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Manipulation and analysis of Mathematical Equations
05-24-2003 #1
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Manipulation and analysis of Mathematical Equations
The title is a bit long, but esentialy it is too the point.
Anyway, I at the begining of the day I was completely out of ideas of what to program, so I decided to try my luck at using C++ to do some mathematical equations.
The problem that I am having is that once I have got the equation in as a string (got a list of variables and their multipliers / dividers) I am stuck as too how to go about manipulating this
What is the best way of doing this. For the basics I just have a whole group of switch and ifelse statements and for loops, but this is getting way out of control.
Any help would be greately apreciated.
What I am planning to do with the equations is find such things as their derivatives and also their solutions for a given set of variables.
Thanks heaps.
Kind of a broad question there. What do you have so far? Post some code so we can see what approach you've taken. For simple arithmatic, just push all of the numbers onto one stack, ops onto
another. Then evaluate them like this:
while(!ops.empty() && stack.count()>1) {
left = stack.pop();
right = stack.pop();
op = ops.pop();
switch(op) {
case '+' : stack.push(left+right); break;
// etc.
total = stack.pop();
if( numeric_limits< byte >::digits != bits_per_byte )
error( "program requires bits_per_byte-bit bytes" );
Thanks for your reply.
That seems like a viable solution for non-algebraic equations.
One program that I have completed uses the Simplex Method for finding solutions to maximisation problems.
The source code is avaliable here
and this is what I am going to be using for the retrieval of the string and looking at it.
Though this is only one of the programs that I am working on at the moment.
To try and slim down the question I am interested in working with algebraic mathematic equations.
Originally posted by WebmasterMattD
To try and slim down the question I am interested in working with algebraic mathematic equations.
What kind of algebraic equations? And are they equations to be solved (i.e. something = something_else, both sides of which may include variables ), expressions to be evaluated, or functions? I
ask because I once wrote a Polynomial class that might help you out, although its input is kind of specific, and the code is ugly because at the time I didn't know STL and the string class.
Still, it may provide some of the functionality you want, or at least some ideas.
Let me know how it works out. I was thinking of writing a set of classes to model math functions, but I thought it would be more trouble than worth (since I had no real use for it, I just thought
it was interesting). Good luck, HTH
Had a look at your PolyNom class.
It has given some good ideas as to tackling the problems I will undoutably encounter, and has also given me a starting point.
Actually, using a stack to evaluate the expression fails to account for one detail... order of operations. A standard solution is to use a tree where each terminal node must be a number or an
identifier (i.e. variable placeholder), and all of the non-terminal nodes are operations (with the number of children corresponding to the number of input variables). This way, the operations
with the highest order can be placed closest to the operands. An example:
The expression:
3*4 + 5*4/5
Has a representation of in the tree of:
* *
3 4 5 /
If you just place it in a stack though, it evaluates as:
(3*4 + 5) * 4/5
For symbolic algebra, you can use a placeholder class for the leaves of the tree. The placeholder can either be a number or variable, based on the internal state, and then you can manipulate the
rest of the tree to solve for the variable.
The word rap as it applies to music is the result of a peculiar phonological rule which has stripped the word of its initial voiceless velar stop.
Thanks for your help.
One prolem that I am running into is when I try to overload the '-' operator.
What I am confused about is which of the classes in the expression
class1 = class2 - class3
does the program use for the pointer 'this'?
I know that it uses class3 as the argument passed to the function when declared as
class operator -( class asdf );
Any help on this would be greately apreciated.
Check out ygf's expression manipulator. He posted the .exe and source on the General Discussions Board.
>>...which of the classes in the expression
class1 = class2 - class3
does the program use for the pointer 'this'?
There could be two overloaded operators in that statement ( '=' & '-' ). For the '-' operator, class2 would be answer, and for the '=' operator, class1 would be the answer.
Originally posted by WebmasterMattD
Thanks for your help.
One prolem that I am running into is when I try to overload the '-' operator.
What I am confused about is which of the classes in the expression
class1 = class2 - class3
does the program use for the pointer 'this'?
Like the Dog said, there are two possibilities, and you will have to cover both. As I understand it, that's basically saying
class1.operator=( class2.operator-( class3 ) )
You'll have to define an operator= and an operator- (or something that allows the object to be converted to something that has those defined).
Sometimes it helps to name the parameter for +,-, etc. "rhs" for "right-hand-side", and think of "this" as the left hand side. Then code the function in terms of the lhs and rhs. HTH.
Much apretiated advice.
I have got the class to the stage where it can do addition and subtraction as well as factorising of quadratics.
What I am stuck on for ideas is how to do long division (in order to do factorising of cubics, quartics and higher degrees. Any other methods would be greatly apreciated if you feel I am going
about this the wrong way.) and how to determine the type of equation that has been entered (currently handles ax^2 + bx + c = 0 style equations) so that it can handle any form of polynomail
If anyone is interested in looking at the code it is avaliable
Thanks for all of your help so far.
What I am stuck on for ideas is how to do long division (in order to do factorising of cubics, quartics and higher degrees. Any other methods would be greatly apreciated if you feel I am going
about this the wrong way.) and how to determine the type of equation that has been entered (currently handles ax^2 + bx + c = 0 style equations) so that it can handle any form of polynomail
I'm using linux. When I type stuff in, it forces me to press enter twice, and it doesn't accept my input: "3x^2+5x+3=0"
Code still verry alpha stage of development.
It will accept the input if you provide spaces as per this example
3x^2 + 2x + 1 = 0
I shall have to go about fixing that part of it up. Thanks for pointing it out.
If your divisor is linear, you can use synthetic division. I believe there are techniques for nonlinear divisors (i.e. for a quadratic, divide once each by two linear factors). The algorithm
isn't too difficult (though it's kinda hard to explain without showing you an example). If you want more info, let me know (or just google it).
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¡Amo fútbol!
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Dec 2002 | {"url":"http://cboard.cprogramming.com/cplusplus-programming/39844-manipulation-analysis-mathematical-equations.html","timestamp":"2014-04-19T15:57:32Z","content_type":null,"content_length":"94176","record_id":"<urn:uuid:f1a30fae-2118-4e16-bee8-e15a506feacf>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00243-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dumb holes - ergo-regions, horizons, and surface gravity
2.5 Dumb holes - ergo-regions, horizons, and surface gravity
Let us start with the notion of an ergo-region: Consider integral curves of the vector If the flow is steady then this is the time translation Killing vector. Even if the flow is not steady the
background Minkowski metric provides us with a natural definition of “at rest”. ThenThis quantity changes sign when [265 , 164 , 422 ]. A trapped surface in acoustics is defined as follows: Take any
closed two-surface. If the fluid velocity is everywhere inward-pointing and the normal component of the fluid velocity is everywhere greater than the local speed of sound, then no matter what
direction a sound wave propagates, it will be swept inward by the fluid flow and be trapped inside the surface. The surface is then said to be outer-trapped. (For comparison with the usual situation
in general relativity see [164 , pages 319-323] or [422 , pages 310-311].) Inner-trapped surfaces (anti-trapped surfaces) can be defined by demanding that the fluid flow is everywhere
outward-pointing with supersonic normal component. It is only because of the fact that the background Minkowski metric provides a natural definition of “at rest” that we can adopt such a simple and
straightforward definition. In ordinary general relativity we need to develop considerable additional technical machinery, such as the notion of the “expansion” of bundles of ingoing and outgoing
null geodesics, before defining trapped surfaces. That the above definition for acoustic geometries is a specialization of the usual one can be seen from the discussion on pages 262-263 of Hawking
and Ellis [164 ]. The acoustic trapped region is now defined as the region containing outer trapped surfaces, and the acoustic (future) apparent horizon as the boundary of the trapped region. That
is, the acoustic apparent horizon is the two-surface for which the normal component of the fluid velocity is everywhere equal to the local speed of sound. (We can also define anti-trapped regions and
past apparent horizons but these notions are of limited utility in general relativity.) The event horizon (absolute horizon) is defined, as in general relativity, by demanding that it be the boundary
of the region from which null geodesics (phonons) cannot escape. This is actually the future event horizon. A past event horizon can be defined in terms of the boundary of the region that cannot be
reached by incoming phonons - strictly speaking this requires us to define notions of past and future null infinities, but we will simply take all relevant incantations as understood. In particular
the event horizon is a null surface, the generators of which are null geodesics.
In all stationary geometries the apparent and event horizons coincide, and the distinction is immaterial. In time-dependent geometries the distinction is often important. When computing the surface
gravity we shall restrict attention to stationary geometries (steady flow). In fluid flows of high symmetry (spherical symmetry, plane symmetry), the ergosphere may coincide with the acoustic
apparent horizon, or even the acoustic event horizon. This is the analogue of the result in general relativity that for static (as opposed to stationary) black holes the ergosphere and event horizon
coincide. For many more details, including appropriate null coordinates and Carter-Penrose diagrams, both in stationary and time-dependent situations, see [13 ].
Because of the definition of event horizon in terms of phonons (null geodesics) that cannot escape the acoustic black hole, the event horizon is automatically a null surface, and the generators of
the event horizon are automatically null geodesics. In the case of acoustics there is one particular parameterization of these null geodesics that is “most natural”, which is the parameterization in
terms of the Newtonian time coordinate of the underlying physical metric. This allows us to unambiguously define a “surface gravity” even for non-stationary (time-dependent) acoustic event horizons,
by calculating the extent to which this natural time parameter fails to be an affine parameter for the null generators of the horizon. (This part of the construction fails in general relativity where
there is no universal natural time-coordinate unless there is a timelike Killing vector - this is why extending the notion of surface gravity to non-stationary geometries in general relativity is so
When it comes to explicitly calculating the surface gravity in terms of suitable gradients of the fluid flow, it is nevertheless very useful to limit attention to situations of steady flow (so that
the acoustic metric is stationary). This has the added bonus that for stationary geometries the notion of “acoustic surface gravity” in acoustics is unambiguously equivalent to the general relativity
definition. It is also useful to take cognizance of the fact that the situation simplifies considerably for static (as opposed to merely stationary) acoustic metrics.
To set up the appropriate framework, write the general stationary acoustic metric in the form
The time translation Killing vector is simply The metric can also be written as Now suppose that the vector Substituting this back into the acoustic line element gives In this coordinate system the
absence of the time-space cross-terms makes manifest that the acoustic geometry is in fact static (there exists a family of spacelike hypersurfaces orthogonal to the timelike Killing vector). The
condition that an acoustic geometry be static, rather than merely stationary, is thus seen to be that is, (since in deriving the existence of the effective metric we have already assumed the fluid to
be irrotational), This requires the fluid flow to be parallel to another vector that is not quite the acceleration but is closely related to it. (Note that, because of the vorticity free assumption,
Once we have a static geometry, we can of course directly apply all of the standard tricks [372] for calculating the surface gravity developed in general relativity. We set up a system of fiducial
observers (FIDOS) by properly normalizing the time-translation Killing vector
The four-acceleration of the FIDOS is defined as and using the fact that That is The surface gravity is now defined by taking the norm so that the surface gravity is given in terms of a normal
derivative byThis is not quite Unruh’s result [376 , 377 , 378 ] since he implicitly took the speed of sound to be a position-independent constant. The fact that prefactor [192 ]. Though derived in a
totally different manner, this result is also compatible with the expression for “surface-gravity” obtained in the solid-state black holes of Reznik [319 ], wherein a position dependent (and
singular) refractive index plays a role analogous to the acoustic metric. As a further consistency check, one can go to the spherically symmetric case and check that this reproduces the results for
“dirty black holes” enunciated in [386 ]. Since this is a static geometry, the relationship between the Hawking temperature and surface gravity may be verified in the usual fast-track manner - using
the Wick rotation trick to analytically continue to Euclidean space [147]. If you don’t like Euclidean signature techniques (which are in any case only applicable to equilibrium situations) you
should go back to the original Hawking derivations [159 , 160 ]. One final comment to wrap up this section: The coordinate transform we used to put the acoustic metric into the explicitly static form
is perfectly good mathematics, and from the general relativity point of view is even a simplification. However, from the point of view of the underlying Newtonian physics of the fluid, this is a
rather bizarre way of deliberately de-synchronizing your clocks to take a perfectly reasonable region - the boundary of the region of supersonic flow - and push it out to “time” plus infinity. From
the fluid dynamics point of view this coordinate transformation is correct but perverse, and it is easier to keep a good grasp on the physics by staying with the original Newtonian time coordinate.
If the fluid flow does not satisfy the integrability condition which allows us to introduce an explicitly static coordinate system, then defining the surface gravity is a little trickier.
Recall that by construction the acoustic apparent horizon is in general defined to be a two-surface for which the normal component of the fluid velocity is everywhere equal to the local speed of
sound, whereas the acoustic event horizon (absolute horizon) is characterised by the boundary of those null geodesics (phonons) that do not escape to infinity. In the stationary case these notions
coincide, and it is still true that the horizon is a null surface, and that the horizon can be ruled by an appropriate set of null curves. Suppose we have somehow isolated the location of the
acoustic horizon, then in the vicinity of the horizon we can split up the fluid flow into normal and tangential components
Here (and for the rest of this particular section) it is essential that we use the natural Newtonian time coordinate inherited from the background Newtonian physics of the fluid. In addition Since
the spatial components of this vector field are by definition tangent to the horizon, the integral curves of this vector field will be generators for the horizon. Furthermore the norm of this vector
(in the acoustic metric) is In particular, on the acoustic horizon
Consider the quantity
To calculate the first term note that Thus And so: On the horizon, where Similarly, for the second term we have On the horizon this again simplifies There is partial cancellation between the two
terms, and so while Comparing this with the standard definition of surface gravity [422 ] we finally have This is in agreement with the previous calculation for static acoustic black holes, and
insofar as there is overlap, is also consistent with results of Unruh [376 , 377 , 378 ], Reznik [319 ], and the results for “dirty black holes” [386]. From the construction it is clear that the
surface gravity is a measure of the extent to which the Newtonian time parameter inherited from the underlying fluid dynamics fails to be an affine parameter for the null geodesics on the horizon.
Again, the justification for going into so much detail on this specific model is that this style of argument can be viewed as a template - it will (with suitable modifications) easily generalise to
more complicated analogue models.
2.5.1 Example: vortex geometry
As an example of a fluid flow where the distinction between ergosphere and acoustic event horizon is critical consider the “draining bathtub” fluid flow. We shall model a draining bathtub by a (3+1)
dimensional flow with a linear sink along the z-axis. Let us start with the simplifying assumption that the background density
In the tangential direction, the requirement that the flow be vorticity free (apart from a possible delta-function contribution at the vortex core) implies, via Stokes’ theorem, that
(If these flow velocities are nonzero, then following the discussion of [401 ] there must be some external force present to set up and maintain the background flow. Fortunately it is easy to see that
this external force affects only the background flow and does not influence the linearised fluctuations we are interested in.) For the background velocity potential we must then have
Note that, as we have previously hinted, the velocity potential is not a true function (because it has a discontinuity on going through
Dropping a position-independent prefactor, the acoustic metric for a draining bathtub is explicitly given by
Equivalently A similar metric, restricted to A=0 (no radial flow), and generalised to an anisotropic speed of sound, has been exhibited by Volovik [404 ], that metric being a model for the acoustic
geometry surrounding physical vortices in superfluid [420], this reference is also useful as background to understanding the Lorentzian geometric aspects of not identical to the metric of a spinning
cosmic string, which would instead take the form [388] In conformity with previous comments, the vortex fluid flow is seen to possess an acoustic metric that is stably causal and which does not
involve closed timelike curves. (At large distances it is possible to approximate the vortex geometry by a spinning cosmic string [404], but this approximation becomes progressively worse as the core
is approached.)
The ergosphere forms at
Note that the sign of
The acoustic event horizon forms once the radial component of the fluid velocity exceeds the speed of sound, that is at
The sign of
2.5.2 Example: slab geometry
A popular model for the investigation of event horizons in the acoustic analogy is the one-dimensional slab geometry where the velocity is always along the
That is
If we set [378 , page 2828, equation (8)], Jacobson [188 , page 7085, equation (4)], Corley and Jacobson [88 ], and Corley [86]. (In this situation one must again invoke an external force to set up
and maintain the fluid flow. Since the conformal factor is regular at the event horizon, we know that the surface gravity and Hawking temperature are independent of this conformal factor [192 ].) In
the general case it is important to realise that the flow can go supersonic for either of two reasons: The fluid could speed up, or the speed of sound could decrease. When it comes to calculating the
“surface gravity” both of these effects will have to be taken into account.
2.5.3 Example: Painlevé-Gullstrand geometry
To see how close the acoustic metric can get to reproducing the Schwarzschild geometry it is first useful to introduce one of the more exotic representations of the Schwarzschild geometry: the
Painlevé-Gullstrand line element, which is simply an unusual choice of coordinates on the Schwarzschild spacetime. In modern notation the Schwarzschild geometry in ingoing (
Equivalently This representation of the Schwarzschild geometry was not (until the advent of the analogue models) particularly well-known, and it has been independently rediscovered several times
during the 20th century. See for instance Painlevé [293], Gullstrand [154], Lemaître [228], the related discussion by Israel [183], and more recently, the paper by Kraus and Wilczek [218]. The
Painlevé-Gullstrand coordinates are related to the more usual Schwarzschild coordinates by Or equivalently With these explicit forms in hand, it becomes an easy exercise to check the equivalence
between the Painlevé-Gullstrand line element and the more usual Schwarzschild form of the line element. It should be noted that the
As emphasised by Kraus and Wilczek, the Painlevé-Gullstrand line element exhibits a number of features of pedagogical interest. In particular the constant time spatial slices are completely flat -
the curvature of space is zero, and all the spacetime curvature of the Schwarzschild geometry has been pushed into the time-time and time-space components of the metric.
Given the Painlevé-Gullstrand line element, it might seem trivial to force the acoustic metric into this form: Simply take
The best we can actually do is this: Pick the speed of sound
So we see that the net result is conformal to the Painlevé-Gullstrand form of the Schwarzschild geometry but not identical to it. For many purposes this is quite good enough: We have an event
horizon, we can define surface gravity, we can analyse Hawking radiation. Since surface gravity and Hawking temperature are conformal invariants [192] this is sufficient for analysing basic features
of the Hawking radiation process. The only way in which the conformal factor can influence the Hawking radiation is through backscattering off the acoustic metric. (The phonons are minimally coupled
scalars, not conformally coupled scalars so there will in general be effects on the frequency-dependent greybody factors.) If we focus attention on the region near the event horizon, the conformal
factor can simply be taken to be a constant, and we can ignore all these complications. | {"url":"http://relativity.livingreviews.org/Articles/lrr-2005-12/articlesu6.html","timestamp":"2014-04-20T11:09:53Z","content_type":null,"content_length":"62383","record_id":"<urn:uuid:e4789fcd-2d61-4421-8be7-998491525c5f>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00451-ip-10-147-4-33.ec2.internal.warc.gz"} |
Final Forecast: 2012 Presidential True Vote/Election Fraud Model
Richard Charnin
Nov 5, 2012
The final 2012 Presidential True Vote/Election Fraud Model exactly forecast Obama’s 332 electoral vote. His projected 51.6% two-party recorded share was close to the actual 51.9%. But Obama actually
did much better in the True Vote forecast (391 EV, 56% two-party). As usual, the systematic fraud factor was in effect, causing a 4-5% red-shift. But Obama overcame the fraud, just as he did in 2008.
The final 2008 Election Model was also right on the money. It forecast Obama would have 53.1% and 365.3 expected EV compared to his actual recorded 52.9% share and 365 EV. But he had a 58.0% True
Vote Model share and 420 EV. His 58.0% share of the unadjusted state exit polls (76,000 respondents) confirmed the True Vote Model.. He won the unadjusted National Exit Poll by 61-37% (17,836
The Presidential True Vote and Monte Carlo Simulation Forecast Model is updated on a daily basis. The election is assumed to be held on the latest poll date.
Final Forecast: 11/06/2012 9am
Obama: 320.7 expected electoral votes; 99.6% win probability (498 of 500 trials).
He has a 332 snapshot EV (actual total).
He leads the state poll weighted average by 49.3-46.2% (51.6% 2-party share).
He leads 50.4-47.0% in 16 of 18 Battleground states with 184 of 205 EV.
Obama leads Romney in the RCP National average: 48.8-48.1%.
Rasmussen and Gallup are Likely Voter (LV) polls which lean to the GOP.
Rasmussen: Romney 49-48%.
Gallup: Romney 50-49%. It was 51-46% a week ago.
Obama leads in the Rand poll 49.5-46.2% (closely matching the state polls). Unlike the national LV polls, the Rand poll doesn’t eliminate respondents but rather weights them on a scale of 1-10 (based
on voter preference and intention to vote).
The 3% Obama margin increase in the Rand poll over the national LV polls illustrates why the LVs understate Obama’s margin by using the Likely Voter Cutoff Model (LVCM). LV polls are a subset of the
registered voter (RV) sample. They always understate the Democratic share. The majority of voters eliminated by the Likely Voter Cutoff Model (LVCM) are Democrats.
The True Vote Model indicates that Obama would have 55.2% of the two-party vote with 371 expected EV in a fraud-free election. Will he be able to overcome the systemic fraud factor?
2012 Presidential True Vote and Monte Carlo Simulation Forecast Model (html)
- The Monte Carlo Electoral Vote Simulation is based on the latest state polls and currently assumes an equal split of undecided voters. The expected electoral vote is the sum of the products of the
state win probabilities and corresponding electoral votes.
- The True Vote Model is based on plausible turnout estimates of new and returning 2008 voters and corresponding vote shares.
The model calculates an estimated True Vote forecast for the National aggregate or any state. The calculation is displayed below the input data section. State poll-based national vote shares,
electoral vote and probabilities are displayed on the right side of the screen.
2008 True Vote 2012 Vote Pct Obama Romney
Obama 76.2 58.0% 72.4 68.8 54.2% 90% 10%
McCain 53.0 40.3% 50.3 47.8 37.7% 7% 93%
Other. 2.20 1.66% 2.10 1.97 1.6% 50% 50%
DNV ...................8.27 6.5% 59% 41%
Total 131.4 100% 124.8 126.8 100% 56.1% 43.9%
..............True Vote........... 71.1 55.7
............. Recorded Vote....... 51.0% 47.2%
............. Projected 2-party... 51.6% 48.4%
............. Electoral Vote
............. Projected Snapshot.. 332 206
............. 500 Simulation Mean. 321 217
............. Expected True EV.... 385 153
............. EV Win Probability.. 99.8%
This worksheet contains the weekly polling trend analysis.
The polling data is from the Real Clear Politics (RCP) and Electoral-vote.com websites. The simulation uses the latest state polls.
View this 500 election trial simulation electoral vote frequency graph.
1988-2008: 274 State exit polls. An 8% Discrepancy
In the six presidential elections from 1988-2008, the Democrats won the average recorded vote by 48-46%. But they led both state and national exit polls by 52-42%. There were approximately 375,000
respondents in the 274 state polls and 90,000 respondents in the six national polls. Overall, an extremely low margin of error.
1988-2008 Unadjusted State and National Exit Poll Database
The Ultimate Smoking Gun that proves Systemic Election Fraud:
1) The Likely Voter Cutoff Model eliminates newly registered Democrats from the LV sub-sample. Kerry had 57-61% of new voters; Obama had 72%.
2) Exit poll precincts are partially selected based on the previous election recorded vote.
3) In the 1988-2008 presidential elections, 226 of 274 exit polls red-shifted to the Republicans. Only about 137 would normally be expected to red-shift. The probability is zero.
4) 126 of the 274 exit polls exceeded the margin of error. Only 14 (5%) would normally be expected. The probability is ZERO.
5) 123 of the 126 exit polls that exceeded the margin of error red-shifted to the Republicans. The probability is ZERO.
No exit polls in 19 states
The National Election Pool (NEP) is a consortium of six corporate media giants which funds the pollster Edison Research to do exit polling in the U.S and abroad. The NEP announced that they would not
exit poll in 19 states, 16 of which are universally thought of as being solid RED states. Or are they?
In 2008, Obama won exit polls in AK, AL, AZ, GA, NE, SD. He came close to winning in TX, KY, SC, TN, MS. These former RED states may have turned PURPLE. View this worksheet in the model.
The bad news is that the NEP decision to eliminate the polls makes it easier for vote margins to be padded and electoral votes flipped. Without the polls, it is much more difficult to calculate the
statistical probabilities of fraud based on exit poll discrepancies. In the 1988-2008 elections, the Democrats led the unadjusted state exit polls by 52-42%, but by just 48-46% in the official
recorded vote. This is a mathematically impossible result which proves systemic election fraud.
The good news is that the post-election True Vote Model should find implausible discrepancies in the recorded state and national votes. After all, that is what it was designed to do.
Sensitivity Analysis
The pre-election TVM built in the 2012 Election Model uses alternative scenarios of 2008 voter turnout and defection rates to derive a plausible estimate of the total final share. The returning voter
assumptions are based on Obama’s 58% True Vote (a plausible estimate) and his 53% recorded share. The latter scenario results in vote shares that are close to the LV polls.
The sensitivity analysis of alternative turnout and vote share scenarios is an important feature in the model. The model displays the effects of effects of incremental changes in turnout rates and
shares of returning voters. The tables display nine scenario combinations of a) Obama and McCain turnout rates and b) Obama/Romney shares of returning Obama and McCain voters. Obama’s vote share,
winning margin and popular vote win probability are displayed for each scenario.
Registered and Likely Voters
Historically, RV polls have closely matched the unadjusted exit polls after undecided voters are allocated and have been confirmed by the True Vote Model.
Likely Voter (LV) polls are a subset of Registered Voter polls and are excellent predictors of the recorded vote – which always understate the Democratic True Vote. One month prior to the election,
the RV polls are replaced by LVs. An artificial “horse race” develops as the polls invariably tighten.
The Likely Voter Cutoff Model (LVCM) understates the voter turnout of millions of new Democrats, thereby increasing the projected Republican share. Democrats always do better in RV polls than in the
LVs. Based on the historical record, the Democratic True Vote share is 4-5% higher than the LV polls indicate. The LVs anticipate the inevitable election fraud reduction in Obama’s estimated 55% True
Vote share.
Media pundits and pollsters are paid to project the recorded vote – not the True Vote. The closer they are, the better they look. They never mention the fraud factor which gets them there, but they
prepare for it by switching to LV polls.
The disinformation loop is closed when the unadjusted, pristine state and national exit polls are adjusted to match the LV recorded vote prediction.
2004 and 2008 Election Models
The 2004 model matched the unadjusted exit polls. Kerry had 51.7% and 337 electoral votes. But the election was stolen. Kerry had 48.3% recorded. View the 2004 Electoral and popular vote trend
The 2008 model exactly matched Obama’s 365 EV. The National model exactly matched his official recorded 52.9% share; the State model projected 53.1%. His official margin was 9.5 million votes.
Obama had 58.0% in the unadjusted, weighted state exit poll aggregate (83,000 respondents) which exactly matched the post-election True Vote Model. Obama’s 23 million True Vote margin was too big to
The National Exit Poll displayed on mainstream media websites (Fox, CNN, ABC, CBS, NYT, etc.) indicates that Obama had 52.9% – his recorded vote. Unadjusted state and national exit polls are always
forced to match the recorded share.
But the media never discussed the fact that Obama had 61% in the unadjusted National Exit Poll (17,836 respondents). View the 2008 Electoral and popular vote trend
This graph summarizes the discrepancies between the 1988-2008 State Exit Polls and the corresponding Recorded Votes.
The True Vote Model
The 2008 True Vote Model (TVM) determined that Obama won in a landslide by 58-40.3%. Based on the historical red-shift, he needs at least a 55% True Vote share to overcome the systemic 5% fraud
factor. The TVM was confirmed by the unadjusted state exit poll aggregate: Obama had an identical 58-40.5% margin (83,000 respondents). He won unadjusted National Exit Poll (17,836 respondents) by an
even bigger 61-37% margin.
In projecting the national and state vote, a 1.25% annual voter mortality rate is assumed. The TVM uses estimated 2008 voter turnout in 2012 and corresponding 2012 vote shares. The rates are applied
to each state in order to derive the national aggregate result.
There are two basic options for estimating returning voters. The default option assumes the unadjusted 2008 exit poll as a basis. The second assumes the recorded vote. It is important to note that
the True Vote is never the same as the recorded vote. The 1988-2008 True Vote Model utilizes estimates of previous election returning and new voters and and adjusted state and national exit poll vote
Monte Carlo Simulation
The simulation consists of 500 election trials. The electoral vote win probability is the number of winning election trials divided by 500.
There are two forecast options in the model. The default option uses projections based on the latest pre-election state polls. The second is based on the state True Vote. The fraud factor is the
difference between the two.
The projected vote share is the sum of the poll and the undecided voter allocation (UVA). The model uses state vote share projections as input to the Normal Distribution function to determine the
state win probability.
In each election trial, a random number (RND) between 0 and 1 is generated for each state and compared to Obama’s state win probability. If RND is greater than the win probability, the Republican
wins the state. If RND is less than the win probability, Obama wins the state. The winner of the election trial is the candidate who has at least 270 electoral votes. The process is repeated in 500
election trials.
Electoral Votes and Win Probabilities
The Electoral Vote is calculated in three ways.
1. The Snapshot EV is a simple summation of the electoral votes. It could be misleading if close state elections favor one candidate.
2. The Mean EV is the average of the 500 simulated election trials.
3. The Theoretical EV is the product sum of the state electoral votes and corresponding win probabilities. A simulation or meta-analysis is not required to calculate the expected EV.
The Mean EV approaches the Theoretical EV as the number of election trials increase. This is an illustration of the Law of Large Numbers.
Obama’s electoral vote win probability is his winning percentage of 500 simulated election trials.
The national popular vote win probability is calculated using the national aggregate of the the projected vote shares. The national margin of error is 1-2% lower than the MoE of the individual
states. That is, if you believe the Law of Large Numbers and convergence to the mean.
The Fraud Factor
The combination of True Vote Model and state poll-based Monte Carlo Simulation enables an analyst to determine if the forecast electoral and popular vote share estimates are plausible. The aggregate
state poll shares can be compared to the default TVM.
The TVM can be forced to match the aggregate poll projection by…
- An incremental change in vote shares. A red flag would be raised if the match required that Obama captured 85% of returning Obama voters and Romney had 95% of returning McCain voters (a 10% net
- Adjusting 2008 voter turnout in 2012. For example, if McCain voter turnout is required to be 10-15% higher than Obama’s, that would raise a red flag.
- Setting the returning voter option to the 2008 recorded vote. The implicit assumption is that the 2008 recorded vote was the True Vote. But the 2008 election was highly fraudulent. Therefore, model
vote shares will closely match the likely voter polls.
Check the simulated, theoretical and snapshot electoral vote projections and corresponding win probabilities.
In 2004, Election Model forecasts were posted weekly using the latest state and national polls. The model was the first to use Monte Carlo simulation and sensitivity analysis to calculate the
probability of winning the electoral vote. The final Nov.1 forecast had Kerry winning 337 electoral votes with 51.8% of the two-party vote, closely matching the unadjusted exit polls.
2004 Election Model Graphs
State aggregate poll trend
Electoral vote and win probability
Electoral and popular vote
Undecided voter allocation impact on electoral vote and win probability
National poll trend
Monte Carlo Simulation
Monte Carlo Electoral Vote Histogram
In the 2006 midterms, the adjusted National Exit Poll was forced to match the House 52-46% Democratic margin. But the 120 Generic Poll Trend Model forecast that the Democrats would have a 56.4% share
– exactly matching the unadjusted exit poll.
The 2008 Election Model projection exactly matched Obama’s 365 electoral votes and was within 0.2% of his 52.9% recorded share. He won by 9.5 million votes. But the model understated his True Vote.
The forecast was based on final likely voter (LV) polls that had Obama leading by 7%. Registered voter (RV) polls had him up by 13% – even before undecided voters were allocated. The landslide was
The post-election True Vote Model determined that Obama won by 23 million votes with 420 EV. His 58% share matched the unadjusted state exit poll aggregate (83,000 respondents).
Exit pollsters and media pundits have never explained the massive 11% state exit poll margin discrepancy or the impossible 17% National Exit Poll discrepancy. If they did, they would surely claim
that the discrepancies were due to reluctant Republican responders. But they will not even try to explain the impossible returning voter adjustments required to force the polls to match the recorded
vote in the 1988, 1992, 2004 and 2008 elections.
2008 Election Model Graphs
Aggregate state polls and projections (2-party vote shares)
Undecided vote allocation effects on projected vote share and win probability
Obama’s projected electoral vote and win probability
Monte Carlo Simulation Electoral Vote Histogram
Published 10/27/12:
Matrix of Deceit: Forcing Pre-election and Exit Polls to Match Fraudulent Vote Counts
Track Record: Election Model Forecast; Post-election True Vote Model
2004 Election Model (2-party shares)
Kerry 51.8%, 337 EV (snapshot)
State exit poll aggregate: 51.7%, 337 EV
Recorded Vote: 48.3%, 255 EV
True Vote Model: 53.6%, 364 EV
2008 Election Model
Obama 53.1%, 365.3 EV (simulation mean);
Recorded: 52.9%, 365 EV
State exit poll aggregate: 58.0%, 420 EV
True Vote Model: 58.0%, 420 EV
2012 Election Model
Obama Projected: 51.6% (2-party), 332 EV snapshot; 320.7 expected; 321.6 mean
Adjusted National Exit Poll (recorded): 51.0-47.2%, 332 EV
True Vote Model 56.1%, 391 EV (snapshot); 385 EV (expected)
Unadjusted State Exit Polls: not released
Unadjusted National Exit Poll: not released
27 responses to “Final Forecast: 2012 Presidential True Vote/Election Fraud Model”
2. In short, “Will he be able to overcome the systemic fraud factor?”
3. I happened upon your blog thru Google. 2 major problems with your data, What polls are you basing these numbers on? The RCP averages show almost a 50-50 split, which reflects the country pretty
accurately based on political affiliation and LV polls.
2. If everything were this much of a fraud and conspiracy, Democrats would never win, which as we see is not happening. Also, a lot of the most populous states have democrat dominated election
commissions . If fraud was true then democrats would have to be in on the conspiracy. Take some meds.
□ I use the latest state polls. Apparently you never viewed the model.
The fraud is systemic. Here is the proof.
□ I would suggest that Obama was ‘allowed’ to win because the republicans knew the economy was about to tank and they didn’t want to be in the drivers seat. Now the economy is improving they
want in again!!
☆ Allowed to win? It was too much to steal.
He won by an approximate 23 million vote landslide.
Massive election fraud reduced his massive landslide by 13 million to 9.5 million.
On the other hand Kerry won by 67-57 million in 2004. It was not enough.
Election fraud also reduced his margin by 13 million
He “lost” by 3 million.
5. Richard, I think what Mags was saying was that by making questionable choices, ie: Sarah Palin, the GOP deliberately tanked the election for the reason Mags stated.
6. Your presidential projections always show the same party winning. That tells me your partisan. A quick search to find the sites you frequent and have been removed from confirms you are partisan.
You can keep your stupid “analysis.” I’m too intelligent for you.
□ Intelligent you may be but not well educated you said, “your partisan” but I think you meant “you’re partisan”!!!
□ Tess, you are too intelligent for me? Well, it must be true because you have just provided an overwhelming set of facts to prove your case. NOT. Yes, you have proved I am partisan. NOT. OK,
tell me specifically what you are looking for. Because you are too intelligent for me, I will try to locate them for you – since you are so intelligent and cannot bother to do so for
Does your information prove that the Democrats did NOT win the TRUE VOTE in 2000 and 2004, and that Bush did NOT steal both elections. I would like to see it.
Because you are so intelligent, you apparently have the proof and can show it to us. And because you are so intelligent, you can also prove that there is no such thing as ELECTION FRAUD.
Yes, Tess, you are much more intelligent than I am. You are so intelligent, that I don’t expect that you will review the FACTS and the mathematical proof that the elections were stolen, based
on the unadjusted exit polls and the True Vote Model and the reams of other evidence. You are too intelligent to have to actually do an investigation and analysis of the FACTS.
Yes, Tess, I admit you are much more intelligent than I am – in your own mind.
7. Tess, you are so intelligent that you have created a new meaning for the word “your”. Way to go!
Keep up the good work Richard!
9. Richard, you present systematic differences between exit polls and recorded votes as proof of election fraud, but in order for that to be proof you must eliminate the possibility of systematic
bias in exit polls. Have you considered that (1) exit polls do not include absentee and mail-in voters, (2) the majority of curmudgeons who refuse to be exit-polled might be Republican, (3) other
potential sources of bias?
Also are you familiar with Choquette and Johnson’s report that claims to prove vote-flipping starting in 2008? The anomalies they’ve identified do not exist in earlier years.
I notice that, in your bar chart showing frequency with which MoE was exceeded, 2008 is much higher than all previous years. So maybe there was already a systematic bias and fraud was introduced
in 2008. I would really like to see a comparison between exit poll results and Choquette and Johnson’s projections of the true vote based on smaller precincts. Have you looked at 2012 (and 2008)
Republican Primary exit polls and found evidence of votes being shifted to Romney??
□ I have looked at Choquette’s work. It’s excellent and thorough. Very convincing.
1) Majority of refusers republican? That’s a worn out disproven myth. In fact the opposite is true. Check the 2005 exit poll evaluation report. Republican response was highest in republican
2) Exit polls do not include mail-in and absentee voters? These appear to be Democratic. Oregon, Washington and Colorado had tiny exit poll discrepancies in 2008- and just happened to be the
states with the highest percentage of early (paper ballot) voting.
Obama had 52% of the first 121 million recorded votes and 59% of the late (absentee, mail-in paper ballots). Kerry and Gore won the late votes recorded after election day by substantial
3) Systematic election fraud takes place in every election. But 2008 was the worst; the margin of error was exceeded in 37 state exit polls.
Compare 2008 to each of the five prior elections:
Finally, I suggest you read this:
12. A couple of things: You say that “Obama had 58.0% in the unadjusted, weighted state exit poll aggregate”. I assume you are referring to the final feed from the NEP, after the polls close and the
exit poll data is weighted, but before it is adjusted to conform to official results. What is your source for this data? As I recall, at the time the data was only available to subscribers who
had to sign severe non-disclosure agreements. Second, assuming that voters who make up their minds at the last minute will be evenly divided is risky. I believe they historically break toward the
challenger–obviously irrelevant in 2008, but relevant this time. Lastly, what is your opinion of the unprecedented level of early voting this year, and how it will affect the accuracy of exit
□ I am referring to the total 82,388 sample unadjusted state exit polls from the NEP. The data source is Roper; it is publicly available.
The unadjusted state and national exit polls for each of the 1988-2008 elections:
Notice that the state exit polls are weighted by votes cast. Pre-weighted and weighted aggregate vote shares are shown. The weighted share is generally 1-2% higher than the unweighted for the
Obama had 61% in the unadjusted National Exit Poll (17836 respondents), but just a 52.87% recorded share. The pollsters had to effectively reduce Obama’s exit poll respondents from 10873 to
9430 (13.3%) in order to force the exit poll to match the recorded vote.
It is true that undecided voters typically break for the challenger.
I assume that the undecided will break fairly evenly. There are very few who are still undecided. In any case, the LV polls understate Obama’s share because of the Likely Voter Cutoff model.
Most rejected registered voters are Democrats.
Since polling samples are at least partially based on prior election recorded votes and not votes cast, they are implicitly biased to favor the Republicans. That is another factor to be
Bottom line is that undecided voters are a minor factor in any case. The pollsters have already allocated undecided voters since the totals are very close to 100% when you consider the third
party vote – which probably hurts Obama more than Romney.
As far as early voting is concerned, I believe it will favor Obama. High voter turnout always favors the Democrat.
Finally, I will reiterate what I said in my daily update: Obama’s True Vote must be at least 55% to overcome the built-in systemic fraud factor.
18. Pingback: Ashley Hastie
19. You produced some excellent points throughout ur posting, “Updated Daily: Presidential True Vote/Election Fraud Forecast
Model Richard Charnin’s Blog”. I may wind up coming back to ur page before long. Thx ,Bette | {"url":"http://richardcharnin.wordpress.com/2012/10/17/update-daily-presidential-true-voteelection-fraud-forecast-model/","timestamp":"2014-04-19T01:53:05Z","content_type":null,"content_length":"118237","record_id":"<urn:uuid:e50263ed-4e6b-4a08-8d75-aa49e2cf367b>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00108-ip-10-147-4-33.ec2.internal.warc.gz"} |
add1, sub1 and + recursive process
13 April 2013 4:56 PM (scheme | guile)
In The Little Schemer you are asked to write the function + using the functions zero?, add1 and sub1, such as the result of (+ 46 12) is 58.
(define add1
(lambda (n) (+ n 1)))
(define sub1
(lambda (n) (- n 1)))
The solution given is:
(define o+
(lambda (n m)
((zero? m) n)
(else (add1 (o+ n (sub1 m)))))))
which is correct but has a little problem.
If you are like me (hopefully not), you might have read twenty times the first chapters of SICP (and haven't gone any further). Remember about recursive and iterative processes and recursive
procedures? In our o+ case we have a recursive procedure that generates a recursive process, as the process generates a chain of deferred add1 operations.
With a recursive process we can easily blow our stack:
scheme@(guile-user)> (o+ 10 100000)
While executing meta-command:
ERROR: Throw to key `vm-error' with args `(vm-run "VM: Stack overflow" ())'.
This is because this will generate:
(add1 (add1 (add1 (add1 .... (add1 10)))))
with as many add1 as m.
So, how to improve this? With a recursive procedure that generates an iterative process:
(define o+
(lambda (n m)
((zero? m) n)
(else (o+ (add1 n) (sub1 m))))))
which will generate:
(o+ 11 99999)
(o+ 12 99998)
(o+ 100010 0)
and doesn't overflow the stack.
scheme@(guile-user)> (o+ 10 100000)
$1 = 100010
Yes, this is a tail-recursive function. | {"url":"http://hacks-galore.org/aleix/blog/archives/2013/04/13/add1-sub1-and-recursive-process","timestamp":"2014-04-18T10:34:12Z","content_type":null,"content_length":"5260","record_id":"<urn:uuid:0d1178c4-72e4-45bd-ac0a-226ac7320f7f>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00106-ip-10-147-4-33.ec2.internal.warc.gz"} |
El Toro Math Tutor
...Because Algebra 2 involves concepts that require knowledge from previous math classes, I ensure that students have a strong background in those more basic concepts before tackling more complex
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22 Subjects: including differential equations, Spanish, grammar, reading | {"url":"http://www.purplemath.com/el_toro_math_tutors.php","timestamp":"2014-04-20T19:19:11Z","content_type":null,"content_length":"23758","record_id":"<urn:uuid:326de064-d291-4072-b972-64f230896588>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00407-ip-10-147-4-33.ec2.internal.warc.gz"} |
George is 8 years more than twice the age of his son Tom. If the difference of their ages is 38, how old is George? - WyzAnt Answers
George is 8 years more than twice the age of his son Tom. If the difference of their ages is 38, how old is George?
Tutors, please
to answer this question.
Hi Jonathon;
Tom's age =x
George' age=2x+8
Let's subtract 8 from both sides...
x=30=Tom's age | {"url":"http://www.wyzant.com/resources/answers/15997/george_is_8_years_more_than_twice_the_age_of_his_son_tom_if_the_difference_of_their_ages_is_38_how_old_is_george","timestamp":"2014-04-18T12:18:49Z","content_type":null,"content_length":"37880","record_id":"<urn:uuid:787d5605-7c82-4917-ad17-a6e1a65381cf>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
Difficult probability problem
August 29th 2011, 07:30 AM #1
Sep 2008
Difficult probability problem
A large white cube is painted red, and then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly. A blind man (who also cannot feel the paint) reassembles the small
cubes into a large one. What is the probability that the outside if this large cube is completely red?
Re: Difficult probability problem
there are 4 types of subcube:
those with no sides painted (1)
those with 1 sides painted (6)
those with 2 sides painted (12)
those with 3 sides painted (8)
Any shuffling of these subcubes within their own group will keep the cube exterior completely red, so there are 1!6!12!8! possible solutions. You should allow for orientations but i think this
has no effect on this part of the calc (i think that for any given position, only 1 orientation is valid).
Step two: find the total number of outcomes (including orientations) and divide one by the other.
PS: I should warn you i dont have the best track record at combinatronics problems so i might have got that spectacularly wrong
Last edited by SpringFan25; August 29th 2011 at 10:28 AM.
Re: Difficult probability problem
August 29th 2011, 10:16 AM #2
MHF Contributor
May 2010
August 30th 2011, 08:07 AM #3 | {"url":"http://mathhelpforum.com/advanced-statistics/186917-difficult-probability-problem.html","timestamp":"2014-04-19T18:12:39Z","content_type":null,"content_length":"37192","record_id":"<urn:uuid:2449ecfc-73aa-47ed-ba88-9ed5237a7448>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00064-ip-10-147-4-33.ec2.internal.warc.gz"} |
Large number division
Date: Wed, 16 Nov 94 10:53:16 PST
From: Keith Averell
Subject: grade 3 math
Dear Dr. Math,
Do you know what 1,223,444,777,888,999 divided by
1,999,888,777,666,555,444,333,222,123 is?
There is one more question: how is it to be a math whiz? I wish I
could be a math whiz like you. How did you learn math so good?
How is it to be a college professor? Is it nice or is it bad? I wish I
could meet you. I have heard about you a little bit.
Clayton Frank Chong
Date: Thu, 17 Nov 1994 00:07:52 -0500 (EST)
From: Dr. Ethan
Subject: Clayton's Question
Hey Clayton,
I am glad you asked. I happen to love division. The problem
you have given will have an answer less than one because the second
number is bigger than the first. The answer turns out to be a very
small decimal. Have you studied decimals?
The answer is .000000000000611756
In regard to your other questions about being a math whiz, I don't know
if I can be much help. I work very hard at math and have become a lot
better, but I am still a long way from being a math whiz. The one thing I
would say is never stop asking good questions and don't just find out the
answer but find out why. I also am not a college professor but I think
that most of them like it.
Ethan - Doctor On Call | {"url":"http://mathforum.org/library/drmath/view/58827.html","timestamp":"2014-04-21T03:19:19Z","content_type":null,"content_length":"6148","record_id":"<urn:uuid:b66550d1-7490-40fb-ab11-833734e8d905>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
188 helpers are online right now
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
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Acute, Obtuse, and Right Angles in Geometry
home | Geometry Sample Video Lessons | Acute, Obtuse, and Right Angles in G . . .
Acute, Obtuse, and Right Angles in Geometry
This clip is just a few minutes of a multi-hour course.
Master this subject with our full length step-by-step lessons!
Lesson Summary:
In this lesson, you'll learn about the differen types of angles that we will frequently encounter in Geometry. We'll learn about Acute, Obtuse, Right, and Straight Angles and how to identify them.
You'll gain practice with these ideas by working example problems and learning to identify the type of angle based on the drawing. This is an important skill because most Geometry problems involve a
figure that the student will need to consult to determine how to solve the problem. Gaining practice with using drawings to solve the problem is very important in Geometry. | {"url":"http://www.mathtutordvd.com/public/Acute_Obtuse_and_Right_Angles_in_Geometry.cfm","timestamp":"2014-04-18T16:56:19Z","content_type":null,"content_length":"39285","record_id":"<urn:uuid:f8c740d2-f9f8-46d0-8c38-a0c6203a39b9>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00356-ip-10-147-4-33.ec2.internal.warc.gz"} |
Term Paper: Contributions of Georg Cantor in Mathematics
Sample Term Paper
Words 2,100
This is a term paper on Georg Cantor’s contribution in the field of mathematics. Cantor was the first to show that there was more than one kind of infinity. In doing so, he was the first to cite the
concept of a 1-to-1 correspondence, even though not calling it such.
Cantor’s 1874 paper, “On a Characteristic Property of All Real Algebraic Numbers”, was the beginning of set theory. It was published in Crelle’s Journal. Previously, all infinite collections had been
thought of being the same size, Cantor was the first to show that there was more than one kind of infinity. In doing so, he was the first to cite the concept of a 1-to-1 correspondence, even though
not calling it such. He then proved that the real numbers were not denumerable, employing a proof more complex than the diagonal argument he first set out in 1891. (O’Connor and Robertson,
What is now known as the Cantor’s theorem was as follows: He first showed that given any set A, the set of all possible subsets of A, called the power set of A, exists. He then established that the
power set of an infinite set A has a size greater than the size of A. consequently there is an infinite ladder of sizes of infinite sets.
Cantor was the first to recognize the value of one-to-one correspondences for set theory. He distinct finite and infinite sets, breaking down the latter into denumerable and nondenumerable sets.
There exists a 1-to-1 correspondence between any denumerable set and the set of all natural numbers; all other infinite sets are nondenumerable. From these come the transfinite cardinal and ordinal
numbers, and their strange arithmetic. His notation for the cardinal numbers was the Hebrew letter aleph with a natural number subscript; for the ordinals he engaged the Greek letter omega. He proved
that the set of all rational numbers is denumerable, but that the set of all real numbers is not and therefore is strictly bigger. The cardinality of the natural numbers is aleph-null; that of the
real is larger, and is at least aleph-one. (Wikipaedia)
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NASA - THE LUNAR LANDER - ASCENDING FROM THE MOON
Calculus Key Topics:
Application of Differentiation - Related Rates
In this exploration activity, students will use the application of differentiation – related rates, to solve problems pertaining to the ascent portion of the Lunar Lander.
Students will
• use the chain rule to find the rates of change of two or more variables that are changing with respect to time; and
• investigate the relationship between the angle of elevation, the rate of change in the angle of elevation and time.
DOWNLOADS > The Lunar Lander - Ascending from the Moon Educator Edition (PDF 497 KB) > The Lunar Lander Student Edition (PDF 321 KB) | {"url":"http://www.nasa.gov/audience/foreducators/mathandscience/exploration/Prob_LunarLander_detail.html","timestamp":"2014-04-24T22:11:00Z","content_type":null,"content_length":"19381","record_id":"<urn:uuid:36c552ff-06cf-4449-bf06-d170a245818d>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
"Mystery bag" problem
September 21st 2009, 04:49 PM #1
Sep 2009
Duluth, California
Hi again MHF,
I'm a bit confused so any help you can provide is greatly appreciated. Thanks a lot. :-)
"Describe how the jester must place the mystery bags and lead weights so that the equation will be a representation of the situation.
Then find the weight of the mystery bag."
a. 5M + 24 = 51 + 2M
b. 43M + 37 = 56M + 24
c. 12M + 15 = 5M + 62
does M equal the mystery weight of the bag?
From what I understand, yes.
From what I understand, they want you to make M the subject of the equation, or, in other words, find the value of M.
To do this, we need to understand the process of undoing/manipulating equations.
The opposite of Addition is Subtraction
The opposite of Multiplication is Division
Hence, to make M the subject in these equations we need to undo some of the operations and shift a few things around.
a. 5M + 24 = 51 + 2M - The aim is to get the M's onto one side, whilst keeping both sides equal (like keeping weights in balance on a scale).
Let's push M onto the left side, as it's mathematical convention for the subject to appear on the left.
a. 5M + 24 = 51 + 2M
1. Get M onto the left side.
3M + 24 = 51 (We substracted (opposite of addition) 2M from both sides (keeping the equation balanced).)
2. Get M by itself.
3M = 27 (We need to undo the addition of 24 (opposite is substraction) before we can deal with the 3 in front of the M).
- M = 27/3 = 9 (We undid the multiplication on M by dividing both sides by 3 (opposite to the multiplication).
If you understand the process, can you complete the rest by yourself?
Yes, I do understand -- and thanks for your assistance once again
September 21st 2009, 05:37 PM #2
Junior Member
Sep 2009
September 21st 2009, 05:56 PM #3
Sep 2009
Duluth, California
September 21st 2009, 06:04 PM #4
Junior Member
Sep 2009
September 22nd 2009, 12:24 PM #5
Sep 2009
Duluth, California | {"url":"http://mathhelpforum.com/algebra/103600-mystery-bag-problem.html","timestamp":"2014-04-16T13:56:44Z","content_type":null,"content_length":"39377","record_id":"<urn:uuid:9f002b50-7324-4a46-b6fd-b56fc57d7a41>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00428-ip-10-147-4-33.ec2.internal.warc.gz"} |
Music Theory Meets Geometry
Posted by midiguru on December 3, 2012
The theory of harmony that serves to organize most of the music of Europe and America (and increasingly of the rest of the world) is built on a couple of basic assumptions — axioms, if you will.
