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1 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO ABSTRACT. Let GX) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdős, we show that logx loglogx loglogloglogx GX) fx), logloglogx) 2 where fx) is a function tending to infinity with X. Our proof combines eisting arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the eistence and distribution of long arithmetic progressions consisting entirely of primes. CONTENTS 1. Introduction 1 2. On arithmetic progressions consisting of primes 5 3. Main construction 9 4. Probability estimates Linear equations in primes with large shifts Proof of Lemma 2.1i) Proof of Lemma 2.1ii) Further comments and speculations 26 Appendi A. Linear equations in primes 26 References INTRODUCTION Write GX) for the maimum gap between consecutive primes less than X. It is clear from the prime number theorem that GX) 1+o1))logX, as the average gap between the prime numbers which are X is logx. In 1931, Westzynthius [32] proved that infinitely often, the gap between consecutive prime numbers can be an arbitrarily large multiple of the average gap, that is, GX)/logX as X. Moreover, he proved the qualitative bound 1 GX) logxlog 3X log 4 X. 1 As usual in the subject, log2 = log, log 3 = loglog, and so on. The conventions for asymptotic notation such as and o) will be defined in Section
2 2 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO In 1935 Erdős [9] improved this to GX) logxlog 2X log 3 X) 2 and in 1938 Rankin [27] made a subsequent improvement GX) c+o1)) logxlog 2Xlog 4 X log 3 X) 2 with c = 1 3. The constant c was subsequently improved several times: to 1 2 eγ by Schönhage [29], then to c = e γ by Rankin [28], c = e γ by Maier and Pomerance [23] and, most recently, c = 2e γ by Pintz [26]. Our aim in this paper is to show that c can be taken arbitrarily large. Theorem 1. Let R > 0. Then for any sufficiently largex, there are at least R logxlog 2Xlog 4 X log 3 X) 2 consecutive composite natural numbers not eceeding X. In other words, we have GX) fx) logxlog 2Xlog 4 X log 3 X) 2 for some functionfx) that goes to infinity asx. Theorem 1 settles in the affirmative a long-standing conjecture of Erdős [10]. Theorem 1 has been simultaneously and independently established by Maynard [25] by a different method relying on the sieve-theoretic techniques related to those used recently in [24] to obtain bounded gaps between primes, rather than results on linear equations between primes). It appears likely that the techniques of this paper and of that in [25] may be combined to establish further results on large prime gaps; we hope to address this in a subsequent work. Based on a probabilistic model of primes, Cramér [6] conjectured that GX) lim sup X log 2 X = 1, and Granville [13], using a refinement of Cramér s model, has conjectured that the limsup above is in fact at least 2e γ = These conjectures are well beyond the reach of our methods. Cramér s model also predicts that the normalized prime gaps p n+1 p n logp n should have eponential distribution, that is, p n+1 p n Clogp n for about e C πx) primes X. Numerical evidence from prime calculations up to [30] matches this prediction quite closely, with the eception of values of C close to logx, in which there is very little data available. In fact, ma X GX)/log 2 X , slightly below the predictions of Cramér and Granville. Unconditional upper bounds forgx) are far from the conjectured truth, the best beinggx) X and due to Baker, Harman and Pintz [2]. Even the Riemann Hypothesis only 2 furnishes the boundgx) X 1/2 logx [5]. All works on lower bounds for GX) have followed a similar overall plan of attack: show that there are at leastgx) consecutive integers inx/2,x], each of which has a very small prime factor. To describe the results, we make the following definition. 2 Some slight improvements are available if one also assumes some form of the pair correlation conjecture; see [20].
3 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS 3 Definition 1. Let be a positive integer. Define Y) to be the largest integer y for which one may select residue classes a p mod p), one for each prime p, which together sieve out cover) the whole interval [y] = {1,...,y}. The relation between this function Y and gaps between primes is encoded in the following simple lemma. Lemma 1.1. Write P) for the product of the primes less than or equal to. Then we have GP) + Y)+) Y) for all. Proof. Set y = Y), and select residue classes a p mod p), one for each prime p, which cover [y]. By the Chinese remainder theorem there is some m, < m + P), with m a p mod p) for all primes p. We claim that all of the numbers m + 1,...,m + y are composite, which means that there is a gap of length y amongst the primes less than m+y, thereby concluding the proof of the lemma. To prove the claim, suppose that 1 t y. Then there is some p such that t a p mod p), and hence m+t a p +a p 0 mod p), and thus p divides m+t. Since m+t > m > p, m+t is indeed composite. By the prime number theorem we have P) = e 1+o1)). It turns out see below) that Y) has size O1). Thus the bound of Lemma 1.1 implies that GX) Y 1+o1))logX ) as X. Theorem 1 follows from this and the following bound for Y, the proof of which is the main business of the paper. Theorem 2. For any R > 0 and for sufficiently largewe have 1.1) Y) R log 3 log 2 ) 2. The function Y is intimately related to Jacobsthal s function j. If n is a positive integer then jn) is defined to be the maimal gap between integers coprime to n. In particular jp)) is the maimal gap between numbers free of prime factors, or equivalently1plus the longest string of consecutive integers, each divisible by some primep. The construction given in the proof of Lemma 1.1 in fact proves that jp)) Y 1+o1))logP) ) = Y 1+o1)) ). This observation, together with results in the literature, gives upper bounds for Y. The best upper bound known isy) 2, which comes from Iwaniec s work [22] on Jacobsthal s function. It is conjectured by Maier and Pomerance that in fact Y) ) 2+o1). This places a serious albeit conjectural) upper bound on how large gaps between primes we can hope to find via lower bounds for Y): a bound in the region of GX) logxloglogx) 2+o1), far from Cramér s conjecture, appears to be the absolute limit of such an approach. We turn now to a discussion of the proof of Theorem 2. Recall that our task is to find y, as large as possible, so that the whole interval [y] may be sieved using congruences a p mod p), one for each prime p. Prior authors divided the sieving into different steps, a key to all of them being to take a common value ofa p for large p, saya p = 0 forz < p < δ, whereδ > 0 is a small constant andz = clog 3 /log 2 for some constant c > 0. The numbers in [y] surviving this first sieving either have all of their prime factors z i.e., they are z-smooth ) or are of the form pm with p prime and m y/δ. One then appeals to bounds for smooth numbers, e.g. [3], to see that there are very few numbers of the first kind, say O/log 2 ). By the prime number theorem there are ylog 2 / unsieved numbers of the second
