[math]/\mathfrak{mg}/[/math]Deutſche Mathematik editionTalk maths, formerly >>16240472
Trying to see if I can generalize the notion of the spectrum of a commutative ring.A ring is a monoid object in the category of abelian groups, so we should look at monoid objects in any monoidal category with a terminal object (maybe we want it to be an abelian category as well). I'm not sure if we need commutativity, but if so, then we look at commutative monoid objects.Given any prime ideal [math] \mathfrak{p} [/math] of a commutative ring [math] R, [/math] we can localize at the [math] \mathfrak{p} [/math] and then mod out by the unique maximal ideal of that local ring to get a map [math] R\to k_{\mathfrak{p}} [/math] where [math] k_{\mathfrak{p}} [/math] is a field. Conversely, given any morphism from [math] R [/math] to a field, we can recover a prime ideal which are all the elements sent to [math] 0. [/math]Hence, prime ideals exactly correspond to morphisms into a field (i.e. simple objects in the category of commutative rings).Therefore, we can define the spectrum of a (commutative) monoid object of a monoidal (abelian) category as the morphisms to simple objects. I guess we might as well consider the spectra of all such objects, and so we consider the collection of morphisms from monoid objects to simple monoid objects. I think this view allows us to compare spectra of two rings, but it may be a little loose.Next, we want to place a topology on this spectrum; I guess we try to mimic the Zariski site, and so we need the notion of inverting a single element. I guess the key is to note that in the setting of commutative rings, if [math] a\in R [/math] and [math] R\to k [/math] where [math] a^{-1}\in k [/math] for some field [math] k, [/math] then this map factors through [math] R\to R[a^{-1}]. [/math] In particular, for any prime ideal [math] \mathfrak{p} [/math] not containing [math] a, [/math] then the corresponding map [math] R\to k_{\mathfrak{p}} [/math] factors through [math] R\to R[a^{-1}]. [/math]
>>16271294Now we only really care about prime elements [math] a\in R [/math] since if [math] a=bc [/math] then [math] b^{-1}=c(bc)^{-1}\in R[a^{-1}] [/math] and similarly [math] c^{-1}\in R[a^{-1}] [/math]Now obviously each prime element [math] a\in R [/math] corresponds to a (minimal) prime ideal, viz. the one generated by [math] a. [/math]Here's where I'm a little unsure, but I think we want to consider all the prime ideals which don't contain the ideal generated by [math] a. [/math] Since they don't contain this ideal, they don't contain [math] a [/math] and so all the corresponding morphisms [math] R\to k_{\mathfrak{p}}\ni a^{-1} [/math] factor through [math] R\to R[a^{-1}]. [/math] I want to say that this factorization satisfies a universal property so that [math] R[a^{-1}] [/math] is some colimit.The difficulty seems to be with choosing which prime ideals we want to consider; in particular identifying a minimal prime ideal. Actually, it is pretty easy to see that if [math] \mathfrak{p}\subset\mathfrak{q} [/math] then [math] k_{\mathfrak{p}}\supset k_{\mathfrak{q}}. [/math] A prime ideal [math] \mathfrak{p} [/math] is minimal then for any field extension [math] k_{\mathfrak{p}}\hookrightarrow k, [/math] any map [math] R\to k [/math] factors through [math] R\to k_{\mathfrak{p}}. [/math]So I guess we want to consider in some sense a "maximal" object [math] k_a [/math] in the category of simple monoid objects, and this will correspond to all the ideals of an element [math] a\in R. [/math] We should then take the category of all simple monoid objects which are not subobjects of [math] k_a, [/math] and the obvious functor from this category to the category of objects under [math] R. [/math] The colimit of this functor will then give our desired object [math] R[a^{-1}]. [/math]Finally, we mimic the Zariski site, taking the topology on the opposite category of monoid objects to have base consisting of isomorphisms and maps [math] R[a^{-1}]\to R. [/math]
>>16271294>>>/x/
>>16271294>>16271296In some sense, I think this seems nice, since it really exhibits the important role that simple objects play, which is something that is possibly overlooked.On the other hand, it feels kind of clunky, having to jump to the category of simple objects and find the maximal such ones. I think there might also be problems if you try to take injections as opposed to honest-to-god embeddings.
>>16271303Sorry, I'll try to stay on topic.I'm really struggling on my calculus homework. My teacher keeps telling me that I can't just plug in the variable when taking the limit. I tried explaining to him that it works everytime to get the correct answer, but the retard just won't listen.
>>16271294>>16271296interesting idea. I don't see why you are trying to recover the spectrum as a set though. It is probably easier to just focus on recovering the site straight away, maybe by generalizing some universal property, e.g. spec is adjoint to [math]\Gamma[/math]. Then you can play around with what categories you have your functors between.
>>16271296>>16271294Mhm. Does this take into account that 1+1 equals 1? No... Well.... pity.
>>16271237Hey I'm kind of a brainlet. Does anyone know a simple or neat example of adjunctions arising from monads (especially in programming contexts)?
Does there exist a space-filling rhombic disphenoid?
bump
I wonder where all the analysis to go. The number theory and algebraic people are way more active in /mg/
>>16272869*analysists
>>16272871Mouf. Now.
>>16271237Why are proofs of spectral theorem and the likes so darn complicated in books? I have seen proofs using induction, minimal polynomials etc. However, I discovered a simple proof using orthogonality of eigenvectors of symmetric which gives way to such a nice enlightening geometry. I just simply don't understand the reason for all this convoluted proofs.
>>16272885Which spectral theorem are you referring to?> I just simply don't understand the reason for all this convoluted proofs.Which proof in particular confuses you? Ask specific questions and we may be able to help.
If you have a fraction so that the denominator is the numerator backwards, do you ever get an integer?
>>16272933Yes. Providing an example is left as a (trivial) exercise for the reader.
>>16272933many such examplesinfinitely many in fact
>>162729331/1
>>16272933In base-1, all of them
>>1627293390/09 = 10
>>16272966Is there an example that is multiple digit and doesn’t end in 0?
>>16272984There are infinitely many.
>>16272933lrn2codethe obvious ones are palindromes and palindromes with any amount of zeros appended, i wondered if there were others and one minute of writing code produced others: 8712
>>16273004Have any examples been explicitly constructed?
>>16273009Yes.The shortest I know of is only 4 digits long. Very easy to find.
>>16273010I wonder how quickly the number of examples below N grow as a function of N. Is it log, like prime numbers? Or worse? Or better?
>>16273015Who cares? It's base dependent. Boring!
>>16273008Interestingly, 8799912 (or any amount of 9s) works.
