text:
|Einstein for Everyone
Back to main course page
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
The arrival of Einstein's general theory of relativity marked a rebirth of interest and work in cosmology, the study of the universe on the largest scale. Within Newtonian theory, cosmology had reduced essentially to one question: just how is matter distributed within an infinite, Euclidean space. Einstein's theory made the question much more interesting. For now cosmologists had to contend with many possible matter distributions, many possible geometries and many possible dynamics for both.
What connected all these together was Einstein's gravitational field equations. They specified just which matter distributions could go with just which geometries and how the whole system might evolve over time. One way to think of these equations is as a law that our universe must satisfy. Another is to imagine them as a selection rule. Among all conceivable universes, only some will satisfy Einstein's equations. These universes are the ones that we designate as possible universes, where "possible" now just means "licensed by Einstein's theory."
|I like to think of these possible universes as each comprising a page of a great book. Metaphorically, Einstein's gravitational field equations are that book. We shall now turn to reading that book. Just what sorts of universes are possible according to Einstein's theory? As we flip from page to page we will see some quite interested universes. Among them we hope to find out own.
Minkowski spacetime is the spacetime in which special relativity holds. This is the simplest all solutions of Einstein's equations. It is the case of no gravity. It arises in when the unsummed curvature of spacetime is zero. What that means is this: if we test for curvature in all possible directions of the spacetime sheets, we find no curvature in each of them. We assemble all these zero results and write them collectively as:
(UNsummed) curvature
of spacetime
This is the unsummed curvature. The summed curvature that arises in Einstein's gravitational field equations comes from adding up these individual curvature terms in a way specified by the Einstein tensor. Since we are adding up many zero terms, we know that the result of the additions must still be zero. That is, if the unsummed curvature is zero, then the summed curvature must also be zero.
Summed curvature
of spacetime
Einstein's equations tell us that this summed curvature equals the matter density. Therefore they will be satisfied if the universe is free of matter. For this reason, Minkowski spacetime is an unrealistic candidate for our universe. We know our universe has matter in it!
A Minkowski spacetime is still physically relevant to our spacetime, however. The situation is similar to one we encounter on our earth's surface. We know that this surface is a sphere in the large and that a non-Euclidean geometry must be used to analyze it.
"Blue marble" image from NASA http://eoimages.gsfc.nasa.gov/images/imagerecords/54000/54388/BlueMarble.jpg
Yet in any small patch--such as an area of land the size of a city--we can ignore the curvature and apply Euclidean geometry without appreciable error.
New York city street grid. (From Google Maps.)
Correspondingly, for many applications, Minkowski spacetime is a sufficiently good approximation in the small.
It will fail, however, whenever, gravitation is strong; that is, when the curvature is large. In the vicinity of the sun, gravitation is not strong. So there we manage to extend the use of Minkowski spacetime by pretending that curvature effects of gravitation are really due to a new field, the gravitational field.
The more interesting spacetimes come from other pages of Einstein's book. We find them by "solving" Einstein's gravitational field equations. That just means that we find spacetimes whose summed curvature matches the matter density in the precise way that Einstein's field equations demand. The activity is called "solving" since it is the same as what is done in algebra. You write an equation for some unknown quantity "x" and solve the equation for x. The resulting value of x is called the "solution."For example:
2x + 1 = 5
2x + 1 -1 = 5 - 1
2x = 4
2x/2 = 4/2
x = 2
In solving the Einstein equations, however, we are writing equations for an unknown universe. When we solve them, the solution is an entire universe.
Sometimes, we write algebraic equations that have more than one solution. For example:
x3 - x = 0
is solved by
x = 1 and x=-1 and x=0.
We might deal with the multiplicity of solutions by saying the we are only interested in positive number solutions. Then we have a unique solution, x=1.
Having multiple solutions is commonly the situation with the Einstein equations. We narrow down the solutions to a few or one in each case by adding extra conditions. We may only be interested in spacetimes that are geometrically the same everywhere (i.e. they are "homogeneous"). Or we may only be interested in solutions that become Minkowskian very far from matter. These extra conditions are known loosely as "boundary conditions," since they are most commonly used to stipulate how the spacetime must be at some boundar such as spatial infinity.
Minkowski spacetime is the simplest solution. Finding others is a formidable mathematical challenge and only the simplest of solutions are easy to find. So when a new solution is found, the new solution is generally named after whoever found it. Finding a solution is really discovering a new, possible universe. So the discoverer's name is then attached to that universe: an Einstein universe, de Sitter spacetime, A Goedel universe, a Reissner-Nordstroem spacetime, and so on.
While the activity of solving Einstein's equations is very hard, the process in conceptual form is quite easy to describe. The Einstein equations specify how spacetime can be locally, that is, at any one point. They say this much curvature always goes with this much matter. To solve the Einstein equations is merely finding a way of distributing curvature and matter over spacetime so neighboring points mesh correctly.
|A rather good analogy is to the solving of a jigsaw puzzle. The Einstein equations give us an endless supply of small pieces. They are how spacetime can be in infinitesimally small patches. Each piece has the right combination of curvature and matter.
Solving the Einstein equations corresponds to
finding a combination of pieces from the supply that can be fitted together.
|The simplest possible solutions are "homogeneous"--that means that they are everywhere the same. In the jigsaw puzzle analogy, that means that the spacetime has to be put together from repetitions of the same piece over and over and over. In the case of a Minkowski spacetime, that piece is just one that has no curvature and no matter.
|This was one of the first interesting, exact solutions
given for Einstein's equations and was computed by the German astronomer Karl
Schwarzschild. It
is of a universe which looks like a Minkowski spacetime as you get close to
infinity in space. It has a central point in
space around which all the curvature is distributed symmetrically. It
is unchanging in time. The conditions then pick out the solution uniquely.
This spacetime is taken to be a good approximation for the spacetime around the sun (as long as we neglect that the sun rotates and has some electric charge).
Even this simple case was difficult to solve. Einstein himself had not found the exact solution for the spacetime surrounding the sun in his work of November 1915. He had then computed the motion of Mercury from an approximation. Schwarzschild found the exact solution and communicated it to Einstein in a letter of December 22, 1915.
The story is tragic. All this happened during the First World War. Schwarzschild was serving in the German army at the Russian front when he found the solution and wrote to Einstein. He died at the front on May 11, 1916.
In the jigsaw puzzle analogy, the solution is put together from pieces that are flat like those of the Minkowski spacetime in regions that are far from the central point. The pieces get more curved as we approach the central point, but their curvature always respects the rotational symmetry of the space around the sun.
The Schwarzschild spacetime is a good approximation of the spacetime around our sun. But does it work for the whole universe? It would work if all the matter of the universe were located in just one island in an otherwise empty space. However, when Einstein started to contemplate these possibilities shortly after completing his general theory of relativity in 1916, he did not like this possibility for two reasons.
First, the astronomical information available to Einstein at that time indicated that the universe was filled with a roughly uniform distribution of stars. So empirically, the model was wrong.
Second, there was a deeper theoretical worry. General relativity had shown how matter fixes the summed curvature of spacetime. Einstein liked that notion a lot. It reduced the arbitrariness of a spacetime. Why did the geometry curve this way here and that way there? It did so because the matter distribution went this way here and that way there. The theory reduced the number of arbitrary stipulations that needed to be made in building a picture of our spacetime.
Einstein liked this idea so much that he elevated the idea to a principle. He demanded that the whole geometry of spacetime must be fixed by the matter distribution. That is stronger than what he had up to then in his field equations. For the Einstein equations only required the matter distribution to fix the summed curvature at each event in spacetime. One value of a summed curvature can correspond to many values of the unsummed curvature.
That possibility left a freedom in the geometry that Einstein now wanted to eliminate. The matter distribution would have to override this freedom. The matter distribution, he insisted, much fix the geometry completely. In 1918, Einstein called this stronger requirement "Mach's principle," since it reminded him of epistemological analyses of space and time undertaken by the physicist-philosopher Ernst Mach.
This page has Mach's classic commentary on Newton's bucket.
It is the principal source for what became "Mach's principle."
|The difficulty is that a Schwarzschild spacetime has a property that is not fixed by the matter distribution. That is its flatness at spatial infinity. There is no matter in realms remote from the center of the spacetime, so there is no matter to determine that flatness. It is something that we have to demand in addition.
In the jigsaw puzzle analogy, the problem is this: near the center, we know that we need to lay down pieces of spacetime that respect the rotational symmetry of spacetime around the central mass. But when we get far away from that central island, what sorts of piece are we supposed to lay down? In a Schwarzschild spacetime, we lay down pieces that look more and more like those found in a Minkowski spacetime. But that is now a choice we are making. Nothing about the matter in the central island forced us to make it. That arbitrariness is what worried Einstein.
If you have been reading attentively, you will have noticed that I have been careful to say only that this arbitrariness worried Einstein. It is not clear that we should we worried about this arbitrariness. Historically, Einstein's imagination was grasped by the idea of the matter distribution fixing spacetime geometry completely. It would be a pretty outcome and one that Einstein found very helpful in guiding his construction of new theories. If those theories survive empirical testing, we might even discover that his idea happens to be true. Somewhere, somehow Einstein came to believe not just that it might happen to be true. He came to believe that it has to be true. This is a much stronger claim. In reptrospect, it hard to see how to justify it. What if the matter of the universe were all collected in one big central island. What is wrong with spacetime remote from the island being flat? The universe has contingent features. Why can't that be one of them?
Einstein eventually lost his enthusiasm for Mach's principle. Writing much later in 1946 for his Autobiographical Notes, he gave this farewell to the principle:
"Mach conjectures that in a truly reasonable theory inertia would have to depend upon the interaction of the masses, precisely as was true for Newton's other forces, a conception that for a long time I considered in principle the correct one. It presupposes implicitly, however, that the basic theory should be of the general type of Newton's mechanics: masses and their interaction as the original concepts. Such an attempt at a resolution does not fit into a consistent field theory, as will be immediately recognized."
