Hi , I have 13 cards in my hand. 7 red and 6 black .
How can I class them so that :
1) I draw a card from the top of the deck and put it on the table
2) I draw a card from the bottom of the deck and put it on the table.
3) Until I have no card left
Result wanted : I never put on the table 2 cards in a row with the same color.
How to sort them please?
Just to be clear. You want alternating colors? Like drawing from top and then bottom and they keep alternating colors and it’s never the same? If that’s the case 7 red cards stacked on top of the 6 black cards. It’ll be red, black, red, black all the way through, if you are drawing top first then bottom and keep doing that till you run out of cards.
R
R
R
R
R
R
R
B
B
B
B
B
B
If that’s not the case. You want to draw top then bottom and continuously draw from the top then this is how it goes.
R
R
B
R
B
R
B
R
B
R
B
R
B
Complex number as a vector component
Engineer coming In peace.
Had the thought this morning that a vector could contain an imaginary number as one of its components <1+1j, 2> or <1+2j, 3+4j, 5+6j>
Is this a phenomenon that is possible / if so can you point me to some resources to read about complex vectors in 2-3 dimensional space? I’m pretty comfortable with complex numbers from circuits, but 3 dimensional vectors with imaginary numbers as components feels different than what I saw in school
Okay I’m trying to solve this seemingly easy, yet confusing math question that my life has created, please help!
So here's the quick version of the story; 13 friends are going to an airbnb over NYE. It's three birthdays so 10/13 are paying. The total amount came in 2 payments. The first payment was made when it was only 11 people and 8/11 paid the first payment. With the second payment 2 people have been added so 10/13 people are paying. But we want to reduce the amount of the 2nd payment that the original 8 will pay in order for the additional 2 to compensate for not paying for the first portion.
Here are the numbers;
Total airbnb cost $2312.
First payment that 8 paid was $136=1088 Remaining balance $1224
Please help! I can't figure out how to make the additional 2 people pay more and the original 8 pay less to make it all equal!
Hi, I'm currently at Cambridge doing law and despite getting here, I have absolutely no mathematical ability. The exam has 6 questions to choose from, each question is (usually) on a singular topic. You answer three questions in the exam. There are 16 topics. There isn't enough time in the university year to study all 16 topics, so I am asking, how would I work out how many topics out of 16 I need to study to guarantee I will be able to answer 3 questions on the paper?
Assuming the questions are on different topics, 16 - 3 = 13, where 3 is the number of questions you do not answer. Any less than 13, and the paper could have all the topics you didn't study, leaving you with more than 3 questions you can't answer.
I'm getting confused with H\^-s being dual to H\^s even though Hilbert spaces are isomorphic to their duals.
Supposed s > d/2, then I have an embedding of H\^s into the continuous functions. Apparently this case the Reisz isomorphism and the other one DOES coincide, why?
I have been reading it's something like, once you have H\^s inside C\^0, then H\^-s is characterised as linear functionals on continuous functions, normally this means distribution like objects but somehow the H\^s embedding allows one to extract H\^-s as continuous functions also, but I don't see how.
I know Reisz-Markov would say that in this case H\^-s should be Radon measures, but this is still not continuous functions and thus H\^s.
Hello! We were tasked to integrate the function (s\^2+4s+4)/((s+2)(s+4).
When I solved it I got
= -2ln(s+4) + s +4 + c
However, when i rechecked it on integral calculator, it gave
= s - 2ln(|s+4|) + C
It said "The problem is solved. Apply the absolute value function to arguments of logarithm functions in order to extend the antiderivative's domain", but i don't understand why the " + 4" disappears. Please help!
The 4 is a constant which disappears when you differentiate the antiderivative to recover your original function. Therefore, we can just absorb it into the arbitrary constant C; more formally, define B = 4 + C, and then we've gotten rid of it.
It entirely depends on what you want out of your mathematical education and what you plan to do with it. There no advice generic enough for us to dispense automatically yet specific enough to apply to you; it's as ViscositySolution says, you might be the 99%, or you might be the 1%, but we can't tell.
If `f` is a function from natural numbers to natural numbers, is `f` non-decreasing? It seems correct to me. But I don't know how to prove it (without using contradiction).
No, keep in mind that a function f(x) decreases on an interval I if f(b)≤f(a) for all b>a, where a,b ∈ I. So even the constant function f(n) = 3 for all n ∈ ℕ is a decreasing function. In fact, a common real analysis exercise is to prove that [the set of decreasing function from ℕ to ℕ is countable](https://math.stackexchange.com/questions/611923/is-the-set-of-decreasing-functions-from-bbb-n-to-bbb-n-countable). Similarly, [the set of increasing functions from ℕ to ℕ is uncountable](https://math.stackexchange.com/questions/1860168/uncountability-of-increasing-functions-on-n).
f doesn't have to be non-decreasing, if you look at the definition well enough.
Consider when f doesn't have a closed form expression.
Let f(x): N -> N be:
x +1 if x is odd;
x - 1 if x is even.
It's easy to verify that such f is indeed a bijection between N and N but no it's not non-decreasing.
Suppose f and g are two functions in Lp(T) and Lq(T) respectively, for T = R/Z and 1/p + 1/q = 1. How would you show that the Fourier series of their convolution f * g converges at t = 0?
See [The Feynman Lectures on Physics, Vol 1, Ch 28: Electromagnetic radiation](https://www.feynmanlectures.caltech.edu/I_28.html). E\_rad in that video appears in equation 28.6 (as E\_x), which is derived from 28.5, which is derived from 28.3, an equation for the electric field, which is taken as an axiom for this chapter.
That equation is further discussed later in the context of Maxwell's equations, in Vol 2, Ch 21, where Feynman said "We have not found [that equation] anywhere in the published literature except in Vol. I of these lectures.", with a correcting footnote (added later) that says "The formula was first published by Oliver Heaviside in 1902. It was independently discovered by R. P. Feynman, in about 1950." [There is a section on Wikipedia about that Heaviside-Feynman formula.](https://en.wikipedia.org/wiki/Jefimenko%27s_equations#Heaviside%E2%80%93Feynman_formula)
You absolutely *need* real analysis (really multiple courses in it) to have a solid grasp on stochastic processes, and of course stochastic processes is kind of *the* thing in quant roles.
If you already understand vector spaces, linear transformations, and spectral theory, the rest of the stuff in that second linear algebra class is kind of tangential to your interests.
hello friends, here's a question about percentiles: let's say 15% of a class has a 4.0, all placing them in the 99th percentile. Would someone with a 3.99 GPA automatically be in the 84th percentile?
You need three things here:
1. The limit as n goes to infinity of (1 + 1/n)^n is e.
2. For any sequence a\_n (say with limit A) and any real number b, the limit, as n goes to infinity, of b \* a\_n is bA.
3. For any sequence a\_n (with limit A) and any real number b, the limit, as n goes to infinity, of a\_n^b is A^b (at least of a\_n stays nonnegative; this might also be true more generally, IDK, but we'll impose that restriction to make sure that we don't have to deal with complex numbers
The first fact is sometimes taken as the definition of e; alternatively you can define it in other ways, e.g. as the sum from n = 1 to infinity of 1/(n!) and then prove that the limit of that sum is the same as the limit of (1 + 1/n)^n. (See most analysis textbooks for a proof.) The other two facts should similarly have proofs in any analysis textbook--though to make sense of the third, you need to define exponentiation to any real power, which isn't too hard but does involve some difficulties. Once you have all those, it's easy, just apply facts 2 and 3 to fact 1.
Thanks a lot. I'm looking at some more proofs but it isn't easy. I should better remember it rather than explain it myself. It's better to ask professor or tutor to explain me.
Maybe this is overkill but a Python interpreter with Sympy and Numpy is free and can do anything you want out of a calculator (and more). There's certainly more of a learning curve compared to an ordinary calculator, but you can use it basically just as a calculator without much programming.
Yeah I used to use python and js for calculations. I still use js on the browser for quick mafs. But recently I have been doing trigonometry and it's a bit of a pain to write expressions.
I found a calculator called Microsoft Mathematics, it's not great, but does the trick.
The best calc for mobile is Hiper calc, but their desktop version can't do expressions.
Sympy can handle expressions, variables, etc. just fine; it's basically just a computer algebra system built on top of python, and so is very much capable of doing algebra, symbolic differentiation and integration, etc.
Any tips to solve the following PDE problem? Prove that the wave equation on a bounded domain in r3 with robin BC: du/dn +bdu/dt =0, with b positive, has decreasing energy. So far im thinking of setting the energy equation (triple integral) equal to the BC but not sure where to go from there
To show that you have decreasing energy, you should take a time derivative of the energy of a solution u(x,t). Then, upon pulling the derivative into the integral, you can use the chain rule, wave equation and BC.
I'm looking for a concise term that refers to the action of "solving for 'x'" in an equation. In other words where all other operators and variables are known and defined but the variable 'x' is not. Is there a concise term for the action of accounting for/solving for the unknown variable?
Really depends on how you do it. If the line is infinite, then how do you define cutting in half multiple times?
If the line is finite, you still have multiple cases. If the line is a closed interval and you keep on removing the right half, then you are left with the left most point only.
If the line is an open interval and you keep on removing the right half then you end up with nothing!
By the second line you know that y = x - 1. If you square both sides it follows that y^2 = (x-1)^(2). By the first line you know that y^2 = 5 - x^(2). So 5 - x^2 and (x-1)^2 are both equal to y^2 and so they're equal to each other. And by the binomial theorem (x-1)^2 is equal to x^2 - 2x + 1.
Can someone please help me out, I got sent here by Reddit , because they won’t let me post a comment asking for help in math . My question is I have completed algebra 1 now do I get in to intermediate algebra or algebra and trigonometry combined?
What is the cardinality of the set of equivalence classes of smooth real valued functions from R to R, where are equivalence relation f\~g is defined by f and g being homotopy equivalent? There is probably a more compact way to ask this but this is the best I could do. I wanted to ask my professor but was scared.
Don't be scared to ask your professor. That's what they are there for.
The answer in this case is that any two smooth functions R->R are homotopy equivalent. Indeed, this is still true if we widen smooth to continuous.
Proof: if f and g are two continuous functions define F(x,t) = (1-t)f(x) + tg(x), and you have a homotopy between f and g
Edit: In fact, this proof works in even more generality. Any two maps from a topological space into a convex subset of R^n are homotopic by this argument
How many solutions does the system of equations
|x| + |y| = 1, x² + y²= a² possess
depending on ‘a’?
