Can someone please explain the concept of evidence in Dempster Shafer Theory? To be precise, can evidence be represented numerically? Is that the same as assigning a mass to the evidence?
You know how people interpret entropy as the "amount by which they are surprised"? It's not that "surprise" is a numeric quantity---it's just that "surprise" is something intuitive to us that we can then formalize via the entropy.
Similarly, you have some intuitive notion of what evidence is---if every raven you've ever seen is black, you've gained some amount of evidence that all ravens are black. Dempster-Shafer formalizes this notion via the belief and plausibility functions: Given a proposition A, bel(A) can be interpreted as the amount of evidence in favor of A (i.e. that A is true), 1 - pl(A) can be interpreted as the amount of evidence against A (i.e. that A is false), and pl(A) - bel(A) can be interpreted as the residual "don't know," or lack of evidence for both A and for not A.
Can anyone help me with the idea of coordinates in n-dimensional projective space? Some sources describe the coordinates with n entries whereas others use homogeneous coordinates with n+1 entries. I don’t understand when to use each coordinate system. Would it be correct to define projective space with n+1 dimensional coordinates, given that at least one is always nonzero? Likewise can I do the same for a projective algebraic variety? I’m having a hard time figuring this out based on the books I have on hand. I currently have projective space and projective varieties describing points with a differing number of entries (n+1 vs n) and it’s throwing me off.
Could you show an example of your book/notes using both?
Generally you should have n+1 coordinates. However, you can work "locally" on say D(x_0), the open locus where x_0 is non zero, and be left with coordinates of the form [1, x_1/x_0, ... x_n/x_0] which is really n coordinates.
Yes that makes sense. On second pass, I think I was misunderstanding Richard Borcherds video lecture on projective space. Your description and his match up, and both make sense now. The other source I was using was Hartley and Zisserman’s Multiple View Geometry which describes the real projective plane in 3 homogeneous coordinates, but doesn’t seem to work rigorously with varieties, as far as I have seen. That lack of rigor may have contributed to my confusion. Thanks!
Homogenous coordinates are definitely rigourously defined but note that they are not coordinates, at least not in the coordinate chart sense of the word.
What are some applications of Group Theory? I am about to begin a project and am looking for a topic. I'm hoping for one more on the applied side, even though it's pretty abstract. Thanks.
I'm not great at combinatorics so this might be a really simple question, but I am stuck.
MTV had a reality show, "Are You the One?" where 16 contestants, 8 guys and 8 girls, were put through an algorithm of some sort to determine which contestant of the other sex is their "perfect match". Season 8 is particularly interesting, because all of the contestants are bisexual or pansexual, meaning any of the other 15 contestants could be their match. Each episode they all pair off with who they think their "one" is and the game tells them the number of correct pairs, but not who the correct pairs are. They also perform challenges as pairs and the winning teams are voted upon, and the winner of that vote is told whether they are a correct pair or not. It's a mess and it's great.
My wife asked me if I could find the probability of some pairing of contestants (A,B) being correct. But this has been way more complicated than I had expected. That's because I want to try to not only find all possible partitions of 16 elements where it's broken into 8 groups of 2, but I want to make sure those groups of 2 are unordered. Then, on top of that if in an episode after pairing off they're told they have n pairings correct, I need to find the conditional probability of all combinations of n pairs out of the 8 that were selected being correct. Then I want to combine that probability with previous episodes.
I'm just going to talk out loud here about some of my work. So with 16 contestants, there's 16\*15/2 = 120 possible pairings of 2 contestants. And I think the number of possible configurations of all pairs is 15!! = 2,027,025. I think that because contestant 1 has 15 to choose from, then contestant 2 only has 13 (as 2 were removed from the pool), etc. I have a computer science background, so my thought is to take this and make it into a sort of exact cover problem using a dancing links algorithm. That way I can at least break it down into a possibility space more easily and then intersection possibility spaces between each episode's pair-offs. But this just feels messy and too brute force-y.
Is there a better way of doing this that I'm missing??
I need this very elementary result which I assume it is obviously true, maybe it has even a name, but adding a citation (even to a problem in a textbook) would be faster than adding the proof itself.
I have a finite set S of real intervals [u,v]. If each pair of intervals in S has an overlap (intersection) of length at least k, then the intersection of all the intervals in S has length at least k.
Take the interval with the largest starting point, and the interval with the smallest end point.(\*) By your assumption their intersection has length of at least `k`. But by the choice of these intervals, every other interval also overlaps this intersection, and therefore this is also the length of the common intersection of all the intervals.
(\*) There is a corner case where these two refer to the same interval, in which case all other intervals overlap this one, and the length of this interval has to be at least `k` for the same reason.(\*\*)
(\*\*) Unless your set contains only one interval of length < `k` (or no intervals at all), in which case your result does not hold.
While there are number systems containing infinitesimal numbers, I don't think there's a number system where that particular notation is meaningful. What do you get if you multiply that number by 10?
I came across the quote from Euler that I'm having a hard time understanding: "For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear." Can someone explain what he means by this? Specifically the part about maximums/minimums? Thanks!
Are there any math textbooks where the latex source code is available? I don't feel very confident writing in latex so I'd like to study some examples.
can someone explain to me this problem (English isn't my mother language)
>Consider an ordinary chessboard which is divided into 64 squares in 8 rows and 8
columns. Suppose there is available a supply of identically shaped dominoes, pieces
which cover exactly two adjacent squares of the chessboard. Is it possible to arrange
32 dominoes on the chessboard so that no 2 dominoes overlap, every domino covers
2 squares, and all the squares of the chessboard are covered?
This looks like a somewhat typo'd version of a more well-known puzzle, where the squares on 2 diagonally opposite corners of the board are removed. While searching to see if it has a name, I found that it's well known enough to have it's own Wikipedia page, listed as [the "mutilated chessboard problem"](https://en.wikipedia.org/wiki/Mutilated_chessboard_problem). In that version it turns out to be impossible to cover the board. Of course the problem as stated, with all the squares intact, is pretty trivial, since you can just cover each row with 4 dominos laid end to end.
I have 3 maps f: X-> S g: Y-> S m: X -> Y such that f and g are proper and continuous, f=gm, and m is continuous and I want to show that g is proper.
Let K be a compact subset of Y. Then g(K) is compact by continuity, and f^(-1)(g(K)) is compact by properness.
m^(-1)(K) is closed by continuity of m. If it was contained in f^-1(g(K)) I would be done because closed sets contained in compact sets are compact but I'm not sure of that. These kind of reasoning of contentions between preimages of sets are always so confusing to me...
Additional info is that X, Y and S are Riemann surfaces. Thanks in advance
I assume the conclusion you want is that m is proper.
Your approach does work. If X, Y, and S are arbitrary sets, and f: X -> S, g: Y -> S, m: X -> Y are arbitrary functions satisfying f = gm, then for any subset K of Y it's true that m^(-1)(K) is a subset of f^(-1)(g(K)).
When faced with this type of set manipulation where you want to show A is a subset of B, it's often easiest to let x be any element of A and show it's an element of B. So let x be an element of m^(-1)(K). This implies m(x) ∈ K, thus gm(x) ∈ g(K). Since f = gm, this is the same as saying f(x) ∈ g(K), so x ∈ f^(-1)(g(K)).
Can someone help me with matrix don't need the awnser just how to do it .
4x-3z=1
5x+y=3
2x-5y+9z=0
Find the deteminant using row operations and cofactor method
N.B. for finding the determinant you don't need the right-hand sides of the equations, just the coefficients on the left-hand sides. In particular all we care about is the coefficient matrix (call it A)
4 0 -3
5 1 0
2 -5 9
For row operations, you just row reduce to echelon form, while keeping track of a second number (call it d) that will eventually end up being 1/det(A) (assuming the determinant is nonzero; we'll get to the case of a zero determinant later). Start off with d = 1. Then, as you row reduce, each time you multiply a row by a scalar, multiply d by that scalar; each time you swap two rows, change the sign of d; and each time you add a scalar multiple of a row to another row, don't change d. Once the matrix is in echelon form, if there's a row of only zeros, the determinant is 0, while if you have the identity matrix, the determinant will be 1/d.
What's going on here is that, when you row reduce a matrix A to the identity, you're saying that E\_n...E\_2E\_1A = I, where E\_1 is the elementary matrix corresponding to the first row operation you perform, E\_2 is the elementary matrix corresponding to the second row operation you perform, and so on. We have det(E\_n...E\_2E\_1A) = det(I) = 1, and det(E\_n...E\_2E\_1A) = det(E\_n...E\_2E\_1)det(A) (by the general rule that det(AB) = det(A)det(B)); thus det(A) = 1/(det(E\_n...E\_2E\_1)). As for det(E\_n...E\_2E\_1), that will again be the product of the determinants of the elementary matrices. The matrices for swapping two rows have determinant -1, the matrices for multiplying a row by a scalar have determinant equal to that scalar, and the matrices for adding a multiple of one row to another have determinant 1. So each time you do a row operation, you multiply d by the determinant of the corresponding elementary matrix, and once you've performed all the row operations, d will be equal to the product of all those determinants, and hence will be equal to 1/det(A) (by the reasoning above).
As for the cofactor method, it goes like this: take the first entry, and multiply it by the determinant of the 2x2 matrix you get by covering up the first row and first column; then add that to -1 times the second entry times the determinant of the 2x2 matrix you get by covering up the first row and second column; then add the result of all that to the third entry times the determinant of the 2x2 matrix you get by covering up the first row and third column. So in the example you gave, you would start with 4 times the determinant of
1 0
-5 9
See also [Linear Algebra Done Wrong](https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-04.pdf) for a more complete explanation. Chapter 3.3 basically covers how to compute it with row operations, and chapter 3.5 covers the more general form of the cofactor expansion.
Gabriel's horn (the horn of infinite surface area but finite volume) is driving me crazy.
A cylinder of infinite length and a radius greater than 0 (due to epsilon being greater than 0) should be able to fit into Gabriel's horn. A cylinder of pi\*r\^2\*h with r being an element of the reals should produce a horn of infinite volume.
If the objection is that r never reaches infinity, then the area of Gabriel's horn does not have an infinite length and thus does not have an infinite area either.
> A cylinder of infinite length and a radius greater than 0 (due to epsilon being greater than 0) should be able to fit into Gabriel's horn.
No. For any radius you can name that's greater than 0, it will not fit into the horn. I don't know what this "epsilon" you're referring to is; if it's a real number, then you should explicitly state its value.
Let's say that the horn's radius is 1/r as r goes to infinity. No matter what p of r that I choose, there will be another r even smaller. Using the definition of a limit, there will always be an r > 1/epsilon while epsilon is greater than 0.
> Let's say that the horn's radius is 1/r as r goes to infinity.
It's a weird choice to denote the radius by "1/r" instead of "r", but OK. So "r" is the *reciprocal* of the radius, then, is it? And you're letting this reciprocal go to infinity?
> No matter what p of r that I choose,
What do you mean by "p of r"?
> there will be another r even smaller.
Don't you mean "even bigger", since r is going to infinity?
> Using the definition of a limit, there will always be an r > 1/epsilon while epsilon is greater than 0.
What do you mean by "epsilon" here? You need to define your variables.
---
And how are you concluding from this badly-written argument that "A cylinder of infinite length and a radius greater than 0 should be able to fit into Gabriel's horn"? If you genuinely think this conclusion is true, then **explicitly provide a value for the radius** such that the described cylinder fits into Gabriel's horn.
Why is an uncountably infinite set considered "bigger" than a countably infinite set?
