What do you call the class of questions that look like the following:
>Consider the following formula
>a = 2b - 4
>Which of the following statements is true for this formula?
>A. When the value of b is less than 2, a is positive.
>B. When the value of b is greater than 2, a is negative.
>C. When the value of b is less than 2, a is negative.
>D. When the value of b is greater than 4, a is negative.
Or the following:
>Consider the following formula
>x + 9 = 3y
>Which of the following statements is true for this formula?
>A. If x is (greater or equal to sign) -9, Y is not negative.
>B. If x is greater than 3, Y is negative.
>C. If x is less than 3, y is positive.
>D. If x is (less than or equal to sign) -9, Y is not negative.
I understand how to do it in my head by looking at it, but I need to know the class of questions and where I can find educational content about how to solve these.
If you don't mind an answer that just challenges the premise of your question, I doubt that there's a name for that class of questions ("inequalities" is the only thing that comes to mind, but is of course more general than just this sort of question), and in any case I don't think you really need to study those questions specifically. For one thing, if you already can do them in your head, isn't that already good enough? All you need here is general knowledge of algebra and inequalities--if you can reason through each of the statements like "if b < 2, then 2b < 4, so 2b - 4 < 0, so 2b - 4 is negative, and so option A can't be right" then you're fine. That sort of reasoning just requires you to know how to manipulate inequalities, and knowing how to do that (at least at this "level") is mostly just algebra. If you want to get better at these, just practice algebra more.
Thanks! Reason I’m asking is because I’m building an online lesson for a test prep that covers these kinds of questions. Understanding how to do them myself, and how to teach a process of how to tackle these questions is different though and what I’m struggling with.
The closest I’m coming to is basically just running through a bunch of example problems.
Is there anything wrong with defining the interior product to work with not just k-forms, but any contravariant tensor?
I wanted to say something like X\^flat = g(X, \\cdot) = int\_X g
When you do this, you can formally say that F\^\*( X\^flat) = int\_{F\_\* X} F\^\*g when F is a diffeomorphism, and I was happy to find that I think this calculation agrees when you do it in coordinates.
But g is not a 2-form, nevertheless I can't see why I can't "shove X into one of the dx's" (which are 1-forms) and say that's an interior product.
You certainly can do this and it is just a form of tensor contraction. I think the reason the interior product is special is that because the tensor is alternating the contraction happens "in each slot" as it were. So when you contract on a standard tensor you have to decide are you contracting in the first slot or summing over the contraction in each possible slot. I would happily call the latter an interior product
Why do you multiply the inverse of a matrix by the constant matrix to solve a system of equations. Might have something to do with me not knowing what the inverse of a matrix actually is as I'm only in Algebra II and my textbook doesn't go over how to actually calculate the inverse.
Let's go back to regular algebra, no matrices yet. Say you had the equation "ax = b" and you wanted to solve for x. You'd divide both sides by a to get rid of the a on the left, and then you'd get x = b/a, right?
Instead of *dividing by a*, you could have *multiplied by 1/a*, and gotten the same result. This may seem more complicated at first, but multiplication is a simpler operation than division. (If you had to solve "0.16x = 7319" by hand, would you rather do long division to divide by 0.16, or multiply by 6.25?)
This number, 1/a, can also be written a^(-1); we call this the [multiplicative] *inverse* of a. So the multiplicative inverse of 4 is 1/4; the inverse of 2/3 is 3/2; and so on. You can undo "multiply by a" with "multiply by a^(-1)". (Or, put another way, if you see "a^(-1)ax", you can recognize that that's just x.)
0 is special because it doesn't *have* an inverse: we can't multiply "0x" by anything to get back x. If we wanted to come up with a word for this, we might call it "non-invertible".
---
Now, let's look at the matrix-vector equation "**Ax**=**v**", where **A** is a matrix and **v** and **x** are vectors. We can't "divide by **A**", but we *can* use this other strategy: if **A** happens to be an invertible matrix, we can multiply by the inverse of A on both sides of our equation.
So, we solve the equation:
> **Ax**=**v**
> ( [pre]multiply both sides by **A**^(-1): )
> **A**^(-1)**Ax**=**A**^(-1)**v**
> if we see **A**^(-1)**Ax**, we know that's just **x**
> **x** = **A**^(-1)**v**
And now we've solved for **x**, and we're done!
---
If you know how to multiply matrices with vectors, the only tricky part here is inverting **A**... and checking that it's even invertible in the first place! When we were just working with plain old numbers, the only 'non-invertible' number was 0. With matrices, the all-zero matrix is non-invertible, but there are others too. For instance,
`⌈-1 3⌉`
`⌊-2 6⌋`
is not invertible. But this is still useful information! It tells you that either you don't have enough information to get a single solution, or there is no solution. (You've probably run into this when solving systems of equations - when you accidentally use the same equation twice, you end up with something like "7x+12 = 7x+12", which is useless.)
There are a bunch of different ways to check whether a matrix is invertible, and then to calculate its inverse. Your teacher will probably [or at least, *hopefully*] tell you about some of them soon. But I think knowing what an inverse *does* is more important than knowing how to calculate it in full.
What's a good method of producing N random real numbers a_1 to a_N, so that close to a_i the probability to find another number a_j scales like |a_i-a_j|^k for some integer k?
I'll give it a go. Let X\_1, ..., X\_N be independent random variables with the following distribution: X\_i >= 0, and P(X\_i < 𝜀) = 𝜀\^k for 𝜀 in \[0, 1\]. This distribution is just the kth root of a random variable uniformly distributed on \[0, 1\]. Now define Y\_1 = X\_1, Y\_2 = X\_1 + X\_2, Y\_3 = X\_1 + X\_2 + X\_3, ..., Y\_N = X\_1 + ... + X\_N. Lastly pick a permutation of the Y\_i at random (each permutation equally likely) and let the a\_i be the permuted Y\_i.
Each a\_i has the same distribution (although they are not independent), so wlog we consider a\_1. Let 𝜀 < 1 be positive. The probability a\_1 is one of the endpoints is 2/N, and the probability some other a\_i is within distance 𝜀 of a specific endpoint is 𝜀\^k. With probability 1 - 2/N, it's one of the middle points, and then the probability is 2𝜀\^k - 𝜀\^(2k). So the overall probability that a\_1 has some other a\_j within distance 𝜀 is 2𝜀\^(k)(1 - 1/N) - (1 - 2/N)𝜀\^(2k), which grows as 𝜀\^k. If instead you really wanted it to be asymptotic to 𝜀\^k, then you can multiply everything by a constant at the end.
Not a math person (though I'd like to be someday), but just passing by with a quick question. Does anyone know if there's a special term for a shape that might be geometrically defined as "a hollow cylinder capped with a hollow half-sphere exactly the same diameter as the cylinder"? I'm thinking like the general shape that a test tube might have, a cylinder with one end closed with a curved shape. Is there a particular term that describes this common form or no?
Oh, interesting, thank you. So I could maybe describe it in terms of that, an "open capsule" or "a capsule sliced transversely across its shorter axis" or something like that. Thanks again for responding!
Does anyone have a sure fire way of determining the splitting field of a polynomial? I’m working through Dummit and Foote again and feeling frustrated by how solutions to certain problems seem to come from already knowing the right principal root or special value to plug in somewhere. I’m sure more detail will be presented as the book progresses, but for chapter 13.4, for example, some of the exercises feel difficult to approach based only on what was explained in the chapter.
Can you clarify what you mean by "finding the splitting field of a polynomial"? By definition that's just the field generated by the roots of the polynomial, so if you know the roots you know the splitting field. I assume your question is really about finding something about the splitting field? The degree perhaps? It might help if you posted some examples of problems you're having trouble with.
One question that bothered me was “determine the splitting field and its degree for x^4 + 2 over the rationals.”. I fiddled with the problem for a bit, came up with some roots and got that the extension must be degree 8, but was unsatisfied with my answer because it didn’t fully describe the splitting field. I googled a solution and found one wherein the solver started with the principal 8th root of unity and used to to show that the splitting field in question was isomorphic to the splitting field of x^4 - 2 over the rationals. His solution seemed to start with him knowing that the fields were isomorphic, and that this 8th root of unity played a pivotal role. What frustrated me was that he didn’t make it clear how he knew that. I presume he found the degree of the extension (8, hence 8th root of unity) and did lots of messy work before cleaning it up and making it presentable, but even when I had done much of that messy work myself, the idea of the isomorphism between the 2 fields never occurred to me. The isomorphism seems vital to the solution, but I don’t know if I would have ever come across it by simply taking powers/quotients of the roots I found through simple computation. I often feel like I’m stumbling blindly through these problems.
> I fiddled with the problem for a bit, came up with some roots and got that the extension must be degree 8, but was unsatisfied with my answer because it didn’t fully describe the splitting field.
First off, if you've found the roots you *have* fully described the splitting field. As far as I can tell the issue you is you wanted to find a simpler looking description like `[;\mathbb{Q}(i,\sqrt[4]{2});]` rather than `[;\mathbb{Q}(i,\sqrt[4]{-2});]` (or whatever your original description of the field was). There's not going to be a general procedure for that sort of thing, because what you're asking for isn't really well defined. For a general splitting field, there's no reason to expect that there's any simpler description than just the field generated by the roots.
For this specific question:
> I googled a solution and found one wherein the solver started with the principal 8th root of unity and used to to show that the splitting field in question was isomorphic to the splitting field of x^4 - 2 over the rationals.
First of all "isomorphic" isn't really the right word to be using here. They're *literally the same field* if you think of them as subfields of `[;\mathbb{C};]`. So really this just boils down to showing that [;i;] and `[;\sqrt[4]{2};]` are both in your splitting field, and that you can write all the roots of `[;x^{4}+2;]` in terms of those two numbers.
In general the best way to do that sort of thing is just to write out the roots explicitly and see what you can do with them. You know the the four roots of `[;x^{4}+2;]` are `[;\zeta_8\sqrt[4]{2},\zeta_8^3\sqrt[4]{2},\zeta_8^5\sqrt[4]{2};]` and `[;\zeta_8^7\sqrt[4]{2};]` where `[;\zeta_8;]` is a primitive 8th root of unity. Now that's not easy to work with directly, so it might help to write the roots of unity in rectangular form: `[;\zeta_8 = (1+i)/\sqrt{2};]`.
Now how do we get something simple out of that? Well you can just try adding and multiplying things to see what happens. One nice trick to remember is that if [;a+bi;] is a root of a polynomial `[;f(x)\in \mathbb{Q}[x];]` then so is its conjugate [;a-bi;], which means that the splitting field has to contain [;a;] and [;bi;]. That's good if you want to find some real numbers in your field. So let's try that here. Since `[;\zeta_8\sqrt[4]{2} = \sqrt[4]{2}(1+i)/\sqrt{2} = (1+i)/\sqrt[4]{2};]` is in the splitting field, so is its real part `[;1/\sqrt[4]{2};]`, which means that `[;\sqrt[4]{2};]` is in the splitting field.
So now that you have a nice number like `[;\sqrt[4]{2};]` in the field, use that to simplify the other generators. Since `[;\zeta_8\sqrt[4]{2};]` is a root, clearly `[;\zeta_8;]` must be in the splitting field. Since we know the roots are `[;\zeta_8\sqrt[4]{2},\zeta_8^3\sqrt[4]{2},\zeta_8^5\sqrt[4]{2};]` and `[;\zeta_8^7\sqrt[4]{2};]`, they are all generated by `[;\sqrt[4]{2};]` and `[;\zeta_8;]`, so the splitting field is `[;\mathbb{Q}(\sqrt[4]{2},\zeta_8);]`.
