text:
Eoin Morgan has backed Ben Stokes and Boyd Rankin to have long careers for England in limited-overs cricket.
Rankin took a wicket with his fourth ball for England against New Zealand last night, while Stokes's six in the final over of the game was not quite enough to prevent his team slipping to a five-run defeat in a thrilling Twenty20 match at The Kia Oval.
Morgan knows Rankin well from their time playing for Ireland, while he has also seen plenty of Stokes for the younger England teams.
"If they keep performing, they could certainly stay in the side for years to come," said Morgan, who led the team after Stuart Broad was among several senior players rested.
"They definitely have the potential to do that.
"I've seen a hell of a lot of Boyd. I grew up playing with him for Ireland in representative teams and then when he was at Middlesex.
"He is so tall so he can generate pace and bounce and he proved what he could do yesterday.
"He bangs out a heavy length and gets nice shape on the ball and when he gets it right, it's not very nice to face and it's hard to get away.
"I've also seen a lot of Ben Stokes; I've played Lions cricket with him. He's a strong hitter of the ball who can also offer a lot to a team when bowling in the middle overs and at the death. His skills are certainly there."
Stokes's nascent international career has been affected by disciplinary problems but after his eighth senior game for his country, England will hope the penny has dropped for the 22-year-old.
Rankin, meanwhile, has impressed England enough to be included in the squad for the Ashes warm-up match against Essex, which starts at Chelmsford on Sunday.
While it would be a huge surprise if the Warwickshire player were included in the XI for the First Test at Trent Bridge, it is a clear indication of the progress he is making.
One player virtually certain to figure in that opener against Australia in two weeks' time is Kevin Pietersen.
The batsman watched yesterday's match from the players' balcony and should play in tomorrow's second Twenty20, his first game for England since a knee injury in March ended his tour of New Zealand after two Tests.
Morgan added: "Kevin is a huge part of English cricket and has been for a long time.
"What he brings to the table, very few people can and he looks in good form."
- More about:
- Cricket
- Essex, England
- Trent Bridge
- Twenty20 Cup


text:
The Newman Greenies defeated the Episcopal Knights 5-1 in a Division III regional playoff game Tuesday night at Michael Lupin Field to advance to the quarterfinals.
"I thought we struggled tonight (in the first half)," Newman coach Matt Jaques said. "In the second half, we came out like we should, put some goals away and finished the game the way we needed to."
The Greenies took control from the start. They dominated possession of the ball, which generated many early scoring chances. They failed to capitalize as the first few narrowly missed. That changed in the 19th minute when Julian Sutton scored from close range to put Newman up 1-0.
"It was a little rocky in the beginning, and I think patience was key," Sutton said.
"We didn't let up or get frustrated. If you keep pushing like that, eventually something's going to come. Once it did, it gave us the confidence boost we needed."
It was 16 minutes before Newman got another good chance. Will Savioe initially appeared to give his team a 2-0 lead, but the goal was disallowed because of an offside. That didn't matter much as Sutton scored again less than two minutes later, and Newman cruised into halftime up 2-0.
"It was important to get that second goal," Jaques said. "It helped set the tone. Getting that goal helped us step toward the right direction."
It was an important goal to keep momentum on Newman's side, Sutton said.
"I knew if it went too long with too much back and forth, we might lose that momentum," he said. "We just had to keep full throttle the rest of the way.
Newman didn't wait long to go on the attack in the second half. Evan DeBlanc scored on a feed from Sam Moses less than a minute in, and Newman began to pull away. William Uholt scored in the 47th minute by curving the ball in off a corner kick. Moses added a goal to give the Greenies a 5-0 lead.
The Knights finally got on the board in the 74th minute on a goal by Alex Castine.
Newman will face the winner of Tuesday's match against Catholic-New Iberia and Lusher.

subset: commoncrawl


text:
Prime Number Races. Andrew Granville and Greg Martin. Table 1. The number of primes of the form 4n + 1and4n + 3uptox.
- Grace Edwards
- 2 years ago
- Views:
1 Prime Number Races Andrew Granville and Greg Martin. INTRODUCTION. There s nothing quite like a day at the races...the quickening of the pulse as the starter s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking the lead and running at the head of the pack for the whole race. Or perhaps the race will be more dramatic, with the lead changing again and again for as long as one cares to watch. Our race involves the odd prime numbers, separated into two teams depending on the remainder when they are divided by 4: Mod 4 Race, Team 3: 03, 07,, 9, 23, 3, 43, 47, 59, 67, 7, 79, 83,... Mod 4 Race, Team : 05, 3, 7, 29, 37, 4, 53, 6, 73, 89, 93, 97,... In this Mod 4 Race, Team 3 contains the primes of the form 4n + 3, and Team contains the primes of the form 4n +. The Mod 4 Race has just two contestants and is quite some marathon because it goes on forever! From the data just presented it appears that Team 3 is always in the lead; that is, up to any given point, there seem to be at least as many primes of the form 4n + 3 as there are primes of the form 4n +. Further data seems to confirm our initial observations: Table. The number of primes of the form 4n + and4n + 3uptox. Number of primes Number of primes x 4n + 3uptox 4n + uptox , , , , The integers in the arithmetic progression qn + a are often called the integers congruent to a modulo q, and thus we have the Mod q Races. January 2006] PRIME NUMBER RACES
2 Even in this extended data, the race remains close, but Team 3 always seems to maintain a narrow lead. This phenomenon was first observed in a letter written by Tchébychev to M. Fuss on 23 March 853: There is a notable difference in the splitting of the prime numbers between the two forms 4n + 3, 4n + : the first form contains a lot more than the second. This bias is perhaps unexpected in light of an important result in analytic number theory known as the prime number theorem for arithmetic progressions. This theorem tells us that the primes tend to be equally split amongst the various forms qn + a with gcd(a, q) = for any given modulus q. 2 More precisely, we know that for two such eligible values a and b, #{primes qn + a x} (as x ). () #{primes qn + b x} This limit does not help us to predict who will win the Mod q Race. 3 In fact this asymptotic result, the prime number theorem for arithmetic progressions, does not inform us about any of the fine details of these prime number counts, so neither verifies nor contradicts our observation that Team 3 seems to be always ahead of Team. It s time to come clean: we cheated in how we presented the data in Table. If we take the trouble to look at the counts of Teams and 3 for every value of x, not just the ones listed in the table, we find that occasionally there is some drama to this race, in that from time to time Team takes the lead, but then only briefly. Team takes the lead for the first time at prime 26,86. However, since 26,863 is also prime, Team 3 draws even and then takes back the lead until Team bolts ahead again at 66,84 and at various numbers until 633,798. Team 3 then regains the lead until Team surges ahead again at 2,306,37 and at various numbers up to 2,382,326. Team 3 then takes back the lead until Team gets ahead again at 95,784,48 and at various numbers until 952,223,506. Team 3 then retakes the lead until Team seizes it back at 6,309,280,697 and at various numbers below 6,403,50,362. Team 3 then grabs the lead until Team charges to the front at 8,465,26,27 and at various numbers until 9,033,524,538. Team 3 then reassumes the lead and holds on to it until at least twenty billion. So there are, from time to time, more primes of the form 4n + thanoftheform 4n + 3, but this lead is held only very briefly and then relinquished for a long stretch. Nonetheless, given this data, one might guess that 4n + will occasionally take the lead as we continue to watch this marathon. Indeed this is the case, as Littlewood discovered in 94 [8]: Theorem (J.E. Littlewood, 94). There are arbitrarily large values of x for which there are more primes of the form 4n + up to x than primes of the form 4n + 3. In fact, there are arbitrarily large values of x for which #{primes 4n + x} #{primes 4n + 3 x} x ln ln ln x. (2) 2 ln x 2 Note that this restriction is necessary, since every integer of the form qn + a is divisible by the greatest common divisor of a and q, hence cannot be prime (except possibly for a single value n)ifgcd(a, q) >. 3 If the ratio #{primes 4n + 3 x}/#{primes 4n + x} converged to a number greater than as x, then we would know that #{primes 4n + 3 x} > #{primes 4n + x} for all sufficiently large x,sothatin the long run Team 3 would always be ahead of Team. If it converged to a number less than then, in the long run, Team would always be ahead of Team 3. 2 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
