region of 55-65%, and some basic practical testing does tend to indicate a
figure of around 59.7% (although this cannot be relied upon due to the extremely limited sample size). Can anyone give me a more accurate number, and if possible, some form of proof?
There is no correlation between events, they are independent.
[note: the basic testing is based upon the following –
- run a simulation at confidence level of X = 50% for a selection of approx 350 events (each event returns an annual cost x(n) for n = [1, 350])
- find an expected value y(n) for each event by multiplying the midpoints of the applicable I and F ranges as described above: y(n) = { I(min) + ½ [I(max) – I(min)] } x { F(min) + ½ [F(max) – F
(min)] }
- find x(n) as a proportion of y(n). Using X = 50% means that x(n) < y(n) – not sure if this is necessarily true, but it is for all of my 350 samples. The average result of x(n) by y(n) is
approximately 0.837
- Divide the original confidence level X by the average of these ratios, in this case 0.837. This gives the predicted confidence level needed to see the simulation match the expected results,
which my tests gave the best estimate as 59.7%]
Is there a particular reason as to why you want to use latin hypercube? I have run 1 zillion ways of simulating and I would recommend using simple monte carlo with antithetic variates to expedite
convergence. If you need more help, feel free to pm me.
I am currently working for a financial analyst using a modelling program which uses Monte Carlo simulations (Latin Hypercube Sampling method) to predict the annual cost of a set of potential
events. I don’t have much formal education or training in advanced probability/statistics so I need some help.
Background –
The model is set up like so:
□ Enter the data set of potential events
□ Specify a set of ranges of potential cost per event (for example, level 1 might be between $0 and $100k per event, level 2 between $100k and $1m, level 3 between $1m and $5m, etc.) – this is
the impact I.
□ Specify a set of ranges of potential number of events per unit time, generally one year (for example, level 1 between 0 and 0.5 events per year, level 2 between 0.5 and 1 events per year,
level 3 between 1 and 4 e.p.y., etc.) – this is the frequency F.
□ Select a particular range for both these categories for each event, for example the first event could have Impact at level 2 and Frequency at level 3 [This would give you a broad estimate of
2.5 occurrences per year costing $550,000 each time leaving a total expected cost of $1.375m per year, assuming likelihood of taking a particular i or f within a range is normally distributed
around the midpoint of that range]
□ The cost per event and events per year are each assigned a distribution curve. In this case, the Impact uses a Gumbel distribution while the Frequency uses a Poisson distribution. This is
used to randomly generate values within the specified range in the MC simulation.
Running the simulation –
As well as a sample size (50,000 simulations as this is enough to guarantee convergence using the LHS method), I specify a confidence level for each simulation. The best way I can describe this
confidence level is as the probability that events of this severity won’t be seen in a given year. For example running simulations at a CL of 0.95, or 95%, will give the costs of these events for
the worst year in twenty; using a CL of 99.93%, or 0.9993, will give the costs of these events for the seventh worst year in ten thousand and so on.
My question is, what confidence level do I use to simulate the ‘average year’? I believe it to be somewhere in the region of 55-65%, and some basic practical testing does tend to indicate a
figure of around 59.7% (although this cannot be relied upon due to the extremely limited sample size). Can anyone give me a more accurate number, and if possible, some form of proof?
There is no correlation between events, they are independent.
[note: the basic testing is based upon the following –
- run a simulation at confidence level of X = 50% for a selection of approx 350 events (each event returns an annual cost x(n) for n = [1, 350])
- find an expected value y(n) for each event by multiplying the midpoints of the applicable I and F ranges as described above: y(n) = { I(min) + ½ [I(max) – I(min)] } x { F(min) + ½ [F(max) – F
(min)] }
- find x(n) as a proportion of y(n). Using X = 50% means that x(n) < y(n) – not sure if this is necessarily true, but it is for all of my 350 samples. The average result of x(n) by y(n) is
approximately 0.837
- Divide the original confidence level X by the average of these ratios, in this case 0.837. This gives the predicted confidence level needed to see the simulation match the expected results,
which my tests gave the best estimate as 59.7%]
Thanks for the reply.
The only reason I am using the LH method is because that is what the original programmer decided! I am not a programmer so couldn't rewrite it even if my company allowed me to.
I'll give a little more detail:
I have been given a set of data and asked to find out "what these events will cost in an average year" but I don't know what confidence level to run the simulations at.
A graph of the confidence level (x-axis) against the probability that the annual results will match that confidence level (y-axis) should give a skewed normal distribution. The modal average is
at the 50% confidence level but this doesn't represent the average year, because, since costs have a lower bound of 0 but only a theoretical upper bound that could be hundreds of times higher
than the average, it is more probable that costs will be slightly above this modal average than below.
The mean (which is the target confidence level) is found at the confidence level C such that on the graph I described, the line x = C divides the area under the curve into to two equal parts; if
the curve is denoted f(x) then this could be written as
{the integral of f(x) evaluated over the interval [0, C]} = {the integral of f(x) evaluated over the interval [C, 100]}.
When I ran simulations, I avoided latin hypercube most of the time. Let me rethink your problem and I shall get back. It is not that hard.
March 9th 2009, 11:34 AM #2
Mar 2009
March 10th 2009, 02:18 AM #3
Mar 2009
March 10th 2009, 07:30 AM #4
Mar 2009
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Reliability, Confidence Level Sample Size
The cumulative Binomial Distribution:
- C.L is the Confidence Level for the Test. This can be thought of as the Probability of more than the Maximum number of allowable failures occurring on Test.
- “N” is the number of unit on test or Sample Size
- “r” is the maximum allowable number of failures on test
- “R” is the reliability of the “System” for the duration of the test
With “no failure”, Cumulative Binomial Distribution equation is given below:
1 – C.L = R^N
Solve for N
N = LN(1 – C.L)/LN(R)
You can also solve top first equation for 1 failure and so own.
If you have any other question, please let me know.
Muhammad Anwar
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17.4 The Product Rule and the Curl
Home | 18.013A | Chapter 17 Tools Glossary Index Up Previous Next
17.4 The Product Rule and the Curl
The product rule applied to the curl takes the form
$∇ ⟶ × ( f * v ⟶ ) = ( ∇ ⟶ f ) × v ⟶ + f * ( ∇ ⟶ × v ⟶ )$
(This is an immediate consequence of the ordinary one dimensional product rule and the linearity of all our products. When the derivatives here act on $f$ they form the first term here, and when they
act on $v ⟶$ the produce the second term. The weird star here denotes ordinary multiplication.)
The approach of the last section for reducing computation of divergence in spherical coordinates can be used just as well for the computation of the curl.
The approach, you will remember, consists of finding vectors pointing in the right directions with $0 ⟶$ curl, expressing a general vector as a combination of these, and using the
product theorem to express the results. With curl $0 ⟶$ one term of the two in the product theorem disappears and we have our formula.
All you need to do it is to find vectors in each appropriate direction with vanishing curl. That is quite easy to do because the gradient of any function will have vanishing curl.
Thus we can take the gradients of $ρ$ and of $θ$ and of $φ$ , and these will be vectors pointing in the right directions, and give us immediately
$curl ⟶ u ^ ρ = 0 ⟶ , curl ⟶ u ^ θ r = 0 ⟶ , curl ⟶ u ^ φ ρ = 0 ⟶$
and so we deduce, via the product theorem
$curl ⟶ ( v 1 u ^ ρ + v 2 u ^ θ + v 3 u ^ φ ) = curl ⟶ ( v 1 u ^ ρ + v 2 r u ^ θ r + v 3 ρ u ^ φ ρ ) = grad ⟶ ( v 1 ) × u ^ ρ + grad ⟶ ( r
v 2 ) × u ^ θ r + grad ⟶ ( ρ v 3 ) × u ^ φ ρ$
Unfortunately I must admit never ever using this result in any context. So you may safely ignore it, I suppose.
17.5 Find a similar expression for the curl in cylindric coordinates.
17.6 Find the curl of $r u ^ ρ$ .
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0844206849 isbn/isbn13 $$ Compare Prices at 110 Bookstores! Learning to Listen in English discount, buy, cheap, used, books & textbooks
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Polyhedrons. Prism, parallelepiped, pyramid
Polyhedron. Convex polyhedron. Prism. Right prism. Oblique prism.
Regular prism. Normal (orthogonal) section of a prism. Parallelepiped.
Right parallelepiped. Right-angled parallelepiped. Cube. Pyramid.
Regular pyramid. Truncated pyramid. Regular truncated pyramid.
Polyhedron is a body, boundary of which consists of pieces of planes (polygons). These polygons are called faces, their sides – edges, their vertices – vertices of polyhedron. Segments, joining two
vertices, which are not placed on the same face, are called diagonals of polyhedron. A polyhedron is called a convex one, if all its diagonals are placed inside of it.
Prism is a polyhedron ( Fig.79 ), two faces of which ABCDE and abcde (bases of prism) are equal polygons with correspondingly parallel sides, and the rest of the faces (AabB, BbcC etc.) are
parallelograms, planes of which are parallel to a straight line (Aa, or Bb, or Cc etc.). Parallelograms AabB, BbcC etc. are called lateral faces; edges Aa, Bb, Cc etc. are called lateral edges. A
height of prism is any perpendicular, drawn from any point of one base to a plane of another base. Depending on a form of polygon in a base, the prism can be correspondingly: triangular,
quadrangular, pentagonal, hexagonal and so on. If lateral edges of a prism are perpendicular to a base plane, this prism is a right prism; otherwise it is an oblique prism. If a base of a right prism
is a regular polygon, this prism is also called a regular one. On Fig.79 an oblique pentagonal prism is shown.
Parallelepiped is a prism, bases of which are parallelograms. So, a parallelepiped has six faces and all of them are parallelograms. Opposite faces are two by two equal and parallel. A parallelepiped
has four diagonals; they all intersect in the one point and they are divided in it into two. If four lateral faces of parallelepiped are rectangles, it is called a right parallelepiped. A right
parallelepiped, all six faces of which are rectangles, is called a right-angled parallelepiped. A diagonal of right-angled parallelepiped d and its edges a, b, c are tied by the relation: d ^2 = a ^2
+ b ^2 + c ^2 . A right-angled parallelepiped, all faces of which are squares, is called a cube. All edges of a cube are equal.
Pyramid is a polyhedron, one face of which (a base of pyramid) is an arbitrary polygon ( ABCDE, Fig.80 ), and all the rest of the faces (lateral faces) are triangles with a common vertex S, called a
vertex of a pyramid. The perpendicular SO, drawn from a vertex of a pyramid to its base, is called a height of pyramid. Depending on a form of polygon in a base, the pyramid can be correspondingly:
triangular, quadrangular, pentagonal, hexagonal and so on. A triangular pyramid is a tetrahedron, a quadrangular one – a pentahedron etc. A pyramid is called a regular one, if its base is a regular
polygon and its height falls into a center of a base. All lateral edges of a regular pyramid are equal; all lateral faces are equal isosceles triangles. A height of lateral face ( SF ) is called an
apothem of a regular pyramid.
If to draw the section abcde, parallel to the base ABCDE ( Fig.81 ) of the pyramid, then a body, concluded between these planes and lateral surface, is called a truncated pyramid. Parallel faces
ABCDE and abcde are called its bases; a distance Oo between them is a height. A truncated pyramid is called a regular one, if a pyramid, from which it was received, is regular. All lateral faces of a
regular truncated pyramid are equal isosceles trapezoids. The height Ff of a lateral face ( Fig.81 ) is called an apothem of a regular truncated pyramid.
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3 sets of 5 pieces of data. Compare and contrast the three sets. Comment briefly on t
November 5th 2010, 11:21 AM
3 sets of 5 pieces of data. Compare and contrast the three sets. Comment on results
Set A Set B Set C
I'm being told to "Use more than one measure of location in the comparisons. There is no need to use measures of dispersion."
So I think that means I need to find the Mean Median and Mode for each set. Which I can do except for the Mode because there are no repeated numbers in each set. I believe I need to make some
kind of frequency distribution graph, but I don't see how I can do that, there are no frequencies.
This question is worth 8 marks so I'm sure they want more from me than just the Mean and Median and a sentence saying "a is greater than b, b and c are very similar". What am I missing?
Any help is very greatly appreciated, thanks.
November 5th 2010, 12:10 PM
All you need to do here is find the median and mean.
A frequency Dist Histogram will be of no value.
November 5th 2010, 12:29 PM
Exactly my thoughts, but that is surely not worth 8 marks. There must be something else.
November 5th 2010, 12:35 PM
You have asnwered the question.
There are many things you can do to measure and compare these data sets. But you can only be assessed on what you have been taught.
November 5th 2010, 12:40 PM
Can you perhaps give me examples of these things I can do? This maths module was well and truly thrust into my course, which we were told it would not be. They say its not degree level, but there
also not really helping us with it. The lecture notes for this exercise only talks about frequency distribution graphs.
Thank you very much for your replies.
November 5th 2010, 07:49 PM
Where do I start, maybe you can perform and ANOVA or test for multivariate normality?
By the sounds of the question finding the mean and median is the way to go. I can't predict or know how marks are given in your course. What level of Math/Stats are you studying?
November 6th 2010, 03:35 AM
That sounds very complicated...
That sounds good, and thanks very much for the answer. But it looks complicated, and perhaps too far. I looked ahead to next weeks lecture notes and they mention box plots, which I had forgotten
about, and to make one for each data set seams like a good way to do it, it allows me to compare them easily. Does that sound like a good idea?
I'm not sure what level it is, its a maths module in a sociology course, it feels like AS level.
November 6th 2010, 12:26 PM
November 6th 2010, 12:42 PM
Box plots are a measure of dispersion? Oh dear. What's best for me to do then? This ANOVA test or multivariate normalization?
Again thanks a lot for the help
November 6th 2010, 12:47 PM
I think you should ask your lecturer/tutor.
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Summary: Math 2020 Spring 2011
Solving Discrete Problems P. Achar
Notes on Chapter 3
Universal Quantifier. A sentence with a universal quantifier can often be rewritten using "Let" or "If
. . . then." The four sentences below are all different ways of expressing the same thing. Note that in the
fourth example, the quantified variable (the subset of N) isn't given a name.
A N, if A is nonempty, then A has a smallest element.
Let A N. If is a nonempty, then A has a smallest element.
If A N is nonempty, then A has a smallest element.
Every nonempty subset of N has a smallest element.
Universal quantifiers in proofs. Remember that:
· when proving a statement involving x, you're not allowed to impose any extra conditions or restric-
tions on x that aren't already in the statement. (But you can break up the proof into parts where you
consider different restrictions on x, as long as they cover all possible cases when taken together.)
· when using a statement involving x in the proof of something else, you get to pick a specific value
for x.
Existential Quantifier. Here are four more sentences that all have the same meaning as each other:
x Z such that x2
- 1 = 0.
There is an integer x such that x2
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Montara SAT Math Tutor
Find a Montara SAT Math Tutor
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...My hours are flexible. For regular students, I am available for help by phone for "emergency" help.Linear Algebra is key to all post-calculus mathematics and includes the study of vectors in
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Trigonometry, Tenth Edition, by Lial, Hornsby, Schneider, and Daniels, engages and supports students in the learning process by developing both the conceptual understanding and the analytical skills
necessary for success in mathematics. With the Tenth Edition, the authors adapt to the new ways in which students are learning, as well as the ever-changing classroom environment.
Table of Contents
Supplements Guide
1. Trigonometric Functions
1.1 Angles
1.2 Angle Relationships and Similar Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions
2. Acute Angles and Right Triangles
2.1 Trigonometric Functions of Acute Angles
2.2 Trigonometric Functions of Non-Acute Angles
2.3 Finding Trigonometric Function Values Using a Calculator
2.4 Solving Right Triangles
2.5 Further Applications of Right Triangles
3. Radian Measure and the Unit Circle
3.1 Radian Measure
3.2 Applications of Radian Measure
3.3 The Unit Circle and Circular Functions
3.4 Linear and Angular Speed
4. Graphs of the Circular Functions
4.1 Graphs of the Sine and Cosine Functions
4.2 Translations of the Graphs of the Sine and Cosine Functions
4.3 Graphs of the Tangent and Cotangent Functions
4.4 Graphs of the Secant and Cosecant Functions
4.5 Harmonic Motion
5. Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
6. Inverse Circular Functions and Trigonometric Equations
6.1 Inverse Circular Functions
6.2 Trigonometric Equations I
6.3 Trigonometric Equations II
6.4 Equations Involving Inverse Trigonometric Functions
7. Applications of Trigonometry and Vectors
7.1 Oblique Triangles and the Law of Sines
7.2 The Ambiguous Case of the Law of Sines
7.3 The Law of Cosines
7.4 Vectors, Operation, and the Dot Product
7.5 Applications of Vectors
8. Complex Numbers, Polar Equations, and Parametric Equations
8.1 Complex Numbers
8.2 Trigonometric (Polar) Form of Complex Numbers
8.3 The Product and Quotient Theorems
8.4 De Moivre's Theorem; Powers and Roots of Complex Numbers
8.5 Polar Equations and Graphs
8.6 Parametic Equations, Graphs, and Applications
Appendix A. Equations and Inequalities
Appendix B. Graphs of Equations
Appendix C. Functions
Appendix D. Graphing Techniques
Solutions to Selected Exercises
Answers to Selected Exercises
Index of Applications
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Here's the question you clicked on:
If P(A or B) = 0.7, P(A) = 0.3 and P(B) = 0.8, determine P(A and B). 0.21 0.56 0.10 or 0.40
Best Response
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\[P(A \cup B)=P(A)+P(B)-P(A\cap B)\] you know every number but \[P(A\cap B) so you can solve for it. is that ok?
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is replying to Can someone tell me what button the professor is hitting...
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Adding and Subtracting Polynomials
Re: Adding and Subtracting Polynomials
Let me look at the analogue of what you are doing. When you are playing pleasant tunes and showing pleasant images, that you had to compose yourself, to your boss, that relaxes you, too?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
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A limit involving a regularizing kernel
up vote 2 down vote favorite
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
Somewhere in the middle of it, I'm stuck at proving a certain limit equality. Maybe it's obvious and I can't get it.
$$ \int_{(\Bbb{R})} \left(\chi(\xi,u)\star \varphi_\varepsilon \right)^2d \xi \to |u| \text{ in } {L}^1_{loc} $$
where $\varphi_\varepsilon(t,x)$ is a regularizing kernel, $u$ satisfies $$\partial_t u +\text{div}A(u)=0 \text{ and }\text{ in }\mathcal{D}^\prime((o,\infty)\times \Bbb{R}^d) $$ and
$$ \chi(\xi,u)=\begin{cases} 1 & {0\leq \xi\leq u} \newline -1 & u \leq \xi \leq 0 \newline 0 & \text{otherwise} \end{cases}$$
Thank you.
[edit] Sorry. I forgot to mention that $u \in L^1_{loc}$.
ap.analysis-of-pdes fa.functional-analysis
2 (a) I think if the equality were to hold, it will hold regardless of the equation. For continuous functions it is easy to check that the limit holds even pointwise, since $|u| = \int_{\mathbb{R}}
\chi(\xi,u)^2 d\xi$ and it is just pushing integrals around and justifying a few Fubini's. (b) If $u\in D'$, you don't necessarily have that $|u|$ is some well-defined object in $L^1_{loc}$. For
example, what is $|\delta'|$? So perhaps what the equation is doing is giving $u$ a priori some regularity so that the absolute value of $u$ is well-defined. – Willie Wong Feb 29 '12 at 14:16
2 Ah, your desired expression appear in the "Proof for (2.2)" step. In (2.2), which is part of the statement of Theorem 2.1. So by hypothesis $u$ is $L^1_{loc}$. So you can probably use the result
for continuous functions and Luzin's theorem to get convergence. – Willie Wong Feb 29 '12 at 14:29
@Willie Wong: Thank you for your comment. I'm starting to understand now. – Beni Bogosel Feb 29 '12 at 16:38
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A new inequality for the von Neumann entropy
Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived
from it. I will describe some work with Andreas Winter in which we prove a new inequality for the von Neumann entropy which is independent of strong subadditivity. This work sheds light on extremal
types of entanglement for multi-party quantum states.
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Odds and Probability
Identifying the odds of something happening is a little different that calculating the probability. It is written as a ratio; however, it is not written as a fraction.
The odds in favor of an event is the ratio of the number of ways the outcome can occur to the number of ways the outcome cannot occur.
# of ways the event CAN occur : # of ways the event CANNOT occur.
This is actually a lot easier than probability. So, let's take a look at an example.
Example 1
Odds and probability is pretty easy! Just remember to use a colon instead of a fraction. Also, remember that you are comparing the number of ways the outcome can occur to the number of ways the
outcome cannot occur (not the total outcomes).
Are you ready to try a problem on your own?
Practice Problem
Answer Key
Great job with odds and probability. This unit is almost over!
Algebra Class Courses
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Travelling Salesman
Problem Description
The travelling salesman problem (TSP) can be formulated as follows: Consider a weighted graph G with n nodes, labeled 1,2,...,n. The nodes represent cities, and the edges represent the roads between
the cities. Each edge from i to j has weight or cost C(i,j), representing the length of the road. The problem is to find the shortest tour that visits all the cities exactly once (except the starting
city, which is also the terminating city). This problem is described in Section 4.7 of [7]. The problem is determined by the cost matrix C, where C(i,j) = C(j,i) (possibly infinity).
This MATLAB program gives the best found tour via the CE method.
Call the program from MATLAB, with the following syntax:
Example: tour=tsp(1000,0.05,0.8,C,0)
where C is a matrix of costs (lengths) between nodes.
N - Number of samples to generate each round
rho - fraction of best samples to take
alpha - smoothing parameter
C - C(i,j) is the cost between node i and node j
traj - 0, node placements
1, node transitions
pi - the best tour found
gtsp0.m This program is used internally to generate tours via node placements.
gtsp1.m This program is used internally to generate tours via node transitions.
stsp.m This program is used internally to evaluate the performance of a particular tour.
minitsp.m This is a program which generates a smaller TSP problem (with a specified number of nodes) from a larger one. This idea is mentioned in Remark 4.13 in [7].
cetoolbox www user
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shortest path around cylinder
August 11th 2008, 04:10 PM #1
Jul 2008
shortest path around cylinder
An ant starts at a point $P$ on the bottom edge of a right circular cylinder of radius $R$ and height $H$. If the ant makes $n$ complete circuits around the cylinder and finishes at a point at
the top edge directly above its starting point, find, with justification, the length of its shortest possible path.
An ant starts at a point $P$ on the bottom edge of a right circular cylinder of radius $R$ and height $H$. If the ant makes $n$ complete circuits around the cylinder and finishes at a point at
the top edge directly above its starting point, find, with justification, the length of its shortest possible path.
Slit the cylinder vertically from the ants starting point and flatten out. Now take n copies of the flattened cylinder and place them side by side.
The shortest path on the cylinder is equivalent to a diagonal on the rectangle made out of n flattened copies of the cylinder.
August 11th 2008, 08:56 PM #2
Grand Panjandrum
Nov 2005
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Lafayette Hill Algebra Tutor
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A question on Wiener Process
up vote 1 down vote favorite
Suppose we have a Wiener process $W$, and $U_x$ is the amount of time spent below the level of $x$ during the time interval $(0,1)$. How can I calculate the probability density function of $U_x$.
Does this has anything to do with local time or arc sine law?
Stephan, could you put it in more details please? Is it something like $\frac{1}{\pi\sqrt{u(1-u)}}e^{-\frac{x^2}{2u}}$?
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Laguna Hills Algebra Tutor
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The Lifting of the Veil in the Operations of Calculation
• The establishment of the Berber-Muslim dynasty of the Almohads in North Africa and Andalusia in the 12th century coincided with the decline in scientific advances in many fields of knowledge,
including medicine. This was not the case with mathematics, and the treatise preserved in this manuscript together with other works by the same author stand as clear proof of the liveliness of
this field under the rule of the Almohads and of the Marinid dynasty that followed. Abū ‛Abbās Ahmad Ibn al-Bannā was born in the second half of the 13th century in Marrakesh and spent most of
his life working as a teacher in the city of Fez. His interests were not limited to mathematics: he produced an introduction to Euclidean geometry and compiled astronomical tables for the
calculation of planetary longitudes, together with treatises on logic, linguistics, and rhetoric. Moreover, he was an active member of the Sufi confraternity of the Hazmīrīya. The biographer
Ahmād ibn Šātir (died, 1375) went so far as to attribute to al-Bannā the performance of miracles. The present work is an extensive commentary in two parts on another treatise by al-Bannā, the
Talhīs ‘amal al-hisāb (The abridgement of the operations of calculation). The complexity of that work was recognized by the famous 14th-century historiographer Ibn Haldūn, who described the
treatise in his Muqaddima (Introduction) as “very difficult for beginners, because of its rigor and the strict sequence of the demonstrations.” In the present commentary, Ibn al-Bannā explains in
detail complex mathematical operations, including combinational computation, extended fractions, arithmetic series, and binomial coefficients and provides a philosophical and theological
framework for his mathematical discourse.
Date Created
Subject Date
Title in Original Language
• كتاب رفع الحجاب عن وجوه اعمال الحساب
Time Period
Additional Subjects
Type of Item
Physical Description
• 54 leaves (25 lines), bound : paper ; 21 x 14 centimeters
• Paper: yellowed cream, with watermarks, in good condition; loose from spine. Red and black ink on title page, text is principally black, with some words in red. Occasional diacritical marks, some
marginal notes, catchwords on rectos. Binding: modern cardboard covered with cloth; leather spine. Naskhī script with titles of chapters in Thuluth.
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Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material,
please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 37
37 3. Forced yield (FY): Pedestrians initiates crossing before the gap is defined as the time needed to cross the width of the vehicle initiated the yield, forcing the driver to slow down crosswalk
at a walking speed of 3.5 ft/s while allowing for a 2-s by entering the crosswalk. safety buffer. This 2-s buffer allows for some pedestrian reaction 4. Crossable gap (CG): Pedestrian encounters a
gap large time before initiating the crossing as well as the safety buffer enough to safely cross the street without the need for a between a completed crossing and the next vehicle arrival. driver
yield. A crossable gap is defined as the time needed Similar to the yield statistics, three gap-related parameters to cross at an assumed walking speed plus a safety buffer. are defined, but only the
last two are used in the analysis: 5. Non-crossable gap (non-CG): Pedestrian encounters a gap between vehicles shorter than the crossable gap threshold. · P(CG): The probability of a gap being
crossable, defined as the number of crossable gaps divided by the number of For event categories 13, the event is associated with the crossable and non-crossable gaps encountered. vehicle (driver)
executing the yielding maneuver. For event · P(CG_ENC): The probability of encountering a CG event, categories 4 and 5, the event is associated with the second of the defined as the number of
crossable gaps divided by the two vehicles that define the gap (the vehicle that "closes" total of all pedestrianvehicle events encountered by the the gap). Conceptually, event type 5 also
represents a vehi- pedestrian. cle that did not yield to the pedestrian. The sum of event · P(GO|CG): The probability of crossable gap utilization, types 1 through 5 corresponds to the total number
of vehicles defined as the number of crossings in a CG divided by the encountered by the pedestrian. The five event categories are total number of CGs encountered by the pedestrian. The used to
define the operational variables below. gap utilization concept is related to other traffic engineer- ing studies that evaluate the numbers of accepted and rejected gaps. Performance Measures Using
the five event outcomes defined above, the NCHRP A walking speed of 3.5 ft/s in the determination of the cross- Project 3-78A analysis framework defines performance mea- able gap threshold is based
on the proposed walking speed in sures to describe the four accessibility criteria: crossing oppor- the latest release of the MUTCD (FHWA 2009). This estimate tunity, opportunity utilization, delay,
and safety. represents the 15th percentile walking speed of the general The first analysis component describes the availability pedestrian population, which is a conservative estimate. As a and
utilization of yields. Initially, all three yield types (rolling, result, the crossable gap threshold used in this project is also stopped, and forced) are combined, but they can also be broken
conservative. It is expected that most sighted pedestrians out for a more detailed assessment. Three performance mea- would likely accept gaps that are shorter than this calculated sures related to
yielding are defined, although only the latter threshold, and this may also be observed for some of the blind two are used in the analysis: study participants. The calculated crossable gap threshold
may therefore introduce a potential analysis bias: The probability · P(Yield): The probability of a driver yielding, defined as the of encountering a crossable gap, P(CG_ENC), may be lower number of
yields divided by the total number of drivers that than what would be perceived by a pedestrian who readily could have yielded. walks at a faster speed and therefore utilizes shorter gaps. Sim- · P
(Y_ENC): The probability of encountering a yield event, ilarly, the probability of utilizing a crossable gap is expected to defined as the number of yields divided by the total of all be high, given
that the threshold for what is considered cross- pedestrianvehicle events encountered by the pedestrian able is high for 85% of the general pedestrian population. until he/she completes the
crossing. Nonetheless, the chosen walking speed is considered a reason- · P(GO|Y): The probability of yield utilization, defined as the able assumption in light of national policy documents like the
number of crossings in a yield divided by the total number MUTCD, and in light of the fact that the threshold is consis- of yields encountered by the pedestrian. tently applied to all sites to allow
for a relative comparison. The same crossable gap definition is also proposed in Chap- The P(Y_ENC) performance measure is different from the ter 6, which talks about extension of the research
results. traditionally used probability of yielding, P(Yield), since it is The combined effect of gap and yield availability and uti- calculated on the basis of all pedestrianvehicle events and
lization is reflected in the delay experienced by pedestrians. not just potential yielders. Figure 14 and the associated dis- Three delay performance measures are defined in the analysis: cussion
provide an example that illustrates the distinction between the two measures. · Observed Delay per Leg (s): The pedestrian delay in sec- The analysis next considers the availability and utilization
onds, defined as the time difference between when the trial of crossable gaps. For the purpose of this analysis, a crossable started and when the pedestrian initiated the crossing.
OCR for page 37
Start of Trial MEASURES Veh. # 1 2 2 3 4 4 5 6 7 7 8 8 9 10 # of Events Cross Yield Cross Cross Yield Cross Cross Cross Yield Cross Yield Cross Cross Cross = 10 Vehicles Vehicle Events (n=10) GO # of
Crossings Pedestrian =1 Crossing Events (n=1) NY Y NY Y NY NY Y Y NY P(Yield) Yield Events = 4/(4+5) = 4/9 (n=9) = 44.4% non-CG CG non-CG CG non-CG CG P(CG) Gap Events = 3/(3+3) = 3/6 (n=6) = 50.0%
YY Y Y P(Y_ENC) Yield Encounters = 4/10 (n=10) = 40.0% CG CG CG P(CG_ENC) CG Encounters = 3/10 (n=10) = 30.0% Rej. Y Rej. Y Rej. Y Rej. Y P(GO|Yield) Yield Utilization = 0/4 (n=4) = 0.0% Rej. CG Rej.
CG Utlz. CG P(GO|CG) CG Utilization = 1/3 (n=4) = 33.3% Delay (sec.) = t(crossing) - t(start Delay trial) First Opportunity Delay>Min (sec.) = t(crossing) - t(first Delay>Min. opportunity) P(Crossing
Opportunity) P(Crossing) = P(Y_ENC) + P(CG_ENC) = P(Y_ENC)*P(GO|Yield) + P(CG_ENC)*P(GO|CG) = 4/10 + 3/10 = 7/10 = (4/10)*0% + (3/10)*33.3% = 70% = 10% This figure shows an illustrative example of
how pedestrianvehicle events are determined in the NCHRP Project 3-78A analysis framework. The figure shows the hypothetical interaction of one pedestrian and 10 vehicles and translates the
different yield and gap events into the performance measures discussed in this chapter. This process is described in detail in the text. Figure 14. Graphical illustration of variable definitions with
example (source: Schroeder and Rouphail 2010).
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Sampling Distribution
Important Questions to consider
• How does the sample size, N, effect the rate at which the sampling distribution(param=sample mean) approaches Normal distribution?
• How does the number of samples taken effect the speed of convergence of the sampling distribution(param=the sample mean) to Normal distribution?
• Are there Central Limit Theorem (CLT) effects generally present for other parameter estimates (e.g., median, SD, range, etc.)? Why?
• Does the shape of the original distribution effect the speed of convergence of the sampling distribution(param=the sample mean) to Normal distribution?
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Summary: A Lower Bound on the Probability of a Finite Union
of Events with Applications \Lambda
Hongyan Kuai, Fady Alajaji and Glen Takahara
Mathematics and Engineering
Department of Mathematics and Statistics
Queen's University
Kingston, Ontario, Canada K7L 3N6
A new lower bound on the probability of
P(A 1 [ \Delta \Delta \Delta [ AN ) is established in terms of only
the individual event probabilities P(A i )'s and
the pairwise event probabilities P(A i `` A j )'s.
This bound is shown to be always at least
as good as two similar lower bounds, one by
de Caen (1997) and the other by Dawson and
Sankoff (1967). Numerical examples for the
computation of this inequality are also pro
vided. Finally, the application of this re
sult to the symbol error probability of an un
coded communication system used in conjunc
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This Article
Bibliographic References
Add to:
Symbolic and Geometric Connectivity Graph Methods for Route Planning in Digitized Maps
May 1992 (vol. 14 no. 5)
pp. 549-565
ASCII Text x
P.D. Holmes, E.R.A. Jungert, "Symbolic and Geometric Connectivity Graph Methods for Route Planning in Digitized Maps," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no.
5, pp. 549-565, May, 1992.
BibTex x
@article{ 10.1109/34.134059,
author = {P.D. Holmes and E.R.A. Jungert},
title = {Symbolic and Geometric Connectivity Graph Methods for Route Planning in Digitized Maps},
journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},
volume = {14},
number = {5},
issn = {0162-8828},
year = {1992},
pages = {549-565},
doi = {http://doi.ieeecomputersociety.org/10.1109/34.134059},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Pattern Analysis and Machine Intelligence
TI - Symbolic and Geometric Connectivity Graph Methods for Route Planning in Digitized Maps
IS - 5
SN - 0162-8828
EPD - 549-565
A1 - P.D. Holmes,
A1 - E.R.A. Jungert,
PY - 1992
KW - symbolic connectivity; obstacle avoidance; geometric connectivity graph methods; route planning; digitized maps; spatial reasoning; 2D route planning; heuristic symbolic processing; A* search;
inference rules; knowledge structure; hierarchical data structure; computational geometry; graph theory; heuristic programming; planning (artificial intelligence); search problems; spatial
reasoning; symbol manipulation
VL - 14
JA - IEEE Transactions on Pattern Analysis and Machine Intelligence
ER -
The results of research involving spatial reasoning within digitized maps are reported, focusing on techniques for 2D route planning in the presence of obstacles. Two alternative approaches to route
planning are discussed, one involving heuristic symbolic processing and the other employing geometric calculations. Both techniques employ A* search over a connectivity graph. The geometric system
produces a simple list of coordinate positions, whereas the symbolic system generates a symbolic description of the planned route. The symbolic system achieves this capability through the use of
inference rules that can analyze and classify spatial relationships within the connectivity graph. The geometric method calculates an exact path from the connectivity information in the graph. Thus,
the connectivity graph acts both as a knowledge structure on which spatial reasoning can be performed and as a data structure supporting geometrical calculations. An extension of the methodology that
exploits a hierarchical data structure is described.
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238-245, vol. 1.
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[14] E. Jungert, "Run length code as an object-oriented spatial data structure," inProc. IEEE Workshop Languages Automat.(Singapore), Aug. 1986, pp. 66-70.
[15] E. Jungert, "Extended symbolic projections as a knowledge structure for spatial reasoning and planning," inPattern Recognition(J. Kittler, Ed.). New York: Springer Verlag, 1988.
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[17] E. Jungert and S. -K. Chang, "An algebra for symbolic image manipulation and transformation," inVisual Database Systems(T. L. Kunii, Ed.). Amsterdam: North-Holland, 1989.
[18] E. Jungert and S. -K. Chang, "An image algebra for pictorial data manipulation," to be published inComput. Vision Graphics Image Processing (CVGIP): Image Understanding.
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1990, pp. 526-533.
[21] E. Jungert and P. D. Holmes, "A hierarchical knowledge structure for heuristic path planning," inProc. Second Int. Conf. Intell. Autonomous Syst.(Amsterdam), Dec. 1989, pp. 230-240.
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[32] J. Pearl,Heuristics: Intelligent Search Strategies for Computer Problem Solving. Reading, Mass: Addison-Wesley, 1984.
