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#proof

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Marlon Joseph Tesoro Battad

I HEREBY SHEW PROOF THAT I CAN PLAY "EASY MUSIC" AND SING N'AT, AND SIMULTANEOUSLY ANNOUNCE, NAY, DEMONSTRATE WHAT I'VE BEEN DOING TO THIS HYMN FOR, HEH, QUITE SOME TIME NOW! WHO KNOWS HOW LONG ITLL TAKE TO FINE TUNE!? TEN YEARS!? ELEVEN!?
MWAHAHAHAHAHAHAHAH!!!
youtu.be/5hxgi8hnRBk

youtu.be- YouTubeEnjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.
#MARLONBATTAD#SANCTUS#BLANKNIGHT
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Markus Redeker

Proof by starvation: This is a proof form in which you first prove that a counterexample to the theorem must have property X, then, using X, prove that it must also have property Y, then that it must also have property Z, ... until you have piled up so many requirements on a counterexample that everybody sees that it cannot exist.

I have done that a few times. It is a nice way to organize one's thoughts.

#Proof#Mathematics
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Akshar Varma

@BernhardWerner My favorite alternative #proof strategy for #induction proofs are #combinatorial (counting) proofs.

I suppose the standard example might be the proof of the coefficients in the binomial theorem expansion, or for the sum of binomial coefficients being powers of 2. These can be proved by induction, of course, but I'm not sure that's common given how easier it is to do a counting proof. It is also much clearer and avoids tedious algebra.

One I like is proving that the sum 1 + 2 + 3 + ⋯ + 𝑛 is 𝑛 + 1 choose 2, the binomial coefficient \(\binom{n+1}{2}\). Bijection proof, counts the same thing in two ways. The thing being counted is the number of ways of choosing two things (distinct, without repetition) from the set {0, 1, ..., 𝑛}. By definition, it is the binomial coefficient we want. The other way to count is to fix the larger number 𝑘, the remaining choices are any of the 𝑘 numbers from 0 to 𝑘 - 1. Thus, across all possible larger numbers, we get the sum from 1 to n.

An alternative alternate proof of the same, slightly more geometric is as follows: arrange dots in a triangle, 1 on row 1, 2 on row 2, and so on up to row n, with n dots. Add a phantom row of n+1 dots below. We want to add up all dots in first n rows: ∑ 𝑖. If you think of all of this as a binary tree/DAG, then every dot has two children (imagine Pascal's triangle). If you pick any two dots in the phantom row, their common ancestor is unique. So counting dots is same as picking two dots in phantom row. Which is the binomial coefficient we want.

Benjamin and Quinn's book on combinatorial proofs is amazing for interpretations of this form (I learned the first proof from it). See also: en.wikipedia.org/wiki/Combinat

en.wikipedia.orgCombinatorial proof - Wikipedia
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C.

and , and every nation in the , should now insist on proof of vaccination against all generally preventable diseases from Americans prior to entry into their country.

If you want Bobby Brainworm in charge of your public health, it's your funeral, but you don't get to spread those germs in our countries to those of our citizens that are too vulnerable to be immunized.

#RFKjr#BobbyBrainworm#vaccine
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getmisch

Flag down on the play, B&H Photo-Video! Was repeatedly assured in-store that buying items & having them shipped would not incur any charges (I bought $200 of stuff!). So when it shipped & I saw a shipping charge, I called right away. Not like them, they're usually more honest.
#BandH #Photo #Video #Store #free #shipping #not #really #called for #credit #back #filter #holder #effects #photography #pictures #instant #color #black& #white #negatives #proof #sheet #roll #film #bellows #camera #shoot

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Chamber of Leather Craft

Here's a bit of how I #work .

I make my patterns myself - the ones I use to cut out pieces for #leather #goods and the ones I use for a #handcarved #decoration .

Two of these #leather #belts are decorated by a pattern that are inspired by #Estonian #folkart .

The other two are decorated by a sort of pattern that makes them #unique as they're cut onto the leather without any prepared drawing.

So as there is no #written #proof of the making process - are they #real or are they #myth ?

This is a photo of a wide leather belt in mahogany. Belt is decorated from top to bottom by hand carved pattern. Pattern is inspired by folkart and consists of rhombuses and triangles. Photo is taken in the darkening forest. Belt is set onto the fallen tree trunk. We also used some small electric lights to decorate the scene so the belt is surrounded by small warm sparkling light-dots. The dark deep green on the background (from the surrounding trees) goes really well with the mahogany color in the center.
This is a photo of a natural leather belt. The color of the belt is light beige as it has not been dyed. Belt is decorated from top to bottom by hand carved pattern. Pattern is inspired by folkart and consists of rhombuses and triangles. Photo is taken in the darkening forest. Belt is set onto the fallen tree trunk. We also used some small electric lights to decorate the scene so the belt is surrounded by small warm sparkling light-dots. The overall background of this photo is dark-grey which let's this light belt to shine in the center.
This is a photo of a narrower leather belt in green. Belt is decorated from top to bottom by hand carved pattern. This pattern makes this belt unique as it is carved onto the leather without any prepared drawing. I'd say there are lot's of lush leaves carved onto the belt. Photo is taken in the forest. Belt is set onto the fallen tree trunk and there are some red-brown colored heather plants on the front. There are also some warm sparkling lights in the right top corner. These come from the electric lights we used to decorate the scene.
This is a photo of a leather belt in blue. Belt is decorated from top to bottom by hand carved pattern. This pattern makes this belt unique as it is carved onto the leather without any prepared drawing. The pattern consists of wavy triangles - could be water streams but could be snowy mountains as well? I have decided to leave this answer to the one who will wear this belt. Photo is taken in the forest. Belt is set onto the fallen tree trunk. There are some warm sparkling lights surrounding the belt. These come from the electric lights we used to decorate the scene. background of this photo is rather dark black and grey but the belt is well lit and the beautiful blue shines in the middle of it all.
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claude

inspired by tavis' deep field #nebulabrot #DeepZoom images on #fractal #fractals forums, I did a little shader that for each c in the complement of the #MandelbrotSet M, colours according to how often z <- z^2 + c hits a given small target disc , weighted by derivative (as a proxy for point density).

it looks as though the hit sources are distributed everywhere near the boundary of M, which i think i can prove for target discs outside a sufficiently large esape circle, but i'm not sure how for discs nearer M. intuitively, by the time any cell pair in binary decomposition of exterior escapes, it covers an annulus with radii R, R^2, so any disc outside R will be hit by some region in every cell pair.

boundary of mandelbrot set and some nearby discs in shades of purple, pixelated.
#math#maths#proof
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