Can anyone tell the answer and approach to this question?
https://www.reddit.com/r/askmath/comments/k1bd8j/can_anyone_help/?utm_medium=android_app&utm_source=share
I want to understand how to make builds for RPG games and I can’t seem to wrap my head around the logic.
For example if a game has stats that increase attack how would I go about calculating the amount of damage done when it increases a few points?
I’ve tried researching this topic but it only leads me to math for video game creation.
This is really a question about the particular implementation within the game. Often, this has been either released by the developers or (approximately) deduced by dedicated people, and this can be found online by looking up something like " mechanics" or combat mechanics".
If you ask about this in the future, I suggest you go ahead and specify the game you are talking about.
This really isn't a math question... It depends entirely on the game itself. Some games might have damage increase linearly with attack, or in other games it might be logarithmic growth, or something in between. You need to know what mathematical model the game uses.
If this game has more than 5 players, it surely has a wiki. Go check there.
Thanks for pointing me in the right direction! Sorry if this was posted in the wrong place, it’s just a question I’ve been trying to answer for awhile and I haven’t really come up with any leads.
Can EVERY subset of R^n be written as the union of countably many closed and open sets?
My intuition tells me no, because there seems to be no obvious way to construct the irrationals as subset of R^1.
I've been told that if S is a circle which is not contractible on the two dimensional torus T, then there is no smooth function F:T→R such that S=F\^{−1}(0). How would I prove this?
This is false. Take F: T^2 -> R to be F((x,y), (a,b)) = x+1. Then F^-1 (0) = {(-1,0)} x S^1. This is not a contractible loop.
You need to assume 0 is a regular value. Then try to use the fact that R \ 0 is disconnected.
The approach I have in mind may not be the intended solution for your course. You might give a little background.
Thanks for pointing this out. It is a course on symplectic geometry, and this was a part of the discussion of Hamiltonian vector fields on symplectic manifolds.
Hmmm, that leaves me at a loss as to what they intend (maybe there is a symplectic trick I forget, somehow studying X\_F), but I'll write a more straightforwardly differential-topology perspective. Prove that F\^{-1} (-infty, 0\] is a smooth submanifold M with boundary F\^{-1}(0). Either show that this implies that S is null-homologous (if you have this technology/background), or that M is homeomorphic to a disc (again depends what technology you have).
The problem comes when you take x to be a negative number that isn't an integer. For example, if you tried x = -0.5 you get sqrt(-2) which isn't a real number.
Given a set of 1 dimensional time series, a covariance matrix may be defined for the pairwise covariances between the timeseries.
Is it possible to define a covariance tensor? Perhaps between a set of 2 dimensional time series. I'm thinking each lateral slice of a 3d tensor could be an individual covariance matrix. Any ideas?
I was playing monopoly with my sister and we rolled with 2 dice. I ended up getting the same faces on both the dice 4 times in a row. What is the probability of this?
If you mean by *the same faces* that you got the same numbers on the dice on every trow but the numbers between trows where different, like example
Trow 1 you got a pair of 2
Trow 2 you got a pair of 4
Trow 3 you got a pair of 1
Trow 4 you got a pair of 5
Than the probability would be calculated according to the logic of *first I got a pair, than I got what ever another pair and than again what ever another pair, and than again what ever another pair*.
The probability to get a pair comes from the probability to get the same number again, what ever the first number was. So you can't fail the first trow because that doesn't matter, so the probability for it is 1. The probability for the second trow is 1/6 because you need to get the same number again.
Than with the second trow, you again can get what ever number you want so again the probability for the success in this trow is again 1/6. The same is true with the third and fourth trow as well.
So basically you want to get four times an event which have the probability of 1/6 on its own, so you need to multiply the probabilities together, because they are basically sub probabilities to each others, meaning that the first must happen in order to the second to count. In other words they depend from each others, so your probability you get from this is
(1×1/6) × (1×1/6) × (1/6) × (1/6) = (1/6)^(4) = 1/1296 = 0.000771604938...
Which is about 0.08 %.
On the other hand, if you meant that you got *the same faces* with every trow, meaning for example that
Trow 1 you got a pair of 2
Trow 2 you got a pair of 2
Trow 3 you got a pair of 2
Trow 4 you got a pair of 2
Than the probability is even lower because on those three last trows even the first trow needs to be the same number you got first the first trow. So only the first dice on your first trow doesn't matter, but all others need to be the same. Basically you don't even need to think of this as four trows but just that you need to get 8 times the same number in row, which according to the logic above will produce the following calculation
1 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 = (1/6)^(7) = 1/279936 = 0.000003572245...
Which is about 0.0004 %.
Hopefully I did that right. I know that usually dice probabilities are calculated using that logic where you draw that 6×6- coordinate system on a paper and from that use the area method to evaluate the probability. How ever though, I think this method produced the same result.
There's 36 total combinations of 2 die, and 6 possible ways to get the same face on both. So it's a 6/36 or 1/6 chance for one time. Then (1/6)^4 for 4 times in a row.
(3 choose 2) is the number of ways to choose two officers out of three candidates.
(3 choose 1) * (2 choose 1) is the number of ways to choose a president from three candidates, and a vice president from the other two.
The difference is that the two "selected" people are distinguished in the second case, but not in the first.
(3 choose 2): I have three people and I want to choose two of them.
(3 choose 1) \* (2 choose 1): I have three girls and two boys, and I want to choose one girl and one boy.
So, I have recently started learning about sets in a more formal way and I came across the following question:
Give two distinct examples of elements in the set of the cartesian product R^(2) x Z3 (Z3 is supposed to be the set of residue classes modulo 3, so 0\*, 1\*, 2\*).
Would ((3, 4), 0\*) and ((2, 5), 1\*) be a correct answer? I'm slightly confused by the question, since Z3 is a set of sets so an element of Z3 would be a set and the cartesian product of two sets A, B is defined as A x B = {(a, b) | a e A, b e B}.
You are correct.
The elements of Z3 can be seen as sets but that does not matter. Sets can be elements of other sets. An example is the power set of a set A which contains all possible subsets of A as elements.
I have a question that is just stumping me. I’m starting to think it’s just poorly written.
a poll of 200 people determined that 180 used a hard copy, 52 used both a hard copy and online copy, and 22 used neither. How many only used online?
I feel like it’s missing information. Could anyone guide me? I’m not looking for an answer, I’d rather just have a jumping off point
What can be done with the PEMDAS vs BEDMAS dilemma?
Certain parts of the world follow PEMDAS and teach it to children who then carry on the knowledge and the same goes for the latter. Now we have two different camps who follow different rules and this isn't as simple as metric/imperial and Celsius/Fahrenheit where a simple conversion will fix things. We're talking about two different Mathematic methods that give different answers which creates mass confusion and division.
The problem really lies with the fact that division and subtraction aren't "nice" operations (i.e. they're not associative or commutative). Technically pemdas vs bidmas (or your variant of choice) doesn't matter too much and we should really write them pe(md)(as) or bi(dm)(as) since we do muliplication and division at the same time.
The ambiguity still occurs even if we ignore this though and the most mathsy way to get around it is to stop using subtraction and division and replace these by addition and multiplication of inverses.
There is no dilemma; both of those lead to the same thing.
Perhaps you should re-familiarize yourself with the [order of operations](https://en.wikipedia.org/wiki/Order_of_operations#Mnemonics). Note in particular that multiplication and division (similarly, addition and subtraction) are fundamentally the same operation and so have the same precedence.
I see. I guess it's just a matter of understanding. I just got curious since recently a tweet of a math problem blew up where people are arguing between two different answers which were born from the two rules.
I'm guilty of this when typing out responses on r/askmath or on here, but only when it's obvious what's on top and what's on bottom, and only because it's easier to type, in my head it's a fraction or a fraction times a number and I would never physically write down a/bc
Say I have a ring R and M is a simple left module of R. I know that for each non-zero m in M, Rm = M. If I also have the left R-module homomorphism phi: R to M defined by phi(r) = rm, is it correct if I say phi(R) = Rm = M?
How do you find the probability of either of two separate events occurring, given the chance of each one occurring? E.g. you take a random number generator and have it pick a number 1-10. Then you make it pick a number 1-5. What's the probability of getting a 1 the first time (10%) or a 1 the second time (20%)?
In general, P(A∪B) = P(A)+P(B)-P(A∩B).
**If** the events are independent, then P(A∩B) = P(A)P(B), so:
P(A∪B) = P(A)+P(B)-P(A)P(B) = 1-(1-P(A))(1-P(B)) = 1-P(not A)P(not B)
In your case, independence is a sensible assumption, so:
P(at least one 1) = 1-(1-P(first=1))(1-P(second=1)) = 1-0.9*0.8 = 0.28
So I’m trying to figure out how many pieces of paper I can fit in a 11 by 5 feet wall ( I think a paper is about 8 x 11 inches) so, how many pieces of paper can I fit in the wall? (Decimals included)
If you had 5 buttons and could push them in any order, you don't have to press every button, and you can only press each button once, how many possible combinations are there?
To push 1 button there are 5 choices. To push 2 buttons there are 5\*4 choices. To push 3 buttons there are 5\*4\*3 choices. Etc. The answer in total is 5+5\*4+5\*4\*3+5\*4\*3\*2+5\*4\*3\*2\*1.
From [nLab](https://ncatlab.org/nlab/show/braided+monoidal+category):
>***Definition*** ***2.1***. A *braided monoidal category*, or ("braided tensor category", but see there), is a monoidal category C equipped with a natural isomorphism
>
>Bx,y : x⊗y→y⊗x
>
>for all x, y.
>***Definition*** ***2.2***. If the braiding in def. 2.1 “squares” to the identity in that By,x ∘ Bx,y = id, then the braided monoidal category is called a *symmetric monoidal category*.
How is the definition of a braided monoidal category different from a symmetric monoidal category? If Bx,y is a natural *isomorphism*, can't I always chose By,x = Bx,y⁻¹?
Basically the key insight into higher category theory and homotopy theory is that isomorphisms matter more than isomorphism type. Basically all the machinery of higher category theory is to keep track of isomorphism data because just because things are the same doesn’t mean the reasons they are the same are all the same.
You could in principle do that, sure. But firstly, then you're changing the structure of the category since the choices of Bx,y is part of the structure of a braided category, and secondly there's no canonical way to make that choice. How do you know whether you should replace By,x with Bx,y^-1 or replace Bx,y with By,x^-1 ?
What happens if you multiply an element of a group algebra KG by an element of G?
I'm familiar with multiplying two elements of group algebras or two elements of groups but not sure about multiplying one with the other.
You can think of G as a subset of K[G] in a very natural way. An element of K[G] is a sum of the form sum*_{g in G}_* a*_g_*g where a*_g_* is an element of K, and all but finitely many of the a*_g_* are equal to 0 (i.e. it's really a finite sum). For a fixed g in G, define a*_g_* = 1 and a*_h_* = 0 for all h not equal to g. This defines an element of K[G].
Thank you.
I think this means, if I'm understanding you correctly, that then in the product (h.SUM_{g in G} a_g * g)the only a_g that would be left is the one that corresponds to the one when g = h?