[33] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
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Index Terms:
symbolic connectivity; obstacle avoidance; geometric connectivity graph methods; route planning; digitized maps; spatial reasoning; 2D route planning; heuristic symbolic processing; A* search;
inference rules; knowledge structure; hierarchical data structure; computational geometry; graph theory; heuristic programming; planning (artificial intelligence); search problems; spatial reasoning;
symbol manipulation
P.D. Holmes, E.R.A. Jungert, "Symbolic and Geometric Connectivity Graph Methods for Route Planning in Digitized Maps," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 5,
pp. 549-565, May 1992, doi:10.1109/34.134059
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Trevor Hastie presents glmnet: lasso and elastic-net regularization in R
May 9, 2013
By Joseph Rickert
by Joseph Rickert
Even a casual glance at the R Community Calendar shows an impressive amount of R user group activity throughout the world: 45 events in April and 31 scheduled so far for May. New groups formed last
month in Knoxville, Tennessee (The Knoxville R User Group: KRUG) and Sheffield in the UK (The Sheffield R Users). An this activity seems to be cumulative. This month, the Bay Area R User’s Group (
BARUG) expects to hold its 52nd and 53rd meet ups while the Sydney Users of R Forum (SURF) will hold its 50th. Everywhere R user groups are sponsoring high quality presentations and making them
available online, but the Orange County R User Group is pushing the envelope with respect to sophistication and reach. Last Friday, I attended a webinar organized by this group where Professor Trevor
Hastie of Stanford University presented Sparse Linear Models with demonstrations using GLMNET. This was a world-class presentation and quite a coup for Orange County to have Professor Hastie present.
The glmnet package written Jerome Friedman, Trevor Hastie and Rob Tibshirani contains very efficient procedures for fitting lasso or elastic-net regularization paths for generalized linear models. So
far the glmnet function can fit gaussian and multiresponse gaussian models, logistic regression, poisson regression, multinomial and grouped multinomial models and the Cox model. The efficiency of
the glmnet algorithm comes from using cyclical coordinate descent in the optimization process and from Jerome Friedman's underlying Fortran code.
Although Professor Hastie’s presentation was primarily concerned with fitting models for the wide problem (the number of explanatory variables is much larger than the number of observations) the
lasso and elastic-net algorithms are just as applicable to data sets with large numbers of observations. It is likely that in the future we will see glmnet implementations for variable selection on
datasets with thousands of variables and hundreds of millions of observations. The following graph shows the regularization paths for the coefficients of a model fit the HIV data from one Professor
Hastie’s examples.
Each curve represents a coefficient in the model. The x axis is a function of lambda, the regularization penalty parameter. The y axis gives the value of the coefficient. The graph shows how the
coefficients “enter the model” (become non-zero) as lambda changes. The following code, based on an example from the webinar, produces the plot and also shows how easy it is to perform
library(glmnet) # load the package
load("hiv.rda") # HIV data
class(hiv.train) # The data are stored as a list
names(hiv.train) # The names of the list elements are x and y
dim(hiv.train$x) # The explanatory data consists of 704 observations of
# 208 binary mutation variables
head(hiv.train[[1]]) # Look at the explanatory data
head(hiv.train[[2]]) # Look at the response data: changes in susceptibility to antiviral drugs
fit=glmnet(hiv.train$x,hiv.train$y) # fit the model
plot(fit,xvar="lambda", main="HIV model coefficient paths") # Plot the paths for the fit
fit # look at the fit for each coefficient
cv.fit=cv.glmnet(hiv.train$x,hiv.train$y) # Perform cross validation on the fited model
plot(cv.fit) # Plot the mean sq error for the cross validated fit as a function
# of lambda the shrinkage parameter
# First vertical line indicates minimal mse
# Second vertical line is one sd from mse: indicates a smaller model
# is "almost as good" as the minimal mse model
tpred=predict(fit,hiv.test$x) # Predictions on the test data
mte=apply((tpred-hiv.test$y)^2,2,mean) # Compute mse for the predictions
points(log(fit$lambda),mte,col="blue",pch="*") # overlay the mse predictions on the plot
legend("topleft",legend=c("10 fold CV","Test"),pch="*",col=c("red","blue"))
Created by Pretty R at inside-R.org
Don’t be content with this partial example. Professor Hastie and The Orange County R User Group have graciously made the slides, code and data available at this link. The webinar is well worth
watching in its entirety.
As you might expect, Professor Hastie gives a masterful presentation: lucid, clear and succinct. This is inspite of the fact that Professor Hastie begins the presentation by commenting that it was
his first webinar ever and that he was a little uncomfortable talking to his screen. (I think anyone who has ever given a webinar can relate to this: you talk to the screen and no energy from the
audience comes back. Nothing is more disruptive to efforts to be enthusiastic than silence.) Nevertheless, Professor Hastie presents a difficult topic with a clarity that carries his audience along,
and he is completely unphased by the inevitable glitch. Watch how he handles the upside down slide. You can download his slides, R scripts and data from the link below.
Trevor Hastie: Sparse Linear Models, with demonstrations using GLMNET
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NWT Literacy Council - Adult Literacy - What's the Problem? For Adults Who Like A Challenge Solving Word Problems - Page 24
PROBLEMS WITH NUMBERS
After each problem tell how to work it by writing A, S, M, or D. If there are two steps, write the letter for each step. Then do the work at the right:
1. Last week when Fred was helping Mr. King he worked 2 hours on Thursday, 2 hours on Friday, and 5 hours on Saturday. If Mr. King paid Fred $7.25 an hour, how much did Fred earn?
2. Jim saves $5.00 from his salary each month. In how many weeks will he have $75.00?
3. Tom and Ned had $7.86 to divide equally. Ned spent $.25 of his share of the money. How much did Ned have left?
4. Ed made a bowling score of 215 last week. His score this week was 198. How much more did he make last week?
5. Last week Joe sold 28 papers on Wednesday, 36 papers on Thursday, 32 papers on Friday, and 51 papers on Saturday. How many papers did he sell altogether on these four days?
6. Two boys picked 5 pails of Saskatoons and sold them for $4.98 each. They divided the money equally. How much did each boy get?
7. Ned had $8.49 when he left home this morning. He paid $2.25 for bus fares, $4.98 for lunch, and $.50 for a newspaper. How much does he have left after paying for these things?
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WeBWorK Help: Interval Notation
Using Interval Notation
• If an endpoint is included, then use [ or ]. If not, then use ( or ). For example, the interval from -3 to 7 that includes 7 but not -3 is expressed (-3,7].
• For infinite intervals, use Inf for ∞ (infinity) and/or -Inf for -∞ (-Infinity). For example, the infinite interval containing all points greater than or equal to 6 is expressed [6,Inf).
• If the set includes more than one interval, they are joined using the union symbol U. For example, the set consisting of all points in (-3,7] together with all points in [-8,-5) is expressed
• If the answer is the empty set, you can specify that by using braces with nothing inside: { }
• You can use R as a shorthand for all real numbers. So, it is equivalent to entering (-Inf, Inf).
• You can use set difference notation. So, for all real numbers except 3, you can use R-{3} or (-Inf, 3)U(3,Inf) (they are the same). Similarly, [1,10)-{3,4} is the same as [1,3)U(3,4)U(4,10).
• WeBWorK will not interpret [2,4]U[3,5] as equivalent to [2,5], unless a problem tells you otherwise. All sets should be expressed in their simplest interval notation form, with no overlapping
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[FOM] The Gold Standard
Harvey Friedman friedman at math.ohio-state.edu
Tue Feb 21 01:58:49 EST 2006
On 2/21/06 12:26 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:
> I have said that impredicative mathematics has no clear
> philosophical basis, whereas predicative mathematics has
> a clear philosophical basis. Here's a sharper formulation:
> Impredicative systems like ZFC and Z_2 lack canonical models.
> Predicative systems like ACA_0 and PA have canonical models.
Set theorist: ZFC has a canonical model. It is called (V,epsilon).
Real line realist: Z_2 has a conical model. It is built out of the acutal
real line.
Finitist: PA has no model at all.
> Is there any reason to expect that ZFC, or even Z_2, has a
> natural model *unless one is a platonist*?
Finitist: Is there any reason to expect aht PA has a natural model, or any
model, *unless one is an infinitist* who believes in such nonsense as
"completed infinite totalities"?
> If one is a platonist and believes in the objective existence
> of a well-defined universe of sets then one simply has to argue
> that ZFC holds in that universe.
Set theorist: This is evident by reflecting on the picture of the
hereditarily finite sets.
>But if one is not a platonist
> it seems that any legitimate model must be in some (possibly
> loose) sense constructed, and then you have a basic difficulty
> in capturing impredicativity.
Set theorist: I don't know how to convert predicativists out of their silly
position to my evident position.
Predicativists: I don't know how to convert set theorists out of their silly
position to my evident position.
> However, that
> doesn't address the question of there being natural models.
> (It also seems sketchy on supporting the truth of arithmetical
> theorems provable in the system.)
Set theorist working with medium large cardinals: But we have natural
models. These are the so called inner models in inner model theory. And we
are trying to find inner models for the large large cardinals. We have to
study them hard in order to find them. Once we find them, all will be clear,
we will have our natural models.
>> ZFC. GOLD STANDARD (rightly or wrongly). Justification:
>> extrapolation from finite set theory.
> Just not a very convincing justification, in my opinion.
> As Arnon Avron points out, lots of properties of finite sets
> fail disastrously for infinite sets.
I just replied to Avron. Let's get to work!
>> Z_2. Justification: realist view of the real number system.
> Fine, if one is a platonist.
The mathematical community likes real numbers in this sense, don't they?
> So I ask: is there any non-platonist justification of the
> assertion that Z_2 has a natural model?
Platonist: Is that your definition of Platonism? Then tautologically, the
answer may well be no. But so what? Why aren't you a Platonist?
Finitist: We know that PA does not have a model. Is there any non-platonist
justification of the assertion that PA is consistent?
Weaver, 2/21/06, 12:34AM:
> I flatly "cannot"
> clearly conceive of possible worlds occupied by structures of type
> <N-union-P(N),epsilon>, but rather that I know of no plausible way
> to conceive of such a world, and think I have good reasons for
> believing this can't be done.
Easy to "conceive". Let A be a FINITE set of absolutely enormous size. Just
ponder P(A). Then pretend that A is so big, you just can't list it. This is
in fact, reality - you can't actually list it.
Weaver, 2/21/06, 12:44AM
> Yes, but Weyl didn't suggest any philosophical reason for
> stopping at arithmetic definability. If I remember right
> he was very clear about the possibility of going further
> but felt that for his purposes (developing 19th century
> real analysis in a predicatively acceptable way) there
> was no need to do this.
Actually, the amount of core math or normal math you pick up by doing higher
but stopping at various proposals for the limits of predicativity is, by
your standards, rather minimal even today.
>In other words his criterion for
> stopping at ACA_0 was esthetic.
I would surmise that Weyl was largely concerned, at least at some point,
with mathematical practice, and not philosophy. So it's not esthetic.
> Friedman claimed that it is "child's play" to come up with
> a coherent foundational stance corresponding to ACA_0. I'd
> still like to see one.
It appeared in my original Gold Standard posting
Harvey Friedman
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Next: About this document Up: CIRCUMSCRIPTION-A FORM OF NONMONOTONIC Previous: REMARKS AND ACKNOWLEDGEMENTS
Amarel, Saul (1971). On Representation of Problems of Reasoning about Actions, in D. Michie (ed.), Machine Intelligence 3, Edinburgh University Press, pp. 131-171.
Chandra, Ashok (1979). Personal conversation, August.
Davis, Martin (1980). Notes on the Mathematics of Non-Monotonic Reasoning, Artificial Intelligence 13 (1, 2), pp. 73-80.
Hayes, Patrick (1979). Personal conversation, September.
Hewitt, Carl (1972). Description and Theoretical Analysis (Using Schemata) of PLANNER: a Language for Proving Theorems and Manipulating Models in a Robot, MIT AI Laboratory TR-258.
McCarthy, John (1959). Programs with Common Sense, Proceedings of the Teddington Conference on the Mechanization of Thought Processes, London: Her Majesty's Stationery Office. (Reprinted in this
volume, pp. 000-000).
McCarthy, John and Patrick Hayes (1969). Some Philosophical Problems from the Standpoint of Artificial Intelligence, in B. Meltzer and D. Michie (eds), Machine Intelligence 4, Edinburgh University.
(Reprinted in B. L. Webber and N. J. Nilsson (eds.), Readings in Artificial Intelligence, Tioga, 1981, pp. 431-450; also in M. J. Ginsberg (ed.), Readings in Nonmonotonic Reasoning, Morgan Kaufmann,
1987, pp. 26-45. Reprinted in (McCarthy 1990).
McCarthy, John (1977). Epistemological Problems of Artificial Intelligence, Proceedings of the Fifth International Joint Conference on Artificial Intelligence, M.I.T., Cambridge, Mass. (Reprinted in
B. L. Webber and N. J. Nilsson (eds.), Readings in Artificial Intelligence, Tioga, 1981, pp. 459-465; also in M. J. Ginsberg (ed.), Readings in Nonmonotonic Reasoning, Morgan Kaufmann, 1987, pp.
46-52. Reprinted in (McCarthy 1990).
McCarthy, John (1979a). Ascribing Mental Qualities to Machines , Philosophical Perspectives in Artificial Intelligence, Martin Ringle, ed., Humanities Press. Reprinted in (McCarthy 1990).
McCarthy, John (1979b). First Order Theories of Individual Concepts and Propositions in Michie, Donald (ed.) Machine Intelligence 9, Ellis Horwood. Reprinted in (McCarthy 1990).
McCarthy, John (1990). Formalizing Common Sense, Ablex.
Reiter, Raymond (1980). A Logic for Default Reasoning, Artificial Intelligence 13 (1, 2), pp. 81-132.
Sussman, G.J., T. Winograd, and E. Charniak (1971). Micro-Planner Reference Manual, AI Memo 203, M.I.T. AI Lab.
Circumscription and the nonmonotonic reasoning formalisms of McDermott and Doyle (1980) and Reiter (1980) differ along two dimensions. First, circumscription is concerned with minimal models, and
they are concerned with arbitrary models. It appears that these approaches solve somewhat different though overlapping classes of problems, and each has its uses. The other difference is that the
reasoning of both other formalisms involves models directly, while the syntactic formulation of circumscription uses axiom schemata. Consequently, their systems are incompletely formal unless the
metamathematics is also formalized, and this hasn't yet been done.
However, schemata are applicable to other formalisms than circumscription. Suppose, for example, that we have some axioms about trains and their presence on tracks, and we wish to express the fact
that if a train may be present, it is unsafe to cross the tracks. In the McDermott-Doyle formalism, this might be expressed
where the properties of the predicate on are supposed expressed in a formula that we may call Axiom(on). The M in (1) stands for ``is possible''. We propose to replace (1) and Axiom(on) by the schema
where Axiom(on) together with on(train,tracks) has a model assuming that Axiom(on) is consistent. Therefore, the schema (2) is essentially a consequence of the McDermott-Doyle formula (1). The
converse isn't true. A predicate symbol may have a model without there being an explicit formula realizing it. I believe, however, that the schema is usable in all cases where the McDermott-Doyle or
Reiter formalisms can be practically applied, and, in particular, to all the examples in their papers.
(If one wants a counter-example to the usability of the schema, one might look at the membership relation of set theory with the finitely axiomatized Gödel-Bernays set theory as the axiom.
It appears that such use of schemata amounts to importing part of the model theory of a subject into the theory itself. It looks useful and even essential for common sense reasoning, but its logical
properties are not obvious.
We can also go frankly to second order logic and write
Second order reasoning, which might be in set theory or a formalism admitting concepts as objects rather than in second order logic, seems to have the advantage that some of the predicate and
function symbols may be left fixed and others imitated by predicate parameters. This allows us to say something like, ``For any interpretation of P and Q satisfying the axiom A, if there is an
interpretation in which R and S satisfy the additional axiom A', then it is unsafe to cross the tracks''. This may be needed to express common sense nonmonotonic reasoning, and it seems more powerful
than any of the above-mentioned nonmonotonic formalisms including circumscription.
The train example is a nonnormal default in Reiter's sense, because we cannot conclude that the train is on the tracks in the absence of evidence to the contrary. Indeed, suppose that we want to wait
for and catch a train at a station across the tracks. If there might be a train coming we will take a bridge rather than a shortcut across the tracks, but we don't want to jump to the conclusion that
there is a train, because then we would think we were too late and give up trying to catch it. The statement can be reformulated as a normal default by writing
but this is unlikely to be equivalent in all cases and the nonnormal expression seems to express better the common sense facts.
Like normal defaults, circumscription doesn't deal with possibility directly, and a circumscriptive treatment of the train problem would involve circumscribing
McDermott, Drew and Jon Doyle (1980). Nonmonotonic Logic I, Artificial Intelligence 13 (1, 2), pp. 41-72.
Reiter, Raymond (1980). A Logic for Default Reasoning, Artificial Intelligence 13 (1, 2), pp. 81-132.
Next: About this document Up: CIRCUMSCRIPTION-A FORM OF NONMONOTONIC Previous: REMARKS AND ACKNOWLEDGEMENTS John McCarthy
Tue May 14 00:04:52 PDT 1996
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- Handbook of Computational Geometry , 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of
arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements
of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and
geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A.
was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was
supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the
Israeli Ac...
- J. Combin. Theory Ser. A , 2004
"... This paper examines the extremal problem of how many 1-entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound,
settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also set ..."
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This paper examines the extremal problem of how many 1-entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a
conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of n-permutations avoiding a fixed
permutation and a related conjecture of Alon and Friedgut [1]. 1
- J. Combin , 2000
"... A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph
is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six co ..."
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A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is
isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and
the number of such Hs is O(c n). The latter part generalizes the Stanley--Wilf conjecture on permutations. Using generalized Davenport--Schinzel sequences, we prove the conjectures with weaker bounds
O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p
, 1996
"... Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the
maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in tim ..."
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Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum
euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(n1+"), for any " ? 0, improving the previous solution of Agarwal et al. [3] by nearly
a factor of O(n).
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turán-type and Ramsey-type extremal problems for
geometric graphs, and discusses their generalizations and applications. ..."
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turán-type and Ramsey-type extremal problems for
geometric graphs, and discusses their generalizations and applications.
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest
subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S
for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most
important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
- IN PAUL ERDÖS, PROC. CONF , 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph
theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
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The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We
survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
, 2001
"... We present a short proof of Füredi's theorem [F] stated in the title. ..."
, 2009
"... In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar
to computational geometers but still foreign to many researchers in (non-geometric) algorithms and data s ..."
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In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to
computational geometers but still foreign to many researchers in (non-geometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures. To
analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object
M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden
substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper
bounds on the size of a P-free object M may be reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure
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Pfenning, Frank - School of Computer Science, Carnegie Mellon University
• Higher-Order Logic Programming Constraint Logic Programming
• Unification via Explicit Substitutions: The Case of Higher-Order Patterns
• Computation and Deduction Frank Pfenning
• Extensions and Applications of Higherorder Unification
• Efficient Resource Management Submitted to Theoretical Computer Science. Comments are welcome!
• Refinement Types as Proof Irrelevance William Lovas and Frank Pfenning
• Avoiding Spurious Causal Dependencies via Proof Irrelevance
• A Linear Logic of Authorization and Knowledge Deepak Garg, Lujo Bauer, Kevin D. Bowers,
• A Learning Algorithm for Localizing People Based on Wireless Signal Strength that Uses Labeled and Unlabeled Data
• A Linear Spine Calculus Iliano Cervesato and Frank Pfenning1
• Verifying Uniqueness in a Logical Framework Penny Anderson1
• Primitive Recursion for Higher-Order Abstract Syntax
• System Description: Twelf --A Meta-Logical Framework for Deductive Systems
• Non-Interference in Constructive Authorization Logic Deepak Garg
• Higher-Order Pattern Complement and the Strict ALBERTO MOMIGLIANO
• Contemporary 1tfathematics Volume 29, 1984
• A Logical Characterization of Forward and Backward Chaining in the Inverse Method
• Using Constrained Intuitionistic Linear Logic for Hybrid Robotic Planning Problems
• Type-Directed Concurrency Deepak Garg and Frank Pfenning
• Implementing the Meta-Theory of Deductive Frank Pfenning and Ekkehard Rohwedder
• The Practice of Logical Frameworks To appear in the proceedings of CAAP'96, Linkoping, Sweden, April 1996
• Partial Polymorphic Type Inference and Higher-Order Unification Frank Pfenning'
• Mode and Termination Checking for Higher-Order Logic Programs
• Problems in Rewriting Applied to Categorical Concepts
• Twelf User's Guide Version 1.3
• Modularity Matters Most Robert Harper
• LEAP: A Language with Eval And Polymorphism Frank Pfenning and Peter Lee
• Optimizing Higher-Order Pattern Unification Brigitte Pientka and Frank Pfenning
• The Occurrence of Continuation Parameters in CPS Terms
• On Equivalence and Canonical Forms in the LF Type Theory
• Modal Types as Staging Specifications for Run-time Code Generation
• On the Unification Problem for Cartesian Closed Categories (Extended Abstract)
• Research on Semantically Based Program-Design Environments
• A Logical Characterization of Forward and Backward Chaining in the Inverse Method
• ARTICLE NO. A Linear Logical Framework1
• Structural Cut Elimination Frank Pfenning
• Unification in a -Calculus with Intersection Types
• A Probabilistic Language based upon Sampling Functions
• de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
• Run-time Code Generation and Modal-ML Philip Wickline Peter Lee Frank Pfenning
• Tutorial on Amy Felty Elsa Gunter
• Decidability Results For Categorical Notions Related To Monads
• Higherorder Unification with Dependent Function Types \Lambda
• HOOTS99 Preliminary Version On proving syntactic properties of CPS programs
• Linear Higher-Order Pre-Unification Iliano Cervesato and Frank Pfenning
• Refinement Types for Logical Frameworks Frank Pfenning
• A concurrent logical framework I: Judgments and properties
• Bibliography on Logical Frameworks Frank Pfenning
• Linear Logical Approximations Robert J. Simmons Frank Pfenning
• [1] Gregor V. Bochmann. Semantics evaluation from left to right. Communications of the ACM,
• Unification and Anti-Unification in the Calculus of Constructions Frank Pfenning
• Automated Techniques for Provably Safe Mobile Code
• Types in Logic Programming Frank Pfenning
• Automated Theorem Proving in a Simple Meta-Logic for LF
• MFPS XV Preliminary Version Relating Natural Deduction and Sequent
• Higher-Order Abstract Syntax Frank Pfenning
• On the Undecidability of Partial Polymorphic Type Reconstruction
• Under consideration for publication in Math. Struct. in Comp. Science A Judgmental Reconstruction of Modal
• Elf: A Language for Logic Definition and Verified Metaprogramming
• Algorithms for Equality and Unification in the Presence of Notational Definitions
• Reasoning About Deductions in Linear Logic (Invited Talk)
• Inductively De ned Types in the Calculus of Constructions
• Dependent Types in Practical Programming (Extended Abstract)
• Electronic Notes in Computer Science 1 (1995) On a Modal -Calculus for S4
• Twelf User's Guide Version 1.3
• Resource Management for the Inverse Method in Linear Logic
• A Focusing Inverse Method Theorem Prover for First-Order Linear Logic
• Systems of Polymorphic Type Assignment in LF Robert Harper
• The MiniML Language Unfortunately one often pays a price for [languages which impose
• Natural Semantics and Some of its Meta-Theory in Elf
• The TPS Theorem Proving System Peter B. Andrews Sunil Issar Dan Nesmith Frank Pfenning
• Efficient Resource Management This paper will appear in the proceedings of the 1996 International Workshop on
• A Framework for Defining Logics Robert Harper* Furio Honselly Gordon Plotkinz
• Problems in Rewriting applied to Categorical Concepts
• Extensions and Applications of Higher-order Unification
• Proof of the Decidability of the Uniform Word Problem for Monads
• Ordered Linear Logic Programming Jeff Polakow and Frank Pfenning 1
• Elf: A Meta-Language for Deductive Systems (System Description)
• A Coverage Checking Algorithm for LF Carsten Schurmann1
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• Natural Deduction for Intuitionistic Non-Commutative Linear Logic
• Focus-preserving Embeddings of Substructural Logics in Intuitionistic Logic University of Pennsylvania
• Reasoning about the Consequences of Authorization Policies in a
• Efficient Intuitionistic Theorem Proving with the Polarized Inverse Method
• Imogen: Focusing the Polarized Inverse Method for Intuitionistic Propositional Logic
• Church and Curry: Combining Intrin-sic and Extrinsic Typing
• Linear Logical Algorithms Robert J. Simmons and Frank Pfenning
• Intuitionistic Letcc via Labelled Deduction Jason Reed 1,2
• Focusing the Inverse Method for Linear Logic Kaustuv Chaudhuri and Frank Pfenning
• Under consideration for publication in J. Functional Programming 1 A Monadic Analysis of Information Flow
• Monadic Concurrent Linear Logic Programming Pablo Lopez
• Under consideration for publication in J. Functional Programming 1 Staged Computation with Names and Necessity
• A Probabilistic Language based upon Sampling Functions Sungwoo Park Frank Pfenning
• Tridirectional Typechecking Joshua Dunfield
• Benjamin C. Pierce. Types and Programming Languages, The MIT Press, Cambridge, Massachusetts, xxi + 623 pp.
• A Symmetric Modal Lambda Calculus for Distributed Computing Tom Murphy VII
• LFM 2004 Preliminary Version Specifying Properties of Concurrent
• Type Assignment for Intersections and Unions in Call-by-Value Languages
• A Type Theory for Memory Allocation and Data Layout Leaf Petersen Robert Harper Karl Crary Frank Pfenning
• LOGICAL FRAMEWORKS--A BRIEF INTRODUCTION FRANK PFENNING (fp+@cs.cmu.edu)
• Human-Readable Machine-Verifiable Proofs for Teaching Constructive Logic
• A Modal Analysis of Staged Computation Rowan Davies
• Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory
• Intersection Types and Computational Effects Rowan Davies
• Termination and Reduction Checking in the Logical Framework
• Properties of Terms in Continuation-Passing Style in an Ordered Logical Framework
• The Relative Complement Problem for Higher-Order Patterns
• A Formalization of the Proof-Carrying Code Architecture in a Linear Logical Framework
• Eliminating Array Bound Checking Through Dependent Types Department of Mathematical Sciences
• A Linear Logical Framework Iliano Cervesato and Frank Pfenning
• A Modal Analysis of Staged Computation Rowan Davies
• Structural Cut Elimination in Linear Logic Frank Pfenning
• Compiler Verification in LF John Hannan
• An Empirical Study of the Runtime Behavior of Higher-Order Logic Programs 1
• A Declarative Alternative to "assert" in Logic Programming
• Refinement Types for ML Tim Freeman
• Logic Programming in the LF Logical Framework Frank Pfenning
• Presenting Intuitive Deductions via Symmetric Simplification
• Towards a Practical Programming Language Based on the Polymorphic Lambda Calculus
• The Ergo Support System: An Integrated Set of Tools for Prototyping
• Single Axioms in the Implicational Propositional Frank Pfenning
• Proof Transformations in Higher-Order Logic
• Analytic and Non-analytic Proofs Frank Pfenning
• A Proof-Carrying File System Deepak Garg
• Proof Theory for Authorization Logic and Its Application to a Practical File System
• Bibliography on Logical Frameworks Frank Pfenning
• Proof-Carrying Code in a Session-Typed Process Calculus
• Functions as Session-Typed Processes Bernardo Toninho1,2
• Invited talk at TLDI'12 Towards Concurrent Type Theory
• Teaching Imperative Programming With Contracts at the Freshmen Level
• Termination in Session-Based Concurrency via Linear Logical Relations (Extended Version)
• Bottom-Up Logic Programming for Multicores Flavio Cruz, Michael P. Ashley-Rollman, Seth Copen Goldstein, Ricardo Rocha, Frank Pfenning
• Under consideration for publication in Math. Struct. in Comp. Science Linear Logic Propositions as Session Types
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15 Subjects: including geometry, reading, English, writing
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Open This for Something Long and Hard
1. Haha, we're doing that crap in class right now.
Haha, we're doing that crap in class right now.
2. math > ??×
math > ??×
3. Long and hard unless you're Asian ;)
Long and hard unless you're Asian ;)
4. I'd be curious to know what the answer actually is. It looks like all of that is legible; any enlightened among us?
I'd be curious to know what the answer actually is. It looks like all of that is legible; any enlightened among us?
5. IMPOSSIBRU!
6. I can't see any of the exponents :/ I doubt it's five though. From experience math problem writers like to use fractions. I think it's because they know I hate fractions.
I can't see any of the exponents :/ I doubt it's five though. From experience math problem writers like to use fractions. I think it's because they know I hate fractions.
7. The answer would most likely contain the variable n.
The answer would most likely contain the variable n.
8. They look like theorems...don't know the names offhand but they aren't really solvable without some given values.
They look like theorems...don't know the names offhand but they aren't really solvable without some given values.
9. It's gonna be a case of "prove that *** = ***"...
It's gonna be a case of "prove that *** = ***"...
10. The awkward moment when you can prove all that mathematic, and it actually isn't 5...
The awkward moment when you can prove all that mathematic, and it actually isn't 5...
11. Proofs are dildoes.
Proofs are dildoes.
12. I think the answer might be "qed".
I think the answer might be "qed".
13. They are Functions (well the middle one definitely is (I would have to look at the graph of the other two to see for sure). They don't really have answers per say. You could either graph these,
or solve them for specific values, but there are no "answers."
They are Functions (well the middle one definitely is (I would have to look at the graph of the other two to see for sure). They don't really have answers per say. You could either graph these,
or solve them for specific values, but there are no "answers."
14. Alexander Sergejev The first and third ones will be "prove by mathematical induction that...", the second one is a cumulation, so you won't be able to plot a graph of it. There's no need to make
yourself look even more stupid by replying.
Alexander Sergejev The first and third ones will be "prove by mathematical induction that...", the second one is a cumulation, so you won't be able to plot a graph of it. There's no need to make
yourself look even more stupid by replying.
15. Bill Hardiman aaaaaaaand we have the first vengeful math geek of the thread.
Bill Hardiman aaaaaaaand we have the first vengeful math geek of the thread.
16. Bill Hardiman You can't graph a function?! Are you serious? I'm sorry ... f(x) means what? For the other two, you aren't actually disagreeing. There is no single answer. You can show that the
mathematical statements are true, and the easiest way to do that is to input values and test it. Perhaps there is some doctorate level way to prove it more theoretically but w/e.
Bill Hardiman You can't graph a function?! Are you serious? I'm sorry ... f(x) means what? For the other two, you aren't actually disagreeing. There is no single answer. You can show that the
mathematical statements are true, and the easiest way to do that is to input values and test it. Perhaps there is some doctorate level way to prove it more theoretically but w/e.
17. Alexander Sergejev In terms of proving them, I'm not sure what level you failed at, but with a AS in maths, I can prove many such terms, using a technique called mathematical induction. As I have
already said. And as for the graph, how about you plot a graph of the sum of the cubes of all real numbers. Enjoy! Whilst it is actually possible, it's a stupid idea to try.
Alexander Sergejev In terms of proving them, I'm not sure what level you failed at, but with a AS in maths, I can prove many such terms, using a technique called mathematical induction. As I have
already said. And as for the graph, how about you plot a graph of the sum of the cubes of all real numbers. Enjoy! Whilst it is actually possible, it's a stupid idea to try.
18. Went no further than differential equations myself.
Went no further than differential equations myself.
19. You guys are all dumb... the asian has already spoken, the answer is 5!
You guys are all dumb... the asian has already spoken, the answer is 5!
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Pittsburgh Teachers Institute
Volume V: Proof in Mathematics: Origin, Practice, Crisis
Proof – What, Why and How?
Adam P. Deutsch
How do we know that what we know is, in fact, true? And how do we communicate what we know so that others will be convinced that it is true? These questions are central to the idea of mathematical
proof and to the motivating force for teaching students of mathematics about proof and how to do proof. The ability to communicate mathematically, more generally, the ability to communicate
logically, forcefully and convincingly is an invaluable skill to which all students should be exposed. Mathematics is an excellent vehicle for teaching students this vital skill.
The curriculum unit described in this document is designed to lead students to an understanding of what proof is, why it is important and how to construct good proofs. Essentially it is a unit about
argument and communication and in this way crosses curricular lines by promoting skills which are universal and useful in the sciences, social sciences and humanities.
Click here to view complete unit.
The Death of Common Sense?
John B. Snodgrass
Most would agree that to possess common sense is a good thing. As a science educator, however, I have found that common sense can be inadequate in explaining or in understanding of many topics. This
paper will examine instances where common sense can be misleading and an inadequate explanatory tool. Explanations and proofs that go beyond common sense are presented to explain topics. Lessons and
instructional strategies that support the explanations are suggested that are suitable for middle school students. Topics covered include freely falling bodies, determining the shape and
circumference of the earth, floating and sinking, vacuums, and logic and mathematic problems. In every case, the topic will be thoroughly discussed and analyzed and, hopefully, interested teachers
can find useful information for their own classes.
Click here to view complete unit.
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Re: classify
Yes, you can see that from the graph.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
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The conversion from Celsius to Fahrenheit and back again can be strange. One way to understand it is on a graph. And you can use this to teach some linear algebra too! … More
Have you ever tried to make a paper box? It’s a challenge to design, but fun to make with this easy template. And if you teach math, you can use the discussion questions in your lesson!… More
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La Habra Heights, CA Science Tutor
Find a La Habra Heights, CA Science Tutor
I have been teaching Anatomy & Physiology for the past 40 years in various healthcare Universities (including medical schools, dental schools, Oriental medical universities, chiropractic school,
community colleges, etc.). I have been considered by many students as one of the best instructors they e...
4 Subjects: including biology, anatomy, physiology, microbiology
...I will ask for feedback so that I can make changes to teaching style as needed to better help the students. I am very flexible with hours, but I do have a 2 hour cancellation policy to
accommodate for any necessary traveling. Students can also reach out to me via email or phone if they have any quick questions about the lesson.
10 Subjects: including physics, biology, chemistry, psychology
...I'm well versed in the subject of statistics whether its probabilities, descriptives, graphing tools, hypothesis testing, parametric & non-parametric tests, correlations (binary/multivariate),
regressions (linear, multivariate, and logistic), ANOVA/ANCOVA/MANOVA, exploratory/confirmatory factor a...
3 Subjects: including ecology, statistics, SPSS
...I took AP Calculus BC in high school and earned a 4 on the AP test with a 4 for my AB sub score. I also took a year of college calculus at UC Berkeley. Earned an A in each of my quarters of
General Chemistry and Organic Chemistry at Cal Poly Pomona in 2010-11.
11 Subjects: including physics, algebra 2, calculus, chemistry
Hello my name is Steven and I am currently a student attending Mt. San Antonio College. I am studying to be a psychology major.
19 Subjects: including sociology, biology, psychology, reading
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. Andreas Lochbihler
Email: >>please click this text<<
In December 2012, I have left the KIT. I am now a postdoc at the ETH Zurich in David Basin's group. Please go to my new homepage
Research interests
Formal semantics
As part of my PhD thesis, I worked on a formal, machine-checked model of Java including threads for both source code and bytecode. The model, which is formalised in the proof assistant Isabelle/HOL,
extends the Jinja language by T. Nipkow and G. Klein and covers now all language and synchronisation primitives for threads and the Java memory model. The semantics for source code and bytecode are
executable and proven to be type safe. The model also incldues a formalised, verified and executable compiler that translates source code programs into bytecode and preserves types and semantics. The
formalisation has been published in the Archive of Formal Proofs under the title Jinja with Threads, where it can also be downloaded.
I have also developed the conversion tool Java2Jinja as an Eclipse plugin to convert Java programs to JinjaThreads' input syntax.
Theorem proving
Code extractors permit to automatically translate formal models into executable prototypes. In this area, I am mainly interested in how to extract efficient prototypes from formal specifications in
the proof assistant Isabelle/HOL. Among others, I focus on how to simplify using efficient data structures in the prototype without cluttering the specification.
Program analysis
The static program analysis slicing determines how different parts of a program might influence each other. This can be used to control in information flow in programs and systems. To improve the
precision of discovery of and information about such influences, I developed in my master's thesis temporal path conditions. By connecting them to a model checker, they are used to automatically
generate "witness" traces for information flow along a specific path.
In this context, I am also interested in ensuring that the analyses themselves are sound. To that end, I work on executable prototypes for verified formalisations of such algorithms.
• Jinja With Threads
Archive of Formal Proofs December 2007 : A. Lochbihler
• On Temporal Path Conditions in Dependence Graphs
7th IEEE Working Conference on Source Code Analysis and Manipulation (SCAM 2007) September 2007, pp. 49--58 (SCAM 2007) : A. Lochbihler, G. Snelting
• Journals: Computer Languages, Systems & Structures; Fundamenta Informaticae; Journal of Automated Reasoning
• Conferences: FSTTCS 2011, ESOP 2012, EMSOFT 2012
Advised thesis projects
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Control algorithm for multiscale flow simulations of water
Publication: Research - peer-review › Journal article – Annual report year: 2009
Control algorithm for multiscale flow simulations of water. / Kotsalis, E. M.; Walther, Jens Honore; Kaxiras, E.; Koumoutsakos, P.
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
, Vol. 79, No. 4, 2009, p. 045701.