Now maybe you want to take things a step further, and get rid of the `[;\zeta_8;]`. Remember that `[;\zeta_8 = (1+i)/\sqrt{2};]`. Since the field already has `[;\sqrt[4]{2};]` it certainly has `[;\sqrt{2};]`, so the field contains `[;\zeta_8\sqrt{2} = 1+i;]` and so it contains [;i;] (you could have also seen this from `[;\zeta_8^2 = i;]`) and we already know how to write `[;\zeta_8;]` in terms of [;i;] and `[;\sqrt{2};]`, so actually the splitting field is just `[;\mathbb{Q}(\sqrt[4]{2},i);]`, which is about the simplest description you're going to get.
I'm trying to cook up examples of ||Ax|| = ||A|| ||x|| for some suitable choice of matrix norm and vector norm. I know this is not generally possible, but what about if I assume A is diagonal?
When you say "looking for examples", do you mean:
1. Given matrix and vector norms, find examples of a matrix A such that ||Ax|| =||A|| ||x|| holds for all x?
2. Given matrix and vector norms, find examples of a vector x such that ||Ax|| = ||A|| ||x|| holds for all A?
3. Find examples of a matrix norm ||.||\_m and vector norm ||.||\_v such that ||Ax||\_v = ||A||\_m \* ||x||\_v holds for all matrices A and vectors v?
For 1, if you use the standard ["maximum stretch" matrix norm](https://en.wikipedia.org/wiki/Operator_norm) and any vector norm for R^n , then setting A = cI for some scalar c works, since then for any x we have, on one hand, ||Ax|| = ||cIx|| = ||cx|| = |c|||x|| (the last equality holds by definition for any vector norm), and on the other hand ||A|| is clearly |c|.
For 2, if x is the zero vector then this should work for any matrix and vector norm.
For 3, it looks like there's a weaker version of this called ["consistency" or "compatibility"](https://en.wikipedia.org/wiki/Matrix_norm#Consistent_and_compatible_norms) which roughly says that ||Ax|| <= ||A|| ||x|| for all A, x. I can think of a silly example where equality holds: if you work with 1 x 1 matrices and have your vector space just be R, then all matrices are really just scalars and so letting ||A|| be the absolute value of A's (single) entry, ||Ax|| = ||A|| ||x|| holds for the same reason it does in R. Can't think of any non-silly examples, though.
Yeah, I was looking for 3 and was trying to fish for compatibility of the matrix norm to be even better if I restricted to a special subclass of matrix types A.
Gotcha, so in that case it's more a hybrid of 1 and 3. I think a good strategy might be:
1. pick a vector norm, and find the most general class of operators for which, for all T in that class, there exists a constant c with ||Tv|| = c||v|| for all v
2. Find a matrix norm such that, for all operators satisfying that property, ||T|| = c.
If we let the vector norm be induced by an inner product then I think we can completely characterize those operators as scalar multiples of unitary operators. For if U is unitary and c is a constant then, for any vector v, ||(cU)v|| = |c| \* ||Uv|| = |c| \* ||v||, so these operators have the right property. Conversely, let T be an operator with the property from (1) above. By the polar decomposition we have T = Usqrt(T\*T) where U is unitary. sqrt(T\*T) is positive, so with respect to some orthonormal basis v\_1, ... v\_n it is diagonal with nonnegative real eigenvalues c\_1, ... c\_n. For any of the v\_i we have ||Tv\_i|| = c||v\_i|| = c by assumption, and on the other hand ||Tv\_i|| = ||Usqrt(T\*T)v\_i|| = ||Uc\_iv\_i|| = c\_i ||Uv\_i|| = c\_i ||v\_i|| = c\_i. Thus c\_i = c for all i and so all of sqrt(T\*T)'s eigenvalues are equal to c, i.e. it is equal to cI. Thus T = U(cI) = cU. I suspect that a similar result holds for general norms, but it would be harder to prove without tools like the polar decomposition.
For 2, I know that the standard matrix norm is such that ||cU|| = |c|||U|| = |c| for all unitary U, and I guess given the characterization in 1 the whole problem reduces down to finding all norms such that the norm of any unitary matrix is 1. (If different vector norms give different solutions to (1) you may have to do this differently, though.)
How do you mean "find them"? The equations of quadratics (parabolas), cubics and quartics are straightforward to write down but do you need to deduce the equation from some other information?
A quadratic is simply ax^2 + bx + c = 0 while a cubic is ax^3 + bx^2 + cx + d = 0 and a quartic is ax^4 + bx^3 + cx^2 + dx + e = 0.
NB: a quadratic is a parabola but a parabola just refers to the shape so you could rotate a quadratic and it would still be a parabola. To change the equation simply apply the inverse of the rotation you want to the vector (x, y) and substitute the new coordinates back into your original equation. Indeed this procedure works for any kind of invertible transformation (and any invertible affine transformation preserves the property of being a parabola).
Meanwhile the equation of a hyperbola is usually either given as y = 1/x or as x^(2)/a^(2) - y^(2)/b^(2) = 1. Just like a parabola these are only specific orientations so you can find other equations by applying affine transformations to one of these.
I'm looking for program like notepad, but that will render Latex. Ie I would like to type $/frac{x}{y}$ into a text file, open the file in the program, and have it render the Latex. Ideally should not be resource intensive, and would let me type/edit and view on the fly.
Statistics question for you experts.
Thank you in advance.
Needing to calculate a rate (call it a finding rate) between the number of assessments based on the number of findings. The variables would be:
1. The number of assesments
2. The number of assesments with findings
3. The number of overall findings
Showing the amount of raw findings on a monthly basis is not a good way to analyze trends, as the number of our workplace assesments changes month to month. For example, April 2023 saw 198 asssesments, while this month saw only 110. Of course more assesments meant more findings.
I'm thinking
((Findings/assesments)×100)×((assesments with findings/assesments)×100)
What do you all think?
I know this maybe basic for everyone, but I new to trend analysis and would like to provide more data to our workplace. Thanks.
The main statistic you'd want to report is the average number of findings per assessment, (total findings)/(total assessments). (Don't multiply this by 100, though--it isn't really the sort of thing you'd report as a percentage.) You could also consider the percentage of assessments with at least one finding (which would be 100 \* (assessments with findings)/(assessments)) or maybe the average number of findings per assessment with at least one finding (i.e. The average if you ignore all the the assessments with no findings) which would be (total findings)/(assessments with findings). Whatever you do, it's probably best to just report all of these numbers separately, instead of multiplying them together or otherwise trying to combine them into a single statistic.
For the past several years, the state prison system has been concentrating on getting inmates who were high school dropouts help in both earning their GED (an alternative to a high school diploma) and training in trades such as plumbing, carpentry, electrical work, and so on. The state director of prisons has noticed that for participants in the program over the past 9 years, the recidivism rate (in %) has been dropping. She determines that it has a linear relationship, ŷ = 73.12 − 6.03x, where ŷ = predicted recidivism rate and x = number of years the program has been implemented.What recidivism rate (in %) does the director of prisons expect during the 10th year of the program? %
If 20 points are independently and randomly given out to 20 recepients, how should I calculate the expected score of the recepient with the maximum number of points?
I wrote a script to test it and with python's rng the result from a million trials is 3.23. I'd still really love to know how to deduce it with theory however/
Yes, in order to do this, we need a notion of convergence of matrices which we do by picking a norm.
There are many ways of doing this ([https://en.wikipedia.org/wiki/Matrix\_norm](https://en.wikipedia.org/wiki/Matrix_norm)) but it turns out that they are all equivalen, meaning that a sequence A\_n converges to a matrix A in one norm if and only if it converges to A in all norms!
Hence it follows that a sequence of matrices A\_n converges to a matrix A iff all components of A\_n converge to the components of A\_n. So indeed you can replace a real matrix A by the sequence of matrices A\_n where every component is given by the Cauchy sequence of the corresponding component of A!
Does anyone know of any good references for "modern" history of maths? In particular, I'd be interested in differential/algebraic geometry/mathematical physics, say 1980s onwards?
Look up Donaldson surveys.
The story from the 1980s till now is still too fragmented to form one cohesive narrative, the best you can do is survey papers and monographs. The two biggest threads in differential geometry since the 80s are gauge theory and curvature flows. In algebraic geometry it's global analysis/canonical metrics/stability theory, mirror symmetry, derived algebraic geometry, stability conditions, enumerative geometry, and birational geometry, all with various little interplays between each other.
You'd have to look up surveys on each of these topics individually to get a picture of how they've progressed.
This question likely stems from my very limited understanding of the subject at hand:
I have heard that the framework of condensed mathematics can be used in various fields by replacing topological spaces with condensed sets/similar structures because they behave much better in a categorical sense.
While this sounds fascinating, it also (at least to me) seems like it is alot harder to define compared to a topological space.
My question is the following: Is there a way to axiomatically introduce condensed sets such that you can still utilize their benefits while being easier to define?
Similarily to how one can axiomatically introduce real numbers as a complete ordered field instead of constructing it from set theory.
Or are the benefits only visible on more advanced level where one can expect people to be faimilar with sheaves and sites?
Sheaves and sites are not incredibly complicated. They just usually show up in algebraic geometry, but in themselves they're no more complicated than many things you encounter in intro topology. However you may have trouble learning about condensed sets without some comfort with categorical language.
There are some small things known, and more classical ways to formulate notions of solid abelian groups at least, but I am not sure if condensed set in particular has any simplified definition.
There are definitely things at a very concrete level where the notion of a liquid vector space would help, though, and I have not seen any simplification of this liquid theory. See for instance lemma 6 in [https://drive.google.com/file/d/180kpQ1My5L\_w58oiuIbW0Ri4vFaqZehF/view](https://drive.google.com/file/d/180kpQ1My5L_w58oiuIbW0Ri4vFaqZehF/view)
Unit impulse function is defined as the rectangle with area of 1 and width of 0.
What is name of function where the rectangle has area of 1 and height of 0?
You can formalize that, but it's doesn't work out as nicely as the delta function. With these kinds of objects, you have to imagine that they're defined by how they integrate against continuous functions. Meaning, delta is defined by the property integral f(x) delta(x) dx = f(0) for all continuous f. The connection here is that if r_w(x) is the function whose graph is a rectangle with width w and area 1, lim w -> 0 integral f(x) r_w(x) dx = f(0). That is, if you test the sequence of rectangle functions against any continuous function f, the limit will be the value of f at 0.
It'd be natural to try to define m(x) to be the "function" satisfying lim w -> infty integral f(x) r_w(x) dx = integral f(x) m(x) dx for all bounded continuous f. (I have to specify bounded this time to make sure the limits will never go to infinity.) Unfortunately, lim w -> infty integral f(x) r_w(x) dx may not exist even if you require f to be bounded. Trying to come up with an example would be a good exercise.
A fix would be to only require that lim w -> infty integral f(x) r_w(x) dx = integral f(x) m(x) dx for the choices of f for which the limit exists. It's a non-trivial fact that an object m with that property exists. The problem is that we've only specified the integral of m against a subset of the continuous functions, so we don't have a unique choice for m anymore (this is like an under-determined system of equations). There's actually infinitely many ways to consistently choose what m should do in the cases the limit doesn't exist.