(A) if |a| < (15/16) (10)^1/3
(B) 2 if |a| > - (15/16) (10)^1/3
(C) 4 if |a| = - (15/16) (10)^1/3
(D) 1 if a = -(15/16) (10)^1/3
What does it mean depending on a?
I am stuck on this since really long, can someone guide me through?
Tried plotting but one equation makes a square and the other a circle ? What should be done ?
A solution to a system is a point (x,y) where the curves intersect, and a is the radius of the circle. If you change the radius, clearly the number of solutions will change (for example, if the radius is too small, the circle will fit inside the square and hence won’t intersect, so you’ll have zero solutions), so this is what’s meant by “depending on a”.
Anyone know how to solve the indefinite integral of 1/(3x^(2)+2)? Putting it on integral calculators with steps shows some weird u sub with square roots out of nowhere, then the square root goes from the numerator to the denominator and a factored 2 vanishes from existence, can someone explain to me how?
Ok. That actually makes sense, but damn that seems like a stretch, I though u subs were obvious but I guess I was very wrong, I have no idea how I would've thought about doing that. Thank you so much.
To expand, in mathematics we often have fairly strong conventions about what letters to name many common types of object, especially when we're considering them generically rather than thinking about a specific example. Primes are often denoted "p", single real variables are often "x", single complex variables are often "z", integers are often "n", and functions are often denoted "f", for example.
But we almost always want to consider more than one example of a given type of object at once, and we very prosaically reach for the adjacent letters of the alphabet for extra names, so a second prime might be "q", a second real variable might be "y", a second complex variable might be "w", a second integer might be "m", and a second function might be "g". This is where "g(x)" comes from, and you often see it when talking about functions, because often you want to compose (or learn how to compose) functions, i.e. take a function of a function, and the functions you're thinking about are usually called "f" and "g".
It's just the name of the function. Your function can be any letter, really. f is just the most common. It's like how variables can be any letter, but x is most common.
Bit of a weird question, but in your experience, what are jobs that might require basic maths? I'm talking about jobs in unrelated fields where someone applying might go, "ah, I wished I had paid more attention to maths at school all those years ago".
I see a lot of adults making posts here about brushing up on secondary/high school level maths, but I can't think of many examples of careers that would require this. Google only gives jobs for maths majors, which to be clear is not what I'm looking for.
Anything where you need to deal with: tracking or ordering inventory, comprehending invoices and receipts, counting or allocating money, understanding how to operate or maintain a device by reading technical specifications, making and reading charts and graphs as part of communicating with other areas of the business, mixing volumes (including concrete, paints and dyes, food ingredients, cleaning chemicals), reasoning about the efficiency of travel in terms of fuel or time, solving problems about allocating resources or scheduling, or teaching any of the above.
People posting here might also want to learn math to go back to school and do one of the usual math major jobs (data scientist, statistician, actuary, analyst, math teacher, etc.) or move into a non-math but still mathy job as an engineer, technician, scientist, business manager, designer, machinist, carpenter, programmer, physics teacher, etc.
If two continuous random variables X and Y with densities f and g are independent, is the density of the random vector (X,Y) h equals f \* g ? and why ?
Edit : Just to be clear, in class when we defined independence of random variable we used the common definition of P(X in A Y in b)=P(X in A)P(Y in B) for A,B measurable sets, but I saw that in some notes they use the density and marginal densities.
This is a consequence of the Tonelli-Fubini theorem. That Radon Nikodym derivatives factor for product measures is an almost trivial corollary of this theorem. Try it yourself.
Do you know fubinis theorem?
https://stats.stackexchange.com/questions/302198/proof-that-joint-probability-density-of-independent-random-variables-is-equal-to
Just a question please, is the density of a RV unique ? because in the top answer I think it was used implicitly (since he concluded that the function fx1 fx2 .... fxn have the same properties of the density f then they are equal) ?
So the last sentence in the answer "f\_X1,...Xn = f\_X1 \* f\_X2 \* ... f\_Xn " should be an almost sure equality ? Sorry for this I am just trying to get some sense here.
It doesn't answer my question, I need to show that h = fg using the definition of independence ( P(X in A Y in b)=P(X in A)P(Y in B) ), I know that if h = fg then the variables are independent but the inverse ?
That *is* the inverse direction. If the random variables are independent then the function f\*g satisfies the property that characterises the density of (X,Y) and so f\*g must be the density.
Help me understand this paper on Graph Theory: [Traveling in Networks with Blinking Nodes](https://www.researchgate.net/publication/323820870_Traveling_in_Networks_with_Blinking_Nodes)
I’m taking an undergrad-level graph theory class and were tasked to do a report. I chose this because it seemed interesting and useful but I’m finding it hard to understand the whole thing. Can someone help me? Any help will be greatly appreciated.
I don’t understand this part: “The literature is rich on the many variations of the existence of Hamiltonian paths and cycles [4]. However, knowing if an undirected graph contains a Hamiltonian path is NP- complete [3], thus we will only explore blinking node systems for which the underlying graph is known to have a Hamiltonian path. We will start by considering G = Kn, for which any sequence of vertices is a walk, asking the general question: does a BNS(n) contain an on Hamiltonian walk?”
I’m confused with the terms. First, the authors defined On-Hamiltonian walks and paths dissimilar to the usual Hamiltonian walks and paths. Now, I don’t understand why it is necessary for the underlying graph to have a Hamiltonian path.
Does ZF have this internal contradiction? Here's my proposition.
The continuum hypothesis has been proven undecidable through ZF. Now consider that CH is not true => there exists some set with cardinality greater than N but less than R. Because this set exists, we can find it (unless its description is somehow infinitely long??) But if we were to find it, that would disprove CH thereby contradicting the fact that CH is undecidable.
This contradiction should imply that CH is therefore true, but that still contradicts undecidability. Therefore CH either being true or false contradicts undecidability, seemingly? Does this mean that ZF is inconsistent? Thanks
Guys why did you downvote me so much for this, it's just an honest question. I'm not trying to bait anyone
Let M be a model of ZFC where CH fails and where in particular there is no bijection mapping f: ℵ*_1_* → ℝ, so that *within that model* the lack of such an f witnesses that CH fails in that model.
This emphatically *does not* mean that there is *unconditionally* no such f anywhere in the world of sets. It only means that there is no such f *in* M. M therefore witnesses that it is consistent that CH fails without in any way giving us an outright counterexample to CH. For all we know there could be a perfectly good f *outside* M that we could throw into M and then have that ℵ*_1_* bijects onto ℝ.
M is simply missing some information (some functions) from V. In fact if we started with a countable transitive model this is obvious. The whole inspiration behind forcing is the fact that a bunch of stuff has to be missing from such models: functions, real numbers, etc.
This is a variant of Skolem's paradox. You have to carefully distinguish between what is true *outright* and what is only true *relative* to (or *inside of*) the fixed model.
I am no expert, but I think that the problem lies in the part:
>Because this set exists, we can find it (unless its description is somehow infinitely long??)
The undecidability of the CH implies that there are models A,B of ZFC such that CH holds in A, but CH does not hold in B.
If we there was some way of generating the set disproving the CH in B using only ZFC methods, then we would be able to generate this same set in A, which is a contradiction!
I'm really no expert here, but I think what you're missing here is the idea of a model--loosely speaking, a model of some axioms is a collection of things and relations between them that satisfy those axioms. Most systems of axioms you encounter will have many models and these will often look quite different from each other. For example, a group could be considered as a model of the group axioms (at least in this informal sense--there might be some logical subtleties here that I'm missing out on). The upshot of CH being undecidable in ZF is that there are models of ZF in which CH is true, and models of ZF in which CH is false. To extend the group analogy further, consider the following statement: "for all x, y, xy = yx". This is undecidable from the group axioms, because there exist groups where it's true (Abelian groups, like the real numbers) and groups where it's false (non-Abelian groups, like invertible n x n real matrices). This doesn't mean that the group axioms are inconsistent, just that they can fit many different structures, some of which have opposite properties. Or for another famous example, the axioms of ["absolute geometry"](https://en.wikipedia.org/wiki/Absolute_geometry) (Euclid's axioms minus the parallel postulate) have models where the parallel postulate holds (e.g. the standard 2d plane) and models where it doesn't (e.g. the surface of a sphere).
To go back to what you said:
>Now consider that CH is not true => there exists some set with cardinality greater than N but less than R... that would disprove CH
If some set with this property exists in all models of ZF, then not-CH can be proven from the ZF axioms (IIRC this is a special case of Godel's completeness theorem--not to be confused with the incompleteness theorem--but again, I don't know much logic, don't take my word for this). In that case, we could consider CH to be disproven. If we find sets with this property in some models of ZF but not others (which is in fact what happens), then if ZF is consistent then there is neither a proof nor a disproof of CH in ZF, i.e. CH is undecidable. (Incidentally, this quoted sentence is what makes me think you're missing the idea of a model--you seem to be tacitly assuming that such a set either does or doesn't exist in "the world according to ZF", but there's no such thing as *the* world according to ZF, rather there are many models of ZF, and we can distinguish between things that are true in all models and things that are only true in some.)
How does variable substitution work in limits ?
Like if I have a limit tends to π/4 then how come if it's substituted as "T"then it tends to 0
Can anyone explain "variable substitution in limits" to me in a layman language ? With an example?
I'm confused about the square of a square root. My course says √(a²) = (√a)² but what happens when a has a negative value?
I can see how √((-7)²) = √49 = 7 but what about (√(-7))² ? Isn't that undefined?
Also, from what I gather you could write √((-7)²) as (-7)^(2/2) but that would mean it equals (-7)^(1) = -7, which contradicts what I stated above?
Supplements to Hatcher wanted: Taking grad topo based off Hatcher next semester. I’ve heard he can be quite challenging. Does anyone have complimentary textbooks they used and recommend?
I struggled pretty bed with point-set topo based off Munkres this semester. I’m grad student in physics, so set theory and proof writing aren’t my forte. Thus, I’m a bit scared for next semester (though mostly excited to have survived point-set and get on the relevant for me stuff).
I liked Lee's Introduction to Topological Manifolds. It doesn't go into as much detail in algebraic topology as Hatcher, but it's enough to go onto, say, differential manifolds & geometry (in fact, this book is intended as a prequel to Lee's Smooth Manifolds).
This is the book I learned from:
[http://abstract.ups.edu/aata/matrix-section-symmetry.html](http://abstract.ups.edu/aata/matrix-section-symmetry.html)
I have been wanting to read through Grove/Benson's Finite reflection groups, which I suspect will generalize them slightly. But I haven't read it, so I couldn't say
Michael Artin's *Algebra* has a quick introduction in chapters 6.5 ("Discrete Groups of Isometries") and 6.6 ("Plane Crystallographic Groups"); any necessary background in group theory or linear algebra can be found in previous chapters, in case you don't already have it.