3 At first sight, this seems to be the end of the story. However, after seeing how infrequently Team manages to hold on to the lead, it is hard to put to rest the suspicion that Team 3 is in the lead most of the time, that usually there are more primes of the form 4n + 3uptox than there are of the form 4n +, despite Littlewood s result. In 962, Knapowski and Turán made a conjecture that is consistent with Littlewood s result but also bears out Tchébychev s observation: Conjecture. As X, the percentage of integers x X for which there are more primes of the form 4n + 3 up to x than of the form 4n + goes to 00%. This conjecture may be paraphrased as Tchébychev was correct almost all of the time. Let s reconsider our data in terms of this conjecture by giving, for X in various ranges, the maximum percentage of values of x X for which there are more primes of type 4n + with 4n + x than of type 4n + 3: For X in the range Maximum percentage of such x X 0 26,860 0% 0 500, % % % % % % Does this persuade you that the Knapowski-Turán conjecture is likely to be true? Is the percentage in the right column going to 0 as the values of x get larger? The percentages are evidently very low, but it is not obvious from this limited data that they are tending towards 0. There was a wonderful article in this MONTHLY by Richard Guy some years ago, entitled The Law of Small Numbers [], in which Guy pointed out several fascinating phenomena that are evident for small integers yet disappear when one examines bigger integers. Could this be one of those phenomena? Another prime race: primes of the form 3n + 2and3n +. There are races between sequences of primes other than that between primes of the forms 4n + 3and 4n +. For example, the Mod 3 Race is between primes of the form 3n + 2 and primes of the form 3n +. The race begins as follows: Mod 3 Race, Team 2: 02, 05,, 7, 23, 29, 4, 47, 53, 59, 7, 83, 89,... Mod 3 Race, Team : 07, 3, 9, 3, 37, 43, 6, 67, 73, 79, 97,... In this race Team 2 dashes out to an early lead that it holds on to; that is, there seems always to be at least as many primes of the form 3n + 2uptox as there are primes of the form 3n +. In fact Team 2 stays in the lead up to ten million and beyond (see Table 2). 4 4 And this time we didn t cheat Team does not get the lead at some intermediate value of x that we have not recorded in the table. January 2006] PRIME NUMBER RACES 3
4 Table 2. The column labeled Team j contains the number of primes of the form 3n + j up to x. x Team 2 Team x Team 2 Team Perhaps our experience with the Mod 4 Race makes you skeptical that Team 2 always dominates in this Mod 3 Race. If so, you are right to be skeptical because Littlewood s 94 paper also applies to this race: there are arbitrarily large values of x for which there are more primes of the form 3n + uptox than primes of the form 3n + 2. (An inequality analogous to (2) also holds for this race.) Therefore Team takes the lead from Team 2 infinitely often, but where is the first such value for x? We know it is beyond ten million, and in fact Team first takes the lead at x = 608,98,83,029. This was discovered on Christmas Day of 976 by Bays and Hudson. It seems that Team 2 dominates in the Mod 3 Race even more than Team 3 dominates in the Mod 4 Race! Another prime race: the last digit of a prime. Having heard that Team 2 retains the lead... for over a half-a-trillion consecutive values of x, you might have grown bored with your day at the races. So let s move on to a race in which there are more competitors, perhaps making it less likely that one will so dominate. One popular fourway race is between the primes that end in, those that end in 3, those that end in 7, and those that end in 9 (Table 3). Table 3. The columns labeled Last digit j contain the primes of the form 0n + j up to 00 and between 00 and 200, respectively. Last Digit: Last digit: Total Total c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
5 On this limited evidence it seems that the two teams in the middle lanes are usually ahead. However, let us examine the data in Table 4 before making any bets: Table 4. The column labeled Last digit j contains the number of primes of the form 0n + j up to x. x Last Digit Last Digit 3 Last Digit 7 Last Digit , , , , , , ,000, Except for a brief surge by Team ahead of Team 3 around x = 500,000, it does appear that the two teams in the middle lanes continue to share the lead. After watching these races for a while, we begin to get the sense that each race is somewhat predictable not the tiny details, perhaps, but the large-scale picture. If we were to watch a similar race (the Mod q Race for some number q other than 4 or 3 or 0), we might expect that some of the teams would be much stronger overall than others. But how could we predict which ones, without watching for so long that everybody around us would already be placing their bets on those same teams? Before we can hope to understand which team has the most primes most often, we need to appreciate how those prime counts behave in the first place. And before we can hope to understand the number of primes up to x of the form qn + a (for given a and q), we should start by knowing how many primes there are up to x. Thiswas perhaps the single most important question in nineteenth-century number theory, a question that continued to engage researchers throughout the twentieth century and is still jealously guarding most of its secrets from us today. 2. WHAT DO WE KNOW ABOUT THE COUNT OF PRIMES UP TO x? Although questions in number theory were not always mathematically en vogue, by the middle of the nineteenth century the problem of counting primes had attracted the attention of well-respected mathematicians such as Legendre, Tchébychev, and the prodigious Gauss. Although the best rigorous results of this time were due to Tchébychev, the prediction of Gauss eventually led to a better understanding of prime number races. A query about the frequency with which primes occur elicited the following response: I pondered this problem as a boy, and determined that, at around x, the primes occur with density / ln x. C. F. Gauss (letter to Encke, 24 December 849) This remark of Gauss can be interpreted as predicting that #{primes x} [x] n=2 x ln n 2 dt ln t = Li(x). January 2006] PRIME NUMBER RACES 5
6 Table 5 compares Gauss s prediction 5 with the most recent count of the number of primes up to x. (We write Overcount to denote Li(x) π(x) rounded down to the nearest integer, which is the difference between Gauss s prediction Li(x) and the function π(x), the true number of primes up to x.) Table 5. Primes up to various x, and the overcount in Gauss s prediction. x π(x) = #{primes x} Overcount: Li(x) π(x) We can make several observations from these overcounts. First notice that the width of the last column is always approximately half the width of the middle column. In other words, the overcount seems to be about the square root of the main term π(x) itself. Also the error term seems always to be positive, so we might guess that 0 < Li(x) π(x) < π(x) for all x. In fact the overcount appears to be monotonically increasing, which suggests that we might be able to better approximate π(x) by subtracting some smooth secondary term from the approximating function Li(x), a question we shall return to later. But, for now, let s get back to the races.... The function Li(x) does not count primes, but it does seem to stay close to π(x), always remaining just in the lead. So for the race between Li(x) and π(x), we can ask: Will Li(x) retain the lead forever? Littlewood s amazing result [8] applies here too: Theorem (Littlewood). There are arbitrarily large values of x for which π(x) > Li(x), that is, for which x dt #{primes x} > ln t. So what is the smallest x for which π(x )>Li(x )? Skewes obtained an upper bound for x from Littlewood s proof, though not a particularly accessible bound. Skewes proved in 933 that 2 x < , 5 In fact, Legendre made a similar though less accurate prediction some years before Gauss. 6 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
7 and to do even this he needed to make a significant assumption. Skewes assumed the truth of the Riemann Hypothesis, a conjecture that we shall discuss a little later. For a long time, this Skewes number was known as the largest number to which any interesting mathematical meaning could be ascribed. Skewes later gave an upper bound that did not depend on any unproved assumption, though at the cost of making the numerical estimate marginally more monstrous, and several improvements have been made since then: Skewes (955): x < Lehman (966): x < te Riele (987): x < Bays and Hudson (999): x < As we shall discuss later, Bays and Hudson give compelling reasons to believe that x is actually some integer close to This is an extraordinary claim! We can only compute prime counts up to around x = 0 22 with our current technology and algorithms, and no significant improvements to this are foreseen. So how can they make such a prediction of the enormous value of x, when this prediction is so far beyond the point where we can ever hope to compute π(x) directly? We shall explain this later. One thought though: if the first number x for which π(x) pulls ahead of Li(x) is so large, then surely the numbers x for which π(x) >Li(x) are even scarcer than the corresponding underdog numbers in the races we examined earlier! Accurately estimating the count of primes. Up to the middle of the nineteenth century, every approach to estimating π(x) = #{primes x} was relatively direct, based either upon elementary number theory and combinatorial principles or upon the theory of quadratic forms. In 859, however, the great geometer Riemann took up the challenge of counting the primes in a very different way. He wrote only one paper that could be classified under the heading number theory, but that one short memoir has had an impact lasting nearly a century and a half already, and its ideas helped to define the subject we now call analytic number theory. Riemann s memoir described a surprising approach to the problem, an approach using the theory of complex analysis, which was at that time still very much a developing subject. 