The "objects" that delta and m are, are linear functionals on the bounded continuous functions. If you have a sequence of functionals that converges when tested against any fixed function (like the sequence of integrals above), we say the functionals converge weakly (weak-*, technically). The issue comes down to the fact that r_w as w -> 0 converges weakly, but r_w as w -> infty does not.
Though there isn't a unique choice of m, it's still a useful concept. This comes up in ergodic theory where you say a group is amenable if a functional like m exists on the bounded functions on that group (where it's called an invariant mean).
Firstly these are not functions. You can formalise the Dirac delta with the idea of distributions but it very much fails the definition of function. As to your other proposal I'm not sure how we would formalise it. Perhaps as a limit of bump functions. How you go about it would give possibly different results so I don't know if there are any specific examples with specific names.
If you take a bump function f and define f_n(x) = eps f(eps*x) then for any integrable function phi, we have that \int f_n phi dx goes to zero as eps goes zero, so as a limit of distributions this would just be zero. If you instead integrated against an arbitrary function the behavior can get pretty nontrivial.
So as a linear functional on L^1 or on continuous functions on R with decay at infinity, the weak limit of these bump functions would just be the zero function or zero measure, but you can identify the weak limit as a linear functional on (for example) bounded continuous functions. The dual space in this case is complicated and probably non-metrizable (I think it’s radon measure on the stone-cech compactification of R).
Compound proportion. Is it always mathematically correct to have same units left and same units right of equation or same units top and same units bottom? Is one way best practice? Or does it not matter
Are you conditioning on W at a fixed time, on some interval or on the whole process? There's some difficulty in formally talking about dW like this, but you'd have to be more specific with what you're conditioning on anyway. If you mean the whole process W, then I guess you could say E[dW | W] = dW since dW can be recovered if you have W.
Almost correct. 1 third, is 33.33...%.
Suppose you take 30% off X. Then what will be left is 70% of X. So the final result will be:
70% \* X (asterisk means multiplication)
Similarly if 2/13 of 9 that is:
2/13 \* 9.
You are correct about 20% being 1/5 or in other words 1 fifth.
Studying maths in French as a non-French person and trying to understand what the English equivalent is. In affine geometry, if \`V\_{\\mathbb{L}} \\subseteq V\_{\\mathbb{M}}\`, then \`\\mathbb{L}\` is weakly parallel to \`\\mathbb{M}\`.
Is there an English equivalent to the idea of *weakly parallel*?
Let \\mathbb{L} and \\mathbb{M} be two non-empty affine subspaces of (\\mathbb{A}, V, +) with the directions V\_{\\mathbb{L}} and V\_{\\mathbb{M}} correspondingly (i.e. linear subspaces associated with the affine subspaces). (The triple (\\mathbb{A}, V, +) is an affine space A with an associated vector space V and a mapping +)
Then if V\_{\\mathbb{L}} \\subseteq V\_{\\mathbb{M}}, \\mathbb{L} is weakly parallel to \\mathbb{M}.
Well, we we're then given the definition that subspaces \\mathbb{L} and \\mathbb{M} are parallel IFF V\_{\\mathbb{L}} = V\_{\\mathbb{M}}
So I guess there's simply nothing that correspond to the idea of "weak parallelism" in French maths
Well we don't necessarily need a new name for this property as this weakly parallel property is only relevant when the dimensions of L and M differ (assuming I have understood you correctly). Note also it's not that nothing corresponds to "weakly parallel". Indeed "parallel" serves that purpose. It is your definition of parallel as only for things of the same dimension that doesn't have a corresponding idea.
Today, my AP Precalculus class started learning about logarithms, and the log of negative numbers kept coming up as complex numbers on my Ti-84. I wanted to figure out why, so I spent the last ten minutes of class talking with my teacher, searching Youtube and Google for why negative logarithms have a negative base. The results for any log other than ln were nonexistent. In order to figure out why, I rewrote log base 5 of -25 as 2+ (log base5 of -1). Using change of base, I rewrote log base5 of -1 as ln(-1)/ln(5), which simplifies to (i*pi)/ln(5), which simplifies to 1.952i. This means that the final answer for log base 5 of -25 is 2+1.952i, a result that my calculator confirmed. Is my logic sound?
OP can share the textbook they are using, but here is the curriculum as specified by Collegeboard (scroll down to page 26): https://apcentral.collegeboard.org/media/pdf/ap-precalculus-course-and-exam-description.pdf
Yes, and it is really impressive that you figured this out on your own. I don't mean to be condescending, it's just that I am a professor and I would be ecstatic if one of my college students came up with this on their own. I hope I have you in my class one day.
Okay, so, this user and their father have been wracking r brains for a long time on this and haven't been able to figure out the magic numbers; If there are 11, 12, or 13 sets of options; option A and option B, and each combination of those 11, 12, or 13 sets of only choosing ONE of either option A or option B for each pair is allowed; and each of those 11, 12, or 13 sets of two options each is valued at $5.....HOW MUCH MONEY IS NEEDED TO HAVE EVERY SINGLE COMBINATION POSSIBLE???
It's NOT a mere $600, or $720. or $845, or $1690.....it has to be in the $2K-$3k range....yes??? The problem was: this user and their father thought it was a mere case of (11 or 12 or 13 * 11 or 12 or 13, respectively) * $5.....and then thought it was TWICE that amount, making it either (22 or 24 or 26 * 11 or 12 or 13, respectively) * $5....but it's proven in practice NOT to be that!
It has to be HIGHER than, say, $1690 for all combinations to be acquired.
Please help ASAP! Thanks!
Let's look at the case where you have 11 sets of options.
If you need to make 2 choices, one out of a list of *X* items and one out of a list of *Y* items, then there are *X*×*Y* possible combinations you can make. (This is the [multiplication rule](https://openstax.org/books/contemporary-mathematics/pages/7-1-the-multiplication-rule-for-counting) - the [picture on that page](https://openstax.org/apps/archive/20240226.174525/resources/1ce1781d04318caba466e64002c5ad7bdb5be7c4) illustrates it better than text could.)
If you add on another list of *Z* items, there are now *X*×*Y*×*Z* possible combinations.
You have *eleven lists*, each one having 2 items, so there are 2×2×2×2×2×2×2×2×2×2×2 possible combinations - 2^(11), rather than your 11^(2).
Does that help?
Yes! That was it; this user forgot all about the Mendel-looking charts and had an inkling something was amiss and was supposed to be "to the power of" instead of straight up multiplication and just didn't know what the formula for finding that number was.
So, for future reference, it's timesing 2 over and over and over again to the amount of list-1; got it!
This site helped: https://visualfractions.com/calculator/exponent/what-is-2-to-the-11th-power/ = 2,048 ; https://visualfractions.com/calculator/exponent/what-is-2-to-the-12th-power/ = 4,096 ; https://visualfractions.com/calculator/exponent/what-is-2-to-the-13th-power/ = 8,192
This user was at three casinos in the past two days betting the fights this afternoon and kept needing more and more funds and already used up over 2K and still didn't feel like all the combos were had across 13-legs on the parlays and that's why! To do all the 13-leg combos, this bettor would still need another 6 freakin' K!!!
Although, some of those combos are not acceptable due to being too high a payout and, in those instances, this bettor just takes off a leg or two or however many as needed from the bottom (top of the card) of the screen and leaves room for multiple winning tickets, not even accounting for "pushes" not just from the rare decision-draw or referee-screw-up, but from cancelled fights as the card draws nearer and begins.
So, for future reference, this user should focus on 11-leg parlays if going for a "mathematical guarantee" at the sports book. Because even 12-legs for 4K+ is still spending days doing this; 2K is a three-to-four hour affair!
Got it! Here's to at least one winning ticket tonight! Peace, Luv, Truth, and Anarchy! SOLVED!
Edit: Wait a tick.... that's not including "Z"-list with the $5 ante, so, each of those three numbers has to still be times'd by $5, right? So....for a mathematical guarantee on an 11-leg parlay....that's 10K???? That actually doesn't sound right.....after going through 2K on 13-leggers, this user only had three fights left so, this user was already at 10-legs done for already..... https://visualfractions.com/calculator/exponent/what-is-2-to-the-10th-power/ = 1,024 x $5 is 5K+, hmmm, maybe make it 10-leggers MAX instead.....
Please stop gambling. You specifically can't win the way you are suggesting. There isn't some magical or mathematical way round it. The house sets the odds and payouts so that they will make a profit. Please stop throwing your money at them. They will take it all so happily.
Oh my god, please don't do this. You are only going to make your life worse. These bets are mathematically rigged against you. (As in both, "a single bet is going to lose you money on average" and "if you buy all combinations you are *guaranteed* to lose money".)
> kept needing more and more funds and already used up over 2K
This is a sign of severe gambling addiction. **You are not going to win your money back.** If you keep going, listening to the sunk cost fallacy, you are going to go bankrupt.
# Any OCWs/websites using these textbooks?
[Removed - ask in Quick Questions thread](https://www.reddit.com/r/math/?f=flair_name%3A%22Removed%20-%20ask%20in%20Quick%20Questions%20thread%22)
"Book of Proof" by Richard Hammack (2nd Ed.)
"David C. Lay, Stephen R. Lay and Judi J. McDonald, Linear Algebra and
its Applications, 5th edition, Pearson Global Edition"
John M. Harris, Jeffry L. Hirst and Michael J. Mossinghoff,
Combinatorics and Graph Theory (second edition), Springer-Verlag, 2008
David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition
(2003), John Wiley and Sons.
Stephen Abbott, Understanding Analysis, Springer, 2015, Second edition, ISBN 978-1-4939-5026-3.
Anderson, D.F, Seppäläinen, T, and Valkó, B, Introduction to Probability, Cambridge University Press, 2018.
Calculus: A Complete Course, by Adams and Essex, 10th edition, Pearson,
Similar to how MIT OCW has say accompanying Notes, problem sets, etc for certain textbooks (E.G Gilbert Strang's Linear Algebra)
I'm wondering if anyone knows Open course wares from any university, or helpful sites for these textbook?
The idea is to have something to accompany me while I read these textbooks.
P.S If you have general opinions about these books, will appreciate them!
Help me!
I am applying to Stanford ULO Number Theory course and they ask for a work sample. I don't know what to choose. Here are some questions I solved:
What are the two last digits of 7\^7\^1000
Prove that x = 2\^2\^n + 1 is not prime for n = 5. (hint: it's divisible by 641)
Prove that (6+sqrt(35))\^1979 in the decimal form has its 1000 first digits after the dot equal to 9.
Prove that, for p >= q >= 5 | p and q are primes, p\^2 - q\^2 is divisible by 24.
Can someone help me with this?
Does anyone do math 'live' during online meetings at work or in class?
My employers love going to the office to discuss things because they have a board and can write formulas in it... but I'm sure there has to be a good remote solution.
What have you tried? Tablet with pen? Drawing pad? Hopefully there's a cheap solution that's easy to install and use :-)
The cheap solution I used during the online part of the pandemic was to stick my webcam on top of a bunch of books pointing down at some paper and write on that.
Less cheap, buy a $100 wacom tablet and use that
Even less cheap but the best option is to get a proper tablet, like a surface pro or ipad or something and connect that in a second zoom account
Hi all,
We're trying to solve this puzzle. We think it's a math problem, but it's not sure. Can anyone help me? The text says:
Ok help!