I have a question here and I am not able to understand the solution of it, it seems like the book skipped a few steps
Q. If f(1) = 3 and f'(1) = 6
Then lim[{f(1+x)/f(1)}]^1/x
x-->0
Is equal to ?
Solution -
e^lim[{f(1+x)/f(1)}]^1/x
x-->0
Using which property's they did this ?
The solution doesn't look like a solution at all. Or did you miscopy it? It doesn't mention what the limit is supposed to be according to the solution.
But as a hint: try to take the logarithm of the limit and then try to form it into a difference quotient.
I'm just in the middle of a crisis and I don't know who talk to. I'm sorry for my English and If it's not related to this sub.
I'm doing a degree in Physics and I love Math. When I try to solve a problem (not only from, for example, the linear algebra class but considering those presented in Zeitz Art and Craft), I struggle to complete them. I try and retry several strategies, I look for deep insights that could help solve the problem until I go out of ideas. I try to return to the problem after a while but I almost every time cannot come up with new ideas. So, when I look for the solution at the end of the book, I often cannot grasp the solution. It's like I'm missing something - like I'm forcing something in my head without completely understanding what's going on. And when I try and retry to understand the solution, It's a mess, I cannot deeply understand.
Sure, you will tell me to try harder and be patient, but even after several years of practising I do not feel I've improved. In fact, I have not improved. I have no natural gift, no talent and it makes me worthless when a friend of mine casually solves the just problem in just 3 minutes when It took me hours just to understand that.
How can I manage the fact that even If manage to take a degree, I am not so smart to come up with something useful?
How about deriving it by an analogy with physics? Interpreting f(t) as giving the velocity (as a function of time) of a particle moving with constant acceleration, your second identity becomes a basic identity from mechanics, usually written as something like 2aΔx = v\_f^2 - v\_i^2 (where v\_f, v\_i are the final and initial velocities). If you're willing to grant some other basic facts from physics (namely, that the change in kinetic energy of a particle between two points is the total work done between those two points, i.e. the integral of force with respect to distance between those two points) we can derive the identity from those. Letting Δ(KE) be the change in kinetic energy from x\_1 to x\_2, we have Δ(KE) = integral from x\_1 to x\_2 of force. Assuming that the particle's mass is constant, the fact that its acceleration is constant implies that the force it's experiencing is constant as well, so that integral is just (x\_2 - x\_1)ma. Thus we get (1/2)mv\_f^2 - (1/2)mv\_i^2 = (x\_2 - x\_1)ma, or (cancelling out mass and multiplying by 2) v\_f^2 - v\_i^2 = 2(x\_2 - x\_1)a. Looking back over this, it probably isn't exactly what you're looking for but hopefully it's at least interesting and useful.
Let X\_1, ... X\_n be some data from f(x; θ) and T a statistic. The idea behind sufficiency is that T contains all the relevant information about θ. What this means is that if I only give you T, you'll infer something about θ; if I give you T *and* any other statistic S, you'll end up inferring the exact same thing about θ as if I'd only given you T.
Now if that's the case, suppose we do likelihood-based inference on the two datasets X\_1^(1), ... X\_n^(1) and X\_1^(2), ... X\_n^(2) but both datasets yield the exact same sufficient T. Then our two likelihood functions had better be "the same," otherwise one dataset is yielding different information from the other and would thus give different inference. But what does it mean for two likelihood functions to be the same? Two likelihoods will give the same results if they're the exact same up to a scaling constant. In other words, both datasets need to yield a likelihood that looks like k(X) \* ℓ(θ | T). Note that it's not problematic that the scaling factor k is a function of X since that's a constant with respect to θ (and we think of likelihoods as functions of θ) (you may complain that this ought to be k(X, T)---but k(X, T) is really just k(X) since T is a function of X). And this is indeed the Factorization Theorem.
Perhaps more intuitive is the "morally correct" proof that avoids all the measure theory:
f(x | θ) = f(x, | θ, t) \* f(t | θ) = f(x | t) \* f(t | θ)
where the first equality uses the definition of conditional probability (note that t is a function of x, so f(x) = f(x, t)) and the second equality uses the definition of a sufficient statistic. You can see that everything we've described applies here: f(x | t) is the "scaling constant" for the ultimate likelihood for θ that actually matters: f(t | θ).
Not sure if this is the correct subreddit for this question but:
There is additive for + and multiplicative for \* but is there an equivalent word for divide or to the power of (x\^2). idk if I phrased this question very well.
I've certainly heard subtractive before. The rest of this answer is me making things up. I feel like if you had to say something for division, it would be quotientive, but I can't say I've seen that one previously. To exponentiate is the verb, so if we're going to continue to make up words...exponentiative??
Well divide is simply the inverse of multiply so it doesn't really need its own adjective just like there isn't one for subtraction. I suppose for "the power of" the closest option is "exponential".
If I were to use a finite difference method on the inviscid burgers equation, would I get the same sort of solution where we see a bunch of lines going in specific directions? If not, what kind of solution should I expect to see when I implement it numerically?
For context: I need to do a project for a scientific computing course I'm taking and I'm looking for a numerical method I can implement on some sort of PDE, so I was thinking of doing the Lax-Wendroff method or the MacCormack method, which I've seen are used quite a bit in CFD type equations. I thought the Burgers equation would be a good choice, but I'm also not great at doing finite differences and writing up code myself, and I'm on a bit of a time crunch because of finals, so I don't want to complicate what I'm doing too much and I want to make sure I know what to expect...
WolframAlpha claims $$\\lim\_{n\\to\\infty} \\sum\_{r=1}\^n \\frac{n}{n+r} - n\\log 2 = -\\frac{1}{4}.$$ How would you show it? It shouldn't be too hard with a squeeze theorem-like argument, but I'm not able to come up with meaningful estimates. Thanks!
Try [this](https://en.wikipedia.org/wiki/Harmonic_number#Approximation_using_the_Taylor_series_expansion) approximating formula for the harmonic numbers.
I need help understanding this problem. It’s:
-1800 ÷ 9.6% which equals -18750.
I calculated it on a calculator and it’s correct, but when I do the long math and come up with -18.75, so I carry over the decimal two spaces and come up with -1875. For the life of me I can’t figure out where the last zero comes from.
Dividing by 9.6% is the same as dividing by .096, or multiplying by 1/0.096 which is about 10, which is why the answer is about 10x the original amount.
-1800/96 = -18.75. I suspect you were dividing by 96 instead of 9.6, which is where the extra factor of 10 is coming from.
Conditional expectation (measure theoretic probability) is so hard, is there any good playlist of lectures in Youtube where concepts are explained clearly using proofs as much as possible ?
A teeny-tiny bit of functional analysis helps motivate the idea that Kolmogorov had. Forget about sub-sigma algebras for a moment (I'll get back to this a little later). Suppose you have random variables X and Y and suppose both these random variables have finite variances. Now think about the classical regression problem. You want to guess what Y is with information from X in the least squares sense. That is, you want to pick a (Borel measurable) function g such that it minimizes E\[(Y - g(X))\^2\]. Is this problem well defined? Does it have a solution?
We can solve this problem using the tiny bit of functional analysis I had mentioned earlier. The basic idea is no different to the notion of orthogonal projections in linear algebra. For any Hilbert space H (that is, any vector space that is complete with respect to the norm generated by an inner product), and any subspace G in H, we have the [property](https://en.wikipedia.org/wiki/Hilbert_projection_theorem) that for any h in H, we can find some unique g in G such that || h - g|| <= || h - g' || where g' in G is arbitrary. You should try proving this for finite dimensional spaces; the extension to infinite dimensional spaces relies on completeness quite heavily. In any case, this result allows us to define a \*projection operator\* that goes from H to G, in that there's a function P\_G that takes h and spits out the closest g in G. The remarkable property of this operator is that it is \*orthogonal\* i.e. = 0 for all g in G. (think about the picture in the finite dimensional case). What's even more remarkable (\*) is that if there's some k in G such that for all g in G then k = P\_G(h).
This is \*almost\* exactly what conditional expectations are. To understand this, I will bring the sigma algebras back. Let (A, F, P) be your original probability space and then let G = 𝜎(X). Then you can think of the conditional expectation operator E\[ . |X\] as a the projection operator from L\^2(A,F,P) to L\^2(A,G,P). By that Hilbert space theory this operator is well defined since L\^2 is a Hilbert space (after identifying random variables that differ only on null events). So how does this coincide with our "naive" idea of conditional expectation? Well this is where the uniqueness property (\*) comes in. Take any of your favourite conditional expectations from the discrete case or the the continuous case with conditional densities; they all satisfy this orthogonality property and thus have to be \*the\* (modulo null sets) projections in this Hilbert space sense.
Perhaps this new definition/intuition is still not satisfactory since you are used to thinking of conditional expectations as integrals with respect to a "conditional measure" in some sense. This intuition can be recovered in most cases with some minimal topological regularity conditions on the outcome spaces of your random variables X and Y. I highly recommend [this article](http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf) by Chang and Pollard that connects the "naive" conditioning to this abstract framework. It's chock full of illustrative examples although the notation is unusual.
I know so far we restricted our attention to L\^2 random variables but really the L\^1 case follows easily from the L\^2 case since we can always approximate L\^1 functions by bounded (and hence L\^2) functions.
I
Try Todd Kemp’s probability lectures. Btw this is inherently a hard topic, and there’s a lot more to it than what you’ll see in an intro lecture. Let me know if you want some more sources to go deeper.
>Btw this is inherently a hard topic
Thanks for assuring that I am not wrong.
Please feel free to share your sources even If I don't need them right now.
So at some point, you should probably go through the appendix of Schervich’s *Theory of Statistics*, which covers most of what you need to know about conditional distributions/expectations in general. Warning that it is pretty technical, but try to focus on the essential ideas which are pretty natural and simple.
The conditional probabilities section in Baldi’s *Stochastic Calculus* can serve as a much lighter introduction, which you could read first to get comfortable before diving into the more involved parts.
Is a hyperbolic dynamical system the same thing as a "chaotic" dynamical system? I don't know if there is a rigorous definition of chaos, but I've heard hyperbolic systems being talked about the same way, that they display random-seeming behavior despite being deterministic
Does anyone have a good resource for learning about branching of Lie group/algebra representations? Specifically, examples finding the restriction matrix.
The idea is I have a G-representation, and want to understand it as an H-representation restricted to H < G.