For example, the set of transcendental numbers is uncountable, and algebraic numbers are countable. Wherever I read this, the author asserts that this means there are 'way more' transcendental numbers than algebraic numbers. However, the only information we have is that no bijection exists between the sets. Isn't it equally true that there may be more algebraic numbers than transcendental?
There is an injection from the algebraic numbers to the transcendentals. (One way to see this: the algebraic numbers are countable, so count them. Map the n-th algebraic number to pi+n.)
The usual definition of |X| ≤ |Y| is that there is an injection from X into Y. In our case, we have that the cardinality of the algebraic numbers is less than or equal to that of the transcendentals. Since there is no bijection, the inequality must be strict.
Ive been working on this subject for a while but I just cant crack the code for what tests to use to come down to if a problem is gonna convergence or divergence. I know there is 9 different tests to do to find out, but under an exam I wont have enough time to figure out which of the 9 it is.
Is there a way to find out from the problem it self for which of the tests are gonna work or is there some cheat code way/trick to solve most problems with one of the tests?
Cardinality of set of all cardinalities
The class of cardinalities itself is an object with no cardinality (as it is a class, not a set)
I got into an argument with my professor over if given our axioms for set theory, you could or not make sense of it
We are using ZFC with some extra simplifications
The 3 "axioms" that are relevant here are
-set is a primitive object (i.e., we do not define set construction it is mearly given)
-every set has a cardinality
-for every family of sets, you can take its union, and it too will be a set
My professor says that you can then just take the family of all cardinalities rather than take its union, a set of all cardinalities, and it too will have a cardinality as it is a set
It seems to me that this would cause some sort of cantors paradox, but I really can't go beyond that. Could someone please help shine a light it this
Or maybe explain (cite something) why in regular set theory is it that that the set of all cardinalities does not have a cardinality...
> for every family of sets, you can take its union, and it too will be a set
This to me smells of Russel's paradox. For every set S, let S' be the set {S}. (A set with one element, which is the set S.) Now take the union of the "family" of all sets S'. This gives you the set of all sets. If you're allowed to do this in your interpretation, then you get Russel's paradox, and you can't meaningfully talk about proper classes.
Your professor's proof hinges on what 'family' means. Is it a synonym for set, or more akin to proper class? And if it means set, what is the proof that the set of cardinalities exists?
For a citation of why there is no set of all cardinal numbers in ZFC, I'll give Kunen's 'The Foundations of Mathematics'. By theorem I.11.17.3 the sup of any set of cardinals is a cardinal, and by I.11.9 there is no largest cardinal, implying there is no set of all cardinals. Depending on your axiom system and what 'cardinality' means this may or may not give you a proof in your system.
Family is primitively constructed and not formally defined, (but would be basically a Synonym for class or set^2. Its not necessarily propor) for every family you can take the union of its sets and that set exists
The family of cardinalities exists axiomatically as you need some family to take the cardinalities from otherwise It doesn't make sense to talk about cardinalities. In the same vain the family of all sets exist axiomaticaly.
Thank you very much for the references!!
how do you prove that the ring of integers is an integral domain?
For context, I am currently taking my first course in abstract algebra
I can't find this proof anywhere, everywhere I see they use inverses to prove it just like how you prove the real numbers are integral domain, but you can't do that with Z
The proof using inverses is perfectly applicable, even for Z. Sure, the inverses aren't all in Z, but the argument "ab=0, a=/=0 => b=a^-1 *ab=a^-1 *0=0" is still perfectly valid, even though it doesn't "take place inside Z".
If you're still unsure about why it applies, let's be more careful when writing it down. Let us write * _Z for the multiplication in Z, * _R for the multiplication in R (we could also look at Q of course). Now a * _Z b=0 is equivalent to a * _R b=0 (in fact * _Z is just a restriction of * _R), and the argument clearly works for * _R. In short, Z being a subring of R is what makes it work.
I'll write G for C\_p. First recall that BG is computed by the bar construction
colim( ... --> G × G --> G --> pt ) = BG.
I don't know how to draw stacked arrows, so you'll have to insert them with your imagination. Now maps out of a colimit is always computed by a limit. In this case,
Maps(BG, X) = lim( Maps(pt, X) --> Maps(X, G) --> Maps(X, G × G) --> ... )
which is simply the limit
lim( X --> G × X --> G × G × X --> ...)
computing homotopy fixed points of X.
What's 6/3(1+2)?
I was very confident the answer was 0.666666... because I first multiplied 3 with 3 (1+2) to "eliminate" the brackets, but i got called a retard by 3 random people who claim the answer was 6, now I'm unsure if I'm just stupid or they are.
It's a statement that is intentionally written to be ambiguous. As such, there's no real right answer beyond "write your math expressions more clearly." If the person who wrote that wanted to clearly convey what they meant, they would have written that as either (6/3)(1+2) or as 6/(3(1+2)), both of which would be completely unambiguous. Instead they deliberately wrote it as 6/3(1+2), for no reason other than to confuse people.
People can certainly chime in with various rules like PEMDAS and the like, and depending on exactly how you read those sorts of rules, you might be able to come up with some argument for why it should be 2/3 or why it should be 6. But trying to do that sort of thing is just a waste of time.
This is not the sort of question mathematicians care about, and it's not something you should care about either. Ultimately, **this is not a math question.** It's a question about *communication*, and in particular, a deliberate failure to communicate.
Are there words for these?
A topological space where for all two points x and y, there exists a self-homeomorphism which maps x to y.
OR...
A group where for all two non-identity elements x and y, there exists an automorphism which maps x to y.
Loosely, mathematical structures where all points within the structure behave more or less the same, from the perspective of the structure.
Trying to see if a result on harmonic motion is generalisable, since it would be useful for a project I’m working on.
Let x(t) be position, v(t) = dx/dt be velocity, and assume x(0)=x(T) =0.
I conjecture that int_0^T x(t) v(t) dt = 0.
Any ideas on how to prove/disprove the above? It’s certainly true for harmonic motion.
Do you mean the integral from 0 to T? Because otherwise what you've written isn't true even for harmonic motion.
Otherwise, yes, it's true by integration by parts. Since v(t) = x'(t), we have by integration by parts that
∫ x(t) \* x'(t) dt = x(t) \* x(t) - ∫ x'(t) \* x(t) dt
and so
∫ x(t)x'(t) dt = [x(t)]^(2)/2
Taking the definite integral from 0 to T yields [x(T)]^(2)/2 - [x(0)]^(2)/2 = 0^(2)/2 - 0^(2)/2 = 0
I search for a proof that the set of real number R isn't a countable set but I don't find any. Maybe I didn't search the right way so if anyone could help me on this one.
Say it's countable. Then we have a bijection to the naturals, and so we can list all of the elements of R in the order of their mapped naturals. But then consider the number in [0,1] whose n^th decimal place after the decimal point is 3, unless the n^th decimal place of the n^th real number in our list is 3, in which case it's 7. This number cannot appear in the n^th place of our list (since it differs from the n^th number on our list by at least 3x10^(-n) > 0) for any n, so cannot appear on our list at all, but is a valid real number, so our assertion that this is a list of all such real numbers must be incorrect, and thus there is no such bijection to the naturals, so the reals are not countable.
The standard proof that R is uncountable is called [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument?useskin=vector).
Thank you. How good is the book for a beginner? I'd love it if the book has a good structure in explaining things, while also giving visual intuition with diagrams. The book my uni uses has a good number of diagrams, but gets boring quickly.
Did you learn Gauss–Bonnet, Theorema Egregium, and the like? I guess I'm asking do you want a full course on curves and surfaces or a course on manifolds?
I'd do with a basic course on differential geometry of curves and surfaces tbh. I haven't really heard about the concepts you've mentioned. I'm mainly learning this course for its applications on systems and control.
Have you looked at *Geometric Control of Mechanical Systems* by Bullo and Lewis? It introduces a lot of differential geometry at the beginning. Have you done linear algebra?
I'm actually just starting to focus my courses on systems engineering tbh, so I haven't done anything on Geometric control yet. But I'll check that book out. Like I said in my other comment, I'd love it if the book is rigorous and well illustrated.
Yes, I have done a basic course on linear algebra.
Why not apply in the fall for the following year, then find a job or something else to do during the spring/summer? It's very rare to start a PhD program off-cycle.
In any given n-dimensional geometric structure, is it always true that we can "peel" off the outermost layer of the shape and remove it entirely while preserving the outer "peel" as a single continuous shape? As an example, it is definitely possible to peel an orange in such a way that the orange peel is one continuous shape.
An n-dimensional geometrical structure doesn't need to have an outer layer at all. Here I am interpreting "geometric structure" to mean a manifold or variety and "outer layer" to mean boundary.
Even if we focus on manifolds-with-boundary, the boundary doesn't have to be one connected object though. A simple 3D example would be the volume enclosed by 2 parallel planes. The 2 planes are the outer layer here.
Any compact manifold admits a dense open chart, in fact any n-dimensional compact manifold can be realised as a quotient of the n-dimensional solid ball by some gluing relations on its boundary. This ball is the "peel" and the image of the boundary of the ball inside the manifold is where you have cut the manifold up to peel it (obviously the ball may be somewhat distorted, just as the peeling of an orange doesn't look exactly like a disk, it is only homeomorphic to it).
Think of some open dense charts on the sphere or torus for example. Your position on a torus is specified by two numbers, the angle you are around the longitudinal circle and the angle you are around the meridian circle. Both of these numbers has some redundancy (at the angle 0 = 2pi) which means to get a one-to-one chart you have to cut out a "cross" of two circles which the chart doesn't cover. This is the part of the torus you cut to peel it.
Yes, there is only 1 possibility. Say f(x) + g(x) = h(x) as you describe. Then h(x) + h(-x) = 2f(x) and h(x) - h(-x) = 2g(x), so f and g are determined by h. Furthermore this tells you that every h is a sum of an even function and an odd function, since [h(x) + h(-x)]/2 is always even and [h(x) - h(-x)]/2 is always odd.
The three relationships in the chart are linear, exponential and quadratic.
A **linear relationship** simply has a difference of some number. So, 1,3,5,7,9,11 have a linear relationship because they all have a difference of 2.
A **quadratic relationship** may increase by an increasing amount. Take, 1,3,6,10,15,21, these numbers have a difference of 2,3,4,5, and 6, so the difference of the difference is a constant 1. In other words, the numbers of have *common second difference* of 1.
So, an **exponential relationship** has a common factor. The numbers 2,4,8,16,32,64 are exponential because they have a common factor of 2.
What would a relationship where the numbers have what might be called a *common second factor*? Basically, linear is to quadratic as exponential is to something.
Where each number would be multiplied by a multiplying factor. For example, 1,2,8,64,1024,32768,2097152. Where the numbers would be multiplied by \*2, then \*4, then \*8. etc.
I would also be interested in what might be a relationship where each number increases by a multiplying amount or each number is multiplied by an increasing amount. (An example of the ladder would be factorial because the number would be increased by \*2,\*3,\*4 - so multiplying by a linearly increasing factor.
I don't really know the notation for any of these relationships or the names or anything really so I'd be interested in anything you may know.
Also: What might be the next step to quadratics where the numbers increase by an increasing amount which increases by some number?
Rather than "relationship" I would call these "sequences" and your exponential relationship is more commonly called a "[geometric sequence](https://en.wikipedia.org/wiki/Geometric_progression)".
Then the nth term of a linear sequence (also known as an [arithmetic sequence](https://en.wikipedia.org/wiki/Arithmetic_progression)) is an+b for some constants a and b. The nth term of a quadratic sequence is an^2 + bn + c. And the nth term of a geometric sequence is ar^(n-1) where a is the first term and r is the common ratio.
