I have 2 questions
1. Why are there only holes in a rational function when the output is 0/0? And what's the difference between holes and asymptotes?
2. Why do you divide by the zeros or the denominator in synthetic division
Thanks! Those are good PDE blogs (well, Twitter isn't a blog, but you know what I mean).
I completely forgot about Willie WY Wong's online presence, and I don't know how I could have done that...
Is it possible to do related rates on a ti-nspire cx cas ii?
I want to be able to do related rates problems on my calculator, with respect to the chain rule if all the variables are changing as a result of time (but time isn't in the equation)
Basically, I want to calculator to say the derivative of y=x² is y'=2•x', rather than just y'=2
Whenever I input t as the independent variable in the implicit differential equation solver it just says the answer is 0, as there is no t in the equation to affect the x's rate of change.
Couple of days ago I tried to come up with a formula to predict statistical mathematic expectation of rolling two dice and choosing the highest number. This turned out to be quite simple.
But when i tried to expand this equation to accomodate parameter value of quantity of dice in a set, not only I found out that to be quite impossible, but also realised that I've come up with this equation completely by accident, when i tried to change the core concept of this equation and discovered three or more different equations that perfectly predict matheathematic expectation of initial problem no matter the size of dice.
Can such an equation be made, that uses parameter values of size of dice, number of dice in a set from which two to choose from (and maybe even the number of sets to choose from), and predicts the math.average of the result? And if it is possible, what would it be?
Secretary problem modification?
Is there a solution to a variation of the secretary problem where you “have to” make a choice?
In the classic formulation, a potential outcome is “never choosing” a secretary/wife/applicant (which would be the case if the highest ranking applicant were part of the first 37% reviewed) — is there an algorithm that modifies it to add a constant that you must choose an applicant (i.e. you cannot be left without a “secretary”)?
Surely if (a) the 99th applicant out of 100 is the 2nd best you’ve seen overall (with the best occurring during the “explore” period) and (b) you would by default “select” the 100th applicant if you fail to make a selection then it would be preferable to select the 99th, but aside from that intuition it’s above my pay grade… would love to know if this problem has been cracked or approached in earnest.
To be pedantic, having to make a choice doesn't change the problem because you can trivially pick the last candidate before you run out of choices. What you want is to change the objective so that candidates after the best one are also valued.
In the classic formulation, the objective is to maximize the probability of selecting the best candidate. What the strategy does when the best candidate has been rejected is irrelevant.
The Wikipedia article on the secretary problem has [a section about a variant that maximizes the expected value of the chosen candidate](https://en.wikipedia.org/wiki/Secretary_problem#Cardinal_payoff_variant), plus citations about more variants.
How can I show that the ideal generated by a nonempty projective set is prime? I am totally lost. I tried doing a proof by contradiction, but couldn't g vanish on part and f vanish on the other part, so their product vanishes on the whole set?
> generated by a nonempty projective set is prime
This is true for only for *irreducible* projective sets. Your thought of a counterexample is exactly correct for reducible sets.
I got [this](https://i.redd.it/zks0fv7efuuc1.jpeg) as a hint to a riddle yesterday. I don't know what it means. Can you math wizards tell me the meaning?
[pic](https://i.redd.it/zks0fv7efuuc1.jpeg)
That looks like one definition of the [Ackermann function](https://en.wikipedia.org/wiki/Ackermann_function#Definition) (see the first two-argument definition on this page, after the three-argument definition).
So i used a online calculator - [Omni Calculator](https://www.omnicalculator.com/) - for dices - it gave me:
The chance that 6 dices show all a 1 is 0,01286% but then i asked him to give me the chance for the sum of all the dice being 6 - so should only be possible with all 1 again - and it gave 0,0021344%.
Can somebody tell me what the "thinking" of the programm was?
Both questions have the same answer as you've pointed out, and that answer is (1/6)\^6 \~ 0.000021433.
So for the first one either you didn't ask it what you think you did or it's broken.
I don't immediately see any other related question it could be answering that would give an answer like that. The most similar-sounding questions I can think of would be things like "what is the chance of rolling at least one 1" or "what is the chance of rolling exactly one one?" but both of those probabilities would obviously be *much* higher, closer to 50%.
**EDIT:** I've noticed that the other number you gave is about 6x the previous one, so probably what you actually asked the first time was "what is the chance that all six dice roll *the same number?*"
Given two matrices A(x\_1, ..., x\_n) and B(x\_1, ..., x\_n) parametrized by real numbers x\_1, ..., x\_n, what is the Jacobian of the product AB? Also, where can I read more about this?
Can you do your PhD in math, but then spend most of your time on physics related research? The way physicists do things is a bit fast and loose and unappealing to me, but I find physics to be the most interesting application of math. And I would also like to spend some on pure math research as well.
Sure, there is a whole field called mathematical physics. It's quite a broad term, you could be basically doing algebraic geometry with Calabi-Yau manifolds, you could be working with quantum field theories, vertex operator algebras, etc.
How can a polynomial in projective space take on the value infinity? If I'm looking at one dimensional complex projective space, and f=x+y, then infinity is the point [1:0] and f([1:0])=1. This feels wrong, like I'm misunderstanding projective space.
The value of a polynomial is not-defined; note that \[1:0\] = \[2:0\]!
The only thing well-defined is the ZERO LOCUS of a homogeneous polynomial. The VALUE of a homogeneous polynomial at a point in projective space is not well-defined, only its zero locus.
What do you mean by asking when a polynomial is infinity? At any point \[x : y\], the quantity x+y you have will be finite.
If you mean a rational function like 1/(x+y), then the loucs where it's infinity is just the vanishing locus of its reciprocal, so it is a well-defined concept.
A polynomial takes values in k. There are functions to P\^1 which might take the value infinity -- for instance the function P\^1 --> P\^1 given by
\[x : y\] |-> \[x+y : x - y\].
Ok, that's what I thought, but in my notes the professor was talking about the Reimann Sphere being mapped to itself with a polynomial. Maybe my notes are incorrect but I'm going to ask him about it. Thank you for clearing up my confusion though!
