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| 5/25 |
| 2006/7/18-20 [Politics/Foreign/MiddleEast/Iraq] UID:43700 Activity:nil |
7/17 We have turned yet another corner in Iraq!
http://www.csua.org/u/ggm
\_ 50 people died today but we're spreading LIBERY FREEDOM AND THE
PURSUIT OF HAPPINESS in Iraq! -jblack #2 fan
\_ We're running out of Iraqi corners
\_ We're just circling the same damn block.
\_ We should build cities with hexagonal blocks. 50% more
corners for the same price!
\_ Not octagons, though. The government would get sued by
UFC.
\_ Can one fill a surface with octagons?
\_ Not in Euclidean or spherical geometry. You can
do it in the hyperbolic plane, see e.g
http://www.geom.uiuc.edu/apps/teich-nav/report/node1.html
Some of Escher's designs were based on this sort
of pattern.
\_ Wow! Thx.
\_ What if they were all round? Is that no corners or
infinite corners?
\_ I'd think of it as infinite corners. But you can't fill
a surface with round shapes.
\- if we think about this in terms of
differentiability i think it is
better to think of round as "no corners".
You may enjoy reading about
THE WEIERSTRASS FUNCTION. --psb
\- if we think about this in terms of differentiability
i think it's better to think of round as no corners.
You may enjoy reading about THE WEIERSTRASS FUNCTION.
I suppose you can also read about LOCAL LINEARITY.
BTW, there are some pretty counter intuitive stuff
in this area [space filling sets and covers]. For a
Berkeley connection, you may
google(dubins, hirsch, scissor) also see
google(dubins, hirsch, scissor) also see e.g.
http://sciboard.louisville.edu/math.html#q12 --psb
\_ Interesting. I was thinking in terms of
circumference and area, which approach 2*pi*r and
pi*r^2 as N approaches infinity. So do the two
ways of thinking contradict to each other? I'm
not a math major. -- PP
ways of thinking contradict each other? I'm
not a math major and I'm only good at high school
math. -- PP |
| 5/25 |
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| www.csua.org/u/ggm -> news.yahoo.com/s/ap/20060718/ap_on_re_mi_ea/iraq_060717155169 AP Gunmen kill 50 in Iraqi market attack By BASSEM MROUE, Associated Press Writer Tue Jul 18, 2:35 AM ET BAGHDAD, Iraq - Gunmen sprayed grenades and automatic weapons fire in a market south of Baghdad on Monday, killing at least 50 people, mostly Shiites. The sectarian attack drew an angry protest from lawmakers who accused Iraqi forces of standing idly by during the rampage. Women and children were among the dead and wounded in the assault in Mahmoudiya, hospital officials said. Late Monday, police said they found 12 bodies in different parts of town -- possible victims of reprisal killings. Iraq , killing 43 people and wounding 65, a Health official said. The bomb went off at about 7:30 am across the street from the gold-domed mosque in Kufa, 100 miles south of Baghdad,, police Capt. The shrine marks the place where Imam Ali, cousin and son-in-law of the Prophet Muhammad, was mortally wounded. Meanwhile, several witnesses, including municipal council members, said the attack began when gunmen -- presumed to be Sunnis -- fired on the funeral of a member of the Mahdi Army, a Shiite militia, killing nine mourners. Assailants then drove to the nearby market area in the town 20 miles south of Baghdad, killing three soldiers at a checkpoint and firing rocket-propelled grenades and automatic rifles at the crowd. After the gunmen sped away, they lobbed several mortar rounds into the neighborhood, the witnesses said. The assault occurred a few hundred yards from Iraqi army and police positions, but the troops did not intervene until the attackers were fleeing, the witnesses said. They spoke on condition of anonymity because of fear of reprisals. The US command announced that three American soldiers were killed in separate attacks Monday -- two in the Baghdad area and one in Anbar province west of the capital. At least 2,554 members of the US military have died since the beginning of the Iraq war in March 2003, according to an Associated Press count. There were conflicting casualty figures in the market attack, with a Shiite television station reporting more than 70 dead. Dawoud al-Taie, director of Mahmoudiya hospital, said 50 people were killed and about 90 were wounded. In Baghdad, Shiite legislator Jalaluddin al-Saghir said Iraqi military authorities had ignored warnings that weapons were being stocked in a mosque near the market. He also said the local police commander refused to order his men to confront the attackers because they lacked weapons and ammunition. Dozens of Shiite lawmakers, including followers of radical anti-American cleric Muqtada al-Sadr, stormed out of a parliament session to protest the performance