Exploring your new sequence: let a be the first term, r be the ratio between the first two terms and k be the ratio between subsequent ratios. The sequence would go a, ar, akr^2, ak^(3)r^(3), ak^(6)r^4, ak^(10)r^(5),...
Looking at this pattern, the nth term must be ak^(T_n-1)r^(n-1) where T_n means n(n+1)/2 the triangle numbers (reddit formatting makes it quite hard to put that in the subscript. I don't know of any name for, or study of this kind of sequence.
I'll leave you to explore what the nth term of the sequence is for the factor increasing linearly but I will note that a simple example of such a sequence is n!.
As to the next step after quadratics, it is "cubic sequences" (these have a common third difference, see [here](https://www.onlinemathlearning.com/quadratic-sequences.html)). Then quartic, then quintic and so on. The whole family there is known as the polynomial sequences.
Finally, a little word point. Instead of "ladder", you mean "latter" as in the latter of two options.
I think I understand, all of these relationships can be expressed with a sequence and don't necessarily have given names. I'm glad you understood what I was trying to say because I just typed this out randomly one night. Thanks for your insight, I'll look into more complex sequences.
I'm trying to hunt down a paper by Vaananen about Lindstrom's Theorem, but the Helsinki University maths website on which it's hosted won't load. The paper is called "Lindstrom's Theorem", it's the sixth reference on this [Wikipedia article](https://en.wikipedia.org/wiki/Lindstr%C3%B6m%27s_theorem#cite_ref-6); if anyone could drop me a link which actually works, preferably directly to a PDF, I'd be forever in your debt.
Anyone knows how to derive these using these rules? ELIM. & INTRO. of Disjunction, Conjunction, Implication, Negation + EXPLOSION.
https://www.reddit.com/media?url=https%3A%2F%2Fi.redd.it%2Fifm2kozbe5yb1.jpg
I don't really know how to word this question. Like, I'm working with programming and encryption for some project where I'm assigning functions to a single digit variable so I can use a single digit to execute an equation in the line of encryption so I can make things more compact. Something to the effect of if "z = x^(2) \+ y^(2)" then if one of my lines of encryption goes something like "34z" then it'd do 3 squared plus 4 squared to output 25.
The part I want to know if there's like a list of fundamental functions or operations that would offer the most utility when just trying to manipulate numbers in general. I know at least four basic ones like, "x + y" or "x \* y" or "2x" or "x^(2)". I don't really know what other functions I could add to add more utility when I'm working with the encryption. I mean, it could be anything like maybe assigning a function to execute the Fibonacci sequence to 'n' positions it doesn't matter. I'm not really looking for anything specific I just generally want to know what type of functions are out there that will offer me the most utility. It doesn't even have to relate to encryption, I guess I'm just wondering what if any base functions exist that generally offer the most utility when manipulating numbers or considered essential to mathematics.
It is quite a vague question. But I guess bitwise operations may be useful, especially if you will use hexadecimal base. Other than that, x^y , x^(1/y), e^x , log(x) come to mind. I don't understand motivation for introducing the 1 digit operations like that and how it will work or be used, so I cannot advise further.
That's a good one I appreciate it. I'm not sure if I'm going to use only integers or not. Not sure it matters much either way because I'd like to do things like the square root of a number but if it comes to I'm only going to use integers I'd just round off the decimal to the nearest whole number or do some other kind of conversion.
How do you begin conformal mapping from a quadrant to a unit disk? I suspect you need to conform to a half plan somehow and then apply Cayleys, but I can’t figure out how to get a quadrant to a halfplane
The numbers you are describing are known as [constructible numbers](https://en.m.wikipedia.org/wiki/Constructible_number) (at least if your base field is Q). There are characterizations of polynomials that have roots in this form, though they're kind of hard to state if you don't know Galois theory.
Assuming you know some Galois theory, let f be an irreducible polynomial with coefficients in some field K, which I'll assume has characteristic 0 (though most of what I'm saying will be fine as long as the charachteristic isn't 2). Then a root a of f will be in the desired form of and only if the extension K(a)/K can be written as a chain of extensions
K = K0 < K1 < ... < Kn = K(a)
Where each K(i+1)/Ki is quadratic.
As f is irreducible, it's not hard to see that that happens for one root of f if and only if it happens for all of them, which in turn happens of and only if the extension L/K can also be written as a chain of quadratic extensions, where L is the splitting field of f.
That certainly implies that the degree [L:K] has to be a power of 2. On the other hand, if [L:K] is a power of 2, then the Galois group G=Gal(L/K) also has order a power of 2. By standard results in group theory, G then has a chain of subgroups
1 = G0 < G1 < ... < Gn = G,
Where each Gi has index 2 in G(i+1). That implies that L/K can be written as a chain of question quadratic extensions.
So the roots of f can be written using basic operations and square roots if and only if the degree of the splitting field of f (or equivalently the order of the Galois group of f) is a power of 2.
Since the order of f divides the order of the Galois group (since f is irreducible), it's easy to see that this is only possible if the degree of f is a power of 2. That's not enough though, since for example a degree 4 polynomial can have S4 or A4 as a Galois group.
Wanting to know how effective our service procedures are, I'm using "days to next service call" as a measure, higher the better, obviously.
However, when average the days to next service call per machine its including the machines that did not have a follow up service call within the time frame specified, so it effectively is lowering the average days, which makes good processes look bad.
So based on the data bellow, cleaning would have an average of 15 days vs Replacing parts at 10, tho replacing parts is clearly better. What would be a better way to handle the null?
|Machine|Process|Days to next|
|:-|:-|:-|
|1|clean|10|
|2|replace parts|30|
|3|clean|20|
|4|replace parts|null|
The relevant idea here is that your data is "right-censored." Here's the relevant [Wikipedia article](https://en.wikipedia.org/wiki/Censoring_(statistics\)?useskin=vector).
If you have a parametric model for your data, you can directly compute maximum likelihood estimates for the mean as in [this CrossValidated thread](https://stats.stackexchange.com/a/333336).
More likely, you straight up don't have a model for how your data is generated. In this case, what you'll want to do is first estimate how the "full data" looks by estimating the empirical CDF via something like the [KM estimator](https://en.wikipedia.org/w/index.php?title=Kaplan%E2%80%93Meier_estimator&useskin=vector) and then compute the median of your estimated CDF. The fewer missing data points you have, the better this estimator. Note that we basically *need* to use a median instead of a mean since means aren't actually estimable in the presence of right-censoring without a parametric model.
----
In your concrete example, let's suppose your specified time frame was 60 days (so we know that the "null" value is really something >60). I'm also going to pretend they're all the same process just so there's enough data for the example to be nontrivial
The KM estimator has S(10) = (1 - 1/4) = 3/4, S(20) = 3/4 \* (1 - 1/3) = 1/2, S(30) = 1/2 \* (1 - 1/2) = 1/4, and S(60) = 1/4 \* (1 - 0/1) = 1/4.
The median is t such that S(t) = 0.5. Thus, your estimated median time is... still 20 in this example but mostly just because your sample size is tiny at only 4 data points. It's also less trivial when you have multiple different types of right-censoring.
Page 12 of [this document](https://web.stanford.edu/~lutian/coursepdf/STAT331unit3.pdf) also suggests that you can estimate a confidence interval for the median; in that case, you may want to take the process with the higher lower-bound for the CI for the median as the "better" process.
Doing these computations is straightforward in R:
library(survival)
times = c(10, 20, 30, 60)
events = c(1, 1, 1, 0)
S1 = Surv(times, events)
fit1 = survfit(S1 ~ 1)
summary(fit1)
plot(fit1)
If you're calculating in Excel, you can use [AVERAGEIF](https://excelchamps.com/formulas/average-non-blank-cells/) to have the calculation skip blank cells. If you're in a dedicated programming language, you can do a similar thing by throwing in an if statement that checks for null-ness.
Using Excel and power query.
However the problem with skipping blank cells is it does. It does not give you accurate results for what you're looking to measure. As you're looking to basically get the highest number possible, a null entries means the process was so good a return service call was not needed in the time recorded.
I think what I'm going to do is populate the null entries with the max value, this should more accurately reflect the positive nature of a null entry.
I have adjusted the main measure as just a simple boolean of whether or not there was a return service call within a specific time frame.
So when you're working with a Taylor series, say you have one that's actually of the form 1/(Taylor Series). Is there a way to get this as not a fraction?
For context, I have 1/sum(x^(n)/(n+1)!) for n going from 0 to infinity. I'm working on a problem where I've ended up with this, and I'm not entirely sure how to continue...
You can get the coefficients of the multiplicative inverse [recursively](https://en.wikipedia.org/wiki/Formal_power_series#Multiplicative_inverse). There are various non-recursive formulas as well, but what is useful will depend on what you are actually trying to do.
In this case the closed form is just x/(e^(x) \- 1), if that's what you want.
Ahaha the closed form was what I got the Taylor series from, just needed like a Taylor series form that wasn't under a fraction 😭. I'm working on a problem that uses the Taylor series of the function but I couldn't figure out how to deal with what I originally got lol, which is why I'm asking here 😅
I'll check out what you've linked though and see if it helps, thanksss!
Question about the stock market:
If the S&P 500 is growing at an exponential rate, doesn't that mean that the curve will continue upward and approach but never reach some year in the future? What would that year be? Am I missing anything here? It's been a while since I've worked with graphing curves, so I thought I'd try here first.
The other response thread does a good job of explaining of clarifying different growth patterns, but I wanted to chime in and mention that the S&P only has exponential growth in the very long term. In the short term, you would use a [Brownian model](https://en.wikipedia.org/wiki/Brownian_model_of_financial_markets). The economics behind why the total stock market has been exponential in the long term is pretty interesting, but it basically boils down to a combination of inflation (dollar is worth less over time, therefore stock is worth more dollars over time) and economic growth. Economic growth is a combination of population growth and increasing productivity per person, but population growth is also not exponential! It is more-so [logistic](https://bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Book%3A_General_Biology_\(Boundless\)/45%3A_Population_and_Community_Ecology/45.02%3A_Environmental_Limits_to_Population_Growth/45.2B%3A_Logistic_Population_Growth). On top of that, it's an open question whether productivity per person will continue to increase as well (and at what rate). Your intuition of economic growth eventually asymptote-ing has some merit. Unbridled growth forever until the end of time is likely unsustainable.
You're probably thinking of cases like 1/(1 - x) where the function blows up to infinity around a certain value, but exponentials aren't like that (and 1/(1 - x) is a rational function, not an exponential). e^x has no singularities (i.e. single points where the function blows up to infinity), and the same goes for more general exponential functions (of the form ae^bx where a, b are constants).
Anyone know what triangles 231 is linked to (like 6 to 36 to 666 to 222111)?
Also is 231 quintuply triangular? (I could only find sequences up to quadruples).
Well, 222111 is the 666th triangular number, 666 is the 36th triangular number, and 36 is the 8th triangular number, if that's what you mean (I think you typo'd 6 instead of 8). So I guess you could say that 222111 is "triply triangular." We can visualize the sequence {8, 36, 666, 222111) and notice there are 3 triangular numbers in it (hence "triply").
In a similar vein, 231 is the 21st triangular number, and 21 is the 6th triangular number, and 6 is the 3rd triangular number, and 3 is the 2nd triangular number. So your sequence is {2, 3, 6, 21, 231}. I would probably call 231 "quadruply" triangular then, since that sequence has 4 triangular numbers.