>>16272966Doesn't count. You can't start a number with a zero
>>16273017Is the asymptotic rate of growth base dependent though?
>>16273004*There are infinity.Saying 'infinitely many' is a bloated and cluttered way of just saying what we already have a word for: Infinity. Anyone that ever uses the phrase 'infinitely many' is a turbo autist.
>>16273456You are wrong.
>>1627293310[__99999...__]89 works For various reasons, the quotient and the first integer of the denominator can only be (2,6), (4,8), and (9,9). We assume the quotient of 1 and 5 aren't allowed if you can't do 0's
>>16273459Wrong. You are wrong.
>>16273530(2,6) do not work, so it's just (4,8) and (9,9)
>>16273530>>16273537Any combination of [1089..1089...1089] works actually, and with the 99999... in the middle. I wasn't too thorough but that might be it for the quotient of 9. I'm not gonna do 4, im done.
Anyone here cracked with Sn rep theory? I'm reading Vershik and Okunkov's paper on their approach and trying to apply it to Khovanov's Heisenberg category, but I'm still getting my feet wet so it seems.
>the reader will enjoy provingok I'm donefuck you book
>>16273761Authors who do this are based. Math should absolutely retain its high barrier to entry, and not pander to the common minds.Just look in any grad department, and the quality of the students is disappointing to say the least. It's not coincidence that this shift in quality coincides with the push to make math more inclusive and "user-friendly".
>>16273763Hard agree.The book very clearly defined its target audience, >>16273761 was not in it, and he acts like it's somehow the author's loss
>>16271305Let f(x)=1 for all x ≠ 0 and f(0)=1. Now your method gives the wrong answer.
How did ya'll get better at maths?
>>16273822We molested the fabric of the universe until it told us more about itself.
>>16273833That's physics, not math! For math you have to molest the fabric of the multiverse of all logically possible universes.
Are there any number theorists on here? How are you supposed to raise a number to a p-adic number? Nobody I've talked to knows how to do it. The only clue I have is that the resulting value should only depend on the residue class of the numbers mod 4
>>16273892Generate a p-adic systemDefine each point as a subset of a set of other points>by definition, each point is generate by a number of points raised to another set of points I actually can only do truth tables, so sorry if my answer doesn't help!
>>16273892https://math.stackexchange.com/questions/436984/raising-a-rational-integer-to-a-p-adic-powermight provide you some insight you'll find helpful
>>16273896>tfw you just make yourself right by definition instead of doing math>truth tables are op
>>16273892aren't they just units in z2, not p-adic numbers?
Is studying Gelfrand's books and Lang's Basic Mathematics really necessary to get into other fields? I can understand Stewart's Precalculus book just fine and I haven't done those
>>16273914Lang is a meme
>>16273916Well what do you recommend for the basics? I find Gelfand dull and obtuse
>>16273912an invertible p-adic number is a p-adic number>>16273898thanks I might be able to get this to work
hello math, I posted this in /sqt/ but haven't got a good enough response yet so I'm posting it again here.math/cs/stats folks itt, I'm sort of an average joe, midwit, autodidact ig and I wish to learn analysis and discrete math, but I only know pre calc algebra. I've got some questions regarding this.For calculus and analysis:>Should I just complete calc1,2,3 b4 doing analysis or can I go straight ahead?>What are the best texts from a theory & problem solving perspective for learning calc 1,2,3 and analysis? I did search a little bit and found baby rudin and apostol being recommended for analysis. I was also able to find some not so often recommended texts like Courant's calculus(for analysis?), Keisler's elementary calculus, thomas' calculus, etc.. What do?>Any good place for finding problem sets & lecture notes that may really help in improving my grasp of applications based calculus and analysis? What about books like schaum series, as in do they have a nice variety of problems? How about previous year GRE papers, putnam, etc.?For discrete math:>Again, what books do I use from theory & problem solving perspective? CL Liu? Kenneth Rosen? Don knuth's concrete math? something else?>where do I find good problem sets & lecture notes that are really useful and again, is schaum's series, previous year GRE papers useful?>Do I have to spent incredible amounts of time trying to study topics like combinatorics & number theory independently if my aim is to do machine learning, theoretical comp sci like algorithms, cryptography, etc.?
>>16274438>>Should I just complete calc1,2,3 b4 doing analysis or can I go straight ahead?about that: it depends on the book. You have books like Amann Escher, Zorich, Apostol which do both simultaneously while Rudin expects prior knowledge in Calc.
>>16274438just do calc 1 up to and including the fundamental theorem of calculus, then jump into analysisreason: calc 2,3 are only needed for engineers. All relevant theorems from calc 2,3 are contained in multivariate analysis courses/books or as preliminaries to differential geometry
>>16274438Look up Evan Chen's notes for Math 55b, Honors Analysis. It has most of the interesting analysis
Are quadratic polynomials over the p-adics classified? Or, say, cubics in two or three or so variables?
>>16274438read a different book on each topic: brezis for func ana, ahlfors for complex ana, gelfand for harmonic ana, bogachev for measure theory, etc. Books that do all these topics at once are incredibly shallow and you won't get any wiser
>>16274946>Evan Chenlegit tranny or was this pic some prank? kek>>16274451>Amann escher, zorich apostol which do both simultaneouslyyou mean single & multi variable calc or were you trying to mean that it has a lot of computation related stuff like tricks for solving integrals. the kind of stuff commonly seen in calc 1 or 2?>>16274938what texts can I use if my prime goal is just calculus 1,2,3 and analysis of single variables? I ask this because I may or may not have time for analysis and was expecting calculus texts good for applications and problems.>>16274956ok
>>16275109>you mean single & multi variable calcyes but also analysis. They have everything a calc book would have but treat the theory rigorously
>>16274438You'll want to learn up to Calc V. In Calc I one learns how to differentiate functions of one variable. In Calc II one learns how to integrate them. In Calc III one does both with finitely many variables. The natural continuation would be Calc IV: differentiation of functions on infinite-dimensional spaces. Finally, Calc V is integration on infinite-dimensional spaces (essentially the theory of Gaussian Feynman path/functional integrals).
How the fuck do you prove the equalizer existence thing in singular cohomology with points? I have functions f,g from A to X and an equalizer of f,g (up to homotopy) h: X to Z. Given a cohomology class in X that pulls back to the same class under f and g, i want to prove that it's a pullback of a class on Z. All cohomology pointed. I've tried defining it at a singular cochain level but didnt get anywhere. This is in an attempt to prove that the pointed singular cohomology functors are homotopy functors (this is one of the two properties).
Should I go for a PhD in theoretical CS?I have a math degree, getting excelent marks on mathematical subjects was no issue, but it turned out my real talent is hidden in algorithms design. Should I pursue it, or go for e.g. algebraic geometry?I'm asking because I'm not sure how rewarding TCS will be in the end.
