.You can see Questions and solutions of IMO 2006 by click Here
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Fri 13 Oct 2006ساعت 2:23 توسط Saeed Hojjati
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When you encounter a finitely generated abelian group g, your professor, or your text book, will tell you that g is the direct product of cyclic subgroups. You can often use this information to disassemble g. For instance, g might be Z3Z5Z72Z43. This is 3 copies of the integers running in parallel, cross the integers mod 5, cross two copies of the integers mod 7, cross the integers mod 43.
Although this theorem is invoked repeatedly, it can be difficult to track down a proof. Well - the proof is here.
If g is not finitely generated, it might not be the direct product or direct sum of cyclic groups. Consider the algebraic closure of the finite field Zp. Its multiplicative group is abelian, but not finitely generated. It has a cyclic subgroup of order p-1, contained in a cycle of order p2-1, contained in a cycle of order p4-1, and so on. This cannot be split into parallel cycles.
A (perhaps) simpler example is the rationals whose denominators are powers of 2. This forms a group under addition, which I will call g. Each cyclic subgroup is infinite. This follows from the fact that Q is an integral domain. So if g is the direct sum of cyclic subgroups then g is the direct sum of copies of Z, and g is a free abelian group. Let x be an element in a basis for g. Divide x by 2 and find another element of g. In a free group, one cannot cut a basis element in half. This is a contradiction, hence g is not the direct sum of cyclic subgroups.
Modules over a PID
Most of the theorems in these pages can be generalized to modules over a pid. That's why this material is under modules, rather than groups. If you only care about abelian groups, the excess baggage can be annoying. What's a module? What's a pid? Who cares? Fair enough, but I didn't want to write all this out twice, for groups and for modules. I'd rather do it once, in its most general setting. So I'm going to talk about modules etc, and if you like, you can think of a module as an abelian group and leave it at that. I won't tell anyone.
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Fri 13 Oct 2006ساعت 2:12 توسط Saeed Hojjati
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Let m be any module, or a left module if you prefer. A chain h of submodules is ascending if hi ⊂hi+1, and descending if hi ⊃ hi+1. In other words, the submodules are getting larger (ascending) or smaller (descending).
If you like set theory, think of a chain as a map from the ordinals into the submodules of m, where the submodules increase or decrease as the ordinals advance. This allows you to build uncountable chains of submodules. Well ok - most of us will never do that, so let's move on.
Noether (biography) investigated modules with no infinite ascending chains; and these modules are now called noetherian. Do read her biography; it wasn't easy for a Jewish woman to succeed in mathematics, in Germany, in the beginning of the 20th century.
At approximately the same time, Artin (biography) explored modules with no infinite descending chains, and these modules are now called artinian.
Another synonym for noetherian is "ascending chain condition", or acc. as you might guess, an artinian module has the "descending chain condition", or dcc. We will usually use the words noetherian and artinian.
If r is a ring, r is a left r module, and the submodules are the ideals. Thus a noetherian ring has no infinite ascending chains of ideals, and an artinian ring has no infinite descending chains of ideals.
A division ring is a rather trivial example. It only has two ideals, 0 and the entire ring. This is obviously noetherian and artinian.
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Fri 13 Oct 2006ساعت 2:10 توسط Saeed Hojjati
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A graph is a set of vertices v, and a set of edges e that connect pairs of points in v. Edges are actually determined by a symmetric irreflexive relation on v cross v. Some graphs may include loops (a point connected to itself) or multiple edges (many lines connecting a pair of points), but they are the exceptions. Unless otherwise indicated, the term "graph" refers to the definition above.
A digraph contains vertices and edges as above, but the edges are directed. In other words, the edge relation need not be symmetric. The edge v1→v2 does not imply the edge v2→v1. Edges are drawn using arrows, rather than line segments. Again, loops and multiple edges are usually prohibited.
A vertex has degree n if there are n edges connecting that vertex to other vertices. The number of edges is half the sum of the degrees.
In a digraph, vertices have indegrees and outdegrees. The number of directed edges is the sum of the indegrees, or the sum of the outdegrees.
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Fri 13 Oct 2006ساعت 1:58 توسط Saeed Hojjati
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If k is an integer, zeta(k), written ζ(k), is the sum of 1/nk as n runs from 1 to infinity. We've already seen that ζ(1) produces the harmonic series, which does not converge, while ζ(2), the sum of
inverse squares, does converge. By dominance, ζ(k) converges for any k > 2.