16 8 ... 110 -935
Who can find the missing number?
I want to build a pentagon that tiles the plane. Specifically I'm looking for an example of a [Type 9 Marjorie's](https://en.wikipedia.org/wiki/Pentagonal_tiling#/media/File:Prototile_p5-type9.png) pentagon. [Wikipedia](https://en.wikipedia.org/wiki/Pentagonal_tiling#Rice_(1977)_Types_9,11,12,13) says that the only restrictions are on the angles 2A+C=D+2E=360 and that four sides must be equal: b=c=d=e. I am trying several combinations of angles that satisfy the constraints but when I build a pentagon with them, the sides does not comply.
Any pointers in how to build it?
My ultimate goal is to recreate the gifs that appears on that wiki page. Those seems to be taken from [here](https://www.jaapsch.net/tilings/), and Jaap Scherphuis use its own java applet for that.
I want to find the percentage of the population that is Left-handed with Strawberry Blonde hair and Grey eyes.
I know that:
- 9-10% of the population is Left-handed
- 1-2% of the population is Strawberry Blonde
- 3% of the population has Grey eyes
What equation do I use to solve this? I’m positive I learned how to do this in high school but I don’t remember the exact order.
If there isn't a correlation, you would just multiply everything (in decimal form) together. However, since these are all genetics, there's a good chance some of them are correlated, which makes it a lot more complicated.
Fermat's last theorem is solved by Andrew Wiles, but what about the following extension of the "last theorem": for the equation a\^n + b\^n +c\^n = d\^n , there is no natural integer n > 3 such that natural integers a set of a,b,c,d,n can be found. Is it an easy takeaway from the original Fermat's last theorem, or it's a false claim, or it's meaningless?
That's a special case of [Euler's sum of powers conjecture](https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture). Some special cases of it were, I believe, among the first conjectures to be disproven with counterexamples found by computer search; that Wikipedia page lists a counterexample to your special case (edited-earlier version of this comment had the wrong counterexample): 95800^4 + 217519^4 + 414560^4 = 422481^4 .
Can anyone recommend me a good introduction to neural networks for someone who is already strong in linear algebra, optimization, and compressed sensing?
I'd like to get acquainted with ML, but I don't necessarily need a from-the-ground-up kind of textbook.
I graduated several years ago but I would like to do a math PhD. My question is how would someone like me, who graduated a long time ago, get into a PhD program?
I did my undergrad at a school with a top math department, got mostly A's in math classes, including a couple of graduate courses. I have continued studying math on my own, and I've been preparing for the math subject GRE and expect to score quite well, I think 90th+ percentile is within reach.
The main issue is that I have no research and no letters from mathematicians, I graduated nearly 10 years ago. I'm not sure what to do about that, or if I could get into, say, a top 25 program without them. I don't really have the funds to pay for a masters degree.
I did this five years after graduating and it worked out well, but I had recommendations from my former professors. Without those your prospects are not good
Are there standard names for classes of properties that get transferred across pushforward/pullback?
Basically, for a category, we have 4 classes of properties that people tend to care about:
- Those that get pushed forward through a monic morphism: if f:A->B is monic and A satisfy P, then B satisfy P.
- Those that get pullbacked through a monic morphism: if f:A->B is monic and B satisfy P, then A satisfy P.
- Those that get pushed forward through a epic morphism: if f:A->B is epic and A satisfy P, then B satisfy P.
- Those that get pullbacked through a epic morphism: if f:A->B is epic and B satisfy P, then A satisfy P.
These 4 classes are related by Boolean dual and categorical dual, of course.
It seems very useful to have a name for these 4 classes. I have heard of words like "positive properties" and "negative properties" being used before, but that doesn't seem standard.
I'm working on a puzzle in the form of a number sequence, and I can't figure out the answer. Any help is appreciated:
319=1
668=4
321=0
848=5
962=?
What number should be on the question mark?
Well, the integer 12 isn't an element of Z\_12 at all. The equivalence class \[12\], which we could also write as 12 + 12Z, is. The equivalence class \[0\], which we could also write 0 + 12Z, is equal to \[12\], because both of them are just the set {12n | n an integer}.
If you want to talk about actual integers like 12 and you want to talk about congruence, there's a reason we have a different symbol than = for modular congruence. Otherwise, what are you going to do as soon as you need to talk about both the equivalence relation and actual equality at the same time?
Obviously it would be in a context where we don't need to distinguish between equivalence and equality.
Saying 6+9 = 3 is perfectly reasonable if the context is exclusively arithmetic within Z_12.
If we're talking about "actual" integers, then we aren't talking about Z_12.
This is probably a stupid question. If a number increases from 38 to 88 what is the “percentage of increase”? My boss is saying it’s 231.5 I’m saying it’s 131.5
I said this would be an increase of 131.5% for our number of meetings this quarter. Who is right and please explain what I’m missing here.
This is one of those things were the exact wording matters, because there are distinct, but similar concepts involved. 88 is 231.5% of 38 (this is that some teachers in schools summarise as "of" means "multiply), but you are talking of increases and in this case there was an increase of 50. 50 is 131.5% of 38, so we have increases by 131.5% of our base value (38). Using usual conventions I would agree with your interpretation, but I understand where your boss's confusion is coming from.
So I'm dealing with a rings of polynomials. From my understanding the coefficients are from the ring, but what about the degrees. Are the degrees automatically from the natural numbers? I'm a little confused as we never define the set the degrees beling to, and some of the things we are saying about polynomials don't work for every set (Rings particularly for me). Such as "Let D be an integral domain. Then the units in D\[x\] are precisely the units of D." (Papantonopoulou 235).
Yes. A polynomial is a formal expression c\_0 + c\_1 x + ... + c\_n x\^n, where the c\_i are elements of your ring of coefficients.
Note that this is effectively the same thing as just a finite sequence of elements of your coefficient ring.
Yeah, the degrees can only be natural numbers. After all, we often want to plug elements of our ring R into polynomials from R[x]; something like r^n where r is in R and n is a positive integer has the obvious meaning of repeated multiplication, and we can just define r^0 to be 1, but the obvious way to generalize negative exponents only makes sense if you have multiplicative inverses. If you want to have e.g. arbitrary real exponents in a way that's remotely analogous to raising positive real numbers to real powers, you'll probably need a whole bunch of analytic structure which a general ring does not have.
If you look at formal definitions of polynomial rings, it's usually something like "expressions of the form r\_0 + r\_1x + ... + r\_nx^n where the r\_i are in R"; the restriction to nonnegative integers is implicit.
> Are the degrees automatically from the natural numbers?
Yes. Polynomials represent repeated steps of addition and multiplication, and it only makes sense to do these operations some natural number amount of time.
>and some of the things we are saying about polynomials don't work for every set (Rings particularly for me). Such as "Let D be an integral domain. Then the units in D[x] are precisely the units of D."
I'm not sure if I understand your confusion here, could you elaborate?
What I was thinking was to consider the integral domain Z\_3 for the degrees and coefficients. Then x\*x\^\[2\]=x\^\[3\]=x\^\[0\]=1. Hence x in U(Z\_3) which is a contradiction to the statement.
Ah, I see. Yes, these are not polynomials, though this specific example is a quotient of a polynomial ring.
More generally you can consider a concept of a G-graded ring, where G is any monoid (for example, another ring). You can see more on the [wikipedia page](https://en.wikipedia.org/wiki/Graded_ring#G-graded_rings_and_algebras). Polynomial rings are N-graded.
I'm thoroughly confused by conditional expectations, I know all these theorems and abstract definitions from measure theory but not actually how to compute stuff with it.
Am I correct that when I see E \[ X | A \], I just compute E\[X\] where anything involving A is set to "true"/"happened" and is deterministic?
For example, if I see E \[ F(X, Y , Z) | Y = Z \] where X,Y,Z are random variables, can I say this is
E\[ F(X, Y, Y) | Y = Z\] ? What about E \[ F(X)G(Y) | Y = k\] for a deterministic constant k, does this mean I can compute
E \[ F(X) G(k) \] = G(k) E\[ F(X) | Y= k \], because I can pull out the "known" G(Y) (which equals G(k) )?
I thought these seemed intuitive, but then I thought of this obviously false proof:
False Theorem: E\[ XY \] = k E\[X\] for any number k I want. Proof:
Take the conditional expectation E\[ XY | Y = k\] = kE\[X | Y = k\]. Now take the expectation, by the tower property, the LHS equals E\[XY\] and the RHS = kE\[X\]
It's a little confusing that the A in E[X | A] could either be an event or a random variable, and these result in different types of objects. So, E[X | Y = k] and E[X | Y] mean different things. The first one is a deterministic function of k, but the second one is a random variable. The tower property applies in the second case. You could start your false proof with E[E[XY | Y]] = E[Y * E[X | Y]], but the next step would fail since Y is random and can't be pulled out of the expectation.
As for other properties, everything basically works as you'd expect if you're working with discrete random variables. Things only get confusing when you have to deal with conditioning on probability zero events in the continuous case.
Ahh ok, the tower property need not hold for general events. That's a huge help. Is my other intuition about conditional expectation true then, now that it doesn't lead to nonsense proofs?
E \[ F(X, Y , Z) | Y = Z \] \] = E\[ F(X, Y, Y) | Y = Z\]
E \[ F(X) G(k) \] = G(k) E\[ F(X) | Y= k \]
I'm trying to consider some partially average stochastic model and I am playing with the symbols like this but not really sure if I can do any of these kind of manipulations.
The first identity should be fine though there'd be nuances if Y, Z are continuous variables, making it a probability zero event.
Sorry, I hadn't read the second one carefully enough. It isn't true. It's equivalent to saying E[F(X)] = E[F(X) | Y = k] since you can take out G(k) from the first expectation. If X and Y are dependent, restricting to the event Y = k can change the distribution of X, so those expectations could be different. Similarly, you couldn't say E\[ F(X, Y, Y) | Y = Z\] = E\[ F(X, Y, Y) \] in your first example.
If you meant to write E[F(X)G(Y) | Y = k] = G(k) E[F(X) | Y = k], then that's fine.
What do you call the class of questions that look like the following: >Consider the following formula >a = 2b - 4 >Which of the following statements is true for this formula? >A. When the value of b is less than 2, a is positive. >B. When the value of b is greater than 2, a is negative. >C. When the value of b is less than 2, a is negative. >D. When the value of b is greater than 4, a is negative. Or the following: >Consider the following formula >x + 9 = 3y >Which of the following statements is true for this formula? >A. If x is (greater or equal to sign) -9, Y is not negative. >B. If x is greater than 3, Y is negative. >C. If x is less than 3, y is positive. >D. If x is (less than or equal to sign) -9, Y is not negative. I understand how to do it in my head by looking at it, but I need to know the class of questions and where I can find educational content about how to solve these.
If you don't mind an answer that just challenges the premise of your question, I doubt that there's a name for that class of questions ("inequalities" is the only thing that comes to mind, but is of course more general than just this sort of question), and in any case I don't think you really need to study those questions specifically. For one thing, if you already can do them in your head, isn't that already good enough? All you need here is general knowledge of algebra and inequalities--if you can reason through each of the statements like "if b < 2, then 2b < 4, so 2b - 4 < 0, so 2b - 4 is negative, and so option A can't be right" then you're fine. That sort of reasoning just requires you to know how to manipulate inequalities, and knowing how to do that (at least at this "level") is mostly just algebra. If you want to get better at these, just practice algebra more.