>>16273761>not trying to prove all the results in the book yourself first anyway
>>16273917I'm working through Basic Mathematics right now and I think it's pretty good. It can be a bit of a slog in the beginning because of the precise language he uses but you get into the flow and really helps you develop a deeper understand of why things work the way they do.
>>16275943>algorithms designSuch as?
>>16275958you'l want to get used to that language. Basically all math is written in a precise manner
>>16275109no evan chen actually trannied out
>>16276006sad. Was he a manlet who got race & heightpilled?
>>16276006>>16275109Please dont use that word. It's uncalled for.
>>16276009didn't mean to sound rude anon, but it breaks my heart to see talented people getting filtered from the genepool because they just so happened to be physically unattractive.
>>16275989That's why I decided to persevere
>>16275829Now that I'm on my computer let me typeset this a bit better.In the category of path connected nondegenerately based spaces we are given[math]f_0, f_1 :A\to X[/math][math]g: X \to Z[/math] s.t. [math]g \circ f_0 \simeq g \circ f_1 [/math]For all spaces [math]Z'[/math] and maps [math]h: X \to Z'[/math], if [math]h \circ f_0 \simeq h \circ f_1[/math], then [math]\exists k:Z \to Z', h \simeq k \circ g[/math]. Let [math]z \in H^q(X, x_0, G)[/math] be such that [math]f_0^*(z) = f_1^*(z) \in H^q(A, a_0, G)[/math]. I want to prove that [math]\exists w \in H^q(Z, z_0, G), g^*(w) = z[/math]H is singular cohomology, G is an arbitrary abelian group.
>>16272908>Which spectral theorem are you referring to?Of matrices.>Which proof in particular confuses you? None of them confuse me and I mentioned which proofs.
>>16276265>Of matricesnormal, real, complex operators? Just be specific. When I hear someone talk about a "general" spectral theorem for linear operators over a field, I assume they're referring to the decomposition of a vector space in generalized eigenspaces and a non-spectral part. This decomposition can look wildly differently depending on the operator
>>16275109bro just start reading and find out yourself, youre wasting your time here
>>16276436Matrices over IR with Euclidean inner product.
>>16276122There's a supposed answer to this on Math SE but it's wrong https://math.stackexchange.com/questions/2342851/proof-that-cohomology-functor-is-a-homotopy-functorIt depends on the specific topology of the equalizer by decomposing it and applying Maier Vietoris sequences. The decomposition doesn't make any sense (perhaps they misunderstood the question). Also it's not given that the equalizer in question is unique and equal to the one in the construction.For example, an equalizer of the two elements [math]f_0 = [id], f_1 = 2[id] \in \pi^1(S^1, p) = [S^1, p; S^1;p][/math] can be any space, since the map that's asserted to exist is explicitly allowed not to be unique.Thus if [math](Z, z_0)[/math] is any space, [math]f: (S^1, p) \to (Z, z_0)[/math], [math]f(x) = z_0[/math] satisfies the condition, since if [math]g:(S^1,p) \to (Z', z_0')[/math] is any map, then [math][g \circ(id)] = [g] \in \pi^1(Z', z_0')[/math] and[math][g \circ(2id)] = 2[g][/math]Thus [math]g \circ f_0 \simeq g \circ f_1 \implies [g] = 0[/math]and any map [math]h: Z \to Z' [/math] will do.I know there are people ITT who claim to have read Spanier and liked it. If you're one of them, please explain, because I'm getting filtered.
>>16276006damn most asian bois make good femboys but he is NOT one of them
>>16276536Nevermind, by the property we pull the cocycle in the standard construction to the random equalizer and the pull it back to the space X, which equals the original cocycle by commutativity. So in fact it does suffices to consider the standard construction.
>>16276559Yup it works out, but not the way that the person wrote in the answer (idk wtf he was thinking) and also using a bit of a different construction than the one Spanier provides.[math]H^q(Z, z_0) \to_m H^q(A) \oplus H^q(X, z_0) \to_n H^q(A) \oplus H^q(A)[/math]Where[math]m(z) = (z|_{A\times\{1/2\}},z|_X)[/math] and [math]n(a,x) = (a - f_0^*(x), a - f_1^*(x))[/math]Which is pretty much exactly what we need. However, to get this decomposition we have to use a different equalizer than the one defined in the book. In the book, Spanier collapses the interval [math]\{z_0\}\times I[/math] but in our construction we don't, so that we can take a nice slice of [math]A[/math] in the middle of our wrapped cylinder. That's why some of the cohomology groups are pointed and others are not.
>>16276614Actually it's not that we can take a nice slice of [math]A[/math] but so that we can take its neighborhood without intersecting the space X. Because if we do, then we also intersect a neighborhood of X which is problematic to deal with, and we can no longer nicely deformation retract onto spaces X and A in our Maier Vietoris decomposition.
>>16276529>>16276265The spectral theorem for symmetric/hermitian matrices is essentially algebraic because the hypotheses are algebraic. You can relate the result to geometric ideas but at some point the key of the proof will depend on an algebraic idea. The approach suggested by your image is to use the quadratic form to show the existence of the maximum or minimum eigenvalue, but how do you justify a priori without the spectral theorem that you can see the quadratic form as an ellipse? Well ignoring that we know that the maximum is achieved irregardless of this let's say by [math]X_1[/math] now, how can we conclude that it is actually an eigenvector? Here is the issue you can basically say [math](AX_1-\lambda_1 X_1,X_1)=0[/math] and then why is it that this implies that it is an eigenvector, that is, [math]X_1\in Ker(A-\lambda_1 I)[\math]? Well here is the part where all boils down to the algebra since this is an if and only if for symmetric semi-definite matrices and I don't see any way to make sense of this part without an algebraic argument. Actually this if and only if is usually justified using the spectral theorem but well.
It's over.
>>16276619No it's not the act of computation is symmetrical.
Can someone explain the structure sheaf and canonical bundle to me? I don't get it at all.