How could one avoid the need to stipulate the properties of spacetime at infinity? In 1917, Einstein came up with an ingenious escape: obliterate spatial infinity!
By adding an extra term to his gravitational field equations, Einstein found a simple solution of his augmented field equations. ("Augmented"? What is that about? It refers to the famous λ. See below for an explanation.) It contains a uniform matter distribution that approximates a uniform distribution of stars. That matter is at rest and the geometry of a spatial slice is unchanging with time. Space, however, curves back onto itself so that it is spherical. That is, space has the geometry of 5NONE with positive curvature. In such a space, there is no infinity at which to stipulate the properties of space and time.
If one pictures just one dimension of space, then the universe looks like a cylinder. Spacetime resides just in the surface of the cylinder. The vertical lines are the world lines of the stars at rest. The one spatial dimension is wrapped back onto itself; the time dimension is not. Each spatial slice at a particular time appears as a circle; if we could represent all three dimensions of space, we would somehow have to replace the circle by a complete sphere of three dimensional space.
The Einstein universe is an especially simple universe. It is homogeneous. That means that, like Minkowski spacetime, it is geometrically the same at every event. It is also spatially isortropic, which means that it is the same in every spatial direction. In the jigsaw puzzle analogy this homogeneity means that the spacetime is assembled from just one sort of piece, used repeated to build the entire spacetime.
Something important passed by rather quickly just now. The Einstein universe turns out not to solve Einstein's gravitational field equations of 1915. In order to accommodate the new cosmology, Einstein had to make what appeared to be a somewhat arbitrary adjustment to his gravitational field equations.
In their original form, they said
summed curvature
of spacetime
We saw in the chapter on general relativity what these equations require inside a uniform matter distribution. That was our first illustration--masses falling in an evacuated tube inside the matter distribution of the earth. There we saw that there is a positive curvature in the space-time sheets of space time that makes the masses accelerated towards each other.
|The situation is the almost exactly the same as with objects within the uniform matter distribution of the Einstein universe. Einstein's original gravitational field equations call for positive curvature in the space-time sheets. That means that his field equations are calling for the same sort of dynamics as we saw for masses falling freely in a tube drilled through the center of the earth. All the matter of the universe should be accelerating towards the other neighboring pieces of matter, just as the neighboring masses in the tube accelerate towards each other. That is, all the matter of this universe should be undergoing everywhere an inward gravitational collapse, perhaps delayed only by an initial outward velocity.
|This difference between this case and the hole in the earth is that the matter of the earth is prevented by the mechanical incompressibility of rock from going into the free-fall motion of of the masses in the tube. The stars comprising cosmic matter are not kept apart by an incompressibility and thus should accelerate towards each other.
The trouble is that there is no curvature in those sheets in an Einstein universe; the spatial slices remain unchanged through time. There is no convergence or divergence of the points of the matter distribution.
Einstein's resolution was to modify his field equations in a way that would no longer call for this particular curvature. That is, he put another term into the equations that supplied the missing curvature. The real justification was essentially only that it gave him the result he wanted, the admissibility of his new universe. The term added was Einstein's celebrated "cosmological term" or just "lambda" λ. It is a constant term added to the equations, which means that it is the same at every event.
His gravitational field equations now read:
summed curvature
of spacetime
As before, each of these
quantities is really a 4x4 table of numbers. The λ table is constant
in the sense that it is the same at every event in spacetime.
At the time, it seemed like a good idea. But Einstein very soon came to regret the addition, which he saw as harming the formal beauty and simplicity of the equations.
|Einstein also almost immediately became embroiled in a dispute with the Dutch astronomer, de Sitter. Einstein had hoped that augmenting his gravitational field equations with the cosmological term would preclude empty universes without matter. De Sitter showed that the augmented equations admitted a cosmology with no matter density, contrary to Einstein's expectations. It was an odd spacetime--now called "de Sitter spacetime"--that is everywhere expanding although there is no matter in it. That means that any two tiny test masses that somehow found their way into the universe would accelerate away from each other, whereever they were located.
|De Sitter's spacetime may seem an elaborate
construction. It turns out, however, to be the simplest spacetime after the flat Minkowski
spacetime. It has constant curvature--that is, it has the same
curvature at every event.
To see how simple it is, recall our original recipe for generating curved spaces (not spacetimes). The simplest case was a flat Euclidean surface. We then generated a two dimensional spherical space by looking at the surface of a three dimensional sphere in a three dimensional space; and we generated a three dimensional spherical space by looking at the surface of a four dimensional sphere embedded in a four dimensional space.
This procedure in the context of a Minkowski spacetime gives us a de Sitter spacetime. We take a five dimensional Minkowski spacetime (one time dimension, four spatial dimensions). In it, we construct the analog of a sphere.
A sphere is the locus of all points that are some fixed distance from a central point. In a Minkowski spacetime, its analog is the locus of all points some fixed spacetime interval from a central point. That locus is the four dimensional hyperboloid shown in the figure. It is a surface of constant curvature, as is a sphere in Euclidean space. The four dimensional surface of that hyperboloid is the de Sitter spacetime.
We can then see how the de Sitter spacetime solves Einstein's gravitational field equations augmented with the cosmological term λ. Since de Sitter spacetime has constant curvature, its summed curvature is everywhere the same. So we generate a solution of Einstein's augmented equation merely by picking that de Sitter spacetime whose summed curvature equals the negative of the contant λ term. Then the two terms on the left hand side of Einstein's equation cancel out to zero; the right hand side is also zero since we assume the spacetime is matter free.
In retrospect, the extra term Einstein added to his equations had a simple interpretation. A uniform mass distribution, if left to itself, ought to collapse under gravitational self-attraction. That is the physical interpretation of the curvature of the space-time sheets that the equations of 1915 were calling for. In adding the cosmological term, Einstein was, in effect, adding a cosmic force of repulsion that would cancel out this natural gravitational self-attraction. That way the matter distribution could remain static.
When de Sitter forms a universe without matter, no gravitational self-attraction of matter opposes λ's powers of repulsion. We can still insert minute test masses into this otherwise empty universe to plumb its properties. With λ's repulsive powers only in effect, we find a universe in which test masses flee everywhere from each other.
|Einstein soon began to have reservations over his introduction of the cosmological term. In 1919, he was already calling it:
"gravely detrimental to the beauty of the theory."
|This improves on the standard English translation which says "gravely detrimental to the formal beauty of the theory." For those who are interested, the original German is "Hierin liegt ein besonders schwerwiegender Schönheitsfehler der Theorie."
Albert Einstein, "Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle?" Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1919, pp. 349-356. On p. 351.
In the 1910s, when Einstein proposed his universe, the natural supposition was that the matter of the universe is static on the largest scale. In the 1920s and 1930s, it became clear that this was not so. In fact the matter of the universe is everywhere expanding rapidly and that expansion was adequate to counter temporarily the gravitational self-attraction called for by Einstein's theory. (A good analogy is a stone tossed into the air. Its initial, upward velocity overcomes the downward pull of gravity, but only temporarily.) When these dynamic cosmologies emerged, Einstein renounced the cosmological term, in a joint paper of 1932 with de Sitter.
|The relevant passage appears about half way down the page:
"Historically the term containing the "cosmological constant" was introduced in the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ."
Einstein's renunciation of the cosmological term has not proven to be fatal to the idea, however. It gives cosmologists, eager to match their models to the latest astronomical data, an extra parameter to adjust, so that they can get a fit of their model to new, recalcitrant data.
In that context, there is a popular re-interpretation of the cosmological term. To see it, take Einstein's augmented gravitational field equations in the case in which there is no ordinary matter, so the term "matter density" is 0.
summed curvature
of spacetime
Now just take the λ term and move it to the other side of the equal sign:
summed curvature
of spacetime
So, where Einstein's original equations used to say "matter density," they now say "-λ." What that means is that Einstein's λ is behaving like an extra sort of matter distributed through space, according to the original equations.
Since we know that λ corresponds to a force of repulsion between matter, it behaves like an odd sort of matter that accelerates the expansion of matter in space. What is odd about it is that all ordinary matter generates attractive gravitational forces. That was the fundamental idea of Newton's original notion of "universal gravitation." As noted above, this now gives some understanding of why the matter-free de Sitter universe is expanding. It is being driven by the repulsions inherent in the cosmological term.
The cosmological constant λ has proven especially useful in recent work in cosmology. Over the last few decades, careful measurements of the motions of distant galaxes have shown motions that do not fit with Einstein's gravitational field equations. Their motion of recession is accelerating faster than Einstein's equations allow. It turns out that we can accommodate those accelerations by adding a cosmological λ term and adjusting its size so that it gives just the observed missing acceleration.
There is an awkwardness in the addition of the term. The great triumph of Einstein's theory had been that it did not need any adjustable parameters to achieve its results. Einstein recovered the anomalous motion of Mercury in 1915 without any tinkering with such a parameter. If the motion of the planet were not anomalous by exactly 43 seconds of arc per century, then his account would have failed.
To make the added term more palatable, the standard approach is to move the λ term to the "matter" side of the gravitational field equations and to declare that it represents a novel form of energy. Since the energy does not radiate light, it is not visible to us. It is called "dark energy."
|To get a sense of the evidential situation, recall the anomalous motion of Mercury. Its perihelion advances by 43 seconds of arc per century. The supposition of extra matter was made as a way of explaining the motion. Leverrier pointed out in 1859 that, if a new planet "Vulcan" were placed in just the right spot, its gravitational pull would explain the anomalous motion. Of course that supposition was not enough. Independent evidence was needed. We had to see the planet. Its position was computed but no telescopic observation ever managed to find it. The supposition of the extra planet Vulcan was abandoned.