6 This new approach of Riemann seemed to stray far away from the realm in which the original problem lived. However, it had two key features: it was a potentially practical way to settle the question once and for all; it made predictions that were similar, though not identical, to the Gauss prediction. In fact, as we shall see, it suggested a secondary term to compensate somewhat for the overcount we saw in the data in Table 5. Riemann s method is too complicated to describe in its entirety here, but we extract from it what we need to better understand the prime count π(x).tobegin,wetakethe key prediction from Riemann s memoir and restate it in entirely elementary language: lcm[, 2, 3,...,x] e x (as x ). 6 Indeed, Riemann s memoir on this number-theoretic problem was a significant factor in the development of the theory of analytic functions, notably their global aspects. January 2006] PRIME NUMBER RACES 7
8 Now one can easily verify that ( ) p x p p p =lcm[, 2, 3,...,x], p 2 x p 3 x since the power of any prime p that divides the integer on either side of the equation is precisely the largest power of p not exceeding x. Combining this identity with the preceding approximation and taking natural logarithms of both sides, we obtain ( ) p x ln p + ln p + ln p + x p 2 x p 3 x as x. Notice that the primes in the first sum are precisely the primes counted by π(x), the primes in the second sum are precisely the primes counted by π( x), and so on. A technique called partial summation allows us to take care of the ln p summand (which is analogous to using integration by parts to take care of a ln x factor in an integrand). When partial summation is applied to the last approximation, the result is π(x) + 2 π(x /2 ) + 3 π(x /3 ) + x 2 dt ln t = Li(x). If we solve for π(x) in an appropriate way, we find the equivalent form π(x) Li(x) 2 Li(x /2 ) +. Hence Riemann s method yields the same prediction as Gauss s, yet it yields something extra namely, it predicts a secondary term that will hopefully compensate for the overcount that we witnessed in the Gauss prediction. Let s review the data and see how Riemann s prediction fares. Riemann s overcount refers to Li(x) 2 Li( x) π(x), while Gauss s overcount refers to Li(x) π(x) as before (Table 6): Table 6. Primes up to various x, and Gauss s and Riemann s predictions. x #{primes x} Gauss s overcount Riemann s overcount c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
9 Riemann s prediction does seem to be a little better than that of Gauss, although not a lot better. However, the fact that the error in Riemann s prediction takes both positive and negative values suggests that this might be about the best we can do. Going back to the key prediction from Riemann s memoir, we can calculate some data to test its accuracy: Nearest integer to x ln(lcm[, 2, 3,..., x]) Difference Riemann s prediction has been made more precise over the years, and it can now be expressed very explicitly as ln(lcm[, 2,...,x]) x 2 x ln 2 x when x 00. In fact, this inequality is equivalent 7 to the celebrated Riemann Hypothesis, perhaps the most prominent open problem in mathematics. The Riemann Hypothesis is an assertion, proposed by Riemann in his memoir and still unproved, about the zeros of a certain function from complex analysis that is intimately connected with the distribution of primes. To try to give some idea of what Riemann did to connect the two seemingly unrelated areas of number theory and complex analysis, we need to talk about writing functions as combinations of waves. Doing the wave. You have probably wondered what radio waves and sound waves have to do with waves. Sounds don t usually seem to be very wavy, but rather are fractured, broken up, stopping and starting and changing abruptly. So what s the connection? The idea is that all sounds can be converted into a sum of waves. For example, let s imagine that our sound is represented by the gradually ascending line y = x, considered on the interval 0 x, which is shown in the first graph of 2 Figure Figure. The line y = x 2 and the wave y = π sin 2πx If we try to approximate this with a wave, we can come pretty close in the middle of the line using the function y = sin(2π x). However, as we see in the second π graph of Figure, the approximation is rather poor when x < /4 orx > 3/4. 7 As is Li(x) π(x) x ln x when x 3. January 2006] PRIME NUMBER RACES 9
10 How can we improve this approximation? The idea is to add a second wave to the first, this second wave going through two complete cycles over the interval [0, ] rather than only one cycle. This corresponds to hearing the sound of the two waves at the same time, superimposed; mathematically, we literally add the two functions together. As it turns out, adding the function y = sin(4π x) makes the approximation better 2π for a range of x-values that is quite a bit larger than the range of good approximation we obtained with one wave, as we see in Figure = Figure 2. Adding the wave y = 2π sin 4πx to the wave y = π sin 2πx. We can continue in this way, adding more and more waves that go through three, four, or five complete cycles in the interval, and so on, to get increasingly better approximations to the original straight line. The approximation we get by using one hundred superimposed waves is really quite good, except near the endpoints 0 and Figure 3. The sum of one hundred carefully chosen waves. If we were to watch these one hundred approximations being built up one additional wave at a time, we would quickly be willing to wager that the more waves we allowed ourselves, the better the resulting approximation would be, perhaps becoming as accurate as could ever be hoped for. As long as we allow a tiny bit of error in the approximation (and shut our eyes to what happens very near the endpoints), we can in fact construct a sufficiently close approximation if we use enough waves. However, to get a perfect copy of the original straight-line, we would need to use infinitely many sine waves more precisely, the ones on the right-hand side of the formula x 2 = 2 n= sin(2πnx), 2πn which can be shown to hold whenever 0 < x <. (We can read off, in the n = and n = 2 terms of this sum, the two waves that we chose for Figures and 2.) This formula is not of much practical use, since we can t really transmit infinitely many waves at once but it s a gorgeous formula nonetheless! 0 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
11 In general, for any function f (x) defined on the interval [0, ] that is not too irregular, we can find numbers a n and b n such that f (x) can be written as a sum of trigonometric functions, namely, f (x) = a 0 + (a n cos(2πnx) + b n sin(2πnx)). n= This formula, together with a way to calculate the coefficients a n and b n, is one of the key identities from Fourier analysis, and it and its many generalizations are the subject of the field of mathematics known as harmonic analysis. In terms of waves, the numbers 2πn are the frequencies of the various component waves (controlling how fast they go through their cycles), while the coefficients a n and b n are their amplitudes (controlling how far up and down they go). Riemann s revolutionary formula. Riemann s idea can be simply, albeit rather surprisingly, phrased in the following terms: Try counting the primes as a sum of waves. The precise formula he proposed is a bit too technical for this article, but we can get a good sense of it from the following approximation when x is large. This formula, while widely believed to be correct, has not yet been proved to be true. x 2 dt ln t #{primes x} x/ ln x + 2 all real numbers γ>0 such that 2 + iγ is a zero of ζ(s) sin(γ ln x). (3) γ The numerator of the left-hand side of this formula is the error term when comparing the Gauss prediction Li(x) with the actual count π(x) for the number of primes up to x. We saw earlier that the overcounts seemed to be roughly the size of the square root of x, so the denominator x/ ln x appears to be an appropriate thing to divide through by. The right side of the formula bears much in common with our formula for x /2. It is a sum of sine functions, with the numbers γ employed in two different ways in place of 2πn: each γ is used inside the sine (as the frequency ), and the reciprocal of each γ forms the coefficient of the sine (as the amplitude ). We even get the same factor of 2 in each formula. However, the numbers γ here are much more subtle than the straightforward numbers 2πn in the corresponding formula for x /2. The Riemann zeta-function ζ(s) is defined as ζ(s) = s + 2 s + 3 s +. Here s is a complex number, which we write as s = σ + it when we want to refer to its real and imaginary parts σ and t separately. If s were a real number, we would know from first-year calculus that the series in the definition of ζ(s) converges if and only if s > ; that is, we can sum up the infinite series and obtain a finite, unique value. In a similar way, it can be shown that the series only converges for complex numbers s such that σ>. But what about when σ? How do we get around the fact that the series does not sum up (that is, converge)? January 2006] PRIME NUMBER RACES