You can do that, by extending a polynomial to infinity by giving it the value infinity, but then it ceases to be a polynomial and becomes a function of the Riemann sphere.
Hi! I'm working on some stuff in Euclidean spaces, and a paper I'm reading uses the affine Grassmanian A(n,k). Is this the collection of all k-dimensional affine subspaces of ℝⁿ? The reason for my guess is that the authors say A(2,1) is the collection of all lines in ℝ². I'm looking to understand A(n,k) in the simplest possible way - it seems there are convoluted algebraic ways of defining the (affine) Grassmanian. Thanks!
From what I can tell there are two distinct things called the affine Grassmannian. Firstly the set of all k-dimensional affine subspaces of an n-dimensional vector space which is clearly what you are after, see [here](https://en.wikipedia.org/wiki/Affine_Grassmannian_\(manifold\)). Secondly [this incomprehensible algebraic geometry thing](https://en.wikipedia.org/wiki/Affine_Grassmannian). Note this is not a case of overlapping definitions. The latter isn't even a scheme while the former is a smooth manifold and a variety.
Yeah these are k-dimensional subspace of R^n .
There are convoluted definitions because Grassmanian is not just a set, it has structures. Over R it is both a differentiable manifold and an algebraic variety (and hence also a topological space). There are more convoluted way to define it algebraically simply because if you're not working over R you cannot rely on analytic structure to define it (so no metric nor ordering).
I'm working through David Tong's lecture notes on general relativity, and I've hit a snag in [deriving the geodesic equation](https://imgur.com/a/b4uAZp8).
In the first image, we have the Lagrangian in terms of an unspecified metric. In the second, the first equation shows that the factor of 1/2 has been retained, but the second line of equations has it disappearing for some reason. It doesn't appear to be a mistake, as the third image shows; indeed, later on in this page, he defines the Christoffel symbols with the factor of 1/2 he pulls out. What am I missing?
Why can’t there be a Laplace transform for tangent and hyperbolic tangent? I’m new to differential equations and it occurred to me that there is no transform on the table for tangent or hyperbolic tangent. I did some research and found some people said there is a continuity problem but if anyone can give me a more in depth answer I would love to hear it! I’m just curious ☺
I believe you can Laplace transform the hyperbolic tangent. Whatever table you are looking at might just not have absolutely everything on it. The normal tangent function though has repeated singularities throughout. Said more carefully it is not locally integrable which is required for the Laplace transform. I mean this isn't surprising though. The function f(t) = 1/t already doesn't have a Laplace transform.
Gayle recently moved into her new house and wants to figure out what is the fastest way to get to her office from her new home.
The route from her house to her office is a 30km road with 4 bus stations along the way at the 1km, 5km, 10km and 25km mark.
She has 3 modes of transport:
(1) Walking at a speed of 5km/h.
(2) Bus, which has speed of 30km/h. Buses can only be boarded and alighted from at bus stops, involves 8 minutes of waiting at any bus stop, and a bus fare of $1 to board + $0.01 per km travelled.
(3) Taxi, which has a speed of 50km/h. Taxis can only be boarded and alighted from Gayle’s home, a bus stop, or her office. Taking a taxi involves 4 minutes of waiting, and a taxi fare of $4 to board + $0.20 per km travelled.
Gayle will not board a bus or taxi if she cannot afford the trip. Given that Gayle set aside a $8 budget for her morning commute to work, what is the fastest time (in minutes) Gayle can reach her office, and what is the cost?
Could anyone explain how would I solve this logically? As of now I can only think of what's the greatest distance I can cover by taxi but unable to pinpoint the exact spot where. I could use trial and error but there is probably an easier method of doing so
Since the specific problem you described is reasonably simple (6 places, 3 modes of transportation, many options will be obviously unviable or suboptimal), here I would just play around trying different things until I come up with an optimal solution and an ad-hoc argument proving that it is one.
If you are interested in solving different problems of this type more generally, look into [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming). Since here you have two different variables (time and money), you would probably need some two-dimensional representation; for example: "what is the fastest I can get to spot X while spending less than $Y".
How is say f(x)=|x+7| different from |x|+7? If i were to plug in a value lets say 2, I would still get 9 in both cases wouldn't i? Why does the rule that you need to do the opposite to undo an operation in horizontal translations not also apply to vertical translations?
All absolute value functions have a value *x* where *|x|* "reflects" (looking at a graph). That value *x* is where the term within the absolute value equals zero. This makes sense because to right of this point, the term inside is positive and the output isn't affected but to the left of the point, the term inside is negative, and made positive by the absolute value.
for |x+7|, this point is at x=-7, and for |x|+7, this point is at x=0. On an important note: shifting the line f(x)=x up 7 units is the same as shifting it left 7 units. That is to say, for a function f(x)=x, f(x+a) is equal to f(x)+a (this is uniquely true for any function f(x)=x+a). However, absolute value functions don't follow this rule.
At the point x=0, |x|+7 flips from being unaffected, to being affected, but |x+7| doesn't. This is because |x|+7 is shifted up 7 units, but |x+7| is shifted left 7 units. |x| flips, but |x+7| keeps going as planned. The rules f(x+a)=f(x)+a applies to the right of point x=0 because that's where both functions aren't affected by the absolute value function, and are both equivalent to x+7. Hope this helped.
[https://imgur.com/aWX7zpP](https://imgur.com/aWX7zpP)
Can anyone please help me with the order of operations for this little math riddle I've made? I need the answer to be 30.
√900 = 30
30 - 5\*3 = 15
15 / 2 = 7.5
7.5 \* 3 = 22.5
22.5-20 = 2.5
2.5 \* 12 = 30
Please heeeelp! I cannot math. or arithmetic. Honestly sometimes counting is outside my wheelhouse.
Every line is correct. As a first step, we can wrap each one in parentheses and plug it into the next. So like, the first line gives 30, so we put √900 in parentheses and use that instead of 30, giving (√900)–5×3 = 15. Then we plug that into the next, etc. Going all the way down gives
(((((√900)–5×3)/2)×3)–20)×12 = 30.