|This declaration has not solved the problem, yet. All we have is an extra term added to the Einstein gravitational field equations and we can put that term on either "curvature" or "matter" side of the equation. To establish that it really does belong on the matter side, we need some independent evidence that there is an exotic form of matter present. Unfortunatley, no such evidence has been forthcoming. There is, of course, no shortage of speculations and claims that this or that novel account solves the problem. But no consensus account has emerged. Until it does, we should treat the idea that the cosmological λ term represents an exotic form of matter with reservation. That is an awkward outcome since standard estimates suggest that around 70% of the matter in the universe is this "dark energy."
λ lives on. Einstein would not be pleased.
Before we turn to pursue the spacetime that best resembles our own in the next chapter, it is interesting to review how Einstein's theory allows us describe universes in which time travel is possible.
The easiest type of time travel universe looks like a trick that is stipulated into existence. However there is nothing illicit about it. And its great simplicity enables us to refine our intuitions about just how time travel can arise.
Einstein showed us through the Einstein universe that we can curve space back onto itself and thus produce a closed space.
That construction proved a little complicated for Einstein since there are three dimensions of space that need to be accommodated. If we want to do it in the time direction, it is much easier. There is only one dimension of time. The simplest case arises if we wrap up the time direction of a Minkowski spacetime. As before, if we consider only one dimension of space, we recover a cylinder. The spacetime is on the surface of the cylinder. For the new case the cylinder is wrapped up in the timelike direction.
You might wonder if a trick like this is really allowed by Einstein's gravitational field equations. It is. Recall that Einstein's gravitational field equations merely fix how each little patch of spacetime must look. A solution is admissible if each patch connects properly with those next to it. That will happen in this spacetime. In any not too big piece, this cylinder universe is exactly the same as a Minkowski spacetime; each piece connects with the one next to it just as they do in a Minkowski spacetime. That is all that is needed for the spacetime to count as a solution of Einstein's gravitational field equations.
The timelike curve on the spacetime represents the life of a traveler who stays at one point in space, but passes through time merely by being. Eventually that worldline will wrap all the way around the spacetime and reconnect. At that point, the traveler will meet his or her former self.
The traditional "grandfather" paradox of time travel arises if the latest stage of the traveler (now imagined to be the grandson of the original traveler) were to kill the original one (the grandfather). A contradiction would ensue. With the assassination complete, there would no traveler to pass through time and commit it. So the assassination happening entails that it doesn't happen.
We have already discussed the difficulties raised by such a paradox in the discussion of tachyons earlier. The difficulty is that we would visit a logical contradiction on our physical theory and that is unsustainable in any cogent physical theory.
|Time traveler goes back in time.
|Time traveler kills grandfather
|Time traveler does not go back in time.
|Time traveler does not go back in time.
|Time traveler does not kill grandfather
|Time traveler goes back in time.
So we have a contradiction:
|Time traveler goes back in time.
|if and
only if
|Time traveler does not go back in time.
Contradictions may be fine in science fiction or when we are writing a screen play for a time travel movie. However contradictions are a mortal threat to a physical theory. From time to time, we will work with an inconsistent theory. We shall see that this happened in the early years of quantum theory. However this tolerance has to be temporary. For an inconsistent theory cannot reliably tell you what is the case. It assures you that some assertion is both true and false at the same time.
The possibility of such paradoxes has led some to conclude that time travel universes are logical impossibilities. That is too hasty. There is an obvious loophole in the paradox. If the assassination attempt fails, then there is no contradiction.
So that is what must happen in a time travel universe. The grandson's bullet must miss; or the gun misfire; or the grandfather ducks; or who knows what. For if the assassination attempt didn't fail, there would be no assassin to attempt it.
That resolution is, as far as I know, admissible. Many find it objectionable since there seems to be no reason in the physics itself that forces the failure of the assassination attempt. What if the grandson takes all due care, aims carefully with a new gun, and so on? How can we be so sure that the attempt will fail.
We can. The intuitions that tell us it will not fail are honed in a type of universe that is quite different from a time travel universe. In the ordinary time travel free universes, such as we presume we inhabit, local constraints prevail. If the gun misfired, for example, it was because something in the state of the gun immediately prior to to the assassination attempt intervened. Perhaps the grandson passed through a rain shower and a component of the gun began to rust.
In a time travel universe, in addition to these sorts of local constraints, we have a new type: global constraints. These are extra constraints that all processes must conform to in order that distant future and distant past mesh when they meet. These global constraints do not arise in time travel free universes. They are what assures us that the assassination attempt must fail.
We can get an idea of how they work from the jigsaw puzzle analogy for solving Einstein's equations.
|First consider a universe without time travel. We
start with a row of pieces that represents space in the present
instant. Then we add successive rows that correspond with space in
successive future times. The pieces we add are constrained only by the local requirement that
each piece mesh with those immediately before and after it in time;
and those around it in space.
These pieces specify the local condition of spacetime: the curvature of its geometry and its matter content. So one piece will have a grandfather; and the successive pieces fitted to it will have the grandfather engendering children, aging and so on.
There will also be pieces in which assassins shoot their victims.
|Now take the case of a time travel universe. All
these constraints apply. But, in addition, as we keep adding the
successive rows, we will eventually end up going all the way round
the space and then the new and powerful constraint will come into
force. The last row we add has to be so perfectly built that it
meshes with the past edge of the first row we put in place. That is a
global constraint. It means that in our planning of which pieces to
lay down, we had to worry about the local meshing of the pieces; and,
in addition, we had to select pieces now so that eventually the final meshing of last and first row would work
If that last piece that we add contains the time traveler attempting to assassinate his grandfather, then that piece must mesh with the past. Those past pieces would contain the time traveler loading his gun and walking down the street to the house of his grandfather.
It must also mesh with the future pieces. Those future pieces will contain a live grandfather who survives, for those pieces will in turn develop forward to give us the time traveler.
That meshing with the future pieces can only happen if the time traveler fails in the assassination attempt.
Here's a simple example in a
different arena of how these sorts of global constraints can work. It is the
arithmetic puzzle, "99." In the puzzle, you are
to start at zero and may add or subtract any number you like between 0 and
10, as many times as you like, provided that the numbers that you are adding
or subtracting are always even numbers. Is there some combination of
additions and subtractions that will get you to 99?
Locally, there is no obstacle to getting to 99. If you could somehow get your sum to 97 or to 95, you could complete the task by adding 2 or 4. Just looking locally at the numbers around 99 reveals no problems.
..., 87, 95, 91, 97, 99.
Globally, however, there is a constraint that necessarily defeats your attempts to arrive at 99. Since you start with zero and may add or subtract even numbers only, your sum must always be an even number. So you can never get to 97 or 95 or any other number that is an even number removed from 99. This global constraint assures your failure to solve the puzzle.
0, 8, 4, 12, 16, 24, 22, 30, ...
That is how things would work also for the time-traveling would-be assassin. The assassin can get to states very close to the successful assassination--the "99" of the puzzle. However those close states would always be the even states, "94," "96", "98," ... They look close to the assassination sought, but physical laws preclude them developing into the successful assassination.
Another simple illustration shows just how powerful these global restriction on a spacetime can be. Consider just about the simplest possible time travel universe: a universe empty of all matter excepting just one mass. Now pick some time slice. What configurations of the particle are possible?
In an ordinary time travel free universe, at some initial moment of time, we can have the worldline of the mass with any initial velocity.
If the spacetime is a time travel, cylinder universe, we are strangely restricted in the possibilities for this time slice. We could choose a mass at rest. That corresponds to the case of a single worldline that eventually wraps back onto itself. But if we have the mass initially moving, then we must also stipulate that clone masses be distributed in space at uniform intervals. These will be the repeated returns of the single mass as it travels all the way round spacetime and back to the present.
The global constraint says that if we have a moving mass here and now, we must also have a moving mass there and now; and there and now; and so on. That sort of constraint would be incomprehensible in a universe without time travel. What reason of physics, we would exclaim, requires it--just as we ask, what reason of physics requires the grandson's assassination attempt to fail!
There is something that looks just a little fishy about the way time travel is arrived at in the cylinder universe. It does not seem to arise from the physics of the spacetime. It comes from a stipulation on our part that the future wrap back onto past. Einstein's theory seems only to get involved in so far as it raises no objection. There is nothing wrong with this way of introducing time travel, of course.
It is nice to know, however, that time travel also can arise more naturally. The Goedel universe is one such example. This solution to the Einstein equations was arrived at by the famous logician, Kurt Goedel, in the 1940s, when he was a colleague of Einstein's at the Institute for Advanced Study in Princeton, and published in 1949.
The Goedel universe is a solution of Einstein's equations with the cosmological term. Its signature property is that it contains closed timelike worldlines. As a result, it is hard to pick out a single timelike direction globally in the spacetime. Rather, we can get a feel for its spacetime properties by taking just a single two dimensional slice of it. It will become clear that this is not a spacelike slice.
If we consider some observer in the middle of this slice, the observer will find all the matter in a great cosmic rotation around them. (For this reason, the Goedel universe cannot be ours. We don't see such rotation.) The reason for the rotation lies with the structure of spacetime itself. As we consider positions in the slice further away from the observer, the light cones start to tip over. So if we consider a large enough chunk of the slice, we can find a timelike curve that loops back onto itself. It forms a closed timelike curve, the hallmark of universes that admit time travel.
The timelike curve is not a geodesic; it represents the trajectory of an accelerating spaceship. To achieve time travel, the spaceship would need to accelerate quite considerably. Most interestingly, the Goedel universe uses no stipulations about past wrapping back onto the future to achieve the possibility of time travel.
There are other universes that admit time travel. Often rotation is involved. Spacetime around an infinitely long, very dense, rapidly rotating tube of matter admits closed timelike curves, for example. Some of the most fascinating of the time travel universes are those in which one part of spacetime is connected to another by a wormhole. That is just a tunnel of spacetime that provides an alternative route from one part of spacetime to another.
Copyright John D. Norton. March 2001; January 2007; February 16, October 15, 27, 2008. March 6, 2013.