Publication: Research - peer-review › Journal article – Annual report year: 2009
Kotsalis, EM
, Walther, JH
, Kaxiras, E & Koumoutsakos, P 2009, '
Control algorithm for multiscale flow simulations of water
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
, vol 79, no. 4, pp. 045701.,
Kotsalis, E. M.
, Walther, J. H.
, Kaxiras, E., & Koumoutsakos, P. (2009).
Control algorithm for multiscale flow simulations of water
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
(4), 045701.
title = "Control algorithm for multiscale flow simulations of water",
publisher = "American Physical Society",
author = "Kotsalis, {E. M.} and Walther, {Jens Honore} and E. Kaxiras and P. Koumoutsakos",
note = "Copyright 2009 American Physical Society",
year = "2009",
doi = "10.1103/PhysRevE.79.045701",
volume = "79",
number = "4",
pages = "045701",
journal = "Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)",
issn = "1539-3755",
TY - JOUR
T1 - Control algorithm for multiscale flow simulations of water
A1 - Kotsalis,E. M.
A1 - Walther,Jens Honore
A1 - Kaxiras,E.
A1 - Koumoutsakos,P.
AU - Kotsalis,E. M.
AU - Walther,Jens Honore
AU - Kaxiras,E.
AU - Koumoutsakos,P.
PB - American Physical Society
PY - 2009
Y1 - 2009
N2 - We present a multiscale algorithm to couple atomistic water models with continuum incompressible flow simulations via a Schwarz domain decomposition approach. The coupling introduces an
inhomogeneity in the description of the atomistic domain and prevents the use of periodic boundary conditions. The use of a mass conserving specular wall results in turn to spurious oscillations in
the density profile of the atomistic description of water. These oscillations can be eliminated by using an external boundary force that effectively accounts for the virial component of the pressure.
In this Rapid Communication, we extend a control algorithm, previously introduced for monatomic molecules, to the case of atomistic water and demonstrate the effectiveness of this approach. The
proposed computational method is validated for the cases of equilibrium and Couette flow of water.
AB - We present a multiscale algorithm to couple atomistic water models with continuum incompressible flow simulations via a Schwarz domain decomposition approach. The coupling introduces an
inhomogeneity in the description of the atomistic domain and prevents the use of periodic boundary conditions. The use of a mass conserving specular wall results in turn to spurious oscillations in
the density profile of the atomistic description of water. These oscillations can be eliminated by using an external boundary force that effectively accounts for the virial component of the pressure.
In this Rapid Communication, we extend a control algorithm, previously introduced for monatomic molecules, to the case of atomistic water and demonstrate the effectiveness of this approach. The
proposed computational method is validated for the cases of equilibrium and Couette flow of water.
KW - water
KW - liquid theory
KW - decomposition
KW - Couette flow
KW - fluid oscillations
KW - flow simulation
UR - http://link.aps.org/doi/10.1103/PhysRevE.79.045701
U2 - 10.1103/PhysRevE.79.045701
DO - 10.1103/PhysRevE.79.045701
JO - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
JF - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
SN - 1539-3755
IS - 4
VL - 79
SP - 045701
ER -
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Grothendieck's mathematical diagram.
up vote 18 down vote favorite
I was going through this article which appeared in the Notices of the AMS, and in it, there's a picture which shows a mathematical diagram drawn by Grothendieck. I would be delighted if anyone could
explain what that is. Thanks in advance.
EDIT: The comments below suggest that the diagram is a dessin d'enfant. And even though I am by no means an expert, I somehow feel that it may not be one. I'd love some clarification on how one may
call it so. Questions I want to ask now, are
1) Is the diagram really a dessin? Or is it something else?
2) Do the shaded parts of the diagram signify something? Are there any special types of dessins which have a similar a structure?
soft-question ag.algebraic-geometry
5 Diagram is on page 936. – supercooldave Jun 28 '10 at 17:22
At least the superimposed image is the mirror image of the one below. But what else? – Unknown Jun 28 '10 at 17:35
5 en.wikipedia.org/wiki/Dessin_d%27enfant – Dan Piponi Jun 28 '10 at 17:53
1 See also mathoverflow.net/questions/1909/what-are-dessins-denfants – Unknown Jun 29 '10 at 7:00
add comment
1 Answer
active oldest votes
First of all, I believe that according to Grothendieck's definition of dessins d’enfants the picture (if I am looking at the right one) indeed seems to show one. At the same time you
have a point that this is not one of the more interesting ones.
On the other hand it might be the very first one Grothendieck ever drew and then one could make a wild guess as to what it shows. I would venture to say that the diagram in question
shows the complex conjugation of the Riemann sphere.
If I understand correctly, Grothendieck's inventing and studying dessins d’enfants was motivated by his goal of finding non-trivial elements of the absolute Galois group $\mathrm{Gal}
(\overline{\mathbb{Q}}/\mathbb Q)$. An obvious (and the only obvious) non-trivial element is complex conjugation of $\mathbb C$, which actually extends to $\mathbb P^1_{\mathbb C}$,
a.k.a. the Riemann sphere.
My guess is that this picture shows that and was perhaps the visual clue that led Grothendieck to make the definition of dessins d’enfants.
Addendum: To answer the question raised in the comments: The picture clearly shows a reflection. Complex conjugation is a reflection. I did not claim I have a proof for this, indeed,
notice the words wild guess above. The only argument I can offer is that
up vote 4 down
vote accepted i) it is reasonable to assume that this picture is or at least has something to do with dessins d’enfants.
ii) it is a very simple drawing for that
iii) there should still be some significance for someone to have put it in the article
iv) it is reasonable to assume that it is an early drawing of dessins d’enfants
v) it is clearly a reflection
vi) complex conjugation is a reflection and has a lot to do with the birth of dessins d’enfants.
As I said, this is a guess, but I wonder if anyone can offer anything other than a guess.
4 Sandor--could you elaborate on how this picture shows complex conjugation? – Daniel Litt Mar 14 '11 at 9:12
1 Hmm...I assumed the two green circles were two hemispheres of the Riemann sphere (hence the two arrows in the diagram would indicate where to glue); then the red lines would be a
graph embedded in the Riemann sphere. But I guess you are suggesting the two green circles are a "before and after" picture, which is why you say this is clearly a reflection? Or am
I misinterpreting your interpretation? – Daniel Litt Mar 14 '11 at 17:57
1 Daniel, you are not misinterpreting :). On the other, I don't think those are hemispheres! I think that those two red arcs that go sort of horizontally are "big circles" of a
sphere, so to me it seems that the picture shows two spheres (i.e., not hemi-). – Sándor Kovács Mar 15 '11 at 0:56
add comment
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Reviews of books, websites, poster sets, movies, and other resources for learning and teaching the history of mathematics.
A collection of problems that should be of interest to teachers at all levels. Frank Swetz shares this observation at the close of his book, Mathematical Expeditions: Exploring Word Problems across
the Ages.
A collection of problems that should be of interest and use to teachers at all levels
Our reviewer finds this collection of translations of Babylonian mathematical tablets to be both fascinating and accessible.
A lively history of number systems and number theory from earliest times up to the notion of "infinity".
A new collection of original source materials in the mathematics of five civilizations.
A superb collection of articles by experts on various areas of the history of analysis, from the Greeks to modern times.
A collection of original texts to help students learn some important areas of mathematics.
A math history class visits the 'Beautiful Science' exhibit at the Huntington Library in Southern California.
A sourcebook of original materials in the history of mathematics from ancient times to the early twentieth century.
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the first resource for mathematics
Iterative hard thresholding for compressed sensing.
(English) Zbl 1174.94008
In the presented paper, it is shown that one of the previously by the authors developed iterative hard thresholding algorithms (termed IHTs) has similar performance guarantees to those of Compressed
Sampling Matching Pursuit. Section 2 of the paper starts with a definition of sparse signal models and a statement of the compressed sensing problem. In Section 3, an iterative hard thresholding
algorithm is discussed. The rest of the paper shows that this algorithm is able to recover, with high accuracy, signals from compressed sensing observations. This result is formally stated in the
theorems of the first subsection of Section 4. The rest of Section 4 is devoted to the proof of the theorems. In fact, the derived result is near-optimal as shown in Section 5. Section 6 takes a
closer look at a stopping criterion for the algorithm, which guarantees certain estimation accuracy. The results of the paper are similar to those for the Compressed Sampling Matching Pursuit
algorithm and a more detailed comparison is given in Section 7. However, as discussed in Section 8, uniform guarantees are not the only consideration and in practice marked differences in the average
performance of different methods are apparent. For many small problems, the restricted isometry property of random matrices is often too large to explain the behavior of the different methods.
Furthermore, it has long been observed that the distribution of the magnitude of the non-zero coefficients also has an important influence on the performance of different methods. Whilst the
theoretical guarantees derived in the presented and similar papers are important to understand the behavior of an algorithm, it is also clear that other facts have to be taken into account in order
to predict the typical performance of algorithms in many practical situations.
94A20 Sampling theory
94A13 Detection theory
[1] Blumensath, T.; Davies, M.: Iterative thresholding for sparse approximations, J. Fourier anal. Appl. 14, No. 5, 629-654 (2008) · Zbl 1175.94060 · doi:10.1007/s00041-008-9035-z
[2] T. Blumensath, M. Davies, A simple, efficient and near optimal algorithm for compressed sensing, in: Proceedings of the Int. Conf. on Acoustics, Speech and Signal Processing, 2009
[3] Candès, E.; Romberg, J.: Quantitative robust uncertainty principles and optimally sparse decompositions, Found. comput. Math. 6, No. 2, 227-254 (2006) · Zbl 1102.94020 · doi:10.1007/
[4] Candès, E.; Romberg, J.; Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE trans. Inform. theory 52, No. 2, 489-509 (2006) ·
Zbl 1231.94017 · doi:10.1109/TIT.2005.862083
[5] Candès, E.; Romberg, J.; Tao, T.: Stable signal recovery from incomplete and inaccurate measurements, Comm. pure appl. Math. 59, No. 8, 1207-1223 (2006) · Zbl 1098.94009 · doi:10.1002/cpa.20124
[6] Dai, W.; Milenkovic, O.: Subspace pursuit for compressive sensing signal reconstruction, IEEE trans. Inform. theory 55, No. 5, 2230-2249 (2009)
[7] Donoho, D.: Compressed sensing, IEEE trans. Inform. theory 52, No. 4, 1289-1306 (2006)
[8] Gorodnitsky, I. F.; George, J. S.; Rao, B. D.: Neuromagnetic source imaging with focuss: A recursive weighted minimum norm algorithm, Neurophysiology 95, No. 4, 231-251 (1995)
[9] N.G. Kingsbury, T.H. Reeves, Iterative image coding with overcomplete complex wavelet transforms, in: Proc. Conf. on Visual Communications and Image Processing, 2003
[10] Mallat, S.; Davis, G.; Zhang, Z.: Adaptive time – frequency decompositions, SPIE J. Opt. eng. 33, No. 7, 2183-2191 (1994)
[11] Mendelson, S.; Pajor, A.; Tomczak-Jaegermann, N.: Reconstruction and subgaussian operators in asymptotic geometric analysis, Geom. funct. Anal. 17, 1248-1282 (2007) · Zbl 1163.46008 ·
[12] Mendelson, S.; Pajor, A.; Tomczak-Jaegermann, N.: Uniform uncertainty principle for Bernoulli and subgaussian ensembles, Constr. approx. 28, No. 3, 277-289 (2008) · Zbl 1230.46011 · doi:10.1007/
[13] Needell, D.; Tropp, J. A.: Cosamp: iterative signal recovery from incomplete and inaccurate samples, Appl. comput. Harmon. anal. 26, No. 3, 301-321 (2009) · Zbl 1163.94003 · doi:10.1016/
[14] Needell, D.; Vershynin, R.: Signal recovery from incomplete and inacurate measurements via regularized orthogonal matching pursuit
[15] Needell, D.; Vershynin, R.: Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit, Found. comput. Math. 9, 317-334 (2009) · Zbl 1183.68739 · doi:10.1007/
[16] Nyquist, H.: Certain topics in telegraph transmission theory, Trans. A.I.E.E., 617-644 (1928)
[17] M. Rudelson, R. Vershynin, Sparse reconstruction by convex relaxation: Fourier and gaussian measurements, in: 40th Annual Conference on Information Sciences and Systems, 2006
[18] Shannon, C. A.; Weaver, W.: The mathematical theory of communication, (1949) · Zbl 0041.25804
[19] Tropp, J. A.; Gilbert, A. C.: Signal recovery from partial information via orthogonal matching pursuit, IEEE trans. Inform. theory 53, No. 12, 4655-4666 (2006)
[20] Vetterli, M.; Marziliano, P.; Blu, T.: Sampling signals with finite rate of innovation, IEEE trans. Signal process. 50, No. 6, 1417-1428 (2002)
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The sum of five integers is 190. If one of these numbers is removed, the average of the four remaining numbers is between 34.5 and 36, inclusive. What is one possible value for the number that was
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Is there a symmetry group lurking behind every WLOG?
up vote 1 down vote favorite
Most of us are introduced to "without loss of generality" before encountering formal group theory. To the uninitiated, the phrase almost seems like cheating, but soon we realize how intuitive and
useful it is for simplifying and shortening proofs.
Perhaps this is a dumb question (in the sense that the answer might well be obvious), but is it true that behind every WLOG there is an implied symmetry group in play?
A Couple of Examples
(Schur's Inequality) If $a,b,c \ge 0 $ and $r \ge 1$, then$$a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r(c-a)(c-b) \ge 0$$
Proof: Without loss of generality, assume $a \ge b \ge c$...
We can do this because the expression at hand is symmetric in $a,b,c$.
The group is $S_3$.
(Fundamental Theorem of Algebra) Every $n^{th}$-degree polynomial $a_n z^n + a_{n-1}z^{n-1} + \dots + a_1 z + a_0$ has a root in $\mathbb{C}$.
Proof: Without loss of generality, assume $a_n = 1$, because we can "divide through" by $a_n$...
The group is $\mathbb{R}- 0$ under multiplication.
2 I can imagine what a "no" answer to this question looks like, but I can't imagine what a "yes" answer would look like. – Qiaochu Yuan Jan 20 '12 at 1:47
16 Sometimes arguments in analysis begin "Let $\{x_n\}$ be a sequence which converges to $x$. Without loss of generality, assume $d(x,x_n) < 2^{-n}$..." Here the issue seems to be extra flexibility
in the definitions rather than symmetry. – Paul Siegel Jan 20 '12 at 2:22
2 You might also want to allow groupoid symmetry since you might say without loss of generality assume the vertices of our graph is $\{1,\ldots,n\}$. – Benjamin Steinberg Jan 20 '12 at 3:49
You can attach a group action to any equivalence relation, but it is usually not canonical. – S. Carnahan♦ Jan 20 '12 at 4:30
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closed as not a real question by Yemon Choi, Benjamin Steinberg, Mariano Suárez-Alvarez♦, Alain Valette, Bill Johnson Jan 20 '12 at 17:11
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Fix a semi-formal language for discourse and proof. (I will let you worry about niggling issues such as consistency and soundness.) Now consider proving your statement with as much detail and
completeness as your semi-formal system allows. Your proof may break into subcases and subproofs, and you may find it convenient to refactor your proof so that you make repeated use of some
lemmas. At some point you may have a situation like wanting to establish in some contexts two statements, say phi(a,b) and phi(b,a), and you find that they have similar proofs. At this point,
you can define an equivalence relation between tuples in the language, and you can construct and investigate symmetries of the syntactic elements of the proof as represented in the
up vote semi-formal system. So you can arrange things so that the answer to your question is yes. Whether it is prudent to take such relations from the semi-formal system back to the realm of your
-2 down objects of study is likely very situation-specific.
Gerhard "Ask Me About System Design" Paseman, 2012.01.19
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Write the set {t: |25+7|>5} as a union of two intervals
October 5th 2012, 09:18 AM #1
Oct 2012
Write the set {t: |25+7|>5} as a union of two intervals
${t: |25+7|>5}$ It's actually equal to or more than 5, but I don't know how to enter the code in this forum, so please bear with me.
Re: Write the set {t: |25+7|>5} as a union of two intervals
October 5th 2012, 10:18 AM #2
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R.J.: Merge and termination in process algebra
Results 1 - 10 of 12
- Formal Aspects of Computing , 2000
"... We investigate different forms of termination in timed process algebras. The integrated framework of discrete and dense time, relative and absolute time process algebras is extended with forms
of successful and unsuccessful termination. The different algebras are interrelated by embeddings and conse ..."
Cited by 155 (25 self)
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We investigate different forms of termination in timed process algebras. The integrated framework of discrete and dense time, relative and absolute time process algebras is extended with forms of
successful and unsuccessful termination. The different algebras are interrelated by embeddings and conservative extensions.
, 1993
"... . We proposed a syntactical format, the path format, for structured operational semantics in which predicates may occur. We proved that strong bisimulation is a congruence for all the operators
that can be defined within the path format. To show that this format is useful we provided many examples t ..."
Cited by 109 (5 self)
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. We proposed a syntactical format, the path format, for structured operational semantics in which predicates may occur. We proved that strong bisimulation is a congruence for all the operators that
can be defined within the path format. To show that this format is useful we provided many examples that we took from the literature about CCS, CSP, and ACP; they do satisfy the path format but no
formats proposed by others. The examples include concepts like termination, convergence, divergence, weak bisimulation, a zero object, side conditions, functions, real time, discrete time,
sequencing, negative premises, negative conclusions, and priorities (or a combination of these notions). Key Words & Phrases: structured operational semantics, term deduction system, transition
system specification, structured state system, labelled transition system, strong bisimulation, congruence theorem, predicate. 1980 Mathematics Subject Classification (1985 Revision): 68Q05, 68Q55.
CR Categories: D.3.1...
, 2007
"... In 1981 Structural Operational Semantics (SOS) was introduced as a systematic way to define operational semantics of programming languages by a set of rules of a certain shape [G.D. Plotkin, A
structural approach to operational semantics, Technical ..."
Cited by 12 (5 self)
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In 1981 Structural Operational Semantics (SOS) was introduced as a systematic way to define operational semantics of programming languages by a set of rules of a certain shape [G.D. Plotkin, A
structural approach to operational semantics, Technical
, 2005
"... In 1981 Structural Operational Semantics (SOS) was introduced as a systematic way to define operational semantics of programming languages by a set of rules of a certain shape [62].
Subsequently, the format of SOS rules became the object of study. Using so-called Transition System Specifications (TS ..."
Cited by 6 (1 self)
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In 1981 Structural Operational Semantics (SOS) was introduced as a systematic way to define operational semantics of programming languages by a set of rules of a certain shape [62]. Subsequently, the
format of SOS rules became the object of study. Using so-called Transition System Specifications (TSS’s) several authors syntactically restricted the format of rules and showed several useful
properties about the semantics induced by any TSS adhering to the format. This has resulted in a line of research proposing several syntactical rule formats and associated meta-theorems. Properties
that are guaranteed by such rule formats range from well-definedness of the operational semantics and compositionality of behavioral equivalences to security- and probability-related issues. In this
paper, we provide an initial hierarchy of SOS rules formats and meta-theorems formulated around them.
- In Branislav Rovan and Peter Vojtás, editors, Proceedings of MFCS 2003 , 2003
"... Abstract. We prove a unique decomposition theorem for a class of ordered commutative monoids. Then, we use our theorem to establish that every weakly normed process definable in ACP ε with
bounded communication can be expressed as the parallel composition of a multiset of weakly normed parallel prim ..."
Cited by 5 (2 self)
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Abstract. We prove a unique decomposition theorem for a class of ordered commutative monoids. Then, we use our theorem to establish that every weakly normed process definable in ACP ε with bounded
communication can be expressed as the parallel composition of a multiset of weakly normed parallel prime processes in exactly one way. 1
- Proceedings of the International School on Formal Methods for the Design of Real-Time Systems (SFM-RT’04), volume 3185 of Lecture Notes in Computer Science , 2004
"... Abstract. We treat theory and application of timed process algebra. We focus on a variant that uses explicit termination and action prefixing. This variant has some advantages over other
variants. We concentrate on relative timing, but the treatment of absolute timing is similar. We treat both discr ..."
Cited by 5 (2 self)
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Abstract. We treat theory and application of timed process algebra. We focus on a variant that uses explicit termination and action prefixing. This variant has some advantages over other variants. We
concentrate on relative timing, but the treatment of absolute timing is similar. We treat both discrete and dense timing. We build up the theory incrementally. The different algebras are interrelated
by embeddings and conservative extensions. As an example, we consider the PAR communication protocol. 1
- In Middeldorp et al , 2005
"... Abstract. There have been several timed extensions of ACP-style process algebras with successful termination. None of them, to our knowledge, are equationally conservative (ground-)extensions of
ACP with successful termination. Here, we point out some design decisions which were the possible causes ..."
Cited by 4 (1 self)
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Abstract. There have been several timed extensions of ACP-style process algebras with successful termination. None of them, to our knowledge, are equationally conservative (ground-)extensions of ACP
with successful termination. Here, we point out some design decisions which were the possible causes of this misfortune and by taking different decisions, we propose a spectrum of timed process
algebras ordered by equational conservativity ordering. 1 The Untimed Past The term “process algebra ” was coined by Jan Bergstra and Jan Willem Klop in [9] to denote an algebraic approach to
concurrency theory. Their process algebra had uniform atomic actions ai for i ∈ I (with I some index set), sequential composition · , choice (alternative composition) + and left-merge � as the basic
composition operators. 1 Much of the core theory of [9] remained intact in the course of more than 20 years of developments in the ACP-school (for Algebra of Communicating
- Dat is dus heel interessant , 1997
"... We introduce an ACP-style discrete-time process algebra with relative timing, that features the empty process. Extensions to this algebra are described, and ample attention is paid to the
considerations and problems that arise when introducing the empty process. We prove time determinacy, soundness, ..."
Cited by 3 (3 self)
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We introduce an ACP-style discrete-time process algebra with relative timing, that features the empty process. Extensions to this algebra are described, and ample attention is paid to the
considerations and problems that arise when introducing the empty process. We prove time determinacy, soundness, completeness, and the axioms of standard concurrency. 1991 Mathematics Subject
Classification: 68Q10, 68Q22, 68Q55. 1991 CR Categories: D.1.3, D.3.1, F.1.2, F.3.2. Keywords: ACP, process algebra, discrete time, relative timing, empty process, time determinacy, soundness,
completeness, axioms of standard concurrency, #,BPA - drt --ID, BPA - drt,# --ID, PA - drt,# --ID, ACP - drt,# --ID, BPA drt,# --ID, PA drt,# --ID, ACP drt,# --ID, RSP(DEP). Note: The investigations
of the second author were supported by the Netherlands Computer Science Research Foundation (SION) with financial support from the Netherlands Organization for Scientific Research (NWO). 3 Contents
1Introduction 5 1.1 Mo...
"... We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove
that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition orde ..."
Cited by 1 (0 self)
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We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that
a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo
bisimulation definable in ACP ε with linear communication, with parallel composition as binary operation, has unique decomposition. We also apply our result to establish that the partial commutative
monoid associated with a well-founded commutative residual algebra has unique decomposition. 1
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Ising model
T It delay fps
GPU based Monte-Carlo simulation of the Ising model on the square 512×512 grid. it is the number of MC iterations per frame. The thermostat algorithm and linear congruential random number generator
are used.
For low temperature you can watch formation of clusters of ordered phase from random state. For T > T[C] = 2.269 you will get desordered phase, for T < 0 the antiferromagnetic model. See Phase
transitions on lattices.
WebCL Demos updated 12 Sep 2011
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determine angle of 3 points
May 19th 2009, 03:17 PM #1
May 2009
determine angle of 3 points
to explain the problem I will first paint a simple picture:
imagine on the x/y axis I have a square who's center is exactly at the origin(0,0)
I now want to work out the angles between the bottom right point and the rest, so bottom right and top right, bottom right and top left etc
you will probably end up with something like 45 degrees, then 140, then 225
I now want to do the same thing, but using 3d space ie the x,y,z axis, this time using a trapezium, still with its center lying on the origin
I used the formula
theta=Acos(dotproduct / sqrt(len)) and it works,
here is the problem..... I cant remember what the technical term for this is, but if I apply the above formulae to the points [1,0,0] [0,1,0] meeting at [0,0,0] it gives me the answer 90 degrees,
which is correct, but I want it to say 270 degrees- in other words measure it counter clockwise, not clockwise
how would I go about doing this
now I know normally I could just take 90 away from 360, but because im working with a plane in 3d space, and trying to get a computer program to work this out for me, if It spits out an answer of
90 degrees, im not sure if its worked it out in a clockwise or anticlockwise direction, and its really important that I know
Once you know the angle, you can check whether it is clockwise or anticlock-wise by applying sine product of vectors.
May 22nd 2009, 09:48 PM #2
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The best known packings of equal circles in a circle
Re: The best known packings of equal circles in a circle
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
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F-Test on comparisons of several variables
February 27th 2009, 08:51 AM #1
F-Test on comparisons of several variables
I am supposed to do an analysis of data in a study of zinc levels in vegetarian vs. non-vegetarian pregnant women.
23 women were monitored, 12 of them were pregnant vegetarian, 6 were non-vegetarian and pregnant, and 5 of them were vegetarian and non-pregnant.
when I compare the residuals there does not appear to be equal standard deviation among the groups, but I don't think I know a test yet that is more robust to departures from equal standard
deviation that involves comparisons among several groups of data.
Mostly what I am confused about is this. I did an ANOVA F-test and came up with a p-value of .98 which pretty much gives extremely convincing evidence that there is not a statistically
significant difference in the means of the three groups, so if there is no evidence for a difference in means between the three groups it isn't necessary to then compare the non-vegetarian mean
to the vegetarian means is it?
Also, I am a little bit confused about the extra group of data about the vegetarian pregnant women. I don't see how it applies. I know it is bad form to throw out any data but in this case it
seems to me that it would make more sense to compare only the data of the pregnant women.
Ooops, sorry. I meant to say that the set of data about the non-pregnant vegetarian women was the set that doesn't seem necessary.
Can you post the anova table?
It sounds like an F_(2,N-4).
I actually ended up making it to the T.A.'s office hours before the class and got my question answered. It was F_(2, N-3) but the result gave a fairly large p-value but apparently just for the
sake of practice we were supposed to also do a 2-sample analysis of the two key groups. Either way thanks for trying to help me.
February 27th 2009, 09:18 AM #2
February 28th 2009, 10:44 PM #3
March 1st 2009, 01:22 PM #4
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Number Generation Question
December 12, 2008 10:06 AM Subscribe
I need to generate the complete series of 5 digit numbers. The catch? Can only use even digits (0,2,4,6,8). I'd like to accomplish this in Excel, if possible.
posted by Rock Steady to Computers & Internet (28 answers total) 1 user marked this as a favorite
Trying to make sure I understand. So you want:
posted by inigo2 at 10:13 AM on December 12, 2008
Correct me if I'm wrong, but you want the even numbers from 20000 to 88888? Or from 00000 to 88888?
posted by jedicus at 10:13 AM on December 12, 2008
inigo2 is right. 46802 is a good result, 46812 is not.
posted by Rock Steady at 10:16 AM on December 12, 2008
That doesn't sound right, because that includes numbers such as 30000 which use an odd digit. It sounds like the OP wants
skipping any numbers that have any odd digits at all.
posted by likedoomsday at 10:17 AM on December 12, 2008
To clarify the leading zero issue, 00802 is good, 00812 bad.
posted by Rock Steady at 10:18 AM on December 12, 2008
You could generate the series of even numbers between 0 and 88888, then in another column use text functions to determine if any of the digits is odd. Filter on that second column, and you've got
your series. You can copy and paste the filtered list into a new sheet if you want the list clean like that.
posted by Dec One at 10:22 AM on December 12, 2008
Well, the quick and easy way is a loop that adds 2 to the iterator, checks for the presence of any odd digits, and then either enters the number (padded with leading zeroes as necessary) or continues
the loop.
A lot of improper numbers will be generated and rejected, but it's simpler than constructing a series of nested loops that generates only the correct numbers.
posted by jedicus at 10:24 AM on December 12, 2008
set y to 1
repeat 88888 times
set y to y+1
if y contains any of "13579"
then repeat else
print y
print return
applescript n00b
posted by Aquaman at 10:24 AM on December 12, 2008
It's not the easiest way, but this will work.
1. Enter the numbers 0,2,4,6,8,10 into a column and then select all five numbers.
2. Click the square at the lower-right hand corner of the selection and drag it down. Excel will follow the pattern and fill in all the even numbers. Hold down the mouse button for a long time (took
me about a minute) until you get to six digit numbers.
3. You now have a list of all the even numbers between 0 and 99,998. To get rid of the numbers that contain 1,3,5,7 or 9, do the following for each number:
a) Go to Find and Replace and go to the Replace tab.
b) Enter *1* into the "Find what" textbox and leave the "Replace with" textbox blank.
c) Click on the Options button and check the "Match entire cell contents" checkbox.
d) Click Replace All.
e) Repeat steps a through d for 3,5,7 and 9.
4. Select the column the numbers are in and go to Data > Sort. Click OK. That will get rid of the empty spaces in the list.
5. Select the column again and go to Format > Cells.
6. Go to the Number tab and select Custom for the Category. Type in 00000 into the Type field. Click OK. (this will add the leading zeroes).
7. Profit.
posted by hootch at 10:29 AM on December 12, 2008
take all the numbers from 00000 to 44444, base 5. Transliterate those (make text, then make numbers from the text) to base ten. Then double every number.
posted by notsnot at 10:30 AM on December 12, 2008
or a series of nested repeats-
n = 0
repeat 5{
repeat 5 {
repeat 5 {
repeat 5 {
repeat 5 { print n; n+2 }
n+20 }
n+200 }
n+2000 }
n+20000 }
posted by noloveforned at 10:35 AM on December 12, 2008
shoot, those should actually be +2, +10, +100, +1000 and +10000... stupid math!
posted by noloveforned at 10:36 AM on December 12, 2008
echo {0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8}posted by nicwolff at 10:38 AM on December 12, 2008
What's the problem? Do you want a xls (or csv) that contains them, or do you want the function to generate them?
To generate, you can use seq and grep : seq -w 00000 99999 | grep '[02468][02468][02468][02468][02468]'
http://www.sharefile.org/showfile-549/5digits.txt has a copy of the output that you can import into excel if you wish.
posted by devbrain at 10:39 AM on December 12, 2008 [1 favorite]
Or, if you want a comma-delimited list for easy import into Excel:
echo {0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8}{0,2,4,6,8},posted by nicwolff at 10:40 AM on December 12, 2008 [2 favorites]
Thanks everyone! Your file did the trick, devbrain. I dig the procedural efforts, but in this case, "do it for me" is just as good as "teach me".
posted by Rock Steady at 10:47 AM on December 12, 2008
Got it for Excel:
1. Select the whole first column and change formatting to text.
2. In a new sheet, put the following in the first few rows of the first column:
3. Select the whole first column again, and do AutoFill Series. (In 2007, it's on the Home Tab under Editing.) This will fill the whole column with even numbers, some higher than 88888, but no
worries. Excel overflows before 20,000 comes up.
4. Select the whole first column, and do Conditional Formatting. Tell it to highlight cells containing the text 1, 3, 5, 7, or 9 and color them red. I had to apply each rule separately for each
5. With the first column selected, filter out all of the red text cells. In 2007, I had to click Filter in the Data tab. This brought up the Filter button on the first cell "00000", which you can
click and say "Filter By Color" and select "Automatic" (Black).
All the numbers you want will be in the first column, and you can copy them to a new sheet if needed. Let me know if any of these steps don't make sense.
posted by ALongDecember at 10:49 AM on December 12, 2008
I emailed you my efforts, Rock Steady, although you have an answer already :).
posted by MadamM at 10:52 AM on December 12, 2008 [1 favorite]
I did it with formulas, which might be what you want. It was a fun little challenge.
You can use excel formatting to get the leading zeroes back. You can also decompose the function to get each digit if that's what you want (it's how I built the formula). Just break it up at the '+'s
You can also replace ROW() with whatever index you want, as long as it's a n=n+1 kinda thing.
posted by teabag at 10:54 AM on December 12, 2008
Ack, noticed a bug in my formula, lemme see if I can fix it...
posted by teabag at 10:57 AM on December 12, 2008
Since Excel doesn't ddo funny bases (at least my vanilla version doesn't) I've made a spreadsheet that figures out the digits using modulo arithmetic. Mefi mail me and I'll send you the spreadsheet.
posted by notsnot at 11:01 AM on December 12, 2008
Fixed, forgot the moduli and my intervals were wrong...
The divisor in the ROUNDDOWN section is 5* the previous divisor, if you're wanting more digits.
posted by teabag at 11:04 AM on December 12, 2008 [1 favorite]
..and since the answer has already been sent to the asker, I'll put in my formulae:
A1=1, A2=A1+1, on down to 3124.
Change the formatting on g1 to "zip code" to display all five numbers.
...and that, folks, is how you crunch out weird bases manually.
posted by notsnot at 11:07 AM on December 12, 2008
Windows cmd shell abuse:
for %a in (0 2 4 6 8) do @for %b in (0 2 4 6 8) do @for %c in (0 2 4 6 8) do @for %d in (0 2 4 6 8) do @for %e in (0 2 4 6 8) do @echo %a%b%c%d%e
posted by milnak at 2:22 PM on December 12, 2008
In Excel, with semi-recursion (start in A2, then C&P to row 3,125):
=0,A1+11112,"Number too high")))))
In JavaScript:
for (i=0; i<9> if (i.toString().search(/.*[13579]/) != 0)
Since you got the answer (the method and the series), can we ask
you are doing this?9>
posted by ostranenie at 2:30 PM on December 12, 2008
That should actually be:
for (i=0; i<99999; i++)
if (i.toString().search(/.*[13579]/) != 0)
posted by ostranenie at 2:32 PM on December 12, 2008
can we ask why you are doing this?
I need to generate passcodes for an electronic door lock. The keypad only has 5 buttons (1|2, 3|4, etc) so I need to stick to even numbers so I don't accidentally give people identical codes (24680
and 24580 are essentially the same number in this system). I wanted the whole series so I can cross out numbers that are used and never re-use them. Thanks again everyone!
posted by Rock Steady at 6:54 AM on December 13, 2008
In theory that's good, but are people really going to be trying others' codes on doors that aren't theirs? Is there a lockout time if the wrong code is input 'n' times in a row? Or an alert system
that tells you when the wrong code is entered? Can users change their own codes or do you have to do it? Is the code database centrally or locally (at the keypad) managed? If it's centrally managed
can you change the code remotely in the case of an emergency? If not, is there a master security code only you know? And so on and so on and so on.
Sorry...I just worry.
posted by ostranenie at 9:54 AM on December 31, 2008
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DMTCS Proceedings
Kℓ--factors in graphs
Daniela Kühn, Deryk Osthus
Let Kℓ- denote the graph obtained from Kℓ by deleting one edge. We show that for every γ>0 and every integer ℓ≥4 there exists an integer n0=n0(γ,ℓ) such that every graph G whose order n≥n0 is
divisible by ℓ and whose minimum degree is at least (ℓ2-3ℓ+1/ℓ(ℓ-2)+γ)n contains a Kℓ--factor, i.e. a collection of disjoint copies of Kℓ- which covers all vertices of G. This is best possible up
to the error term γn and yields an approximate solution to a conjecture of Kawarabayashi.
Full Text:
GZIP Compressed PostScript PostScript PDF original HTML abstract page
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Matches for:
Translations of Mathematical Monographs
1997; 336 pp; hardcover
Volume: 157
ISBN-10: 0-8218-0374-3
ISBN-13: 978-0-8218-0374-5
List Price: US$129
Member Price: US$103.20
Order Code: MMONO/157
The study of natural and social phemomena indicates that the future development of many processes depends not only on their present state, but also on their history. Such processes can be described
mathematically by using the machinery of equations with aftereffect.
This book is a comprehensive, up-to-date presentation of control theory for hereditary systems of various types. Topics covered include background of the theory of hereditary equations, their
applications in modeling real phenomena, optimal control of deterministic and stochastic systems, optimal estimation of systems with delay, and optimal control with uncertainties. The exposition is
illustrated by examples, figures, and tables.
Graduate and postgraduate students in applied mathematics, mechanics, engineering, automation, cybernetics, and biomathematics; researchers and applied mathematicians interested in control theory,
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• Elements of the theory of systems with aftereffect
• The dynamic programming method
• Optimality conditions for deterministic systems with aftereffect
• Investigation of self-adjusting systems with reference model
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• Optimal estimation
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• Bibliography
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Confusing with bitwise NOT operator
10-16-2009, 07:23 PM
Confusing with bitwise NOT operator
I don't know whether anyone pay attention with bitwise logical NOT operator, but I am confusing with this.