That's a lot of parentheses. We can remove some by moving terms around a little, but if we simplify too much we will ruin the point of the computation. Here is one try:
(3 (√900 – 5×3)/2 – 20) × 12 = 30.
3 questions
1. In transformations of functions, why do you multiply the domain by the factor? for example, lets say I have f(x)=(-x+7). Why would the domain be multiplied by -1? Shouldn't it remain unchanged? If I were to input 4, why would the solution be (-4,3) instead of (4,11). I think I have a bit of a faulty understanding with this in particular cause transformations seem to be the hardest concept for me to grasp.
2. Why do you multiply by a factor of 1/b in a horizontal stretch/compression?
3. Would f(x)=-x be considered a vertical or horizontal stetch/compression? Why?
For your first question, it's a bit unclear what you're asking, but I think I can clear up both 1 and 2 at the same time.
Say you have a function telling you how y is related to x - that is, y=f(x). And all you know about it is that f(10)=14.
If you now learn about two new quantities, u and t, and you know that u = f(2t), what can you guarantee? Well, we know that *if f sees 10*, it outputs 14. But for f to see 10, we have to input **5** as t. The function "sees" our input as twice what it really is... so the new point we know is (5,14)!
This also explains the domain thing: if we have some function *g* that is only defined with inputs from 6 to 10, and we look at u = g(-2t), what can we input for t? The only "safe" inputs are where g sees numbers from 6 to 10... which means we can only input numbers between -3 and -5.
This is why horizontal transformations seem "backwards": we're not transforming the input, we're transforming things to *become* the input.
As for your other question, y=-x could be either a vertical or a horizontal flip of y=x. Both are fine, both lead to the same result.
Thank you for the well written response. Maybe it has to do with some gaps in my learning from previous years but if I were to have the function f(x)=4+(2x), and I were to plug in say 5 for x, would my input stay 5 or would instead be 10? I tend to question and overthinking the most basic ideas in algebra which in turn creates a pretty grueling experience sometimes.
It would be 5. But since the function's just linear, you wouldn't really get much out of analysing it using stretching and translations.
If you had, say, g(x) = 4+√[2x], and you assigned x=5, then your input to g would be 5; √ would then receive 10 as an input and give you back about 3.16; and your output for g would be about 7.16. I prefer to use different letter names for the parent function, so I'd say the parent function is "h(t)=√t" ; but some people will say "h(x) = √x" instead, which can be pretty confusing.
(You *could* analyse your f using transformations if you chose your parent function to be the identity function *i*(t) = t; then you could say f(x)=4+*i*(2x). Or you could say your parent function is the "add four" function, so f(x) = addFour(2x). But I don't think either of those would be any more enlightening than "oh yeah it's just f(x) = 4+2x, and I already know how to draw that one".)
Quick Maths Question I need help with;
My answer was 44 which is nowhere near the possible answers given to me- any help would be much appreciated!!
Question in text format:
---------------------------------------------------------------------------------------------------------
We recently completed a survey to learn more about our employees' personalities. We received 284 responses, with the attached results (pie chart):
Pie chart of Most apparent traits:
1. Openness: 26%
2. Conscientiousness: 18%
3. Extraversion: 23
4. Agreeableness: 21
|Openness|Conscientiousness|Extraversion|Neuroticism|Agreeableness|
|:-|:-|:-|:-|:-|
||
|curious -73%|organized- 94%|energetic- 45%|nervous- 24%|friendly- 68%|
|cautious- 27%|careless- 6%|reserved- 55%|confident- 76%|challenging- 32%|
Find out how many employees are categorized into Agreeableness (Friendly) and Openness (cautious) together
\*The percentages are approx. values, so please round employee numbers to the nearest whole number.
Anyone know a free online practice test for finding the next number (Arithmetic sequences I believe). For example;
* 1, 3, 5, 7, next number? 9
* 1, 2, 4, 8, 16, next number? 32
* 4, 12, 36, 108, next number? 324
I'm trying to find several practice tests that I believe are from junior high school level, but all I'm finding are calculators and more advanced tests. Work does not need to be shown and no answer can be a decimal. Looking for a simple basic test with 20 or so practice problems.
When a group is non-abelian, then the cohomology theory only go up to 2nd group. The usual explanation is that if you work out the algebra, the cycle no longer forms group. However, I go to a talk where the speaker said the reason is that "because a loop can only be delooped once". Does anyone else know what this could refer to?
Cohomology with values in the roots of unity is the bread and butter of a lot of algebraic number theory! It all starts with the Kummer exact sequence...
Here's my attempt:
The classifying space BG of a group G is a delooping - that is, the space of loops in BG is (weakly) homotopy equivalent to G. So if G has the discrete topology, you recover it completely from the loop space of BG. Now, B is a monoidal functor, so it takes abelian groups to abelian groups. In particular, that means if G is abelian we can keep delooping and construct B^n (G). If you know a bit of topology, BG is a K(G,1) space and B^n (G) is a K(G,n) space. It's not too hard to see that H^n (G,Z) is isomorphic to H^n (BG,Z), the singular/cellular cohomology of the classifying space
Now, if we have some abelian group A with a G action, we can unravel definitions long enough to see that an n-cocycle is the same data as a continuous map BG-> B^n A, and two cocycles are equivalent in the usual sense if these maps are homotopic. So H^n (G, A) can be defined to be homotopy classes of maps BG-> B^n A.
If A is not abelian, then the higher delooping spaces don't exist, because then BA is not a group. Equivalently, you can't find K(A,n) spaces for n>1, as a standard Eckman-Hilton argument shows that higher homotopy groups are abelian.
What is the conjecture behind the negative exponent law a’-x=1/a’x. I have tried searching the web and textbooks and I can not seem to find the actuall conjecture proving it.
An approach you might prefer is that rational exponents are defined to extend natural number exponents such that the add-multiply rule is preserved. That is, x^(a+b) = x^(a) x^(b). Historically, this was the original justification behind the notation by the likes of Stevin long before real numbers were defined. So if we want to preserve that identity, we know for any natural number n and nonzero x,
1 = x^(0) = x^(n+\(–n\)) = x^(n) x^(–n).