4 4 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO kind. By contrast, if one were to take a random choice for a p for z < p < δ, then with high probability, the number of unsifted integers in [y] would be considerably larger, about ylogz/. One then performs a second sieving, choosing a p for small p z. Using a greedy algorithm, for instance, one can easily sieve out all but ylog 2 p z 1 1 ) e γ ylog 2 p logz of the remaining numbers. There are alternative approaches using eplicit choices for a p ; we will choose our a p at random. The set V of numbers surviving this second sieving has about the same size in each case.) If V π) πδ), the number of very large primes, then we perform a rather trivial) third sieving as follows: each v V can be matched with one of these primes p, and one may simply take a p = v. This is the route followed by all authors up to and including Rankin [28]; improvements to G) up to this point depended on improved bounds for counts of smooth numbers. The new idea introduced by Maier and Pomerance [23] was to make the third sieving more efficient and less trivial!) by using many p δ,] to sift not one but two elements of V. To do this they established a kind of twin primes on average result implying that for mostp δ,], there are many pairs of elements ofv that are congruent modulop. Then the authors proved a crucial combinatorial result that disjoint sets V p eist, each of two elements congruent modulo p, for a large proportion of these primes p; that is, for a large proportion of p, a p mod p will sift out two elements of V, and the sifted elements are disjoint. Pintz [26] proved a best possible version of the combinatorial result, that in fact one can achieve a nearly perfect matching, that is, disjoint setsv p for almost all primesp δ,], and this led to the heretofore best lower bound forgx). Heuristically, much more along these lines should be possible. With y comparable to the right-hand side of 1.1), the setv turns out have epected cardinality comparable to a large multiple of/. Assuming that V is a random subset of [y], for every prime p δ,] there should in fact be a residue class a mod p) containing /log 2 ) O1) elements of V. Roughly, the heuristic predicts that the sizes of the sets V a mod p)) are Poisson distributed with parameter V /p.) Whilst we cannot establish anything close to this, we are able to use almost all primesp /2,] to siever elements ofv, for any fied r. Where Maier and Pomerance appealed to in fact proved) a result about pairs of primes on average, we use results about arithmetic progressions of primes of length r, established in work of the second and fourth authors [16], [15] and of these authors and Ziegler [18]. Specifically, we need results about progressions q,q+r!p,q+2r!p,...,q+r 1)r!p; if one ignores the technical factorr!, these are progressions of primes with prime common difference. By taking a p = q, the congruence a p mod p) allows us to sift out all r elements of such a progression, and it is here that we proceed more efficiently than prior authors. Ensuring that many of these r-element sifted sets are disjoint or at least have small intersections) is a rather difficult problem, however. Rather than dealing with these intersections directly, we utilize the random choice of a p in the second step to prove that with high probability, V has a certain regularity with respect to intersections with progressions of the form q, q + r!p, q + 2r!p,..., q +r 1)r!p. We then prove that most elements of V survive the third sieving with uniformly small probability Acknowledgments. The research of SK was partially performed while he was visiting KF at the University of Illinois at Urbana Champaign. Research of KF and SK was also carried out in part at the University of Chicago. KF and SK are thankful to Prof. Wilhelm Schlag for hosting these visits. KF also thanks the hospitality of the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences.
5 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS 5 Also, the research of the SK and TT was partially performed while SK was visiting the Institute for Pure and Applied Mathematics IPAM) at UCLA, which is supported by the National Science Foundation. KF was supported by NSF grant DMS BG was supported by ERC Starting Grant , Approimate algebraic structure. TT was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS Finally, all four of us wish to thank Andrew Granville and James Maynard for helpful discussions Notational conventions. We use f = Og) and f g to denote the claim that there is a constant C > 0 such that f ) Cg ) for all in the domain of f. We adopt the convention that C is independent of any parameter unless such dependence is indicated by subscript such as u, ecept that C may depend on the parameter r which we consider to be fied) in Sections 2 4 and 6 7. In Sections 2 4 and 6 7, the symbol o1) will stand for a function which tends to 0 as, uniform in all parameters ecept r unless otherwise indicated. The same convention applies to the asymptotic notationf) g), which meansf) = 1+o1))g). In Sections 5 and the Appendi,ogN)) refers to some functionhn) satisfyinglim N hn)/gn) = 0. The symbolsp,q will always denote prime numbers. In Sections 2 4 and 6 7,sdenotes a prime, whereas in Sections 5 and the Appendi, s is a positive integer which measures the compleity of a system of linear forms. Finally, we will be using the probabilistic method and will thus be working with finite probability spaces. Generically we write P for probability, and E for epectation. If a finite set A is equipped with the uniform probability measure, we write P a A and E a A for the associated probability and epectation. Variables in boldface will denote random real-valued scalars, while arrowed boldface symbols denote random vectors, e.g. a. We also use #A to denote the cardinality of A, and for any positive real z, we let [z] := {n N : 1 n z} denote the set of natural numbers up to z. 2. ON ARITHMETIC PROGRESSIONS CONSISTING OF PRIMES A key tool in the proof of Theorem 2 is an asymptotic formula for counts of arithmetic progressions of primes. In fact, we shall be interested in progressions of primes of length r whose common difference is r! times a prime 3, for positive integer values of r. The key technical result we shall need is Lemma 2.4 below. This is a relatively straightforward consequence of Lemma 2.1 below, which relies on the work on linear equations in primes of the second and fourth authors and Ziegler. We turn to the details. Let y be a sufficiently large quantity which goes to infinity for the purposes of asymptotic notation), and let be a quantity that goes to infinity at a slightly slower rate than y; for sake of concreteness we will impose the hypotheses 2.1) y. In fact the analysis in this section would apply under the slightly weaker hypotheses ylog O1) y oy), but we will stick with 2.1) for sake of concreteness since this condition will certainly be satisfied when applying the results of this section to prove Theorem 2. From 2.1) we see in particular that logy, 3 One could replacer! here if desired by the slightly smaller primorialpr); as observed long ago by Lagrange and Waring [8], this primorial must divide the spacing of any sufficiently large arithmetic progression of primes of length r. However, replacing r! bypr) would lead to only a negligible savings in the estimates here.
6 6 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO so we will use and logy more or less interchangeably in what follows. Let P denote the set of all primes in the interval /2,], and Q denote the set of all primes in the interval /4,y]; thus from the prime number theorem we have 2.2) #P 2 ; #Q y. In other words,p andqboth have density 1 inside/2,] and /4,y] respectively. Let r 1 be a fied natural number. We define a relation between P and Q as follows: if p P and q Q, we write p q if the entire arithmetic progression {q,q + r!p,...,q + r 1)r!p} is contained inside Q. One may think of the r relations p q ir!p for i = 0,...,r 1 as defining r different but closely related) bipartite graphs between P and Q. Note that if p q, then the residue class q mod p) is guaranteed to contain at least r primes from Q, which is the main reason why we are interested in these relations particularly for somewhat large values of r). For our main argument, we will be interested in the typical degrees of the bipartite graphs associated to the relationsp q ir!p. Specifically, we are interested 4 in the following questions for a given0 i r 1: i) For a typical p P, how many q Q are there such that p q ir!p? Note that the answer to this question does not depend on i.) ii) For a typical q Q, how manyp P are there such that p q ir!p? If P and Q were distributed randomly inside the intervals /2, ] and /4, y] respectively, with cardinalities given by 2.2), then standard probabilistic arguments using for instance the Chernoff inequality) would suggest that the answer to question i) is y log r, while the answer to question ii) is 2log r. As it turns out, the local structure of the primes for instance, the fact that all the elements of P and Q are coprime tor!) will bias the answers to each of these two questions; however as one may epect from double counting considerations), they will be biased by eactly the same factorα r defined in 2.3) below), and the net effect of this bias will cancel itself out at the end of the proof of Theorem 2. One can predict the answers to Questions i) and ii) using the Hardy-Littlewood prime tuples conjecture [21]. If we apply this conjecture and ignore any issues as to how uniform the error term in that conjecture is with respect to various parameters), one soon arrives 5 at the prediction that the answer to Question i) y should be α r log r for all p P, and similarly the answer to Question ii) should be α r2log r for all q Q, where for the rest of the paper α r will denote the singular series 2.3) α r := ) p r 1 p r)p r 1 p 1 p 1) r. p r p>r The eact form ofα r is not important for our argument, so long as it is finite, positive, and does not depend onory; but these claims are clear from 2.3) note that the second factor p r)pr 1 p 1) is non-zero and behaves r asymptotically as 1+O1/p 2 )). As mentioned previously, this quantity will appear in two separate places in the proof of Theorem 2, but these two occurrences will eventually cancel each other out. The Hardy-Littlewood conjecture is still out of reach of current technology. Note that even the much weaker question as to whether the relation p q is satisfied for at least one pair of p and q for any given r is at least as hard as establishing that the primes contain arbitrarily long arithmetic progressions, which was 4 Actually, for technical reasons we will eventually replace the relation by slightly smaller relation, which will in turn be randomly refined to an even smaller relation a ; see below. 5 See also Sections 6, 7 for some closely related computations.
7 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS 7 only established by the second and fourth authors in [14]. However, for the argument used to prove Theorem 2, it will suffice to be able to answer Question i) for almost all p P rather than all p P, and similarly for Question ii). In other words, we only need a special case of) the Hardy-Littlewood prime conjecture on average. This is easier to establish; for instance, Balog [1] was able to use the circle method or linear Fourier analysis ) to establish the prime tuples conjecture for most tuples in some sense. The results in [1] are not strong enough for our applications, because of our need to consider arbitrarily long arithmetic progressions which are well-known to not be amenable to linear Fourier-analytic methods for r 4, see [12]) rather than arbitrary prime tuples. Instead we will use a modification of) the more recent work of the second and fourth authors [16]. More precisely, we claim the following bounds. Lemma 2.1. Let,y,r,P,Q, and be as above. Let0 i r 1. i) For all but o/) of thep P, we have the estimate y #{q Q : p q ir!p} α r log r. ii) For all but oy/) of theq Q, we have #{p P : p q ir!p} α r 2log r. iii) For all p P, we have the upper bounds #{q Q : p q ir!p} y log r. iv) For all q Q, we have the upper bounds #{p P : p q ir!p} 2log r. Parts iii) and iv) follow from standard sieve-theoretic methods e.g. the Selberg sieve); we omit the proof here, referring the reader instead 6 to [19] or [11]. The more interesting bounds are i) and ii). As stated above, these two claims are almost relatively straightforward consequences of the main result of the paper [16] of the second and fourth authors. However, some modifications of that work are required to deal with the fact that and y are of somewhat different sizes. In Section 5 below we state the modified version of the main result of [16] that we need, Theorem 7. The deductions of parts i) and ii) of Lemmas 2.1 are rather similar to one another, and are given in Sections 6 and 7 respectively. Finally, a proof of Theorem 7 can be obtained by modifying the arguments of [16] in quite a straightforward manner, but in a large number of places. We record these modifications in Appendi A. As presently defined, it is possible for the bipartite graphs given by the p q ir!p to overlap, thus it may happen thatp q ir!p andp q jr!p for somep P,q Q, and0 i < j r 1. For instance, this situation will occur if Q has an arithmetic progression q,q +r!p,...,q +r r!p of length r +1 with p P. For technical reasons, such overlaps are undesirable for our applications. However, these overlaps are rather rare and can be easily removed by the following simple device. We define the modified relation between P and Q by declaring p q if the progression{q, q + r!p,..., q +r 1)r!p}is contained inside Q, butq+r r!p does not lie in Q. From construction we have the following basic fact: Lemma 2.2. For anyp P andq Q that there is at most one0 i r 1 such that p q ir!p. We can then modify Lemma 2.1 slightly by replacing the relation with its slightly perturbed version : 6 One could also deduce these bounds from Proposition 6.4 in Appendi A.
8 8 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO Lemma 2.3. Let,y,r,P,Q, and be as above. Let0 i r 1. i) For all but o/) of thep P, we have the estimate y #{q Q : p q ir!p} α r log r. ii) For all but oy/) of theq Q, we have #{p P : p q ir!p} α r 2log r. iii) For all p P, we have the upper bounds #{q Q : p q ir!p} y log r. iv) For all q Q, we have the upper bounds #{p P : p q ir!p} 2log r. Proof. Parts iii) and iv) are immediate from their counterparts in Lemma 2.1, since is a subrelation of. To prove i), we simply observe from Lemma 2.1iii) withr replaced byr+1) that #{q Q : p q ir!p butp q ir!p} y log r+1, and the claim then follows from Lemma 2.1i) and the triangle inequality. The claim ii) is proven similarly. For technical reasons, it will be convenient to reformulate the main results of Lemma 2.3 as follows. Lemma 2.4. Let,y,r,P,Q, and be as above. Then there eist subsetsp 0,Q 0 ofp,q respectively with 2.4) #P 0 2 ; #Q 0 y, such that y 2.5) #{q Q : p q ir!p} α r log r for all p P 0 and0 i r 1, and similarly that 2.6) #{p P 0 : p q ir!p} α r 2log r for all q Q 0 and 0 i r 1. Proof. From Lemma 2.3i) we may already find a subset P 0 of the desired cardinality obeying 2.5). If the P 0 in 2.6) were replaced by P, then a similar argument using Lemma 2.3ii) and taking the union bound for the eceptional sets for each 0 i r 1) would give the remainder of the lemma. To deal with the presence of P 0 in 2.6), it thus suffices to show that ) #{p P\P 0 : p q ir!p} = o log r for all but oy/) of theq Q. By Markov s inequality, it suffices to show that #{p,q) P\P 0 ) Q : p q ir!p} = o log r y ).
9 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS 9 But this follows by summing Lemma 2.3iii) for allp P\P 0, since the setp\p 0 has cardinalityo/). 3. MAIN CONSTRUCTION We now begin the proof of Theorem 2. It suffices to establish the following claim: Theorem 3 First reduction). Let r 13 be an integer. Take to be sufficiently large depending on r and going to infinity for the purposes of asymptotic notation), and then define y by the formula 3.1) y := r 6logr log 3 log 2 ) 2. Then there eists a residue class a s mod s) for each prime s, such that the union of these classes contains every positive integer less than or equal to y. The numerical values of13 and6in the above theorem are only of minor significance, and can be ignored for a first reading. Observe that,y obey the condition 2.1) from the previous section. If Theorem 3 holds, then in terms of the quantity Y) defined in the introduction, we have Y) y which by 3.1) will imply Theorem 2 by takingr sufficiently large depending onr. It remains to prove Theorem 3. Set 3.2) z := log 3 /3log 2 ), and partition the primes less than or equal to into the four disjoint classes S 1 := {s prime : s or z < s /4} S 2 := {s prime : < s z} S 3 := P = {s prime : /2 < s } S 4 := {s prime : /4 < s /2}. We are going to sieve [y] in four stages by removing at most one congruence class a s mod s) for each prime s S i, i = 1,2,3,4. If we can do this in such a way that nothing is left at the end, we shall have achieved our goal. We first dispose of the final sieving process involving S 4 ), as it is rather trivial. Namely, we reduce Theorem 3 to Theorem 4 Second reduction). Let r,,y be as in Theorem 3, and let S 1,S 2,S 3 be as above. Then there eists a residue class a s mod s) for each s S 1 S 2 S 3, such that the union of these classes contains all but at most 1 5 +o1)) of the positive integers less than or equal to y. Indeed, if thea s mod s) fors S 1 S 2 S 3 are as in Theorem 4, then from the prime number theorem, the number of integers less than y that have not already been covered by a residue class is smaller than the number of primes in S 4. Thus, we may eliminate each of these surviving integers using a residue class a s mod s) from a different elementsfroms 4 and selecting residue classes arbitrarily for anys S 4 that are left over), and Theorem 3 follows. It remains to prove Theorem 4. For this, we perform the first sieving process using up the primes from S 1 ) and reduce to
10 10 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO Theorem 5 Third reduction). Let r,,y,s 2,S 3 be as in Theorem 4, and as in the previous section) let Q denote the primes in the range/4,y]. Then there eists a residue classa s mod s) for eachs S 2 S 3, such that the union of these classes contains all but at most 1 5 +o1)) of the elements of Q. Proof of Theorem 4 assuming Theorem 5. We take a s := 0 for all s S 1. Write R [y] for the residual set of elements which survive this first sieving, that is to say R consists of all numbers in [y] that are not divisible by any primesins 1. Taking into account that /4) > y from 3.1), we conclude that R = Q R err, where R err contains only z-smooth numbers, that is to say numbers in [y] all of whose prime factors are at mostz. Let u denote the quantity u := logy logz, so from 3.2) one hasu 3 log 2 log 3. By standard counts for smooth numbers e.g. de Bruijn s theorem [3]), #R err ye ulogu+ouloglogu+2)) y = log 3+o1) = log 2+o1) = o/). Thus the contribution of R err may be absorbed into the eceptional set in Theorem 4, and this theorem is now immediate from Theorem 5. Remark 1. One can replace the appeal to de Bruijn s theorem here by the simpler bounds of Rankin [27, Lemma II], if one makes the very minor change of increasing the 3 in the denominator of 3.2) to 4, and to similarly increase the6in 3.1) to8. It remains to establish Theorem 5. Recall from 2.2) that Q has cardinality y/. This is significantly larger than the error term of o1)) permitted in Theorem 5; our sieving process has to reduce the size of Q by a factor comparable to y/. The purpose of the second sieving, by congruences a s mod s) with s S 2, is to achieve almost all of this size reduction. Our choice of the a s for s S 2 will be completely random which is why we are using the boldface font here): that is, for each prime s S 2 we selecta s uniformly at random from{0,1,...,s 1}, and these choices are independent for different values of s. Write a for the random vector a s ) s S2. Observe that if n is any integer not depending on a), then the probability that n lies outside of all of the a s mod s) is eactly equal to γ := s S ). s This quantity will be an important normalizing factor in the arguments that follow. From Mertens theorem and 3.1), 3.2) we see that 3.3) γ log 2 logz 3log 2) 2 log 3 r 2logr y.
11 LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS 11 Set Description Epected cardinality P Primes in/2,] 2 P 0 Primes inp connected to the epected # of primes in Q 2 P 1 a) Primes inp 0 connected to the epected # of primes in Q a) 2 P 1 a,q;i) Primes inp 1 a) i-connected to a given primeq Q 1 a) γ r 1 α r 2log r Q Primes in/4,y] y Q 0 Primes inqconnected to the epected # of primes in P 0 y Q a) Randomly refined subset ofq r 2logr Q a,p) Primes inq a) connected to a given primep P 1 a) γ r y α r Q 0 a) Intersection ofq a) withq 0 r 2logr Q 1 a) Primes inq 0 a) connected to the epected # of primes in P 1 a) r 2logr Q 1 a, q) Randomly refined subset ofq 1 a) 1 2logr log r TABLE 1. A brief description of the various P and Q-type sets used in the construction, and their epected size. Roughly speaking, the congruence classes from S 1 are used to cut down [y] to approimately Q, the congruence classes from S 2 are used to cut Q down to approimately Q 0 a), the congruence classes from S 3 = P are used to cut Q 0 a) down to approimately Q 1 a, q), and the congruence classes in S 4 are used to cover all surviving elements from previous sieving. WriteQ a) for the random) residual set of primesq inqthat do not lie in any of the congruence classes a s mod s) fors S 2. We will in fact focus primarily on the slightly smaller set Q 0 a) := Q a) Q 0 whereq 0 is the subset of Q constructed in Lemma 2.4. From linearity of epectation we see that 3.4) E#Q a) = γ#q and thus from 3.3), 2.2) 3.5) E#Q a) r 2logr. Similarly, from Lemma 2.4 we have ) y #Q\Q 0 ) = o and thus from linearity of epectation and 3.3) we have E#Q a)\q 0 a)) = o γ y ) ) = o. In particular, from Markov s inequality we have 3.6) #Q a)\q 0 a)) = o with probability 1 o1). We have an analogous concentration bound for #Q a): γ y ) ) = o
12 12 KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO Lemma 3.1. With probability1 o1), we have In particular, from 3.6) we also have with probability 1 o1). #Q a) r 2logr γ y. #Q 0 a) r 2logr γ y This lemma is proven by a routine application of the second moment method; we defer that proof to Section 4. It will now suffice to show Theorem 6 Fourth reduction). Let,y,r, a,p 0,Q 0 a) be as above, and let ε > 0 be a quantity going to zero arbitrarily slowly as, thus ε = o1). Then with probability at least ε in the random choice of a, we may find a length r arithmetic progression {q p + ir!p : 0 i r 1} for each p P 0, such that the union of these progressions contains all but at most 1 5 +o1)) of the elements ofq 0 a). Theo1) decay in the conclusion may depend on ε.) Indeed, from this theorem and taking ε going to zero sufficiently slowly) we may find a such that the conclusions of this theorem hold simultaneously with 3.6), and by combining the residue classes from a with the residue classes q p mod p) for p P 0 from Theorem 6 and selecting residue classes arbitrarily forp P\P 0 ), we obtain Theorem 5. It remains to establish Theorem 6. Note now from Lemma 3.1) that we only need to refine the surviving r logr set Q 0 a) by a constant factor comparable to. Recall from the previous section that we had the relation ), rather than by a factor like y/ that goes to infinity as betweenp andq. We now refine this relation to a random) relation between P 0 and Q a) as follows. If p P 0 and q Q a), we write p a q if p q and if the arithmetic progression {q,q + r!p,...,q + r 1)r!p} is contained in Q a) i.e. the entire progression survives the second sieving process). Intuitively, if p P 0 and q Q are such that p q, we epect p a q to occur with probability close to γ r. The following lemma makes this intuition precise compare with Lemma 2.4): Lemma 3.2. Let ε > 0 be a quantity going to zero arbitrarily slowly as. Then with probability at least ε, we can find random) subsetsp 1 a) of P 0 andq 1 a) of Q 0 a) obeying the cardinality bounds 3.7) #P 1 a) 2 ; #Q 1 a) #Q 0 a) r 2logr, such that for all p P 1 a) and0 i r 1, and such that #{q Q a) : p a q ir!p} γ r y α r log r #{p P 1 a) : p a q ir!p} γ r 1 α r 2log r for all q Q 1 a) and0 i r 1. The impliedo1) errors in the notation may depend onε.)
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Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negativeMore information
Collinear Points in Permutations
Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,More information
The degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote theMore information
A Sublinear Bipartiteness Tester for Bounded Degree Graphs
A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartiteMore information
Probabilities and Random Variables
Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments soMore information


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My Asus tf700t has been locking up and either shutting down and restarting on its own or requiring me to hold down the power button until it restarts. Anyone else having this problem or know a fix? I've deleted several recently added apps and checked for updates but that hasn't worked. Love my tablet but this is frustrating and annoying.
Sent from my ASUS Transformer Pad TF700T using Android Central Forums

subset: commoncrawl


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The West Central School Board members did approve a reduction in force proposal at their recent meeting. Superintendent Charles Mellon said there was only one this year.
"That person is going on our recall list," stated Mellon. "He is currently teaching science in the middle school, but he does have an elementary license so if we had a class size increase – which I hope we do – or someone else from the elementary leaves the school system for whatever reason, he has the opportunity to come back. He will be on the recall list for two years. There is still a possibility for him."


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Music has always been important at St Mary's, which pioneered the liturgical performance of 16th and 17th century choral music in the mid 20th century.
The church has a Director of Music, Choral Scholars, Girl Choristers and a Church Choir. St Mary's is affiliated to the Royal School of Church Music.
Choral Services
Regular choral services are offered, including Sung Eucharist, with a Latin Mass or English Communion setting and communion motet sung by the Choral Scholars on the 1st Sunday of each month. Special choral services sung by the scholars, sometimes with the girl choristers, include the candlelit Advent and Christmas Carol Services, Midnight Mass, Maundy Thursday Sung Eucharist, Good Friday Devotion, Easter Sunday Sung Eucharist, and an evening Ascension Day Services.
Choral Evensongs are sung several times a year, and solo anthems are offered at some services of Evensong.
The scholars and choristers also sing at wedding services.
Please see the list of forthcoming choral services on our website.
If you are an enthusiastic singer aged 10-18, come and sing with our choristers.
They rehearse at 7.50-8.30am on Wednesdays and Thursdays at Hinchley Wood School. They sing at the 3rd Sunday Family Services, and also alongside the professional choral scholars for major services and choral evensongs. Choristers are presented with Royal School of Church Music medals and ribbons, and have the opportunity to be admitted as Head Chorister. They also get to read lessons in Carol Services.
Frobenius Organ
It was the focus on early choral music that inspired the commissioning of the superb organ installed in the church by Frobenius, Organ builders of Lyngby, Denmark, in 1975.
This outstanding instrument was voiced in the church by Erik Frobenius, and is widely praised for its light, responsive touch and the delightful sweetness and clarity of its sound. It is a smaller cousin of the renowned organ of Queen's College, Oxford, installed by Frobenius a decade earlier.
The organ is a joy to play and has attracted a number of eminent organists to play, give recitals and make recordings on St Mary's organ, including James O'Donnell, James Dalton, Simon Preston, Dame Gillian Weir, Susi Jeans and Ian leGrice. A CD recording of the organ was recently recorded by Emma Gibbins, a previous Director of Music of St Mary's, and is available for sale at the church.
The specification can be found on the National Pipe Organ here
With grateful thanks to the Ouseley Church Music Trust www.ouseleytrust.org.uk for their support.


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Pigeon Forge & Gatlinburg Cabin Rentals at its finest. Pigeon Forge cabin rentals by Majestic Mountain Vacations offer the best in Smoky Mountain accommodations. Enjoy the Smoky Mountain views & elegance of a 1 to 12 bedroom luxury Gatlinburg TN cabin rental for your honeymoon, family vacation or other event. Gatlinburg, Pigeon Forge, and the majestic Great Smoky Mountains are waiting.
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Are you planning a getaway for a church group, youth group, wedding or corporate retreat? These luxurious 6 to 12 bedroom Tennessee cabin rentals were designed specifically for groups like yours. Beautifully crafted and fully equipped with every modern convenience, these log cabin rentals will make your Smoky Mountain cabin vacation an extravagant affair. And best of all these premier mountain cabin rentals offer spectacular views of the stunning Smoky Mountains and are only minutes from Pigeon Forge TN & downtown Gatlinburg.
We maintain the highest standards of rental unit quality, impeccable cleanliness and maintenance and reliable customer service available locally 24 hours a day.


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Background: JPMorgan Chase & Co. (NYSE:JPM) is a leading global financial services firm. The company trades an average of 12.8 million shares per day, and the market cap is $212 billion.
Price To Book: 1
JPMorgan is forecast to report WEAK second-quarter earnings before the market opens on July 15, 2014. The consensus estimate is currently $1.29 a share, falling 31 cents (19.4%) from $1.6 during the same period last year. Estimates from analysts range from a low of $1.16 per share, up to the highest estimate of $1.38 per share.
The whisper numbers I'm finding range from $1.33 to $1.37. A significant beat, along with an improving outlook, should propel the shares higher in my opinion. The stock is priced for zero or negative growth with a forward price-to-earnings ratio of only 9.5.
When a P/E ratio is under 10, it means investors have walked away and more or less given up. They have a point though, JPMorgan is so big and the government isn't exactly making life easy for banks and financial firms lately. Still, buying on dips with the newly raised dividend is a safe play.
In comparison, Bank of America (NYSE:BAC) is 10.5, Citigroup (NYSE:C) is 8.8, Wells Fargo (NYSE:WFC) is 12.1, and U.S. Bancorp (NYSE:USB) is 12.8. Compared to other industries, banks are cheap, but there are good reasons why. The list starts with government red tape and mountains of new (and old) regulations.
If the shares go nowhere, you get an effective yield of 2.86% for waiting. It's a classic heads you win, tails you break even type of investment for a long-term hold. That includes the difference in dividends shareholders receive. Bank of America and Citigroup pay four cents a year, while Wells Fargo pays $1.40 (2.67% yield), and JP Morgan investors collect $1.60 per year.
After last quarter's 12 cent miss, the shares decreased only slightly 3.9%, or $2.22 from the close before earnings release to the closing after. The final price preceding the report was $57.40, and concluded at $55.18 the next trading session. The reason, is again, the shares are already fully discounted for little or no growth. A miss just isn't going to beat this stock up, while a beat will give it legs.
Traders and investors are expecting $2.24 change higher or lower, or about a 4% change, during the next 10 days based on current option pricing. That's a range of $53.52 to $58, just for one standard deviation. The price of options makes buying or selling premium about equal if you get the direction right. Otherwise, selling premium is slightly superior using historical volatility as the judge.
Shares have traded slightly lower in the last month of trading, and are about break even at 1.5% less than a month ago. That implies a lackluster report one way or another, but highly subject to change in the days leading up to the report.
Unlike Bank of America and Citigroup, I think JPMorgan is worth looking at with less than highly speculative capital. In my articles for Bank of America and Citigroup, I was reminded how much potential they have compared to Wells Fargo.
It's true both Bank of America and Citigroup have potential, but nothing says potential wasted like having the government involved in your activities. JPMorgan isn't immune to various agencies sticking their hands into the bank's wallet, but it's nowhere close to that of Bank of America and Citigroup.
Once Bank of America and Citigroup are able to escape from the leg irons of big brother, I will likely change my opinion, or at least take another close examination. That could wind up taking a lot longer than many realize though.
I also wrote an earnings preview for Wells Fargo describing what to expect and why I think it makes a good safe investment, especially if you're dividend-focused. Wells Fargo makes the top of my list along with U.S. Bancorp, but JPMorgan is my next big bank value.
CEO Jamie Dimon is undergoing treatment for cancer, but that should have zero impact on this report and I don't see it becoming a material issue in future quarters.
That said, if you're worried about earnings and are contemplating an exit, I suggest you sell calls against the shares. You can always buy them back if you change your mind. That will give you downside protection without paying for it. Otherwise, keep the seat belt fastened and enjoy the ride while you get paid to do so.
Disclosure: The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.


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Veronica Mars creator Rob Thomas has revealed that a big screen version of the show is "close" to being completed.
The series, which starred Kristen Bell as a student who moonlights as a private investigator, ran from 2004 until 2007.
Speaking to Entertainment Weekly, Thomas said that he had almost finished writing the project, saying: "It's close. A few key details left to solve.
"I'm preparing the pitch now - literally now. I hope to go in and talk to [executive producer] Joel Silver and [Warner Bros.] in the next week or two to see if they're interested."
However, he stressed that this did not mean that the movie was a done deal. "I'm going to give it my best shot and let the chips fall where they may...It's entirely possible I won't have an answer at the end of the pitch.
"They will want to consider [the pitch], look at some numbers - whatever it is people on the business side do - and then they'll give me some sort of response. Then, if they want to move forward, there's the deal-making that'll need to go on."


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|By Mageddon725
|Part of the Sons and Daughters continuity.
About Sons and Daughters
S&D is the fanon prelude to the Legend of Korra, telling the story of the children of Team Avatar and their adventures in Republic City.
The Story So Far...
Tenzin and Sora have brought their friends to Republic City and have settled in. Meanwhile, Aang is put on trial in the Spirit World for his murder of Koh and acquitted, but he realizes that certain spirits would rather see him dead.
Next Chapters
Chapter 14: Opening Act
Percent Complete: 100%
Chapter 15: Final Curtain
Percent Complete: 100%
Chapter 16: The Procession
Percent Complete: 100%
Chapter 17: Epilogue
Percent Complete: 100%
See more
For the collective works of the author, go here.


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Horn Pins
Horn Pins
"Embellishments from Africa."
Village artisans use discarded horn from domestic cattle to craft these extraordinary shawl pins. We are assured that these elegant embellishments are imported following all current protected-species regulations.
With these lovely pins, you're not only not only adding a touch of beauty to your knitted pieces, you're also supporting an emerging economy in a developing country.
Please note that each piece is one-of-a-kind and may not look exactly like the photo due to natural color variations in the horn. Not only that, but your pin's color will vary as you move through different light!
Clean with unscented soap and water. An occasional light coat of oil will keep these natural gems looking tip-top.
Horn pin - safety pin shape.
Horn pin - tear drop shape.


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We found this hotel for our stay in Florence after reading other reviews on TA. Have to say, the location was great, owners/staff couldn't be nicer. Not modern, old but clean, good free wifi, and good breakfast. We stayed for 3 nights, left for Cinque Terre for 2 nights, then returned for one more night before heading to Rome. They let us leave 2 bags with them till our return. When we did return, our room was ready and there were our bags inside! The ladies who make breakfast couldn't be sweeter. Definitely good value and would stay there again.
- Official Description (provided by the hotel):
- The Casci is a small family managed hotel located in an ancient palace right in the heart of Florence, just a few steps from the Duomo and the Accademia (David), within walking distance from all museums and sights. It can be easily reached from the main railway station and by car, too, with a valet garage nearby. All rooms have private bathroom, air conditioning, flat screen TV with DVD, free WI-FI, hairdryer, fridge and safety-box. ... more less
- Reservation Options:
- TripAdvisor is proud to partner with Booking.com, Travelocity, Priceline, Hotwire, Hotels.com, Expedia and TripOnline SA so you can book your Hotel Casci reservations with confidence. We help millions of travelers each month to find the perfect hotel for both vacation and business trips, always with the best discounts and special offers.
- Also Known As:


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Cummins Inc., CMI is in the exchange NYSE and its industry is Diversified Machinery in the sector of Industrial Goods. Based in USA, Cummins Inc., CMI has a market cap of 19705.46. Since its IPO date on the 12/18/1984, Cummins Inc., CMI performance year to date is 32.75%. Today Cummins Inc., CMI has gained 0.00%, with a current price of 114.7.
Ownership of the company is 0.20% for insider ownership while institutional ownership is 87.90%. The management of the company have seen the company have a payout ratio of 49.00%. Return of assets are at 8.70%, with return on investment at 16.60%.
In terms of debt levels and profit levels, Cummins Inc., CMI is seeing a long-term debt/equity of 0.23. While Total debt/equity is 0.26. With a profit margin of 7.10%, this is combined with a gross margin of 25.70%, and operating margin of 10.60%. Cummins Inc. ability to meet debt levels, with a current ratio of 1.9, while the quick ratio is 1.2.
For the last year Cummins Inc., CMI has seen a EPS growth of -13.00%. A performance for the year of -6.93%. The 52-week high is -9.53%, and the 52-week low is 46.26%. The average volume for Cummins Inc., CMI is 0.
With a target price of 110.58, can Cummins Inc., CMI reach this target? Looking at the value indicators of Cummins Inc., CMI. Cummins Inc. has a P/E of 15.2 and a forward P/E of 14.59. Perhaps the more useful indicator than P/E, is PEG which has a value of *TBA. Cummins Inc. also has a P/S and a P/B of 1.05 and 2.85 respectively. For P/cash, Cummins Inc. has a value of 15.47, while it is 27.29 for P/free cash flow.
At the current price of 114.7, Cummins Inc. has a dividend yield of 3.40%. We see a return on equity of 17.80%.
Looking more long-term Cummins Inc., is projected to get an EPS growth for the next five years of -0.44%. In the short-term an EPS growth of -0.89% in the next year is forecasted. This is after a EPS growth of -13.00% for this year and for the last five years a 8.30% growth has been seen.


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Hand fed playful and friendly blue parakeets. Male and female handled everyday.$60
Eva is almost 5 years old now. She is very temperate macaw with beautiful coloring. She does know how to say "Hello" and her name! We are selling her due…$1,800
macaw parrot in terrific condition old Adores people and human company Can be shy at 1st but warms to people very quickly and talks when comfortable Good…No Price Listed
90//Good talker and companion Macaw contact us via text to 912 x 333 x 85 x 03 if interested and for pictures$500
Young red black canary. It sings. I have seven that sing.SOLD
Red black female canary . I year.SOLD
I have more colors available then whats seen on the photo. I have parakeets I need to find a home for. They have not been played with much. There are…SOLD
Both birds, 2 cages, bird food, and all their toys. They are loads of fun but my wife and I need to downsize. They like lots of attention.SOLD
I have 5 beautiful hand fed male & female parakeets. There are a variety of colors available. We have blue, purple, white or light, green and many more…SOLD


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Wow, here's a really nice Harris Teeter deal for you guys and gals. Through 11/3, Frosted Flakes Cereal is on sale BOGO ($3.55 for two.) We have a great printable coupon that will double, an eVIC coupon to stack and a MobiSave cashback offer. Check out the deal below.
At Harris Teeter:
BUY TWO FROSTED FLAKES CEREALS, on sale BOGO $3.55
Get $0.75 cashback from MobiSave for buying two Frosted Flakes
Use the $0.50/1 Frosted Flakes Cereal eVIC store coupon
and use two $0.50/1 Frosted Flakes printable manufacturer's coupons (preclipped) – doubles to $1 each
Price at Register: $1.05, but get $0.75 cashback from MobiSave
NET PRICE FOR ALL: $0.30 (or 15 cents each!)
Thanks and photo credit, Moola Saving Mom.Extreme Couponing Tip: You can usually get two prints of most internet coupons. To find them in the easiest possible way, use our preclipped links that take you right to the coupon. Then come back and click the same link to get your 2nd print. This will save you the time spent searching for the coupon.
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|Message view
|« Date » · « Thread »
|« Date » · « Thread »
|Shane Curcuru <...@shanecurcuru.org>
|Re: What is a PMC Chair?
|Thu, 31 May 2012 14:40:17 GMT
On 2012-05-31 10:31 AM, Yong Lin Ma wrote: > How often a new chair will be selected? Yearly or depends on ... Up to the project. Many TLPs have bylaws that state they'll have an annual election, although in many cases this is ignored (or forgotten, more likely), and PMC chairs sit for... however long the project wants them. In many cases, the existing chair either gets overloaded (typically with real-life changes outside the project) or simply wants to give someone else a chance, and they call for nominations themselves. It's really up to the PMC to decide this, and then submit resolutions to the board (which are required to change chairs). Presuming the PMC is acting in a healthy manner, the board simply passes the resolution at the next meeting. For a project like AOO, I personally think it would be valuable to have an expectation of an annual nomination/vote process for the chair. - Shane > > > On Thu, May 31, 2012 at 9:56 PM, Ross Gardler > <firstname.lastname@example.com> wrote: >> Since discussion has turned to graduation I'd like to invite people to >> consider who they would like to have as PMC chair. The first part of >> this is understanding what the role of a PMC chair is. >> >> First and foremost the position of chair does not bring any additional >> authority over the project, at least not in normal circumstances. It >> is true that in the event of a deadlock the chair has a casting vote, >> however I have never seen this happen. In reality the chair is just >> the same as any other PMC member except that they are expected to do a >> certain amount of "paperwork" for the PMC and, more importantly, they >> are a community facilitator. You can find a full description of the >> responsibilities at [1]. In summary they are: >> >> - Subscribe to and monitor board@ (and board meeting minutes) and >> infrastructure@ at lists, ensuring the community takes any necessary >> actions >> >> - Submit quarterly reports >> >> - Maintain PMC membership records >> >> - Ensure everyones voice is heard >> >> Before calling for nominations (and people can self-nominate if they >> so desire) I would like to take a few days to allow people to ask any >> questions about the role and the type of individual that is best >> suited to be a chair. >> >> Ross >> >> [1] http://www.apache.org/dev/pmc.html#chair >> >> >> >> >> >> >> -- >> Ross Gardler (@rgardler) >> Programme Leader (Open Development) >> OpenDirective http://opendirective.com
|View raw message


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K&N 57 Series Generation II Air Intake, Jeep (2006-10) Grand Cherokee, V8-6.1L
The 57 Series Generation II air intake, also referred to as an FIPK®, is K&N's ultimate performance product. These performance intakes are designed to increase power by as much as 10%. 57 Series FIPK's are CARB exempt, street legal in all 50 US states, come with a Million Mile Limited Warranty and are guaranteed to increase power. K&N air intake systems replace your vehicle's restrictive factory air filter and air intake housing. K&N intakes are designed to dramatically reduce intake restriction as they smooth and straighten air flow. This translates into more power and torque for your vehicle. The filters on these kits are washable and reusable.
Estimated Horsepower Gain: 12 HP @ 5000 RPM
Results will vary.
Estimated Horsepower Gain: 12 HP @ 5000 RPM
Results will vary.
Was: $557.99
Sale: $307.99
Save: 45%
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Replacement Filter: RF-1042
- 57-1555
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text: Nader pushes single-payer health care in CU talk By on April 24, 2009 By Richard Haugh THE COLORADO STATESMAN Consumer activist and corporate gadfly Ralph Nader urged adoption of a single-payer system of delivering health care to Americans last week when he spoke to an overflow audience in a club room at the University of Colorado's football stadium. Since universal health coverage was first proposed by President Harry […]This content is only available to subscribers. Login or Subscribe Share this:EmailTwitterFacebook Current News, Default


text:
Barry Walsh
Vice's Shane Smith talks revolution
The founder and CEO of the media co presented the MacTaggart Memorial Lecture at the Edinburgh International Television Festival.
Our House preps culinary competition for Blue Ant
So You Think You Can Be a Chef begins production in Toronto in September. (Our House Media president Joe Houlihan pictured.)
eOne shares buoyed by more takeover talk
On the heels of eOne's rejection of ITV's takeover bid last week, reports circle that private equity firm KKR & Co. is considering its own offer.
USA, Syfy cutting staff numbers in restructure
Following the recent formation of NBCU Cable's Entertainment Networks Group, approximately 25 staff members from Syfy and USA Network will be let go. (Pictured: ENG president Chris McCumber)
IMAX makes major VR moves
IMAX will develop a cinema-grade VR camera with Google, and partner with StarVR headset company Starbreeze on VR exhibits.
Morley Safer passes away at 84
The veteran journalist and 60 Minutes correspondent had officially retired from the long-running CBS newsmagazine last week.
Remembering Canadian TV pioneer Ralph C. Ellis
The award-winning independent producer and distributor, and founder of Ellis Entertainment, has died at the age of 91.
CBC launches short docs platform
The new site will focus on shareable content that is less than 30 minutes.
Entertainment One acquires Renegade83
eOne has acquired a 65% stake in the California-based prodco behind Naked and Afraid (pictured).
NBCU forms Entertainment Networks group with USA, Syfy
The restructuring will see USA topper Chris McCumber (pictured) oversee the newly formed group as president and Syfy president David Howe take on the newly created role of president, strategy and commercial growth.
RIDM responds to 'of the North' controversy
The film from Quebec filmmaker Dominic Gagnon, which screened twice during the Montreal-based doc festival, uses various clips of Inuit culled from video sites.
Pilgrim Studios will still have "full creative autonomy" under its current management team, headed by founder Craig Piligian (pictured).


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November 4, 2010
Photo by S. Dixon for WorldRedEye.com
Legendary emcee Pharoahe Monch performed at Wall Lounge Miami at The W hotel a couple night ago. The World Red Eye folks were there to catch the action. Guests were at hand to see him perform while DJ Ross One was on the 1's and 2's. Click here for all the photos.


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Set The Agenda Items10 textbooks were found
Beyond Liberalism and Fundamentalism: How Modern and Postmodern Philosophy Set the Theological Agenda (Rockwell Lecture)RentList Price:$35.00
From the Earth Summit to Local Agenda 21: Working Towards Sustainable Development (Earthscan Library Collection: International Environmental Governance Set)List Price:$125.00
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Taking Minutes of Meetings: Set the Agenda; Identify What to Note; Write Accurate Minutes (Creating Success)RentList Price:$14.95
RentList Price:$33.99
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From the Earth Summit to Local Agenda 21: Working towards sustainable development (International Environmental Governance Set)List Price:$54.95
Confronting Globalization: The Quest for a Social Agenda (Bulletin of Comparative Labour Relations Series Set)RentList Price:$213.00
California and the World Ocean '97: Taking a Look at California's Ocean Resources: An Agenda for the Future: Conference Proceedings, 2 vol. setList Price:$164.00
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Being productive is a function of many things. One of which is the speed of your Productivity App. With the help of our user community, we are always looking for ways to improve the speed of getting "stuff" into IQTELL. And ultimately getting "stuff done"!
With this latest new feature/improvement, anytime you click on any Date field, e.g., Start Date, Due Date, Follow-up Date, etc., you get the "EZ Date Picker".
Notice, that you can also use the shortcut keys. For example, if I want to quickly set the Due Date as 2 weeks from today, I'll simply click "2" on my keyboard.
Hope you enjoyed this one!
P.S – To those using our Android or iOS apps, Productivity apps, be sure to check out the IQTELL Web app. Everything is auto-synced between IQTELL's apps.


text: This is the way we should respond to things like this I am very proud of my LGBT family and friends. Even Jesus said show the love to those who persecute you. As a gay Christian I say to my LGBT church and are supporters we need to get louder and show the great love and unity that we are being denied by the far right. I love you all greatly.


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The Cabinet is the US President's appointed & confirmed advisors. There are many Cabinet positions, each the head of their respective Executive Department.
Other democratic nations also have a cabinet.
- Secretary of State
- Secretary of Defense
- Attorney General
- Secretary of the Treasury
- Secretaries of a lot of stuff like Agriculture & Education
- Secretary of Homeland Security (the only one that really matters)


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If you are looking for good pizza in the Northern Virginia area, look no further than The Don's; you will NOT be disappointed. Many of the gourmet ingredients, like the olive oil, come straight from the motherland, Italy. They have a ton of fresh toppings and even offer Gluten free dough. My favorite is the Sicilian Hit. Recently, they also started offering sandwiches that are HUGE and super tasty and they have Junior's cake and Cheesecakes straight from NY. The owner, Waleed and his family are so nice and the customer service is great. They are very involved in the local community as well. I tried Fireworks Pizza in Leesburg for the first time last weekend and honestly, the Don's is hands down MUCH better in my opinion.
Own or manage this property? Claim your listing for free to respond to reviews, update your profile and much more.


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The Pennsylvania Attorney General's office, through its website, announced Saturday that Penn State athletic director Tim Curley and university senior administrator Gary Schultz had been charged with perjury and failure to report suspected child abuse.
The charges stem from testimony the two made to a grand jury investigating the Sandusky case.
The attorney general's office lays out its cases against Sandusky, Curley and Schultz AT THIS LINK (PDF).
Sandusky was arraigned Saturday in Centre County Saturday and released on $100,000 unsecured bail. He was ordered to have no contact with minors.
The Harrisburg Patriot-News reported that Curley and Schultz would turn themselves in to officials in Harrisburg on Monday. They are being prosecuted in Dauphin County.
Curley has been athletic director at Penn State since 1993.
Click here for more coverage of the case by FightOnState.com.


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Devo - [I Can't Get No] Satisfaction (Video) MP3
2009 WMG [I Can't Get No] Satisfaction (Video)
Devo - (I Can't Get No) Satisfaction MP3
From their first album: Q: Are We Not Men A: We Are Devo! I do not own the copyrights to any of the content in this video.
Devo I Can't Get No Satisfaction MP3
Devo (I Can't Get No) Satisfaction Q: Are We Not Men? A: We Are Devo!
Devo - Satisfaction | Live in Santa Ana | Moshcam MP3
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Devo I Can't Get No Satisfaction MP3
Devo (/ˈdiːvoʊ/, originally /diːˈvoʊ/)[4] is an American rock band formed in 1972, consisting of members from Kent and Akron, Ohio. The classic line-up of ...
DEVO - Satisfaction - Live 1978 Chorus TV Paris MP3
DEVO performs 'Satisfaction' (Rolling Stones cover) Live on Chorus TV - 1978 Paris, France http://www.clubdevo.com/
Devo - (I Can't Get No) Satisfaction (Live 1978) MP3
DEVO - Satisfaction - live 1977 MP3
9 July 1977 : Devo in live at Max's Kansas City, in New York. Satisfaction (R. Stones cover) in 1977 they played it more slowly than in 1978 !
DEVO - I Can' t Get No Satisfaction (Casino soundtrack) MP3
Devo 'Satisfaction' live at The Fillmore, SF, 14 Jan 2012 MP3
Wide video shot, audio recorded, mixed and mastered, and all video post-production by Brian Albers at Humorless Productions. Front row video by lairygirl.
Satisfaction (original single version) - Devo MP3
Created with http://www.mp32tube.com.
Devo I Can't Get No (Satisfaction) MP3
An experimental video. Fotos and still images captured using the software Eyesweb. It's about me.
Devo - Satisfaction (I Can´t Get No ) - HD MP3
DEVO - Satisfaction - live 1988 MP3
DEVO outdoor daytime show at UC Berkeley in California in 1988. Satisfaction ( best Rolling Stones cover ever ! ) David Kendricks (drums) Bob (guitar solo) and ...
Devo - Satisfaction (I can't get no) - Drum Cover MP3
Same simple beat through the song but it is fun to play and it's just a great song. All music property of Devo.
Devo - (I can't get no) Satisfaction - Karaoke - Instrumental cover MP3
Your comments are much appreciated. I made this just for fun, no commercial purposes whatsoever. For all of you who prefers (like me) this version to the ...
Devo - Live In Japan 5-28-79 (Part 1) MP3
Intro, "Secret Agent Man," "Satisfaction," "Too Much Paranoia"
Devo - Satisfaction [Deconstruction] MP3
Not a remix, instead I broke down the rhythm elements of this distinctive version of the Stones classic. BabyBabyBabyBabyBabyBabyBaby.
DEVO Satisfaction EZ Listening MP3
Devo - Satisfaction - bass cover MP3


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Non-Latex Flex-Band Three-Pack ENGLISH PACKAGING includes 1 each: light (orange), regular (lemon) & extra-strength (lime) non-latex Flex-Bands
Made with TPE (Thermo Plastic Elastomer) these bands are the perfect alternative to traditional Flex-Bands. This economical tool can help exercisers simulate many of the moves performed on the ever-popular Reformer. Includes 3 Flex-Bands; light (orange), regular (lemon) and extra-strength (lime) and the Ultimate Body Sculpting DVD. Ideal for customers with latex allergies or sensitivities. Each band is 6.5' long. Before starting any rigorous exercise program, it is recommended that you consult your physician for a complete check-up. Be sure to follow all enclosed instructions carefully for proper use of the equipment.


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Term 1 ends at 3.00pm for Year 1 - 6 and at 3.20pm for Years 7 & 8. read more
The Junior Enrichment Programme with Cathy Chambers takes place at 9.00am today. For further details please see your son's class teacher. read more
The Parents' Association will be running a sausage sizzle at lunchtime. Please see your son's class teacher for further information. read more
The CPSSA Swimming Championships will be taking place today for those boys who qualified from the ISSA Swimming Championships. Please contact Mr G Taylor, Sports Director, for further information. read more
The Term 1 Parents' Association AGM Meeting will take place at 6.00pm. All parents are welcome to attend. For more information please contact Jane Lattimore, Parents' Association President, on 021 387 747. read more
The Trust Board Meeting will commence at 5.30pm. read more
Art's Week takes place this week, with lots of class and year level activities. Please contact Mrs Sandra Johnson, Head of Art, for further information. read more
The Boarding Experience Weekend at Christ's College takes place this weekend. Please see Mr M Hocquard, Director of Boarding, or Matron for full details. read more


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Fujifilm is kicking off the New Year right, with a slew of new point and shoot cameras and another new X-mount lens. The new cameras hail from the long zoom and "tough" camera side of the Fuji point-and-shoot spectrum.
Following the successes of the RX1 and RX100 cameras, Sony has just announced the Cyber-shot DSC-RX10, an advanced compact camera that borrows features and design from both cameras but differs in several significant ways.
Sony has just released several updates and new models in their lines of photographic equipment. Two interchangeable-lens cameras, new lenses and Cyber-shot point-and-shoots are available. These cameras build upon Sony's extensive history of making cameras with high imaging performance that are enjoyable to use regularly.
Fujifilm has recently introduced several new point-and-shoot cameras that possess a broad array of features and serve to improve upon their existing line of non-interchangeable lens X series and FinePix digital cameras. At the forefront of this announcement are the highly anticipated X100S and X20 cameras, which feature retro styling combined with high imaging performance.
The Canon G series has long been considered an advanced, high end point-and-shoot camera, and for good reasons. The PowerShot G15 carries on in this tradition and in this five-minute overview, Larry Becker covers all the reasons why this camera is so powerful.
Even with the dropping prices of DSLRs and the rising appeal of mirrorless interchangeable lens cameras, the so called "bridge" camera is as popular as ever. "Bridge cameras" are big-bodied, big zoom point-and-shoot cameras that have the look and heft of a more expensive and feature-heavy camera but still maintain the simplicity of the standard and compact point-and-shoot cameras.
This video covers Nikon's high-end COOLPIX P7100 10.1mp camera.
The Nikon COOLPIX P7700 is the successor to the high end, point-and-shoot P7100, and features several refinements for greater control and imaging capabilities. Much of the technology has been retained from the predecessor, such as the 1/1.7"-sized sensor, large 3.0" articulating LCD monitor, 28-200mm (35mm equivalent) NIKKOR ED lens and full manual controls.
Panasonic has just announced a range of new cameras spanning the spectrum from compact point and shoot to an interchangeable lens mirrorless camera, with multiple choices in between.
Sony's newly announced DSC-RX100 is a unique compact camera in design and performance. While it takes on the demeanor of a small, simple-to-use camera, it also provides a great deal of appeal for even the professional photographer.