of the security forces. In the first 17 days of July, at least 617 Iraqis have been killed in war-related violence, at least 527 civilians and 90 police and security forces, according to an AP count. In the nearly two months since the unity government took office on May 20, more than 1,850 Iraqis have been killed, including at least 1,585 civilians and 267 security forces. The July figure represents a marked increase over the same period last year when an AP count showed at least 450 Iraqis killed, at least 306 civilians and 144 police and security forces. The 617 killed so far this July is already near the total killed in all July last year: 714. In Mahmoudiya, long a flashpoint of Shiite-Sunni tension, tempers boiled as frantic relatives milled about the hospital, scuffling with guards and Iraqi soldiers who tried to keep them outside so doctors could treat the wounded. "You are strong men only when you face us, but you let them do what they did to us," one man shouted at a guard. The Shiite television station Al-Forat broadcast strident quotes from Shiites who blamed the attack on Sunni religious extremists. They expressed outrage that Sunni politicians could not rein in the militants. The main Sunni bloc in parliament said the attack may have been retaliation for the kidnapping of seven Sunnis whose bodies were found Sunday in Mahmoudiya. The bloc accused Shiite-dominated Iraqi security forces of failing to control the situation. The events also raised doubts about the effectiveness of the US strategy of handing over large areas of the country to Iraqi control, while keeping US troops in reserve. US troops of the 101st Airborne Division reported hearing detonations and gunfire, the US command said. But Iraqi troops are responsible for security in Mahmoudiya, and American soldiers do not intervene unless asked by the Iraqis. Four soldiers and a former soldier from the division are accused of raping and murdering a teenage girl near Mahmoudiya on March 12. A sixth soldier is accused of failing to report the crime. The Mahmoudiya attack was part of a rising tide of tit-for-tat killings and intimidation that many Iraqis fear is the prelude to civil war. The campaign of intimidation and attacks is slowly transforming Baghdad into sectarian zones under the tacit control of armed groups that protect members of their sect and drive away others. On July 9, Shiite militiamen swept through the mostly Sunni neighborhood of Jihad in western Baghdad, dragging Sunnis from their cars and shooting them in the street. Faced with such massacres, Iraqis are turning to sectarian militias to protect themselves because government forces cannot. Some Sunnis, who form the backbone of the insurgency, now say privately they want American troops to remain in Iraq to protect them from Shiite militias. Carlos Gutierrez came to Baghdad Monday and signed an agreement with the Iraqis to encourage foreign investment and lay the foundation for a market economy after decades of state control. "We are convinced that Iraq is ready for recovery," Gutierrez told reporters, later acknowledging that "clearly, security is still the No. Also Monday, the final group of Japanese troops left Iraq and arrived in Kuwait, ending Japan's two-year humanitarian mission in southern Iraq. The rest of the Japanese contingent, which had numbered more than 600, departed over the past two weeks. Relatives of a victim killed by armed gunmen in a Mahmoudiya market mourn during a funeral procession, Monday, July 17, 2006, in the holy city of Najaf, southern Iraq. Dozens of heavily armed attackers raided the open air market in Mahmoudiya, south of Baghdad, killing at least 41 people and wounding about 90, Iraqi and US officials said. The information contained in the AP News report may not be published, broadcast, rewritten or redistributed without the prior written authority of The Associated Press. |
| www.geom.uiuc.edu/apps/teich-nav/report/node1.html Navigating Teichmueller Space Introduction One way of examining topological surfaces is by cutting them up'' and flattening them out on a plane. The easily-visualized example of this is the genus 1 case, the one-holed torus. Making one straight cut produces a cylinder, which can be cut open yet again to produce a rectangle with opposite sides identified. This is a common depiction of the torus, in which movement on the surface is simulated by a line off one side being wrapped around and coming back in on the opposite. To get an even better picture of the torus we can tile the plane with the rectangle, so that we do not have to wrap our movements around, but rather can move anywhere in the plane and still be on a copy of the original tile, ie on the surface. We can even change the