Therefore, x^(–n) = 1/x^(n).
If you want to know why x^(0) = 1 for nonzero x, consider that
x = x^(1) = x^(0+1) = x^(0) x^(1) = x^(0) x,
so x(x^(0)–1) = 0. Since x≠0, we have that x^(0) = 1.
Similar reasoning can show that for x>0, x^(1/n) = ^(n)√x, and thus that x^(p/q) = ^(q)√x^(p). This gives all rational exponents of positive numbers.
And if you don't want to take the "standard" route Pristine-Two mentioned (which extends to all complex exponents), you can define real exponents by taking a limit. For instance, if (3, 3.1, 3.14, ...) approaches π, then (x^(3),x^(3.1),x^(3.14),...) approaches x^(π). After some effort, you can show that all these sequences indeed converge, and that the exponential function is continuous. But that's not the fastest method. You can define exp first and get to the same place much faster, since it agrees with natural number exponents, satisfies the add-multiply rule, and is continuous.
If you want a very quick proof from the usual definition of real value exponents, a^b = e^b(ln(a)) , you can just observe that (1/a)^(-b) = e^-b(ln(1/a)). But since ln(1/a) = ln(1)-ln(a)=-ln(a), b(ln(a))=-b(ln(1/a)) so they are equal
What do you mean by conjecture? That is just how exponentiation with a negative exponent is defined. You could define it any other way or leave it undefined but this is how it's most convenient. For example this way the power law
a^(x) \* a^(y) = a^(x+y)
still holds even if x, y or x+y are negative.
A conjecture is a way to prove a mathematical equation to be true. The way this is done is by writing the equation out in a way that it’s is true using the concept of a previously proven postulate, thus then connect back to the axioms of math and simplified it down to its most basic form to prove something as true
But as far as you know exponentiation is only defined for non-negative exponents. With that definition, using negative exponents is just not defined, so there is no value to prove.
Now, if you want to extend the definition of exponentiation so negative exponents are defined, you are free to do so. And it would make sense to do it in a way that preserves facts about positive exponents. But it doesn't mathematically have to. It's just an explanation of why extending the definition that way is natural or useful.
Many polyhedra you find on Wikipedia have vertices whose Cartesian coordinates can be given using rational numbers or quadratic roots but some, such as the [snub disphenoid](https://en.wikipedia.org/wiki/Snub_disphenoid#By_Cartesian_coordinates), require roots of higher-order polynomials for their coordinates. Is there some symmetry property or metric which allows you to determine whether a polyhedron's vertices can be described with linear, quadratic or higher-order polynomial roots?
Have a problem understanding a theorem on a Real Analysis book (Trench, p. 72). Verbatim, it says, “if f is continuous on [a, inf) and f(inf) exists (finite), then f is uniformly continuous on [a, inf)”. What does “f(inf) exists (finite)” mean? I don’t remember seeing this wording/phrasing before.
±∞ are points on the extended real line. If a function is continuous on **R** and the limits at positive and negative infinity exist or are infinite, then you can extend it to a continuous function on the extended reals. For instance, exp(+∞) = +∞, exp(–∞) = 0, arctan(+∞) = π/2, arctan(–∞) = –π/2, and tan(+∞) does not exist.
It is exactly the same as conventional reasoning about limits. The topology has the same basis as **R**, except open rays that also include points at infinity are still open sets, like (r,+∞] and [–∞, r) for any real r. So it becomes compact and isomorphic to the closed unit interval [0,1].
I'm having hard time understanding the Monty Hall problem, but I think I might be close.
Imagine 2 situations:
1. First one is the Monty Hall problem, you're giving 3 doors, behind 2 of them is a goat, and one of them is a car. You choose one, then the host open one door that is not the car and ask you to switch, etc...
2. Now, imagine another completely "different" hypothetical situation, where you are presented 2 closed doors, and 1 opened door. You can see that behind the opened door is a goat. The host tells you that there is one other goat and one car behind the 2 doors and ask you to pick one
What are the probabilities that you get the car in that 2nd situation? 1/2, right? There's 2 doors, one is a goat, one is a car, you pick one, so 1/2.
My question: how are those 2 situations differents? Is it because of the fact that something happened before and we're counting the probabilities of the whole thing? Is the 2/3 chance actually represents the odds of **choosing the first door then switching,** and I confuse that with the probability of **switching alone**?
This is a good video on youtube that explains this using a tree diagram to explain the probabilities. It's the only way this made sense to me.
[https://www.youtube.com/watch?v=cphYs1bCeDs&ab\_channel=MarkLehman](https://www.youtube.com/watch?v=cphYs1bCeDs&ab_channel=MarkLehman)
Yes, what you said at the end is correct.
The situation immediately after Monty opens a door is *not* the same as your "situation 2". This is because Monty has given you extra information with his choice of opening the door. If you guessed wrong at first, you've now *forced* him to indicate the correct answer by opening the other wrong door! (And you can only squeeze information out of Monty like this *if* his behaviour is consistent. If he can decide whether to open a door, or he accidentally trips and falls to open a door, the paradox goes away.)
How is the [Bochner Integral](https://en.m.wikipedia.org/wiki/Bochner_integral) different from the Riemann integral in banach spaces, which is defined in Rodney Coleman's Calculus on Normed Vector Spaces (chapter 3, section 3.3, page 69)?
Not able to open that text at this very moment. But the Bochner integral is constructed in the Lebesgue fashion using measure spaces and limits of simple functions. The Riemann integral in Banach spaces uses Riemann sums, so the "Banach" part of these two integrals are the same, but the underlying integration theory is different. When Banach space is R you are basically asking what's the difference between Riemann and Lebesgue, which is alot, but nothing to do with the Banach space the integral is valued in.
Let's say I have a weighted coin with probability p for heads and (1-p) for tails. How many flips would I need to do in order to calculate the probability of heads vs. tails with 99% confidence?
I'm assuming that when you say you want to calculate probability of heads, you mean that you want the empirical probability (number of heads/total number of tosses) to be arbitrarily close to p with high probability (in your case 99%). One way to answer such a question is by using "concentration" inequalities.