text:
Economics Departments, Institutes and Research Centers in the World
Social Policy and Development Centre (SPDC)Location: Karachi, Pakistan
Homepage: http://www.spdc.org.pk/
Phone: (092-21) 111-113-113
Fax: (092-21) 453 4285
Postal: 15, Maqbool Cooperative Housing Society, Block 7 & 8, Karachi 75350
Handle: RePEc:edi:spdccpk
Areas or Functions: Welfare Economics Areas or Functions: Welfare Economics, Think Tanks
Additional details:
People who have registered with RePEc and have claimed to be affiliated with this institution:Find all publications of all authors listed below conveniently compiled on IDEAS. Authors with publications are listed below with . We have lost contact with authors marked with ?, and they do not count towards the ranking of this institution.
This listing is provided to you by EDIRC at the Economic Research Department, Federal Reserve Bank of St. Louis.
Please send additions and corrections to Christian Zimmermann.

subset: commoncrawl


text:
Pune, MH
Job Code:
- Banking & Finance
- Insurance
Applying for this job will take you to an external site
Job Details
Dear Candidate,GOLDEN OPPORTUNITY TO EARN AS MUCH AS YOU WANT!!!!!!!!!We are having urgent openings in LIFE Insurancefor Different Positions and locations.Position-Manager levelCompensation Offered- 1.80LakhLocation ALL OVER INDIAJob SummaryA Position OverviewPosition TitleAssurbanking Business Development ManagersDepartmentAgencyLevel/ BandMANAGER.Role SummaryProvide support in Sales of HEALTH Insurance business through bank customers at bank branches across assigned locations B Organizational RelationshipsReports ToTerritory Manager/ Area ManagerSupervisesNAC Job DimensionsGeographic Area CoveredBranches / ClusterInternal StakeholdersTrainingBranch OperationsDistribution OperationsExternalChannel Partner D Key Result AreasOrganization ProcessKey ContributionsSales Assist bank in acquiring banking customers through insurance product saleAchieve pre-set business targets on ANP & Product Mix, case count, active branch, active sales staff and various KPIs for designated branchesWorking jointing with the Branch Manager on Business Implementation planDriving and motivating the branch staff along with Bank RM and providing support for logging in applications sourcedTapping the right database of the branch and ensure authentic documentationBuild relationship with the Bank Sales/ operations team and ensure the business targets and Productivity targets of the allocated branches are met effectivelyProspect and meet customers within and outside the Branch as when requiredTo develop, agree and implement short term and long term plans to achieve sales targetsTo achieve branch RM activation targetsWork collectively with Bank Sales / operations team and conduct joint sales calls as a team to achieve business nos.Seek commitment from the partner towards achievement of business objectiveRelationship ManagementManaging the relationship between internal team and channel partner so as toFoster sales for the company and maintain utmost levels of responsiveness to requirements from the Channel PartnerProvide all possible support to the channel partner as a face of parent companyEffectively execute all Marketing activities as per Partners requirementEnsuring timely issuance of policies with resolving all pending etc.Adhere to the customer touch-point engagement in order to service his portfolio of customersTo maintain the desired persistency ratioPrompt post-sales service with respect to all domainsStrategize and interact closely with the RM on business plan execution Ensure ComplianceEnsure compliance to internal sales process & other compliance standardsShould be process & product champion to ensures adherence to policies and procedures to cultivate a compliance culture in the teamMISAdhering to the Training Road map.Providing timely reports to the Supervisor as per the requirement.Adhering to the Tata AIA Life code of conduct.E Skills RequiredTechnicalProduct/Subject matter expertiseBusiness perspective & planningFinance / Insurance industry awarenessSelf managementProblem solving skillsPeer level co-ordination and influencing skillsBehavioralEssentialDesiredInterpersonal skills Communication skills Creative thinking skills Supervising/Leadership skills Teamwork Skills Influencing skills Relationship Building skills Decision making skills Language SkillsEnglish, Local language F Incumbent Characteristics EssentialDesiredQualificationGraduatePGExperience0.5 - 4 years in Sales (Insurance / Financial Service Background / Direct Sales) If interested to apply for any of above mentioned profile , share your CV email@example.com free to call 8237089427 or sms JOBS on for further discussion . Regards MeghaHr Recruiter8237089427
Degree: M.Com. (Commerce) | M.Pharm. (Pharmacy) | M.Sc. (Science) | MA (Arts) | MBA/ PGDM | MCA/ PGDCA
Additional Degree: BA (Arts) | B.Com. (Commerce) | B.Pharm. (Pharmacy) | B.Plan. (Planning) | B.Sc. (Science) | BBA/ BBM/ BBS | BCA (Computer Application) | Diploma-Other Diploma | B.Ed. (Education) | BHM (Hotel Management)
Experience: 1-6
Degree: M.Com. (Commerce) | M.Pharm. (Pharmacy) | M.Sc. (Science) | MA (Arts) | MBA/ PGDM | MCA/ PGDCA
Additional Degree: BA (Arts) | B.Com. (Commerce) | B.Pharm. (Pharmacy) | B.Plan. (Planning) | B.Sc. (Science) | BBA/ BBM/ BBS | BCA (Computer Application) | Diploma-Other Diploma | B.Ed. (Education) | BHM (Hotel Management)
Experience: 1-6
RequirementsBroker | Channel Sales | Corporate Sales | Direct/Commission Sales
Applying for this job will take you to an external site


text:
Viewing Blog
View All Blogs | View Blogs by The Ithaka | View Blogs in Cruising Log
How do you handle paying your bills?
By The Ithaka - Published November 21, 1999 - Viewed 1016 times
How do you handle paying your bills?
From Bernadette : There are excellent professional services that many cruisers use to handle their mail and the payment of regular bills. You can find out about them by looking at the back of magazines such as Cruising World; most advertise in the classifieds. In our case, we have a rental house at home, so we need a bit more service than those companies provide (in case something goes wrong in the house, our tenant needs someone to call). So we have a friend we trust in Rhode Island tend to our mail and bills, we've given our tenant her name and phone number, and we pay her by the hour. She goes to our post-office box once a week, opens our mail, emails us about anything unusual, pays items that need attention, and makes deposits for us.
We've set up all our recurring bills, however, to be paid through the online banking feature in our checking account, so that is maintenance free. And we have our credit cards paid through auto-deduction from our checking account, which we set up while we were home. (We can check our balances from internet cafes while we're traveling, which is handy.) This leaves only a few items that need personal attention every month.
One thing to keep in mind, is that you need to keep a significant amount of money in your checking account when you go cruising (we keep $5,000 in it at all times) so that you have access to enough cash to cover emergencies quickly or to buy a couple of airline tickets, fast, in cash. This means that you should never, under any circumstances, carry a debit card around with you on a regular basis, or use a debit card for anything other than withdrawing funds at a legitimate bank or money machine – in other words, don't use it to pay your bill at a restaurant, or in a store when you're traveling. Keep it tucked safely away aboard until the day you take it out to use it to obtain some cash from a machine or bank. Then put it back in its hiding place when you're finished using it.
Also, keep in mind that debit cards have no transaction fees, so the best idea we've heard is to get one especially for use with ATM machines, and keep ONLY a minimal amount in that account, say $2,000. This limits your financial exposure if your card or card number is stolen.
To get our mail while we're cruising, we wait till we know someone who is flying down, and we ask our friend at home to FedEx our important mail to them to carry down to us. Whenever we meet a cruiser who has a guest flying back to the States, we give the guest our flat, stamped mail. All cruisers are happy to do these tasks for one another, and none of us has ever flown to the States and back without carrying back down with us spares and mail for the cruisers anchored around us. It's a very helpful and supportive community where everyone helps everyone else.
Blog Comments
There are 0 blog comments.
Sorry there are no blog comments.
|Post Blog Comments
Sorry but you must be logged in to submit comments.


text:
Maps & Driving Directions
Our hospital is conveniently located at 8640 Elk Grove Blvd Elk Grove, CA. 95624.
Coming from HWY 99 take the Elk Grove Blvd exit and go east down Elk Grove Blvd.
Once you pass East Stockton Blvd, which is the first intersection, look for us on your right hand side.
Find Other Nearby VCA Locations


text:
The Indianapolis Colts have one of the busiest offseasons in the NFL. With a new offensive coordinator, a solid group of rookies and more than a few new faces via free agency, this team has been very busy the past few months.
Even after a 11-5 season, it was pretty clear that the Colts' front office wasn't satisfied with being just a playoff team. They've been aggressive throughout the entire offseason, trying to fix as many holes in the roster as possible.
With so much changing in so short of a time, there have definitely been some winners and losers. These are determined by whether they have proved themselves, how they now fit with this new group of players and if someone has come in at their position to potentially take their spot.
With that being said, here are the 10 total biggest winners and losers up to this point in the offseason.