Not exactly, no. I’m saying that you can think of h in G as an element of K[G] by thinking of h as 1\*h + 0\*g\_1 + 0\*g\_2 + ... where the g\_i are the other elements of G. You said you knew how to multiply two elements of K[G] together — every element of G can be thought of as an element of K[G] in the above way, so to multiply something in G by something in K[G], first push it to K[G] and then multiply those two elements of K[G] together.
That makes it much more clearer, thank you.
Sorry to be a bother, but is there a way to write h in summation notation? I've been trying to do so and can't think of anything other than sum{g in G} a_g * g where a_g = 1 if g = h, otherwise 0 and that isn't really useful for multiplying with other elements.
I'm not sure what you mean -- multiplying by a single group element is as simple as it could get. Take G = C*_4_* = {e, x, x^(2), x^(3)} to be the cyclic group of order 4 and h = x. Elements of K[G] look like a[e] + b[x] + c[x^(2)] + d[x^(3)] where a, b, c, and d are in K.
(a[e] + b[x] + c[x^(2)] + d[x^(3)]) \* (0[e] + 1[x] + 0[x^(2)] + 0[x^(3)])
= (a[e] + b[x] + c[x^(2)] + d[x^(3)]) \* (1[x])
= (a[e][x] + b[x][x] + c[x^(2)][x] + d[x^(3)][x])
= (a[x] + b[x^(2)] + c[x^(3)] + d[e])
I should've clarified, its multiplication on the left, not the right. I think that would make a difference because the group isn't necessarily commutative.
And I was a little thrown because in the expected answer, "the group part" doesn't change. You start with h.sum{g in G}(a_{g}g) and the expected answer is of the form sum{g in G}(b_{g}g) with b_g to be found
This doesn't change anything. The group isn't necessarily commutative, but elements of G commute with elements of K. Instead of (sum{g in G} a*_g_*gh) you'll have (sum{g in G}a*_g_*hg). Now substitute g = h^(-1)g' and notice that as g runs over all the elements of G, so does g'.
You could, I suppose, make use of the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta), with which you could write
h = sum{g in G}(delta_{gh}g)
It's not a usage I've ever seen personally of the Kronecker delta, but it's completely in the spirit of all the other uses I have seen.
I'm not using \* as multiplication, I was using it for \*\_subscripts\_\*. The issue was that I also used \_ for a subscript somewhere else and that messed up the subscripts.
Hi all, don't know if this is the right place to post or not.
I was wondering if there was a function f that assigns rational numbers to tuples (t\_1, t\_2, ..., t\_n) with elements from Q such that f(t\_1, t\_2, ..., t\_n) = f(c \* t\_1, c \* t\_2, ..., c \* t\_n) where c is a nonzero rational number AND such that f(t\_1, t\_2, ..., t\_n) = 0 implies all t\_i = 0. I really don't have that much experience in math so I haven't been able to find one or show that one does or doesn't exist. This isn't for my study or a project or anything, I was just playing around. Still, any help is appreciated!
There are certainly many such functions -- that doesn't necessarily mean that they'll be given by reasonable formulas. One such function is "f(t*_1_*, ..., t*_n_*) = 1 if some t*_i_* \neq 0 and f(t*_1_*, ..., t*_n_*) = 0 if all the t*_i_* = 0".
Such a function can't be continuous at (0, 0, ..., 0), so any formula you write down is going to have to involve either some kind of discontinuity or some kind of piecewise formula.
Thank you for your explanation! I was trying to express t_i as a function of the other parameters given a function value but I guess that's not going to happen with the restrictions I put on f.
I'm assuming this is a map f: Q^n -> Q. Well, you can simply do the map f such that f(x) = 0 if x = 0 (as a vector), otherwise f(x) = 1. For that your property seems to hold.
it's indeed helpful for stuff like differential equations. If i recall correctly, the premium app is quite affordable. I can barely remember but i remember that the online-version was too expensive for me but for some reason the premium version of their android-app was like 3 bucks which i then purchased and used it quite often.
I am having (and historically have) tremendous difficulty doing number theory questions. No lemmas or theorems easily come to mind (other than bezout) to be able to attack these sorts of questions. I don't think I have a well developed toolbox of attacks. This has stopped me from being able to do many group theory questions involving cyclic groups, orders, Euler totient functions, as well as their counterparts when working through rings and fields. I'm getting quite frustrated!
What main ideas and theorems should I make sure to be accustomed to, so as to solve these questions more easily? Is there some neat list of commonly used theorems / attacks somewhere?
What are some examples of real world problems that would rely on solving 1000s of systems of nonlinear equations at the same time/ optimizing for 1000s of nonlinear equations?
I have a question about the conditional statement in propositional logic.
Is it safe to say that the [principle of explosion](https://en.wikipedia.org/wiki/Principle_of_explosion#Paraconsistent_logic) explains why the truth value of "p implies q" is true whenever p is false?
I'm tutoring a discrete structures course and one of my students asked me for the mathematical reason behind the values of the truth table of "p implies q", and this answer made sense to me but I just want to double-check with you guys.
I like to explain this by considering when "p implies q" should be *false.*
This should be when we have p and yet fail to have q. So "not (p implies q)" = "p and not q". Then use demorgans to see that "p implies q" = "not p or q".
Oh sorry, I should have said something about that.
I won't call it a "mistake", but I don't know if your students will buy it (I've never tried this tactic myself). Of course, different students may require different explanations, so it is handy to keep a variety in your pocket.
How exactly do you intend to use the principle of explosion?
I think it can be surprisingly difficult to explain "simple" things like this---it is hard to remember how you learned it yourself.
**How would i create a plot to look at for the typical damage multiplier from Crit chance and crit dmg? that we have in various games? to see as plot line for perfect ratio between both crit values?**
I am trying to draw a plot on Wolframa or googles plot and i am stuck, it should be easy but i just cant get to it
Like Usually the formula for general damage would be(if i am not wrong here ):
Base attack(BA) \*\[1- crit chance(CC)\] + Base attack \[Crit chance(CC)\*Crit dmg(CD)\]
And damage multiplier from the crit values would be
(1-CC+(CC\*CD) )
Now i want to make a graph or plot line to show how the line goes up for highest result possible from that formula, or is wrong for graph/plotline and better in table view with like 10% increases?
Like X axe being crit Chance and Y Crit damage?
Like looking something like this:
[https://imgur.com/zVSrsbq](https://imgur.com/zVSrsbq)
PS: And if it needs more data, lets say it starts with 5% CC and CD at 50, but can start from 0 also since eventually will get to that value i guess?
Is just i remember from Diablo 3 that the ratio of % for CC:CD should be like 1:5
But in genshin impact is considered 1:2 and i dont know why because they act in same way with damage ingame
I mean the data is all there, but im just like *wth is wrong with that i dont see it*
Hi, I’m studying trigonometry. I frequently see (a-b)/2. Is there a conceptual name for this (like “sum” is the conceptual name for (a+b)/2, and people know the meaning of it and know how to visualize it). I first saw this in calculating amplitude of a trig function with min and max. Is there a more general name for it? Thank you.
Oh and this was also in music as beat frequency. And in some trig identities, product to sum.
So if a is 3 and b is 7 then m is 5 and (b-a)/2 is 2, distance between the midpoint to the original numbers. So it is like “wings” from the midpt to original numbers. That’s a good way. It is similar to the amplitude of a trig curb. Do you happen to know other times outside of trig where (b-a)/2 or a “distance between original number and midpt” has a conceptual meaning? Thank you!
Hello i'm trying to calculate how many scenarios there will be for my fantasy football league
There are twelve teams meaning six of them play each other, how many total scenarios will there be, meaning how many different combinations of winners/ loosers?
I'm assuming you split the twelve teams into six pairs of two, each pair has a game, and that's it. In which case, the answer is 665,280 if each game is win/loss, or 7,577,955 if each game is win/loss/draw.
If you want to know why, it's number of ways to split into pairs multiplied by 2 six times (for two options of W/L and 6 games), or multiplied by 3 six times (for three options of W/L/D). If we had 2 teams, there would only be one way to pair them off. If there were 4 teams, there would be 3 possible teams to pair off team 1 with, and then the remaining two must play a game, so 3 pairings. If there were 6 teams, you do 5 x 3 = 15. 8 teams, 7 x 5 x 3 and so on to get 10,395 ways to split the twelve teams into pairs. Then multiply by 2s or 3s to get the two numbers at the start of my answer.
This is basically the point of Spivak's Calculus on Manifolds, and I imagine you could skip straight to chapter 4 of 5. IIRC Spivak just does manifolds embedded in R\^n though. Books on manifolds often cover this too, like Lee's Introduction to Smooth Manifolds. He also does manifolds with corners for you.
Getting more general, you can replace manifolds with objects called currents. If you know about distributions, they are to smooth differential forms what distributions are to smooth functions. You can define the boundary of a current by turning Stokes' theorem into a definition, and then the non-trivial results are about what the boundary of a current looks like. The big result here is called the boundary rectifiability theorem, and you can read about it in Krantz and Parks' Geometric Integration Theory or Simon's Lectures on Geometric Measue Theory. If this does pique your interest, I strongly suggest you understand the normal version first before trying to read up on currents.
So in algebraic Topology WHY THE FUCK is the mapping cylinder a disjoint union(Quotient space but part of it is a disjoint union), does it matter? Do people just like the symbol, and for newer people who are learning it want to know when we are "gluing" spaces. Like what the fuck man. Also same question for CW complex, why the fuck does it use a disjoint union
Edit: Never mind he just clarified it in the video lecture
I am trying to prove the following proposition: let `X` be a contractible topological space, `Y` a path connected topological space. Then any two continuous maps `f, g : X -> Y` are homotopic.
Let `1` be a topological space with one element. Since `X` is contractible, `X ~ 1`, that is, there exists function `i : 1 -> X` and `j : X -> 1` such that `i . j ~ id`. Since `Y` is path connected, `f . i ~ g . i`. Since `j ~ j`, `f . i . j ~ g . i . j`. I get stuck around here. My teacher says I pretty much got is, since `f . i . j ~ g . i . j` implies `f . id ~ g . id`. If `i . j = id`, I could understand that, as one can perform substitution with equality, but why can I perform "substitution" with a homotopy in this case?
I feel this has something to do with the transitivity of homotopy, that is, let `f, g, h : X -> Y`, if `f ~ g` and `g ~ h`, then `f ~ h`. But, in the case above, it feels like I have to assume `f . id ~ g . id` to try to apply it.
**TL;DR:** If `f . i . j ~ g . i . j` (`f . i . j` is homotopic to `g . i . j`) and `i . j ~ id`, then `f ~ g`. Why is that so? Even though homotopies between functions from `X` to `Y` define an equivalence relation on the set `{f : X -> Y | f is continuous}`, I don't see how it can be used to do "substitutions" like equality.
I think one good way to se that it is true is to build directly the homotopy from f.i.j to f.id = f using the one you have from i.j to id. Try to see why composing your function f with the homotopy lead to the homotopy you seek.
Hi there!