First, there are exactly 12 pitch classes (A, B-flat, B, C, C-sharp, and so on). Second, the pitches relate to one another within a one-dimensional space. That is, they’re laid out in a line, which
conventionally runs from side to side with lower pitches to the left and higher pitches to the right.
The relations among pitch classes, which are what harmony theory is about, all take place within this one-dimensional matrix. Octave equivalence (transposition) and the Circle of Fifths introduce
wrinkles, but the wrinkles can easily be mapped onto the one-dimensional layout.
To be sure, the frequency spectrum, in which pitches are defined by their number of vibrations per second, is one-dimensional. The fact that harmony theory defines relations in one dimension is not
wrong. But it’s a limitation conceptually.
Guitarists play in two dimensions. Because of the layout of the fretboard, they sometimes discover scale and chord relationships that keyboard players fail to notice. They do it by moving geometrical
patterns across from one string to another, as well as up or down the neck of the guitar.
Harmony theory in two dimensions turns out to be quite interesting. When we add the possibility that there may be more than 12 pitch classes within the octave, matters become very interesting indeed.
The ancestor of my new Z-Board is the Z-tar, a MIDI interface designed for guitarists by Harvey Starr. The Z-Board goes a bit further than the Z-tar in that it has 12 horizontal rows of keys, not
just six. But the idea is similar: You can move up in pitch either by moving to the right, or by moving across from one row of keys to the next.
Scales and chord voicings are two-dimensional structures. They can be moved (transposed) vertically, horizontally, or diagonally. Of course, any movement translates into an upward or downward shift
in absolute pitch. In theory, you could accomplish exactly the same thing on a one-dimensional keyboard. But you wouldn’t, because the patterns would be hard to see and too spread out to play.
Any key on the Z-Board can be assigned any MIDI note number, so the nature of the 2D pitch grid is entirely arbitrary. As a starting point, it might be laid out exactly like a guitar fretboard, with
half-steps from left to right and intervals of a perfect fourth (five half-steps) vertically.
Because I’m interested in microtonal scales with more than 12 equal-tempered steps per octave, it occurred to me that a slightly different layout might be more useful. So I created a key map in which
the vertical intervals are five chromatic steps (I call them chroma-steps), as on a guitar, but the left-to-right interval is two chroma-steps per key rather than one. Playing a chromatic scale on
this layout requires a zigzag movement, but how often do you play chromatic scales? On a 5×2 grid, chord voicings and useful scales lie more neatly under the hand.
[Edit: Having experimented with this grid for a few days, I backed up and tried the default 5x1 grid, as on a guitar tuned in fourths. The guitar grid, I've found, is much easier to wrap one's brains
around, at least in 19-tone. The diagram below, then, is deprecated.]
The 19-note equal-tempered tuning has embedded within it a very familiar-sounding diatonic major scale. In this case, however, the whole-steps are three chroma-steps wide, while the half-steps (more
accurately, 2/3 steps) are two chroma-steps wide. When you map this major scale onto the 5×2 grid, it looks like this:
The pattern is not hard to see: Ascending whole-steps are diagonals upward and to the left, while “half-steps” are horizontal moves to the right.
But why limit ourselves to familiar major scales? The strength of two-dimensional harmony theory is that any pattern is potentially interesting. Eighty years of jazz have taught us that almost any
combination of pitches within the 12-note tuning can be exciting. Jazz has largely erased the notions of consonance and dissonance, which guided earlier classical theory.
An added attraction of a 19-note equal-tempered scale is that 19 is a prime number. In the 12-note tuning we all know and love, moving up or down by any number of chroma-steps except 5 or 7 repeats
at the octave, and without using all of the notes. If we move up by 3′s, we get a diminished 7th chord, because 12 is divisible by 3 — and also by 2, 4, and 6. But in 19-tone, no matter what interval
our pattern uses for repetition, the pattern has to cycle through all 19 pitch classes before it returns to its starting point.
As a result, scale patterns on the 2D grid generally spiral off into space harmonically rather than repeating at the octave. Evocative? I think so.
There is, as yet, no coherent harmony theory that would describe the 19-tone tuning at all, much less describe it in terms of patterns on a two-dimensional matrix. But that’s part of the fun, isn’t
it? How many entirely new chords did you discover this week? I discovered several dozen. | {"url":"http://midiguru.wordpress.com/2012/12/03/music-theory-meets-geometry/","timestamp":"2014-04-18T23:38:37Z","content_type":null,"content_length":"56651","record_id":"<urn:uuid:1ad27ebb-c5b8-4f8a-a6f9-498daaf5517b>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
reciprocal cm?
November 2nd 2009, 01:17 AM #1
Feb 2009
reciprocal cm?
hi all,
im confused with the way an answer is given in reciprocal cm format the questions was:
The temperature $\theta C$ measured at a distance $x cm$ is given by:
for distances $x\geq 1$
Find the temeparture gradient (i.e. the rate at which temperature decreases with distance) 10cm from the flame.
my answer was
$t=f(d)=16+450x^{-\frac{3}{2}}$, where t(emp), d(istance)
$=-\frac{6.75}{\sqrt{10}} \equiv -0.675\sqrt{10}$
$=-2.13C/cm$ to 3 sf
but the book (Edexcel A level maths} gives the answer as
can someone please explain what the reciprocal cm is all about.
hi all,
im confused with the way an answer is given in reciprocal cm format the questions was:
The temperature $\theta C$ measured at a distance $x cm$ is given by:
for distances $x\geq 1$
Find the temeparture gradient (i.e. the rate at which temperature decreases with distance) 10cm from the flame.
my answer was
$t=f(d)=16+450x^{-\frac{3}{2}}$, where t(emp), d(istance)
$=-\frac{6.75}{\sqrt{10}} \equiv -0.675\sqrt{10}$
$=-2.13C/cm$ to 3 sf
but the book (Edexcel A level maths} gives the answer as
can someone please explain what the reciprocal cm is all about.
Your book has got it wrong. The unit is either $^0 \text{C/cm}$ or $^0 \text{C cm}^{-1}$but not a mixture of both.
thanks for yor answer MF. it was me who got it wrong sorry. the book does say $^0\text{C cm}^{-1}$ now it makes sense. what an idiot!
November 2nd 2009, 01:39 AM #2
November 2nd 2009, 01:54 AM #3
Feb 2009 | {"url":"http://mathhelpforum.com/calculus/111905-reciprocal-cm.html","timestamp":"2014-04-17T18:25:40Z","content_type":null,"content_length":"39814","record_id":"<urn:uuid:8d06c289-b6a6-4da2-9676-73158bf89185>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00276-ip-10-147-4-33.ec2.internal.warc.gz"} |
PLaneT Package Repository : Home > williams > science.plt > package version 3.5
#lang scheme
;;; PLT Scheme Science Collection
;;; gamma-sf-plot-example.ss
;;; Copyright (c) 2004-2008 M. Douglas Williams
;;; This library is free software; you can redistribute it and/or
;;; modify it under the terms of the GNU Lesser General Public
;;; License as published by the Free Software Foundation; either
;;; version 2.1 of the License, or (at your option) any later version.
;;; This library is distributed in the hope that it will be useful,
;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
;;; Lesser General Public License for more details.
;;; You should have received a copy of the GNU Lesser General Public
;;; License along with this library; if not, write to the Free
;;; Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
;;; 02111-1307 USA.
;;; -------------------------------------------------------------------
(require (planet "gamma.ss" ("williams" "science.plt")
(require (lib "plot.ss" "plot"))
(plot (line gamma)
(x-min 0.001) (x-max 6) (x-label "X")
(y-min 0) (y-max 120) (y-label "Y")
(title "Gamma Function, gamma(x)")) | {"url":"http://planet.racket-lang.org/package-source/williams/science.plt/3/5/examples/gamma-sf-plot-example1.ss","timestamp":"2014-04-19T22:54:11Z","content_type":null,"content_length":"6504","record_id":"<urn:uuid:c141dd2e-5066-4045-af59-b3d33121b25d>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00283-ip-10-147-4-33.ec2.internal.warc.gz"} |
Transforms from one geometry to another geometry using a strategy.
template<typename Geometry1, typename Geometry2>
bool transform(Geometry1 const & geometry1, Geometry2 & geometry2)
Type Concept Name Description
Geometry1 const & Any type fulfilling a Geometry Concept geometry1 A model of the specified concept
Geometry2 & Any type fulfilling a Geometry Concept geometry2 A model of the specified concept
True if the transformation could be done
#include <boost/geometry/geometry.hpp>
#include <boost/geometry/algorithms/transform.hpp>
The function transform is not defined by OGC.
Case Behavior
Spherical (degree) / Spherical (radian) Transforms coordinates from degree to radian, or vice versa
Spherical / Cartesian (3D) Transforms coordinates from spherical coordinates to X,Y,Z, or vice versa, on a unit sphere
Spherical (degree, with radius) / Spherical (radian, with Transforms coordinates from degree to radian, or vice versa. Third coordinate (radius) is untouched
Spherical (with radius) / Cartesian (3D) Transforms coordinates from spherical coordinates to X,Y,Z, or vice versa, on a unit sphere. Third coordinate (radius) is taken into
Shows how points can be transformed using the default strategy
#include <iostream>
#include <boost/geometry.hpp>
int main()
namespace bg = boost::geometry;
// Select a point near the pole (theta=5.0, phi=15.0)
bg::model::point<long double, 2, bg::cs::spherical<bg::degree> > p1(15.0, 5.0);
// Transform from degree to radian. Default strategy is automatically selected,
// it will convert from degree to radian
bg::model::point<long double, 2, bg::cs::spherical<bg::radian> > p2;
bg::transform(p1, p2);
// Transform from degree (lon-lat) to 3D (x,y,z). Default strategy is automatically selected,
// it will consider points on a unit sphere
bg::model::point<long double, 3, bg::cs::cartesian> p3;
bg::transform(p1, p3);
<< "p1: " << bg::dsv(p1) << std::endl
<< "p2: " << bg::dsv(p2) << std::endl
<< "p3: " << bg::dsv(p3) << std::endl;
return 0;
p1: (15, 5)
p2: (0.261799, 0.0872665)
p3: (0.084186, 0.0225576, 0.996195) | {"url":"http://www.boost.org/doc/libs/1_50_0/libs/geometry/doc/html/geometry/reference/algorithms/transform/transform_2.html","timestamp":"2014-04-21T00:02:06Z","content_type":null,"content_length":"18653","record_id":"<urn:uuid:ff9ecacf-03cc-4baa-9cc2-0b3ba3c07f55>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00199-ip-10-147-4-33.ec2.internal.warc.gz"} |
Your company/institution subscribes to Transactions (A,B,C,D,).
If you subscribe to the Transactions personally as an IEICE member, you can access to the online article by login with your ID and PW.
A Simplification Algorithm for Calculation of the Mutual Information by Quantum Combined Measurement
Shogo USAMI Tsuyoshi Sasaki USUDA Ichi TAKUMI Masayasu HATA
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences Vol.E82-A No.10 pp.2185-2190
Publication Date: 1999/10/25
Online ISSN:
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Quantum Information
quantum information theory , channel capacity , square-root measurement , mutual information , superadditivity ,
Full Text: PDF(658.3KB)
Recently, the quantum information theory attracts much attention. In quantum information theory, the existence of superadditivity in capacity of a quantum channel was foreseen conventionally. So far,
some examples of codes which show the superadditivity in capacity have been clarified. However in present stage, characteristics of superadditivity are not still clear up enough. The reason is as
follows. All examples were shown by calculating the mutual information by quantum combined measurement, so that one had to solve the eigenvalue and the eigenvector problems. In this paper, we
construct a simplification algorithm to calculate the mutual information by using square-root measurement as decoding process of quantum combined measurement. The eigenvalue and the eigenvector
problems are avoided in the algorithm by using group covariancy of binary linear codes. Moreover, we derive the analytical solution of the mutual information for parity check codes with any length as
an example of applying the simplification algorithm. | {"url":"http://search.ieice.or.jp/bin/summary.php?id=e82-a_10_2185&category=A&lang=E&year=1999&abst=","timestamp":"2014-04-21T02:05:59Z","content_type":null,"content_length":"21186","record_id":"<urn:uuid:06e68519-93cf-42e9-936a-eb0d41a51a3d>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00178-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prim's Algorithms
November 13th 2007, 08:02 AM #1
Prim's Algorithms
I'm slightly confused as i hav 2 differing answers when using two different algorithms (Kruskal and Prim)
For the first part i have used Prim's algorithm [tabular method] for a distance matrix (see attachment) to find the minimum spanning tree for the graph.
My answer gives me: AB,AC,DE with AB=6, AC=7 and DE=8. this gives me a total weight of 21.
However this is where i started to get confused as i have used Kruskals algorithm in a previous question and also Prims (the non tabular method) and got different answers with AB,AC,CD,DE with a
total weight of 30.
is this possible, the reason i ask is that in my initial answer AB,AC,DE DE is not linked to the others which makes me think that i have two seperate trees and not one spanning tree. do i need
the extra edge or is it enough to have all the vertexes? Thanks in advance MHF!
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/advanced-applied-math/22643-prim-s-algorithms.html","timestamp":"2014-04-18T09:45:33Z","content_type":null,"content_length":"29996","record_id":"<urn:uuid:8fa2e077-e253-44bb-942a-4be02b4e9a8b>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00277-ip-10-147-4-33.ec2.internal.warc.gz"} |
Control Of Electronic Expansion Valves
September 3, 2004
My article in the Aug. 2 issue of
The News
reviewed conventional thermostatic expansion valves (TEVs or TXVs) and covered the basic operation of electronic expansion valves (EEVs). This column will focus on control schemes of EEVs.
EEVs have small stepper motors that open and close their valve port. They do this in response to signals sent to them by an electronic controller. Sensors like thermistors and pressure transducers
are wired to an electronic controller and act as feedback devices to the controller telling it what is happening out in the actual refrigeration and air system in a refrigerated case. (See Figure 1.)
Stepper motors are driven by a gear train and can run at 200 steps per second and can return to their exact position very quickly.
This gives the EEV very accurate control of refrigerant flow. Evaporator superheat and the refrigerated case's discharge air temperature can be controlled very precisely.
EEVs consist of many components often referred to as hardware. This hardware consists of the stepper motor itself, its wiring, and the controller. A controller with a built-in microprocessor controls
the EEV.
There are also many transistors involved in the control of most EEVs. Transistors are nothing but solid-state switches. They receive a small electrical signal from the microprocessor to their base
lead. This allows current flow from the emitter to the collector.
The microprocessor actually sequences signals to the base lead of the transistor. This sequencing turns the transistors on and off in pairs, which steps the EEV either open or shut.
Controllers and their software or algorithms have many different variations or schemes in which they control. Three types of control schemes are proportional, integral, and derivative.
In proportional control, actual superheat temperatures will try to approach the superheat set point but may not reach it. The difference between the actual superheat temperature and the set point is
called the offset. Some means of predicting offset must be used because it changes with time and heat loads.
With integral control, the control program or algorithm calculates the deviation or the amount of offset that is changing and is added to the set point. It does this by calculating the area under the
curve of time versus temperature. This type of control scheme is often referred to as reset.
A derivative control scheme looks at the rate of change in temperature versus time graph or its slope. If the rate of change is great, the software of algorithm steps the valve faster in order to
satisfy the set point.
In order for the hardware to operate, a set of instructions or software must be given to the microprocessor. As mentioned earlier, this set of instructions is often referred to as an algorithm. Below
is an example of an algorithm for an EEV trying to control evaporator superheat.
If superheat is 20 degrees F, then open the EEV 200 steps.
If superheat is 10 degrees, then open the EEV 150 steps.
If superheat is 5 degrees, then open the EEV 0 steps.
If superheat is 2 degrees, then close the EEV 25 steps.
If superheat is 0 degrees, then close the EEV 1,000 steps.
Notice that the last line of the algorithm closed the valve 1,000 steps. This is a process referred to as overdriving. Most EEVs have been made to handle overdriving without damage to the valve.
Overdriving makes sure the EEV is closed so there will not be any damage to the compressor from flooding or slugging refrigerant. This algorithm, however, will only allow the EEV to control 5 degrees
of superheat. Any higher superheat will open the valve and any lower superheat will close the valve.
A more advanced algorithm uses "let" statements and "input" statements. There are also variables like X and Y for numbers to be assigned. "If-then" statements are also used, and each line of the
algorithm is numbered.
The line numbering allows for loops to be used in the programming. Below is an algorithm that will eventually allow the EEV to reach its superheat set point. It is referred to as a proportional
algorithm because it will change the EEVs output (steps) directly in relation to the input (evaporator superheat).
20 Let ‘X' be evaporator superheat set point
30 Input ‘X'
40 Let ‘Y' be actual evaporator superheat
50 If X=Y, then close the valve 0 steps
60 If X>Y, then close the EEV 1 step
70 If X 80 Go to line 50
Because proportional control will experience overshoot and undershoot of the set point, adding the integral feature can enhance proportional algorithms as is the case with the one above. Remember,
integral control senses the actual deviation of actual evaporator superheat from the set point. An offset can then be applied to the valve. Below is an example of an integral algorithm enhancement.
If average evaporator superheat for 40 seconds is 3 degrees high, then open the EEV 15 steps.
Another enhancement that can be added to the proportional and integral control algorithm is a derivative feature. The rate of change of evaporator superheat is sensed by estimating the slope of the
curve for the change in evaporator superheat. A greater slope means faster changes to the EEV steps. Below is an example of a derivative algorithm enhancement.
100 If the evaporator superheat had decreased 0.5 degrees in 10 seconds, then close the EEV 15 steps
101 If the evaporator superheat had decreased 5 degrees in 3 seconds, then close the EEV 150 steps.
Notice that the derivative enhancement to the software or algorithm dealt with rates of changes of evaporator superheat.
John Tomczyk is a professor of HVACR at Ferris State University, Big Rapids, Mich., and the author of Troubleshooting and Servicing Modern Air Conditioning & Refrigeration Systems, published by ESCO
Press. To order, call 800-726-9696. Tomczyk can be reached by e-mail at tomczykj@tucker-usa.com.
Publication date: 09/06/2004 | {"url":"http://www.achrnews.com/articles/print/control-of-electronic-expansion-valves","timestamp":"2014-04-20T03:21:09Z","content_type":null,"content_length":"8929","record_id":"<urn:uuid:305d8ff1-ead9-4269-bfb4-f8e90d049d54>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00139-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
November 22nd 2010, 08:49 AM #1
Junior Member
Mar 2010
coin tossing
find the sample space for
(a)tossing a coin repeatedly until the first head appears . The quantity is the number of tosses before the first head?!?!
(b) two dies are tossed, the quantity of interest is the sum of dots
im completely lost with a but is just= {2,3,4,5,6,7,8,9,10,11,12}???
any help appreciated!!
The part in blue above is correct.
For part (a) the outcome space is the set of non-negative integers: $\{0,1,2,3,\cdots\}$
November 22nd 2010, 08:55 AM #2 | {"url":"http://mathhelpforum.com/advanced-statistics/164075-coin-tossing.html","timestamp":"2014-04-16T17:41:40Z","content_type":null,"content_length":"33361","record_id":"<urn:uuid:a1ee340a-c7c6-4eaf-a8ba-8976403b4128>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
46-XX Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx}
46Gxx Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) [See also 28-XX, 46Txx]
46G05 Derivatives [See also 46T20, 58C20, 58C25]
46G10 Vector-valued measures and integration [See also 28Bxx, 46B22]
46G12 Measures and integration on abstract linear spaces [See also 28C20, 46T12]
46G15 Functional analytic lifting theory [See also 28A51]
46G20 Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]
46G25 (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60]
46G99 None of the above, but in this section | {"url":"http://www.ams.org/msc/msc.html?t=46G12&btn=Current","timestamp":"2014-04-16T11:03:10Z","content_type":null,"content_length":"13341","record_id":"<urn:uuid:549cf1ff-15d3-4d12-a7a9-d5e531493fb5>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00008-ip-10-147-4-33.ec2.internal.warc.gz"} |
Forward kinematics data modeling
July 26th 2013, 01:18 PM
Forward kinematics data modeling
I have built a simple robotic arm using 3 RC Servos and an Arduino.
I just want to play around with it and learn something about robotics.
Currently, I am trying to compute the position of the tip of the robotic arm using the three angular positions of the servos.
"Forward kinematics" I think is the technical term for this.
Btw the tip of the arm is a pen, I thought I might try to draw soemthing with it later on.
In the movement range of the arm I set up a cartesian coordinate system and recorded 24 (angle => position) samples.
table header: servo1, servo2, servo3, x, y, z 111,61,33,0,0,0 96,50,71,0,0 - Pastebin.com
Now, I am trying to model this data, but I am a bit out of my depth here.
Here is my approach so far:
I use the Denavit–Hartenberg equations found on Wikipedia Denavit?Hartenberg parameters - Wikipedia, the free encyclopedia.
I then try to determine the parameters using least squares optimization.
I also added linear terms to the input and output of the model to compensate for possible distortions (e.g. phase-shift in the servo angle):
My Python code: [Python] Forward kinematics optimization - Pastebin.com
But it just won't work, none of my attempts have converged on a good solution.
I also tried a simple 3x4 matrix multiplication, which tbh does not make much sense as a model, but oddly it didnt do worse than the more sophisticated model above.
I hope there is some smart guy out there who can help. Thanks in advance. | {"url":"http://mathhelpforum.com/advanced-applied-math/220846-forward-kinematics-data-modeling-print.html","timestamp":"2014-04-19T23:30:14Z","content_type":null,"content_length":"5157","record_id":"<urn:uuid:a6f5eb40-bc2e-465c-b8a3-b28aafe0c46c>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00250-ip-10-147-4-33.ec2.internal.warc.gz"} |
Devault Science Tutor
Find a Devault Science Tutor
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...It is such a shame that English is not rigorously taught at most schools today. I can remember strict teachers drilling proper English usage into my head: diagramming sentences, looking up
words in the dictionary, re-writing papers that my teachers knew I didn't put much effort into. It's no wonder that my English skills exceed those of most of today's English teachers.
23 Subjects: including ACT Science, English, calculus, geometry | {"url":"http://www.purplemath.com/Devault_Science_tutors.php","timestamp":"2014-04-19T20:06:51Z","content_type":null,"content_length":"23777","record_id":"<urn:uuid:bd04dd5d-a6ee-4151-bfb2-6212f6c00bc8>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00584-ip-10-147-4-33.ec2.internal.warc.gz"} |
Philip Voc words
If your printer has a special "duplex" option, you can use it to automatically print double-sided. However for most printers, you need to:
1. Print the "odd-numbered" pages first
2. Feed the printed pages back into the printer
3. Print the "even-numbered" pages
If your printer prints pages face up, you may need to tell your printer to reverse the order when printing the even-numbered pages. | {"url":"http://quizlet.com/4959781/print","timestamp":"2014-04-20T06:25:41Z","content_type":null,"content_length":"219034","record_id":"<urn:uuid:542400ae-c330-4be6-9d0b-27f9974fc24f>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00302-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why does everyone say algebra 2 is such a hard class?
I was terrible in geometry last year. Most people say geometry is easier than algebra two, but i was hoping that algebra 2 would be easier for me. Everyone, even the really smart kids say algebra 2
is really hard and i was looking for it to be easier than geometry. I dont want to get scared. Why exactly is it that everyone says its so hard? What are some good ideas/habits/ways i can study so i
can do better in this math class?
Suggestion by Michael
In my opinion it isn’t hard because I like Algebra I. If your good in Algebra I then you’ll love that class. Also if you like the teacher that’s even better.
Suggestion by KANDI
i took geometry last year and am taking algebra 2 this year… many ppl say its hard because theyre not good at math or they try to psych you out and when you get there its just the fear of difficulty
thats got you…
Give your answer to this question below!
These tests would be more beneficial to my algebra 2 honors class. What is capitalized?
I know I have to capitalize Algebra 2, but what about honors?
Suggestion by James
Grammatically, you are correct. “Algebra 2″ must be capitalized. However, “honors” is not capitalized, since it is an adjective.
“Honors” would only be capitalized if it is part of the official class name. For example “Honors Algebra 2.”
Hope this helps!
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Factoring by Grouping to Solve a Polynomial Equation
Date: 05/26/2005 at 21:06:47
From: Kate
Subject: re: Factoring and Solving Polynomial Equations
How do you factor polynomials when there is an x cubed in the equation
and it doesn't fit into a perfect square form?
For example: x^3 + 3x^2 - x - 3 = 0
I think that you would first have to add 3 to both sides:
x^3 + 3x^2 - x = 3
Then you would be able to take an x out of the equation:
x(x^2 + 3x - 1) = 3
The problem is that there are no factors of -1 which will work
together to give you a positive 3 in the middle. Would you then have
to use the quadratic formula to solve it? That seems way too messy.
Date: 05/26/2005 at 23:11:14
From: Doctor Peterson
Subject: re: Factoring and Solving Polynomial Equations
Hi, Kate.
To solve an equation by factoring, the other side has to be zero, so
you don't want to change the right side by adding 3. In factoring,
you are paying attention only to the one expression you are factoring,
even if it is part of an equation.
In general, a quadrinomial (four-term polynomial) is very hard to
factor. So if you are given one, you can expect it to be one you can
do using one of the few available techniques. They are:
1. Factor out a common factor (as you tried to do with the x)
2. Factor by grouping (see below)
3. Factor three terms as a trinomial, and then combine that with the
other term as a difference of squares
The last possibility is a rarity. Your problem can be done by
Then idea is to group the terms into two pairs, and factor out the
greatest common factor in EACH PAIR. If it works right, the resulting
two factored expressions will then have a common factor that you can
pull out. Here's an example:
2x^3 - 3x^2 - 4x + 6
We group the first two terms and the last two terms, and factor out
the GCF from each pair:
(2x^3 - 3x^2) - (4x - 6)
x^2(2x - 3) - 2(2x - 3)
Now we see that the two "terms" both have a factor of 2x-3, so we
factor that out:
(2x - 3)(x^2 - 2)
If we were lucky, we could now factor that second term (if it were a
difference of squares); since we can't, we're done.
Can you do that with your problem?
If you have any further questions, feel free to write back.
- Doctor Peterson, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/67772.html","timestamp":"2014-04-18T03:06:16Z","content_type":null,"content_length":"7412","record_id":"<urn:uuid:93360aed-0ee9-405d-9732-1e414dcaa132>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00446-ip-10-147-4-33.ec2.internal.warc.gz"} |
Toulouse 2004 Meeting
We show some finite time stabilization results concerning nonlinear oscillators subject to dry friction. Applications to physical sciences and decision sciences are given.
The main object of the talk is the geometric analysis of attainable sets in time T > 0 for nonlinear control systems y¢(t) = f (y(t),u(t) ), obtaining sufficient conditions for these sets to
satisfy a uniform interior sphere condition. This result is then used to recover the semiconcavity of the value function of time optimal control problems with a general target.
We study the asymptotic convergence of the flow associated with a dynamical system obtained by coupling the continuous steepest descent method with a penalty scheme for a convex program. The
penalty parameter involved in this dynamical system may be seen as a control variable. We establish the convergence of the primal trajectories towards an optimal solution whose characterization
depends on the behaviour of the penalty parameter, and we also discuss the convergence of the associates dual paths.
This is joint work with M. Courdurier (Univ. Washington, Seattle).
We present here a type of constrained dynamics, given by an ODE with discontinuous and nonlinear righthand side. Differential inclusions related to this type of equation, which we call projected
differential equation (PrDE), appeared in the mathematical literature as early as 1973, then continued to be studied in the '80s. However, the definition in the form we use today was introduced
in the early '90s, on Euclidean space.
Although with discontinuities, a PrDE has solutions in the class of absolutely continuous functions. We then associate a dynamics, given by these solutions, and obtain the now widely used
projected dynamical system (PDS). A projected dynamics is essentially a positional control problem, since the solutions of any PrDE are constrained to evolve within and on the boundary of a
closed, convex subset of the underlying space.
A crucial trait of a PrDE is that its critical points (the zeros of the righthand side) coincide with the solutions to a variational inequality problem (VI), thus making the associated projected
dynamics extremely useful in applications, for example: spatial price equilibria, financial equilibria and transportation.
In this talk we present a brief history of this topic, highlighting the researchers who contributed to its advance, as well as the contribution of the author, together with collaborators, to the
very recent developments in this area.
Nous étudions la régularité des solutions d'équations intégro-différentielles. Ces équations peuvent être interprétées comme des équations de Bellman-Isaacs de certains problèmes de contrôle
Les équations que nous étudions sont des équations d'Hamilton-Jacobi d'ordre 1 perturbées par un opérateur non local, le laplacien fractionnaire. Il existe déjà une littérature importante à ce
sujet et la théorie des solutions de viscosité permet de donner un sens faible, non seulement aux dérivées mais aussi à l'opérateur.
Nous prouvons que sous des hypothèses naturelles sur l'Hamiltonien (celles qui assure l'unicité de la solution de l'équation non perturbée), la solution de viscosité est deux fois différentiable
en espace et une fois en temps. Pour ce faire, nous construisons tout d'abord une solution de viscosité bornée et Lipschitzienne, puis nous utilisons une représentation intégrale de l'équation et
des méthodes de point fixe pour prouver que la solution est en fait C^2.
Nous estimons enfin la différence entre la solution de l'équation non perturbée et la solution de l'équation perturbée par un opérateur non local évanescent.
We are concerned with second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. We prove a comparison principle between semicontinuous viscosity sub- and super-solutions
growing at most quadratically.
As an application, we consider a finite horizon stochastic control problem with unbounded controls and we prove that the value function is the unique viscosity solution of the corresponding
dynamic programming equation.
This is a joint work with Francesca Da Lio (Universita di Torino, Italy).
We are interested in the local stabilization of parabolic equations or systems around an unstable stationary solution. The feedback control is an internal or a boundary control. We are mainly
concerned with the case where the nonlinear term of the state equation is unbounded in the space of initial data. We prove different local stabilization results by linear feedback laws (obtained
by solving some algebraic Riccati equations), or by nonlinear ones (by considering Hamilton-Jacobi-Bellman equations).
Applications to the Navier-Stokes equations and to other nonlinear parabolic equations will be given.
We consider a class of nonlinear optimal control problems where the cost functional depends on the arrival time of the trajectory on a given target set. A well known example is the minimum time
We study the conditions under which the value function of the problem is semiconcave. In contrast to previous works (e.g.Cannarsa-Sinestrari [1995]) which required an interior sphere condition on
the target, in the result we present here the target can be completely general. On the other hand, the dynamics of the system is assumed to satisfy suitable regularity properties. In particular,
the set of admissible velocities for the system must have a smooth boundary at every point in a neighborhood of the target.
In the '80s Crandall and Lions introduced the concept of viscosity solution in order to get existence and/or uniqueness results for Hamilton-Jacobi equations. We first investigate the Dirichlet
and Cauchy-Dirichlet problems for such equations, where the Hamiltonian is associated to a problem of calculus of variations, and prove that if the data are analytic then the viscosity solution
is moreover subanalytic. We then extend this result to Hamilton-Jacobi equations stemming from optimal control problems, in particular from sub-Riemannian geometry, which are generalized eikonal
We consider the measure-driven differential inclusion
ï dx Î F æ x(t) ö dt + G æ x(t) ö dm(t)
í è ø è ø \leqno(DI)
where F : R^n \rightrightarrows R^n and G :R^n \rightrightarrows M[n×m] are given multifunctions with nonempty, compact, and convex values, and m is a Borel vector-valued measure with values in a
closed convex cone K Í R^m. We will discuss the nature of a "solution" to (DI) and how it can be obtained via a graphical limit of Euler-polygonal arcs. We shall also introduce and provide local
characterizations of weak and strong invariance. | {"url":"http://cms.math.ca/Events/Toulouse2004/abs/ss8.html","timestamp":"2014-04-21T09:54:55Z","content_type":null,"content_length":"20899","record_id":"<urn:uuid:d83873ca-ccff-44ce-a8a4-7a795a9135d5>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00492-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why, and How, Should Geologists Use Compositional Data Analysis/Dealing With Zero Values
There is almost not a case in exploration geology, where the studied data does not includes below detection limits and/or zero values, and since most of the geological data responds to lognormal
distributions, these “zero data” represent a mathematical challenge for the processing.
The method that I am proposing takes into consideration the well-known relationships between some elements. For example, in copper porphide deposits there is always a significant direct correlation
between the copper values and the molybdenum ones. However, while copper will always be above the limit of detection, many of the molybdenum values will be “rounded zeros”. In such a case, I will
take the lower quartile of the real molybdenum values and establish a regression equation with copper, and then I will estimate the “rounded” zero values of molybdenum by their corresponding copper
One can apply this method to any type of data, provided we establish first their correlation dependency.
One of the main advantages of this method is that we do not obtain a fixed value for the “rounded zeros”, but one that depends on the value of the other variable.
Are there any zeros in the house?Edit
We need to start by recognizing that there are zero values in geology. For example the amount of quartz in a syenite is zero, since quartz cannot co-exists with nepheline (Trusova and Chernov, 1982).
In binomial distributions, like for example the drilling of an ore body, you either will intersect the ore body (1) or not (0). Another common zero is a North azimuth, however we can always change
that zero for the value of 360°. These are the “Essential Zeros” (Aitchison, 2003) or “Real zeros”. They are not a problem for as long as their population does not respond to a lognormal
distribution, since you cannot take the logarithm of a zero.
Then in geology, especially in geochemistry, we also have “Rounded Zeros”. In some cases, laboratories report below detection limit (b.d.l.) as zeros or non-existent, while in most cases they just
put the b.d.l. as the value for that parameter. These b.d.l. values are a similar problem to the “Rounded Zeros”. Let us illustrate with the example proposed in Table 28.
Table 28. CLR transformed data for the used example. The original b.d.l. value was 0.5 ppm of Mo.
Au Cu Mo Au Cu Mo
0.23342 0.72138 0.0452 0.22814 0.19547 0.57639
0.32663 0.61146 0.06191 0.47232 0.48404 0.04363
0.12652 0.16198 0.71149 0.21663 0.27648 0.50689
0.20133 0.28139 0.51728 0.207 0.24005 0.55295
0.20796 0.3302 0.46184 0.30235 0.24198 0.45567
0.41506 0.51552 0.06942 0.20662 0.12838 0.665
0.20034 0.21824 0.58142 0.25618 0.3309 0.41292
Au Cu Mo Au Cu Mo
0.13951 0.18003 0.68046 0.26629 0.29193 0.44178
0.14029 0.12599 0.73372 0.50983 0.45371 0.03645
0.12876 0.17744 0.6938 0.26377 0.25831 0.47792
0.13442 0.28513 0.58045 0.50258 0.46207 0.03536
0.22589 0.15577 0.61834 0.61358 0.3443 0.04212
0.18861 0.13306 0.67834 0.34444 0.32052 0.33504
0.26028 0.28088 0.45884 0.38402 0.1694 0.44658
0.19831 0.31928 0.48241 0.24623 0.25965 0.49412
0.37797 0.58109 0.04094 0.26424 0.17672 0.55904
0.47982 0.46952 0.05065 0.42291 0.14872 0.42837
0.17888 0.314 0.50712 0.2371 0.18903 0.57387
0.26791 0.17539 0.5567 0.27013 0.17273 0.55714
0.64782 0.3135 0.03868 0.51564 0.45322 0.03113
The ternary diagram on Fig. 51 shows these results.
Figure 51. Ternary diagram of the studied data. To the left the CLR transformed data, to the right the same data after being centered using CoDaPack software,
It is clear that even centering does not solve the “problem” of the b.l.d. data, which remain grouped along the AB axis.
Zero, zero… What shall I do with you?Edit
Geologists, even those that are not knowledgeable of compositional data analysis, have being dealing with this problem for quite some time (Kashdan et al., 1979). One of the most frequently use
technique is amalgamation (Aitchison, 1986). Amalgamation, e.g. adding Na[2]O and K[2]O, as total alkalis is a solution, but sometimes we need to differentiate between a sodic and a potassic
alteration, and therefore amalgamation is not an option.
Pre-classification into groups is another solution, but it requires a good knowledge of the distribution of the data and the geochemical characteristics of the groups that is not always available.
Considering the zero values equal to the limit of detection of the used equipment, or substituting it by some other constant (e.g. half the limit of detection) will generate spurious distributions,
especially in ternary diagrams as we show in Fig. 51.
Same situation will occur if we replace the zero values by a very small amount (Bacon-Shone, 2003) using non-parametric or parametric techniques (imputation). Even if we add the same small value to
all of the analyzed parameters, we will get the same spurious distribution.
How do I deal with spurious distributions?Edit
The method that I am proposing takes into consideration the existence of well-known relationships between some elements. For example, in copper porphide deposits, there is always a clear dependency
between the copper values and the molybdenum ones (Fig. 52), but while copper will always be above the limit of detection, many of the molybdenum values will include b.d.l. values (“Rounded Zeros”).
Figure 52. As in all Cu porphide deposits, there is a strong correlation between Cu and Mo values. We can use such correlation to estimate the values b.d.l. for Mo.
In this case, I will take the lower quartile of the real molybdenum values (Table 29) and establish a regression equation with copper, and then we will estimate the “Rounded Zero” values of
molybdenum by those estimated from their corresponding copper values (Table 30).
Table 29. Values of the lower quartile of real data for Mo from this study.
Au Cu Mo
0.41506 0.51552 0.06942
0.34444 0.32052 0.33504
0.25618 0.3309 0.41292
0.42291 0.14872 0.42837
0.26629 0.29193 0.44178
0.38402 0.1694 0.44658
0.30235 0.24198 0.45567
0.26028 0.28088 0.45884
0.20796 0.3302 0.46184
0.26377 0.25831 0.47792
Table 30. Results of the regression analysis for the lower quartile of real molybdenum data from the studied case.
Regression Statistics
Multiple R 0.792759158
R Square 0.628467082
Standard Error 0.079131742
Observations 10
df SS MS F Significance F
Regression 1 0.084737701 0.084737701 13.53241 0.006231332
Residual 8 0.050094661 0.006261833
Total 9 0.134832361
Coefficients Std. Error t Stat P-value
Intercept 0.674594109 0.079027652 8.536178017 2.73E-05
X Variable 1 -0.954714886 0.259529113 -3.678642738 0.006231
Therefore, according to Table 30, we could use regression equation (34) in order to estimate the b.d.l. values for Mo.
Equation 34. Regression equation for Mo.
Fig. 53 shows that the obtained results are close to the predicted line.
So, did we get ride of the spurious effect?
As Fig. 54 clearly shows, only one value of Mo was really close to zero, while the rest has now a value that is a geological reflection of the geochemical characteristics of the data.
Figure 54. The ternary diagram of the left clearly shows that only one sample has low value of Mo. The ternary diagram to the right compares the original data in red with the new estimated values of
Mo in green.
Last modified on 16 February 2011, at 22:04 | {"url":"http://en.m.wikibooks.org/wiki/Why,_and_How,_Should_Geologists_Use_Compositional_Data_Analysis/Dealing_With_Zero_Values","timestamp":"2014-04-19T19:51:20Z","content_type":null,"content_length":"29717","record_id":"<urn:uuid:01d36542-432a-4a3c-bf85-5795ac54b2dd>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00163-ip-10-147-4-33.ec2.internal.warc.gz"} |
Tech Tuesday: Data Structures (Arrays)
If you have been a faithful reader of Tech Tuesday, you may be asking yourself just how many more posts on Data Structures could possibly be coming up as this is now the sixth such post (including
the introduction). The answer is that after today there will be at least one more before moving on to a different subject for now. But that’s not for a lack of material — there is always more to
say about data structures. And if nothing else one important takeaway from these posts should be that understanding the keywords of a programming language is only a tiny part of what it takes to be
effective at programming.
The posts so far have dealt with data structures that let us combine different attributes into a single thing (structs, maps, JSON, objects). But what if we have many of the same thing? Keeping
with our canonical example the point. What data structure can we use to keep track of many points? In the examples so far we used two separate variables to hold our point A and point B. What if we
had a 1,000 points that together make up an image of size 20 x 50? Using 1,000 different variables would be incredibly cumbersome.
Instead, most programming languages (but not all) have built in support for arrays. We already encountered a version of the array in the post on maps, which are also known as associative arrays.
But what we are interested in now are arrays that use integers as indices and can be multidimensional as in the following example, which makes a 20 x 50 image of random colors with each color
represented by a value between 0 and 255:
int points[20][50]; /* this is our two dimensional array */
int x, y;
for (x = 0; x < 19; x++) {
for (y = 0; y < 50; y++) {
points[x][y] = rand() % 256; /* modulo to get values 0 to 255 */
You can see this code running here. The notation point[x][y] lets us access the array element at the position x and y. As a longstanding programming convention, the indices always start at 0. Which
means that for an image of size 20 x 50, we go from 0 to 19 for x and from 0 to 49 for y as you see in the code. This convention has been the source of many a programming bug. If we want the array
element that represents the point at x = 2 and y = 3, we need to access point[1][2] instead of point[2][3]. This is an instance of what is known as the off-by-one bug.
Where then does this convention come from? In some languages, such as the C code above, arrays with integer indices have super efficient access. In those languages you declare the size of the array
upfront along with the size of each element as we did with the line “int points[20][50];”. A 20 by 50 array of bytes (instead of integers in the example) would take up only 1,000 bytes, or less than
1 kilobyte which is technically 1,024 or 2 ^ 10 bytes. Using a byte we could represent up to 256 different colors in our point example.
Now assume the whole array starts at a say memory location 5000 (usually memory locations would be denoted in hexadecimal but I will use decimal to make this example easier to follow). The program
can then reserve memory locations 5000 to 5999 for the array which will consist of 50 rows (y) of 20 columns (x) each. So point[1,2] can then be found at memory location 5041. In general point[x,y]
can be found at memory location 5000 + y * 20 + x. You can now see immediately why it makes sense for the array indices to be “zero based” (meaning starting at 0). point[0,0] is located at the
beginning of the reserved memory region as 5000 + 0 * 20 + 0 = 5000 and point[19,49] is located at 5000 + 49 * 20 + 19 = 5999 which is the end of the array.
That approach to laying out arrays in memory does, however, not work for dynamic languages. If you don’t know in advance how large an array is going to be and if you don’t even know the size of each
element (as that might change from one element to the next), you can not have a simple formula for locating an element in memory. Instead, you need to have some intermediate lookup mechanism. Some
table that contains the known indices and points to memory locations. That approach also handles another issue: so-called sparse arrays. Imagine that instead of the image where every point has a
color, we are dealing with a huge matrix of one million by one million elements but of which only about one hundred have values and all the others are 0. Reserving vast amounts of memory would be
terribly inefficient.
Next Tuesday, in what will be the last post on Data Structures (at least for now), we will take a look at another set of data structures that let us keep track of many things in the form of lists and
other ways of linking elements to each other.
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Books on Classical/analytical Mechanics
French, Newtonian Mechanics if you're coming to the subject for the first time. (Sorry, didn't see your last message until now. Still a good book for reviewing topics like orbital mechanics.)
Fowles, Analytical Mechanics is a good compromise between simplicity and coverage. I found it great for review and getting up to speed on a topic. Symon is also very good, a little more
sophisticated, but also more wordy.
Goldstein is the "canonical" graduate text, and thus tries to cover all the bases. I find it a bit too dense and fat for self study, even for graduate students, though good for reference. (I remember
disliking the book intensely in graduate school, but I'm not sure that was altogether a rational response).
Landau is at the same level and blessedly concise. Some nice worked problems in Landau, too. Not everyone responds to the style, but if you already feel comfortable with the basics, I think this
would be the most pleasant book to spend your free time with. | {"url":"http://www.physicsforums.com/showthread.php?p=2205548","timestamp":"2014-04-18T23:20:30Z","content_type":null,"content_length":"39981","record_id":"<urn:uuid:f18469c7-6a34-47e9-a6ff-68a4c755303b>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00244-ip-10-147-4-33.ec2.internal.warc.gz"} |
More Complicated Brauer Computations
Let’s wrap up some of our Brauer group loose ends today. We can push through the calculation of the Brauer groups of curves over some other fields using the same methods as the last post, but just a
little more effort.
First, note that with absolutely no extra effort we can run the same argument as yesterday in the following situation. Suppose ${X}$ is a regular, integral, quasi-compact scheme of dimension ${1}$
with the property that all closed points ${v\in X}$ have perfect residue fields ${k(v)}$. Let ${g: \text{Spec} K \hookrightarrow X}$ be the inclusion of the generic point.
Running the Leray spectral sequence a little further than last time still gives us an inclusion, but we will usually want more information because ${Br(K)}$ may not be ${0}$. The low degree terms
(plus the argument from last time) gives us a sequence:
$\displaystyle 0\rightarrow Br'(X)\rightarrow Br(K)\rightarrow \bigoplus_v Hom_{cont}(G_{k(v)}, \mathbb{Q}/\mathbb{Z})\rightarrow H^3(X, \mathbb{G}_m)\rightarrow \cdots$
This allows us to recover a result we already proved. In the special case that ${X=\text{Spec} A}$ where ${A}$ is a Henselian DVR with perfect residue field ${k}$, then the uniformizing parameter
defines a splitting to get a split exact sequence
$\displaystyle 0\rightarrow Br(A)\rightarrow Br(K)\rightarrow Hom_{cont}(G_k, \mathbb{Q}/\mathbb{Z})\rightarrow 0$
Thus when ${A}$ is a strict local ring (e.g. ${\mathbb{Z}_p}$) we get an isomorphism ${Br(K)\rightarrow \mathbb{Q}/\mathbb{Z}}$ since ${Br(A)\simeq Br(k)=0}$ (since ${k}$ is ${C_1}$). In fact, going
back to Brauer groups of fields, we had a lot of trouble trying to figure anything out about number fields. Now we may have a tool (although without class field theory it isn’t very useful, so we’ll
skip this for now).
The last computation we’ll do today is to consider a smooth (projective) curve over a finite field ${C/k}$. Fix a separable closure ${k^s}$ and ${K}$ the function field. First, we could attempt to
use Leray on the generic point, since we can use that ${H^3(K, \mathbb{G}_m)=0}$ to get some more information. Unfortunately without something else this isn’t enough to recover ${Br(C)}$ up to
Instead, consider the base change map ${f: C^s=C\otimes_k k^s\rightarrow C}$. We use the Hochschild-Serre spectral sequence ${H^p(G_k, H^q(C^s, \mathbb{G}_m))\Rightarrow H^{p+q}(C, \mathbb{G}_m)}$.
The low degree terms give us
$\displaystyle 0\rightarrow Br(k)\rightarrow \ker (Br(C)\rightarrow Br(C^s))\rightarrow H^1(G_k, Pic(C^s))\rightarrow \cdots$
First, ${\ker( Br(C)\rightarrow Br(C^s))=Br(C)}$ by the last post. Next ${H^1(G_k, Pic^0(C^s))=0}$ by Lang’s theorem as stated in Mumford’s Abelian Varieties, so ${H^1(G_k, Pic(C^s))=0}$ as well.
That tells us that ${Br(C)\simeq Br(k)=0}$ since ${k}$ is ${C_1}$. So even over finite fields (finite was really used and not just ${C_1}$ for Lang’s theorem) we get that smooth, projective curves
have trivial Brauer group. | {"url":"http://hilbertthm90.wordpress.com/2012/12/27/more-complicated-brauer-computations/","timestamp":"2014-04-17T21:45:21Z","content_type":null,"content_length":"84862","record_id":"<urn:uuid:e8b35d0e-7daf-4a2e-8903-333a3e41ead9>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
digitalmars.D.announce - Statistics library
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for arrays of
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number generation.
A struct to generate all possible permutations of a sequence.
On the other hand, I'm a scientist, not a full-time programmer, and although I
can write working code, I have no clue what it takes to get code up to the
gold standard of "production." Also, this library is very D2-dependent, and I
have no interest in back-porting it. Of course if by some chance someone else
wanted to back-port it, they'd be more than welcome.
Most of the code is covered somehow or another by unit tests, although I
cheated a lot by having some unit tests depend on multiple functions.
Is there any interest in this from others in the D community? Do other people
think that D would benefit from having a decent statistics library? Other
Oct 23 2008
dsimcha, I think the struct to generate permutations is out of place there, and
more fit in a module like the comb (combinatorics) of mine.
Beside that detail, I like the idea of having a standard module with basic
statistics, so I am interested :-)
Oct 23 2008
Reply to dsimcha,
Since there's really no good comprehensive statistics library for D
(Tango has a little bit, the beginnings of a few are on dsource, but
nothing much), Ive been rolling my own statistics functions as
necessary. Almost by accident, it seems like I've built up the
beginnings of a decent statistics library. I'm debating whether it
might be interesting enough to people to be worth releasing, and
whether enough community help would be available to really make it
production quality, or to merge it with other people's efforts in this
Well for starters, just ask and I'll get you access to put it on scrapple. That's if you don't want to go to the trouble of having your own project (it's not much trouble BTW)
Oct 23 2008
== Quote from BCS (ao pathlink.com)'s article
Reply to dsimcha,
Since there's really no good comprehensive statistics library for D
(Tango has a little bit, the beginnings of a few are on dsource, but
nothing much), Ive been rolling my own statistics functions as
necessary. Almost by accident, it seems like I've built up the
beginnings of a decent statistics library. I'm debating whether it
might be interesting enough to people to be worth releasing, and
whether enough community help would be available to really make it
production quality, or to merge it with other people's efforts in this
That's if you don't want to go to the trouble of having your own project (it's not much trouble BTW)
Sounds good at least for now. It's only about 1500 lines of code including unittests, comments, etc. I'll put it up on scrapple with a permissive license, and people can make suggestions, and
integrate it into Phobos and Tango as they see fit.
Oct 24 2008
Reply to dsimcha,
== Quote from BCS (ao pathlink.com)'s article
Reply to dsimcha,
Since there's really no good comprehensive statistics library for D
(Tango has a little bit, the beginnings of a few are on dsource, but
nothing much), Ive been rolling my own statistics functions as
necessary. Almost by accident, it seems like I've built up the
beginnings of a decent statistics library. I'm debating whether it
might be interesting enough to people to be worth releasing, and
whether enough community help would be available to really make it
production quality, or to merge it with other people's efforts in
this area.
scrapple. That's if you don't want to go to the trouble of having your own project (it's not much trouble BTW)
including unittests, comments, etc. I'll put it up on scrapple with a permissive license, and people can make suggestions, and integrate it into Phobos and Tango as they see fit.
If you don't already have access send me your username and I'll add you.
Oct 24 2008
== Quote from BCS (ao pathlink.com)'s article
Reply to dsimcha,
== Quote from BCS (ao pathlink.com)'s article
Reply to dsimcha,
Since there's really no good comprehensive statistics library for D
(Tango has a little bit, the beginnings of a few are on dsource, but
nothing much), Ive been rolling my own statistics functions as
necessary. Almost by accident, it seems like I've built up the
beginnings of a decent statistics library. I'm debating whether it
might be interesting enough to people to be worth releasing, and
whether enough community help would be available to really make it
production quality, or to merge it with other people's efforts in
this area.
scrapple. That's if you don't want to go to the trouble of having your own project (it's not much trouble BTW)
including unittests, comments, etc. I'll put it up on scrapple with a permissive license, and people can make suggestions, and integrate it into Phobos and Tango as they see fit.
Username: dsimcha Yes, I realize that it's best to do things like this off the newsgroup, but your email address doesn't seem to work.
Oct 25 2008
Reply to dsimcha,
Yes, I realize that it's best to do things like this off the
newsgroup, but your email address doesn't seem to work.