For example, I want to use NOT operator with an integer variable which has value 1, this means (let me take 8 bits instead of 4 bytes) :
If we run a program for this problem, we get -2 which is not equal to 1111 1110 at all
The thing that I know is that after 1 is converted to 1's complement, the result is converted back to 2's complement and keep the MSB bit unchanged as the sign bit, that's how we get -2. But why
is it like that? Does this have any thing related to performance purpose or it's the rule of representing binary number in form of machine code? :(
Thanks and pardon my ignorance about the very basic thing!
10-16-2009, 07:36 PM
The advantages of twos complement can found on wikipedia.
10-16-2009, 09:03 PM
so it's all about the sake of performance.
Thanks for the useful topic on Wiki!
10-16-2009, 09:06 PM
Not all about performance. That wiki example mentions other advantages.
10-16-2009, 11:06 PM
I don't know whether anyone pay attention with bitwise logical NOT operator, but I am confusing with this.
For example, I want to use NOT operator with an integer variable which has value 1, this means (let me take 8 bits instead of 4 bytes) :
1 is 0000 0001
~1 is 1111 1110
If we run a program for this problem, we get -2 which is not equal to 1111 1110 at all
The thing that I know is that after 1 is converted to 1's complement, the result is converted back to 2's complement and keep the MSB bit unchanged as the sign bit, that's how we get -2. But why
is it like that? Does this have any thing related to performance purpose or it's the rule of representing binary number in form of machine code? :(
Thanks and pardon my ignorance about the very basic thing!
It's because the number is a signed number. So the leading 1 says the number is negative.
1111 1111 = -1
1111 1110 = -2
1111 1101 = -3
If the number were unsigned (not supported in Java), then 1111 1110 would equal 254.
I've only used the bitwise not operation when working with bitmasks to clear a specific flag. For example "bitmask & ~flag" will clear the specific flag in the bitmask.
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|
, pa. 45, col. 2, l. 40, for 800, read 680.—l. 46, for 28 1/3 read 26.—l. 48, for balloon read parachute.—l. 51 and 52, for 28 1/3 read 26, and for 13 read 12.—l. 55, read 2 feet 3 inches. |
Pa. 46, col. 2, at the end of the article on Aerostation, add, See an ingenious and learned treatise on the mathematical and physical principles of Airballoons, by the late Dr. Damen, professor of
philosophy and mathematics in the University of Leyden, entitled, Physical and Mathematical Contemplations on Aerostatic Balloons, &c; in 8vo, at Utrecht, 1784.
Pa. 70, col. 1, l. 4, dele -√(3 - 1 = 2).-l. 5, at the end add -√(3 - 1) = 2.
Pa. 71, col. 1, l. 9, for y^2 + 2y - 7 read ―(y^2 + 2y-7).
AFFECTED Equations, add (from Francis Maseres, Esq.)—“This expression of Affected Equations seems to require some further explanation. It was introduced by the celebrated Vieta, the great father and
restorer of Algebra. He has many expressions peculiar to himself, and which have not been adopted by subsequent Algebraists. Amongst these are the following ones. He calls a set of quantities in
continual geometrical proportion, (such as the quantities 1, x, x^2, x^3, x^4, x^5, x^6 x^7, &c,) a set of scalar quantities, or magnitudines scalares; and, when there are several of these scalar
quantities mentioned together, (as in the compound quantity x^5 + ax^4-b^2x^3,) he calls the highest quantity, or that which is farthest in the scale of quantities 1, x, x^2, x^3, x^4, x^5, x^6, x^7,
&c. (to wit, the quantity x^5 in the said compound quantity x^5 + ax^4-b^2x^3,) the power of the fundamental quantity x, or of the second term in the said scale; and he calls the lower scalar
quantities which are involved in the second and third terms of the said compound quantity x^5+ax^4 -b^2x^3, to wit, the quantities x^4 and x^3, (or, in our present language, the inferior powers of x,
) scalar quantities of a parodic degree to x^5, or the power of the fundamental quantity x. This word parodic I take to be derived (though Vieta does not tell us so) from the Greek words pxrx\ and o
(do\s, which signify near and a way or road, because these inferior scalar quantities x^3 and x^4 lie in the way as you pass along in the scale of the aforesaid quantities 1, x, x^2, x^3, x^4, x^5, x
^6 x^7, &c, from 1 to x^5, which he calls the power of x in the said compound quantity x^5 + ax^4-b^2x^3. These inferiour scalar quantities x^3 and x^4 are therefore parodic, or situated in the way
to, or are leading to, the higher scalar quantity x^5. He then proceeds to define a pure power and an affected power, and tells us that a pure power is a scalar quantity that is not affected with any
parodic, or inferiour scalar quantity, and that an affected power is a scalar quantity that is connected by addition, or subtraction with one, or more, inferiour, or parodic, scalar quantities,
combined with co-efficients that raise them to the same dimension as the power itself, or make them homogeneous to it, and consequently capable of being added to it, or subtracted from it. Thus x^5
alone is a pure power of x, namely, its fifth power; and x^5 + ax^4-b^2x^3 is an affected power of x, namely, its fifth power affected by, or connected with, the two parodic, or inferiour, scalar
quantities x^3 and x^4, which are multiplied into bb and a, in order to make them homogeneous to, or of the same dimension with, x itself, and capable of being added to it or subtracted from it. See
Schooten's Edition of Vieta's works, published at Leyden in Holland in the year 1646, pages 3 and 4.
“This, then, being the meaning of the expression, a pure power and an affected power, the meaning of the corresponding expressions of a pure equation and an affected equation follows from it of
course: a pure equation signifying an equation in which a pure power of an unknown quantity is declared to be equal to some known quantity; such as the equation ; and an affected equation signifying
an equation in which a power of an unknown quantity affected by, or connected, either by addition or subtraction, with, some inferiour powers of the same unknown quantity, (multiplied into proper
co-efficients in order to make them homogeneous to the said highest power of the said unknown quantity,) is declared to be equal to some known quantity; such as the equation . This I take to be the
original meaning of the expression an affected equation. But, as the language of Vieta has not been adopted by subsequent writers of Algebra, I should think it would be more convenient to call them
by some other name. And, perhaps those of binomial, trinomial, quadrinomial, quinquinomial, and, in general, that of multinomial equations, would be as convenient as any. Thus, , might all be called
binomial equations, because they would be equations in which a binomial quantity, or quantity consisting of two terms that involved the unknown quantity x, is declared to be equal to a known
quantity; and, for a like reason, the equations , might be called trinomial equations. And the like names might be given to equations of a greater number of terms. Dr. Hutton, I observe, in his
excellent new Mathematical and Philosophical Dictionary, just now published, (Feb. 2. 1795,) calls them compound equations; which is likewise a very proper name for them, and less obscure than that
of affected equations.”
Pa. 76, col. 1, l. 25, for √(3 + 1) - √(3 - 1), read √(3 + 1) - √(3 - 1).
Pa. 94, col. 2, l. 34, for Spaniard, read Portuguese.
Pa. 95, col. 2, after l. 21, or the end of the paragraph relating to Dr. Barrow, add as follows:—Of these lectures, the 13th deserves the most special notice, being entirely employed upon Equations,
delivered in a very curious way. He there treats of the nature and number of their roots, and the limits of their magnitudes, from the description of lines accommodated to each, viz, treating the
subject as a branch of the doctrine of maxima and minima, which, in the opinion of some persons, is the right way of considering them, and far preferable to the so much boasted invention of the
generation of Equations from each other discovered by Harriot and Descartes.
Pa. 97, col. 2, after l. 3, add—Dr. Waring and the Rev. M. Vince, of Cambridge, have both given many | improvements and discoveries in series and in other branches of analysis. Those of Mr. Vince are
chiefly contained in the latter volumes of the Philosophical Transactions; where also are several of Dr. Waring's; but the bulk of this gentleman's improvements are contained in his separate
publications, particularly the Meditationes Algebraicæ, published in 1770; the Proprietates Algebraicarum Curvarum, 1772; and the Meditationes Analyticæ, 1776; an account of the chief contents of
which, a friend has favoured me with, as follows.
Of Dr. Waring's Meditationes Algebraicæ.
The first chapter treats of the transformation of algebraical equations into others, of which the roots have given algebraical relation to the roots of the given equations.
The general resolution of this problem requires the finding the aggregates of each of the values of algebraical functions of the roots of the given equation: for this purpose the author begins with
finding the sum of the m^th power of each of the roots of the equation by a series proceeding according to the dimensions of p the sum of the roots: this series (when continued in infinitum and
converges) finds also the sum of any root of the above-mentioned quantities. From this series is deduced the law of the reversion of the series , which finds x in terms of y; and also the law of a
series, which expresses the greatest or least roots, and their powers or roots of a given algebraical equation, and which may be applied whether that root is possible or impossible, if the root be
much greater or less than each of the remaining ones. All the powers and roots of this series, when continued in infinitum, observe the same law.
On this subject are further added some elegant theorems; of which, one finds the sum of all quantities of this kind a^ab^bg^c, &c; where a, b, g, &c, denote the roots of the given equation. This has
been since published by the celebrated mathematician Mr. le Grange in the Academy of Sciences at Paris.
There is also added a method of considerable utility in these matters; viz, the assuming equations whose roots are known, and thence deducing the coefficients of the equations sought: and also from
the terms of an inferior equation deducing the terms of a superior.
The second chapter principally treats of the limits and number of impossible and affirmative and negative roots of algebraical equations.
Some new properties are added, of the limiting equations resulting from multiplying the successive terms of the given equation into an arithmetical series; and a method of finding limits between each
of the roots of a given equation, since published in the Berlin Acts, and also some new methods of finding equations whose roots are limits between the roots of other equations. In theor. 4 and 5 are
contained quantities which are always greater than certain others, when they are all possible; from whence may be deduced Newton's and several other rules for finding the number of impossible roots:
these rules may be rendered somewhat more general by multiplying the given equations into others, whose roots are all possible, and finding whether im- possible roots may be deduced by the rule in
the resulting equation, which cannot from it be discovered in the given one. A rule is given, deduced from each successive four terms of the given equation, and consequently much more general than
rules deduced from each successive three terms. The former always discovers the true number of impossible roots contained in quadratic and cubic equations, the latter in quadratic only. There is also
a rule given for finding the number of impossible roots from an equation, of which the roots are the squares, &c, of the roots of a given equation; and a second from an equation of which the roots
are the squares of the differences of the roots of a given equation; and a third rule for finding an equation, of which the root is ; if be the given equation, &c, these latter resolutions always
discover the true number of impossible roots contained in cubic, biquadratic and sursolid equations; and also whether or not any impossible roots are contained in any given equation; and also from
the last term whether the number of impossible roots contained be 2, 6, 10, &c, or 0, 4, 8, &c. The principle of a 4th rule is given by finding when two roots once, twice, thrice, &c, or four, &c,
roots become equal. From a method given of finding the number of impossible roots contained in an equation involving only one unknown quantity, is deduced a method of discovering limits between which
are contained any number of impossible roots in an equation involving two or more unknown quantities. From the number of impossible, affirmative and negative roots contained in a given equation, is
delivered a method of finding the number of impossible, &c roots contained in an equation of which the roots have a given algebraical relation to the roots of the given equation.
The principles are subjoined of finding the number of affirmative and negative roots contained in an algebraical equation: but this necessarily supposes a method of finding the number of its
impossible roots known. It is demonstrated, that if the equation be multiplied by x - a, then every change of signs in the given, will have one, or three, or five, &c in the resulting equation; and
if it be multiplied by x + a, then every continuation from + to + or - <*>o -, will produce one, or three, or five, &c such continuations in the resulting, whence every equation will contain at least
so many changes of signs in its successive terms as there are affirmative roots, and so many continued progresses from + to + and - to -, as there are negative. In a biquadratic , of which two roots
are impossible, and s an affirmative quantity, then it is demonstrated that the two possible ones will be both negative or both affirmative, according as p^3 - 4pq + 8r is an affirmative or negative
quantity, if the signs of the coefficients, p, q, r, s are neither all affirmative, nor alternately - and +. The number of impossible and affirmative and negative roots contained in the equation is
likewise given, &c. If , then the content of all the values | of the quantity w will be to the content of all the values of the quantity v :: ± l^n : h^m, from whence are deduced some properties of
parabolic curves. Ex. gr. Let the equation expressing the relation between the absciss x and ordinate y be . Then will the content under the (n - 1) greatest ordinates be to the square of the content
of all the distances between any two points in which the absciss cuts the curve :: a^n-1 : n^n-2. The quotient of the content of all the sines divided by the content of all the cosines to the points
in which the absciss cuts the curve, will be to the content of all the abovementioned greatest ordinates :: n^na : 1. Similar propositions are deduced concerning the ordinates to the points of
contrary flexure, &c.
The third chapter is versant, concerning, 1st finding the roots of equations or irrational quantities, which have given relations to each other: this is performed by substitution or division and
finding the common divisors of the quantities resulting; and 2d concerning more (n) equations containing a less number (m) of supposed unknown quantities, which consequently require n-m equations,
since named equations of condition; these are likewise deduced from the method of finding common divisors. 3dly, Concerning the resolution of equations; in this case is given, 1. The reduction or
resolution of some recurring equations. 2. Some properties of the roots of the equation . 3. Resolution of a biquadratic , by reducing it to an equation . 4. A resolution of the biquadratic by adding
(p^2 + 2n) x^2 + 2pnx + n^2 to both sides of the equation, so as to complete the square; and the deducing that the values of n are (ab + gd)/2, (ag + bd)/2, (ad + bg)/2; the values of √ (q + p^2 + 2n
) are (a + b - g - d)/2, (a + g - b - d)/2, &c, and the values of √ (s + n^2) are (ab - gd)/2, (ag - bd)/2, &c; if a, b, g, d, are the roots of the given equation. 5. A resolution of equations as
general as any yet discovered, viz, the assuming ; and exterminating the irrational quantities, viz, from assuming are deduced different resolutions of cubic; from different resolutions of biquadric;
from the equations , are deduced De Moivre's equation, and several others of new formula not before delivered. 6. The resolution , first given by Euler, shewn to be a very particular; but this is
rendered here much more general by assuming a more general resolution. 7. The resolution and reduction of equations from exterminating irrational quantities. 8. Reduction of some equations, when they
are deduced from others by reducing them to the original equations. 9. The finding a quantity, which multiplied into a given irrational will produce a rational quantity, and thence deducing from a
given equation involving irrational quantities the dimensions to which the equation freed from them will ascend. 10. Let P = a series either ascending or descending according to the dimensions of x,
from thence is deduced the sum of a series consisting of its alternate terms, or terms at (n) distance from each other. 11. It is proved, that Cardan's resolution of a cubic, is a resolution of an
equation of 9 dimensions or three different cubics: similar principles are applied to some other equations. 12. General principles are given for the deducing the function of the roots of the given,
which constitute the coefficients or roots of the transformed equation. E. g. Let a cubic equation , thence is shewn the function of the roots of x, which constitute z, and further the cases of the
cubic, which are resolvable by the transformed equation, whose root is z: the same principles are applied to biquadratics. 13. The correspondent impossible roots of a given irrational quantity are
deduced; and also the different roots of a given resolution. 14. The biquadratic of the formula is distinguished into two quadratic equations involving only possible quantities, and thence every
algebraic equation is proved to consist of simple and quadratic divisors involving only possible quantities. 15. A method is delivered of transforming irrational quantities into others; but it is
cautioned, that in reduction and transformation correspondent roots should be used, otherwise it is probable that we shall fall into errors, of which examples are given. 16. The convergency of a root
found by the common method of approximations is given; and it is discovered that the convergency principally depends on the quantity assumed for the root being much more near to one root than to any
other; and independent of it, not on how near it is to a root.
The fourth chapter is principally conversant concerning more algebraical equations and their reductions to one. 1. It gives the law of the resolution of any number of simple equations; and the
reduction of n simple equations to n - 1 by means of others. 2. The method of reducing more (n) equations into one so as to exterminate n - 1 unknown quantities by the method of common divisors, and
further delivers the principles of investigating the roots or values of the unknown quantities, which result from this, or, which is much the same, from the common method of Erasmus Bartholinus, and
which are not contained in the given equations. 3. If two algebraical equations of n and m dimensions of the unknown quantities x and y are reduced to one so as to exterminate one of the unknown
quantities, the principles are given of finding the dimensions to which the other will ascend: if it ascends to n X m dimensions; then the sum of the roots depends on the terms of n and n - 1
dimensions in the one, and m and m - 1 in the other, and similarly of the products of every two; &c. From this principle are deduced several properties of algebraical curves. | The same principles
are applied to more equations involving more unknown quantities. 4. Some two equations of given formulæ are reduced to one so as to exterminate one unknown quantity. 5. Two equations are likewise
reduced to one so as to exterminate unknown quantities by means of insinite series. 6. A method of finding whether some equations contain the same roots of the unknown quantities as others. 7. From
the correspondent roots of the unknown quantities in given equations are found the constitution of their coefficients; and from thence the aggregates of the functions of the roots of two or more
equations. 8. Some things are given concerning the transformations of more equations than one, of their impossible roots, of their roots which have a given relation to each other. 9. Some reductions
and resolutions of more equations involving more unknown quantities. 10. If two equations similarly involve two unknown quantities x and y; then the equation of which the root is x or y is
demonstrated to have twice the dimensions of the equation whose root is any rational function of x + y or x^2 + y^2 or any rational recurring function of x and y; and if for y be substituted - y;
then in the equation whose root is the resulting quantity the dimensions will be the same as in the equations whose root is x or y, but its formula will be of half the number of dimensions. The same
principles are applied to more equations similarly involving more unknown quantities. 11. If there are two equations involving two unknown quantities, one deduced from the other, by some
substitutions investigated from equations similarly involving two unknown quantities; then the equation whose root is one of the unknown quantities will be recurring. 12. Let A and B be functions of
x and y, a method is given of finding, whether A is a function of B. 13. Methods of approximations to the roots of equations when they are unequal, or two or more nearly equal, possible or
impossible; and also some remarks on the increments or decrements of the roots, in passing from one equation to others of the same number of dimensions are given.
The fifth chapter treats of rational and integral values of the unknown quantities of given equations.
1. It finds the rational and integral simple, quadratic, &c divisors (by a method different to Waessaner's) of a given equation, which involves one or more unknown quantities. 2. If two equations
involve two unknown quantities x and y; the same irrationality, which is contained in x will likewise be contained in its correspondent value of y, unless two or more values of the quantity (x or y)
are equal, &c. 3. A method is given of finding integral correspondent values of the unknown quantities of two or more equations involving as many unknown quantities. 4. A method is also delivered of
deducing when a given equation can be resolved by means of square, cube, &c roots; and when by similar methods it can be reduced to equations of 1/2, 1/4, &c, its dimensions. 5. A method is given of
finding a quantity or number, in which are contained all the divisors of any given rational or integral quantities. 6. A method different from Schooten's, Newton's, and Euler's, of extracting the
root of a binomial surd a + √b is given, and the principle demonstrated on which all the rules are founded given by Schooten, viz, the multiplying the binomial surd so that the n^th root of A^2 - B
can be extracted, where A + √B is the resulting surd; and it is further proved that multiplying the given surd a + √b into 2^n will render Newton's resolution as general as the others; and lastly the
extraction of the (m^th) root of the quantity A + B√^np + C√^n(p^2) + ... + √^n(p^n-1) is given. 7. The law of Dr. Wallis's approximations in terms of the successive quotients, as also of continual
fractions is deduced. 8. A method of deducing the integral values of each of the unknown quantities x, y, z, v, &c, contained in the equation in terms of quantities, for which may be assumed any
whole numbers. 9. Two or more equations are reduced to one, so as to exterminate unknown quantities; and if the unknown quantities of the resulting equations be integral or fractional, then the
unknown quantities of the given equations will also be integral or fractional. 10. Principles are delivered of deducing equations of which the unknown quantities admit of correspondent and known
integral or rational values. 11. Correspondent integral or rational values of the unknown quantities in several equations are given, and from some values of the abovementioned kind given, are deduced
others. 12. A method of denoting any numbers either by fours, fives, fixes, &c, and their powers; and similar properties deduced as in decimal arithmetic. 13. It is demonstrated that the sum of the
divisors of the number 1. 2. 3 ... x = N has to N a greater ratio than the sum of the divisors of any number L less than N has to L; and some other similar properties. 14. In the Philosophical
Transactions are given properties similar to Mr. Euler's of the sum of divisors of the natural numbers, and some others. 15. Let , where a, b, r, p and q are whole numbers, then N2m + 1 and N2m + 2
can be compounded by (m + 1) different ways of the quantities p^2 + rq^2; the different ways were first given in the Medit. 16. Every number consists of 1, 2, 3 or 4 squares, and of 1, 2, 3, 4, .. 9
cubes, and therefore if a number N is equal to 3 squares or 8 cubes, the problem may not be possible. 17. Let x and z be any whole numbers, and a and b numbers prime to each other, then ax + bz can
constitute any number, which exceeds a X b - a - b. 18. Let r the greatest common divisor of m and n - 1, where n is a prime number; the number of remainders from the division of the number 1^m, 2^m,
3^m, &c, in infinitum by n will be (n - 1)/r + 1: from which are deduced several propositions. 19. Sir John Wilson's property delivered and demonstrated, viz, 1. 2. 3 ... n - 1 + 1 will be divisible
by n, if n be a prime number. 20. The sum of the powers 1^r + 2^r + 3^r + ... x^r are found divisible by x. ―(x + 1), if r be a whole number; from whence is deduced an elegant property of all
parabolas correspondent to the property of Archimedes of the inscribed triangles in a conical parabola. 21. Some properties of exponential equations; several other new properties of algebraical
quantities and equations are given in these Meditations. They were sent to the Royal Socicty in 1757, and since published in the years 1760, 62, and 69, |
Properties of Algebraical Carves.
The equation expressing the relation between the absciss and its correspondent ordinates of a curve is transformed into another which expresses the relation between different abscissæ and their
ordinates, from which is deduced, that there may be n and not more different diameters in a curve of n - 1 order, which cuts its ordinates in a given angle; and likewise that a diameter can have no
more than n - 1 different inclinations of its ordinates, unless the diameter be a general one. 2. The formula of the equations to curves, all whose diameters are parallel, or cut each other in a
given point, or which have a general diameter to which the lines any how inclined are ordinates. 3. It is proved that there cannot be more than n/m different inclinations of parallel ordinates, which
cut the curve in n - m points only, possible or impossible. 4. Something is added concerning diameters, which cut their ordinates on both sides into equal parts. 5. It is demonstrated that there are
curves of any number of odd orders, that cut a right line in 2, 4, 6, &c, points only; and of any number of even orders that cut a right line in 3, 5, 7, &c points; and consequently that the order of
the curve cannot be denounced from the number of points, in which it cuts a right line. 6. The principles are delivered of finding the asymptotes, parabolical legs, ovals, points, &c, of a curve, of
which the equation marking the relation between the absciss and its ordinates is given; and also given the number of asymptotes, parabolical legs of different kinds, ovals, points of different kinds,
the least order of a curve, which receives them, is deduced. 7. An equation expressing the relation between an absciss and its ordinates, is transformed into an equation expressing the relation
between the distances from two or more points, the latter may be varied an infinite number of ways; and thence are deduced some properties. Many resolutions of this kind are only resolutions of a
particular case contained in it; and consequently can never be deduced from any general reasoning; they are often deduced from some particular cases, which are known to answer several conditions of
the problem. Transformations of a given curve into others by substitutions, and properties of the loci of some points are deduced, from which Mr. Cotes's property of algebraical curves, and others of
a similar and somewhat different nature are derived. 8. Let a curve of n dimensions have n asymptotes, then the content of the n abscissæ will be to the content of the n ordinates, in the same ratio
in the curve and asymptotes, the sum of their (n) subnormals to ordinates perpendicular to their abscissæ will be equal to the curve and the asymptotes; and they will have the same central and
diametrial curves. 9. Some propositions are added concerning the construction of equations, and some equations are constructed from the principles of Slusius.—If two curves of n and m dimensions have
a common asymptote; or the terms of the equations to the curves of the greatest dimensions have a common divisor, then the curves cannot intersect each other in n X m points, possible or impossible.
If the two curves have a common general centre, and intersect each other in n X m points, then the sum of the affirmative abscissæ &c to those points will be equal to the sum of the negative; and the
sum of the n subnormals to a curve which has a general centre will be proportional to the distance from that centre. 10. Something is added on the description of curves. 11. No curve which has an
hyperbolical leg of the conical kind can in general be squared. 12. It is demonstrated that no oval sigure, which does not intersect itself in a given point, can in general be expressed in finite
algebraical terms. 13. Given an algebraical equation, and similarly equations expressing a relation between x and y, &c; and also a fluxional quantity which is an algebraical function (z) of x and y
and their fluxions; a method is given of deducing an equation whose root is z; and thence some properties of curves. 14. Properties similar to the subsequent of conic sections, are extended to curves
of superior orders, viz, if lines be drawn from given points in them in given angles to four lines inscribed in the conic section, then will the rectangle under two of those lines be to the rectangle
under the other two in a given ratio. Several properties are added, which follow from the application of algebraical propositions invented in the Medit. Algebr. to curve lines.
The second chapter treats of curvoids and epicurvoids, or curves generated by the rotation of given curves on right lines or curves, and gives a method of rectifying and squaring them; and from the
radii of curvature of the generating curves being given, it deduces the length and radius of curvature of the curve generated at the correspondent point; it also asserts that from them may be deduced
the construction of the fluxional equations of the different orders.
The third chapter treats of algebraical solids. 1. It deduces the equation to every section of a solid generated by the rotation of a curve round its axis; and from thence the different sections
generated by the rotation of conic sections round their axis. 2. The equation to solids contains the relation between the two abscissæ and their ordinates, and the order of the solid may be
distinguished according to the dimensions of the equation; or the solid may be defined by two equations expressing the relation between the three abovementioned quantities, and a fourth which may be
the axis of the section: there is further given a method of deducing the equation to any section of these solids, and from it the equation to the curve projected on a plane by a given curve. 3. A
method of deducing the projection of a curve or solid on each other. 4. If the equation be x - a = 0, (x being the distance from a given point) then it may denote the periphery of a circle if one
plane, or the surface of a globe if it refers to a solid. 5. Let x and y denote the distances from two respective points, then an equation expressing the relation between x and y designs the
periphery of a curve, if contained in the same plane, or the surface of a solid generated by the rotation of a curve round its axis, passing through the two given points, if a solid. 6. An equation
expressing the relation between lines drawn from three or more points may denote an equation to a solid. 7. If x, z and y denote the two abscisses and correspondent ordinates to a solid, and the
terms of x and y, or x and z, or y and z; or x, z and y be similarly involved; then may the solid be divided into two | or six similar and equal parts; and if no unequal power of x or y or z; or x
and y, &c; or x, y and z be contained in the equation, then the curve may further be divided in general into twice, four or six times the preceding number of equal parts. 7. Curves of double
curvature are designed by two equations expressing the relation between two abscissæ and correspondent ordinates, or between lines drawn from three or more points; similar properties may be deduced
from these as from the equations to curves.
Chapter the 4th treats of the maxims and minims of polygons inscribed and circumscribed about curves, and thence deduces certain quantities equal to each other, when maxims and minims are contained
at every point of the curve: it further contains several properties of conic sections. 1. If any rectilinear figure circumscribes an ellipse, the content under the alternate segments of the line made
by the points in which the line touches the ellipse will be equal. 2. If a right line cuts a conic section, and the parts of the line without the conic section on both sides are equal; and any
rectilinear figure, which begins and ends at the bounds of the abovementioned line, be described round the conic section, then the contents under the alternate segments of the circumscribing lines as
divided in the points of contact will be equal. 3. If two polygons be circumscribed about an ellipse, and the sides are cut by the points of contacts in the same ratios in the one as in the other;
then will the areas of the two polygons be equal. 4. If two lines cut a conic section proportionally, i. e. they are divided by the conic section in the same ratio in the one as in the other, and if
polygons be described round the conic section, terminated at the ends of those lines, of which the sides are divided by the points of contact in the same ratio in the one as in the other, then will
the area of the two polygons be equal, as likewise the curvilinear area. 5. If all the sides of two polygons inscribed in an ellipse make the two angles at the same point equal, and two polygons of
this kind be inscribed in the curve, then will the sum of the sides of the one polygon be equal to the sum of the sides of the other. Several other similar properties are added, as also properties of
solids generated by the rotation of a conic section round its axis; to which I shall mention the three or four following. 1. The diagonals of a parallelogram circumscribing an ellipse or hyperbola
will be conjugate diameters. 2. The sections of a solid generated by the rotation of a conic section round its axis, which pass through its focus, will have that point for the focus of all the
sections. 3. If 4 perpendiculars be drawn from any point in an hyperbola to its periphery; and two lines from the same point to the asymptotes and the ordinates from the 4 points of the curve and the
2 of the asymptotes be drawn to the absciss; then will the sum of the resulting abscissæ to the former be double to the sum of the abscissæ to the latter. 4. If an arc of the periphery of a circle be
divided into n equal parts, a, 2a, 3a, &c, and p = chord of the arc 180 - na, and a and b be the roots of the quadratic and radius 1: then will a^n+b^n=chord of the arc 180 - na, from whence may be
deduced the divisors of the quantity x^2n - Ax^n + 1; and also the equation whose roots are the distances of a point in the circle from those points of equal division, and further may be deduced the
sum of all the values of any algebraical function of those lines.
Most of the properties of circles given by Archimedes are extended to conic sections, and some of the algebraical and geometrical properties of Pappus are rendered more general; and the principles
invented applied to many other cases. In the first edition of this book published in 1762 were nearly enumerated the lines of the fourth order on the same principles as Newton's enumeration of lines
of the third order; but this has since been rejected by the author as not sufficiently distinguishing the curve, and as being of no great utility.
Meditationes Analyticæ.
The first chapter treats of finding the fluxion of a fluent, when the quantity or fluent is considered as generated by motion; or the parts from the whole when the whole or quantity is considered as
consisting of innumerable parts. It further gives the law of a series, which expresses the fluxion of an exponential of any order.
Chapter 2, is versant about the fluents of fluxions. 1. It finds the general fluent of a fluxion Px^., when P is any algebraical function of x however irrational but not exponential; for which intent
it investigates the common divisors of any two quantities contained under the different vincula; and thence the common divisors of the resulting divisors, and so on; and likewise all the equal
divisors contained in any of the abovementioned quantities; whence it so reduces the quantity P, that no equal nor common divisors may be contained in any of the resulting quantities under the
different vincula; and from the common method deduces the terms of a series to the number, which the series is shewn to consist of, when it does not proceed in infinitum. 2. It demonstrates, that if
the dimensions of x in the denominator of P exceed its dimensions in the numerator by 1, then the fluent cannot be expressed in finite terms; and also if one factor of P be (A ± (A^2 + a)^1/2)^l,
where a is an invariable quantity, and in some other cases the substitution required must be somewhat different. 3. The fluents of some fluential and exponential fluxions, or fluxions involving
fluents and exponential quantities, are given. 4. A general method of discovering whether the fluent of any fluxion of any order involving one, two or more variable quantities, and their fluxions,
can be expressed in terms of the variable quantities and their fluxions. 5. The correction of fluents of all orders, and thence the fluent contained between any values of the variable quantities and
their fluxions, is given; in these corrections the same roots of the irrational quantities are to be used in the correction as in the fluent. 6. From the transformation of equations and the
principles before delivered, are deduced fluents equal to each other. 7. Some exponential quantities given which continually change from possibility to impossibility, and from impossibility to
possibility. 8. Is a method of finding whether the fluent of any fluxion contained between any limits are finite or not. 9. The sum of the fluents of a fluxion which is. an algebraical function of
the letter x multiplied into x^. can always be expressed by finite terms, circular arcs and logarithms, the extraction of the roots of equations being granted. | 10. Some fluxions involving
irrational quantities are reduced to others, in which no irrationality is contained. 11. The general principles of deducing whether the fluent of a given fluxion can generally be expressed by finite
algebraical terms, their circular arcs and logarithms. 12. Some equal correspondent fluents are found by substitutions deduced from equations in which two variable quantities are similarly involved.
13. Some necessary corrections are given of finding the fluents of all the fluxions of the formula x^pn ± sn - 1 x^. X R^m ± l X S^o ± m X T^t X n X &c, (where s, l, m, n, &c denote any whole
numbers, and from a + b + g + &c, independent fluents; but perhaps not from a + b + g + &c fluents, which have different values of the quantities, s, l, m, n, &c. 14. The number of independent
fluents of the formulæ x^q + an + bm X (a + bx^n + cx^m)^l + p X x^., where a, b and p denote whole affirmative numbers, &c; and the number of independent fluents of the formulæ X^.∫Yx^., where X^.
is a fluxion of which the fluent can be sound, from which can be deduced all of the same formula, is immediately known from the number of independent fluents of the formula Yx^. and XYx^. which
determine all of those formulæ. 15. Let , and from some fluents of the fluxions of the formulæ p X x^mn - 1 x^., where m is a whole affirmative number, are determined the remaining ones of the same
formula. 16. Something is added concerning finding the value of a fraction, when both the numerator and denominator vanish; and lastly from the fluents of some fluxions being given, the method of
deducing the fluents of others.
Chapter 3, principally treats of algebraical and fluxional equations. 1. It gives the method of transforming two or more fluxional equations into one so as to exterminate one or more variable
quantities and their fluxions, and finds the order of the resulting equation. 2. It reduces some fluxional equations into more. 3. A method of reducing fluxional equations involving fluents so as to
exterminate the fluents. 3. Some cases are given, in which the two variable quantities contained in a given equation are expressed in terms of a third. 4. Given an algebraical equation expressing the
relation between x and y; a method is given of finding the fluent of yx^.^n or other fluxions in finite terms of x and y, if they can be expressed by such; or else by infinite series; this was first
taught in the Philosophical Transactions in the year 1764. 5. Something is added concerning the correction of fluxional equations. 6. A method of investigating, whether a given equation is the
general fluent of a given fluxional equation. 7. The method of deducing, whether a given equation is a particular or general fluent of a given fluxional equation. In both by substituting for the
fluxions their values deduced from the fluential equation their values &c in the fluxional, the fluxional must result = 0; and in the general fluent there must be contained so many invariable
quantities to be assumed at will independently as is the order of the fluent; and in both all the variable quantities must necessarily be variable, and no function of them vanish out of the fluxional
equation from the substitution; for then all the conditions of the fluxional equation are answered by the fluential. 8. An investigation, when fluxional equations are integrable. 9. From some fluents
are deduced others, e. g. if the area between any two ordinates to one abscissa can in general be found, then the area between any two ordinates of any other abscissa can be found &c. 10. From given
fluxional equations and the fluents of some fluxions are deduced the fluents of many others. 11. The fluent of the first order of a fluxional equation of the nth order will have (n) different values
and n different multipliers; and the fluent of the second order n.(n - 1)/2 different values, &c. 12. Let a = 0, b = 0, g = 0, &c, (n) general fluents of the fluxional equation, l = 0, then will any
function of the fluents a, b, g, &c be a fluent of the same fluxional equation l = 0. 13. From assuming equations, which contain only simple powers of the invariable quantities to be assumed at will,
may easily be deduced fluxional equations, of which the general resolutions are known: 2. From assuming the values of any variable quantities and substituting then their fluxions for the variable
quantities, &c. in any functions p, r, &c of the variables assumed, let the quantities resulting be A, B, &c; then generally will p = A, r = B, &c. be fluxional equations, of which the particular
fluentials are known. It may be observed in this place as before, that from no general reasoning can particular fluents be deduced. 14. In the resolution of fluxional equations it is observed, that
from the logarithmic and exponential quantities contained in the fluxional, may be deduced by chapter 1 the exponentials &c contained in the fluential: 2, and in a similar manner from the irrational
quantities and denominators contained in it, the correspondent irrational quantities and denominators contained in the fluential: 3, the greatest dimensions of y multiplied into x^. must be greater
than those of y into y^. by unity; when there are two of this kind &c, (dx^. + ey^.) the refolution is given; and so of more. 15. In the given equation, if the fluxion of the greatest order does not
ascend to one dimension only; then by extraction &c so reduce the equation, that it may ascend to one dimension only; and thence find the fluent of any fluxion P^ny^. + Q^n - 1y^. + &c, + R′^nz^. + &
c. 16. Let a fluxional equation be given involving x and y, in which x flows uniformly, a method is given of finding whether it admits of a multiplier, which is a function of x ∴ and similarly of
multipliers of other formulæ. 17. The method of deducing the multipliers of fluxional equations by infinite series. 18. Some fluxional equations are reduced by substitutions, which substitutions are
commonly easily deducible from the fluxional equation given. 19. Somewhat concerning the reduction of some fluxional equations to homogeneous, and concerning homogeneous equations of different
orders; and of reducing an homogeneous fluxional equation of n order to a fluxional equation of n - 1 order: and | also of reducing m fluxional equations of n order to one of mn - 1 orders, and so of
all others to one degree less than the order generally occurring if they had not been homogeneous. 20. The substitution of an exponential for a variable quantity in equations which contain no
exponential quantity; for sometimes n has been substituted for a quantity which flows uniformly, and then w supposed to flow uniformly, which leads to a false resolution. 21. A caution is given not
to substitute homogeneous functions of no dimensions for variable quantities; and in the general resolution to observe, that there is contained an invariable quantity to be assumed at will, which is
not contained in the fluxional equation. 22. Something more added concerning the fluents of , where p, q, r, &c. are functions of x, and so of some other fluxional equations. 23. Fluxional equations
are deduced, of which the variable quantities cannot be expressed in terms of each other, but both may be expressed in terms of a third. 24. Every fluxion or fluent which is a function x, y, z, and
x, y. &c. is expressed in terms of partial differences. 25. The resolution of some equations expressing the relation between partial differences &c is given. 26. Some observations on finding the
fluents of fluxions, when the variable quantities become infinite.