angles of the recangle to make it a parallelogram, and still tile with it nicely. When we move up one genus to the two-holed torus, however, cutting the surface open creates an octagon. To see that this is indeed the case imagine beginning with an octagon and gluing together the first and third, second and fourth, fifth and seventh, and the sixth and eighth sides. We see that the octagon is a more interesting case when we begin trying to tile the plane. The sum of the interior angles of an octagon in the Euclidean plane is 1080 degrees. But we cannot fit an integer number of such octagons around a vertex! If we take two copies of the octagon and join them along one edge we will already have 270 degrees around a vertex, so we cannot fit even one more copy into the remaining 90 degrees. And to achieve a tiling that will simulate the genus-two surface we would like to fit eight tiles around each vertex! The solution to this problem is that we must attempt our tiling in a space that allows octagons to have interior angle sums of 360 degrees; in the regular case, we need an octagon with eight equal sides and eight angles of 45 degrees each. Hyperbolic geometry arose from the idea that perhaps a valid system of theorems and manipulations could be based on a variation of the standard postulates of Euclid. The only difference between the two base sets of assumptions is in the parallel axiom. Euclid's original postulate proposed that for a line and a point not on the line, there is only one line through that point that is parallel to the original. Alternatively, in hyperbolic geometry there are many lines through a point that do not intersect a line not containing that point. To visualize the hyperbolic plane we will use what is known as the Poincare disk model. This is constructed on the unit disk, with straight lines given as arcs of circles orthogonal to the boundary of the disk, or in some cases as diameters. In this model, it is not intuitively obvious what the distance is between any two points, because there is a hyperbolic metric on the disk. An arc (line) segment that is close to the boundary appears shorter than one oif the same length near to the center of the disk. Furthermore, if we are considering the intersection of two arcs (lines) we see that by pulling the point at which they intersect toward the boundary, we can make the angle arbitrarily smaller. Thus we can see that if we take a regular polygon, with all sides of equal length, and we pull the vertices toward the boundary of the disk, we can decrease the angle at each to 45 degrees. Though we will have lengthened the sides, they will still be equal to one another. Using polygons in such a space, we can examine how to depict different Riemann surfaces visually as tilings. Figure: A tiling of the Poincare disk by regular octagons. In addition to the Poincare disk model of hyperbolic space, we will also be using the Minkowski or hyperboloid model to do a number of calculations within. There is a simple conversion that will take a point in the Poincare disk model to the Minkowski, and vice versa. The advantage to working within this model of hyperbolic space is that the Minkowski metric is much simpler than that of the Poincare disk, and thus many of the calculations become simpler in it. Once we have hyperbolic space to work within, our question is which polygons are actually tileable. Each octagon can be expressed by sixteen values, two coordinates for each vertex. So the space of all possible octagons is sixteen-dimensional. But we must place certain restrictions on exactly how the sixteen values are chosen to define the octagon so that it will tile. If we imagine gluing up our polygon into a surface embedded in three space, we can see what some of these conditions will be. When the octagon is glued together the vertices will all become one point on the surface. Thus, the interior angles of the octagon must sum 360 degrees so that when it is glued up this will be true. In gluing up the octagon we also see that the sides that are glued together (identified with each other) must be of equal length. This gives us one angle condition and four side-pairing conditions, which restricts the sixteen-dimensional space of all octagons to an eleven-dimensional space of octagons that will work in tiling the Poincare disk. How does this space relate to the Teichmueller space of the two-holed torus? We will see that it actually projects onto the Teichmueller space. From this we know that each Riemann surface (each point in the Teichmueller space) is given by at least one tiling in hyperbolic space. Thus there is a tiling that corresponds to every conformal structure on a surface- in fact, there are many. Three of these extra