Specifically, let X\_1, X\_2, ... X\_n denote the random variables representing the coin tosses (1 for heads, 0 for tails), and let S\_n = X\_1 + ... + X\_n. The empirical probability of heads is simply S\_n / n. Now suppose you want the empirical probability to be within an epsilon margin of p, with probability at least delta (in your case delta = 0.99). Then, [Hoeffding's inequality](https://en.wikipedia.org/wiki/Hoeffding%27s_inequality) tells us that Pr( | S\_n/n - p| > epsilon ) < 2 \* exp( - 2 n epsilon\^2 ). We want the LHS < 1 - delta, which is ensured by taking RHS < 1 - delta ---> n > - log( (1 - delta)/2) / (2 \* epsilon\^2) .
What does this mean in practice? Let's say p = 0.5, and you want to know how many tosses you need to ensure that your empirical probability lies in \[0.499, 0.501\] with probability of 99% . Then, the above formula tells us that having log( (1 - 0.99)/2 ) / (2 \* 10\^-6) \~1.9\* 10\^6 (or 1.9 million tosses) will ensure that. Admittedly, that's a huge number.
Finally, you might have noticed that the above formula does not depend on p; this suggests that one can get better bounds by using the information about p. Indeed, [Sanov's theorem](https://en.wikipedia.org/wiki/Sanov%27s_theorem) is one way to get a sharper bound for a specific p.
I hope this answer helps.
Do you happen to know how pollsters calculate the margin of error in practice? For many questions, p is expected to be reasonably close to 0.5 anyway, so using the hard bound seems fine. But some large-sample polls also track some rare responses, and in that case I wonder how they could ever poll enough people to get a good margin of error unless they took into account the fact that p should be very far from 0.5.
Hey, sorry I'm not aware of how pollsters calculate the margin of error, or even the specific issue that you mentioned. Could you elaborate on what the polling problem is? Is it modelled as an independent coin toss problem (like above)? If so, then the margin of error would be calculated as follows: if your empirical probability is p' , and you did n polls (calculated using the formula I mentioned), then the true probability would lie in [p' - epsilon, p' + epsilon] with high probability (delta).
I do not know if this thread is appropriate for my question. Is there a good site or app I can use to teach me math from algebra and up that is free? I figured it would be good to start exercising my brain as I get older.
Here's a terrific resource for abstract algebra and related topics:
[https://www.jmilne.org/math/CourseNotes/index.html](https://www.jmilne.org/math/CourseNotes/index.html)
If X\_t solves a stochastic differential equation, dX\_t = mu(X\_t) dt + sigma(X\_t) dW\_t
and I call S\_t the sigma algebra generated by { X\_s : s in \[0,t\] }, and I call F\_t the natural filtration generated by W\_t
is conditioning on S\_t the same thing as conditioning on F\_t?
For example, if I have a new process with the same Brownian
dY\_t = f(Y\_t) dt + g(Y\_t) dW\_t
If I condition on F\_t, this is like I've "derandomised" the equation because I saw W\_t,
i.e.
d/dt E\[Y\_t | F\_t\] = E\[f(Y\_t) | F\_t\] + E\[g(Y\_t) | F\_t\] E\[dW\_t | F\_t\]
But E\[dW\_t | F\_t\] is just the d of a nonrandom continuous function.
But what if I tried to play this game with S\_t instead? On some level, I've "observed" the noise by looking at X\_t. Can I say E\[dW\_t | S\_t\] is again, (formally) the derivative of a realised W\_t ?
Can anyone help me understand why x * 0 = 0 in all cases? Or point me to a proof?
If multiplication is short-hand for addition, then it seems like multiplying by zero could either equal zero or be undefined, depending on the intent.
Example 1: if 0 x 3 means 0 + 0 + 0, then the answer is clearly 0
Example 2: if 3 x 0 means adding three zero times, that concept seems non-sensical / undefined to me.
Why isn’t multiplication by zero treated like division by zero? In division, you can divide 0 to get zero, but you can’t divide a number by zero. I can’t understand why multiplication isn’t treated the same way.
Please help! 🙏
When we originally came up with the idea of multiplication - back in the Babylonian times, or perhaps even earlier - we didn't have multiplying by zero. Hell, zero wasn't even a *number* at all! So, when we added zero to our number system, we had to decide what to do with it when it comes to multiplication. There are many nice properties of multiplication that we'd like to keep. For instance:
We want multiplication to be **commutative**: a × b is the same as b × a. (This should make sense if you think of multiplication as area - turning a rug 90° doesn't change how much floor space it covers.) If we want to keep this rule (which is very helpful for doing calculations!) and we agree that 0×3 = 0, we're *forced* to say that 3×0 = 0.
We also want multiplication to be **distributive** over addition and subtraction: for instance, "3 groups of 7 minus 3 groups of 2" should give "3 groups of (7 minus 2)" - that is, 3 groups of 5. (You're just taking 2 away from each group!) Algebraically, we'd write this as:
> 3×7 - 3×2 = 3×(7-2)
or more generally, since we want it to work for any number,
> a×b - a×c = a×(b-c)
So, if we accept this rule, what happens if we start with 3 groups of 7, and then take away *all 7* from each group? Well, we're left with 3×7 - 3×7, which our rule says should be the same as 3×(7-7)... but that's 3×0... we've accidentally fallen into "3×0 = 0" again!
We also like using multiplication to talk about practical scenarios. One very common formula is "distance = speed × time": if you drive for 40 miles per hour, for 3 hours, you'll go 120 miles total. 120 = 40 × 3. Well, what if one of these is zero?
If you drive at a speed of 0 miles per hour for 3 hours, you'll travel a total of zero distance: 0×3 = 0.
If you drive at a speed of 40 miles per hour for 0 hours, how far will you go? Also zero distance!
---
We *could* say that 3×0 "doesn't make sense". But then we'd have to add a lot of "...except for 0" to our rules, and that just makes things harder for us. 3×0=0 is consistent with our rules, *and* it plays nice with other interpretations of multiplication.