>>16276788The way I think of the structure sheaf (I assume you're referring to this iteration of it) of a suitable variety is the set of the "allowed" rational functions on an open subset U. If you take an open ball on a surface for instance, you cannot have rational functions that assume zeros of the bottom polynomial inside that set allowed in your set of "allowed" (regular) functions. So this creates a picture where when you restrict your set more and more up to a single point (this is the stalk), you end up with a ton of rational functions, and it's actually the same as localizing away from the prime ideal that point represents in the coordinate ring. But when you look at the entire variety, you're not allowed basically ANY rational functions at all. This creates a picture that resembles a sheaf of wheat, if you're able to picture it. If you pinch the bottom of the sheaf of wheat, you get a big plumage on top, i.e. lots of allowed regular functions, and if you make the plumage at the bottom of the sheaf very large, you get only a small number of regular functions. This particular sheaf is one we get "for free" and so we use it a lot.The key here is that in general the structure sheaf tracks what rational (again we use the term regular) functions you are allowed to have, i.e. no zeros of the bottom polynomial, in any particular open subset of the variety.As for the canonical line bundle, you have to be familiar with the cotangent bundle first. In that case it's very similar to the volume form of a manifold, if you know what that is. It's actually something you can think of pictorially quite effectively if you consider 3d space as a motivating example.Hope that helps, i'm kinda retarded
>>16275975New and faster algorithms for well known graph problems. I'm not comfortable doxing myself by sharing my most cited paper.
>>16275109>I may or may not have time for analysisbut you sure do have time for shitposting on /sci/
>>16276618
>>16277067Read Lang's Algebra, it's so much better
>>16277234Kill yourself.
>>16275943consider specializing in computational aspects of algebraic geometry. have a look at Gröbner basis theory, for example
>>16277237not that anon but Lang's section on spectral decomposition is genuinely a nice read. Great exercises too
Chapter 10 of this book (spectral theory for unbounded operators) is owning me hard, anyone got some good sources for learning about it? I think Grandpa Rudin Covers the spectral theorem for unbounded operators towards the end, is it worth giving it a shot?
>>16271296>>16271294This is your brain on commutative algebra
>>16277390I think it is pointless without knowing some measure theory. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf Here are some notes wich are more foccused but the writting is sometimes quite shit and has some errors.
>>16275601>The natural continuation would be Calc IV: differentiation of functions on infinite-dimensional spaces. Finally, Calc V is integration on infinite-dimensional spaces (essentially the theory of Gaussian Feynman path/functional integrals).is that graduate level stuff?
>>16277734late undergrad/grad and nobody calls it "calc IV/V"
hi /math/how do you take notes for future reference in a clean and efficient manner? pen and paper? digitally with OneNote or other note-taking app? pen and paper and then LaTeX + pdf? going through a book right now and I'd like to keep my solutions as reference
>>16277784I usually sketch a solution on paper and then type it up in latex. I like this because when writing a proof out in detail I often catch mistakes I made before.
I'm reading through this book about Differential Geometry, where a surface at a 3D point p can be parametrized by a function X that maps parts of R^2 to the surface around p. The author is going through a proof that doing a coordinate change from one parametrization to another is diffeomorphic. The proof is shown in the pic, and my abbreviated version is in the bottom in red. In my short version, I don't talk about neighborhoods - it's more a gist of the proof.My issue is, he goes on to say that in that proof, it is vital that X^{-1} and Y^{-1} are continuous, and honestly, I'm not seeing it. At the top, he does say that (X^{-1} o Y) is a homeomorphism, and that the rest of the proof is to show it is also a diffeomorphism. But what if X^{-1} isn't continuous? I don't see how the proof fails. The Inverse Function Theorem (IFT) concerns the differentiability of X and Y, not that their inverses are continuous. Like, does it have something to do with the neighborhoods of X(q)? I wish he just explicitly mentioned the importance during the proof instead of later on
>>16277784I just use pen and paperTeXing notes is way too much fucking work unless you intend to hand them out to other people and I don't own a tabletFor a long time I didn't take any notes at all but eventually you reach the point where people start telling you things that aren't written down anywhere and I kept forgetting them
>>16277772what is called then?
>>16277657I do know measure theory, not sure what makes you think I don't.
>>16273778this is just the constant function 1. The signum function defined by f(x)=-1 for x<0, f(x)=0 for x=0 and f(x)=1 for x>0 would show the failure of anon's method.
>>16273456Infinity is a more elusive concept than its use in this particular context
>>16272933let d1=...=dn. Then the number d1...dn/dn...d1 is an integer.
>>16277871Lmao I was just stating it because it is a requirement. I don't know who the fuck are you or what you actually know.
>>16276875Cool, what type of methods do you use?
>>16277827by chain rule, the local inverse to a C^k function is C^k. you care about the inverses being continuous because otherwise the change of coordinates might not be a homeomorphism, which would just be weird: the underlying space remains the same, so a change of local coordinates should retain the local geometric structure (i.e. be a C^k isomorphism AKA diffeomorphism)
I love mathematics but I am stupid :(
>>16278469gotchu covered pal
>>16278472based helpful anon this looks great thank you
>>16278475I told you I was stupid :(
>>16278469https://sheafification.com/the-fast-track/
>>16277854Functional Analysis and Quantum Field Theory. Zinn-Justin's book handles functional integrals rather thoroughly
Do we have any serious algebraists/number theorists (preferably ~PhD-level) who would be willing to tutor me in local class field theory for free?
>>16276615>>16276614>>16276559>>16276536This doesn't work because the uncollapsed space is not an equalizer, because the inclusion maps are not point-homotopic.At this point I'm at a loss as to how to proceed. If anyone who has a clue about elementary algebraic topology would like to help, please do.
>>16278913Are you self-studying? What's your skill level?
>>16279062My goal is to learn localCFT from this book https://link.springer.com/book/10.1007/978-0-387-72488-1The author follows the approach of Hasse-Noether via CSAs and the Hasse invariant. I've self-studied all of the other relevant chapters last year>Absolute values on fields (completion, Hensel's lemma, extending AVs)>Local fields (classification, ramification)>Semisimplicity and Artin-Wedderburn theory>CSAs and the Brauer group>Brauer group of a local fieldbut the actual chapter on CFT kicked my ass, so I dropped it, now I want to come backI'm a a first-year master's, reasonably well-versed in algebra and number theory
>>16279071Too bad nobody here actually knows any math.
>>16279102I figure as much, but I posted on the off-chance. Screw you for making me list out my credentials if you weren't planning on helping
So imagine you're starting at (0,0) and you started running in a straight line to any location on this grid, I want to know where you were when you crossed a 1 or -1 on either axis (the grey dotted box in my picture). Example, if you ran straight to (4,4), you would've crossed 1 at (1,1). If you ran straight to (-4,0), you would've crossed 1 at (-1, 0). How do you deduce this mathematically in the simplest way possible? Say I ran to (3.23, 7.4), where was I when I crossed a 1 on either axis?
I didn't save the picture with the grey dotted box, goddamn it.
What book would you recommend for module theory, about direct sums, free, projective and injective modules in particular?Keep in mind that I'm dumb.