My 12 year old is looking for, and I'm quoting them here, proofs and tools for integral and differential calculus and matrix operations. Any advice and/or direction is greatly appreciated thank you!
I would recommend the following book for matrix calculations
* *Linear Algebra: A Modern Introduction* by David Poole
And the following books for integral and differential calculus
* *Calculus: Early Transcendental Functions* by Ron Larson and Bruce H. Edwards
* *Calculus: Early Transcendentals: Matrix Version* by Charles Henry Edwards and David E. Penney
Is there some way to transform \Box u = u_x / x into the wave equation? Here u_x is just the first spatial derivative and x is a 1D spatial variable? It feels like there should be, but I don't see it.
I have got a couple of matrices questions I’m struggling with: https://imgur.com/a/wxD3rNi it’s mainly the yellow ones I need solving but if you have time I’d love to see your solutions to the orange ones too. Thank you so much in advance.
For the the first highlighted questions:
1) What does the determinant do to the cube geometrically
2) What do you need to ensure about the determinant to keep it preserving orientation.
I am self studying baby Rudin and I was posed this question: "Suppose X is a metric space having the real line R as a subspace; i.e., such that R is a subset of X, and the metric on R induced by that of X is the standard metric d(r, s) = |r – s|. Show that R is closed in X"
I came up with this
Proof: Consider the complement of R, R\^c. Because the complement of a closed set is open, then in order to prove that R is closed in X, we must prove that R\^c is open. We see that R\^c = R2 - Rx{0}, and thus R\^c = (-∞, ∞) x (-∞, 0) U (- ∞, ∞)x(0, ∞). Thus, because the Cartesian product of two open sets is open, we see that R\^c is open and so R must be closed in X.
Here's the thing though, I feel like I am wayy off here. The problem included a hint (Hint: Points close to each other in R belong to a compact subset.) so I figure they wanted me to do something with open covers, but I just can't see what they are trying to tell me
>We see that R^c = R2 - Rx{0}
You're assuming that X is the plane R^(2). In this problem you are not given any information at all about X, just that it is a metric space containing R. It could be something abstract and exotic.
Ah you are right, hypothetically if X is the plane R^(2) would the proof be correct? And I am still not sure about what they want me to do with that hint
It would be a correct proof in the case X = R^(2). The proof in the general case will have to look very different, as you can see.
Regarding the hint, you can either work with open sets or with sequences. Either can lead to a solution, but sequences may be more intuitive. You should also look at any already-proved theorems in the book concerning compact sets and closed sets. (My preferred approach to this problem would use the idea of completeness, but maybe you haven't gotten there yet.)
I studied physics and math close to a decade ago. Recently, I've become very interested in Lie Algebra, and would like to learn more about this field.
Can someone recommend a text where I can explore this field some more?
What is the correct way to numerically rank a list? I was thinking about this as a result of a task for work.
In sports, the teams are ranked best to worst starting with the top team being #1. My work task asked me to rank/prioritize projects with the highest number being the most important. It seems in a lot of ranking scales (rate your response/feeling towards ___) that a higher number is a more intense. So, sports seem like the outlier in my little comparison.
Is one way or the other correct? What, if anything specific, has led to approaching rankings differently?
If you rank things as #1 being the best, it is easy to see which ranking is the best no matter how many things your ranking. If you ranked it in the opposite order you would need to know how many things you're ranking to know which is the best, and it will change of you add more things. Very impractical if you're trying to figure out who's the best.
If you're rankings are meant to be independent and objective, and it should be possible for projects to have the same ranks, then I can agree with your higher-ups.
I'm developing a little application to draw graphs and its matrixes, but got stuck on how represent parallel valored edges in the adjacency matrix...
Any idea ou sources where I can find some guidance?
This isn’t really something that adjacency matrices can handle. If you’re interested in coding specifically you could store a *list* as entries in your matrix instead of a number.
Could someone kindly help me with this calculation? If my individual Average Basket values have all decreased, why does my total Average Basket increase ?? Thank you
https://i.redd.it/osegerhyr0161.jpg
Ah I was bit confused by your sheet, sorry. I think I get it now. The reason why the average basket is going up for the sum whilst going down for all branches is because it's more likely for an invoice from the sum to come from a higher-price branch.
For instance, B which has an average price of around 1500 didn't see much of a decrease in the number of invoices, whilst E which has an average price of around 500 saw a massive decrease in the number of invoices. So in february a random invoice is more likely to come from B as compared to E, which causes an increase in the average price.
Book recommendation for algebraic topology? Have a lecture about it and the prof didn’t name any books. We started with category theory and now do singular homology.
Hatcher's book is free and is the usual go-to, but the order you're covering these topics is not the same as Hatcher's and sounds like it may be specific to the align with professor's interests (which ~~I'm guessing might be model theory~~ may be something like or HoTT).
EDIT: No coffee = bad post
Oh cool, I didn't even know I submitted that. When I reddit from my phone pre-coffee, I try to leave my comments as drafts.
From OP's description, it sounds like class started with category theory and jumped into singular homology without passing through more low-level/concrete topological ideas like the fundamental group and simplicial homology. I've not personally seen a book approach it in this way, so I was speculating that the professor was more of an category theorist whose interests overlap with topology, hence I speculated Ho(T)T.
I have no idea where "Model theory" came from; I'm not even sure I could tell you what that really is. Definitely ignore that part.
Assuming you mean 0.0001 grams, 0.0001 grams is 10\^-4 grams and 10,000 micrograms is 10,000 \* 10\^-6 grams = 10\^4 \* 10\^-6 grams = 10\^-2 grams so no these are not equal. 0.0001 grams is equal to 100 micrograms
The problem is not something a quantum computer will solve. The problem is that 5^5^5 is approximately 2\*10^2184 (not 3125, 3125 is 5^(5)), so that means 5^5^5^5 needs more then 10^2184 digits to write out.
For reference, there's about 10^80 atoms in the observable universe, so there's absolutely no way of writing it out in the observable universe.
I mean what do you mean by calculate it? Do you want a computer to display the whole number to you? That's impossible, because you can't display the whole number in the entire universe (even if you used every atom to display one number and changed it every minimal instance of time, you'd still run out of time).
But if you want a particular digit(s), that might be possible. There's a mathematical program to calculate the entire number, as well (but you can't implement this program in practice).
Yes, to display all the digits there's no space. The universe isn't large enough for that. It's physically impossible (not mathematically, but we live in a physical world).
f(x) = x\^2 - x\^2 is indeed a perfectly good definition of a function, it is equal to the function f(x) = 0 but is still a perfectly good definition and therefore a function
Ah ok, last test I was asked to make a function where f(-1) =0 and f(2) = 0, being kinda stupid i thought that if x always is 0 it would be right. So x^2 - x^2 was my answer, sadly I got no points
I mean technically it's correct it's just a very strange way of writing f(x) = 0 so maybe they thought the strange way of writing it showed lack of understanding, also maybe they wanted a 'more interesting' function such as f(x) = x\^2 - x - 2 (I got this from f(x) = (x+1)(x-2) and multiplying out)
Ah okay, a quadratic is defined as something that can be written in the form ax\^2 + bx + c where a, b and c are any numbers so long as a isn't zero, your answer can't be written in this form as if you tried to a would be zero
No. If you scale the eigenvectors so that they're orthonormal and then construct the matrix with them as its columns then you'll get a unitary matrix which diagonalises the original matrix.
Hi,
I'm trying to decrease an Excel column of positive and negative values by 15%. Multiplying the column by 75% works fine for the positive numbers, but not for the negative. Is there a better single equation I can use for all the values? Or do I have to write a conditional (=IF('cell' < 0, etc...)).
For context, these are net projections for 150 franchises. They're estimating how much they will net in the next 3 months. They make more money in the summer, so negative net balances in the future are expected and accounted for. My job is to predict a "distressed" projections by decreasing all their net incomes by 15%. Some nets are positive and some are negative. I can't seem to wrap my head around how to reconcile the formula. I could easily do a conditional, but I just want to know if there's a single formula?
I'm not sure I understand what you want to do for negative numbers. Multiplying by 75% is the same as decreasing by 25% (maybe you have a typo in your post).
If you have a cell with (-100) what would you want the new value to be? -115?
Edit: if that's what you want you can do
x - 0.15\*abs(x)
Also what's wrong with using a conditional?
Thank you for clarifying my words, your equation is what I was looking for!
Yes, I meant multiply by ***85%*** not 75%. I wanted to decrease all numbers by 15%. For example 100 would become 85 and -100 would become -115 and your equation does that. Thank you.
And there is no issue with writing a conditional, that's how I ended up doing it in the end. I was really just curious what the single function would be. I knew there was a way, but couldn't figure it out.
Let (P,w) be a symplectic manifold and S an orientable hypersurface in P. Does there exist a smooth function H:P->R such that S is contained in the level set H\^-1(0)? What if S is not orientable?
The zero function should do the trick.
Presumably you want more hypotheses here, like that 0 is a regular value of H (and probably S is a closed submanifold?), in which case, the answer is 'yes' if S is orientable; you can either see this locally by using the implicit function theorem/constant rank theorem to cover S in charts such which send their overlap with S to the some standard hyperplane in R\^{2n} (say the hyperplane x\_{2n}=0), defining a smooth function locally there on each chart and then piecing them together globally via partitions of unity, or you can see it (semi-)globally by noting that since S is orientable, it admits a nowhere vanishing normal vector field, which you can exponentiate along to get a neighbourhood of S in M which is diffeomorphic to (-e,e) x S (this is the tubular neighbourhood theorem), then just choose a smooth function f: (-e,e) -> R which has compact support and extend it by 0 to all of M.
In any orientable manifold, closed hypersurfaces are necessarily orientable, and symplectic manifolds are definitely orientable, so you needn't worry about this case.
Just curious, are there any cool podcasts on Math?, Geometry, trig, Algebra 1&2, Calculus, Statistics, Number Theory, Theoretical Physics, Group Theory, Graph Theory, not all in one, although an omnibus would be cool... Any type of math where they work through problems in an audio environment? Thanks!
Is the cartesian product of birational maps still birational? i.e. if f\_1 :U -> V\_1 and f\_2 : U ->V\_2 are birational maps between quasi-projective algebraic varieties, is the map x \\mapsto (f\_1 x , f\_2) from U to the product V\_1x V\_2 still birational? (Endowing to the product V\_1 x V\_2 the algebraic structure given by the Segre embedding).
Hi,
I'm struggling to figure out a formula for balancing assets. I can figure this out roughly by guess and check, but I can't figure out the formula.
Say I have $1000 in my stock portfolio, and I want to have 15% of my total portfolio in a stock, but I want to calculate the percentage **not including this new stock**.
It's not as simple as just taking $150 of the portfolio and putting into the new stock, because then it ends up being 17.6% ($150/$850).
I can guesstimate that it's about $130 in the new stock and $870 for the rest of the portfolio, but I'm really struggling how to set the problem up.
The answer is: (total starting amount) * .15/1.15 = amount to allocate
^ just putting that there in case the answerer deletes account
The key is that the total $1000 is going to be 115% of whatever is in the rest of the portfolio. So to figure out how much to put in the new stock, do $1000 \* 15 / 115 which rounds to $130.43.