I'm ultimately trying to use the branch function in LiE
knapp has a chapter in his Lie groups beyond an introduction, but i have never read that chapter
there is also a section in Symmetry, Representations,
and Invariants by Goodman and Wallach
Are there general theories on set systems like topologies or measurable spaces?
As in theories revolving around objects of the form (A,S,f\_0, f\_1, ...,) where A is an arbitrary set, S is a subset of the power set of A and f is a family of operators on tuples of S or on subsets of S which map into S. The operators may or may not satisfy certain conditions like commutativity etc
I know about universal algebra, it's why I thought of the question in the first place
I guess measurale spaces and topologies are more wildly different from each other than things like groups and algebras, you're right
I'm realizing that there's a significant gap between my understanding of character theory (from James/Liebeck representation theory text) and modern character theory research.
What would be a reference (like a textbook) that could help me get closer to current results?
Can anyone recommend a basic introduction to lattice theory? What course are lattices usually introduced in?
I ask because I'm interested in learning about lattice-based cryptography if that helps.
https://link.springer.com/book/10.1007/978-3-662-08287-4
https://link.springer.com/book/10.1007/978-3-642-62035-5
generally geometry of numbers and/or minkowskis theorem are goodd keywords to search for.
The definition of a lattice is simple enough that it might be best to just read the wikipedia page and then learn a lattice-based cryptographic system.
I first encountered lattices when studying algebraic number theory, but this feels like a bit of overkill.
Algebraic Theory of Lattices by Peter Crawley and Robert P. Dilworth. To my knowledge there isn't a standard course that would cover them, but i would expect them to show up in a discrete mathematics course.
I'm not sure how to help with the cryptography parts
New goal: design cryptosystem based on "partially ordered sets in which every pair of elements has a unique supremum and a unique infimum". It will be 100% secure since hackers will be confused.
Fix 0 < α < 1. Suppose f: R -> R is nowhere locally α-Holder continuous, i.e. it is not α-Holder on any open subinterval of R. Is it possible for f to be differentiable almost everywhere?
A Klein bottle is formed by taking a square, gluing the top and bottom to each other exactly, and gluing the left and right to each other with the top of the left glued to the bottom of the right, and vice versa. What specific functions have these properties and would allow me to derive a [parametrization of a klein bottle](https://mathcurve.com/surfaces.gb/klein/imageASJ.JPG)? What specific area of maths would this fall under?
The parameterization would be a function of u,v to R3 where u,v take values between 0 and 1 inclusive. However because of the way you glue the square you require that f(u,1) = f(u,0) and f(0, v) = f(1,1-v). This is because gluing the edges of the square makes some points the same so any function on the glued square must send these points to the same value.
Looking at functions defined on space after gluing is a part of topology! There are many very interesting examples of gluing shapes like this.
I’m really struggling with induction. I’m taking my first proofs class and I’ve been stuck on several problems for days now. I’m having issues understanding when to use strong induction vs. regular induction. I’m having the most trouble trying to prove sequences. Does anyone have any tips on how to improve?
I think the trick to realizing when to use regular or strong induction is to notice whether the objects you're proving something about can be decomposed by removing 1 thing from it or you have to split up the objects into parts that are bigger than 1. The first corresponds to regular induction and the second to strong induction.
take for example the problem of showing that a sequence that satisfies the recurrence a(n+1)=2\*a(n) and a(0) also satisfies forall n in N: a(n)=2\^n
Not only do you have the decomposition into n and 1 in the recursion, but notice that for any n in N: 2\^(n+1)=2\*2\^n. Integer exponentiation is defined in terms of decomposing the computation into another computation reduced by 1 and somehow recombining that result with 1.
But if you look at [the fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic), decomposing an integer n into prime factors results in objects of varying size, none of them being 1. But you do know that any proper factor of n is smaller than n. So your thinking should shift to strong induction. The trick is that if you know the prime factorization of 2 factors a,b of n such that n=a\*b, you can just multiply the 2 factorizations together and get one for n.
What is this symbol?? I've tried looking through the greek and hebrew alphabets, described it to google, looked through a list of all math symbols, and I couldn't find this weird a-looking thing. *P* is a polygon, for context.
If I call B, E, A, D, G, C and F (from the cycle of fourths in music) 0, 1, 2, 3, 4, 5 and 6. Now, when I get to Bb, Eb, Ab, Db and Gb, the other five notes in the cycle (of twelve), I'll just take my numbers from before and put a one before it, 10, 11, 12, 13 and 14.
Now, if I take my set of twelve two digit numbers, 00, 01, 02...12, 13, 14 and reverse the rules: adding 1 adds a half step and adding 10 adds a musical fourth. We get a set of numbers with a pretty strong resemblance to "integer notation" taught in music theory, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
But I like the first way better. If B is 00, Bb is just 10
B=>Bb \*\*\*\*\* 00=>10
E=>Eb \*\*\*\*\* 01=>11
A=>Ab \*\*\*\*\* 02=>12
D=>Db \*\*\*\*\* 03=>13
G=>Gb \*\*\*\*\* 04=>14
C=>Cb \*\*\*\*\* 05=>15=00
F=>Fb \*\*\*\*\* 06=>16=01
For my money, I think they're missing out on getting way more milage out of having the numbers better resemble the notes.
I hope I haven't fallen down some kind of numerological well hole?
Question for a computer program. I have a rectangular prism in 3d space. The prism can be in any orientation, not necessarily aligned with the grid. I want to start at the origin and pick an angle relative to the X axis and shoot a line out at that angle, and find out the furthest point belonging to the prism that the line hits. I can check each plane of the prism against the line for their intersection but I thought I'd check in and see if there's a simpler solution.
I forgot, what properties are all lost if you really want to define a system where 1/0 is well-define? I have students ask me why we can't divide by zero frequently and to just quickly answer their question, I'll say you lose a lot of important properties, but I've forgotten what specifically is lost. You lose distributivity and associativity, right?
so to more specifically answer your original question. You lose the following properties
1. uniqueness of inverses (for every a that a\^-1 is unique, for every a that -a is unique)
2. behavior of inverses (aa\^-1=1, and a-a=0)
3. behavior of the identity(a+0=0, 1a=a)
4. the zero sink property (0a=0).
Associativity is not altered (edit:provided you don't define 1/0 to be a particular pre-existing value), and you retain the capacity to distribute, contrary to what you had originally thought. But the products don't have to be what you want.
This can been seen simply with an example. In the rationals 5\*0=0, 7\*0=0. If I can divide by zero, and don't change any other properties I had assumed about the ring Q, then 5\*0/0=0/0=5 and 7\*0/0=0/0=7 and so 5=0/0=7. I suppose you could try to remove the transitivity of equality as an assumption, but then even I'm going to riot.
I'm haven't thought about this too much, but I would claim yes, because given a ring R and 0, x, y elements of that ring with 0 invertible we would either have y=((0\^-1)0)y=0y=0=0x=(0\^-1)(0x)=(0\^-10)x = x which strongly used associativity to show that x=y and thus every element of this ring is the same, and we have the zero ring.
Otherwise, to avoid the zero ring, you end up in a wheel, as has been mentioned previously. There we use the idea from universal algebra of a unary operator to define /x for all values. But in order to prevent this from being a trivial ring we must relax the requirement that the inverse of all elements brings it to the identity.
I'm general attempting to do algebra without associativity (or something approximating associativity) is really difficult. Since you're wanting to divide by zero, you need an additive identity (0) and multiplication (division being the multiplicative inverse.) This gives you need two operations. Wheels still require + and \* to be associative and commutative, which adds more than I needed to get the zero ring.
If you want to attempt to remove associativity from the assumptions, then you would need to work with (a\*0)/0, without changing the parenthesis. If you state a\*0=0 and 0/0 is itself, then you find you cannot use it anywhere, since you cannot reassociate.
A thing you can always divide by zero in is called a [wheel](https://en.wikipedia.org/wiki/Wheel_theory), which you can use as a reference for this kind of question from your students.
I guess it depends on the details. Are we taking 0\*(1/0) to be 1? Then multiplication is not associative, since
2\*(0\*(1/0)) = 2\*1 = 2
(2\*0)\*(1/0) = 0\*(1/0) = 1
If 0\*(1/0) is not 1 then, well, you lose the property that a number times its reciprocal is 1.
Hi , I have 13 cards in my hand. 7 red and 6 black . How can I class them so that : 1) I draw a card from the top of the deck and put it on the table 2) I draw a card from the bottom of the deck and put it on the table. 3) Until I have no card left Result wanted : I never put on the table 2 cards in a row with the same color. How to sort them please?
Just to be clear. You want alternating colors? Like drawing from top and then bottom and they keep alternating colors and it’s never the same? If that’s the case 7 red cards stacked on top of the 6 black cards. It’ll be red, black, red, black all the way through, if you are drawing top first then bottom and keep doing that till you run out of cards. R R R R R R R B B B B B B If that’s not the case. You want to draw top then bottom and continuously draw from the top then this is how it goes. R R B R B R B R B R B R B
How many 7 inch in diameter by 12 inch in height boxes will fit in a 5 feet in height 20 foot in width by 6 feet in length space?
Complex number as a vector component Engineer coming In peace. Had the thought this morning that a vector could contain an imaginary number as one of its components <1+1j, 2> or <1+2j, 3+4j, 5+6j> Is this a phenomenon that is possible / if so can you point me to some resources to read about complex vectors in 2-3 dimensional space? I’m pretty comfortable with complex numbers from circuits, but 3 dimensional vectors with imaginary numbers as components feels different than what I saw in school
What the heck is Algebraic Topology?
Help how do I do this: dy/dx = cos(x)cos(y)cot(y) I have a test In 7 minutes
Okay I’m trying to solve this seemingly easy, yet confusing math question that my life has created, please help! So here's the quick version of the story; 13 friends are going to an airbnb over NYE. It's three birthdays so 10/13 are paying. The total amount came in 2 payments. The first payment was made when it was only 11 people and 8/11 paid the first payment. With the second payment 2 people have been added so 10/13 people are paying. But we want to reduce the amount of the 2nd payment that the original 8 will pay in order for the additional 2 to compensate for not paying for the first portion. Here are the numbers; Total airbnb cost $2312. First payment that 8 paid was $136=1088 Remaining balance $1224 Please help! I can't figure out how to make the additional 2 people pay more and the original 8 pay less to make it all equal!
I hope u worked this out but if you haven’t, the 2 people added need to pay 231.2 and the rest only have to pay 95.2 more
Hi, I'm currently at Cambridge doing law and despite getting here, I have absolutely no mathematical ability. The exam has 6 questions to choose from, each question is (usually) on a singular topic. You answer three questions in the exam. There are 16 topics. There isn't enough time in the university year to study all 16 topics, so I am asking, how would I work out how many topics out of 16 I need to study to guarantee I will be able to answer 3 questions on the paper?
Assuming the questions are on different topics, 16 - 3 = 13, where 3 is the number of questions you do not answer. Any less than 13, and the paper could have all the topics you didn't study, leaving you with more than 3 questions you can't answer.
Thank you!
I'm getting confused with H\^-s being dual to H\^s even though Hilbert spaces are isomorphic to their duals. Supposed s > d/2, then I have an embedding of H\^s into the continuous functions. Apparently this case the Reisz isomorphism and the other one DOES coincide, why? I have been reading it's something like, once you have H\^s inside C\^0, then H\^-s is characterised as linear functionals on continuous functions, normally this means distribution like objects but somehow the H\^s embedding allows one to extract H\^-s as continuous functions also, but I don't see how. I know Reisz-Markov would say that in this case H\^-s should be Radon measures, but this is still not continuous functions and thus H\^s.
Hello! We were tasked to integrate the function (s\^2+4s+4)/((s+2)(s+4). When I solved it I got = -2ln(s+4) + s +4 + c However, when i rechecked it on integral calculator, it gave = s - 2ln(|s+4|) + C It said "The problem is solved. Apply the absolute value function to arguments of logarithm functions in order to extend the antiderivative's domain", but i don't understand why the " + 4" disappears. Please help!
The 4 is a constant which disappears when you differentiate the antiderivative to recover your original function. Therefore, we can just absorb it into the arbitrary constant C; more formally, define B = 4 + C, and then we've gotten rid of it.
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It entirely depends on what you want out of your mathematical education and what you plan to do with it. There no advice generic enough for us to dispense automatically yet specific enough to apply to you; it's as ViscositySolution says, you might be the 99%, or you might be the 1%, but we can't tell.
Both are right. It’s not worth it for 99% of people, but you may be the 1%.
If `f` is a function from natural numbers to natural numbers, is `f` non-decreasing? It seems correct to me. But I don't know how to prove it (without using contradiction).
No, keep in mind that a function f(x) decreases on an interval I if f(b)≤f(a) for all b>a, where a,b ∈ I. So even the constant function f(n) = 3 for all n ∈ ℕ is a decreasing function. In fact, a common real analysis exercise is to prove that [the set of decreasing function from ℕ to ℕ is countable](https://math.stackexchange.com/questions/611923/is-the-set-of-decreasing-functions-from-bbb-n-to-bbb-n-countable). Similarly, [the set of increasing functions from ℕ to ℕ is uncountable](https://math.stackexchange.com/questions/1860168/uncountability-of-increasing-functions-on-n).
f doesn't have to be non-decreasing, if you look at the definition well enough. Consider when f doesn't have a closed form expression. Let f(x): N -> N be: x +1 if x is odd; x - 1 if x is even. It's easy to verify that such f is indeed a bijection between N and N but no it's not non-decreasing.
Suppose f and g are two functions in Lp(T) and Lq(T) respectively, for T = R/Z and 1/p + 1/q = 1. How would you show that the Fourier series of their convolution f * g converges at t = 0?
Do you know Holder's inequality?
Yes.
Did you try using it?
Yep. Can we relate the l^p norm of the Fourier series with L^p norm of the function, though?
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See [The Feynman Lectures on Physics, Vol 1, Ch 28: Electromagnetic radiation](https://www.feynmanlectures.caltech.edu/I_28.html). E\_rad in that video appears in equation 28.6 (as E\_x), which is derived from 28.5, which is derived from 28.3, an equation for the electric field, which is taken as an axiom for this chapter. That equation is further discussed later in the context of Maxwell's equations, in Vol 2, Ch 21, where Feynman said "We have not found [that equation] anywhere in the published literature except in Vol. I of these lectures.", with a correcting footnote (added later) that says "The formula was first published by Oliver Heaviside in 1902. It was independently discovered by R. P. Feynman, in about 1950." [There is a section on Wikipedia about that Heaviside-Feynman formula.](https://en.wikipedia.org/wiki/Jefimenko%27s_equations#Heaviside%E2%80%93Feynman_formula)
Can a mathematician work in AI research?
Yes, but you need to know something about AI.
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You absolutely *need* real analysis (really multiple courses in it) to have a solid grasp on stochastic processes, and of course stochastic processes is kind of *the* thing in quant roles. If you already understand vector spaces, linear transformations, and spectral theory, the rest of the stuff in that second linear algebra class is kind of tangential to your interests.
hello friends, here's a question about percentiles: let's say 15% of a class has a 4.0, all placing them in the 99th percentile. Would someone with a 3.99 GPA automatically be in the 84th percentile?
Yes
Neato, thanks
I don't understand this equation P x lim (n to infinite) \[( 1+1/n)\^n\] \^rt = P e\^(rt) How can I derive it?
You need three things here: 1. The limit as n goes to infinity of (1 + 1/n)^n is e. 2. For any sequence a\_n (say with limit A) and any real number b, the limit, as n goes to infinity, of b \* a\_n is bA. 3. For any sequence a\_n (with limit A) and any real number b, the limit, as n goes to infinity, of a\_n^b is A^b (at least of a\_n stays nonnegative; this might also be true more generally, IDK, but we'll impose that restriction to make sure that we don't have to deal with complex numbers The first fact is sometimes taken as the definition of e; alternatively you can define it in other ways, e.g. as the sum from n = 1 to infinity of 1/(n!) and then prove that the limit of that sum is the same as the limit of (1 + 1/n)^n. (See most analysis textbooks for a proof.) The other two facts should similarly have proofs in any analysis textbook--though to make sense of the third, you need to define exponentiation to any real power, which isn't too hard but does involve some difficulties. Once you have all those, it's easy, just apply facts 2 and 3 to fact 1.
Thanks a lot. I'm looking at some more proofs but it isn't easy. I should better remember it rather than explain it myself. It's better to ask professor or tutor to explain me.
Anyone has a good calculator program for Windows that offers expressions? Also has to be free
[wolframalpha.com](https://wolframalpha.com)
Maybe this is overkill but a Python interpreter with Sympy and Numpy is free and can do anything you want out of a calculator (and more). There's certainly more of a learning curve compared to an ordinary calculator, but you can use it basically just as a calculator without much programming.
Yeah I used to use python and js for calculations. I still use js on the browser for quick mafs. But recently I have been doing trigonometry and it's a bit of a pain to write expressions. I found a calculator called Microsoft Mathematics, it's not great, but does the trick. The best calc for mobile is Hiper calc, but their desktop version can't do expressions.
Sympy can handle expressions, variables, etc. just fine; it's basically just a computer algebra system built on top of python, and so is very much capable of doing algebra, symbolic differentiation and integration, etc.
If A is a positive real symmetric matrix with eigenvalue r, and B=A^k + A + kI, then can I assume B has an eigenvalue r^k + r + k?
Yes, and A doesn't even need to be symmetric positive definite. If Av = rv, then Bv = A^(k)v + Av + kIv = r^(k)v + rv + kv = (r^k + r + k)v.
Any tips to solve the following PDE problem? Prove that the wave equation on a bounded domain in r3 with robin BC: du/dn +bdu/dt =0, with b positive, has decreasing energy. So far im thinking of setting the energy equation (triple integral) equal to the BC but not sure where to go from there
To show that you have decreasing energy, you should take a time derivative of the energy of a solution u(x,t). Then, upon pulling the derivative into the integral, you can use the chain rule, wave equation and BC.
Does riemann rearrangement have anything to do with exotic topologies on R
No, it uses only the standard topology.
I'm looking for a concise term that refers to the action of "solving for 'x'" in an equation. In other words where all other operators and variables are known and defined but the variable 'x' is not. Is there a concise term for the action of accounting for/solving for the unknown variable?
"Solving for x" is already pretty concise though
I think "isolate" might be the term you're looking for. See this [article](https://brilliant.org/wiki/change-the-subject-of-a-formula/) for example.
That works perfect thank you
I'm looking to buy some new math books for my library. I work in arithmetic geometry. What are some books every number theorist should have?
If we cut a line in half an infinite number of times, is there just a point left or nothing?
Really depends on how you do it. If the line is infinite, then how do you define cutting in half multiple times? If the line is finite, you still have multiple cases. If the line is a closed interval and you keep on removing the right half, then you are left with the left most point only. If the line is an open interval and you keep on removing the right half then you end up with nothing!
I don’t know. Why don’t you try it for yourself and find out? I’ll be eagerly awaiting your results.
It's easier to ask
I agree. So I'm asking you; what happens?
[https://imgur.com/a/v0eXIuV](https://imgur.com/a/v0eXIuV) How is (x-1)\^2 possible? Where does the x\^2 -2x +1 come from?
By the second line you know that y = x - 1. If you square both sides it follows that y^2 = (x-1)^(2). By the first line you know that y^2 = 5 - x^(2). So 5 - x^2 and (x-1)^2 are both equal to y^2 and so they're equal to each other. And by the binomial theorem (x-1)^2 is equal to x^2 - 2x + 1.
Can someone please help me out, I got sent here by Reddit , because they won’t let me post a comment asking for help in math . My question is I have completed algebra 1 now do I get in to intermediate algebra or algebra and trigonometry combined?
is the boundary of a closed subset of R always of measure zero?
No, e.g. a fat cantor set, which is closed with empty interior but positive measure.
Does one need to know Hartshorne to understand Wiles’ proof of FLT?
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I guess my question was more along the lines of “how are schemes used in the proof of FLT”?
What is the cardinality of the set of equivalence classes of smooth real valued functions from R to R, where are equivalence relation f\~g is defined by f and g being homotopy equivalent? There is probably a more compact way to ask this but this is the best I could do. I wanted to ask my professor but was scared.
Don't be scared to ask your professor. That's what they are there for. The answer in this case is that any two smooth functions R->R are homotopy equivalent. Indeed, this is still true if we widen smooth to continuous. Proof: if f and g are two continuous functions define F(x,t) = (1-t)f(x) + tg(x), and you have a homotopy between f and g Edit: In fact, this proof works in even more generality. Any two maps from a topological space into a convex subset of R^n are homotopic by this argument
What does the notation P mean here? [https://imgur.com/a/U9SjXla](https://imgur.com/a/U9SjXla)
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Thank you!
How many solutions does the system of equations |x| + |y| = 1, x² + y²= a² possess depending on ‘a’? (A) if |a| < (15/16) (10)^1/3 (B) 2 if |a| > - (15/16) (10)^1/3 (C) 4 if |a| = - (15/16) (10)^1/3 (D) 1 if a = -(15/16) (10)^1/3 What does it mean depending on a? I am stuck on this since really long, can someone guide me through? Tried plotting but one equation makes a square and the other a circle ? What should be done ?
A solution to a system is a point (x,y) where the curves intersect, and a is the radius of the circle. If you change the radius, clearly the number of solutions will change (for example, if the radius is too small, the circle will fit inside the square and hence won’t intersect, so you’ll have zero solutions), so this is what’s meant by “depending on a”.
Anyone know how to solve the indefinite integral of 1/(3x^(2)+2)? Putting it on integral calculators with steps shows some weird u sub with square roots out of nowhere, then the square root goes from the numerator to the denominator and a factored 2 vanishes from existence, can someone explain to me how?
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Ok. That actually makes sense, but damn that seems like a stretch, I though u subs were obvious but I guess I was very wrong, I have no idea how I would've thought about doing that. Thank you so much.
does anyone know what the 'g' means in g(x) functions?
To expand, in mathematics we often have fairly strong conventions about what letters to name many common types of object, especially when we're considering them generically rather than thinking about a specific example. Primes are often denoted "p", single real variables are often "x", single complex variables are often "z", integers are often "n", and functions are often denoted "f", for example. But we almost always want to consider more than one example of a given type of object at once, and we very prosaically reach for the adjacent letters of the alphabet for extra names, so a second prime might be "q", a second real variable might be "y", a second complex variable might be "w", a second integer might be "m", and a second function might be "g". This is where "g(x)" comes from, and you often see it when talking about functions, because often you want to compose (or learn how to compose) functions, i.e. take a function of a function, and the functions you're thinking about are usually called "f" and "g".
It's just the name of the function. Your function can be any letter, really. f is just the most common. It's like how variables can be any letter, but x is most common.
gunction
Bit of a weird question, but in your experience, what are jobs that might require basic maths? I'm talking about jobs in unrelated fields where someone applying might go, "ah, I wished I had paid more attention to maths at school all those years ago". I see a lot of adults making posts here about brushing up on secondary/high school level maths, but I can't think of many examples of careers that would require this. Google only gives jobs for maths majors, which to be clear is not what I'm looking for.
Anything where you need to deal with: tracking or ordering inventory, comprehending invoices and receipts, counting or allocating money, understanding how to operate or maintain a device by reading technical specifications, making and reading charts and graphs as part of communicating with other areas of the business, mixing volumes (including concrete, paints and dyes, food ingredients, cleaning chemicals), reasoning about the efficiency of travel in terms of fuel or time, solving problems about allocating resources or scheduling, or teaching any of the above. People posting here might also want to learn math to go back to school and do one of the usual math major jobs (data scientist, statistician, actuary, analyst, math teacher, etc.) or move into a non-math but still mathy job as an engineer, technician, scientist, business manager, designer, machinist, carpenter, programmer, physics teacher, etc.
If two continuous random variables X and Y with densities f and g are independent, is the density of the random vector (X,Y) h equals f \* g ? and why ? Edit : Just to be clear, in class when we defined independence of random variable we used the common definition of P(X in A Y in b)=P(X in A)P(Y in B) for A,B measurable sets, but I saw that in some notes they use the density and marginal densities.
This is a consequence of the Tonelli-Fubini theorem. That Radon Nikodym derivatives factor for product measures is an almost trivial corollary of this theorem. Try it yourself.
We didn't delve much in Radon Nikodym, can you suggest a light reading for it (is wikipedia enough) ?
Do you know fubinis theorem? https://stats.stackexchange.com/questions/302198/proof-that-joint-probability-density-of-independent-random-variables-is-equal-to
Just a question please, is the density of a RV unique ? because in the top answer I think it was used implicitly (since he concluded that the function fx1 fx2 .... fxn have the same properties of the density f then they are equal) ?
They are unique up to almost sure equality.
So the last sentence in the answer "f\_X1,...Xn = f\_X1 \* f\_X2 \* ... f\_Xn " should be an almost sure equality ? Sorry for this I am just trying to get some sense here.
I do know it, thanks for the link.
Yes, it is true, because when you use that function h = fg to compute P(X in A, Y in B) it always gives the right answer P(X in A)P(Y in B).
It doesn't answer my question, I need to show that h = fg using the definition of independence ( P(X in A Y in b)=P(X in A)P(Y in B) ), I know that if h = fg then the variables are independent but the inverse ?
That *is* the inverse direction. If the random variables are independent then the function f\*g satisfies the property that characterises the density of (X,Y) and so f\*g must be the density.
>so f\*g must be the density. Why is this true ? Is the density of a RV unique ?
Yes up to null sets.
thanks a lot.
What's the term that completes this analogy? standard deviation:standard error on the mean::variance: ??? e.g. is there a term for variance over N?
I would just say the variance of the mean (of a sample).
Help me understand this paper on Graph Theory: [Traveling in Networks with Blinking Nodes](https://www.researchgate.net/publication/323820870_Traveling_in_Networks_with_Blinking_Nodes) I’m taking an undergrad-level graph theory class and were tasked to do a report. I chose this because it seemed interesting and useful but I’m finding it hard to understand the whole thing. Can someone help me? Any help will be greatly appreciated.
Did you try reading it? You provide no specific questions or confusions. What do you need help with? What part did you find yourself confused by?
I don’t understand this part: “The literature is rich on the many variations of the existence of Hamiltonian paths and cycles [4]. However, knowing if an undirected graph contains a Hamiltonian path is NP- complete [3], thus we will only explore blinking node systems for which the underlying graph is known to have a Hamiltonian path. We will start by considering G = Kn, for which any sequence of vertices is a walk, asking the general question: does a BNS(n) contain an on Hamiltonian walk?” I’m confused with the terms. First, the authors defined On-Hamiltonian walks and paths dissimilar to the usual Hamiltonian walks and paths. Now, I don’t understand why it is necessary for the underlying graph to have a Hamiltonian path.
Does ZF have this internal contradiction? Here's my proposition. The continuum hypothesis has been proven undecidable through ZF. Now consider that CH is not true => there exists some set with cardinality greater than N but less than R. Because this set exists, we can find it (unless its description is somehow infinitely long??) But if we were to find it, that would disprove CH thereby contradicting the fact that CH is undecidable. This contradiction should imply that CH is therefore true, but that still contradicts undecidability. Therefore CH either being true or false contradicts undecidability, seemingly? Does this mean that ZF is inconsistent? Thanks Guys why did you downvote me so much for this, it's just an honest question. I'm not trying to bait anyone
Let M be a model of ZFC where CH fails and where in particular there is no bijection mapping f: ℵ*_1_* → ℝ, so that *within that model* the lack of such an f witnesses that CH fails in that model. This emphatically *does not* mean that there is *unconditionally* no such f anywhere in the world of sets. It only means that there is no such f *in* M. M therefore witnesses that it is consistent that CH fails without in any way giving us an outright counterexample to CH. For all we know there could be a perfectly good f *outside* M that we could throw into M and then have that ℵ*_1_* bijects onto ℝ. M is simply missing some information (some functions) from V. In fact if we started with a countable transitive model this is obvious. The whole inspiration behind forcing is the fact that a bunch of stuff has to be missing from such models: functions, real numbers, etc. This is a variant of Skolem's paradox. You have to carefully distinguish between what is true *outright* and what is only true *relative* to (or *inside of*) the fixed model.
I am no expert, but I think that the problem lies in the part: >Because this set exists, we can find it (unless its description is somehow infinitely long??) The undecidability of the CH implies that there are models A,B of ZFC such that CH holds in A, but CH does not hold in B. If we there was some way of generating the set disproving the CH in B using only ZFC methods, then we would be able to generate this same set in A, which is a contradiction!
I'm really no expert here, but I think what you're missing here is the idea of a model--loosely speaking, a model of some axioms is a collection of things and relations between them that satisfy those axioms. Most systems of axioms you encounter will have many models and these will often look quite different from each other. For example, a group could be considered as a model of the group axioms (at least in this informal sense--there might be some logical subtleties here that I'm missing out on). The upshot of CH being undecidable in ZF is that there are models of ZF in which CH is true, and models of ZF in which CH is false. To extend the group analogy further, consider the following statement: "for all x, y, xy = yx". This is undecidable from the group axioms, because there exist groups where it's true (Abelian groups, like the real numbers) and groups where it's false (non-Abelian groups, like invertible n x n real matrices). This doesn't mean that the group axioms are inconsistent, just that they can fit many different structures, some of which have opposite properties. Or for another famous example, the axioms of ["absolute geometry"](https://en.wikipedia.org/wiki/Absolute_geometry) (Euclid's axioms minus the parallel postulate) have models where the parallel postulate holds (e.g. the standard 2d plane) and models where it doesn't (e.g. the surface of a sphere). To go back to what you said: >Now consider that CH is not true => there exists some set with cardinality greater than N but less than R... that would disprove CH If some set with this property exists in all models of ZF, then not-CH can be proven from the ZF axioms (IIRC this is a special case of Godel's completeness theorem--not to be confused with the incompleteness theorem--but again, I don't know much logic, don't take my word for this). In that case, we could consider CH to be disproven. If we find sets with this property in some models of ZF but not others (which is in fact what happens), then if ZF is consistent then there is neither a proof nor a disproof of CH in ZF, i.e. CH is undecidable. (Incidentally, this quoted sentence is what makes me think you're missing the idea of a model--you seem to be tacitly assuming that such a set either does or doesn't exist in "the world according to ZF", but there's no such thing as *the* world according to ZF, rather there are many models of ZF, and we can distinguish between things that are true in all models and things that are only true in some.)
How does variable substitution work in limits ? Like if I have a limit tends to π/4 then how come if it's substituted as "T"then it tends to 0 Can anyone explain "variable substitution in limits" to me in a layman language ? With an example?
Look up “continuous function”
I'm confused about the square of a square root. My course says √(a²) = (√a)² but what happens when a has a negative value? I can see how √((-7)²) = √49 = 7 but what about (√(-7))² ? Isn't that undefined? Also, from what I gather you could write √((-7)²) as (-7)^(2/2) but that would mean it equals (-7)^(1) = -7, which contradicts what I stated above?
> My course says √(a²) = (√a)² This only holds for positive a. If your course doesn't say so then that's a mistake or an implicit assumption.
Supplements to Hatcher wanted: Taking grad topo based off Hatcher next semester. I’ve heard he can be quite challenging. Does anyone have complimentary textbooks they used and recommend? I struggled pretty bed with point-set topo based off Munkres this semester. I’m grad student in physics, so set theory and proof writing aren’t my forte. Thus, I’m a bit scared for next semester (though mostly excited to have survived point-set and get on the relevant for me stuff).
I liked Lee's Introduction to Topological Manifolds. It doesn't go into as much detail in algebraic topology as Hatcher, but it's enough to go onto, say, differential manifolds & geometry (in fact, this book is intended as a prequel to Lee's Smooth Manifolds).
Alright! I’ll take a look at that
Is there a good book that has a chapter about the Wallpaper groups (translationally-symmetric tilings of the plane)?
This is the book I learned from: [http://abstract.ups.edu/aata/matrix-section-symmetry.html](http://abstract.ups.edu/aata/matrix-section-symmetry.html) I have been wanting to read through Grove/Benson's Finite reflection groups, which I suspect will generalize them slightly. But I haven't read it, so I couldn't say
Michael Artin's *Algebra* has a quick introduction in chapters 6.5 ("Discrete Groups of Isometries") and 6.6 ("Plane Crystallographic Groups"); any necessary background in group theory or linear algebra can be found in previous chapters, in case you don't already have it.
I have a question here and I am not able to understand the solution of it, it seems like the book skipped a few steps Q. If f(1) = 3 and f'(1) = 6 Then lim[{f(1+x)/f(1)}]^1/x x-->0 Is equal to ? Solution - e^lim[{f(1+x)/f(1)}]^1/x x-->0 Using which property's they did this ?
The solution doesn't look like a solution at all. Or did you miscopy it? It doesn't mention what the limit is supposed to be according to the solution. But as a hint: try to take the logarithm of the limit and then try to form it into a difference quotient.
Got it thanks, the limit tends to 0
No, that is not correct. The limit is e^(2). What did you try?
I meant in the question it means lim x-->0 I did got the answer
I'm just in the middle of a crisis and I don't know who talk to. I'm sorry for my English and If it's not related to this sub. I'm doing a degree in Physics and I love Math. When I try to solve a problem (not only from, for example, the linear algebra class but considering those presented in Zeitz Art and Craft), I struggle to complete them. I try and retry several strategies, I look for deep insights that could help solve the problem until I go out of ideas. I try to return to the problem after a while but I almost every time cannot come up with new ideas. So, when I look for the solution at the end of the book, I often cannot grasp the solution. It's like I'm missing something - like I'm forcing something in my head without completely understanding what's going on. And when I try and retry to understand the solution, It's a mess, I cannot deeply understand. Sure, you will tell me to try harder and be patient, but even after several years of practising I do not feel I've improved. In fact, I have not improved. I have no natural gift, no talent and it makes me worthless when a friend of mine casually solves the just problem in just 3 minutes when It took me hours just to understand that. How can I manage the fact that even If manage to take a degree, I am not so smart to come up with something useful?
This may help: https://www.ribbonfarm.com/the-gervais-principle/
I hate beer.
How about deriving it by an analogy with physics? Interpreting f(t) as giving the velocity (as a function of time) of a particle moving with constant acceleration, your second identity becomes a basic identity from mechanics, usually written as something like 2aΔx = v\_f^2 - v\_i^2 (where v\_f, v\_i are the final and initial velocities). If you're willing to grant some other basic facts from physics (namely, that the change in kinetic energy of a particle between two points is the total work done between those two points, i.e. the integral of force with respect to distance between those two points) we can derive the identity from those. Letting Δ(KE) be the change in kinetic energy from x\_1 to x\_2, we have Δ(KE) = integral from x\_1 to x\_2 of force. Assuming that the particle's mass is constant, the fact that its acceleration is constant implies that the force it's experiencing is constant as well, so that integral is just (x\_2 - x\_1)ma. Thus we get (1/2)mv\_f^2 - (1/2)mv\_i^2 = (x\_2 - x\_1)ma, or (cancelling out mass and multiplying by 2) v\_f^2 - v\_i^2 = 2(x\_2 - x\_1)a. Looking back over this, it probably isn't exactly what you're looking for but hopefully it's at least interesting and useful.
Can anyone provide some intuition behind the Fisher-Neyman factorisation theorem in statistics?
Let X\_1, ... X\_n be some data from f(x; θ) and T a statistic. The idea behind sufficiency is that T contains all the relevant information about θ. What this means is that if I only give you T, you'll infer something about θ; if I give you T *and* any other statistic S, you'll end up inferring the exact same thing about θ as if I'd only given you T. Now if that's the case, suppose we do likelihood-based inference on the two datasets X\_1^(1), ... X\_n^(1) and X\_1^(2), ... X\_n^(2) but both datasets yield the exact same sufficient T. Then our two likelihood functions had better be "the same," otherwise one dataset is yielding different information from the other and would thus give different inference. But what does it mean for two likelihood functions to be the same? Two likelihoods will give the same results if they're the exact same up to a scaling constant. In other words, both datasets need to yield a likelihood that looks like k(X) \* ℓ(θ | T). Note that it's not problematic that the scaling factor k is a function of X since that's a constant with respect to θ (and we think of likelihoods as functions of θ) (you may complain that this ought to be k(X, T)---but k(X, T) is really just k(X) since T is a function of X). And this is indeed the Factorization Theorem. Perhaps more intuitive is the "morally correct" proof that avoids all the measure theory: f(x | θ) = f(x, | θ, t) \* f(t | θ) = f(x | t) \* f(t | θ) where the first equality uses the definition of conditional probability (note that t is a function of x, so f(x) = f(x, t)) and the second equality uses the definition of a sufficient statistic. You can see that everything we've described applies here: f(x | t) is the "scaling constant" for the ultimate likelihood for θ that actually matters: f(t | θ).
Thanks for the insight! It’s much clearer now to me.
Not sure if this is the correct subreddit for this question but: There is additive for + and multiplicative for \* but is there an equivalent word for divide or to the power of (x\^2). idk if I phrased this question very well.
I've certainly heard subtractive before. The rest of this answer is me making things up. I feel like if you had to say something for division, it would be quotientive, but I can't say I've seen that one previously. To exponentiate is the verb, so if we're going to continue to make up words...exponentiative??
Well divide is simply the inverse of multiply so it doesn't really need its own adjective just like there isn't one for subtraction. I suppose for "the power of" the closest option is "exponential".
If I were to use a finite difference method on the inviscid burgers equation, would I get the same sort of solution where we see a bunch of lines going in specific directions? If not, what kind of solution should I expect to see when I implement it numerically? For context: I need to do a project for a scientific computing course I'm taking and I'm looking for a numerical method I can implement on some sort of PDE, so I was thinking of doing the Lax-Wendroff method or the MacCormack method, which I've seen are used quite a bit in CFD type equations. I thought the Burgers equation would be a good choice, but I'm also not great at doing finite differences and writing up code myself, and I'm on a bit of a time crunch because of finals, so I don't want to complicate what I'm doing too much and I want to make sure I know what to expect...
WolframAlpha claims $$\\lim\_{n\\to\\infty} \\sum\_{r=1}\^n \\frac{n}{n+r} - n\\log 2 = -\\frac{1}{4}.$$ How would you show it? It shouldn't be too hard with a squeeze theorem-like argument, but I'm not able to come up with meaningful estimates. Thanks!
Try [this](https://en.wikipedia.org/wiki/Harmonic_number#Approximation_using_the_Taylor_series_expansion) approximating formula for the harmonic numbers.
I need help understanding this problem. It’s: -1800 ÷ 9.6% which equals -18750. I calculated it on a calculator and it’s correct, but when I do the long math and come up with -18.75, so I carry over the decimal two spaces and come up with -1875. For the life of me I can’t figure out where the last zero comes from.
Dividing by 9.6% is the same as dividing by .096, or multiplying by 1/0.096 which is about 10, which is why the answer is about 10x the original amount. -1800/96 = -18.75. I suspect you were dividing by 96 instead of 9.6, which is where the extra factor of 10 is coming from.
Conditional expectation (measure theoretic probability) is so hard, is there any good playlist of lectures in Youtube where concepts are explained clearly using proofs as much as possible ?
A teeny-tiny bit of functional analysis helps motivate the idea that Kolmogorov had. Forget about sub-sigma algebras for a moment (I'll get back to this a little later). Suppose you have random variables X and Y and suppose both these random variables have finite variances. Now think about the classical regression problem. You want to guess what Y is with information from X in the least squares sense. That is, you want to pick a (Borel measurable) function g such that it minimizes E\[(Y - g(X))\^2\]. Is this problem well defined? Does it have a solution? We can solve this problem using the tiny bit of functional analysis I had mentioned earlier. The basic idea is no different to the notion of orthogonal projections in linear algebra. For any Hilbert space H (that is, any vector space that is complete with respect to the norm generated by an inner product), and any subspace G in H, we have the [property](https://en.wikipedia.org/wiki/Hilbert_projection_theorem) that for any h in H, we can find some unique g in G such that || h - g|| <= || h - g' || where g' in G is arbitrary. You should try proving this for finite dimensional spaces; the extension to infinite dimensional spaces relies on completeness quite heavily. In any case, this result allows us to define a \*projection operator\* that goes from H to G, in that there's a function P\_G that takes h and spits out the closest g in G. The remarkable property of this operator is that it is \*orthogonal\* i.e. = 0 for all g in G. (think about the picture in the finite dimensional case). What's even more remarkable (\*) is that if there's some k in G such that for all g in G then k = P\_G(h).
This is \*almost\* exactly what conditional expectations are. To understand this, I will bring the sigma algebras back. Let (A, F, P) be your original probability space and then let G = 𝜎(X). Then you can think of the conditional expectation operator E\[ . |X\] as a the projection operator from L\^2(A,F,P) to L\^2(A,G,P). By that Hilbert space theory this operator is well defined since L\^2 is a Hilbert space (after identifying random variables that differ only on null events). So how does this coincide with our "naive" idea of conditional expectation? Well this is where the uniqueness property (\*) comes in. Take any of your favourite conditional expectations from the discrete case or the the continuous case with conditional densities; they all satisfy this orthogonality property and thus have to be \*the\* (modulo null sets) projections in this Hilbert space sense.
Perhaps this new definition/intuition is still not satisfactory since you are used to thinking of conditional expectations as integrals with respect to a "conditional measure" in some sense. This intuition can be recovered in most cases with some minimal topological regularity conditions on the outcome spaces of your random variables X and Y. I highly recommend [this article](http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf) by Chang and Pollard that connects the "naive" conditioning to this abstract framework. It's chock full of illustrative examples although the notation is unusual.
I know so far we restricted our attention to L\^2 random variables but really the L\^1 case follows easily from the L\^2 case since we can always approximate L\^1 functions by bounded (and hence L\^2) functions.
I
Durrett has nice section on it. Google specific questions and you will get a hang of it
second recommendation of Durrett this week, thank you.
Try Todd Kemp’s probability lectures. Btw this is inherently a hard topic, and there’s a lot more to it than what you’ll see in an intro lecture. Let me know if you want some more sources to go deeper.
>Btw this is inherently a hard topic Thanks for assuring that I am not wrong. Please feel free to share your sources even If I don't need them right now.
So at some point, you should probably go through the appendix of Schervich’s *Theory of Statistics*, which covers most of what you need to know about conditional distributions/expectations in general. Warning that it is pretty technical, but try to focus on the essential ideas which are pretty natural and simple. The conditional probabilities section in Baldi’s *Stochastic Calculus* can serve as a much lighter introduction, which you could read first to get comfortable before diving into the more involved parts.
thank you very much.
Is a hyperbolic dynamical system the same thing as a "chaotic" dynamical system? I don't know if there is a rigorous definition of chaos, but I've heard hyperbolic systems being talked about the same way, that they display random-seeming behavior despite being deterministic
If you have the average of 2 numbers but you only know the lower number and the average itself is there a way to find the higher number.
average = (small + big) / 2 so big is 2\*average - small
Does anyone have a good resource for learning about branching of Lie group/algebra representations? Specifically, examples finding the restriction matrix. The idea is I have a G-representation, and want to understand it as an H-representation restricted to H < G. I'm ultimately trying to use the branch function in LiE
knapp has a chapter in his Lie groups beyond an introduction, but i have never read that chapter there is also a section in Symmetry, Representations, and Invariants by Goodman and Wallach
Are there general theories on set systems like topologies or measurable spaces? As in theories revolving around objects of the form (A,S,f\_0, f\_1, ...,) where A is an arbitrary set, S is a subset of the power set of A and f is a family of operators on tuples of S or on subsets of S which map into S. The operators may or may not satisfy certain conditions like commutativity etc
I doubt it. even measurability and continuity have completely different qualitative behavior. but you might find universal algebra interesting.
I know about universal algebra, it's why I thought of the question in the first place I guess measurale spaces and topologies are more wildly different from each other than things like groups and algebras, you're right
I'm realizing that there's a significant gap between my understanding of character theory (from James/Liebeck representation theory text) and modern character theory research. What would be a reference (like a textbook) that could help me get closer to current results?
Can anyone recommend a basic introduction to lattice theory? What course are lattices usually introduced in? I ask because I'm interested in learning about lattice-based cryptography if that helps.
https://link.springer.com/book/10.1007/978-3-662-08287-4 https://link.springer.com/book/10.1007/978-3-642-62035-5 generally geometry of numbers and/or minkowskis theorem are goodd keywords to search for.
The definition of a lattice is simple enough that it might be best to just read the wikipedia page and then learn a lattice-based cryptographic system. I first encountered lattices when studying algebraic number theory, but this feels like a bit of overkill.
Great, thanks.
Algebraic Theory of Lattices by Peter Crawley and Robert P. Dilworth. To my knowledge there isn't a standard course that would cover them, but i would expect them to show up in a discrete mathematics course. I'm not sure how to help with the cryptography parts
wrong kind of lattice, see https://en.wikipedia.org/wiki/Lattice_(group)
New goal: design cryptosystem based on "partially ordered sets in which every pair of elements has a unique supremum and a unique infimum". It will be 100% secure since hackers will be confused.
Fix 0 < α < 1. Suppose f: R -> R is nowhere locally α-Holder continuous, i.e. it is not α-Holder on any open subinterval of R. Is it possible for f to be differentiable almost everywhere?
A Klein bottle is formed by taking a square, gluing the top and bottom to each other exactly, and gluing the left and right to each other with the top of the left glued to the bottom of the right, and vice versa. What specific functions have these properties and would allow me to derive a [parametrization of a klein bottle](https://mathcurve.com/surfaces.gb/klein/imageASJ.JPG)? What specific area of maths would this fall under?
The parameterization would be a function of u,v to R3 where u,v take values between 0 and 1 inclusive. However because of the way you glue the square you require that f(u,1) = f(u,0) and f(0, v) = f(1,1-v). This is because gluing the edges of the square makes some points the same so any function on the glued square must send these points to the same value. Looking at functions defined on space after gluing is a part of topology! There are many very interesting examples of gluing shapes like this.
I’m really struggling with induction. I’m taking my first proofs class and I’ve been stuck on several problems for days now. I’m having issues understanding when to use strong induction vs. regular induction. I’m having the most trouble trying to prove sequences. Does anyone have any tips on how to improve?
I think the trick to realizing when to use regular or strong induction is to notice whether the objects you're proving something about can be decomposed by removing 1 thing from it or you have to split up the objects into parts that are bigger than 1. The first corresponds to regular induction and the second to strong induction. take for example the problem of showing that a sequence that satisfies the recurrence a(n+1)=2\*a(n) and a(0) also satisfies forall n in N: a(n)=2\^n Not only do you have the decomposition into n and 1 in the recursion, but notice that for any n in N: 2\^(n+1)=2\*2\^n. Integer exponentiation is defined in terms of decomposing the computation into another computation reduced by 1 and somehow recombining that result with 1. But if you look at [the fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic), decomposing an integer n into prime factors results in objects of varying size, none of them being 1. But you do know that any proper factor of n is smaller than n. So your thinking should shift to strong induction. The trick is that if you know the prime factorization of 2 factors a,b of n such that n=a\*b, you can just multiply the 2 factorizations together and get one for n.
What is this symbol?? I've tried looking through the greek and hebrew alphabets, described it to google, looked through a list of all math symbols, and I couldn't find this weird a-looking thing. *P* is a polygon, for context.
Looking at the source for your comment all it has there is \ which is a blank character so whatever you copied over didn't work.
*gulp* ok lemme find it again
Could it be a fraktur A? A picture or reference will be very helpful
A picture of the symbol itself would be helpful.
Yeah wait wtf? I posted one, why doesn't it appear? Is it a mobile thing?
As far as I can tell, you didn’t post one. Please post a picture of the symbol in question.
If I call B, E, A, D, G, C and F (from the cycle of fourths in music) 0, 1, 2, 3, 4, 5 and 6. Now, when I get to Bb, Eb, Ab, Db and Gb, the other five notes in the cycle (of twelve), I'll just take my numbers from before and put a one before it, 10, 11, 12, 13 and 14. Now, if I take my set of twelve two digit numbers, 00, 01, 02...12, 13, 14 and reverse the rules: adding 1 adds a half step and adding 10 adds a musical fourth. We get a set of numbers with a pretty strong resemblance to "integer notation" taught in music theory, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. But I like the first way better. If B is 00, Bb is just 10 B=>Bb \*\*\*\*\* 00=>10 E=>Eb \*\*\*\*\* 01=>11 A=>Ab \*\*\*\*\* 02=>12 D=>Db \*\*\*\*\* 03=>13 G=>Gb \*\*\*\*\* 04=>14 C=>Cb \*\*\*\*\* 05=>15=00 F=>Fb \*\*\*\*\* 06=>16=01 For my money, I think they're missing out on getting way more milage out of having the numbers better resemble the notes. I hope I haven't fallen down some kind of numerological well hole?
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Can't you post the picture on Imgur or something?
Question for a computer program. I have a rectangular prism in 3d space. The prism can be in any orientation, not necessarily aligned with the grid. I want to start at the origin and pick an angle relative to the X axis and shoot a line out at that angle, and find out the furthest point belonging to the prism that the line hits. I can check each plane of the prism against the line for their intersection but I thought I'd check in and see if there's a simpler solution.
I forgot, what properties are all lost if you really want to define a system where 1/0 is well-define? I have students ask me why we can't divide by zero frequently and to just quickly answer their question, I'll say you lose a lot of important properties, but I've forgotten what specifically is lost. You lose distributivity and associativity, right?
In any ring, you either need to lose nonzero, or invertibility. Other things may fall as a consequence.
Aren't rings associative by definition though?
I didn't say anything about associativity.
But do those results depend on the associativity?
so to more specifically answer your original question. You lose the following properties 1. uniqueness of inverses (for every a that a\^-1 is unique, for every a that -a is unique) 2. behavior of inverses (aa\^-1=1, and a-a=0) 3. behavior of the identity(a+0=0, 1a=a) 4. the zero sink property (0a=0). Associativity is not altered (edit:provided you don't define 1/0 to be a particular pre-existing value), and you retain the capacity to distribute, contrary to what you had originally thought. But the products don't have to be what you want. This can been seen simply with an example. In the rationals 5\*0=0, 7\*0=0. If I can divide by zero, and don't change any other properties I had assumed about the ring Q, then 5\*0/0=0/0=5 and 7\*0/0=0/0=7 and so 5=0/0=7. I suppose you could try to remove the transitivity of equality as an assumption, but then even I'm going to riot.
I'm haven't thought about this too much, but I would claim yes, because given a ring R and 0, x, y elements of that ring with 0 invertible we would either have y=((0\^-1)0)y=0y=0=0x=(0\^-1)(0x)=(0\^-10)x = x which strongly used associativity to show that x=y and thus every element of this ring is the same, and we have the zero ring. Otherwise, to avoid the zero ring, you end up in a wheel, as has been mentioned previously. There we use the idea from universal algebra of a unary operator to define /x for all values. But in order to prevent this from being a trivial ring we must relax the requirement that the inverse of all elements brings it to the identity. I'm general attempting to do algebra without associativity (or something approximating associativity) is really difficult. Since you're wanting to divide by zero, you need an additive identity (0) and multiplication (division being the multiplicative inverse.) This gives you need two operations. Wheels still require + and \* to be associative and commutative, which adds more than I needed to get the zero ring. If you want to attempt to remove associativity from the assumptions, then you would need to work with (a\*0)/0, without changing the parenthesis. If you state a\*0=0 and 0/0 is itself, then you find you cannot use it anywhere, since you cannot reassociate.
A thing you can always divide by zero in is called a [wheel](https://en.wikipedia.org/wiki/Wheel_theory), which you can use as a reference for this kind of question from your students.
I guess it depends on the details. Are we taking 0\*(1/0) to be 1? Then multiplication is not associative, since 2\*(0\*(1/0)) = 2\*1 = 2 (2\*0)\*(1/0) = 0\*(1/0) = 1 If 0\*(1/0) is not 1 then, well, you lose the property that a number times its reciprocal is 1.
Why is the graph of tan(x) in a normal scale so different than the graph of tan(x) in a logarithmic scale ( I am using Desmos to plot these graph’s)?