The second book treats of increments and their integrals. 1. Some new laws of the increments are given. 2. The fluxion of the increment of P will be equal to the increment of the fluxion; where P is
any function of x, if only the fluxion of the increment of x be equal to the increment of the fluxion. 3. Increments are reduced to others of given formulæ e. g. a + b/x + g/(x(x+x[.])) +, &c, and it
is observed that if b be not = 0, then the integral cannot be found in finite terms of the variable quantity, &c. It may be observed, that Taylor, Monmort, &c, first found the integral of the two
increments x.―(x-x[.]).―(x-2x[.]) ... ―(x-―(n-1) x[.]) and 1/(x.―(x-x[.]) ...―(x-―(n-1)x[.])) but did not proceed much further (correspondent to the finding the fluxion of the fluent x^n); the
increments of fluents have been since deduced, &c. In this book are discovered propositions correspondent to most of the inventions in fluxions, e. g. a method of finding the integral of any
increment expressed in algebraical or exponential terms of the variable quantity or quantities, and when the fluent cannot be expressed: it is observed that they cannot be expressed in finite terms
of the variable x, &c, if the dimensions of x, &c, in the denominator exceed its dimensions in the numerator by 1; or if any factor in the denominator of the fraction reduced to its lower terms have
not another contained likewise in the denominator, distant by a whole number, multiplied into the increment of x. —The increments of some integrals are deduced from the integrals of other increments;
the integrals of some incremental equations from different methods; their general integrals, and particular corrections, &c, &c; but here it is to be observed, that the general problem of increments
cannot be extended beyond the particular of fluxions, but somewhat more may be added, when both are joined together. The third book is versant concerning infinite series. 1. It gives the ratio of the
apparent and real convergency. 2. A method of finding limits between which the sum of the series consists; and also whether the sum of the series is sinite or not from the terms being given or
equation between the terms. 3. The convergency of the whole series is judged from the ratio of convergency of the terms at an insinite distance. 4. The series from the fluent converges, if the series
from the fluxion does, there are several propositions on insinite series deducible from the common algebra. 5. Let an equation ; and b/a much greater than c/b, c/b than d/c; &c. then will all the
roots be possible, and a/b an approximation to the least root, b/c to the next, &c: if an equation , and if one root be much less than any m root, but much greater than the remaining; or if the
equation be , then will the approximation to the above root be i/h - (k/i - (gi^2)/h^3) + &c. 6. Somewhat on the approximations when the approximation given is much more near to one, two, or more
roots than to any other, and on the degree of convergency of the subsequent approximations deduced; and their ultimate approximations. 7. Given approximations to m roots of a given equation are
deduced more near approximations to them. 8. The incremental equation given and applied to approximations. 9. From given approximations to two or more unknown quantities contained in two or more
equations are deduced more near approximations to them, either when the approximations given are more near to one, or to two, or more roots of one or more of the unknown quantities than to any
others, and so of infinite equations. 10. New series are given for the fluents of different fluxions. 1. . The sine of the arc A±e is S ± Ce - 1/2 Se^2, &c, and cosine of the same arc = C± Se - 1/
(2.3) Ce^2 ± &c. S and C being the sine and cosine of A, the fluent of the fluxion of an elliptical arc √((1-cx^2)x^.)/√(1 - x^3) which differs little from the arc of a circle when e is a very small
quantity = A′ - c/2 X (1.A - xP)/2 - &c, where , | , and A = arc of a circle of which the sine is x.
A similar series may be applied from the arc of an hyperbola or ellipse, to find a correspondent arc of an hyperbola or ellipse not much different from the preceding. In this method the series
proceeds according to the dimensions of some small quantities, and the first term of the series is generally a near value of the quantity sought. These series properly instituted will generally
converge the swiftest. 11. Something new is added concerning the fluent of the fluxional equation ; E and F being any quantities to be assumed at will; and of correspondent equations to logarithms,
and finding their values when z is increased by e. 12. A series for the increase of the arc from a small increase of the tangent, fine, &c. 12. When the terms a and x of the binomial a±x are equal,
the cases are given in which the series or the series a^mx = (m/2)a^m-1 x + &c, &c. will ultimately converge. 13. If any algebraical quantity V a function of x be reduced into a series proceeding
according to the dimensions of x, a general method of finding what are the limits between which it converges; or the series from ∫ Vx^., &c; and the method of interpolations so as to render them
converging. 14. The convergency of different series are compared together. e. g. is given : there is an erratum contained in this example, for a - is sometimes printed instead of a +: this series is
easily deduced from Bernouilli's method of deducing infinite series, and has been since printed in the Philosophical Transactions. 15. Given algebraical or fluxional equations, and a fluxional
quantity, a method is given of finding a series, which expresses the fluent of the fluxional quantity, from which principles are deduced new series for the area of a segment of a circle, the
periphery of the ellipse, hyperbola, &c. 16. It is shewn, that serieses proceeding according to the dimensions of a quantity x always diverge, when serieses for the same purpose proceeding according
to the reciprocal of its dimensions converge; unless sometimes in the case when they both become the same. 17. As series proceeding in infinitum according to the dimensions of the quantity x were
first invented or used for the finding the fluents of fluxions, it being reduced into terms, whose fluents were known: so in finding integrals of increments it may be necessary to reduce the quantity
into an infinite series of terms, whose integrals are known, and which converges. Examples of formulæ of serieses of this kind are given. 18. Methods are given of finding the value of one unknown
quantity contained in one or more equations involving more unknown quantities, and the law of their convergencies and the interpolations necessary to render serieses for finding fluents converging,
similar principles may be applied to incremental and fluxional equations. 19. It is observed, that in finding the value of any variable quantity in a series proceeding according to the dimensions of
another, there will occur in a fluxional or incremental equation of (n) order in the series n invariable quantities to be assumed at will; and also the fluxional equations, &c. from whence they will
arise. 20. The finding the integral of ż/z, &c. 21. From the correspondent relation between the sums of two series resulting, which are functions of a variable quantity y, when the relation between x
and z two values of y are given, is given a method of finding the coefficients of the series. 22. The rule generally called the reductio ad absurdum extended to more substitutions.
The fourth book treats of the summation of series, a method of correspondent values and several other problems. 1. Of finding the sum of a series expressed by a rational function of z into x^n2;
where z denotes successively the numbers 1, 2, 3, &c, in infinitum. 2. Given an equation expressing the relation between the successive sums, the relation between the successive terms is known, and
the vice versa, &c. 3. It is found from an equation expressing the relation between the successive sums, terms and z the distance from the first term of the series, whether the sum of the series is
finite or not. 4. The difference between z^-0 and ―(z+1)[-0], where z denotes the distance from the first term of the series, will be — 0 X z^-0-1, which is greater than the simple ratio let 0 be as
small as possible, and consequently the sum of the series finite. 5. If a series a + bx + cx^2 + x^3, of which at an infinite distance the preceding coefficients have to the subsequent the ratio of r
:1, be multiplied into a function = 0, when x=d, then if a be greater than r the series will diverge; if less converge. 6. From adding several terms of one or more series together may be formed a
series, of which the sum from the sums of the preceding series is known. 6. Serieses are formed, of which the sums are known from varying the divisors, &c. 7. From given series are deduced others, of
which the sums are known, and the sum of many series are deduced from finding the fluxions of fluents and fluents of fluxions. 8. From the relation between the different terms given is deduced the
correspondent fluxional equation. 9. The finding the terms of any series, which can be deduced from given series; and thence deducing many series of which the sums can be found from the sum of the
given series. 10. Series are given of which the sums can be found from finite terms, circular arcs, logarithms, elliptical and hyperbolical arcs. 11. From a general expression, when algebraical,
fluxional, incremental, &c, for the sum of a series can be deduced a similar expression for the sum of every second, third, &c, terms. 12. An infinite series may be a particular resolution of
infinite fluxional equations. 13. The terms of some series may be infinite and their sums known. 14. The general fluent of is given by a series of the same kind, and the same of some other fluxional
equations. 15. A quantity is found which multiplied into a series | more swiftly converging gives a given series. 16. The first differences of the terms of some series are given; if the terms are in
geometrical ratio to each other the abovementioned differences will also be in geometrical ratio to each other: whence it appears, that the series from this method of differences will converge least
when the given series converges swiftest, &c, but not always the contrary. Several other propositions are added concerning the method of differences applied to series. 17. A parabolico-hyperbolical
curve is drawn through any number of points, as also an algebraical solid —. 18. Something is given concerning the convergency &c. of series deduced from the differences of the numerators of a given
series, of which the denominators constitute a geometrical progression. 19. A rule is given for rendering series converging, in which it is observed that the sum of so many terms should be found that
z the distance from the first term of the series may exceed the greatest root of the equation resulting from the quantity which expresses the term made = 0. 20. An equation expressing the relation
between the sums and terms is reduced to an infinite fluxional equation expressing the relation between the sum or term, its fluxions, and z the distance from the first term of the series. 21. From a
method being known of finding the sum of a series, which involves one variable only, is given a method of finding the sum of series which involve more variable quantities: and from assuming sums of
serieses of this kind are deduced their terms. 22. The sums of series are found consisting of irrational terms. 23. The principle of the convergency of the approximations found in drawing parabolical
curves through given points. 24. Something new is given concerning the interpolations of quantities. 25. . if a, b, g, &c, are the roots of , &c. 26. Something is added concerning series from . 27.
Nandens's Problems are somewhat extended. 28. Something is added on changing continual fractions into others. 29. A method of transforming series into continual factors. 30. A rule for finding the
sine and cosine of n/m the arc; and transforming an algebraical equation into an equation expressed in terms of sines and cosines, and thence from an approximation to the sine is found one more near;
the same might have been performed by tangents, cotangents, secants, cosecants, &c. 31. From some-fluents given have been found others, and consequently by reducing the fluents to infinite series
from some infinite series given may others be deduced. 32. The fluent of (xa x^.)/(1 ± x^n) is found by approximation, where a is an irrational quantity, which method of finding approximations to the
indices may be applied to other cases. 33. The sum of the fractions are found when the denominators = 0, and consequently each particular in- finite. 34. It is asserted, that the sum of certain
fractions given become = 0, when the terms are expressed by a fraction of which the denominator is a rational function of the distance from the first term of the series. 35. ∫x^a - b - 1x^.∫x^b - g -
1x^. ∫x^g - d - 1x^. X P, where , &c, will be to ∫x^b - a - 1x^. ∫x^a - g - 1x∫x^g - d - 1x^.∫&c. X P :: x^z : x^b if the fluents are contained between the same values of x. 36. Are given some series
consisting of two, of which the one converges, when the other diverges, and consequently the sum of both diverges; &c. 37. From the law of a series being given, the law of the series which expresses
the square, or some function of the given series, is found.
1. A method of differences, which deduces from the sums given any successive sums, e. g. Let S^1, S^2, S^3, S^4, be the logarithms of the ratios r : r + p, r : r + 2p, r : r + 3p, r : r + 4p, then
will the logarithm of r : r + 5p be 5 X (S^4 - S^2) + 10(S^2 - S^3) nearly: then rules are given in general, and likewise their errors from the true values.
2. A method of correspondent values is given, e. g. Let a, b, c, d, &c, be values of x; and S^a, S^b, S^c, S^d, &c, correspondent values of y; then may .
3. If the formula of the series be ; if the formula of the series be , which answers to Briggs's or Newton's method of interpolations; or the series will be x^h/a^h X ((x^k - b^k) (x^k - c^k) (x^k -
d^k) &c)/((a^k - b^k) (a^k - c^k) (a^k - d^k) &c) X S^a + x^h/b^h X ((x^k - a^k) (x^k - c^k) (x^k - d^k) &c)/((b^k - a^k) (b^k - c^k) (b^k - d^k) &c) X S^b + &c; if the formula of the series be a
general formula, which includes the preceding.
5. The series is given for deducing others when the number of correspondent values given are either even or odd, and the values of x are equidistant from each other. 6. And also from correspondent
values of x and y to a number of equidistant values of x is deduced the value of y to the next successive or any successive value of x. 7. Some arithmetical theorems are deduced from the preceding
propositions. 8. Another method is given of resolving the preceding problem. 9. A method of correcting the solution from a solution | given which finds (n) values of y to (n) given values of x true,
and m false to (m) other values. 10. A similar resolution is added from correspondent values of x, y, z, &c given; and more general resolutions. 11. Given the resolution of some cases, and formula in
which the general is contained, a method is given in some cases of deducing it. 12. The principles of a method of deductions and reductions are added.
In a Pamphlet published at Cambridge, algebraical quantities are translated into probable relations, and some theorems on probabilities thence deduced; to which are adjoined,
1. The theorem ; this becomes the binomial theorem when l = 0; and it will afford answers to similar cases when the whole number of chances are increased or diminished constantly by l, as the
binomial does when they remain the same, a similar multinomial theorem is given. In the same pamphlet are further added some new propositions on chances, on the values of lives, survivorships, &c. In
these books are also contained the inventions of others on similar subjects, which in the prefaces are ascribed to their respective authors.
In the Philosophical Transactions are given some properties of numbers, &c, of which some have been published in the books above mentioned; to which may be subjoined something in mixed mathematics,
viz, a paper on central forces, which extends not only to central forces, but also to forces applied in any other direction, as in the direction of the tangent, and consequently includes resistances,
&c. It gives a rule for finding the forces tending to two or more given points when the curve described and velocity of the body in every point of it is given, e. g. Let the curve be an ellipse, and
the velocity the same at every point, and the two centres of force be the foci of the ellipse; then will the forces tending to the two foci be equal, and vary as the square of the sine of the angle
contained between the distance from the centre of force to the point in which the body is situated, and the tangent to the curve at that point.
The method of deducing the fluxional equations which express the curve described by a body acted on by any forces tending to given points, or applied in any given directions: some other propositions
are contained on similar subjects. 2. A paper on the fluxions of the attractions of lines, surfaces, and solids, and from the different methods of deducing them are found different fluents equal to
each other: a third paper gives a solution of Kepler's problem of cutting the area of a circle described round a point by approximations, which also is applied to other cases; this like- wise
contains some other problems. Many of these discoveries have since been published, some in the London, and other foreign transactions.
Let , then will l denote the log. of N to the modulus e. If e the modulus = 10, then will the system be the common or Briggs's system of logarithms. Logarithms, and the sums of some other serieses,
of the formulæ ax^h + bx^h + k + &c may be deduced in a manner similar to that which was used by the Ancients for finding the sines of the arcs of circles.
To particularise the numerous propositions contained in these works, would exceed the limits of our design. Besides those already mentioned, others are interspersed through the whole works.
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Redwood Estates ACT Tutor
Find a Redwood Estates ACT Tutor
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CGTalk - A form of aim constrain [Math]
I am currently working on a project which involves a lot of rigid bodies. And I have found a problem, I need to aim my rigid bodies (a cannon) so it can hit a moving target.
What I have found out that best best thing would be to find out if the cannon is aiming at the target, in front of target or behind the target. I can than use impulses to rotate the cannon. Or atlest
that is what I think.
I have tryed to do this with some trigometrik, but I most admit that math is not my strongest field, even though I find it interesting. So any help would be great, or a link to a page explaing the
consept about what I am trying to do.
Thanks in advance
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etale map
etale map
Étale morphisms
The notion of étale map is an abstraction of that of local homeomorphism in topology. The concept is usually found in places with a geometric or topological flavour.
Between topological spaces
An étale map between topological spaces is a local homeomorphism; see étalé space (which is the total space of such a map viewed as a bundle).
Between smooth manifolds
An étale map between smooth spaces is a local diffeomorphisms, which is in particular a local homeomorphism on the underlying topological spaces.
Between schemes (affine schemes / rings)
For an étale map between schemes see étale morphism of schemes.
Restricted to affine schemes, this yields, dually, a notion of étale morphisms between rings. Étale maps between noncommutative rings have also been considered.
Between analytic spaces
• Étale maps between analytic spaces are closely related to étale maps between schemes, while also a special case of an étale map between smooth spaces.
Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth
Between toposes
The idea of étale morphisms can be axiomatized in any topos. This idea goes back to lectures by André Joyal in the 1970s. See (Joyal-Moerdijk 1994) and (Dubuc 2000).
Axiomatizations of the notion of étale maps in general toposes are discussed in
• Eduardo Dubuc, Axiomatic etal maps and a theory of spectrum, Journal of pure and applied algebra, 149 (2000)
Revised on April 12, 2014 00:38:59 by
Urs Schreiber
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Applications of Microsoft® Excel in Analytical Chemistry,
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Applications of Microsoft® Excel in Analytical Chemistry, 2nd Edition | 9781285087955
ISBN-13: 9781285087955 See more
Author(s): Holler/Crouch
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Additional product details
ISBN-10 1285300491, ISBN-13 9781285300498
ISBN-10 128508795X, ISBN-13 9781285087955
Author(s): Holler/Crouch
Publisher: Cengage Learning
Copyright year: © 2014 Pages: 480
This supplement can be used in any analytical chemistry course. The exercises teaches you how to use Microsoft® Excel® using applications from statistics, data analysis equilibrium calculations,
curve fitting, and more. Operations include everything from basic arithmetic and cell formatting to Solver, Goal Seek, and the Data Analysis Toolpak. The authors show you how to use a spreadsheet to
construct log diagrams and to plot the results. Statistical data treatment includes descriptive statistics, linear regression, hypothesis testing, and analysis of variance. Tutorial exercises include
nonlinear regression such as fitting the Van Deemter equation, fitting kinetics data, determining error coefficients in spectrophotometry, and calculating titration curves. Additional features
include solving complex systems of equilibrium equations and advanced graphical methods: error bars, charts with insets, matrices and determinants, and much more.
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A Golf Club Has A Lightweight Flexible Shaft With ... | Chegg.com
A golf club has a lightweight flexible shaft with a heavy block of wood or metal (called the head of the club) at the end. A golfer making a long shot off the tee uses a driver, a club whose 300 g
head is much more massive than the 46 g ball it will hit. The golfer swings the driver so that the club head is moving at 43 m/s just before it collides with the ball. The collision is so rapid that
it can be treated as the collision of a moving 300 g mass (the club head) with a stationary 46 g mass (the ball); the shaft of the club and the golfer can be ignored. The collision takes 4.4 ms, and
the ball leaves the tee with a speed of 75 m/s.
What is the speed of the club head immediately after the collision?
a.) 32 m/s
b.) 28 m/s
c.) 22 m/s
d.) 12 m/s
If we define the kinetic energy of the club head before the collision as "what you paid" and the kinetic energy of the ball immediately after as "what you get," what is the efficiency of this energy
a.) 0.54
b.) 0.46
c.) 0.37
d.) 0.27
(Please show work)
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A Venturi Meter Is A Device For Measuring The Speed ... | Chegg.com
A Venturi meter is a device for measuring the speed of a fluid within a pipe. The drawing shows a gas flowing at speed v2 through a horizontal section of pipe whose cross-sectional area is A2 =
0.0689 m2. The gas has a density of ? = 1.30 kg/m3. The Venturi meter has a cross-sectional area of A1 = 0.0212 m2 and has been substituted for a section of the larger pipe. The pressure difference
between the two sections is P2 - P1 = 176 Pa. Find (a) the speed v2 of the gas in the larger original pipe and (b) the volume flow rate Q of the gas
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Assorted Triples 5: Europe, Europa, Rail Europe [COMPLETE]
Commander62890, Ljex, and Dustine win Assorted Triples 5
This is the fifth installment in a series of short triples tournaments.
This tournament will feature 8 teams battling it out in a single-elimination format. All games will be sequential and have automatic deployment.
There will be 3 rounds, with a different map and settings used in each round. No map will be used twice, and the settings will be mixed in order to challenge the participants.
There will be 1 game per round.
In round 1, you will generally find medium-sized maps. It could also be a quirky map; though, if it is, it will be one of the smaller/less complicated ones.
In round 2, you will generally find large maps. Maps with lots of territories, but probably not a lot of quirkiness, will be found here.
In round 3, you will generally find more complicated maps. In this category are included some of the toughest of CC's maps.
For this tournament, the maps and settings we will be using are:
Round 1: Europe --- No Spoils, Adjacent, Foggy
Round 2: Europa --- Escalating, Unlimited, No Fog
Round 3: Rail Europe --- Escalating, Chained, No Fog
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A Thought Experiment with Time Dilation
A thought experiment:
Reference frame: A surface in space with no atmosphere.
A rocket ship is approaching just above the level of the surface. Its measured velocity is a constant 0.8c (0.8 times the speed of light) relative to the surface. Special relativity predicts that
time in the space ship runs at 0.6 times that on the space surface.
The space ship has a very small window which will face the space surface when it arrives, as well as two light detectors, one at the window and a long one 1 m away and parallel to the window. A laser
beam projects a beam perpendicular to the space surface and in the path of the coming spaceship window. A very short flash of light falls on the window detector and then on the detector 1m away as
the space ship passes.
The detectors in the space ship show the beam to take a skewed path because of the ships 0.8c motion relative to the light beam. The length of that path is observed to be (0.8^2+1^2)^1/2 = 1.28 m.
Since the time dilation is 0.6 relative to the surface, the time difference at the two detectors measured by a clock in the spaceship should be: 0.6x1.28/c where c is in m/sec. This yields 2.56x10^-9
seconds. Dividing by the distance 1.28 m gives a time it took the beam to travel a meter of 2.00x10^-9, yielding a measured c of 5.0x10^8 m/sec if the time dilation were correct. Of course this is
nonsense. (Note that 5.0x10^8/0.6 = 3.0x10^8 suggesting maybe there is no time dilation in the transverse direction.)
A person on the space surface, knowing that the window and screen are 1 m apart and recognizing that the very short beam will travel in a straight line calculates that the light beam takes 1/c
seconds to traverse the distance between the window and screen from his reference position which is 3.33x10^-9 sec. A person in the space ship sees the light beam traveling 1.28 m in his reference
frame. Since he must, according to relativity, measure c at 3.0x10^8 m/sec he should experience a time delay between the two light signals of 1.281/3.0x10^8 or 4.27x10^9 sec. Thus the time dilation
factor would be 0.781 in the transverse direction and not 0.6 as special relativity would predict for the direction of travel.
How can time be dilated differently in one direction (the direction of travel) from the transverse direction? A clock cannot distinguish directions. Perhaps someone can straighten me out, or time
dilation in special relativity crumbles.
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Topic: pwelch power correction factor
Replies: 6 Last Post: Dec 7, 2012 10:33 AM
Messages: [ Previous | Next ]
Re: pwelch power correction factor
Posted: Nov 19, 2010 12:31 AM
On Nov 19, 5:37 am, "Henik Vargas" <hvarg...@gmail.com> wrote:
> > What makes you think you need a correction factor?
> > Instead of replicating what matlab does, read up on
> > the method you play with.
> > Rune
> Rune,
> The motivation for the correction factor comes from the fact that windowing reduces the power of the signal, therefore, the power of the psd is much lower than that of the unwindowed signal.
Another way of looking at it is that the correction factor makes the power spectrum estimate asymptotically unbiased (See Hayes, Statistical Digital Signal Processing and Modeling for a reference).
> Also, I read the documentation for the function, and I think my question is a fair one. I could just blindly use the function as is, but then I lose a learning opportunity.
My point is that in the applications where spectrum estimation
methods like Welch's method are used, the main focus is to get
an *impression* of the spectrum shape. The exact numbers are
As I said, read up on the methods.
Date Subject Author
11/15/10 pwelch power correction factor Henik Vargas
11/16/10 Re: pwelch power correction factor Rune Allnor
11/18/10 Re: pwelch power correction factor Henik Vargas
11/19/10 Re: pwelch power correction factor Rune Allnor
11/19/10 Re: pwelch power correction factor Henik Vargas
11/29/10 Re: pwelch power correction factor Henik Vargas
12/7/12 Re: pwelch power correction factor Jeremy
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Using Vinogradov's theorem for finding prime solutions to a linear equation (an exercise from Vaughan's book)
up vote 4 down vote favorite
I'm trying to solve an exercise from Vaughan's book, "The Hardy-Littlewood Method" (ex. 3 in chapter 3: Goldbach's problems, p.36), because I want to use the result stated in it. It is a variation of
Vinogradov's theorem on every large enough odd integer being the sum of 3 primes.
Here I need to show that there are "many" triples of prime numbers less than or equal to $N$ that solve the equation with integer coefficients: $b_1p_1+b_2p_2+b_3p_3-b_4=0$. More specifically, I need
to show that if $R(N)=\sum_{(p_1,p_2,p_3)\mid p_i\leq n, b_1p_1+b_2p_2+b_3p_3-b_4=0} (\log p_1)(\log p_2)(\log p_3)$ then $R(N)=J(N)\mathfrak S +O(N^2/\log ^AN)$ where $J(N)$ is the number of integer
solutions to $b_1m_1+b_2m_2+b_3m_3-b_4=0$ satisfying $m_i\leq N$.
Following the Proof as given in Vaughan and Nathanzon, I have defined $F_i(x)=\sum _{p\leq N}\log p\cdot e(b_ipx)$ so that $R(N)=\int _0 ^1 F_1(x)F_2(x)F_3(x)e(-b_4x)dx$.
When integrating over the major arcs $\mathcal M$, I guess I should approximate $F_i(x)$ by $G_i(x)=\frac{c_q(b_i)}{\phi (q)}u_i \left(x-\frac{a}{q}\right)$ where $u_i(y)=\sum _{m\leq N}e(mb_iy)$ and
evaluate the integral $\int _\mathcal M G_1(x)G_2(x)G_3(x)e(-b_4x)dx$.
So, I start by integrating $G_1(x)G_2(x)G_3(x)e(-b_4x)$ over a singel major arc $(\frac {a}{q}-\frac {Q}{N}, \frac {a}{q}+\frac {Q}{N})$. After a change of variables I end up with the integral $\int
_\frac {-Q}{N}^\frac {Q}{N}u_1(y)u_2(y)u_3(y)e(-b_4y)dy$, and I want to bound the difference between the last integral to: $J(N)=\int _\frac {-1}{2}^\frac {1}{2}u_1(y)u_2(y)u_3(y)e(-b_4y)dy$. I guess
this is how I get the $J(N)$ in the main term of the desired expression for $R(N)$.
If so, I want to bound $\int _\frac {-1}{2}^\frac {-Q}{N}u_1(y)u_2(y)u_3(y)e(-b_4y)dy$ and $\int _\frac {Q}{N}^\frac {1}{2}u_1(y)u_2(y)u_3(y)e(-b_4y)dy$. In the proof of Vinogradov's Theorem, this
difference is $O(Q^2/N^2)$ where $Q=\log ^BN$. But now, in the case dealt with in the exercise, this difference seems to be huge: $O(N^3)$. Since, when $b_iy$ is an integer $e(b_iy)=1$ which gives
$u_i(y)=N$ and $|u_1(y)u_2(y)u_3(y)|=O(N^3)$.
To summarize, my questions are: How do I bound $\int _\frac {Q}{N}^\frac {1}{2}u_1(y)u_2(y)u_3(y)e(-b_4y)dy$? Does anyone know of a place where this claim is proved?
nt.number-theory analytic-number-theory
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3 Answers
active oldest votes
I too would be very curious about this bound and will think further about this. The nice thing about the $u_i(y)$, as I recall my advisor having told me, is that the estimates for geometric
sums can be independent of their length. So, similar to an estimate for geometric sums found in Chapter 25 of H. Davenport, Multiplicative Number Theory, Third Edition, one has that
$$ u_i(y) \ll \min\left(N, \frac{1}{\|b_iy\|}\right). $$
Following the perspective of this enlightening post, Montgomery's uncertainty principle, from the blog of Professor Tao, another nice way to think of it is that the $u_i(y)$ is the Fourier
transform of a function $f: \mathbb{Z} \rightarrow \{0,1\}$ that avoids $p-1$ residue classes modulo $p$ for each prime $p$ dividing $b_i$.
A technique for the estimate you require which initially occurred to me was one I learnt from:
S. Baier, L. Zhao, Primes in Quadratic Progressions on Average, Math. Ann., Vol. 338, 2007, No. 4, pp. 963-982.
The analogous situation to the bound you need is treated there at equations (4.2) and (4.3). But there is a difference between that problem and this one. Over there, one only has the case
$b_i=1$ and the maximum value of $\sum_{m\leq N}e(m y)$ can be avoided by avoiding $(-Q/N,Q/N)$. But here, when $b_i>1$, the interval $(Q/N, 1/2)$ still contains values where $u_i(y)$
attains its maximum.
So one observes that by naively applying Hölder's inequality with $\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}=1$ to obtain
$$ \int_{Q/N}^{1/2}\left|u_1(y)u_2(y)u_3(y)\right|dy \ll \prod_{i=1}^3\left(\int_{Q/N}^{1/2}\left|u_i(y)\right|^{n_i}dy\right)^{1/n_i}, $$
where, for each $u_i(y)$, if one ignores that one is integrating from $Q/N$ and instead integrates from $0$, one has that
$$ \int_{0}^{1/2}\left|u_i(y)\right|^{n_i}dy \ll b_i\left(\int_{0}^{1/(b_iN)}N^{n_i} dy + \int_{1/(b_iN)}^{1/2b_i}\frac{1}{(b_iy)^{n_i}}dy\right) \ll N^{n_i-1} $$
which only gives $O(N^2)$ for the term you need to estimate. It can be observed that throwing away the part from $(0,Q/N)$ will not save much like this. Letting $b=\max(b_1,b_2,b_3)$ and
applying Hölder's inequality to the interval $(Q/N,1/(2b))$ gives
$$ \int_{Q/N}^{1/(2b)}u_1(y)u_2(y)u_3(y)dy \ll \prod_{i=1}^3\left(\int_{Q/N}^{1/(2b)}\left|\frac{1}{b_iy}\right|^{n_i}dy\right)^{1/n_i} \ll \frac{1}{b_1b_2b_3}\left(\frac{N}{Q}\right)^2, $$
but this is only the easy part of the interval to treat like this.
Here are some special cases:
up vote 5
down vote Case 1 Amongst, $b_1,b_2,b_3$, at least one $b_i$=1. Without loss of generality, suppose $b_1=1$. Then by Cauchy's inequality and Parserval's identity,
$$ \int_{Q/N}^{1/2}u_1(y)u_2(y)u_3(y)dy \ll \sup_{y\in (Q/N,1/2)}u_1(y)\left(\int_0^1\left|u_2(y)\right|^2dy\right)^{1/2} \left(\int_0^1\left|u_3(y)\right|^2dy\right)^{1/2} $$
which is $O\left(N^2/Q\right)$.
Case 2 $b_1,b_2,b_3$ are pairwise relatively prime and $|b_i|<R$, say, for $R$ sufficiently small. The interval $(Q/N,1/(2b))$ was dealt with, so here we consider $(1/(2b),1/2)$. For $i=
1,2,3$, let
$$ \mathfrak{M}_{b_i} = \bigcup_{k=0}^{b_i-1}\left[\frac{k}{b_i}-\frac{1}{b_iN},\frac{k}{b_i}+\frac{1}{b_iN}\right]. $$
Then the conditions for this case imply that the $\mathfrak{M}_{b_i}$ only intersect each other at an interval centered at zero that is contained in $(-1/(2b),1/(2b))$. The idea is that the
$u_i(y)$ are not simultaneously large on $(1/(2b),1-1/(2b))$. Without loss of generality, consider $u_1(y)$ and $u_2(y)$ on $(1/(2b),1-1/(2b))$ and suppose that $u_1(y) \gg N^{1/2}$. Then
$y$ is of the form
$$ y= \frac{k \pm \beta}{b_1}, $$
$k \not=0$, $\beta>0$ and $1/\beta > N^{1/2}$. But then,
$$ \frac{1}{\|b_2y\|} = \frac{b_1}{(kb_2 \bmod b_1) + \beta} = O(b_1) $$
since $k\not=0$. Therefore, this implies that on $(1/(2b),1/2)$, one has that $|u_1u_2u_3|\ll NR^2 + N^{3/2}$.
There are some papers that deal with this kind of problem using the circle method. They are:
A. Balog, Linear Equations in Primes, Mathematika (1992), 39 : pp 367-378
M.C. Liu, K.M. Tsang, Small prime solutions of linear equations, Théorie des nombres (Quebec, PQ, 1987), 595–624, de Gruyter, Berlin, 1989.
M.C. Liu, K.M. Tsang, Small prime solutions of some additive equations, Monatsh. Math. 111 (1991), no. 2, 147–169.
Thank you for a very detailed answer. I think the result you mentioned about $u_i(y)\ll \min (N,\frac {1}{||b_iy||})$ can be used to obtain: $$\int_\frac {Q}{N} ^\frac {1}{2b} u_1(y)u_2
(y)u_3(y)e(-b_4y)dy\ll (b_1b_2b_3)^{-1} (\frac{Q}{N})^2$$since if $y\in (\frac{Q}{N},\frac{1}{2b})$, then $$y\leq \frac{1}{2b}\Rightarrow by\leq \frac {1}{2} \Rightarrow b_iy\leq \frac
{1}{2}$$which gives $||b_iy||=|b_iy|$ and therefore $u_i(y)\ll \frac {1}{|b_iy|}$. But, like you said, this is the easy part of the interval. Would be happy to know if you have more
ideas. – Tal Horesh Jan 20 '12 at 13:22
You are welcome. Some special cases have been added. Let me know if you have more ideas too. – Timothy Foo Jan 21 '12 at 1:17
I tried to use your idea for "case 2" to get the general case. Does this seem true to you? – Tal Horesh Jan 21 '12 at 19:48
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First, we may assume $(b_1,b_2,b_3)=1$. That gives that for each $y\in (0,1)$ at most two of $\{b_1y,b_2y,b_3y}$ can be integers. Now, take $$J^{(i)}_k=\left[\frac {k}{b_i}-\frac {1}{b_iN
^\frac{1}{3}},\frac {k}{b_i}+\frac {1}{b_iN^\frac{1}{3}}\right]$$ for all $1\leq k\leq b_i-1$. Assume $N$ to be large enough such that no two of these intervals (for all $i=1,2,3$ and all
$k$) intersect each other, unless when they are both centered around the same point $y_0$, i.e both $b_iy_0$ and $b_jy_0$ are integers. If so, at most two of these intervals can intersect
each other.
Also, define: $$I^{(i)}_k=\left[\frac {k}{b_i}-\frac {1}{b_iN^\frac{1}{2}},\frac {k}{b_i}+\frac {1}{b_iN^\frac{1}{2}}\right]\subset J^{(i)}_k.$$
On $I^{(i)}_k$ we have that $|u_i(y)|\sim N$, so if $y\in I^{(i)}_{k_1}\cap I^{(j)}_{k_2}$ then $|u_i(y)u_j(y)|\sim N^2$.But, for $l\neq i,j$ we have that $|u_l(y)|\ll N^\frac {1}{3}$ as
$y\notin J^{(l)}_k$ for all $k$. Over a certain $I^{(i)}_k$ we can therefore bound $|u_1(y)u_2(y)u_3(y)| $ by $$|u_1(y)u_2(y)u_3(y)|\ll N^2N^\frac {1}{3}.$$ As $|I^{(i)}_k|=\frac{2}{b_iN^
\frac{1}{2}}$, we have that: $$\int _{I^{(i)}_k} |u_1(y)u_2(y)u_3(y)|dy \ll N^2N^\frac {1}{3}\frac{2}{b_iN^\frac{1}{2}}=\frac{2}{b_i}N^2N^\frac {-1}{6}$$
There are at most $|b_i|$ such $I^{(i)}_k$'s, so the integral over all of them is bounded by $2N^2N^\frac {-1}{6}$.
Now, say we're in $J^{(i)}_k\setminus I^{(i)}_k$. Then at least one of the $u_i$'s is bounded by $N^\frac{1}{3}$, and the other two are at bounded by $N^\frac{1}{2}$. The length of each
up vote 2 $J^{(i)}_k$ is $\frac{2}{b_iN^\frac{1}{3}}$, so: $$\int _{J^{(i)}_k\setminus I^{(i)}_k} |u_1(y)u_2(y)u_3(y)|dy \ll N^\frac {1}{2}N^\frac {1}{2}N^\frac {1}{3}\frac{2}{b_iN^\frac{1}{3}}=\
down vote frac{2}{b_i}N.$$ Again, the integral over the total of intervals $J^{(i)}_k\setminus I^{(i)}_k$ is bounded by $2N$.