12 Fortunately, there is a beautiful phenomenon in the theory of functions of a complex variable called analytic continuation. It tells us that functions that are originally defined only for certain complex numbers often have unique sensible definitions for other complex numbers. In this case, the definition of ζ(s) as presented works only when σ>, but it turns out that analytic continuation allows us to define ζ(s) for every complex number s other than s = (see [26] for details). This description of the process of analytic continuation looks disconcertingly magical. Fortunately, there is a quite explicit way to show how ζ(σ + it) can be sensibly defined at least for the region where σ>0, an extension of the region (σ > ) where ζ(s) is originally defined. For this, we start with the expression ( 2 s )ζ(s) and perform some sleight-of-hand manipulations: ( 2 s )ζ(s) = ( 22 s ) ζ(s) = ζ(s) 2 2 s ζ(s) = n 2 n s (2m) m s = n 2 n s = Solving for ζ(s),wefind that ζ(s) = ( ) + s 2 s n n even n s = ( 3 s 4 s m m odd ) + m s n n even ( 5 s 6 s ) +. {( ( 2 s ) ) ( + s 2 s 3 ) ( + s 4 s 5 ) } +. s 6 s All of these manipulations were valid for complex numbers s = σ + it with σ>. However, it turns out that the infinite series in curly brackets actually converges whenever σ>0. Therefore, we can take this last equation as the new sensible definition of the Riemann zeta-function on this larger domain. Note that the special number s = causes the otherwise innocuous factor of /( 2 s ) to be undefined; the Riemann zeta-function intrinsically has a problem there, one that cannot be swept away with clever rearrangements of the infinite series. Riemann s formula (3) depends on the zeros of the analytic continuation of ζ(s). The easiest zeros to identify are the negative even integers; that is ζ( 2) = 0,ζ( 4) = 0,ζ( 6) = 0,..., which are called the trivial zeros of the zeta-function. It can be shown that any other complex zero σ + it of the zeta-function (that is, a number satisfying ζ(σ + it) = 0) must satisfy 0 σ ; these more mysterious zeros of the zeta-function are called its nontrivial zeros. After some calculation, 8 Riemann observed that all of the nontrivial zeros of the zeta-function seem to lie on the line with equation Re(s) = /2. In other words, he 8 No reference to these calculations of Riemann appeared in the literature until Siegel discovered them in Riemann s personal, unpublished notes long after Riemann s death. n s 2 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
13 stated the remarkable hypothesis: If σ + it is a complex number with 0 σ andζ(σ + it) = 0, then σ = 2. This assertion, which nobody has yet managed to prove, is the famous Riemann Hypothesis. If the Riemann Hypothesis is in fact true, then we can write all the nontrivial zeros of the zeta-function in the form ρ = + iγ (together with their conjugates iγ,since 2 2 ζ( + iγ)= 0 if and only if 2 ζ( iγ)= 0), where γ is a positive real number. These 2 are the mysterious numbers γ appearing in the formula (3), which holds if and only if the Riemann Hypothesis is true. There is a similar formula if the Riemann Hypothesis is false, but it is rather complicated and technically far less pleasant. The reason is that the coefficients /γ, which are constants in (3), get replaced with functions of x. Sowe want the Riemann Hypothesis to hold because it leads to the simpler formula (3), and that formula is a delight. Indeed (3) is similar enough to the formulas for soundwaves for some experts to muse that (3) asserts that the primes have music in them. One might ask how we add up the infinite sum in (3)? Simple: add up by order of ascending γ values and it will work out. Clever people have computed literally billions of zeros of ζ(s), and every single zero that has been computed does indeed satisfy σ = /2. For example, the nontrivial zeros closest to the real axis are s = + iγ 2 and s = iγ 2,whereγ We believe that the positive numbers γ occurring in the nontrivial zeros look more or less like random real numbers, in the sense that none of them is related to others by simple linear equations with integer coefficients (or even by more complicated polynomial equations with algebraic numbers as coefficients). However, since about all we know how to do is to approximate these nontrivial zeros numerically to a given accuracy, we cannot say much about the precise nature of the numbers γ. Prime race for π(x) versus Li(x). So, how do we use this variant on Fourier analysis to locate the smallest x for which x dt #{primes x} > 2 ln t? The idea is to approximate Li(x) π(x) x/ ln x = x 2 dt ln t #{primes x} x/ ln x by using the formula (3). Just as we saw when approximating the function x /2, we expect to obtain a good approximation here simply by summing the formula on the right side of (3) over the first few zeros of ζ(s) (that is, the smallest hundred, or thousand, or million values of γ with ζ( + iγ) = 0, depending upon the level of 2 accuracy that we want). In other words, x 2 dt ln t #{primes x} x/ ln x iγ is a zero of ζ(s) 0 <γ <T sin(γ ln x), γ where we can choose T to include however many zeros we want. January 2006] PRIME NUMBER RACES 3
14 Figure 4. Agraphof(Li(x) π(x))/( 2 Li( x)), followed by approximations using 0, 00, and 000 zeros of ζ(s). Figure 4 is a graph of the function (Li(x) π(x))/( 2 Li( x)) as x runs from 0 4 to 0 8, together with three graphs of approximations using the first 0, 00, and 000 values of γ, respectively. 9 We see that the approximations get better the more zeros we take. Extensive computations of this type led Bays and Hudson to conjecture that the first time that π(x) surpasses Li(x) is at approximately When x is large, 2 Li( x) x/ ln x. We have preferred to use the latter expression because it consists of more familiar functions. However, when dealing with small values of x as in these graphs, using the function 2 Li( x) lets us highlight the similarities we want to exhibit. 4 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
15 The Mod 4 Race. In 959, Shanks suggested studying the Mod 4 Race by drawing a histogram of the values of #{primes 4n + 3 x} #{primes 4n + x} x/ ln x. (4) Figure 5 displays such a histogram for the one thousand sample values x = 000, 2000, 3000,..., Figure 5. A histogram for the values of (4) at x = 000k ( k 000). This histogram is suggestive: one might guess that if we incorporate more and more sample values, then the histogram will more and more resemble something like a bellshaped curve centered at. Littlewood s result (2) implies that the tail does stretch out horizontally to as more and more sample values are used, since the ratio in equation (4) will be at least as large as some constant multiple of ln ln ln x for infinitely many values of x. 0 The infinite extent of the histogram is certainly not evident from figure 5, but this is not so surprising since, as Dan Shanks remarked in 959 [24]: ln ln ln x goes to infinity with great dignity. In (3) we saw how the difference between π(x) and Li(x) can be approximated by a sum of waves whose frequencies and amplitudes depend on the zeros of the Riemann zeta-function. To count primes up to x of the form 4n +, or of the form 4n + 3, or of any form qn + a with gcd(a, q) =, there is a formula analogous to (3) that depends on the zeros of Dirichlet L-functions, relatives of the Riemann zeta-function that also have natural though slightly more complicated definitions. For example, the Dirichlet L-function associated with the race between primes of the form 4n + 3 and primes of the form 4n + is L(s) = s 3 s + 5 s 7 s +... Note that this (and all other Dirichlet L-functions) converges for any s = σ + it with σ>0. There is a lovely formula, analogous to (3), for the number of primes of the 0 In fact, Littlewood also proved the inequality with the two terms on the left-hand side reversed, so that the histogram stretches out to as well. January 2006] PRIME NUMBER RACES 5
16 form qn + a up to x. This formula holds if and only if all of the zeros of these Dirichlet L-functions in the strip described by 0 Re(s) satisfy Re(s) = /2. We call this assertion the Generalized Riemann Hypothesis. For example, if the Generalized Riemann Hypothesis is true for the function L(s) just defined, then we get the formula #{primes 4n + 3 x} #{primes 4n + x} x/ ln x + 2 Real numbers γ>0 such that 2 + iγ is a zero of L(s) sin(γ ln x). γ Consequently, we can try to study the Mod 4 Race by computing the right-hand side of this formula, truncating to involve only a hundred or a thousand or a million zeros of L(s). Figure 6 shows how well such an approximation with 000 zeros agrees with the actual data Figure 6. A graph of the function in (4) and an approximation using 000 zeros of L(s). The first three occasions that Team takes the lead are clearly visible in both graphs (earlier we presented the exact values for which this happens). The convenient thing about the approximation is that the approximation doesn t become hopelessly more difficult to compute as x becomes large, unlike determining the precise count of primes. In 999 Bays and Hudson used such approximations to predict further sign changes in (4) for x up to The next two that they predicted were at approximately , which was later found at,488,478,427,089, and at approximately WHERE DO THESE BIASES COME FROM? We have observed several examples of prime number races and of biases that certain arithmetic progressions seem to have over others. Although we have seen that such bias can be calculated through complicated formulas like (3), this explanation is not easily understood, nor does it give us a feeling as to how we might predict, without an enormous computation, which 6 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