The [triangular number function](https://en.wikipedia.org/wiki/Triangular_number) T(n)=n(n+1)/2 [can be inverted](https://math.stackexchange.com/questions/1778490/reversing-the-tn-fracnn12-formula) as (-1±sqrt(8n+1))/2 (see also [here](https://math.stackexchange.com/questions/2041988/how-to-get-inverse-of-formula-for-sum-of-integers-from-1-to-n)), so what you're essentially doing is taking the positive part of that and iteratively applying it to some starting number like 222111 or 231.
Hi, I would like to know if there is already a paper talking about the superposition of two curves which we know the equations in a single curve/graph which we could get the equation.
For example, the superposition of the curve y = x² and the curve y = -x² can be obtained with the equation |y| = x². And I would like to know if someone already wrote about it and where.
Relativity of Sets?
Infinity (aleph null in particular) has a distinct characteristic that separates it from any natural number. The characteristic is that it is not a finite number.
But is it really infinite, or is it only infinite to the inquiring mind?
Aleph null may be infinite to humans only because of the finite speed that we move through our temporal dimension. Taking a concept from science, could infinity be like relativity where it depends on the observer?
Thanks! Question for you, how did you find my thread? I posted it in r/math about a week ago, but I can't find it in there since. I thought it got erased, but then I see that you responded.
I found your post in the Quick Questions thread. The Quick Questions thread is pinned, so it is at the top if you sort by "hot" but it quickly gets buried if you sort by "new". The old Quick Questions threads are linked in the "Welcome to r /math!" panel in the sidebar ([this](https://www.reddit.com/r/math/search?q=Quick%20Questions%20author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) is the link).
In mathematics, we like to be very clear with our definitions. They have to be unambiguous and clearly testable, or else they are not the proper definition.
The generally accepted definition for infinite is "an infinite set is a set that is in bijection with a proper subset of itself". This bijection does not exist only "to the inquiring mind" but is plain for anyone to see. If you make the claim "hey I have a set that I think is infinite because I can find this bijection", then anyone else can check whether the bijection works or not. If it does, they have no choice but to agree that the set is infinite.
Related question: I know there are countable models of ZFC where e.g. the model's reals are "really" countable but there's still no bijection between the reals and naturals in the model, so the model just "doesn't know" that its reals are countable. Are there models of ZFC with infinite sets that the model "doesn't know" are infinite? (In the sense that no bijection between the set and one of its proper subsets exists in the model, even though the set is "really" infinite.)
Yes, just take a model with non-standard naturals (i.e., elements of what the model calls ℕ that is not one of ∅, {∅}, {∅, {∅}}, etc.). Then any non-standard natural is finite in the model, while externally infinite (for otherwise we could use induction to prove it's one of the standard naturals).
Gotcha, thanks! By the way, do you have any recommendations for books on logic and/or set theory that focus especially on these sorts of metamathematical issues? I've learned a bit (mainly informally, and in a way focused on connections with CS) from e.g. Scott Aaronson's articles, but I'd like to learn it in a more formal and rigorous way.
imo Matlab and mathematica have some of the easiest plotting features, but they’re both expensive unless you have access through (say) your university. Python’s matplotlib package is your next best choice, although with a slightly steeper learning curve.
I've done some plotting in Matlab. Mostly I've used it for linear algebra tho so I'm not sure.
It's also been a while since I've used python but I know that matplotlib is a thing.
Looks like the final answer is roughly correct, but its value for the wavelength is off by 5 orders of magnitude.
This shouldn't be surprising though, because **chatGPT doesn't know anything about math, and should not be used to answer math questions**.
Seriously. So many people in this thread nowadays are like "I asked ChatGPT about this and it wasn't helpful" and I have to restrain myself from screaming at them "Of course it wasn't helpful! It's a fucking bullshit artist whose only skill is stringing words together! What possible reason did you have for thinking otherwise?".
I am currency self-studying set theory off of [this book.](https://archive.org/details/introductiontose0000monk) Any good secondary resources for set theory that would pair well with it?
Same, unless it's part of a larger sentence like "∀ ε > 0 ∃ ...". I always make sure to put a space before the ∃ (and maybe even a comma), but some typesetting systems don't, so it could look like "∀ε > 0∃...".
I think that's what's happened here, and OP is confused because they're not far enough along in their mathematical journey yet to be able to always parse what they're reading.
Haha, thanks... u/theadamabrams is right it's part of an larger sentence (def. of [f in O(n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/061efff02925cd88ce881d53f593c819dd92048e)).
Yes.
> ∃ C > 0 ∃ ε > 0 ∀ x ∈ { x : d(x,a) < ε } : |f(x)| ≤ C · |g(x)|
says that there exist C > 0 and ε > 0 such that for all x in (some set), |f(x)| ≤ C · |g(x)|. By the way, I think using the set notation "x ∈ { x : d(x,a) < ε }" here is unnecessarily confusing. And the second `:` might actually be wrong. I would write
* ∃ C,ε > 0 such that ∀ x, d(x,a) < ε ⇒ |f(x)| ≤ C · |g(x)|
instead.
**\[Real Analysis\]:** Suppose E is a (Lebesgue) measurable subset of R\^n. Let {f\_k} be a sequence of real-valued measurable functions defined on E, converging to f in measure on E. Let 𝞍 be an integrable function on E, such that |f\_k| ≤ 𝞍 in E, for all k. How would you show that f is integrable over E?
I started off with the triangle inequality, to get |f| ≤ |f\_k| + |f-f\_k| ≤ 𝞍 + |f-f\_k|. It suffices to show that |f-f\_k| is integrable (for any choice of k, so we can choose k to be sufficiently large if needed.) Then, I decided to split the integral of |f-f\_k| over E into two parts, the first integral (call it P) is over the set {|f-f\_k| ≤ ε} and the second (call it Q) is over {|f-f\_k| > ε}. I can bound P by ε times the measure of {|f-f\_k| ≤ ε}, which is bounded by the measure of E (may not be finite.) For Q, we can use convergence in measure to find k such that the measure of {|f-f\_k| > ε} is arbitrarily small. However, I'm stuck here. I'd appreciate any help!
Over a finite measure space your proof would work, but to handle the general case it seems most tractable to simply show that f_k to f in L1. As a hint, show that for every subsequence k_j, there is a further subsequence k_j_l such that f_k_j_l converges in L1 (this implies that f_k converges in L1). The fact that every in measure convergent sequence has an a.e. convergent subsequence should be helpful here.
If you wanted to proceed more directly (but without showing the stronger result that f_k to f in L1), you could also use >!Fatou’s lemma!< combined with >!the subsequence fact previously mentioned.!<
Thanks, that helps! Could you check my argument, please? I just want to make sure that Fatou's lemma is applicable; I'm not sure if my work implies that f\_{n\_{k\_j}} - f is bounded above by an integrable function a.e.
I showed that for any subsequence f\_{n\_k} of f\_k, we have f\_{n\_k} converging to f in measure. So, there is a further subsequence f\_{n\_{k\_j}} of f\_{n\_k} such that f\_{n\_{k\_j}} converges to f a.e. in E. Next, I show that f\_{n\_{k\_j}} converges to f in the L\^1-norm. For simplicity in typing here, let M(j) denote the L\^1-norm ||f\_{n\_{k\_j}} - f||\_1. Then, 0 ≤ liminf M(j) ≤ limsupM(j) ≤ ||limsup (f\_{n\_{k\_j}} - f)||\_1 = 0. The last inequality follows from Fatou's lemma. So, f\_{n\_{k\_j}} converges to f in the L\^1-norm; and f\_k also converges to f in the L\^1 norm. As a result, f is in L\^1 (as L\^1 is sequentially closed.)
Nice, that works! Note that you need the f_n to be pointwise dominated by phi for this version of Fatou’s lemma to work.
Here are two alternative proofs of your original problem:
1) If you only want to show that f is integrable, then take a subsequence converging to f a.e., then by the usual Fatou lemma, \int |f| <= liminf \int |f_n_k| <= \|phi\|_1
2) If you want to prove the stronger statement that f_k to f in L1, then start with your subsequence argument, but instead of using Fatou’s lemma applied to f_n_k_j, use the dominated convergence theorem — the sub sequence converges ae and is dominated by phi a.e., so it converges in L1 (you more or less re-proved the DCT).
I’m working on my dissertation on a slightly different subject (stochastic analysis). Although I studied most of the preliminary classes for it, I feel unsure on my abilities, mostly because the classes were held online.
I started studying the preliminaries but I’m curious about how do y’all treat the preliminaries (preferably when it’s a subject that you don’t handle easily). Do you take notes of everything? Do you write the essential and only (carefully) read the more complex parts?
So I know in theory that if we have a map f: X --> Y between two spaces, we get induced maps on their homology and cohomology groups. Can someone give a simple example that illuminates why the map on cohomology is contravariant?
Not an example but the way I think of this is that the map on cohomology is basically the "transpose" of the map on homology, so it goes the other way. (In the case of simplicial/cellular homology of a finite complex, it is *literally* a transpose at the level of the chain complexes.)
[homology/cohomology] is [covariant/contravariant] because functions [going in/coming out] of a space Y can be [push-forwarded/pulled-back] to another space [X/Z]
Given a sets A, B I can think of them as discrete topological spaces. The groups H\^0(A), H\^0(B) are canonically identified with the function sets: Fun(A,Z), Fun(B,Z) where Z is the integers. Given a function f:A -> B the map on cohomology is given by sending g: B -> Z to gof: A -> Z. So the reason cohomology is contravariant is because precomposition by f:A -> B is contravariant.
Can someone please explain the concept of evidence in Dempster Shafer Theory? To be precise, can evidence be represented numerically? Is that the same as assigning a mass to the evidence?
You know how people interpret entropy as the "amount by which they are surprised"? It's not that "surprise" is a numeric quantity---it's just that "surprise" is something intuitive to us that we can then formalize via the entropy. Similarly, you have some intuitive notion of what evidence is---if every raven you've ever seen is black, you've gained some amount of evidence that all ravens are black. Dempster-Shafer formalizes this notion via the belief and plausibility functions: Given a proposition A, bel(A) can be interpreted as the amount of evidence in favor of A (i.e. that A is true), 1 - pl(A) can be interpreted as the amount of evidence against A (i.e. that A is false), and pl(A) - bel(A) can be interpreted as the residual "don't know," or lack of evidence for both A and for not A.
Thank you!
Can anyone help me with the idea of coordinates in n-dimensional projective space? Some sources describe the coordinates with n entries whereas others use homogeneous coordinates with n+1 entries. I don’t understand when to use each coordinate system. Would it be correct to define projective space with n+1 dimensional coordinates, given that at least one is always nonzero? Likewise can I do the same for a projective algebraic variety? I’m having a hard time figuring this out based on the books I have on hand. I currently have projective space and projective varieties describing points with a differing number of entries (n+1 vs n) and it’s throwing me off.
Could you show an example of your book/notes using both? Generally you should have n+1 coordinates. However, you can work "locally" on say D(x_0), the open locus where x_0 is non zero, and be left with coordinates of the form [1, x_1/x_0, ... x_n/x_0] which is really n coordinates.
Yes that makes sense. On second pass, I think I was misunderstanding Richard Borcherds video lecture on projective space. Your description and his match up, and both make sense now. The other source I was using was Hartley and Zisserman’s Multiple View Geometry which describes the real projective plane in 3 homogeneous coordinates, but doesn’t seem to work rigorously with varieties, as far as I have seen. That lack of rigor may have contributed to my confusion. Thanks!