>>16279140dude this is like high school math desu , you shouldn't be having much trouble with it unless you are in high school. make the equation of the line from the two points (0,0) and (3.23 , 7.4) slope= 7.7/3.23 , y-intercept is zero . So equation is y= 7.7/3.23 x , and now pug in x=1 and x=-1 you get y=2.38 and y=-2.38 , so the points are (1,2.38) and (-1 ,2.38) for y-axis . the points are (0.42,1) and (.0.42,-1) .
>>16279162Matsumura Commutative Ring theory
>>16279162Bro, direct sums are basic shit. Projective/injective modules are also basic, the only real "challenge" is proving every module has an injective resolution. That said, I have 2 options for you1) The relevant chapters of Jacobson's Basic Algebra 1&2: in BA1 he introduces modules and their direct sums, and proves the decomposition theorem for modules over PID, the rest he does in BA2. Jacobson's writing is absolutely impeccable, he gets across everything you need to know in a very clear manner.2) Weintraub's "Algebra -- An Approach through Module Theory": does what it says on the tin, he introduces modules right after groups and does all "basic" algebra through their lens. The book definitely stands out from others in the field and is pretty well written, but it has nothing on Jacobson imoDon't listen to this guy >>16279263, he's likely trolling (I haven't read Matsumura myself, but I don't expect you'd need to read a text of this level to learn about modules)
>>16279162this is what I learned from
>>16278414But where in the proof is this even required? The Jacobian dF is nonzero in the neighborhood of q which leads to F^{-1} to be in C^{\inf} in that neighborhood. Doesn't that automatically mean that x^{-1} is in C^0? I don't see why it is vital beforehand that x^{-1} is continuous.
>>16279162>>16279267>Bro, direct sums are basic shitIn fact, I'll lay them out for you right now. I assume that you know what a ring is, what a (left) module over a ring is, and what a module homomorphism is -- that is all you're going to need. In the following let [math]R[/math] be a fixed ring, we're considering (left) modules over it (I won't be saying "[math]R[/math]-module" every time). Module homomorphism are referred to simply as morphisms.First, let [math]M_i[/math] be a collection of modules indexed by some set [math]I[/math], then their (external) direct sum [math]\bigoplus_iM_i[/math] is defined as the subset of the cartesian product [math]\bigtimes_iM_i[/math] of all those tuples [math](x_i)_{i\in I}[/math] such that [math]x_i=0[/math] for almost all [math]i\in I[/math] (i.e. every tuple has at only finitely many non-zero components). Check that this set, with the operations defined component-wise, is a module. Check that for all [math]j\in I[/math] the map [math]\iota_j:M_j\to\bigotimes_iM_i[/math], where [math]x\in M_i[/math] is taken to the tuple whose every component is zero except for the [math]j[/math]-th one, which is [math]x[/math], is an injective morphism.The (external) direct sum has the following (two-part) universal property1) Given a module [math]N[/math] and collection of morphisms [math]f_i:M_i\to N[/math], there exists a *unique* morphism [math]f:\bigoplus_iM_i\to N[/math] such that [math]f\circ\iota_i=f_i[/math] for all [math]i\in I[/math]2) The direct sum is the unique module with this property, i.e. if [math]M'[/math] is another module with this property, then there is a unique isomorphism [math]\bigotimes_iM_i\to M'[/math].Prove this.(If you know category theory, this means that the direct sum is the coproduct in the category of modules)
>>16279299I got testicular torsion from reading your post due to the direct sum twisting into a tensor product.
>>16279299Now for internal direct sums: if [math]M[/math] is a fixed module and [math]M_i[/math] are its submodules, then their sum [math]\sum_iM_i[/math] is defined as the subset of all finite sums [math]x_{i_1}+\cdots+x_{i_n}[/math] with [math]x_{i_r}\in M_{i_r}[/math]. Check that1) This is a submodule.2) This is the smallest submodule containing all the [math]M_i[/math].The following are equivalent for the submodules [math]M_i[/math]:i) Whenever [math]x_{i_1}+\cdots+x_{i_n}=0[/math] for some [math]x_{i_r}\in M_{i_r}[/math], then [math]x_{i_1}=\cdots=x_{i_n}=0[/math].ii) For all [math]j\in I[/math] holds [math]M_j\cap\sum_{i\neq j}M_i=\{0\}[/math].Prove this.If any of the above equivalent conditions hold, we say that the submodules [math]M_i[/math] are independent. If [math]M=\sum_iM_i[/math] and the [math]M_i[/math] are independent, then [math]M[/math] is said to be the internal direct sum of the [math]M_i[/math]. Prove that [math]M[/math] is the internal sum of the [math]M_i[/math] if and only if [math]M[/math] is isomorphic to [math]\bigoplus_iM_i[/math].>>16279303Kek, my bad
>>16279299>>16279308Finally, concerning free modules and direct sums: a (possibly infinite) subset [math]\{x_i:i\in I\}[/math] of a module [math]M[/math] is called1) Linearly independent, if *finite* non-trivial linear combinations of its elements are non-zero, i.e. [math]a_1x_{i_1}+\cdots+a_nx_{i_n}=0[/math] implies [math]a_1=\cdots=a_n=0[/math].2) Spanning, if [math]M=\sum_iRx_i[/math], i.e. every [math]x\in M[/math] is equal to a linear combination of its elements.A basis of a module is a linearly independent spanning subset and a free module is one that has a basis.Prove that [math]M[/math] is free if and only if there exists an index set [math]I[/math] such that [math]M\cong\oplus_{i\in I}R[/math], i.e. [math]M[/math] is isomorphic to the [math]I[/math]-indexed direct sum of [math]R[/math] with itself.And you're pretty much done, that is all you need to know about direct sums up until you start learning about projective modules and exact sequences.
>>16279267Seconding for Jacobson. Absolutely outstanding resource, as he writes in a concrete way that you'd want for a first encounter of algebra, while still getting across key concepts that display the bigger picture down the line.
I was thinking about a solution to pic related and came up with a way to generalize it. Take any nonzero commutative ring R and[math]f_1(x_1), f_2(x_2), \ldots, f_n(x_n) \in R' = R[x_1,...,x_n][/math]Prove that there is no set of polynomials [math]a_i(x_1,\ldots, x_n) \in R'[/math] s.t.[math]\sum_{i=1}^n a_i(x_1,\ldots, x_n)f_i(x_i) = 1[/math].Seems like there should be an elementary, combinatorial way to prove this but I am unable to without extending the coefficient ring (after which it becomes trivial). Would anyone like to have a go at it?
>>16280216Oh I forgot to note that f_i are all supposed to have a positive degree.