As we shall see, this function generalizes in several different ways. For instance, the exponent k could be a real or complex number, and that's just a start.
The zeta function, in all its generality, is essential for analytic number theory - so perhaps I should describe it there. But it is also used in algebraic number theory. On the other hand, complex calculus is used to evaluate the zeta function, so perhaps it belongs there. Finally, the function is defined as a convergent series, so maybe it belongs here, under sequences and series. Apparently I found this last point compelling, because here it is.
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Fri 13 Oct 2006ساعت 1:55 توسط Saeed Hojjati
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You can see Results of 47th International Mathematical Olympiad by click Here
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Thu 20 Jul 2006ساعت 21:5 توسط Saeed Hojjati
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A paradox first enunciated by Einstein
et al. (1935), who proposed a
thought experiment that appeared to demonstrate quantum mechanics to be an incomplete theory. The usual view of quantum mechanics says that a wave function determines the probabilities of an actual experimental result and that it is the most complete possible specification of the quantum state. Einstein
et al. believed the predictions of quantum mechanics to be correct, but only as the result of statistical distributions of other unknown but real properties of the particles.
Bohm (1951) presented a paper in which he described a modified form of the Einstein-Podolsky-Rosen thought experiment which he believed to be conceptually equivalent to that suggested by Einstein et al. (1935), but which was easier to treat mathematically. Bohm suggested using two atoms with a known total spin of zero, separated in a way that the spin of each atom points in a direction exactly opposite to that of the other. In this situation, the angular momentum of one particle can be measured indirectly by measuring the corresponding vector of the other particle. Bell (1964) subsequently formulated Bell's inequalities, which seemed to be a physically reasonable condition of locality which imposed restrictions on the maximum correlations of the measurements of a pair of spin 1/2 particles formed somehow in the singlet state and moving freely in opposite directions. This inequality can be tested in a laboratory experiment because the statistical predictions of quantum mechanics are incompatible with any local hidden variables theory apparently satisfying only the natural assumptions of "locality," as shown by the predictions of Bell's inequalities.
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Thu 20 Jul 2006ساعت 20:39 توسط Saeed Hojjati
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You can see problems of IMO(International Mathematical Olympiad)2005 by click Here
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Tue 18 Jul 2006ساعت 13:45 توسط Saeed Hojjati
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Here is an image of Mobius Strips : 
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Tue 18 Jul 2006ساعت 13:37 توسط Saeed Hojjati
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It is a great honour and at the same time a necessity for me to round out and develop my thoughts on the foundations of mathematics, which was expounded here one day five years ago and which since then have constantly kept me most actively occupied. With this new way of providing a foundation for mathematics, which we may appropriately call a proof theory, I pursue a significant goal, for I should like to eliminate once and for all the questions regarding the foundations of mathematics, in the form in which they are now posed, by turning every mathematical proposition into a formula that can be concretely exhibited and strictly derived, thus recasting mathematical definitions and inferences in such a way that they are unshakeable and yet provide an adequate picture of the whole science. I believe that I can attain this goal completely with my proof theory, even if a great deal of work must still be done before it is fully developed.
No more than any other science can mathematics be founded by logic alone; rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of representation, certain extra-logical concrete objects that are intuitively present as immediate experience prior to in thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediate, given intuitively, together with the objects, is something that neither can be reduced to anything else nor requires reduction. This is the basic philosophical position that I regard as requisite for mathematics and, in general, for all scientific thinking, understanding, and communication. And in mathematics, in particular, we consider is the concrete signs themselves, whose shape, according to the conception we have adopted, is immediately, clear and recognisable. This is the very least that must be presupposed; no scientific thinker can dispense with it, and therefore everyone must maintain it, consciously, or not.
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Sat 8 Jul 2006ساعت 17:59 توسط Saeed Hojjati
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RSA numbers are composite numbers having exactly two prime factors (i.e., so-called semiprimes) that have been listed in the Factoring Challenge of RSA Security® and have been particularly chosen to be difficult to factor.
While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.
RSA Laboratories sponsors the RSA Factoring Challenge to encourage research into computational number theory and the practical difficulty of factoring large integers, and because it can be helpful for users of the RSA encryption public-key cryptography algorithm for choosing suitable key lengths for an appropriate level of security. A cash prize is awarded to the first person to factor each challenge number.