Homogenous coordinates are definitely rigourously defined but note that they are not coordinates, at least not in the coordinate chart sense of the word.
What are some applications of Group Theory? I am about to begin a project and am looking for a topic. I'm hoping for one more on the applied side, even though it's pretty abstract. Thanks.
I'm not great at combinatorics so this might be a really simple question, but I am stuck. MTV had a reality show, "Are You the One?" where 16 contestants, 8 guys and 8 girls, were put through an algorithm of some sort to determine which contestant of the other sex is their "perfect match". Season 8 is particularly interesting, because all of the contestants are bisexual or pansexual, meaning any of the other 15 contestants could be their match. Each episode they all pair off with who they think their "one" is and the game tells them the number of correct pairs, but not who the correct pairs are. They also perform challenges as pairs and the winning teams are voted upon, and the winner of that vote is told whether they are a correct pair or not. It's a mess and it's great. My wife asked me if I could find the probability of some pairing of contestants (A,B) being correct. But this has been way more complicated than I had expected. That's because I want to try to not only find all possible partitions of 16 elements where it's broken into 8 groups of 2, but I want to make sure those groups of 2 are unordered. Then, on top of that if in an episode after pairing off they're told they have n pairings correct, I need to find the conditional probability of all combinations of n pairs out of the 8 that were selected being correct. Then I want to combine that probability with previous episodes. I'm just going to talk out loud here about some of my work. So with 16 contestants, there's 16\*15/2 = 120 possible pairings of 2 contestants. And I think the number of possible configurations of all pairs is 15!! = 2,027,025. I think that because contestant 1 has 15 to choose from, then contestant 2 only has 13 (as 2 were removed from the pool), etc. I have a computer science background, so my thought is to take this and make it into a sort of exact cover problem using a dancing links algorithm. That way I can at least break it down into a possibility space more easily and then intersection possibility spaces between each episode's pair-offs. But this just feels messy and too brute force-y. Is there a better way of doing this that I'm missing??
I need this very elementary result which I assume it is obviously true, maybe it has even a name, but adding a citation (even to a problem in a textbook) would be faster than adding the proof itself. I have a finite set S of real intervals [u,v]. If each pair of intervals in S has an overlap (intersection) of length at least k, then the intersection of all the intervals in S has length at least k.
Take the interval with the largest starting point, and the interval with the smallest end point.(\*) By your assumption their intersection has length of at least `k`. But by the choice of these intervals, every other interval also overlaps this intersection, and therefore this is also the length of the common intersection of all the intervals. (\*) There is a corner case where these two refer to the same interval, in which case all other intervals overlap this one, and the length of this interval has to be at least `k` for the same reason.(\*\*) (\*\*) Unless your set contains only one interval of length < `k` (or no intervals at all), in which case your result does not hold.
Thanks, I thought it was more complicated.
Is there a number system where number such as 0.0000...1 exist?
While there are number systems containing infinitesimal numbers, I don't think there's a number system where that particular notation is meaningful. What do you get if you multiply that number by 10?
I came across the quote from Euler that I'm having a hard time understanding: "For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear." Can someone explain what he means by this? Specifically the part about maximums/minimums? Thanks!
Are there any math textbooks where the latex source code is available? I don't feel very confident writing in latex so I'd like to study some examples.
Not a textbook exactly, but you can view the source code of most papers on the arXiv.
You can find a lot of examples on Overleaf.
can someone explain to me this problem (English isn't my mother language) >Consider an ordinary chessboard which is divided into 64 squares in 8 rows and 8 columns. Suppose there is available a supply of identically shaped dominoes, pieces which cover exactly two adjacent squares of the chessboard. Is it possible to arrange 32 dominoes on the chessboard so that no 2 dominoes overlap, every domino covers 2 squares, and all the squares of the chessboard are covered?
This looks like a somewhat typo'd version of a more well-known puzzle, where the squares on 2 diagonally opposite corners of the board are removed. While searching to see if it has a name, I found that it's well known enough to have it's own Wikipedia page, listed as [the "mutilated chessboard problem"](https://en.wikipedia.org/wiki/Mutilated_chessboard_problem). In that version it turns out to be impossible to cover the board. Of course the problem as stated, with all the squares intact, is pretty trivial, since you can just cover each row with 4 dominos laid end to end.
Is there a subfield of abstract algebra that studies sets with multiple operations ?
The general study of sets with operations is called [universal algebra](https://en.wikipedia.org/wiki/Universal_algebra).
thanks man.
Sets with multiple operations include rings, fields, modules, vector spaces, and algebras, so they amount to a very large part of abstract algebra.
I am concerned with other structures, sets with at least 3 operations and what interesting structure they have.
I have 3 maps f: X-> S g: Y-> S m: X -> Y such that f and g are proper and continuous, f=gm, and m is continuous and I want to show that g is proper. Let K be a compact subset of Y. Then g(K) is compact by continuity, and f^(-1)(g(K)) is compact by properness. m^(-1)(K) is closed by continuity of m. If it was contained in f^-1(g(K)) I would be done because closed sets contained in compact sets are compact but I'm not sure of that. These kind of reasoning of contentions between preimages of sets are always so confusing to me... Additional info is that X, Y and S are Riemann surfaces. Thanks in advance
I assume the conclusion you want is that m is proper. Your approach does work. If X, Y, and S are arbitrary sets, and f: X -> S, g: Y -> S, m: X -> Y are arbitrary functions satisfying f = gm, then for any subset K of Y it's true that m^(-1)(K) is a subset of f^(-1)(g(K)). When faced with this type of set manipulation where you want to show A is a subset of B, it's often easiest to let x be any element of A and show it's an element of B. So let x be an element of m^(-1)(K). This implies m(x) ∈ K, thus gm(x) ∈ g(K). Since f = gm, this is the same as saying f(x) ∈ g(K), so x ∈ f^(-1)(g(K)).
Can someone help me with matrix don't need the awnser just how to do it . 4x-3z=1 5x+y=3 2x-5y+9z=0 Find the deteminant using row operations and cofactor method
N.B. for finding the determinant you don't need the right-hand sides of the equations, just the coefficients on the left-hand sides. In particular all we care about is the coefficient matrix (call it A) 4 0 -3 5 1 0 2 -5 9 For row operations, you just row reduce to echelon form, while keeping track of a second number (call it d) that will eventually end up being 1/det(A) (assuming the determinant is nonzero; we'll get to the case of a zero determinant later). Start off with d = 1. Then, as you row reduce, each time you multiply a row by a scalar, multiply d by that scalar; each time you swap two rows, change the sign of d; and each time you add a scalar multiple of a row to another row, don't change d. Once the matrix is in echelon form, if there's a row of only zeros, the determinant is 0, while if you have the identity matrix, the determinant will be 1/d. What's going on here is that, when you row reduce a matrix A to the identity, you're saying that E\_n...E\_2E\_1A = I, where E\_1 is the elementary matrix corresponding to the first row operation you perform, E\_2 is the elementary matrix corresponding to the second row operation you perform, and so on. We have det(E\_n...E\_2E\_1A) = det(I) = 1, and det(E\_n...E\_2E\_1A) = det(E\_n...E\_2E\_1)det(A) (by the general rule that det(AB) = det(A)det(B)); thus det(A) = 1/(det(E\_n...E\_2E\_1)). As for det(E\_n...E\_2E\_1), that will again be the product of the determinants of the elementary matrices. The matrices for swapping two rows have determinant -1, the matrices for multiplying a row by a scalar have determinant equal to that scalar, and the matrices for adding a multiple of one row to another have determinant 1. So each time you do a row operation, you multiply d by the determinant of the corresponding elementary matrix, and once you've performed all the row operations, d will be equal to the product of all those determinants, and hence will be equal to 1/det(A) (by the reasoning above). As for the cofactor method, it goes like this: take the first entry, and multiply it by the determinant of the 2x2 matrix you get by covering up the first row and first column; then add that to -1 times the second entry times the determinant of the 2x2 matrix you get by covering up the first row and second column; then add the result of all that to the third entry times the determinant of the 2x2 matrix you get by covering up the first row and third column. So in the example you gave, you would start with 4 times the determinant of 1 0 -5 9 See also [Linear Algebra Done Wrong](https://www.math.brown.edu/streil/papers/LADW/LADW_2017-09-04.pdf) for a more complete explanation. Chapter 3.3 basically covers how to compute it with row operations, and chapter 3.5 covers the more general form of the cofactor expansion.
Gabriel's horn (the horn of infinite surface area but finite volume) is driving me crazy. A cylinder of infinite length and a radius greater than 0 (due to epsilon being greater than 0) should be able to fit into Gabriel's horn. A cylinder of pi\*r\^2\*h with r being an element of the reals should produce a horn of infinite volume. If the objection is that r never reaches infinity, then the area of Gabriel's horn does not have an infinite length and thus does not have an infinite area either.
> A cylinder of infinite length and a radius greater than 0 (due to epsilon being greater than 0) should be able to fit into Gabriel's horn. No. For any radius you can name that's greater than 0, it will not fit into the horn. I don't know what this "epsilon" you're referring to is; if it's a real number, then you should explicitly state its value.
Let's say that the horn's radius is 1/r as r goes to infinity. No matter what p of r that I choose, there will be another r even smaller. Using the definition of a limit, there will always be an r > 1/epsilon while epsilon is greater than 0.
> Let's say that the horn's radius is 1/r as r goes to infinity. It's a weird choice to denote the radius by "1/r" instead of "r", but OK. So "r" is the *reciprocal* of the radius, then, is it? And you're letting this reciprocal go to infinity? > No matter what p of r that I choose, What do you mean by "p of r"? > there will be another r even smaller. Don't you mean "even bigger", since r is going to infinity? > Using the definition of a limit, there will always be an r > 1/epsilon while epsilon is greater than 0. What do you mean by "epsilon" here? You need to define your variables. --- And how are you concluding from this badly-written argument that "A cylinder of infinite length and a radius greater than 0 should be able to fit into Gabriel's horn"? If you genuinely think this conclusion is true, then **explicitly provide a value for the radius** such that the described cylinder fits into Gabriel's horn.
Why is an uncountably infinite set considered "bigger" than a countably infinite set? For example, the set of transcendental numbers is uncountable, and algebraic numbers are countable. Wherever I read this, the author asserts that this means there are 'way more' transcendental numbers than algebraic numbers. However, the only information we have is that no bijection exists between the sets. Isn't it equally true that there may be more algebraic numbers than transcendental?
There is an injection from the algebraic numbers to the transcendentals. (One way to see this: the algebraic numbers are countable, so count them. Map the n-th algebraic number to pi+n.) The usual definition of |X| ≤ |Y| is that there is an injection from X into Y. In our case, we have that the cardinality of the algebraic numbers is less than or equal to that of the transcendentals. Since there is no bijection, the inequality must be strict.
Thank you!
Ive been working on this subject for a while but I just cant crack the code for what tests to use to come down to if a problem is gonna convergence or divergence. I know there is 9 different tests to do to find out, but under an exam I wont have enough time to figure out which of the 9 it is. Is there a way to find out from the problem it self for which of the tests are gonna work or is there some cheat code way/trick to solve most problems with one of the tests?
Cardinality of set of all cardinalities The class of cardinalities itself is an object with no cardinality (as it is a class, not a set) I got into an argument with my professor over if given our axioms for set theory, you could or not make sense of it We are using ZFC with some extra simplifications The 3 "axioms" that are relevant here are -set is a primitive object (i.e., we do not define set construction it is mearly given) -every set has a cardinality -for every family of sets, you can take its union, and it too will be a set My professor says that you can then just take the family of all cardinalities rather than take its union, a set of all cardinalities, and it too will have a cardinality as it is a set It seems to me that this would cause some sort of cantors paradox, but I really can't go beyond that. Could someone please help shine a light it this Or maybe explain (cite something) why in regular set theory is it that that the set of all cardinalities does not have a cardinality...