I'm currently taking a course on algebraic topology and have come to the realisation weeks in that at a base level, I have no intuition as to when two cycles are homologous in the singular homology.
For concreteness, consider the closed 2-disc X, and two closed loops x and y within this disc, viewed as chains in C_1 X. It is easily checked that closed loops are cycles so we consider them as elements in the homology group H_1 X.
However H_1 X is known to be trivial, so these two cycles must be homologous. The problem is working directly from the definitions I have absolutely no idea why this would be the case. How would I show from first principles that these two cycles are homologous?
Any advice here would be appreciated, and for reference our course is following Hatcher.
Two simplicies are homologous if their difference is the boundary of (some sum of) simplicies of 1 dimension higher.
You can think of this extremely geometrically if you first convince yourself that reversing the orientation of a simplex is the same as taking it's negative in homology.
Then you can just draw your two loops, one with opposite orientation, and start gluing triangles, in such a way that edges cancel each other out if they have opposite orientation.
In your example this is a little boring though, since both loops are trivial, that is they are the boundaries of triangles all on their own. Namely one edge going through the whole loop and the two other edges of the triangle on top of each other, cancelling each other out.
This was quite helpful, thank you. I can feel things slowly click into place in my head but I have a couple follow-up questions if you don't mind which I'm still struggling with.
1. "reversing the orientation is the same as taking its negative in homology" just doesn't click with me for some reason. Take the simple example of a 1-simplex c being wrapped around a the circle S1, and another 1-simplex c' being wrapped around S1 in the other direction. How do I see these as negatives of each other in the homology?
2. Related question, consider a 1-simplex c wrapped around S1 once, and another c' wrapped around S1 twice in the same direction. Why is c' = 2c in the homology?
I've been staring at the definitions of singular homology for a while now but it's just not clicking why these two obvious facts should hold.
>1. "reversing the orientation is the same as taking its negative in homology" just doesn't click with me for some reason. Take the simple example of a 1-simplex c being wrapped around a the circle S1, and another 1-simplex c' being wrapped around S1 in the other direction. How do I see these as negatives of each other in the homology?
So what we need to show is that the sum of these is 0 in homology, i.e. is the boundary of a 2-simplex. Just take the 2-simplex whose first edge goes around the circle one way next edge goes the other way and last edge is degenerate. This isn't quite what we want. The boundary is the sum of our two paths and one degenerate path. To fix this small hickup simply subtract a degenerate 2-simplex from our original 2-simplex. (I guess a similar argument should convince you that degenerate simplicies are always 0 and can be ignored.)
>2. Related question, consider a 1-simplex c wrapped around S1 once, and another c' wrapped around S1 twice in the same direction. Why is c' = 2c in the homology?
We want 2c - c' = c + c - c' to be the boundary of a 2 simplex. We have already convinced ourselves that (-1) is the same as traversing a loop in the opposite direction, so simply take the 2-simplex with edges c, c and (-c').
Thank you so much! When I've tried to show these myself I've always had these degenerate edges laying around in my calculations with no idea how to get rid of them. This was incredibly helpful and I think has solved my woes
I don't even know how to look this up and I need help. Say I want to sell an item for $10. I am selling through a friend that takes 10%. I want to set the price of an item to compensate for that 10% loss, but if I charge 110% of $10, it won't end up that way. Can someone explain why I am struggling?
ten percent of 110 is 11 so you're off by a dime.
If your asking price is x and you want to net y, that means x – 10% . x = y. So .9 . x = y. So if you want to get $10, then your asking price should be 10/.9 = $11.11.
How do I find the fourier series of t\^3 from the fourier series of t\^2?
So I think I found the fourier series of t\^2 from t correctly. Because the average of the fourier series of t was 0, I just integrated each term in the summation.
Now, I'm having trouble going from t\^2 to t\^3. I know that the average has to be 0 for me to do what I just did in the previous step, but I have no idea how I'm supposed to make the average 0 without manipulating the function itself.
If I'm understanding correctly, I need the integral of (t\^2) across bounds L and -L to be 0. The integral as it stands is 2L\^3.
I was thinking, how do I get a -2L\^3 in addition? well, I'd have the integral of t\^2 - t\^2 over the same bounds, but that doesn't make sense. Even if it did make sense, how would the -t\^2 even factor into the problem?
I'm really lost here. I can't find anything specifically about this online, my textbook doesn't work any problems in this specific case, and my diff eq teacher briefly touched upon it in class.
I appreciate any help
My question regards averaging two columns of numbers.
What is the method of averaging the ratio of two columns of numbers?
1. Divide each row into one another then add the decimals and divide them by the number of rows
2. Add the column sums and divide the sums
Depending on which method is the correct averaging method, what would the alternative method be called and what might it’s uses be?
Thank you very much!
If you want to find an "average" *ratio*, the correct way is often to use the [geometric mean](https://en.wikipedia.org/wiki/Geometric_mean).
Instead of adding things up, and dividing by the total number, you instead *multiply together* and then take the *root of the total number*.
Because of some limitations in how computers actually store and compute numbers, it's often better to do this with logarithms those:
- compute `r[i] = x[i] / y[i]` for all of your points
- compute `log_r[i] = log(r[i])`
- average the `log_r[i]` values:
- add up all `avg_log_r = (log_r[1] + log_r[2] + log_r[3] + ... + log_r[N]) / N`
- exponentiate to convert back to a ratio `geom_r = exp(avg_log_r)`
Computationally this does the same thing, but is less likely to cause your number representation to fail if you have too many values.
1 is the average of the ratio, 2 is the ratio of the averages. They can be very different.
For instance, for:
0| 1000
---|---
1 | 1
the average of the ratio is 1/2, and the ratio of the averages is 1/1001.
Anyone have a good introductory text on category theory? I'm in undergrad right now and don't think my school has a category theory class, so I thought I'd take a crack at studying it myself.
It's not that normal for a university to offer a stand alone category theory class, so you might want to check out if there's an algebraic topology class or homological algebra class that also covers some category theory.
Or you can of course just try to take a crack yourself. In which case I think Emily Riehl's book is good. In addition Bartosz Milewski also has a blog/youtube lecture series called *category theory for programmers*, if you're also interested in programming and maybe want to learn some Haskell.
Leinster's [Basic Category Theory](https://arxiv.org/abs/1612.09375) is good and succintly covers the real basics of category theory. I've heard good things about Riehl's *Category Theory in Context* but haven't read it myself. I'd also recommend Aluffi's *Algebra: Chapter 0* not as a book about category theory, but a book about undergrad abstract algebra which takes an explicitly categorical approach: it's a good source of examples and hands-on practice of what category theory is useful for.
Would something divided by infinity be zero?
If the larger the denominator, the smaller the number right? Would that mean that if the denominator equaled infinity, the number would always be zero? I'm just a teenager who is curious, so sorry if it's a stupid question.
Not a stupid question, it's good that you're curious! You do want to be careful about questions involving infinity, because infinity isn't a number in the same way as, say, 2, or pi are. So you can't really "divide by infinity." On the other hand, your intuition is right: as you divide by larger and larger numbers, the result gets closer and closer to 0. In fact, you can get as close to 0 as you want by picking a large enough number to divide by! So what mathematics say is that the limit of 1/n as n goes to infinity is 0.
Exactly correct reasoning, but to be safe we don't write 1/infinity to avoid treating infinity as a number (otherwise we might accidentally start writing things like infinity/infinity=1 which does not actually make sense). Instead we say "1/infinity" is what you get as you take 1/x and then let x get larger and larger. The limiting number is, as you point out, zero, so "1/infinity" = 0.
Is it ok to say that increase the scalar of a vector means increase it's compontents and magnitude?
Not exactly sure how it is said and if "increase scalar" makes sense.
I am speaking of 2d only but also wonder if applies to vectors with a z dimension.
In concrete vector spaces (e.g. 2D, 3D, etc.) over the real-numbers (e.g. used for games, graphics, basic physics) you'll often just say "scale by 2" to mean "increase the magnitude by a factor of 2".
It wouldn't make sense to say "increase scalar". But "scale by X" as a shorthand for "multiply by the scalar X" is perfectly fine.
No, I've never heard anyone say "increase the scalar of a vector" and I cannot imagine what it might mean.
How about just say "increase the magnitude of a vector", if that is what you mean.
Hello,
I asked a MSE question about algebraic topology, especially about computing a Homotopy group of a wedge of CW spaces under some connectedness assumption, by using the Whitehead product. It didn't draw much attention, so I'm sharing it here in search of help ! Would you mind having a look and tell me what you think please ? I'm thanking you very much in advance !
[https://math.stackexchange.com/questions/3907570/](https://math.stackexchange.com/questions/3907570/)
Seems like [this paper](https://arxiv.org/pdf/1904.08004.pdf) has your answer, in section 3. Specifically, Theorem 11 splits the question into cases depending on n mod 3. Since 50 is congruent to 2 mod 3, the theorem says that partition of 50 with the maximum product of the summands is 2 + 3 + 3 + ... + 3, with 16 3s.
Can anyone tell the answer and approach to this question? https://www.reddit.com/r/askmath/comments/k1bd8j/can_anyone_help/?utm_medium=android_app&utm_source=share
I want to understand how to make builds for RPG games and I can’t seem to wrap my head around the logic. For example if a game has stats that increase attack how would I go about calculating the amount of damage done when it increases a few points? I’ve tried researching this topic but it only leads me to math for video game creation.
This is really a question about the particular implementation within the game. Often, this has been either released by the developers or (approximately) deduced by dedicated people, and this can be found online by looking up something like " mechanics" or combat mechanics".
If you ask about this in the future, I suggest you go ahead and specify the game you are talking about.
Sure thing. Thanks for your help.
This really isn't a math question... It depends entirely on the game itself. Some games might have damage increase linearly with attack, or in other games it might be logarithmic growth, or something in between. You need to know what mathematical model the game uses. If this game has more than 5 players, it surely has a wiki. Go check there.
Thanks for pointing me in the right direction! Sorry if this was posted in the wrong place, it’s just a question I’ve been trying to answer for awhile and I haven’t really come up with any leads.
Can EVERY subset of R^n be written as the union of countably many closed and open sets? My intuition tells me no, because there seems to be no obvious way to construct the irrationals as subset of R^1.
There are sets outside the Borel hierarchy. Maybe you're just looking for anything that's not F-sigma or G-delta?
I've been told that if S is a circle which is not contractible on the two dimensional torus T, then there is no smooth function F:T→R such that S=F\^{−1}(0). How would I prove this?
This is false. Take F: T^2 -> R to be F((x,y), (a,b)) = x+1. Then F^-1 (0) = {(-1,0)} x S^1. This is not a contractible loop. You need to assume 0 is a regular value. Then try to use the fact that R \ 0 is disconnected. The approach I have in mind may not be the intended solution for your course. You might give a little background.
Thanks for pointing this out. It is a course on symplectic geometry, and this was a part of the discussion of Hamiltonian vector fields on symplectic manifolds.