Outside of all the intervals $J^{(i)}_k$, for all $i=1,2,3$ and $1\leq k\leq b_i-1$, we know that $|u_i(y)|\ll N^\frac {1}{3}$ and therefore $|u_1(y)u_2(y)u_3(y)|\ll N$. Everything is
inside $(0,1)$, so the total of the integral "outside of the $J^{(i)}_k$'s" is bounded by $N$.
To summarize: $$\int _\frac {Q}{N}^\frac{1}{2}|u_1(y)u_2(y)u_3(y)|dy\ll 2N^2N^\frac {-1}{6}+2N+N=O(N^{1\frac {5}{6}}).$$
But, this is true only when we are far enough from 0. At 0, all three functions $u_i$ obtain their maximum simultaneously, so the main contribution to the integral will be near 0. take
$b$ such that $\frac{1}{bN^{\frac{1}{3}}}\leq\frac{1}{2b_{i}}$ for all $i=1,2,3$. Write: $$\mathcal S= \left[\frac {Q}{N},\frac{1}{bN^{\frac{1}{3}}} \right].$$ We estimate the integral on
$\mathcal S$. Note that if $y\in \mathcal S$ then $b_{i}y\leq\frac{1}{2}$ and therefore $\left\Vert b_{i}y\right\Vert =\left|b_{i}y\right|$ for all $i=1,2,3$. Thus, $u_{i}\left(y\right)\
ll\frac{1}{\left\Vert b_{i}y\right\Vert }=\frac{1}{\left|b_{i}y\right|}$ for $y\in\mathcal S$. Now: $$\int_{\mathcal S}|u_1(y)u_2(y)u_3(y)|dy \ll \frac{1}{\left|b_{1}b_{2}b_{3}\right|} \
int_{\mathcal S}\frac{1}{y^{3}}dy=O\left(\frac{N^{2}}{Q^{2}}\right).$$
We get that The integral of $u_1(y)u_2(y)u_3(y)$ on $\left[\frac{Q}{N},\frac{1}{2}\right]$ is $O\left(\frac{N^{2}}{Q^{2}}\right)$, and this bound is tight.
Yes, that seems quite right, nice. Thanks! – Timothy Foo Jan 22 '12 at 1:19
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I wanted to suggest some simplifications/improvements.
First of all you can avoid doing case-analysis by using the elementary estimate
$v_1v_2v_3 \leq \min(v_1,v_2,v_3) \max(v_1,v_2,v_3)^2$,
which is valid for all positive real numbers $v_1,v_2,v_3$. In your problem you end up with the bound
$\int_{Q/N}^{1/2} |u_1(y)u_2(y)u_3(y)|dy$
$\leq \left(\sup_{y \in [Q/N,1/2]}\min_{i=1,2,3}|u_i(y)| \right) \int_{Q/N}^{1/2}\max_{i=1,2,3}(|u_i(y)|^2)dy$
$\leq \left(\sup_{y \in [Q/N,1/2]}\min_{i=1,2,3}|u_i(y)|\right) \int_{-1/2}^{1/2}\max_{i=1,2,3}(|u_i(y)|^2)dy.$
And by Parseval's identity
up vote 0 down
vote $\leq \left(\sup_{y \in [Q/N,1/2]}\min_{i=1,2,3}|u_i(y)|\right) N$
If $b_1,b_2,b_3$ are coprime (as we have to assume for this estimate), most values of $y$ are at least $(2b_1b_2b_3)^{-1}$ away from a point of the form $a/b_i$ for some $i \in \{1,2,3
\}$. For those $y$ we have the estimate
$\sup_{y \in [1/(b_1b_2b_3),1/2]}\min_{i=1,2,3}|u_i(y)| \ll b_1b_2b_3$.
For the remaining values of $y$, which are close to $0$, we have to use the weaker estimate
$\sup_{y \in [Q/N,1/(b_1b_2b_3)]}\min_{i=1,2,3}|u_i(y)| \ll N/Q$.
Altogether we obtain
$\int_{Q/N}^{1/2} |u_1(y)u_2(y)u_3(y)|dy \ll N^2/Q$.
I hope I didn't screw things up somewhere...
Thank you very much. Perhaps I'm wrong here, but by Parseval's identity we get $$\int_{-\frac{1}{2}}^{\frac{1}{2}}\text {max}\left(|u_{i}\left(y\right)|^{2}\right)dy\leq N^{2}$$
Don't we? I mean, with $N^2$ instead of $N$... – Tal Horesh Jan 28 '12 at 11:45
I think I got it correct. It is the number of solutions to $b_im = b_in$ for $n,m \leq N$, which is $N$ (if $b_i \neq 0$). The maximum is taken over $i = 1,2,3$, not over $y$. This
might have confused you. – Eugen Keil Jan 28 '12 at 15:56
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Model A
From Wikisocion
Model A is the socionic model of the psyche elaborated by Aushra Augusta. It includes:
1. rules for positioning the 8 different information elements in a framework of as many different slots (called functions) to create 16 distinct type formulas
2. general characteristics of each function that apply to the information element in that position regardless of type
Part 1 (the syntax) is rigorously defined and is often referred to alone as "Model A," but is an empty shell without part 2 (the semantics), which imbues the model with empirical content. There can
be no standard description of part 2, so it is formulated differently by different authors. Augusta's foundational works set down the basics for socionists' understanding of Model A.
With Model A as an abstract framework, intertype relations can be understood and analyzed, and type descriptions can be generated. However, as a rule some practical observation and introspection is
necessary to understand the model.
The relationship structure in socionics is a direct consequence of Model A.
Model A is composed of 8 functions, which are filled by 8 information elements. Functions are designated by cells that have a standard numbering from 1 to 8. Elements are designated by geometric
Each type perceives and processes all of the aspects of reality, but with varying degrees of clarity, depth, and comfort. Model A describes the perceptual and behavioral characteristics associated
with each position. By combining the characteristics of an element with those of one of the eight positions of Model A, we can generalize traits and attitudes shared by the two socionic types with
the element in that position. In practice, the true nature of the functions is subtle and difficult to discern without a thorough understanding of each and every type in its particularity. By
observation one may gradually form a rich and realistic picture of the functions (and information elements) themselves.
Across the socion
Information elements fill functions to make 16 valid Models A. These represent the 16 types of the socion.
To view a description of how a type uses an information element, click on the symbol in that type's Model A.
Model A divides into four blocks (or rows) containing two functions apiece. Each block contains one rational and one irrational element, one extroverted and one introverted element. Traditionally it
is thought that the functions of each block are somehow connected and codependent with each other. Other socionists maintain that the functions manifest themselves separately.
Augusta chose the terms Ego, Super-ego, Id, and Super-id by analogy with Freud's model of the psyche. However, the meaning of the terms is somewhat different than in psychoanalysis.
Positioning rules
There are certain rules for positioning the elements in Model A. As soon as you place an element in one of the functions, you automatically place three others and are left with only two options for
the rest of the functions. This is because there are certain relationships between the elements. For instance, part of what makes a leading function is a certain suggestiveness to and subdued use
of 'rival' elements and .
Here are the rules for positioning elements in Model A:
• The accepting functions must contain either rational or irrational elements, and the producing functions the opposite.
• The mental functions must contain either static or dynamic elements, and the vital functions the opposite.
• The Ego and Super-Id elements complement each other. They conflict, respectively, with the Super-ego and Id elements.
Given these rules, each element occupies one function — that is, a type gives a one-to-one mapping between functions and elements.
See also
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Combinations and Permutations "Rules"
Re: Combinations and Permutations "Rules"
Hmmm... I use Boraxo, 20 mule team of course. Truthfully, I have never heard that term and do not consider it correct. You know how I feel about jargon. If you need authoritative evidence in addition
to the two links provided just let me say that I am considered to be the best combinatoricist on my block. True it is a small block just 4 buildings but still...
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
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Niwot ACT Tutor
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st: Fw: Calculating a confidence interval for population variance based
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st: Fw: Calculating a confidence interval for population variance based upon sample.
From "Carl Mastropaolo" <accuscience@comcast.net>
To <statalist@hsphsun2.harvard.edu>
Subject st: Fw: Calculating a confidence interval for population variance based upon sample.
Date Mon, 1 Feb 2010 09:47:18 -0500
I have STATA10.
I can not figure out how to calculate a simple confidence interval for a
population variance
based upon sample data. STATA will calculate the sample standard deviation,
but I can not figure out how to compute a confidence interval for the
population standard deviation. Is there a command which will do this?
Any help?
* 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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Rings and Fields query
November 23rd 2007, 01:07 PM #1
Nov 2007
Just trying to work on my rings and fields.
If its not too much trouble could someone explain how to do this for me? Its not so much a solution I'm looking for as the method as I really need to understand it
Let R={a+b ROOT(-14) e C:a,b e Z}
where C is complex and Z is the real no's
(i) Prove that U(R)= {+-1}
(ii) Show that the elements 3,5, (1+ROOT(-14)) and (1- ROOT(-14))
are irreducible in R
(iii) By considering the equatioion 3.5 = (1+ROOT(-14))(1- ROOT(-14)) show that 3 is not prime in R
(iv) Is R a unique factorisation domain?
I know there are a few concepts involved but I really am in trouble in this subject and help would be much appreciated
Let R={a+b ROOT(-14) e C:a,b e Z}
where C is complex and Z is the real no's
(i) Prove that U(R)= {+-1}
The Euclidean norm of an element is $a^2+14b^2$ this is equal to one if and only if $a=\pm 1 , b=0$. Thus, $\pm 1$ are the only units.
(ii) Show that the elements 3,5, (1+ROOT(-14)) and (1- ROOT(-14))
are irreducible in R
(iii) By considering the equatioion 3.5 = (1+ROOT(-14))(1- ROOT(-14)) show that 3 is not prime in R
(iv) Is R a unique factorisation domain?
No, it is not a UFD. Once you have have shown that these elements are irreducible elements the equation $3\cdot 5 = (1+\sqrt{-14})(1-\sqrt{-14})$ shows there is a different factorization.
November 24th 2007, 02:06 PM #2
Global Moderator
Nov 2005
New York City
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What is an Option?
An option is a contract that gives the right, but not the obligation, to buy or sell 100 shares of the underlying stock at a particular price.
Option contracts expire. The value of the option is based on how many days it has to expiration and how near the profit target (strike price) is.
As an example:
SPY stock is trading at $140 per share and you believe the stock is headed up. You could buy 100 shares of the stock ($140 x 100 shares = $14,000) or you could buy an option that has a strike price
of $140, as an example, for $200 that expires in the next 17 days. That means instead of investing $14,000 you can participate in the stock’s up move with just $200. Controlling $14,000 worth of
stock option with $200 is LEVERAGE.
If you are right and the stock rises to $144 per share before option expires in 17 days, it will be worth $14,400 ($144 x 100 shares).
Here is how you would calculate the profit:
Sale price – Purchase price – premium paid for the option.
$14,400 - $14,000 = $400 - $200 = $200 profit or 100% profit.
Most options are not held until the expiration date, but are sold at a profit when the value of it appreciates. Conversely, options are sold at a loss when the value of it depreciates.
Here is an example of how options work using purchase of a house: You are not sure if the housing market is going up or down, and you want to buy a house only if it is going to appreciate in value.
You approach the seller and offer to buy the house six months in the future, but you want to lock in the current price of the house with no obligation to buy it in the future. For the seller to agree
to such terms asks for a $6,000 premium. That means, if you buy the house in six months it will cost you $306,000 instead of $300,000.
If the price of the house appreciates to $320,000, you would buy the house for $300,000 plus the $6,000 option premium. Since the value of the house has gone up to $320,000, you will make a profit of
However, if the value of the house declines down to $280,000 then you wouldn’t buy the house for the agreed price of $300,000, and forgo the $6,000 premium. In this situation instead of losing
$20,000, you only would lose $6,000.
Basic Option Facts
Options are quoted in per share prices, but only sold in 100 share lots. For example, a call option might be quoted at $2 per share, but you would pay $200 because options are always sold in
100-share lots.
If SPY option expires 1 cent in profit then you will be obligated to buy 100 shares of SPY. If at the time of the expiration SPY is trading at $140 per share it will cost $140 x 100 shares = $14,000.
So make sure if the option is trading in profit to exit your option unless you want to own the shares. We trade options so we are not interested in buying the stock.
● The premium price is based on how many days are left to expiration and each day the premium devalues. As a simplified example if an options has 20 days to expire and it costs $2 then each day the
value of the option will decline 10 cents ($2 / 20 days).
● Options increase in value when the underlying stock goes up. Inversely, options decline in value when the underlying stock goes down.
● Options have volatility. When the volatility of the stock goes up the option premium goes up and when the volatility goes down the option premium goes down.
● The maximum amount an option can lose is the premium price.
● An option can increase in value many fold.
More option information: www.eoption.com/option_basics.html
Why Options can Generate Big Profits?
Options are leveraged investment tool, and they limit the loss. The disadvantage of buying an option is the premium that one has to pay. The premiums erode with time and eats away at the profit and
that is why it is important to know…
… when to buy an option
… and if the stock is more likely to go up or down
The most attractive parts of options are the leverage, and the fact that the losses can be limited if there is a major adverse price move. Even though the premium erodes the profits, due to the high
leverage one can earn more money using options then stocks. Key to success in options trading is timing. If the timing is not right then the option's value will erode with time before a profit can be
On the other hand, if the timing of the option purchase is right the value of the investment can gain 100%, 200%, 300% or more within days.
Three major factors will influence the price of the option.
• Value of the stock.
• Time value - number of days left to expiration.
• Volatility.
We have explained how options get effected by Stock Appreciation and Time. Here we explain the effect volatility has on options. The indicator below the chart shows how much volatility can fluctuate
in a short period of time. At the lower end of the indicator the volatility was $1.80 per day and at the upper end of the indicator the volatility jumped to $6.50. When the volatility is greater the
premium on option goes up.
Best Time to Buy Options
Timing: To minimize the erosion of the option premium due to time it is important to buy an option just before it turns back up. Our trading system for many years has been an excellent timing tool.
With over 70% of the time it has been able to predict the day a stock will turn within a day or two. For illustration on how well our trading system predicts the turning points of a stock see the
blue arrows on the chart.
Low Volatility: To minimize the cost of option due to high volatility, buy options when the relative volatility of the stock is low. An option could be twice as expensive when the volatility is high.
That means during low volatility you could buy, as an example, twice as many options, and earn twice as much money for the same risk.
Near Expiration Date: Since the option premium erodes with time near the expiration date the option costs very little, but it still controls 100 shares of the underlying stock; therefore, if the
market rallies soon after the purchase of the stock it will increase in value substantially.
What Exactly are Strike Prices?
The strike price of an options contract is the price that the underlying asset is agreed to be traded at. For example, a strike price of $50 allows you to buy the underlying stock at $50 any time
prior to expiration no matter what price that stock is trading at.
The further away the strike price of an option is from the current price of the stock the less expensive the option is. If a stock is trading at $50, then an option with a strike price of $52 will be
less expensive than an option with a strike price of $51.
If you have a brokerage account it is very likely they give you real time options data access. However, if you do not have real time data feed, finance.yahoo.com provides 15 minute delayed data at
Call and Put Options
A call option, often simply labeled a "call" gives you the right to buy the underlying option. If you think a stock is going up, you would by a call option.
When the price of the underlying instrument surpasses the strike price, the option is said to be "in the money". When the price of the underlying instrument is below the strike price, the option is
said to be "out of the money". When the price of the underlying instrument is equal the strike price, the option is said to be "on the money".
Here are three examples:
If as stock is trading at $140, then an option with the strike price of $139 is “in the money” one dollar.
If as stock is trading at $140, then an option with the strike price of $141 is “out of the money” one dollar.
If as stock is trading at $140, then an option with the strike price of $140 is “on the money”.
A put option, often simply labeled a “put” gives you the right to sell the underlying option. If you think a stock is going down you would by a put option.
To learn more about options you could visit the below link
Contact Us | FAQ |Disclosure | Privacy | Affiliates Make Money
Copyright © 2012 Michael Dylan & 1 Option Profit.com, All Rights Reserved
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Assessing risk communication in breast cancer: Are continuous measures of patient knowledge better than categorical?
To compare the performance of categorical and continuous measures of patient knowledge in the context of risk communication about breast cancer, in terms of statistical and clinical significance as
well as efficiency.
Twenty breast cancer patients provided estimates of 10-year mortality risk before and after their oncology visit. The oncologist reviewed risk estimates from Adjuvant!, a well-validated and commonly
used prognostic model. Using the Adjuvant! estimates as a gold standard, we calculated how accurate the patient estimates were before and after the visit. We used 3 novel continuous measures of
patient accuracy, the absolute bias, Brier, and Kullback-Leibler scores, and compared them to a categorical measure in terms of sensitivity to intervention effects. We also calculated the sample size
required to replicate the primary study using the categorical and continuous measures, as a means of comparing efficiency.
In this sample, the Kullback-Leibler measure was most sensitive to the intervention effects (p=0.004), followed by Brier and absolute bias (both p=0.011), and finally the categorical measure (0.125).
The sample size required to replicate the primary study was 18 for the Kullback-Leibler measure, 23 for absolute bias and Brier, and 37 for the categorical measure.
The continuous measures led to more efficient sample sizes and to rejection of the null hypothesis of no intervention effect. However, the difference in sensitivity of the continuous measures was not
statistically significant, and the performance of the categorical measure depends on the researcher’s categorical cutoff for accuracy. Continuous measures of patient accuracy may be more sensitive
and efficient, while categorical measures may be more clinically relevant.
Researchers and others interested in assessing the accuracy of patient knowledge should weigh the trade-offs between clinical relevance and statistical significance while designing or evaluating risk
communication studies.
Keywords: Risk communication, patient knowledge, realistic expectations, shared decision making, physician-patient relations, decision aids, Adjuvant!, breast cancer, software, patient education,
measurement, information theory
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This Article
Bibliographic References
Add to:
Design of a Radix 4 Division Unit with Simple Selection Table
December 1992 (vol. 41 no. 12)
pp. 1606-1611
ASCII Text x
P. Montuschi, L. Ciminiera, "Design of a Radix 4 Division Unit with Simple Selection Table," IEEE Transactions on Computers, vol. 41, no. 12, pp. 1606-1611, December, 1992.
BibTex x
@article{ 10.1109/12.214670,
author = {P. Montuschi and L. Ciminiera},
title = {Design of a Radix 4 Division Unit with Simple Selection Table},
journal ={IEEE Transactions on Computers},
volume = {41},
number = {12},
issn = {0018-9340},
year = {1992},
pages = {1606-1611},
doi = {http://doi.ieeecomputersociety.org/10.1109/12.214670},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Computers
TI - Design of a Radix 4 Division Unit with Simple Selection Table
IS - 12
SN - 0018-9340
EPD - 1606-1611
A1 - P. Montuschi,
A1 - L. Ciminiera,
PY - 1992
KW - remainder updating; radix 4 division unit; selection table; radix 4 division architecture; digit selection; concurrent substeps; digital arithmetic; dividing circuits; number theory.
VL - 41
JA - IEEE Transactions on Computers
ER -
A radix 4 division architecture is presented which partially overlaps the updating of the remainder with the digit selection procedure. It is obtained by separating the radix 4 digit selection
process into two concurrent substeps. The proposed unit requires a simple selection table and involves a small extra expense for the additional hardware compared to the usual radix 4 division units.
Four possible implementations are derived from the general model, with different types of substeps. The high level evaluation shows that the proposed architectures offer an efficient alternative.
[1] D. E. Atkins, "Higher-radix division using estimates of the divisor and partial remainders,"IEEE Trans. Comput., vol. C-17, pp. 925-934, Oct. 1968.
[2] B. K. Bose, L. Pei, G. S. Taylor, and D. A. Patterson, "Fast multiply and divide for a VLSI floating-point unit," inProc. 8th IEEE Symp. Comput. Arithmet., Como, Italy, May 1987, pp. 87-94.
[3] C. W. Clenshaw, "Polynomial approximations to elementary functions,"Math. Tables Aids Comput., 1954.
[4] M. D. Ercegovac and T. Lang, "A division algorithm with prediction of quotient digits," inProc. 7th Symp. Comput. Arithmet., Urbana, IL, June 1985, pp. 64-71.
[5] M. D. Ercegovac and T. Lang, "On-the-fly conversion of redundant into conventional representations,"IEEE Trans. Comput., vol. C-36, no. 7, pp. 895-897, July 1987.
[6] M. D. Ercegovac and T. Lang, "Division," Internal Rep. CS252, Comput. Sci. Dep., U.C.L.A., 1988.
[7] M. D. Ercegovac and T. Lang, "On-the-fly rounding for division and square root," inProc. 9th Symp. Comput. Arithmetic, 1989, pp. 169-173.
[8] M. D. Ercegovac and T. Lang, "Simple radix-4 division with operands scaling,"IEEE Trans. Comput., vol. C-39, pp. 1204-1208, Sept. 1990.
[9] J. Fandrianto, "Algorithm for high speed shared radix 4 division and radix 4 square-root," inProc. 8th IEEE Symp. Comput. Arithmet., Como, Italy, May 1987, pp. 73-79.
[10] J. Fandrianto, "Algorithm for high speed shared radix 8 division and radix 8 square root," inProc. 9th Symp. Comput. Arithmetic, Sept. 1989, pp. 68-75.
[11] S. Kuninobu, T. Nishiyama, H. Edamatsu, T. Taniguchi, and N. Takagi, "Design of high speed MOS multiplier and divider using redundant binary representation," inProc. 8th IEEE Symp. Comput.
Arithmet., Como, Italy, May 1987, pp. 80-86.
[12] J. E. Robertson, "A new class of digital division methods,"IRE Trans. Electron. Comput., vol. EC-7, pp. 218-222, Sept. 1958.
[13] G. S. Taylor, "Radix 16 SRT dividers with overlapped quotient selection stages," inProc. 7th Symp. Comput. Arithmet., Urbana, IL, June 1985, pp. 64-71.
[14] K. S. Trivedi and J. G. Rusnak, "Higher radix on-line division," inProc. 4th IEEE Symp. Comput. Arithmet., Santa Monica, CA, Oct. 1978, pp. 183-189.
[15] P. K.-G. Tu and M. D. Ercegovac, "A radix 4 on-line division algorithm," inProc. 8th IEEE Symp. Comput. Arithmet., Como, Italy, May 1987, pp. 181-187.
[16] J. H. Zurawski and J. B. Gosling, "Design of a high-speed square root multiply and divide unit,"IEEE Trans. Comput., vol. C-36, pp. 13-23, Jan. 1987.
Index Terms:
remainder updating; radix 4 division unit; selection table; radix 4 division architecture; digit selection; concurrent substeps; digital arithmetic; dividing circuits; number theory.
P. Montuschi, L. Ciminiera, "Design of a Radix 4 Division Unit with Simple Selection Table," IEEE Transactions on Computers, vol. 41, no. 12, pp. 1606-1611, Dec. 1992, doi:10.1109/12.214670
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Even Microsoft Needs Help Avoiding Spreadsheet Errors
The cells didn't add up in a Microsoft Excel ad posted in the New York subway system. Here's why.
Both return an answer of $9000, so the spreadsheet rookie might argue that the original answer is correct and just as good as using the AutoSum button.
Although both methods return the same answer, the AutoSum is the right way to do this. Excel has smarts built in that make sure the erroneous answer in the ad will not happen if you use AutoSum. When
you insert new rows in the budget, even at the end of the budget, the AutoSum formula will automatically adjust. As soon as you type in a number in B6, the Total formula becomes =SUM(B1:B6)
Fill in the number in B7 and the total adjusts again:
These adjustments only happen if you use the SUM version of the formula. If you used the original formula and add new rows, even rows in the middle of the budget, the formula will never adjust, and
you will end up with the wrong result shown in the Microsoft ad.
Crazy things happen when marketing people try to use Excel. This is just the latest example, one in which Microsoft’s ad agency produced an interim result that “looked” OK, but could not adapt.
Bill Jelen is a contributing editor and the host of MrExcel.com. He points out that his book, “Don’t Fear the Spreadsheet,” warns against this exact type of formula.
8 thoughts on “Even Microsoft Needs Help Avoiding Spreadsheet Errors”
1. Microsoft Excel is dumb and it is not capable of think and make errors right by its own therefore, we must ensure our arguments and logic are perfectly all right based on these only excel gives
its feedback. I have experienced some companies really go nuts with spreadsheets as their logic and functions are inaccurate and eventually giving a flawed answer. However, if you use it
accurately then it may be the best tool available in the market by considering its functionality.
2. Even better than summing rows 1 through 5 would be to leave a blank space at the top and bottom of the area to be totaled and include these within the =Sum formula. This allows for added rows,
even adding one below the last function.
A check sum of some sort is also helpful, whether its percentages which should total 100% or tabulating rows and columns and then checking that the sum of the rows equals the sum of the columns.
3. It may well have been a deliberate mistake. The human mind is sensitive to things not quite right. After all, look at the free PR they’ve gotten because of the mistake.
4. This is a funny story about marketing people, but it does help make a more serious point about checking your final output before it causes reputational damage. For some people, the damage can be
real, whether in financial loss or even safety calculations in engineering. The European Spreadsheet Risk Interest Group (EuSpRIG) takes a professional interest in risks in end-user computing and
has an annual conference to discuss ways to prevent & detect errors and improve productivity and efficiency in the use of spreadsheets. The next is in Delft in July 2014, see http://
5. …or they were trying to be too clever and used =ROUNDDOWN(SUM(B1:B8),-3) to display the total down to the nearest thousand
…or maybe the SUM formula is right, but they’ve used a custom format to round down (is that possible?)
Do you think we deserve an explanation from Microsoft. Not sure this is the sort of free PR that they want for their flagship office tool
6. They would not be the first organisation to get this wrong. Putting in effective controls over spreadsheets is a key part of using them. http://finsburysolutions.com
7. So typical of non financial people to not 2x check their work. Real Accountants and Finance people are taught that immediately: ALWAYS 2x check your work. This is a easy mistake to make because
the formula only adds the cells you tell it to and that doesn’t include a series of cells (=sumA3.A21) if that is the result that is desired. Who the hell cares about a graph if the answer is
8. Excel is as smart as the user. Try Excel add-in QIMacros for Excel for a low cost and easy way to make the data relevant and informative
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Re: Common subexpression analysis (summary)
Re: Common subexpression analysis (summary)
Dik.Winter@cwi.nl (Dik T. Winter)
Mon, 13 Jul 1992 22:32:04 GMT
From comp.compilers
| List of all articles for this month |
Newsgroups: comp.compilers
From: Dik.Winter@cwi.nl (Dik T. Winter)
Organization: CWI, Amsterdam
Date: Mon, 13 Jul 1992 22:32:04 GMT
References: 92-06-135 92-07-028
Keywords: optimize
In article 92-07-028 Bruce.Hoult@bbs.actrix.gen.nz writes:
At the end:
> I'll bet if there are any compilers that can do this, they're probably
> for FORTRAN :-)
I do not think so. The reason comes later.
> buzzard@eng.umd.edu (Sean Barrett) writes:
> > TURN:
> > while (y >= 0)
> > if (x*x + y*y >= r*r)
> > --y;
> > else
> > ++x;
> > INTO:
> > r2 = r*r;
> > x2 = x*x;
> > y2 = y*y;
> > while (y >= 0)
> > if (x2 + y2 > r2)
> > --y, y2 = y*y;
> > else
> > ++x, x2 = x*x;
Well, I just checked the two very aggresive compilers, Cray C and Cray
Fortran. The Fortran compiler leaves it as is. The C compiler only puts
the r*r calculation outside the loop. On the other hand the Fortran
compiler sometimes performs amazing optimizations; but this one is not
worthwile. It would save 2 cycles from a loop that takes quite a bit
more. The optimization is only worthwile if multiplication is (much)
slower than addition, which is not always the case!
> Why didn't you do strength reduction while you were at it?
This solution replaces one multiply by three adds, so is only worthwile
if a multiply is three times as slow as an add.
But the best optimization depends on whether x, y and r are ints or
floats. I think they are ints, in that case a full optimization is:
if(y >= 0) {
if(x < r && x >= -r) x = r;
y = 0;
So there must be more. Perchance Sean Barrett is calculating the number
of lattice points in a circle? That means that the value of x is used
within the loop, and Bruce Hoult's second optimized must be reviewed.
(Optimization is a bit more complex when the variables are float, but the
result is similar.)
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
home: bovenover 215, 1025 jn amsterdam, nederland
Post a followup to this message
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Proposition 7
Incommensurable magnitudes do not have to one another the ratio which a number has to a number.
Let A and B be incommensurable magnitudes.
I say that A does not have to B the ratio which a number has to a number.
If A does have to B the ratio which a number has to a number, then A is commensurable with B.
But it is not, therefore A does not have to B the ratio which a number has to a number.
Therefore, incommensurable magnitudes do not have to one another the ratio which a number has to a number.
This proposition is the contrapositive of the last one. It is used in X.11.
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17 Subjects: including statistics, calculus, precalculus, geometry
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A High School Math-Science Journal
In my first year of teaching, fresh from my haze from history grad school, I remember approaching the history and English department chairs about creating a high school level journal for those
subjects. I mean, our school has a literary magazine, and also even a publication for works in foreign languages (seriously!). But nothing for amazing critical analyses and interpretations in English
and history. I figured having something like this might encourage students to revise already excellent work for publication, and also make the audience of their paper be an audience of more than one.
I even contacted the literary magazine student editors to see if they would feel like the journal would encroach on their domain (they said no). For reasons that are still quite beyond my
understanding (because I still think it’s an amazing idea), both department heads rebuffed my idea. (Also, if they said yes, they would have gotten an enthusiastic first year teacher who would have
taken on all this work!)
And so, I let this idea pass. One of many that I have, think are awesome, and then languish and die, either due to my own laziness or due to external circumstances beyond my control.
Until last year. When I was thinking: I’m a math teacher. Why not start a math and science journal? It’s so obvious that I don’t know why the idea didn’t hit me over the head years ago. So I found a
science teacher compatriot who I knew would be interested, and we came up with an initial plan. And at the end of last year, we presented it to some students who we thought might have been interested
(as this was something that is something that has to be for them, by them… if they don’t want it, there’s not point in doing it… it’s not about us…). They were, and we were officially off to the
We shared with the students the following document we made, with a brief outline of one vision for the journal. But with the understanding that this was their thing so their ideas reign supreme. This
was, in some sense, a mock-up that the science teacher and I made to show them one possibility. The one thing that the science teacher and I were really aiming for in our mock-up was that the journal
shouldn’t just be for superstar students. We wanted to come up with an journal that has a low barrier of entry for students submitting to the journal, and that if a student has interest or a passion
for math or science, that’s really all they need to get started. To do this, original and deep research wasn’t really the primary focus of the journal. So here’s our brief proposal:
The additional benefit of having this journal is hopefully it will cause curricular changes. Teachers will hopefully feel moved to create assignments that go “outside of the box” — and that could
result in things being submitted. Students who express an interest in some math-y or science-y idea (like why is 0/0 undefined… something that came up in calculus this week) could have a teacher say
“hey, that’s great… why don’t you look it up and do a 3 minute presentation on what you find tomorrow?” … and if they do a good job, encourage them to write it up for the journal. Or a teacher might
assign a group project on nuclear disasters, and encourages the students who do extraordinary work to submit their project to the journal. (Which can be showcased by teachers the following year!) Or
a student who notices a neat pattern, or comes up with an innovative explanation for something, or who wants to try to create their own sudoku puzzle, or decides to research fractions that satisfy $\
frac{1}{a}+\frac{1}{b}=\frac{2}{a+b}$. Or whatever. Knowing there is a publication you can direct the student to, as a way to say “hey, you’re doing something awesome… seriously… so awesome I think
you kind of have to share it with others!” is going to be so cool for teachers. (As a random aside, I was thinking I could enlist the help of the art and photography teachers, because of the overlap
between math and art… They might make an assignment based around something mathematical/geometrical, which students can submit…)
I honestly have no idea how this is going to turn out. What’s going to happen. How the word is going to get out. If anything will be submitted. If kids get excited about it. Lots of questions. But I
have a deep feeling that the answers will come and good things are going to happen with this.
I’m soliciting in the comments any thoughts you might have about this. If your school does a math journal, a science journal, or a math-science journal, what does it look like? What works and what
doesn’t? Do you have a website/sample we could look at? If you don’t have one, and you are inspired and think of awesome things kids could put in there (e.g. kids submitting their own puzzles! kids
writing book reviews of popular math/science books, or biographies of mathematicians/scientists! getting kids to create photographs or computer images of science or data visualization or just making
geometrical graphing designs! trust me — brainstorming this is super fun!) I’d love to share any and all ideas with the kids involved with this project at my school.
13 thoughts on “A High School Math-Science Journal”
1. I am going to leave my own first comment. First! Anyway, it seems St. Ann’s was already starting this last year, when we were just coming up with the idea… here is their (awesome) math journal:
2. (Replying only so I can subscribe to comments. I’m intrigued.)
3. Sounds like a good idea, as long as no teacher is so foolish as to require submissions, which would kill the pleasure of it. Incidentally, $1/a + 1/b = 2/(a+b)$ has only complex solutions ($a = \
pm i b$). Do your students get taught enough complex numbers to get that?
□ I think some would. They are taught complex number basics in Algebra II and then more with them in Precalculus (when doing vectors/polar).
4. I think this is a great idea and similar to what we were trying to do with capstonelearning.org. I especially like the idea of it being peer reviewed by high school students. Ages ago, my school
had a national literary journal where the creative writing class was in charge of reviewing submissions they got from high schools all over the nation. I think such a project could work equally
well for a math and science journal.
5. I’d love to get my students rolling at this level: I’ve got some candidates now who just need a little motivating. Please keep us posted on how this is developing!
6. Don’t know if I’ve told you or not, but we’ve got a math Journal going on at Saint Ann’s called “Peer Points.” It’s part great, part weak. I’d love to chat about it sometime.
□ Paul – I just looked at Saint Ann’s journal … I love it! I look forward to following the work to see how it grows! I’m an elementary principal turned 9th grade algebra teacher this year; used
to sponsor a literary magazine; would love to get creative students writing about math!
☆ Hey! So glad you found it and enjoyed it. It’s been neat to see the journal develop, and I’d say we have a long way to go before it’s a really thriving consistently beautiful thing, but
we’re off to a great start. The key is that we (at least some of us) are making mathematical writing a key component of what it means to do math. I have to say though, my favorite
sections of Peer Points are always written by kids 8th grade and below. Especially grades 3-5. :)
7. I didn’t know how to reach you via email so I am reaching out in this way. I am the founder and CEO of an educational development company that works with educational publishers to develop
content. We look for highly skilled writers and editors on a regular basis. I’m hoping that you might have some suggestions for people who would be interested in doing freelance work as math
writers and/or editors. Please feel free to reach out to me at sourcing@apasseducation.com
Andy Pass
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Help on a convergent sequence limit problem
January 6th 2010, 06:40 PM #1
Junior Member
Aug 2008
I tried using the method of doing a limit to infinity which gave me infinity/infinity (an indeterminate) so I used L'hopital's rule and took the derivative of top and bottom. However that still
leaves me with that segment. Some help would be nice, thanks.
Really? You know L'hopital's rule but you don't know that $|k|<1\implies\lim k^n=0$?
i do not know that sorry =/
Yes you do, though you might not immediately recognize it. What he is saying is that if the absolute value of "k" is less than 1, then the limit as n approaches infinity of k to the n, is 0.
In your above problem:
$a_{n}=\frac{3^n}{4^n} \Longrightarrow a_{n}=\left(\frac{3}{4}\right)^{n}$
Where k is equal to $\frac{3}{4}$
oh yea that does make sense since the bottom would keep getting smaller than the top. However my teacher says I have to do it in an algebraic way and show work. Is there really a way with work to
I mean you can prove it by the definition. Or note that $\lim_{n\to\infty}\frac{\left(\frac{3}{4}\right)^{n +1}}{\left(\frac{3}{4}\right)^n}=\frac{3}{4}<1$ so that $\sum_{n=1}^{\infty}\left(\frac
{3}{4}\right)^n$ converges which means that $\left(\frac{3}{4}\right)^n\to 0$
hmm I see... alrighty. Thanks for the help!
January 6th 2010, 06:43 PM #2
January 6th 2010, 06:44 PM #3
Junior Member
Aug 2008
January 6th 2010, 06:50 PM #4
January 6th 2010, 06:52 PM #5
Super Member
Jul 2009
January 6th 2010, 06:53 PM #6
Junior Member
Aug 2008
January 6th 2010, 06:57 PM #7
January 6th 2010, 07:04 PM #8
Junior Member
Aug 2008
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Spherical coordinate system
668pages on
this wiki
The spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates, (ρ, φ, θ), where ρ represents the radial distance of a point
from a fixed origin, φ represents the zenith angle from the positive z-axis and θ represents the azimuth angle from the positive x-axis. The geographic coordinate system is similar to the spherical
coordinate system.