text: Versace shirt, $725, versace.com. Team Versace W's art and photo assistant is ready to play. June 19, 2014 9:00 AM | by Biel Parklee As a die-hard soccer fan, I have been looking for a way to stand out from the masses of World Cup fans. Fortunately, Versace has created this over-the-top commemorative T-shirt. Score. See More Most WantedVersaceWorld CupMost Wanted,Versace,World Cup don't miss a thing Sign up for the latest from W editors. sign upThank You For Subscribing.invalid email Facebook Twitter Email Related Stories Around The Web Previous The Row Resort 2015 Next New York Fashion Week Men's: The Highlights So Far


text:
Orders are shipped using USPS. Large orders are sometimes shipped UPS. Orders are shipped Monday through Friday and are usually shipped within one to three business days of us receiving the order. During the holiday season please allow one to two extra days for delivery.
All prices include shipping charges.
We now ship Internationally. We have partnered with Bongo International. All International orders must be placed on our website and checkout thru Bongo.
We do ship to APO addresses and enclose a gift for there service.
You may set up your personal address book allowing you to ship to multiple recipients and/or multiple addresses on the same order.
We suggest that you ship your orders with Signature Delivery (please see below).
Return Policy for coffee:
Blue Mountain Coffee values each customer and will do our best to give the best customer service in the business.
We guarantee:
Blue Mountain Coffee is not responsible for:
Return Policy for all other items:
Coffee Grinder returns or exchanges will follow the guidelines of the manufacturer. All problems should be addressed with manufacturer. The manufacturer will do everything they can to help the customer.
Please call Blue Mountain Coffee, Inc. for all other product returns. All items are subject to a 15% restocking fee. All items returned must be in like new condition. If any item is defective Blue Mountain Coffee, Inc. must be called before returning item. Defective or incorrect items will receive a full refund if said item is returned unopened.
Items ordered incorrectly or shipped to incorrect or insufficient addresses because of customer error will be the sole responsibility of purchaser.
Blue Mountain Coffee, Inc.
4250 Keith Bridge Rd. Unit 220
Cumming, GA 30041
Buy Coffee
About Our Coffee
About Us
In the News
Mavis Bank Coffee Factory
How Coffee is Cultivated
Contact Us
My Account
My Cart
Free E-Cards


text:
The Style Gallery is a great new way for you to view, love, and share outfit photos. It's meant to showcase
all the expressive, creative, and inspirational personalities of our community!
Love you too!
Your email has been sent!
Strike some serious poses in these dapper, menswear-inspired Oxford flats. Their smoothly rounded toe, charming lace-up silhouette, and pretty pinhole detailing give these sepia brown, vegan faux leather shoes attitude and charm. So don't just stand there! Get to it and pair them with high-waisted trousers, a cropped tank top, and an oversized blazer. For a full-on fashionable look like that - there's nothing to it!
Final Sale: This item cannot be returned or exchanged.
Sorry this item isn't available anymore


text:
Joe Scarborough: "You have a lot of people running around, saying harsh things that sell books and push ratings and lose elections, he said on Monday. Conservatism is a racket for a lot of people to get very, very rich. With no thought of winning elections."
Karl Rove and other republican leaders accused of putting self before party. (photo: Getty Images)
We are concerned about a recent drift towards vitriol in the RSN Reader comments section. There is a fine line between moderation and censorship. No one likes a harsh or confrontational forum atmosphere. At the same time everyone wants to be able to express themselves freely. We'll start by encouraging good judgment. If that doesn't work we'll have to ramp up the moderation.
General guidelines: Avoid personal attacks on other forum members; Avoid remarks that are ethnically derogatory; Do not advocate violence, or any illegal activity.
Remember that making the world better begins with responsible action.
- The RSN Team
The anger-bots crave the stoking of their inflamed outrage, the need for scapegoats and boogie-men. These clowns are giving them more than they feel they pay for.
And even if there is fraud, the government could not go after them. It would only prove that the government is out to silence conservatives and turn America into Beck's Stalinist-Hitle rist state. Besides, if HSBC and big banks can get away with crimes that would put any one of us in jail or Gitmo for life, because they are too big to fail, same goes for Limbaugh. So much for corporations as people.
Well why not the Owner-media that goes along with it, mostly conservative in spite of the far right's continuously hysterical whining about the "Liberal Press".
THE NEW STREAMLINED RSN LOGIN PROCESS: Register once, then login and you are ready to comment. All you need is a Username and a Password of your choosing and you are free to comment whenever you like! Welcome to the Reader Supported News community.


text:
WATCH: RIhanna Glitters With "Diamonds," And Queerty Is Going On Tour With Her!
Rihanna has just released the video for "Diamonds," her new song and the fastest-rising single of her career to date. It's also the first track off RiRi's upcoming album Unapologetic, coming out Monday, November 19.
Queerty will be all over the launch of Unapologetic in a really incredible way: Writer Rich Lopez will be tagging along with Rihanna for her incredible 777 tour next week—Seven concerts in seven cities over seven days. We'll be delivering exclusive stories, photos and updates throughout the week, so be sure to keep it locked on Queerty for all your Rihanna 411.
well i am looking forward to this. i've always enjoyed rich's writing in the dallas observer and dallas voice. sounds like it will be a fun ride!


text:
|Thread Tools
|Display Modes
Weekend on the Nantahala (and the big ones that didn't get away)
On Saturday morning my Dad and I decided that we were going to head up and fish the Tuckaseegee and Nantahala like we had last weekend. We ended up leaving around 10:30. After about a 2.5 hour drive, I could hear the roar of nantahala rapids outside our car window. My Dad and I geared up and headed into the river. I walked a few yards downstream of the pull off we parked on. There was a small, deep pool with a run going down one side. I looked down in the slack water and saw there were many fish in there. I got excited and made my firsd drift through. BANG first hit and a miss. I made another drift and BANG again except this time he did not get away. To my surprise the fish i hooked was bigger than the ones I had seen. He definately put a nice little bend in my 7" 3 wt! After a few minutes of him horsing me around the pool, I netted him and got a nice sized grin on my face. It was a solid 15-16" rainbow.
I released the fish after snapping a few photos. I decided that he hadnt spooked out the pool and threw my nymph rig back in. I got another hit and stripped in a small (anout 5") wild rainbow. I didnt think there would be wild fish in this section because of all the fish stocked. I threw him back in. I deepened my rig to the depth I thought i was seeing the fish. I ended up with 2 more of these guys.
I decided to leave the small fish and head upstream to meet up with my dad. As I waded around a corner i saw my dad upstream from me hollering bloody murder at me. He told me to come up fast. Not knowing what happened and thinking of all the bad things that could happen on a trout stream, I ran up the road to him. To my relief, the reason he was yelling was because he left his net in the car and had a big fish. He kept fighting it for a few more minutes and I then netted the beast. It was a healthy 19" rainbow.
We snapped some photos and i then revived the fish for him. This was the start of a long day of coldddddddddd (very cold) hands. I got my hands out of the water and they were blood red. I tried to warm them up, but that just was not going to happen due to the 30 degree temperatures. I fished upstream with no luck in some awesome areas. I guess the fish havent fully spread out yet. I made my way up and found a good run along the road where i figured they had stoocked. I positioned myself and then casted. BANG fish on! It got off...too much slack. I drifted through again and caught a fish. I stayed here about 15 minutes and ended up landing 8 rainbows with the biggest aroung 14". I do not have pictures of them because of low camera battery. I decided to save if for bigger fish. I headed upstream and fished at a nice brown in the bridge pool. I could not get him to take. Smart fish. I kept goin up stream and caught a few more here and there. All rainbows. I got a tangle and cut my tippet to re-rig, but my hands hit a point where they were so cold i couldn't even tie a knot. I decided to call it quits. I walked downstream to find my dad. I found him in the run where i caught the 8 rainbows. He had the biggest smile on his face. He said he had just let go his biggest brook trout ever. About 16". He was completely stoked...even more so than with the rainbow he caught earlier. I told him i was ready to quit for the day. He said he was good to go too. We headed toward Almond and stopped and ate dinner at the NOC center. It was pretty good. Then we went and stayed the night in our airstream at Tumbling Waters.
Day two:
After a bad night of sleep I got up and got my stuff packed for the day. We headed to the river at about 8:30. We parked in a pull off a little further up stream than the one from the day before. I decided to trudge straight up stream there i had caught the 8 rainbows and my dad the big brook. As i was climbing down to it i fell probably 3 times adding on to my collection of bruises. The rocks were covered in a thin layer of frost/ice. I finally got down and started fishing. Immediately i hooked a fish that got off right away. Then after another few drifts i hooked up with a very very solid fish. I set the hook and some big splashes followed. The fish jumped and i saw it was a big rainbow. I thought to myself 'how am i supposed to land this fish on this tiny 3 wt.?' Luckily for me he didn't run down stream, but rather stayed deep in the run. I had fought him for about 5 minutes before i got him up to where i could see him. He then took off downstream, but i turned him. After a few more minutes i netted the fish. I was one proud guy. It was by far my best 3 wt. fish! He was about 18"
After letting this fish go, my dad showed up. I told him about the fish. He said he hadn't got anything yet. He fished the lower end of the pool i was fishing. I walked up a few yards to the next run. I immediately spotted another big fish. I did a near perfect cast an then the indicator dove. Another big fish on! This fish immedialely ran downstream and took a bunch of drag. I dropped back down into the pool I got the other big one from. A few minutes later I netted this one. Two big fish in 15 minutes...not bad! He was at about 17"
This fish had a weird tumor-like object on on its lower jaw.
I let this fish go and went upstream. Due to the fish not being spread out yet, i didn't do so well. I made my way up to a big pool and saw a bunch in. I casted and immediately was rewarded with a smaller stocker. I stripped it in and let it go. I casted again and got another one about the same size. (about 10") I left and headed upstream. I caught a few more here and there. I walked back down to try the bridge pool. I tried my natural nymphs with no luck. I walked downstream and waded across some when i noticed about 7 brook trout stacked up along a log. They were very nice sized. I could not get them to hit my nymphs, so i stooped down to a lower level...i tied on a y2k.... Immediately they gave some looks. I then saw one of the descent ones suck it in. I stuck him and pulled him away from the other fish, so he wouldn't spook them. I landed my first brookie of the trip. He looked like this.
I quickly released the fish. I wasn't getting many more looks on my orange y2k, so i switched to a pink one. This got some more looks. I witnessed anther descent one suck it in and i snagged it. I got it in. He looked exactly like the last one. I casted the pink one back out and had a real big brookie suck it in. I managed to pull it out of its mouth. I tied on a yellow/pink/orange y2k and got him to hit again. Same story...pulled it out. I couldn't get him to take it again. I switched again to a neon green BH y2k. I thought there was no way he had seen this color yet. I got a good cast in and saw him suck it in again. Dumb stocker. I stuck him and he ran downstream taking some float line with him. I fought him a few minutes and then netted my biggest brook trout to date.
Even though he was a dumb stocker i was proud. I used my y2k trick further upstream in the bridge pool where i couldn't get a fish to hit. There must have been an egg hatch or something because they were ALL over it. I honestly caught about 15 more fish standing on one rock. One brook and the rest rainbows. Dumb stockers.
My dad walked up to me and said he had only caught one. I chuckled and told him how many fish i had caught. I told him to come down and fish here. As he was coming down, i put out another drift. I said, "I think i fished this hole out." Of course once i said that my indicator dove. I then set the hook on my first brown of the trip. I completed my slam.
After letting this fish go i called it quits and let my dad throw a few. He missed a couple and then decided to head on home. You all really need to get up there! The river is fishing great. Put something big and flashy up top with a natural dropper. Hint: all the big rainbows hit a natural-looking fly. Thanks for reading!
Last edited by Rainbow Warrior; 11-08-2010 at 07:09 AM..
Awesome report man I could read these all night long thanks rainbowwarrior for entertaining me tonight. I have never fished there but rafted the river a couple of times. That area doesn't look like the rafting section too little water or they haven't opened the dam maybe.
Adam Beal
Hey Jack (JAB)...
Nantahala DH Waters
This is the part of the Nantahala above the point where the pipe which carries the largest part of the water from the River (and the lake 8 or nine miles above) empties through the turbine. If you are traveling from Topton it would be a left turn up Wayah Road, there must be close to 10 miles of water up there. Nice post by the way Rainbow Warrior!
Umm..there are two miles of DH, two and a half max. There's more water there, but the remainder(past White Oak Creek) is marginal trout water and listed as hatchery supported, while the headwaters near Standing Indian hold wild fish which are much smaller than the nice fish shown here. Nice pics though and great fish RW! It is fishing well this year. so far....
www.owljones.com - OwlJones.com - The Internet's Only "Fishertainment" Website
Last edited by Owl; 11-08-2010 at 04:17 AM..
Great photos and trip report.
-Shawn MadisonEvery human has four endowments- self awareness, conscience, independent will, & creative imagination.
FB https://www.facebook.com/ClinchRiverTN
YTB http://www.youtube.com/user/MadisonBoats?feature=mhee
These give us the ultimate human freedom... The power to choose, to respond, to change.
Looks like it. But so what? Are you going to imply that he snagged it and then placed the fly precisely into the corner of the fish's mouth for the perfect photo op? Without being there, I don't know how anyone could imply anything other than what the fisherman suggests...that he caught the fish. It's not like the hooking both the point fly and dropper on a hook up doesn't happen.
www.owljones.com - OwlJones.com - The Internet's Only "Fishertainment" Website
It's 4 miles from the 1st bridge to Whiteoak creek.
That's what NCWRC considers the delayed harvest section.
Very nice pictures Rainbow Warrior. I bet those fish put up a good fight.
You guys have been hittin' up the DH streams big time this year. Glad you're getting to enjoy them.
|Thread Tools
|Display Modes
Linear Mode
All times are GMT -4. The time now is 10:54 AM.