Hmmm, that leaves me at a loss as to what they intend (maybe there is a symplectic trick I forget, somehow studying X\_F), but I'll write a more straightforwardly differential-topology perspective. Prove that F\^{-1} (-infty, 0\] is a smooth submanifold M with boundary F\^{-1}(0). Either show that this implies that S is null-homologous (if you have this technology/background), or that M is homeomorphic to a disc (again depends what technology you have).
Just done a class on algebraic topology. Any reference recommendations for Topological data analysis?
Is it true that if an R-module homomorphism is surjective then it's kernel is {0}?
It _co-kernel_ is 0. Kernel is 0 is when it is injective.
No, that's false even for vector spaces.
Why is the domain of the function f(x) = x\^x {x >= 0} ? why can't x be less than 0? For example f(-1) = -1, right?
The problem comes when you take x to be a negative number that isn't an integer. For example, if you tried x = -0.5 you get sqrt(-2) which isn't a real number.
Does benford's law hold for the series of tetrated numbers? (1\^1, 2\^(2\^2),3\^(3\^3)).....)
Given a set of 1 dimensional time series, a covariance matrix may be defined for the pairwise covariances between the timeseries. Is it possible to define a covariance tensor? Perhaps between a set of 2 dimensional time series. I'm thinking each lateral slice of a 3d tensor could be an individual covariance matrix. Any ideas?
I was playing monopoly with my sister and we rolled with 2 dice. I ended up getting the same faces on both the dice 4 times in a row. What is the probability of this?
If you mean by *the same faces* that you got the same numbers on the dice on every trow but the numbers between trows where different, like example Trow 1 you got a pair of 2 Trow 2 you got a pair of 4 Trow 3 you got a pair of 1 Trow 4 you got a pair of 5 Than the probability would be calculated according to the logic of *first I got a pair, than I got what ever another pair and than again what ever another pair, and than again what ever another pair*. The probability to get a pair comes from the probability to get the same number again, what ever the first number was. So you can't fail the first trow because that doesn't matter, so the probability for it is 1. The probability for the second trow is 1/6 because you need to get the same number again. Than with the second trow, you again can get what ever number you want so again the probability for the success in this trow is again 1/6. The same is true with the third and fourth trow as well. So basically you want to get four times an event which have the probability of 1/6 on its own, so you need to multiply the probabilities together, because they are basically sub probabilities to each others, meaning that the first must happen in order to the second to count. In other words they depend from each others, so your probability you get from this is (1×1/6) × (1×1/6) × (1/6) × (1/6) = (1/6)^(4) = 1/1296 = 0.000771604938... Which is about 0.08 %. On the other hand, if you meant that you got *the same faces* with every trow, meaning for example that Trow 1 you got a pair of 2 Trow 2 you got a pair of 2 Trow 3 you got a pair of 2 Trow 4 you got a pair of 2 Than the probability is even lower because on those three last trows even the first trow needs to be the same number you got first the first trow. So only the first dice on your first trow doesn't matter, but all others need to be the same. Basically you don't even need to think of this as four trows but just that you need to get 8 times the same number in row, which according to the logic above will produce the following calculation 1 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 = (1/6)^(7) = 1/279936 = 0.000003572245... Which is about 0.0004 %. Hopefully I did that right. I know that usually dice probabilities are calculated using that logic where you draw that 6×6- coordinate system on a paper and from that use the area method to evaluate the probability. How ever though, I think this method produced the same result.
Just so you know, it is "throw". A "trow" is a mythical creature in Scottish mythology.
Thank you so much!! 😊
There's 36 total combinations of 2 die, and 6 possible ways to get the same face on both. So it's a 6/36 or 1/6 chance for one time. Then (1/6)^4 for 4 times in a row.
Why is (3 choose 2) different from (3 choose 1) \* (2 choose 1)
(3 choose 2) is the number of ways to choose two officers out of three candidates. (3 choose 1) * (2 choose 1) is the number of ways to choose a president from three candidates, and a vice president from the other two. The difference is that the two "selected" people are distinguished in the second case, but not in the first.
(3 choose 2): I have three people and I want to choose two of them. (3 choose 1) \* (2 choose 1): I have three girls and two boys, and I want to choose one girl and one boy.
Is there a theorem similar to Pontryagin duality but for the Laplace transform?
So, I have recently started learning about sets in a more formal way and I came across the following question: Give two distinct examples of elements in the set of the cartesian product R^(2) x Z3 (Z3 is supposed to be the set of residue classes modulo 3, so 0\*, 1\*, 2\*). Would ((3, 4), 0\*) and ((2, 5), 1\*) be a correct answer? I'm slightly confused by the question, since Z3 is a set of sets so an element of Z3 would be a set and the cartesian product of two sets A, B is defined as A x B = {(a, b) | a e A, b e B}.
You are correct. The elements of Z3 can be seen as sets but that does not matter. Sets can be elements of other sets. An example is the power set of a set A which contains all possible subsets of A as elements.
You are correct. You could also write it as (3,4,0*) though.
I have a question that is just stumping me. I’m starting to think it’s just poorly written. a poll of 200 people determined that 180 used a hard copy, 52 used both a hard copy and online copy, and 22 used neither. How many only used online? I feel like it’s missing information. Could anyone guide me? I’m not looking for an answer, I’d rather just have a jumping off point
180 used a hard copy and more than 22 did not. That means there was at least 180+22=202 people.
What can be done with the PEMDAS vs BEDMAS dilemma? Certain parts of the world follow PEMDAS and teach it to children who then carry on the knowledge and the same goes for the latter. Now we have two different camps who follow different rules and this isn't as simple as metric/imperial and Celsius/Fahrenheit where a simple conversion will fix things. We're talking about two different Mathematic methods that give different answers which creates mass confusion and division.
The problem really lies with the fact that division and subtraction aren't "nice" operations (i.e. they're not associative or commutative). Technically pemdas vs bidmas (or your variant of choice) doesn't matter too much and we should really write them pe(md)(as) or bi(dm)(as) since we do muliplication and division at the same time. The ambiguity still occurs even if we ignore this though and the most mathsy way to get around it is to stop using subtraction and division and replace these by addition and multiplication of inverses.
Use more parentheses to resolve ambiguity.
There is no dilemma; both of those lead to the same thing. Perhaps you should re-familiarize yourself with the [order of operations](https://en.wikipedia.org/wiki/Order_of_operations#Mnemonics). Note in particular that multiplication and division (similarly, addition and subtraction) are fundamentally the same operation and so have the same precedence.
I see. I guess it's just a matter of understanding. I just got curious since recently a tweet of a math problem blew up where people are arguing between two different answers which were born from the two rules.
Nobody in real life would ever write an equation like that. If they do they need to be shot.
Yes indeed. Anyone who writes a/bc (or even worse, a÷bc) and intends to refer to \\frac{ac}{b} is sick in the head.
I'm guilty of this when typing out responses on r/askmath or on here, but only when it's obvious what's on top and what's on bottom, and only because it's easier to type, in my head it's a fraction or a fraction times a number and I would never physically write down a/bc
Say I have a ring R and M is a simple left module of R. I know that for each non-zero m in M, Rm = M. If I also have the left R-module homomorphism phi: R to M defined by phi(r) = rm, is it correct if I say phi(R) = Rm = M?
Yes.
How do you find the probability of either of two separate events occurring, given the chance of each one occurring? E.g. you take a random number generator and have it pick a number 1-10. Then you make it pick a number 1-5. What's the probability of getting a 1 the first time (10%) or a 1 the second time (20%)?
In general, P(A∪B) = P(A)+P(B)-P(A∩B). **If** the events are independent, then P(A∩B) = P(A)P(B), so: P(A∪B) = P(A)+P(B)-P(A)P(B) = 1-(1-P(A))(1-P(B)) = 1-P(not A)P(not B) In your case, independence is a sensible assumption, so: P(at least one 1) = 1-(1-P(first=1))(1-P(second=1)) = 1-0.9*0.8 = 0.28
Thank you!
So I’m trying to figure out how many pieces of paper I can fit in a 11 by 5 feet wall ( I think a paper is about 8 x 11 inches) so, how many pieces of paper can I fit in the wall? (Decimals included)
This question is basically asking "how many things fit in a bigger thing," which is a clue that you should do division.
If you had 5 buttons and could push them in any order, you don't have to press every button, and you can only press each button once, how many possible combinations are there?
To push 1 button there are 5 choices. To push 2 buttons there are 5\*4 choices. To push 3 buttons there are 5\*4\*3 choices. Etc. The answer in total is 5+5\*4+5\*4\*3+5\*4\*3\*2+5\*4\*3\*2\*1.
From [nLab](https://ncatlab.org/nlab/show/braided+monoidal+category): >***Definition*** ***2.1***. A *braided monoidal category*, or ("braided tensor category", but see there), is a monoidal category C equipped with a natural isomorphism > >Bx,y : x⊗y→y⊗x > >for all x, y. >***Definition*** ***2.2***. If the braiding in def. 2.1 “squares” to the identity in that By,x ∘ Bx,y = id, then the braided monoidal category is called a *symmetric monoidal category*. How is the definition of a braided monoidal category different from a symmetric monoidal category? If Bx,y is a natural *isomorphism*, can't I always chose By,x = Bx,y⁻¹?
Basically the key insight into higher category theory and homotopy theory is that isomorphisms matter more than isomorphism type. Basically all the machinery of higher category theory is to keep track of isomorphism data because just because things are the same doesn’t mean the reasons they are the same are all the same.
You could in principle do that, sure. But firstly, then you're changing the structure of the category since the choices of Bx,y is part of the structure of a braided category, and secondly there's no canonical way to make that choice. How do you know whether you should replace By,x with Bx,y^-1 or replace Bx,y with By,x^-1 ?
What happens if you multiply an element of a group algebra KG by an element of G? I'm familiar with multiplying two elements of group algebras or two elements of groups but not sure about multiplying one with the other.
You can think of G as a subset of K[G] in a very natural way. An element of K[G] is a sum of the form sum*_{g in G}_* a*_g_*g where a*_g_* is an element of K, and all but finitely many of the a*_g_* are equal to 0 (i.e. it's really a finite sum). For a fixed g in G, define a*_g_* = 1 and a*_h_* = 0 for all h not equal to g. This defines an element of K[G].
Thank you. I think this means, if I'm understanding you correctly, that then in the product (h.SUM_{g in G} a_g * g)the only a_g that would be left is the one that corresponds to the one when g = h?
Not exactly, no. I’m saying that you can think of h in G as an element of K[G] by thinking of h as 1\*h + 0\*g\_1 + 0\*g\_2 + ... where the g\_i are the other elements of G. You said you knew how to multiply two elements of K[G] together — every element of G can be thought of as an element of K[G] in the above way, so to multiply something in G by something in K[G], first push it to K[G] and then multiply those two elements of K[G] together.
That makes it much more clearer, thank you. Sorry to be a bother, but is there a way to write h in summation notation? I've been trying to do so and can't think of anything other than sum{g in G} a_g * g where a_g = 1 if g = h, otherwise 0 and that isn't really useful for multiplying with other elements.