dimensions of tilings beyond those that represent the Teichmueller space of the surface are given by the Moebius transformations that allow rotations and translations of the polygon. Moving within the space of all octagons that tile, we will find some tileable polygon that may actually represent the same Riemann surface as another valid tiling, yet be rotated by some amount, or translated in some direction so that they appear to be different. So with these three translations/rotations, and our previous five conditions of angle sum and side-pairings, we have accounted for eight of the sixteen dimensions in the space of all octagons. For two last degrees of freedom, consider the glued-together'' version of our example- the octagon. Remember that the vertices will all end up being identified to the same point on the surface; this is the point at which we begin cutting up the surface to create the polygon. We are free to move this initial cutting point around on the surface, which is two-dimensional. This means that in looking at various tilings we may find two different octagons that both represent the same Riemann surface, ie the same point in the Teichmueller space, simply produced by different initial cutting locations. When considering the higher genus cases, these basic concepts will still hold true. If we take into account that we will be seeing some tilings that are actually just rotations, translations, and different base cutting points of the same Riemann surface. |
| sciboard.louisville.edu/math.html#q12 Mathematics Click on a question to know the answer Q What is the current status of Fermat's last theorem? Q I have this complicated symbolic problem (most likely a symbolic integral or a DE system) that I can't solve. Q The existence of a projective plane of order 10 has long been an outstanding problem in discrete mathematics and finite geometry. Q Is there a formula to determine the day of the week, given the month, day and year? Q What is the formula for the Surface Area of a sphere in Euclidean N - Space. Is it possible to cut a sphere into a finite number of pieces and reassemble into a solid of twice the volume? Q Is there a theory of quaternionic analytic functions, that is, a four- dimensional analog to the theory of complex analytic functions? Return to Question and Answers Q What is the current status of Fermat's last theorem? FLT was proven in 1993 by Andrew Wiles working at Princeton. It is not intended to be 'deep',but rather is intended for non-specialists. The first case assumes (abc,n) = 1 The second case is the general case. They all used extensions of the Wiefrich criteria and improved upon work performed by Gunderson and Shanks&Williams. The first case has been proven to be true for an infinite number of exponents by Adelman, Frey, et. Second Case: It has been proven true up to n = 150,000 by Tanner & Wagstaff. The work used new techniques for computing Bernoulli numbers mod p and improved upon work of Vandiver. The work involved computing the irregular primes up to 150,000. FLT is true for all regular primes by a theorem of Kummer. In the case of irregular primes, some additional computations are needed. Update: Fermat's Last Theorem has been proved true up to exponent 2,000,000 in the general case. The method used was that of Wagstaff: enumerating and eliminating irregular primes by Bernoulli number computations. The computations were performed on a set of NeXT computers by Richard Crandall. Conjectures There are many open conjectures that imply FLT. These conjectures come from different directions, but can be basically broken into several classes: (and there are interrelationships between the classes) conjectures arising from Diophantine approximation theory such as The ABC conjecture, the Szpiro conjecture, the Hall conjecture, etc. For an excellent survey article on these subjects see the article by Serge Lang in the Bulletin of the AMS, July 1990 entitled "Old and new conjectured diophantine inequalities". Masser and Osterle formulated the following known as the ABC conjecture: Given epsilon > 0, there exists a number C(epsilon) such that for any set of non-zero, relatively prime integers a,b,c such that a+b = c we have max( |a|, |b|, |c|) <= C (epsilon) N (abc) ^ ( 1 + epsilon ) where N ( x ) is the product of the distinct primes dividing x It is easy to see that it implies FLT asymptotically. The conjecture was motivated by a theorem, due to Mason tha essentially says the ABC conjecture is true for polynomials. There are also a number of inter-twined conjectures involving heights on elliptic curves that are related to much of this stuff. conjectures arising from the study of elliptic curves and modular forms. There is a very important and