RSA numbers were originally spaced at intervals of 10 decimal digits between 100 and 500 digits, and prizes were awarded according to a complicated formula. These original numbers were named according to the number of decimal digits, so RSA-100 was a hundred-digit number. As computers and algorithms became faster, the unfactored challenge numbers were removed from the prize list and replaced with a set of numbers with fixed cash prizes. At this point, the naming convention was also changed so that the trailing number would indicate the number of digits in the binary representation of the number. Hence, RSA-640 has 640 binary digits, which translates to 193 digits in decimal.
RSA numbers received widespread attention when a 129-digit number known as RSA-129 was used by R. Rivest, A. Shamir, and L. Adleman to publish one of the first public-key messages together with a $100 reward for the message's decryption (Gardner 1977). Despite widespread belief at the time that the message encoded by RSA-129 would take millions of years to break, it was factored in 1994 using a distributed computation which harnessed networked computers spread around the globe performing a multiple polynomial quadratic sieve (Leutwyler 1994). The result of all the concentrated number crunching was decryption of the encoded message to yield the profound plaintext message "The magic words are squeamish ossifrage." (For the benefit of non-ornithologists, an ossifrage is a rare predatory vulture found in the mountains of Europe.) The corresponding factorization (into a 64-digit number and a 65-digit number) is
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Thu 8 Jun 2006ساعت 15:36 توسط Saeed Hojjati
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The Fields Medals are commonly regarded as mathematics' closest analog to the Nobel Prize (which does not exist in mathematics), and are awarded every four years by the International Mathematical Union to one or more outstanding researchers. "Fields Medals" are more properly known by their official name, "International medals for outstanding discoveries in mathematics."
The Field Medals were first proposed at the 1924 International Congress of Mathematicians in Toronto, where a resolution was adopted stating that at each subsequent conference, two gold medals should be awarded to recognize outstanding mathematical achievement. Professor J. C. Fields, a Canadian mathematician who was secretary of the 1924 Congress, later donated funds establishing the medals which were named in his honor. Consistent with Fields' wish that the awards recognize both existing work and the promise of future achievement, it was agreed to restrict the medals to mathematicians not over forty at the year of the Congress. In 1966 it was agreed that, in light of the great expansion of mathematical research, up to four medals could be awarded at each Congress.
The Fields Medal is the highest scientific award for mathematicians, and is presented every four years at the International Congress of Mathematicians, together with a prize of 15000 Canadian dollars. The first Fields Medal was awarded in 1936 at the World Congress in Oslo. The Fields Medal is made of gold, and shows the head of Archimedes (287-212 BC) together with a quotation attributed to him: "Transire suum pectus mundoque potiri" ("Rise above oneself and grasp the world"). The reverse side bears the inscription: "Congregati ex toto orbe mathematici ob scripta insignia tribuere" ("the mathematicians assembled here from all over the world pay tribute for outstanding work").
Nobel prizes were created in the will of the Swedish chemist and inventor of dynamite Alfred Nobel, but Nobel, who was an inventor and industrialist, did not create a prize in mathematics because he was not particularly interested in mathematics or theoretical science. In fact, his will speaks of prizes for those "inventions or discoveries" of greatest practical benefit to mankind. While it is commonly stated that Nobel decided against a Nobel prize in math because of anger over the romantic attentions of a famous mathematician (often claimed to be Gosta Mittag-Leffler ) to a woman in his life, there is no historical evidence to support the story. Furthermore, Nobel was a lifelong bachelor, although he did have a Viennese woman named Sophie Hess as his mistress (Lopez-Ortiz).
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Thu 8 Jun 2006ساعت 15:35 توسط Saeed Hojjati
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Several prizes are awarded periodically for outstanding mathematical achievement. There is no Nobel Prize in mathematics, and the most prestigious mathematical award is known as the Fields medal. In rough order of importance, other awards are the Wolf Prize of the Wolf Foundation of Israel, the Leroy P. Steele Prize of the American Mathematical Society, followed by the Bôcher Prize, Cole Prizes in algebra and number theory, and the Delbert Ray Fulkerson Prize, all presented by the American Mathematical Society.
The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven "Millennium Prize Problems," selected by focusing on important classic questions in mathematics that have resisted solution over the years. A $7 million prize fund has been established for the solution to these problems, with $1 million allocated to each. The problems consist of the Riemann hypothesis, Poincaré conjecture, Hodge conjecture, Swinnerton-Dyer conjecture, solution of the Navier-Stokes equations , formulation of Yang-Mills theory , and determination of whether NP-problems are actually P-problems.
