Thomae's function edition[eqn] f(x) = \begin{cases}\frac{1}{q} & \text{if } x = \frac{p}{q}, \text{where } p,q \in \mathbb{Z} \text{ are co-prime} \\0 & \text{if }x \text{ is irrational}\end{cases}[/eqn]previous thread>>16432396
Define a positive rational exponent as: [math]x^{a+b/c} = x^a x^{b/c} = x^a\sqrt[c]{x^b}[/math]Then we consider exponent addition:[math]x^{a+b/c}x^{d+e/f}[/math]First add the whole numbers, no need for proof:[math]=x^{a+d}x^{b/c}x^{e/f}[/math]Then get the same denominator:[math]x^{b/c}x^{e/f}=x^{fb/fc}x^{ec/fc}[/math]The exponents remain equal during the last step:[math]x^{eb/ec}=({\sqrt[ec]{x}})^{eb} = (({\sqrt[ec]{x}})^{e})^b = ({\sqrt[c]{x}})^{b} [/math] Rewrite the numerator based on [math](ab)^{1/x}=a^{1/x}b^{1/x}[/math][eqn]=\sqrt[fc]{x^{fb}x^{ec}}[/eqn]And perform the final addition:[eqn]=\sqrt[fc]{x^{fb+ec}}=x^{(fb+ec)/fc}[/eqn]So by splitting the process into more believable steps we understand why exponent addition holds for rational powers.
Define a positive rational exponent as: [math]x^{a+b/c} = x^a x^{b/c} = x^a\sqrt[c]{x^b}[/math]Then we consider exponent addition:[math]x^{a+b/c}x^{d+e/f}[/math]First add the whole numbers, no need for proof:[math]=x^{a+d}x^{b/c}x^{e/f}[/math]Then get the same denominator:[math]x^{b/c}x^{e/f}=x^{fb/fc}x^{ec/fc}[/math]The exponents remain equal during the last step:[math]x^{eb/ec}=({\sqrt[ec]{x}})^{eb} = (({\sqrt[ec]{x}})^{e})^b = ({\sqrt[c]{x}})^{b} [/math] Rewrite the numerator based on [eqn](ab)^{1/x}=a^{1/x}b^{1/x}[/eqn][eqn]=\sqrt[fc]{x^{fb}x^{ec}}[/eqn]And perform the final addition:[eqn]=\sqrt[fc]{x^{fb+ec}}=x^{(fb+ec)/fc}[/eqn]So by splitting the process into more believable steps we understand why exponent addition holds for rational powers.
can someone prove that vectors have a morphism to topologies which is closed under transposition
>>16475814do your own homework
>>16475814>>16475815
Is there a way to turn the problem of palacking 17 unit squares into a polynomial with 51 terms (17 tuples of x-coord, y-coord, angle) or is it impossible?
>>16475762Do you guys have recommendations for textbooks on graph theory?
>>16475958It wouldn't make sense as a degree 51 polynomial but it might make sense as a linear equation of 51 variables.
>>16475775>>16475777Cool! Now prove the Recusion Theorem (so that integer exponents always make sense) and the existence of nth roots (from the completeness axiom of the real numbers)
>>16475959The standard is Dietsel's Graph Theory in GTM. It's the only graph theory textbook I have any real familiarity with, as the other graph books I've read have been focusing on applications (primarily PGM's and Markov Random Fields).
>>16476042Diestel*
>>16475770People always post this, but they should point to Misha Verbitsky's Problem Course in Undergraduate Mathematics too.http://shenme.de/listki/>This page is home to an English translation a of course that was taught by Misha Verbitsky and Dmitry Kaledin at the Independent University of Moscow in the fall of 2004 (also known under the name “Trivium”). The course was targeted for first-year students, so the prerequisites include only elementary high-school mathematics
>>>>/pol/488535575What is written on the blackboard?
>>16476352Idk but there’s a massive clue here if you care to look it up
How is the Suslin hypothesis independent of ZFC if it fails in L? How can a Suslin line cease to be?
>>16476062is this as much of a meme as the chart you replied to?
>>16476981>Misha VerbitskyIs the same guy from the meme listhttp://imperium.lenin.ru/~verbit/MATH/programma.html
>>16476042This wasn't exactly what I was looking for. Turns out I need a textbook on *algebraic* graph theory.
>>16476909>1. R does not have a least nor a greatest element;>2.the order on R is dense (between any two distinct elements there is another);> 3. the order on R is complete, in the sense that every non-empty bounded subset has a supremum > and an infimum; and> 4. every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R),If you admit more sets (go beyond L), then the suslin line may cease to satisfy 3, because there are more nonempty bounded subsets of the line.It could also start to fail 4 because there may be more collections of mutually disjoint non-empty open intervals in R, which happen not to be countable.
What are some mathematical systems in which 45=47?
[eqn]x^2+y^2=25[/eqn][eqn]2x+2yy'=0[/eqnwhyyyyyyyyyyyyyyyyyyyyymuh just take da derivative !!
[eqn]x^2+y^2=25[/eqn][eqn]2x+2yy'=0[/eqn]whyyyyyyyyyyyyyyyyyyyyymuh just take da derivative !!
>>16477217 >>16477214Are you retardeded?>>16477194Arithmetic modulo 2
>>16477217lolllll physics-pilled
Elliptic curves for retards/"undergraduates"?
>>16477322Silverman and Taint (lol) or nah?
>>16477324lmao boomer yellow books
Currently reviewing basic geometry on Khan Academy because I have terminal imposter syndrome and I'm never going to be confident in my abilities. My current neurotic obsession is that my fundamentals are mediocre so I have to systematically go through all my high school math to make sure I know it.