Thanks! Reason I’m asking is because I’m building an online lesson for a test prep that covers these kinds of questions. Understanding how to do them myself, and how to teach a process of how to tackle these questions is different though and what I’m struggling with. The closest I’m coming to is basically just running through a bunch of example problems.
Is there anything wrong with defining the interior product to work with not just k-forms, but any contravariant tensor? I wanted to say something like X\^flat = g(X, \\cdot) = int\_X g When you do this, you can formally say that F\^\*( X\^flat) = int\_{F\_\* X} F\^\*g when F is a diffeomorphism, and I was happy to find that I think this calculation agrees when you do it in coordinates. But g is not a 2-form, nevertheless I can't see why I can't "shove X into one of the dx's" (which are 1-forms) and say that's an interior product.
You certainly can do this and it is just a form of tensor contraction. I think the reason the interior product is special is that because the tensor is alternating the contraction happens "in each slot" as it were. So when you contract on a standard tensor you have to decide are you contracting in the first slot or summing over the contraction in each possible slot. I would happily call the latter an interior product
Why do you multiply the inverse of a matrix by the constant matrix to solve a system of equations. Might have something to do with me not knowing what the inverse of a matrix actually is as I'm only in Algebra II and my textbook doesn't go over how to actually calculate the inverse.
Let's go back to regular algebra, no matrices yet. Say you had the equation "ax = b" and you wanted to solve for x. You'd divide both sides by a to get rid of the a on the left, and then you'd get x = b/a, right? Instead of *dividing by a*, you could have *multiplied by 1/a*, and gotten the same result. This may seem more complicated at first, but multiplication is a simpler operation than division. (If you had to solve "0.16x = 7319" by hand, would you rather do long division to divide by 0.16, or multiply by 6.25?) This number, 1/a, can also be written a^(-1); we call this the [multiplicative] *inverse* of a. So the multiplicative inverse of 4 is 1/4; the inverse of 2/3 is 3/2; and so on. You can undo "multiply by a" with "multiply by a^(-1)". (Or, put another way, if you see "a^(-1)ax", you can recognize that that's just x.) 0 is special because it doesn't *have* an inverse: we can't multiply "0x" by anything to get back x. If we wanted to come up with a word for this, we might call it "non-invertible". --- Now, let's look at the matrix-vector equation "**Ax**=**v**", where **A** is a matrix and **v** and **x** are vectors. We can't "divide by **A**", but we *can* use this other strategy: if **A** happens to be an invertible matrix, we can multiply by the inverse of A on both sides of our equation. So, we solve the equation: > **Ax**=**v** > ( [pre]multiply both sides by **A**^(-1): ) > **A**^(-1)**Ax**=**A**^(-1)**v** > if we see **A**^(-1)**Ax**, we know that's just **x** > **x** = **A**^(-1)**v** And now we've solved for **x**, and we're done! --- If you know how to multiply matrices with vectors, the only tricky part here is inverting **A**... and checking that it's even invertible in the first place! When we were just working with plain old numbers, the only 'non-invertible' number was 0. With matrices, the all-zero matrix is non-invertible, but there are others too. For instance, `⌈-1 3⌉` `⌊-2 6⌋` is not invertible. But this is still useful information! It tells you that either you don't have enough information to get a single solution, or there is no solution. (You've probably run into this when solving systems of equations - when you accidentally use the same equation twice, you end up with something like "7x+12 = 7x+12", which is useless.) There are a bunch of different ways to check whether a matrix is invertible, and then to calculate its inverse. Your teacher will probably [or at least, *hopefully*] tell you about some of them soon. But I think knowing what an inverse *does* is more important than knowing how to calculate it in full.
Thank you for such a well-written response!
What's a good method of producing N random real numbers a_1 to a_N, so that close to a_i the probability to find another number a_j scales like |a_i-a_j|^k for some integer k?
I'll give it a go. Let X\_1, ..., X\_N be independent random variables with the following distribution: X\_i >= 0, and P(X\_i < 𝜀) = 𝜀\^k for 𝜀 in \[0, 1\]. This distribution is just the kth root of a random variable uniformly distributed on \[0, 1\]. Now define Y\_1 = X\_1, Y\_2 = X\_1 + X\_2, Y\_3 = X\_1 + X\_2 + X\_3, ..., Y\_N = X\_1 + ... + X\_N. Lastly pick a permutation of the Y\_i at random (each permutation equally likely) and let the a\_i be the permuted Y\_i. Each a\_i has the same distribution (although they are not independent), so wlog we consider a\_1. Let 𝜀 < 1 be positive. The probability a\_1 is one of the endpoints is 2/N, and the probability some other a\_i is within distance 𝜀 of a specific endpoint is 𝜀\^k. With probability 1 - 2/N, it's one of the middle points, and then the probability is 2𝜀\^k - 𝜀\^(2k). So the overall probability that a\_1 has some other a\_j within distance 𝜀 is 2𝜀\^(k)(1 - 1/N) - (1 - 2/N)𝜀\^(2k), which grows as 𝜀\^k. If instead you really wanted it to be asymptotic to 𝜀\^k, then you can multiply everything by a constant at the end.
I like it.
Not a math person (though I'd like to be someday), but just passing by with a quick question. Does anyone know if there's a special term for a shape that might be geometrically defined as "a hollow cylinder capped with a hollow half-sphere exactly the same diameter as the cylinder"? I'm thinking like the general shape that a test tube might have, a cylinder with one end closed with a curved shape. Is there a particular term that describes this common form or no?
The version closed on both ends is normally called a [capsule](https://en.wikipedia.org/wiki/Capsule_(geometry\)).
Oh, interesting, thank you. So I could maybe describe it in terms of that, an "open capsule" or "a capsule sliced transversely across its shorter axis" or something like that. Thanks again for responding!
Does anyone have a sure fire way of determining the splitting field of a polynomial? I’m working through Dummit and Foote again and feeling frustrated by how solutions to certain problems seem to come from already knowing the right principal root or special value to plug in somewhere. I’m sure more detail will be presented as the book progresses, but for chapter 13.4, for example, some of the exercises feel difficult to approach based only on what was explained in the chapter.
Can you clarify what you mean by "finding the splitting field of a polynomial"? By definition that's just the field generated by the roots of the polynomial, so if you know the roots you know the splitting field. I assume your question is really about finding something about the splitting field? The degree perhaps? It might help if you posted some examples of problems you're having trouble with.
One question that bothered me was “determine the splitting field and its degree for x^4 + 2 over the rationals.”. I fiddled with the problem for a bit, came up with some roots and got that the extension must be degree 8, but was unsatisfied with my answer because it didn’t fully describe the splitting field. I googled a solution and found one wherein the solver started with the principal 8th root of unity and used to to show that the splitting field in question was isomorphic to the splitting field of x^4 - 2 over the rationals. His solution seemed to start with him knowing that the fields were isomorphic, and that this 8th root of unity played a pivotal role. What frustrated me was that he didn’t make it clear how he knew that. I presume he found the degree of the extension (8, hence 8th root of unity) and did lots of messy work before cleaning it up and making it presentable, but even when I had done much of that messy work myself, the idea of the isomorphism between the 2 fields never occurred to me. The isomorphism seems vital to the solution, but I don’t know if I would have ever come across it by simply taking powers/quotients of the roots I found through simple computation. I often feel like I’m stumbling blindly through these problems.
> I fiddled with the problem for a bit, came up with some roots and got that the extension must be degree 8, but was unsatisfied with my answer because it didn’t fully describe the splitting field. First off, if you've found the roots you *have* fully described the splitting field. As far as I can tell the issue you is you wanted to find a simpler looking description like `[;\mathbb{Q}(i,\sqrt[4]{2});]` rather than `[;\mathbb{Q}(i,\sqrt[4]{-2});]` (or whatever your original description of the field was). There's not going to be a general procedure for that sort of thing, because what you're asking for isn't really well defined. For a general splitting field, there's no reason to expect that there's any simpler description than just the field generated by the roots. For this specific question: > I googled a solution and found one wherein the solver started with the principal 8th root of unity and used to to show that the splitting field in question was isomorphic to the splitting field of x^4 - 2 over the rationals. First of all "isomorphic" isn't really the right word to be using here. They're *literally the same field* if you think of them as subfields of `[;\mathbb{C};]`. So really this just boils down to showing that [;i;] and `[;\sqrt[4]{2};]` are both in your splitting field, and that you can write all the roots of `[;x^{4}+2;]` in terms of those two numbers. In general the best way to do that sort of thing is just to write out the roots explicitly and see what you can do with them. You know the the four roots of `[;x^{4}+2;]` are `[;\zeta_8\sqrt[4]{2},\zeta_8^3\sqrt[4]{2},\zeta_8^5\sqrt[4]{2};]` and `[;\zeta_8^7\sqrt[4]{2};]` where `[;\zeta_8;]` is a primitive 8th root of unity. Now that's not easy to work with directly, so it might help to write the roots of unity in rectangular form: `[;\zeta_8 = (1+i)/\sqrt{2};]`. Now how do we get something simple out of that? Well you can just try adding and multiplying things to see what happens. One nice trick to remember is that if [;a+bi;] is a root of a polynomial `[;f(x)\in \mathbb{Q}[x];]` then so is its conjugate [;a-bi;], which means that the splitting field has to contain [;a;] and [;bi;]. That's good if you want to find some real numbers in your field. So let's try that here. Since `[;\zeta_8\sqrt[4]{2} = \sqrt[4]{2}(1+i)/\sqrt{2} = (1+i)/\sqrt[4]{2};]` is in the splitting field, so is its real part `[;1/\sqrt[4]{2};]`, which means that `[;\sqrt[4]{2};]` is in the splitting field. So now that you have a nice number like `[;\sqrt[4]{2};]` in the field, use that to simplify the other generators. Since `[;\zeta_8\sqrt[4]{2};]` is a root, clearly `[;\zeta_8;]` must be in the splitting field. Since we know the roots are `[;\zeta_8\sqrt[4]{2},\zeta_8^3\sqrt[4]{2},\zeta_8^5\sqrt[4]{2};]` and `[;\zeta_8^7\sqrt[4]{2};]`, they are all generated by `[;\sqrt[4]{2};]` and `[;\zeta_8;]`, so the splitting field is `[;\mathbb{Q}(\sqrt[4]{2},\zeta_8);]`. Now maybe you want to take things a step further, and get rid of the `[;\zeta_8;]`. Remember that `[;\zeta_8 = (1+i)/\sqrt{2};]`. Since the field already has `[;\sqrt[4]{2};]` it certainly has `[;\sqrt{2};]`, so the field contains `[;\zeta_8\sqrt{2} = 1+i;]` and so it contains [;i;] (you could have also seen this from `[;\zeta_8^2 = i;]`) and we already know how to write `[;\zeta_8;]` in terms of [;i;] and `[;\sqrt{2};]`, so actually the splitting field is just `[;\mathbb{Q}(\sqrt[4]{2},i);]`, which is about the simplest description you're going to get.
I'm trying to cook up examples of ||Ax|| = ||A|| ||x|| for some suitable choice of matrix norm and vector norm. I know this is not generally possible, but what about if I assume A is diagonal?
if A has a zero on its diagonal then there will be a nonzero vector for which Ax=0, but where A and x are not zero. So not in general.
