Talk Nerdy To Me ;)
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28-01-2017, 05:19 PM
RE: Talk Nerdy To Me ;)
(28-01-2017 08:58 AM)Kernel Sohcahtoa Wrote:  *Please note that this was originally posted in the random thoughts thread and accidentally in the "would you rather thread" (my apologies for this). However, IMO, it is more logical to post it here in Jennybee's talk nerdy thread.

Hello TTA members and anyone else. I decided to work a random proof in my math book and thought some posters here might be interested in it. Here it is:

Prove that 1^2+2^2+…+ (n-1)^2 < n^3/3 < 1^2+2^2+…+ n^2 is true for all positive integers n.

Hint: use mathematical induction.

* Please note that I've placed spoiler tags on the background info and proof for the following reasons: First, I don't want to bore those who are familiar with induction; second, this is just my attempt at the proof and I'd like to give readers the chance to do the proof for themselves if they wish.

Background on Mathematical induction

Suppose we wanted to prove that a particular statement was true for all positive integers. Manually going through each integer would take forever and there are probably better things you could do with your time. As a result, mathematical induction is a tool that allows one to prove such a statement via the following process. First, we need to plug any positive integer into the equation (usually the integer 1) in order to verify that it is true (this is the basis step). Once we have verified the basis step, then the next step is the inductive step. First, we make an inductive hypothesis by assuming that the particular statement is true for some integer, say k. Next, we must show that our inductive hypothesis implies that k+1 is also true for this same statement.

Now, induction makes more sense if we think of all of the integers as dominoes, which are set up in such a way that if you knocked any one down then all of them would go down. Hence, by assuming that the statement is true for k and verifying that it is true for k+1, this shows that k knocks down k+1, and as a result, all of the positive integers n must also get knocked down.

Proof

We will use mathematical induction to prove this statement

Basis step. Let n=1. Then (1-1)^2 < 1^3/3 < 1^2 or 0 < 1/3 < 1 which is true.

Inductive Step.

Inductive Hypothesis. Let’s assume that 1^2+2^2+…+ (k-1)^2 < k^3/3 < 1^2+2^2+…+ k^2 is true for a positive integer k. As a result, we must show that this implies 1^2+2^2+…+ (k+1-1)^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2 or 1^2+2^2+…+ k^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2. Once we demonstrate this then the proof will be complete.

Observe that 1^2+2^2+…+ (k+1-1)^2 = 1^2+2^2+…+ k^2. Now, since 1^2+2^2+…+ (k-1)^2 < k^3/3, then it must be the case that 1^2+2^2+…+ k^2 < ( k+1)^3/3. Furthermore, since k^3/3 < 1^2+2^2+…+ k^2, then it must be that (k+1)^3/3 < 1^2+2^2+…+ (k+1)^2.

Now, we can also say that 1^2+2^2+…+ (k+1-1)^2 = 1^2+2^2+…+ k^2 ≥ k^3/3 < (k+1)^3/3 ≥ 1^2+2^2+…+ k^2 < 1^2+2^2+…+ (k+1)^2.
Hence, 1^2+2^2+…+ k^2 < ( k+1)^3/3 < 1^2+2^2+…+ (k+1)^2.

Consequently, 1^2+2^2+…+ (n-1)^2 < n^3/3 < 1^2+2^2+…+ n^2 is true for all positive integers n.


Jenny is soaking wet now.....




After her shower......

"If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality.
The very idea of God is a product of the human imagination."
- Paul Dirac
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31-01-2017, 06:27 PM
RE: Talk Nerdy To Me ;)
This is so Keyser Söze don't feel so lonely. Big Grin