>>16280216>>16280239Oh yeah another condition is that for each f_i(x), not all the coefficients of x^i for i>0 are nilpotent, so that f_i(x) is not a unit in R[x]
Why are there no good textbooks on proof theory?
A few years back I saw a video maybe on numberphile where some mathematician wrote a book where he came up with some new axioms/system of writing math and he developed it. I can't seem to find it, does anyone know what it is?
As someone who's an absolute brainlet that learns from seeing other people solve problems are there any good videos on linear programming/the basics of graph theory in relation to Operations Research?
>>16280812If you're thinking of law of forms it isn't really what it's described as
I study math because I want anime to be real. What's your reason for studying math?
I'm trying kind of a "backwards proof" for the kernel representation of the fractional brownian motion, using pic related and trying to calculate the variance of (1) using the properties of the distribution of the Wiener integral, and hoping to find [math] t^{2H} [/math], but I'm not seeing it, I've tried substituting by [math] x=\frac{u-s}{t-s} [/math] but I'm not seeing anything interesting, if you guys could give me some pointers (or just tell me that it was a shitty idea from the get go) , I'd appreciate it. >>16277784I use goodnotes on my ipad, it has handwriting recognition, so it's useful when I want to quickly look up a theorem or proof.I was at some point interested in typing notes in latex using vim snippets by following this tutorial https://castel.dev/post/lecture-notes-1/ , but it proved to be too much of a hassle just for making my notes just a bit cleaner looking
How do we respond?
Currently writing my thesis and I have a (not strictly mathematical) question:If I have a single line of text between the end of a proof and the start of the next theorem, should that single line of paragraph be indented? Or should I use \noindent to push it towards the left?
>>16281160No bro it's a video of a guy who wrote a book about all the derivations that came from his system, and it was cutesy and shit with lines and dots or something. It was some casual thing he did to begin with but then derived some more interesting things and put them all in a book and he was describing it from the start of the system. It was an actual mathematician dude, and it was on numberphile or some other channel like that.
>>16281270I've also tried to substitute by [math]x=u/s [/math] in the inner integral, and got an incomplete beta function, but still no idea on how to go from here
>>16281369All paragraphs should be indented, no matter how smallIf that feels awkward it may be a sign that there is a better way to format what you're writing. If it's a comment on one of the theorems, or how the proofs relate to each other, it should probably go inside a remark
>>16279545>>16279315Nta but nice explanations thanks. If you feel like explaining more of module concepts I would read it (but I am not asking you to do free teaching labor of course).
>>16281270If I'm not mistaken this is done in lemma 3.1 in https://link.springer.com/article/10.1023/A:1008634027843.There's some different scaling with their constant [math]V_H[/math] and your constant [math]c_H[/math], but it appears to be the same.Also, it's probably very difficult to directly evaluate your integral (at least, I can't think of something nice and the article doesn't do it this way either), so it might not be a very fruitful idea.
>>16281578gotcha, yeah I was starting to feel that maybe I was being a little overconfident, thanks for the link
>>16281416I see, thanks.
>>16281369You should never use formatting commands in the main document (including commands like [math] \texttt{\mathbf} [/math] and [math] \texttt{\vspace} [/math]). All formatting should be automated by the preamble. Otherwise, you'd be going against the point of [math] \mathrm \LaTeX [/math]. Equations are sometimes an exception since it is hard to automate their formatting.
>>16282192Good point.
>>16282192Actually, now that you mention that, I do actually have one more question to pose: should I make it a habit of leaving a blank line in the TeX file before and after a [math]\texttt{\begin{align}}[/math] and [math]\texttt{\end{align}}[/math], respectively? Or should whatever text is before or after not have a blank space in between? Because I think for the case of the [math]\texttt{itemize}[/math] environment, for example, leaving a space actually indents the following paragraph but not doing so has the same effect as [math]\texttt{\noindent}[/math].
>>16282223I don't think you should leave a space. In all use-cases I can think of, an equation is connected with the surrounding text. Leaving a blank line would start a new paragraph. Sometimes an equation ends a paragraph, in which case you should end the equation with a [math] \texttt{\mathpunct.} [/math] and if the situation calls for it, [math] \texttt{\qedhere} [/math]. Then, leave a blank line. Leaving a blank line after itemize is fine, but typically not before. You can always leave a commented blank line if you think the code looks nicer that way.
can anyone here help me with a problem I'm stuck on from Basic Mathematics by Lang. I'm trying to teach myself so getting stuck is terrible.The prompt is to rationalize the numerator and remove any square root. it's in the chapter on real numbers. I took a picture to show the start, the textbook answer and the answer I got. is the textbook answer simply mistaken??? the parenthesis don't make any sense to me because without them I think i would be right.
>>16282331No, you're correct. It's a misprint.
>>16282341THANK U
>>16282331What a weird exercise. There's basically never a reason to rationalize the numerator if it's going to make the denominator irrational. Then again, Lang's books are a meme for a reason.
>>16282348thank you. I'm trying to teach myself math so I just used textbook chart guides on /lit/ to find where to start and some of them listed this textbook but now im wondering if I should be spending my time with a better resource
>>16282350You say no. Fag.
>>16282353what does this mean
>>16282354His drinksees. OMFG YOU ARE SOOOO DOOPID, KYAD. physical stuff is not heaven faggy kid.
>>16282354Fart in my mouf now
>>16275601>The natural continuationWhat about, you know, doing calculus on manifolds, or learning actual measure theory? That seems the most "natural" continuation to me
>>16282350Lang's books are more of a 4chan meme than a real recommendation. Given what you were studying in >>16282331 It looks like you're trying to complete a precalculus curriculum. I suggest following the AP Precalculus curriculum, any of the study guides will be suitable for learning (e.g. Princeton Review, Barron's, Shuam's Outlines, etc.) and then you can use practice exams to self-assess how well you know the material. Here's the site from college board for AP Precalculus. https://apcentral.collegeboard.org/courses/ap-precalculus/course
>>16282373thank you!! yes I will check all this out. My ultimate goal is to understand enough to code 3d graphics impressively using trigonometry and stuff
>>16282373no, Lang isn't a meme. Reading long ass books on precalc is.
Started reading about constructive math recently to see what's all the fuss about, I don't think I'm going back bros, the arguments for it are just too strong
>>16282398like?
>>16282377 http://pastebin.com/sy2MbenC
You may be mathematicians, but can you solve the TORMENT on POKEMON EMERALD?You know when they ask you interview questions and you have to select words to make a phrase? That stuff is a secret hard cryptology game which can be worked out by studying the game. If you succeed, a secret room opens where you can fight all previous Pokémon.