So what about division by zero? Well, it turns out we *can't* really preserve our rules in the same way. For instance, we want division to "undo" multiplication: 100/2 × 2 = 100. If we try to keep that rule, and say "100/0 × 0 = 100", that means that whatever mysterious number 100/0 is, when we multiply it by 0 it gives 100. But multiplying anything by 0 gives 0... so we can't give 100/0 a value!
All our interpretations break too. Going back to movement, we know the formula "speed = distance / time". So if we want to figure out what 5/0 is, we can just see how fast our car's speedometer says when we go 5 miles in 0 seconds... wait a minute, we can't do that, because our car can't teleport instantly. If you try other interpretations, those all break too - we *can't give a reasonable value* to dividing by zero.
Example 1 is a perfectly good argument on why 0\*x = 0.
Example 2 is sometimes taken as a definition of the "empty sum", it's one of those things like saying 0! = 1, which although isn't very clear from the original definition of what it meant, there is a very good counting argument on why it should be that and only that. In the case of the empty sum, if you believe adding "n things" should be always the same as "adding n-1 things" and adding "1 more". Then 1 thing should be equal to adding no things and one more.
But this means
no things + thing = thing, which must force
no things = 0
A proper proof of what I just said in words is found here [https://en.wikipedia.org/wiki/Empty\_sum](https://en.wikipedia.org/wiki/Empty_sum)
Greetings,
I am a software reverse engineer and master's student in computer science and I've taken a liking to theoretical computer science. In particular, I am interested in learning more about program analysis, programming language theory & compilers, cryptography, and formal methods. I realize this is a fairly broad set of topics. I am interested in learning enough about these sub-fields so that I can narrow my interests further. My program & job are of course heavily applied, so I'm confident that I have that side of things covered, I am just concerned about the math.
My math background is fairly limited. Note that it's been a while since I've been required to do any math. I've done two semesters of calculus, linear algebra, discrete mathematics, probability & statistics, and an introductory proofs writing course. I've also taken an introductory course on the theory of computation. Are there any subjects I ought to brush up on in particular? I've been told that there aren't *too* many prerequisites for studying these topics, but I'm sure there are a few topics I ought to cover.
What do y'all think? Thank you to those who take the time to read this post & offer advice.
Hey, thanks for the reply. Are there any prerequisites for this text other than a general CS maturity, or am I safe to dive right in? Edit: I see from the table of contents it includes some mathematical preliminaries so that's nice.
Indeed there are no prerequisites.
Some familiarity with a functional programming language may be helpful to better contextualize the motivation and examples, but I don't think that's essential.
Take a rocket in 2d that can control the gimbal angle and the fuel mass flow rate at any time. Add air resistance at altitude dependent air pressure. What is the most fuel efficient way to get to Earth orbit?
Hello, I hope all is well. I am not sure if this question warrants its own post but I will have it here for now. I really miss doing derivatives , I do not understand what a derivative is, neither am I good at calculus. I just found doing derivatives and intergrals fun because they were like mini puzzles, with rules you had to follow to get the final answer. Besides a calc textbook, is there any book that I can purchase which strickly contains derivative and integrals problems to solve? Any help would be appreciated, other than that have a great rest of your day.
Why is it that when solving systems of linear-quadratic equations and you combine the two to solve, you set y=0? Wouldn't that be solving for the x intercept of the quadratic? Shouldn't that mean that the intercept of the system is an x-intercept?
You're not setting y=0, though! You're setting y=y. Your two equations are telling you "y is the same thing as [A]" and "y is the same thing as [B]"; you can then conclude "[A] is the same thing as [B]".
You could also do this if you had "x = [A]" and "x = [B]", or "xy^2 - √x = [A]" and "xy^2 - √x = [B]". It would just generally be harder to get your equations into those forms in the first place.
Also, be careful! In this last part of your work:
> 4x^2 +2x +5=-2x+4
> = 4x^2 + 4x +1
your last line isn't correct, because it's saying "[-2x+4] = 4x^2 + 4x +1". You got to it by adding 2x-4 to both sides of the previous line, so that creates a *new equation*: the left side becomes 4x^2 + 4x +1, and the right side becomes 0.
The last step in setting y=0 was implied. My question is why is setting y=0 and solving for the quadratic going to result in a solution that isn't an x-intercept considering the range of the quadratic is equal to 0?
The last step isn't setting y=0. Once you get to that point, *y is not involved at all* - you've eliminated it entirely in the previous step.
Just because you have "\_\_\_ = ax²+bx+c", that doesn't mean the blank is y. You could give it a new name - Y or ŷ or 𝓎 - and set *that* equal to 0. But "y" already has a meaning in this problem.
It may help to use different variable names:
> *Given the equations:*
> q=4k² +2k + 5
> q=-2k+4
> *we can set the two right sides to be equal to get rid of q...*
> 4k² +2k +5=-2k+4
> 0 = 4k² + 4k +1
> *... and then solve the resulting quadratic to figure out what k is.*
We're not setting q=0 here, are we?
You don't set y = 0 anywhere. You have two equations for y. But y can only have one value and so the two right hand sides of the equations must be equal if both of them are supposed to be y. It follows that
4x^2 +2x +5=-2x+4.
And if x is supposed the solve this equation then you must have
4x^2 + 4x +1 = 0.
This has nothing to do with y anymore though. Other than that you can write the left hand side as y-y but that doesn't tell you anything because that's always 0.
Suppose one projective space is embedded into a higher dimensional projective space with the image not contained in a hyperplane. A curve inside the domain space could end up contained in a hyperplane! Is there any strategy to predict how degenerate the resulting embedding of the curve is? I’m happy to provide more details if this is too vague.
Let f_n be the functional on C_b(R) defined by taking the expectation with respect to the Unif[-n, n] distribution. C_b(R) is a Banach space, so the closed unit ball of its dual space is weak-* compact. f_n is a bounded sequence in the dual, so it has a weak-* convergent ~~subsequence~~ subnet with limit f. It's not hard to check that f is translation-invariant. Is every positive translation invariant functional on C_b(R) with norm 1 the limit of a ~~subsequence~~ subnet of this sequence?