Coordinate system definition and notation
The spherical coordinate system represents points as a tuple of three components. Typically in America, the components are notated as (ρ, φ, θ) for distance, zenith and azimuth, while elsewhere the
notation is reversed for zenith and azimuth as (ρ, θ, φ). The former has the advantage of being most compatible with the notation for the two-dimensional polar coordinate system and the
three-dimensional cylindrical coordinate system, while the latter has the broader acceptance geographically. The notation convention of the author of any work pertaining to spherical coordinates
should always be checked before using the formulas and equations of that author. This article uses "American" notation.
The three coordinates (ρ, φ, θ) are defined as:
• 0 ≤ ρ is the distance from the origin to a given point P.
• 0 ≤ φ ≤ 180° is the angle between the positive z-axis and the line formed between the origin and P.
• 0 ≤ θ ≤ 360° is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.
φ is referred to as the zenith or colatitude, while θ is referred to as the azimuth.
According to this system, φ and θ lose significance when ρ = 0 and θ loses significance if sin(φ) = 0 (at φ = 0 and φ = 180°.
To plot a point from its spherical coordinates, go ρ units from the origin along the positive z-axis, rotate φ about the y-axis in the direction of the positive x-axis and rotate θ about the z-axis
in the direction of the positive y-axis.
Coordinate system conversions
As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.
Cartesian coordinate system
The three spherical coordinates are converted to Cartesian coordinates by:
${x}=\rho \, \sin\phi \, \cos\theta \quad$
${y}=\rho \, \sin\phi \, \sin\theta \quad$
${z}=\rho \, \cos\phi \quad$
Conversely, Cartesian coordinates may be converted to spherical coordinates by:
${\rho}=\sqrt{x^2 + y^2 + z^2}$
${\phi}=\cos ^{ - 1} \left( {\frac{z}{{\sqrt {x^2 + y^2 + z^2 } }}} \right)$
${\theta}=\tan ^{-1} \left( {\frac{y}{x}} \right)$
Geographic coordinate system
The geographic coordinate system is an alternate version of the spherical coordinate system, used primarily in geography though also in mathematics and physics applications. In geography, ρ is
usually dropped or replaced with a value representing elevation or altitude.
Latitude is the complement of the zenith or colatitude, and can be converted by:
${\delta}=90^\circ - \phi$, and
${\phi}=90^\circ - \delta$,
though latitude is typically represented by φ as well. This represents a zenith angle originating from the xy-plane with a domain -90° ≤ φ ≤ 90°. The longitude is the azimuth angle shifted 180° from
θ to give a domain of -180° ≤ θ ≤ 180°.
Cylindrical coordinate system
The cylindrical coordinate system is a three-dimensional extrusion of the polar coordinate system, with an h coordinate to describe a point's height above or below the xy-plane. The full coordinate
tuple is (r, θ, h).
Spherical coordinates may be converted to cylindrical coordinates by:
$r = \rho \sin \phi \,$
$\theta = \theta \,$
$h = \rho \cos \phi \,$
Cylindrical coordinates may be converted to spherical coordinates by:
${\theta}=\theta \quad$
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how to relate power and sound give some formulas
can some1 teach me some formula on how to relate power(mW/W) to sound(dBv/dBSPL)?i need some formula about this two topics and relate them because we are having a feasibility study on how to convert
sound to electricity and we need some formulas,i know its kind a hard to transform sound to electricity but we really need some help..hope many of you would reply need some feedback asap
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Rice Mathematics Grads - in their own words
Choose a Field:
Research on Infectious Diseases | Biology | Environment | Chemistry | Computers | Economics | Finance | Energy Financial Products Trading | Business | Physics | Electrical Engineering | Statistics |
Non-profit | Technology | Innovative Research | Community Involvement |
Computational Biology
Research on Infectious Diseases
I actually knew I wanted to go to med school but figured if I was going to do biological stuff the rest of my life (had no idea I'd end up doing NIH funded research), I should do "something else"
while I could. In those days, one couldn't get into med school with humanities so math seemed the way to go, and I liked it. The other thing I have always told people is that, while I have not used
math directly, what it really trained me to do was "sit and focus" on solving a problem. One can't be jumping up every 10 min. getting coffee, checking email (of course, that did not exist then) and
expect to "solve" a math problem. That being "forced to focus", and learning how to do so, has been extremely helpful throughout. More information from the Rice News. (BEM - class of 1969)
My path is "conventional" in that I'm a faculty member. But, I study neuroscience rather than math, and I have a "wet lab" in which one sees live mice and lots of fancy equipment. But I nevertheless
use math (sometimes very simple, sometimes more sophisticated) on a daily basis. For example, one of the things we do is build new and better microscopes for looking at the brain. The manuscript
draft I'm working on right now (the editor window is open next to this email) is tackling the problem, "what performance metrics of our optical system do we need to collect so that we can better
understand what it does to the light rays emitted from the sample?" It turns out to be a problem that uses a smattering of differential geometry, and for which it's actually useful to prove a theorem
or two based on the fundamental physics of light propagation. Other times we think about signal analysis, image registration (which results in huge, nonlinear PDEs that need to be solved
numerically), and clustering.
I'd characterize most of the pure math we use as "late 19th/early 20th century," with many of the applied/numerical methods rather more up-to-date than that. Neuroscience is increasingly getting to
the point where data collection will not be the rate limiting step; instead, people who have good ideas about how to analyze data (invariably a mathematical problem) will make contributions of
increasing importance. (TH - Class of 1991)
I graduated from Rice with a Math / CAAM B.A. in 2005. During my final semester, I took a seminar course in soil science, and decided I would like to work in soil and groundwater remediation. After
graduation, I returned to California and applied for an internship with the Groundwater Department of the Alameda County Water District in Fremont. I was not very optimistic, since the website stated
that they were looking for engineering or geology graduates. It turned out to be my lucky day - one of the groundwater engineers was writing his own modelling program to simulate the intrusion of
seawater into the aquifer, and I was hired as his intern. I helped him with a finite element model of aquifer head levels, and a finite volume model for both head levels and seawater and contaminant
transport. I did not write any of the code itself, but I prepared input files, designed grids, calibrated the model, and ran the model under various scenarios. I often reviewed my notes from courses
I took at Rice, particularly CAAM 452: Numerical Methods for PDE's. I am glad I soldiered through that class even though it got tough! It was quite exhilerating to apply things I learned in school to
do something so eminently useful - help preserve a community's water supply.
In addition to working on the models, I helped oversee soil and groundwater remediation work, and operated a pilot ultra-filtration device at one of ACWD's treatment plants. When my internship
expired after 1.5 years (coincidentally, on the same date as my graduation from Rice two years earlier), I applied to San Jose State University's Civil and Environmental Engineering Department to
pursue a master's degree. That's where I am now. I will graduate in a couple years and hopefully go on to work in municipal water supply.
One reason I decided to study mathematics (besides the fact that it's fun) was that I didn't yet know what I wanted to do, and I hoped a math degree would allow me entry into whatever field I
eventually chose. This turned out to be a pretty good plan! (MA - Class of 2005)
It's nice to hear from the Math Dept after so many years (I graduated in 2002). As for career paths ... I was a double major in Chemistry (which was more my primary major) ... I proceded to go to MIT
for a year before joining an MD-PhD program at the University of Illinois at Urbana-Champaign getting my PhD in physical chemistry working in an electrical engineerig / bioengineering lab. So
certainly not your typical pathway for a math major. (FN - Class of 2002)
You can certainly tell your students that a math degree comes in very handy for what I do, which is rendering engineering for video games. I work with the Wii, XBox, PC, DS. Although of course it's
primarily a programming discipline, there's a lot of 3D math involved, and I do use a surprising amount of trig, discrete math, even a little bit of topology. I recommend pairing a Math degree with a
C.S. degree, as I did, and at least in '98 the Math department made that an easy and worthwhile fit. Although CAAM seems a natural fit, I actually found that specializing in both "pure" disciplines
was the way to go because I got a really solid grounding in each. Certainly certain individual CAAM classes were very helpful though. (AH - Class of 1998)
I got a Ph.D. in economics (from Maryland) and now work for the Antitrust Division of the Department of Justice. We have a number of math majors in our economics department here and at least one math
Ph.D. (JL - Class of 1996)
After leaving Rice, I worked as a computer programmer until last year when I became a quantitative analyst in the Anti-Money Laundering (AML) department of PNC Bank (in Pittsburgh, PA).
Regrettably, I haven't used much "pure" math in my old or current job. However, I feel the training and ways of thinking that I was exposed to helped me a great deal in making sense of voluminous
amounts of informationand classifying it and breaking things down into useful pieces.
In programming, I used a lot of elementary logic (De Morgan's Law, etc ....)and a little bit of optimization to make my programs run faster.
In my current job, I do a lot of data mining and use a lot of descriptive statistics to present my findings to my users. I also occasionally use probability theory (things like Chebyshev's
inequality) to determine"normal" behavior and to help set thresholds that trigger anti-money laundering alerts that are then investigated by the department.
When dealing with new questions that come up, the process is similar to research in general: observations lead to hypotheses that are then tested and either supported, eliminated or further
tweaked.Having a degree in math has been a great asset and generally has made my ideas and suggestions well received. (AG - Class of 1997)
Energy Financial Products Trading
I'm a Rice alum who graduated in 2003 (Math, CAAM, English). I received your newsletter asking alumni to provide updates as to what they're up to so I thought I'd write and let you know.
I currently work for an energy trading firm, specifically in the role of quantitative analytics. I sit on the trade floor and work for our electricity & natural gas options trader pricing deals,
developing new ways of modeling stochastic processes like price volatility or correlation and building out data systems & databases that make these tasks easier. It's frantic and unregimented but
rewarding, sort of like Rice. If you have any undergraduates who are interested in the more quantitative aspect of finance or just curious about financial mathematics in general I'd be happy to talk
to them. (JT - Class of 2003)
I started by getting my math Ph.D., with the sure intent of being a professor:
1994 - 1998 Ph.D. in Math at Duke, Probability Theory
However, in my 4th/final Ph.D. year, I decided to pursue a career in business and began taking MBA courses and applied to consulting firms. After graduating, I took a job with the Boston Consulting
Group (a management consulting firm) in a position as if I had an MBA. BCG recruits a fair number of Ph.D.s and lots of generally smart folks from the undergrad ranks regardless of major, thus this
is a very accessible path for a math major.
So from there my career has been
1998 - 2001 BCG
2001 - 2003 El Paso Energy (I went to work for my client)
2003 - 2008 Kohlberg, Kravis, Roberts & Co (KKR), the global buyout firm
For the last 5 years I have worked for KKR in a group called KKR Capstone helping improve the operations of companies that KKR buys. For anyone interested in the private equity world, I'd be happy to
talk with them about the possibilities.
I would be very happy to talk to any Rice undergrad in math about how to pursue strategy/management consulting with a Bain, BCG, or McKinsey or how to pursue private equity. (CF - Class of 1994)
To keep you updated on where your alumni are ...
After getting a PhD in Physics and wintering-over for two years at the Amundsen-Scott South Pole Station, I'm now an Asst. Professor of Physics and Astronomy at Oberlin College in Oberlin, OH. So,
sorry to say that I won't be able to help you gain additional perspective on where math majors can go beyond academia. :-) (CM - Class of 1994)
Electrical Engineering
My own story, for what it's worth: I was a math major at Rice, and added a dual major with CAAM early on when I discovered a whole lot of interesting stuff going on there. In fact, Steve Cox
encouraged me to get a Master's in CAAM during my senior year, and I even had my own office in the basement of Herman Brown. I worked with Danny Sorensen on large-scale numerical linear algebra,
which led to a summer job at the Mathworks.
Senior year, I took a couple of DSP courses from Sid Burrus in ECE and found a whole other set of interesting math problems I had never heard about in the math department. So, I went to grad school
at Princeton for a PhD in electrical engineering. Now I'm a tenured associate prof in electrical, computer, and systems engineering at RPI.
I really value the mathematical education I got at Rice, and it's still very useful in my professional life. RR - Class of 1996)
I graduated with a math degree in '02, but wasn't terribly interested in a programming job with Google or Microsoft like so many math majors I knew were. While the prospect of a career in research
did pique my interest, I instead decided to give the corporate world a try by becoming an actuary. I interned with Towers Perrin here in Houston the summer after my junior year, and wound up sticking
with them after I graduated. I hadn't really heard much about the actuarial career while at Rice (I know UT has the big program which churns out actuaries on a regular basis) so I pretty much learned
on the fly. I started taking the requisite exams during my internship, and finished up the full slate (roughly 12 of them) this past spring.
If you have any math majors who would like additional information on actuarial careers or corporate life in general, please feel free to have them contact me. We're always looking for talent and I
know your department has plenty of it. ( DS - Class of 2002)
I graduated in 2005, and I too wondered "What am I going to do as a math major?" I wasn't sure at all whether I wanted to work at Microsoft or the NSA, or teach, or pursue a PhD, or..... etc. But for
now, I joined Teach for America and am just finishing up my second year teaching 9th grade in Donna, Texas (in the Rio Grande Valley). So that's my story! Turns out I love teaching, and plan to keep
doing it for a few years, but I do see graduate school in my future, and who knows after that. (AK - Class of 2005)
I received my BA in Math in 1976 and my MA in Math in 1979 (both from Rice)....I took on a position with the MITRE Corporation in June 1979 (then working with NASA), and have been with this company
ever since. Right now I'm working at Ft. Bliss (El Paso) and the White Sands Missile Range on integrating some new technology into the US Army's tactical networks and information systems....I've
never regretted studying mathematics instead of some more directly "applicable" field. Plus, it's easy to wow one's colleagues with a few derivations of basic formulae that they never understood, but
just committed temporarily to memory! (RM - Class of 1979)
Innovative Research
Professionally, I continue love the work I do as a health care venture capitalist. I became a Managing Director in my firm about 4 years ago and have gotten to work on some really wonderful
companies. The most recent one is called iPierian, which is an iPS cell (“induced pluripotent stem cell”) and cellular reprogramming focused company co-founded with George Daley, Doug Melton, Deepak
Srivastava (a Rice grad!), and Lee Rubin based in South San Francisco. It’s a pretty exciting space and we are getting a lot of press coverage, in part because of the science, in part because of the
people, and in part given the way this technology could change the face of medicine (also, Al Gore has been very demonstrative about our company and the potential of this area, which certainly
excites the media). (AL - Class of 1998)
Update from the Rice News ...
Community Involvement
Thank you for sending these newsletters. I appreciate hearing about what is going on in the department after having been gone for so long (I graduated in 1995).
I appreciate the education I received at Rice and strong foundation I received as a math major. I taught high school math in Houston for two years after graduating for Rice and then caught the .com
fever of the late 90's. I started with IBM in 1997 and have been here since then.
I have been working on project at IBM for the past 5 years that I have thoroughly enjoyed called World Community Grid. It is a volunteer computing initiative that is funded by IBM's philanthropic
arm. Like Seti@Home, it asks volunteers to register on our website and download a small software client. The client downloads packets of work and runs the research at lowest priority on the system.
Collectively, our volunteers are providing about 340TFlops of computing power (or about 2000 years of processor time per week). The research that runs on World Community Grid is selected from
research proposals submitted to us. The research is selected based on it having a humanitarian benefit, suitability to run on a volunteer computing grid and that the research is performed by a
research institution capable of seeing the project to its conclusion. (KR - Class of 1995)
Computational Biology
My group's research is focused on developing predictive models of molecular recognition using high-resolution structural modeling. We are currently working to predict the specificity of protein-DNA
and protein-peptide interactions. We develop and apply new algorithms for molecular modeling within the framework of the Rosetta software package, a set of tools for the prediction and design of
protein structures and interactions. (PB - Class of 1995)
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Cloud-based graphical tools simplify motion design
Many positioning stages used in process applications — such as semiconductor manufacturing, microelectronic assembly, lab automation, and light machining — are characterized by attributes such as
light inertial loads, short travel, small footprints, and low cost. But these same tasks may demand high throughput, high jerk, high acceleration, high velocity, high precision, and smoothness of
motion, all with high reliability.
It can be tough to design a motion system that meets both sets of constants. The challenge lies in optimizing the system while keeping costs down. System analysis tools such as Mathcad and Matlab may
help by modeling performance as a function of various system parameters and operating conditions.
But, by turning to the Internet cloud, it’s possible to develop a simplified graphical tool resembling Mathcad and Matlab in concept that engineers can use interactively to quickly analyze various
positioning stage configurations.
The tool can quickly estimate answers to questions such as:
What is the settling time of a Parker LCR30 ball-screw stage and LCR30 belt-driven stage, on a 3-mm move, with 3-kg load, to an accuracy of 150 m?
Getting such answers with tools like Matlab may require expert modeling and costly, licensed software. It may be time consuming and expensive to test individual stages. On the other hand, cloud-based
tools may give good estimates within a few minutes of trial-and-error runs.
Such tools have several advantages. First, in many cases the tool is free to use on the Web. Second, operation is usually self-explanatory with 24/7 availability in the cloud. Third, the tool is
usually accessible by PC, cell phone, or even tablet. Combined with a manufacturer’s database of related components, the tool can help choose the optimum product for an application.
Model description
At Parker, we model our positioning-system analysis tool as a closed loop. It consists of two blocks, one for the controller/amplifier and another for the stage that includes motor, actuator, and a
moving load. The operation and interaction of these block is defined by the controller transfer function:
the stage transfer function:
the open-loop transfer function:
and the closed-loop transfer function:
where ωnat = natural frequency, ζ = damping coefficient, and n = sample point. The four equations simulate the time response of the stage with a Runge Kutta fourth-order numerical integration.
For the frequency-response analysis, the gain and the phase angles in the equations are:
where G = gain, R = real component of the complex transfer function in the four equations, I = imaginary component of the complex transfer functions, and Ph = phase.
System parameters
The stage is characterized by its moving mass, stiffness, structural damping coefficient, and motor constant. The controller is characterized by its proportional, derivative, and integral gain
settings and the step function setpoint.
These parameters may be changed interactively by trialand- error iterations to either represent known system parameters of potential stages or as trial parameters. Trial parameters may help reach and
confirm desirable system characteristics or a desirable performance variable.
Operation of the application is controlled by a set of operating buttons. Once the system parameters are entered, either manually or by clicking the Example button, a click on the Run button lets the
engineer observe the results. A Clear button erases the input variables permitting a new start from scratch.
System variables
The application output first shows the output variables that the tool generates, including stage characteristics and controller-filter characteristics. Stage-characteristic variables include natural
frequency and structural-damping coefficient. The controller-filter characteristics include a corner frequency for the integral gain and a corner frequency for the derivative gain.
Note that once the moving weight is known, the damping coefficient and the stiffness may be estimated to generate the desired stage characteristics by iteration. Here are the typical stage
characteristics derived from the working example:
The choice of PID variables automatically determines the corner frequencies of the controller. The corner frequencies assist the tuning process in selecting the PID gains for the desired stage
System performance in the time domain
With the response plotted in the time domain, the graph shows the position response of the stage to a step input. It should be noted that the user needs to indicate the time interval for the step
response and the settling window for which settling time will be calculated in the yellow boxes. The results in the blue boxes include the maximum amplifier current required to make this move, along
with the maximum power that the motor needs to generate. The results also indicate the maximum values of the jerk, acceleration, and velocity of the stage. Finally, the results show the settling time
to the desired precision window.
Settling time is an important variable of positioning systems. Many positioning applications, such as auto focus in scanning microscopes, may operate in repetitive small steps many thousands of
times an hour. This requirement means that throughput critically depends on settling time and that optimum performance demands a minimal settling time. The tool lets the user quickly try out various
stage configurations, with various controller parameters, in an attempt to optimize the desired move. Users can optimize for minimum settling time subject to constraints such as maximum current,
maximum acceleration, maximum velocity, or maximum jerk.
Plant-frequency response
The plant-frequency response is a quick graphical means of presenting the stage characteristics. As shown in the gain chart, the corner frequency is close to 10 Hz. The exact value is 13 Hz. The
structural damping is high as indicated by the low amplification. Up to the corner frequency the gain — which is the ratio of the current amplitude and the output position amplitude — is relatively
constant. For frequencies higher than the corner frequency, the gain drops at a rate of 40 dB per decade. The phase, as expected for a second-order system, starts at 0° in low frequencies, passes
through –90° at the corner frequency and ends up at –180° at high frequencies.
PID tuning may not be an easy task. However, it is critical to optimize system performance. Although the literature is saturated with examples of PID analysis, it is quite difficult for a nonservo
expert to follow its principles in a relatively short time. It should be noted that most modern servocontrollers have one form or another of PID filters. And many widely used controllers, such as the
Parker ACR, include additional filters such as feed-forward observers and notch filters, which are outside the scope of this article.
The first observation is that the PID gain has a trough shape. The bottom is determined by the proportional gain. For example, a proportional gain of 10,000 sets the bottom of the Gain graph at 80
dB. Recalling that the definition of a dB = 20 × log of the gain, it implies that dB = 20 × Log 10,000, which is equal to 20 × 4 or 80 dB.
The next observation is that the left side of the trough is a low-pass filter which boosts the gain of low-frequency inputs. This result is from the integrator gain. The integrator operates over a
long period; therefore, the lower the input frequency the longer the time before position change takes place and, therefore, the higher the gain. It should also be noted that the integral gain drops
at a rate of 20 dB/decade, crossing the bottom of the trough, as set by the proportional gain, at the corner frequency around 10 Hz. The exact value in the example is 19 Hz.
Similarly, the right side of the trough is the derivative gain. It is in fact a high pass filter, which amplifies the high frequency gains. Intuitively, the higher the frequency the higher the
rate-of-change of position and, therefore, the resulting derivative gain is higher. Notice that the integral gain ramps up at a rate of 20 dB per decade. The corner frequency of the derivative gain
is around 100 Hz. Once again, the exact value of the derivative corner frequency is 31.8 Hz.
A closer look at the phase chart reveals that it starts at –90°and gradually increases to 90°. Between these two limits, the phase chart goes through –45° at the integral corner frequency. It then
continues through 0°, at the center of the proportional gain region, and through 45° at the derivative corner frequency.
We may use these corner frequencies of the integral and derivative gains and the bottom trough of the proportional gain as cornerstones to shape up the PID frequency response and reach the desired
Open-loop transfer function
The open-loop gain is the sum of the plant gain and the controller gain. Similarly, the open-loop phase chart is the algebraic sum of the plant phase and the controller phase.
The important system performance variables drawn from the open-loop diagram are the position bandwidth and the stability margins. The position bandwidth is the cross over frequency at which the gain
crosses the –3 dB line. In the example it’s about 20 Hz. The stability of the system is measured by the phase margin and gain margin. The phase margin is estimated to be 40°, the difference between
the phase at the crossover frequency (–140°) and the -180° point. The preferred practice is a minimum phase margin of 45° for a robust design. Robust servosystems are defined as systems that display
minimal sensitivity to changes in parameters.
The gain margin is another stability variable determined from the open loop diagram as the distance between the gain at –180° phase and the 0-dB line. In the example it is not relevant because the
phase diagram does not cross the –180° line.
In summary, the objective of PID controller tuning is to shape the open-loop diagram to maximize the position bandwidth while maintaining good stability margins. Different stage types produce
differing position bandwidths with times to settle within a 1- m position.
Closed-loop transfer function
The closed-loop frequency response of the system shows the gain for lower frequencies is close to zero up to the corner frequency. Then it gradually changes its form and starts resembling the shape
of the open-loop chart at higher frequencies.
Similarly, at lower frequencies the phase is close to zero and, as the frequency rises beyond the corner frequency, it starts to resemble the shape of the openloop chart.
The following is an example of a positioning stage for semiconductor metrology application.
The positioning specification is for a stage moving a load of 15 kg, a step size of 2 mm and settling to 0.05 m in less than 140 msec with an amplifier current that does not exceed 15 A. The
simulation ran as shown using the tool at optineer.com/Analysis.
Within 10 min, trial-and-error testing determined the least-expensive motor that will do the job. It has a force constant of 54 N/A, equivalent to the Parker Trilogy 310-4. The simulation assumes
~92-Hz natural frequency for the stage with a typical linear motor structural damping of ~0.03. The resulting maximum current was ~12.5 A and the move and settle of a 2-mm step to 50 nm completed in
81.2 msec. The simulation also shows the PID gains used to get this performance. The results indicate a bandwidth around 15 Hz.
As demonstrated, simple analytic tools may reside on an internet cloud server and be available for users 24/7 as free promotional support. It should be made clear, however, that such tools depend on
the assumptions used. Therefore, these tools, such as the one presented in this article, must be used with caution and their results should always be validated by other tools or actual testing.
Edited by Robert Repas. For more information, visit the Parker Hannifin Daedal Div. website.
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North Brunswick Science Tutor
Find a North Brunswick Science Tutor
...I am a PhD Computational Scientist, and in the course of my research I have written hundreds of Perl programs to solve computational problems. I have an excellent knowledge of the language, and
I have experience training graduate students to program in Perl. I have been a user of Mathematica in...
15 Subjects: including biology, calculus, chemistry, precalculus
...I like teaching all kids and teens, special or not, In science, math and French. I was born in France, French language and culture is part of who I amBut I am also a PhD Scientist who tutors,
taught at the university for many years prior working in US industry. I am also very sensitive to special needs students.
14 Subjects: including nutrition, biochemistry, biology, chemistry
...I believe students benefit if they have a sense of how to begin and how to structure their thoughts to develop a coherent essay. I also teach political science courses such as European
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16 Subjects: including anthropology, English, reading, grammar
...It is my goal to find it so that every child is capable of learning.I have been working with children of all ages and abilities for a number of years. I currently hold certification as a School
Psychologist in both NJ and NY. I have experience working with special needs students as part of my job.
12 Subjects: including psychology, English, algebra 1, reading
...All the spectrum of subjects from mechanics, optics, electrostatics, magnetostatics, thermodynamics were in my curriculum. I also tutored high school students. I have a PhD in physics so I had
to use calculus, algebra and of course pre-algebra during all my education and also during my career.
9 Subjects: including physical science, physics, calculus, algebra 1
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How Many Calories Burned Riding a Bike
The amount of calories you burn biking in 20 minutes depends on how much you weigh and how fast you cycle. A leisurely bike ride of 10 mph or less will burn fewer calories than cycling faster in a
20-minute period. A 150-lb. person biking at less than 10 mph will burn 90 calories in 20 minutes, according to the Calories Burned Biking website. By increasing the speed to 12 to 14 mph, that
person will burn 198 calories in 20 minutes. A 200-lb. person will burn 120 calories with a leisurely bike ride, but traveling between 12 and 14 mph will burn 264 calories in 20 minutes. A 240-lb.
person will burn 144 calories in 20 minutes, while more vigorous biking will burn 317 calories.
Cycling for 20 minutes a day is beneficial, as the Department of Health and Human Services recommends at least 2 1/2 hours per week of moderate activity, such as leisurely bike riding, or 1 1/4 hours
of vigorous aerobic exercise, such as biking between 14 and 16 mph. In addition, a strength-training exercise is recommended at least twice a week. For weight loss, you may need to increase your
activity more, according to MayoClinic.com.
You must know your time as well as distance covered to estimate how many calories you burn riding a bike three miles. Use your time and distance to figure your average speed. To calculate average
speed, divide the distance covered by your time in minutes and multiply by 60. For example, if you ride three miles in 18 minutes, use this formula: 3/18 x 60 = 10 mph. Next, look on a chart of
calories burned over time such as those provided by MayoClinic.com or NutriStrategy.com to estimate calories burned. These charts usually list calories burned for a full hour. To find the calories
burned for other amounts of time, multiply the calories-burned estimate for your weight and speed by your time in minutes divided by 60. Suppose you ride a road bike for 18 minutes at 10 mph — three
miles — and you weigh 155 lbs. You will burn 281 calories multiplied by 18/60, or about 84 calories.
How Much Do You Weigh?
When calculating the number of calories burned on an average bike ride, you’ll need to consider your weight. When it comes to biking–or virtually any other form of exercise–the more you weigh, the
more calories you are likely to burn while engaged in cycling.
How Fast do You Ride?
The speed at which you operate your bicycle is another determinant factor in the number of burned calories you accumulate while biking. The faster you ride, the more calories you will burn. So when
cycling, try to keep a swift, steady pace.
How Long are Your Bike Rides?
Lastly, you’ll need to take the length of your bike rides into consideration when determining how many calories you burned. Much like the previously discussed factors, longer bike rides will result
in greater numbers of burned calories. So if you’re looking to lose weight, try to take long rides, albeit not excessively long ones.
Having considered the above-mentioned factors, you’re ready to determine the number of calories you burn while cycling. For example, if you weigh 120-175 pounds and bike at an average rate of 10 mph,
you will burn 215-240 calories per half hour. If you weigh 200-250 pounds and bike at the same speed, you will burn 300-360 calories per half hour. If you weigh 300-350 pounds and bike at the same
speed, you will burn 420-500 calories per half hour.
Related posts
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Math Help
September 8th 2009, 03:57 AM #1
Senior Member
Jan 2009
Obtain an expression of f(x) in ascending power of 1/x up to the term in 1/x^3 . Determine the set of values of x which this expansion is valid .
My work .
When i continue simplifying , i got $\frac{5}{x^2}+\frac{5}{x^3}$
Am i correct ?
The other thing is to find the set of values of x , is it
$<br /> <br /> |-\frac{3}{x}|<1<br />$
and $|\frac{2}{x}|<1$ ??
Last edited by thereddevils; September 8th 2009 at 05:39 AM.
Obtain an expression of f(x) in ascending power of 1/x up to the term in 1/x^3 . Determine the set of values of x which this expansion is valid .
My work .
When i continue simplifying , i got $\frac{5}{x^2}+\frac{5}{x^3}$
Am i correct ?
The other thing is to find the set of values of x , is it
$<br /> <br /> |-\frac{3}{x}|<1<br />$
and $|\frac{2}{x}|<1$ ??
your function
if your question at the first form I think the solution will be
the denominator should not equal zero in each fraction
so $xe 0$ and
$1+\frac{2}{x}e 0 \Rightarrow xe -2$ and
$1-\frac{3}{x} e 0 \Rightarrow xe 3$
so the values of x all real numbers expect {-2,0,3}
$f(x)=\frac{1}{x-3} - \frac{1}{x+2}$
and the values of x is the same as the first form
your function
if your question at the first form I think the solution will be
the denominator should not equal zero in each fraction
so $xe 0$ and
$1+\frac{2}{x}e 0 \Rightarrow xe -2$ and
$1-\frac{3}{x} e 0 \Rightarrow xe 3$
so the values of x all real numbers expect {-2,0,3}
$f(x)=\frac{1}{x-3} - \frac{1}{x+2}$
and the values of x is the same as the first form
Well , that doesn't answer my question . However , the answer to this question is $1+\frac{5}{x^2}+\frac{5}{x^3}$ , i just cant figure out where is my '1' .
And also the ranges of x which makes this expansion valid ??
September 8th 2009, 04:33 AM #2
September 8th 2009, 05:42 AM #3
Senior Member
Jan 2009
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Chapter 6 Lecture Ppt Presentation
Chapter 6: Engine Design :
Chapter 6: Engine Design BAE 599 - Lecture 6
Engine Design Process :
Engine Design Process Load Factor Displacement Bore/Stroke Ratio
Load Factor :
Load Factor Load factor is the ratio of average power output to maximum power output. Gasoline engines typically are designed with a load factor of 0.3 while diesel engines are designed with load
factors close to 1.0.
Selection of Displacement :
Selection of Displacement After rated speed and power is specified, the next step is to select the displacement. Typical pbme pressures range from 700 to 900 kPa are reasonable.
Bore/Stroke Ratios :
Bore/Stroke Ratios Typical bore/stroke ratios range from 0.84 to 0.96. Extreme values range from 0.79 to 1.30. Lower bore/stroke ratios facilitate higher compression ratios – more air flow supports
higher power output.
Engine Timing and Firing Order :
Engine Timing and Firing Order Engine cylinders are numbered from front to back on in-line engines. For “V” engines, the cylinders are numbered front to back on the left then right banks. Alternate
numbering scheme for “V” engines is the order (front to back) in which pistons are connected to the crank.
Direction of Rotation :
Direction of Rotation Standard direction of crankshaft rotation is clockwise when the crank is viewed from the front of the engine.
Fig. 6.1: Two-Cylinder Firing Order :
Fig. 6.1: Two-Cylinder Firing Order Special slide for Brandon!!!!
Fig. 6.1: Four-Cylinder Firing Order :
Fig. 6.1: Four-Cylinder Firing Order
Fig. 6.1: Six-Cylinder Firing Order :
Fig. 6.1: Six-Cylinder Firing Order
Fig. 6.1: Eight-Cylinder (V) Firing Order :
Fig. 6.1: Eight-Cylinder (V) Firing Order
Fig. 6.3: Valve Timing :
Fig. 6.3: Valve Timing
Engine Balance :
Engine Balance Rotating Masses Reciprocating Masses Rotational Speed Fluctuations Crankshaft Twist
Piston Crank Dynamics :
Piston Crank Dynamics The piston position as a function of crank angle, where S is the piston position from HDC, R is the crank throw, and L is the connecting rod length.
Fig. 6.4: Crankshaft with Counterweights :
Fig. 6.4: Crankshaft with Counterweights
Piston Crank Dynamics :
Piston Crank Dynamics The previous equation can be simplified to, using a binomial series expansion.
Piston Crank Dynamics :
Piston Crank Dynamics Differentiating the previous equation, velocity becomes, where
Piston Crank Dynamics :
Piston Crank Dynamics Differentiating the previous equation, acceleration becomes, where a is acceleration.
Piston Inertial Force :
Piston Inertial Force Using Newton’s Law and the previous equation, where m is piston/connecting rod mass, and F is the force generated when accelerating this mass.
Fig. 6.5: Piston-Crankshaft Dynamics :
Fig. 6.5: Piston-Crankshaft Dynamics
Effective Mass of Connecting Rod :
Effective Mass of Connecting Rod The portion of the connecting rod mass added to the piston mass is, where b and L are specified in Fig. 6.5.
Reciprocating Unbalance in Single-Cylinder Engines :
Reciprocating Unbalance in Single-Cylinder Engines The oscillating force in the x-direction (vertical) becomes, where mp is the mass of the piston, mc1 and mc2 are masses of the connecting rod, and
me is the mass of the crank pin.
Effective Mass of Crankpin :
Effective Mass of Crankpin The effective mass of the crankpin is determined as, where mcp is the mass of the crankpin, mca is the mass of the material supporting the crankpin, mcb is the mass of the
counterweight opposite the crankpin, R is the crankpin radius (1/2 the stroke), and Ra and Rb are the radii to the respective crankshaft masses.
Reciprocating Unbalance in Single-Cylinder Engines :
Reciprocating Unbalance in Single-Cylinder Engines The oscillating force in the y-direction (horizontal) becomes,
Primary and Secondary Shaking Forces :
Primary and Secondary Shaking Forces The oscillating forces (both x and y directions) have components at two frequencies. The primary shaking force occurs at engine speed. The secondary shaking force
occurs at twice the frequency of engine speed. Counterweights are sized to cancel half of the vertical primary shaking force, larger masses would increase the lateral primary shaking force to an
unacceptable level.
Reciprocating Unbalance in Multi-Cylinder Engines :
Reciprocating Unbalance in Multi-Cylinder Engines The Fx forces are added together for multi-cylinder engines – proper phase angle must be included. “Lanchester Balancers” are often used in
4-cylinder engines to cancle the secondary shaking force (used as the tractor frame) to protect the operator from vibration
Table 6.1: Amplitude of Shaking Forces :
Table 6.1: Amplitude of Shaking Forces
Fig. 6.6: Lanchester Balancer :
Fig. 6.6: Lanchester Balancer
Inertial Couples :
Inertial Couples Inertial couples (Table 6.1) tend to make the engine rock about the y-axis (coinciding with the centerline of the crankshaft). Three-cylinder engines have provisions for balancing
half of the primary couple by use of a counterweighted front pulley. Unfortunately, the counterweighted pulley generates a new yawing couple about the z-axis.
Instantaneous Torque and Flywheels :
Instantaneous Torque and Flywheels Instantaneous torque is a product of Qt and R. The end result is summarized as,
Fig. 6.5: Piston-Crankshaft Dynamics :
Fig. 6.5: Piston-Crankshaft Dynamics
Fig. 6.7: Instantaneous Torque :
Fig. 6.7: Instantaneous Torque
Radial Force at Crankshaft :
Radial Force at Crankshaft If the forces at the top of the piston is known (Fp), then the radial force (Qr) at the crankshaft becomes,
Flywheel Design :
Flywheel Design Instantaneous torque is less than average torque (Tave) for most of the cycle.
Flywheel Design :
Flywheel Design Average output torque is equal to the load torque on the engine. When the average torque drops below load torque, the engine stalls! When the instantaneous torque is greater than load
torque, the flywheel accelerates and stores energy. When the instantaneous torque is less than load torque, the flywheel gives up kinetic energy.