>>16282331It seems learning math isn't really making you intelligent because if you were intelligent you would have just tested it for different values instead of coming here and asking. I think you should just not bother.
>>16282401Computational meaning for all existence proofs, that's like, so kosherAlso intuitionist logic just makes sense
>>16282408that's a good idea, next time I'll try that first! i was getting too confuzzled
>>16282331you can always plot simple shit like this on desmos. You're right btw. Also focus on finishing Lang and starting with Linear Algebra soon (e.g. Shilov or Gorodentsev's book) if you care about graphics design.
>>16282412Don't let that nigga discourage you, I know him personally and he is a certified BITCH!
>>16282350these charts and the /sci/ wiki usually have you work through 300 proof, precalc, calc, trig, euclidean geometry, and olympiad problem books before doing any remotely serious math. Just try reading a book on the exact thing you want to learn and go back from there. You'll have a much better picture of what you actually have to learn.
>>16282327Thanks, that's very insightful. I will admit that I'm not too familiar with the usage of [math]\texttt{\mathpunct}[/math], but I can read up on it.
>>16282401>>16282398>>16282411Despite being being a PDE/FA guy myself, I admit I too have been heavily influenced by constructivism, particularly the strain called Ultrafinitism. If you haven't read this essay yet, you're missing out: "Real" Analysis is a Degenerate Case of Discrete Analysishttps://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf
>>16282331Are you a girl?
>>16282442I'm an adult woman, but yes, still a girl
>>16282415From one meme book to another.>if you care about graphics design.This is just the cherry on top.
>>16282448ywnbaw
any chance the author of these pictures still posts here? (1/2)
>>16282531he said he'd compile his notes in a book/pdf but haven't heard from him yet :((2/2)
>>16282448Do you have a boyfriend?
>>16282562What does it matter? Trannies are always willing to add one more male to their harem of beta male orbiters
>>16282567>>>/pol/ nazi trash
>>16282562No I do not math anon
>>16282578>nazi"nazi" is what trannies call normal people, anon
I have a rank-one matrix [math]M = xx^T[/math].I fix some basis and replace the diagonal entries of M with ones.Is there some trick to compute the eigenvalues of the resulting matrix easily?
>>16282426This is not really much like constructivism. It doesn't use intuitionistic logic or anything.
never ask a mathematician what is 843 times 2349 or what is the definition of a set
>>16282426>zeilbergerLove this based jew like you wouldn't believe. But I don't really agree with his position. There's always been a tension bteween concepts and calculations in math. I think classical mathematicans push it too far to one side by not caring at all about how meaningful their existence statements are. They only care about consistency(can't even prove that!) and to me this is not enough and it's very dangerous, with the risk of turning mathematics into a mere language game. On the other hand, finitists push it too far into the other direction: only focusing on what is "realistic", you risk turning math into a mere cookbook, whrere you really don't understand what's going on. I think constructivism in the sense of bishop is the best approach so far (it has been expanded upon since he dropped his banger "constructive analysis", where you get the best of both worlds imo.>ultrafinitismAnother point on this: while I do have a certain respect for the sentiment (as things get bigger, the less meaningful they are), it's in my view an informalizable aspect. Suppose there's an absolute hard computational limit: You would merely put an asterisk in from of every established result saying "ermmm only valid till this big ass number!!", which is kinda unimportant. Then we would have to realize that ACTUAL computational limits are relative to the context you are working with, so even this theoretical absolute limit would be useless because of that too.
>>16282673No. You will be paid off and repaired. Yes. It's a problem. But it's not too much of a problem nor is it difficult for me to fix. The extra life was done this way. Morals on me
>>16282676what in the goddamn...
>>16282682It's all legal and financial faggot. Get with the times.
>>16282682It's a fixed fee and service order, or hell. Take your meds
>>16282686& Lesser hell. Like a jester. Involved in transact 1 of the former statement.
>>16282682You should really just filter out any namefags, at best they're midwits, at worst they're nonsensical
>>16282475>t. doesn't read books
>>16282693MOUF. Now fag.
Some OC
>>16282700variant
How would you explain to normies the features of “mathematical understanding” that you just don’t get with a “casual understanding” of something like statistics or ML for example
>>16282723Ask them to explain what they understand in detail, and hit them with "so you don't actually understand it. That's the difference between you and a mathematician." when they inevitably stumble.
>>16282682Kek
>>16282676>>16282684>>16282686>>16282688
>>16282723For ML there is no difference. It's probably the least mathematically rigorous field imaginable that still technically uses "math"
>>16282742I explained poorly, I don’t mean math vs ML, I mean the way a mathematician understands ML vs how a normie understands ML at a very shallow level.
>>16282728Yes trying to explain it to them without making them feel stupid (because they pay me 250k a year)
>>16282744Still very little difference. ML is mostly just matrix multiplys, approximating gradient descent, and the intuition for that is simple.The truth is even the mathematician only understands things at a shallow level. Nobody knows why architecture X is slightly better than architecture Y for some task, or why xavier uniform initialization is sometimes better than normal initialization. The only real advantage to being a mathematician is being able to understand notation a bit easier
>>16282331c z e c h i azechiaAnyway as the other anon said focus on finishing one book, there's no need to hunt for the "perfect resource" especially with precalculus, just focus on getting the intuitive idea behind as much of the covered material as possible. You won't be usually doing operations like rationalizing the numerator but doing such exercises mainly helps with getting mileage and understanding the underlying concepts better. If you want to do 3D graphics then linear algebra will be a good next step, take a look at Gilbert Strang's book "Linear Algebra and Its Applications" and the accompanying course (leftures are on YouTube) are touted as one of the holy grails of computational linear algebra resourcesnso take a look at those. Really depending on how far you're into Lang's boom you can probably try one of the linear algebra resources and see how if goes, then go back to Lang as needed if it's too much or whatever.
Am I overthinking or is the last part of 15 quite complicated to prove?I did it by first reducing all the terms in the relation to ones of the form (a - mu_ij(a), by using the fact that the connecting maps are homomorphisms. Then I reduceng to the case where the only term in the finite sum of relations is nonzero, is the term with the greatest index, by taking an upper bound j of all the indices and replacing m with m - (m - mu_{ij}(m)).Then I reduced m into two parts. One which shows up as a_ij term in the sum, which I showed that eventually maps to zero. The other which is a sum of terms mu_{ji}(b_j). To show that this term equals zero, I minimized the number of terms in the sum and essentially used graph reduction to show all the corresponding indices are mutually incomparable, then, that they do not connect to anything lower, and then finally I got that they're all zero.Seems like a lot of work for something that seems so trivial.
>>16282761>c z e c h i a>z>e>c>h>i>aCan someone explain why I keep seeing this (usually with london instead of czechia)? What does it mean? Is it all the same schizo spamming it?