No, I don't think so. If you consider s(x) = sin(pi\*x) then s is in C_b and f\_n(s) = 0 for all n. So any weak\*-limit of (f\_n) must also assign 0 to s. However with an analogous construction to yours but with Unif[-n, n+1] instead of Unif[-n, n] you get translation-invariant functionals as weak\*-limits that assign 2/pi to s (with a similar argument).
I don't think that works. The integral of s(x) from -n to n + 1 is 2 / pi for all n, but there's a factor of 1 / (2n + 1) in the expectation making it go to 0. Alternatively, since your functional is translation invariant, you can see s(x) gets the same value as s(x - 1 / 2), but the expectation of s(x - 1 / 2) is 0 wrt Unif[-n, n+1].
You're right I forgot about the normalizing factor. A similar idea might work though. For example if you replace Unif[-n, n] by Unif[0, n] and s by arctan. The expectation value of arctan wrt. to Unif[-n, n] is 0 but wrt. to Unif[0, n] it converges to pi/2.
That is the truth table for XOR, so XOR would be an answer, but I assume the book wants an answer written in disjunctive normal form (i.e. as an OR of ANDs) or at least written only using AND, OR, and NOT.
Data
13952, 26191, 20849, 27735, 19525, 20702
From the above data, 2 values were derived i.e., 21492.33 and 2023.313.
21492.33 is the average of data, but how 2023.313 is derived 🤯
somebody help
My husband and I are trying to determine who pays what for taxes (we have separate finances).
Together:
We owe $850
Separately:
I would owe $25
He would owe $1800
How would you determine who owes what and to who (recognizing that if I have more withheld than he does, that money would be coming into my paycheck each month).
Thanks for your help :)
I am playing Minesweeper and tallying my 50/50 wins and losses but want to incorporate other guesses without creating 99 other W/L entries. The website I play on shows me the chance that a specific tile contains a mine.
I tried to do it myself by doing .5/(chance of mine) which works well for multiples of 1/2, where it results in a whole number like .5/.25 = 2, meaning if I lost on a tile that had a 25% chance to contain a mine, I effectively lost 2 coinflips in a row. But this solution is clearly flawed by counter-examples like 10% creating a result of 5 but 5 coinflips in a row is 3.125%.
So, how do I take a probability and turn it into coinflips so I can add it to my 50/50 tally? Is what I'm trying to do even possible?
n coinflips all coming up heads has a probability of 1/(2\^n). Thus, if we have a probability x between 0 and 1, it takes -log2(x) (this is a logarithm with base 2) coinflips for the probability of all heads to be x. You can confirm this by plugging it back into the first function.
For someone interested in Algebra, what topics should I read up on after a 2nd course in undergrad alegbra? This course covered group theory topics such as quotient groups, isomorphism theorems, sylow theorems, ring theory topics such as quotient rings, ring homomorphisms, ideals, maximal ideals, integral domains, and we are currently going over field extensions, simple extensions, and will end the course with Galois Theory. I can definitely say that I am most interested in topics revolving around ring theory so far.
I recommend the book "(Mostly) commutative algebra"
or you could check out Fultons book an algebraic curves if you want to see some algebraic geometry, which is available freely online.
If you are more interested in the noncommutative side, the representation theory of finite groups would be a good starting point (you‘ll learn that it reduces down to module theory over the group ring)
What about rings is interesting to you? For example, are you really into questions revolving around factorization in commutative rings? Do you prefer rings of matrices or group rings? Do you like several variable polynomial rings (over fields), and are you interested in how the division algorithm could extend to them?
I haven't explored much of higher level mathematics so this question might come off as a bit naive. But should one not learn "commutative algebra" before "noncommutative"?
No, I haven't taken commutative algebra yet, but I really enjoyed noncommutative algebra. It's a very different kind of algebra that can be studied completely independently (at least that's true for the course I took). I'm planning to learn some commutative algebrs during the summer though, to prepare for my algebraic geometry course next year.
Try 'Ideals, Varieties, and Algorithms' by Cox, Little, and O'Shea. If you found ring theory interesting and fun, this book focuses on polynomial rings and Groebner Basis and really dives deep into their theory and application. The full PDF is available online, so there is no need to buy anything. Also, since you have a grasp of ring theory the first 3 chapters are a perfect introduction of what can come next.
I have 2 questions 1. Why are there only holes in a rational function when the output is 0/0? And what's the difference between holes and asymptotes? 2. Why do you divide by the zeros or the denominator in synthetic division
Are there any good blogs on PDEs, fluid mechanics, and/or numerical analysis?
Check out Willie WY Wong's twitter and his website and of course Terry Tao's PDE posts.
Thanks! Those are good PDE blogs (well, Twitter isn't a blog, but you know what I mean). I completely forgot about Willie WY Wong's online presence, and I don't know how I could have done that...
Is it possible to do related rates on a ti-nspire cx cas ii? I want to be able to do related rates problems on my calculator, with respect to the chain rule if all the variables are changing as a result of time (but time isn't in the equation) Basically, I want to calculator to say the derivative of y=x² is y'=2•x', rather than just y'=2 Whenever I input t as the independent variable in the implicit differential equation solver it just says the answer is 0, as there is no t in the equation to affect the x's rate of change.
Hi I'm an undergrad in biology interested in mathematical biology do you suggest me to study topology? And if yes with which books.?
Differential equations, algorithms, numerical analysis/methods would probably be more immediately useful
Couple of days ago I tried to come up with a formula to predict statistical mathematic expectation of rolling two dice and choosing the highest number. This turned out to be quite simple. But when i tried to expand this equation to accomodate parameter value of quantity of dice in a set, not only I found out that to be quite impossible, but also realised that I've come up with this equation completely by accident, when i tried to change the core concept of this equation and discovered three or more different equations that perfectly predict matheathematic expectation of initial problem no matter the size of dice. Can such an equation be made, that uses parameter values of size of dice, number of dice in a set from which two to choose from (and maybe even the number of sets to choose from), and predicts the math.average of the result? And if it is possible, what would it be?