Sorry. I figure I get enough SPAM as it is. Besides, there are about 2 dozen other ways to get it if you are persistent enough Oh. Your in, have fun (I really ought to make up some boiler plate like
this: http://en.wikipedia.org/wiki/Sudo#Design ;)
Oct 26 2008
Andrei Alexandrescu <SeeWebsiteForEmail erdani.org>
dsimcha wrote:
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for arrays of
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number generation.
A struct to generate all possible permutations of a sequence.
On the other hand, I'm a scientist, not a full-time programmer, and although I
can write working code, I have no clue what it takes to get code up to the
gold standard of "production." Also, this library is very D2-dependent, and I
have no interest in back-porting it. Of course if by some chance someone else
wanted to back-port it, they'd be more than welcome.
Most of the code is covered somehow or another by unit tests, although I
cheated a lot by having some unit tests depend on multiple functions.
Is there any interest in this from others in the D community? Do other people
think that D would benefit from having a decent statistics library? Other
If the community is interested, I'd be glad to take over your code and put it in Phobos. Andrei
Oct 23 2008
Reply to Andrei,
dsimcha wrote:
Since there's really no good comprehensive statistics library for D
(Tango has a little bit, the beginnings of a few are on dsource, but
nothing much), Ive been rolling my own statistics functions as
necessary. Almost by accident, it seems like I've built up the
beginnings of a decent statistics library. I'm debating whether it
might be interesting enough to people to be worth releasing, and
whether enough community help would be available to really make it
production quality, or to merge it with other people's efforts in
this area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the
Kendall tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for
arrays of numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs,
hypergeometric, Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number
A struct to generate all possible permutations of a sequence.
On the other hand, I'm a scientist, not a full-time programmer, and
although I can write working code, I have no clue what it takes to
get code up to the gold standard of "production." Also, this library
is very D2-dependent, and I have no interest in back-porting it. Of
course if by some chance someone else wanted to back-port it, they'd
be more than welcome.
Most of the code is covered somehow or another by unit tests,
although I cheated a lot by having some unit tests depend on multiple
Is there any interest in this from others in the D community? Do
other people think that D would benefit from having a decent
statistics library? Other comments?
put it in Phobos. Andrei
Even better would be getting it in both Phobos and Tango. Shouldn't be hard as I can't think it should depend on much.
Oct 23 2008
== Quote from BCS (ao pathlink.com)'s article
Even better would be getting it in both Phobos and Tango. Shouldn't be hard
as I can't think it should depend on much.
First, Tango needs to be ported to D2 (I realize that this is happening) or my code needs to be ported to D1. Anyhow, here are the dependencies: Non-trivial, i.e. in several places: std.math,
std.traits, std.functional, some custom sorting functions I wrote, which could just be included Trivial, i.e. in only one or two small places, pretty sure Tango has a drop-in replacement std.bigint
(for factorial, although all functions that actually use a factorial are calculated in log space, and therefore don't depend on this), std.algorithm (for swap, isSorted), std.random
Oct 23 2008
dsimcha wrote:
== Quote from BCS (ao pathlink.com)'s article
Even better would be getting it in both Phobos and Tango. Shouldn't be hard
as I can't think it should depend on much.
First, Tango needs to be ported to D2 (I realize that this is happening) or my code needs to be ported to D1. Anyhow, here are the dependencies: Non-trivial, i.e. in several places: std.math,
std.traits, std.functional, some custom sorting functions I wrote, which could just be included
Trivial, i.e. in only one or two small places, pretty sure Tango has a drop-in
std.bigint (for factorial, although all functions that actually use a factorial
are calculated in log space, and therefore don't depend on this), std.algorithm
(for swap, isSorted), std.random
This is another instance where we need a common namespace. Practically nothing in tango.math.* has dependencies on other parts of Tango, other than on the stuff which is now in Core. All the modules
you mention would ideally be in the common namespace.
Oct 30 2008
Andrei Alexandrescu wrote:
If the community is interested, I'd be glad to take over your code and
put it in Phobos.
I'm interested.
Oct 23 2008
Dejan Lekic <dejan.lekic tiscali.co.uk>
If my vote counts - I am all for it. :)
Oct 27 2008
Now that I've figured out how the heck to use SVN, it's up on scrapple.
Everything basically has unittests and has been dogfooded by me, but if there
any bugs I missed, please file. I'm actually kind of surprised at the level of
interest in this project.
Just a reminder: The current version makes pretty heavy use of D2 features, so
is completely incompatible with D1. D1 compatibility was not even a
in the design of anything. I have no intention of backporting to D1, since D2
getting pretty close to ready anyhow.
Also, I'm not a lawyer, but the intent of the license I put in the header of the
files is to be very permissive to allow integration into Phobos and Tango of
whatever functionality Andrei and Don see fit. If you see a problem with the
I have worded the license, let me know and I'll change it.
Oct 27 2008
"Bill Baxter" <wbaxter gmail.com>
On Fri, Oct 24, 2008 at 7:43 AM, dsimcha <dsimcha yahoo.com> wrote:
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for arrays of
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number generation.
A struct to generate all possible permutations of a sequence.
I don't know what a lot of those things are, but statistics to me means you will probably have (or eventually want) things like covariance which are best represented as matrices. Does your package
also have a matrix library? --bb
Oct 23 2008
== Quote from Bill Baxter (wbaxter gmail.com)'s article
On Fri, Oct 24, 2008 at 7:43 AM, dsimcha <dsimcha yahoo.com> wrote:
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for arrays of
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number generation.
A struct to generate all possible permutations of a sequence.
means you will probably have (or eventually want) things like covariance which are best represented as matrices. Does your package also have a matrix library? --bb
No, it doesn't have a matrix library right now. I make no claim that it is in any way complete right now, but I do think it has some pretty useful stuff that's not likely to be anywhere else for D.
Oct 23 2008
== Quote from Bill Baxter (wbaxter gmail.com)'s article
Ok, so it's mainly for 1d statistics then?
Oct 23 2008
On Fri, Oct 24, 2008 at 9:39 AM, dsimcha <dsimcha yahoo.com> wrote:
== Quote from Bill Baxter (wbaxter gmail.com)'s article
On Fri, Oct 24, 2008 at 7:43 AM, dsimcha <dsimcha yahoo.com> wrote:
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for arrays of
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs.
Inverse normal distribution, and normally distributed random number generation.
A struct to generate all possible permutations of a sequence.
means you will probably have (or eventually want) things like covariance which are best represented as matrices. Does your package also have a matrix library? --bb
No, it doesn't have a matrix library right now. I make no claim that it is in any way complete right now, but I do think it has some pretty useful stuff that's not likely to be anywhere else for D.
Ok, so it's mainly for 1d statistics then? --bb
Oct 23 2008
Don <nospam nospam.com.au>
dsimcha wrote:
Since there's really no good comprehensive statistics library for D (Tango has
a little bit, the beginnings of a few are on dsource, but nothing much), Ive
been rolling my own statistics functions as necessary. Almost by accident, it
seems like I've built up the beginnings of a decent statistics library. I'm
debating whether it might be interesting enough to people to be worth
releasing, and whether enough community help would be available to really make
it production quality, or to merge it with other people's efforts in this
area. The following functionality is currently available:
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs. Inverse normal distribution,
Most of these are in Tango (not Kolmogorov). Are yours different in some way?
A struct to generate all possible permutations of a sequence.
Correlation (Pearson, Spearman rho, Kendall tau). Note that the
tau correlation is a very efficient O(N log N) version.
Mean, standard deviation, variance, kurtosis, percent variance for
numeric values.
Shannon entropy, mutual information.
Kolmogorov-Smirnov tests
Sounds good.
On the other hand, I'm a scientist, not a full-time programmer,
Me too!
Is there any interest in this from others in the D community? Do other people
think that D would benefit from having a decent statistics library?
Yes. Which is why I put the existing stuff into Tango.
Oct 24 2008
== Quote from Don (nospam nospam.com.au)'s article
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs. Inverse normal distribution,
They calculate the exact log factorial using a caching scheme. Not sure how much accuracy this actually buys, though it costs some memory. I should probably change the logFactorial function to a
gamma approximation at least for large N. Also, Tango doesn't have hypergeometric.
A struct to generate all possible permutations of a sequence.
> Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall > tau correlation is a very efficient O(N log N) version. > > Mean, standard deviation, variance, kurtosis, percent variance
for arrays of > numeric values. > > Shannon entropy, mutual information. > > Kolmogorov-Smirnov tests Sounds good.
This is more the part that I thought might be useful.
On the other hand, I'm a scientist, not a full-time programmer,
Is there any interest in this from others in the D community? Do other people
think that D would benefit from having a decent statistics library?
BCS has offered me Scrapple access, I'll post the code there under a permissive license. From there, Tango and Phobos devs can look at it and do as they see fit.
Oct 24 2008
dsimcha wrote:
== Quote from Don (nospam nospam.com.au)'s article
Binomial, hypergeometric, normal, Poisson, Kolmogorov CDFs, hypergeometric,
Poisson, binomial PDFs. Inverse normal distribution,
They calculate the exact log factorial using a caching scheme. Not sure how much accuracy this actually buys, though it costs some memory. I should probably change the logFactorial function to a
gamma approximation at least for large N.
That shouldn't be necessary. If logGamma() isn't giving an accurate factorial (within a couple of bits of precision), that's a problem with logGamma. Please generate a bug report.
Also, Tango doesn't have hypergeometric.
You're right. It's still on my hard disk, I wasn't quite happy it.
A struct to generate all possible permutations of a sequence.
> Correlation (Pearson, Spearman rho, Kendall tau). Note that the Kendall > tau correlation is a very efficient O(N log N) version. > > Mean, standard deviation, variance, kurtosis, percent variance
for arrays of > numeric values. > > Shannon entropy, mutual information. > > Kolmogorov-Smirnov tests Sounds good.
This is more the part that I thought might be useful.
On the other hand, I'm a scientist, not a full-time programmer,
Is there any interest in this from others in the D community? Do other people
think that D would benefit from having a decent statistics library?
BCS has offered me Scrapple access, I'll post the code there under a permissive license. From there, Tango and Phobos devs can look at it and do as they see fit.
Oct 25 2008 | {"url":"http://www.digitalmars.com/d/archives/digitalmars/D/announce/Statistics_library_13966.html","timestamp":"2014-04-19T12:33:01Z","content_type":null,"content_length":"48368","record_id":"<urn:uuid:3e5598c7-6051-41b2-9402-7875e5bedbd9>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00269-ip-10-147-4-33.ec2.internal.warc.gz"} |
Stefan Finsterle
Stefan Finsterle's Publications
Journal Articles
1. Poskas, P., A. Narkuniene, D. Grigaliuniene, and S. Finsterle, Comparison of radionuclide releases from a conceptual geological repository for RBMK-1500 and BWR spent nuclear fuel, Nuclear
Technology, 185(3), 322-335, doi:10.1382/NT13-52, 2014.
2. Freedman, V.L., X. Chen, S. Finsterle, M.D. Freshley, I. Gorton, L.J. Gosink, E.H. Keating, C.S. Lansing, W.A.M. Moerglein, C.J. Murray, G.S.H. Pau, E. Porter, S. Purohit, M. Rockhold, K.L.
Schuchardt, C. Sivaramakrishnan, V.V. Vesselinov, and S. R. Waichler, A high-performance workflow system for subsurface simulation, Environmental Modelling & Software, doi:10.1016/
j.envsoft.2014.01.030, 55, 176–189, 2014.
3. Rechard, R.P., H.H. Liu, Y.W. Tsang, and S. Finsterle, Characterization of natural barrier of Yucca Mountain disposal system for spent nuclear fuel and high-level radioactive waste, Reliability
Engineering & System Safety, 122, 32-52, doi:10.1016/j.ress.2013.06.020, 2014.
4. Finsterle, S., and E.L. Sonnenthal, Foreword to the Special Issue on the TOUGH Symposium 2012, Computers & Geosciences, 65, 1, doi:10.1016/j.cageo.2014.01.003, 2014.
5. Finsterle, S., E.L. Sonnenthal, and N. Spycher, Advances in subsurface modeling: The TOUGH suite of simulators, Computers & Geosciences, 65, 2-12, doi:10.1016/j.cageo.2013.06.009, 2014.
6. Wellmann, J.F., S. Finsterle, and A. Croucher, Integrating structural geological data into the inverse modelling framework of iTOUGH2, Computers & Geosciences, 65, 95-103, doi:10.1016/
j.cageo.2013.10.014, 2014.
7. Pau, G.S.H., Y. Zhang, S. Finsterle, H. Wainwright, and J. Birkholzer, Reduced order modeling in iTOUGH2, Computers & Geosciences, 65, 118-126, doi:10.1016/j.cageo.2013.08.008, 2014.
8. Commer, M., M.B. Kowalsky, J. Doetsch, G. Newman, and S. Finsterle, MPiTOUGH2: A parallel parameter estimation framework for hydrological and hydrogeophysical applications, Computers &
Geosciences, 65, 127-135, doi:10.1016/j.cageo.2013.06.011, 2014.
9. Wainwright, H., S. Finsterle, Y. Jung, Q. Zhou, and J.T. Birkholzer, Making sense of global sensitivity analyses, Computers & Geosciences, 65, 84-94, doi:10.1016/j.cageo.2013.06.006, 2014.
10. Akhavan, M., P.T. Imhoff, S. Andres, and S. Finsterle, Model evaluation of impacts on denitrification under rapid infiltration basin systems, J. Contam. Hydrol., 152, 18-34, doi:10.1016/
j.jconhyd.2013.05.007, 2013.
11. Pau, G., Y. Zhang, and S. Finsterle, Reduced order models for many-query subsurface flow applications, Computational Geosciences, 17(4), 705-721, doi:10.1007/s10596-013-9349-z, 2013.
12. Wainwright, H.M., S. Finsterle, Q. Zhou, and J.T. Birkholzer, Modeling the performance of large-scale CO2 storage systems: A comparison of different sensitivity analysis methods, International
Journal of Greenhouse Gas Control, 17, 189-205, doi:10.1016/j.ijggc.2013.05.007, 2013.
13. Doetsch, J., M.B. Kowalsky, C. Doughty, S. Finsterle, J.B. Ajo-Franklin, C.R. Carrigan, X. Yang, S.D. Hovorka, and T.M. Daley, Constraining CO[2] simulations by coupled modeling and inversion of
electrical resistance and gas composition data, International Journal of Greenhouse Gas Control, 18, 510-522, doi:10.1016/j.ijggc.2013.04.011, 2013.
14. Finsterle, S., Y. Zhang, L. Pan, P. Dobson, and K. Oglesby, Microhole arrays for improved heat mining from enhanced geothermal systems, Geothermics, 47, 104-115, doi:10.1016/
j.geothermics.2013.03.001, 2013.
15. Oldenburg, C.M., B.M. Freifeld, K. Pruess, L. Pan, S. Finsterle, and G.J. Moridis, Numerical simulations of the Macondo well blowout reveal strong control of oil flow by reservoir permeability
and exsolution of gas, Proceedings of the National Academy of Sciences, 109(50), 20254-20259, doi:10.1073/pnas.1105165108, December 2012.
16. Shi, X., M. Ye, S. Finsterle, and J. Wu, Comparing nonlinear regression and Markov Chain Monte Carlo methods for assessment of prediction uncertainty in vadose zone modeling, Vadose Zone J., 11
(4), doi:10.2136/vzj2011.0147, 2012.
17. Finsterle, S., M.B. Kowalsky, and K. Pruess, TOUGH: Model use, calibration, and validation, Transactions of the ASABE, 55(4), 1275-1290, 2012.
18. Akhavan, M., P.T. Imhoff, S. Finsterle, and A.S. Andres, Application of a coupled overland flow-vadose zone model to rapid infiltration basin systems, Vadose Zone J., 11(2), doi:10.2136/
vzj2011.0140, 2012.
19. Kowalsky, M. B., S. Finsterle, K. H. Williams, C. Murray, M. Commer, D. Newcomer, A. Englert, C. I. Steefel, and S. S. Hubbard, On parameterization of the inverse problem for estimating aquifer
properties using tracer data, Water Resour. Res., 48, W06535, doi:10.1029/2011WR011203, 2012.
20. Zhang, Y., L., Pan, K. Pruess, and S. Finsterle, A time-convolution approach for modeling heat exchange between a wellbore and the surrounding formation, Geothermics, 40(4), 261-266, doi:10.1016/
j.geothermics.2011.08.003, 2011.
21. Williamson, M., J. Meza, D. Moulton, I. Gorton, M. Freshley, P. Dixon, R. Seitz, C. Steefel, S. Finsterle, S. Hubbard, M. Zhu, K. Gerdes, R. Patterson, and Y.T. Collazo, Advanced Simulation
Capability for Environmental Management (ASCEM): An overview of initial results, Technology and Innovation, 13, 175–199, doi:10.3727/194982411X13085939956625, 2011.
22. Finsterle, S., and Y. Zhang, Solving iTOUGH2 simulation and estimation problems using the PEST protocol, Environmental Modelling and Software, 26, 959-968, doi:10.1016/j.envsoft.2011.02.008,
23. Finsterle, S., and M.B. Kowalsky, A truncated Levenberg-Marquardt algorithm for the calibration of highly parameterized nonlinear models, Computers & Geosciences, 37, 731-738, doi:10.1016/
j.cageo.2010.11.005, 2011.
24. Finsterle, S., and Y. Zhang, Error handling strategies in multiphase inverse modeling, Computers & Geosciences, 37, 724-730, doi:10.1016/j.cageo.2010.11.009, 2011.
25. Jung, Y., P. Imhoff, and S. Finsterle, Estimation of landfill gas generation rate and gas permeability of refuse using inverse modeling, Transport in Porous Media, 90(1), 41-58, doi:10.1007/
s11242-010-9659-9, 2011.
26. Xu, T., R. Senger, and S. Finsterle, Bentonite alteration due to thermal-hydro-chemical processes during the early thermal period in a nuclear waste repository, Nuclear Technology, 174(3),
438-451, 2011.
27. Zhang, Y., S. Hubbard, and S. Finsterle, Factors governing sustainable groundwater pumping near a river, Ground Water, 49(3), 432-444, doi:10.111/j.1745-6584.2010.00743.x, 2011.
28. Takeda, M., T. Hiratsuka, K. Ito, and S. Finsterle, An asymmetric diffusion experiment for the determination of diffusion and sorption coefficients of rock samples, Journal of Contaminant
Hydrology, 123, 114–129, doi:10.1016/j.jconhyd.2010.12.012, 2011.
29. Kowalsky, M.B., E. Gasperikova, S. Finsterle, D. Watson, and S.S. Hubbard, Coupled modeling of hydrogeochemical and electrical resistivity data for exploring the impact of recharge on subsurface
contamination, Water Resour. Res., 47, W02509, doi:10.029/2009WR008947, 2011.
30. Zhang, Y., B. Freifeld, S. Finsterle, M. Leahy, J. Ennis-King, L. Paterson, and T. Dance, Single-well experimental design for studying residual trapping of supercritical carbon dioxide,
International Journal of Greenhouse Gas Control, 5, 88-98, doi:10.1016/j.ijggc.2010.06.011, 2011.
31. Neerdael, B., and S. Finsterle, The use of numerical models in support of site characterization and performance assessment studies for geological repositories, Nuclear Engineering and Technology,
42(2), 145-150, 2010.
32. Lehikoinen, A., J.M.J. Huttunen, S. Finsterle, M.B. Kowalsky, and J.P. Kaipio, Dynamical inversion for hydrological process monitoring with electrical resistance tomography under model
uncertainties, Water Resour. Res., 46, W04513, doi:10.1029/2009WR008470, 2010.
33. Zhang, Y., C.M. Oldenburg, and S. Finsterle, Percolation-theory and fuzzy rule-based probability estimation of fault leakage at geologic carbon sequestration sites, Env. Earth Sci., 59,
1447–1459, doi:10.1007/s12665-009-0131-4, 2010.
34. Lehikoinen, A., S. Finsterle, A. Voutilainen, M.B. Kowalsky, and J.P. Kaipio, Dynamical inversion of geophysical ERT data: state estimation in the vadose zone, Inverse Problems in Science and
Engineering, 17(6), 716–736, doi:10.1080/17415970802475951, 2009.
35. Mukhopadhyay, S., Y.W, Tsang, and S. Finsterle, Parameter estimation from flowing fluid temperature logging data in unsaturated fractured rock using multiphase inverse modeling, Water Resour.
Res., 45, W04414, doi:10.1029/2008WR006869, 2009.
36. Senger, R., T. Xu, P. Marschall, and S. Finsterle, Investigation of two-phase flow phenomena associated with corrosion in an SF/HLW repository in Opalinus clay, Switzerland, Physics and
Chemistry of the Earth, doi:10.1016/j.pce.2008.10.034, 33, S317–S326, 2008.
37. Xu, T., S. Senger, and S. Finsterle, Corrosion-induced gas generation in a nuclear waste repository: Reactive geochemistry and multiphase flow effect, Appl. Geochem., 23, 3423–3433, doi:10.1016/
j.apgeochem.2008.07.012, 2008.
38. Kowalsky, M.B., J. Birkholzer, J. Peterson, S. Finsterle, S. Mukhopadhyay, and Y. Tsang, Sensitivity analysis for joint inversion of ground-penetrating radar and thermal-hydrological data from a
large-scale underground heater test, Nuclear Technology, 164(2), 169–179, 2008.
39. Freifeld, B.M., S. Finsterle, T.C. Onstott, T. Toole, and L.M. Pratt, Ground surface temperature reconstructions: Using in situ estimates for thermal conductivity acquired with a fiber-optic
distributed thermal perturbation sensor, Geophysical Research Letter, 35, L14309, doi:10.1029/2008GL034762, 2008.
40. Kiryukhin, A.V., N.P. Asaulova, and S. Finsterle, Inverse modeling and forecasting for the exploitation of the Pauzhetsky geothermal field, Kamchatka, Russia, Geothermics, 37, 540–562,
doi:10.1016/j.geothermics.2008.04.003, 2008.
41. Finsterle, S., C. Doughty, M.B. Kowalsky, G.J. Moridis, L. Pan, T. Xu, Y. Zhang, and K. Pruess, Advanced vadose zone simulations using TOUGH, Vadose Zone J., 7:601–609, doi:10.2136/vzj2007.0059,
42. Salve, R., N.Y. Krakauer, M.B. Kowalsky, and S. Finsterle, A qualitative assessment of microclimatic perturbations in a tunnel, Int. J. Climatol., 28(15), 2081-U3, doi:10.1002/joc.1697, 2008.
43. Finsterle, S., and M. B. Kowalsky, Joint hydrological-geophysical inversion for soil structure identification, Vadose Zone J., 7:287-293, doi:10.2136/vzj2006.0078, 2008.
44. Revil, A., N. Linde, A. Cerepi, D. Jougnot, S. Matthäi, and S. Finsterle, Electrokinetic coupling in unsaturated porous media, J. Colloid Interface Sci., 313, 315–327, doi:10.1016/
j.jcis.2007.03.037, 2007.
45. Zhang, Y. C. M. Oldenburg, S. Finsterle and G. S. Bodvarsson, System-level modeling for economic evaluation of geological CO[2] storage in gas reservoirs, Energy Conservation and Management, 48
(6), 1827–1833, doi:10.1016/j.enconman.2007.01.018, 2007.
46. Lehikoinen, A., S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, Approximation errors and truncation of computational domains with application to geophysical
tomography, Inverse Problems and Imaging, 1(2), 371–389, 2007.
47. Finsterle, S., Comment on “Seepage into drifts and tunnels in unsaturated fractured rock” by Dani Or, Markus Tuller, and Randall Fedors, Water Resour. Res., 42, W07603, doi:10.1029/2005WR004777,
48. Linde, N., S. Finsterle, and S. Hubbard, Inversion of tracer test data using tomographic constraints, Water Resour. Res., 42(4), W04410, doi:10.1029/2004WR003806, 2006.
49. Zhang, Y., H. H. Liu, Q. Zhou, and S. Finsterle, Effects of diffusive property heterogeneity on effective matrix diffusion coefficient for fractured rock, Water Resour. Res., 42, W04405,
doi:10.1029/2005WR004513, 2006.
50. Finsterle, S., Demonstration of optimization techniques for groundwater plume remediation using iTOUGH2, Environmental Modelling and Software, 21(5), 665–680, doi:10.1016/j.envsoft.2004.11.012,
51. Kowalsky, M., S. Finsterle, J. Peterson, S. Hubbard, Y. Rubin, E. Majer, A. Ward, and G. Gee, Estimation of field-scale soil hydraulic parameters and dielectric parameters through joint inversion
of GPR and hydrological data, Water Resour. Res., 41, W11425, doi:10.1029/2005WR004237, 2005.
52. Finsterle, S., and C.M. Oldenburg, Research advances in vadose zone hydrology through simulations with the TOUGH codes, Vadose Zone J., 3: 737, 2004.
53. Finsterle, S., Multiphase inverse modeling: Review and iTOUGH2 applications, Vadose Zone J., 3: 747–762, 2004.
54. Ghezzehei, T. A., R. C. Trautz, S. Finsterle, P. J. Cook, and C. F. Ahlers, Modeling coupled evaporation and seepage in ventilated tunnels, Vadose Zone J., 3: 806–818, 2004.
55. Gallagher, P. M., and S. Finsterle, Physical and numerical model of colloidal silica injection for passive site stabilization, Vadose Zone J., 3: 917–925, 2004.
56. Kitterød, N.-O., and S. Finsterle, Simulating unsaturated flow fields based on saturation measurements, Journal of Hydraulic Research, 42, 121–129, 2004.
57. Kowalsky, M.B., S. Finsterle, and Y. Rubin, Estimating flow parameter distributions using ground-penetrating radar and hydrological measurements during transient flow in the vadose zone, Adv.
Water Resour., 27(6), doi:10.1016/j.advwatres.2004.03.003, 583–599, 2004.
58. Unger, A., S. Finsterle, and G. S. Bodvarsson, Transport of radon gas into a tunnel at Yucca Mountain—estimating large-scale fractured tuff hydraulic properties and implications for the
ventilation system, Journal of Contam. Hydrol., 70, doi:10.1016/j.jconhyd.2003.07.001, 152–171, 2004.
59. Vasco, D.W., S. Finsterle, Numerical trajectory calculations for the efficient inversion of flow and transport observations, Water Resour. Res., 40, W01507, doi:10.1029/2003WR002362, 2004.
60. Engelhardt, I., S. Finsterle, and C. Hofstee, Experimental and numerical investigation of flow phenomena in nonisothermal, variably saturated bentonite/crushed rock mixtures, Vadose Zone J., 2:
239–246, 2003.
61. Engelhardt, I., and S. Finsterle, Thermal-hydrologic experiments with bentonite/crushed rock mixtures and estimation of effective parameters by inverse modeling, Applied Clay Science, 23,
111–120, doi:10.1016/S0169-1317(03)00093-0, 2003.
62. Houseworth, J. E., S. Finsterle, and G. S. Bodvarsson, Flow and transport in the drift shadow in a dual-continuum model, J. Contam. Hydr., 62–63, 133–156, doi:10.1016/S0169-7722(02)00172-9, 2003.
63. Finsterle, S., C. F. Ahlers, R. C. Trautz, and P. J. Cook, Inverse and predictive modeling of seepage into underground openings, J. Contam. Hydr., 62–63, 89–109, doi:10.1016/S0169-7722(02)00174-2
, 2003.
64. Mays, D. C., B. Faybishenko, and S. Finsterle, Information entropy to measure temporal and spatial complexity of unsaturated flow in heterogeneous media, Water Resour. Res., 38(12), 1313,
doi:10.1029/2001WR001185, 2002.
65. Liu, H. H., G. S. Bodvarsson, and S. Finsterle, A note on unsaturated flow in two-dimensional fracture networks, Water Resour. Res., 38(9), 1176, doi:10.1027/2001WR000977, 2002.
66. Finsterle, S., J. T. Fabryka-Martin, and J. S. Y. Wang, Migration of a water pulse through fractured porous media, J. Contam. Hydr., 54 (1–2), 37–57, 2002.
67. Finsterle, S., and R. C. Trautz, Numerical modeling of seepage into underground openings, Mining Engineering, 53(9), 52–56, 2001.
68. Finsterle, S., Using the continuum approach to model unsaturated flow in fractured rock, Water Resour. Res., 36(8), 2055–2066, 2000.
69. Moridis, G. J., S. Finsterle, and J. Heiser, Evaluation of alternative designs for an injectable barrier at the Brookhaven National Laboratory Site, Long Island, New York, Water Resour. Res., 35
(10), 2937–2953, 1999.
70. Ahlers, C. F., S. Finsterle, and G. S. Bodvarsson, Characterization of subsurface pneumatic response at Yucca Mountain, J. Contam. Hydr., 38(1–3), 47–68, 1999.
71. Wang, J. S. Y., R. C. Trautz, P. J. Cook, S. Finsterle, A. L. James, and J. Birkholzer, Field tests and model analyses of seepage into drift, J. Contam. Hydr., 38(1–3), 323–347, 1999.
72. Finsterle, S., and B. Faybishenko, Inverse modeling of a radial multistep outflow experiment for determining unsaturated hydraulic properties, Adv. in Water Resour., 22(5), 431–444, 1999.
73. Finsterle, S., and J. Najita, Robust estimation of hydrogeologic model parameters, Water Resour. Res., 34(11), 2939–2947, 1998.
74. Finsterle, S., and P. Persoff, Determining permeability of tight rock samples using inverse modeling, Water Resour. Res., 33 (8), 1803–1811, 1997.
75. Finsterle, S., and K. Pruess, Solving the estimation-identification problem in two-phase flow modeling, Water Resour. Res., 31 (4), 913–924, April, 1995.
Book Chapters
1. Saibi, H., S. Finsterle, R. Bertani, and J. Nishijima, Geothermal Energy, in: K. Lee, and J. Kauffman (Eds.), Handbook of Sustainable Engineering, Springer, New York, 2013.
2. Bodvarsson, G. S., S. Finsterle, H. H. Liu, C. M. Oldenburg, K. Pruess, E. Sonnenthal, and Y.-S. Wu, Flow and transport modeling of subsurface systems, in: B. B. Looney and R. W. Falta (eds.),
Vadose Zone Science and Technology Solutions, Battelle Press, Columbus, Ohio, 2000.
3. Finsterle, S., K. Pruess, G. Björnsson, and A. Battistelli, Evaluation of geothermal well behavior using inverse modeling, in: Faybishenko (ed.), Dynamics of Fluids in Fractured Rocks,
Geophysical Monograph 122, pp. 377–387, American Geophysical Union, Washington DC, 2000.
4. Faybishenko, B., and S. Finsterle, Tensiometry in fractured rocks, in: Zhang, D., and Winter, C. L., eds., Theory, Modeling, and Field Investigation in Hydrogeology: A Special Volume in Honor of
Shlomo P. Neuman’s 60^th Birthday, Boulder, Colorado, Geological Society of America Special Paper 348, p. 161–174, 2000.
5. Finsterle, S., Direct and inverse modeling of multiphase flow systems, In: R. Helmig et al. (eds.), Modeling and Computation in Environmental Sciences, Friedr. Vieweg & Sohn Verlagsgesellschaft
mbH, Braunschweig/Wiesbaden, Germany, 148-157,1997.
Conference Papers and Abstracts
1. Rinaldi, A.P., J. Rutqvist, S. Finsterle, and H.H. Liu, Forward and inverse modeling of ground surface uplift at In Salah (Algeria), ARMA, 2014.
2. Freshley, M., T, Scheibe, D. Moulton, S.S. Hubbard, H. Murakami, S. Finsterle, C.I. Steefel, V. Freedman, G. Flach, R. Seitz, P. Dixon, and J. Marble, Advanced Simulation Capability for
Environmental Management initial user release, Waste Management, Phoenix, Arizona, March 2–6, 2014.
3. Commer, M., M.B. Kowalsky, G.A. Newman, and S. Finsterle, Hydrogeophysical joint inversion capabilities and impact of petrophysical assumptions, SEG, Houston, Texas, 22–27 Sept., 2013.
4. Rinaldi, A.P., J. Rutqvist, S. Finsterle, and H.H. Liu, Modeling ground displacement at InSalah CO2 injection site, 12th Annual Conference on Carbon Capture Utilization & Sequestration,
Pittsburgh, PA, May 13–16, 2013.
5. Wainwright, H.M, S. Finsterle, Q. Zhou, and J.T. Birkholzer, Improved global sensitivity analysis: Applications to CO2 storage systems, MODFLOW and More, Golden, Colorado, June 2–5, 2013.
6. Seitz, R., M. Freshley, P. Dixon, S.S. Hubbard, V. Freedman, G. Flach, B. Faybishenko, I. Gorton, S. Finsterle, J.D. Moulton, C.I. Steefel, and J. Marble, Advanced simulation capability for
environmental management – current status and Phase II demonstration results, Waste Management 2013 Conference, Phoenix, Arizona, February 14-28, 2013.
7. Imhoff, P.T., M. Akhavan, S. Finsterle, and S. Andres, Modeling engineered approaches to enhance denitrification under rapid infiltration basins, AGU Fall Meeting, San Francisco, California,
December 3–7, 2012.
8. Commer, M., M.B. Kowalsky, J. Doetsch, G.A. Newman, and S. Finsterle, Investigation of approaches for hydrogeophysical joint inversion using a parallel computing platform, AGU Fall Meeting, San
Francisco, California, December 3–7, 2012.
9. Pau, G., Y. Zhang, and S. Finsterle, Gaussian process regression approach for modeling subsurface flow, AGU Fall Meeting, San Francisco, California, December 3–7, 2012.
10. Druhan, J., S. Finsterle, N.T. Vandehey, R. Boutchko, J.P. O’Neil, W.W. Moses, and P.S. Nico, Direct observations of flow path evolution during reactive transport in porous media using clinical
nuclear imaging tomography, AGU Fall Meeting, San Francisco, California, December 3–7, 2012.
11. Takeda, M., T. Hiratsuka, M. Manaka, S. Finsterle, and K. Ito, A sequence of laboratory experiments for the determination of chemico-osmotic, hydraulic and diffusion parameters of rock sample,
AGU Fall Meeting, San Francisco, California, December 3–7, 2012.
12. Doetsch, J., M.B. Kowalsky, C. Doughty, S. Finsterle, J.B. Ajo-Franklin, X. Yang, C.R. Carrigan, and T.M. Daley, Structural and fully coupled joint inversion for CO2 migration monitoring, AGU
Fall Meeting, San Francisco, California, December 3–7, 2012.
13. Akhavan, M., P.T. Imhoff, S. Andres, and S. Finsterle, Importance of overland flow in denitrification of wastewater applied to rapid infiltration basins, TOUGH Symposium, Berkeley, California,
Sept. 17–19, 2012.
14. Moridis, G.J., C.M. Freeman, S. Webb, and S. Finsterle, The RealGas and RealGasH2O options of the TOUGH+ code for the simulation of coupled fluid and heat flow in tight/shale gas systems, TOUGH
Symposium, Berkeley, California, Sept. 17–19, 2012.
15. Kowalsky, M.B., M. Commer, K.H. Williams, and S. Finsterle, On parameterizing heterogeneity and incorporating geophysical measurements in hydrogeophysical inverse modeling, TOUGH Symposium,
Berkeley, California, Sept. 17–19, 2012.
16. Commer, M., M.B. Kowalsky, S. Finsterle, and G.A. Newman, Advances in hydrogeophysical joint inversion, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
17. Doetsch, J., M.B. Kowalsky, S. Finsterle, C. Doughty, J.B. Ajo-Franklin, and T.M. Daley, Geophysical data improve stability and convergence of hydrological property estimation: a synthetic CO2
injection study, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
18. Wainwright, H., S. Finsterle, Y. Jung, and J. Birkholzer, iTOUGH2 global sensitivity analysis module: Applications to CO2 storage systems, TOUGH Symposium, Berkeley, California, Sept. 17–19,
19. Wellmann, F., A. Croucher, and S. Finsterle, Adding geology to the equation: towards integrating structural geological data into inverse modeling with iTOUGH2. TOUGH Symposium, Berkeley,
California, Sept. 17–19, 2012.
20. Pau, G., Y. Zhang, and S. Finsterle, Reduced order models for subsurface flow in iTOUGH2, TOUGH Symposium, Sept. 17–19, 2012.
21. Keen, N., G. Paul, S. Finsterle, and E. Sonnenthal, TOUGHLIB: A new library to improve TOUGH parallel development, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
22. Takeda, M., T. Hiratsuka, K. Ito, and S. Finsterle, Development and application of a chemical osmosis simulator based on TOUGH2, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
23. Zhang, Y., L. Pan, P.F. Dobson, K. Oglesby, and S. Finsterle, Simulating microhole-based heat mining from enhanced geothermal systems, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
24. Finsterle, S., What’s new in iTOUGH2?, TOUGH Symposium, Berkeley, California, Sept. 17–19, 2012.
25. Doetsch, J., M.B. Kowalsky, C. Doughty, S. Finsterle, J.B. Ajo-Franklin, X. Yang, C.R. Carrigan, and T.M. Daley, Fully coupled hydrogeophysical inversion of CO2 migration in a deep saline
aquifer, SEG-AGU Hydrogeophysics workshop, Boise, ID, 2012.
26. Wainwright, H., S. Finsterle, Q. Zhou, J. Birkholzer, Uncertainty Quantification of the CO2 Storage System for a Hypothetical GCS Project in the Southern San Joaquin Basin in California, XIX
International Conference on Computational Methods in Water Resources, Urbana-Champaign, Illinois, June 17–22, 2012.
27. Vesselinov, V.V., G. Pau, and S. Finsterle, AGNI: Coupling model analysis tools and high-performance subsurface flow and transport simulators for risk and performance assessment, XIX
International Conference on Computational Methods in Water Resources, Urbana-Champaign, Illinois, June 17–22, 2012.
28. Schuchardt, K., D. Agarwal, S. Finsterle, C. Gable, E. Keating, J. Myer, A. Shoshani, L. Gosink, C. Lansing, W. Moeglein, G. Pau, E. Porter, S. Purohit, M. Rockhold, and C. Sivaramakrishnan,
ASCEM—An open environment for advances simulation, XIX International Conference on Computational Methods in Water Resources, Urbana-Champaign, Illinois, June 17–22, 2012.
29. Commer, M., M.B. Kowalsky, S. Finsterle, and G.A. Newman, MPiTOUGH2-EMGeo—A massively parallel data inversion framework for joint hydrogeophysical real-world applications, XIX International
Conference on Computational Methods in Water Resources, Urbana-Champaign, Illinois, June 17–22, 2012.
30. Zhang, Y., L. Pan, P. Dobson, K. Oglesby, and S. Finsterle, Microholes for improved heat extraction from EGS reservoirs: Numerical evaluation, Stanford Geothermal Workshop, Stanford, Calif., Jan.
30–Feb. 1, 2012.
31. Freshley, M.D., S.S. Hubbard, V. Freedman, G. Flach, I. Gorton, J.D. Moulton, S. Finsterle, C.I. Steefel, and P. Dixon, Advanced Simulation Capability for Environmental Management (ASCEM):
Development and demonstrations, Waste Management 2012.
32. Freedman, V., X. Chen, E. Keating, D. Higdon, M. Rockhold, S. Finsterle, K. Schuchardt, I. Gorton, and M. Freshley, Assessing uncertainty in subsurface transport predictions using the ASCEM
toolset, AGU Fall Meeting, San Francisco, Dec. 5–9, 2011.
33. Finsterle, S., and M.B. Kowalsky, Soil structure identification through inverse modeling, AGU Fall Meeting, San Francisco, Dec. 5–9, 2011.
34. Kowalsky, M.B., M. Commer, J. Ajo-Franklin, C. Doughty, T. Daley, and S. Finsterle, Feasibility of coupled hydrogeophysical inversion for characterization and monitoring of subsurface CO2
injection, AGU Fall Meeting, San Francisco, Dec. 5–9, 2011.
35. Druhan, J.L., N.T. Vandehey, R. Buchko, J.P. O’Neil, W.W. Moses, S. Finsterle, C. Steefel, and P.S. Nico, Observing the coupled behavior of geochemistry and flow path evolution during
bioreduction using clinical nuclear imaging tomography, AGU Fall Meeting, San Francisco, Dec. 5–9, 2011.
36. Finsterle, S., Joint inversion in Hydrogeology: Standing the test of practical application, MODFLOW and More 2011: Integrated Hydrologic Modeling, Colorado School of Mines, Golden Colorado, June
5–8, 2011.
37. Murakami, H., S. Finsterle, Q. Zhou, and J.T. Birkholzer, Uncertainty quantification and global sensitivity analysis of CO2 migration and pressure buildup for a hypothetical GCS project in the
Southern San Joaquin Basin in California, Tenth Annual Conference on Carbon Capture and Sequestration, Pittsburgh, PA, May 2–5, 2011.
38. Akhavan, M., P. Imhoff, S. Andres, and S. Finsterle, Minimizing contaminant transport to groundwater from rapid infiltration basins: Model evaluation of soil heterogeneity, Ground Water Summit,
Baltimore, Maryland, May 1–5, 2011.
39. Meza, J.C., S. Finsterle, M.D. Freshley, I. Gorton, S.S. Hubbard, M., J.D. Moulton, and C.I. Steefel, Advances Simulation Capability for Environmental Management (ASCEM): Early Site
Demonstration, Waste Management Conference 2011, Phoenix, Arizona, Feb. 27–March 3, 2011.
40. Akhavan, M., P.T. Imhoff, S. Andres, and S. Finsterle, Modeling hydrologic and geochemical aspects of rapid infiltration basins, AGU Fall meeting, San Francisco, Dec. 13–17, 2010.
41. Commer, M., M.B. Kowalsky, S. Finsterle, and G.A. Newman, Enhanced subsurface fluid characterization using joint hydrological and geophysical imaging, AGU Fall meeting, San Francisco, Dec. 13–17,
42. Kowalsky, M.B., E. Gasperikova, S. Finsterle, D. Watson, G. Baker, and S.S. Hubbard, Coupled modeling of hydrogeochemical and electrical resistivity data for exploring the impact of recharge on
subsurface contamination, AGU Fall meeting, San Francisco, Dec. 13–17, 2010.
43. Birkholzer, J.T., Q. Zhou, A. Cortis, and S. Finsterle, On the quantification of Regional Pressure Buildup from Large-Scale CO2 Storage Projects, Energy Procedia, 4, 4371–4378, 2011.
44. Akhavan, M., P. Imhoff, S. Andres, and S. Finsterle, Disposing treated wastewater in rapid infiltration basins: Simulations to optimize soil-aquifer treatment, paper presented at the 6th
International Conference on Sustainable Water Environment, U. of Delaware, Newark, Delaware, July 29–31, 2010.
45. Zhang, Y., B. Freifeld, S. Finsterle, M. Leahy, J. Ennis-King, L. Paterson, and T. dance, Estimating CO2 residual trapping from single-well test: Experimental design calculations, Ninth Annual
Conference on Carbon Capture and Sequestration, Pittsburgh, PA, May 10–13, 2010.
46. Jadoon, K.Z., M.B. Kowalsky, S. Finsterle, S.S. Hubbard, H. Vereecken, and S. Lambot, Coupled hydrogeophysical inversion of time-lapse off-ground GPR and hydrological data, EGU General Assembly,
Vienna, Austria, 2010.
47. Leahy, M., J. Ennis-King, L. Paterson, T. Dance, Y. Zhang, B. Freifeld, and S. Finsterle, Design of a single-well residual saturation test for the CO2CRC Otway Project, Sixth IMA Conference on
Modelling Permeable Rocks, University of Edinburgh, March 29–April 1, 2010.
48. Finsterle, S., and M.B. Kowalsky, What’s new with iTOUGH2?, Proceedings of TOUGH Symposium 2009, Lawrence Berkeley National Laboratory, Berkeley, Calif., September 14–16, 2009.
49. Jung, Y., P. Imhoff, and S. Finsterle, Estimation of landfill gas generation rates and gas permeability field of refuse using inverse modeling, Proceedings of TOUGH Symposium 2009, Lawrence
Berkeley National Laboratory, Berkeley, Calif., September 14–16, 2009.
50. Kowalsky, S. Finsterle, G. Moridis, and S.S. Hubbard, Hydrogeophysical approaches with the TOUGH family of codes, Proceedings of TOUGH Symposium 2009, Lawrence Berkeley National Laboratory,
Berkeley, Calif., September 14–16, 2009.
51. Zhou, Q., L. Pan, J. Hylen, B.G. Lundberg, R.K. Plunkett, S.H. Pordes, and S. Finsterle, Modeling of Multiphase diffusive processes of tritium in an underground accelerator facility, Proceedings
of TOUGH Symposium 2009, Lawrence Berkeley National Laboratory, Berkeley, Calif., September 14–16, 2009.
52. Zhang, Y., L. Paterson, B. Freifeld, M. Leahy, J. Ennis-King, and S. Finsterle, Single-well experimental design for studying CO2 trapping mechanisms at Otway, Proceedings of TOUGH Symposium 2009,
Lawrence Berkeley National Laboratory, Berkeley, Calif., September 14–16, 2009.
53. Finsterle, S., Joint multiphase inverse modeling: Standing the test of practical application, Workshop on “Energy, Wind and Water: Algorithms for Simulation, Optimization and Control”, New
Zealand Institute for Mathematics & its Application (NZIMA), University of Auckland, New Zealand, February 9–12, 2009.
54. Árnason, K., and S. Finsterle, Joint inversion of resistivity and thermo-hydraulic data for geothermal reservoir modeling, I-GET Final Conference, Geology and Geophysics in Geothermal
Exploration, GFZ Potsdam, Germany, February 23/24, 2009.
55. Zhang, Y., C.M. Oldenburg, S. Finsterle, P. Jordan, and K. Zhang, Probability estimation of CO2 leakage through faults at geologic carbon sequestration sites, 9th International Conference on
Greenhouse Gas Technologies, Washington DC, November 16–20, Energy Procedia, 1, 41-46, 2009.
56. Finsterle, S., Multiphase inverse modeling in structured soils and fractured rocks, inivted paper presented at the 2008 SSSA-GSA Joint Meeting, Houston, TX, October 5–9, 2008.
57. Finsterle, S., Hydrogeological Modeling in Support of Site Characterization and Performance Assessment of the Yucca Mountain Nuclear Waste Site, Colloquium at Sacramento State University, October
30, 2008.
58. Jung, Y., P. Imhoff, S. Finsterle, R. Yazdani, and D. Augenstein, Estimation of gas permeability field in landfills using inverse modeling, XVII International Conference on Computational Methods
in Water Resources, San Francisco, July 6–10, 2008.
59. Zhang, Y., M.B. Kowalsky, S. Finsterle, and S. Hubbard, Impact of groundwater pumping on near-river hydrology, XVII International Conference on Computational Methods in Water Resources, San
Francisco, July 6–10, 2008.
60. Kowalsky, M.B., S. Finsterle, D. Watson, and S. Hubbard, Use of coupled hydrological-geophysical modeling framework for exploring the impact of recharge on TDS, XVII International Conference on
Computational Methods in Water Resources, San Francisco, July 6–10, 2008.
61. Linde, N., O. Genoni, J. Doetsch, and S. Finsterle, Modeling and inversion of self-potential signals associated with flooding events at the Thur River, Switzerland, XVII International Conference
on Computational Methods in Water Resources, San Francisco, July 6–10, 2008.
62. Finsterle, S., Joint multiphase inverse modeling: Standing the test of practical application, Dept. of Civil and Environmental Engineering, Environmental Fluid Mechanics and Hydrology, Stanford
University, April 28, 2008
63. Xu T., Finsterle S., Senger R., 2007. Water-Steel Canister Interaction and H[2] Gas Pressure Buildup in a Nuclear Waste Repository, in Water-Rock Interaction WRI-12 (D. Bullen & Y. Wang, eds.),
p. 673–676, Taylor & Francis Publishers, London, 2007.
64. Finsterle, S., M.B. Kowalsky, C. Oberdörster, and A. Lehikoinen, Joint inversion of hydrogeological and geophysical data: Too much information?, AGU Fall Meeting, San Francisco, Dec. 10–14, 2007.
65. James, A., J.J. McDonnell, S. Finsterle, H.J. Tromp van Meerveld, Physics-based hydrologic-response simulation at the hillslope scale: An experimentalist’s view and a cautionary tale, AGU Fall
Meeting, San Francisco, Dec. 10–14, 2007.
66. Senger, R., T. Xu, P. Marschall, and S. Finsterle, Modeling approaches of two-phase flow phenomena associated with corrosion of SF/HLW canisters in a proposed repository in Opalinus clay,
Switzerland, Clays in Natural & Engineered Barriers for Radioactive Waste Confinement, 3rd International Meeting, Lille, France, Sept. 17–20, 2007.
67. Lehikoinen, A., S. Finsterle, A. Voutilainen, M.B. Kowalsky, and J. P. Kaipio, Dynamical Geophysical ERT Using State Estimation Approach, ModelCare2007, Copenhagen, Denmark, September 9–13, 2007.
68. Lehikoinen, A., J.M.J. Huttunen, S. Finsterle, M.B. Kowalsky, and J. Kaipio, An extension of a nonstationary inversion method with approximation error analysis applied to hydrological process
monitoring, AGU Joint Assembly, Acapulco, Mexico, May 22–25, 2007.
69. Freifeld, B., C. Doughty, J. Walker, L. Kryder, K. Gilmore, S. Finsterle, and J. Sampson, Characterization of rapid, localized groundwater transport in the Crater Flat Tuffs, Yucca Mountain,
Nevada, Devils Hole Workshop, Death Valley, California, May 2–3, 2007.
70. Lehikoinen, A., A. Voutilainen, J. P. Kaipio, S. Finsterle, and M.B. Kowalsky, A nonstationary inversion approach for imaging fluid flow in unsaturated porous media, Inver Problems, Design and
Optimization Symposium, Miami, Florida, April 16–18, 2007.
71. Finsterle, S., Forward and inverse modeling of non-isothermal multiphase flow in fractured porous media, presented at Summer Workshop of the New Zealand Institute of Mathematics and Applications
“Partial Differential Equations: Analysis, Applications, and Inverse Problems”, Waitangi, New Zealand, January 8–13, 2007.
72. Finsterle, S., Joint hydrological-geophysical inversion for soil structure identification, presented at Summer Workshop of the New Zealand Institute of Mathematics and Applications “Partial
Differential Equations: Analysis, Applications, and Inverse Problems”, Waitangi, New Zealand, January 8–13, 2007.
73. Lehikoinen, A., S. Finsterle, A. Voutilainen, M.B. Kowalsky, and J. P. Kaipio, A nonstationary geophysical inversion approach for imaging and characterization of unsaturated porous media, paper
presented at the 5th World Congress on Industrial Process Tomography, Bergen, Norway, September 3–6, 2006.
74. Kowalsky, M.B., E. Majer, J.E. Peterson, S. Finsterle, and A. Mazzella, Coupled geophysical-hydrological modeling of controlled NAPL spill, AGU Fall Meeting, San Francisco, Dec. 11–15, 2006.
75. Andrews, J.L., M. O. Saar, and S. Finsterle, Heat and helium transport: The effect of subsurface processes on near-surface signals, AGU Fall Meeting, San Francisco, Dec. 11–15, 2006.
76. Lehikoinen, A., S. Finsterle, A. Voutilainen, M.B. Kowalsky, and J. P. Kaipio, A nonstationary geophysical inversion approach with modeling-error correction for imaging fluid flow, AGU Fall
Meeting, San Francisco, Dec. 11–15, 2006.
77. Freifeld, B., J. Walker, C. Doughty, L. Kryder, K. Gilmore, S. Finsterle, and J. Sampson, Evidence of rapid localized groundwater transport in volcanic tuffs beneath Yucca Mountain, Nevada, AGU
Fall Meeting, San Francisco, Dec. 11–15, 2006.
78. Lehikoinen, A., S. Finsterle, A. Voutilainen, M.B. Kowalsky, and J. P. Kaipio, A Bayesian nonstationary inversion approach for imaging fluid flow, 28th New Zealand Geothermal Workshop, University
of Aukland, Nov. 15–17, 2006.
