42 F, 22 mph wind….brisk, to say the least. And yes, it drizzled a bit at times, and there were a couple of claps of thunder. But that is seasonal around here for early April; no room to complain.
I did wear a long sleeve t-shirt though for my hilly 6.4 mile (10.3 km) run which took 1:03:53 today. Up until last week, this would have been one of my faster times but today, this represented a “keep it easy” effort. Seriously; I barely broke a sweat. Then again, at 10 mpm (just under), I shouldn’t have.
I was 9:48 at 1.03, 24:4x at the 2.5 turn around, 36:09 at the start of the second loop, 13:3x for the second loop and 9:40 for the trip home.
On the way down Cornstalk (the first time) I saw a bespandexed lady come down from the building and get on the course right in front of me. I saw her build (slender), her gait (quick and efficient, compared to mine) and told myself “don’t EVEN think about it” (meaning: “don’t try to keep up with her”). Sure enough when I got down to the base of the hill, she was out of sight; I couldn’t even tell which way she had turned.
Being old, fat and slow SUCKS!!!! :-)
Seriously, I enjoyed the run and thought about this weekend.
Foot: not an issue, but it isn’t fully well. I need to ice it after EVERY workout.
Workout notes I felt better this morning so I decided to run.
It was warm (compared to recent weather) 67 F, 79 percent humidity. Yeah, that is sweater weather in Texas. :-)
I decided to try my Cornstalk classic 4.2 mile course (hilly)
It felt more difficult than expected; I felt slightly sick during the first mile so I told myself to relax; that came at 9:08 (for 1.03)
It was 3:19 down to the loop, 12:03 for the 1.3 (+) loop and 3:59 back up the hill, 8:45 back home. Time: 37:17, (8:52 mpm) for my fastest since July 2004 and my 5’th fastest since 2003. Ok, this used to be 33-34 when I was in 20 minute 5K shape. But last year it was a struggle to get 40:00.
But no, it was NOT an easy effort; it was work.
Now to shower, eat and do math. Still, I overdid it a bit; I am not 100 percent.
This is only for my records: last night saw me getting up a LOT and a LOT of pink bismuth. Otherwise, I don’t feel that bad; I was told by my spouse (who lovingly gave me this bug :-) ) that this should “pass” in a couple of more days.
I might walk again today and try something approaching a full workout tomorrow.
So, just a bit of humor:
Workout notes Great weather; walked my Cornstalk classic course in about 58 minutes (by time of day; I had to wait to cross streets, etc.) This was about 13:30 mpm or so on a hilly course; it was just hard enough to get slightly damp with sweat.
I am feeling better, but this mini-workout took something (just a little) out of me. There is no way in Hades I could have done this 7 times in a row (enough to make 30 miles) today. And yes, I’ve walked 50 miles in a row at a faster pace…a LONG time ago.
Later, my wife tells me that one of her former students (in his early 30’s) ran his first half marathon in 1:54. “That’s good, right?” she asks. I reminded her that when I was 39 and 40, I had run a 1:42 (windy; a month after a marathon) and a 1:35 (peaked) and she had yelled “get going Lard-Butt!” at me as I finished (25-30 minutes behind the winner). So, is he (her former student) a lard-butt? “No…that’s different.”
I got this e-mail message from Rick Santorum:
Grab the popcorn folks; this will be fun. :-)
See the earth through Saturn’s rings…and Saturn, with rings, from the earth via the moon:
The view from the other direction:
Cross posted in my math blog:
In my non-math life I am an avid runner and walker. Ok, my enthusiasm for these sports greatly excedes my talent and accomplishments for these sports; I once (ONCE) broke 40 minutes for the 10K run and that was in 1982; the winner (a fellow named Bill Rodgers) won that race and finished 11 minutes ahead of me that day! :-) Now I’ve gotten even slower; my fastest 10K is around 53 minutes and I haven’t broken 50 since 2005. :-(
But alas I got a minor bug and had to skip today’s planned races; hence I am using this morning to blog about some math.
Real Analysis and Calculus
I’ve said this before and I’ll say it again: one of my biggest struggles with real analysis and calculus was that I often didn’t see the point of the nuances in the proof of the big theorems. My immature intuition was one in which differentiable functions were, well, analytic (though I didn’t know that was my underlying assumption at the time). Their graphs were nice smooth lines, though I knew about corners (say, at .
So, it appears to me that one of the way we can introduce the big theorems (along with the nuances) is to have a list of counter examples at the ready and be ready to present these PRIOR to the proof; that way we can say “ok, HERE is why we need to include this hypothesis” or “here is why this simple minded construction won’t work.”
So, what are my favorite examples? Well, one is the function is a winner. This gives an example of a function that is not analytic (on any open interval containing 0 ).
The family of examples I’d like to focus on today is , fixed, .
Note: henceforth, when I write I’ll let it be understood that I mean the conditional function that I wrote above.