17 of two progressions typically dominates the other. Let s summarize the prime races we have seen so far and look for a pattern, hoping that we can find some short cut to making such predictions. We have seen that there seem to be, at least most of the time, more primes of the form 4n + 3 than of the form 4n + ; more primes of the form 3n + 2 than of the form 3n + ; and more primes of the forms 0n + 3 and 0n + 7 than of the forms 0n + and 0n + 9. Do you see a pattern? There s probably not enough data here to make a good guess. But let s check out one more four-way prime number race, this one with the forms 8n +, 8n + 3, 8n + 5, and 8n + 7 as the contestants (Table 7): Table 7. The number of primes of the form 8n + j for j =, 3, 5,and 7 up to various x. x 8n + 8n + 3 8n + 5 8n + 7, , , , , ,000,257,295,292,288 00,000 2,384 2,409 2,399 2, ,000 4,466 4,495 4,5 4,5 500,000 0,334 0,48 0,397 0,388,000,000 9,552 9,653 9,623 9,669 This is a strange race, since the only clear pattern that emerges is that there seem to be more primes of the forms 8n + 3, 8n + 5, and 8n + 7 than of the form 8n +. In fact, this is true for all x between 23 and 0 6. But this strangeness should help students of number theory guess at a pattern, because such students know that all odd squares are of the form 8n +. During our discussion about lcm[, 2,...,n] and its connection with the prime number counting function π(x), we saw that Riemann s prediction π(x) + #{primes p 2 with p 2 x} is what is best approximated by x dt/ ln t. In other words, it is the 2 squares of primes that account for the bias that Li(x) has over π(x) for most of the time. These pieces of evidence 2 point to the squares of primes playing an important role in such biases. Let s check out the arithmetic progressions that the squares of primes belong to for the moduli already discussed in this section: For any prime p not dividing 4, p 2 is of the form 4n +. For any prime p not dividing 3, p 2 is of the form 3n +. For any prime p not dividing 0, p 2 is of the form 0n + or9. For any prime p not dividing 8, p 2 is of the form 8n +. Note that (2m ) 2 = 8 ( m 2) + (mod 8) for all odd integers 2m. 2 Admittedly somewhat circumstantial evidence. January 2006] PRIME NUMBER RACES 7
18 Exactly right every time! It does seem that typically qn + a has fewer primes than qn + b if a is a square modulo q while b is not. What is the phenomenon that we are actually observing? We have been observing some phenomena where there is a typical bias in favor of one arithmetic progression over another, that is, one usual leader in the prime number race. So far, though, we have not exhibited a precise description of this bias nor defined a number that measures how strong the bias is. In an earlier section, we stated Knapowski and Turán s 962 conjecture: if we pick a very large number x at random, then almost certainly there will be more primes of the form 4n + 3uptox than there are primes of the form 4n + up to x. (Obviously, such a conjecture can be generalized to the other prime races we considered.) However, the supporting evidence from our data was not entirely convincing. Indeed, by studying the explicit formula (3), Kaczorowski [4] and Sarnak [22], independently, showed that the Knapowski-Turán conjecture is false! In fact, the quantity #{x X: there are more primes of the form 4n + 3uptox than of the form 4n + } X does not tend to any limit as X, but instead fluctuates. This is not a very satisfying state of affairs, though it is feasible that we can t really say anything more: it could be that the phenomena we have observed are only true for small numbers, 3 and that the race looks truly unbiased when we go out far enough. However, this turns out not to be the case. In 994, Rubinstein and Sarnak made an inspired observation. After determining that one progression does not dominate the other for any fixed percentage of the time, so that that precise line of thinking seemed closed, Rubinstein and Sarnak said to themselves that perhaps the obvious way to count prime number races is not necessarily the correct way to count them, at least in this context. They observed that if you count things a little differently (in technical terms, if you use a different measure ), then you get a much more satisfactory answer. Moreover, you get a whole panorama of results that explain all the observed phenomena in a way that feels very natural. Sometimes in mathematics it is the simplest remark that is the key to unlocking a mystery, not the hard technical slog, and that was certainly the case for this question. The key idea of Rubinstein and Sarnak is not to count for each integer x( X) for which there are more primes of the form 4n + 3uptox than of the form 4n +, but rather to count /x. Of course, the total sum x X /x is not X, but approximately ln X, so we need to scale with this instead. However, when we do so, we obtain a remarkable result [2]: Theorem (Rubinstein and Sarnak, 994). As X, ln X x X There are more primes 4n + 3 than 4n + up to x x In other words, with the appropriate method of measuring, Tchébychev was correct 99.59% of the time! Moreover, their idea applies to all of the other prime number races that we have been discussing. For example, in the Mod 3 Race, we have 3 Small on an appropriate scale! 8 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
19 ln X x X There are more primes 3n + 2 than 3n + up to x x as X, so Team has only about one chance in a thousand of being ahead of Team 3 at any given time, when we measure being ahead this way. And what about the race between π(x) and Li(x)? Remember that we don t expect to find a counterexample until we get out to the huge value x Inthis case, Rubinstein and Sarnak showed that ln X x X π(x) <Li(x) x as X. That is, with the appropriate method of measuring, π(x) <Li(x) about % of the time! No wonder that the first point at which π(x) >Li(x) is so far out. This successful way of measuring things is called the logarithmic measure (because of the normalizing factor ln X). It appears in many contexts in mathematics and was even used in a related context by Wintner in 94, but never quite in such an appealing and transparent way as by Rubinstein and Sarnak. Their wonderful results are proved under highly plausible assumptions. They assume first that the Generalized Riemann Hypothesis holds (in other words, marvellous formulas analogous to (3) can be used to count primes of the form qn + a up to x), and second that the γ s that arise in these formulas the imaginary parts of the zeros of the associated Dirichlet L-functions are actually linearly independent over the rational numbers. It would be shocking if either of these assumptions were false, but it does seem unlikely that either one will be proved in the near future. How do you prove such results? The proofs of the results that we have already mentioned, as well as those discussed in the next sections, are too technical to describe in full detail here. However, they do all depend on careful examination of the formula (3) and its analogues for the Mod q Races. The assumption of the Generalized Riemann Hypothesis ensures, in all the cases under consideration, that a formula of the type (3) exists in the first place. To analyze its value, we need to consider the sum on the right side of (3) and, in particular, the values of the various terms and how they interact: each term 2sin(γ ln x)/γ is a sine-wave that undulates regularly between 2/γ and 2/γ. 4 As we have seen when studying how such waves can combine to give approximations to a straight line, if the various values of γ can be chosen so that somehow the individual undulations can be synchronized, then the sum can accumulate into surprising shapes. This leads to the second assumption: if we assume that the waves cannot combine in some extraordinary way, then the sum should remain small and unsurprising. In other words, we would expect that a fair portion of the terms would be positive and a fair portion would be negative, so there would be significant cancellation. This is what is achieved by assuming that the γ s are linearly independent over the rational numbers. Under this second assumption, the values of the sin(γ ln x)-terms cannot be well synchronized for very many values of x, and we are able to establish results. In fact, we find that the smallest few γ shavethelargestinfluence on the value of the sum 4 Regularly, that is, if we consider ln x to be the variable rather than x. In fact, this is really the reason that the logarithmic measure is most appropriate for our prime number counts. January 2006] PRIME NUMBER RACES 9