Shut up and code!
Waterfall, scrum, XP, agile, test-driven, release early and often, design-driven, human centered, lean and mean, domain-driven, formal methods and their “retrenchment” (no, seriously), behavior-driven, model du jour, … for pete’s sake I could code up twenty different doohickeys before I could list all of the different ways of telling me how I should do it. To be fair, all of these structured methods were designed with good intentions in mind and are remarkably successful at modeling the thing they were invented to model. Now before you start calling me a bloody anarchist cowboy coder gone off range, yes structured methods are necessary for coordinating and specifying any system of any significant size and complexity or planes will fall out of the air and micirowave ovens will cook the cooker and blah blah blah. That’s all well and good but what about the little guy in the street who just wants to tell his thingamajiggy to ask the whatchamacallit to do thisandthat when wotthehellwosthat goes off? For projects small enough to be built by a singlecouplefew programming motherfuckers (Zed A. Shaw – date unclear), designed for a single “you got one job” purpose, all of these paradigms and philosophies and methods and models and formal processes and “you gotta do it the right way” just get in the way of “let’s get this shit done”. Which is really the only reason you’re bothering to code this up in the first place right? You got some shit you want to get done yesterday. When your main goal is unfettered by any external constraints, the only way to go is “stream of consciousness coding”. This starts with the barest minimum of specifications, the “you got 5 sentences, keep ‘em short” approach to system requirements. That whole process of thinking it through carefully before you sit down to write a line of code, yeah that’s bullshit. Once you got your 5 sentence spec it’s time to just code motherfucker code. Now I’m not suggesting this is for amateurs, this next step is strictly for professionals and should not be attempted at home. You spend zero time considering which tools or IDEs or languages or libraries to use. You grab the very first things you think of and just start coding. Like I said, don’t try this at home, need a few tools under your belt for this to work. I’m guessing you got ‘em if you made it this far. Okay I’m sitting down, what do I do now master? Not a damn thing. You go away and let someone who knows what they’re doing take over. I don’t care how you find your happy place, just go away. They’ll let you know when you can come back. Then and only then can you consider and review what they’ve done. What you’ll find is that no matter how ugly, no matter how unmaintainable, no matter how unsecure, no matter what its friggin’ computational complexity is, you will find in there a model of beauty in its simplicity and efficiency. You couldn’t have done it better yourself. There ain’t no Camel case or any other naming convention at all. Variables are declared in an ad hoc as needed basis. No time is wasted considering functions and methods and classes and properties, they form spontaneously as more and more scaffolding is built on the barest of skeletons. (In WPF the poor naked beast about to twisted and mauled into tesseracts and manifolds and and Klein bottles and Moebius Strips is just called MainWindow. Seems a bit ignomonius if you ask me.) What you will find is that for each obstacle you subconsciously encountered, you tackled it in the quickest and most naïve way you knew how and that is a beautiful thing. Now I better go back and clean that shit up before somebody sees it. Now shut up and code!

#sigh
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31-01-2017, 06:44 PM
RE: Talk Nerdy To Me ;)
One of my favorite fish families is the genus Hypoplectrus.

These fish are sympatric - multiple species evolve from a single ancestral species while inhabiting the same geographic region.

The practice assortative mating - they self-select based on color pattern even though all Hypoplectus species can interbreed, hybrids account for only 4% of the population.

They are simultaneous hermaphrodites - having both male and female reproductive organs. To avoid self-fertilization, they “egg trade”. The individual acting as a male releases sperm while the one acting as a female releases eggs, then they reverse roles!

Practice aggressive mimicry - Hamlet species have evolved via their sympatric mating choices to have similar color patterns (aggressive mimicry) to non-predatory fish (model).
They blend in with their “model” to sneak up & catch their preferred food.

“I am quite sure now that often, very often, in matters concerning religion and politics a man’s reasoning powers are not above the monkey’s.”~Mark Twain
“Ocean: A body of water occupying about two-thirds of a world made for man - who has no gills.”~ Ambrose Bierce
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31-01-2017, 08:26 PM
RE: Talk Nerdy To Me ;)
(31-01-2017 06:27 PM)GirlyMan Wrote:  This is so Keyser Söze don't feel so lonely. Big Grin