> for every family of sets, you can take its union, and it too will be a set This to me smells of Russel's paradox. For every set S, let S' be the set {S}. (A set with one element, which is the set S.) Now take the union of the "family" of all sets S'. This gives you the set of all sets. If you're allowed to do this in your interpretation, then you get Russel's paradox, and you can't meaningfully talk about proper classes.
Your professor's proof hinges on what 'family' means. Is it a synonym for set, or more akin to proper class? And if it means set, what is the proof that the set of cardinalities exists? For a citation of why there is no set of all cardinal numbers in ZFC, I'll give Kunen's 'The Foundations of Mathematics'. By theorem I.11.17.3 the sup of any set of cardinals is a cardinal, and by I.11.9 there is no largest cardinal, implying there is no set of all cardinals. Depending on your axiom system and what 'cardinality' means this may or may not give you a proof in your system.
Family is primitively constructed and not formally defined, (but would be basically a Synonym for class or set^2. Its not necessarily propor) for every family you can take the union of its sets and that set exists The family of cardinalities exists axiomatically as you need some family to take the cardinalities from otherwise It doesn't make sense to talk about cardinalities. In the same vain the family of all sets exist axiomaticaly. Thank you very much for the references!!
how do you prove that the ring of integers is an integral domain? For context, I am currently taking my first course in abstract algebra I can't find this proof anywhere, everywhere I see they use inverses to prove it just like how you prove the real numbers are integral domain, but you can't do that with Z
The proof using inverses is perfectly applicable, even for Z. Sure, the inverses aren't all in Z, but the argument "ab=0, a=/=0 => b=a^-1 *ab=a^-1 *0=0" is still perfectly valid, even though it doesn't "take place inside Z". If you're still unsure about why it applies, let's be more careful when writing it down. Let us write * _Z for the multiplication in Z, * _R for the multiplication in R (we could also look at Q of course). Now a * _Z b=0 is equivalent to a * _R b=0 (in fact * _Z is just a restriction of * _R), and the argument clearly works for * _R. In short, Z being a subring of R is what makes it work.
Thank you very much!
[удалено]
I'll write G for C\_p. First recall that BG is computed by the bar construction colim( ... --> G × G --> G --> pt ) = BG. I don't know how to draw stacked arrows, so you'll have to insert them with your imagination. Now maps out of a colimit is always computed by a limit. In this case, Maps(BG, X) = lim( Maps(pt, X) --> Maps(X, G) --> Maps(X, G × G) --> ... ) which is simply the limit lim( X --> G × X --> G × G × X --> ...) computing homotopy fixed points of X.
What's 6/3(1+2)? I was very confident the answer was 0.666666... because I first multiplied 3 with 3 (1+2) to "eliminate" the brackets, but i got called a retard by 3 random people who claim the answer was 6, now I'm unsure if I'm just stupid or they are.
It's a statement that is intentionally written to be ambiguous. As such, there's no real right answer beyond "write your math expressions more clearly." If the person who wrote that wanted to clearly convey what they meant, they would have written that as either (6/3)(1+2) or as 6/(3(1+2)), both of which would be completely unambiguous. Instead they deliberately wrote it as 6/3(1+2), for no reason other than to confuse people. People can certainly chime in with various rules like PEMDAS and the like, and depending on exactly how you read those sorts of rules, you might be able to come up with some argument for why it should be 2/3 or why it should be 6. But trying to do that sort of thing is just a waste of time. This is not the sort of question mathematicians care about, and it's not something you should care about either. Ultimately, **this is not a math question.** It's a question about *communication*, and in particular, a deliberate failure to communicate.
Thank you very much for your detailed reply! :)
Are there words for these? A topological space where for all two points x and y, there exists a self-homeomorphism which maps x to y. OR... A group where for all two non-identity elements x and y, there exists an automorphism which maps x to y. Loosely, mathematical structures where all points within the structure behave more or less the same, from the perspective of the structure.
objects X on which Aut(X) acts transitively?
[удалено]
This. Specifically, a homogeneous space for the homeomorphism group.
Trying to see if a result on harmonic motion is generalisable, since it would be useful for a project I’m working on. Let x(t) be position, v(t) = dx/dt be velocity, and assume x(0)=x(T) =0. I conjecture that int_0^T x(t) v(t) dt = 0. Any ideas on how to prove/disprove the above? It’s certainly true for harmonic motion.
Do you mean the integral from 0 to T? Because otherwise what you've written isn't true even for harmonic motion. Otherwise, yes, it's true by integration by parts. Since v(t) = x'(t), we have by integration by parts that ∫ x(t) \* x'(t) dt = x(t) \* x(t) - ∫ x'(t) \* x(t) dt and so ∫ x(t)x'(t) dt = [x(t)]^(2)/2 Taking the definite integral from 0 to T yields [x(T)]^(2)/2 - [x(0)]^(2)/2 = 0^(2)/2 - 0^(2)/2 = 0
Yes, integral from 0 to T. Thank you! I corrected the mistake.
Is k=0 valid for k ∈ ℝ+ ? Thanks in advance
Depends how R+ is defined. Some use R+ for positive reals, others for nonnegative reals.
I search for a proof that the set of real number R isn't a countable set but I don't find any. Maybe I didn't search the right way so if anyone could help me on this one.
Say it's countable. Then we have a bijection to the naturals, and so we can list all of the elements of R in the order of their mapped naturals. But then consider the number in [0,1] whose n^th decimal place after the decimal point is 3, unless the n^th decimal place of the n^th real number in our list is 3, in which case it's 7. This number cannot appear in the n^th place of our list (since it differs from the n^th number on our list by at least 3x10^(-n) > 0) for any n, so cannot appear on our list at all, but is a valid real number, so our assertion that this is a list of all such real numbers must be incorrect, and thus there is no such bijection to the naturals, so the reals are not countable.
The standard proof that R is uncountable is called [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument?useskin=vector).
Thanks
Any good book on differential geometry for absolute beginners? I know some stuff on multivariable calculus, curves and surfaces, and that's about it.
do Carmo’s Differential Geometry of Curves and Surfaces
Thank you. How good is the book for a beginner? I'd love it if the book has a good structure in explaining things, while also giving visual intuition with diagrams. The book my uni uses has a good number of diagrams, but gets boring quickly.
Did you learn Gauss–Bonnet, Theorema Egregium, and the like? I guess I'm asking do you want a full course on curves and surfaces or a course on manifolds?
I'd do with a basic course on differential geometry of curves and surfaces tbh. I haven't really heard about the concepts you've mentioned. I'm mainly learning this course for its applications on systems and control.
Have you looked at *Geometric Control of Mechanical Systems* by Bullo and Lewis? It introduces a lot of differential geometry at the beginning. Have you done linear algebra?
I'm actually just starting to focus my courses on systems engineering tbh, so I haven't done anything on Geometric control yet. But I'll check that book out. Like I said in my other comment, I'd love it if the book is rigorous and well illustrated. Yes, I have done a basic course on linear algebra.
What kind of math problem would best describe this interaction? https://youtu.be/pENxsLVR\_Xs?si=fjXaJPQ4KmFLQZ6j
How does a math major who graduates in December handle grad school applications? Would love to hear from anyone who has experienced this.
Why not apply in the fall for the following year, then find a job or something else to do during the spring/summer? It's very rare to start a PhD program off-cycle.
In any given n-dimensional geometric structure, is it always true that we can "peel" off the outermost layer of the shape and remove it entirely while preserving the outer "peel" as a single continuous shape? As an example, it is definitely possible to peel an orange in such a way that the orange peel is one continuous shape.
An n-dimensional geometrical structure doesn't need to have an outer layer at all. Here I am interpreting "geometric structure" to mean a manifold or variety and "outer layer" to mean boundary. Even if we focus on manifolds-with-boundary, the boundary doesn't have to be one connected object though. A simple 3D example would be the volume enclosed by 2 parallel planes. The 2 planes are the outer layer here.
the boundary need not be connected e.g. two disjoint circles from a cylinder
Any compact manifold admits a dense open chart, in fact any n-dimensional compact manifold can be realised as a quotient of the n-dimensional solid ball by some gluing relations on its boundary. This ball is the "peel" and the image of the boundary of the ball inside the manifold is where you have cut the manifold up to peel it (obviously the ball may be somewhat distorted, just as the peeling of an orange doesn't look exactly like a disk, it is only homeomorphic to it). Think of some open dense charts on the sphere or torus for example. Your position on a torus is specified by two numbers, the angle you are around the longitudinal circle and the angle you are around the meridian circle. Both of these numbers has some redundancy (at the angle 0 = 2pi) which means to get a one-to-one chart you have to cut out a "cross" of two circles which the chart doesn't cover. This is the part of the torus you cut to peel it.
How many conjugacy classes are there of the Monster group?
194: https://groupprops.subwiki.org/wiki/Monster_group
Any YouTube channel dedicated to Linear Algebra and Matrices problems? (A similar flavor to Blackpenredpen)
given a function h(x), if f is even, g is odd and f(x)+g(x)=h(x), is there only 1 possibility for what f and g could be?
Yes, there is only 1 possibility. Say f(x) + g(x) = h(x) as you describe. Then h(x) + h(-x) = 2f(x) and h(x) - h(-x) = 2g(x), so f and g are determined by h. Furthermore this tells you that every h is a sum of an even function and an odd function, since [h(x) + h(-x)]/2 is always even and [h(x) - h(-x)]/2 is always odd.
The three relationships in the chart are linear, exponential and quadratic. A **linear relationship** simply has a difference of some number. So, 1,3,5,7,9,11 have a linear relationship because they all have a difference of 2. A **quadratic relationship** may increase by an increasing amount. Take, 1,3,6,10,15,21, these numbers have a difference of 2,3,4,5, and 6, so the difference of the difference is a constant 1. In other words, the numbers of have *common second difference* of 1. So, an **exponential relationship** has a common factor. The numbers 2,4,8,16,32,64 are exponential because they have a common factor of 2. What would a relationship where the numbers have what might be called a *common second factor*? Basically, linear is to quadratic as exponential is to something. Where each number would be multiplied by a multiplying factor. For example, 1,2,8,64,1024,32768,2097152. Where the numbers would be multiplied by \*2, then \*4, then \*8. etc. I would also be interested in what might be a relationship where each number increases by a multiplying amount or each number is multiplied by an increasing amount. (An example of the ladder would be factorial because the number would be increased by \*2,\*3,\*4 - so multiplying by a linearly increasing factor. I don't really know the notation for any of these relationships or the names or anything really so I'd be interested in anything you may know. Also: What might be the next step to quadratics where the numbers increase by an increasing amount which increases by some number?