well known conjecture known as the Taniyama-Weil-Shimura conjecture that concerns elliptic curves. This conjecture has been shown by the work of Frey, Serre, Ribet, et. to imply FLT uniformly, not just asymptotically as with the ABC conj. The conjecture basically states that all elliptic curves can be parameterized in terms of modular forms. Sha, the Tate-Shafarevich group on elliptic curves of rank 0 or 1 By the way. An interesting aspect of this work is that there is a close connection between Sha, and some of the classical work on FLT. For example, there is a classical proof that uses infinite descent to prove FLT for n = 4 It can be shown that there is an elliptic curve associated with FLT and that for n=4, Sha is trivial. It can also be shown that in the cases where Sha is non-trivial, that infinite-descent arguments do not work; The diophantine and elliptic curve conjectures all involve deep properties of integers. Until these conjecture were tied to FLT, FLT had been regarded by most mathematicians as an isolated problem; Now it can be seen that it follows from some deep and fundamental properties of the integers. This theorem was proved with the aid of a computer in 1976. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations. Largest known Mersenne prime It is 2 ^ 756839 - 1 It was discovered by a Cray-2 in England in 1992. Largest known prime The largest known prime was 391581 * 2 ^ 216193 - 1 See Brown, Noll, Parady, Smith, Smith, and Zarantonello, Letter to the editor, American Mathematical Monthly, vol. Now the largest known prime is the Mersenne prime described above. Largest known twin primes The largest known twin primes are 1706595 * 2 ^ 11235 +- 1 See B K Parady and J F Smith and S E Zarantonello, Smith, Noll and Brown. largest Fermat number with known factorization F11 = ( 2 ^ ( 2 ^ 11 ) ) + 1 which was factored by Brent & Morain in 1988. Current status on Mersenne primes Mersenne primes are primes of the form 2 ^ p - 1 For 2 ^ p - 1 to be prime we must have that p is prime. nr p year by 1-5 2,3,5,7,13 in or before the middle ages 6-7 17,19 1588 Cataldi 8 31 1750 Euler 9 61 1883 Pervouchine 10 89 1911 Powers 11 107 1914 Powers 12 127 1876 Lucas 13-14 521,607 1952 Robinson 15-17 1279,2203,2281 1952 Lehmer 18 3217 1957 Riesel 19-20 4253,4423 1961 Hurwitz & Selfridge 21-23 9689,9941,11213 1963 Gillies 24 19937 1971 Tuckerman 25 21701 1978 Noll & Nickel 26 23209 1979 Noll 27 44497 1979 Slowinski & Nelson 28 86243 1982 Slowinski 29 110503 1988 Colquitt & Welsh jr. Find a friend with access to a computer algebra system like MAPLE, MACSYMA or MATHEMATICA and ask her/him to solve it. If packages cannot solve it, then (and only then) ask the net. There are other Computer Algebra packages available that may better suit your needs. Maple Purpose: Symbolic and numeric computation, mathematical programming, and mathematical visualization. edu DOE-Macsyma Purpose: Symbolic and mathematical manipulations. Contact: National Energy Software Center Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60439 Phone: (708) 972-7250 Pari Purpose: Number-theoretic computations and simple numerical analysis. Available for Sun 3, Sun 4, generic 32-bit Unix, and Macintosh II. fr Mathematica Purpose: Mathematical computation and visualization, symbolic programming. COM Matlab Purpose: matrix laboratory' for tasks involving matrices, graphics and general numerical computation. com Cayley Purpose: Computation in algebraic and combinatorial structures such as groups, rings, fields, modules and graphs. According to some Calculus textbooks, 0 ** 0 is an "indeterminate form". When evaluating a limit of the form 0 ** 0, then you need to know that limits of that form are called "indeterminate forms", and that you need to use a special technique such as L'Hopital's rule to evaluate them. Otherwise, 0 ** 0=1 seems to be the most useful choice for 0 ** 0 This convention allows us to extend definitions in different areas of mathematics that otherwise would require treating 0 as a special case. Notice that 0 ** 0 is a discontinuity of the function x ** y Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f( x ) ** g( x ) approaches 1 as x approaches 0 from the right. Graham, D Knuth, O Patashnik): "Some textbooks leave the quantity 0 ** 0 undefined, because the functions x ** 0 and 0 ** x have different limiting values when x decreases to 0 But this is a mistake. We must define x ** 0 = 1 for all x, if the binomial theorem is to be valid when x = 0 , y = 0, and/or x = -y. The theorem is too important to be arbitrarily restricted! " Using the well-known epsilon - delta definition of limit, one can easily show that this limit is 1 The statement that 09999... An "informal" argument could be ... |