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| Profil stúpania na Ostrvu (vľavo) a na Veľkú Svišťovku (vpravo). Zdroj mapy.cz |
Z detských časov si pamätám ako Slavkovský štít vypĺňal podstatnú časť severovýchodného horizontu. A jeho vrchol vyzeral veľmi, veľmi vysoko. V tej dobe som tiež absolvoval niekoľko „veľkých túr“ zo Štrbského Plesa na Popradské. Od Popradského je vidieť serpentíny výstupu do Sedla pod Ostrvou – je to pohľad, pri ktorom má človek pocit, že tam určite nechce…
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| Bismarck, by Bundesarchiv, Bild 193-04-1-26 / CC-BY-SA 3.0, CC BY-SA 3.0 de, https://commons.wikimedia.org/w/index.php?curid=5438172 |
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| Somár, kúsok od značky. Pripomienka pravej podstaty trailových bežcov. |
Nejaký rok dozadu som sa rozhodol ísť dobrovoľníčiť na trailové podujatie. Realizoval som to na tohtoročnom Štefánik Trail s hlavnou trasou zo Sološnice do Rače. Na Babe sme rozložili občerstvovačku, poobdivovali a vypravili kategóriu človek + pes. Potom aj nordic walking. Postupom času začali pribiehať drsní ultratrailisti. Vidieť ich "z druhej strany" bolo inšpirujúce: vyžarovali nadšenie, odhodlanie, únavu. Občas pochybnosti. Na odbyt išli hlavne voda a kola. Našťastie nám nemuseli asistovať pripravení zdravotníci.
Z kruháča medzi Pod Lesom a Novou Lesnou – lokálpatrioti prepáčia nejasnosť vyjadrenia – je to na Slavkovský štít 10.4 km. Prvých tri a pol kilometra vedie k dolnej stanici pozemnej lanovky v Starom Smokovci. Toto je prvá tretina vzdialenosti na vrchol, ale len 12% prevýšenia. Vtipné je, že to lokálne dá zabrať i tak, napr. úsek od liečebne k Timrave je luxusný. Koniec-koncov Memoriál Jána Stilla tento úsek obchádza. Tak, či onak kilometre s prevýšením nad 50 metrov si nohy všimnú a je treba sa tomu prispôsobiť. Tu som sa šetril, vediac, čo ma ďalej čaká.
Búrka ustupuje a po týždni ignorovania bežeckých topánok ma to ťahá von. Na obzore sa týči Jura, prezentujúca vhodný cieľ. Plán: St-Geni – Crêt de la Neige, necelých 15 km jedným smerom s 1500 metrami prevýšenia. Pozerám na staré výkony, usudzujem, že tempo okolo 8:00 min/km je dostatočne pesimistický odhad. Ak nepôjdem úplne mimo, odhad splním a nebudem mať problém s vlakom.
One such entity made its appearance only once. First, we were reminded about the basics of Riemann integration[1] – cut the integration interval into sub-intervals, take the maximum of the function on each of them multiplied by the length of the corresponding sub-interval, add them together. Then send the number of sub-intervals to infinity and evaluate the limit.
A simile followed: This is the same approach as the one used by a cashier. When making the daily total, they take value of take each coin[2] and add it to the total.[3] Common sense, right? A revelation followed, though. A smart cashier (šikovný pokladník) does it in a different way. First, he counts all the coins of a particular denomination and create a subtotal by multiplying the number of coins by the denomination. In the end, he adds the subtotals. Clever indeed, right? If you are in education or mathematics, you might guess what followed. The smart cashier's approach was used to approximate the area under the curve in Fig. 1. This was our introduction to the Lebesgue integration by our Calculus lecturer, Mirko Rokyta.
Fig. 2: Picture by User:Svebert, CC0, via Wikimedia Commons. The main idea of Lebesgue integration is to look “horizontally” and count “how often” the function reaches a certain value. The resulting construction is mathematically very powerful.
I have not heard of the smart cashier since Fall of 2001.
A month or so ago, I came across
a challenge in my work. I had a dataset with colours reported by users and I wanted
to analyse if the colours were happy, neutral or sad. We were interested in the aggregate value of happiness, as well as partial sums per sets.
My original approach was straightforward: go through ever entry, assign it a value (1, 0 or -1) and process these at the end. Just for fun, try doing so. My main difficulty at this point was not the number of entries but the need to make a choice – often in a situation where I had no strong opinion. Is brick red (left) a happy colour or is it neutral? What about byzantium (right) – is the “dark enough” to be considered sad? Or is it regally happy? When faced with the task, I did what seemed the best: I asked a colleague to rate as well, and then we reconciled the differences. A few weeks later, an onerous task became a seemingly insurmountable one:
Short of employing an algorithmic approach (or paying someone to do it), the only option seems to be going entry by entry. Entry by entry..., like Riemann. Riemann is like a cashier..., oh, what would the clever cashier do? I do not know yet, but let’s see. Wait, if you order the first column, you’ll have four instances of mauve, three of electric blue and three of ivory. So you can count the mauves in the first column, assign it a value (1?) and proceed. Soon afterwards I realise that ivory and mauve are also present in the other columns. Now the solution is clear: you only need to rate the unique entries! Lookups and sums will take care of the rest. In the end, the work needed was reduced by a factor of 3. (That’s a big deal.)
The Clever Cashier emerges victorious. More importantly, the episode shows how a simple beautiful[4] concept can be reused in a different setting decades after being learnt. Allow me to end with a big appreciation to Mirko Rokyta for teaching in this way.
[1] I ask the experts to be generous with the non-rigorous language.
[2] We will consider only coins: they are more visual, especially when stacked.
[3] The cashier uses a discrete approximation.
[4] This is the right combination of adjectives. Henri Lebesgue’s idea is much better than simple and its undeniable elegance gives it beauty.