text:
Hair in cycles
General discussion about the development of the open source Blender
Moderators: jesterKing, stiv
1 post • Page 1 of 1
Hair in cycles
When will hair/particle strands be included in cycles render? I realize that you can convert hair to curves. But I think that is glitchy and doesn't work well.
1 post • Page 1 of 1
Who is online
Users browsing this forum: No registered users and 1 guest


text: Ravens rookie Kapron Lewis-Moore signs four-year dealSixth-round defensive end Kapron Lewis-Moore became the first member of the Ravens' rookie draft class to agree to terms as he's signing a four-year contract, according to a source. "I just try to have a positive mind about it," Lewis-Moore said. It was a… (more)Sources: baltimoresun.com profootballtalk.com


text:
Dannii Minogue in first interview since split with partner of four years Kris Smith in Marie Claire magazine
- From: The Daily Telegraph
- May 08, 2012
DANNII Minogue is rolling with the punches following her bust-up with Kris Smith, the unlucky-in-love TV star has said in her first post-split interview.
Following a life lived in the limelight, the Australia's Got Talent judge has told magazine marie claire she is well-versed in dealing with showbiz.
"How do I juggle everything now? Some days with ease, some days with difficulty," she said.
"I have found that over the years, whatever happens, you have to roll with it. Fame and work have always had an impact on my family since I went into this business at seven years of age. It's all I know."
Kris Smith opens his heart about break-up with Dannii Minogue
Dannii Minogue and Kris Smith split after four years
Have your say
- Skip to:
- Read comments
- Add comments


text:
There's a live conference scheduled today for 11:15am PST, 2:15pm EST. TMZ has the live feed.
The Beverly Hills Police Department is expected to make a statement this morning concerning the death of Whitney Houston.
The BHPD responded to the scene at the Beverly Hilton hotel Saturday where Whitney's body was found, submerged in a bath tub.
Cops say first responders were told … "Ms. Houston was discovered in her bathtub by a member of her personal staff at approximately 3:30 pm on Saturday February 11th. She was underwater and apparently unconscious. Ms. Houston was pulled from the tub by members of her staff and hotel security was promptly notified."
The news conference is scheduled for 11:15am PST.
Watch the live feed here.
via TheWordEyeHeard


text:
Student Proposal Goes Viral
By Brett Wilson | January 8, 2014
The internet swarmed when Regent University School of Communication & the Arts (SCA) alumna April Albritton '13 said "yes" to SCA student Beecher Reuning's marriage proposal. Shortly after a video Reuning posted documenting the couple's relationship—and ultimately their engagement—Beecher and Reuning's major life-decision has become a major hit on the web.
It wasn't long after the proposal video, which features Albritton and Reuning's "greatest hits" from their relationship history, was made public to friends and family members on various social media sites that it began to receive thousands of views. A few days later it was picked up on popular online video sites such as GodVine, GodTube, BuzzFeed, Yahoo Canada and even MSN Austrailia.
"I know it happened because our friends were awesome enough to share our story," said Albritton.
On Sunday, Dec. 29, the couple shared their story via satellite on Fox News's Fox & Friends at the network's Atlanta headquarters.
"I thought 'what better way to capture this?'" said Reuning during the interview. "And it was also an added plus that I could surprise her."
And though the interview had the potential to be broadcast to many more people than even saw the initial video, Albritton explained that she and Reuning were "too excited to be nervous."
"We both felt so overwhelmingly blessed by the opportunity that we were just thrilled to be on air to talk about our relationship, which has already been a blessing in itself," said Albritton. "It was crazy to think that behind this camera were potentially millions of people watching us live as we shared our story."
Much of Albriton and Reuning's story has come together through their shared experiences at Regent. The video features reenactments from their relationship around the Virginia Beach campus.
"Regent has had a huge role in our story," said Albritton, explaining that without the support from their group of friends and persevering in the midst of a personal tragedy, the couple might never have met. "Regent will always be very close to our hearts because God used the University to bring us together."
Watch their Fox & Friends spot.
Mindy Hughes, Public Relations
Phone: 757.352.4095 Fax: 757.352.4888
E-mail: email@example.com
Latest News
Regent Law Alumnus Finds Success With Own Firm in Norfolk, Virginia
by Brett Wilson Tubbs | November 29, 2016
Shakespeare's Much Ado About Nothing Opens Friday
by Brett Wilson Tubbs | November 28, 2016
Regent University School of Law Professor's Viral Video Sparks Book Idea
by Brett Wilson Tubbs | November 28, 2016
Showcasing Faculty Books, Articles and Performances
By Brennan Smith | November 23, 2016
Admiral Vern Clark, USN, (Ret.) Speaks at Veterans Prayer Breakfast
by Brett Wilson Tubbs | November 23, 2016
Regent University School of Law Alumnus Selected to Supreme Court of Wisconsin
by Brett Wilson Tubbs | November 22, 2016