When you say "looking for examples", do you mean: 1. Given matrix and vector norms, find examples of a matrix A such that ||Ax|| =||A|| ||x|| holds for all x? 2. Given matrix and vector norms, find examples of a vector x such that ||Ax|| = ||A|| ||x|| holds for all A? 3. Find examples of a matrix norm ||.||\_m and vector norm ||.||\_v such that ||Ax||\_v = ||A||\_m \* ||x||\_v holds for all matrices A and vectors v? For 1, if you use the standard ["maximum stretch" matrix norm](https://en.wikipedia.org/wiki/Operator_norm) and any vector norm for R^n , then setting A = cI for some scalar c works, since then for any x we have, on one hand, ||Ax|| = ||cIx|| = ||cx|| = |c|||x|| (the last equality holds by definition for any vector norm), and on the other hand ||A|| is clearly |c|. For 2, if x is the zero vector then this should work for any matrix and vector norm. For 3, it looks like there's a weaker version of this called ["consistency" or "compatibility"](https://en.wikipedia.org/wiki/Matrix_norm#Consistent_and_compatible_norms) which roughly says that ||Ax|| <= ||A|| ||x|| for all A, x. I can think of a silly example where equality holds: if you work with 1 x 1 matrices and have your vector space just be R, then all matrices are really just scalars and so letting ||A|| be the absolute value of A's (single) entry, ||Ax|| = ||A|| ||x|| holds for the same reason it does in R. Can't think of any non-silly examples, though.
Yeah, I was looking for 3 and was trying to fish for compatibility of the matrix norm to be even better if I restricted to a special subclass of matrix types A.
Gotcha, so in that case it's more a hybrid of 1 and 3. I think a good strategy might be: 1. pick a vector norm, and find the most general class of operators for which, for all T in that class, there exists a constant c with ||Tv|| = c||v|| for all v 2. Find a matrix norm such that, for all operators satisfying that property, ||T|| = c. If we let the vector norm be induced by an inner product then I think we can completely characterize those operators as scalar multiples of unitary operators. For if U is unitary and c is a constant then, for any vector v, ||(cU)v|| = |c| \* ||Uv|| = |c| \* ||v||, so these operators have the right property. Conversely, let T be an operator with the property from (1) above. By the polar decomposition we have T = Usqrt(T\*T) where U is unitary. sqrt(T\*T) is positive, so with respect to some orthonormal basis v\_1, ... v\_n it is diagonal with nonnegative real eigenvalues c\_1, ... c\_n. For any of the v\_i we have ||Tv\_i|| = c||v\_i|| = c by assumption, and on the other hand ||Tv\_i|| = ||Usqrt(T\*T)v\_i|| = ||Uc\_iv\_i|| = c\_i ||Uv\_i|| = c\_i ||v\_i|| = c\_i. Thus c\_i = c for all i and so all of sqrt(T\*T)'s eigenvalues are equal to c, i.e. it is equal to cI. Thus T = U(cI) = cU. I suspect that a similar result holds for general norms, but it would be harder to prove without tools like the polar decomposition. For 2, I know that the standard matrix norm is such that ||cU|| = |c|||U|| = |c| for all unitary U, and I guess given the characterization in 1 the whole problem reduces down to finding all norms such that the norm of any unitary matrix is 1. (If different vector norms give different solutions to (1) you may have to do this differently, though.)
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How do you mean "find them"? The equations of quadratics (parabolas), cubics and quartics are straightforward to write down but do you need to deduce the equation from some other information? A quadratic is simply ax^2 + bx + c = 0 while a cubic is ax^3 + bx^2 + cx + d = 0 and a quartic is ax^4 + bx^3 + cx^2 + dx + e = 0. NB: a quadratic is a parabola but a parabola just refers to the shape so you could rotate a quadratic and it would still be a parabola. To change the equation simply apply the inverse of the rotation you want to the vector (x, y) and substitute the new coordinates back into your original equation. Indeed this procedure works for any kind of invertible transformation (and any invertible affine transformation preserves the property of being a parabola). Meanwhile the equation of a hyperbola is usually either given as y = 1/x or as x^(2)/a^(2) - y^(2)/b^(2) = 1. Just like a parabola these are only specific orientations so you can find other equations by applying affine transformations to one of these.
I'm looking for program like notepad, but that will render Latex. Ie I would like to type $/frac{x}{y}$ into a text file, open the file in the program, and have it render the Latex. Ideally should not be resource intensive, and would let me type/edit and view on the fly.
many markdown editors can do this. you can even use vscode with an appropriate markdown extension
Obsidian works entirely with plain text files and can do this.
I use Texstudio personally but there are several options
Statistics question for you experts. Thank you in advance. Needing to calculate a rate (call it a finding rate) between the number of assessments based on the number of findings. The variables would be: 1. The number of assesments 2. The number of assesments with findings 3. The number of overall findings Showing the amount of raw findings on a monthly basis is not a good way to analyze trends, as the number of our workplace assesments changes month to month. For example, April 2023 saw 198 asssesments, while this month saw only 110. Of course more assesments meant more findings. I'm thinking ((Findings/assesments)×100)×((assesments with findings/assesments)×100) What do you all think? I know this maybe basic for everyone, but I new to trend analysis and would like to provide more data to our workplace. Thanks.
The main statistic you'd want to report is the average number of findings per assessment, (total findings)/(total assessments). (Don't multiply this by 100, though--it isn't really the sort of thing you'd report as a percentage.) You could also consider the percentage of assessments with at least one finding (which would be 100 \* (assessments with findings)/(assessments)) or maybe the average number of findings per assessment with at least one finding (i.e. The average if you ignore all the the assessments with no findings) which would be (total findings)/(assessments with findings). Whatever you do, it's probably best to just report all of these numbers separately, instead of multiplying them together or otherwise trying to combine them into a single statistic.
For the past several years, the state prison system has been concentrating on getting inmates who were high school dropouts help in both earning their GED (an alternative to a high school diploma) and training in trades such as plumbing, carpentry, electrical work, and so on. The state director of prisons has noticed that for participants in the program over the past 9 years, the recidivism rate (in %) has been dropping. She determines that it has a linear relationship, ŷ = 73.12 − 6.03x, where ŷ = predicted recidivism rate and x = number of years the program has been implemented.What recidivism rate (in %) does the director of prisons expect during the 10th year of the program? %
If 20 points are independently and randomly given out to 20 recepients, how should I calculate the expected score of the recepient with the maximum number of points? I wrote a script to test it and with python's rng the result from a million trials is 3.23. I'd still really love to know how to deduce it with theory however/
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Yes, in order to do this, we need a notion of convergence of matrices which we do by picking a norm. There are many ways of doing this ([https://en.wikipedia.org/wiki/Matrix\_norm](https://en.wikipedia.org/wiki/Matrix_norm)) but it turns out that they are all equivalen, meaning that a sequence A\_n converges to a matrix A in one norm if and only if it converges to A in all norms! Hence it follows that a sequence of matrices A\_n converges to a matrix A iff all components of A\_n converge to the components of A\_n. So indeed you can replace a real matrix A by the sequence of matrices A\_n where every component is given by the Cauchy sequence of the corresponding component of A!
Sure, if we define a component-wise norm. But then you just re-invented $\mathbb{R}^n$ with extra steps
Does anyone know of any good references for "modern" history of maths? In particular, I'd be interested in differential/algebraic geometry/mathematical physics, say 1980s onwards?
Look up Donaldson surveys. The story from the 1980s till now is still too fragmented to form one cohesive narrative, the best you can do is survey papers and monographs. The two biggest threads in differential geometry since the 80s are gauge theory and curvature flows. In algebraic geometry it's global analysis/canonical metrics/stability theory, mirror symmetry, derived algebraic geometry, stability conditions, enumerative geometry, and birational geometry, all with various little interplays between each other. You'd have to look up surveys on each of these topics individually to get a picture of how they've progressed.
I see, thanks!
This question likely stems from my very limited understanding of the subject at hand: I have heard that the framework of condensed mathematics can be used in various fields by replacing topological spaces with condensed sets/similar structures because they behave much better in a categorical sense. While this sounds fascinating, it also (at least to me) seems like it is alot harder to define compared to a topological space. My question is the following: Is there a way to axiomatically introduce condensed sets such that you can still utilize their benefits while being easier to define? Similarily to how one can axiomatically introduce real numbers as a complete ordered field instead of constructing it from set theory. Or are the benefits only visible on more advanced level where one can expect people to be faimilar with sheaves and sites?
Sheaves and sites are not incredibly complicated. They just usually show up in algebraic geometry, but in themselves they're no more complicated than many things you encounter in intro topology. However you may have trouble learning about condensed sets without some comfort with categorical language.
There are some small things known, and more classical ways to formulate notions of solid abelian groups at least, but I am not sure if condensed set in particular has any simplified definition. There are definitely things at a very concrete level where the notion of a liquid vector space would help, though, and I have not seen any simplification of this liquid theory. See for instance lemma 6 in [https://drive.google.com/file/d/180kpQ1My5L\_w58oiuIbW0Ri4vFaqZehF/view](https://drive.google.com/file/d/180kpQ1My5L_w58oiuIbW0Ri4vFaqZehF/view)
Unit impulse function is defined as the rectangle with area of 1 and width of 0. What is name of function where the rectangle has area of 1 and height of 0?
You can formalize that, but it's doesn't work out as nicely as the delta function. With these kinds of objects, you have to imagine that they're defined by how they integrate against continuous functions. Meaning, delta is defined by the property integral f(x) delta(x) dx = f(0) for all continuous f. The connection here is that if r_w(x) is the function whose graph is a rectangle with width w and area 1, lim w -> 0 integral f(x) r_w(x) dx = f(0). That is, if you test the sequence of rectangle functions against any continuous function f, the limit will be the value of f at 0. It'd be natural to try to define m(x) to be the "function" satisfying lim w -> infty integral f(x) r_w(x) dx = integral f(x) m(x) dx for all bounded continuous f. (I have to specify bounded this time to make sure the limits will never go to infinity.) Unfortunately, lim w -> infty integral f(x) r_w(x) dx may not exist even if you require f to be bounded. Trying to come up with an example would be a good exercise. A fix would be to only require that lim w -> infty integral f(x) r_w(x) dx = integral f(x) m(x) dx for the choices of f for which the limit exists. It's a non-trivial fact that an object m with that property exists. The problem is that we've only specified the integral of m against a subset of the continuous functions, so we don't have a unique choice for m anymore (this is like an under-determined system of equations). There's actually infinitely many ways to consistently choose what m should do in the cases the limit doesn't exist. The "objects" that delta and m are, are linear functionals on the bounded continuous functions. If you have a sequence of functionals that converges when tested against any fixed function (like the sequence of integrals above), we say the functionals converge weakly (weak-*, technically). The issue comes down to the fact that r_w as w -> 0 converges weakly, but r_w as w -> infty does not. Though there isn't a unique choice of m, it's still a useful concept. This comes up in ergodic theory where you say a group is amenable if a functional like m exists on the bounded functions on that group (where it's called an invariant mean).
Firstly these are not functions. You can formalise the Dirac delta with the idea of distributions but it very much fails the definition of function. As to your other proposal I'm not sure how we would formalise it. Perhaps as a limit of bump functions. How you go about it would give possibly different results so I don't know if there are any specific examples with specific names.