>>16282723From my experience, mathematicians don't understand shit about statistics. They just want to do math. It's people who actually work with data that know how statistics actually work.
>>16282789It means the poster hopes the woman is in the said country.
>>16282835How in the world do you know that? :o
How do I know if I know math? I have a serious imposter complex regarding my abilities, despite having a BSc in mathematics.
>>16282789>>16282861it's an old /fit/ meme
>>16282865Math is possibly the easiest field to test your abilities in. Here's a few ways to test your ability1. Go through a serious math book and do all the exercises (examples: Hartshorne, Spanier)2. Do the same except prove all the theorems yourself instead of reading them.3. Do the same for papers.4. Solve some open problems and publish your proof in reputable journals.
>>16282754> Nobody knows why architecture X is slightly better than architecture Y for some taskPlenty of people do. The only reason it isn’t emphasized today is that it’s easier to train data science monkeys to iterate through a bunch of models programmatically than to actually sit and explain to them which family of models is likely to work best on each type of problem
>>16283042>Plenty of people do. Then prove it. Write a mathematical proof demonstrating that some architecture is the best at a certain task. Until that becomes common place, ml research will remain a meme>iterate through a bunch of models programmaticallyThat basically is the current state of ml research right now, unless things have changed in the last few years. Take a pre-existing model, use a new pooling/transfer function or architecture or whatever, and publish a paper if it produces slightly better results. It's just code monkeys throwing shit at the wall
>>16271237Was ist die Unterschied zwischen deutsche Physik und andere Physik?
How correct is the following "proof"? Can it be improved? How original is it?-------------------------------------Natural numbers can store an arbitrary amount of information, while real numbers can store an infinite amount of information.The first one is pretty obvious. To encode a sequence of 1s and 0s into a natural, you can pick the convention of choosing the natural whose binary representation always starts with a 1 and only then is equal the binary sequence that you want to represent, to avoid the issues of leading zeros getting truncated. So if you can convert your information to a sequence of ones and zeros (which, afaik, that is roughly* what is meant as "information" is mathematically) they you can pack as much information as you want into a natural.The second one is a bit more subtle. Reals can contain an infinite amount of information because of the way they are defined. Real numbers can be defined by infinite Cauchy sequences. The terms of this infinite Cauchy sequence do not need to be computable. They just need to be arbitrary rational numbers such that for any ε there exists an α that guarantees |xᵢ-xⱼ|<ε given i,j>α.If any two Cauchy sequences converge to the same real number, we say that they are both equal, so we cannot rely on the exact terms of the sequence to encode information. But we can use the convergence requirement to our advantage.Let μ=1/2ʰ.Let l(ₑ,ₖ)=2kμ+eμ k∈ℕ, e∈{0,1}For any real number r, we define bₕ as follows:bₕ=1 if there exists a k∈ℕ, β∈ℕ such that l(ₒ,ₖ)<xₚ<l(1,ₖ) for all p>β, and bₕ=1 if there exists a k∈ℕ, β∈ℕ such that l(ₒ,ₖ)+μ<xₚ<l(1,ₖ)+μ for all p>β.In this way we can construct a number such that bₕ is either 0 or 1 depending on the terms of the Cauchy sequence used to define it up to the β term, for any natural number h. Thus we have proven that we can encode an infinite amount of binary digits in a single real number.
* The reason it may not be exactly that, is you can have a trit of information (choosing one out of three elements) which stores an amount of information that is strictly between 1 and 2 bits. But you can define a convention, such as storing the base at the beginning of the sequence and then adding redundancy such that some sub-sequences of the binary sequence are considered equivalent.
>>16283058The problem is that mathematical proofs take time and there’s not enough people competent enough to do it and not enough funding to get those people to dedicate their time to it. There’s tens of millions of ML ants now that can import scikitlearn and run a loop over a bunch of models they don’t understand for some problem no one cares about than there are qualified mathematicians to make a proof about the absolute best model to train on some problem that no one cares about.
>>16283155I didn’t read all of it but basically yes both statements are true. That’s why storing large numbers or arbitrarily precise numbers requires an increasing amount of computer memory. Anyone who wants to have fun can open a Python console and input 10 ** (10 ** 10) and have fun watch their computer crawl to a halt. Watch your system resources as this happens too
I'm looking for a PDF of this book in particular:https://www.amazon.com/Lectures-Modern-Mathematics-T-Saaty/dp/0471748269I've looked online, but I couldn't find a PDF in the usual places.
>>16283438That's funny, I found a pdf from the usual place immediately.
>>16283438https://archive.org/details/lecturesonmodern0000saat_m3k0/page/n5/mode/1up
I'm going insane. A while ago I saw a video about a statistician(?) giving a lecture about conspiracy theories and the math behind them. I think one example he used was how ancient sites being on special Ley Lines isn't special, because there are just so many ancient sites so by pure probability some will fall on a line. I think he then demonstrated this by showing how a bunch of Tesco stores are also on Ley Lines or something.Does anyone know who I'm talking about, I can't find the video
>>16283493Oh and I think another example he used, is how you can pick any random number sequence and correlate it to words in the bible and make it spell almost anything you want to
>>16283493>>16283496https://www.youtube.com/watch?v=sf5OrthVRPANvm found it
>>16282761Strang is pure garbage and nobody should ever be subjected to this
>>16283444It's unavailable though. Nothing on libgen or scihub
>>16284631What do you mean unavailable , it’s right there on the page, just read it
Whats the most advanced kind of math that you would use in a business context? I think something like stochastic calculus maybe.
>>16284661Depends what business you’re in. Can’t tell you how many times I’ve been asked to solve reformulations of the halting problem
>>16284452WhyI've read that his is a computationally focused approach etc. and to be fair I think I've seen some people share views similar to yours now that I think about it?
>>16284708Not him, but Strang never gives intuition for his results and is mostly just blind algorithms, but to be fair most pure math books don't give either; they'll use purely algebraic shit like minimal polynomials to prove purely geometric results like spectral decomposition. I have found Halmos's proofs to be quite enlightening despite being a very terse book. Surprisingly, Lang also has a few geometric proofs and is very close to my preferred proof of spectral decomposition, but his could be better (it's Lang after all). If you want to apply linear algebra, geometric intuition is quite important because a lot of the results based on linear algebra are motivated by geometry. For instance, Principal Component Analysis is completely motivated by elliptical shape of Normal contours, but most statistics book will just blindly apply eigenvectors for no apparent reason. That said, most people hate Strang because they think they are real badasses for wanting all math to be soulless le heckin Bourbaki style.