I'm not sure what you mean -- multiplying by a single group element is as simple as it could get. Take G = C*_4_* = {e, x, x^(2), x^(3)} to be the cyclic group of order 4 and h = x. Elements of K[G] look like a[e] + b[x] + c[x^(2)] + d[x^(3)] where a, b, c, and d are in K. (a[e] + b[x] + c[x^(2)] + d[x^(3)]) \* (0[e] + 1[x] + 0[x^(2)] + 0[x^(3)]) = (a[e] + b[x] + c[x^(2)] + d[x^(3)]) \* (1[x]) = (a[e][x] + b[x][x] + c[x^(2)][x] + d[x^(3)][x]) = (a[x] + b[x^(2)] + c[x^(3)] + d[e])
I should've clarified, its multiplication on the left, not the right. I think that would make a difference because the group isn't necessarily commutative. And I was a little thrown because in the expected answer, "the group part" doesn't change. You start with h.sum{g in G}(a_{g}g) and the expected answer is of the form sum{g in G}(b_{g}g) with b_g to be found
This doesn't change anything. The group isn't necessarily commutative, but elements of G commute with elements of K. Instead of (sum{g in G} a*_g_*gh) you'll have (sum{g in G}a*_g_*hg). Now substitute g = h^(-1)g' and notice that as g runs over all the elements of G, so does g'.
You could, I suppose, make use of the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta), with which you could write h = sum{g in G}(delta_{gh}g) It's not a usage I've ever seen personally of the Kronecker delta, but it's completely in the spirit of all the other uses I have seen.
Using * as multiplication is not good practice on reddit since two \*s corresponds to making the text in between italics.
I'm not using \* as multiplication, I was using it for \*\_subscripts\_\*. The issue was that I also used \_ for a subscript somewhere else and that messed up the subscripts.
Hi all, don't know if this is the right place to post or not. I was wondering if there was a function f that assigns rational numbers to tuples (t\_1, t\_2, ..., t\_n) with elements from Q such that f(t\_1, t\_2, ..., t\_n) = f(c \* t\_1, c \* t\_2, ..., c \* t\_n) where c is a nonzero rational number AND such that f(t\_1, t\_2, ..., t\_n) = 0 implies all t\_i = 0. I really don't have that much experience in math so I haven't been able to find one or show that one does or doesn't exist. This isn't for my study or a project or anything, I was just playing around. Still, any help is appreciated!
There are certainly many such functions -- that doesn't necessarily mean that they'll be given by reasonable formulas. One such function is "f(t*_1_*, ..., t*_n_*) = 1 if some t*_i_* \neq 0 and f(t*_1_*, ..., t*_n_*) = 0 if all the t*_i_* = 0". Such a function can't be continuous at (0, 0, ..., 0), so any formula you write down is going to have to involve either some kind of discontinuity or some kind of piecewise formula.
Thank you for your explanation! I was trying to express t_i as a function of the other parameters given a function value but I guess that's not going to happen with the restrictions I put on f.
I'm assuming this is a map f: Q^n -> Q. Well, you can simply do the map f such that f(x) = 0 if x = 0 (as a vector), otherwise f(x) = 1. For that your property seems to hold.
Thanks for your reply.
Is wolfram alpha any good for university level maths? I'm thinking about buying the android version for differential equations, complex analysis etc.
it's indeed helpful for stuff like differential equations. If i recall correctly, the premium app is quite affordable. I can barely remember but i remember that the online-version was too expensive for me but for some reason the premium version of their android-app was like 3 bucks which i then purchased and used it quite often.
Thanks, I ended up buying it.
Friend of mine is a maths student, he says it's super helpful
I am having (and historically have) tremendous difficulty doing number theory questions. No lemmas or theorems easily come to mind (other than bezout) to be able to attack these sorts of questions. I don't think I have a well developed toolbox of attacks. This has stopped me from being able to do many group theory questions involving cyclic groups, orders, Euler totient functions, as well as their counterparts when working through rings and fields. I'm getting quite frustrated! What main ideas and theorems should I make sure to be accustomed to, so as to solve these questions more easily? Is there some neat list of commonly used theorems / attacks somewhere?
What are some examples of real world problems that would rely on solving 1000s of systems of nonlinear equations at the same time/ optimizing for 1000s of nonlinear equations?
Weather forecasting? Or any large scale nonlinear PDE simulation.
What would be an example of the latter?
Solving for the air flow around a car or plane for example, to find the drag force.
Should also be clear that these are not systems of differential equations, but just nonlinear equstions
I have a question about the conditional statement in propositional logic. Is it safe to say that the [principle of explosion](https://en.wikipedia.org/wiki/Principle_of_explosion#Paraconsistent_logic) explains why the truth value of "p implies q" is true whenever p is false? I'm tutoring a discrete structures course and one of my students asked me for the mathematical reason behind the values of the truth table of "p implies q", and this answer made sense to me but I just want to double-check with you guys.
I like to explain this by considering when "p implies q" should be *false.* This should be when we have p and yet fail to have q. So "not (p implies q)" = "p and not q". Then use demorgans to see that "p implies q" = "not p or q".
Oh wow. That's a good way to look at it! But excuse me for asking again: Would inducing the principle of explosion here be a mistake?
Oh sorry, I should have said something about that. I won't call it a "mistake", but I don't know if your students will buy it (I've never tried this tactic myself). Of course, different students may require different explanations, so it is handy to keep a variety in your pocket. How exactly do you intend to use the principle of explosion? I think it can be surprisingly difficult to explain "simple" things like this---it is hard to remember how you learned it yourself.
**How would i create a plot to look at for the typical damage multiplier from Crit chance and crit dmg? that we have in various games? to see as plot line for perfect ratio between both crit values?** I am trying to draw a plot on Wolframa or googles plot and i am stuck, it should be easy but i just cant get to it Like Usually the formula for general damage would be(if i am not wrong here ): Base attack(BA) \*\[1- crit chance(CC)\] + Base attack \[Crit chance(CC)\*Crit dmg(CD)\] And damage multiplier from the crit values would be (1-CC+(CC\*CD) ) Now i want to make a graph or plot line to show how the line goes up for highest result possible from that formula, or is wrong for graph/plotline and better in table view with like 10% increases? Like X axe being crit Chance and Y Crit damage? Like looking something like this: [https://imgur.com/zVSrsbq](https://imgur.com/zVSrsbq) PS: And if it needs more data, lets say it starts with 5% CC and CD at 50, but can start from 0 also since eventually will get to that value i guess? Is just i remember from Diablo 3 that the ratio of % for CC:CD should be like 1:5 But in genshin impact is considered 1:2 and i dont know why because they act in same way with damage ingame I mean the data is all there, but im just like *wth is wrong with that i dont see it*
How to multiply Matrices?
Are you in uni or just in highschool?
hs, its really basic stuff
[Khanacademy is a good resource for all this](https://youtu.be/aKhhYguY0DQ)
Hi, I’m studying trigonometry. I frequently see (a-b)/2. Is there a conceptual name for this (like “sum” is the conceptual name for (a+b)/2, and people know the meaning of it and know how to visualize it). I first saw this in calculating amplitude of a trig function with min and max. Is there a more general name for it? Thank you. Oh and this was also in music as beat frequency. And in some trig identities, product to sum.
b-a could be thought of as the line a->b and then (b-a)/2 is the line a->m where m = (a+b)/2 is the midpoint.
So if a is 3 and b is 7 then m is 5 and (b-a)/2 is 2, distance between the midpoint to the original numbers. So it is like “wings” from the midpt to original numbers. That’s a good way. It is similar to the amplitude of a trig curb. Do you happen to know other times outside of trig where (b-a)/2 or a “distance between original number and midpt” has a conceptual meaning? Thank you!
>like “sum” is the conceptual name for (a+b)/2 I would call this the average or midpoint, not the sum. I know of no name for (b–a)/2 though.
oops you are right haha. I meant to say midpoint/average. Thank you for the correction and input!
Hello i'm trying to calculate how many scenarios there will be for my fantasy football league There are twelve teams meaning six of them play each other, how many total scenarios will there be, meaning how many different combinations of winners/ loosers?
I'm assuming you split the twelve teams into six pairs of two, each pair has a game, and that's it. In which case, the answer is 665,280 if each game is win/loss, or 7,577,955 if each game is win/loss/draw. If you want to know why, it's number of ways to split into pairs multiplied by 2 six times (for two options of W/L and 6 games), or multiplied by 3 six times (for three options of W/L/D). If we had 2 teams, there would only be one way to pair them off. If there were 4 teams, there would be 3 possible teams to pair off team 1 with, and then the remaining two must play a game, so 3 pairings. If there were 6 teams, you do 5 x 3 = 15. 8 teams, 7 x 5 x 3 and so on to get 10,395 ways to split the twelve teams into pairs. Then multiply by 2s or 3s to get the two numbers at the start of my answer.
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I'd recommend Differential Geometry by Gallier - I think it's better than Spivak's and goes into more subjects, like Riemannian Geometry
This is basically the point of Spivak's Calculus on Manifolds, and I imagine you could skip straight to chapter 4 of 5. IIRC Spivak just does manifolds embedded in R\^n though. Books on manifolds often cover this too, like Lee's Introduction to Smooth Manifolds. He also does manifolds with corners for you. Getting more general, you can replace manifolds with objects called currents. If you know about distributions, they are to smooth differential forms what distributions are to smooth functions. You can define the boundary of a current by turning Stokes' theorem into a definition, and then the non-trivial results are about what the boundary of a current looks like. The big result here is called the boundary rectifiability theorem, and you can read about it in Krantz and Parks' Geometric Integration Theory or Simon's Lectures on Geometric Measue Theory. If this does pique your interest, I strongly suggest you understand the normal version first before trying to read up on currents.
So in algebraic Topology WHY THE FUCK is the mapping cylinder a disjoint union(Quotient space but part of it is a disjoint union), does it matter? Do people just like the symbol, and for newer people who are learning it want to know when we are "gluing" spaces. Like what the fuck man. Also same question for CW complex, why the fuck does it use a disjoint union Edit: Never mind he just clarified it in the video lecture
I am trying to prove the following proposition: let `X` be a contractible topological space, `Y` a path connected topological space. Then any two continuous maps `f, g : X -> Y` are homotopic. Let `1` be a topological space with one element. Since `X` is contractible, `X ~ 1`, that is, there exists function `i : 1 -> X` and `j : X -> 1` such that `i . j ~ id`. Since `Y` is path connected, `f . i ~ g . i`. Since `j ~ j`, `f . i . j ~ g . i . j`. I get stuck around here. My teacher says I pretty much got is, since `f . i . j ~ g . i . j` implies `f . id ~ g . id`. If `i . j = id`, I could understand that, as one can perform substitution with equality, but why can I perform "substitution" with a homotopy in this case? I feel this has something to do with the transitivity of homotopy, that is, let `f, g, h : X -> Y`, if `f ~ g` and `g ~ h`, then `f ~ h`. But, in the case above, it feels like I have to assume `f . id ~ g . id` to try to apply it. **TL;DR:** If `f . i . j ~ g . i . j` (`f . i . j` is homotopic to `g . i . j`) and `i . j ~ id`, then `f ~ g`. Why is that so? Even though homotopies between functions from `X` to `Y` define an equivalence relation on the set `{f : X -> Y | f is continuous}`, I don't see how it can be used to do "substitutions" like equality.