Fig. 6.8: Instantaneous Toque for 4 and 6 Cylinder Engines :
Fig. 6.8: Instantaneous Toque for 4 and 6 Cylinder Engines
Required Mass Moment of Inertia :
Required Mass Moment of Inertia Mass moment of inertia can be estimated as, where DE is the kinetic energy transfer and k is the speed variation coefficient.
Required Mass Moment of Inertia :
Required Mass Moment of Inertia The kinetic energy transfer can be estimated as, where W is the indicated work per revolution and l is a ratio from Table 6.2.
Table 6.2: Approximate Flywheel Constants :
Table 6.2: Approximate Flywheel Constants
Required Mass Moment of Inertia :
Required Mass Moment of Inertia The speed variation is obtained from the following relationship, where p is the percent of allowable speed variation.
Required Mass Moment of Inertia :
Required Mass Moment of Inertia Please recall the indicated work can be estimated as,
Vibration Dampeners :
Vibration Dampeners Instantaneous torques tend to twist the crankshaft. Higher harmonics from these torque fluctuations can approach the natural frequency of the crankshaft. More problematic with
long crankshafts (in-line 6 cylinder engines). Problem is corrected by mounting a small flywheel to the front of the crankshaft – connection is flexible.
Fig. 6.9: Vibration Dampeners :
Fig. 6.9: Vibration Dampeners
Homework Set No. 5 :
Homework Set No. 5 Do the even problems at the end of Chapter 6 for next Tuesday.
Footnote: 2001 smart fortwo cdi :
Footnote: 2001 smart fortwo cdi In view of the limited space available, installing a hybrid drive unit into the smart fortwo cdi, was an enticing prospect for the DaimlerChrysler engineers. This
smallest prototype from DaimlerChrysler hybrid vehicles is fitted with a 20 kW/28 hp electric motor, which together with the three-cylinder diesel engine constitutes a space-saving unit developing 30
kW/41 hp. The drive unit is a winner on all three counts: reduced fuel consumption, enhanced ride comfort and favorable acceleration with practically no interruption to tractive force.
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Selected Publications
Health Effects of Outdoor Air Pollution
• Acute mortality effects of PM[2.5] constituents [PDF]
• Reduced Bayesian hierarchical models [PDF]
• Spatial misalignment in time series studies [PDF]
• Chemical composition of PM[2.5] and hospital admissions [PDF]
• Coarse particulate matter (PM[10-2.5]) and hospital admissions [PDF]
• Fine particulate matter (PM[2.5]) and hospital admissions [PDF]
• Exposure-response curve and ozone [PDF]
• Bayesian hierarchical distributed lag time series model [PDF]
• Nickel/vanadium and mortality in NMMAPS [PDF]
• Trends in the short-term effect of PM[10] on mortality in the U.S. (1987–2000) [PDF]
• Model choice in time series studies of air pollution and mortality [PDF]
• Seasonal analyses of air pollution and mortality [PDF]
• Statistical methods for environmental epidemiology in R: A case study in air pollution and health [Book]
Climate Change, Weather, and Health
• Calculating the heat index [PDF]
• Heat-related emergency hospitalizations for respiratory diseases in the Medicare population [PDF]
• Flexible distributed lag models using random functions with application to estimating mortality displacement from heat-related deaths [PDF]
• Extending distributed lag models to higher degrees [PDF]
• Heat waves and mortality in Chicago [PDF]
• Bayesian model averaging, temperature, and mortality [PDF]
Methods for Reproducible Research
• Commentary in Science [PDF]
• Biostatistics editorial [PDF]
• Cacher package for R [PDF]
• Distributed reproducible research using cached computations [PDF]
• stashR package for R [PDF]
• Reproducible epidemiologic research (commentary) [PDF]
• Visualizing multivariate time series data [PDF]
• The NMMAPSdata package [PDF]
• Point process modeling in R [PDF]
• Wildfire point process modeling [PDF]
• Quantitative analysis of literary styles [PDF]
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Arcadian Functor
M Theory Lesson 1
It's high time we began a set of lessons in
M theory
. Today we will begin to look at the combinatorial structure of moduli spaces of Riemann surfaces, as studied by
Consider the following fractional transformations on the upper half plane. T is the map taking z to z + 1. A fundamental region for this map is a strip of width 1. We take the strip between -1/2 and
1/2 on the real line. The map S transforms the inside of the unit circle to the outside via z --> -1/z. Note that SS = 1 and (TS)^3 = 1. A fundamental region for the group so generated is the part of
the selected strip above the unit circle. Maps compose via matrix multiplication for fractional transformations. The group generated by the map S fixes i and TS fixes the point w = exp(pi i/3), a
root of unity.
By gluing this region into a cylinder with 2 singular points we obtain M(1,1), the orbifold moduli of the one punctured torus. The J invariant gives J(i) = 1 and J(w) = 0. We also take J(i oo) = oo.
Then it is possible to describe the equivalence between elliptic curves by the relation J(tau) = J(tau'), where tau is the complex parameter which characterises the curve. So the moduli M(1,1) is
parameterised by either tau in the upper half plane, or by z in the Riemann sphere CP^1 without the points 0,1,oo, which is the moduli M(0,4), obtained via a quotient of H using gamma(2).
5 Comments:
Mahndisa S. Rigmaiden said...
Teichmuller would be proud! And thanks for the lesson.
Hi Mahndisa
I'm glad that at least you are here. It would be pretty lonely otherwise.
Hello there:
Keep your chin up. Due to you and Matti, I have learned quite a bit and my next few posts will invoke some neat ideas. I will likely post them for Monday hopefully:)
BTW I am going over that series of notes by Mulase. The material is quite dense, but readable:)
Do keep up the lessons. Anything with 3rd roots of unity can't be useless.
I'm making little progress on book because the Hotel / casino I'm staying at in Curacao has both internet and Venezuelan TV with Spanish subtitled bad American movies.
It turns out that one can learn very useful foreign language phrases this way. If I just see this last movie a couple more times I will learn the Spanish for "Death to all the a**holes of the
world," something that you can use almost anywhere.
Hi Carl
Please tell us this useful phrase if you manage to work it out!
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Comparing large prime powers.
November 8th 2010, 05:32 AM
Comparing large prime powers.
I have two very large primes (a, b) , each raised to a large (non-prime) integer powers (p,q) .
I want to evaluate the comparison:
a^p >= b^q
However, I don't want to attempt to calculate a^p, b^q since the results will be astronomically huge, and I only need to know which is the largest.
Obviously if a>b and p>q then the answer is trivial, however in the general case I'm stumped...
Any suggestions ?
November 8th 2010, 05:16 PM
What about looking at their logarithms? It suffices to compare $p\log a$ and $q\log b$.
November 8th 2010, 11:55 PM
Thanks, but...
Thanks for the reply, however I am dealing with very large ( hundreds of digits ) numbers, and although this approach would be a lot easier to calculate, it still involves a considerable amount
of computation for very large numbers...
I was looking at the following approach to break up the numerator and iteratively
approach the answer in a loop:
if a^p < b^q
then (a^p) mod D =a^p // where D = b^q
= ( (a mod D ) * (a^(p-1) mod D ) ) mod D
Then the algorithm would run something like:
In a loop of 'p' iterations keep a result 'r' (initially 1 ? ). For each iteration multiply 'r' by (a mod D ). If the result is greater than D then result = result - D.
Obviously r will always be < 2D.
The problem here is that we still need 'p' big multiplications etc, the size of the integers involved would be governed by the size of D.
November 9th 2010, 06:57 AM
Wikipedia says that a logarithm can be computed in $O(\log n \ M(n))$ time, where $M(n)$ is the speed of your multiplication. That does not seem slow at all.
Did you actually try the logarithm approach for whatever numbers you are working with?
Regarding your proposal, working modulo $b^q$ actually doesn't help. You still have to compute $b^q$, and in the worst case scenario, all of your multiplications of $a$ will be run as integers.
So in other words, your method is just a complicated way of using brute force.
November 9th 2010, 07:07 AM
Thanks, I'm definately going to look into it at the weekend
Thanks for the reply, I'm definately going to look into the log approach at the weekend when I've got some time to sit down and look at it analytically.
I'm thinking that I may be able to evaluate either side as a series, taking the difference of the series and see if it converges to a positive or negative to indicate which side of the inequality
is greater.
Thanks for pointing me in this direction. I'll post again here when I've got a bit further in a few days time. I'm a programmer, not a mathematician, so there's a good chance that I'll miss the
obvious solution if I don't know the maths!
November 9th 2010, 07:08 AM
Sounds good. Best of luck!
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April 4th 2009, 05:15 PM #1
Junior Member
May 2008
Letting R be a commutative ring, and letting I = {a in R | a^n =0 for some n in S} where S is the set of positive integers.
Prove I is an ideal of R.
This is just saying that the nilradical is an ideal.
note if $x^n=0$ then $\forall a \in R$$(ax)^n=0$
Also if $x^n=y^n=0$ then
$(x+y)^{m+n-1}= \sum_{k=0}^{m+n-1}{n \choose k} x^{m+n-1-k}y^k=0.$
April 4th 2009, 05:30 PM #2
Dec 2008
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Interactive Voronoi Diagram Generator with WebGL
1. Make sure you are running a browser with WebGL enabled.
2. Click in black box to add another point to the set of input points.
3. Click and keep mouse down to temporarily add a point. Drag your mouse around to watch how the new input point influences the Voronoi diagram. On release, the new point will be added.
4. To make a query point, such that the Voronoi diagram will display the stolen area from the nearest neighbors, check the NNI query point box before adding the point.
5. If you would like to save a diagram for reuse later, copy the data from the data field and simply reload it in later.
Cone Radius:
NNI Query point?
Speed (Pixels/sec):
User defined:
Define your own speed function below with Javascript in terms of time t (optional array, length 4, of random variables rand):
• Sinusoidal: 5 * Math.sin(t)
• Random linear speed: rand[0]* 15
• Random quadratic speed: 2 * rand[1] * t + rand[0] * 10
• Possibly cubic: Math.round(rand[1]) * Math.pow(t,2) + rand[0] * 10
Interactive Voronoi Diagram Generator with WebGL
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ReSharper SDK Adventures Part 7 — Factoring expressions
Welcome to another part of the SDK adventures. Most of the sample plugins shown so far have been fairly simple, with implementation times ranging from a few minutes to an hour, at most. In this part
of the series I want to show what developing a fully-fledged, complicated feature is like.
Since complex features take a long time to develop, this post will only show the initial phases of development of a factoring context action that we will later refine and improve. Let’s get started!
When writing numeric expressions, you may come accross something like the following:
The above expression is worth analyzing because it’s not perfectly efficient: it’s doing one more multiplication than is needed to get the result. Here’s a better form:
Wouldn’t it be great if ReSharper could help factor out terms in similar expressions? Of course it would, and this is what we’re going to do here. However, as it turns out, this problem is a lot more
complicated than it looks.
Checking Availability
If you open up an expression such as the one above in ReSharper’s PSI Viewer (you’ll need to be running in Internal mode for this), you’ll see that it’s basically a combination of IBinaryExpressions
of type IAdditiveExpression and IMultiplicativeExpression. The operator precedence has already been applied, but there’s a caveat that additive expressions represent either addition or subtraction,
whereas multiplicative ones represent either multiplication or division. We’ll ignore this distinction for now, as a proper solution for all of these is even more complicated than what you’re about
to see.
Given that an expression such as a+b+c is really represented as (a+b)+c, we have good reason to implement a flattening algorithm that turns a binary tree into a flat list:
We can now think about checking the applicability of our context action. The analysis is deep, but it begins with finding the ‘root’ additive expression:
At this point, we need to do two things:
• Flatten the top-level addition so that (a+b)+c becomes just a+b+c. In fact, we keep it as a simple list.
• Do the same thing for the inner multiplications, so that (a*b)*c can be treated as just a*b*c.
The actual implementation uses our Flatten<T>() function as well as a special FlattenMultiplications() function.
Flattening multiplications is a similar operation that gives us even better data: it returns a list-of-list-of-expressions, i.e. data that we can work with to be able to actually analyze the
commonality of terms:
What we do now is build a frequency table of the terms — what they are, where they appear and how often. For example, for the expression a*x*x + b*x + c we would get a table such as:
│ │a│x│b│c│
The implementation is not particularly difficult:
Having the terms in a histogram of sorts, we find out the one that’s used most:
Once we have the term, we find out how many times we’re going to extract if (for example, in a*x*x*x + b*x*x it needs to be taken out twice), and we can finally return true. Note that we also keep a
record of affected terms, because even a term with a zero in it may either be a multiplicand of the term we’ve extracted, or not.
Applying the Action
The first thing we output is the multiplier term and the opening brace to our subexpression:
Then, we set up a lambda that outputs the sum-of-products in a neat fashion:
And then, of course, we generate the new expression and replace the old one:
Seeing it in Action
Let’s try a simple example. Say we have this method that computes a polynomial:
Executing the factoring action once, we get
Firing it again within the braces we get
And this is what our action is ultimately about.
Our approach works, but suffers from a few methodological deficiencies, namely
• We currently do not handle subtraction and division well
• We can only factor out one variable
• The algorithm is not recursive, so we have to invoke it several times to fully optimize the expression
In the next part of the series, we’ll take a look at fixing these issues. Meanwhile, check out the source code and stay tuned for more! ■
One Response to ReSharper SDK Adventures Part 7 — Factoring expressions
1. This is a really great post. It’s great to get more insight into how R# works behind the scenes. Keep posts like this coming!
This entry was posted in News and Events and tagged plugins, ReSharper. Bookmark the permalink.
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Physics 117 - Principles of Physics I - Mechanics
Physics 117 is an introduction (or re-introduction) to mechanics: the study of the motion of objects. We will learn about kinematics, forces, energy, and momentum. We'll be reviewing these concepts
while also learning about how to write proper solutions, to solve problems, and to think analytically and critically.
The course goals are:
· To acquaint students with calculus-based Newtonian Mechanics,
· To help students understand the difference between an exercise and a problem,
· To help students become a better problem solver,
· To introduce students to estimating to obtain meaningful results,
· To introduce students to methods of effectively communicating scientific information.
The main body of information for this course is located on this webpage.
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Excel VLOOKUP Formula Examples – Including how to create dynamic vlookup formulasExcel VLOOKUP Formula Examples - Including how to create dynamic vlookup formulas
In part 1 of this series we took a look at VLOOKUP formula basics. If you want to learn how a VLOOKUP formula works, or know someone else who is struggling with VLOOKUPs, check out the first article
“Discover a simple way to understand how VLOOKUP formulas work in Excel“, and come back to this article.
Here in part 2, we have a 14 minute video that explains three different ways to write the VLOOKUP formula for exact matches. If you have never used the COLUMNS function or MATCH function with
VLOOKUP, you are in for a treat because you will find out how to turn the standard VLOOKUP into a dynamic formula. As usual I have included a sample workbook with a table of fictional employee data
for you to download and follow along with the examples.
Image Credits:
CGI waves courtesy of “gerard79“.
Abstract circle shape courtesy of “iprole“.
Download the VLOOKUP Employee Table Sample Workbook
In order to follow along with the video and article please download the Sample Workbook “VLOOKUP_Employee_Table.xlsx” by clicking here
Note: the sample workbook has been tested on Excel 2010 (32-bit, English language version) and you may get different results in other version of Excel. If you find that you can’t use this workbook on
your version of Excel, please let us know using the comment section below.
The sample download workbook contains slightly different cell references compared to the workbook used in the video, but the data itself is identical and provides a convenient way for you to practise
your VLOOKUP formulas.
VLOOKUP Tutorial Video Part 2
Watch our 14 minute video to help you understand three different ways of writing the VLOOKUP formula for exact matches. Click on the video below to play it and listen to my audio commentary.
View count
Video Transcript
Note: The video was transcribed by SpeechPad.com and then edited into the form you see below.
We’re going to have a look at three different ways to write VLOOKUP formulas. One is a straightforward, simple approach, the second is using the columns function, and the third is using the MATCH
Here we have an employee LOOKUP table with the employee ID on the left. You see we’ve got employee full name, SSN, department, start date, and earnings for each of these employees.
What we’re going to do is use a VLOOKUP formula to populate this top table, so that whenever we change the data validation drop-down, say we change it from 3 to 2, we shall see the corresponding line
pulled out at the top.
Note about the Data
None of the names here are real people, they’re all computer-generated names. So if you see someone’s name that you recognize, it isn’t that person.
Learn more about Data Validation
By the way, if you want to learn more about data validation, you should check out my book “Power Tips for Excel.” That takes you through some data validation tips and tricks.
What we will cover in this video
Now I’m going to take you through three different ways of writing the VLOOKUP formula.
1. Simple way – type in numbers for column index number
2. COLUMN function – use the COLUMN function to specify column index number
3. MATCH function – use the MATCH function to specify column index number
The first is the standard approach, the second uses the COLUMN function, and the third uses the MATCH function. While the first way is the simple way, it is actually not the most efficient in this
case. We’ll go through method one, then we’ll check out the column method and the match method which in my opinion are usually more useful.
VLOOKUP Method One – Type in numbers for column index number (not dynamic)
The simple method is to use the VLOOKUP formula without any other functions. Go to cell C5 and we will set up the VLOOKUP to pull up “Full Name” using the employee ID. In the formula bar, type =
VLOOKUP, then type out the full VLOOKUP formula. When I click on E5 I also press the F4 key (three times) to lock this to absolute references so it doesn’t move from column B ($B5).
The table array is down here, but I’m not going to select the column headings. I will just select the data. Again I press the F4 key once to lock it to absolute reference ($B$10:$G$59).
Next I enter the column index, which is 2 for “Full Name”. Finally I type False, to specify that I want an exact match.
Press the Enter key, and you’ll notice that employee EMP002 gets pulled up, Lucian Franklin. And what I should be able to do now is copy that across into each of these cells (D7:G7). What I’ll do is
go Control + C for copy, and Control + Alt + V for paste special because I don’t want to overwrite the formatting, so I’m going to select paste formulas.
Now, you see it’s just put the full name in each of these boxes, which we don’t want.
Instead, I need to go up here, and change the column index number from a 2 to a 3. Press tab, and then for Department, I have to change it from a 2 to a 4. Press tab again to move right. And then in
Start Date, I need that to be a 5. Press tab. And in Earnings, I need that to be a 6. (See screenshot below)
So now we have the LOOKUP table completed using the simple approach. But it does mean that you have to manually enter 2, 3, 4, 5, and 6 as the column offset because when you copy and paste the
VLOOKUP formula the col_index_num does not automatically update for you.
VLOOKUP Method Two – Use the COLUMN function to specify column index number (semi-dynamic)
Method two uses the columns function, which you can see here. So the formula is going to end up like this, it’s going to be a VLOOKUP with the standard parameters. But it’s also going to use the
function COLUMNS with an array, and I explain that later, with a FALSE at the end for an exact match.
Go to cell C5 and type =VLOOKUP(. Select cell B5 as the lookup_value, and press F4 to lock the cell reference to $B$5. That means when it is copied across, it will always reference cell B5.
For the table_array, I will select the data excluding the column headings (as before, this is good practice when writing VLOOKUP formulas). Press Control + Shift + Right arrow, Control + Shift + Down
arrow, and that will select all the data in the table (this works because there are no empty data cells). Press the F4 key to lock that to absolute references.
And now, for the column index number, for “Full Name”, I want it to come out with a 2. And what the function COLUMNS does is it counts the number of columns in a particular array, where an array is
just a group of cells on the worksheet.
We want the array to start at B9 and go up to C9. I’m going to lock the range reference to the column B, by inserting a dollar sign ($B9:C9). And so when I copy this across, you’ll that the first
cell in the range reference stays in column B while the second cell in the range reference moves from column C to column D to column E etc. depending on how far I copy across (e.g. $B9:E9)
The last argument in the VLOOKUP formula is the range_lookup, and I set that to FALSE for an exact match.
So for employee EMP004 it has found “Denton Q Dale”. And now I’m going to copy the VLOOKUP formula, using Control + Alt + V to copy and paste special formulas only, so I don’t overwrite the existing
cell formats.
Now to explain what happened. The COLUMNS function looks at the range of cells you give it, and counts the number of columns in that range. As you copy the formula across, you notice that the range
stays fixed on column B, so it stays at $B9, but as I copy it across the second part of the range changes to C9, then D9, then E9, then F9 and finally G9 depending on how far to the right I copy.
So in cell G5, the COLUMNS function looks at the range $B9 to G9, and counts how many columns are there. So that would be one column, two columns, three columns, four columns, five columns, six
columns, and when it counts six columns, it will give the VLOOKUP formula, the right parameter for column index number.
VLOOKUP Method Three – Use the MATCH function to specify column index number (Dynamic)
Method three uses the MATCH function instead of the columns function to give you the column index number. The MATCH function returns the relative position of an item in an array that matches a
specified value.
So for example, if I want to find out where “Full Name” was in this array, what I do is type “=MATCH(“. First function argument is lookup_value so I select C4 (“Full Name”. The second function
argument is lookup_array so I select B9:G9 (the data table headings). The third and last function argument is match_type and we want an exact match so type 0 or FALSE.
The formula above gives a value of 2, which is correct, because the first column (“Employee ID”) does not contain “Full Name”, whereas the second column does contain “Full Name”.
I will modify the MATCH formula to make the range B9:G9 into absolute references by selecting it and pressing F4. The reason is that we want to copy this across and have it update with the correct
range, so copy the formula using CONTROL + C, then paste it to the cells on the right, pressing CONTROL + ALT + V to paste special as formulas.
When the formula is copied across to the right, you can see the results are 2, 3, 4, 5 and 6. What it’s done is evaluates the correct column number to use in our lookup formula by matching the
contents of row 4 to the contents of row 9.
Time to combine the MATCH function and VLOOKUP function
Now we now work through the VLOOKUP formula to see why the MATCH function can be very useful.
This is the combined VLOOKUP + MATCH formula: “=VLOOKUP($B$5, $B$10:$G$59, MATCH(C$4, $B$9:$G$9, 0), FALSE)”
For the third function argument column_index, we use the MATCH function to lookup up the value “Full Name” in cell C4, and find its position in the range of column headings in B9:G9.
Note the use of absolute references to lock the first argument of MATCH lookup_value to row 4, and lock the second argument of MATCH lookup_array to B9:G9. Set the match_type to 0 for exact match.
Copy the VLOOKUP + MATCH formula from cell C5 using the keyboard combination CONTROL + C, then paste special as formulas into cells D5:G5 using the keyboard combination CONTROL + ALT + V for the
paste special dialog box.
This is the result:
You will see that by writing one VLOOKUP + MATCH formula and copying it across, you did not need to manually change the col_index_num inside the VLOOKUP formula. And if you were to change the value
to cell C4 from “Full Name” to “Department”, the VLOOKUP formula in cell C5 dynamically knows to look up the “Department” instead of the “Full Name”. This demonstrates the flexibility of the VLOOKUP
+ MATCH combination.
What could go wrong with Dynamic VLOOKUPs?
Here’s a brief list of things that could go wrong with your dynamic VLOOKUP formulas. It’s by no means a comprehensive list, but covers a few fundamental errors:
• Using COLUMNS function (Method 2) when the order of your lookup table headers does not match the order of your data table headers (e.g. “Full Name, SSN, Department, Start Date” vs. “Full Name,
Department, SSN, Start Date”)
• When using the MATCH function (Method 3) your lookup table headers do not match your data table headers (e.g. misspelling “Depatment” instead of “Department”)
• Your absolute references might not be correct, and as you copy your dynamic VLOOKUP formula into other cells the COLUMNS() or MATCH() function could be looking up the wrong cells.
Time for you to practise your VLOOKUP formulas
I recommend you go to Excel and play around with VLOOKUP functions and try to insert COLUMN and MATCH to make the VLOOKUP formulas dynamic, because that way, you’ll master them quicker. All right?
Have fun!
If you found the video and article helpful share it with three or more of your colleagues and friends using the sharing buttons on this page, or email this link directly to them http://
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Results 1 - 10 of 414
- MACHINE LEARNING, FUNCTIONAL GENOMICS SPECIAL ISSUE , 2003
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- In Proceedings of the 21st National Conference on Artificial Intelligence , 2006
"... Relationships between concepts account for a large proportion of semantic knowledge. We present a nonparametric Bayesian model that discovers systems of related concepts. Given data involving
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Relationships between concepts account for a large proportion of semantic knowledge. We present a nonparametric Bayesian model that discovers systems of related concepts. Given data involving several
sets of entities, our model discovers the kinds of entities in each set and the relations between kinds that are possible or likely. We apply our approach to four problems: clustering objects and
features, learning ontologies, discovering kinship systems, and discovering structure in political data. Philosophers, psychologists and computer scientists have proposed that semantic knowledge is
best understood as a system of relations. Two questions immediately arise: how can these systems be represented, and how are these representations acquired? Researchers who start with the
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"... Motivation: Clustering is a useful exploratory technique for the analysis of gene expression data. Many different heuristic clustering algorithms have been proposed in this context. Clustering
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Motivation: Clustering is a useful exploratory technique for the analysis of gene expression data. Many different heuristic clustering algorithms have been proposed in this context. Clustering
algorithms based on probability models offer a principled alternative to heuristic algorithms. In particular, model-based clustering assumes that the data is generated by a finite mixture of
underlying probability distributions such as multivariate normal distributions. The issues of selecting a 'good' clustering method and determining the 'correct' number of clusters are reduced to
model selection problems in the probability framework. Gaussian mixture models have been shown to be a powerful tool for clustering in many applications.
- In Advances in Neural Information Processing Systems 17 , 2005
"... Statistical approaches to language learning typically focus on either short-range syntactic dependencies or long-range semantic dependencies between words. We present a generative model that
uses both kinds of dependencies, and can be used to simultaneously find syntactic classes and semantic topics ..."
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Statistical approaches to language learning typically focus on either short-range syntactic dependencies or long-range semantic dependencies between words. We present a generative model that uses
both kinds of dependencies, and can be used to simultaneously find syntactic classes and semantic topics despite having no representation of syntax or semantics beyond statistical dependency. This
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- J. of Graph Alg. and App. bf , 2004
"... Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We
propose here a measure of similarities between vertices based on random walks which has several important advan ..."
Cited by 94 (2 self)
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Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose
here a measure of similarities between vertices based on random walks which has several important advantages: it captures well the community structure in a network, it can be computed efficiently,
and it can be used in an agglomerative algorithm to compute efficiently the community structure of a network. We propose such an algorithm, called Walktrap, which runs in time O(mn 2) and space O(n
2) in the worst case, and in time O(n 2 log n) and space O(n 2) in most real-world cases (n and m are respectively the number of vertices and edges in the input graph). Extensive comparison tests
show that our algorithm surpasses previously proposed ones concerning the quality of the obtained community structures and that it stands among the best ones concerning the running time.
, 2003
"... Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high
similarity and points in different clusters having low similarity. In this paper, we derive a new cost fu ..."
Cited by 92 (4 self)
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Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high
similarity and points in different clusters having low similarity. In this paper, we derive a new cost function for spectral clustering based on a measure of error between a given partition and a
solution of the spectral relaxation of a minimum normalized cut problem. Minimizing this cost function with respect to the partition leads to a new spectral clustering algorithm. Minimizing with
respect to the similarity matrix leads to an algorithm for learning the similarity matrix. We develop a tractable approximation of our cost function that is based on the power method of computing
eigenvectors. 1
- In Proc. 17th NIPS , 2003
"... When clustering a dataset, the right number k of clusters to use is often not obvious, and choosing k automatically is a hard algorithmic problem. In this paper we present an improved algorithm
for learning k while clustering. The G-means algorithm is based on a statistical test for the hypothesis t ..."
Cited by 85 (6 self)
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When clustering a dataset, the right number k of clusters to use is often not obvious, and choosing k automatically is a hard algorithmic problem. In this paper we present an improved algorithm for
learning k while clustering. The G-means algorithm is based on a statistical test for the hypothesis that a subset of data follows a Gaussian distribution. G-means runs k-means with increasing k in a
hierarchical fashion until the test accepts the hypothesis that the data assigned to each k-means center are Gaussian. Two key advantages are that the hypothesis test does not limit the covariance of
the data and does not compute a full covariance matrix. Additionally, G-means only requires one intuitive parameter, the standard statistical significance level α. We present results from experiments
showing that the algorithm works well, and better than a recent method based on the BIC penalty for model complexity. In these experiments, we show that the BIC is ineffective as a scoring function,
since it does
, 2007
"... Unsupervised image segmentation is an important component in many image understanding algorithms and practical vision systems. However, evaluation of segmentation algorithms thus far has been
largely subjective, leaving a system designer to judge the effectiveness of a technique based only on intui ..."
Cited by 78 (2 self)
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Unsupervised image segmentation is an important component in many image understanding algorithms and practical vision systems. However, evaluation of segmentation algorithms thus far has been largely
subjective, leaving a system designer to judge the effectiveness of a technique based only on intuition and results in the form of a few example segmented images. This is largely due to image
segmentation being an ill-defined problem—there is no unique ground-truth segmentation of an image against which the output of an algorithm may be compared. This paper demonstrates how a recently
proposed measure of similarity, the Normalized Probabilistic Rand (NPR) index, can be used to perform a quantitative comparison between image segmentation algorithms using a hand-labeled set of
ground-truth segmentations. We show that the measure allows principled comparisons between segmentations created by different algorithms, as well as segmentations on different images. We outline a
procedure for algorithm evaluation through an example evaluation of some familiar algorithms—the mean-shift-based algorithm, an efficient graph-based segmentation algorithm, a hybrid algorithm that
combines the strengths of both methods, and expectation maximization. Results are presented on the 300 images in the publicly available Berkeley Segmentation Data Set.
- In IJCAI , 2003
"... We present a simple, easily implemented spectral learning algorithm which applies equally whether we have no supervisory information, pairwise link constraints, or labeled examples. In the
unsupervised case, it performs consistently with other spectral clustering algorithms. In the supervised case, ..."
Cited by 71 (5 self)
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We present a simple, easily implemented spectral learning algorithm which applies equally whether we have no supervisory information, pairwise link constraints, or labeled examples. In the
unsupervised case, it performs consistently with other spectral clustering algorithms. In the supervised case, our approach achieves high accuracy on the categorization of thousands of documents
given only a few dozen labeled training documents for the 20 Newsgroups data set. Furthermore, its classification accuracy increases with the addition of unlabeled documents, demonstrating effective
use of unlabeled data. By using normalized affinity matrices which are both symmetric and stochastic, we also obtain both a probabilistic interpretation of our method and certain guarantees of
performance. 1
- In ICML ’05: Proceedings of the 22nd international conference on Machine learning , 2005
"... This paper views clusterings as elements of a lattice. Distances between clusterings are analyzed in their relationship to the lattice. From this vantage point, we first give an axiomatic
characterization of some criteria for comparing clusterings, including the variation of information and the unad ..."
Cited by 69 (3 self)
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This paper views clusterings as elements of a lattice. Distances between clusterings are analyzed in their relationship to the lattice. From this vantage point, we first give an axiomatic
characterization of some criteria for comparing clusterings, including the variation of information and the unadjusted Rand index. Then we study other distances between partitions w.r.t these axioms
and prove an impossibility result: there is no “sensible” criterion for comparing clusterings that is simultaneously (1) aligned with the lattice of partitions, (2) convexely additive, and (3)
bounded. 1.
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Logic gates and boolean algebra
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Logic gates and boolean algebra
Logoc gates, boolean algebra, truth tables, karnaugh maps... this stuff just doesn't click for me at all. So if anyone hear understands it, help would be muchly appreciated.
Here is the circuit... sorry it looks like a little kids drawing. i had to draw it in paint as i have no scanner to scan the official question paper with.
What I have to do is...
1. Determine the Boolean function implemented by this circuit. Express that you determine it from the circuit above.
2. Create a truth table for this circuit.
3. Simplify the Boolean function found from above (you may use karnaugh mapping or algebraic methods to simplify the expression)
4. Draw a simpler circuit.
Thats it. Any answer or absolutely any tips or whatever will be awesome!
cheers guys
Last edited by morik (2006-04-30 02:15:54)
Re: Logic gates and boolean algebra
That circuit isn't valid. One of the lines that goes into each and gate doesn't have a source.
Unless they all branch out from B? If a wire splits (i.e. goes off in two directions), you represent that by a dot where the split occurs. For example:
Is that what you mean?
Last edited by Ricky (2006-04-30 02:39:34)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: Logic gates and boolean algebra
yes, that line you thought was invalid does branch out from B. Sorry, my fault for forgetting to pu the dot to represent the split in.
Re: Logic gates and boolean algebra
Three input variables, so 2^3 is eight squares in the Karnaugh map.
You can make it a 2 by 4 grid or a 4 by 2 grid or a 2 X 2 X 2 cube, if you like symmetry and 3-D.
If you want to use a 4 by 4 grid, just remember that a variable D is a don't care value, either one or
zero, so you have two boxes for each of your permutations of A,B,and C.
It takes a while to catch on to this stuff, but I'll help more soon.
igloo myrtilles fourmis
Re: Logic gates and boolean algebra
truth tables and Karnaugh maps are two ways of describing input combinations to outputs of false or true, or don't care.
"Don't care" values can be assigned 0 or 1 so the circuit or boolean expression ends up to your advantage.
In your example, the middle and gate is doing nothing for the circuit because a zero and a one are AND'd
together making a zero that is then put into the final 3-input or gate. A zero into an OR gate means that the
other inputs are considered. Why this is true requires explanation for which I might attempt later.
A logic one (true value) into an AND gate means that the other inputs will be considered.
igloo myrtilles fourmis
Re: Logic gates and boolean algebra
Also, tell me which way you want to letter your C, B, A on a Karnaugh map.
C on left and B A on top edge or A on left and B C on top edge?
Then go 0 1 down for rows and 00 01 11 10 for columns across?
Is that how you were taught?
igloo myrtilles fourmis
Real Member
Re: Logic gates and boolean algebra
Karnaugh map???
IPBLE: Increasing Performance By Lowering Expectations.
Real Member
Re: Logic gates and boolean algebra
This picture is some electric-like logic diagramq as I see.
Let's see what could be it...
1. Determine the Boolean function implemented by this circuit. Express that you determine it from the circuit above.
That means that the funtion will be obviously:
The triangle-circle block has 1 in and 1 out, so it may be f(x)=!x.
What does the half-circle means?
Is it "AND" or "OR"?
And what is the cutted-circle left F?
Union(...) or Intersection(...)?
IPBLE: Increasing Performance By Lowering Expectations.
Re: Logic gates and boolean algebra
The triangle-circle block has 1 in and 1 out, so it may be f(x)=!x.
Great deduction krassi!
1. Boolean function
I'm not entirely sure what you are looking for here. Like for example, would a function be add or subtract, or NOR, or NAND or something similar? If so, I don't know of any standard function that
takes 3 inputs.
2. Create a truth table for this circuit.
You should be able to do this one, it isn't too hard. Just start with the three inputs A, B, C being 0, 0, 0, and add 1 (in binary) each time. So first is 0, 0, 0 then it's 0, 0, 1, then it's 0, 1,
0. Each time, just go through and see which one is true or false.
3. Simplify the Boolean function found from above
I think the easiest way to do this is through boolean algebra. Just take the sum of products to start out with:
^ is "and"
v is "or"
~ is "not"
(~A ^ B ^ ~C) v (A ^ B ^ ~C) v (A ^ B ^ C)
What I did was I took every entry in the truth table where F is 1 (true), and I anded all the conditions together. For example, one line in my truth table is:
A B C F
So that's ~A since A is 0, B since B is 1, and ~C since C is 0. And these three conditions together, then do the same for every other line that F turns out to be 1. Now or all those conditions
together. And now, simplify.
From the above, you can take a B out of each since B must be true in each condition. So it becomes:
B ^ ( (~A ^ ~C) v (A ^ ~C) v (A ^ C) )
And you can simplify this futher. Use the distributive property first, then use the reverse distributive property. It will end up being:
B ^ (~C v A)
Now use this for 4.
4. Draw a simpler circuit.
Draw the ciruit in the order that you would read the expression. For example, you read the parantheses first. So take C, not it, and then or that with A. Then take the output from this, and and it
with B. The output from this gate is F. See how much simpiler your circuit is? It only uses 2 gates instead of 4.
Last edited by Ricky (2006-04-30 09:08:10)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: Logic gates and boolean algebra
Oh, and for those who like to play around with these sort of things, Logisim is great fun.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: Logic gates and boolean algebra
Hey guys,
In reply to John E. Franklin's question... I was taught to do karanugh maps with A on the left edge and B and C on the top edge with, like you say, 0 and 1 down and 00 01 11 10 across the top.
Thanks a lot for all your replies. They have certainly improved my understanding of the subject and have taken off a stress that I am suffering about the upcoming exam on it.
And Ricky, that piece of freeware is awesome! Thanks tons for that.
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