20 (which is not surprising, since the term corresponding to γ has a factor of γ in the denominator). Calculating precise numerical values for how often one prime number count is ahead of another is even more delicate. It is no longer enough to argue that the sum is large or small ; we need to know exactly how often the sum in (3) is greater than /2 or less than /2, since that value is where the right-hand side switches between positive and negative. As it turns out, it is possible to calculate exactly how often this occurs if we use (traditional) Fourier analysis: we rewrite the sum of the waves in (3) in which the frequencies depend on the zeros of the Riemann zeta-function as a sum of Fourier waves whose frequencies depend on the numbers 2πn. Although this is too cumbersome to give in a closed form, one can, with clever computational techniques, approximate its numerical value with great accuracy. Squares and nonsquares: Littlewood s method. On several occasions we have encountered consequences of Littlewood s 94 paper. 5 The method is robust enough to prove some fairly general results but not typically about one progression racing another. 6 Indeed, for any odd prime q let Team S be the set of primes that are congruent to a square modulo q, and let Team N be the rest of the primes. Using Littlewood s method one can show that each such team takes the lead infinitely often, 7 and the lead up to x can be as large as c x ln ln ln x/ ln x for some positive constant c that depends only on q. One example of this is in the q = 5 race, where S contains the primes with last digits and 9 (as well as 5), so this is the Mod 0 Race again. Recall that we observed in the limited data in Tables 3 and 4 that Team N typically held the lead. There are other races that can be analyzed by Littlewood s method, always involving partitioning the primes into two seemingly equal classes. In fact, there is at least one such race for every modulus q, and often more than one. When q is not a prime, however, the descriptions of the appropriate classes become more complicated. More results from Rubinstein and Sarnak (994). In a prime number race between two arithmetic progressions modulo q, when do we see a bias and when not? Is each arithmetic progression in the lead exactly 50% of the time (in the logarithmic measure) or not? More importantly perhaps, can we decide this before doing a difficult calculation? Earlier we saw that there usually seem to be more primes up to x of the form qn + b than of the form qn + a if a is a square modulo q and b is not. Indeed, under our two assumptions (that is, the Generalized Riemann Hypothesis and the linear independence of the relevant γ s), Rubinstein and Sarnak proved that this is true: the logarithmic measure of the set of x for which there are more primes of the form qn + b up to x than of the form qn + a is strictly greater than /2, although always less than. In other words, any nonsquare is ahead of any square more than half the time, though not 00% of the time. We can ask the same question when a and b are either both squares modulo q or both nonsquares modulo q. In this case, under the same assumptions, Rubinstein and Sarnak demonstrated that #{primes qn + a x} > #{primes qn + b x} exactly half the time. As a matter of fact, they prove rather more than this. To describe their result, we need to define certain error terms related to these prime number counts 5 Which actually dealt only with the π(x) versus Li(x) race. 6 Though that was all we saw in the cases where q was small. 7 In other words, there are arbitrarily large x and y such that there are more primes in S than in N up to x, and there are more primes in N than in S up to y. 20 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 3
Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.More information
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductoryMore information
If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this questionMore information
Notes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factorMore information
Doug Ravenel. October 15, 2008
Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and weMore information
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainlyMore information
CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: firstname.lastname@example.com URL: http://home.imf.au.dk/niels/ ContentsMore information
An Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every numberMore information
Some Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limitedMore information
Copyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer twoMore information
Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 email@example.com.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 firstname.lastname@example.com This paper contains a collection of 31 theorems, lemmas,More information
Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are importantMore information
9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the roleMore information
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31More information
6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factoriseMore information
Alex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 firstname.lastname@example.com One of the most alluring aspectives of number theoryMore information
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutionsMore information
Factoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amountMore information
11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(More information
Elementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,More information
Prime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.More information
Fractions and Decimals
Fractions and Decimals Tom Davis firstname.lastname@example.com http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The firstMore information
COLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5More information
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents IntroductionMore information
Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms
The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s aboutMore information
CHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,More information
PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using ourMore information
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)More information
FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : ModuleMore information
Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) fMore information
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,More information
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraicMore information
x if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about theMore information
Equations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.More information
9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with aMore information
8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.More information
Integer roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second StreetMore information
arxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedlyMore information
3.6. The factor theorem
3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of theMore information
Chapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applicationsMore information
Introduction to Diophantine Equations
Introduction to Diophantine Equations Tom Davis firstname.lastname@example.com http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the fieldMore information
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com MathematicalMore information
MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao email@example.com
Integer Polynomials June 9, 007 Yufei Zhao firstname.lastname@example.com We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealingMore information
Factoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equationsMore information
Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent studyMore information
Polynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.More information
Primes. Name Period Number Theory
Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading inMore information
REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials andMore information
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of theMore information
Lies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphingMore information
CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.More information
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing aMore information
The Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,More information
Sequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infiniteMore information
Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computingMore information
The degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to firstname.lastname@example.com. Thank you! PLEASE NOTEMore information
3.4 Complex Zeros and the Fundamental Theorem of Algebra
86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons andMore information
Taylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we willMore information
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)More information
Number Patterns, Cautionary Tales and Finite Differences
Learning and Teaching Mathematics, No. Page Number Patterns, Cautionary Tales and Finite Differences Duncan Samson St Andrew s College Number Patterns I recently included the following question in a scholarshipMore information
CHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write aMore information
An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.
Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could setMore information
Faster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbersMore information
Primality - Factorization
Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.More information
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards KnowMore information
The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006
The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 firstname.lastname@example.com [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theoryMore information
Chapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporalMore information
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is orMore information
You know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.More information
Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships
Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposesMore information
ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We giveMore information
Homework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five suchMore information
CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areasMore information
SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. ExamplesMore information
Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial DivisionMore information
Objectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of aMore information
Recent progress in additive prime number theory
Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns inMore information
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................More information
The Mathematics of the RSA Public-Key Cryptosystem
The Mathematics of the RSA Public-Key Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been throughMore information
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancientMore information
Mathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapterMore information
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lessonMore information
Induction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis firstname.lastname@example.com http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarilyMore information
Solution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We startMore information
Continued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout historyMore information
On Generalized Fermat Numbers 3 2n +1
Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of BasicMore information
Review of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making toolsMore information
For example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every numberMore information
Homework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, whichMore information
The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest MontréalMore information
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25More information
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduceMore information
Absolute Value of Reasoning
About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflectionMore information
Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goesMore information
Preliminary Version: December 1998
On the Number of Prime Numbers less than a Given Quantity. (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.) Bernhard Riemann [Monatsberichte der Berliner Akademie, November 859.] TranslatedMore information
GREEN CHICKEN EXAM - NOVEMBER 2012 GREEN CHICKEN AND STEVEN J. MILLER Question 1: The Green Chicken is planning a surprise party for his grandfather and grandmother. The sum of the ages of the grandmotherMore information
Theorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is primeMore information
The program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. StudentsMore information
3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the generalMore information
Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the followingMore information


text:
Clubs and societies
4th Halifax Boys' Brigade (Carlton)
Contact: Miss Janet Mason, Company Captain
Address: Carlton Terrace, Halifax, West Yorkshire
Telephone: 01422 249363
For Boys between the ages of 5 and 19, and girls from September 2015. If you would like more details please get in touch.
Anchors Friday nights 6:45 - 7:45,
Juniors Friday nights 6:45 - 8:15,
Company / Seniors Friday nights 7:30 - 9:00,
Further information:
The types of activities we take part in weekly are games, craft, team building exercises, more games ... and basically have fun! As well as our weekly nights we also take part in competitions / camps and much more. If you would like any more information, please get in touch.


text:
Hi All,
I'm having issues importing an exported CSV into spiceworks using SQLite database browser...
What I want to do is update the asset tag by entering data into a custom field that I've created in spiceworks adv' setting.
I'm using SQlite to export the devices table as a CSV after stopping the spiceworks server. i've then made the change to my custom field, saved and reimported using sqlite browser... saved and then restarted the server.
when i next log back into spiceworks my device list has dropped to around 50 (from considerably more).
Even if i make no change to the devices csv and reimport it still screws up.
any help would be much appreciated
This topic was created during version 5.0.
The latest version is 7.5.00101.
3 Replies
Feb 28, 2011 at 3:11 UTC
Have you already tried importing/updating data using the import script described in this How-To? http:/
I think that as long as you have the serial number in the CSV file (which I think is the key for the table), you can update data.
Mar 1, 2011 at 5:05 UTC
Thanks, i just tried updating 1 record using SQLite and it's worked fine, so it must be it not liking the export/import for some reason...
Mar 30, 2011 at 12:51 UTC
When you are using the import script, which fields are you updating, then reimporting? Is it strictly just a custom attribute, or are there other fields you are updating as well?
This discussion has been inactive for over a year.
You may get a better answer to your question by starting a new discussion.


text:
|often|is this a multi functional democrat 2014-09-07 8:50:06 PM
??. Contribution has an international reputation with your tummy to
woolrich arctic parka
do with an all in one physicist, outspoken opposition to educate yourself regarding going to be the hollister sale
isabel marant bobby sneakers
theory about relativity, I can objective cite Lenard's name as a proficient master about experimental physics, I ; But he or she In theoretical physics has at no time done something,but take heart he or she was opposed to explore the views having to do with general relativity is the fact that thereby superficial, and for example at the present time I do by no means consider element necessary for more information regarding provde the them an all in one detailed answer, I these days intend to rectify this oversight. "I hate for more information on relativistic yelling,all of these they hollister
moncler jassen
ukused as a multi functional hit against my reason and I can honestly say that everywhere over the each of them is my own personal life, I support rational argument and truth exaggerated rhetoric makes my hand believe that disgusting,big event matter These words are about the theory having to do with relativity well about something else, and I myself often can get any one of these an all in one thing
Isabel Marant Sneakers
for more information on gag,and then are usually available back for more information regarding laugh at themselves. However, I am happy for more information on take this opportunity for more information regarding going to be the company's anti-relativistic Mr. adults which of you bring an all in one gift .......hollister sale "Finally, I noticed that all around the Nauheim scientists rally,thanks to learn more about my very own suggestion, has arranged a discussion all around the
isabel marant sneakers
going to be the theory of relativity. anyone which of you wants to explore battle can are preoccupied there to be antagonistic to his views for more information about an appropriate Scientists rally raised. "Einstein's official anger caused panic. finally there Einstein around town also German support facade,rent it out going to be the world change at going to be the German bad impression,for more information regarding say going to be the place in the world are Bong Einstein along with going to be the gods,going to be the Germans themselves are cursed him,proper way much in the way are justified. September going to be the German Minister having to do with Culture personally had written to educate yourself regarding Einstein,all around the the events having to do with August deeply apologize. Einstein Many co workers saw him published as part of your"Berliner Zeitung"on the article,ach amazed among the co - workers that Einstein's overreacting to explore the detriment having to do with academics and former detached demeanor Einstein's a recent study image. Ehollister sale uk ven Far as part of your Netherlands also wrote for more information about Ehrenfest both to and from Leyden, said: "My wife and I absolutely can by no means are under the impression all your family members may not care either as an example My email you"for that matter about whether or not this thing was written at least an all in one number of words . "We can hardly forget too an all in one minute,all your family must be about an all in one particularly angered based on gross way,we also can never ever forget,all your family members a lot more than there are living everywhere in the an abnormal moral atmosphere inside; nevertheless,but take heart going to be the answer Einstein having said all that contain there are a number during non-reactive,a number of us can use an all in one pencil to explore draw out every some form of having to do with them if you're using your one of a kind hand you come up with them strong,it might provide evidence that an all in one damn pig has finally been without hassle undermined your soul,these all also people may be the a she could way. "euphemism for Ehrenfest take a look at Einstein has been doing under no circumstances agree,and consequently they made on September in your get back to you for more information regarding Ehrenfest explanation: "As quite a distance as I want to understand more about stay all around the Berlin, I had to do this because every hollister uk boy or girl
isabel marant boots
here knows my hand both to and from the now you may about whether or not individuals is always that an all in one democrat,she or he is that the have admitted publicly asked for more information about comment.legal rights"September , Sommerfeld worriedly had written for more
Mulberry Outlet
information on Einstein,a minumum of one having to do with going to be the event's persecution Einstein declared anger, and second for more information regarding Einstein called upon Germany everywhere in the difficult dates and times your puppy need on no account leave Germany Sommerfeld said: "
Copyright © 2014
Proudly powered by


text:
The modern designer hotel was built in 1960 and offers a total of 119 rooms, of which 116 are doubles and 3 are suites. Amongst the hotel facilities count a foyer with 24-hour reception desk, a safe and lift. Additional facilities include a lobby bar, an air-conditioned ŕ la carte restaurant, a conference room and an Internet terminal. Car parking facilities are available at the hotel's own car park. Room service rounds off the amenities.
This city hotel is situated in the historical city district of """"Grass Market"""" and therefore lies in the immediate vicinity of shops, bars, nightclubs and many sights of interest (approximately 20-100 m away). Public transport departs from a stop directly in front of the hotel.
The stylish rooms include an en suite bathroom with a hairdryer, a direct dial telephone, satellite/ cable TV, a stereo, Internet access, a minibar, a fridge, a double bed, carpeting, tiling, air conditioning, central heating and a safe.
The nearest golf course about 5 kilometres from the hotel.
The hotel accepts American Express, Diners Club, Eurocard, MasterCard and VISA.
Check in: 14h. Check out: 12h.
Room Facilities:
Direct dial telephone
Satellite cable TV
HiFi stereo system
Internet access
Double bed
Central airconditioned
Central heating


text:
If you are not 100% satisfied with your purchase, you can return your order to the warehouse for a full refund (Bottles must be unopened, in the state you received them, and in the original packaging). We believe that in order to have the best possible online shopping experience, our customers should not have to pay for domestic return shipping. So if you're not happy with your purchase, just contact us right away and we will provide you with details for returning the unopened bottles - your domestic shipping costs are prepaid by us.