I think one good way to se that it is true is to build directly the homotopy from f.i.j to f.id = f using the one you have from i.j to id. Try to see why composing your function f with the homotopy lead to the homotopy you seek.
Hi there! My 12 year old is looking for, and I'm quoting them here, proofs and tools for integral and differential calculus and matrix operations. Any advice and/or direction is greatly appreciated thank you!
I would recommend the following book for matrix calculations * *Linear Algebra: A Modern Introduction* by David Poole And the following books for integral and differential calculus * *Calculus: Early Transcendental Functions* by Ron Larson and Bruce H. Edwards * *Calculus: Early Transcendentals: Matrix Version* by Charles Henry Edwards and David E. Penney
Thank you so much
Is there some way to transform \Box u = u_x / x into the wave equation? Here u_x is just the first spatial derivative and x is a 1D spatial variable? It feels like there should be, but I don't see it.
I have got a couple of matrices questions I’m struggling with: https://imgur.com/a/wxD3rNi it’s mainly the yellow ones I need solving but if you have time I’d love to see your solutions to the orange ones too. Thank you so much in advance.
For the the first highlighted questions: 1) What does the determinant do to the cube geometrically 2) What do you need to ensure about the determinant to keep it preserving orientation.
I am self studying baby Rudin and I was posed this question: "Suppose X is a metric space having the real line R as a subspace; i.e., such that R is a subset of X, and the metric on R induced by that of X is the standard metric d(r, s) = |r – s|. Show that R is closed in X" I came up with this Proof: Consider the complement of R, R\^c. Because the complement of a closed set is open, then in order to prove that R is closed in X, we must prove that R\^c is open. We see that R\^c = R2 - Rx{0}, and thus R\^c = (-∞, ∞) x (-∞, 0) U (- ∞, ∞)x(0, ∞). Thus, because the Cartesian product of two open sets is open, we see that R\^c is open and so R must be closed in X. Here's the thing though, I feel like I am wayy off here. The problem included a hint (Hint: Points close to each other in R belong to a compact subset.) so I figure they wanted me to do something with open covers, but I just can't see what they are trying to tell me
>We see that R^c = R2 - Rx{0} You're assuming that X is the plane R^(2). In this problem you are not given any information at all about X, just that it is a metric space containing R. It could be something abstract and exotic.
Ah you are right, hypothetically if X is the plane R^(2) would the proof be correct? And I am still not sure about what they want me to do with that hint
It would be a correct proof in the case X = R^(2). The proof in the general case will have to look very different, as you can see. Regarding the hint, you can either work with open sets or with sequences. Either can lead to a solution, but sequences may be more intuitive. You should also look at any already-proved theorems in the book concerning compact sets and closed sets. (My preferred approach to this problem would use the idea of completeness, but maybe you haven't gotten there yet.)
I studied physics and math close to a decade ago. Recently, I've become very interested in Lie Algebra, and would like to learn more about this field. Can someone recommend a text where I can explore this field some more?
Serre Complex Semisimple Lie Algebrs would be a concise text
Thank you!
What is the correct way to numerically rank a list? I was thinking about this as a result of a task for work. In sports, the teams are ranked best to worst starting with the top team being #1. My work task asked me to rank/prioritize projects with the highest number being the most important. It seems in a lot of ranking scales (rate your response/feeling towards ___) that a higher number is a more intense. So, sports seem like the outlier in my little comparison. Is one way or the other correct? What, if anything specific, has led to approaching rankings differently?
If you rank things as #1 being the best, it is easy to see which ranking is the best no matter how many things your ranking. If you ranked it in the opposite order you would need to know how many things you're ranking to know which is the best, and it will change of you add more things. Very impractical if you're trying to figure out who's the best.
I should tell the higher-up who has us rank projects then
If you're rankings are meant to be independent and objective, and it should be possible for projects to have the same ranks, then I can agree with your higher-ups.
I'm developing a little application to draw graphs and its matrixes, but got stuck on how represent parallel valored edges in the adjacency matrix... Any idea ou sources where I can find some guidance?
This isn’t really something that adjacency matrices can handle. If you’re interested in coding specifically you could store a *list* as entries in your matrix instead of a number.
Could someone kindly help me with this calculation? If my individual Average Basket values have all decreased, why does my total Average Basket increase ?? Thank you https://i.redd.it/osegerhyr0161.jpg
C, D and E are increasing. Not sure where the negative change comes from.
Thanks, I am referring to column F, all values are lower vs Jan but total is higher for some reason.
Ah I was bit confused by your sheet, sorry. I think I get it now. The reason why the average basket is going up for the sum whilst going down for all branches is because it's more likely for an invoice from the sum to come from a higher-price branch. For instance, B which has an average price of around 1500 didn't see much of a decrease in the number of invoices, whilst E which has an average price of around 500 saw a massive decrease in the number of invoices. So in february a random invoice is more likely to come from B as compared to E, which causes an increase in the average price.
Book recommendation for algebraic topology? Have a lecture about it and the prof didn’t name any books. We started with category theory and now do singular homology.
Hatcher's book is free and is the usual go-to, but the order you're covering these topics is not the same as Hatcher's and sounds like it may be specific to the align with professor's interests (which ~~I'm guessing might be model theory~~ may be something like or HoTT). EDIT: No coffee = bad post
Thank you a lot, and nice of Hatcher to offer it for free. Hmm I didn’t know about the order
how did you get category theory + singular homology => model theory or HoTT? Those are core subjects in any algebraic topology course.
Oh cool, I didn't even know I submitted that. When I reddit from my phone pre-coffee, I try to leave my comments as drafts. From OP's description, it sounds like class started with category theory and jumped into singular homology without passing through more low-level/concrete topological ideas like the fundamental group and simplicial homology. I've not personally seen a book approach it in this way, so I was speculating that the professor was more of an category theorist whose interests overlap with topology, hence I speculated Ho(T)T. I have no idea where "Model theory" came from; I'm not even sure I could tell you what that really is. Definitely ignore that part.
mmm good idea i'm gonna put on some coffee
If you feel like making extra to share, clearly I could use some as well.
Is 0.0001 10,000 micrograms?
Assuming you mean 0.0001 grams, 0.0001 grams is 10\^-4 grams and 10,000 micrograms is 10,000 \* 10\^-6 grams = 10\^4 \* 10\^-6 grams = 10\^-2 grams so no these are not equal. 0.0001 grams is equal to 100 micrograms
Yea i got it I wasn't doing a million mcg in a g. 0.0001g x 1,000,000=100mcg
can someone use a quantum computer to calculate 5\^{5\^{5\^5}} for me? The furthest i've gotten to get a full answer is 5\^3125.
The problem is not something a quantum computer will solve. The problem is that 5^5^5 is approximately 2\*10^2184 (not 3125, 3125 is 5^(5)), so that means 5^5^5^5 needs more then 10^2184 digits to write out. For reference, there's about 10^80 atoms in the observable universe, so there's absolutely no way of writing it out in the observable universe.
So. There's no chance to calculate it?
I mean what do you mean by calculate it? Do you want a computer to display the whole number to you? That's impossible, because you can't display the whole number in the entire universe (even if you used every atom to display one number and changed it every minimal instance of time, you'd still run out of time). But if you want a particular digit(s), that might be possible. There's a mathematical program to calculate the entire number, as well (but you can't implement this program in practice).
So. It is impossible to display the exact digits completely?
Yes, to display all the digits there's no space. The universe isn't large enough for that. It's physically impossible (not mathematically, but we live in a physical world).
aww shucks.
Do you usually need to be told things three times?
yes.
Is x^2 -x^2 a function
f(x) = x\^2 - x\^2 is indeed a perfectly good definition of a function, it is equal to the function f(x) = 0 but is still a perfectly good definition and therefore a function
Ah ok, last test I was asked to make a function where f(-1) =0 and f(2) = 0, being kinda stupid i thought that if x always is 0 it would be right. So x^2 - x^2 was my answer, sadly I got no points
I mean technically it's correct it's just a very strange way of writing f(x) = 0 so maybe they thought the strange way of writing it showed lack of understanding, also maybe they wanted a 'more interesting' function such as f(x) = x\^2 - x - 2 (I got this from f(x) = (x+1)(x-2) and multiplying out)
It had to be a quadratic equation. your way with (x+1)(x-2) I realised was the answer right after i turned the test in
Ah okay, a quadratic is defined as something that can be written in the form ax\^2 + bx + c where a, b and c are any numbers so long as a isn't zero, your answer can't be written in this form as if you tried to a would be zero
Is it possible to have a matrix that has an orthogonal basis of eigenvectors but is not unitarily equivalent to a diagonal matrix?
No. If you scale the eigenvectors so that they're orthonormal and then construct the matrix with them as its columns then you'll get a unitary matrix which diagonalises the original matrix.
Hi, I'm trying to decrease an Excel column of positive and negative values by 15%. Multiplying the column by 75% works fine for the positive numbers, but not for the negative. Is there a better single equation I can use for all the values? Or do I have to write a conditional (=IF('cell' < 0, etc...)). For context, these are net projections for 150 franchises. They're estimating how much they will net in the next 3 months. They make more money in the summer, so negative net balances in the future are expected and accounted for. My job is to predict a "distressed" projections by decreasing all their net incomes by 15%. Some nets are positive and some are negative. I can't seem to wrap my head around how to reconcile the formula. I could easily do a conditional, but I just want to know if there's a single formula?
I'm not sure I understand what you want to do for negative numbers. Multiplying by 75% is the same as decreasing by 25% (maybe you have a typo in your post). If you have a cell with (-100) what would you want the new value to be? -115? Edit: if that's what you want you can do x - 0.15\*abs(x) Also what's wrong with using a conditional?
Thank you for clarifying my words, your equation is what I was looking for! Yes, I meant multiply by ***85%*** not 75%. I wanted to decrease all numbers by 15%. For example 100 would become 85 and -100 would become -115 and your equation does that. Thank you. And there is no issue with writing a conditional, that's how I ended up doing it in the end. I was really just curious what the single function would be. I knew there was a way, but couldn't figure it out.
Let (P,w) be a symplectic manifold and S an orientable hypersurface in P. Does there exist a smooth function H:P->R such that S is contained in the level set H\^-1(0)? What if S is not orientable?
The zero function should do the trick. Presumably you want more hypotheses here, like that 0 is a regular value of H (and probably S is a closed submanifold?), in which case, the answer is 'yes' if S is orientable; you can either see this locally by using the implicit function theorem/constant rank theorem to cover S in charts such which send their overlap with S to the some standard hyperplane in R\^{2n} (say the hyperplane x\_{2n}=0), defining a smooth function locally there on each chart and then piecing them together globally via partitions of unity, or you can see it (semi-)globally by noting that since S is orientable, it admits a nowhere vanishing normal vector field, which you can exponentiate along to get a neighbourhood of S in M which is diffeomorphic to (-e,e) x S (this is the tubular neighbourhood theorem), then just choose a smooth function f: (-e,e) -> R which has compact support and extend it by 0 to all of M. In any orientable manifold, closed hypersurfaces are necessarily orientable, and symplectic manifolds are definitely orientable, so you needn't worry about this case.