>>16477344Everything in high school geometry is some kind of affine translation.
>>16477194The field of order 1048576.
>>16477344Don't forget gelfand
If the universe entirely* consisted of math and you are some angel being able to look at it from an outside perspective, would probability even be a coherent concept?
>>16479947Yes since probability is just mapping binary outcomes to partitions of (0,1). It would be much less important to them than it is to us, however.
What is the difference between space and a field and R^3?
>>16479947>>>/x/
>>16480021>SpaceMetric space? Vector space? Just "space" is a rather vague concept, just like saying "number" instead of rational number or complex number.>FieldTwo meanings, one in algebra (ring with nonzero elements invertible under multiplication), the other in vector calculus (vector field, scalar field). Returning to algebra, the geometric line has field operations (real addition and multiplication) and the geometric plane too (complex addition and multiplication). Im not sure, but maybe its not possible to define field operations for points in 3D space (quaternion operations define a 4D algebra that is not commutative under multiplication, therefore not a field)>R^3Is an example of vector space. Tangentially, is an example of metric space, measure space and topological space too.>physicsR^3 is a model for physical space according to classical newtonian physics
What springer books r u guys picking up during this sale?I'm eyeing some of Pierre Brumaud's older texts, but Im also trying to find a good text on algebraic combinatorics.
>>16481171Damn some books are almost 75% off. I assume they're trying to get rid of some old inventory.Anyways, I got the problem books by polya and szego. I've heard good things about them.
I posted a thread about 0^0, hoping that seeing people argue about exponentiation would better help me visualize the riemann surface of x^y and I'm really disappointed by the amount of people in the thread who are instead arguing about multiplication.Have any of you ever gotten really good at visualising exponential surfaces?
Consider using a single cup for tea, and you never swap the cup.Instead of completely finishing up the tea, you generally drink until some of the tea ends up cold, leaving a fraction of the tea left. Instead of dumping the tea and replacing it with new cold water, you keep that tea in there while throwing the teabag and placing a new teabag, and then filling up the rest of the cup with new boiling water for the remainder.Concretely, with raw numbers, as an example you would drink your cup of tea until 20% remains, you will then throw the tea bag and replace it with a new, and place 80% of the cup with new boiling water, meaning 20% is tea and 80% is new boiling water.If you do this an infinite amount of times, what will the potency of the tea end up being? For any given fraction left undrank, although it is always the same fraction X that is left undrank. Do not consider any physical factors such as evaporation
I was stuck on the logic behind implicit differentiation:[eqn]x^3+y^3=6xy[/eqn] implies that[eqn]3x^2+3y^2y'=6xy'+6y[/eqn]until I rewrite the equation as[eqn]f(x)=g(x)[/eqn]so[eqn]f'(x)=g'(x)[/eqn]
Are math spergs ever bad at humanities or is it always the other way around?
>>16483163We are good at implementing algorithms in Python
>>16481171Are they still on sale?
How do i git gud at math again after not using it except for budgeting and percentages for years? I remember exponents, probability and i can read a graph. Biz major.
>>16483163I'm an EE, not a math sperg, but in general most of the EE's I know don't really read or pay attention to the humanities. There's two other people in my lab (two of the only white people in the department) who also read and pay attention to the arts, but most EE grad students I know are spending all of their free time on vidya or something instead of reading/music/the arts/etc.
>>16481171Pierre Bremaud's Fourier Analysis and Probability Theory books are both pretty good and cheap through the UTX series. I'm kind of meh on his mathematics for signal processing book, but it's good in trying to synthesize the applied Fourier analysis used in DSP with Lebesgue integral methods.My favorite book of Bremaud's is, unfortunately, not on sale for the physical copy. His discrete probability models book is actual gold. It covers basically everything you'd need for discrete probability and even covers Markov Random Fields and a good amount of discrete source information theory. Really good stuff. As far as I'm aware that one is only available through hardcover and isn't on sale because of this though.
im trying to make a lock-in amplifier but i dont know what the fuck a Hilbert space is.let [math]f(t)[/math] be a pulse signal of some frequency and duty cycle, and [math]g(t)[/math] be a similar pulse signal but smaller in magnitude and with a fuck ton of noise. the idea is to get rid of the noise by multiplying to two signals and integrating over some period.1. do the integrals over one period for both functions absolutely need to be 0? it makes things easier if not.2. do the integrals over one period of the square of both functions absolutely need to be 1? it makes things easier if not.
>>16485720What about you? If an english professor asked you to write a short story with correct grammar that made them feel something could you do it? You have one hour.
>>16475762x=-1/-1 =1 which maps the function to -1x=1/1 =1 which maps the function to 1Your setup maps one x value to two different values in the range, so you actually don’t have a function at all. 1 and -1 are coprime with all integers in case you didn’t know. Try requiring q to be a natural number.
>>16485976Yeah, the [math] q [/math] needs to be restricted to [math] \mathbb{N} [/math]
why did he do it?
Is it possible to position a unit sphere in R^3 such that it has no rational points?
>>16486140Consider(x - pi)^2 + y^2 + z^2 = 1
>>16485821I'd like to think so? I don't read as much literature as I used to in undergrad, but I still try to read at least one or two works of fiction per semester.Recently I finished re-reading Zen and the Art of Motorcycle Maintenance (which I first read in HS). I've been going back and forth between starting Shogun and Before the Dawn. Neither are high-art literature but they aren't slop either.
>>16485728Hm, I'll try and pick it up used.
Why isn't a flattened weierstass function (like 0.01W or sqrtW or 0.01(sqrtW)) lipschitz? It looks like it avoids the forbidden steepness regions of the plane around each point.