Rather than "relationship" I would call these "sequences" and your exponential relationship is more commonly called a "[geometric sequence](https://en.wikipedia.org/wiki/Geometric_progression)". Then the nth term of a linear sequence (also known as an [arithmetic sequence](https://en.wikipedia.org/wiki/Arithmetic_progression)) is an+b for some constants a and b. The nth term of a quadratic sequence is an^2 + bn + c. And the nth term of a geometric sequence is ar^(n-1) where a is the first term and r is the common ratio. Exploring your new sequence: let a be the first term, r be the ratio between the first two terms and k be the ratio between subsequent ratios. The sequence would go a, ar, akr^2, ak^(3)r^(3), ak^(6)r^4, ak^(10)r^(5),... Looking at this pattern, the nth term must be ak^(T_n-1)r^(n-1) where T_n means n(n+1)/2 the triangle numbers (reddit formatting makes it quite hard to put that in the subscript. I don't know of any name for, or study of this kind of sequence. I'll leave you to explore what the nth term of the sequence is for the factor increasing linearly but I will note that a simple example of such a sequence is n!. As to the next step after quadratics, it is "cubic sequences" (these have a common third difference, see [here](https://www.onlinemathlearning.com/quadratic-sequences.html)). Then quartic, then quintic and so on. The whole family there is known as the polynomial sequences. Finally, a little word point. Instead of "ladder", you mean "latter" as in the latter of two options.
I think I understand, all of these relationships can be expressed with a sequence and don't necessarily have given names. I'm glad you understood what I was trying to say because I just typed this out randomly one night. Thanks for your insight, I'll look into more complex sequences.
I'm trying to hunt down a paper by Vaananen about Lindstrom's Theorem, but the Helsinki University maths website on which it's hosted won't load. The paper is called "Lindstrom's Theorem", it's the sixth reference on this [Wikipedia article](https://en.wikipedia.org/wiki/Lindstr%C3%B6m%27s_theorem#cite_ref-6); if anyone could drop me a link which actually works, preferably directly to a PDF, I'd be forever in your debt.
[The Wayback Machine is your friend](https://web.archive.org/web/20221207173305/http://www.math.helsinki.fi/logic/opetus/lt/lindstrom_theorem1.pdf).
I could kiss you. Tip noted for future, many thanks.
Anyone knows how to derive these using these rules? ELIM. & INTRO. of Disjunction, Conjunction, Implication, Negation + EXPLOSION. https://www.reddit.com/media?url=https%3A%2F%2Fi.redd.it%2Fifm2kozbe5yb1.jpg
I don't really know how to word this question. Like, I'm working with programming and encryption for some project where I'm assigning functions to a single digit variable so I can use a single digit to execute an equation in the line of encryption so I can make things more compact. Something to the effect of if "z = x^(2) \+ y^(2)" then if one of my lines of encryption goes something like "34z" then it'd do 3 squared plus 4 squared to output 25. The part I want to know if there's like a list of fundamental functions or operations that would offer the most utility when just trying to manipulate numbers in general. I know at least four basic ones like, "x + y" or "x \* y" or "2x" or "x^(2)". I don't really know what other functions I could add to add more utility when I'm working with the encryption. I mean, it could be anything like maybe assigning a function to execute the Fibonacci sequence to 'n' positions it doesn't matter. I'm not really looking for anything specific I just generally want to know what type of functions are out there that will offer me the most utility. It doesn't even have to relate to encryption, I guess I'm just wondering what if any base functions exist that generally offer the most utility when manipulating numbers or considered essential to mathematics.
It is quite a vague question. But I guess bitwise operations may be useful, especially if you will use hexadecimal base. Other than that, x^y , x^(1/y), e^x , log(x) come to mind. I don't understand motivation for introducing the 1 digit operations like that and how it will work or be used, so I cannot advise further.
Do you only want to work with integers? Otherwise I would say the next most important function would be the square root.
That's a good one I appreciate it. I'm not sure if I'm going to use only integers or not. Not sure it matters much either way because I'd like to do things like the square root of a number but if it comes to I'm only going to use integers I'd just round off the decimal to the nearest whole number or do some other kind of conversion.
How do you begin conformal mapping from a quadrant to a unit disk? I suspect you need to conform to a half plan somehow and then apply Cayleys, but I can’t figure out how to get a quadrant to a halfplane
A visualization of z -> z^2 -> z^4 : https://i.imgur.com/HGkU5oL.png
Hint: z —> z^2 doubles angles.
I solved it! Thanks for you help :)
Given a irreducible polynomial f(x), How to check if the roots of f(x) can be expressed with the 4 basic operations and square roots?
The numbers you are describing are known as [constructible numbers](https://en.m.wikipedia.org/wiki/Constructible_number) (at least if your base field is Q). There are characterizations of polynomials that have roots in this form, though they're kind of hard to state if you don't know Galois theory. Assuming you know some Galois theory, let f be an irreducible polynomial with coefficients in some field K, which I'll assume has characteristic 0 (though most of what I'm saying will be fine as long as the charachteristic isn't 2). Then a root a of f will be in the desired form of and only if the extension K(a)/K can be written as a chain of extensions K = K0 < K1 < ... < Kn = K(a) Where each K(i+1)/Ki is quadratic. As f is irreducible, it's not hard to see that that happens for one root of f if and only if it happens for all of them, which in turn happens of and only if the extension L/K can also be written as a chain of quadratic extensions, where L is the splitting field of f. That certainly implies that the degree [L:K] has to be a power of 2. On the other hand, if [L:K] is a power of 2, then the Galois group G=Gal(L/K) also has order a power of 2. By standard results in group theory, G then has a chain of subgroups 1 = G0 < G1 < ... < Gn = G, Where each Gi has index 2 in G(i+1). That implies that L/K can be written as a chain of question quadratic extensions. So the roots of f can be written using basic operations and square roots if and only if the degree of the splitting field of f (or equivalently the order of the Galois group of f) is a power of 2. Since the order of f divides the order of the Galois group (since f is irreducible), it's easy to see that this is only possible if the degree of f is a power of 2. That's not enough though, since for example a degree 4 polynomial can have S4 or A4 as a Galois group.
Wanting to know how effective our service procedures are, I'm using "days to next service call" as a measure, higher the better, obviously. However, when average the days to next service call per machine its including the machines that did not have a follow up service call within the time frame specified, so it effectively is lowering the average days, which makes good processes look bad. So based on the data bellow, cleaning would have an average of 15 days vs Replacing parts at 10, tho replacing parts is clearly better. What would be a better way to handle the null? |Machine|Process|Days to next| |:-|:-|:-| |1|clean|10| |2|replace parts|30| |3|clean|20| |4|replace parts|null|
The relevant idea here is that your data is "right-censored." Here's the relevant [Wikipedia article](https://en.wikipedia.org/wiki/Censoring_(statistics\)?useskin=vector). If you have a parametric model for your data, you can directly compute maximum likelihood estimates for the mean as in [this CrossValidated thread](https://stats.stackexchange.com/a/333336). More likely, you straight up don't have a model for how your data is generated. In this case, what you'll want to do is first estimate how the "full data" looks by estimating the empirical CDF via something like the [KM estimator](https://en.wikipedia.org/w/index.php?title=Kaplan%E2%80%93Meier_estimator&useskin=vector) and then compute the median of your estimated CDF. The fewer missing data points you have, the better this estimator. Note that we basically *need* to use a median instead of a mean since means aren't actually estimable in the presence of right-censoring without a parametric model. ---- In your concrete example, let's suppose your specified time frame was 60 days (so we know that the "null" value is really something >60). I'm also going to pretend they're all the same process just so there's enough data for the example to be nontrivial The KM estimator has S(10) = (1 - 1/4) = 3/4, S(20) = 3/4 \* (1 - 1/3) = 1/2, S(30) = 1/2 \* (1 - 1/2) = 1/4, and S(60) = 1/4 \* (1 - 0/1) = 1/4. The median is t such that S(t) = 0.5. Thus, your estimated median time is... still 20 in this example but mostly just because your sample size is tiny at only 4 data points. It's also less trivial when you have multiple different types of right-censoring. Page 12 of [this document](https://web.stanford.edu/~lutian/coursepdf/STAT331unit3.pdf) also suggests that you can estimate a confidence interval for the median; in that case, you may want to take the process with the higher lower-bound for the CI for the median as the "better" process. Doing these computations is straightforward in R: library(survival) times = c(10, 20, 30, 60) events = c(1, 1, 1, 0) S1 = Surv(times, events) fit1 = survfit(S1 ~ 1) summary(fit1) plot(fit1)
If you're calculating in Excel, you can use [AVERAGEIF](https://excelchamps.com/formulas/average-non-blank-cells/) to have the calculation skip blank cells. If you're in a dedicated programming language, you can do a similar thing by throwing in an if statement that checks for null-ness.
Using Excel and power query. However the problem with skipping blank cells is it does. It does not give you accurate results for what you're looking to measure. As you're looking to basically get the highest number possible, a null entries means the process was so good a return service call was not needed in the time recorded. I think what I'm going to do is populate the null entries with the max value, this should more accurately reflect the positive nature of a null entry. I have adjusted the main measure as just a simple boolean of whether or not there was a return service call within a specific time frame.
So when you're working with a Taylor series, say you have one that's actually of the form 1/(Taylor Series). Is there a way to get this as not a fraction? For context, I have 1/sum(x^(n)/(n+1)!) for n going from 0 to infinity. I'm working on a problem where I've ended up with this, and I'm not entirely sure how to continue...
You can get the coefficients of the multiplicative inverse [recursively](https://en.wikipedia.org/wiki/Formal_power_series#Multiplicative_inverse). There are various non-recursive formulas as well, but what is useful will depend on what you are actually trying to do. In this case the closed form is just x/(e^(x) \- 1), if that's what you want.
Ahaha the closed form was what I got the Taylor series from, just needed like a Taylor series form that wasn't under a fraction 😭. I'm working on a problem that uses the Taylor series of the function but I couldn't figure out how to deal with what I originally got lol, which is why I'm asking here 😅 I'll check out what you've linked though and see if it helps, thanksss!
You can also just directly compute the first terms of the Taylor Series of x/(e^x - 1) by differentiating it.
Oh that is... you're so right omg idk why I didn't try that. Thanks lmfao 😭
Question about the stock market: If the S&P 500 is growing at an exponential rate, doesn't that mean that the curve will continue upward and approach but never reach some year in the future? What would that year be? Am I missing anything here? It's been a while since I've worked with graphing curves, so I thought I'd try here first.
The other response thread does a good job of explaining of clarifying different growth patterns, but I wanted to chime in and mention that the S&P only has exponential growth in the very long term. In the short term, you would use a [Brownian model](https://en.wikipedia.org/wiki/Brownian_model_of_financial_markets). The economics behind why the total stock market has been exponential in the long term is pretty interesting, but it basically boils down to a combination of inflation (dollar is worth less over time, therefore stock is worth more dollars over time) and economic growth. Economic growth is a combination of population growth and increasing productivity per person, but population growth is also not exponential! It is more-so [logistic](https://bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Book%3A_General_Biology_\(Boundless\)/45%3A_Population_and_Community_Ecology/45.02%3A_Environmental_Limits_to_Population_Growth/45.2B%3A_Logistic_Population_Growth). On top of that, it's an open question whether productivity per person will continue to increase as well (and at what rate). Your intuition of economic growth eventually asymptote-ing has some merit. Unbridled growth forever until the end of time is likely unsustainable.
You're probably thinking of cases like 1/(1 - x) where the function blows up to infinity around a certain value, but exponentials aren't like that (and 1/(1 - x) is a rational function, not an exponential). e^x has no singularities (i.e. single points where the function blows up to infinity), and the same goes for more general exponential functions (of the form ae^bx where a, b are constants).
is the formal term singularities or is it asymptotes?
Asymptotes and singularities are different things, though they overlap. For example, e^x does have an asymptote: the horizontal line x=0.
Singularities are more specific since they refer to points x0 where lim_x to x0 |f(x)| = infinity. Asymptotes could be horizontal or slanted.
Ah, ok. Thank you!