79. Zhang, Y., H. Liu, S. Finsterle, and Q. Zhou, A simplified representation of matrix diffusion effects incorporating diffusive property heterogeneity using particle tracking. Western Pacific
Geophysics Meeting (WPGM), Beijing, China. 2006.
80. Zhang, Y., C. M. Oldenburg, S. Finsterle, and G. S. Bodvarsson, A study on the economic feasibility of carbon sequestration with enhanced gas recovery (CSEGR) at Rio Vista, California. GoldSim
Conference, Vancouver, BC, Canada. 2006.
81. Finsterle, S., and M. B. Kowalsky, Joint hydrological-geophysical inversion for soil structure identification, Proceedings of TOUGH Symposium 2006, Lawrence Berkeley National Laboratory,
Berkeley, Calif., May 15–17, 2006.
82. Zhang, Y., C. M. Oldenburg, S. Finsterle, and G. S. Bodvarsson, System-level modeling for geological storage of CO[2], Proceedings of TOUGH Symposium 2006, Lawrence Berkeley National Laboratory,
Berkeley, Calif., May 15–17, 2006.
83. Moridis, G. J., M. B. Kowalsky, S. Finsterle, and K. Pruess, TOUGH+: The new generation of object-oriented family of codes for the solution of problems of flow and transport in the subsurface,
Proceedings of TOUGH Symposium 2006, Lawrence Berkeley National Laboratory, Berkeley, Calif., May 15–17, 2006.
84. Kiryukhin, A. V., N. P. Asaulova, S. Finsterle, T. V. Rychkova, and N. Obora, Modeling the Pauzhetsky Geothermal Field, Kamchatka, Russia, Using iTOUGH2, Proceedings of TOUGH Symposium 2006,
Lawrence Berkeley National Laboratory, Berkeley, Calif., May 15–17, 2006.
85. Kowalsky, M. B., J. T. Birkholzer, S. Finsterle, J. Peterson, S. Mukhopadhyay, and Y. Tsang, Joint Inversion of Ground-Penetrating Radar and Thermal-Hydrological Data Collected During a
Large-Scale Heater Test, Proceedings of TOUGH Symposium 2006, Lawrence Berkeley National Laboratory, Berkeley, Calif., May 15–17, 2006.
86. Kowalsky, M. B., and S. Finsterle, Joint inversion of ground-penetrating radar and thermal-hydrological measurements: Overview and recent progress, EGU General Assembly, Vienna, Austria, April
2–7, 2006.
87. Finsterle, S., and M.B. Kowalsky, Model structure identification through joint inversion of hydrological and geophysical data, AGU Fall Meeting, San Francisco, Dec. 5–9, 2005.
88. Kowalsky, M.B., J. Peterson, J. Birkholzer, S. Finsterle, S. Mukhopadhyay, and Y. Tsang, Joint inversion of ground-penetrating radar and thermal-hydrological data collected during a large-scale
heater test, AGU Fall Meeting, San Francisco, Dec. 5–9, 2005.
89. Zhang, Y., H. H. Liu, S. Finsterle, and Q. Zhou, How dual-scale diffusive property heterogeneity affects the effective matrix diffusion coefficient in fractured rock, AGU Fall Meeting, San
Francisco, Dec. 5–9, 2005.
90. Didita, L., S. Finsterle, P. Ilie, and A. Danchiv, Safety assessment methodology for Romanian LILW Repository, ICEM’05, The 10^th International Conference on Environmental Remediation and
Radioactive Waste Management, Glasgow, Scotland, Sept. 4–8, 2005.
91. Linde, N., J. Chen, S. Hubbard, and S. Finsterle, Inferring the relation between radar velocity and permeability: a comparison between tracer test and flowmeter data, Paper presented at the 2^nd
General Assembly of the European Geosciences Union, Vienna, Austria, April 25–29, 2005.
92. Welker, A.L., P.M. Gallagher, S. Finsterle, and C. Conlee, Design of an experiment to assess the effectiveness of passive site stabilization, Geo-Frontiers, Austin, Texas, January 24–26, 2005.
93. Kowalsky, M.B., S. Finsterle, J. Peterson, S. Hubbard, Y. Rubin, E. Majer, A. Ward, and G. Gee, Estimating field-scale soil hydraulic properties and petrophysical models through joint GPR/
hydrological measurement inversion, AGU Fall Meeting, San Francisco, 2004.
94. Kowalsky, M.B., S. Finsterle, J. Peterson, S. Hubbard, Y. Rubin, E. Majer, A. Ward, and G. Gee, Estimating field-scale soil hydraulic properties through joint inversion of cross-hole GPR travel
times and hydrological measurements, Abstract No. 80539, GSA Annual Meeting, Denver, November 7–10, 2004.
95. Didita, L., P. Ilie, A. Danchiv, S. Finsterle, Safety assessment for Romanian LILW repository, International Symposium on Disposal of Low Activity Radioactive Waste, Cordoba, Spain, December
13–17, 2004.
96. Linde, N., J. Chen, M. Kowalsky, S. Finsterle, Y. Rubin, and S. Hubbard, Improved characterization through joint hydrogeophysical inversion: Examples of three different approaches, paper
presented at NATO Advanced Research Workshop, Soil and Groundwater Contamination: Improved Risk Assessment, St. Petersburg, Russia, July 26–30, 2004.
97. Linde, N., S. Finsterle, and S. S. Hubbard, Inversion of hydrological tracer test data using tomographic constraints, CGU, AGU, SEG, and EEGS Joint Assembly, Montreal, Canada, May 17–21, 2004.
98. Finsterle, S., The role of model conceptualization and parameterization in multiphase inverse modeling, Paper presented at the 1^st General Assembly of the European Geosciences Union, Nice,
France, April 25–30, 2004.
99. Finsterle, S., and T. A. Ghezzehei, Effect of heterogeneity structure on estimation uncertainty for a calibrated seepage model, AGU 2003 Fall Meeting, San Francisco, Calif., December 8–12, 2003.
100. Ghezzehei, T. A., and S. Finsterle, Film flow along tunnel wall, AGU 2003 Fall Meeting, San Francisco, Calif., December 8–12, 2003.
101. Kowalsky, M. B., Y. Rubin, J. Peterson, and S. Finsterle, Estimation of flow parameters using crosshole GPR travel times and hydrological data collected during transient flow experiments, AGU
2003 Fall Meeting, San Francisco, Calif., December 8–12, 2003.
102. Unger, A., S. Finsterle, and G. S. Bodvarsson, Transport of radon gas into a tunnel at Yucca Mountain—Estimating large-scale fractured tuff hydraulic properties and implications for the
operation of the ventilation system, Paper No. 250-3, GSA Annual Meeting, Seattle, November 2–5, 2003.
103. Finsterle, S., T. A. Ghezzehei, R. C. Trautz, C. F. Ahlers, and P. J. Cook, Evaporation from a seepage face, in: ESD Research Review 2003, Lawrence Berkeley National Laboratory, Berkeley,
Calif., 2003.
104. Ahlers, C. F., T. Ghezzehei, and S. Finsterle, Development and testing of a method for efficient simulation of evaporation from a seepage face, Proceedings, TOUGH Symposium 2003, Berkeley,
Calif., May 12-14, 2003.
105. Bodvarsson, G.S., J. Birkholzer, S. Finsterle, H.-H. Liu, J. Rutqvist, and Y.-S. Wu, The use of TOUGH2/iTOUGH2 in support of the Yucca Mountain Project: Successes and limitations, Proceedings,
TOUGH Symposium 2003, Berkeley, Calif., May 12–14, 2003.
106. Finsterle, S., iTOUGH2: From parameter estimation to model structure identification, Proceedings, TOUGH Symposium 2003, Berkeley, Calif., May 12–14, 2003.
107. Ghezzehei, T.A., S. Finsterle, and R. C. Trautz, Evaluating the effectiveness of liquid diversion around an underground opening when evaporation is non-negligible, Proceedings, TOUGH Symposium
2003, Berkeley, Calif., May 12–14, 2003.
108. Kim, J., and S. Finsterle, Application of Automatic Differentiation in TOUGH2, Proceedings, TOUGH Symposium 2003, Berkeley, Calif., May 12–14, 2003.
109. Kowalsky, M., S. Finsterle, and Y. Rubin, Estimating flow parameters using Ground-Penetrating Radar and hydrological data during transient flow in the vadose zone, Proceedings, TOUGH Symposium
2003, Berkeley, Calif., May 12–14, 2003.
110. Unger, A., S. Finsterle, and G.S. Bodvarsson, Estimating large-scale fractured rock properties from radon data collected in a ventilated tunnel, Proceedings, TOUGH Symposium 2003, Berkeley,
Calif., May 12–14, 2003.
111. Finsterle, S., Testing and modeling of the high-level nuclear waste disposal site at Yucca Mountain, Nevada, Invited lecture, Institute of Hydrodynamic and Water Resources, Swiss Federal
Institute of Technology (ETH), Zürich, Switzerland, September 14, 2002.
112. Finsterle, S., Forward and inverse modeling in Earth Sciences, Invited Speaker, Workshop: Modelling of Partially Saturated Soils, DFG Research Group Mechanics of Partially Saturated Soils,
University of Stuttgart, Stuttgart, Germany, September 18, 2002.
113. Kowalsky, M., Y. Rubin, and S. Finsterle, Inversion of hydrogeological and time-lapsed GPR data for flow parameters, AGU 2002 Fall Meeting, San Francisco, Calif., December 6–10, 2002.
114. Seol, Y., T. Kneafsey, K. Ito, and S. Finsterle, Two-D simulations of flow and transport on a meter-sized unsaturated fractured tuff block, AGU 2002 Fall Meeting, San Francisco, Calif., December
6–10, 2002.
115. Houseworth, J. A., S. Finsterle, and G. S. Bodvarsson, Dual-continuum modeling of flow and transport in the drift shadow, GSA 2002 Annual Meeting, Denver, Colorado, October 27–30, 2002.
116. Liu, H. H., G. S. Bodvarsson, and Finsterle, Unsaturated flow in two-dimensional fracture networks, GSA 2002 Annual Meeting, Denver, Colorado, October 27–30, 2002.
117. Ghezzehei, T., S. Finsterle, C. F. Ahlers, R. C. Trautz, and P. J. Cook, Inverse modeling of seepage into underground opening, GSA 2002 Annual Meeting, Denver, Colorado, October 27–30, 2002.
118. Finsterle, S., R. C. Trautz, C. F. Ahlers, P. J. Cook, Modeling mesoscale experiments at Yucca Mountain, Nevada, Subsurface Science Symposium, Boise, Idaho, Oct. 13–16, 2002.
119. Finsterle, S., Inverse modeling in multiphase flow through porous media, invited paper presented at the SIAM 50^th Anniversary and 2002 Annual Meeting, Philadelphia, PA, July 8–12, 2002.
120. Engelhardt, I., S. Finsterle, and C. Hofstee, Thermoconductivity experiments with backfill material and estimation of effective parameters by inverse modeling, Paper presented at: Specialist
Workshop on Clay Microstructure and its Importance to Soil Behaviour, Lund, Sweden, October 15–17, 2002.
121. Ahlers, C. F., R. C. Trautz, P. J. Cook, and S. Finsterle, Testing and Modeling of Seepage into Underground Openings in a Heterogeneous Fracture System at Yucca Mountain, Nevada, paper presented
at the IAHR International Groundwater Symposium ”Bridging the Gap between Measurements and Modeling in Heterogeneous Media” Berkeley, California, March 25–29, 2002.
122. Kitterød, N.-O., and S. Finsterle, Simulating unsaturated flow fields based on Ground Penetrating Radar and saturation measurements, paper presented at the IAHR International Groundwater
Symposium ”Bridging the Gap between Measurements and Modeling in Heterogeneous Media” Berkeley, California, March 25–29, 2002.
123. Mays, D. C., B. Faybishenko, and S. Finsterle, Information Entropy to measure temporal and spatial complexity of unsaturated flow in heterogeneous media, paper presented at the IAHR
International Groundwater Symposium ”Bridging the Gap between Measurements and Modeling in Heterogeneous Media” Berkeley, California, March 25–29, 2002.
124. Kim, J.G., and S. Finsterle, Application of Automatic Differentiation for the simulation of nonisothermal, multiphase flow in geothermal reservoirs, Proceedings, Twenty-Seventh Stanford
Geothermal Workshop, Stanford University, Stanford, California, January 28–30, 2002.
125. Finsterle, S., Impact of parameter uncertainty, variability, and conceptual model errors on predictions of flow through fractured porous media, AGU 2000 Fall Meeting, San Francisco, Calif.,
December 13–17, 2000.
126. Finsterle, S., Multiphase inverse modeling—Possibilities and limitations, in: Groundwater Updates, Proceedings of the IAHR International Symposium 2000 on Groundwater, Sonic City, Omiya, Japan,
May 8–10, 2000, p. 479, Springer-Verlag, Tokyo, 2000.
127. Finsterle, S., and R. C. Trautz, Numerical modeling of seepage into underground openings, Proceedings (CD-ROM): 2000 SME Annual Meeting and Exhibit, Salt Lake City, Utah, February 28 – March 1,
128. Finsterle, S., and B. Faybishenko, What does a tensiometer measure in fractured rock?, In: M. Th. van Genuchten, F. J. Leij, and L. Wu (eds.), Characterization and Measurement of the Hydraulic
Properties of Unsaturated Porous Media, Part 2, pp. 867–875, University of California, Riverside, Calif., 1999.
129. Finsterle, S., and R. C. Trautz, Drift seepage in unsaturated fractured rock, AGU 1999 Fall Meeting, San Francisco, Calif., December 13–17, 1999.
130. Kitterød, N.-O., and S. Finsterle, Inverse modeling of unsaturated flow parameters combined with stochastic simulation by Empirical Orthogonal Functions, Groundwater 2000, June 6–8, 2000,
Copenhagen, Denmark.
131. Faybishenko, B., and S. Finsterle, On the physics of tensiometry in heterogeneous soils and rocks, AGU Spring Meeting ‘99.
132. Webb, S. W., K. Pruess, J. M. Phelan, and S. Finsterle, Development of a mechanistic model for the movement of chemical signatures from buried landmines/UXO, Proceedings of the SPIE – The
International Society of Optical Engineering, 3710, 270–282, 1999.
133. Finsterle, S., K. Pruess, G. Björnsson, and A. Battistelli, Characterization of fractured geothermal reservoirs using inverse modeling, Proceedings, Dynamics of Fluids in Fractured Rocks, pp.
152–154, Lawrence Berkeley National Laboratory, Berkeley, Calif., February 10–12, 1999.
134. Finsterle, S., and K. Pruess, Automatic calibration of geothermal reservoir models through parallel computing on a workstation cluster, Proceedings, Twenty-Fourth Workshop on Geothermal
Reservoir Engineering, Stanford University, Stanford, Calif., 123–130, January 25–27, 1999.
135. Finsterle, S., C. Satik, and M. Guerrero, Analysis of boiling experiment using inverse modeling, Proceedings, TOUGH Workshop '98, May 4–6, Lawrence Berkeley National Laboratory, Berkeley,
Calif., pp. 281–287, 1998.
136. Finsterle, S., T. O. Sonnenborg, and B. Faybishenko, Inverse modeling of a multistep outflow experiment for determining hysteretic hydraulic properties, Proceedings, TOUGH Workshop '98, May 4–6,
Lawrence Berkeley National Laboratory, Berkeley, Calif., pp. 250–256, 1998.
137. James, A. L., C. M. Oldenburg, and S. Finsterle, Analysis of uncertainty for 2-D fracture flow and seepage into an excavated drift, Proceedings, TOUGH Workshop '98, May 4–6, Lawrence Berkeley
National Laboratory, Berkeley, Calif., pp. 205–209, 1998.
138. Ahlers, C. F., S. Finsterle, and G. S. Bodvarsson, Characterization and prediction of subsurface pneumatic pressure variations at Yucca Mountain, Nevada, Proceedings, TOUGH Workshop '98, May
4–6, Lawrence Berkeley Laboratory, Berkeley, Calif., pp. 222–227, 1998.
139. Webb, S. W., S. Finsterle, K. Pruess, and J. M. Phelan, Prediction of the TNT signature from buried landmines, Proceedings, TOUGH Workshop '98, May 4–6, Lawrence Berkeley National Laboratory,
Berkeley, Calif., pp. 124–129, 1998.
140. Hodges, R. A., R. W. Falta, and S. Finsterle, Three-dimensional simulation of DNAPL transport at the Savannah River Site, Proceedings, TOUGH Workshop '98, May 4–6, Lawrence Berkeley National
Laboratory, Berkeley, Calif., pp. 136–141, 1998.
141. Finsterle, S., Multiphase inverse problem: An overview, Proceedings, Geothermal Program Review XVI, pp. 3-3–3-9, Berkeley, Calif., April 1–2, 1998.
142. Guerrero, M. T., C. Satik, S. Finsterle, and R. Horne, Inferring relative permeability from dynamic boiling experiments, Proceedings, Twenty-Third Workshop on Geothermal Reservoir Engineering,
Stanford University, Stanford, California, 389–396, January 26–28, 1998.
143. Finsterle, S., and B. Faybishenko, What does a tensiometer measure in fractured rock?, Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media, Riverside,
Calif., October 22–24, 1997.
144. Wu, Y. S., S. Finsterle, and G. S. Bodvarsson, Field-Scale Modeling Studies to Characterize Hydraulic Properties for Water, Air and Heat Flow in the Unsaturated Zone of Yucca Mountain, Book
Chapter, in Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media, edited by M. Th. Van Genuchten, Leij, F. J. and Wu, L., Published by the University of
California, Riverside, pp. 1537-1548, 1999.
145. Wu, Y. S., S. Finsterle, and G. S. Bodvarsson, A field-scale modeling study of characterizing hydraulic properties for Water, Air and Heat Flow in the Unsaturated Zone of Yucca Mountain, Nevada,
AGU Fall Meeting, December 8–12, San Francisco, Calif., 1997.
146. James, A. L., C. M. Oldenburg, and S. Finsterle, Analysis of uncertainty for 2-D fracture flow and seepage into an excavated niche, AGU Fall Meeting, December 8–12, San Francisco, Calif., 1997.
147. Finsterle, S., Inverse modeling for multiphase flow in porous media, Fourth SIAM conference on Mathematical and Computational Issues in the Geosciences, Albuquerque, June 16–18, 1997.
148. Ruskauff, G. J., P. S. Domski, R. Beauheim, and S. Finsterle, Numerical inversion of a two-phase hydraulic test, Fourth SIAM conference on Mathematical and Computational Issues in the
Geosciences, Albuquerque, June 16–18, 1997.
149. Finsterle, S., Problems with the multiphase inverse problem, paper presented at the Third Tracer Workshop, Austin, Texas, May 20–22, 1997.
150. Finsterle, S., and K. Pruess, Development of inverse modeling techniques for geothermal applications, Proceedings, Geothermal Program Review XV, pp. 2-47–2-54, San Francisco, Calif., March
24–26, 1997.
151. Pruess, K., C. M. Oldenburg, G. J. Moridis, and S. Finsterle, S., Water injection into vapor- and liquid-dominated reservoirs: Modeling of heat transfer and mass transport, Proceedings, DOE
Geothermal Program Review XV, San Francisco, Calif., March 25–26, 1997.
152. Finsterle, S, C. M. Oldenburg, A.L. James, K. Pruess, G.J. Moridis, Mathematical modeling of permeation grouting and subsurface barrier performance, Proceedings, International Containment
Technology Conference and Exhibition, St. Petersburg, Florida, February 9–12, pp. 438–444, 1997.
153. Finsterle, S., K. Pruess, D. P. Bullivant, and M. J. O'Sullivan, Application of inverse modeling to geothermal reservoir simulation, Proceedings, Twenty-Second Workshop on Geothermal Reservoir
Engineering, Stanford University, Stanford, California, January 27–29, pp. 309–316, 1997.
154. Hodges, R. A., R. W. Falta, and S. Finsterle, Simulation of DNAPL emplacement and redistribution using geostatistically generated heterogeneous permeability distributions, presented at SA GSA,
155. Buesing, J. L., S. Finsterle, and M. Pantazidou, The role of capillary pressure curve selection in modeling LNAPL transport in the vadose zone, Non-Aqueous Phase Liquids (NAPLs) in the
Subsurface, ASCE Specialty Conference, Nov. 10–14, 1996.
156. Roeder, E., C. L. Wright, S. Finsterle, S. E. Brame, R. W. Falta, and C. M. Cindy, Inversion of a 3-dimensional tracer test, presented at the GSA national meeting, Denver, CO, October 28–31,
157. Moridis, G., L. Myer, P. Persoff, S. Finsterle, J. Apps, D. Vasco, S. Muller, P. Yen, and K. Pruess, A field test of permeation grouting in heterogeneous soils using a new generation of barrier
liquids, presented at ER '95 Committed to Results, Colorado Convention Center, Denver, CO, August, 1995.
158. Ahlers, C.F., S. Finsterle, Y.S. Wu, and G.S. Bodvarsson, Determination of pneumatic permeability of a multi-layered system by inversion of pneumatic pressure data, presented at the AGU Fall
Meeting, December 11–15, San Francisco, Calif., 1995.
159. Finsterle, S., and K. Pruess, Automatic history matching of geothermal field performance, Proceedings, 17th New Zealand Geothermal Workshop, November 8–10, pp. 193–198, Auckland, New Zealand,
160. Finsterle, S., Direct and inverse modeling of multiphase flow systems, Invited paper, Modeling and Computation in Environmental Sciences, Proceedings of the First GAMM-Seminar at ICA Stuttgart,
October 12–13, 1995, pp. 146–157, 1995.
161. Finsterle, S., C.F. Tsang, S.M. Benson, and G.J. Moridis, Use of blocking agent to improve the effectiveness of soil vapor extraction, presented at the ACE, I&EC Symposium Emerging Technologies
in Hazardous Waste Management VII, September 17–20, Atlanta, Georgia, 1995.
162. Faybishenko, B., and S. Finsterle, Design and analysis of single-core experiments for the determination of unsaturated hydraulic properties, presented at the Vadose Zone Hydrology Conference,
September 6–8, Davis, Calif., 1995.
163. Persoff, P. S. Finsterle, G. J. Moridis, J. Apps, K. Pruess, and S.J. Muller, Injectable barriers for waste isolation, presented at the ASME/AIChE National Heat Transfer Conference, August 5–9,
Portland, Oregon, 1995.
164. Finsterle, S., G. S. Bodvarsson, and G. Chen, Inverse modeling as a step in the calibration of the LBL-USGS site-scale model of Yucca Mountain, presented at the International High Level
Radioactive Waste Management Conference, May 1–5, Las Vegas, Nevada, pp. 154–156, 1995.
165. Cox, B. L., S. Finsterle, and J. S. Y. Wang, Experimental and numerical aqueous flow through a partially saturated fracture, presented at the International High Level Radioactive Waste
Management Conference, May 1–5, Las Vegas, Nevada, 1995.
166. Finsterle, S., and K. Pruess, ITOUGH2: Solving TOUGH inverse problems, Proceedings, TOUGH Workshop '95, March 20–22, pp. 287–292, Report LBL-37200, Lawrence Berkeley Laboratory, Berkeley,
Calif., 1995.
167. Finsterle, S., and K. Pruess, Using simulation-optimization techniques to improve multiphase aquifer remediation, Proceedings, TOUGH Workshop '95, March 20–22, pp. 181–186, Report LBL-37200,
Lawrence Berkeley Laboratory, Berkeley, Calif., 1995.
168. Witherspoon, P. A., P. Fuller, and S. Finsterle, Three-dimensional multiphase effects in aquifer gas storage, Proceedings, TOUGH Workshop '95, March 20–22, pp. 137–156, Report LBL-37200,
Lawrence Berkeley Laboratory, Berkeley, Calif., 1995.
169. Persoff, P. S. Finsterle, G. J. Moridis, J. Apps, K. Pruess, and S.J. Muller, Designing injectable Colloidal Silica barriers for waste isolation at the Hanford site, Proceedings, Thirty-Third
Hanford Symposium on Health and the Environment—In-Situ Remediation: Scientific Basis for Current and Future Technologies, Nov. 7–11, 1994, G. Gee and N. R. Wing (eds.), Battelle Press, Richland,
WA, 87–102, 1994.
170. Finsterle, S., G. J. Moridis, K. Pruess, and P. Persoff, Development of an in-situ containment technology, AGU Fall Meeting, December 5–9, San Francisco, Calif., 1994.
171. Senger, R. K., S. Finsterle, P. Vinard, Gas and water transport in a fractured-porous medium: Field-scale characterization of two-phase flow parameter and pressure buildup associated with gas
release from a L/ILW repository, presented at the Chapman Conference on Aqueous Phase and Multiphase Transport in Fractured Rock, September 12–15, Burlington, Vermont, 1994.
172. Vinard, P., J.-M. Lavanchy, S. Finsterle, J. Ph. McCord, Hydrodynamics of a fractured, gas-bearing marl aquitard characterized by large-scale sub-normal pressure conditions, AAPG Hedberg
Research Conference, Denver, Colorado, June 8–10, 1994.
173. Finsterle, S., and K. Pruess, Estimating two-phase hydraulic properties by inverse modeling, Proceedings, Fifth Annual International High Level Radioactive Waste Management Conference, Las
Vegas, Nevada, April 1994, pp. 2160–2167, American Nuclear Society, La Grange Park, Ill., 1994.
174. Pruess, K., S. Finsterle, P. Persoff, and C. Oldenburg, Phenomenological studies of two-phase flow processes for nuclear waste isolation, Proceedings, Fifth Annual International High Level
Radioactive Waste Management Conference, Las Vegas, Nevada, April 1994, pp. 2639–2647, American Nuclear Society, La Grange Park, Ill., 1994.
175. Finsterle, S., G. J. Moridis, K. Pruess, and P. Persoff, Physical barriers formed from gelling liquids: Numerical design of laboratory and field experiments, AGU Spring Meeting, May 23–27,
Baltimore, Maryland, 1994.
176. Persoff, P., S. Finsterle, G. J. Moridis, and K. Pruess, Physical barriers formed from gelling liquids: Experimental investigations, AGU Spring Meeting, May 23–27, Baltimore, Maryland, 1994.
177. Finsterle, S., Determining two-phase hydraulic properties by inverse modeling, Proceedings, IX. International Conference on Computational Methods in Water Resources, June 9–12, Denver, Colorado,
178. Finsterle, S., S. Mishra, and J.-M. Lavanchy, Hydrogeologic site characterization under two-phase flow conditions, Proceedings, NEA Workshop on Gas Generation and Release From Radioactive Waste
Repositories, September 23–26, Aix-en-Provence, France, 1991.
1. Kowalsky, M.B., J. Doetsch, M. Commer, S. Finsterle, C. Doughty, Q. Zhou, J. Ajo-Franklin, J. Birkholzer, T. Daley. Coupled Inversion of Hydrological and Geophysical Data for Improved Prediction
of Subsurface CO2 Migration; NRAP-TRS-III-XXX-2013; NRAP Technical Report Series; U.S. Department of Energy, National Energy Technology Laboratory: Morgantown, WV, 2013.
2. Wainwright, H., S. Finsterle, Q. Zhou, Q., and J. Birkholzer, Modeling the Performance of Large-Scale CO2 Storage Systems: A Comparison of Different Sensitivity Analysis Methods, Report
NRAP-TRS-III-002-2012, NRAP Technical Report Series, pp. 24, U.S. Department Energy, National Energy Technology Laboratory, Morgantown, WV, 2012.
3. Finsterle, S., D. Hawkes, G. Moridis, S. Mukhopadhyay, C. Oldenburg, L. Pfeiffer, J. Rutqvist, E. Sonnenthal, N. Spycher, C. Valladao, and L. Zheng (Eds.), Proceedings of the TOUGH Symposium
2012, Report LBNL-5808E, Lawrence Berkeley National Laboratory, Berkeley, Calif., 2012.Freifeld, B., and S. Finsterle, Imaging Fluid Flow in Geothermal Wells Using Distributed Thermal
Perturbation Sensing, Report LBNL-4588E, Lawrence Berkeley National Laboratory, Berkeley, Calif., December 2010.
4. Finsterle, S., iTOUGH2 Universal Optimization Using the PEST Protocol, Report LBNL-LB3698E, Lawrence Berkeley National Laboratory, Berkeley, Calif., July 2010.
5. Finsterle, S., iTOUGH2-IFC: An Integrated Flow Code in Support of Nagra's Probabilistic Safety Assessment—User's Guide and Model Description, Report LBNL-1441E, Lawrence Berkeley National
Laboratory, Berkeley, Calif., January 2009.
6. Mukhopadhyay, S., N. Spycher, G.X. Zhang, E.L. Sonnenthal, and S. Finsterle, THC sensitivity study of heterogeneous permeability and capillarity effects, ANL-NBS-HS-000047 REV 01, Sandia National
Laboratories, OCRWM Lead Laboratory for Repository Systems, Las Vegas, Nevada, 2007.
7. Finsterle, S., and M.B. Kowalsky, iTOUGH2-GSLIB User’s Guide, Report LBNL/PUB-3191 , Lawrence Berkeley National Laboratory, Berkeley, Calif., June 2007.
8. Andrews, J.L., Finsterle, S., and M.O. Saar, Mass- and Temperature Dependent Diffusion Coefficients of Light Noble Gases for the TOUGH2- Module, Report LBNL-62595, Lawrence Berkeley National
Laboratory, Berkeley, Calif., April 2007.
9. Finsterle, S., M. Conrad, B.M. Kennedy, K. Pruess, T. Kneafsey, R. Salve, G. Su., and Q. Zhang, Mobility of Tritium in Engineered and Earth Materials at the NuMI Facility, Fermilab, Report
LBNL-61798, Lawrence Berkeley National Laboratory, Berkeley, Calif., March 2007.
10. Freifeld, B, Doughty, C., and S. Finsterle, Preliminary Estimates of Specific Recharge and Transport Velocities near Borehole NC-EWDP-24PB, Report LBNL-60740, Lawrence Berkeley National
Laboratory, Berkeley, Calif., June 2006.
11. Finsterle, S., and Y. Seol, Preliminary Evaluation of Drift Seepage Model Using Information form the ESF South Ramp at Yucca Mountain, Nevada, Report LBNL-58698, Lawrence Berkeley National
Laboratory, Berkeley, Calif., August 2005.
12. Finsterle, S., Seepage Calibration Model and Seepage Testing Data, MDL-NBS-HS-00004, REV 03, LBID-2470, 2004.
13. Finsterle, S., and G.S. Bodvarsson, Technical Basis Document No.3: Seepage into Drifts, LBID-2523, 2003.
14. Pavelescu, M., L. Didita, S. Finsterle, and G. S. Bodvarsson, Scoping Calculations of Radionuclide Release and Transport Through the Unsaturated Zone at the Saligny Site, Final Report, NATO
Linkage Grant CLG.976709/2000, Pitesti, Romania, and Berkeley, Calif., 2002.
15. Finsterle, S., Demonstration of Optimization Techniques for Groundwater Plume Remediation, Report LBNL-46746, Lawrence Berkeley National Laboratory, Berkeley, Calif., 2000.
16. Faybishenko, B., and S. Finsterle, On Tensiometry in Fractured Rocks, Report LBNL-43864, Lawrence Berkeley National Laboratory, Berkeley, Calif., 1999.
17. Finsterle, S., iTOUGH2 User's Guide, Report LBNL-40040, Lawrence Berkeley National Laboratory, Berkeley, Calif., 1999a.
18. Finsterle, S., iTOUGH2 Command Reference, Report LBNL-40041, Lawrence Berkeley National Laboratory, Berkeley, Calif., 1999b.
19. Finsterle, S., iTOUGH2 Sample Problems, Report LBNL-40042, Lawrence Berkeley National Laboratory, Berkeley, Calif., 1999c.
20. Finsterle, S., G. Björnsson, and K. Pruess, Characterization of fractured geothermal reservoirs using inverse modeling, Earth Sciences Division Annual Report 1998–1999, pp. 99–100, Report
LBNL-43816, Lawrence Berkeley National Laboratory, Berkeley, Calif., 1999.
21. Finsterle, S., Parallelization of iTOUGH2 Using PVM, Report LBNL-42261, Lawrence Berkeley National Laboratory, Berkeley, Calif., October 1998.
22. Finsterle, S., ITOUGH2 V3.2, Verification and Validation Report, Report LBNL-42002, Lawrence Berkeley National Laboratory, Berkeley, Calif., June 1998.
23. Finsterle, S., A. L. James, and J. S. Y. Wang, Numerical modeling water migration in fracture-matrix system, Earth Sciences Division Annual Report 1997, pp. 61–62, Report LBNL-42452, Lawrence
Berkeley National Laboratory, Berkeley, Calif., 1997.
24. Pruess, K., S. Finsterle, G. Moridis, C. Oldenburg and Y.-S. Wu, General-purpose reservoir simulators: the TOUGH2 family, GRC Bulletin, 53–57, February 1997.
25. Finsterle, S., and K. Pruess, Design and Analysis of a Well Test for Determining Two-Phase Hydraulic Properties, Report LBNL-39620, Lawrence Berkeley National Laboratory, Berkeley, Calif.,
December 1996.
26. Finsterle, S., K. Pruess, and P. Fraser, ITOUGH2 Software Qualification, Lawrence Berkeley National Laboratory, Report LBNL-39489, Berkeley, Calif., October 1996.
27. Moridis, G., P. Yen, P. Persoff, S. Finsterle, P. Williams, L. Myer, and K. Pruess, A Design Study for a Medium-Scale Field Demonstration of the Viscous Barrier Technology, Lawrence Berkeley
National Laboratory, Report LBNL-38916, Berkeley, Calif., September 1996.
28. Pruess, K., S. Finsterle, G. Moridis, C. Oldenburg, E. Antunez, and Y. S. Wu, Advances in the TOUGH2 Family of General-Purpose Reservoir Simulators, Report LBL-38573, Lawrence Berkeley
Laboratory, Berkeley, Calif., April, 1996.
29. Eugster, S., and S. Finsterle, GTS-TPF: Design Calculations for an Extended Gas Threshold Pressure Test at Grimsel Test Site, Nagra Internal Report, Nagra, Wettingen, Switzerland, September,
30. Falta, R. W., K. Pruess, S. Finsterle, and A. Battistelli, T2VOC User's Guide, Report LBL-36400, Lawrence Berkeley Laboratory, Berkeley, Calif., March, 1995.
31. Finsterle, S., Design of a Welltest for Determining Two-Phase Hydraulic Properties, Report LBL-37448, Lawrence Berkeley Laboratory, Berkeley, Calif., January 1995.
32. Finsterle, S., G. J. Moridis, and K. Pruess, A TOUGH2 Equation-of-State Module for the Simulation of Two-Phase Flow of Air, Water, and a Miscible Gelling Liquid, Report LBL-36086, Lawrence
Berkeley Laboratory, Berkeley, Calif., May 1994.
33. Finsterle, S., Inverse Modeling of Test SB4-VM2/216.7 at Wellenberg, Report LBL-35454, Lawrence Berkeley Laboratory, Berkeley, Calif., March 1994.
34. Finsterle, S., ITOUGH2 User's Guide, Version 2.2, Report LBL-34581, Lawrence Berkeley Laboratory, Berkeley, Calif., (Accn: MOL.19941026.0075), August 1993.
35. Finsterle, S., and K. Pruess, FLG: Design Calculations for a Combined Ventilation and Brine Injection Experiment at the Grimsel Rock Laboratory, Report LBL-34460, Lawrence Berkeley Laboratory,
Berkeley, Calif., July 1993.
36. Finsterle, S., Inverse Modellierung zur Bestimmung hydrogeologischer Parameter eines Zweiphasensystems, Mitteilung Nr. 121 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie (VAW),
Eidgenössische Technische Hochschule (ETH), Zürich (also Nagra Technical Report NTB 93-20, Nagra, Wettingen, Switzerland), February, 1993.
37. Finsterle, S., Wellenberg: One-Dimensional Stress Release Model Assuming Two-Phase Flow Conditions, Nagra Internal Report, Wettingen, Switzerland, September, 1992.
38. Finsterle, S., ITOUGH User's Guide, Version 1.0, Nagra Internal Report, Nagra, Wettingen, Switzerland, August, 1992.
39. Finsterle, S., S. Mishra, P. Vinard, and S. Vomvoris, Wellenberg: Regional Gas Depletion Model—One-Dimensional Scoping Calculations, Nagra Internal Report, Wettingen, Switzerland, May, 1992.
40. Finsterle, S., and S. Vomvoris, Inflow to Stripa Validation Drift Under Two-Phase Conditions: Scoping Calculations, Nagra Internal Report, Wettingen, Switzerland, May, 1991.
41. Finsterle, S., S. Mishra, and J.-M. Lavanchy, Interpretation of Selected OBS-1 Tests Using the Two-Phase Numerical Simulator TOUGH, Nagra Internal Report, Wettingen, Switzerland, April, 1991.
42. Finsterle, S., E. Schlueter, and K. Pruess, Exploratory Simulations of Multiphase Effects in Gas Injection and Ventilation Tests in an Underground Rock Laboratory, Report LBL-28810, Lawrence
Berkeley Laboratory (also Nagra-DOE-Co-Project Report NDC-13), Berkeley, Calif., June, 1990.
43. Finsterle, S., FLG: Design Calculations for a Gas Test at Grimsel Rock Laboratory, Nagra Internal Report, Nagra, Wettingen, Switzerland, July, 1989.
44. Finsterle, S., Auswertung der GPS Messkampagne Naters, Mitteilung Nr. 151 des Institutes für Geodäsie und Photogrammetrie, Eidgenössische Technische Hochschule, Zürich, Switzerland, 1987. | {"url":"http://esd.lbl.gov/about/staff/stefanfinsterle/publications.html","timestamp":"2014-04-18T23:16:37Z","content_type":null,"content_length":"97059","record_id":"<urn:uuid:356bdfb8-c7ad-489f-b4f0-53beeaeeaaac>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00611-ip-10-147-4-33.ec2.internal.warc.gz"} |
Extensibility of real analytic function of several variables to complex domain
up vote 1 down vote favorite
My question relates to the extensibility of a real analytic function of several variables to a specific complex domain.
In order to formulate the question, let me define the following complex domains:
$$\Delta_\theta := \{t = \rho e^{i\gamma} \in \mathbb{C} : \rho > 0 \text{ and } \gamma \in (-\theta,\theta)\}$$ (an open sector),
$$\Delta_\theta^{T} := \Delta_\theta \cap \{t\in \mathbb{C} : 0 < \text{Re}\ t < T\}$$ (an open triangle),
$$\Delta_\theta^{T',T} := \bigcup_{0\leq \xi \leq T-T'} (\xi + \Delta_\theta^{T'})$$ (an open pencil shape),
$$\mathcal{X}^{r} := \{z=x+iy \in \mathbb{C}^n : |y|_\infty < r\}$$ (a strip).
Suppose $f : [0,1]\times \mathbb{R}^n \mapsto \mathbb{R}$ and $f\in C¹(\overline{(0,T)\times\mathbb{R}^n})$ and $f$ is (jointly) real analytic on $(0,1)\times \mathbb{R}^n$.
Let $\Omega := \mathcal{X}^{r_0}\times \Delta^{T_0,T}_{\theta_0}$. Then I am interested in conditions on $f$ (e.g. on the coefficients of its power series expansion), considered as a real-valued
function on $[0,1]\times \mathbb{R}^n$, which allow it to be extended to $\Omega$, such that:
(i) $f$ is holomorphic on $\Omega$,
(ii) $f$ is bounded on $\Omega$,
(iii) $f\in C^1(\overline{\Omega})$,
for some (arbitrary) $r_0\in(0,\infty)$, $0<T_0\leq T < \infty$ and $\theta_0\in (0,\pi/2)$.
I know that every real analytic function can be extended into a neighborhood of the real axis in the complex domain. However, I am unclear about conditions, which allow me to conclude that the
extension actually includes $\Omega$ and further, that this extension is bounded and in $C^1(\overline{\Omega})$.
Are there any well known ways of approaching these type of extension problems and / or any literature recommendations?
real-analysis complex-analysis analytic-functions analytic-continuation
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Calculus with Parametric Curves - Boundless Open Textbook
Parametric equations are equations which depend on a single parameter. You can rewrite y=x such that x=t and y=t where t is the parameter.
A common example occurs in physics, where it is necessary to follow the trajectory of a moving object. The position of the object is given by x and y, signifying horizontal and vertical displacement,
respectively. As time goes on the object flies through its path and x and y change. Therefore, we can say that both x and y depend on a parameter t, which is time .
A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.
This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise. Thus, one can describe the velocity of a particle following
such a parametrized path as:
$v(t)=r'(t)=(x'(t),y'(t),z'(t))=(-a\sin(t),a \cos(t),b)$
where v is the velocity, r is the distance, and x, y, and z are the coordinates. The apostrophe represents the derivative with respect to the parameter.
The acceleration can be written as follows with the double apostrophe signifying the second derivative:
$a(t)=r''(t)=(x''(t),y''(t),z''(t))=(-a\cos(t),-a \sin(t),b)$
Writing these equations in parametric form gives a common parameter for both equations to depend on. This makes integration and differentiation easier to carry out as they rely on the same variable.
Writing x and y explicitly in terms of t enables one to differentiate and integrate with respect to t. The horizontal velocity is the time rate of change of the x value, and the vertical velocity is
the time rate of change of the y value. Writing in parametric form makes this easier to do. | {"url":"https://www.boundless.com/calculus/differential-equations-parametric-equations-and-sequences-and-series/parametric-equations-and-polar-coordinates/calculus-with-parametric-curves/","timestamp":"2014-04-16T13:46:31Z","content_type":null,"content_length":"62721","record_id":"<urn:uuid:6ec7696c-1f84-4ee3-a7bb-973e77771d5a>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00337-ip-10-147-4-33.ec2.internal.warc.gz"} |
<mathematics> A mapping of a metric space onto another or onto itself so that the distance between any two points in the original space is the same as the distance between their images in the second
space. For example, any combination of rotation and translation is an isometry of the plane.
Last updated: 1997-12-13
Try this search on Wikipedia, OneLook, Google
Nearby terms: isolated « ISO Latin 1 « isometric joystick « isometry » isomorphic » isomorphism » isomorphism class
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The Ethics Conditions Formulas flow, one to the next, with the first step of one formula directly following the final step of the previous formula.
But what do you do if your stat graph indicates you’ve moved up a condition before you even have a chance to finish a formula? Do you just drop that formula and start on the next one? The answer is “
No.” One completes the formula he has begun.
Here’s an example. An Executive Director in looking over his statistics sees that they are in Emergency. He immediately sees to it that the “Promote” step of the Emergency Formula is begun. Once that
is well in hand, he begins to “Change his operating basis.” He gets on-the-job training actions being done on some of his sales staff and puts three more personnel into one of his major production
But before he has a chance to do each of the remaining steps of the Emergency Formula, the income and delivery statistics move up into Normal Operation.
What does he do? Well, he is now in a Condition of Normal by statistics. But the Normal Formula would also cause him to complete the Emergency Formula, because in the Normal Formula you drop out what
is unsuccessful and you push what was successful; what was successful here was the Emergency Formula. Thus, this Executive Director can get continued improvement on the graph by completing the
Emergency Formula as the actions on the Emergency Formula are what got him to Normal so quickly. So he would push them until they were completed fully. This doesn’t mean he is still in an Emergency
Condition—the stats are now rising and the condition is Normal. It’s a bit of an oddball thing.
As another example, suppose someone is doing a Junior Danger Formula. The person goes step by step through the procedure and writes up his or her overts and withholds and any known out-ethics
situation and starts applying the Personal Danger Formula. But before he completes the formula, his stats rise. It would be dangerous indeed for this person to not finish the Danger Formula (e.g.,
getting done the “Reorganize your life” and “Formulate and adopt firm policy” steps of the Danger Formula).
That one’s stats rise before completing a formula doesn’t mean he can’t go into the higher condition his stats now indicate. However, it would be a grave fault not to complete the undone steps of an
earlier formula. So, as in the above examples, one has to complete the earlier formula, then complete the next formula and continue on as his graph dictates.
Completing a formula is very vital. One doesn’t just name a formula. He gets it completed. | {"url":"http://www.scientologycourses.org/courses-view/conditions/step/read-completing-conditions-formulas.html","timestamp":"2014-04-16T13:42:00Z","content_type":null,"content_length":"29466","record_id":"<urn:uuid:fd404495-4f9a-473e-a74d-c04ba0c93c8f>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00181-ip-10-147-4-33.ec2.internal.warc.gz"} |
Coral Springs, FL Algebra 2 Tutor
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Parallel Lines intersect in Square
July 16th 2008, 03:34 PM #1
Jul 2008
Parallel Lines intersect in Square
Hey guys, I have an idea on how to do this problem for certain cases - does anyone know how to show this for a general case (since thats what its asking for I think):
"If we have 4 points A,B,C, D which are given on a straight line (in that order), show how to construct a pair of paralllel lines through 'A' and 'B' and another pair of parallel lines through
'C' and 'D' so that the 4 lines intersect in a square"
For these lines to intersect, they must not be vertical neither horizontal. So it seems by inductive reasoning. Thus they must be oblique or slanted parallel lines.
With a quick sketch, the attachment of this post is what I have in mind.
Hey guys, I have an idea on how to do this problem for certain cases - does anyone know how to show this for a general case (since thats what its asking for I think):
"If we have 4 points A,B,C, D which are given on a straight line (in that order), show how to construct a pair of paralllel lines through 'A' and 'B' and another pair of parallel lines through
'C' and 'D' so that the 4 lines intersect in a square"
If the resulting figure is to be a square, then the A,B,C,D must be equally spaced.
Get the midpoint of BC. Call it M.
With M as center, draw semicircles with A and B. (A and D must be on the larger semicircle, while B and C must be on the smaller semicircle.)
Draw a line through M that is perpendicular to the original line. This line will bisect the two semicircles.
Draw a line through A that will pass through the the point of intersection of the larger semicircle and the perpendicular line from M.
Do that too from D.
Draw a line through B that will pass through the the point of intersection of the smaller semicircle and the perpendicular line from M.
Do that too from C.
What is the figure formed now by those four lines?
If the resulting figure is to be a square, then the A,B,C,D must be equally spaced.
Get the midpoint of BC. Call it M.
With M as center, draw semicircles with A and B. (A and D must be on the larger semicircle, while B and C must be on the smaller semicircle.)
Draw a line through M that is perpendicular to the original line. This line will bisect the two semicircles.
Draw a line through A that will pass through the the point of intersection of the larger semicircle and the perpendicular line from M.
Do that too from D.
Draw a line through B that will pass through the the point of intersection of the smaller semicircle and the perpendicular line from M.
Do that too from C.
What is the figure formed now by those four lines?
wow! thanks for the help you made it so easy. and yes it works out perfect! thanks again
Hey guys, I have an idea on how to do this problem for certain cases - does anyone know how to show this for a general case (since thats what its asking for I think):
"If we have 4 points A,B,C, D which are given on a straight line (in that order), show how to construct a pair of paralllel lines through 'A' and 'B' and another pair of parallel lines through
'C' and 'D' so that the 4 lines intersect in a square"
1. It doesn't matter how the 4 points are placed on the line segment. If the order is A, B, C, D then you always can construct a square.
2. The vertices of the square are located on semi-circles over AC, AD, BC, BD.
3. I haven't found the most important property to place one vertix definitely on it's semi-circle.
July 16th 2008, 03:49 PM #2
July 16th 2008, 05:22 PM #3
MHF Contributor
Apr 2005
July 17th 2008, 04:49 AM #4
Jul 2008
July 17th 2008, 05:47 AM #5 | {"url":"http://mathhelpforum.com/geometry/43860-parallel-lines-intersect-square.html","timestamp":"2014-04-17T19:56:57Z","content_type":null,"content_length":"45851","record_id":"<urn:uuid:4db7677e-794e-45ad-88c4-b4b272f9beb3>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00324-ip-10-147-4-33.ec2.internal.warc.gz"} |
Student Support Forum: 'Matrices and for loops' topic
Author Comment/Response
If I understand what you want, take a look at what this does:
Clear[f, a, b, i, j];
A = Table[{i, j}, {j, 1, 3}, {i, 1, 10}]
A // TraditionalForm
f[a_, b_] := a + b;
B = Table[f[i, j], {j, 1, 3}, {i, 1, 10}]
B // TraditionalForm
Hopefully that should be relatively self-explanatory. You can do a lot of things when you get comfortable with list manipulation in Mathematica, and you can basically view a matrix as a
nested list.
You can look up Table, List, //, and TraditionalForm in the help to get a better understanding of what is going on here.
Note that a ; at the end of a line suppresses output. Leaving the ; off allows output to be generated.
You can use lines like
f[a_, b_] := a + b;
to define functions. The underbars have to be there in the function definition for the definition to apply to any arguments you stick into f[,]. The : on the := is probably safest to keep
there in function definitions if you don't understand what it does, although depending on what you're doing it might slow things down some. Also, the := has the side effect that you don't
really need a ; to suppress output from this line.
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Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.
up vote 4 down vote favorite
In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such
that $\mathcal{B}_n\cap\mathcal{P}$ is embedded into the $(n-1)$ dimensional unit $L_1$ sphere: $\mathcal{B}_{n-1}: |x_1|+|x_2|+\ldots+|x_{n-1}|\leq 1, x_n=0$ after the rotation?
geometry linear-algebra convex-polytopes
1 Oh, I misread your question. Well, try counting extreme points of ${\mathcal B}_n\cap {\mathcal P}$ when $a_1=\dots=a_n=1$; I think that may lead towards an answer (but I haven't checked the
details) – Yemon Choi Apr 5 '11 at 20:00
Hint: what is the shape of the $\ell_1^3$ sphere, and what is the shape of a cross-section parallel to one of its faces? – David Eppstein Apr 5 '11 at 23:24
Thanks for the hint. In 3 dimensional case, the sphere is a diamond shape and the cross-section parallel to one of this faces is a regular hexagon, which can be proved inside a $\ell_1^2$ sphere.
On high dimensional case, it is same as the example given by Yemon Choi. However, I'm not sure how to embed it in $\ell_1^{n-1}$. – Chao Li Apr 6 '11 at 0:03
add comment
1 Answer
active oldest votes
Thanks for the comments above. I have just proofed this problem is only true when $n\leq 4$.
When $n\leq 4$, without loss of generality, assume $|a_n|\geq |a_1|, \ldots, |a_{n-1}|$. Let $Q$ be the rotation on the plane spanned by $(a_1, \ldots, a_n)$ and $(0,\ldots, 0, 1)$ such
that applying $Q$ to $(a_1, \ldots, a_n)$ rotates it to $(0,\ldots, 0, 1)$. Apply the rotation to $\mathcal{B}\cap\mathcal{P}$ will inscribe it into $\mathcal{B}_{n-1}$.
up vote 1
down vote For $n=5$, the intersection between $\mathcal{B}_n$ and $\mathcal{P}: a_1+a_2+a_3+a_4+a_5=0$ can not be inscribed in to the 4-dimension $L_1$ sphere. The intuition of the proof is that
accepted the $\mathcal{B}_n\cap\mathcal{P}$ contains the intersection between the 3-dimension $L_1$ sphere and hyperplane $a_1+a_2+a_3+a_4=0$. This intersection can only be inscribed into the
4-dimensional $L_1$ sphere under two rotations (with symmetry cases ignored). One can verified that non of those two rotations can fit $\mathcal{B}_n\cap\mathcal{P}$ into a
4-dimensional $L_1$ sphere.
add comment
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Finding Affine Transformations
So this is my problem… I have two unstructured point clouds, containing different numbers of points, which have been sampled in a noisy manner. I want to find the “best” affine transformation between
these sets of points. I’ve tried ICP, but the points are too noisy and I often fail to find enough pairs of points for the algorithm to succeed. I have a procedure to combine the sets once they are
registered as closely as possible, but solving the registration problem in the presence of noise is very difficult. Does anyone have any suggestions, ideas, code? Thanks!
Maybe you could do some kind of PCA of the data and derive the affine transformation that maps one’s principal components into the others. If that’s not good enough, it could be a starting point for
some kind of refinement. I’m not sure what ICP is.