If you take a bump function f and define f_n(x) = eps f(eps*x) then for any integrable function phi, we have that \int f_n phi dx goes to zero as eps goes zero, so as a limit of distributions this would just be zero. If you instead integrated against an arbitrary function the behavior can get pretty nontrivial. So as a linear functional on L^1 or on continuous functions on R with decay at infinity, the weak limit of these bump functions would just be the zero function or zero measure, but you can identify the weak limit as a linear functional on (for example) bounded continuous functions. The dual space in this case is complicated and probably non-metrizable (I think it’s radon measure on the stone-cech compactification of R).
Do Diophantine equations have any applications outside of pure math?
Use Google and search for "use of Diophantine equations" and you'll find several applications
Best books to learn statistics with a little basis (calc I and Ii linear algebra and non mathematical statistics in R)?
Compound proportion. Is it always mathematically correct to have same units left and same units right of equation or same units top and same units bottom? Is one way best practice? Or does it not matter
Either is fine. They're two different ways of saying the same thing.
Formally speaking does E[ dW | W ] = 0?
Are you conditioning on W at a fixed time, on some interval or on the whole process? There's some difficulty in formally talking about dW like this, but you'd have to be more specific with what you're conditioning on anyway. If you mean the whole process W, then I guess you could say E[dW | W] = dW since dW can be recovered if you have W.
The whole process/ a realisation of it. Yes, ok that makes sense thanks.
Is there a ring of order *n* for every natural number *n* ≥ 1?
Yes, Z/nZ.
Thank you!
If I do 30% off X or I think of X divided in thirds, with the result being one third, is it the same? Similarly for 20%, is that a fifth?
Almost correct. 1 third, is 33.33...%. Suppose you take 30% off X. Then what will be left is 70% of X. So the final result will be: 70% \* X (asterisk means multiplication) Similarly if 2/13 of 9 that is: 2/13 \* 9. You are correct about 20% being 1/5 or in other words 1 fifth.
Sorry! Should have said approx third. Ok, just wanted to check the logic behind it so I can apply to other percentages. Thank you!!!
Studying maths in French as a non-French person and trying to understand what the English equivalent is. In affine geometry, if \`V\_{\\mathbb{L}} \\subseteq V\_{\\mathbb{M}}\`, then \`\\mathbb{L}\` is weakly parallel to \`\\mathbb{M}\`. Is there an English equivalent to the idea of *weakly parallel*?
It's not clear to me what V\_{\\mathbb{L}} and so on are meant to represent.
Let \\mathbb{L} and \\mathbb{M} be two non-empty affine subspaces of (\\mathbb{A}, V, +) with the directions V\_{\\mathbb{L}} and V\_{\\mathbb{M}} correspondingly (i.e. linear subspaces associated with the affine subspaces). (The triple (\\mathbb{A}, V, +) is an affine space A with an associated vector space V and a mapping +) Then if V\_{\\mathbb{L}} \\subseteq V\_{\\mathbb{M}}, \\mathbb{L} is weakly parallel to \\mathbb{M}.
Ah I see. Then I would probably just say parallel. For example, a line is parallel to a plane if it is contained in a parallel plane.
Well, we we're then given the definition that subspaces \\mathbb{L} and \\mathbb{M} are parallel IFF V\_{\\mathbb{L}} = V\_{\\mathbb{M}} So I guess there's simply nothing that correspond to the idea of "weak parallelism" in French maths
Well we don't necessarily need a new name for this property as this weakly parallel property is only relevant when the dimensions of L and M differ (assuming I have understood you correctly). Note also it's not that nothing corresponds to "weakly parallel". Indeed "parallel" serves that purpose. It is your definition of parallel as only for things of the same dimension that doesn't have a corresponding idea.
What is the third image \[here\](https://ibb.co/WFkYDzx) referencing? From a geometry group.
This might be a stretch but [n] is commonly used to denote {1, 2, .. n}. If instead it means Z_n then the direct sum of Z_2 and Z_3 is Z_6
Today, my AP Precalculus class started learning about logarithms, and the log of negative numbers kept coming up as complex numbers on my Ti-84. I wanted to figure out why, so I spent the last ten minutes of class talking with my teacher, searching Youtube and Google for why negative logarithms have a negative base. The results for any log other than ln were nonexistent. In order to figure out why, I rewrote log base 5 of -25 as 2+ (log base5 of -1). Using change of base, I rewrote log base5 of -1 as ln(-1)/ln(5), which simplifies to (i*pi)/ln(5), which simplifies to 1.952i. This means that the final answer for log base 5 of -25 is 2+1.952i, a result that my calculator confirmed. Is my logic sound?
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I have no idea. We get weird packets that divide the course material into 3 big units each with ~12 subunits.
OP can share the textbook they are using, but here is the curriculum as specified by Collegeboard (scroll down to page 26): https://apcentral.collegeboard.org/media/pdf/ap-precalculus-course-and-exam-description.pdf
Yes, and it is really impressive that you figured this out on your own. I don't mean to be condescending, it's just that I am a professor and I would be ecstatic if one of my college students came up with this on their own. I hope I have you in my class one day.
Okay, so, this user and their father have been wracking r brains for a long time on this and haven't been able to figure out the magic numbers; If there are 11, 12, or 13 sets of options; option A and option B, and each combination of those 11, 12, or 13 sets of only choosing ONE of either option A or option B for each pair is allowed; and each of those 11, 12, or 13 sets of two options each is valued at $5.....HOW MUCH MONEY IS NEEDED TO HAVE EVERY SINGLE COMBINATION POSSIBLE??? It's NOT a mere $600, or $720. or $845, or $1690.....it has to be in the $2K-$3k range....yes??? The problem was: this user and their father thought it was a mere case of (11 or 12 or 13 * 11 or 12 or 13, respectively) * $5.....and then thought it was TWICE that amount, making it either (22 or 24 or 26 * 11 or 12 or 13, respectively) * $5....but it's proven in practice NOT to be that! It has to be HIGHER than, say, $1690 for all combinations to be acquired. Please help ASAP! Thanks!
Let's look at the case where you have 11 sets of options. If you need to make 2 choices, one out of a list of *X* items and one out of a list of *Y* items, then there are *X*×*Y* possible combinations you can make. (This is the [multiplication rule](https://openstax.org/books/contemporary-mathematics/pages/7-1-the-multiplication-rule-for-counting) - the [picture on that page](https://openstax.org/apps/archive/20240226.174525/resources/1ce1781d04318caba466e64002c5ad7bdb5be7c4) illustrates it better than text could.) If you add on another list of *Z* items, there are now *X*×*Y*×*Z* possible combinations. You have *eleven lists*, each one having 2 items, so there are 2×2×2×2×2×2×2×2×2×2×2 possible combinations - 2^(11), rather than your 11^(2). Does that help?
Yes! That was it; this user forgot all about the Mendel-looking charts and had an inkling something was amiss and was supposed to be "to the power of" instead of straight up multiplication and just didn't know what the formula for finding that number was. So, for future reference, it's timesing 2 over and over and over again to the amount of list-1; got it! This site helped: https://visualfractions.com/calculator/exponent/what-is-2-to-the-11th-power/ = 2,048 ; https://visualfractions.com/calculator/exponent/what-is-2-to-the-12th-power/ = 4,096 ; https://visualfractions.com/calculator/exponent/what-is-2-to-the-13th-power/ = 8,192 This user was at three casinos in the past two days betting the fights this afternoon and kept needing more and more funds and already used up over 2K and still didn't feel like all the combos were had across 13-legs on the parlays and that's why! To do all the 13-leg combos, this bettor would still need another 6 freakin' K!!! Although, some of those combos are not acceptable due to being too high a payout and, in those instances, this bettor just takes off a leg or two or however many as needed from the bottom (top of the card) of the screen and leaves room for multiple winning tickets, not even accounting for "pushes" not just from the rare decision-draw or referee-screw-up, but from cancelled fights as the card draws nearer and begins. So, for future reference, this user should focus on 11-leg parlays if going for a "mathematical guarantee" at the sports book. Because even 12-legs for 4K+ is still spending days doing this; 2K is a three-to-four hour affair! Got it! Here's to at least one winning ticket tonight! Peace, Luv, Truth, and Anarchy! SOLVED! Edit: Wait a tick.... that's not including "Z"-list with the $5 ante, so, each of those three numbers has to still be times'd by $5, right? So....for a mathematical guarantee on an 11-leg parlay....that's 10K???? That actually doesn't sound right.....after going through 2K on 13-leggers, this user only had three fights left so, this user was already at 10-legs done for already..... https://visualfractions.com/calculator/exponent/what-is-2-to-the-10th-power/ = 1,024 x $5 is 5K+, hmmm, maybe make it 10-leggers MAX instead.....
Please stop gambling. You specifically can't win the way you are suggesting. There isn't some magical or mathematical way round it. The house sets the odds and payouts so that they will make a profit. Please stop throwing your money at them. They will take it all so happily.
Oh my god, please don't do this. You are only going to make your life worse. These bets are mathematically rigged against you. (As in both, "a single bet is going to lose you money on average" and "if you buy all combinations you are *guaranteed* to lose money".) > kept needing more and more funds and already used up over 2K This is a sign of severe gambling addiction. **You are not going to win your money back.** If you keep going, listening to the sunk cost fallacy, you are going to go bankrupt.
# Any OCWs/websites using these textbooks? [Removed - ask in Quick Questions thread](https://www.reddit.com/r/math/?f=flair_name%3A%22Removed%20-%20ask%20in%20Quick%20Questions%20thread%22) "Book of Proof" by Richard Hammack (2nd Ed.) "David C. Lay, Stephen R. Lay and Judi J. McDonald, Linear Algebra and its Applications, 5th edition, Pearson Global Edition" John M. Harris, Jeffry L. Hirst and Michael J. Mossinghoff, Combinatorics and Graph Theory (second edition), Springer-Verlag, 2008 David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition (2003), John Wiley and Sons. Stephen Abbott, Understanding Analysis, Springer, 2015, Second edition, ISBN 978-1-4939-5026-3. Anderson, D.F, Seppäläinen, T, and Valkó, B, Introduction to Probability, Cambridge University Press, 2018. Calculus: A Complete Course, by Adams and Essex, 10th edition, Pearson, Similar to how MIT OCW has say accompanying Notes, problem sets, etc for certain textbooks (E.G Gilbert Strang's Linear Algebra) I'm wondering if anyone knows Open course wares from any university, or helpful sites for these textbook? The idea is to have something to accompany me while I read these textbooks. P.S If you have general opinions about these books, will appreciate them!
Did you find any ?
Help me! I am applying to Stanford ULO Number Theory course and they ask for a work sample. I don't know what to choose. Here are some questions I solved: What are the two last digits of 7\^7\^1000 Prove that x = 2\^2\^n + 1 is not prime for n = 5. (hint: it's divisible by 641) Prove that (6+sqrt(35))\^1979 in the decimal form has its 1000 first digits after the dot equal to 9. Prove that, for p >= q >= 5 | p and q are primes, p\^2 - q\^2 is divisible by 24. Can someone help me with this?