Use of this example:
1. Squeeze theorem in calculus: of course, ; this is one time we can calculate a limit without using a function which one can merely “plug in”. It is easy to see that .
2. Use of the limit definition of derivative: one can see that ; this is one case where we can’t merely “calculate”.
3. provides an example of a function that is differentiable at the origin but is not continuously differentiable there. It isn’t hard to see why; away from 0 the derivative is and the limit as approaches zero exists for the first term but not the second. Of course, by upping the power of one can find a function that is times differentiable at the origin but not continuously differentiable.
4. The proof of the chain rule. Suppose is differentiable at and is differentiable at . Then we know that is differentiable at and the derivative is . The “natural” proof (say, for non-constant near looks at the difference quotient: which works fine, so long as . So what could possibly go wrong; surely the set of values of for which for a differentiable function is finite right? :-) That is where comes into play; this equals zero at an infinite number of points in any neighborhood of the origin.
Hence the proof of the chain rule needs a workaround of some sort. This is a decent article on this topic; it discusses the usual workaround: define . Then it is easy to see that since the second factor of the last term is zero when and the limit of exists at .
Of course, one doesn’t have to worry about any of this if one introduces the “grown up” definition of derivative from the get-go (as in: best linear approximation) and if one has a very gifted class, why not?
5. The concept of “bounded variation” and the Riemann-Stiltjes integral: given functions over some closed interval and partitions look at upper and lower sums of and if the upper and lower sums converge as the width of the partions go to zero, you have the integral . But this works only if has what is known as “bounded variation”: that is, there exists some number such that for ALL partitions . Now if is differentiable with a bounded derivative on (e. g. is continuously differentiable on then it isn’t hard to see that had bounded variation. Just let be a bound for and then use the Mean Value Theorem to replace each by and the result follows easily.
So, what sort of function is continuous but NOT of bounded variation? Yep, you guessed it! Now to make the bookkeeping easier we’ll use its sibling function: . :-) Now consider a partition of the following variety: . Example: say . Compute the variation: . This leads to trouble as this sum has no limit as we progress with more points in the sequence of partitions; we end up with a divergent series (the Harmonic Series) as one term as points are added to the partition.
6. The concept of Absolute Continuity: this is important when one develops the Fundamental Theorem of Calculus for the Lebesgue integral. You know what it means for to be continuous on an interval. You know what it means for to be uniformly continuous on an interval (basically, for the whole interval, the same works for a given no matter where you are, and if the interval is a closed one, an easy “compactness” argument shows that continuity and uniform continuity are equivalent. Absolute continuity is like uniform continuity on steroids. I’ll state it for a closed interval: is absolutely continuous on an interval if, given any there is a such that for where are pairwise disjoint intervals. An example of a function that is continuous on a closed interval but not absolutely continuous? Yes; on any interval containing is an example; the work that we did in paragraph 5 works nicely; just make the intervals pairwise disjoint.
The morning came up icy (just below freezing, an inch of snow lead to slick morning roads). So I passed up the outdoor 5K run and did a lifting workout followed by a 10K on a track (50 laps of an indoor track).
I did my usual routine at a slightly slower pace.
rotator cuff (pulley), rows (Hammer Machine), curls (pulley), pull downs (lat machine)
row: 10 x 180, 10 x 230, 10 x 230
curl: 10 x 42, 10 x 47, 10 x 50
pull down: 10 x 140, 10 x 160, 10 x 160
Bench press plus sets of 20 sit ups (one after each bench press set)
10 x 135, 3 x 175, 5 x 170
Pull ups followed by sit ups (20 at the highest incline)
7, 7, 6, 6. The 7’s were done with knuckles facing me, the 6’s were done in a shoulder friendly position (knuckles facing sideways)
Incline bench press with one more set of sit ups:
7 x 135, 6 x 135
I finished with dumbbell rotator cuff exercises and two sets of 15 x 40 lb. dumbbell military presses.
Then I stretched and ran 10K on the track: my time was 56:50 for 50 laps (about 6.25 miles, slightly longer than 10K)
9:38, 9:21 (18:59), 9:12 (28:11), 8:57 (37:09), 8:47 (45:57), 8:40 (54:38), 2:12 (56:50)
Note that I was mostly by myself for the first 4-4.5 miles and then I paced off of one young woman for about .5 miles (4.5 to 5) and then took it in on my own.
Evidently I am almost over my illness; this 56:50 was only 30 seconds slower than what I did at a 10K race this September, and I had a warm up prior to that and I didn’t lift weights first. Then again, this was a springy track with no wind, hills, pot holes, etc. But it wasn’t a bad workout.
One funny note: near the track there is a mat where people can stretch or work with an exercise ball. As I completed one of the middle laps I saw a woman with a larger butt doing this (facing away from the track, almost perpendicular to it:
(click on the thumbnail to see the full size photo).
I got distracted for just a moment and regretted not having a camera. Well, not really.
(click on the thumbnail for the full size photo)
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