That’s a good idea, maybe I’ll give it a try. ICP stands for Iterative Closest Point; a Google search will provide a ton of references. Basically, however, you find pairs of points from the different
point clouds, and then find a transformation the minimizes the sum of squared distances. If you iterate this process (find close pairs of points, estimate the transformation in a least-squares sense,
transform the points and repeat) you will eventually converge on a transformation. The problem is that when the point sets are noisy, as in my case, the independence of the sample noise makes it
difficult to find pairs of points that agree on a distance minimizing transformation.
Yeah, that sounds like it would be difficult to tune. Perhaps using the PCA approach to provide the initial guess for ICP will yield better results.
I recently read something about obtaining a matrix with noisy data as input, and then creating the best (orthonormal) rotation matrix from it. To behonest, I skipped that part, maybe it is totally
unrelated to your problem, but if you care I can try to find the text.
That sounds interesting… do you remember the title or the authors ?
Yes, I found it: A Flexible New Technique for Camera Calibration, Appendix C. Now that I re-read it, I’m even less sure that it will be useful, anyway:
The problem considered in this section is to solve the best rotation matrix R to approximate a given
3 x 3 matrix Q. Here, “best” is in the sense of the smallest Frobenius norm of the difference R-Q.
Did you try combining your method with a flexible optimization technique? Algorithms like Levenberg-Marquardt are quite accepting with regard to noise. | {"url":"http://devmaster.net/posts/15013/finding-affine-transformations","timestamp":"2014-04-18T05:40:12Z","content_type":null,"content_length":"22284","record_id":"<urn:uuid:a119282f-5f19-49ba-a005-5b7c34f95da0>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00567-ip-10-147-4-33.ec2.internal.warc.gz"} |
Stirling's approximation
July 19th 2011, 07:06 PM
Stirling's approximation
Hi, I was just checking out the proof of this in William Feller's book, and it jumps from the arithmetic mean of (nlog(n)-n) and ((n+1)log(n+1)-n) to ((n+(1/2))log(n)-n). I'm not seeing it. The
book is certainly a good read, but nonetheless I'm missing out. Any hints?
July 20th 2011, 01:44 AM
Re: Stirling's approximation
Prove that $L=\dfrac{1}{2}\displaystyle\lim_{n \to{+}\infty}{\frac{n\log n+(n+1)\log (n+1)-2n}{(n+1/2)\log n-n}}=\ldots=1$
Hint: Divide numerator and denominator by $(n+1)\log (n+1)$ .
July 20th 2011, 05:14 AM
Re: Stirling's approximation
It's not equal, it's only an approximation of the approximation.
And this is where the ~ comes from then?
July 20th 2011, 05:19 AM
Re: Stirling's approximation | {"url":"http://mathhelpforum.com/advanced-statistics/184852-stirlings-approximation-print.html","timestamp":"2014-04-17T03:09:45Z","content_type":null,"content_length":"5422","record_id":"<urn:uuid:41809d63-7a43-446a-8268-5c999efd8c25>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00187-ip-10-147-4-33.ec2.internal.warc.gz"} |
Performance Analysis of Relay Subset Selection for Amplify-and-Forward Cognitive Relay Networks
The Scientific World Journal
Volume 2014 (2014), Article ID 548082, 10 pages
Research Article
Performance Analysis of Relay Subset Selection for Amplify-and-Forward Cognitive Relay Networks
^1Department of Electrical Engineering, Air University, Islamabad, Pakistan
^2Department of Electronics Engineering, IIU, Islamabad, Pakistan
Received 31 August 2013; Accepted 9 October 2013; Published 6 March 2014
Academic Editors: G. De Marco and A. Svigelj
Copyright © 2014 Kiran Sultan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Cooperative communication is regarded as a key technology in wireless networks, including cognitive radio networks (CRNs), which increases the diversity order of the signal to combat the unfavorable
effects of the fading channels, by allowing distributed terminals to collaborate through sophisticated signal processing. Underlay CRNs have strict interference constraints towards the secondary
users (SUs) active in the frequency band of the primary users (PUs), which limits their transmit power and their coverage area. Relay selection offers a potential solution to the challenges faced by
underlay networks, by selecting either single best relay or a subset of potential relay set under different design requirements and assumptions. The best relay selection schemes proposed in the
literature for amplify-and-forward (AF) based underlay cognitive relay networks have been very well studied in terms of outage probability (OP) and bit error rate (BER), which is deficient in
multiple relay selection schemes. The novelty of this work is to study the outage behavior of multiple relay selection in the underlay CRN and derive the closed-form expressions for the OP and BER
through cumulative distribution function (CDF) of the SNR received at the destination. The effectiveness of relay subset selection is shown through simulation results.
1. Introduction
Enabling secondary transmissions ensuring minimum quality of service (QoS) with constrained transmission power is a major design challenge faced by underlay CRNs and it requires fine tuning and
adjustment of the transmit power of the SUs. The purpose of limiting the transmit power is to keep the primary communication undisturbed [1]. In [2], the authors suggested a transmit power allocation
scheme for dual-hop CRNs operating in AF mode, under transmit power constraints and interference constraints. First, the optimization problem was simplified by relaxing the transmit power constraint
to obtain a suboptimal solution, which was then further utilized to propose a power allocation scheme in order to satisfy both constraints all the time.
The problem highlighted above becomes more complicated when the secondary source-destination pair is unable to communicate directly due to deep fading or shadowing and so forth. Cooperative
communication [3] is an effective means of increasing the spatial diversity of a signal in wireless communication networks. Such a communication strategy efficiently improves system throughput,
combats channel fading, reduces power consumption, and increases transmission reliability and coverage area [4–6]. Cooperative communication techniques follow such approaches as collaborative signal
processing, cooperative coding, and relaying [7]. The literature review reveals that amplify-and-forward (AF) is the simplest and the most widely employed relaying protocol, in which the relay just
scales the received message and forwards it to the destination without performing any regenerating action, thus requiring less processing and low power consumption at the relay [8]. Relay-assisted
CRNs have emerged as a potential solution to cope with the challenges faced by underlay networks. However, it may not be a feasible idea to use all the relays in a cognitive radio system to assist
SUs, because the interference produced by the relays may exceed the interference threshold of the PUs, which forces the secondary network to transmit at very low power, reducing the signal-to-noise
ratio (SNR) at the destination. Relay(s) selection comes as a fascinating solution to this problem; however, selection of subset of multiple relays satisfying interference and transmit power
constraints is more complex than single relay selection in underlay networks and limited effort has been done in this context so far. The interference threshold can be defined by average or
instantaneous interference power received at the primary receiver [9]. The instantaneous or peak interference power requires knowledge about instantaneous channel gains of the interference channels
and it is suitable for real-time traffic. The average interference power applies to nonreal time traffic where the average SNR determines the QoS.
Some of the research contributions in the area of best relay selection are as follows. In [10], Fredj and Aïssa presented a scenario in which a secondary transmitter used the services of intermediate
relays to communicate to its receiver. In this scenario, best relay was selected from the potential relay set to enable secondary communication under interference constraints. Furthermore, end-to-end
SNR statistics were derived and bit error rate (BER) was evaluated for different modulation schemes. In [11], Seyfi et al. proposed a best relay selection scheme for dual-hop cognitive relay network
under transmit power constraints and interference constraints. Furthermore, the outage probability of the secondary network with relay selection was derived while considering the effect of PU
interference. The derived results were tested through simulations. In [12], Bao et al. proposed best relay selection and considered tight lower bound of the end-to-end SNR to derive the closed-form
expressions for cumulative distribution function (CDF) and probability density function (PDF) over nonidentical Rayleigh fading channels. The derived results were used to investigate the outage
probability and average symbol error probability of proposed system. The performance was evaluated against some key parameters. The asymptotic analysis of the scenario showed that interference
constraint does not affect the diversity gain. Li investigated best relay selection based on full and partial channel state information (CSI) in [13] and compared the performance of both schemes by
deriving the closed-form expressions for outage probability. For this purpose, a cluster of cognitive relays assisting a single source-destination pair was considered. It was proved that
partial-CSI-based relay selection was outperformed by the full-CSI-based relay selection.
Research contributions in the area of multiple relay selection are, however, quite limited. In our correspondence, we will use “relay subset selection” or “multiple relay selection” interchangeably.
Multiple relay selection schemes to maximize the SNR at the destination in an underlay CRN were proposed in [14], and their performance was compared against different levels of source transmit power,
considering different sizes of potential relay network and different interference threshold levels. Naeem et al. considered a dual-hop CRN and proposed a multiple relay selection scheme with
interference awareness for underlay CR systems in [15]. It was proved through simulations that the performance of the proposed scheme approached exhaustive search technique while having low
implementation complexity.
These prior works have significantly improved our understanding of relay-assisted CRNs and are selected for discussion because all the contributions were built on some common assumptions which are as
These prior contributions have significantly improved our understanding of relay-assisted CRNs, and all of these were built on some common assumptions which are explained as follows. First, underlay
spectrum sharing model was assumed for each scenario. Second, all schemes assumed severe shadowing on the line-of-sight path between source-destination pair, thus making direct communication
impossible. Third, all system models were built up using single-antenna terminals. Fourth, AF relaying was assumed at the cognitive relay network. Fifth, all the highlighted contributions for best
and multiple relay selection assume the availability of CSI of the interference channels. Each of the proposed schemes has been analyzed with interference and transmit power constraints. Furthermore,
these contributions have been highlighted due to the reason that performance analysis in terms of outage behavior and BER has been carried out only for single (best) relay selection schemes. However,
the effect of multiple relay selection on the OP and BER of the secondary system operating in an underlay spectrum sharing environment is not presently available in the literature, to the best of our
In this paper, the deficiencies highlighted in the performance evaluation of multiple relay selection schemes have been focused on. This paper investigates, for the first time, the outage behavior
and BER of the secondary network for multiple relay selection, which has not been done so far, to the best of our knowledge, for AF based underlay cognitive relay networks. Furthermore, similar
derivation has been carried out for the single best relay selection scheme for fair comparison. A dual-hop relay-assisted CRN is considered for this purpose, and a multiple relay selection scheme is
proposed, aiming to maximize the secondary system performance, while satisfying the interference threshold of the primary network. We have carried out the comparison between the proposed multiple
relay selection scheme and the best relay selection scheme and proved that the multiple relay selection outperforms the best relay selection in terms of OP and BER.
The remaining paper is structured as follows. The system model and the mathematical formulation of the problem have been explained in Section 2. The algorithm proposed for multiple relay selection is
explained in Section 3. The performance analysis has been carried out in Section 4 followed by Section 5 which presents the simulation results. The whole work is concluded in Section 6.
2. System Model and Problem Formulation
Figure 1 shows the system model comprising a secondary source , destination , and a PU . The end-to-end secondary communication is entirely dependent on a potential relay set consisting of candidates
having cognitive radio capabilities due to a large physical separation involved between source-destination pair. The entire relay network exists near a PU in an underlay spectrum sharing mode, while
the source being far away from the PU does not interfere with the primary signals.
Rayleigh flat-fading scenario is assumed, in which the independent and identically distributed (i.i.d.) source-relay, relay-destination, and relay-PU channel coefficients are designated as , , and ,
respectively, where is treated as the interference channel. It is further assumed that the instantaneous CSI is available at each potential relay, and the instantaneous value of interference
threshold is computed to perform relay selection. Single-antenna terminals are assumed at the primary and secondary networks, and relays employ AF protocol with adjustable gains. The half-duplex mode
of communication takes place at the relay network which is completed in two time slots. The source transmits a symbol in time slot 1 and the received signal at the th relay is given as where denotes
the source transmit power and represents additive white Gaussian noise (AWGN) at the th relay with zero mean and variance . In time slot 2, the destination receives the scaled version of the received
message from the relay network while the source is silent. The signal received at the destination is expressed as where is modeled as AWGN with variance , received at the destination. The signal
amplified according to AF scheme is given as where represents the transmit power of the th relay in the above equation and is defined in AF relaying as where is the randomly selected amplification
factor of the th relay according to the proposed algorithm.
Substituting (1) and (3) in (2) and solving the resulting expression, end-to-end SNR of the th relay link can be expressed as [16] Or in compact form where and denote the SNR achieved at the
source-relay and relay-destination links, respectively, with noise variance normalized to one.
The end-to-end SNR at the secondary destination due to relaying links is then given by We explain our proposed relay subset selection problem as follows. Let represent the transmit power vector of
the potential relays in the network; that is, . The number of all nontrivial subsets is given by . The th subset is denoted as , where . The cardinality of th subset is . Next is to compute total
interference power due to each th subset of relays towards the PU, where interference offered by each th relay in any subset is defined as .
Let be the number of subsets out of , denoted as , which satisfy the interference threshold towards the PU. The interference constraint for th such subset can be given as The mathematical formulation
of this optimization problem is as follows: satisfying the constraint
In order to investigate the performance of the overall system in terms of outage probability and average probability of error, we need to know the distribution of which is not mathematically
tractable. To overcome this problem, tight upper and lower bounds for in (6) exist in the literature [17]; that is, , where and . The bounds on show that the minimum value of occurs when , and, for
this case, , and, if is increased further through transmit power control, the upper bound is approached. Keeping the behavior of under consideration, we aim to maximize through controlled transmit
power allocation to each relay so that of each relay link tends to approach its upper bound causing an overall favorable impact on , while keeping the sum interference constraint satisfied. Thus, the
relay subset selection algorithm aims to pick up that subset of relays, which maximizes combined SNR of relay-links, , where, for each th subset , is defined as .
Thus, the mathematically tractable form of our optimization problem is given as
3. The Proposed Algorithm
Let be the initial set of potential relays. The proposed algorithm works as follows. Transmit power of each relay is initialized, followed by selecting all possible subsets of relays which are able
to satisfy the sum interference power threshold set by the PU. For all such subsets, combined SNR of relay-destination links is computed and finally that subset is declared as the selected subset
which maximizes the SNR. The pseudocode is provided in Algorithm 1. For more clarity, the flowchart of the proposed algorithm is also presented in Figure 2.
4. Performance Analysis
In this section, the performance of the proposed multiple relay selection scheme is investigated and has been compared with the best relay selection scheme. The criterion for relay selection is kept
the same for both schemes for fair comparison. Performance evaluation is carried out in terms of outage probability and average probability of error. We consider both cases separately as follows.
4.1. Multiple Relay Selection
As mentioned earlier, Rayleigh distributed channel coefficients are assumed for the considered network with their squared amplitudes being exponential random variables. Therefore, the PDFs of and ,
being independent and exponentially distributed, are given by and the corresponding CDFs are given by where denotes the average second-hop SNR for th relaying link and is the average strength of
interference channel from the th relay and PU.
Given subsets for selection, the conditional PDF of the SNR of the finally chosen subset according to the proposed relay subset selection scheme where is given as In order to simplify further
analysis of above PDF, we assume that, during a hop transmission, instantaneous SNRs have the same average values for all relays. Hence, . Using this assumption, (12) is rewritten as In the above
equation, the first part is the PDF of combined SNR of final selected subset being evaluated at . Since each element in the selected subset is exponentially distributed, then the PDF of the combined
SNR , being the sum of exponential random variables with same mean, will be Erlang distributed and is given by where is the cardinality of selected set .
The second part of (13), that is, , is the CDF of SNR of th subset being evaluated at . As mentioned above, the SNR of each th subset follows Erlang distribution; thus the CDF of will be expressed as
Therefore, the PDF of selected relay subset in (13) will take the form of An important consideration is that the PDF given above is conditioned over , that is, the number of subsets which are able to
satisfy the interference constraints. The value of may vary from to . If , communication between secondary source-destination pair is not possible. This situation occurs if sum interference threshold
imposed by PU is too low that no subset of relays is able to meet the requirement without amplification, thus making secondary communication impossible. But this is not the case in our scenario as
the relay network is assumed to be far away from the PU. Thus, takes the values between and . If , there would be no relay subset selection and if , the destination will decide which relay subset is
the one satisfying the proposed criteria. The interference constraint can be satisfied by each subset in with a probability , where dictates the Erlang distribution following the same assumption for
interference strengths of relayed links; that is, . Thus, the PDF can be obtained in the same way as And the corresponding CDF is written as Thus, the probability of availability of subsets out of
subsets which satisfy interference threshold follows binomial distribution The unconditional PDF of SNR at the destination due to selected subset can be found by using (18) and (19) in (16) as The
corresponding CDF can be obtained by integrating the above PDF w.r.t. and using [18, Equation (3.381.1)]; thus, where denotes the incomplete gamma function given in [18, Equation (8.354.1)] as The
outage event occurs in a communication system if the SNR received at the destination falls below a set threshold . The probability of this event can be directly obtained from the CDF of the received
SNR given in (21) evaluated at ; that is, .
Average bit error probability is usually evaluated using the probability of error conditioned over a given SNR in AWGN. This conditional probability of error is defined in terms of standard function
and its average is taken over the PDF of received SNR. Therefore where and is a constant and its selection depends on the modulation scheme employed. Referring to the technique in [19], the above
equation takes the form Solving the above equation using [18, Equation 3.461.2], we obtain
4.2. Best Relay Selection
In order to verify the effectiveness of the proposed multiple relay selection scheme, a similar derivation has been carried out for best relay selection. Based on the same criteria for multiple relay
selection, the relay which is able to maximize the SNR of relay-destination link, while satisfying the primary interference threshold, is declared the best relay by the destination. Thus, the
optimization problem formulated in (11a) and (11b) can be expressed as where denotes the index of the best relay selected for communication.
In order to investigate the system performance for best relay selection, we follow the same assumptions for channel conditions as stated in the above section. Thus, the PDFs and CDFs of and will be
exponentially distributed as given in (11a) and (11b) for each candidate relay satisfying interference threshold.
Given relays for selection out of potential relays, such that, the interference offered by each th relay is below the interference level set by the PU, the conditional PDF of , that is, the SNR of
the final selected relay, where , is given according to the proposed relay subset selection scheme as Assuming the same average values of instantaneous SNRs for all relays to simplify further
analysis, (27) can be rewritten as In the above equation, the first part is the PDF of SNR of best chosen relay being evaluated at .
The second part of the equation, that is, , is the CDF of SNR of th relay being evaluated at . Since the SNR of each th relay follows the exponential distribution as mentioned above, then the
conditional PDF of selected relay using (11a) and (11b) will take the form as given by An important consideration is that the PDF obtained in the above equation is conditioned over , that is, the
number of relays which satisfies the interference constraints. The value of may take any value from to . If , communication between secondary source-destination pair is not possible. This situation
occurs if the relay network experiences a too high interference threshold level set by the PU which is not satisfied by even a single relay, thus making secondary communication impossible. But this
is not the case in our scenario as the relay network is assumed to be far away from the PU. Thus, takes the values between and . If , there would be no relay subset selection, and if , the
destination picks up the best relay satisfying the proposed criteria. Each member of the potential relay set can satisfy the interference constraint with a probability , where dictates the
exponential distribution and following the same assumption for interference strengths of relayed links; that is, .
Thus, the probability of availability of relays out of relays which satisfy interference threshold follows binomial distribution The unconditional PDF of SNR due to the best selected relay can be
found by using (11a), (11b), and (30) in (29) as The corresponding CDF can be obtained by integrating the above PDF w.r.t. . The resulting CDF is Outage probability can be directly obtained from the
CDF of the received SNR given in (32) evaluated at ; that is, .
Average bit error probability using the same technique as those employed for multiple relay selection and using [18, Equation (3.321.3)] is given by In the next section, the results derived for the
single best relay selection and multiple relay selection have been investigated for a well-defined range of certain parameters for the primary and secondary networks.
5. Simulation Results
This section verifies the effectiveness of the proposed scheme for selecting the subset of relays. For all simulations, source transmit power is set to 10. Zero mean unit variance AWGN is assumed for
each link. Furthermore, for the relay subset selection algorithm, and represent the number of potential relays and selected relays, respectively. Binary phase shift keying (BPSK) with is the
modulation scheme employed. The interfering channels towards the PU are generated by setting . Table 1 provides the parameter settings for the performance evaluation.
Figure 3 provides the comparison of best relay selection, multiple relay selection, and all relays Participation schemes in terms of SNR achieved at the relay-destination links against different
levels of interference threshold . Figure 3(a) shows that the multiple relay selection algorithm outperforms both the best relay selection and all the relay techniques due to freedom of selecting the
best subset of relays which can maximize secondary system performance through controlled transmit power allocation to the relay network keeping in view the privilege of PUs. However, in order to
allow all relays to participate in transmission, source transmit power needs to be suppressed keeping in view the interference constraint, which in turn produces negative effect on the power received
at the relay network, eventually decreasing the SNR received at the destination. Furthermore, a single best relay is also unable to maximize the secondary performance through single best relay. The
corresponding total number of selected relays is shown in Figure 3(b). There is a very strong observation that if the interference threshold is made too tight, the multiple relay selection problem
reduces to single best relay selection, whereas, on the other hand, relaxing the interference threshold adds more relays to the network, and eventually, maximum cooperative diversity is achieved for
dB. Moreover, the greater the number of candidate relays in the potential relay network, the higher the flexibility added to the system to allow more relays to participate in the communication,
which are favorable for secondary communication and not harmful for primary communication at the same time, as shown in the case for dB. Thus, the multiple relay selection scheme is the optimal
choice for medium levels of interference threshold.
In Figures 4 and 5, outage probability and bit error rate of the best and multiple relay selection schemes are investigated, respectively, by varying the average SNR per hop for the different number
of potential relays . and are both set to , respectively [20, 21]. As obvious from Figure 4, the outage probability is maximum for the single best relay selection and significantly decreases in the
case of proposed relay subset selection due to the fact that spatial diversity enhances system performance by improving SNR received at the destination. An important observation is the improved
system performance in the case of proposed multiple relay selection scheme, because in order to design an underlay network with full cooperative diversity, transmit power of the source needs to be
suppressed even if the relays just forward the received signal without any further amplification. On the other hand, in multiple relay selection, increasing the number of potential relays generates
more subsets which are able to satisfy the interference threshold set by the PU, thus giving more freedom to choose the optimal combination of relays which exhibit good channel conditions towards the
destination. Furthermore, relay selection gives priority to those relays that exhibit good channel conditions towards secondary destination and allows them to transmit at high power to improve
secondary throughput. Similar trends are observed in Figure 4 due to the same reasons.
6. Conclusion
The major contribution of this paper is the derivation of the outage probability and bit error rate for multiple relay selection. For this purpose, a multiple relay selection algorithm is proposed
for CRNs operating in an underlay environment near a PU. In this scenario, we select the optimal combination of relays from the potential relay set aiming to maximize the SNR received at the
destination, keeping in view the interference threshold of the primary network. The proposed scheme proves the effectiveness of multiple relay selection in energy-constrained CRNs. Finally, the
outage probability and average probability of error have been derived in closed forms through the CDF of the received SNR at secondary destination, which has not been done in the literature so far
for multiple relay selection. Performance evaluation shows that multiple relay selection outperforms best relay and all relay techniques. Simulation results recommend different operating points for
the entire system under different levels of interference threshold and number of potential relays. In future research, this work will be extended to include the line-of sight path between
source-destination pair, and also considering the interference from the concurrent primary transmissions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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'12), pp. 1784–1788, 2012. View at Publisher · View at Google Scholar | {"url":"http://www.hindawi.com/journals/tswj/2014/548082/","timestamp":"2014-04-18T11:59:24Z","content_type":null,"content_length":"373645","record_id":"<urn:uuid:1ffb9f6d-fe79-4862-a315-80d64606ff0b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00570-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: July 2004 [00523]
[Date Index] [Thread Index] [Author Index]
Re: Further information about size limits for Normal[SparseArray[<>]]?
• To: mathgroup at smc.vnet.net
• Subject: [mg49616] Re: Further information about size limits for Normal[SparseArray[<>]]?
• From: ab_def at prontomail.com (Maxim)
• Date: Sat, 24 Jul 2004 03:48:38 -0400 (EDT)
• References: <cdqr0s$kop$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com
Scott Morrison <scott at math.berkeley.edu> wrote in message news:<cdqr0s$kop$1 at smc.vnet.net>...
> More generally, does anyone have a definitive (or even partial) list of
> functions which know how to operate directly on SparseArrays, without
> having to resort to their Normal forms?
Most of the basic operations have been extended to handle SparseArray
objects, so, for example, there is no need to add any fixes to
LinearAlgebra`MatrixManipulation` to make it compatible with
SparseArray, because those fixes have already been made to the basic
Mathematica functions like Part. However, there are a few exceptions:
in some cases a function has to do the parsing of the input expression
and forgets about SparseArray; for example:
{y'[x] == SparseArray[{{1, 1} -> 1, {2, 2} -> 1}],
y[0] == IdentityMatrix[2]},
y, {x, 0, 1}]
NDSolve::ndnum: Encountered non-numerical value for a derivative at x
== 0.`.
NDSolve[{y'[x] == SparseArray[<2>, {2, 2}], y[0] == {{1, 0}, {0, 1}}},
y, {x, 0, 1}]
{y'[x] == IdentityMatrix[2],
y[0] == SparseArray[{{1, 1} -> 1, {2, 2} -> 1}]},
y, {x, 0, 1}]
NDSolve::ndinnt: Initial condition SparseArray[<2>, {2, 2}] is not a
number or a rectangular array of numbers.
NDSolve[{y'[x] == {{1, 0}, {0, 1}}, y[0] == SparseArray[<2>, {2, 2}]},
y, {x, 0, 1}]
Both examples work as expected after converting SparseArray to a
matrix. This can lead to more unpleasant consequences; each of the
following two examples crashes the Mathematica kernel (at least, in
version 5.0):
FindRoot[Norm[x - SparseArray[{{1, 1} -> 1, {2, 2} -> 1}]], {x,
FindMinimum[Norm[x - SparseArray[{{1, 1} -> 1, {2, 2} -> 1}]], {x,
Next, sometimes it's hard to make this construct work seamlessly in
all situations; I suppose there was a good reason to make SparseArray
an atomic object, but this leads to some unexpected things; suppose we
a = SparseArray[{i_, j_} -> 2i + j - 3, {2, 2}];
(*Normal@a is {{0,1},{2,3}}*)
Map[Print, a, {2}]
MapIndexed[Print[#]&, a, {2}]
Print /@ Level[a, {2}]
What will be the output in those three cases? All three will be
different: the first one will output 1,2,3,0 because Map goes over the
rules in SparseArray and 0, as the default value, is processed last;
on the other hand, MapIndexed has to convert the SparseArray to List
(i.e., do Normal) and thus Print will give 0,1,2,3. Finally, even
though Map goes into SparseArray, Level gives {} because SparseArray
is an atom! So the third Print line won't print anything at all. I
don't think this is very logical.
Maxim Rytin
m.r at inbox.ru
• Follow-Ups: | {"url":"http://forums.wolfram.com/mathgroup/archive/2004/Jul/msg00523.html","timestamp":"2014-04-20T03:31:33Z","content_type":null,"content_length":"37362","record_id":"<urn:uuid:89260f26-1e9f-4c2e-ad52-2c3a6c713787>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00369-ip-10-147-4-33.ec2.internal.warc.gz"} |
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 355
lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 355 Investigation of Global and Local Flow
Details by a Fully Three- dimensional Seakeeping Method V.Bertram (HSVA, Germany) H.Yasukawa (Mitsubishi Heavy Industries, Japan) ABSTRACT A fully-three-dimensional Rankine panel method in the
frequency domain is validated for local pressures, motions, and added resistance. Previous formulae for added resistance contained errors resulting in large differences to experiments. This has now
been remedied. The method is linearized with respect to wave height. The steady flow contribution is captured completely by solving the fully nonlinear wave-resistance problem first and linearizing
the seakeeping problem around this solution. The same grids on the hull are taken for both steady and seakeeping computation. On the free surface different grids are used, either following
quasi-streamlined grids or rectangular grids with cut-outs for the hull. The results from the steady solution are interpolated on the new free-surface grid. The method is applied to various test
cases. Motions are in good agreement with experiments, but this is also the case for strip method results. Local pressures, especially for shorter waves, are much better predicted than by strip
method. The added resistance is sensitive to higher derivatives of the potential and a numerical differentiation of these terms may be preferable to using higher-order panels. 1. INTRODUCTION The
most commonly used tools to determine seakeeping properties are based on strip theory. The strip method approach is cheap, fast, and for most cases also quite accurate. However, strip methods do not
perform so well for high- speed ships, full hullforms (tankers), ships with strong flare, and generally for low encounter frequencies which typically occur in following seas. They are also
questionable with regard to local pressures which are needed as input for finite- element analyses. Approaches to improve predictions of seakeeping properties should capture: – 3-D effects of the
flow 3-D effects are important for low encounter frequencies and full hull forms. 3-D diffraction at the bow region of tankers contributes considerably to added resistance, [1]. – Forward-speed
effects Strip methods include forward speed by the change in encounter frequency. But forward speed enters the ship motion problem in additional ways: the local steady flow field, the steady wave
pattern of the ship, and the change of the hull form and wetted surface due to squat (dynamic sinkage and trim). We will present here a 3-d Rankine singularity method (RSM) which captures all
forward-speed effects. The method is ‘fully three-dimensional', i.e. both steady and unsteady flow contributions are captured three-dimensionally. For a recent survey of Rankine singularity methods
for forward-speed seakeeping, we refer to [1], [2]. 2. THEORY 2.1. Physical model We consider a ship moving with mean speed U in a harmonic wave of small amplitude h. We assume an ideal flow. Then
the fundamental field equation is Laplace's equation. In addition, boundary conditions are postulated: 1. No water flows through the ship's surface. 2. At the trailing edge of the ship, the pressures
are equal on both sides. (Kutta condition) 3. No water flows through the free surface. (Kinematic free-surface condition) the authoritative version for attribution.
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version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 356 4. There is atmospheric pressure at the
free surface. (Dynamic free-surface condition) 5. Far away from the ship, the disturbance caused by the ship vanishes. 6. Waves created by the ship move away from the ship. For certain combinations
of frequency of incident wave and speed of the ship, waves created by the ship propagate only downstream. (Radiation condition) 7. Waves created by the ship should leave artificial boundaries of the
computational domain without reflection. They may not reach the ship again. (Open-boundary condition) 8. Forces on the ship result in motions. (Average longitudinal forces are assumed to be
counteracted by corresponding propulsive forces, i.e. the average speed U remains constant.) 2.2. Mathematical model All coordinate systems here are right-handed Cartesian systems. The inertial Oxyz
system moves uniformly with velocity U. x points in the direction of the body's mean velocity U, z points vertically downward. The Oxyz system is fixed at the body and follows its motions. When the
body is at rest position, x, y, z coincide with x, y, z. The angle of encounter µ between body and incident wave is defined such that µ=180° denotes head sea and µ=90° beam sea. The body has 6
degrees of freedom for rigid body motion. We denote corresponding to the degrees of freedom: u1 surge motion of O in x-direction, relative to O u2 sway motion of O in y-direction, relative to O u3
heave motion of O in z-direction, relative to O u4 angle of roll=angle of rotation around x-axis u5 angle of pitch=angle of rotation around y-axis u6 angle of yaw=angle of rotation around z-axis The
motion vector is and the rotational motion vector are given by: (1) (2) All motions are assumed to be small of order O(h). Then for the 3 angles αi, the following approximations are valid: sin(αi)=
tan(αi)=αi, cos(αi)=1. The theory has been described rather extensively by [2]. In the following, we will therefore only briefly review the theory except in cases where changes to [2] justify a more
detailed discussion. We decompose potential and free surface elevation into steady and time-harmonic parts: (3) (4) The superposition principle can be used within a linearised theory. Therefore the
radiation problems for all 6 degrees of freedom of the rigid-body motions and the diffraction problem are solved separately. The total solution is a linear combination of the solutions for each
independent problem. The harmonic potential is divided into the potential of the incident wave the diffraction potential and 6 radiation potentials. It is convenient to divide and into symmetrical
and antisymmetrical parts to take advantage of the (usual) geometrical symmetry: (5) The conditions satisfied by the steady flow potential are: The particle acceleration in the steady flow is: We
define an acceleration vector For convenience we introduce an abbreviation: At the steady free surface: On the body surface: The combined, linearized free-surface condition is at z=ζ(0): (6) The last
term in (6) is explicitly written: (7) the authoritative version for attribution.
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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 357 With the abbreviation the boundary
condition at is: (8) The Kutta condition requires that at the trailing edge the pressures are equal on both sides. For monohulls, this is automatically fulfilled on the centreplane for the symmetric
contributions. Then only the antisymmetric pressures have to vanish, compare (17): (9) This yields on points at the trailing edge: (10) For catamarans, the Kutta condition requires for both symmetric
and antisymmetric contributions, that the pressures on both sides of the trailing edge are the same. This is enforced by selecting pairs of collocation points at the trailing edge and matching the
pressures. The 2 unknown diffraction potentials and the 6 unknown radiation potentials are determined by approximating the unknown potentials by a superposition of a finite number of Rankine
higher-order panels on the ship and above the free surface. For the antisymmetric cases, in addition Thiart elements (semi-infinite dipole strips on the plane y=0), [2], [3], are arranged and a Kutta
condition is imposed on collocation points at the last column of collocation points on the stern. The l.h.s. of the four systems of equations for the symmetrical cases and the l.h.s. for the four
systems of equations for the antisymmetrical cases share the same coefficients each. Thus four systems of equations can be solved simultaneously using Gauss elimination. Radiation and open-boundary
conditions are fulfilled by the ‘shifting' technique (adding one row of collocation points at the upstream end of the free-surface grid and one row of source elements at the downstream end of the
free- surface grid), [4]. This technique works only well for τ>0.4, as also demonstrated by [5]. Elements use mirror images at y=0. For the symmetrical cases, all mirror images have same strength.
For the antisymmetrical case, the mirror images on the negative y-sector have negative element strength of same absolute magnitude. Each unknown potential is then written as: (11) mi is the strength
of the ith element, φ the potential of an element of unit strength including all mirror images. φ is real for the Rankine elements and complex for the Thiart elements. The same grid on the hull is
used as for the steady problem. The grid on the free surface is created new. The quantities on the new grid are linearly interpolated within the new grid from the values on the old grid. Outside the
old grid in the far field, all quantities are set to uniform flow on the new grid. The interpolation of results introduces only small differences as observed in various test cases. Structured grids
on the free surface are generated by one of the following techniques: 1. The longitudinal grid lines follow quasi streamlines around the hull. The transverse grid lines are equidistantly spaced on
lines y=const. A maximum entrance angle of 30° is kept which results in zones not covered by the grid near the bow and stern of blunt ships. 2. A rectangular grid is created consisting of lines x=
const. and y=const. Panels within the waterline are deleted. The first technique is well suited for slender ships, the second technique better for blunt ships. The second technique, called ‘cut-out'
technique was proposed for the steady wave-resistance problem by [6] and [7]. Both grid generation options are available for both steady and seakeeping free-surface grid generation. While it is in
principle possible to switch from one grid type to the other between steady and seakeeping computations, it is recommended to use consistently cut-out grids for full hulls like tankers and the
standard grid option (quasi streamlines) for slender ships. After the potential (i=1…8) have been determined, only the motions ui remain as unknowns. The forces and moments acting on the body result
from the body's weight and from integrating the pressure over the instantaneous wetted surface S. The body's weight is: (12) m is the body's mass. and are expressed in the inertial system ( is the
inward unit normal vector): (13) the authoritative version for attribution. (14)
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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 358 is the center of gravity. The pressure
is given by Bernoulli's equation: (15) (16) (17) Eqs. (13) and (14) yield: (18) (19) The ship is in equilibrium for steady flow. Therefore the steady forces and moments are all zero. The first-order
parts give (r.h.s. quantities are now all functions of): (20) (21) where and Eqs. (18) and (19) (steady equilibrium) have been used. Note: The difference between instantaneous wetted surface and
average wetted surface still has not to be considered as the steady pressure p(0) is small in the region of difference. The instationary pressure is divided into parts due to the incident wave,
radiation and diffraction: (22) Again the incident wave and diffraction contributions can be decomposed into symmetrical and antisymmetrical parts: (23) (24) Using the unit motion potentials and the
pressure equation (17) the pressure parts pi are derived: (25) The individual terms in the integrals (20) and (21) are expressed in terms of the motions ui, using the vector identity (26) (27) The
relation between forces, moments and motion acceleration is: (28) (29) Mass distribution symmetrical in y is assumed. etc. are the moments of inertia and the centrifugal moments with respect to the
origin of the body-fixed Oxyz-system: (30) the authoritative version for attribution. We introduce the abbreviations: (31) (32) (33) (34) Recall that the instationary pressure contribution is: (35)
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version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 359 (36) Then we can rewrite (26), (27) and
(28): (37) The weight terms and contribute with W=mg: (38) The mass terms and contribute: (39) with M being (40) where the radii of inertia have been introduced: etc. Combining Eqs. (36) and (37)
yields a linear system of 6 equations in the unknown ui that is quickly solved using Gauss elimination. 2.3. Added resistance Following a similar approach as for the first-order forces, a formula for
the added resistance can be derived that uses only quantities computed so far. The added resistance is the negative time-averaged value of the x-component of the second-order force. If t1 and t2 are
time-harmonic quantities, the time-average of t1t2 is where is the conjugate complex of (41) The force in x-direction is given by: The integral over the wetted surface can be expressed as a double
integral over a body-fitted curvilinear coordinate system. One coordinate follows rather longitudinal lines from stern to bow, the other coordinate follows the hull contour from the free surface down
to the keel. One of the longitudinal coordinate lines follows the contour of the steady wave profile and this is the ‘zero' line for the other ‘section' coordinate. This modified waterline contour C
accounts also for steady trim and sinkage and differs usually the still waterline contour. The contour line C splits at the stern and both sides run from stern to bow. The ‘section' coordinate runs
from the actual free surface Z to the keel K. Then we can re-write any integral over the wetted surface as: (42) (43) (44) We can thus split the integration into one integral over the average wetted
surface S(0) and a correction double integral. the authoritative version for attribution. If we apply this to the second-order time-average longitudinal force on the ship, we obtain: (45) Combining
(20) and (28), yields: (46)
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version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 360 Thus: (47) The second-order pressure p
(2) is: (48) The term containing the time-derivative of the second-order potential vanishes in the time-average: (49) The term ∇ p(1) involves again second derivatives of the potential on the hull:
(50) Z is the first-order difference between average (steady) and instantaneous wave profile on the hull: (51) with: (52) The curvilinear ‘section' coordinate s can be approximated to first order in
the vicinity of the steady wave profile by a tangential straight line: (53) z′ is a vertical coordinate with origin at the height of the steady surface pointing downwards. Let n′ be a modified
normal: (54) Let N be the unit normal on the contour in the x-y-plane. Then (55) p(0)(z′) We develop in a Taylor series around z′=0 (average free surface=steady free surface): (56) z′ As and z point
both downwards, the derivation is interchangable. The ‘steady' pressure is zero at the ‘steady' free surface. Thus: (57) The p(1) term is simple. A Taylor series gives p(1)(z′)≈p(1)(0). Then:
Similarly the other first-order quantity is simply multiplied by Z in the integration over z′. Thus eventually we get for the time-averaged second-order longitudinal force: (58) The added resistance
is: (59) The integrals in the Ti are evaluated numerically over the starboard half only and multiplied by 2 for symmetrical/ symmetrical and antisymmetrical/antisymmetrical pressure-normal
combinations only. (Antisymmetrical/symmetrical combinations yield zero contributions.) The decomposition into the authoritative version for attribution.
OCR for page 355
lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 361 symmetrical and antisymmetrical parts
complicates evaluating the added resistance. The individual terms are decomposed into symmetrical and antisymmetrical parts: (60) (61) (62) (63) (65) (64) Note that the second component of each of
these vectors contains y-derivatives of the ‘other' potential to ensure consistently symmetrical (i.e. f(y)=f(−y)) and antisymmetrical (f(y)=−f(−y)) behavior. The second derivatives of the harmonic
potentials are neglected in the expression for for simplicity (‘desperation rather than physical insight'). We introduce the abbreviation (66) (67) (68) with (69) (70) Retaining only symmetrical
terms in T1 and T2 yields: (71) Due to symmetry, the above integrals are twice the value of the integrals over the starboard half only. 3. APPLICATIONS 3.1. Local pressures So far, applications of
the present RSM were shown only for relatively high Froude numbers, which for most angles of encounters and wave lengths of interest result in sufficiently high τ values. For these cases, good
agreement with experiments for motions was demonstrated, [2], [3], [8]–[11]. Numerical studies showed that the influence of the steady flow on the results for motions is significant, for moderate
wave lengths, but negligible for short and long waves. This was explained by purely numerical investigations of local pressures. A research cooperation allowed now to investigate local pressures for
a VLCC, Table I, at Fn=0.131. The exact geometry of the test case is confidential. The pressures are the amplitudes of the pressure fluctuation, i.e. pressures without hydrostatic and steady
hydrodynamic pressures. The tanker was discretised using 495 ele the authoritative version for attribution.
OCR for page 355
lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 362 ments. First the steady fully nonlinear
wave resistance problem was solved. The grid on the free-surface was generated using the ‘cut-out' technique. This technique generates a structured grid consisting of rectangular elements. Elements
which are partially or totally inside the hull are then eliminated and then the shifting technique is applied. This technique is known to give better results for full hulls than streamlining a grid
around the hull. The fully nonlinear method used 3 iterations which reduced the error at the free surface by 4 orders of magnitude. The same grid for the hull was employed for the seakeeping
computations (at the dynamic trim and sinkage). Pressure integrations considered only the area submerged in the steady case. The free surface in the seakeeping computations was discretised with
typically 1400 elements. Again the ‘cut-out' technique was employed and the steady results interpolated from the ‘steady' grid to the ‘unsteady' grid. Test computations for two wave lengths with
free-surface grids involving approximately 4200 elements yielded results that were only 5% different. This may be interpreted as that the coarser discretisation is sufficient. The computational
results are compared to measurements of [12] and MHI strip method results. The strip method is based on standard STF method with Lewis section representation, but includes an empirical correction
[13]. The results include motions and pressures on the hull at a location x= −0.078Lpp (23.95m behind amidships). The motions agree rather well for both head sea and oblique sea with µ=150°, Figs. 1
and 2. However, strip method also predicts heave and pitch motions well. In fact, for long waves strip methods gives better results than the RSM. This is not surprising. Strip methods are known to
predict heave and pitch motions well for usual ships and ship speeds. The present RSM uses the shifting technique which deteriorates in performance for τ<0.4…0.5. Sway and yaw are also well
predicted, the maximum of the roll motion is underpredicted. This may be due to the deterioration of the shifting technique, as for a fast containership with Fn=0.275 Bertram [2] obtained significant
overprediction for roll resonance as expected for a method that does not include empirical corrections for nonlinear roll damping. Figs. 3 and 4 compare pressures. Starboard is the weather side. For
head waves the computed pressures are of course symmetrical to the midship plane (90°). One point on the port side was then plotted on its corresponding position on the starboard side. Pressures
computed by the RSM agree well with measured pressures for λ/L<1.25 for µ=180° and λ/L<1.0 for µ=150°. These limits correspond for the investigated low Froude number to τ-values around 0.35…0.4. For
short waves, the computations underpredict the pressures at the bottom of the ship compared to measurements. However, as the pressures should decay exponentially with depth like all wave effects, for
short waves the near-zero values of the computation appear to be more plausible and we assume that they reflect in this case reality better than the measured values. For waves of moderate length 0.5
<λ/L<0.75, measured and computed pressures at the ship bottom agree well. The strip method results for pressures are worse for short waves λ/L=0.2, 0.3 where diffraction effects are stronger than
radiation effects. In summary, the RSM predicted pressures and motions well, the strip method predicted pressures in short waves badly, but motions well. The RSM is currently limited in practice to
approximately τ>0.4. Unless techniques are developed to extend it to smaller τ-Values, the RSM will remain a research tool of limited functionality. We see hybrid methods matching an inner RSM
solution to an outer Green function method or Fourier-Kochin solution as most promising approach to extend the method to low τ-values, but at present no such research is planned due to lack of funds.
Table I: Test case VLCC Lpp 307.00 m zg 4.333 m B 54.00 m kx 19.193 m T 19.50 m ky 73.987 m CB 0.813 kz 76.750 m KG 15.17 m kxz 0m xg 10.045 m 3.2. Added resistance We show here applications to the
ITTC standard test case S-175 containership in head seas. Computations are compared to experiments of Mitsubishi Heavy Industries. Fig. 5 shows results for Fn=0.25 and Fig. 6 for Fn=0.3. For all
three motions the agreement is good. In fact, the computational results for heave for long waves appear to be more plausible, as they tend as expected monotonously to 1. The discrepancies between
experiments and computations for the higher Froude number Fn=0.3 are most likely due to nonlinear damping effects in the experiments. For Fn=0.275, [2] showed that the phase information is also
correctly captured, at least the authoritative version for attribution.
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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 363 for wave lengths where the RAOs are not
approximately zero. The added resistance is similarly well captured. The differences for the higher Froude number are explained by the differences in capturing the motions. The derived formula seems
to be debugged now and can be recommended for other ‘fully three-dimensional' methods. ACKNOWLEDGMENT We are grateful for the support of our younger colleagues Shuji Mizokami and ‘Cowboy' Tanaka in
preparation of the results. REFERENCES 1. Bertram, V. and Yasukawa, H., “Rankine source methods for seakeeping problems,” Jahrbuch der Schiffbautechnischen Gesellschaft, Springer, 1996, pp. 411–425
2. Bertram, V., “Numerical investigation of steady flow effects in 3-d seakeeping computations”, 22. Symp. Naval Hydrodyn., Washington, 1998 3. Bertram, V. and Thiart, G., “A Kutta condition for ship
seakeeping computations with a Rankine panel method”, Ship Technology Research 45, 1998, pp. 54–63 4. Bertram, V., “Fulfilling open-boundary and radiation condition in free-surface problems using
Rankine sources”, Ship Technology Research 37, 1990, pp. 47–52 5. Iwashita, H. and Ito, A., “Seakeeping computations of a blunt ship capturing the influence of the steady flow”, Ship Technology
Research 45, 1998, pp. 159–171 6. Jensen, G., “Berechnung der stationären Potentialströmung um ein Schiff unter Berücksichtigung der nichtlinearen Randbedingung an der freien Wasseroberfläche”, IfS
Report 484, Univ. Hamburg, 1988 7. Nakatake, K. and Ando, J., “Rankine source method using rectangular panels on water surface”, 11. Workshop Water Waves and Floating Bodies, Hamburg, 1996 8.
Bertram, V., “Vergleich verschiedener 3D-Verfahren zur Berechnung des Seeverhaltens von Schiffen”, Jahrbuch Schiffbautechnische Gesellschaft, Springer, 1997, pp. 594–600 9. Bertram, V. and Thiart,
G., “A Rankine panel method for ships in oblique waves”, Euromech 374, Poitiers, 1998, pp. 221–229 10. Bertram, V. and Thiart, G., “Fully three-dimensional ship seakeeping computations with a
surge-corrected Rankine panel method”, J. Marine Science and Technology, 1998, pp. 94–101 11. Bertram, V. and Thiart, G., “Fully 3-d seakeeping computations for real ship geometries”, Jahrbuch
Schiffbautechnische Gesellschaft, Springer, 1998, pp. 244–249 12. Tanizawa, K., Taguchi, H., Saruta, T. and Watanabe, I., “Experimental study of wave pressure on VLCC running in short waves”, J. Soc.
Nav. Arch. Japan 174, 1993, pp. 233–242 13. Mizoguchi, S., “Exciting forces on a high speed container ship in regular oblique waves—Frequency selections for calculating exciting forces by the strip
method—”, J. Kansai Soc. Nav. Arch. Japan 187, 1982, pp. 71–83 the authoritative version for attribution.
OCR for page 355
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks
are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally
inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 2: Like Fig. 1, but for µ=150° Fig. 1: Motions for VLCC, Fn=0.131, µ=180°; • experiment,
○ RSM, · strip method INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 364
OCR for page 355
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks
are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally
inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 4: Like Fig. 3, but for µ=150° Fig. 3: VLCC, Fn=0.131, µ=180°; unsteady pressure angle;
90°=bottom, 0° starboard CWL; • exp., ○ RSM, · strip method INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD at x=−0.078Lpp plotted over circumference 365
OCR for page 355
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks
are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally
inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 6: As Fig. 5, but for Fn=0.3 Fig. 5: RAOs for motions and added resistance for S175, Fn=
0.25, µ=180°, • exp., ○ RSM INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 366
OCR for page 355
lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print
version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original
typesetting files. Page breaks are true to the original; line INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 367 DISCUSSION H.Chun Pusan National
University, Korea 1) I think that you solved iteratively the nonlinear free surface boundary condition. In that case, how did you treat the coefficients (such as added mass, damping, restoring force)
of the motion equation? 2) You mentioned that the panel shift method can not meet the radiation condition for some cases. Do you have any alternative idea to satisfy the radiation condition for such
cases? AUTHORS' REPLY 1) We treated the linearized (unsteady) ship motion problem based on fully nonlinear steady flow[2]. Then, since the ship motion problem is linearized, we can define all
coefficients such as added mass, damping, restoring force and exciting forces. Of course, the effect of steady wave elevation and flow on the coefficients is included in the computations. 2) We see
so-called hybrid methods matching an inner RSM solution to an outer Green function method or Fourier- Kochin solution as most promising approach to extend the method to low values. DISCUSSION
M.Kashiwagi Kyushu University, Japan Firstly I wish to commend you for completing complicated calculations of the added resistance with the pressure integration method. I have a couple of questions
concerning the grids. I understand that what you call ‘cut-out' technique is used for a tanker and possibly S-175 container ship as well. Why do you think the ‘cut-out' technique goes well? If you
use the alternative quasi-streamline grid, how is the result going to be? Does the computation break down or are obtained results much different from experiments? Are there some criteria of which
grid should be used for given values of the block coefficient and the Froude number? AUTHORS' REPLY We used ‘cut-out' method for the computation of VLCC. The reason is that quasi-streamlined grids
either give very distorted grids near the bow and stern or do not locate collocation points on the water surface near the ship ends. Distorted cells lead to problems with the radiation condition. The
‘cut-out' method is robuster in this respect for full waterline forms. We recommend the cut-out method for ships with block-coefficients above 0.7. DISCUSSION M.Ohkusu Kyushu University, Japan
Apparently, a pressure transducer is located on the water line in your experiment. This transducer is naturally out of water some duration during one period of the motion. Then the time history of
the pressure measured will be of not a sinusoidal but a truncated sinusoidal curve. So I wonder how you treated with the truncated curve to derive your value of pressure. If you take the first
harmonic components of this curve, you will obtain much smaller amplitude of the pressure. Nevertheless, your pressure at the water line looks consistent with the pressure at other locations.
AUTHORS' REPLY As you pointed out, we observed the time history of a truncated sinusoidal curve in the pressure measurement in the vicinity of the water line. From the time history data, amplitude
was defined as variation between zero-level and the positive peak value[12]. So the experimental data plotted in Figs. 3 and 4 is not the first harmonic components of the curve. In the computations,
we do not take the truncated effect into account. The reason why the calculated accuracy is insufficient in the vicinity of water line may be due to treatment of the truncated effect. the
authoritative version for attribution. | {"url":"http://www.nap.edu/openbook.php?record_id=10189&page=355","timestamp":"2014-04-21T15:08:01Z","content_type":null,"content_length":"198833","record_id":"<urn:uuid:b169719d-c8a1-4fec-b3ec-53721f62cc80>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00284-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Mathematical Methods for Computer Science
Principal lecturers: Peter Robinson & Neil Dodgson
Taken by: Part IB
Course material
This course introduces two areas of mathematics that are relevant for many applications in Computer Science. There will be six lectures on probability theory in the Michaelmas Term and a further six
lectures on Fourier methods in the Lent Term. The two halves have previously been given in the opposite order, so don't be surprised if occsional references to Part I and Part I appear the wrong way
Probability theory (Michaelmas Term)
Lectures will be on Tuesdays and Thursdays from 10:00 to 11:00 in Lecture Theatre 1 at the William Gates Building, starting on Thursday 12 November.