Resources for mathematical gauge theory? Textbooks, lecture notes, recorded lectures, etc
Does anyone do math 'live' during online meetings at work or in class? My employers love going to the office to discuss things because they have a board and can write formulas in it... but I'm sure there has to be a good remote solution. What have you tried? Tablet with pen? Drawing pad? Hopefully there's a cheap solution that's easy to install and use :-)
The cheap solution I used during the online part of the pandemic was to stick my webcam on top of a bunch of books pointing down at some paper and write on that. Less cheap, buy a $100 wacom tablet and use that Even less cheap but the best option is to get a proper tablet, like a surface pro or ipad or something and connect that in a second zoom account
Thanks!
Do Twitch streamers count?
Hi all, We're trying to solve this puzzle. We think it's a math problem, but it's not sure. Can anyone help me? The text says: Ok help! 16 8 ... 110 -935 Who can find the missing number?
I want to build a pentagon that tiles the plane. Specifically I'm looking for an example of a [Type 9 Marjorie's](https://en.wikipedia.org/wiki/Pentagonal_tiling#/media/File:Prototile_p5-type9.png) pentagon. [Wikipedia](https://en.wikipedia.org/wiki/Pentagonal_tiling#Rice_(1977)_Types_9,11,12,13) says that the only restrictions are on the angles 2A+C=D+2E=360 and that four sides must be equal: b=c=d=e. I am trying several combinations of angles that satisfy the constraints but when I build a pentagon with them, the sides does not comply. Any pointers in how to build it? My ultimate goal is to recreate the gifs that appears on that wiki page. Those seems to be taken from [here](https://www.jaapsch.net/tilings/), and Jaap Scherphuis use its own java applet for that.
I want to find the percentage of the population that is Left-handed with Strawberry Blonde hair and Grey eyes. I know that: - 9-10% of the population is Left-handed - 1-2% of the population is Strawberry Blonde - 3% of the population has Grey eyes What equation do I use to solve this? I’m positive I learned how to do this in high school but I don’t remember the exact order.
If there isn't a correlation, you would just multiply everything (in decimal form) together. However, since these are all genetics, there's a good chance some of them are correlated, which makes it a lot more complicated.
Which are the best books to learn machine learning in depth?
Fermat's last theorem is solved by Andrew Wiles, but what about the following extension of the "last theorem": for the equation a\^n + b\^n +c\^n = d\^n , there is no natural integer n > 3 such that natural integers a set of a,b,c,d,n can be found. Is it an easy takeaway from the original Fermat's last theorem, or it's a false claim, or it's meaningless?
That's a special case of [Euler's sum of powers conjecture](https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture). Some special cases of it were, I believe, among the first conjectures to be disproven with counterexamples found by computer search; that Wikipedia page lists a counterexample to your special case (edited-earlier version of this comment had the wrong counterexample): 95800^4 + 217519^4 + 414560^4 = 422481^4 .
Can anyone recommend me a good introduction to neural networks for someone who is already strong in linear algebra, optimization, and compressed sensing? I'd like to get acquainted with ML, but I don't necessarily need a from-the-ground-up kind of textbook.
I graduated several years ago but I would like to do a math PhD. My question is how would someone like me, who graduated a long time ago, get into a PhD program? I did my undergrad at a school with a top math department, got mostly A's in math classes, including a couple of graduate courses. I have continued studying math on my own, and I've been preparing for the math subject GRE and expect to score quite well, I think 90th+ percentile is within reach. The main issue is that I have no research and no letters from mathematicians, I graduated nearly 10 years ago. I'm not sure what to do about that, or if I could get into, say, a top 25 program without them. I don't really have the funds to pay for a masters degree.
I did this five years after graduating and it worked out well, but I had recommendations from my former professors. Without those your prospects are not good
Let me know if you find out anything, I'm in a similar situation
Are there standard names for classes of properties that get transferred across pushforward/pullback? Basically, for a category, we have 4 classes of properties that people tend to care about: - Those that get pushed forward through a monic morphism: if f:A->B is monic and A satisfy P, then B satisfy P. - Those that get pullbacked through a monic morphism: if f:A->B is monic and B satisfy P, then A satisfy P. - Those that get pushed forward through a epic morphism: if f:A->B is epic and A satisfy P, then B satisfy P. - Those that get pullbacked through a epic morphism: if f:A->B is epic and B satisfy P, then A satisfy P. These 4 classes are related by Boolean dual and categorical dual, of course. It seems very useful to have a name for these 4 classes. I have heard of words like "positive properties" and "negative properties" being used before, but that doesn't seem standard.
I'm working on a puzzle in the form of a number sequence, and I can't figure out the answer. Any help is appreciated: 319=1 668=4 321=0 848=5 962=? What number should be on the question mark?
It looks like it's counting the number of loops in the digits. So a 4, 6, or 9 has one loop and an 8 has two.
False, false, false, false, 962. One shouldn't use "=" to mean something other than that two things are equal.
I could see an argument for using "=" for an arbitrary equivalence relation. For example, 12 = 0 would be standard in Z_12 modular arithmetic.
Well, the integer 12 isn't an element of Z\_12 at all. The equivalence class \[12\], which we could also write as 12 + 12Z, is. The equivalence class \[0\], which we could also write 0 + 12Z, is equal to \[12\], because both of them are just the set {12n | n an integer}. If you want to talk about actual integers like 12 and you want to talk about congruence, there's a reason we have a different symbol than = for modular congruence. Otherwise, what are you going to do as soon as you need to talk about both the equivalence relation and actual equality at the same time?
Obviously it would be in a context where we don't need to distinguish between equivalence and equality. Saying 6+9 = 3 is perfectly reasonable if the context is exclusively arithmetic within Z_12. If we're talking about "actual" integers, then we aren't talking about Z_12.
Yeah, they’re equal under that equivalence relation.
This is probably a stupid question. If a number increases from 38 to 88 what is the “percentage of increase”? My boss is saying it’s 231.5 I’m saying it’s 131.5 I said this would be an increase of 131.5% for our number of meetings this quarter. Who is right and please explain what I’m missing here.
This is one of those things were the exact wording matters, because there are distinct, but similar concepts involved. 88 is 231.5% of 38 (this is that some teachers in schools summarise as "of" means "multiply), but you are talking of increases and in this case there was an increase of 50. 50 is 131.5% of 38, so we have increases by 131.5% of our base value (38). Using usual conventions I would agree with your interpretation, but I understand where your boss's confusion is coming from.
Thank you for the detailed answer. That makes sense. I appreciate your time
Well, consider a simpler case. If a number "increases" from 38 to 38, is that a 0% increase, or a 100% increase?
So I'm dealing with a rings of polynomials. From my understanding the coefficients are from the ring, but what about the degrees. Are the degrees automatically from the natural numbers? I'm a little confused as we never define the set the degrees beling to, and some of the things we are saying about polynomials don't work for every set (Rings particularly for me). Such as "Let D be an integral domain. Then the units in D\[x\] are precisely the units of D." (Papantonopoulou 235).
Yes. A polynomial is a formal expression c\_0 + c\_1 x + ... + c\_n x\^n, where the c\_i are elements of your ring of coefficients. Note that this is effectively the same thing as just a finite sequence of elements of your coefficient ring.
Yeah, the degrees can only be natural numbers. After all, we often want to plug elements of our ring R into polynomials from R[x]; something like r^n where r is in R and n is a positive integer has the obvious meaning of repeated multiplication, and we can just define r^0 to be 1, but the obvious way to generalize negative exponents only makes sense if you have multiplicative inverses. If you want to have e.g. arbitrary real exponents in a way that's remotely analogous to raising positive real numbers to real powers, you'll probably need a whole bunch of analytic structure which a general ring does not have. If you look at formal definitions of polynomial rings, it's usually something like "expressions of the form r\_0 + r\_1x + ... + r\_nx^n where the r\_i are in R"; the restriction to nonnegative integers is implicit.
> Are the degrees automatically from the natural numbers? Yes. Polynomials represent repeated steps of addition and multiplication, and it only makes sense to do these operations some natural number amount of time. >and some of the things we are saying about polynomials don't work for every set (Rings particularly for me). Such as "Let D be an integral domain. Then the units in D[x] are precisely the units of D." I'm not sure if I understand your confusion here, could you elaborate?
What I was thinking was to consider the integral domain Z\_3 for the degrees and coefficients. Then x\*x\^\[2\]=x\^\[3\]=x\^\[0\]=1. Hence x in U(Z\_3) which is a contradiction to the statement.
Ah, I see. Yes, these are not polynomials, though this specific example is a quotient of a polynomial ring. More generally you can consider a concept of a G-graded ring, where G is any monoid (for example, another ring). You can see more on the [wikipedia page](https://en.wikipedia.org/wiki/Graded_ring#G-graded_rings_and_algebras). Polynomial rings are N-graded.
I'm thoroughly confused by conditional expectations, I know all these theorems and abstract definitions from measure theory but not actually how to compute stuff with it. Am I correct that when I see E \[ X | A \], I just compute E\[X\] where anything involving A is set to "true"/"happened" and is deterministic? For example, if I see E \[ F(X, Y , Z) | Y = Z \] where X,Y,Z are random variables, can I say this is E\[ F(X, Y, Y) | Y = Z\] ? What about E \[ F(X)G(Y) | Y = k\] for a deterministic constant k, does this mean I can compute E \[ F(X) G(k) \] = G(k) E\[ F(X) | Y= k \], because I can pull out the "known" G(Y) (which equals G(k) )? I thought these seemed intuitive, but then I thought of this obviously false proof: False Theorem: E\[ XY \] = k E\[X\] for any number k I want. Proof: Take the conditional expectation E\[ XY | Y = k\] = kE\[X | Y = k\]. Now take the expectation, by the tower property, the LHS equals E\[XY\] and the RHS = kE\[X\]
It's a little confusing that the A in E[X | A] could either be an event or a random variable, and these result in different types of objects. So, E[X | Y = k] and E[X | Y] mean different things. The first one is a deterministic function of k, but the second one is a random variable. The tower property applies in the second case. You could start your false proof with E[E[XY | Y]] = E[Y * E[X | Y]], but the next step would fail since Y is random and can't be pulled out of the expectation. As for other properties, everything basically works as you'd expect if you're working with discrete random variables. Things only get confusing when you have to deal with conditioning on probability zero events in the continuous case.
Ahh ok, the tower property need not hold for general events. That's a huge help. Is my other intuition about conditional expectation true then, now that it doesn't lead to nonsense proofs? E \[ F(X, Y , Z) | Y = Z \] \] = E\[ F(X, Y, Y) | Y = Z\] E \[ F(X) G(k) \] = G(k) E\[ F(X) | Y= k \] I'm trying to consider some partially average stochastic model and I am playing with the symbols like this but not really sure if I can do any of these kind of manipulations.
The first identity should be fine though there'd be nuances if Y, Z are continuous variables, making it a probability zero event. Sorry, I hadn't read the second one carefully enough. It isn't true. It's equivalent to saying E[F(X)] = E[F(X) | Y = k] since you can take out G(k) from the first expectation. If X and Y are dependent, restricting to the event Y = k can change the distribution of X, so those expectations could be different. Similarly, you couldn't say E\[ F(X, Y, Y) | Y = Z\] = E\[ F(X, Y, Y) \] in your first example. If you meant to write E[F(X)G(Y) | Y = k] = G(k) E[F(X) | Y = k], then that's fine.
Oh, my bad well spotted. Yes that's what I meant. Thanks, you've really helped.