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Thu 8 Jun 2006ساعت 15:34 توسط Saeed Hojjati
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A quartic surface which is the locus of zeros of the determinant of a symmetric matrix of linear forms. A general symmetroid has 10 ordinary double points (Jessop 1916, Hunt 1996).
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Thu 8 Jun 2006ساعت 15:33 توسط Saeed Hojjati
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The pedal curve of the cissoid, when the pedal point is on the axis beyond the asymptote at a distance from the cusp which is four times that of the asymptote is a cardioid.
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Thu 8 Jun 2006ساعت 15:30 توسط Saeed Hojjati
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The fundamental theorem of combinatorics, and of probability and statistics, is also the most obvious. If there are six possible outcomes from the throw of a die, and two outcomes from the toss of a coin, Then the two events taken together have twelve possible outcomes. This is really a statement about the cardinality of the cross product of two finite sets, and its proof is best left in set theory, where it belongs.
There is a catch. The individual events must be independent. If someone decides to drop the coin flat on the table, heads up, whenever the die comes up 6, there are only 11 possible outcomes, not 12. This disclaimer is obvious, but in the real world, seemingly unrelated events are often connected in subtle ways.
The branch of mathematics known as combinatorics combines independent events and counts, or at least estimates, the number of outcomes. Probability brings in the concept of randomness, so that one of these outcomes can be chosen at random. Then we can compute the odds of a royal flush, 4 chances in 2598960. That will come later. For now, let's start counting events.
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Thu 8 Jun 2006ساعت 1:13 توسط Saeed Hojjati
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As the name suggests, algebraic number theory employs modern algebraic techniques to solve problems in number theory. We are therefore interested in integral extensions of Z (the integers), and field extensions of Q (the rationals). However, other extensions will prove useful as well.
A global field is a finite separable extension of Q or of Zp(t). We are extending the rationals, or the quotients of polynomials in t over the integers mod p.
A number field is a finite extension of Q. Since Q has characteristic 0, extensions of q are automatically separable. Thus a number field is a global field. In contrast, Zp(t) supports inseparable extensions, such as the pth root of t. We want to avoid those; hence a global field is declared separable.
within the context of a global field, an algebraic number is algebraic over the base field, either Q or Zp(t). Of course every element in a finite extension is algebraic, so everything in a global field qualifies as an algebraic number.
An algebraic integer is integral over the base ring, Z or Zp[t]. For example, sqrt(2) is an algebraic integer, while 1/3 is not.
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Thu 8 Jun 2006ساعت 1:12 توسط Saeed Hojjati
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Although calculus is not a requirement for this topic, an understanding of differentiation, and ordinary differential equations, is recommended. As you will see, there are similarities between difference equations (presented here), and differential equations (described under calculus). The former is a quantized version of the latter, or if you prefer, the latter is a continuous version of the former.
Difference equations are easier to manipulate and solve, because you don't need an underlying theory of limits and continuity. The functions are discrete sequences, with no δ's or ε's in sight. However, the methods employed to solve these equations often come directly from calculus, and some of the intuition may be lost if you have never solved a differential equation before
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Thu 8 Jun 2006ساعت 1:10 توسط Saeed Hojjati
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Please send me your hot qustions about Mathematical olympiads (Like IMO).
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Thu 8 Jun 2006ساعت 0:43 توسط Saeed Hojjati
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Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.
The definition of topology leads to the following mathematical joke (Renteln and Dundes 2005):
Q: What is a topologist? A: Someone who cannot distinguish between a doughnut and a coffee cup.
There is more to topology, though. Topology began with the study of curves, surfaces, and other objects in the plane and three-space. One of the central ideas in topology is that spatial objects like circles and spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are "represented" or "embedded" in space. For example, the statement "if you remove a point from a circle, you get a line segment" applies just as well to the circle as to an ellipse, and even to tangled or knotted circles, since the statement involves only topological properties.
Topology has to do with the study of spatial objects such as curves, surfaces, the space we call our universe, the space-time of general relativity, fractals, knots, manifolds (which are objects with some of the same basic spatial properties as our universe), phase spaces that are encountered in physics (such as the space of hand-positions of a clock), symmetry groups like the collection of ways of rotating a top , etc.
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Thu 8 Jun 2006ساعت 0:9 توسط Saeed Hojjati
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