Fourier methods (Lent Term)
Lectures will be on Tuesdays and Thursdays from 11:00 to 12:00 in Lecture Theatre 1 at the William Gates Building. The first lecture is on 14 January. There are no lectures in the following week:
students are expected to read the handwritten notes (see below and handout in the first lecture) during that week. The remaining five lectures will start on Tuesday 26 January.
Copies of all the above will be provided to all students in the first lecture. The handout also contains a ten page extract from E. Oran Brigham's book "The fast Fourier transform and its
applications", copied under the University's CLA license. This extract cannot be put on this website owing to copyright.
Some Fourier Transform applets | {"url":"http://www.cl.cam.ac.uk/teaching/0910/MathMforCS/","timestamp":"2014-04-18T05:37:37Z","content_type":null,"content_length":"12603","record_id":"<urn:uuid:4b46d1f6-11bf-44ba-a4ec-72469a9f2c0c>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00357-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is hyperfocal distance for 210mm lens @f:64 (4x5 format) [Archive] - Large Format Photography Forum
View Full Version : What is hyperfocal distance for 210mm lens @f:64 (4x5 format)
14-Nov-2003, 14:00
I don't seem to be able to find a chart for it. Thank you.
Jay DeFehr
14-Nov-2003, 14:10
24.32' Here's a calculator:
Witold Grabiec
14-Nov-2003, 14:40
It should be around 24 ft.
Download this free f/calc applet from:
This way you've got one on your PC for a quick reference.
jerry brodkey
14-Nov-2003, 14:48
I had downloaded fcalc and so my fcalc gives a value of 15 feet. The circle of confusion suggested by fcalc is 0.15mm. One of these programs seems off.....
14-Nov-2003, 15:21
Thanks, Guys.
Bob Salomon - HP Marketing
14-Nov-2003, 15:23
And you will be well into diffraction too.
Witold Grabiec
14-Nov-2003, 16:38
I've just checked f/calc which I downloaded a couple of days ago. It gives me 24 ft. Make sure you have all veriables in order. Mine gives me a COC of ~ 0.09. COC of 0.15 does indeed give you some 15
ft on HD, but I had to enter this value manually.
By the way f/calc calculates COC as f/1730 and author calls it a "zeiss formula". COC is the key factor that affects HD or DOF calculations. As you can see from above it will make a substantial
difference. I'm assuming however, that you need the HD to get maximum DOF. The higher the COC the more DOF you get, thus the closer the HD will be. Your COC of 0.15 is a bit high as it is usually
listed in the 0.10 range, value also stated in the Applied Photographic Optics.
jerry brodkey
14-Nov-2003, 18:28
We probably have different versions of fcalc. My version is running on linux and when I click on 4X5 it puts in the value of 0.15 for the COC. When I manually enter 0.10 I get the same value as you
14-Nov-2003, 18:35
Thanks again, Guys. Please don't spend a lot of time on this. Basically I wanted the figure to check myself visually (since the GG image is so difficult to see distinctly when stopping the Dagor down
to f:64). Visually, I got 25' which is "close enough for government work" as they say.
Witold Grabiec
14-Nov-2003, 19:30
It might help if you emailed the author of f/calc (something he's asking for) about the error. Yu can find his contact in the Help section of f/calc. I say error because the COC value of ~ 0.10 is
the most frequently quoted for 4x5 in major publications on this subject (and my Windows version actually gets it right). It is likely a simple typo in the code. I would guess that all of your COC
are off. For 6x6 you should be getting ~ 0.05.
jerry brodkey
14-Nov-2003, 20:11
Witold, Several years ago I tried fcalc and found that he was using a wrong formula for magnification. I wrote him an email and asked about this problem. He wrote back and said he didn't care. I got
so mad I took it off my computer. When people started talking about it recently I put it back on to see if it had changed. I don't know if he has updated the magnification formula or not and frankly
I'll get rid of the program again. If I need to find the hyperfocal distance I'll just plug the values into HD = f*f/Nc.
There is a lesson here. People write programs and you really have no idea if their formulas and methods are correct. It is better to calculate what you need yourself.Then you know it's correct.
Witold Grabiec
14-Nov-2003, 20:36
Yep, I was not aware of this part. If that's how he responded then, oh well. The help section kind of goes over what he uses for calculations. I'll be checking on that in the near future. He has some
other strange things there too.
I agree the HD formula, as all other for what we might need in LF calculations, is simple and easy. This is just a little applet that cost nothing to download.
Darin Cozine
14-Nov-2003, 23:17
OK, I'm a bit confused..
I used the mountainstorm website and plugged in my 65mm lens and f-16 for 4x5 format. The calculator gave me 9.3 feet. <br/>
Question 1: Do I focus at full aperture on 9.3 feet, then stop down to f16 to get infinity in focus?<br/>
Question 1a: If so, how close can objects get and still be in focus?
Question 2: Why does the format make a difference in the hyperfocal distance?
Witold Grabiec
15-Nov-2003, 05:07
HD is a special case of Depth of Field, where the far focusing end of DOF falls on infinity and the near focus end is the Hyperfocal Distance, or the distance you focus your lens on, or the distance
of closest focus. So for (and based on your figures):
1. Yes
1a. 9.3 ft
2. DOF and thus HD calculations depend on the ASSUMED size of the Circle of Confusion (COC), that will give adequate sharpness, which determines how well the detail is resolved in a given plane of
"focus". This depends on the viewing distance of the print, which is considered as the diagonal of the negative. As the negative size goes up, so does the COC.
As you well know, a standard lens for ANY format is the one with a focal length approximately equal to the diagonal of the negative. A term "cone of vision" is derived from the structure of the human
eye and how an image is "printed" on its retina. It is stated, that a human eye resolves about 5 lp mm (lines per mm) which translates to a dot size of 0.2 mm which, in turn, also translates to an
8x10 inch print with the smallest dot of 0.2 mm viewed at its diagonal. (And by the way, photographic paper is said to be capable of around 6 lp mm, so paper is not an issue in these considerations
as it resolves better)
Imagine a cone projecting out from your eye, with its apex at the eye ball and lets see if I can explain this important concept:
a - comfortable viewing angle of around 45-55 degrees (as per standard lens) gives an approximate comfortable viewing distance equal to negative's diagonal, this would be the angle of the "cone of
vision" or COV, comfort being stated as seeing it in a natural perspective
b - when the negative is enlarged (or you're viewing a larger print), it will then have to be viewed from further away in order for it to match this COV, so as to make a "point of detail" on the
print come to a point at the eye ball.
c - after the light passes the eye ball entrance, it will then project on the retina and the ~50 degree angle of projection will end up being that 0.2 mm dot on the retina
d - now, if we take the value of 0.2 mm and print multiplication factor (based on 8x10 standard as this applies to), we will arrive at the COC size that matches our eye's resolving ability (based of
course on the viewing distance of prints diameter, if you view this print at a closer range you will see it more out of focus)
e - to clarify point "d", the 0.2 mm COC is as for an 8x10 viewed at its diagonal, so if you make a contact print from an 8x10 negative, then the COC would be 0.2 mm, if you use a smaller negative
you need to work backwards ( a 35 mm frame is about 8 times smaller, so the COC for a 35 mm negative is 0.2 / 8 = ~ 0.025 mm) and if you use a larger one, you would multiply (an 11x14 negative is
some 2 times larger, so the COC for it would come to around 2 x 0.2 = 0.4 mm)
This is a subjective matter because no human eye is created equal. So some may accuse of you of producing out of focus prints, while others may disagree. The maximum researched threshold for a human
eye resolving ability has been found at 9 l pm or 0.1 mm dot, almost twice the average.
phil sweeney
15-Nov-2003, 05:25
as format size goes up depth of field goes down. The CoC is 1/1500 (or 1/1720 or whichever the user decides) multiplied by the diagonal of the film. 4 x 5 film diagonal is 6.403 inches. CoC of 0.1 mm
is a good all around number for enlarging.
Focus on the hyperfocal distance of 9.3 feet (your example) and about 1/2 times 9.3 (4.65 feet) to infinity should be in focus at f16.
Witold Grabiec
15-Nov-2003, 06:51
Looking at my expalnation above, I was so proud of for a moment, reveals some contradictions. Without going into detailed calculations: - as stated, a COC of 0.2 mm is used as minimum perceiveable
detail when viewed at a comfortable viewing distance - comfortable viewing distance refers to a comfortable "near distance of distinct vision" which was established to be around 250 mm (this is
considered closest an eye can focus on comfortably), in other words if a dot of 0.2 mm is viewed at 250 mm it should be resolved by an average human eye, which ultimately approximates an 8x10 with a
minimum dot of 0.2 mm viewed at its diagonal
- this lead to establishing what a standard lens should be, given a negative size, and all of it fills nicely in the concept of a Cone of Vision as stated earlier, and the ~ 50 degree angle of view
for such,
- depending on the final print size, it is easiest to simply compare it's magnification ratio against an 8x10 and adjust the 0.2 mm accordingly
All other numbers are correct, as are examples of COCs. And the 4x5 does lead to a COC of 0.1 mm based on this.
Since there is never a gurantee a print will be viewed at its "proper" distance, there is never a guarantee it will be perceived as intended. This is especially the case with larger prints.
While I am now comfortable with above, I welcome any comments on this.
Witold Grabiec
15-Nov-2003, 07:05
To clarify the Hyperfocal Distance concept:
Hyperfocal Distance is what you set your lens on AND, this is the closest subject in focus (not half of that) with far end of the resulting Dpeth of Field falling on infinity. So again, if your HD is
9.3 ft, then everything closer then 9.3 ft will be outside of your depth of field limits (as per chosen COC value, and other factors).
Witold Grabiec
15-Nov-2003, 07:26
I stand corrected, it must be this damn coffee this morning:
Phil is correct - HD is put to work so you get your far end of DOF on infinity and the near end will be half of HD. Talk about Circle of Confusion huh?
Darin Cozine
15-Nov-2003, 13:15
Thank you all for your clarifacatinos (and confusions :)
Alan Davenport
16-Nov-2003, 15:24
Talk about Circle of Confusion huh? LOL!
Maybe I'm out of line here, but I'm mostly wondering how many people ever worry about hyperfocal distance with a large format camera. Once you get the camera on a tripod and start using movements, is
HD focusing really meaningful? Are that many folks still using press cameras handheld?
phil sweeney
17-Nov-2003, 02:57
Alan: probably not often however, for me, I do when using a 90mm lens for 4 x 5. See this small article on prefocus.
Depth of field with a 90mm lense on a 4 x 5 (http://home.att.net/~shipale/DOF.html)
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The tread life is normally distributed with a standard deviation of 295.5 kilometers
Number of results: 27,486
It is claimed that an automobile is driven on the average more than 20,000 kilometers per year. Random sample of 10 automobile owners are asked to keep a record of the kilometers they travel. If the
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Friday, June 3, 2011 at 5:35pm by Ali
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Friday, December 9, 2011 at 10:47am by Ray
statistics probability
The scores on a test are normally distributed with a mean of 140 and a standard deviation of 28, what is the score that is one standard deviation above the mean?
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Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15.
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Tuesday, April 12, 2011 at 1:57pm by kaleb
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ivy tech
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cost of producing is 50 cents each and they sell for 1.25 each. demand for programs at each game is normally distributed with a mean of 2500 and standard deviation if 200. how many shouild be
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Thursday, May 5, 2011 at 1:53am by Confused
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Monday, October 3, 2011 at 1:50am by Katty
Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
Monday, October 3, 2011 at 6:12pm by Jenifer
Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be115 or more?
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The Cause of X-ray crystal Diffraction
This is a multi-part question that stems from understanding how X-ray diffraction occurs in crystals (eg. protein crystals).
1. Diffraction occurs when Bragg's Law is satisfied, but I'm sure the waves aren't actually being reflected. The x-ray's are scattering. What type of scattering is it? (I think it is Rutherford
scattering or Rayleigh scattering)
It is elastic scattering, or something very close to elastic. The scattered photon doesn't lose significant amounts of energy. If the photon did change energy, the scattered wave would change
wavelength. The scattered wave would probably change phase. The scattered wave from an atom couldn't interfere with scattered waves from other atoms.
2. When this scattering occurs, is the electron absorbing and re-emitting the x-ray? If so, is this the type of absorption that occurs as electrons shift energy shells, or is the absorption and
re-emission a result of electrons moving in the electron cloud (eg. oscillating)?
No. The electron can't be absorbing and remitting the x-ray energy. If it did, the result "scattered" wave from one atom can't interfere with the scattered wave from other atoms.
There are x-rays that are created by the absorption and emission x-ray energy. This is called x-ray fluorescence. X-ray fluorescence occurs at a longer wavelength then the incident x-rays. Because
the phase is randomized, there is no diffraction pattern associated with x-ray fluorescence. X-ray fluorescence is important in chemical analysis, but not in crystallographic analysis.
One caveat. Quantum mechanical analysis, which you probably won't need for a while, includes both real and virtual photons.
I have been talking about real photons. There is a concept that scattering occurs by absorption and re-emission of virtual photons. With virtual photons, energy is created and destroyed on a time
scale where the Heisenberg uncertainty principle. However, the energy that is created and destroyed can't be measured.
One can say that the electrons in the atoms of the crystal, when the incident xray wave hits them, exchanges virtual photons. This can confuse discussions with experts between field. Virtual photon
exchange is strictly an internal process within the atom, and doesn't directly effect the diffraction pattern.
The real photons contributing to the diffraction pattern do not change energy. Real photons that change energy can't contribute to the diffraction pattern.
Virtual photons are unimportant for crystal analysis. Quantum mechanics is generally unnecessary for determining the structure of a crystal.
3. I assume that the electrons are re-emitting x-rays and those x-rays that are in phase result in diffraction. Electrons can emit x-rays in all directions, why is it that diffraction is strongest in
the direction of the x-ray source/ x-ray generator. That is, why is the strongest diffraction around the beam stop on the detector? Why don't you get low resolution, high intensity diffraction
anywhere else (eg. 90 degrees to the x-ray source)?
It is a little unclear what you mean by the diffraction is strongest in the direction of the x-ray source and generator. A diffraction pattern can be seen over a broad range of angles. I think that a
high resolution diffraction pattern, if not high intensity, can even be seen at 90 degrees. I conjecture that you are talking about the extinction component.
The extinction component is the "shadow" of the atoms. The photons that are scattered can't proceed in the direction of the source. For the limit of measurements done on a classical scale, the
extinction profile is the shadow of the atoms. Ray optics is usually used to analyze the shadow of large objects. However, fine angular measurements would show that the shadow contains a lot of
structure that can't be determined by ray optics.
The extinction component is a diffraction phenomenon. The conventional shadow of large objects is merely the first dark band of the diffraction pattern. The extinction component is caused by the
interference between the scattered light and the incident radiation.
There are three components of energy flow in a scattering experiment. They are the incident flux, the scattered flux, and the extinction flux. The incident flux comes from the source. Diffraction
patterns are associated with the aperture. The scattered flux comes only from the atoms of the object. Interference between waves scattered from different atoms dominates the diffraction pattern of
the scattered flux. The extinction flux is the shadow of the atoms. The diffraction pattern of the scattered flux is dominated by interference between the waves scattered from the atoms and the
incident wave.
Most of what you learn about in classes on diffraction probably concerns the scattered radiation. Most diffraction experiments are done at extreme scattering angles. I think that a lot of experiments
are done at 90 degrees. However, at very small scattering angles the extinction flux is important. Since it is hard to analyze, most diffraction experiments include a baffle to block off the
extinction flux.
The baffle at small angles has two purposes. First, it blocks off the shadow of the atoms. The shadow of the atoms has some information on the source, which crystallographers don't care about.
Second, the baffle blocks off the incident radiation. The incident radiation contains information about the source and contains no information about the crystal. Therefore, the baffle can be
considered a "filter" that screens out information about the source of xray radiation.
Beer's Law comes from the extinction flux. The incident flux minus the extinction flux is the transmitted beam. It decreases due to both absorption of radiation and scattering. It may be useful to
consider the experimental differences between using Beer's Law and using the law of diffraction.
An extinction coefficient is determined by the extinction flux. Beer's Law is the exponential decay of energy in a beam with time. The exponential drop in intensity of xray radiation from one end of
the crystal can be described on two scales of angular resolution. One one hand, the extinction coefficient is determined by the interference between incident radiation and scattered radiation. When
measuring the extinction coefficient, one doesn't use a baffle.
The scattered radiation may peak at small scattering angles. However, small scattering angles is also where the shadow lies. Actual experiments involve a compromise. Angles should be small enough to
get intense scattering but large enough to avoid the shadow.
Note all the incident photons are scattered by the crystal. If the sample is thin and the xray photons very high in energy, most photons pass through the crystal without scattering. The atoms are
small so there is plenty of space between the atoms. The value of the cross section is small, and the spacing between atoms high. Most photons just pass through. Hence, the total amount of light
passing through the sample at small angles will mostly be due to incident flux. However, there will be structure to the diffraction of the crystal added by the shadow of the atoms. If all you want is
the structure of the crystal, the transmitted beam is just unwanted background. | {"url":"http://www.physicsforums.com/showthread.php?p=4177811","timestamp":"2014-04-21T04:43:21Z","content_type":null,"content_length":"94244","record_id":"<urn:uuid:342d2c25-7b51-417c-a312-07242ad35981>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00499-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability of detection and minimal value of SNR
From the point of view of the user, an EO detection sensor is characterized by its false alarm rate and by its probability of detection. We have seen that the first criterion, FAR, is the one that
defines the threshold value to be imposed upon the output. In what follows, it is shown how the designer must take into account the second one, i.e. the probability of detection, in order to evaluate
the minimum value of the signal to noise ratio.
By setting :
then :
f and g are said to be reciprocal functions from each other, and there comes :
and hence :
On the other hand, if the gaussian hypothesis is being representative of the « signal + noise » probability density function, the probability of detection is equal to :
and hence :
For a further simplification, we will consider here the case where the incoming signal is not fluctuating from sample to sample (signal is then said to be « certain »or « stationary »), and weak
enough so that its presence or absence does not modify the noise level significantly. This approach can be justified in a vast majority of thermal infrared systems, for which the permanent background
flux is much larger than the flux variation brought by the presence of the target to be detected. Consequently, the average output voltage is almost not modified by the presence of the target. If the
noise is essentially due to the background radiation (in which case the sensor is said to be « BLIP », Background Limited Infrared Photodetector), the noise level is about the same in the absence or
in the presence of the signal, which allows to write :
One concludes then that the minimum signal to noise ratio necessary for the sensor to satisfy the probability of detection and probability of false alarm (after conversion from the false alarm rate
as explained above) :
The table below makes the correpondence between reciprocal functions y and g(y) for several values of y, in case of gaussian noise :
Tableau 03 : Correspondence between reciprocal functions y and g(y) for a Gaussian distribution [zoom...]Info
The abacus that follows (Figure 17) may then be used in order to evaluate the minimum value of signal to noise ratio that is necessary to obtain the couple of specifications requested for Pd and PFA.
This abacus is meant to be used only for the detection of stationary signals embedded in noise (the signal must not fluctuate from sample to sample), so that the « noise alone » and « signal + noise
» probability density functions have the same rms value.
Figure 17 : Relationships between Pd, PFA and signal to noise ratio (case of stationary signal) [zoom...]Info
For example, if the required probability of detection Pd is 90%, and if the probability of false alarm PFA is 10-10, then the minimum value of the video SNR is 8.
This abacus may also be read the other way around, if simulation of the sensor shows that the expected value for its signal to noise ratio is at best SNRmax. In these conditions, the optimal
performances of the sensor are all aligned along the vertical line of abscissa SNRmax. For example, let us suppose that the maximum value that is predicted by the sensor model for the signal to noise
ratio is SNRmax = 6. Then, by consulting the vertical line drawn from that limiting value, one may predict that performances of the sensor will be limited to a set of (Pd, PFA) values such as : Pd =
70% et PFA=10-7, Pd =85% et PFA=10-6, Pd=93% et PFA =10-5, Pd = 98% et PFA=10-4, ... | {"url":"http://optique-ingenieur.org/en/courses/OPI_ang_M05_C05/co/Contenu_16.html","timestamp":"2014-04-21T07:13:16Z","content_type":null,"content_length":"24571","record_id":"<urn:uuid:4fcf52f6-9607-4d18-ad43-ffa4b01190a2>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00585-ip-10-147-4-33.ec2.internal.warc.gz"} |
Glenview, IL Algebra Tutor
Find a Glenview, IL Algebra Tutor
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Carmichael numbers
- ANN. OF MATH , 1982
"... ..."
- Graph Theory Notes N. Y
"... The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many
developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1 ..."
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The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many
developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1
, 2008
"... A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in
number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening ..."
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A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in number
theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdős and Lemke. Kleitman and Lemke then made more general
conjectures for finite groups, strengthening the requirements of zerosum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that
for every sequence (gk) |G| k=1 of |G | elements of a finite abelian group G there is a nonempty subsequence (gk)k∈K such that ∑ k∈K gk = 0G and ∑ k∈K 1/|gk | ≤ 1, where |g | is the order of the
element g ∈ G.
, 2004
"... A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in
number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening ..."
Add to MetaCart
A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in number
theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdős and Lemke. Kleitman and Lemke then made more general
conjectures for finite groups, strengthening the requirements of zerosum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that
for every sequence (gk) |G| k=1 of |G | elements of a finite abelian group G there is a nonempty subsequence (gk)k∈K such that ∑ k∈K gk = 0G and ∑ k∈K 1/|gk | ≤ 1, where |g | is the order of the
element g ∈ G.
, 2013
"... We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at
least x 1/5 when x is large enough (depending on m). ..."
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We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1
/5 when x is large enough (depending on m).
"... Abstract. The Miller-Rabin pseudo primality test is widely used in cryptographic libraries, because of its apparent simplicity. But the test is not always correctly implemented. For example the
pseudo primality test in GNU Crypto 1.1.0 uses a fixed set of bases. This paper shows how this flaw can be ..."
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Abstract. The Miller-Rabin pseudo primality test is widely used in cryptographic libraries, because of its apparent simplicity. But the test is not always correctly implemented. For example the
pseudo primality test in GNU Crypto 1.1.0 uses a fixed set of bases. This paper shows how this flaw can be exploited to break the SRP implementation in GNU Crypto. The attack is demonstrated by
explicitly constructing pseudoprimes that satisfy the parameter checks in SRP and that allow a dictionary attack. This dictionary attack would not be possible if the pseudo primality test were
correctly implemented. Often important details are overlooked in implementations of cryptographic protocols until specific attacks have been demonstrated. The goal of the paper is to demonstrate the
need to implement pseudo primality tests carefully. This is done by describing a concrete attack against GNU Crypto 1.1.0. The pseudo primality test of this library is incorrect. It performs a trial
division and a Miller-Rabin test with a fixed set of bases. Because the bases are known in advance an | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=3371449","timestamp":"2014-04-18T01:17:28Z","content_type":null,"content_length":"23373","record_id":"<urn:uuid:0c6aeec7-cf95-4fda-a429-ed94cc7dfc39>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00560-ip-10-147-4-33.ec2.internal.warc.gz"} |
Partial Differential Equation using D'Alembert's approach
February 23rd 2009, 01:02 PM #1
Jan 2008
Partial Differential Equation using D'Alembert's approach
Solve by using D'Alembert's solution with the even extensions of f(x) and g(x)
$\mu_{tt} = c^2\mu_{xx}$ where $0\leq x <\infty$
$\mu(x,0) = f(x)$, $\mu_t(x,0) = g(x)$ where $0\leq x <\infty$
$\mu_x(0,t) = 0$ where $t\ge 0$
Since we have the boundary condition $u_x(0,t)=0$ we would consider even extensions. Let $f_1(x)\text{ and }g_1(x)$ be even extensions, and we will also assume that these extensions are
well-behaved to satisfy the condition of D'Alembert's solution. Thus, we have that $u_tt = c^2u_{xx}$ for $(x,t) \in \mathbb{R}^2$. Thus, the solution is given by:
$u(x,t) = \frac{1}{2}[f_1(x+ct) - f_1(x-ct)] + \frac{1}{2a}\int_{x-ct}^{x+ct}g_1(\xi) d\xi$.
Notice that $u(-x,t) = u(x,t)$ therefore $u_x(0,t) = 0$, this is precisely what we want.
February 23rd 2009, 02:55 PM #2
Global Moderator
Nov 2005
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Yes, that's what we are doing this year and have done last year...
basic and further integration (inverse trig functions, partial fractions and integration, integration by parts....),
the same with differentiation (implicit differentiation, parametric equations, logarithmic differentiation....)
complex numbers,
sequences and series,
Yes, we have done area under a curve in the first integration unit.
Sorry if i rambled a bit, but just thought i'd explain some of the topics we have done.
For learning everything you know from books some of your solutions are excellent. Even if you had gone to school i'd be impressed.
As for the problems, the sphere one was one my teacher showed me. However, the rather difficult logic/story problem about the walkers came from a small local maths paper copied for me by my teacher.
So unfortunately they didn't come from a book as such. Apologies | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=25080","timestamp":"2014-04-18T11:01:08Z","content_type":null,"content_length":"18997","record_id":"<urn:uuid:cabbf402-fe5d-438d-9121-d5793b2e88f2>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra 2- reposted no one answered
Number of results: 155,503
Algebra 2- reposted no one answered
Thank you, I think I was just over thinking the questions. One more; how to I find the y intercept of the function?
Friday, November 2, 2007 at 8:53am by Allisen
Algebra 2- reposted no one answered
The function is not a straight line, but since you are talking about the y-axis intercept, that is where x = 0, and they tell you that the value of y there is 5.
Friday, November 2, 2007 at 8:53am by drwls
Algebra 2- reposted no one answered
look at the function and answer the questions. x y -3 2 -1 0 0 5 1 2 2 7 What is x when f(x)=2 ? What is a zero of this function?
Friday, November 2, 2007 at 8:53am by Allisen
I posted earlier but no one answered so I was hoping if I reposted someone could help
Sunday, September 18, 2011 at 5:03pm by Elllie
Algebra 2- reposted no one answered
Look at your table and see what x is when y=2. It is right there at the top. A "zero of the function" is where y = 0. Look at your second row of the table, below (-3,2) to see the x value at that
Friday, November 2, 2007 at 8:53am by drwls
physics help!
This question was answered when it was reposted later by "perry".
Wednesday, September 2, 2009 at 10:12pm by drwls
AP Statistics
This question was answered after it was reposted later
Friday, January 9, 2009 at 9:40pm by drwls
I'm sorry... I accidently reposted this--it is not answered yet though.
Monday, January 4, 2010 at 4:03pm by coug
Barbara, we do not do your homework for you. I am sure that when you have answered this and reposted, teachers will be happy to make suggestions.
Saturday, July 26, 2008 at 5:30pm by GuruBlue
English-The Five People you meet in heaven
I know. When I reposted I didn't realize I did. I thought someone erased my previous post. I found it after I reposted. My mistake.
Monday, August 27, 2007 at 10:24pm by brie
I posted this earlier but it didn't get answered. So I'm posting it again with hopes that, this time, it will be answered. Please solve for x: |x-6|>7 I got x=6 as a final answer, but that doesn't
make sense because 6 is not greater than 7. Please help!! Thanks :) No one ...
Tuesday, April 17, 2012 at 7:55pm by Marie
answer a question
Why has no one answered my two math questions i put on here. Please look again UNDER your postings because I see answers there! Ashley, Be sure to look above this post. I reposted them so the math
teachers might find them more easily. =)
Friday, January 19, 2007 at 12:40pm by Ashley
Geometry..... reposted ques by me
the maths tutor where online because ques which were posted after me were answered, and why reposts will be deleted
Friday, September 6, 2013 at 7:59am by leena
OMG, sorry reiny, i didnt think anyone was going to help me out on this one so i reposted it (again with the wrong equation)... the correct one is.. -3x=4y=-6 5x-3z=-22 (3z instead of 3y) 3y+2z=-1
Saturday, November 13, 2010 at 11:20pm by Carry
i reposted it btw ok? :):):) just so you know.
Tuesday, February 17, 2009 at 4:41pm by Melanii
Spanish-Please ignore you nswered it earlier
Please disregard-answered earlier-I reposted as I think you were answering earlier post
Friday, November 25, 2011 at 6:16pm by Mia
see original post
You reposted this question within one minute of the previous. It usually takes more than one minute for this type of question to be answered by a human being. See: http://www.jiskha.com/display.cgi?
Thursday, October 21, 2010 at 12:57am by MathMate
College Algebra
They didn't line up correctly. I reposted the correct way.
Wednesday, December 11, 2013 at 4:54pm by Sarah
Just realized that Steve had already answered this question before you reposted it http://www.jiskha.com/display.cgi?id=1324054249 Have patience and always check if your question has been answered
before re-posting it. It saves unnecessary work on our part.
Friday, December 16, 2011 at 1:33pm by Reiny
Pre Calc
Disregard this question, i reposted a updated one.
Sunday, September 12, 2010 at 6:18pm by Emma M
arggghhh- Algebra
Just noticed that your same question has now been asked and answered 6 times As a matter of fact I answered this same one in Oct 2011, doing it exactly the same way. Who would have thought ??
Tuesday, January 29, 2013 at 9:22pm by Reiny
Wow, you waited all of 6 minutes before your reposted. Check your previous post.
Sunday, January 13, 2013 at 5:07pm by Reiny
Tuesday, April 14, 2009 at 9:40pm by Tanisha HOW COME NO ONE ANSWERED ANY OF MY QUESTI
8th grade Algebra I (junior level math)
We could help you better, if you reposted with a specific question that is causing you problems.
Tuesday, December 1, 2009 at 7:14pm by PsyDAG
Sorry, Damon. I removed her question here while you were answering it. I've reposted your answer above.
Friday, July 22, 2011 at 7:22pm by Ms. Sue
the last few times I've posted questions no one has answered them...I'm starting to get kind of upset. let's see if this one gets answered: the perimeter of a square is 44. find the length of a
Monday, March 18, 2013 at 8:42pm by 21 Guns
Business Communication
I forgot them in this one so I reposted it above this forum. And yes, I'm trying to identify the strengths and weaknesses
Thursday, November 22, 2007 at 10:26pm by Cupcake
Just noticed that Steve answered the same question when you reposted it just a few entries above this Please avoid reposting the same question, especially when it still shows near the top of the list
of questions like it did here. It avoids unnecessary duplication of solutions.
Sunday, March 23, 2014 at 1:26am by Reiny
Tuesday, April 14, 2009 at 9:40pm by Tanisha HOW COME NO ONE ANSWERED ANY OF MY QUESTI
I post more than one because I usually only get one answered because no one is usually on!
Friday, July 27, 2012 at 7:22pm by Anna&Karen
Algebra - Correction
Good thing you reposted it. I misread the problem, and solved it for having 3 times as much 3% as 2%. See bobpursley's solution for the correct answer. Having read my algebra carefully, though, you
should have been able to do it right.
Thursday, December 29, 2011 at 9:51pm by Steve
I answered your same question on Friday at 10:01 drwls answered your same re-posted question at 11:20 Please go back and check if your question has been answered before you post it multiple times.
Friday, May 23, 2008 at 11:23pm by Reiny
ALGEBRA 2 W/TRIG
answered in your last post. Please do not repeat the same question without checking if it has been answered.
Wednesday, August 25, 2010 at 9:16pm by Reiny
Please don't repost a problem that you posted before, without checking if it has been answered. I answered your question.
Monday, October 4, 2010 at 4:39pm by Reiny
This should already be answered. I answered later posts which appear above this one.
Wednesday, January 23, 2008 at 10:10pm by SraJMcGin
cultural diversity-reposted question
I am still of the opinion that C is the correct answer. Regardless of her parents' deaths, she is being raised by family members "as one of their own"
Saturday, October 27, 2007 at 2:51pm by GuruBlue
algebra 1
I answered one for you. Now it's your turn. I'll be glad to check your answer.
Thursday, February 3, 2011 at 1:47pm by Ms. Sue
algebra 1
I answered one for you. Now it's your turn. I'll be glad to check your answer.
Thursday, February 3, 2011 at 1:48pm by Ms. Sue
Science help.
Damon, I posted almost the same questions as her and way before her too. How come you answered hers, but no one answered mine?
Tuesday, February 4, 2014 at 3:32pm by Nicole M
Perhaps it is different names, but the questions are remarkably similar. Having answered one, I thing WLS has answered them all. Acentripetal = v^2/r
Thursday, August 11, 2011 at 12:01pm by Damon
for # 10 I answered scientific for #9 I answered the last one Family,religion, education goeverment and economy
Wednesday, May 8, 2013 at 4:08pm by mace Ms Sue pls check
Thanks for those who answered my previous question! I have just one more: x^2-625=0, what is the GCF? How is it deterimined? It does not appear to me that there is one. How would it then be factored
and have the zero-product powers applied? Then, what about 120x^2-34x-21=0 (...
Monday, November 10, 2008 at 5:14pm by Rachelle
I just answered that same question you posted about one hour ago.
Tuesday, September 9, 2008 at 7:20pm by Reiny
Algebra II
look at the condition I stated for V.A.' s in the other post I answered for you, and see if you can answer this one.
Wednesday, July 6, 2011 at 10:55pm by Reiny
algebra story problem
I just answered this one above for someone that posted as "college student"
Friday, February 15, 2013 at 3:19pm by JJ
MAths algebra
simple Pythagoras, look at the same kind of questions I answered for you above this one.
Tuesday, April 2, 2013 at 2:58am by Reiny
There is only one table, and that is what you have written. I answered this question elsewhere. You posted it more than once.
Thursday, September 11, 2008 at 8:58pm by drwls
Algebra B
You need to use parentheses. x-4/x+3 = x-1/x+2 Is the problem, x - 4/(x+3) = (x-1)/x + 2 x - 4/x + 3 = x - 1/(x+2) Or x - 4/x + 3 = x - 1/x + 2 Can you see the difference in these equations? This is
probably why no one answered your question.
Thursday, January 27, 2011 at 11:51pm by helper
3rd grade math
on one test, chi answered 76 out of 100 questions correctly. on another test, he answered test, he answered 7 out of 10 questions correctly. write these scores as decimals and compare them.
Tuesday, January 29, 2013 at 6:40pm by mythreyee
College Mathematics
I answered this question for you 20 minutes before you reposted it http://www.jiskha.com/display.cgi?id=1363627561 What part of my answer did you not understand ?
Monday, March 18, 2013 at 2:09pm by Reiny
algebra interest word problem
this problem has the same structure as the one I answered for you 4 minutes ago. The only difference is that one has to conclude that the amount of interest is $166 let me know what rate you got.
Monday, March 9, 2009 at 7:23pm by Reiny
to HELPER advanced funtions
that answer wasnt the one I looked for. i didnt want to be mean so i reposted it. can you please help me! THANK YOU SO MUCH !! describe the similarities and differences between y=x^2 and a power
funtion with a even degree higher than two?
Wednesday, February 9, 2011 at 8:43pm by balasaba
answered twice already
I have answered this one twice as well ^ means exponent times 2 for two spheres forcing third
Saturday, January 1, 2011 at 4:31pm by Damon
If your question is answered, you'll see a post right underneath it and to the right a little bit -- and it should be answered by one of the tutors.
Wednesday, February 9, 2011 at 12:20pm by Writeacher
I posted this last night but it didn't get answered. So I'm posting it again with hopes that, this time, it will be answered. Please solve for x: |x-6|>7 I got x=6 as a final answer, but that doesn't
make sense because 6 is not greater than 7. Please help!! Thanks :)
Tuesday, April 17, 2012 at 7:29pm by Marie
Problem reposted and answered here: http://www.jiskha.com/display.cgi?id=1255001673
Thursday, October 8, 2009 at 7:30am by MathMate
this question is still not answered will some one walk through it step by step
Thursday, October 9, 2008 at 9:37pm by Not answered
Reiny: I read your post, then glanced at mine, said oh my, erased my post, then said oh my again, and reposted. Such a life when one is tired.
Saturday, March 16, 2013 at 8:56pm by bobpursley
Dr.Bob the question Ihad reposted (NOT the old one
Thank your help Dr.Bob
Sunday, March 9, 2008 at 8:46pm by ~christina~
Juanita answered 21, and Anna answered three fewer than this, so Anna must have answered 18. Earl answered two more than Anna, so Earl must have answered... what?
Thursday, September 18, 2008 at 7:14pm by David Q
This is almost identical to the one I just answered, just with new numbers. You can do it.
Thursday, January 17, 2008 at 9:37pm by Damon
For Alex - Math
Alex , I had answered your question before you reposted it http://www.jiskha.com/display.cgi?id=1352503674 By not checking your old posting you are causing other tutors to do unnecessary and extra
Friday, November 9, 2012 at 7:24pm by Reiny
Great! Thank you. Pretty much I really only need help on the circles and that other post below (the one that hasn't been answered) and then I should be done- Finally!=)
Saturday, September 15, 2007 at 4:19pm by Jamie
Algebra 2
looks like the same kind of question as the one I just answered, try to follow the same method.
Monday, September 28, 2009 at 10:59am by Reiny
Thanks, it didn't look right thats why I reposted
Thursday, January 27, 2011 at 7:27pm by Hannah
I posted this earlier in the day but no one answered. Can anyone help with these problems? Factor completely: 1. 8x3 + 2x3 – 12x –3 2. 27x3+8 3.125 – 8x3
Sunday, January 13, 2008 at 6:22pm by Ashlynn
English (Summer Reading Essay)
I reposted it by the way becasue it was just terrible before when it was all one paragraph becasue I couldn't post multiple paragraphs I had to make it all one and even now it wouldn't let me so I
had to post sevearl paragaphs in there on post. At least now it'll be more ...
Wednesday, July 16, 2008 at 10:54am by Dylan
Answered below. Please don't repost without checking your original post first for an answer. Also, tutors volunteer their time. You must have patience. If your question is not answered after 4 hours
or so, then repost your question.
Friday, February 11, 2011 at 10:29pm by helper
i still didnt get it.. so i reposted
Sunday, March 9, 2008 at 10:57am by Hokoslavia
sorry accidentally reposted!!
Tuesday, November 1, 2011 at 1:34pm by Alison
I'm confused on when to use a two capital letters, two lowercase letters or one of each. This is for traits. We just started it in Biology. It has something to do with heterozygous and homozygous
dominant and homozygous recessive and so on. For example Pp PP pp I don't know ...
Monday, April 7, 2008 at 7:12pm by Joe Smith Bob
graphing math (last ones) ill figure out the rest
The first question I answered below at the original post. The second question is done the same way that I answered for x=4 except this one has a vertical line parallel to the y axis and passing
through the point (1/4,0).
Sunday, October 4, 2009 at 10:30pm by DrBob222
Good heavens! This is the one problem that has been answered! Obviously you haven't even checked to see if it's been answered. Please do not post any more questions without indicating what you know
and don't know about the problem.
Monday, August 13, 2012 at 5:51pm by Ms. Sue
Can you help me with this one? Each auditorium has both floor seats, and balcony seats. Auditoruim A has 200 floor seats and 500 balcony seats. Auditorium B has 100 floor seats and 200 balcony seats.
Let f represent the cost of a floor seat and b represent the cost of a ...
Thursday, November 5, 2009 at 8:19pm by newcomb
English 3
You posted these questions before, and no one answered. I have not answered because I have not read these works or have not read them recently enough. Sorry.
Thursday, March 29, 2012 at 11:53am by Writeacher
college organic chemistry
It is not a good idea to piggy back questions because it looks as though the original question has been answered, so one looks at the next. I have answered the hexyl benzene question above.
Thursday, November 11, 2010 at 8:08am by Dr Russ
What is the average summer temperature in the desert biome? Select one: a. 38 degrees celcius b. 48 degrees celcius c. 58 degrees celcius Question 2 Not yet answered Marked out of 1.00 Flag question
Question text What is the average winter temperature in the Savanna biome? ...
Friday, March 15, 2013 at 1:36pm by paragonnova
None of my questions are being answered
From Cabe I found nothing either although there was a question about an organization CABE. There were 2 for math (one was algebra) by Jake and Eliza also posted math. I try to answer every one that
HAS no answer if I can, but I do NOT do math in any form! Sra (aka Mme)
Tuesday, November 1, 2011 at 11:40pm by SraJMcGin
A simple __________ expresses one complete thought (one independent clause) with one subject and one predicate. A) clause B) phrase C) sentence I answered c, although I think it could be B also
Monday, October 15, 2012 at 5:18pm by Marie
Discrete Math
Idk what happened at 7:24 I think my computer had a glitch or something and it reposted. This one is throwing me for a loop: Solve the recurrence relation a_n = 2a_n - 1 – a_n -2, n ≥ 2, given a₀ =
40, a₁ = 37. Characteristic polynomial: x^2 - 2 + 1 How do I ...
Tuesday, April 5, 2011 at 4:11pm by Francesca
help please:)
i post it more then once because no one answered that one and i thought no one will pay attention to that because its all the way on the bottom. and this is like my first time doing that.
Wednesday, September 28, 2011 at 4:17pm by ciara
Thank you both for your suggestions. I appreciate it. I reposted the essay.
Sunday, February 3, 2008 at 11:34pm by Bizzy
i did try to but i still didnt understand it thats why i reposted it
Sunday, March 15, 2009 at 10:29pm by cb
Geometry..... reposted ques by me
i did not made the diagram like this.......
Friday, September 6, 2013 at 7:59am by leena
If you piggy back one question on another you are unlikely to get an answer, because it looks as though the question has been answered. You should show you question as a new problem.
Monday, August 31, 2009 at 11:42pm by Dr Russ
On one test, Chi answered 76 out of 100 questions correctly . On another test , he answered 7 out of 10 questions corrctly. Write these scores as decimals and compare them. I don't know..
Sunday, February 16, 2014 at 9:11pm by Nawaf
Can someone please help me answer these two questions? I have already answered one, and need it checked. Give any three ordered pairs that are solutions of the equation: 1. x-y=6 (12,6) (10,4) (15,9)
2.-3x - 2y = 0 ??? 3.2x- 3y = 12 I have only one ordered pair for this: (9,2...
Wednesday, May 6, 2009 at 10:30am by mysterychicken
Do you really read what you've posted. More importantly do you read the responses? Bob Pursley has answered. Damon has answered. I have answered. In addition I pointed out some errors and you keep
repeating those. What's the problem?
Saturday, March 1, 2014 at 6:56pm by DrBob222
Critical Thinking
Ok Ms. Sue I have reposted my question
Monday, September 29, 2008 at 1:11pm by Mike
Thank you-I reposted a new post so please disregard it-thank you again
Tuesday, November 30, 2010 at 8:00pm by Julie
Gr. 9 math
I reposted a better structure at 1:58 thank you for your help so far.
Sunday, December 12, 2010 at 11:58am by olav
I posted a few questions and they all disappeared so i reposted.
Monday, February 13, 2012 at 6:15pm by Katie
I posted a few questions and they all disappeared so i reposted.
Monday, February 13, 2012 at 6:15pm by Katie
Math help
Thank you both for answering,sorry i reposted same question.
Tuesday, March 6, 2012 at 8:22pm by Ashley k
Please only post your questions once. Repeating posts will not get a quicker response. In addition, it wastes our time looking over reposts that have already been answered in a previous post. Thank
you. I answered your question in one of your later posts.
Thursday, February 5, 2009 at 4:17pm by PsyDAG
Math HELPPP
In a survey of 2000 adults 50 years and older of whom 40% were retired and 60% were pre-retired, the following question was asked: Do you expect your income needs to vary from year to year in
retirement? Of those who were retired, 33% answered no, and 67% answered yes. Of ...
Tuesday, October 16, 2012 at 5:20pm by Bill
Bob received 6 points for each question he answered correctly on Part 1 of a test and 4 points for each question he answered correctly on Part 2. If he answered 33 questions correctly and received a
total of 168 points, how many questions did he answer correctly on Part 1?
Friday, June 1, 2012 at 3:57pm by Kevin
grade 12 chemisstry
I think this was reposted above with answers and I responded.
Tuesday, June 3, 2008 at 4:37pm by DrBob222
Re: Math
I reposted because I asked this about 2 hours ago just in case
Tuesday, September 28, 2010 at 7:53pm by Amy~
Hint: recurrence --> fibonacci. This question is reposted from brilliant.
Wednesday, July 10, 2013 at 11:00am by exactly
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/search/index.cgi?query=Algebra+2-+reposted+no+one+answered","timestamp":"2014-04-18T08:49:19Z","content_type":null,"content_length":"35097","record_id":"<urn:uuid:be108695-8bc7-48c5-8835-ce16504426e7>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00442-ip-10-147-4-33.ec2.internal.warc.gz"} |
data flow
Next: REC/C operators and predicates Up: The rectbl.h header and Previous: REC header
Figure 7 shows the data flow in REC/C, which is not very extensive. There is essentially one array, a pushdown list of ample dimension, which will hold some complex numbers. There data can be
introduced, following which partial results of computations can be stored, and eventually used as coordinates for the graphing operators. These operators display points or lines in the Plane Window
in various forms. | {"url":"http://delta.cs.cinvestav.mx/~mcintosh/comun/rec-c/node13.html","timestamp":"2014-04-18T05:29:47Z","content_type":null,"content_length":"3085","record_id":"<urn:uuid:67a59381-5ac6-4463-997c-da2ad30f566c>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
Communication Effects for Message-Based Concurrency
, 1991
"... We present the first algorithm for reconstructing the types and effects of expressions in the presence of first class procedures in a polymorphic typed language. Effects are static descriptions
of the dynamic behavior of expressions. Just as a type describes what an expression computes, an effect de ..."
Cited by 109 (6 self)
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We present the first algorithm for reconstructing the types and effects of expressions in the presence of first class procedures in a polymorphic typed language. Effects are static descriptions of
the dynamic behavior of expressions. Just as a type describes what an expression computes, an effect describes how an expression computes. Types are more complicated to reconstruct in the presence of
effects because the algebra of effects induces complex constraints on both effects and types. In this paper we show how to perform reconstruction in the presence of such constraints with a new
algorithm called algebraic reconstruction, prove that it is sound and complete, and discuss its practical import. This research was supported by DARPA under ONR Contract N00014-89-J-1988. 1
, 1997
"... We address the type and effect inference in higher-order concurrent functional programming languages à la Concurrent ML. We present three extensions of the type and effect discipline. First, the
discipline is extended to deal with infinite but recursive effects. Second, the inferred effects are stru ..."
Add to MetaCart
We address the type and effect inference in higher-order concurrent functional programming languages à la Concurrent ML. We present three extensions of the type and effect discipline. First, the
discipline is extended to deal with infinite but recursive effects. Second, the inferred effects are structured, i.e., we keep track of the structure of effects (sequencing, choice, parallel
composition, and recursion) instead of using an AC1I (associative, commutative, unitary, and idempotent) effect cumulation operator. Third, for the sake of flexibility, a subtyping relation is
considered on the type and effect algebras. This is much more powerful than the classical subeffecting technique. This is meant to avoid type mismatches that may arise in some typing contexts between
subexpressions that have similar type structure but different effect annotations. We present the language syntax together with its static semantics. The latter consists of the typing rules and an
inference algorithm that...
"... Abstract. Monadic effect systems provide a unified way of tracking effects of computations, but there is no unified mechanism for tracking how computations rely on the environment in which they
are executed. This is becoming an important problem for modern software – we need to track where distribut ..."
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Abstract. Monadic effect systems provide a unified way of tracking effects of computations, but there is no unified mechanism for tracking how computations rely on the environment in which they are
executed. This is becoming an important problem for modern software – we need to track where distributed computations run, which resources a program uses and how they use other capabilities of the
environment. We consider three examples of context-dependence analysis: liveness analysis, tracking the use of implicit parameters (similar to tracking of resource usage in distributed computation),
and calculating caching requirements for dataflow programs. Informed by these cases, we present a unified calculus for tracking context dependence in functional languages together with a categorical
semantics based on indexed comonads. We believe that indexed comonads are the right foundation for constructing context-aware languages and type systems and that following an approach akin to monads
can lead to a widespread use of the concept. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2043523","timestamp":"2014-04-17T20:03:10Z","content_type":null,"content_length":"17660","record_id":"<urn:uuid:69159968-80d4-4699-b297-44976d092d9b>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00586-ip-10-147-4-33.ec2.internal.warc.gz"} |
Excel/OpenOffice spreadsheet template for expected and actual values
up vote 0 down vote favorite
I would like to create a spreadsheet to generate graphs about cost items (anything with a cost, doesn't matter what) in order to have:
• expected values: these would be created using some formulas to simulate future costs.
• actual values: these would be put in manually based on actual costs after billing.
At the end of a given time period (weekly, monthly...) the user would "freeze" the expected value that was calculated earlier, and add the corresponding actual value (and these should never be
changed again).
The process would repeat itself for the next time periods while re-using either the same or new calculations for the expected values.
Over time, graphs will show the evolution of and different between the expected and actual values so as to allow for improvements (are the input values for calculations too optimistic / pessimistic?,
were any cost items forgotten or not properly defined?...).
Can someone point me to a template from which I could create such a spreadsheet?
1 Does this work for you? These kinds of things are so particular to your line of business that using a pre-canned template is rarely acceptable, and a proper financial forecasting system is often
the best solution, rather than using Excel/OO. – allquixotic Nov 28 '12 at 16:20
Thanks. The idea is there but that model is way too static (if I want to add an item I have a lot to update for it to work): I'm gonna continue my quest and will post whatever I find in an answer.
– user359650 Nov 28 '12 at 16:41
Most likely anything that is not "way too static" is an information system -- before you start looking at MS Access, maybe investigate real financial software first. – allquixotic Nov 28 '12 at
add comment
Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.
Browse other questions tagged microsoft-excel openoffice.org spreadsheet simulation modeling or ask your own question. | {"url":"http://superuser.com/questions/511884/excel-openoffice-spreadsheet-template-for-expected-and-actual-values?answertab=votes","timestamp":"2014-04-23T23:56:26Z","content_type":null,"content_length":"66209","record_id":"<urn:uuid:1efe0f3c-a325-42ab-ba04-6c6a6ac21668>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Grundlehren der Mathematischen Wissenschaften 338. Berlin: Springer (ISBN 978-3-540-71049-3/hbk). xxii, 973 p. EUR 99.95/net; SFR 166.00; $ 159.00; £ 79.00 (2009).
As the title suggests, the book is aimed to old and new problems of optimal transport. After the publication of his [Topics in optimal transportation. Providence, RI: American Mathematical Society
(AMS) (2003; Zbl 1106.90001)], the present book offered to the author a good opportunity to approach differently the whole theory, with alternative proofs and a more probabilistic presentation, and
to incorporate new results. Among these new results was John Mather’s minimal measures which has a lot to do with optimal transport, and the fact that optimal transport could provide a robust
synthetic approach to Ricci curvature bounds. In comparison with the previous book, this approach is oriented more on probability, geometry, and dynamical systems, and less on analysis and physics.
According to the author’s recommendation, both books can be read independently, or together, and their complementarity can have pedagogical value.
In order to give the reader a feeling for the content and applicability of the results we point out in the sequel the titles of the parts and the chapters: Introduction: (1) Couplings and changes of
variables; (2) Three examples of coupling techniques; (3) The founding fathers of optimal transport; Part I – Qualitative description of optimal transport: (4) Basic properties; (5) Cyclical
monotonicity and Kantorovich duality; (6) The Wasserstein distances; (7) Displacement interpolation; (8) The Monge-Mather shortening principle; (9) Solution of the Monge problem. I – Global approach;
(10) Solution of the Monge problem. II – Local approach; (11) The Jacobian equation; (12) Smoothness; (13) Qualitative picture; Part II Optimal transport and Riemannian geometry: (14) Ricci
curvature; (15) Otto calculus; (16) Displacement convexity. I; (17) Displacement convexity. II; (18) Volume control; (19) Density control and local regularity; (20) Infinitesimal displacement
convexity; (21) Isoperimetric-type inequalities; (22) Concentration inequalities; (23) Gradient flows. I; (24) Gradient flows. II – Qualitative properties; (25) Gradient flows. III – Functional
inequalities; Part III Synthetic treatment of Ricci curvature: (26) Analytic and synthetic points of view; (27) Convergence of metric-measure spaces; (28) Stability of optimal transport; (29) Weak
Ricci curvature bounds. I – Definition and stability; (30) Weak Ricci curvature bounds. II – Geometric and analytic properties; Conclusions and open problems.
This meticulous work is based on very large bibliography (846 titles) that is converted into a very valuable monograph that presents many statements and theorems written specifically for this
approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes. Disseminated throughout the book, several appendices contain either some domains of
mathematics useful to non-experts, or proofs of important auxiliary results.
Very useful instruments such as List of short statements, List of figures, Index, and Some notable cost functions are accessible at the end of the book.
53-02 Research monographs (differential geometry)
49-02 Research monographs (calculus of variations)
90-02 Research monographs (optimization)
93-02 Research monographs (systems and control) | {"url":"http://zbmath.org/?format=complete&q=an:1156.53003","timestamp":"2014-04-20T09:03:17Z","content_type":null,"content_length":"24320","record_id":"<urn:uuid:0d4f8a88-302e-4268-9420-7e8c525b7fda>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00211-ip-10-147-4-33.ec2.internal.warc.gz"} |
Books request on nonlinear recurrence relations.
up vote 3 down vote favorite
Hi, do you have some sort of a bibliography on advanced techniques in recurrence equations, such as nonlinear ones and others?
As I see it recurrence equations are quite similar to differential equations.
Obviously we can go from ODE to a recurrence relation by plugging suitable power series solution to the ODE, but can I also reverse this proccess; i.e, have a recurrence relation and then find a
suitable ODE for this relation, is this always the case?
Thanks in advance, Alan.
recurrences ca.analysis-and-odes
add comment
1 Answer
active oldest votes
There's a rather old book by Levy and Lessman, Finite Difference Equations.
up vote 1 down vote
Thanks, I'll check it. – Alan Mar 19 '13 at 8:05
add comment
Not the answer you're looking for? Browse other questions tagged recurrences ca.analysis-and-odes or ask your own question. | {"url":"http://mathoverflow.net/questions/124851/books-request-on-nonlinear-recurrence-relations","timestamp":"2014-04-18T10:57:26Z","content_type":null,"content_length":"50402","record_id":"<urn:uuid:7817f732-75a4-4c7e-ae56-d8891e5d8e9b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00339-ip-10-147-4-33.ec2.internal.warc.gz"} |
Implicit Sampling
Multiphysics in Materials and Industrial Devices
Simulation of Vertical-Axis Wind Turbines
Rolling Tires and Failure
About the Group
The Mathematics Group at LBNL develops new mathematical models, devises new algorithms, explores new applications, exports key technologies, and trains young scientists in support of DOE. We use
mathematical tools from a variety of areas in mathematics, physics, statistics, and computer science, including statistical physics, differential geometry, asymptotics, graph theory, partial
differential equations, discrete mathematics, and combinatorics. The problems we attack are both technologically interesting and mathematically challenging, and form a set of interrelated computing
methodologies and applications in support of the DOE energy mission.
One set of topics focuses on optimizing the manufacture and operation of engineering and industrial processes. Applications include semiconductors, coating rollers, inkjet printing technologies and
microfluid effects, foams in manufacturing processes, new metals, granular mixers, coal hoppers, rolling tires, mode-locked lasers, wind turbines, vibrating RF MEMS devices for wireless
communications, and dynamic fracture in bulk metallic glasses. Another set of topics focuses on tools for the analysis of energy processes, and includes stochastic methods in environmental science,
data analysis for meteorological data, data synthesis for wind energy and large-scale ocean currents, seismic imaging, image processing and analysis for analyzing cellular structures, and complex
fluid-membrane solvers for understanding the dynamics behind cellular development in new biofuels. and path planning for determining chemical accessibility in new materials such as zeolite and metal
organic frameworks for gas separation sieves in carbon sequestration.
Tackling these problems with enough accuracy to be useful requires some of the most advanced computational resources. To that end, part of our work is aimed at developing the mathematics behind
higher order accurate algorithms that naturally lend themselves to new architectures, where attention to communication, data exchange, and decompositions offer the opportunity for tremendous speedup.
Our program consists of faculty, postdocs, graduate students, and visitors. The four principal investigators (Chorin, Persson, Sethian, and Wilkening) are all faculty at UC Berkeley. However, the
close proximity of LBNL to campus and the far greater wealth of resources at LBNL makes it the attractive center of applied computational mathematics at Berkeley. This is reflected in the fact that
all the graduate students, postdocs and faculty have a shared seminar space and offices. Together with the presence of NERSC and other computational resources, this makes LBNL a natural focal point
and magnet. | {"url":"http://crd.lbl.gov/groups-depts/math/","timestamp":"2014-04-17T05:26:27Z","content_type":null,"content_length":"25469","record_id":"<urn:uuid:7240a5cd-8120-4056-871b-654d4f2445cd>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00040-ip-10-147-4-33.ec2.internal.warc.gz"} |
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12 Subjects: including algebra 1, calculus, statistics, algebra 2 | {"url":"http://www.purplemath.com/rolling_meadows_algebra_1_tutors.php","timestamp":"2014-04-17T15:56:09Z","content_type":null,"content_length":"24289","record_id":"<urn:uuid:9d983f6f-ce70-48ac-a420-89e0445dd858>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00629-ip-10-147-4-33.ec2.internal.warc.gz"} |
What Happened to Euclidean Geometry in Modern Mathematics Curriculums?
April 22nd 2012, 07:31 PM
What Happened to Euclidean Geometry in Modern Mathematics Curriculums?
It has occurred to me over the course of my first year here at UCLA that mathematicians used to be very good at Euclidean geometry (take the Principia as a prime example, or the early solutions
to the Brachistochrone problem)...and over time that has faded. I personally have never taken a course in the subject, and beyond simple facts, I rarely have ever needed it. Though I can see its
used in modeling/solving some problems; for example, it was useful in geometric optics/physics problems (but these are mostly lower division/basic problems not requiring a lock of postulates/
Yet in looking through some of these forums I've noticed that a good number of people here are actually pretty good at it, and there are many questions pertaining to the subject. So I guess this
is now a two-part discussion. Is it worthwhile to become more skilled in Euclidean geometry (i.e. take a rigorous course in foundations of geometry)? And why do you suppose hardly anyone knows
the subject anymore?
April 26th 2012, 12:39 PM
Re: What Happened to Euclidean Geometry in Modern Mathematics Curriculums?
Well, I can't speak for today's schools but I took a geometry course in High School many years ago and have, in fact, taught Euclidean geometry in college- although as a remedial course. You will
certainly need to know basic Euclidean geometry for "Cartesian coordinates" which are a fundamental part of pre-Calculus and Calculus courses. What reason do you have to believe that "hardly
anyone knows the subject anymore"? | {"url":"http://mathhelpforum.com/math/197752-what-happened-euclidean-geometry-modern-mathematics-curriculums-print.html","timestamp":"2014-04-21T05:15:35Z","content_type":null,"content_length":"4945","record_id":"<urn:uuid:88a0fc23-35f1-44af-b4d8-2fe95113872a>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00043-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mc Lean, VA Algebra Tutor
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Obsessed With Obsessions
Paul Auster , whose eyes burn with the intensity of a star gone supernova, was staring me down, and not in a seductive manner. More like I was threatening his children.
Auster, a writer whose work I admire, was taking questions following a reading at my graduate school, and I’d asked him about his apparent obsession with money in his fiction. I imagine he was so
sick of addressing this particular issue that he turned into a literary anti-superhero and tried killing his questioner with laser beams springing forth from his pupils.
Funny, I don’t mind being asked about my obsessions—in fact, I can talk obsessively about them.
They are broad and minuscule, sublime and ridiculous: The Mayor of Casterbridge, the Cubs, Rachmaninoff’s Piano Concerto No. 3, Elvis (Presley and Costello), certain unknown bands from my hometown,
the nature (and unfaithfulness) of memory, Jackie Chan, evolutionary psychology, Spinal Tap, certain movie stars, astrophysics, criminalizing Bud Light.
Also, big numbers. Not the kind of numbers used in counting money, or grains of sand, or even atoms in the universe. Larger than that. Numbers like Graham’s number, considered the largest number
ever used in a mathematical proof. (And if you will stick with this mess, I’ll actually show how this relates to writing. Or fiction. Or Paul Auster’s glare. Or all of them.)
Graham’s number is named after one R.L. Graham, who used it in a proof dealing with an really, really hard mathematical problem. What is fascinating about Graham’s number is despite the fact it was
actually put to use, it is beyond human comprehension.
Yes, you can use it but not comprehend it.
The only way I can explain Graham’s number is via exponentials. (And if I mess up the math, please feel free to correct me).
Starting with: 3^3 = 3x3x3 = 27, which you already knew. But add another 3 to the top of the exponential string–just one–and you get the following: 3^3^3 =3^27=7,652,600,000,000 (That’s 7.65
By adding an additional three to the stack, and it’s: 3^3^3^3 = 3^3^27= 3^^7,652,600,000,000.
Don’t try this at home
Which means multiplying 3 by 3 more than 7.6 trillion times. In other words, stacking four 3s on top of each other creates a really, really big number. So big you can’t calculate it—3^3^3^3 is by far
more than all the atoms in the observable universe (10^70).
Simply stack “3″s that another 61 times (yes, for a total of 64 times) and you get Graham’s number, which is so far beyond the realm of human comprehension that it beggars the imagination.
Compare it to something most of us are familiar with: the googol (the number, not the site engine that deposited you here in search of pr0n-o-graphic pictures). A googol is 10^100—a 10 with 100 zeros
behind it.
A googolplex is 10^10^100—a googol a googol number of times. A googolplex is so enormous that if you tried writing out the number, and put each “0″ on an atom, there wouldn’t be enough material in
the entire universe to do so.
Now consider this: Graham’s number is far, far, far larger than a googolplex, which is to Graham’s number as an electron is to the Milky Way.
In a sense, Graham’s number bigger than infinity, for infinity is at least comprehensible. To me, Graham’s number is another way of saying that numbers, which are a human invention, exist in their
own dimension independent of comprehension. We understand the universe in ways that we cannot understand. Is that the coolest, most amazing, facinating thing ever?
What, you say that it’s not?
The most humbling thing in the universe
OK, so I’m a nerd. In a different life, I would have been a mathematician, if that life actually included aptitude with mathematics. Instead, I became a writer, but one of my preoccupations is
describing the cool, the offbeat, the indescribable—like Graham’s number or Elvis Presley or the Cubs.
Like Paul Auster’s fixation with money, I am obsessed.
And if we aren’t obsessed with something, we really shouldn’t be writing.
The word for today’s post is erudite. I think. Didn’t you just have a birthday? I think Facebook said you did.
You had me at the Cubs. But the rest of this is hooky as well.
Written: Erudition? I think I caught it from a guy on the bus, who was coughing heavily.
As far as birthdays, mine was several months ago, and ’twas not on FB. At least, “bookfraud” isn’t on Facebook. At least that’s the idea.
Adam: Glad that I had you at Cubs, and the rest was hooky as well. Which I think is a good thing, as in hooked you into reading. I hope.
Yes, good thing. Reading. No johns or tackle boxes involved.
I admit it: I skipped all the parts about numbers. But it doesn’t matter because OMG I hate light beer too so let’s be best friends!
Oh, and uh, I was using oh em gee ironically. And stuff. | {"url":"http://bookfraud.com/2011/01/27/obsessed-with-obsessions/","timestamp":"2014-04-20T08:26:28Z","content_type":null,"content_length":"74405","record_id":"<urn:uuid:2e6cfccb-d8e3-4ff2-9f7a-01d05856decb>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
Glendale Heights Prealgebra Tutor
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Proceedings of the fifteenth Southeastern Conference on Combinatorics, Graph Theory and Computing held at Louisiana State University, March 5-8, 1984
Proceedings of the fifteenth Southeastern Conference on Combinatorics, Graph Theory and Computing held at Louisiana State University, March 5-8, 1984, Volume 3
Ralph G. Stanton
We haven't found any reviews in the usual places.
H Niederhausen y 5
Palem and D S Fussel 18
J H Ristroph and C E Ashley 27
20 other sections not shown
Bibliographic information | {"url":"http://books.google.com/books?id=2yklAQAAIAAJ&q=two-graph&dq=related:UOM39015046537521&source=gbs_word_cloud_r&cad=6","timestamp":"2014-04-19T13:08:00Z","content_type":null,"content_length":"116665","record_id":"<urn:uuid:b8707358-2781-41c9-b4bd-69274a71434c>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00637-ip-10-147-4-33.ec2.internal.warc.gz"} |
velocity vector
September 26th 2008, 05:38 AM
Carl Feltham
velocity vector
A level maths mechanics:
a particle has an initial velocity of i + 2j and is accelerating in the direction i + j. if the magnitude of the acceleration is 5root2, find the velocity vector and the speed of the particle
after 2 seconds
September 26th 2008, 05:47 AM
mr fantastic
Same idea as your question at http://www.mathhelpforum.com/math-he...n-vectors.html.
a = (5root2) (i + j)/root2 = 5(i + j).
Now do the necessary integrations. | {"url":"http://mathhelpforum.com/math-topics/50703-velocity-vector-print.html","timestamp":"2014-04-17T14:40:47Z","content_type":null,"content_length":"4510","record_id":"<urn:uuid:05fb4b96-082c-4b2c-995a-87e333c9711b>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
monoids-0.1.33: Monoids, specialized containers and a general map/reduce framework Source code Contents Index
Portability portable
Data.Group Stability experimental
Maintainer Edward Kmett <ekmett@gmail.com>
Extends Monoid to support Group operations
module Data.Monoid.Additive
class Monoid a => Group a where Source
Minimal complete definition: gnegate or minus
minus :: a -> a -> a Source
gsubtract :: a -> a -> a Source
Produced by Haddock version 2.4.2 | {"url":"http://hackage.haskell.org/package/monoids-0.1.33/docs/Data-Group.html","timestamp":"2014-04-19T20:30:24Z","content_type":null,"content_length":"8980","record_id":"<urn:uuid:b8792f2b-a80c-4d42-8a7c-d4b72712a389>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00222-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: How to specify that a variable is an Integer?
Replies: 1 Last Post: Oct 25, 1996 10:38 PM
Messages: [ Previous | Next ]
How to specify that a variable is an Integer?
Posted: Oct 17, 1996 12:34 AM
I have an indefinite integral of a function that contains a and b as
parameters. Mathematica is able to compute the indefinite integral when it
is supplied integer values of a and b but cannot solve it when they are
left as parameters.
The question is: Is there any way of telling mathematica that the
parameters a and b are Positive Integer numbers?
Juan M. Ruiz
Boston University
Department of Economics
email: jruiz@bu.edu
Date Subject Author
10/17/96 How to specify that a variable is an Integer? Economics Department
10/25/96 Re: How to specify that a variable is an Integer? Paul A. Rubin | {"url":"http://mathforum.org/kb/thread.jspa?threadID=224093","timestamp":"2014-04-17T21:45:35Z","content_type":null,"content_length":"17454","record_id":"<urn:uuid:e90a0181-91fd-4865-9b2a-447f0dfcef05>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Arbitrage, rationality, and equilibrium
"... Abstract. The Modern Portfolio Theory of Markowitz maximized portfolio expected return subject to holding total portfolio variance below a selected level. Digital Portfolio Theory is an
extension of Modern Portfolio Theory, with the added dimension of memory. Digital Portfolio Theory decomposes the ..."
Cited by 1 (1 self)
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Abstract. The Modern Portfolio Theory of Markowitz maximized portfolio expected return subject to holding total portfolio variance below a selected level. Digital Portfolio Theory is an extension of
Modern Portfolio Theory, with the added dimension of memory. Digital Portfolio Theory decomposes the portfolio variance into independent components using the signal processing decomposition of
variance. The risk or variance of each security’s return process is represented by multiple periodic components. These periodic variance components are further decomposed into systematic and
unsystematic parts relative to a reference index. The Digital Portfolio Theory model maximizes portfolio expected return subject to a set of linear constraints that control systematic, unsystematic,
calendar and non-calendar variance. The paper formulates a single period, digital signal processing, portfolio selection model using cross-covariance constraints to describe covariance and
autocorrelation characteristics. Expected calendar effects can be optimally arbitraged by controlling the memory or autocorrelation characteristics of the efficient portfolios. The Digital Portfolio
Theory optimization model is compared to the Modern Portfolio Theory model and is used to find efficient portfolios with zero calendar risk for selected periods. Key words: portfolio optimization,
portfolio theory, digital signal processing, calendar anomalies 1.
, 2005
"... The notion of Net Present Value (NPV) is thought to formally translate the notion of economic profit, where the discount rate is the cost of capital. The latter is the expected rate of return of
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The notion of Net Present Value (NPV) is thought to formally translate the notion of economic profit, where the discount rate is the cost of capital. The latter is the expected rate of return of an
equivalent-risk alternative that the investor might undertake and is often found by making recourse to the Capital Asset Pricing Model. This paper shows that the notions of disequilibrium NPV and
economic profit are not equivalent: NPV-minded agents are open to framing effects and to arbitrage losses, which imply violations of Modigliani and Miller’s Proposition I. The notion of
disequilibrium (present) value, deductively derived from the CAPM by several authors and widely used in applied corporate finance, should therefore be dismissed.
, 2005
"... Abstract. This paper shows that a decision maker using the CAPM for valuing firms and making decisions may contradict Modigliani and Miller’s Proposition I, if he adopts the widely-accepted
disequilibrium NPV. As a consequence, CAPM-minded agents employing this NPV are open to arbitrage losses and m ..."
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Abstract. This paper shows that a decision maker using the CAPM for valuing firms and making decisions may contradict Modigliani and Miller’s Proposition I, if he adopts the widely-accepted
disequilibrium NPV. As a consequence, CAPM-minded agents employing this NPV are open to arbitrage losses and miss arbitrage opportunities. As a result, even though the use of the disequilibrium NPV
for decision-making is deductively drawn from the CAPM, its use for both valuation and decision should be rejected.
"... This paper generalizes the Debreu/Gorman characterization of additively decomposable functionals and separable preferences to infinite dimensions. The first novelty concerns the very definition
of additively decomposable functional for infinite dimensions. For decision under uncertainty, our result ..."
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This paper generalizes the Debreu/Gorman characterization of additively decomposable functionals and separable preferences to infinite dimensions. The first novelty concerns the very definition of
additively decomposable functional for infinite dimensions. For decision under uncertainty, our result provides a state-dependent extension of Savage's expected utility. A characterization in terms
of preference conditions identifies the empirical content of the model; it amounts to Savage's axiom system with P4 (likelihood ordering) dropped. Our approach does not require that a (probability)
measure on the state space be given a priori, or can be derived from extraneous conditions outside the realm of decision theory. Bayesian updating of new information is still possible, even though no
prior probabilities are given. The finding suggests that the sure-thing principle, rather than prior probability, is at the heart of Bayesian updating. 1. Introduction. Separability | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=5241020","timestamp":"2014-04-16T09:14:24Z","content_type":null,"content_length":"20522","record_id":"<urn:uuid:9fec47be-8262-49e3-97bb-21bcd8c56eea>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00637-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: April 2010 [00038]
[Date Index] [Thread Index] [Author Index]
Re: Speed Up of Calculations on Large Lists
• To: mathgroup at smc.vnet.net
• Subject: [mg108828] Re: Speed Up of Calculations on Large Lists
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Fri, 2 Apr 2010 05:20:31 -0500 (EST)
On 4/1/10 at 5:59 AM, sheaven at gmx.de (sheaven) wrote:
>I am new to Mathematica and try get a understanding of its power. I
>plan to use Mathematica mainly for financial data analysis (large
>Currently, I am trying to optimize calculation time for calculations
>based on some sample data. I started with with a moving average of
>share prices, because Mathematica already has a built in moving
>average function for benchmarking.
>I know that the built-in functions are always more efficient than
>any user built function. Unfortunately, I have to create functions
>not built in (e.g. something like "moving variance") in the future.
>I have tried numerous ways to calc the moving average as efficiently
>as possible. So far, I found that a function based on Span (or
>List[[x;;y]]) is most efficient. Below are my test results.
>Unfortunately, my UDF is still more than 5x slower than the built in
>Do you have any ideas to further speed up the function. I am already
>using Compile and Parallelize.
>This is what I got so far:
>1. Functions for moving average:
<function code snipped>
>2. Create sample data: data = 100 + # & /@
>Accumulate[RandomReal[{-1, 1}, {10000}]];
a side point here. The plus function works on lists. That is:
data = 100 + Accumulate[RandomReal[{-1,1}, 10000]];
will produce the same result as your code but be a bit faster.
Note, the difference in speed here will be quite small and is
clearly not the thrust of your message. But I point this out
since such small difference can add up to something significant
in more complex code.
>3. Test if functions yield same results: Test1 = movAverageC[data,
>30, 250, 10]; (*Moving average for 30 days to 250 days in steps of
OK. Here is the timing results I get for you compiled code based
on Span
In[1]:= movAverageOwn2FC =
Compile[{{dataInput, _Real,
1}, {days, _Integer}, {length, _Integer}},
N[Mean[dataInput[[1 + # ;; days + #]]]] & /@
Range[0, length - days, 1]];
In[2]:= data = 100 + Accumulate[RandomReal[{-1, 1}, {10000}]];
In[3]:= Timing[Table[movAverageOwn2FC[data, 20, Length@data], {100}];]
Out[3]= {1.45855,Null}
Now here is a definition using ListConvolve
In[4]:= newMoveAverage[data_, windowLen_] :=
Module[{ker = Table[1, {windowLen}]/windowLen},
ListConvolve[ker, data]]
In[5]:= Timing[Table[newMoveAverage[data, 20], {100}];]
Out[5]= {0.103379,Null}
So, on my machine using a single core without Compile, using
ListConvolve improves the speed by more than 10X. Using both
parallel processing with both cores should improve this result
for very large data arrays. Note, ListConvolve is so fast, the
overhead of setting up parallel processes will probably degrade
times for small data arrays. I have not tested this to verify my
guess here.
Compile also might improve things somewhat. But this probably
won't be significant. Compile can offer significant improvement
in some code particularly when procedural programming is used.
But compile seldom offers improvement in code with one or two
function calls and no procedural structures such as For. In
fact, there are times when using Compile will actually degrade
the execution speed.
Finally, to demonstrate the code with ListConvolve does the same
as your code:
In[6]:= movAverageOwn2FC[data, 20, Length@data] ==
newMoveAverage[data, 20]
Out[6]= True | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Apr/msg00038.html","timestamp":"2014-04-16T19:06:58Z","content_type":null,"content_length":"28446","record_id":"<urn:uuid:376458c6-2839-4a95-9b8f-082fe8ede004>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dispersive estimates for the Schrödinger equation
Files in this item
Green_William.pdf (1000KB) (no description provided) PDF
Title: Dispersive estimates for the Schrödinger equation
Author(s): Green, William R.
Director of Erdogan, Burak
Committee Laugesen, Richard
Committee Erdogan, Burak; Hundertmark, Dirk; Bronski, Jared
Department / Mathematics
Discipline: Mathematics
Granting University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Schrodinger operators dispersive estimates oscillatory integrals
In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in
\mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely
continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$.
Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V
Abstract: (x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in
dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on
the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}
dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce
this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in
$x$ and in the Fourier transform of $x$.
Issue Date: 2010-08-31
URI: http://hdl.handle.net/2142/16998
Rights Copyright 2010 William R. Green
Date 2010-08-31
Available in 2012-09-07
Date 2010-08
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A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 735128, 10 pages
Research Article
A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations
^1Islamic Azad University, Shabestar Branch, Shabestar 5381637181, Iran
^2Department of Applied Mathematics, University of Tabriz, Tabriz 5166616471, Iran
^3Mathematics Department, Institute for Advanced Studies in Basic Sciences, Zanjan 45137-66731, Iran
^4Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela (USC), 15782 Santiago de Compostela, Spain
^5Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 12 June 2013; Accepted 28 August 2013
Academic Editor: Marcia Federson
Copyright © 2013 K. Ivaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the
proposed algorithms are given, and their applicability is illustrated with some examples.
1. Introduction
Hybrid systems are devoted to modeling, design, and validation of interactive systems of computer programs and continuous systems. That is, control systems that are capable of controlling complex
systems which have discrete event dynamics as well as continuous time dynamics can be modeled by hybrid systems. The differential systems containing fuzzy valued functions and interaction with a
discrete time controller are named hybrid fuzzy differential systems.
The Hukuhara derivative of a fuzzy-number-valued function was introduced in [1]. Under this setting, the existence and uniqueness of the solution of a fuzzy differential equation are studied by
Kaleva [2, 3], Seikkala [4], and Kloeden [5]. This approach has the disadvantage that it leads to solutions which have an increasing length of their support [2]. A generalized differentiability was
studied in [6–8]. This concept allows us to resolve the previously mentioned shortcoming. Indeed, the generalized derivative is defined for a larger class of fuzzy-number-valued functions than the
Hukuhara derivative. Some applications of numerical methods in FDE and hybrid fuzzy differential equation (HFDE) are presented in [9–19]. Some other approaches to study FDE and fuzzy dynamical
systems have been investigated in [20–22].
In engineering and physical problems, Trapezoidal rule is a simple and powerful method to solve numerically related ODEs. Trapezoidal rule has a higher convergence order in comparison to other one
step methods, for instance, Euler method.
In this work, we concentrate on numerical procedure for solving FDEs and HFDEs, whenever these equations possess unique fuzzy solutions.
In Section 2, we briefly present the basic definitions. Trapezoidal rule for solving fuzzy differential equations is introduced in Section 3, and convergence and stability of the mentioned method are
proved. The proposed algorithm is illustrated by solving two examples. In Section 4 we present Trapezoidal rule for solving hybrid fuzzy differential equations.
2. Preliminary Notes
In this section the most basic definition of ordinary differential equations (ODEs) and notation used in fuzzy calculus are introduced. See, for example, [23].
Consider the first-order ordinary differential equation where and . A linear multistep method applied to (1) is with , , given starting values . In the case , the corresponding methods (2) are
explicit and are implicit otherwise. The constant step size leads to time discretizations with respect to the grid points . The value is an approximation of the exact solution at . The special case
of explicit methods, , , , , , and , corresponds to the Midpoint rule: and the especial case of implicit methods, , , , and, corresponds to the Trapezoidal rule: For an explicit method, (2) yields
the current value directly in terms of , , , which, at this stage of the computation, have already been calculated. An implicit method will call for the solution, at each stage of computation, of the
the equation where is a known function of previously calculated values , , . When the original differential equation in (1) is linear, then (5) is linear in , and there is no problem in solving it.
When is nonlinear, for finding solution of (1), we can use the following iteration:
Definition 1. Associated with the multistep method (2), we define the first characteristic polynomial as follows:
Theorem 2. A multistep method is stable if the first characteristic polynomial satisfies the root condition, that is, the roots of lie on or within the unit circle, and further the roots on the unit
circle are simple.
According to Theorem 2, we know the Midpoint rule and Trapezoidal rule are stable.
Definition 3. The difference operator and the associated linear multistep method (2) are said to be of order if for the following equation: we have ,, where and , for .
According to Definition 3, Midpoint rule and Trapezoidal rule are second-order methods.
We now recall some general concepts of fuzzy set theory; see, for example, [2, 24].
Definition 4. Let be a nonempty set. A fuzzy set in is characterized by its membership function, and is interpreted as the degree of membership of an element in fuzzy set for each .
Let us denote by the class of fuzzy subsets of the real axis, that is, satisfying the following properties: (i)is normal, that is, there exists such that ,(ii) is a convex fuzzy set (i.e., , ), (iii)
is upper semicontinuous on , (iv) is compact, where denotes the closure of a subset. The space is called the space of fuzzy numbers. Obviously, . For , we denote Then from (i)–(iv), it follows that
the -level set is a nonempty compact interval for all . The notation denotes explicitly the -level set of . The following remark shows when is a valid -level set.
Remark 5. The sufficient conditions for to define the parametric form of a fuzzy number are as follows: (i) is a bounded monotonic increasing (nondecreasing) left-continuous function on and
right-continuous for ,(ii) is a bounded monotonic decreasing (nonincreasing) left-continuous function on and right-continuous for .(iii), .
For and , the sum and the product are defined by , , , where means the usual addition of two intervals (subsets) of , and means the usual product between a scaler and a subset of .
The metric structure is given by the Hausdorff distance by The following properties are well known: , ,, , ,, ,and is complete metric spaces.
Let be a real interval. A mapping is called a fuzzy process and its -level set is denoted by A triangular fuzzy number is defined by an ordered triple with , where the graph of is a triangle with
base on the interval and vertex at . An -level of is always a closed, bounded interval. We write ; then for any .
Definition 6. Let . If there exists such that , then is called the H-difference of and , and it is denoted by .
In this paper the sign “” stands always for H-difference, and let us remark that . Usually we denote by , while stands for the H-difference.
Definition 7. Let be a fuzzy function. We say is Hukuhara differentiable at if there exists an element such that the limits exist and are equal to . Here the limits are taken in the metric space .
Definition 8. Let . The fuzzy integral is defined by provided the Lebesgue integrals on the right exist.
Remark 9. Let . If is Hukuhara differentiable and its Hukuhara derivative is integrable over , then for all values of , , where .
Theorem 10. Let , , be the observed data, and suppose that each of the is a triangular fuzzy number. Then for each , the fuzzy polynomial interpolation is a fuzzy-value continuous function , where ,,
and such that .
Proof. See [25].
3. Fuzzy Differential Equations
Consider the first-order fuzzy differential equation , where is a fuzzy function of , is a fuzzy function of crisp variable and fuzzy variable , and is Hukuhara fuzzy derivative of . If an initial
value is given, a fuzzy Cauchy problem of first order will be obtained as follows: By Theorem 5.2 in [11] we may replace (21) by equivalent system The parametric form of (22) is given by for . In
some cases the system given by (23) can be solved analytically. In most cases analytical solutions may not be found, and a numerical approach must be considered. Some numerical methods such as the
fuzzy Euler method, Nyström method, and predictor-corrector method presented in [7, 10, 11, 13, 15]. In the following, we present a new method to numerical solution of FDE.
3.1. Trapezoidal Rule for Fuzzy Differential Equations
In the interval we consider a set of discrete equally spaced grid points . The exact and approximate solutions at , , are denoted by and , respectively. The grid points at which the solution is
calculated are Let , which is triangular fuzzy number. We have By fuzzy interpolation, Theorem 10, we get where , interpolates with the interpolation data given by the value , and , .
For we have From (16) and (25) it follows that where According to (25), if (26) and (27) are situated in (31), (27) and (28) in (32), we obtain By integration we have By (16) deduce Similarly we
obtain Therefore, Trapezoidal rule is obtained as follows: for .
3.2. Convergence and Stability
Suppose the exact solution is approximated by some . The exact and approximate solutions at , are denoted by and , respectively. Our next result determines the pointwise convergence of the
Trapezoidal approximates to the exact solution. The following lemma will be applied to show convergence of these approximates; that is,
Lemma 11. Let a sequence of numbers satisfy for some given positive constant and . Then
Proof. See [15].
Let and be the functions and of (22), where and are constants and . The domain where and are defined is therefore
Theorem 12. Let and belong to , and let the partial derivatives of , be bounded over . Then for arbitrary fixed , the Trapezoidal rule approximate of (37) converges to the exact solutions , uniformly
in , for .
Proof. It is sufficient to show that By using Taylor’s theorem, we get where , . Consequently, Denote and . Then where and , and is a bound for partial derivatives of and in , . Thus, If we put and ,
then Then by Lemma 11 and , we have If , then , which concludes the proof.
Remark 13. According to Definition 3, Trapezoidal rule is a second-order method. In fact we may consider the definition of convergence order given in Definition 3 for system of ODEs.
Theorem 14. Trapezoidal rule is stable.
Proof. For Trapezoidal rule exists only one characteristic polynomial , and it is clear that satisfies the root condition. Then by Theorem 2, the Trapezoidal rule is stable.
3.3. Numerical Results
In this section we apply Trapezoidal rule for numerical solution of two linear fuzzy differential equations. We compare our results with Midpoint rule. The authors in [13] have presented the Midpoint
rule for numerical solution of FDEs as follows: The Midpoint rule is a second-order and stable method [13].
In the following two examples, the implicit nature of Trapezoidal rule for solving linear fuzzy differential equation is implemented by solving a linear system at each stage of computation.
Example 15 (see [13]). Consider the initial value problem The exact solution at for is given by A comparison between the exact solution, , and the approximate solutions by Midpoint method [13], , and
Trapezoidal method, , at with , is shown in Table 1 and Figure 1.
Example 16. Let us consider the first-order fuzzy differential equation where .
The exact solution at is given by A comparison between the exact solution, , and the approximate solutions by Midpoint method, , and Trapezoidal method, , at with , is shown in Table 2 and Figure 2.
4. Hybrid Fuzzy Differential Equations
Consider the hybrid fuzzy differential equation where is strictly increasing and unbounded, denotes , is continuous, and each is a continuous function. A solution to (55) will be a function
satisfying (55). For , let , where . The hybrid fuzzy differential equation in (55) can be written in expanded form as and a solution of (55) can be expressed as We note that the solution of (55) is
continuous and piecewise differentiable over and differentiable on each interval for any fixed and .
Theorem 17. Suppose for that each is such that If for each there exists such that for all , then (55) and the hybrid system of ODEs are equivalent.
Proof. See [19].
4.1. Trapezoidal Rule for Hybrid Fuzzy Differential Equations
For each , to numerically solve (55) in , replace each interval , by a set of regularly spaced grid points (including the endpoints). The grid point on will be , , at which the exact solution will be
approximated by some . We set , and , if .
According to Section 3, by similar computation we obtain the Trapezoidal rule for solving (60) as follows: for , .
Next, we give the algorithm to numerically solve (55) in .
First Step. will be a numerical solution generated by (61) for as follows: is a numerical solution of (60) over .
Second Step. For each , will be numerical solution generated by (61) for where , is a numerical solution of (60) over for each .
For arbitrary fixed and , we can prove that the numerical solution of (55) converges to the exact solution; that is, The Trapezoidal rule is a one-step method as the Euler method. Therefore, the
proof of the convergence closely follows the idea of the proof of Theorem 3.2 in [18] and Theorem 4.1 in [19].
Theorem 18. Consider the system of (55). Suppose for some fixed and that , where is obtained by (61). Then
Proof. See [19].
Example 19. Consider the following hybrid fuzzy system: where is a triangular fuzzy number having -level sets , By [19, Example ], we know (66) has a unique solution and the exact solution on is
given by To numerically solve the hybrid fuzzy initial value problem (66) we apply the Trapezoidal rule for hybrid fuzzy differential equations.
A comparison between the exact solution and the approximate solutions by Midpoint method and Trapezoidal method at with is shown in Table 3 and Figure 3.
5. Conclusion
We have presented Trapezoidal rule for numerical solution of first-order fuzzy differential equations and hybrid fuzzy differential equations. Also convergence and stability of the method are
studied. To illustrate the efficiency of the new method, we have compared our method with the Midpoint rule in some examples. We have shown the global error in Trapezoidal rule is much less than in
Midpoint rule.
For future research, we will apply Trapezoidal rule to fuzzy differential equations and hybrid fuzzy differential equations under generalized Hukuhara differentiability. Also one can apply
Trapezoidal rule and Midpoint rule as a predictor-corrector method to solve FDE and HFDE.
The first author would like to thank the financial support of the Islamic Azad University, Shabestar Branch. The research of the third author has been partially supported by Ministerio de Economia y
Competitividad (Spain), Project MTM2010-15314, and cofinanced by the European Community fund FEDER. This research was completed during the visit of the second author to the USC.
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the encyclopedic entry of hexadecimal number system
In mathematics, a number system is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication.
Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.
For a history of number systems, see number. For a history of the symbols used to represent numbers, see numeral.
Logical construction of number systems
Natural numbers
Simply put, the natural numbers consist of the set of all whole numbers greater than or equal to zero. The set is denoted with a capital
. (In some books, the natural numbers omit 0 and begin with 1. (There is no general agreement on this subject.)
Giuseppe Peano
developed these
for the
natural numbers
Axiom one: There is a natural number 0.
Axiom two: Every natural number a has a successor, denoted by S(a).
Axiom three: There is no natural number whose successor is 0.
Axiom four: Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b).
Axiom of induction: If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers.
From these five axioms, all of the properties of the natural numbers can be deduced. The number one is defined as 1 = S(0).
Most number systems include the idea of equality. In mathematics, equality is an equivalence relation, meaning it obeys the three axioms of equality:
Reflexive axiom: a = a.
Symmetric axiom: a = b implies b = a.
Transitive axiom: a = b and b = c implies a = c.
(Taken together, the symmetric and transitive axioms imply Euclid's Common Notion One: "Things equal to the same thing are equal to each other.")
Operations can be defined on the natural numbers. Addition is essentially repeated application of the successor function, defined by
a + 0 = a
a + S(b) = S(a + b).
(This implies that S(
) = S(
+ 0) =
+ S(0) =
+ 1, so S(
) is written
+ 1 from now on.) The axiom of induction allows us to conclude that this defines
for all natural numbers
. We proceed to define
of natural numbers as repeated addition, formally
a · 0 = 0
a · (b+1) = a · b + a,
which inductively defines multiplication for all natural numbers
. Multiplication
a · b
is also written
, or simply
Exponentiation can then be defined similarly using repeated multiplication,
a^0 = 1
a^b+1 = a^b · a.
The exponentiation
a^ b
is often written
, especially in media where superscripting is impossible or undesirable.
The process of defining further operators in this way is covered in the Tetration article.
Operations may have inverses. Subtraction is defined as the inverse of addition. By definition, a − b = c means b + c = a. Division is similarly the inverse of multiplication. By definition a / b = c
means b · c = a. Note that under these definitions, subtraction and division are undefined for many pairs of natural numbers. In the natural number system, no meaning is assigned to 3 − 5 or to 5/3.
Exponentiation has two inverses, extraction of roots and logarithms. To say that the bth root of a is c, ^b√a = c, means that c^ b = a. To say that the logarithm to base b of a is c, log[b]a = c,
means that b^c = a. In other words, if x^y = z, x is the root ^y√z, and y is the logarithm log[x]z. As with division, ^b√a and log[b]a are not defined for all values of b and a.
An alternative way to approach the operations is using axiomatics. In these axioms, the usual order of operations is assumed.
Axioms of operations:
Commutative axioms: a + b = b + a; a · b = b · a.
Associative axioms: a + (b + c) = (a + b) + c; a · (b · c) = (a · b) · c.
Distributive axioms: a · (b + c) = a · b + a · c; (b · c) ^ n = b^ n · c ^ n.
Identity axioms: a + 0 = a; a · 1 = a; a^ 0 = 1 .
Inverse axioms: − a exists and has the property that a − a = 0; if a is non-zero, then 1/a exists and has the property that a · (1/a) = 1.
All of these axioms can be proven as theorems, starting with inductive definitions and the Peano Axioms, except the inverse axioms.
The natural numbers can be extended to the number system called the integers (denoted with Z) as follows. For every non-zero natural number a, there exists an integer denoted −a, which is not a
natural number. As a special case −0 is defined as the natural number 0. The successor function can be extended to the integers by the rule S(−a) = −(a − 1).
Addition can be defined on the integers inductively as follows. If a and b are natural numbers, then −a + −b = −(a + b). If a is any integer, then a + 0 = a. If b is a non-zero integer, then a + b =
(a − 1) + S(b). It is then necessary to show that addition is well-defined in the case where b is a natural number. The definition of subtraction extends to the integers unchanged, and now it can be
proven that a − b is defined for all integers a and b. To justify the use of − for both "minus" and "negative", one proves that a − b = a + −b. Multiplication can be defined as follows. For all
natural numbers a and b, −a · b = a · −b = −(a · b). It follows as a theorem that −a · −b = a · b. The definition of division extends to the integers unchanged, but division is not defined in every
case. At this point we can define natural number powers of integers in exactly the same way we defined natural number powers of natural numbers. However, we need a larger number system to define
negative number powers of integers. It is also notable that our definition of roots becomes ambiguous in the integers: √4 can mean either 2 or −2. It is customary to consider the positive root when
two roots exist.
From these definitions, it can be proven that all of the axioms of operations hold for integers except the multiplicative inverse. A number system with this property is called a commutative ring with
Rational numbers
The rational numbers (denoted with Q) are the number system that extends the integers to include numbers which can be written as fractions. It allows division to be defined for all pairs of numbers
except for division by zero. It also allows the definition of exponents to be extended to negative integer exponents, and to some, but not all, rational exponents.
We define a fraction a / b to be an ordered pair, where a is any integer and b is any non-zero natural number. We define equality of fractions by a / b = c / d if and only if a · d = b · c, and
define a/1 = a, which embeds the integers in the set of all fractions. These definitions of equality partition the set of fractions and integers into equivalence classes. The canonical representative
of an equivalence class is an element a / b where b is positive and relatively prime to a, or the integer a if b=1. Finding the canonical representative of an equivalence class of rational numbers is
also called reducing to lowest terms. The set of rational numbers $mathbb\left\{Q\right\}$ is defined to be either the set of equivalence classes or the set of canonical representatives.
We justify using the same symbol for fractions and division by proving that the fraction a / b = c = c/1 just when a = b · c. We define addition of fractions by (a / b) + (c / d) = (a · d + c · b)/(b
· d). We define multiplication of fractions by (a / b) · (c / d) = (a · c)/(b · d). Since b and d are non-zero, b · d is also non-zero. We can then define subtraction as the inverse of addition and
division as the inverse of multiplication, just as we did for integers, and prove that for any rational a / b and c / d, if c is non-zero, then (a / b)/(c / d) = (a · d)/(b · c). Thus division is now
defined for any two rational numbers, provided the divisor is not zero.
We can now extend the definition of exponentiation to include negative exponents, by defining (for any natural number n and nonzero rational number a) a^ −n = 1/(a^n). We can also define the use of
rational exponents in some cases, by defining a^m / n = b to mean a^m = b^n. In other words, a^m / n = ^n√(a^m), provided the root exists. This is unambiguous if n is odd. If n is even and m is odd,
this definition would be ambiguous, and taking the positive root is again customary.
With these definitions, all of the axioms of operations hold without exception. A number system in which addition and multiplication are defined for all pairs of numbers, and in which the axioms of
operations hold, is called a field.
Polynomials are not usually called numbers, but they share many properties with numbers. All of the axioms of operations hold for polynomials except for the axiom of multiplicative inverses.
Polynomials do not, in general, have multiplicative inverses. Thus the set of polynomials, like the integers, is a commutative ring (with identity).
Algebraic numbers
The algebraic numbers are a number system that includes all of the rational numbers, and is included in the set of real numbers. The construction of the algebraic numbers requires an understanding of
the definition and properties of an extension field. Roughly speaking, one extends the rational numbers by appending all zeroes of polynomials with integer coefficents. This, however, would append
complex numbers, which are usually excluded from the algebraic numbers, unless the set is called the complex algebraic numbers. It is, therefore, traditional to construct the real numbers first, and
then define the algebraic numbers as a subset of the reals. The algebraic numbers form a field.
Real numbers
There are many ways to construct the
real number
) system: equivalence classes of
Cauchy sequences
, transcendental extension fields, and
Dedekind cuts
, to mention just three. But the most elementary definition is that the real numbers are all numbers that can be written as decimals.
A decimal can only have finitely many digits to the left of the decimal point, but to the right of the decimal point there are three cases to consider. A decimal may terminate, repeat, or continue
forever without ever becoming an infinite sequence of repeating strings of digits (in brief, a non-repeating decimal). In the first two cases, the decimal is rational, that is, it can be changed to a
fraction. In the third case, the decimal is irrational.
Irrational numbers may be either algebraic or transcendental (non-algebraic). There is no easy way to tell whether a non-repeating decimal is algebraic or transcendental. In fact, there are many open
questions on this subject.
The real numbers are (up to isomorphism) the only complete ordered field.
The number forty-two is a real number because it can be written as a decimal: 42.0. The number one half is both a rational number and a real number because it can be written 0.5. The number one third
is both a rational number and a real number because it can be written 0.333... . The square root of two is an algebraic real number, which as a decimal is 1.4142135... . The ratio of the
circumference of a circle to its diameter,
, is a transcendental real number, which as a decimal is 3.1415927... . The square root of negative one,
, is not a real number, and cannot be written as a decimal, because the square of any decimal is never negative.
Equivalence classes of decimal numerals
Just as the fractions 1/2 and 2/4 are equal, the real numbers 1.0 and
are equal. The easiest way to see this is to start with the equation 1/3 = 0.333... and multiply both sides by three.
In general, any real number whose decimal form has an unending string of repeating nines is equal to the decimal obtained to removing all of the nines in the unending string that lie to the right of
the decimal point, and increasing the rightmost non-nine digit by one. If all of the digits are nine, then it may be necessary to append a leading 0. Thus 123.789999999... = 123.79 and 999.999... =
0999.999... = 1000.0.
This fact is a consequence of the definition of the limit of an infinite series.
Complex numbers
The complex numbers are numbers which can be written in the form a + b · i, where a and b are real numbers and i is the square root of minus one -- that is, a number whose square is minus one. The
complex numbers can be viewed as abstract symbols, as representing points in the Argand plane, as an extension field of the real numbers, as a two-dimensional vector space with basis {1, i}, and in
many other ways. Originally thought to be a pure abstraction, they have proved enormously useful in many practical applications, particularly in electrical engineering.
The complex numbers are a complete field, but cannot be ordered in any way that is consistent with the usual properties of inequalities. That is, there is no meaningful answer to the question, "Which
is greater, 1 or i?"
More advanced number systems
The word
has no generally agreed upon mathematical meaning, nor does the word
number system
. Instead, we have many examples. Thus there is no rule to say what is a number and what is not. Some of the more interesting examples of abstractions that can be considered numbers include the
, the
, and the
transfinite numbers
• Richard Dedekind, 1888. Was sind und was sollen die Zahlen? ("What are and what should the numbers be?"). Braunschweig.
• Edmund Landau, 2001, ISBN 082182693X, Foundations of Analysis, American Mathematical Society.
• Giuseppe Peano, 1889. Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method). Bocca, Torino. Jean van Heijenoort, trans., 1967. A Source Book of
Mathematical Logic: 1879-1931. Harvard Univ. Press: 83-97.
• B. A. Sethuraman (1996). Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility. Springer. ISBN 0-387-94848-1.
• Solomon Feferman (1964). The Numbers Systems : Foundations of Algebra and Analysis. Addison-Wesley.
• Stoll, Robert R., 1979 (1963). Set Theory and Logic. Dover.
See also | {"url":"http://www.reference.com/browse/hexadecimal+number+system","timestamp":"2014-04-17T16:39:48Z","content_type":null,"content_length":"104362","record_id":"<urn:uuid:05c587e0-2c72-487e-a27a-aacaf32795d9>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00005-ip-10-147-4-33.ec2.internal.warc.gz"} |
A self-evaluating evaluator
A self-evaluating evaluator
December is always a busy month, so the number of blog posts have been low for a while. To make up for it, I have been digging in the archives and found a self-evaluating evaluator.
Erann Gat
back in 2002 asked the following question in comp.lang.scheme:
The topic of self-generating programs (a program whose output is itself) comes up periodically here, but I've never seen the topic of a self-evaluating program discussed. Of course, meta-circular
interpreters are ubiquitous, but the usual metacircular interpreter you find in textbooks typically evaluates a slightly different dialect than it is actually written in.
My question is: what is the shortest meta-circular
interpreter that is actually capable of evaluating itself?
It was quite fun to write a short meta-circular interpreter. The shortness-constraint creates a tension between powerful constructs and easily-implementable constructs, when choosing language
features to include in the language. My solution below has the following features:
• conditional construct
• functions (of one argument)
• application
• quotation
This set of features is pretty minimal. The requirement that the evaluator must be self-evaluating means, it must be possible to pass the code of the evaluator to the evaluator, so quotation is just
the right tool. Note that variable assignment (set!) is missing.
Since only functions of one argument is supported, it is necessary to curry the normal Scheme functions in the initial environment given to the evaluator. This can be seen in the following version of
; Support code for fak-example
(define (plus a) (lambda (b) (+ a b)))
(define (mult a) (lambda (b) (* a b)))
(define (one? x) (= x 1))
(define (sub1 n) (- n 1))
(define fact '(lambda (f)
(lambda (n)
[(one? n) 1]
[#t ((mult n) ((f f) (sub1 n)))]))))
(define fact-code (list (list fact fact) 5))
The standard definition of factorial is recursive. The chosen languages supports neither recursive bindings nor assignment, so instead we use the same trick, the Y-combinator uses. To calculate
factorial of 5, one must write: ((fact fact) 5).
Since the evaluator must be self-evaluating, the definition of the evaluator uses the same trick as the factorial example. To call the evaluator on the factorial program, one writes:
(((jas-eval jas-eval) expression) initial-env)
And if we want to let the evaluator evaluate
(jas-eval (fak 5) initial-env)
we write
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote fac-code)) initial-env))
(((jas-eval jas-eval) expression) initial-env)
After posting version 1 of the evaluator,
Joe Marshall
asked a very interesting question:
For the sake of curiosity, how much slower is the extra interpretation layer?
Can you go another level deeper?
The timings on my current computer is as follows:
; jas-eval evaluating "(fact 5)"
cpu time: 0 real time: 0 gc time: 0
; jas-eval evaluating "(jas-eval (fact 5))"
cpu time: 15 real time: 16 gc time: 0
; jas-eval evaluating "(jas-eval (jas-eval (fact 5)))"
cpu time: 922 real time: 954 gc time: 0
; jas-eval evaluating "(jas-eval (jas-eval (jas-eval (fact 5))))"
cpu time: 63891 real time: 66109 gc time: 12122
Exponential regression reveals the slow down to be 65 per level (R squared is 0.9999).
; Self evaluating evaluator, version 2
; Jens Axel Soegaard, oct 2002
(define first car)
(define second cadr)
(define third caddr)
(define rest cdr)
; Support code for fak-example
(define (plus a) (lambda (b) (+ a b)))
(define (mult a) (lambda (b) (* a b)))
(define (one? x) (= x 1))
(define (sub1 n) (- n 1))
(define fact '(lambda (f)
(lambda (n)
[(one? n) 1]
[#t ((mult n) ((f f) (sub1 n)))]))))
(define fact-code (list (list fact fact) 5))
; Conventions: n name, v value, r environment, e expression
(define code-jas-eval
'(lambda (ev)
(lambda (e)
(lambda (r)
[(symbol? e) (r e)]
[(not (pair? e)) e]
[(pair? e) (cond
[((ceq? (first e)) 'quote) (first (rest e))]
[((ceq? (first e)) 'cond)
[(((ev ev) (first (second e))) r) (((ev ev) (second (second e))) r)]
[#t (((ev ev) ((ccons 'cond) (rest (rest e)))) r)])]
[((ceq? (first e)) 'lambda) (lambda (x)
(((ev ev) (third e))
(lambda (n)
[((ceq? (first (second e))) n) x]
[#t (r n)]))))]
[#t ((((ev ev) (first e)) r)
(((ev ev) (second e)) r))])])))))
; setup initial environment;
; currying the primitives used in the evaluator
(define (ceq? a)
(lambda (b)
(eq? a b)))
(define (ccons a)
(lambda (b)
(cons a b)))
(define initial-env eval)
; use Scheme-eval to get jas-eval
(define jas-eval (eval code-jas-eval))
(define initial-env eval)
; use jas-eval to evaluate (fact 5)
(define expression fact-code)
(time (((jas-eval jas-eval) expression) initial-env))
; use jas-eval to evaluate (jas-eval (fact 5) initial-env)
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote fact-code)) initial-env))
(time (((jas-eval jas-eval) expression) initial-env))
; use jas-eval to evaluate (jas-eval (jas-eval (fact 5) initial-env))
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote fact-code)) initial-env))
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote expression)) initial-env))
(time (((jas-eval jas-eval) expression) initial-env))
; use jas-eval to evaluate (jas-eval (jas-eval (jas-eval (fact 5) initial-env)))
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote fact-code)) initial-env))
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote expression)) initial-env))
(define expression (list (list (list code-jas-eval code-jas-eval) (list 'quote expression)) initial-env))
(time (((jas-eval jas-eval) expression) initial-env))
1 Comments:
Hi Jens,
I tried out your nifty self-evaluating evaluator and I got the following timing results:
cpu time: 0 real time: 0 gc time: 0
cpu time: 32 real time: 32 gc time: 0
cpu time: 1156 real time: 1156 gc time: 0
You'll notice I don't have a fourth time. Something was taking exponential time to complete, and I got worried that it was going to crash DrScheme before it reached and answer, so I stopped the
In the name of good sport, I'm going to post my core-form evaluator that is the result of an exercise in TSPL. I'll see if I can get it to run four levels deep and report the times. Mind you it's
not my code, just my transformation to core form, so regardless, Dybvig will still take all the credit (or blame.)
Links to this post: | {"url":"http://blog.scheme.dk/2006/12/self-evaluating-evaluator.html","timestamp":"2014-04-21T02:02:05Z","content_type":null,"content_length":"40997","record_id":"<urn:uuid:7b61ae21-a762-4b4d-96ba-d55a6843f27e>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00085-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help Isolating a Variable
November 2nd 2009, 03:06 PM #1
Oct 2009
Help Isolating a Variable
It's been awhile since I was in math, and I'm in a Calculus class right now. I need to find dx/dy of an implied function, and I'm having trouble re-arranging the function to do so. The equation I
have is:
sin (X^2+y^2) = xy + 1
how do I isolate for y?
You can't. You're expected to get dy/dx using implicit differentiation.
November 2nd 2009, 03:10 PM #2 | {"url":"http://mathhelpforum.com/calculus/112032-help-isolating-variable.html","timestamp":"2014-04-18T14:05:40Z","content_type":null,"content_length":"33077","record_id":"<urn:uuid:fd97bc9c-a18d-479a-bbc3-f45ade3cb202>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: RE: funny results with -cluster- in -somersd-
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st: RE: funny results with -cluster- in -somersd-
From "Newson, Roger B" <r.newson@imperial.ac.uk>
To "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject st: RE: funny results with -cluster- in -somersd-
Date Thu, 19 Mar 2009 19:07:06 +0000
These results are not unexpected.
In the first example, the denominator of Kendall's tau-a, as defined in Equations (9), (10), (11), (12) (13), (14) and (15) of Newson (2006), will be equal to 4*4=16, because it is the mean of 10 values of 16 for each of the 10 clusters of 4 observations each. The numerator of Kendall's tau-a will be 4*3=12 (because the numerator terms comparing the same observation with itself are zero), and will once again be the sample mean of 12 values of 12 for all clusters. Therefore, the standard errors, and the covariance, of both the numerator and the denominator of Kendall's tau-a will be zero, and the estimate of Kendall's tau-a will be 12/16=0.75. By the way, Al need not have specified the wstrata(id) option, because funtype(wclass) ensures that comparisons are limited to within-cluster comparisons.
In the second example, Al has specified no funtype() option, causing the funtype() option to default to funtype(bcluster). As Al has also specified wstrata(id), Kendall's tau-a will be limited to comparisons that are both between clusters (because of funtype(bcluster)) an within clusters (because of wstrata(id) cluster(id)). As no comparisons are both between clusters and within clusters, both the numerator and the denominator of Kendall's tau-a will be zero.
I hope this helps.
Best wishes
Newson R. Confidence intervals for rank statistics: Somers' D and extensions. The Stata Journal 2006; 6(3): 309-334. Download pre-publication draft from
Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
Opinions expressed are those of the author, not of the institution.
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Feiveson, Alan H. (JSC-SK311)
Sent: 19 March 2009 14:15
To: statalist@hsphsun2.harvard.edu
Subject: st: funny results with -cluster- in -somersd-
Hi - I've been using -somersd- to make within-stratum comparisons and also use the -cluster- option to account for repeated observations for each value of id. In this example, there are 10 vlaues of id, each replicated 4 times. Also in this data there is another variable u, which is a random uniform number - so all the values of u are distinct. Therefore, I would expect that if I make comparisons within strata, I should get tau_uu = 1. I thought that to account for the clusters in within-stratum comparisons I had to use -funtype(wcluster)-. But results are strange. Here is what I tried:
. somersd u u , funtype(wcluster) cluster(id) wstrata(id) taua
Within-cluster Kendall's tau-a with variable: u
Transformation: Untransformed
Within strata defined by: id
Valid observations: 40
Number of clusters: 10
Symmetric 95% CI
(Std. Err. adjusted for 10 clusters in id)
| Jackknife
u | Coef. Std. Err. z P>|z| [95% Conf. Interval]
u | .75 . . . . .
u | .75 . . . . .
. somersd u u , cluster(id) wstrata(id) taua
Kendall's tau-a with variable: u
Transformation: Untransformed
Within strata defined by: id
Valid observations: 40
Number of clusters: 10
Symmetric 95% CI
(Std. Err. adjusted for 10 clusters in id)
| Jackknife
u | Coef. Std. Err. z P>|z| [95% Conf. Interval]
u | (dropped)
u | (dropped)
. somersd u u , funtype(wcluster) wstrata(id) taua
Within-cluster Kendall's tau-a with variable: u
Transformation: Untransformed
Within strata defined by: id
Valid observations: 40
Symmetric 95% CI
| Jackknife
u | Coef. Std. Err. z P>|z| [95% Conf. Interval]
u | (dropped)
u | (dropped)
. somersd u u , wstrata(id) taua
Kendall's tau-a with variable: u
Transformation: Untransformed
Within strata defined by: id
Valid observations: 40
Symmetric 95% CI
| Jackknife
u | Coef. Std. Err. z P>|z| [95% Conf. Interval]
u | 1 . . . . .
u | 1 . . . . .
In case A, I get tau_uu = 0.75 (not 1).
In cases B and C (if either funtype or cluster are missing), I get zilch.
In case D, without any references to cluster, I get the "correct" value. But I want to account for the clusters- what am I missing?
Al Feiveson
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Loaves and fishes
I once caught a fish WITH a loaf...
To add, can someone calculate how much energy would need to be condensed to make about 1000 fish/loaves of bread?
Assume an average loaf of bread masses about 0.5 kg, and an average trout-sized fish masses about 2 kg.
1000 of each gives you 2500 kg of mass.
Apply Einstein's energy/mass equivalence equation: E = mc^2, where 'c' is approximately 3 x 10^8 m/s:
E = 2500 kg x (3x10^8 m/s)^2
= 2500 kg x (9x10^16 m^2/s^2)
= 2.25 x 10^20 kg/m^2/s^2
1 kg/m^2/s^2 is one Joule, so the energy requirement is 225 quintillion Joules.
To put that into perspective, that's about 62.5 trillion kilowatt-hours, or about as much electricity as the entire United States would use in about 17,000 years, based on 2010 rates of use.
Of course this just gives the energy required for the requisite mass, and doesn't take into consideration how much energy might be necessary to effect the actual conversion.
That Jesus, man, he was one POWERFUL dude. | {"url":"http://whywontgodhealamputees.com/forums/index.php/topic,25620.29.html","timestamp":"2014-04-21T07:24:16Z","content_type":null,"content_length":"36475","record_id":"<urn:uuid:ac25c911-4b04-46f3-bbae-1f75151c0a59>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00261-ip-10-147-4-33.ec2.internal.warc.gz"} |
probability help
May 17th 2009, 06:28 AM #1
Super Member
Sep 2008
probability help
A fairground game involves trying to hit a moving target with a gunshot. A round consists of up
to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player
misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of
one another.
(a) Find the probability that Linda scores 30 points in a round.
$0.6^3 = 0.216$
The random variable X is the number of points Linda scores in a round.
(b) Find the probability distribution of X.
x = 0, 10 , 20 , 30
p(x) 0.4 0.24 0.144 0.216
(c) Find the mean and the standard deviation of X.
$E(X) = 11.76$
$SD = 11.7$
A game consists of 2 rounds.
(d) Find the probability that Linda scores more points in round 2 than in round 1.
I need help woth the last question, 'd', How do I start?
(a) Find the probability that Linda scores 30 points in a round.
$0.6^3 = 0.216$
The random variable X is the number of points Linda scores in a round.
(b) Find the probability distribution of X.
x = 0, 10 , 20 , 30
p(x) 0.4 0.24 0.144 0.216
(c) Find the mean and the standard deviation of X.
$E(X) = 11.76$
$SD = 11.7$
A game consists of 2 rounds.
(d) Find the probability that Linda scores more points in round 2 than in round 1.
I need help woth the last question, 'd', How do I start?
Hi Tweety,
For (d), use your results from (b).
Let's say the result of the first round is X and the result of the second round is Y. There are only 4 possibilities for each, so you should find it easy to list all the pairs (x,y) where x < y.
Use independence of X and Y, along with your results from (b), to find the probability of each (x,y) pair. Then add the probabilities up.
May 17th 2009, 01:15 PM #2 | {"url":"http://mathhelpforum.com/statistics/89342-probability-help.html","timestamp":"2014-04-21T08:37:11Z","content_type":null,"content_length":"35710","record_id":"<urn:uuid:a5afefb6-1765-457d-a847-c1e110d2cf83>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dosage Calculation Questions
1. 0
May 31, '09 by
Can someone please help me with the following two problems. I have tried several times to work them and cannot come up with the answer that is correct:
Tylenol Elixir grV is ordered. The bottle reads 25 mg/cc. How much should the nurse administer? The answer is 12 ccs, but I can't figure out how to get that number.
A vial contains 2 GM of powder. After being reconstituted with 4 cc saline, how many mls are needed to administer a 500 mg dose? The answer is 1 cc, but again, can't figure out how to arrive at
I'm sure I'm not setting up the problems correctly.
Any help would be greatly appreciated!!! I'm doing this as a review for a dosage calculation test and I need to know how to set these up.
3. 0
tylenol elixir grv is ordered. the bottle reads 25 mg/cc. how much should the nurse administer? the answer is 12 ccs, but i can't figure out how to get that number.
the first thing you need to do is convert grains to mg. 1 grain consists of 60 mg (some references state 65). so, you can solve this using ratio : proportion. x : 5 = 60: 1 x = 300 next, use d /h
x q to determine how many ml you need. d = the dose ordered; h = the quantity on hand, and q = the total volume of the quantity on hand x = 325 / 25 x 1 x = 12
a vial contains 2 gm of powder. after being reconstituted with 4 cc saline, how many mls are needed to administer a 500 mg dose? the answer is 1 cc, but again, can't figure out how to arrive
at that.
first, you need to convert g to mg, then you can use d/h x q to solve this problem as well. x = 500 / 2000 x 4 x = 0.25 x 4 x = 1 hope this helps.
4. 0
May 31, '09 by
Tylenol Elixir grV is ordered. The bottle reads 25 mg/cc. How much should the nurse administer? The answer is 12 ccs, but I can't figure out how to get that number.
on hand: 25mg/cc
ordered: gr 5---> 60mg=gr---> 5x60=300mg ordered
I do ratio and proportion to figure the dosage calculations out
25mg/cc=300mg/Xcc, solve for X (cross multiply)
x=12 cc
A vial contains 2 GM of powder. After being reconstituted with 4 cc saline, how many mls are needed to administer a 500 mg dose? The answer is 1 cc, but again, can't figure out how to arrive at
does 2 GM mean 2 grams? Thats what I am assuming. Anyways, 2 grams=2000mg (1000mg to 1 g)
on hand: 2000mg/4cc
ordered: 500mg
ratio and portion again:
2000mg/4cc=500mg/Xcc, solve for X (cross multiply)
x=1 cc
Last edit by melmarie23 on Jun 1, '09
5. 0
tylenol elixir grv is ordered. the bottle reads 25 mg/cc. how much should the nurse administer? the answer is 12 ccs.
dose desired:
tylenol elixir grain v (5)
dose on hand:
tylenol elixir 25 mg/cc
amount the dose on hand comes in:
1 cc
conversion factor:
1 grain = 60 mg
by dimensional analysis (factor label) method:
5 grains
(dose desired)
/25 mg
(dose on hand)
x 1 cc
(amount the dose on hand comes in)
/1 x 60 mg/1 grain
(conversion factor)
= 12 cc
(amount the nurse will administer)
if you do the conversion first (5 grains = 300 mg), the problem simply becomes:
300 mg (dose desired)/ 25 mg (dose on hand) x 1 cc (amount the dose on hand comes in) = 12 cc (amount the nurse will administer)
a vial contains 2 gm of powder. after being reconstituted with 4 cc saline, how many mls are needed to administer a 500 mg dose? the answer is 1 cc.
dose desired:
500 mg
dose on hand:
2 grams
amount the dose on hand comes in:
4 cc
conversion factor:
1 gram = 1000 mg
by dimensional analysis (factor label) method: 500 mg (dose desired)
/2 grams
(dose on hand)
x 4 cc
(amount the dose on hand comes in)
/1 x 1 gram/1000 mg
(conversion factor)
= 1 cc
(amount the nurse will administer)
if you do the conversion first (2 grams = 2000 mg), the problem simply becomes:
500 mg (dose desired)/2000 mg (dose on hand) x 4 cc (amount the dose on hand comes in) = 1 cc (amount the nurse will administer)
6. 0
from wannabenursetx
A vial contains 2 GM of powder. After being reconstituted with 4 cc saline, how many mls are needed to administer a 500 mg dose? The answer is 1 cc, but again, can't figure out how to arrive
at that.
Don't know if you would be interested in trying to solve this a less mathematical and more verbal way; but, I suspect you are getting hung up on the math to the point you are having difficulty
thinking it through.
So, for the above question look at it this way:
The vial contains 2GM of powder & you need to administer 500mg dose; so, 2 GM is the same as 2000mg (converting that to what is asked for in the dose to be given).
Next, looking at the liquid portion, you have 4 cc. The 4 cc in that vial will hold the 2000mg, after being reconstituted.
So, now you will have 2000mg of powder floating around in that 4cc of saline.
Got to administer 500mg dose. But, the full vial of 4cc has 2000mg. 1/2 of that vial is 2cc & 1/2 of the powder floating would be 1000mg. But, you need to give 500mg---and, that wouldn't be 1/2
(1000mg in 2cc), it WOULD be 1/4 of that.
1/4 of 2000mg is 500mg & 1/4 of 4cc would be 1cc.
So, to give the dose of 500mg you would have to draw up and administer 1cc (1/4 of the 2000mg floating around in the 4cc of saline). Try to visualize it. | {"url":"http://allnurses.com/nursing-student-assistance/dosage-calculation-questions-396577.html","timestamp":"2014-04-16T11:13:10Z","content_type":null,"content_length":"39618","record_id":"<urn:uuid:8ffd09a6-96c6-49fd-a401-0106def5b07e>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00158-ip-10-147-4-33.ec2.internal.warc.gz"} |
Programming Praxis - Emirps
Programming Praxis – Emirps
In today’s Programming Praxis, our task is to enumerate all the non-palindrome prime numbers that are still prime when their digits are reversed. Let’s get started, shall we?
Since we are to focus on maximum speed, the obvious choice is to use a library.
import Data.Numbers.Primes
I tried a couple different versions, and settled on a simple filter. This version can calculate all the emirps under one million in about 650 ms. One other version where the isPrime test was replaced
by first putting all the primes in a Set and doing membership testing was a fraction faster (about 10 ms), but it is about twice as long and requires specifying the maximum up front, which is less
convenient then the current infinite stream. I also tried to make it faster via parallelism, but I could only get it to go slower, so it looks like the benefits of doing everything in parallel don’t
weigh up against the additional overhead. Or maybe I’m just not using the library right.
emirps :: [Integer]
emirps = [p | p <- primes, let rev = reverse (show p)
, isPrime (read rev), show p /= rev]
All that’s left is to print all the emirps under a million.
main :: IO ()
main = print $ takeWhile (< 1000000) emirps
As mentioned, just evaluating the list takes about 650 ms, which is about the same as the Scheme version.
Tags: bonsai, code, emirps, Haskell, kata, praxis, primes, programming
Grégory LEOCADIE Says:
November 3, 2010 at 3:46 pm | Reply
You use the Data.Numbers.Primes package, but if you did your own prime numbers generation, would the bench be horrible ?
++ greg
Remco Niemeijer Says:
November 3, 2010 at 6:35 pm | Reply
That would depend on the algorithm used. Obviously a naive algorithm, like the one I used in the previous exercise, is going to be pretty slow. I tried using the basic wheel factorization method from
http://bonsaicode.wordpress.com/2009/05/08/programming-praxis-%E2%80%93-wheel-factorization/ (Data.Numbers.Primes also relies on wheel factorization, albeit a more complex version) and it calculates
the whole list in about 1.5 seconds. That’s on a different computer than the original solution though, so the two aren’t directly comparable. Still, it’s not too far off and that’s with an algorithm
that hasn’t exactly been optimized for speed (for instance, my primes function was simply 2 : filter isPrime [3,5..]). So the short answer is no, it wouldn’t be a whole lot worse. But why bother
reinventing the wheel (factorization)? :)
Grégory LEOCADIE Says:
November 3, 2010 at 9:54 pm | Reply
I was looking for comments/tips about an efficient implementation of the prime numbers generation, and you answered it well.
And I agree, no need to reinvent the wheel :D (good one ;))
Thanks again | {"url":"http://bonsaicode.wordpress.com/2010/11/02/programming-praxis-emirps/","timestamp":"2014-04-18T13:06:09Z","content_type":null,"content_length":"52148","record_id":"<urn:uuid:7adaf519-2304-45ab-9919-d476aa661da0>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find Coordinates of point given length and formula of line
November 16th 2010, 09:15 PM #1
Nov 2010
Find Coordinates of point given length and formula of line
Okay, this is a coordinate geometry question I think. It is question 11 from IGCSE Additional Mathematics - October/November 2007 Paper 2.
This is part two of the question. In part one, I found the length of a line OB. It starts from the origin and goes until it meets point B. I have to find the coordinates of B. The formula of the
line is y=2x and the length is 2√5. As I said before, it goes through the origin.
It is part of a right angled triangle BOA, with BA being the hypotenuse, OA being the second longest side and OB the shortest side. Length of OA is 3√5 and coordinate A is (6,-3). BO and AO meet
to form the 90deg angle which makes it a right angled triangle.
Please help me figure this out. I have tried but cannot find a solution. I need to know in less than an hour
Thanks in advance.
The distance from the origin is $2\sqrt5$
The line concenrned is y = 2x.
At any point (x, y) on the line, the distance is $d^2 = x^2 + y^2$
Using what you've got, you have:
$(2\sqrt5)^2 = x^2 + y^2$
$20 = x^2 + y^2$
Now, you have got the line y = 2x.
Use simultaneous equations to find the x coordinate of B.
Depending on where point B lies, relative of the origin, you can have 2 different answers.
Thanks alot. I was able to do it.
You're welcome
November 16th 2010, 09:44 PM #2
November 16th 2010, 09:53 PM #3
Nov 2010
November 16th 2010, 09:55 PM #4 | {"url":"http://mathhelpforum.com/geometry/163516-find-coordinates-point-given-length-formula-line.html","timestamp":"2014-04-17T18:48:58Z","content_type":null,"content_length":"39279","record_id":"<urn:uuid:6bd38bd4-07b5-4f38-b47c-8b7e0acc4833>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00209-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bond Tutor: Free Interactive Textbook and Calculators3.8 Calculating Forward Rates from Spot RatesMyCSSMenu Save Document
3.8 Calculating Forward Rates from Spot Rates
The concept of a forward rate applies to any future period. Suppose you want to know what the implied forward rate (annualized) is for the period starting in Year 10 and running through to the end of
Year 20. How would you calculate this?
It turns out that the general formula for computing forward rates is straightforward to derive and is constructed from the definition of a spot rate. Recall that the spot interest rate for n-periods
is the geometric average of the n one period spot and forward rates:
Expanding both sides by the power of n, the n-period spot rate is:
and the n1 period spot rate is:
The n-period forward rate is obtained by dividing the n-period spot rate raised to the power of n by the n1 period spot rate raised to the power of n-1. You can check that all terms in the product of
the spot and forward rates cancel except for the period n forward rate as follows:
This mathematical relationship can be generalized to provide any forward rate from the ratio of two spot rates that are raised to the power equal to their life. For example, the t period forward rate
defined over Period n-t to Period t, is computed directly from the n-period spot rate and the n-t period spot rate as follows:
This provides the forward rate for (n-t,n), denoted as:
This result allows us to price an important financial contract that is called, not surprisingly, a foward contract. | {"url":"http://www.bondtutor.com/btchp3/topic8/topic8.htm","timestamp":"2014-04-20T18:35:02Z","content_type":null,"content_length":"34966","record_id":"<urn:uuid:4062e508-fb92-45e7-9281-a68b965abd11>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
Excel New Feature - INQUIRE
> Excel New Feature – INQUIRE
Excel New Feature – INQUIRE
Excel 2013 included some great new features for BI. However, one really cool feature that flew under the radar was Inquire. For all intents and purposes, this is formula tracing at the GYM, on
steroids and the only the only runner in the race. It’s a great inclusion to understand the structure of the workbook including dependencies between books (including formula references), record the
formulas and document just about anything in the book.
So let’s start with a quick recap of formula auditing. This is still available and shows the how a cell is derived. You typically activate the cell and then press the Trace Precedents (if you want to
see what cells are used in the cells formula) or Trace Dependents (if you want to see what other cells have a formula which is dependent on the selected cell). These are in the formulas ribbon. We
can see how this works in the following image where the dependencies of cell B7 are shown as B5 and B6 (note the blue line connecting the cells).
When the formula was a linked value from another book, a grid would show to indicate and external reference. In the prior image, the cells A1 and A2 refer to cells in another workbook (called book 2)
and so tracing the Precedents would show the following.
Now let’s compare this to Inquire
Firstly, Inquire must be activate as an Excel add-in. Go to File à Options, then chose add-ins from the options pane, then manage click Go from the Excel Add-Ins dropdown.
Ensure that the Inquire add-in is checked. Then an Inquire ribbon tab should be present.
The ribbon gives us many options, but let’s focus on the structure of the work book. If we click the Workbook analysis button, a new window will open which allows us to chose what we want to look at.
For example, we could list formulas by selecting the All formulas from the Formulas node in the tree (as shown). Note that all formulas are listed which includes a reference to an external book.
Don’t like the window? We can even export the results for any number of selected nodes (to a new workbook) by hitting the ‘Excel Export’ button (in this window).
We can investigate the relationships (both Precedents and Dependencies) of a particular cell (just click the Cell Relationship button in the ribbon). Here a new window opens with our choices (we can
chose the direction and number of expansion levels (for investigation)).
This post has just scratched the surface – there are a few other interesting options to explore (eg file comparisons and passwords), however, this feature should be very useful for tracing (and
perhaps more importantly documenting) formulas in books.
1. December 12, 2013 at 1:05 am |
2. January 7, 2014 at 12:20 pm |
Awesome post, Paul. I’ll be digging deeper into this hidden gem. Thanks for sharing.
3. January 9, 2014 at 9:59 am |
Great article, just a quick comment that my version of Excel has the add in under COM not under Excel.
1. January 7, 2014 at 2:35 pm | | {"url":"https://paultebraak.wordpress.com/2013/12/10/excel-new-feature-inquire/","timestamp":"2014-04-18T08:02:09Z","content_type":null,"content_length":"59406","record_id":"<urn:uuid:d23a76f6-d6c4-429d-886b-86d9298545e4>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00648-ip-10-147-4-33.ec2.internal.warc.gz"} |
Points on Cocentric Circles
November 23rd 2011, 01:04 AM #4
November 22nd 2011, 06:41 PM #3
November 22nd 2011, 04:43 AM #2
Super Member
November 21st 2011, 09:25 PM #1
Re: Points on Cocentric Circles
Dear Soroban
Thanks for the reply.
There is a problem in my case i.e. cernter of the circles O is not the origin so what changes be made to accommodate it?
Re: Points on Cocentric Circles
Hello, tariqchpk!
Place center $O$ at the origin.
The larger circle has equation:
. . $x^2 + y^2 \:=\:R^2 \quad\Rightarrow\quad x \:=\:\pm\sqrt{R^2 - y^2}$
Let $m = MO.$
The horizontal line has equation:. $y = m$
The line and the circle intersect at:
. . $A'\left(\sqrt{R^2-m^2},\:m\right)\,\text{ and }\,B'\left(\text{-}\sqrt{R^2-m^2},\:m\right)$
And we have:. $C'(0,\,\text{-}R)$
Points on Cocentric Circles
Re: Points on Cocentric Circles | {"url":"http://mathhelpforum.com/geometry/192453-points-cocentric-circles.html","timestamp":"2014-04-16T08:02:42Z","content_type":null,"content_length":"43822","record_id":"<urn:uuid:853445f2-b45c-455b-822d-0819927a31de>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00568-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posted by jasmine20 on Friday, April 20, 2007 at 4:54pm.
how do i convert
-2.25 into a fraction?
OK, if dis is a calculator question, den all u gotta do is follow the steps...
1. Type into calculator -2.25 and press the equals button.
2. Press the Shift button
3. Press the button dat has ab/c
And dis should giv you a fraction
You can think of this in two parts.
-2.25 has numbers before and after the decimal.
Before the decimal, you have -2. Just leave that like it is.
After the decimal, you have 25. This is the part that really needs converted. To do that, let's look at the names of the places after the decimal:
.x = tenths
.0x = hundreths
.00x = thousandths
Since there are 2 numbers after the decimal, you are dealing with hundreths. There are 25 hundreths.
Take the fraction 25/100
So the answer (there is one more step) is
-2 25/100
25/100 = 1/4
So you have
-2 1/4
pa kabar
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Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 386253, 15 pages
Research Article
Implicit-Relation-Type Cyclic Contractive Mappings and Applications to Integral Equations
^1Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Chhattisgarh, Raipur 492101, India
^2Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Beograd, Serbia
^3Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
Received 28 July 2012; Accepted 5 September 2012
Academic Editor: Sehie Park
Copyright © 2012 Hemant Kumar Nashine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to
support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.
1. Introduction and Preliminaries
It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in
classical functional analysis. The Banach contraction principle is a very popular tool which is used to solve existence problems in many branches of mathematical analysis and its applications. It is
no surprise that there is a great number of generalizations of this fundamental theorem. They go in several directions modifying the basic contractive condition or changing the ambient space. This
celebrated theorem can be stated as follows.
Theorem 1.1 (see [1]). Let be a complete metric space and let be a mapping of into itself satisfying: where is a constant in . Then, has a unique fixed point .
There is in the literature a great number of generalizations of the Banach contraction principle (see, e.g., [2] and references cited therein).
Inequality (1.1) implies continuity of . A natural question is whether we can find contractive conditions which will imply existence of a fixed point in a complete metric space but will not imply
On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al. [3].
Definition 1.2 (see [3, 4]). Let be a metric space. Let be a positive integer and let be nonempty subsets of . Then is said to be a cyclic representation of with respect to if
(i) , are nonempty closed sets, and
(ii) , .
Kirk et al. [3] proved the following result.
Theorem 1.3 (see [3]). Let be a metric metric space and let be a cyclic representation of with respect to . If holds for all , where , and , then has a unique fixed point and .
Notice that, while contractions are always continuous, cyclic contractions might not be.
Following [3], a number of fixed point theorems on cyclic representations of with respect to a self-mapping have appeared (see, e.g., [4–12]).
In this paper, we introduce a new class of cyclic contractive mappings satisfying an implicit relation in the framework of metric spaces and then derive the existence and uniqueness of fixed points
for such mappings. Suitable examples are provided to demonstrate the validity of our results. Our main result generalizes and improves many existing theorems in the literature. We also give an
application of the presented results in the area of integral equations and prove an existence theorem for solutions of a system of integral equations in the last section.
2. Notation and Definitions
First, we introduce some further notations and definitions that will be used later.
2.1. Implicit Relation and Related Concepts
In recent years, Popa [13] used implicit functions rather than contraction conditions to prove fixed point theorems in metric spaces whose strength lies in its unifying power. Namely, an implicit
function can cover several contraction conditions which include known as well as some new conditions. This fact is evident from examples furnished in Popa [13]. Implicit relations on metric spaces
have been used in many articles (for details see [14–19] and references cited therein).
In this section, we define a suitable implicit function involving six real nonnegative arguments to prove our results, that was given in [20].
Let denote the nonnegative real numbers and let be the set of all continuous functions satisfying the following conditions: : is non-increasing in variables ; : there exists a right continuous
function , , for , such that for , or implies ; : , , for all .
Example 2.1. , where .
Example 2.2. , where .
Example 2.3. , where is right continuous and , for .
Example 2.4. , where , , and .
We need the following lemma for the proof of our theorems.
Lemma 2.5 (see [21]). Let be a right continuous function such that for every . Then , where denotes the times repeated composition of with itself.
Next, we introduce a new notion of cyclic contractive mapping and establish a new results for such mappings.
Definition 2.6. Let be a metric space. Let be a positive integer, let be nonempty subsets of , and . An operator is called an implicit relation type cyclic contractive mapping if
(*) is a cyclic representation of with respect to ;
(**) for any , (with ), for some .
Using Example 2.2, we present an example of an implicit relation type cyclic contractive mapping.
Example 2.7. Let with the usual metric. Suppose , , and ; note that . Define such that Clearly, and are closed subsets of . Moreover, for , so that is a cyclic representation of with respect to .
Furthermore, if is given by then . We will show that implicit relation type cyclic contractive conditions are verified. We will distinguish the following cases:(1), .(i)When and , we deduce and
inequality (2.3) is trivially satisfied.(ii)When and , we deduce and then . Inequality (2.3) holds as it reduces to .(2), .(i)When and , we deduce and inequality (2.3) is trivially satisfied.(ii)When
and , we deduce and Then . Inequality (2.3) holds as it reduces to .
Hence, is an implicit relation type cyclic contractive mapping.
3. Main Result
Our main result is the following.
Theorem 3.1. Let be a complete metric space, , nonempty closed subsets of , and . Suppose is an implicit relation type cyclic contractive mapping, for some . Then has a unique fixed point. Moreover,
the fixed point of belongs to .
Proof. Let (such a point exists since ). Define the sequence in by We will prove that If for some , we have , then (3.2) follows immediately. So, we can suppose that for all . From the condition , we
observe that for all , there exists such that . Then, from the condition , we have and so Now using , we have and from , there exists a right continuous function , , , for , such that for all , If we
continue this procedure, we can have and so from Lemma 2.5,
Next we show that is a Cauchy sequence. Suppose it is not true. Then we can find a and two sequences of integers , , with We may also assume by choosing to be the smallest number exceeding for which
(3.9) holds. Now (3.7), (3.9), and (3.10) imply and so
On the other hand, for all , there exists such that . Then (for large enough, ) and lie in different adjacently labelled sets and for certain . Using the triangle inequality, we get which, by (3.12),
implies that Using (3.2), we have Again, using the triangle inequality, we get Passing to the limit as in the above inequality and using (3.16) and (3.14), we get Similarly, we have Passing to the
limit as and using (3.2) and (3.14), we obtain Similarly, we have
Using the condition (2.3) for and , we have and so Now letting and using (3.12), (3.14), and (3.18)–(3.21), we have, by continuity of , that a contradiction with since we have supposed that . Thus,
is a Cauchy sequence in . Since is complete, there exists such that We will prove that From condition , and since , we have . Since is closed, from (3.25), we get that . Again, from the condition ,
we have . Since is closed, from (3.25), we get that . Continuing this process, we obtain (3.26).
Now, we will prove that is a fixed point of . Indeed, from (3.26), for all , there exists such that . Applying with and , we obtain and so letting from the last inequality, we also have which is a
contradiction to . Thus, and so ; that is, is a fixed point of .
Finally, we prove that is the unique fixed point of . Assume that is another fixed point of , that is, . By the condition , this implies that . Then we can apply for and . Hence, we obtain Since and
are fixed points of , we can show easily that . If , we get which is a contradiction to . Then we have , that is, . Thus, we have proved the uniqueness of the fixed point.
In what follows, we deduce some fixed point theorems from our main result given by Theorem 3.1.
If we take and in Theorem 3.1, then we get immediately the following fixed point theorem.
Corollary 3.2. Let be a complete metric space and let satisfy the following condition: there exists such that for all . Then has a unique fixed point.
Corollary 3.3. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that
(*)' is a cyclic representation of with respect to ;
(**)' for any , with , where . Then has a unique fixed point. Moreover, the fixed point of belongs to .
Remark 3.4. Corollary 3.3 is an extension to Theorem 2.1 in [3, 4].
Corollary 3.5. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that
(*)' is a cyclic representation of with respect to ;
(**)' for any , with , where is right continuous and for . Then has a unique fixed point. Moreover, the fixed point of belongs to .
Remark 3.6. Taking in Corollary 3.5, with , we obtain a generalized version of Theorem 3 in [3, 8].
Corollary 3.7. Let be a complete metric space, , nonempty closed subsets of , , and . Suppose that there exists such that(*)' is a cyclic representation of with respect to ;(**)' for any , with ,
where , , . Then has a unique fixed point. Moreover, the fixed point of belongs to .
The following example demonstrates the validity of Theorem 3.1.
Example 3.8. Let with the usual metric. Suppose , , and . Define by , for all . Clearly, are closed subsets of . Moreover, for so that is a cyclic representation of with respect to . Moreover,
mapping is implicit relation type cyclic contractive, with defined by Indeed, to see this fact we examine the following cases.
Inequality (2.3) reduces to (I) For , :(i)suppose and . Then inequality (2.3) holds as it reduces to ;(ii)suppose and . Then inequality (2.3) holds as it reduces to ;(iii)suppose and . Then
inequality (2.3) holds as it reduces to ;(iv)suppose and . Then inequality (2.3) holds as it reduces to ;(v)suppose and . Then inequality (2.3) holds as it reduces to .(II) For , :(i)suppose and .
Then inequality (2.3) holds as it reduces to ;(ii)suppose and . Then inequality (2.3) holds as it reduces to ;(iii)suppose and . Then inequality (2.3) holds as it reduces to .(III) For , , inequality
(2.3) trivially holds.
Similarly other cases can be verified. Hence, is an implicit relation type cyclic contractive mapping. Therefore, all conditions of Theorem 3.1 are satisfied and so has a fixed point (which is ).
We illustrate Theorem 3.1 by another example which is obtained by modifying the one from [22].
Example 3.9. Let and we define by and let , , and be three subsets of .
Define by Let the function be defined by where , , , , , and , for all . Then is an implicit type cyclic contractive mapping for for . Therefore, all conditions of Theorem 3.1 are satisfied and so
has a fixed point (which is ).
4. An Application to Integral Equations
In this section, we apply Theorem 3.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.
We consider the following nonlinear integral equation, where , and are continuous functions.
Let be the set of real continuous functions on . We endow with the standard metric It is well known that is a complete metric space. Define the mapping by
Let , such that We suppose that for all , we have We suppose that for all , is a decreasing function, that is, We suppose that Finally, we suppose that for all , for all with and or and , where .
Now, define the set We have the following result.
Theorem 4.1. Under the assumptions (4.4)–(4.9), Problem (4.1) has one and only one solution .
Proof. Define the closed subsets of , , and by We will prove that Let , that is, Using condition (4.7), since for all , we obtain that The above inequality with condition (4.5) imply that for all .
Then we have .
Similarly, let , that is, Using condition (4.7), since for all , we obtain that The above inequality with condition (4.6) imply that for all . Then we have . Finally, we deduce that (4.12) holds.
Now, let , that is, for all , This implies from condition (4.4) that for all , Now, using conditions (4.8) and (4.9), we can write that for all , we have This implies that Using the same technique,
we can show that the above inequality holds also if we take .
Now, all the conditions of Corollary 3.3 are satisfied (with ) and we deduce that has a unique fixed point ; that is, is the unique solution to (4.1).
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC Grant no. 55000613).
Moreover, the second author is grateful to the Ministry of Science and Technological Development of Serbia and the third author gratefully acknowledges the support provided visitor project by the
Department of Mathematic and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of
Alberta, Canada’ as a visitor for the short-term research.
1. S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
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at Google Scholar | {"url":"http://www.hindawi.com/journals/aaa/2012/386253/","timestamp":"2014-04-20T18:52:03Z","content_type":null,"content_length":"729389","record_id":"<urn:uuid:3dc73c6d-0537-4585-af3e-7159ca6cf001>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
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OCR for page 16
16 Whether using proportionate or disproportionate sam- degree of precision and reflects the spread of observed val- pling, weights are developed for each group (strata) in the ues that would be seen
if the survey were repeated numerous sample once the surveying is completed. Weights are most times. The confidence level (95%) is how often the observed often based on ridership. The weight for each
stratum is cal- transfer rate would be within 3 percentage points of the true culated based on the ratio of total ridership for the strata and transfer rate if the survey were repeated numerous
times. the number of surveys collected from that group. Total ridership may be for a given route, route and time-of-day The sample size that is needed for a given survey depends combination, or
station. As an example in a rail survey, on the population size and level of precision desired. If the WMATA weighted surveys to daily ridership by mezzanine. researcher wants to be within 3
percentage points, for exam- ple, a sample size of 1,066 is required for a large population, For weighting by boardings, transit agencies measure but approximately one-half that number for a
population of total boardings in a variety of ways. SANDAG, for example, about 1,000. Table 8 provides sample sizes needed for three used automatic passenger counters to determine the number levels
of precision (10%, 5%, and 3%) at a 95% confidence of passengers boarding for each route and time-of-day com- level for various population sizes. bination. Ridership can also be based on entries as
recorded by bus fare boxes or turnstiles. Another method is for survey In transit surveys, it is often desired to achieve a given level workers to count the total number of passengers entering, of
precision for each of a number of major routes or for each whether or not they accepted a survey form. (See chapter five of several time periods. In this situation, the sample size needs for further
discussion of the fieldwork protocols for these to be computed for each subgroup; for example, each route or counts.) day part. A Transit Authority of River City (TARC) survey, for example, developed
the sample plan based on achieving a Although most responding transit agencies weighted sampling error of 8 percentage points for routes with 1,000 or surveys by ridership, more complex methods are
sometimes more average weekday boardings and 12 percentage points for used, particularly for O&D surveys. A good example is routes with fewer weekday boardings. In addition, the bottom PATH, which
used an advanced statistical technique called 10 routes in terms of ridership were sampled as one unit with iterative biproportional fitting to weight response by station a sampling error of 5
percentage points. entry and exit and time of day. For surveys with stratified sampling, as in the TARC survey, calculating the sampling error for the entire survey must take MINIMIZING SAMPLING AND
NONRESPONSE account of the stratified sample design. One cannot simply use ERROR IN SURVEY the overall number of responses to calculate the sampling error. Stratification may change the efficiency of
the sample-- The precision of a survey is determined by the amount of in some cases improving efficiency (as when the strata are error created in the process of taking a sample and conduct-
relatively homogeneous) or reducing efficiency (when the vari- ing data collection. Sampling error, which arises from ance of each strata are about the same). The specific situation surveying a
sample of the study population rather than the will affect the sampling error of the total sample. entire population, is often the focus of discussion of survey error issues. There are other sources
of error, however, including nonresponse error, coverage error (discussed Nonresponse Error earlier), and measurement error (discussed in chapter four). Another major source of error is nonresponse
error, which results from failure to obtain completed surveys from some Sampling Error Virtually all on-board and intercept transit surveys involve TABLE 8 taking a sample of the study population and
are thus subject SAMPLE SIZES NEEDED FOR VARIOUS POPULATION SIZES to sampling error. Because surveys rely on a sample of the AT VARIOUS LEVELS OF PRECISION population, survey results are likely to be
somewhat differ- Sampling Error for 95% Confidence Level ent than if the entire population was interviewed. Population ±10% ±5% ±3% 200 65 132 169 The sampling error is an expression of the
difference 400 78 196 291 between the true (but unknown) value and observed values 1,000 88 278 517 if, hypothetically, the survey were repeated numerous times. To use an example, suppose that an
on-board bus survey 6,000 95 361 906 found that 20% of riders transferred to another bus on the 20,000 96 377 1,013 trip. The sampling error might be stated as plus or minus 1,000,000 96 384 1,066 3
percentage points with a 95% confidence level. The con- Note: Sample size needed for each sampling error; responses with frequency fidence interval (plus or minus 3 percentage points) is the of 50%.
OCR for page 16
17 portion of the population selected in the sample. It is inevitable However, one can attempt to evaluate and possibly compen- that some riders refuse to take a survey, never return a survey sate
for nonresponse. The likely impact from nonresponse can that they took, or refuse to be interviewed. These respondents be evaluated by comparing characteristics of respondents with might have
responded in the same way as respondents who did those of the entire population or those within the sampling complete the survey, or they might not have. In contrast frame. The comparison is
sometimes made for rider charac- to sampling error that can be computed, there is no standard- teristics such as gender, race, and place of residence, or for trip ized way to compute the error that
arises from nonresponse. characteristics such as on and off locations. | {"url":"http://www.nap.edu/openbook.php?record_id=13866&page=16","timestamp":"2014-04-19T12:39:37Z","content_type":null,"content_length":"43428","record_id":"<urn:uuid:bff87235-44a8-40a5-9d3f-69fd442a06d0>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00535-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?
up vote 3 down vote favorite
The fundamental group $\mathcal{F}(N \subset M)$ of an inclusion of $II_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \subset M) \} $
(see here)
- It's $\mathbb{R}_{+}^{*}$ for every finite index finite depth irreducible subfactor.
- It's trivial for uncountably many subfactors of the form $R^{\mathbb{Z}_{2}} \subset R⋊\mathbb{Z}_{3}$ (Bisch-Nicoara-Popa)
A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
- Every $2$-supertransitive subfactors: $A_{n}$-subfactor, Haagerup subfactor...
- Every group-subgroup subfactor $(R^{G} \subset R^{H})$ such that $(H \subset G)$ is a maximal subgroup
(i.e. $\pi_{H}(G)$ is a primitive permutation group with $\pi_{H} : G \to S_{X}$ canonical for $X = G/H$).
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$ ?
Remark : For being relevant, we need to restrict to factors with full fundamental group. And for being reasonable, we can start by inclusion of hyperfinite $II_{1}$ factors. Next, we could enlarge to
every factors with full fundamental group (for example $L(\mathbb{F}_{\infty})$).
2 I highly doubt this is known. I don't have much intuition about what being a maximal subfactor gives you, but I think the only results about fundamental groups of subfactors is what you give
above. (Well almost, by Popa's work any amenable subfactor, not just finite depth, has full fundamental group, and as mention in the BNP article you cite it is true for any stable subfactor). As
far as results where it is not all of $\mathbb{R}^+$, I think that the BNP examples are all that is known. – Owen Sizemore Jul 11 '13 at 22:01
Intuitively, a maximal subfactor is a kind of quantum generalization of a prime number, because $R^{G} \subset R$ is maximal iff $G = \mathbb{Z}/p\mathbb{Z}$ with $p$ a prime number. – Sébastien
Palcoux Jul 12 '13 at 8:21
I had read about stable subfactor in the BNP: << i.e. splits a common copy of the hyperfinite $II_{1}$ factor $R$ >>, but I don't understand what does it mean. – Sébastien Palcoux Jul 12 '13 at
1 Stable means the that $P\subset Q\simeq P\otimes\mathcal{R}\subset Q\otimes \mathcal{R}$. – Owen Sizemore Jul 13 '13 at 4:48
Thank you again ! – Sébastien Palcoux Jul 13 '13 at 9:31
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starters question about handling data types.
I am following this book and it wants me to encrypt a 4-digit integer that the user inputs. so the user Inputs 4789 for example now I want to perform actions on each of the digits separately. How can
I work with the 4 / 7 / 8 / 9 separately?
Arithmetic -- dividing by ten, or hundred, or thousand; taking remainders when dividing by ten, or hundred, or thousand....
Oh okay, thanks lol... should of figured that out myself I think
So what stage of the book are you at (what is the name of the chapter, what sort of things have you been tought in that chapter)? I can think of at least three very different methods to do that, but
which one to suggest depends on whether you are at one of the first three chapters or further on in the chapter. -- Mats
Uhm Im learning about control statements now.... If else while etc... although I already know about these because I also program some PHP... but PHP does not require to declare variable types...
So what variable types do you think you need (and which ones have you been told about so far)? Also, I take it that you have an encryption where you "rotate" a number 3 steps like on one of those
bicycle/briefcase locks, so 1 rotated 2 steps makes 3, 9 rotated 3 steps makes 2, etc, yeah? -- Mats
uhm the asignement is like this : /* (Cryptography) A company wants to transmit data over the telephone, but is concerned that its phones could be tapped. All of the data are transmitted as
four-digit integers. The company has asked you to write a program that encrypts the data so that it can be transmitted more securely. Your program should read a four-digit integer and encrypt it as
follows: Replace each digit by (the sum of that digit plus 7) modulus 10. Then, swap the first digit with the third, swap the second digit with the fourth and print the encrypted integer. Write a
separate program that inputs an encrypted fourdigit integer and decrypts it to form the original number. */ I know about INT and string... but just to use, not advanced... btw i still did not find a
So start with a program that can read in an integer and print it. Then add functionality to print each digit out of the number using the method of dividing and using modulo to find the remainder.
Then add the functionality to modify each digit, and then put it back together into a single integer again. -- Mats
I dont quite understand how to use the modulo... How should I work it to get each digit? first digit, divide by 1000 but how do I get the second,third and forth one? this is what I got till now:
Code: int Encrypt (int value) { // do it } int main () { int to_be_encrypted,encrypted; std::cout <<"Please enter a 4 digit integer number for encryption"; std::cin >> to_be_encrypted; encrypted =
Encrypt(to_be_encrypted); std::cout <<"The original number: " << to_be_encrypted << std::endl; std::cout <<"The encrypted number: " << encrypted << std::endl; std::cin.get(); return 0; }
int Encrypt (int value) { // do it } int main () { int to_be_encrypted,encrypted; std::cout <<"Please enter a 4 digit integer number for encryption"; std::cin >> to_be_encrypted; encrypted = Encrypt
(to_be_encrypted); std::cout <<"The original number: " << to_be_encrypted << std::endl; std::cout <<"The encrypted number: " << encrypted << std::endl; std::cin.get(); return 0; }
Hint: 4789 / 1000 = 4 4789 % 1000 = 789
ok thnx, I got it now... can I modify my program to run better? or is this pretty much what was expected from me? Code: #include <iostream> int Encrypt (int value) { // compute seperate digits. int
First,Second,Third,Forth; First = value/1000; Second = (value%1000) / 100; Third = (value%1000) % 100 / 10; Forth = value % 10; First = (First+First+7) % 10; Second = (Second+Second+7) % 10; Third =
(Third+Third+7) % 10; Forth = (Forth+Forth+7) % 10; int Encrypted_number = (Third*1000)+(Forth*100)+(First*10)+(Forth); return (Encrypted_number); } int main () { int to_be_encrypted,encrypted;
std::cout <<"Please enter a 4 digit integer number for encryption"; std::cin >> to_be_encrypted; encrypted = Encrypt(to_be_encrypted); std::cout <<"The original number: " << to_be_encrypted <<
std::endl; std::cout <<"The encrypted number: " << encrypted << std::endl; std::cin.get(); std::cin.get(); return 0; }I am going to build the decryption application tomorrow, for I am tired :D
#include <iostream> int Encrypt (int value) { // compute seperate digits. int First,Second,Third,Forth; First = value/1000; Second = (value%1000) / 100; Third = (value%1000) % 100 / 10; Forth = value
% 10; First = (First+First+7) % 10; Second = (Second+Second+7) % 10; Third = (Third+Third+7) % 10; Forth = (Forth+Forth+7) % 10; int Encrypted_number = (Third*1000)+(Forth*100)+(First*10)+(Forth);
return (Encrypted_number); } int main () { int to_be_encrypted,encrypted; std::cout <<"Please enter a 4 digit integer number for encryption"; std::cin >> to_be_encrypted; encrypted = Encrypt
(to_be_encrypted); std::cout <<"The original number: " << to_be_encrypted << std::endl; std::cout <<"The encrypted number: " << encrypted << std::endl; std::cin.get(); std::cin.get(); return 0; }
>> (value%1000) % 100 That's the same as (value % 100). Your encryption function is not quite right. You are adding the digit to itself before adding seven to it, but that messes up the
encryption. There's also a bug when you recombine the encrypted digits (where's second?).
And it's actually a good idea to spell things right: Fourth should probably be the name of that variable. -- Mats
Then there's the easiest encryption: Code: #include <stdio.h> char EncryptChar( char c ) { return ~c; } void EncryptStr( const char* in, char* out ) { while ( *in ) { *out = EncryptChar( *in ); ++in;
++out; } *out = '\0'; } int main() { const char str[] = "Hello World"; char encryptStr[30]; EncryptStr( str, encryptStr ); printf( "Original: (%s)\nEncrypted: (%s)\n", str, encryptStr ); return 0; }
#include <stdio.h> char EncryptChar( char c ) { return ~c; } void EncryptStr( const char* in, char* out ) { while ( *in ) { *out = EncryptChar( *in ); ++in; ++out; } *out = '\0'; } int main() { const
char str[] = "Hello World"; char encryptStr[30]; EncryptStr( str, encryptStr ); printf( "Original: (%s)\nEncrypted: (%s)\n", str, encryptStr ); return 0; } | {"url":"http://cboard.cprogramming.com/cplusplus-programming/105419-starters-question-about-handling-data-types-printable-thread.html","timestamp":"2014-04-20T16:32:42Z","content_type":null,"content_length":"17599","record_id":"<urn:uuid:984eb07e-79b4-414f-908c-dd7c52439622>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00397-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Help with geometry problems.
Replies: 3 Last Post: Dec 23, 2009 8:31 AM
Messages: [ Previous | Next ]
Re: Help with geometry problems.
Posted: Aug 6, 2009 10:12 PM
> Help needed for geometry problems....
> Solve this equation by using the quadratic form.
> 2/x-5-1/x+1=1
> ....
I've just found your question which has been languishing here for some time. It's actually about algebra, not geometry, so you may have better luck with such questions in the much busier news group
Anyway, I'll try to answer this one now. Typing mathematical formulae is tricky, and yours hasn't got enough spaces or brackets/parentheses to make it clear. If it means
2/x - 5 - 1/x + 1 = 1
then it just amounts to 1/x = 5 which is easy to solve.
However, I suspect you may have meant
2/(x-5) - 1/(x+1) = 1
in which case multiply both sides of the equation by (x-5)(x+1) to get rid of the fractions, then switch all terms to one side and you'll have an ordinary quadratic equation.
Ken Pledger.
Date Subject Author
6/15/09 Help with geometry problems. courtney
7/13/09 Re: Help with geometry problems. andrea
8/6/09 Re: Help with geometry problems. Ken Pledger
12/23/09 Re: Help with geometry problems. tina harris | {"url":"http://mathforum.org/kb/message.jspa?messageID=6807784","timestamp":"2014-04-19T20:34:59Z","content_type":null,"content_length":"20285","record_id":"<urn:uuid:06e6f474-a546-47d2-95d8-67e951bc1fef>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00464-ip-10-147-4-33.ec2.internal.warc.gz"} |
K-Trimmed mean formula.. is it K ripley function?
June 27th 2011, 06:46 AM #1
Jun 2011
K-Trimmed mean formula.. is it K ripley function?
On this pdf paper, authors show formulas for calculating Trimmed Mean and
Winsorized Mean on unsorted data:
Regarding formulas (2) , (3) and (4) ... they show that formula (4) is a way
to compute formula (2) and they use min() and max() functions practically to
solve K() on formula (2) .. BUT what is K() ?
Formula (3) is Trimmed Mean and they don't give another version of it. Should
be converted to the same form of formula (4) by using min() and max() and if
yes, how should it be written correctly?
Is K() the K-Function by Ripley? And if yes, which version?
Of course there are formulas (5) and (6) where authors give another way of computing the same Trimmed Mean on unsorted data and for which they provide their C code implementation.
However I'd like to understand how K() should be solved and written on formula (3) for Trimmed Mean which uses Winsorized Mean of formula (4).
Thanks to anyone that can explain it to me.
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