text:
Other energy - June 11
Click on the headline (link) for the full text.
Many more articles are available through the Energy Bulletin homepage
Blair signs deal with France to pave way for new wave of nuclear reactors
Brendan Carlin , Telegraph
Tony Blair signalled his determination to order a new generation of nuclear power stations by signing a formal agreement yesterday that could lead to French companies building them.
The move could "definitely lead" to contracts for the French when the new atomic plants came to be built, British officials said privately.
The creation of the Franco-British Nuclear Forum, agreed at the Anglo-French summit in Paris, came as the Prime Minister once again indicated that Britain had to stay nuclear.
Despite insisting that he was not prejudging the energy review, to be published next month, Mr Blair made clear that approving a new wave of nuclear power stations would be an essential part of his legacy.
(10 June 2006)
Related from the UK Times: Brown's backing clears way for a nuclear future.
Railroads Struggle to Ship Coal in U.S.
Bob Moen, Associated Press via Yahoo!News
...Power plants around the country have seen their coal stockpiles dwindle, mainly because of problems with shipping coal out of Wyoming and increasing worldwide demand for energy.
The result has been higher electric bills in some areas because power companies were forced to replace coal with more expensive natural gas to feed their plants.
"People call us the Saudi Arabia of coal. But if you don't get it to the power plants, it doesn't matter," said Mike Grisso, executive director of the Alliance for Rail Competition, a shippers' organization.
...Anthony Hatch, an independent transportation analyst in New York, said he believes railroads will meet future demands for shipping coal. But it will take time because of the enormous task of expanding an industry that until only a few years ago was abandoning track as its business dwindled.
...With plentiful coal reserves and alternative fuels still too costly or years away from becoming reality, coal is seen by many as the most practical means to meet the nation's and world's growing power needs.
"The economy is still rolling along so everybody expects production and demand to keep increasing," Fred Freme, industry statistician with the U.S. Energy Department's Energy Information Administration. "It is the cheapest as far as electric generation goes."
(9 June 2006)
Super Battery (capacitors)
Victor Limjoco, ScienCentral News
As our portable devices get more high-tech, the batteries that power them can seem to lag behind. But Joel Schindall and his team at M.I.T. plan to make long charge times and expensive replacements a thing of the past--by improving on technology from the past.
They turned to the capacitor, which was invented nearly 300 years ago. Schindall explains, "We made the connection that perhaps we could take an old product, a capacitor, and use a new technology, nanotechnology, to make that old product in a new way."
Rechargable and disposable batteries use a chemical reaction to produce energy. "That's an effective way to store a large amount of energy," he says, "but the problem is that after many charges and discharges ... the battery loses capacity to the point where the user has to discard it.
But capacitors contain energy as an electric field of charged particles created by two metal electrodes. Capacitors charge faster and last longer than normal batteries. The problem is that storage capacity is proportional to the surface area of the battery's electrodes, so even today's most powerful capacitors hold 25 times less energy than similarly sized standard chemical batteries.
The researchers solved this by covering the electrodes with millions of tiny filaments called nanotubes. Each nanotube is 30,000 times thinner than a human hair. Similar to how a thick, fuzzy bath towel soaks up more water than a thin, flat bed sheet, the nanotube filaments on increase the surface area of the electrodes and allow the capacitor to store more energy. Schindall says this combines the strength of today's batteries with the longevity and speed of capacitors.
"It could be recharged many, many times perhaps hundreds of thousands of times, and ... it could be recharged very quickly, just in a matter of seconds rather than a matter of hours," he says.
(8 June 2006)
Video available at original.
Malaysians urged to change energy use patterns
Bernama, Business Times (Malaysia)
MALAYSIANS need to change their energy consumption and production patterns in order to adapt to global challenges, Minister of Energy, Water and Communications Datuk Seri Dr Lim Keng Yaik says.
The minister said energy security and sustainability, including ensuring a minimum impact on environment, will require everyone to play a crucial role so that "all of us can still enjoy these privileges in future".
"The energy revolution that is characterised by the continued high prices of petroleum, requires that we all begin to give unprecedented emphasis on ensuring the security of our supplies, as well as the sustainability of these supplies," he said at the the Malaysian Oil Scientists' and Technologies' Association annual dinner in Kuala Lumpur yesterday.
Dr Lim said the final commercial energy demand in Malaysia is expected to increase by 47 per cent by 2010. It has already increased by 31 per cent, to 1,632 petajoules in 2005 from 1,244 petajoules in 2000.
(10 June 2006)
Questions About the World's Biggest Natural Gas Field
Dave, The Oil Drum
Without much fanfare, Qatar announced a moratorium on new development of the natural gas North Field basin, a decision that had actually been taken in 2005. At the same time, in a recent presentation by Matt Simmons entitled Tight Oil Supplies, we run into this intriguing slide (#44, big pdf warning).
This report will go into considerable detail about the future role of the North Field/South Pars natural gas field, it's size and importance, the reasons for the moratorium and finally important questions about both the geology and proven reserves of the field. As Simmons notes in his slide, there is a "large degree of uncertainty regarding [the] true potential [of this field]". The topic is important regarding the uncommon phrase "peak natural gas" on a global scale. As we know, natural gas production has already peaked in North America.
[TOD editor's note, by Dave] Generously, Matt Simmons and SCI have given me permission to reference their report Simmons Oil Monthly - Qatar, by Robert A. Kessler, April 24, 2006. Matt wanted me to "alert readers that SCI made a major exception to their strict rules of only sending their institutional research to their institutional clients". My thanks to Matt and SCI for making this report available. I believe it will appear soon on the Simmons & Company International website. I'll do an update when that happens.]
(9 June 2006)
Green bloggers on China
China, Coal and Green Leapfrogging by Alex Steffen at WorldChanging.
Pollution and the Chinese Future by Alex Steffen at WorldChanging.
Duh, China's big by John McGrath at Gristmill.
(June 2006)
What do you think? Leave a comment below.
Sign up for regular Resilience bulletins direct to your email.
Take action!
Find out more about Community Resilience. See our COMMUNITIES page
Start your own projects. See our RESOURCES page.
Help build resilience. DONATE NOW.
This is a community site and the discussion is moderated. The rules in brief: no personal abuse and no climate denial. Complete Guidelines.


text:
21 June 2008
an underground journey
So rather than taking pictures of the beautiful day outside, I decided to instead get familiar with my new camera and take some pictures inside the flourescent jungle that is the NYC subway system. There is some blur in the pictures below as I didn't have enough time to get a steady photo and still capture the movement with a better ISO and slower shutter - i.e. I was in a rush to board a train. Anyone have any suggestions about this?
With pictures like the ones above, I wanted to evoke the feeling of entering some sort of dungeon.
But there is life in this dungeon. note: how many asian accordionists have you ever seen?
I like this composition a lot.
Posted by Chester at 10:42 PM 4 comments:
20 June 2008
a sign of the times
Posted by Chester at 3:21 PM 2 comments:
19 June 2008
The Superest (tea edition)
this self described "continually running game of My Team, Your Team" (see link in title) is truly one of the more constantly updated delights of the web. For those who are unfamiliar with the concept, the site consists of drawings of superheroes, each designed to defeat the previous superhero. For those of you who have yet to visit, it is a good waste of time filled with safe for work humor. Some selections below:
this is of course defeated by this character. Of some resemblance to yours truly:
Further down the chain of superheroes, we get this lovely character:
now please do go and waste some time.
Posted by Chester at 4:36 PM No comments:
Labels: tea art, webcrawling
Tea & Sympathy (a review)
go here to view my review criteria.
Charming. That is the only acceptable one word description for any patron of Tea & Sympathy. It is not, I daresay, a fairy-wand-malfunction caused restaurant interior kind of charming, but a home away from home I only wished I had so that I could be a more interesting person kind of charming. Cramped may be another that comes to mind, though for reverse claustrophobics such as myself (claustrophilics? that sounds too much like a sexual disorder), that isn't too much of a problem. All this charm is necessary for us non-Brits who unfortunately do not have the rose-tinted glasses for the oh so comforting but plain English fare. Apparently, the food isn't spectacular, but mind-blowing food is not the reason that people spend their money here. This is a place for the homesick. A place for those who want to feel smothered with the cozy feel of low light, hanging portraits of the queen, and eat food that feels like its been made with love (perhaps an effect of fat content?). That is a charm I can appreciate. Even the waitstaff is charming, though in that seemingly short-tempered though its probably because I am not British and I can't relate kind of way (I swear that will be the last near sentence long descriptive phrase I will use in this review).
Having said all that, I do wish you potential patrons luck in getting a seat at this place. Its popularity is well known so a moderate wait is more than likely. Those who are looking for a venue to hold a small tea party between friends could hold it here though there are really better and more appropriate venues for that kind of frouffy activity - not that I disapprove of course. Really, those in the know go here either for the food (which I have yet fully try and thus cannot review) or the cream tea which consists of two scones/biscuits, clotted cream, jam, and a pot of tea:
At a little over 10 dollars for the entire setup, it might seem a little expensive for those on a budget (please tip well). However, the experience is well worth it as there are few places in the city where you can even find clotted cream, jam, and scones at the same and particularly charming place. My only complaint is that the scones were rather cool the first time I received them - warm or at least room temperature would have been nice. The afternoon tea on the other hand is pricey at best and a waste of caloric intake at worst. I might have been better satisfied had I been allowed to share the tower with a friend, albeit with a second pot of tea. Unfortunately, sharing is not an option.
Regardless, the above selection is rather lackluster - though perhaps I shouldn't have expected any better. The cakes, though decent, lacked excitement, or even any hint of luxuriousness which isn't very difficult to work into a cake. The scones, despite my good review above, could not hold up the rest of the service especially at that price point. Really, most of the fault lies in their fingerling sandwiches which really depressed the entire experience with their oh so bland flavor. True, I should not have expected more out of mayo, bread, and assorted ingredients (my way of relating its forgettability). I just couldn't shake the feeling of wanting something more. Perhaps I have been spoiled by other establishments and their more resplendent offerings at lower prices.
The tea can be best described as of consistent quality - and by that I mean that it is forgivable. Those expecting at least "Far Too Good For Ordinary People" grade leaf should stay away as it will definitely not be your cup of tea. However, those of us willing to come down from our lofty perches will not recoil in horror. The leaf grades used in these lovely porcelain pots are not of the best quality - though not my personal taste, the pots are rather an entertaining and necessary part of the atmosphere. Although there is a respectable collection of flavored and brand name bagged teas, most of the single region selections are few and far between and of regular quality. The tea is also served loose in the pot (traditional I suppose) and comes with an English tea strainer to catch the leaves. While being a slightly better way of serving tea than using crumpled up removable strainers, I still have a large problem with serving tea in this way. Not only does it rely on the customer's knowledge of tea strength (which is sometimes absent), it also puts those customers who do have such knowledge on an unnecessary clock by forcing them to be mindful of the remaining leaves. The risk of an overbrewed second cup is so high that it might as well be unavoidable. For some concoctions which consist mostly of herbals or rooibos, this might not be a problem. However, for traditional blends such as earl grey and average grade single leaf varieties, bitterness is an all too often result of this method of serving tea. Milk is the only option left for those not wanting to waste liquid. As a purist, however, this is an all too depressing outcome (I don't drink Assams much either). Yet, it must be remembered that Tea & Sympathy is less about the tea as it is about having a British dining experience - which is why it lacks the focus and meticulousness of other tea specialty establishments.
The store, Carry On Tea & Sympathy is equally, if not more charming than the actual eating establishment. With a large stock of consumables, tea, porcelain and other tea brewing paraphernalia, shopping here can be a lovely experience.
Regular tea purchasers should note that the store does not carry any bulk loose-leaf tea. However, this deficiency is somewhat vindicated by the large selection of prepackaged bagged and loose-leaf varieties on sale. Though hardly breaking the boundaries of the traditional British implements, the sheer volume of porcelain and other paraphernalia is still impressive. A sample of what is available can be found on the website (see link within title of post). In addition to tea related goods, there is an impressive selection of British goods on sale such as candies and souvenir type baubles for those looking to purchase. All in all, the store, like the restaurant, is more of a collection of British related goods than it is strictly a tea store. That being said, the store still carries an impressive amount of tea related goods as tea is very important aspect of that culture. Those looking for a regular shop to fulfill their tea-drinking needs could be more than satisfied shopping here, though there are many better options around the city and on the Internet.
Despite the charm of the establishment, the tea and the afternoon tea set can leave those looking for a top notch tea experience wanting for more. The tea is of your standard quality so those expecting high end leaf grades should either lower their expectations or turn away. Sadly, those looking for a perfect afternoon tea are likely to be disappointed as well due to lackluster tea sandwiches and your average quality cakes. A shining light can be found, however, in the scone/biscuit service which is generally well priced for the experience. All in all, the selling point of Tea and Sympathy is its charm. The waitstaff is lovely and the decor is cozy. Those willing to stomach a moderate wait and a loss of personal space to feel like they waltzed into a small English diner should not hesitate to make a visit. In that respect, I can guarantee that both Tea & Sympathy and Carry on Tea & Sympathy will deliver. 3.49 stars (rounded down below).
link to yelp for restaurant and store.
Disagree? leave comments.
Posted by Chester at 11:17 AM 2 comments:
Labels: good, greenwich village, review, tea rooms, tea vendors, tea ware, west village
Subscribe to: Posts (Atom)