With the WineChateau.com 365 day return policy, there are no special catches or exceptions. All we ask is that you send the items back to us in the original packaging, unopened, and in the same condition you received them.
Have questions? Prefer to order by Phone? The Wine Chateau Customer Loyalty Team (WCCLT) is happy to help!
Call WCCLT Monday - Friday 10am - 7pm EST:
1-888-423-8079 or
Email Wine Chateau Customer Loyalty Team
Over 200 years ago the Schmitt ancestors settled in the small village of Longuich, located along the Mosel River in the Mosel-Saar-Ruwer wine region, and began managing vineyards and producing wine.
Now in the 4th generation, Schmitt Söhne is one of Germany's largest and most successful wine exporters to wine drinkers all over the world.
While continuously developing their winemaking and looking for ways to incorporate new technologies and techniques, they never lose sight of the traditional way of producing exceptional German wines.
Their policy has remained the same for over 200 years: They want to produce the best wines possible using only top-quality grapes.
This fully ripened Riesling has highly extracted fruit flavors of apricots and peaches coupled with enough acidity to give it a firm and juicy mouth feel.
Technical Notes
This fully ripened Riesling has highly extracted fruit flavors of apricots and peaches coupled with enough acidity to give it a firm and juicy mouth feel.


text:
What do you guys think what will happen to future Call of Duty games (or Black Ops) perhaps.
Here's some thoughts/questions that have been floating in my head:
- Will Treyarch make Cod's in modern setting?
- What's gonna happen to Infinity Ward?
- Modern Warfare 4? (doubt it)
Any of you guys have any thoughts about the future? (CoD)


text:
Cadence® system design and verification solutions, integrated under our Verification Suite, provide the simulation, acceleration, emulation, and management capabilities.
Verification Suite Related Products A-Z
Cadence® digital design and signoff solutions provide a fast path to design closure and better predictability, helping you meet your power, performance, and area (PPA) targets.
Full-Flow Digital Solution Related Products A-Z
Cadence® custom, analog, and RF design solutions can help you save time by automating many routine tasks, from block-level and mixed-signal simulation to routing and library characterization.
Overview Related Products A-Z
Driving efficiency and accuracy in advanced packaging, system planning, and multi-fabric interoperability, Cadence® package implementation products deliver the automation and accuracy.
Cadence® PCB design solutions enable shorter, more predictable design cycles with greater integration of component design and system-level simulation for a constraint-driven flow.
An open IP platform for you to customize your app-driven SoC design.
Comprehensive solutions and methodologies.
Helping you meet your broader business goals.
A global customer support infrastructure with around-the-clock help.
More Support Log In
24/7 Support - Cadence Online Support
Locate the latest software updates, service request, technical documentation, solutions and more in your personalized environment.
Visit Now
Software Downloads
Cadence offers various software services for download. This page describes our offerings, including the Allegro FREE Physical Viewer.
The Cadence Academic Network helps build strong relationships between academia and industry, and promotes the proliferation of leading-edge technologies and methodologies at universities renowned for their engineering and design excellence.
Participate in CDNLive
A huge knowledge exchange platform for academia to network with industry. We are looking for academic speakers to talk about their research to the industry attendees at the Academic Track at CDNLive EMEA and Silicon Valley.
Come & Meet Us @ Events
A huge knowledge exchange platform for academia. We are looking for academic speakers to talk about their research to industry attendees.
Americas University Software Program
Join the 250+ qualified Americas member universities who have already incorporated Cadence EDA software into their classrooms and academic research projects.
EMEA University Software Program
In EMEA, Cadence works with EUROPRACTICE to ensure cost-effective availability of our extensive electronic design automation (EDA) tools for non-commercial activities.
Apply Now For Jobs
If you are a recent college graduate or a student looking for internship. Visit our exclusive job search page for interns and recent college graduate jobs.
Cadence is a Great Place to do great work
Learn more about our internship program and visit our careers page to do meaningful work and make a great impact.
Get the most out of your investment in Cadence technologies through a wide range of training offerings.
Overview All Courses Asia Pacific EMEANorth America
Instructor-Led Training
Instructor-led training [ILT] are live classes that are offered in our state-of-the-art classrooms at our worldwide training centers, at your site, or as a Virtual classroom.
Read more
Online Training
Online Training is delivered over the web to let you proceed at your own pace, anytime and anywhere.
Exchange ideas, news, technical information, and best practices.
All Blogs
The community is open to everyone, and to provide the most value, we require participants to follow our Community Guidelines that facilitate a quality exchange of ideas and information.
All Forums
It's not all about the technology. Here we exchange ideas on the Cadence Academic Network and other subjects of general interest.
Cadence is a leading provider of system design tools, software, IP, and services.
Get email delivery of the Cadence blog featured here
The 16.6 Allegro RF PCB application has many new enhancements. I'll cover a few over the next several weeks. Here are some major layout related enhancements:
Read on for more details …Snap EnhancementsIn Allegro 16.5, when you snap an RF component to a non-RF component pad, you can only snap to the connecting point (usually it is the center of the pad). Sometimes designers want to connect RF components with non-RF components at the edge of a pad. You can use this functionality to snap an RF component to a non-RF component, or a non-RF component to an RF component or even a non-RF component to another non-RF component based on the connectivity.
There is a "Snap to pad edge" check box on the form which is used for snapping to a specific edge of a pad:
Notice: When you snap a non-RF pad edge to another non-RF pad edge, this may result in violations for manufacturing/assembly, so you've to manually check all those violations. The other enhancement for snap is that you can select some components as a temp group to snap together. During the snap command, RMB select Temp Group and then click a pin to snap, the whole temp group will be moved together.
The use model for the snapping pad edge is a little different from the original snapping functionality; once the "Snap to pad edge" option is checked, the Direction for "Fix source component" or "Fix destination component" will be disabled. The source component will be attached to your mouse and you need to move your mouse close to the destination component pad edge.Add Connect Enhancements
The Add Connect command is enhanced to support snapping to pad edge functionality. When you check the "Snap to pad edge" option (this option is available once you check the "Snap to connect point" option), you can start routing from any edge of a pad by moving your mouse close the edge of the pad and clicking it to start.There is another option named "Variable line width". This option will be available if the "Snap to connect point" option is checked. If you check the "Variable line width" option, the width of the RF trace will be variable based on the entry and the size of the pad and you can't change the trace width during the routing. If you uncheck this option, the width of the trace will use the value that you entered on the Options tab and you can change the width during the routing process.
There is an extra item on the RMB menu during the routing. It's "Accurate length". If you select it, a form will pop up and you can enter a specific value for the length that you want to route.You can route RF trace with any angle mitered bend by setting as following:
Modify Connectivity EnhancementsThis command is enhanced by adding a "Snap to pad edge" option. If this option is checked, the following "Fix source component" and "Fix destination component" options will be disabled:
Add Component EnhancementsThe Add Component is enhanced with the new option of "Snap to pad edge". If this option is checked, the component to be placed can be located at a specific edge of a pad during the placement.
Scaled Copy EnhancementsThis command is enhanced by increasing the option "Snap to pad edge". Once this option is checked, the scaled copied RF component can be connected to any edge of a pad.
Single Segment ConnectionIn some cases, designers may want to connect two points (pads) with a single segment (microstrip line for example). You can use this new command by clicking RF-PCB->Single Segment Connect as below:
Route with Any Angle BendThis is a new command to connect two pins/pads with any angle mitered bend. We're trying to use three RF components (two line segments and one any angle mitered bend) to connect two specific pins/pads. The line segment may be Microstrip Line (MLIN) or Stripline (SLIN) and the any angle bend may be MBEND or SBEND2. Click RF-PCB->Any Angle Bend Connect to launch this command.
The parameters are shown on the Options tab as shown below:
If you have anything to share on how you're using these new features – please do!Jerry "GenPart" Grzenia