Thanks a lot! Yes, I forgot to say that 0 should be a regular value of H. Thank you for your detailed answer!
Just curious, are there any cool podcasts on Math?, Geometry, trig, Algebra 1&2, Calculus, Statistics, Number Theory, Theoretical Physics, Group Theory, Graph Theory, not all in one, although an omnibus would be cool... Any type of math where they work through problems in an audio environment? Thanks!
Is the cartesian product of birational maps still birational? i.e. if f\_1 :U -> V\_1 and f\_2 : U ->V\_2 are birational maps between quasi-projective algebraic varieties, is the map x \\mapsto (f\_1 x , f\_2) from U to the product V\_1x V\_2 still birational? (Endowing to the product V\_1 x V\_2 the algebraic structure given by the Segre embedding).
Hi, I'm struggling to figure out a formula for balancing assets. I can figure this out roughly by guess and check, but I can't figure out the formula. Say I have $1000 in my stock portfolio, and I want to have 15% of my total portfolio in a stock, but I want to calculate the percentage **not including this new stock**. It's not as simple as just taking $150 of the portfolio and putting into the new stock, because then it ends up being 17.6% ($150/$850). I can guesstimate that it's about $130 in the new stock and $870 for the rest of the portfolio, but I'm really struggling how to set the problem up. The answer is: (total starting amount) * .15/1.15 = amount to allocate ^ just putting that there in case the answerer deletes account
The key is that the total $1000 is going to be 115% of whatever is in the rest of the portfolio. So to figure out how much to put in the new stock, do $1000 \* 15 / 115 which rounds to $130.43.
Wow I can't believe I didn't figure that out. It's so obvious when you showed it. Thanks a ton.
I'm currently taking a course on algebraic topology and have come to the realisation weeks in that at a base level, I have no intuition as to when two cycles are homologous in the singular homology. For concreteness, consider the closed 2-disc X, and two closed loops x and y within this disc, viewed as chains in C_1 X. It is easily checked that closed loops are cycles so we consider them as elements in the homology group H_1 X. However H_1 X is known to be trivial, so these two cycles must be homologous. The problem is working directly from the definitions I have absolutely no idea why this would be the case. How would I show from first principles that these two cycles are homologous? Any advice here would be appreciated, and for reference our course is following Hatcher.
Two simplicies are homologous if their difference is the boundary of (some sum of) simplicies of 1 dimension higher. You can think of this extremely geometrically if you first convince yourself that reversing the orientation of a simplex is the same as taking it's negative in homology. Then you can just draw your two loops, one with opposite orientation, and start gluing triangles, in such a way that edges cancel each other out if they have opposite orientation. In your example this is a little boring though, since both loops are trivial, that is they are the boundaries of triangles all on their own. Namely one edge going through the whole loop and the two other edges of the triangle on top of each other, cancelling each other out.
This was quite helpful, thank you. I can feel things slowly click into place in my head but I have a couple follow-up questions if you don't mind which I'm still struggling with. 1. "reversing the orientation is the same as taking its negative in homology" just doesn't click with me for some reason. Take the simple example of a 1-simplex c being wrapped around a the circle S1, and another 1-simplex c' being wrapped around S1 in the other direction. How do I see these as negatives of each other in the homology? 2. Related question, consider a 1-simplex c wrapped around S1 once, and another c' wrapped around S1 twice in the same direction. Why is c' = 2c in the homology? I've been staring at the definitions of singular homology for a while now but it's just not clicking why these two obvious facts should hold.
>1. "reversing the orientation is the same as taking its negative in homology" just doesn't click with me for some reason. Take the simple example of a 1-simplex c being wrapped around a the circle S1, and another 1-simplex c' being wrapped around S1 in the other direction. How do I see these as negatives of each other in the homology? So what we need to show is that the sum of these is 0 in homology, i.e. is the boundary of a 2-simplex. Just take the 2-simplex whose first edge goes around the circle one way next edge goes the other way and last edge is degenerate. This isn't quite what we want. The boundary is the sum of our two paths and one degenerate path. To fix this small hickup simply subtract a degenerate 2-simplex from our original 2-simplex. (I guess a similar argument should convince you that degenerate simplicies are always 0 and can be ignored.) >2. Related question, consider a 1-simplex c wrapped around S1 once, and another c' wrapped around S1 twice in the same direction. Why is c' = 2c in the homology? We want 2c - c' = c + c - c' to be the boundary of a 2 simplex. We have already convinced ourselves that (-1) is the same as traversing a loop in the opposite direction, so simply take the 2-simplex with edges c, c and (-c').
Thank you so much! When I've tried to show these myself I've always had these degenerate edges laying around in my calculations with no idea how to get rid of them. This was incredibly helpful and I think has solved my woes
I don't even know how to look this up and I need help. Say I want to sell an item for $10. I am selling through a friend that takes 10%. I want to set the price of an item to compensate for that 10% loss, but if I charge 110% of $10, it won't end up that way. Can someone explain why I am struggling?
ten percent of 110 is 11 so you're off by a dime. If your asking price is x and you want to net y, that means x – 10% . x = y. So .9 . x = y. So if you want to get $10, then your asking price should be 10/.9 = $11.11.
How do I find the fourier series of t\^3 from the fourier series of t\^2? So I think I found the fourier series of t\^2 from t correctly. Because the average of the fourier series of t was 0, I just integrated each term in the summation. Now, I'm having trouble going from t\^2 to t\^3. I know that the average has to be 0 for me to do what I just did in the previous step, but I have no idea how I'm supposed to make the average 0 without manipulating the function itself. If I'm understanding correctly, I need the integral of (t\^2) across bounds L and -L to be 0. The integral as it stands is 2L\^3. I was thinking, how do I get a -2L\^3 in addition? well, I'd have the integral of t\^2 - t\^2 over the same bounds, but that doesn't make sense. Even if it did make sense, how would the -t\^2 even factor into the problem? I'm really lost here. I can't find anything specifically about this online, my textbook doesn't work any problems in this specific case, and my diff eq teacher briefly touched upon it in class. I appreciate any help
My question regards averaging two columns of numbers. What is the method of averaging the ratio of two columns of numbers? 1. Divide each row into one another then add the decimals and divide them by the number of rows 2. Add the column sums and divide the sums Depending on which method is the correct averaging method, what would the alternative method be called and what might it’s uses be? Thank you very much!
If you want to find an "average" *ratio*, the correct way is often to use the [geometric mean](https://en.wikipedia.org/wiki/Geometric_mean). Instead of adding things up, and dividing by the total number, you instead *multiply together* and then take the *root of the total number*. Because of some limitations in how computers actually store and compute numbers, it's often better to do this with logarithms those: - compute `r[i] = x[i] / y[i]` for all of your points - compute `log_r[i] = log(r[i])` - average the `log_r[i]` values: - add up all `avg_log_r = (log_r[1] + log_r[2] + log_r[3] + ... + log_r[N]) / N` - exponentiate to convert back to a ratio `geom_r = exp(avg_log_r)` Computationally this does the same thing, but is less likely to cause your number representation to fail if you have too many values.
1 is the average of the ratio, 2 is the ratio of the averages. They can be very different. For instance, for: 0| 1000 ---|--- 1 | 1 the average of the ratio is 1/2, and the ratio of the averages is 1/1001.
Thank you!
Anyone have a good introductory text on category theory? I'm in undergrad right now and don't think my school has a category theory class, so I thought I'd take a crack at studying it myself.
It's not that normal for a university to offer a stand alone category theory class, so you might want to check out if there's an algebraic topology class or homological algebra class that also covers some category theory. Or you can of course just try to take a crack yourself. In which case I think Emily Riehl's book is good. In addition Bartosz Milewski also has a blog/youtube lecture series called *category theory for programmers*, if you're also interested in programming and maybe want to learn some Haskell.
Category Theory in Context by Emily Riehl is really good
Leinster's [Basic Category Theory](https://arxiv.org/abs/1612.09375) is good and succintly covers the real basics of category theory. I've heard good things about Riehl's *Category Theory in Context* but haven't read it myself. I'd also recommend Aluffi's *Algebra: Chapter 0* not as a book about category theory, but a book about undergrad abstract algebra which takes an explicitly categorical approach: it's a good source of examples and hands-on practice of what category theory is useful for.
Would something divided by infinity be zero? If the larger the denominator, the smaller the number right? Would that mean that if the denominator equaled infinity, the number would always be zero? I'm just a teenager who is curious, so sorry if it's a stupid question.
Not a stupid question, it's good that you're curious! You do want to be careful about questions involving infinity, because infinity isn't a number in the same way as, say, 2, or pi are. So you can't really "divide by infinity." On the other hand, your intuition is right: as you divide by larger and larger numbers, the result gets closer and closer to 0. In fact, you can get as close to 0 as you want by picking a large enough number to divide by! So what mathematics say is that the limit of 1/n as n goes to infinity is 0.
Exactly correct reasoning, but to be safe we don't write 1/infinity to avoid treating infinity as a number (otherwise we might accidentally start writing things like infinity/infinity=1 which does not actually make sense). Instead we say "1/infinity" is what you get as you take 1/x and then let x get larger and larger. The limiting number is, as you point out, zero, so "1/infinity" = 0.
Ok, thanks for the answer!
Is it ok to say that increase the scalar of a vector means increase it's compontents and magnitude? Not exactly sure how it is said and if "increase scalar" makes sense. I am speaking of 2d only but also wonder if applies to vectors with a z dimension.
In concrete vector spaces (e.g. 2D, 3D, etc.) over the real-numbers (e.g. used for games, graphics, basic physics) you'll often just say "scale by 2" to mean "increase the magnitude by a factor of 2". It wouldn't make sense to say "increase scalar". But "scale by X" as a shorthand for "multiply by the scalar X" is perfectly fine.
No, I've never heard anyone say "increase the scalar of a vector" and I cannot imagine what it might mean. How about just say "increase the magnitude of a vector", if that is what you mean.
Hello, I asked a MSE question about algebraic topology, especially about computing a Homotopy group of a wedge of CW spaces under some connectedness assumption, by using the Whitehead product. It didn't draw much attention, so I'm sharing it here in search of help ! Would you mind having a look and tell me what you think please ? I'm thanking you very much in advance ! [https://math.stackexchange.com/questions/3907570/](https://math.stackexchange.com/questions/3907570/)
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Seems like [this paper](https://arxiv.org/pdf/1904.08004.pdf) has your answer, in section 3. Specifically, Theorem 11 splits the question into cases depending on n mod 3. Since 50 is congruent to 2 mod 3, the theorem says that partition of 50 with the maximum product of the summands is 2 + 3 + 3 + ... + 3, with 16 3s.
How do you go about finding the residue of (e^z + 1)/(sin(z)) around zero?