text:
Looking for an affordable dinner option? Papa Murphys has three different Large Pizzas for $5 each! Get a pepperoni, Sausage or Cheese Pizza as part of their $5 Faves Pizza deal. (Update – some locations have raised their $5 fave prices to $5.99, so be sure to contact your local store directly to find out what they are currently charging for this special).
What is the $5 Fave Pizza? They are lighter (meaning less hearty) versions of their regular pizzas. The toppings tend to be less than your standard pizza, but my kids love them, and they are a great way to feed a crowd! My kids end up picking off many of the toppings anyway, so I don't think they notice the lighter toppings at all. I will often pick one up for my kids, so the adults can enjoy a pizza of our choice without any complaining. Personally, I like them better then other pizza chains $5 options since I can bake them at home at our convenience.
You might also want to check out Papa Murphys $10 Tuesday Special that they are running now during the summer. We love this offer too! It's any size, any kind of pizza for $10 on Tuesdays.
This is available at participating locations – so be sure to call ahead and confirm that your store will be participating. Leave a comment below if you can confirm your location will be participating!
Find more great restaurant deals here!


text:
On Tap Today: Water, a gang of Pirates and a heroes' journey
Water made Milwaukee a destination from the beginning, and new photography exhibits opening Wednesday at Marquette University' Haggerty Museum of Art - "Dark Blue" and "Perimeter" - pay heed to water's power and pull.
Speaking of Marquette, the surprisingly soaring Golden Eagles take on the Pirates of Seton Hall in Big East basketball action at the BMO Harris Bradley Center at 8 p.m. »Read Full Blog Post
Singers blend magically in MSO's Baroque music concert
A pared-down MSO, two remarkable voices and an extremely communicative conductor joined forces in an absolutely captivating evening of Baroque music on Friday.
Guest conductor Nicholas McGegan led about 40 Milwaukee Symphony Orchestra strings, plus harpsichord/continuo organ, through an engaging program of Handel, Pergolesi and Scarlatti. They were joined by soprano Yulia Van Doren and countertenor Daniel Taylor. »Read Full Article


text:
Welcome to Amulet Vineyard & Winery
Amulet Vineyard is located in the Beechworth wine region. The wines are hand crafted on site 11km south of Beechworth amidst rolling hills and bushland. Amulet is also the home of Beechworth Cider.
Cellar door is open daily from 10am-5pm. Come and enjoy our hospitality, modern function area and cellar door as well as our awesome views.
Amulet offers seasonal a la carte lunches featuring regional produce, Fridays through to Sundays from 12 noon. Lunches are also available on public holidays and other days during holiday periods; please phone 03 5727 0420 for confirmation. At any time of the year, lunch can be provided by prior arrangement; please phone 03 5727 0420.
The summer menu features main courses such as Ocean trout carpaccio, Minute steak, Risotto verde, Chorizo with pepperonata salad, Baked figs with cheese & prosciutto, Pulled pork, and the Chef's selection tasting plate. Desserts include Honeycomb pannacotta with biscotti and Table Top figs in syrup.
Bookings are highly recommended: 03 5727 0420.
Amulet has a wonderful space for your next family gathering – birthday, engagement, wedding, anniversary, or corporate event. Feel free to contact us to discuss your requirements and we'll be happy to help.
|Copyright Amulet Vineyard and Winery | Site by WangNET


text:
I have two thoughts.
1. As to finding the poles: Take the equation and go as follows:
Use the quadratic formula to solve for , then take the logarithm and divide by .
2. What contour are you planning on using?
I must calculate .
Attempt: First I wanted to find the singularities of . I didn't succeeded in it. I know that and using this equality I found out that the singularities of f are when and , simultaneously (when z=a+ib). Which seems way too hard to solve in my opinion. So I'm guessing I'm missing a much simpler way to find the singularities of f.
I've also noticed that is the real part of but I don't see how it can help.
I'd like some help. Thank you.
Thanks a lot to both. I'm going to try to solve the problem via both ways.
I reach or . Taking the complex logarithm and noting that both numbers are real numbers (hence their argument is worth 0) and dividing by i, I get that or which I doubt is right.
I'm also not sure I can do the rest. I think I should still find the poles of the integrand? If so, I reach some non sense ( and ).
See here for my explanation of contour integration. You don't actually have to use the contour enclosing all the singularities. Refer to posts 6 through 8 in the link for my recommendations on choosing contours. You do need to know what singularities are actually there, of course.
Fortunately, you have a relatively easy way of checking your answer: plug those numbers you got into the cosine function, and see if you get -2. I think you have some sign errors there....which I doubt is right.
If you plan on pursuing my proposed method of contour integration, you're fairly soon going to be doing something rather similar to Opalg's very fine approach: choosing a contour and parametrizing it. What'll you pick?
Ok thanks a lot to both.
Edit: I could accessto an old version of the page with google chrome. I will copy it here.
andOriginally Posted by AckbeetEdit 2: Latex cannot be copied... A few questions regarding your post #8. Can be the whole real axis? In the case of my integral, can it be the part of the real axis starting from 0 and ending at ? If so, I would certainly pick a contour including this part of path. Probably a semi circle, I'm not sure to include 0, 1 or 2 poles. I also need to know where they are located (I will do it right now). But if I enclose at least 1 pole, I won't know the value of since I'd have to calculate the residue of it. I think I'm somehow confused.If you make no mistakes, there's loads of freedom in choosing contours. Let me outline the general idea of the solution. You pick a contour such as I have described. Let's say that the contour has various "legs of the journey", or is partitioned into the following contours: , where one of them, , corresponds to the original integral, and . You've got an original integrand . Now, let's suppose further that you know what the value of
is. What you do is set
The hope is that all the , except the one you're interested in, are very easy to do. You might be able to eliminate them on the basis of an ML estimate, for example. Then, you simply shove everything except the integral in which you're interested over to the other side:
Everything on the RHS is either known or easy to compute. Voila! You're done. That's the high-level view of contour integration as it's used to evaluate real-valued integrals. Make sense?
, in the context of post # 8 there, has to be the interval you started out with, or something so close to it that finding it will directly allow you to write down the value of the original integral. It can be the whole real axis, or it can be just from to .
Including zero poles is the easiest way to go, if you can. Here's what I would do: expand your original interval to to . Since the integrand is even, you can simply divide by two. That is,
Then, as you've intimated, I would use a semi-circle from to . There are no poles inside this contour.
Where would you go from here?
Ok thanks Ackbeet. I'm actually stuck on finding the singularities. I keep finding the same non sense result. I really don't know where I'm wrong: . Solving for gives or .
Taking the logarithm, I get that or or . I'm guessing I made an error when taking the logarithm of the complex exponential function? If so, I don't see it. I'll answer your question when I find the singularities.
(1/2)(|-2+ sqrt 3| +1/|sqrt 3 -2|) - Wolfram|Alpha.
To Ackbeet: Indeed, I trust you that both singularities are outside the contour you've described. The problem is that if I can't realize it myself, I wouldn't know it when I'm alone. But as of now, I'll just move on and assume that the singularities are outside the contour. Let me think how I'd continue.
Edit: The contour is the semi circle going from -pi to pi. Since no pole nor singularity is enclosed, the contour integral is worth 0 according to Cauchy's integral theorem. Hmm seems way too easy...
I found your problem, though. Take the principal value logarithm on both sides of
[LaTeX ERROR: Convert failed]
On the left hand side we get just . However, since the right hand side is NEGATIVE, we must do this:
[LaTeX ERROR: Convert failed] , since the number is on the negative axis and has argument of [LaTeX ERROR: Convert failed]
Similarly for the other root, since it's negative. In the end, it's your answers with [LaTeX ERROR: Convert failed] added:
Plugging the 1st one in the equation gives: