text:
The quest for narrow admissible tuples July 2, 2013Posted by Scott Morrison in polymath.
Tags: admissible tuples, polymath8, zhang
(A guest post by Andrew Sutherland.)
With more than 400 comments tacked on to the previous blog post, it's past time to rollover to a new one. As just punishment for having contributed more than my fair share of those comments, Scott has asked me to write a guest post summarizing the current state of affairs. This task is made easier by Tao's recent progress report on the polymath project to sharpen Zhang's result on bounded gaps between primes. If you haven't already read the progress report I encourage you to do so, but for the benefit of newcomers who would like to understand how our quest for narrow admissible tuples fits in the bounded prime gaps polymath project, here goes.
The Hardy-Littlewood prime tuples conjecture states that every admissible tuple has infinitely many translates that consist entirely of primes. Here a tuple is simply a set of integers, which we view as an increasing sequence ; we refer to a tuple of size as a -tuple. A tuple is admissible if it does not contain a complete set of residues modulo any prime . For example, 0,2,4 is not an admissible 3-tuple, but both 0,2,6 and 0,4,6 are. A translate of a tuple is obtained by adding a fixed integer to each element; the sequences 5,7,11 and 11,13,17 are the first two translates of 0,2,6 that consist entirely of primes, and we expect that there are infinitely more. Admissibility is clearly a necessary condition for a tuple to have infinitely many translates made up of primes; the conjecture is that it is also sufficient.
Zhang proved a weakened form the prime tuples conjecture, namely, that for all sufficiently large , every admissible -tuple has infinitely many translates that contain at least 2 primes (as opposed to ). He made this result explicit by showing that one may take , and then noted the existence of an admissible -tuple with diameter (difference of largest and smallest elements) less than 70,000,000. Zhang's -tuple consists of the first primes greater than , which is clearly admissible. As observed by Trudgian, the diameter of this -tuple is actually less than 60,000,000 (it is precisely 59,874,954).
Further improvements to Zhang's bound came rapidly, first by finding narrower admissible -tuples, then by optimizing and the critical parameter on which it depends (this means making larger; is proportional to ). Since it began on June 4, the polymath8 project has been working along three main lines of attack: (1) improving bounds on and a related parameter , (2) deriving smaller values of from a given pair , and (3) the search for narrow admissible -tuples.You can see the steady progress that has been made on these three interlocking fronts by viewing the list of world records.
A brief perusal of this list makes it clear that, other than some quick initial advances made by tightening obvious slack in Zhang's bounds, most of the big gains have come from improving the bounds on (edit: as pointed out by v08ltu below, reducing the dependence of on from to was also a major advance); see Tao's progress report and related blog posts for a summary of this work. Once new values of and have been established, it is now relatively straight-forward to derive an optimal (at least within 1 or 2; the introduction of Pintz's method has streamlined this process). There then remains the task of finding admissible -tuples that are as narrow as possible; it is this last step that is the subject of this blog post and the two that preceded it. Our goal is to compute , the smallest possible diameter of an admissible -tuple, or at least to obtain bounds (particularly upper bounds) that are as tight as we can make them.
A general way to construct a narrow admissible -tuple is to suppose that we first sieve the integers of one residue class modulo each prime and then choose a set of survivors, preferably ones that are as close together as possible. In fact, it is usually not necessary to sieve a residue class for every prime in order to obtain an admissible -tuple, asymptotically an bound should suffice. The exact number of residue classes that require sieving depends not only on , but also on the interval in which one looks for survivors (it could also depend on the order in which one sieves residue classes, but we will ignore this issue).
All of the initial methods we considered involved sieving residue classes 0 mod , and varied only in where to look for the survivors. Zhang takes the first survivors greater than 1 (after sieving modulo primes up to ), and Morrison's early optimizations effectively did the same, but with a lower sieving bound. The Hensley-Richards approach instead selects survivors from an interval centered at the origin, and the asymmetric Hensley-Richards optimization shifts this interval slightly (see our wiki page for precise descriptions of each of these approaches, along with benchmark results for particular values of interest).
But there are sound practical reasons for not always sieving 0 mod . Assuming we believe the prime tuples conjecture (which we do!), we can certainly find an optimally narrow admissible -tuple somewhere among the primes greater than , all of which survive sieving 0 modulo primes . However, the quantitative form of the prime tuples conjecture tells us roughly how far we might need to search in order to find one. The answer is depressingly large: the expected number of translates of any particular admissible -tuple to be found among the primes is thus we may need to search through the primes in an interval of size exponential in in order to have a good chance of finding even one translate of the -tuple we seek.
Schinzel suggested that it would be better to sieve 1 mod 2 rather than 0 mod 2, and more generally to sieve 1 mod for all primes up to some intermediate bound and then switch to sieving 0 mod for the remaining primes. We find that simply following Schinzel's initial suggestion, works best, and one can see the improvement this yields on the benchmarks page (unlike Schinzel, we don't restrict ourselves to picking the first survivors to the right of the origin, we may shift the interval to obtain a better bound).
But sieving a fixed set of residue classes is still too restrictive. In order to find narrower admissible tuples we must relax this constraint and instead consider a greedy approach, where we start by picking an interval in which we hope to find survivors (we know that the size of this interval should be just slightly larger than ), and then run through the primes in order, sieving whichever residue class is least occupied by survivors (we can break ties in any way we like, including at random). Unfortunately a purely greedy approach does not work very well. What works much better is to start with a Schinzel sieve, sieving 1 mod 2 and 0 mod primes up to a bound slightly smaller than , and then start making greedy choices. Initially the greedy choice will tend to be the residue class 0 mod , but it will deviate as the primes get larger. For best results the choice of interval is based on the success of the greedy sieving.
This is known as the "greedy-greedy" algorithm, and while it may not have a particularly well chosen name (this is the downside of doing math in a blog comment thread, you tend to throw out the first thing that comes to mind and then get stuck with it), it performs remarkably well. For the values of listed in the benchmarks table, the output of the greedy-greedy algorithm is within 1 percent of the best results known, even for , where the optimal value is known.
But what about that last 1 percent? Here we switch to local optimizations, taking a good admissible tuple (e.g. one output by the greedy-greedy algorithm) and trying to make it better. There are several methods for doing this, some involve swapping a small set of sieved residue classes for a different set, others shift the tuple by adding elements at one end and deleting them from the other. Another approach is to randomly perturb the tuple by adding additional elements that make it inadmissible and then re-sieving to obtain a new admissible tuple. This can be done in a structured way by using a randomized version of the greedy-greedy algorithm to obtain a similar but slightly different admissible tuple in approximately the same interval, merging it with the reference tuple, and then re-sieving to obtain a new admissible tuple. These operations can all be iterated and interleaved, ideally producing a narrower admissible tuple. But even when this does not happen one often obtains a different admissible tuple with the same diameter, providing another reference tuple against which further local optimizations may be applied.
Recently, improvements in the bounds on brought below 4507, and we entered a regime where good bounds on are already known, thanks to prior work by Engelsma. His work was motivated by the second Hardy-Littlewood conjecture, which claims that for all , a claim that Hensley and Richards showed is asymptotically incompatible with the prime tuples conjecture (and now generally believed to be false). Engelsma was able to find an admissible 447-tuple with diameter 3158, implying that if the prime tuples conjecture holds then there exists an (infinitely many in fact) for which , which is greater than . In the process of obtaining this result, Engelsma spent several years doing extensive computations, and obtained provably optimal bounds on for all , as well as upper bounds on for . The quality of these upper bounds is better in some regions than in others (Engelsma naturally focused on the areas that were most directly related to his research), but they are generally quite good, and for up to about 700 believed to be the best possible.
We have now merged our results with those of Engelsma and placed them in an online database of narrow admissible -tuples that currently holds records for all up to 5000. The database is also accepts submissions of better admissible tuples, and we invite anyone and everyone to try and improve it. Since it went online a week ago it has processed over 1000 submissions, and currently holds tuples that improve Engelsma's bounds at 1927 values of , the smallest of which is 785. As I write this stands at (subject to confirmation), which happens to be one of the places where we have made an improvement, yielding a current prime gap bound of 6,712 (but I expect this may drop again soon).
In addition to supporting the polymath prime gaps project, we expect this database will have other applications, including further investigations of the second Hardy-Littlewood conjecture. As can be seen in this chart, not only have we found many examples of admissible -tuples whose diameter satisfies , one can view the growth rate of the implied lower bounds on in this chart.
- 1. v08ltu - July 2, 2013
I would say that improving the k_0 bound from 1/omega^2 to 1/omega^{3/2} [step (2) in your list] was a big gain also, at least back then it was almost a factor of 100. The introduction of the Pintz method has also streamlined the computation process to pass from omega to k_0. Since then it has largely been improving omega.
- 2. Andrew Sutherland - July 2, 2013
@v08ltu: Good point, this was an oversight on my part. I've made appropriate edits and asked Scott to post them — thanks for the correction.
- 3. M Flax - July 3, 2013
I'm curious: have you tried a one-step lookahead as a variant of the greedy algorithm? Suppose that the current prime is p_1 and the next is p_2. You could accumulate the number of residues mod p_1, mod p_2, and mod (p_1 p_2) into three arrays A, B, C (of sizes p_1, p_2 and p_1 p_2 respectively of course). From this you could determine which pair of residues (e.g. first mod p_1 and then mod p_2) would minimize the number of elements removed. (E.g. the r_1 and r_2 such that A_r_1 + B_r_2 – C_r_1,r_2 is least.)
You could then actually sieve by the mod p_1 residue and then repeat with p_2 and p_3. This might prevent the greedy step from being too shortsighted?
- 4. Andrew Sutherland - July 3, 2013
@Flax: I actually did implement a one-step look-ahead and found that it rarely changed the choices, and that, moreover, the local optimizations were generally able to correct the same sort of mistakes that a one-step look-ahead might avoid. Given that we are running the algorithm in a loop where we iteratively apply random perturbations, I concluded that it made more sense to keep the core component as fast as possible, so that it could process more iterations in a given period of time.
But that's not to say that a deeper look-ahead might not pay dividends, and I would certainly encourage experimentation along these lines.
There is a trade-off between designing for an algorithm that can fairly quickly get to a good solution versus an algorithm that is willing to burn a lot of time trying to find the best solution possible. Since the polymath8 project started, I don't think the value of k0 has ever remained stable for more than 4 days at a time, and especially when k0 was much larger than it is now, it was useful to have an algorithm that could quickly get to a good, it not great, solution. This has also been very useful in building the database of admissible tuples for k up to 5000 and being able to quickly populate it with reasonably narrow tuples.
But once things settle down and we have a "final" value for k0, it will be time to buckle down and focus a lot of computation on a particular problem instance (assuming k0 does not drop below 342). I don't think we're quite there yet (it looks like k0 could soon drop to around 720). But that doesn't mean we shouldn't be thinking about better algorithms now. Another thing worth considering is designing an algorithm that targets a specific diameter, say 2 less then the best value currently known.
- 5. Andrew Sutherland - July 4, 2013
A quick update: it looks like we now have a new target k0=720,
http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489, which gives a prime gap bound of 5414. This bound comes from Engelsma's original tables and is close to (but not quite in) the region where they are believed to be optimal (but perhaps Thomas can chime in here).
- 6. Pace Nielsen - July 5, 2013
Can someone comment on how difficult it would be to extend the range of provable optimality?
- 7. Andrew Sutherland - July 5, 2013
@Pace: Thomas Engelsma is surely the best person to answer this question, but I am running some experiments now that might shed some light on it (the short answer is that I think going beyond k=342 is certainly feasible, but getting to k=720 almost certainly isn't).
In the mean time, let me note the lower bounds that we do have for H(720). Having fixed a parity issue in my code, I can now confirm all the lower bounds computed by Avishlay using the partitioning method with Engelsma's bounds for k = 4832.
We can improve this by using one of two additional lower bounds that Engelsma computed. Let k(w) be the largest k for which there exists an admissible k-tuple of diameter strictly less than w. In addition to determining k(w) for all w up to 2329 (where k(2329)=342), Engelsma determined k(2529)=369 and k(2655)=384 (see http://www.opertech.com/primes/k-tuples.html). (NB: these don't imply H(369)=2528 and H(384)=2654, but they do imply H(370) > 2529 and H(385) > 2655, and also that k(w) <= 384 for all w <= 2655 and k(w) <= 369 for all w <= 2529).
If we partition an interval of width (diameter+1) w=4924 into intervals of width 2269 and 2655, we get
k(4924) 4924, and by parity, H(720) >= 4926.
But this is well short of the upper bound H(720) <= 5414.
- 8. Andrew Sutherland - July 5, 2013
WordPress apparently ate part of #7. Tthe end of the second paragraph should say that H(720) >= 4832 follows from a partitioning bound using Engelsma's optimal bounds for k <= 342. A complete table of such bounds for k = 2330, but we believe H(343)=2340).
- 9. Andrew Sutherland - July 5, 2013
Bah, wordpress won't let me repost the link to Avishlay's table for some reason, the original post is at https://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24223.
Finally, my last point was that improving lower bounds on H(k) for any k > 342 (or on k(w) for w > 2329) could potentially improve the lower bound for H(720), as well as many others. So an initial goal might be to prove H(343)=2340. We currently only know that H(343) >= 2330.
- 10. xfxie - July 5, 2013
@Pace: I believe Thomas and Klas should have much more to say on your question. Here are only some thoughts based on my limited experience.
I did implement a recursive depth-first search strategy, constructing from 0 with feasible values and simple pruning based on and the current best $H$ value, to do exact computation. In my realization, the performance was highly dependently on the set of residues at small primes (due to the combinatorial nature) that are used by optimal solutions. For example, it needed around 105.2 s to solve =30, but almost needed no time to solve =50. Besides, there might have many optimal solutions for the same value. (For example, for =32, the first, second, and third solutions can be found in 0.0s, 24.23s, and 24.26s.) I believe that the search might be somewhat accelerated (but only for some small instances) if using some randomness in selecting the next node.
The problem might be tackled much more efficiently if some nice properties about possible optimal substructures can be established, for pruning the search space at very early stages.
BTW: Maybe the discussion about the problem property in the previous blog, e.g., #391, #392, #405, and #406 (by Terence and Andrew), can stimulate more positive clues in the near future.
- 11. Andrew Sutherland - July 6, 2013
Last night I ran a program to try and naively assess the feasibility of computing optimal bounds on H(k). Given k and a target upper bound d on H(k), the program attempts to prove that H(k) >= d by sieving the interval [0,d-2] and verifying that you cannot get k survivors (if this verification fails, it outputs an admissible k-tuple of diameter less than d, thereby proving H(k) < d. It assumes (wlog) that 0 is unseived, so for each prime p <= k it checks the residue classes from 1 to p-1 mod p, pruning the search by always requiring the number of survivors to stay above k.
Running overnight on my laptop, the program was able to verify all of Engelsma's lower bounds for k up to 200. The longest computation was for k=198, which took about 1.5 hours and required about 7.5*10^11 sieving operations; it reached a maximum search depth of 24 primes (up to p=89), with an "average" search depth of about 11.5.
The chart below shows a plot of k vs the base-10 log of the # of sieving operations needed to compute a tight lower bound on H(k). A crude back-of-the-envelope argument (see below) suggests that the "feasible" limit of this approach is around 10^20 sieving operations. Depending on how you fit the curve (it's clearly subexponential) this corresponds to k around 400-500. So I think it is definitely feasible to get tight bounds for k well above 342, but 720 is clearly out of reach with this approach (something more clever would be required).
Optimistic back-of-the-envelope calculation that intentionally ignores many technical details: With a straight-forward C implementation, the "average" sieving operation takes about 70-80 nanosecs on my laptop (including the time to update the survivor count), with the interval size around 1000. As k increases the interval size goes up but the cost of sieving goes down (sieving mod bigger primes is faster), so let's call it a wash and suppose the time to sieve one residue class is a constant. One can gain a factor of 2 by taking advantage of symmetry (I didn't), and I'm willing to believe that a careful assembly language implementation that avoids branching inside the critical loops might gain another factor of 5 or 10. So say 5-10 nanosecs per sieving operation on a sequential non-SIMD architecture. It's not unreasonable to suppose we might gain a factor of 500 or even 1000 due to parallelism on a GPU (combination of SIMD and multi-threading). So for the sake of argument let's suppose we can do 10^11 sieving ops/sec/GPU, and that we are willing to throw 100 GPUs at the problem and let them grind away for, say, 10^7 seconds (about 4 months). That gets us 10^20 sieving operations.
- 12. Thomas J Engelsma - July 6, 2013
@Scott: Much to my wife's demise, I took the exhaustive search folders and notes with me on vacation. A little history on the exhaustive search program – The first program was tuned to search widths less than 2048. Then it was rewritten to search larger widths, but had a restriction that it could only work with primes less than 256. For the widths of interest, this was not a concern, a simple VB program tested the remaining primes whenever a pattern was found. The current version has the ability to break the search into segments (5760 separated branches) so in the event of power loss the search was not lost. Also, the program has four search parameters, which may be useful to this project.
The parameters are search for:
1) pi(w)+1 elements
2) k(w) elements
3) kb(w)+1 elements
4) kb(w) elements
Type 4 searches are my main objective, as the number of unique patterns is of major interest for current study, but the type 2 searches could be modified to k(w)+1 and may prove beneficial for this project. Quick note is k(w) is first occurrence, while kb(w) has an additional restriction that both the first and last element of the tuple must be a prime, this allows pre-pruning the search tree, one residue class can be removed because of the first element, and an additional element can be removed when the width is a multiple of the search prime plus one. Right now I am looking into making the program able to be distributed, where a width and a block of the 5760 branches can be run on a computer and returns the results.
@xfxie: WOW, love the work you are doing. I have been following your progress, and occasionally testing the intermediate widths for kb(w) values (the gray erratic line in http://www.opertech.com/primes/trophy.bmp ) Impressive! The best fit line is not implying linearity, but is a metric to measure improvement. When searching stopped, the rate was an additional packed prime for every increase of 267 in width. The work you have done in the past week has this now at 265 !!! (If you keep this up the k-tuples conjecture may be in jeopardy) I have not integrated your work into the current patterns database yet, due to the nature of embedded patterns, I plan to purchase a separate hard drive and add a single pattern at a time so a single core residue set does not dominate. Also, here real soon I hope to do a quick internal gap study, as I believe your search routines use a 'Greedy' algorithm, which tends to produce two large gaps instead of many small gaps. If you are using existing tuple core residue sets, this may not be true (hopefully you are).
@Drew: Having some deja-vu here. You are running into one of the reasons I insisted to find and retain ALL patterns for a specific kb(w) search. To save some serious time, original searching needed an good approximate start value for k, to do this tuples were partitioned similar to your 'lower bound'. Values calculated by adding two tuples eg. kb(2269)=335 and kb(2655)=385 implying kb(4925)<=370 could be improved by using existing patterns one at a time to extend the width and verifying that no section in the new tuple violated a maximum density condition determined by the known smaller tuple data. I wrote a spaghetti code program to do this, and probably could duplicate it to achieve a better lower bound for a specific k0.
@Drew: There was mention of embedded tuples in a previous post. Many 'best' tuples are extensions of an existing smaller tuple, and an attempt to use this was tried. But of more interest was the largest embedded tuple that was smaller than half the target tuple width. Also, a study was done on multiple smaller tuple patterns that were embedded, in an attempt to find start search criteria.
Not sure who this is for, as I cannot find the post now, but the Merten's theorem is of major relevance and excited to see work heading in that direction.
I'm sure I've missed some postings, I will get caught up soon.
- 13. Thomas J Engelsma - July 6, 2013
@Drew: Here are some rough timings for exhaustive search of kb(w), your search for k(w)+1 would be faster, as there would be at least one less branch to search, allowing an even earlier exit condition.
10/8/2008 >> w=2203 completed
1/7/2009 >> w=2263 completed (30 widths)
2/1/2009 >> w=2303 completed (20 widths)
2/7/2009 >> w=2311 completed (4 widths)
9/12/2009 >> w=2331 completed (10 widths)
Problem with the above is there were many idle times in the above, but this could give you a rough benchmark for time consumed. Also, as you have already seen (aka k=198 in above post) every width has it's own time fingerprint.
As for using SIMD and tight assembly code. the inner loop in the program consumed somewhere around 68 clock cycles. (It purred nicely).
- 14. Thomas J Engelsma - July 6, 2013
@Andrew: Ran a test for you, here is an output log for doing a type 2 test – search for k(w)+1. Using the program as is requires checking each width because of the restriction that both first and last elements are prime locations. The program could be altered to remove that restriction, thereby allowing checking only w: 1257 to verify that k:198 cannot happen earlier. This would cost 20% more time max. Would be major reduction in time. For some reason I cannot get the compiler on the laptop to recognize the SIMD Boolean instructions or I'd start altering it today.
Using program as is : approx 48 minutes to verify all widths
Altering program : approx 5 minutes to verify next smallest width
When in the larger widths this could be major.
06 14:16:46 2) w: 1237 k: 197 (0)
06 14:17:58 2) w: 1237 k: 197 failed (1-5760)
06 14:19:19 2) w: 1239 k: 198 failed (1-5760)
06 14:21:05 2) w: 1241 k: 198 failed (1-5760)
06 14:23:05 2) w: 1243 k: 198 failed (1-5760)
06 14:25:22 2) w: 1245 k: 198 failed (1-5760)
06 14:28:38 2) w: 1247 k: 198 failed (1-5760)
06 14:32:09 2) w: 1249 k: 198 failed (1-5760)
06 14:38:08 2) w: 1251 k: 198 failed (1-5760)
06 14:42:49 2) w: 1253 k: 198 failed (1-5760)
06 14:50:08 2) w: 1255 k: 198 failed (1-5760)
06 14:58:20 2) w: 1257 k: 198 failed (1-5760)
06 15:02:38 2) w: 1259 k: 199 failed (1-5760)
- 15. Andrew Sutherland - July 7, 2013
@Thomas: Thanks for the timings. It would be interesting to know how they scale with k/w (i.e. how does the shape of the curve compare to the chart in #11, say as k goes from 2 to 200 with w=H(k)+1).
- 16. Thomas J Engelsma - July 8, 2013
Overnight, the exhaustive search program was set to iterate through a list of w's and search a single sub-branch of the tree. The w's are two smaller than the smallest known value for each k.
(eg. a pattern of 342 exists for a width of 2329) Verifying that no pattern of 342 exists in a width of 2327 would thereby set 2328 as the optimal diameter for k0=342.
Complete timings are erratic, but show some of the overall growth. If time permits I may redo the test using a different branch, this test was branch 642 or 5760. The pattern tested probably had an early exit for width 2409, other patterns will probably have timings closer to those of 2405.
Width K0 1 branch Complete
w:2327 k:342 9m:33s 38.2 days
w:2339 k:343 30m:52s 123.5 days
w:2341 k:344 38m:47s 155.1 days
w:2353 k:345 33m:49s 135.3 days
w:2357 k:346 34m:13s 136.9 days
w:2363 k:347 15m:44s 62.9 days
w:2375 k:348 48m:42s 194.8 days
w:2381 k:349 19m:55s 79.7 days
w:2387 k:350 23m:01s 92.1 days
w:2393 k:351 1h:18m:59s 315.9 days
w:2399 k:352 15m:27s 61.8 days
w:2405 k:353 36m:33s 146.2 days
w:2409 k:354 9m:01s 36.1 days
w:2417 k:355 14m:56s 59.7 days
w:2439 k:356 2h:23m:02s 572.1 days
w:2443 k:357 1h:46m:02s 424.1 days
w:2451 k:358 3h:16m:02s 784.1 days
w:2459 k:359 1h:03m:13s 252.9 days
w:2473 k:360 49m:09s 196.6 days
w:2479 k:361 1h:26m:24s 345.6 days
w:2483 k:362 1h:46m:50s 427.3 days
w:2489 k:363 1h:14m:59s 299.9 days
w:2495 k:364 19m:17s 77.1 days
w:2501 k:365 1h:23m:28s 333.9 days
w:2515 k:366 2h:25m:19s 581.3 days
w:2519 k:367 1h:03m:32s 254.1 days
w:2521 k:368 1h:08m:01s 272.1 days
- 18. Thomas J Engelsma - July 13, 2013
Have a couple more conditional jump statements to optimize and the exhaustive search program is ready to target k0 values.
Is there a specific k0 that we want targeted ??
The program can currently handle widths up to 3400, by changing a constant this value can be increased. It tests primes up to 255, secondary testing can prove admissibility for remainder of the primes.
(Note: for the 5414 width, the largest destructive prime is 239)
- 19. Andrew Sutherland - July 13, 2013
The current target remains k0=720 (width 5414), but it is possible that this could come down to something close to k0=630 (width 4660). I suggest holding off on any major computations until this is nailed down, but if it is easy to get timings for 1 branch just to see what things look like, that would be interesting.
- 21. Joe Fitzsimons - July 14, 2013
I'm afraid I haven't been following this closely, and only started reading through the relevant posts this evening, so please excuse me is what I am about to say is appallingly misinformed. Apologies in advance for the lack of latex. I don't quite know how to do it in wordpress comments.
It seems to me that the residues you sieve with don't matter in the grand scheme of things, and only the location you look for the survivors counts. The reason I say this is that given some set of pairs of residues and primes, (r_i,p_i), we could always compute some number C by solving C = -r_i mod p_i using the Chinese Remainder Theorem, and then adding this constant to the elements of H to form a new admissible set of the same cardinality, where now all of the residues are non-zero.
So, this seems to suggest that we may as well focus on sieving out the zero residues, and instead simply choose where to look strategically. But this should be fairly simple: for any given interval of T integers, each prime p_i eliminates either ceil(T/p_i) or floor(T/p_i) integers. Clearly we would like as few integers to be eliminated as possible, meaning we want to make sure that different primes eliminate the same integer as often as possible. But there is no reason to believe that we should find this around 0, but rather around the product of all the primes less than or equal to k_0. Let X = \prod_(i:p_i<=k0) p_i. Then if we look at intervals centred on X, it should minimize the number of integers eliminated, by maximizing the number of times any tuple of primes less than or equal to k_0 eliminate the same integer, and hence minimize the total interval required to find an admissible set. The size of the interval required should then be straight-forward to calculate directly from k0.
- 22. Joe Fitzsimons - July 14, 2013
On closer inspection, it seems this is effectively the same as the Hensley-Richards approach, so please feel free to ignore/delete my comments.
- 23. Joe Fitzsimons - July 14, 2013
Also, I should have suggested intervals starting at X-1 to maximize the number of times tuples occur (centering minimizes them), though this has the disadvantage of ensuring that ceil(T/p_i) integers are eliminated, so it was incorrect for me to claim it necessarily minimized the interval, although it does appear to be within |{p_i}_i| of the actual minimum.
- 24. Andrew Sutherland - July 14, 2013
@xfxie: I notice that the submission server was down for several hours today due to a communications problem with the back end. It's back up now and your recent submissions have been processed.
@Joe: You are entirely correct that choosing the interval to sieve and choosing the residue classes to sieve are mathematically equivalent. Without loss of generality we could assume we always sieve the interval [0,d] (where d is the diameter), in which case the only choice is the set of residue classes S to sieve, or we could alternatively assume that we always sieve an interval [s,s+d] using the zero residue classes, in which case the only choice is choosing the location s (the approximate value of d is easy to determine, both asymptotically and in practice). But as I noted in the blog post, the integer s corresponding to a particular set of residue classes S (via the CRT) will, in general, be extremely large, making it impractical to test even a miniscule fraction of the possible values of s.
Remarkably, using a small value of s (on the order of k0) combined with a particular method of constructing S (start with a Schinzel sieve and extend it greedily), works surprisingly well — this is the basis of the so-called "greedy-greedy" algorithm, it chooses s and S together, rather than fixing one or the other. As Terry noted in https://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24283, there is some theoretical justification for why this works, and it may be possible to prove that it gives a small asymptotic advantage over other methods that have been analyzed, at least on a heuristic basis (see Section 1 of the progress report at http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/).
- 25. Wouter Castryck - August 23, 2013
A minor remark about Avishay's note: one can refine Claim 1.3 slightly by making some considerations modulo 6. Because is constant in fairly large regions, the gain one gets from this is disappointingly limited, though. For our main case of interest , it doesn't even help at all (but maybe it could if Engelsma's tables become extended). Here the lower bound reads (which improves to if one uses Drew's trick from #7). For Avishay's bound of improves to (or from to using Drew's trick).
In any case, here's an example of such a finer analysis:
Let be an admissible set of size . We may assume that $0 \in S$. Let and . Clearly and are admissible, so and , from which Avishay's claim follows.
Now suppose that . Since , this implies that and . This gives the following candidate boundary points of and , respectively:
This already implies that . But in many cases mod 3 one can do better. For instance, assume that moreover . Then
By admissibility this means that either is contained in the smaller interval or is contained in the smaller interval , from which
If one makes a similar analysis for all cases , one obtains the following (I just copy-paste the Magma function I used; an output [[a,b],[c,d]] should be read as , while an output [[a,b]] should be read as ; here as soon as ).
function boundingpair(d1,d2)
// WE ASSUME d1 + d2 NOT TOO SMALL
case [d1 mod 2, d2 mod 2]:
when [0,0]:
case [d1 mod 3, d2 mod 3]:
when [0,1]: return [[d1-3,d2-1],[d1-1,d2-3]];
when [1,1]: return [[d1-3,d2-1],[d1-1,d2-3]];
when [1,2]: return [[d1-3,d2-1],[d1-1,d2-5]];
when [1,0]: return [[d1-1,d2-3]];
when [2,1]: return [[d1-1,d2-3]];
else return [[d1-1,d2-1]];
end case;
when [0,1]:
case [d1 mod 3, d2 mod 3]:
when [0,0]: return [[d1-3,d2],[d1-1,d2-2]];
when [1,0]: return [[d1-3,d2],[d1-1,d2-2]];
when [1,1]: return [[d1-3,d2],[d1-1,d2-4]];
when [1,2]: return [[d1-1,d2-2]];
when [2,0]: return [[d1-1,d2-2]];
else return [[d1-1,d2]];
end case;
when [1,0]:
case [d1 mod 3, d2 mod 3]:
when [0,1]: return [[d1-2,d2-1],[d1,d2-3]];
when [2,1]: return [[d1-2,d2-1],[d1,d2-3]];
when [0,2]: return [[d1-2,d2-1],[d1,d2-5]];
when [0,0]: return [[d1,d2-3]];
when [1,1]: return [[d1,d2-3]];
else return [[d1,d2-1]];
end case;
when [1,1]:
case [d1 mod 3, d2 mod 3]:
when [0,2]: return [[d1-2,d2-2],[d1,d2-4]];
when [2,2]: return [[d1-2,d2-2],[d1,d2-4]];
when [0,0]: return [[d1-2,d2-2],[d1,d2-6]];
when [0,1]: return [[d1,d2-4]];
when [1,2]: return [[d1,d2-4]];
else return [[d1,d2-2]];
end case;
end case;
end function;
- 26. Wouter Castryck - August 23, 2013
* the formula that did not parse should be
* … while an output [[a,b]] should be read as …
- 27. Wouter Castryck - August 23, 2013
Maybe one more lower bound: for this gives (but this should be close to the point where inclusion-exclusion takes over).
- 28. Andrew Sutherland - August 23, 2013
@Wouter: thanks for this. My guess it that for 1783 one can do better than 12240 using inclusion-exclusion, but I haven't tried this yet. I'll do this when I get a chance (but if someone else wants to run this computation, please do!).
- 29. Wouter Castryck - August 26, 2013
Hi Drew, if you want I can gather data for the lower bound part of the paper. I've now also implemented Avishay's inclusion-exclusion method in Magma. It works reasonably fast, but an implementation in C / assembly would of course allow one to push things further. As I don't master these languages I'd have to leave that to someone else. But maybe it's not too important to push the very limits here?
In any case, you're right that exclusion-inclusion will improve the bound for 1783. When combined with an exhaustive search up to or (if we are patient) , it looks like one will end up with a lower bound of about 12700 (vs. 12240).
- 30. Andrew Sutherland - August 27, 2013
@Wouter: That would be great! I am playing around with a C implementation that combines inclusion-exclusion with a pruned search (as in #11), but I don't know yet how well it will work. In any case it would be good to have a baseline to compare against. I would also be curious to know the running times for various choices of parameters.
In addition to generating lower bound data for the paper, it would also be nice to fill in the blank spots in tables on the Wiki, even in cases where inclusion-exclusion doesn't give as good a bound as partitioning (just for the sake of comparison).
- 31. Andrew Sutherland - August 27, 2013
Hi Wouter. I did manage to get my inclusion/exclusion C code working, but its still not as fast as it should be — I need to play with it some more, but I want to make sure it is correct first.
I get the same answers you posted for 342 (2046 at 13, 2072 at 17, and I get 2096 at 19), and for 632 (3976 at 13 and I get 4020 at 17).
But for k=2000 I get a better lower bound than Avishay's 14082 at 17, I get 14176 (using inc/exc primes up to 173). And at 13 I get 14046 (using inc/exc primes up to 167). I would be curious to know what you get for k=2000 using p_exh=13 and 17.
And for 1783 I get 12408 at 13 (largest inc/exc prime 157) and 12520 at 17 (largest inc/exc prime 167).
There are two optimizations that I am using (besides pruning and early abort), that are worth mentioning in case you aren't already using them in your Magma code. (1) wlog I assume 0 is always in the admissible tuple, so I never consider residue classes 0 mod p. (2) As soon as the inc/exc depth 2 bound for a given p drops goes negative I don't bother trying any larger p's. In theory this might give me a worse bound, but so far I haven't seen this.
- 32. Wouter Castryck - August 27, 2013
Hi Drew, I aborted my computations because I wanted to make a couple of adjustments. But for p_exh = 13, k = 2000 I would have ended up between 14000 and 14048, and for p_exh = 13, k = 1783 I would have found a value between 12400 and 12410. So your outcomes look compatible.
I used the same optimizations as you did (and I also aborted as soon as the contribution of LB_p dropped to negative), but I should rethink the way I nest the loops. One other thing that frustrates me is that I don't find a satisfactory way of exploiting the symmetry x -> -x. Did you?
I indeed temporarily copy-pasted Avishay's bounds to the p_exh = 17 row, but it was to be expected that they are not optimal (Avishay computed upper bounds for for a couple of smaller values of d, and then used partitioning to bridge the gap).
In fact, one adjustment I wanted to make is the following way of zooming to the "right" value of d, given k. But maybe you already do this. We start from a value of d that's clearly too big to be useful in proving a lower bound on H(k). As before, we iterate over all tuples of residue classes mod 2, 3, …, p_exh (never considering 0 as you said), and for each we compute the corresponding inclusion-exclusion bound . If , we gradually drop d by 2 and reconsider the same $\tau$ until . We then proceed with the next tuple. The point is that there is no need to reconsider the previous $\tau$'s.
Now I'll call it a night, but tomorrow I'll try to confirm your lower bounds.
- 33. Andrew Sutherland - August 28, 2013
I updated the partition bounds in the Wiki for k_0 > 5000 to use the lower bounds at 370 and 385 (but these do not use your mod 6 optimization). A complete list of bounds for k_0 up to 4000000 is available at
- 34. Andrew Sutherland - August 28, 2013
We can improve the partition bounds slightly by using H(343) >= 2334. This can proved as follows:
1) Engelsma proved that there are exactly 16 admissible 342-tuples with diameter 2328, and I was able to compute these, see
http://math.mit.edu/~drew/sequences_342_2328.txt. One can check that in all 16 cases adding either the element 2330 or 2332 does not yield an admissible 343-tuple.
2) It follows that any admissible 343-tuple with diameter 2332 would (up to translation) have to look like 0 … 2330 2332, but this is inadmissible mod 3.
@Wouter: can you rerun your mod 6 optimization using H(343) >= 2334 to see if it changes any of the bounds in the Wiki? I note that it already improves the bound for 10719 to 74076.
- 35. Wouter Castryck - August 28, 2013
Hi Drew, there was a subtle bug in my code: my lower bound was sometimes smaller than yours. But now my outcomes match with yours (at least in the p_exh = 13 cases, p_exh = 17 is still running, so I'll double-check that too). I already filled them out on the wiki.
So now I can gradually start filling the grid. It would be good to check a few more cases against each other.
- 36. Andrew Sutherland - August 28, 2013
I agree with your bounds for k=672 and k=1000 with p_exh = 13. With p_exh = 17 I get 4310 and 6660 respectively.
- 37. Wouter Castryck - August 28, 2013
Ok, I filled them in. The new bound on H(343) doesn't help for the first few k's in the table (up to k = 1000). I'm currently running a bigger computation.
- 38. Andrew Sutherland - August 29, 2013
As a generalization of Wouter's mod 6 suggestion in #25, one can use the following. Let be the maximum for which there exists an admissible -tuple of diameter . Let denote the set of all tuples obtained by sieving the interval of one residue class modulo each prime (note that may vary within . Then for any bound we have
So far I haven't been able to make large enough to improve the bounds for H(632), but using this does improve the lower bound for H(1000) to 6810, for example.
- 39. Andrew Sutherland - September 1, 2013
Just a quick follow-up on Wouter's suggestion in #32 — I implemented this and it significantly speeds up the process of obtaining lower bounds via inclusion/exclusion. This is what I am currently using to fill in the expanded Benchmarks table on the wiki at
- 40. Andrew Sutherland - September 3, 2013
@Wouter: for k0=6329 and p_exh=17 I get the inclusion-exclusion bound 49,356. This is the only case where I haven't seen an improvement over the bounds Avishay originally posted (his bound for 6329 is posted at
https://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24008). I'm wondering if maybe there is a typo in this post. Have you been able to reproduce this bound?
- 41. Wouter Castryck - September 3, 2013
Hi, my slowish implementation is still running on this entry, and it might take a few more days to decide. But currently it already tells me that the lower bound will be at most 49,356. So, me too, I find an inconsistency with Avishay's post. I would also guess/hope that there's a typo?
- 42. Andrew Sutherland - September 3, 2013
I think it probably makes sense to go with the more conservative value 49,356 that we appear to agree on (but let me know if your bound changes). If avishay wants to correct us on this he is certainly welcome to.
- 43. Andrew Sutherland - September 3, 2013
Regarding #38, I realized my implementation was incorrectly assuming h_k=d after sieving [0,d], but this need not be the case, h_k will be smaller than d in cases where d gets sieved. I've updated the wiki to reflect the slight improvements one gets by correcting this mistake (so far just for k0 up to 34429, I'll update the others as the computations finish).
- 44. Andrew Sutherland - September 6, 2013
Just to follow up on #41 and #42, it looks like we both eventually arrived at the same lower bound 49,344 for k=6329 using p_exh=17, so I have updated the wiki accordingly.
Sorry comments are closed for this entry


text:
Points of Pride
The College of Business and Management at the University of Illinois at Springfield is accredited by AACSB International — The Association to Advance Collegiate Schools of Business. Read More>>
Two online graduate degree programs at the University of Illinois at Springfield – the M.S. in Computer Science and the M.S. in Management Information Systems – were rated among the top dozen programs in a recent national survey conducted by GetEducated.com, the only consumer-oriented, online degree clearinghouse in the country. Read More>>
Dr. David Bertaina, associate professor of History, was named University Scholar for the 2014-2015 academic year.
StaLynn Davis, a History graduate student is a fall 2014 recipient of the President's Research in Diversity Travel Award.
- Conference: Reworking Freedom: Graduate Student Workshop on Re-Centering the Enslaved in Histories of the Americas
- Individual Paper: "Alas! The sunny face of the slave is not always an indication of sunshine in the heart:" 19th Century Enslaved Women and Religion

subset: commoncrawl


text:
The videos below are both of the same documentary on the 7:7 2005 terrorist attacks in London. The documentary is a very moving account of people caught up in those terrible attacks ten years ago.
The videos below are both of the same documentary on the 7:7 2005 terrorist attacks in London. The documentary is a very moving account of people caught up in those terrible attacks ten years ago.
Originally posted on TIME:
Australian counterterrorism officials say they have foiled an imminent terrorist attack after the arrest of two men at a house in western Sydney.
The suspects have been charged with terrorist offenses, Reuters reports.
Police say a homemade ISIS flag was found at the house, as well as a machete, a hunting knife and "a video which depicted a man talking about carrying out an attack."
New South Wales Deputy Police Commissioner Catherine Burn accused the suspects of "preparing to do this act yesterday."
Australia's national threat level has been on "high" since last September, when news broke out that militants were planning to publicly behead a random member of the public.
The country is also concerned with homegrown militancy. Dozens of its citizens are said to be fighting for ISIS in Syria and Iraq, and last December a gunman with ISIS sympathies held up a central Sydney café, leading to…
View original 10 more words
Originally posted on Quartz:
The men who French police allege murdered 12 people in or around the Charlie Hebdo offices were killed in a standoff with security forces on Friday, seemingly bringing some resolution to this week's wave of dramatic violence and fear.
The deaths of the suspects, brothers Chérif and Saïd Kouachi, ended a three-day manhunt that began in Paris and spanned hundreds of square miles of France. Here's a look at how it played out:
Originally posted on TIME:
Three terror suspects and four hostages were killed in France on Friday, as police brought to an end two separate hostage incidents relating to the deadly terror attacks on satirical newspaper Charlie Hebdo.
In an industrial estate outside Paris, police killed the two brothers who allegedly mounted Wednesday's devastating assault that killed 12 people and set off the biggest manhunt in modern French history.
Just a 15-minute drive away, SWAT teams killed another gunman who had holed up all afternoon in a kosher supermarket in a city suburb with several hostages. The siege left five dead in total, but the gunman's alleged co-conspirator remained on the run Friday night, having escaped as police stormed the store.
[time-gallery id="3661290″]
Saïd and Chérif Kouachi, 34 and 32, the two French-born brothers who authorities say stormed Charlie Hebdo with Kalashnikovs and then fled in a stolen vehicle, holed up in a commercial…
View original 675 more words
Originally posted on TIME:
Al Qaeda in the Arabian Peninsula (AQAP) took responsibility on Friday for the attack on the Paris satirical magazine Charlie Hebdo. A statement provided to the Associated Press said "the leadership of AQAP directed the operations and they have chosen their target carefully." The attack, which killed 12 people, was intended to as "revenge for the honor" of the Prophet Mohammad, the depiction of whom is forbidden by Islamic tradition. The magazine had repeatedly mocked him in cartoons.
The claim, which the unnamed spokesman said was delayed two days after Wednesday's attack for "security reasons," did not come as a shock. But one of the gunmen in the Charlie Hebdo attack shouted, "You can tell the media it was Al-Qaeda in Yemen" during the assault, according to a witness. And a the French news channel BFM Television reported that Said and Cherif Kouachi made the same claim in a…
View original 710 more words


text:
- Home
- Consultations
- Information
- About
- Contact us
We provide case evaluations remotely for veterinarians all over the world as well as in-person consultations two days a week at the Animal Referral Hospital in Homebush.
We provide education for veterinarians in the form of continuing educations / CPD seminars all over the world.
We conduct and publish clinical studies to help advance the state of knowledge of animal cancer and its management.
We're all about providing the best possible quality of life for pet animals with cancer and their human families.
Information, conversion charts and templates to help you manage treatment in your patients.
We understand how difficult it can be to have a pet diagnosed with cancer so we've put together some answers to common questions.
Veterinary Oncology Consultants
379 Lake Innes Drive
Lake Innes, NSW 2446
+61 (2) 6585 3192


text:
No Equipment Necessary: Mountain Climbers
Mountain Climbers: The Move That Does It All
by Emily Bibb
Like us on Facebook
If you're a fan of exercises that work your body without machines, weights, or equipment, try doing mountain climbers! A POPSUGAR Fitness favorite, this exercise works the lower body and the core and is a great form of cardio. Learn how to do the complete move below.
- Start in a traditional plank — shoulders over hands and weight on just your toes.
- With your core engaged, bring your right knee forward under your chest, with the toes just off the ground. Return to your basic plank. Switch legs, bringing the left knee forward.
- Keep switching legs and begin to pick up the pace until it feels a little like running in place in a plank position.
- Continue "running" in your plank for one minute.
Fit tip: for a stint of cardio during weight workouts, use this exercise as an active break in between exercises or reps.
Image Source: POPSUGAR Studios
Strength Training
POPSUGAR, the #1 independent media and technology company for women. Where more than 75 million women go for original, inspirational content that feeds their passions and interests.
Kelly15226183 1 year
Lynda Blesss
Caroline15221522 1 year
This is a nice schedule and all, but what beginner really has the stamina for a 20 minute run or a 60 minute cycling class? That doesn't sound particularly useful for a beginner to me - I know I wouldn't be able to do that in the first week.
veronicaraye 7 years
def. makes your calfs feel good.
vanessa_j 7 years
i LOVE these!!!
fashionhore 7 years
Mountain climbers kill me! I love doing them though because if I do enough, I don't have to do cardio for that day. It's much easier, for me at least, than running.
ElectroPopTart 7 years
I want arms and shoulders like that boy, lol.
lovekailua 7 years
this is exactly what i need! i just quit my gym b/c i have no time literally to use it and saves me 35 a month. i need to get more of these type things (esp for butt and thighs!). awesome!
syako 7 years
I love moves in the plank position. They are so incredibly tough but so effective!
aimeeb 7 years
I've done these before, not easy!
adelka 7 years
I just had a conversation yesterday about how you can get a wonderful and butt-kicking workout without the gym! I am a fan of these, plies, squats, lunges, puch-ups, and other moves as well. Throw some running into the mix and you're all set!
bealotus 7 years
these are tough!
squirrellypoo 7 years
ahh, we do those in my Boot Camp class, only we call them Runner's Starts, since they echo the pose the elite track runners use at the starting gate...


text:
The Metropolitan Council, the regional transit planning agency for Minnesota's Twin Cities of Minneapolis and St. Paul, Wednesday approved a 14-mile Southwest Corridor light rail line linking Minneapolis to Eden Prairie, Minn.
The locally preferred alternative (LPA) was recommended to the council by the Hennepin County Regional Rail Authority and the communities that the corridor will serve. The route also will serve Minnetonka, Hopkins, and St. Louis Park. The route is projected to cost as much as $1.25 billion.
Met Council amended the region's 2030 Transportation Policy Plan (TPP) to include LRT as the mode of choice for the route, making the project eligible for federal funding.
"World class cities have growing, vibrant transit systems," said Council Chair Peter Bell. "The selection of the mode and alignment for this corridor is an important step forward. It moves the project to the next level and continues the process of building out the region's rail corridor network."
The LRT route will link with the existing Hiawatha LRT line at Target Field Station in Minneapolis, offering connections to Northstar commuter rail services at that station and access to the Central Corridor LRT route serving St. Paul, now beginning construction.
Met Council will submit a New Starts applicationto the Federal Transit Administration this summer, asking for permission to begin preliminary engineering on the project. The route is expected to generate 28,000 rider trips per day by 2030, which Met Council says is comparable to current ridership on Hiawatha LRT.
New York's Metropolitan Transportation Authority Wednesday locked in roughly $1 billion in capital funding by signing an oft-delayed contract with Related Companies, joined by Oxford Properties Group, to develop the MTA's Hudson Yards on Manhattan's West Side.
Related and Oxford are making a $21.75 million deposit in conjunction with the signing of the contract, and are expected to make additional deposits of roughly $11 million each, subject to specified fluctuations in economic conditions. Critics of the deal have said MTA should be more stringent in its expectations for payment.
The site, west of Pennsylvania Station, is among the lastremaining underdeveloped parcels of land in Manhattan, would be developed to include 12 million square feet of commercial and residential space. MTA will lease the site to the developers for 99 years, with purchase options. Construction will occur over ongoing rail operations, which include MTA Long Island Rail Road's West Side Yard and Amtrak's Northeast Corridor.
"This is a tremendously exciting development project that together with the extension of the 7 line [subway] will turn this area into a vibrant residential and commercial neighborhood," said MTA Chairman and CEO Jay H. Walder. "We were also able to maximize value for the MTA and provide a new revenue stream to support many of our vital capital projects."
Jay Cross, president of Related Hudson Yards, said, "Related and our new partner Oxford Properties Group look forward to continuing the excellent working relationship we have developed with the MTA and LIRR over the last two years as we all work to transform the Yards into New York's new 21st century neighborhood."
Russian Railways, Siemens AG, and Russia's Aeroexpress signed a Memorandum on Production, Supply and Servicing of Modern Russian Trains on Wednesday in Sochi, Russia, site of the next Winter Olympics in 2014.
The joint venture will produce new-generation Desiro RU Selectric trains with asynchronous traction drive. These trains will be built by a joint venture in Russia, based on the Olympic electric trains that Siemens will supply to Russian Railways in 2012-2013 under a contract signed between the two companies in December 2009.
Serial production of the modern electric trains is set to begin at the joint venture in 2014, and the share of associated parts and components of the train produced in Russia is anticipated to reach 80% by 2017. The joint venture plans to reach production capacity of 200 vehicles per year.
Russian Railways President Vladimir Yakunin signed on behalf of the railroad; also signing were Hans-Jörg Grundmann, executive director of Seimens Mobility and vice president of Siemens AG, and Aeroexpress board member Maxim Liksutov.
Moscow-based Aeroexpress Ltd. provides air-to-rail links in several locations within Russia; Russian Railways owns 50% of the company.
Efforts to (re-)establish the first streetcar route in Washington, D.C. were first dashed, then resurrected, in a 24-hour period, as the City Council first cut $47 million in funding, then restored the amount Wednesday following a public outcry, fueled in part by email and blog activity.
The H Street Streetcar, estimated to cost $60 million, already is under construction along its namesake street and on Benning Road in the city's Northeast quadrant.
Funds originally were cut when D.C. Council Chairman Vincent Gray said the streetcar line, part of a larger, long-range plan, it needed more planning, including power source options given the ban on overhead wires in much of the city. But news of the cut, reported by the Greater Greater Washington blog, generated substantial public outcry, prompting the council to restore funding.
"What a splendid relief this is," said Councilman Jim Graham, a supporter of the project. "The streetcars are back on track. We will not have a delay."
The H Street Streetcar currently is slated to begin service in 2012, though six more streetcars are needed to supplement three cars already acquired from Czech Republic-based Inekon Group a.s.
House Rep. Earl Blumenauer (D-Ore.), considered on of thetop U.S. proponents of light rail transit and streetcars, reportedly called council members urging the body to reconsider any cuts in funding. "We're working with a dozen cities that are in some stage of streetcar programs. None has more promise than the District of Columbia," Blumenauer said.
The Railway Systems Suppliers, Inc. C&S Exhibition, held May 17-19 in Omaha, Neb., drew 1,712 attendees, "not including spouses," RSSI said in a statement Wednesday.
Of those attendees, 989 were supplier representatives, while the remainder 723, represented railroad, transit, or government organizations.
The annual event also sported 334 booth spaces, staffed by 164 exhibitors.
RSSI also said its Innovation Theatre had 12 member companies participating "with varying results." The theatre drew more than 100 participants. "We have had several requests to continue this service," RSSI said.
RSSI said 32 companies exhibited at the annual event for the first time.
In 2011, RSSI will be joining forces with RSI (Railway Supply Institute), REMSA (Railway Engineering-Maintenance Suppliers Association) and AREMA (American Railway Engineering and Maintenance-of-way Association) for the first-ever Railway Interchange
2011 in Minneapolis, Minn,. Sept. 18 - 21. The event will feature
hundreds of exhibits, both indoor and outdoor, and technical sessions with AREMA and CMA (Coordinated Mechanical Associations). For more information, visit www.railwayinterchange.org.
U.S. Class I railroad employment totaled 149,749 in mid-April, up 1.21% from March but down 2.93% from April 2009, according to a Surface Transportation Board count posted Wednesday on STB's website.
The biggest numerical increase in April was in the transportation (train and engine) category, where the number of operating crew members on the job increased 1.74% to 58,814.
The biggest percentage increase was in the maintenance of way and structures group, where employment rose 2.2% to 34,202.
The April Class I roster also showed 24,170 maintenance of equipment and stores workers on the job, up 0.48% from March; 13,155 professional and administrative, down 0.66; 8,935 executives, officials, and staff assistants, down 0.68%; and transportation (other than train and engine), 6,473, up 0.86%.
Bombardier Transportation said Wednesday it has signed a cooperation agreement with the Augsburg Transport Authority (Stadtwerke Augsburg Verkehrs GmbH) to "install the contactless and catenary-free BOMBARDIER PRIMOVE system for trams as apilot project in the city of Augsburg," Germany.
The Primove system will be installed on a 0.8-kilometer(one-half mile) section of Line 3 to the Augsburg fairground. The pilot project intends to demonstrate the technical capability and electromagnetic compatibility Primove offers light rail vehicles in an urban environment, where overhead wires are deemed impractical or otherwise undesirable. Installation work will start this summer.
Bombardier says Primove involves cables laid beneath theground and connected to the power conditioning and supply network. They are only energized when fully covered by the vehicle, which ensures safe operation. A pick-up coil underneath the vehicle turns the magnetic field created by thecables in the ground into an electric current that feeds the vehicle traction system.
We are delighted to extend our long and successfulpartnership with the Augsburg Transport Authority to include this innovative development in electric mobility. The Primove catenary-free technology is now mature for installation into a demanding urban tram network. Over timecatenary-free operation will become a standard element of light rail systems and we are confident that Primove will be the clear choice for many cities inthe future", said Eran Gartner, president, Systems Division, Bombardier Transportation.
Josef Doppelbauer, chief technical officer, Bombardier Transportation, said, "By dispensing with the need for overhead catenary, the PRIMOVE system will open upnew opportunities for urban planners in the design of integrated transportation systems for cities."
Norbert Walter, managing director of Stadtwerke Augsburg, said, "Our expectation for the Primove pilot project is to gain further insights into newdevelopments in energy management and energy savings in tramway operations. Wewill also cooperate with the Augsburg University of Applied Sciences for this. Public transport stands to benefit from this innovative project in the field of electric mobility."
Bombardier already has a contract to supply a fleet of 27 Flexity Outlook vehicles to Augsburg Transport Authority; delivery began in 2009 andwill continue through to the end of 2010.
The Chlorine Institute (CI) isn't yielding any ground on a CI-sponsored Positive Train Control cost-benefit study purporting to show that contrary to railroad arguments, the government-mandated installation of PTC by 2015 will provide strong economic benefits to the rail industry.
If its findings are accepted by the Federal Railroad Administration, the study would seriously undercut the railroads' position that the federal government should help pay the multibillion-dollar bill for installing the safety system. It could also relieve shippers of hazardous materials from responsibility for indirectly funding PTC through higher rates.
CI on Tuesday sent the FRA what it called the final report of the study, which it said confirms that a regulation implementing the mandate "vastly understates the technology's benefits to the railroad industry and public."
Based on initial findings of the study, the institute asked the FRA in March to reissue the rule with a new cost-benefit analysis. CI said Tuesday that while it "strongly supports PTC, the faulty cost-benefit analysis in the current rule fosters a situation that could allow railroads to impose on shippers of chlorine and other toxic inhalation hazard (TIH) chemicals an unfairly large share of the costs of applying PTC technology."
The CI study was conducted by L. E. Peabody & Associates, Inc., Alexandria, Va.
The fate of 233 miles of rail in Maine's Aroostook and Penobscot counties may be determined by a referendum on June 8 in which the state will seek approval to issue bonds to buy the line if the Surface Transportation Board approves the abandonment application of the Montreal, Maine & Atlantic Railway.
In view of this, the STB issued a decision Tuesday slightly altering the schedule of the mediation that is now under way between the State of Maine, which thinks continued rail service is essential, and the railroad, which says it's a losing proposition.
In Tuesday's decision, the board said: "On April 22, 2010, following a meeting attended that day by MMA, the state, and board mediation staff, MMA filed a motion requesting that the Board extend the procedural schedule for a period of 3 weeks so that MMA and the State could enter into confidential mediation concerning the future of the line. In a decision served on April 26, 2010, the board granted this request and made MMA's rebuttal due on May 26.
"Mediation between MMA and the State is progressing. Theboard believes that a negotiated settlement would be in the best interests of the parties and the public in this case. The board understands that the parties may not be able to reach a final agreement concerning the sale of the lines without knowing the outcome of the June 8 referendum. Accordingly, MMA and the state are directed to submit a joint status report outlining their progress toward an agreement on the sale by June 17, 2010. If the parties have not reached an agreement or have not made substantial progress toward reaching an agreement by June 17, 2010, the board will resume its consideration of the merits of this abandonment proceeding and prepare a final Board decision."
St. Paul, Minn.'s Central Corridor light rail transit project Tuesday "received permission today to enter final design, the last step before award of a federal Full Funding Grant Agreement for the biggest public works project in Minnesota history," according to the Metropolitan Council, the prime governmental advocate for the controversial light rail route linking the state capital with neighboring Minneapolis and the existing Hiawatha LRT line.
In a statement, Met Council said, "Immediately upon receiving permission to enter final design, the project office will submit all documentation to the Federal Transit Administration for award of a Full Funding Grant Agreement. An FFGA represents the federal government's commitment to reimburse project partners for half the cost of building the $957 million, 11-mile line connecting St. Paul and Minneapolis. "
Anticipated funding for the route is 50% from federal sources, 30% from the Counties Transit Improvement Board, 10% from the state of Minnesota, 7% from Ramsey County (which includes St. Paul), 3% from Hennepin County (which includes Minneapolis), and residual sums from St. Paul, Metropolitan Council, and the Central Corridor Funders Collaborative.
"The FTA's approval represents another important step in transforming this project from a 20-year-old dream into reality," said Peter Bell, chair of the Met Council. "It will mean improved access for thousands of metro area residents to employment, educational, and economic opportunities all along the corridor and beyond."
Remember Me


text:
Religion briefs for Feb. 23
Hamilton is a best-selling author and senior pastor at United Methodist Church of the Resurrection in Leawood, Kan.
Hamilton's first lecture, "Can We Agree to Disagree? Discussing Sexuality in the Church," will be at 1 p.m. in OCU's Bishop W. Angie Smith Chapel, 2501 N Blackwelder. At Church of the Servant, 14343 N MacArthur Blvd., he will present "When Christians Get it Wrong" at 4 p.m. and "Why? Making Sense of God's Will" at 7 p.m.
For more information, call Janet Boone at 208-5020 or email email@example.com.
Gospel singing event is scheduled
DEL CITY — The Del City Gospel Singing will be at 7 p.m. March 2 at Del City First Church of the Nazarene, 4701 SE 26. Guest singers include Harmony Quartet and Sherman Andrus.
There is no admission charge but a love offering will be taken. For more information, call 677-2330.
Parish plans Lenten mission
The annual Lenten Mission at St. Eugene Catholic Church is set for March 4-6 at the church, 2400 W Hefner Road. The Rev. Nick Rice from the Archdiocese of Louisville will be guest presenter.
Morning sessions, set for 9 a.m. each day, will feature Mass, followed by teaching. Evening sessions will be from 7 to 8:30 p.m. each day. Childcare will be available for all sessions. Confession will only be heard after morning sessions.
Rice is the founder of Mass of the Air and a 20-year panelist on "The Moral Side of the News" radio show in Louisville, Ky. For more information, call 751-7115.
Women's conference is scheduled
NICHOLS HILLS — Christ the King Catholic Church will present its annual "Ladies of Grace" conference from 9 a.m. to 2:30 p.m. March 9 at 8005 Dorset Drive.
Laura Nelson, a Catholic speaker, author, teacher and blogger from Grapevine, Texas, will present "Wonder Woman, Wondering Woman or Woman of Wonder: Which one are you?" Registration of $25 includes lunch and must be made by March 4.
To register, call 242-4521 or email firstname.lastname@example.com.
From staff and wire reports


text:
Minor League Scores – 4/23/14
Zach Davies struggles in his most recent start for Bowie
Minor League Scores – 4/22/14
Hunter Harvey makes his fourth start for the Shorebirds
Minor League Scores – 4/21/14
Tyler Wilson puts up a strong outing at Bowie
Tim Berry Game Scouting Report – 4/17/14
Tim Berry Game Scouting Report from his 4/17/14 Start
Minor League Players of the Week: 4/14 – 4/20
Position Player of the Week – Drew Dosch (Delmarva) The Shorebirds' third baseman went 10 for 23 this week with 2 doubles, a triple, 2 RBI and a walk. Dosch is .311/.344/.393 on the season with an OPS of .737. Starting Pitcher of the Week– Hunter Harvey (Delmarva) In case you haven't heard, Harvey pitched […]
Minor League Scores – 4/20/14
Kevin Gausman pitches five strong innings for the Tides
Shorebirds and Keys Notes 4/12 – 4/15
The rest of the ON staff scouting notes from the Shorebirds and Keys games they witnessed this week.
Minor League Scores – 4/17/14
Steven Brault struck out 9 batters in 7.0 scoreless innings for Delmarva on Thursday Night
Shorebirds Notes 4/12 – 4/13
Shorebirds notes on Hector Veloz, Jon Keller and Sebastian Vader from the weekend
Minor League Scores – 4/14/14
Josh Hart went 3-4 with a RBI and SB for the Shorebirds.


text:
The Blood Type Diet Archives Volume 8
Re: A New Germ Theory
Posted By: Tom Norris
Ed (I hope this is going to you Ed, haven't figure out this board ),
I have Lyme disease. I'm on antibiotics and will be for some time if I follow my alliopathic doctor's advice.
It settled in my CNS, and my nerves are so sensitive, I find the cultures in yogurt to be irritating to them. I tried various herbal preparations, but they also irritated the nerves. Much as I have pursued the alternative models (homeopathy, acupuncture, etc.) , I must confess, that, while slow, I've found the tetracycline I've been taking to very slowly give me more benefits. I've been following ER4YT at the same time and have consulted with Peter D'Adamo's dad. He said to follow the diet and take the ABX.
I've had Lyme disease for 25 years and was mis-diagnosed and mis-treated by both alliopathic and alternative practitioners. I did find one natural healer who said that I had a parasite, but again, the herbal preparations were too irritating to my nerves. I tolerate the ABX better than most things.
Messages in This Thread
Contact D'Adamo Personalized Nutrition customer service for support on where to begin.
A Daily Webzine Celebrating The Wonder Of Individuality & The Blood Type Diet®
Center of Excellence in Generative Medicine (COEGM) is Dr. D'Adamo's main clinical facility. To schedule an appointment please visit:
The statements made on our websites have not been evaluated by the FDA (U.S. Food & Drug Administration).
Our products and services are not intended to diagnose, cure or prevent any disease. If a condition persists, please contact your physician.
Copyright © 2015, Hoop-A-Joop, LLC, Inc. All Rights Reserved


text:
#229 – Minneapolis Moline "R"
Warning: Missing argument 2 for wpdb::prepare(), called in /home/content/69/8670569/html/mm/wp-content/themes/Mineapolismolinevector2/functions.php on line 55 and defined in /home/content/69/8670569/html/mm/wp-includes/wp-db.php on line 1037
Minneapolis Moline R
4 cylinder gas engine
Very good tires
For more pictures click on title and/or picture.
This piece/unit is located in Canada 3 miles from USA border at Blaine, Wash right off I-5 freeway. Call Al @ 206-227-6673 or 604-813-5500 Email any questions and/or requests for shipping quotes to the following;


text:
Order lapels, sweaters, hats and more at your student store and show your University pride.
[ Skip Main Nav ]
There's a personal story behind each graduate's journey to the diploma he or she holds on commencement day. You want to be there to hear it. And your graduate wants you there, too. Please join us in the celebration.
Date: Saturday, August 29, 2015
Time: 10:00 am CDT
Venue: Municipal Auditorium
Address: 417 4th Ave. N. Nashville, TN 37201
If you have any questions, please email email@example.com.


text:
My family has been invited to take the Listerine Healthy Habits for a Lifetime Oral Care Challenge and I gladly accepted because we are not doing so well in the healthy habits department when it comes to our teeth and gums. My husband and I both need dental work, but just cannot afford to have it done all at once. We definitely do not want our children (currently 8, 5, and 5) going down this road, but when it is a struggle just to make sure they are brushing at least twice a day, doing more seems nearly impossible. After hearing what dental experts had to say about oral care in young children, I realized that we must have our children floss, brush, and rinse twice a day every day. And as the adults, my husband and I must lead by example. That means no more working until 1am and then just throwing myself into bed without brushing because I am too tired!
In this challenge, my family has agreed to floss, brush, and rinse with mouthwash at least twice a day for 3 weeks. Here are some dental facts I had learned that convinced me more than ever that succeeding at this challenge is a necessity:
- Brushing alone only cleans 25% of the germs in your mouth.
- 75% of adults suffer from some form of gum disease
- 50% of children have their first cavity or filling between the ages of 5 and 9 years old.
- It is recommended that a child should visit a pediatric dentist as soon as his or her first tooth erupts.
- It is recommended that parents begin to floss their children's teeth as soon as there are two adjacent teeth.
It will take a few days for the children to become accustomed to the floss and the mouthwash, but based on how well things went this morning, I am extremely optimistic that we will successfully complete this challenge. As a reward, if all 5 of us complete this challenge, we will take the children to a family movie of their choice.
Over the next 3 weeks, look for updates on how our family is doing in the Listerine Healthy Habits for a Lifetime Oral Care Challenge on Facebook and Twitter. Once our 3 weeks is over, I will update our 3G2S readers about what has worked for us and what has not. For tips about how to improve oral care habits for your family, visit Walmart.com, who is doing their part by selling Listerine and Reach products at very affordable prices.
I received products and compensation from Johnson & Johnson and The Motherhood
as part of my participation in the Oral Care Challenge. All thoughts and opinions expressed in this post are my own.


text:
A brush with the hawk
All this week we will be turning over the cover of G2 to some of Britain's leading artists. David Hockney kicks off our series with an exclusive self portrait. He tells Andrew Marr why a brush captures reality better than a camera - and then paints him.
G2 Comment & features p2
'I had swapped one prison for another'Under the Taliban, Hala risked her life to give girls an education. And then she was caught. Here she gives a gripping account of her escape from Afghanistan and her suffering at the hands of US immigration officials.
Comment & features p8
Safety in numbersWells, Norfolk.


text:
"Guidelines recommend that all adults over the age of 20 should receive risk factor screening from their family doctor every two to five years. There are more specific guidelines for patients with type 2 diabetes as their risk of a heart attack is twice as high as the general population."
The analysis from the SHIELD study (Study to Help Improve Early evaluation and management of risk factor Leading to Diabetes) was performed to see if ongoing recommendations for improved screening in the USA were resulting in more patients being diagnosed with heart disease before they became symptomatic.
It forms part of the larger SHIELD survey of 211,097 US households who responded to a national questionnaire.
More than 18,400 people who had participated in the baseline and first follow-up SHIELD surveys - and had diabetes or one of the five cardiometabolic risk factors - were sent questionnaires and 13,877 (75%) responded. Of these, 1,573 (11% of the total) had been diagnosed with heart disease.
Just under two-thirds had heart disease on its own (62%). They were predominantly male (59%) and white (91%) with an average age of 67. The remaining 38% had heart disease and type 2 diabetes. They were slightly more likely to be women (50.7%), 88% were white and their average age was 69.
The average time since diagnosis was 11.7 years in CHD patients without diabetes and 10.7 years in individuals with diabetes and CHD. Almost a third of the non diabetes group (31%) and 36% of the diabetes group had been diagnosed since 2001 when a number of consensus statements and guidelines on screening and prevention were published.
"Despite increased knowledge and awareness of the risk factors for CHD, many individuals are not diagnosed with heart disease until they are sy
|Contact: Annette Whibley


text:
The Style Gallery is a great new way for you to view, love, and share outfit photos. It's meant to showcase
all the expressive, creative, and inspirational personalities of our community!
Love you too!
Your email has been sent!
Just one stride past the ornately framed mirror in the ballroom assures that the union of you and this floor-length dress are one of the most elegant sights here. From the sweetheart bodice swirling with sequins, to the handkerchief-layers of the black chiffon skirt, this captivating frock reflects pure magnificence to all those who see you in it!
Final Sale: This item cannot be returned or exchanged.
All measurements in inches.
Sorry this item isn't available anymore
Remarkable fashion & decor, inspired by our community, & delivered with uncommon care.
Copyright ©2002-2015 ModCloth.com. All Rights Reserved.
Site Map
CA Notice
Privacy & Security
Terms of Use


text:
The Organization for Economic Cooperation and Development (OECD) released a peer review of Canada's international aid programs on June 19. The review was complimentary of parts of Canada's foreign aid service, such as the country's 'field presence' in countries where aid is being delivered. The organization did have some issues, however, with the clarity and coherence of Canada's aid plan.
Particularly, the review questioned a decision by the Canadian International Development Agency (CIDA) to partner with mining companies – profitable corporations – saying that CIDA has shown no rationale for working with these commercial interests. The agency says they are supporting economic growth and international trade in these countries.
The report also comments on the deteriorating relationship between the Canadian government and certain NGOs. The OECD says the relationship between the government and NGOs or civil society groups seems to have become 'selective.'
The truth is that the Harper government is selective about the groups it works with, and has a history of cutting funding to groups that disagree with the Tory mandate. They also badmouth these groups and call them names.
The OECD suggests in this review that given the economic outlook of this country – that which the Harper government is often bragging about – Canada could afford to increase aid to countries in need. The Tories actually plan to cut aid funding by $389 million this year. They have also reduced the number of countries that receive aid from Canada – a move the OECD actually notes as a good thing, saying more concentrated aid is better.
Brian Tomlinson, an aid analyst in Nova Scotia, told the Toronto Star the CIDA's partnership branch, which has helped to distribute aid money in the past has been cut by 15 percent, and because of this:
"It's possible that this year there will be no new calls for aid project proposals because of the cuts," Tomlinson said. "That's never happened before."
The cuts to Canada's aid funding comes as the Minister of International Development, Bev Oda, who is responsible for CIDA, is in the news for her ministerial spending habits. Perhaps CIDA and its Minister should be more transparent about where their priorities lie.
Related Stories:
A Canadian MP in Hot Water over Foreign Aid (and One Word)
Will A Cabinet Shuffle Make Things Better Or Worse?
Canadian Government Quietly Guts Planned Parenthood Funding?
Read more: bev oda, canada, canadian international development agency, foreign aid, oecd, politics
Photo Credit: Vince Alongi
I am so ashamed of the Canadian government's budget cuts. These affect primarily the weakest in society. The next 4 years, with a Conservative Party majority, we will likely see more and deeper cuts, unfortunately. A majority of Canadians did not support the Harper Conservatives, but with our first past the post system and 3 to 4 major parties, they won a majority of seats and so have control of both Parliament and the Senate, they can do pretty well whatever they want until the next election.
What happened to Canada? Oh yeah. Global Corporation Organization. That .0001% is international, they want to control EVERYTHING and EVERYONE!
Harper's priorities are all about making money for the boys at the top especially the oil industry in his home province of Alberta. He has lost his popularity completely and will lose the next election- The NDP just need to have a candidate that is warm and walking. Hopefully some of this damage can be undone but it is a long wait.
I believe the UN target for foreign aid for each nation is 0.7% of the nation's gross domestic production. This is what we all should be demanding of our governments.
humanismo é o futuro do mundo os ricos tem o dever de ajudar os menos afortunados
We shouldnt be cutting aid..
With two billion of the world's seven billion people living in extreme poverty, nearly 900 million of these populations living on the brink of starvation, now is not the time to reduce aid. We need to help these people grow their own foods, or we could end up having to fight them as part of some hostile force seeking retaliation against the rich, developed nations, The U.S. with five percent of the worlds population consumes over 24 percent of the world's resources. On a percentage basis, Canadians are also big consumers. To starving populations, we are "targets" living in "target rich" nations.
Canada should definitely not cut aid!!! It seems maybe they are doing some type of population control.
We have a duty to help those less fortunate.
Disclaimer: The views expressed above are solely those of the author and may
not reflect those of
Care2, Inc., its employees or advertisers.
What a nice story. Glad this mommy and her babies found a rescuer.
Good to read to see how many are really improved mindsets and willing to turn to greener pastures.
Problem on this page? Briefly let us know what isn't working for you and we'll try to make it right!


text:
DDO Store Sales: October 9th - 15th
Upgrades your Account-based Bag Deposit Box with 250 additional item slots of storage space. You can put anything into your Bag Deposit Box that goes into a bag or quiver, with the exception of Bound to Character items.
Shared Bank: Item Storage Share items among all characters in your account on the same server! Expand your Shared Bank up to 80 slots with an initial 20 base slots and ten 10-slot upgrades from the DDO Store. Your shared bank is the only way to trade bound-to-account items (which cannot be mailed or auctioned).
VIPs gain access to the first 20 slots for free as long as VIP status is maintained.
Easily and efficiently store stacks of gems, collectibles or ingredients in one inventory slot!
• Small, Medium, Large & Huge Gem Bags
• Small, Medium, Large & Huge Collectibles Bags
• Small, Medium, Large, X-Large, Huge, Gargantuan & Colossal Ingredients Bags
• Small, Medium, Large, Huge, Gargantuan, & Colossal Augment Bags
• Small & Large Cookie Jars
• Bound to character
• Auto-collect, automatically placing items you acquire into the appropriate bag!
The base Shared Bank: Platinum Vault is FREE FOR VIP!


text:
I should be somewhere between 5weeks 3days - 5weeks 6days. This is also my 4th pregnancy. My symptoms seem to be disappearing. I am no longer fatigued, and this morning I noticed my boobs no longer hurt AT ALL! The only time I had this experience was with my mc in May... with the first 2 pregnancies I had my symptoms throughout the whole pregnancies. I am so scared I will miscarry again. Has anyone had a similar experience? I think I just need some reassurance.
5 weeks and my boobs stopped hurting all of a sudden! *Update*
Bump i'm 5 weeks today I believe, having no symptoms today, besides feeling tired...hope it's common!
I'm still waiting for a BFP this time, but with my last pregnancy I had sore boobs for a few weeks then nothing. I never had morning sickness, was never tired until the end of the 3rd trimester. So, please try not to stress about it. As you know, all prenancies are different. GL!
Jessica & Mariano ~ 04.05.06
Kylee ~ 10.20.00 Alyssa ~ 10.22.07 Brooklyn ~ 03.03.13
From what I have read, it is common for symptoms to come and go, change and transform in intensity, but not to disappear altogether. Good luck.
Me + DH = DD1 + DD2 + DS Due March 3, 2013!
Every pregnancy is different. With my m/c I had terrible symptoms that never went away until I started bleeding. With my DD, my symptoms were barely noticeable and completely disappeared very early. Stay positive!
V & B hitched '04
3 angels, DD 11/14/11 w/ loss of twin at 11.5 wks #2 EDD 2/28/13
"Give me a burger, onion rings, and a list of people who killed their parents and got away with it. I need some heroes."
Good luck! Let us know how your levels were!
With my best friend since 2008 married 2013
DD born 9/2013
Our family is complete with lil man 4/2015
I'm having this same thing - really no symptoms.... Just a tad tired and my boobs feel back to pre-preggo normal state...I did get my HCG level checked today and it was 2522 and progesterone is 80. I'm just shy of 5 weeks I think.
I'm choosing to view this as us being one of the lucky ones so far!!!


text:
Children's Core for Biomedical Statistics (CCBS) is pleased to provide the following information for data management programs used at Seattle Children's, including how to access and links to resources.
REDCap (Research Electronic Data Capture) is a secure web application created for researchers by researchers for building and managing online surveys and databases.
REDCap Office Hours
CCBS office hours will be held every first and third Monday of every month from 12–1 p.m. at main campus in the PCRC Conference Room OA.8.151.
REDCap consultations also are available by appointment. Please email us to request an appointment.
DatStat is a commercially available electronic data capture software. Although not widely available at Seattle Children's, it is in use by several ongoing clinical studies within the research institute.
Medidata is a commercially available, web-based electronic data capture software. Although not widely available at Seattle Children's, it is in use by the Therapeutics Development Network (TDN).


text:
Skinny Frozen Peanut Butter Pie.
This pie. Wow.
When I was brainstorming birthday cake recipes to make for Kevin's birthday, I got a brainfreeze. It may OR may not have been from the orange creamsicle I was inhaling, but I'm pretty sure it was because I was SO overwhelmed with all the sprinkle-y, chocolate buttercream-y, and death-by-chocolate thoughts swirling in my head.
I threw up my hands and asked friends on my Facebook what I should make. I got so many wonderful ideas, so thank you for sending them in!
Lucky for my tastebuds, Kevin loves peanut butter & chocolate as much as I do, so I knew his birthday treat HAD to involve the mighty duo. Since I have no patience for baking up a fancy schmancy layered birthday cake on a weeknight, a no-bake treat was definitely top priority here.
AND I love a skinny dessert as much as the next girl!
Using some lighter alternatives, I made one home run pie that I honestly think will be demanded at each and every birthday going forward. And on holidays. And well, probably on a regular basis. And I'm more than OK with that. Fat free frozen whipped topping, light cream cheese, a sprinkle of sugar, a generous pour of sugar-free chocolate syrup. All frozen together on top of a homemade Oreo cookie crust.
I couldn't stop slicing off little slivers, which was embarrassing because it wasn't even my own birthday. I couldn't stop licking the spatula as I swirled on each layer. I washed it between each layer, don't worry! I couldn't stop crushing up more Reese's Pieces to sprinkle on top. They're one of my favorite ice cream toppers. Anyone agree?!
I wanted more of this pie the second it was all gone.
This pie. It's legendary. Creamy, thick, melty, full of peanut butter, chocolate, a Reese's Pieces crunch, buttery oreo cookie crust, and I don't even have to turn on the oven.
The pie may or may not have been served to the birthday boy with a slice already taken out for picture purposes. #bloggerproblems
Skinny Frozen Peanut Butter Pie
A lightened up no-bake peanut butter pie. Incredible!
Yield: serves 8-10
Oreo Cookie Crust
- 24 reduced fat Oreos (or regular Oreos if you prefer)
- 1/4 cup light butter, melted (I used Land O' Lakes Light)
- 1/2 cup + 2 Tablespoons creamy peanut butter
- 1/2 cup granulated sugar
- 1/4 cup milk
- 8 ounces reduced fat cream cheese, softened (may also use fat free)
- 8 ounces fat-free thawed whipped topping (I used Cool Whip Free)1
- 1/4 cup chocolate syrup
- 1 cup crushed Reese's Pieces, divided
- First, make the Oreo cookie crust by crushing the Oreos into a fine crumb using a food processor or blender. Using a rubber spatula or wooden spoon, mix the crushed Oreo crumbs and melted butter together. It may look like there is not enough butter, but it will come together.
- Place into a 9 or 10 inch pie pan, pressing the crumbs onto the bottom and up the sides as much as it will fit. Try to make the crust 1/8 inch thick all around. It does not have to be perfect. Set in the freezer as you make the filling.
- Prepare the pie filling in a large bowl by mixing the peanut butter, sugar, cream cheese, and milk. I used an electric mixer, but a rubber spatula or wooden spoon would do the trick. Mix until the mixture is completely smooth. Gently fold in the whipped topping.
- Remove pie crust from the freezer and layer 1/2 of the pie filling onto the crust. Sprinkle with 1/2 of the crushed Reese's Pieces. Layer with the rest of the pie filling, swirling to make it completely smooth on top. Top with chocolate syrup and remaining Reese's Pieces.
- Freeze for at least 4 hours or overnight. 20 minutes before serving, remove from the freezer and let sit out to gently thaw. This made the pie much easier to cut.
- Make ahead tip: Pie can be made ahead, covered tightly, and frozen up to 2-3 days.
Additional Notes:
Feel free to use any candy topping of your choice. Peanut butter cups, crushed cookies, candy bars, coconut, chocolate chips, etc. Layer with caramel sauce, use a graham cracker crust, or use less or more sugar depending on your taste. This is a highly adaptable pie.
1Instead of the whipped topping, you can use 1 cup of fresh whipped cream. Combine 1 cup of heavy cream with 3-4 Tablespoons of sugar and whip on high for 3-4 minutes until you have whipped cream.
Adapted from Skinny Taste
© Sally's Baking Addiction. All images & content are copyright protected. Please do not use my images without prior permission. If you want to republish this recipe, please re-write the recipe in your own words, or link back to this post for the recipe.
Similar lightened-up recipes:
Skinny Frozen Mint Chocolate Chip Pie
Skinny Peanut Butter Swirl Brownies
Lightened-Up Peanut Butter Ice Cream Pie
Skinny Double Chocolate Chip Muffins
Related Posts
97 Responses to "Skinny Frozen Peanut Butter Pie."
Mmm, this reminds me of a Jello no-bake peanut butter dessert my friends and I used to whip up in high school! Peanut butter pie is one of my favorite desserts, and I love the lighter ingredients you use. Definitely could see this becoming a regular at many holiday parties, birthdays, etc.
So… I got all inspired by the latest creation by the latest creation by Fiber One (CHOCOLATE!!!) make this pie tonight. I used the new 80 Cal Chocolate Fiber One cereal as the crust. I also swapped the peanut butter for PB2, used Stevia, and threw in a layer of thinly sliced frozen bananas in the middle layer. (Peanut butter? Chocolate? Banana? Yes, please!) EPIC!!!!!! Thank you for continuing to be such an awesome inspiration girl! Keep em' coming!
I made your Peanut Butter Pie and it turned out delicious. I substituted Chocolate Fiber One for the crust instead of Oreos and then used Naturally More Peanut Butter, Fat Free Whip Cream and Cream Cheese and Almond Milk. I also drizzled the Naturally More Peanut Butter with Sugar Free Chocolate Syrup and Dark Chocolate Chips on top before freezing. My husband and little boy couldn't get enough of it.
I made this yesterday when I got home from work and it was perfectly frozen 4 hours later for dinner. This recipe was so easy and so good. Pretty decadent too…my fiancé was thrilled because he always complains we never have desserts in the house. It was a refreshing treat and now I'm brining the leftovers to the office!
hi Sally. This looks so good, but here in Norway we can't get cool whip and such. I found a recipe for making my own though. Do you think this would work?:
* 1 cup of cold milk
* 1 tsp of vanilla
* 3 T of sugar
* 1 tsp of Xanthan Gum or Guar Gum
In a mixing bowl, pour in the milk and vanilla. Turn on the mixer to a low to medium speed and slowly add in the dry ingredients. You want to make sure that they don't clump, so don't just drop it all in there at once. Once everything has mixed well for a minute or two, turn up the mixer to a medium to high speed. Do this for about 6-8 minutes or until you have a nice whipped and thick consistency.
I was thinking tp just weigh out the correct amount and replace it for the cool whip. Thanks!
Sally, you and I must be twins (or were in some other life!). Peanut butter and chocolate is my absolute kryptonite. Nothing makes me happier than a Reese's or ANYTHING with peanut butter or chocolate (ESPECIALLY when they are together)!! And baking too, so that's even more evidence!! :p
I made this recipe for a friend for our dessert and I couldn't wait to tell you about it! I actually used a chocolate graham cracker crust (I unfortunately was limited when it came to the crust so I was in a pinch and had to buy store bought :(, but next time homemade for sure!!). I used reduced fat cream cheese (I figured it would be creamier and have more flavor) and Fat free cool whip. Also, instead of Reese's Pieces on top. I actually used…wait for it…
CHOCOLATE CHIPS AND PEANUT BUTTER CHIPS! Sally…it was incredible. I even made it so the chocolate was on the outside and the middle was swarmed with those peanut butter chips… like a Reese's! The creaminess of the chips paired wonderfully with the texture of the cake. And of course I drizzled it with Reduced Hershey's Syrup in a beautiful cross-hatch pattern. My friend deemed it a "Christmas Miracle" to give you an idea…so thank you for the wonderful recipe. It is so versatile. You could experiment so much, especially with all the variations! I can't wait to try all of them.
SALLY! I just totally killed this shit. I needed a birthday dessert for my sister and was in a total panic because 1) it was the night before her bday and all the grocery stores were closed and 2) SHE NEVER KNOWS WHAT SHE WANTS. Every year she shrugs at the birthday dessert question. SHRUGS. Like a cycloptic freakshow from another planet! Who doesn't know what dessert they want?! What kind of MUTANT am I related to?
Anyway. I dug into your blog and emerged with this little gem. I had Reese's Pieces on hand (who doesn't?); I had plenty of peanutbutter (because I can't live on chocolate alone [LIES!]); I even had a nearly untouched bag of Oreos (good thing you didn't call for Cheetos, I was down to crumbs in that department). But the only thing I didn't have was Cream Cheese1 DRAT IT ALL!
Luckily, a quick dash to Walgreens (after midnight, asking for cream cheese, in my pajamas, getting the hairy eyeball from the night manager but NEVERMIND) solved that little snag. I whipped up this pie in NO TIME at all. Of course, I ran out of Reese's Pieces and had to crush up some fun size Snickers bars (which I also always have on hand because, hello, it's called: Knowing What You're About) but they worked like a dream within a dream – a harmonious blend of crunch & creamy!
The next day, this pie even outshone the birthday girl and was all ANYONE could talk about. I sat around smirking and accepting compliments and ignoring my sister's black looks of JEALOUSY. Of course, don't you fret about that, once she took a bite, all was immediately forgiven. And then she wouldn't let anyone have a second piece and took the rest of the pie home for herself. (PIG!)
My point is this: You saved me. This glorious, nearly sin-free pie was SCRUMPTIOUS in a way that never existed before. Or, more simply put, YOU ROCK. :o)
Pingback: skinny frozen peanut butter pie. | Cooking Pics
Pingback: Great Low Calorie Recipes « Two Moms and a Journey