Shut up and code!
Waterfall, scrum, XP, agile, test-driven, release early and often, design-driven, human centered, lean and mean, domain-driven, formal methods and their “retrenchment” (no, seriously), behavior-driven, model du jour, … for pete’s sake I could code up twenty different doohickeys before I could list all of the different ways of telling me how I should do it. To be fair, all of these structured methods were designed with good intentions in mind and are remarkably successful at modeling the thing they were invented to model. Now before you start calling me a bloody anarchist cowboy coder gone off range, yes structured methods are necessary for coordinating and specifying any system of any significant size and complexity or planes will fall out of the air and micirowave ovens will cook the cooker and blah blah blah. That’s all well and good but what about the little guy in the street who just wants to tell his thingamajiggy to ask the whatchamacallit to do thisandthat when wotthehellwosthat goes off? For projects small enough to be built by a singlecouplefew programming motherfuckers (Zed A. Shaw – date unclear), designed for a single “you got one job” purpose, all of these paradigms and philosophies and methods and models and formal processes and “you gotta do it the right way” just get in the way of “let’s get this shit done”. Which is really the only reason you’re bothering to code this up in the first place right? You got some shit you want to get done yesterday. When your main goal is unfettered by any external constraints, the only way to go is “stream of consciousness coding”. This starts with the barest minimum of specifications, the “you got 5 sentences, keep ‘em short” approach to system requirements. That whole process of thinking it through carefully before you sit down to write a line of code, yeah that’s bullshit. Once you got your 5 sentence spec it’s time to just code motherfucker code. Now I’m not suggesting this is for amateurs, this next step is strictly for professionals and should not be attempted at home. You spend zero time considering which tools or IDEs or languages or libraries to use. You grab the very first things you think of and just start coding. Like I said, don’t try this at home, need a few tools under your belt for this to work. I’m guessing you got ‘em if you made it this far. Okay I’m sitting down, what do I do now master? Not a damn thing. You go away and let someone who knows what they’re doing take over. I don’t care how you find your happy place, just go away. They’ll let you know when you can come back. Then and only then can you consider and review what they’ve done. What you’ll find is that no matter how ugly, no matter how unmaintainable, no matter how unsecure, no matter what its friggin’ computational complexity is, you will find in there a model of beauty in its simplicity and efficiency. You couldn’t have done it better yourself. There ain’t no Camel case or any other naming convention at all. Variables are declared in an ad hoc as needed basis. No time is wasted considering functions and methods and classes and properties, they form spontaneously as more and more scaffolding is built on the barest of skeletons. (In WPF the poor naked beast about to twisted and mauled into tesseracts and manifolds and and Klein bottles and Moebius Strips is just called MainWindow. Seems a bit ignomonius if you ask me.) What you will find is that for each obstacle you subconsciously encountered, you tackled it in the quickest and most naïve way you knew how and that is a beautiful thing. Now I better go back and clean that shit up before somebody sees it. Now shut up and code!

It might work, but you end up with write-only code. Drinking Beverage

Skepticism is not a position; it is an approach to claims.
Science is not a subject, but a method.
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01-02-2017, 09:20 AM
RE: Talk Nerdy To Me ;)
(31-01-2017 08:26 PM)Chas Wrote:  
(31-01-2017 06:27 PM)GirlyMan Wrote:  Shut up and code!

It might work, but you end up with write-only code. Drinking Beverage

For single purpose software think that's fine. For example, I just finished writing a test harness for an experimental psychologist's experiment. Nobody's ever going to look at that code. The FSM (not the flying spaghetti monster FSM, the other FSM) is a simple little thing specific to this experiment. So I just hardcoded it in. I'm not interested in making a general purpose FSM editor. The experimental material is also hard-coded in. If they want to change the state machine or material I'll go in and change the code to hard-code it in as well. Took 6 hours to code it up and get it working. Took the damn longwinded PhD longer than that to regale me with all the fantastic things this experiment was designed to investigate like I gave a shit or something.

#sigh
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01-02-2017, 10:08 AM
RE: Talk Nerdy To Me ;)




[Image: 03052c6ebc73a54679d754180b62e990.png]

[Image: 20cad83ad8d757191e2878b0f4bf05a9.png]
"Don't answer that. A rhetorical question."
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01-02-2017, 10:42 AM
RE: Talk Nerdy To Me ;)
(01-02-2017 10:08 AM)ELK12695 Wrote:  



[Image: 03052c6ebc73a54679d754180b62e990.png]

omg woooooooow (O.o)
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01-02-2017, 11:35 AM
RE: Talk Nerdy To Me ;)
(01-02-2017 10:42 AM)kineo Wrote:  
(01-02-2017 10:08 AM)ELK12695 Wrote:  



[Image: 03052c6ebc73a54679d754180b62e990.png]

omg woooooooow (O.o)

It is the origin of this meme
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[Image: 20cad83ad8d757191e2878b0f4bf05a9.png]
"Don't answer that. A rhetorical question."
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01-02-2017, 12:46 PM
RE: Talk Nerdy To Me ;)



Don't let those gnomes and their illusions get you down. They're just gnomes and illusions.

--Jake the Dog, Adventure Time

Alouette, je te plumerai.
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01-02-2017, 11:29 PM
RE: Talk Nerdy To Me ;)
Here's a video of Arthur Benjamin who is a Mathematician, Mathemagician, and gentleman.




"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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