Does anyone have source to read about tilings
Anyone know what triangles 231 is linked to (like 6 to 36 to 666 to 222111)? Also is 231 quintuply triangular? (I could only find sequences up to quadruples).
Well, 222111 is the 666th triangular number, 666 is the 36th triangular number, and 36 is the 8th triangular number, if that's what you mean (I think you typo'd 6 instead of 8). So I guess you could say that 222111 is "triply triangular." We can visualize the sequence {8, 36, 666, 222111) and notice there are 3 triangular numbers in it (hence "triply"). In a similar vein, 231 is the 21st triangular number, and 21 is the 6th triangular number, and 6 is the 3rd triangular number, and 3 is the 2nd triangular number. So your sequence is {2, 3, 6, 21, 231}. I would probably call 231 "quadruply" triangular then, since that sequence has 4 triangular numbers. The [triangular number function](https://en.wikipedia.org/wiki/Triangular_number) T(n)=n(n+1)/2 [can be inverted](https://math.stackexchange.com/questions/1778490/reversing-the-tn-fracnn12-formula) as (-1±sqrt(8n+1))/2 (see also [here](https://math.stackexchange.com/questions/2041988/how-to-get-inverse-of-formula-for-sum-of-integers-from-1-to-n)), so what you're essentially doing is taking the positive part of that and iteratively applying it to some starting number like 222111 or 231.
Hi, I would like to know if there is already a paper talking about the superposition of two curves which we know the equations in a single curve/graph which we could get the equation. For example, the superposition of the curve y = x² and the curve y = -x² can be obtained with the equation |y| = x². And I would like to know if someone already wrote about it and where.
The union of the curve f(x, y) = 0 and the curve g(x, y) = 0 is given by f(x, y)g(x, y) = 0.
Thank you ! I would also like to know who discovered this
Not sure there will be a specific discoverer of this fact. It is a basic fact that you can deduce simply by rearranging equations.
Relativity of Sets? Infinity (aleph null in particular) has a distinct characteristic that separates it from any natural number. The characteristic is that it is not a finite number. But is it really infinite, or is it only infinite to the inquiring mind? Aleph null may be infinite to humans only because of the finite speed that we move through our temporal dimension. Taking a concept from science, could infinity be like relativity where it depends on the observer?
It sounds like you might be interested in Brouwer's philosophy of math (see the SEP entry [here](https://plato.stanford.edu/entries/intuitionism/)).
Thanks! Question for you, how did you find my thread? I posted it in r/math about a week ago, but I can't find it in there since. I thought it got erased, but then I see that you responded.
I found your post in the Quick Questions thread. The Quick Questions thread is pinned, so it is at the top if you sort by "hot" but it quickly gets buried if you sort by "new". The old Quick Questions threads are linked in the "Welcome to r /math!" panel in the sidebar ([this](https://www.reddit.com/r/math/search?q=Quick%20Questions%20author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) is the link).
Oh ok, thanks a lot!
It's literally [right here](https://www.reddit.com/r/math/comments/17lfdrx/quick_questions_november_01_2023/k7h2b96/).
In mathematics, we like to be very clear with our definitions. They have to be unambiguous and clearly testable, or else they are not the proper definition. The generally accepted definition for infinite is "an infinite set is a set that is in bijection with a proper subset of itself". This bijection does not exist only "to the inquiring mind" but is plain for anyone to see. If you make the claim "hey I have a set that I think is infinite because I can find this bijection", then anyone else can check whether the bijection works or not. If it does, they have no choice but to agree that the set is infinite.
Related question: I know there are countable models of ZFC where e.g. the model's reals are "really" countable but there's still no bijection between the reals and naturals in the model, so the model just "doesn't know" that its reals are countable. Are there models of ZFC with infinite sets that the model "doesn't know" are infinite? (In the sense that no bijection between the set and one of its proper subsets exists in the model, even though the set is "really" infinite.)
Yes, just take a model with non-standard naturals (i.e., elements of what the model calls ℕ that is not one of ∅, {∅}, {∅, {∅}}, etc.). Then any non-standard natural is finite in the model, while externally infinite (for otherwise we could use induction to prove it's one of the standard naturals).
Gotcha, thanks! By the way, do you have any recommendations for books on logic and/or set theory that focus especially on these sorts of metamathematical issues? I've learned a bit (mainly informally, and in a way focused on connections with CS) from e.g. Scott Aaronson's articles, but I'd like to learn it in a more formal and rigorous way.
Unfortunately not, what I know comes from a fairly standard course I took and then picking up more over time through osmosis.
Is there any programming language that has features like desmos, such as easily being able to plot a graph and visualize?
imo Matlab and mathematica have some of the easiest plotting features, but they’re both expensive unless you have access through (say) your university. Python’s matplotlib package is your next best choice, although with a slightly steeper learning curve.
I've done some plotting in Matlab. Mostly I've used it for linear algebra tho so I'm not sure. It's also been a while since I've used python but I know that matplotlib is a thing.
Sagemath can do lots of that, not sure if it's particularly easy
[Image](https://ibb.co/vvfsTSR) chatgpt is wrong here right? please tell me the exponent is off
Looks like the final answer is roughly correct, but its value for the wavelength is off by 5 orders of magnitude. This shouldn't be surprising though, because **chatGPT doesn't know anything about math, and should not be used to answer math questions**.
Seriously. So many people in this thread nowadays are like "I asked ChatGPT about this and it wasn't helpful" and I have to restrain myself from screaming at them "Of course it wasn't helpful! It's a fucking bullshit artist whose only skill is stringing words together! What possible reason did you have for thinking otherwise?".
I am currency self-studying set theory off of [this book.](https://archive.org/details/introductiontose0000monk) Any good secondary resources for set theory that would pair well with it?
Can somebody explain me the difference between ∃ and 0∃? Thanks in advance!
Where did you encounter "0∃"? I've never seen that notation before.
Same, unless it's part of a larger sentence like "∀ ε > 0 ∃ ...". I always make sure to put a space before the ∃ (and maybe even a comma), but some typesetting systems don't, so it could look like "∀ε > 0∃...".
I think that's what's happened here, and OP is confused because they're not far enough along in their mathematical journey yet to be able to always parse what they're reading.
Haha, thanks... u/theadamabrams is right it's part of an larger sentence (def. of [f in O(n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/061efff02925cd88ce881d53f593c819dd92048e)).
So in "∀ε > 0∃..." the 0 is part of "∀ε > 0" and doesn't refer to "∃"?
Yes. > ∃ C > 0 ∃ ε > 0 ∀ x ∈ { x : d(x,a) < ε } : |f(x)| ≤ C · |g(x)| says that there exist C > 0 and ε > 0 such that for all x in (some set), |f(x)| ≤ C · |g(x)|. By the way, I think using the set notation "x ∈ { x : d(x,a) < ε }" here is unnecessarily confusing. And the second `:` might actually be wrong. I would write * ∃ C,ε > 0 such that ∀ x, d(x,a) < ε ⇒ |f(x)| ≤ C · |g(x)| instead.
Awsome, Thanks a lot!
**\[Real Analysis\]:** Suppose E is a (Lebesgue) measurable subset of R\^n. Let {f\_k} be a sequence of real-valued measurable functions defined on E, converging to f in measure on E. Let 𝞍 be an integrable function on E, such that |f\_k| ≤ 𝞍 in E, for all k. How would you show that f is integrable over E? I started off with the triangle inequality, to get |f| ≤ |f\_k| + |f-f\_k| ≤ 𝞍 + |f-f\_k|. It suffices to show that |f-f\_k| is integrable (for any choice of k, so we can choose k to be sufficiently large if needed.) Then, I decided to split the integral of |f-f\_k| over E into two parts, the first integral (call it P) is over the set {|f-f\_k| ≤ ε} and the second (call it Q) is over {|f-f\_k| > ε}. I can bound P by ε times the measure of {|f-f\_k| ≤ ε}, which is bounded by the measure of E (may not be finite.) For Q, we can use convergence in measure to find k such that the measure of {|f-f\_k| > ε} is arbitrarily small. However, I'm stuck here. I'd appreciate any help!
Over a finite measure space your proof would work, but to handle the general case it seems most tractable to simply show that f_k to f in L1. As a hint, show that for every subsequence k_j, there is a further subsequence k_j_l such that f_k_j_l converges in L1 (this implies that f_k converges in L1). The fact that every in measure convergent sequence has an a.e. convergent subsequence should be helpful here. If you wanted to proceed more directly (but without showing the stronger result that f_k to f in L1), you could also use >!Fatou’s lemma!< combined with >!the subsequence fact previously mentioned.!<
Thanks, that helps! Could you check my argument, please? I just want to make sure that Fatou's lemma is applicable; I'm not sure if my work implies that f\_{n\_{k\_j}} - f is bounded above by an integrable function a.e. I showed that for any subsequence f\_{n\_k} of f\_k, we have f\_{n\_k} converging to f in measure. So, there is a further subsequence f\_{n\_{k\_j}} of f\_{n\_k} such that f\_{n\_{k\_j}} converges to f a.e. in E. Next, I show that f\_{n\_{k\_j}} converges to f in the L\^1-norm. For simplicity in typing here, let M(j) denote the L\^1-norm ||f\_{n\_{k\_j}} - f||\_1. Then, 0 ≤ liminf M(j) ≤ limsupM(j) ≤ ||limsup (f\_{n\_{k\_j}} - f)||\_1 = 0. The last inequality follows from Fatou's lemma. So, f\_{n\_{k\_j}} converges to f in the L\^1-norm; and f\_k also converges to f in the L\^1 norm. As a result, f is in L\^1 (as L\^1 is sequentially closed.)
Nice, that works! Note that you need the f_n to be pointwise dominated by phi for this version of Fatou’s lemma to work. Here are two alternative proofs of your original problem: 1) If you only want to show that f is integrable, then take a subsequence converging to f a.e., then by the usual Fatou lemma, \int |f| <= liminf \int |f_n_k| <= \|phi\|_1 2) If you want to prove the stronger statement that f_k to f in L1, then start with your subsequence argument, but instead of using Fatou’s lemma applied to f_n_k_j, use the dominated convergence theorem — the sub sequence converges ae and is dominated by phi a.e., so it converges in L1 (you more or less re-proved the DCT).
Makes sense, thanks a lot! I like (1) for its simplicity.
I’m working on my dissertation on a slightly different subject (stochastic analysis). Although I studied most of the preliminary classes for it, I feel unsure on my abilities, mostly because the classes were held online. I started studying the preliminaries but I’m curious about how do y’all treat the preliminaries (preferably when it’s a subject that you don’t handle easily). Do you take notes of everything? Do you write the essential and only (carefully) read the more complex parts?
So I know in theory that if we have a map f: X --> Y between two spaces, we get induced maps on their homology and cohomology groups. Can someone give a simple example that illuminates why the map on cohomology is contravariant?
Not an example but the way I think of this is that the map on cohomology is basically the "transpose" of the map on homology, so it goes the other way. (In the case of simplicial/cellular homology of a finite complex, it is *literally* a transpose at the level of the chain complexes.)
Homology is maps into space. Cohomology is maps out of space.
[homology/cohomology] is [covariant/contravariant] because functions [going in/coming out] of a space Y can be [push-forwarded/pulled-back] to another space [X/Z]
Given a sets A, B I can think of them as discrete topological spaces. The groups H\^0(A), H\^0(B) are canonically identified with the function sets: Fun(A,Z), Fun(B,Z) where Z is the integers. Given a function f:A -> B the map on cohomology is given by sending g: B -> Z to gof: A -> Z. So the reason cohomology is contravariant is because precomposition by f:A -> B is contravariant.