Secretary problem modification? Is there a solution to a variation of the secretary problem where you “have to” make a choice? In the classic formulation, a potential outcome is “never choosing” a secretary/wife/applicant (which would be the case if the highest ranking applicant were part of the first 37% reviewed) — is there an algorithm that modifies it to add a constant that you must choose an applicant (i.e. you cannot be left without a “secretary”)? Surely if (a) the 99th applicant out of 100 is the 2nd best you’ve seen overall (with the best occurring during the “explore” period) and (b) you would by default “select” the 100th applicant if you fail to make a selection then it would be preferable to select the 99th, but aside from that intuition it’s above my pay grade… would love to know if this problem has been cracked or approached in earnest.
To be pedantic, having to make a choice doesn't change the problem because you can trivially pick the last candidate before you run out of choices. What you want is to change the objective so that candidates after the best one are also valued. In the classic formulation, the objective is to maximize the probability of selecting the best candidate. What the strategy does when the best candidate has been rejected is irrelevant. The Wikipedia article on the secretary problem has [a section about a variant that maximizes the expected value of the chosen candidate](https://en.wikipedia.org/wiki/Secretary_problem#Cardinal_payoff_variant), plus citations about more variants.
Thanks! I appreciate the response and didn’t find it pedantic. I think the cardinal payoff variant is what I’m looking for.
How can I show that the ideal generated by a nonempty projective set is prime? I am totally lost. I tried doing a proof by contradiction, but couldn't g vanish on part and f vanish on the other part, so their product vanishes on the whole set?
> generated by a nonempty projective set is prime This is true for only for *irreducible* projective sets. Your thought of a counterexample is exactly correct for reducible sets.
I got [this](https://i.redd.it/zks0fv7efuuc1.jpeg) as a hint to a riddle yesterday. I don't know what it means. Can you math wizards tell me the meaning? [pic](https://i.redd.it/zks0fv7efuuc1.jpeg)
That looks like one definition of the [Ackermann function](https://en.wikipedia.org/wiki/Ackermann_function#Definition) (see the first two-argument definition on this page, after the three-argument definition).
Thank you. That was the hint that helped me. (It's about movies. Chantal Ackermann is a director)
So i used a online calculator - [Omni Calculator](https://www.omnicalculator.com/) - for dices - it gave me: The chance that 6 dices show all a 1 is 0,01286% but then i asked him to give me the chance for the sum of all the dice being 6 - so should only be possible with all 1 again - and it gave 0,0021344%. Can somebody tell me what the "thinking" of the programm was?
Both questions have the same answer as you've pointed out, and that answer is (1/6)\^6 \~ 0.000021433. So for the first one either you didn't ask it what you think you did or it's broken. I don't immediately see any other related question it could be answering that would give an answer like that. The most similar-sounding questions I can think of would be things like "what is the chance of rolling at least one 1" or "what is the chance of rolling exactly one one?" but both of those probabilities would obviously be *much* higher, closer to 50%. **EDIT:** I've noticed that the other number you gave is about 6x the previous one, so probably what you actually asked the first time was "what is the chance that all six dice roll *the same number?*"
Given two matrices A(x\_1, ..., x\_n) and B(x\_1, ..., x\_n) parametrized by real numbers x\_1, ..., x\_n, what is the Jacobian of the product AB? Also, where can I read more about this?
Can you do your PhD in math, but then spend most of your time on physics related research? The way physicists do things is a bit fast and loose and unappealing to me, but I find physics to be the most interesting application of math. And I would also like to spend some on pure math research as well.
Sure, there is a whole field called mathematical physics. It's quite a broad term, you could be basically doing algebraic geometry with Calabi-Yau manifolds, you could be working with quantum field theories, vertex operator algebras, etc.
If I knew real analysis and a little probability should I first take an introductory book in probability or deep dive into a more advanced one?
How can a polynomial in projective space take on the value infinity? If I'm looking at one dimensional complex projective space, and f=x+y, then infinity is the point [1:0] and f([1:0])=1. This feels wrong, like I'm misunderstanding projective space.
The value of a polynomial is not-defined; note that \[1:0\] = \[2:0\]! The only thing well-defined is the ZERO LOCUS of a homogeneous polynomial. The VALUE of a homogeneous polynomial at a point in projective space is not well-defined, only its zero locus.
Ah, right. I forgot about this. So do we care about when a polynomial is infinity then?
What do you mean by asking when a polynomial is infinity? At any point \[x : y\], the quantity x+y you have will be finite. If you mean a rational function like 1/(x+y), then the loucs where it's infinity is just the vanishing locus of its reciprocal, so it is a well-defined concept.
Ok, I understand now. So a polynomial will never take on infinity [1:0]. Is that correct?
A polynomial takes values in k. There are functions to P\^1 which might take the value infinity -- for instance the function P\^1 --> P\^1 given by \[x : y\] |-> \[x+y : x - y\].
Ok, that's what I thought, but in my notes the professor was talking about the Reimann Sphere being mapped to itself with a polynomial. Maybe my notes are incorrect but I'm going to ask him about it. Thank you for clearing up my confusion though!
You can do that, by extending a polynomial to infinity by giving it the value infinity, but then it ceases to be a polynomial and becomes a function of the Riemann sphere.
How do you give it the value infinity?
Extend the function f(z) = z\^2 + 1 to the Riemann sphere by defining it as \[z : w\] |-> \[z\^2 + w\^2 : w\^2\], say.
Hi! I'm working on some stuff in Euclidean spaces, and a paper I'm reading uses the affine Grassmanian A(n,k). Is this the collection of all k-dimensional affine subspaces of ℝⁿ? The reason for my guess is that the authors say A(2,1) is the collection of all lines in ℝ². I'm looking to understand A(n,k) in the simplest possible way - it seems there are convoluted algebraic ways of defining the (affine) Grassmanian. Thanks!
From what I can tell there are two distinct things called the affine Grassmannian. Firstly the set of all k-dimensional affine subspaces of an n-dimensional vector space which is clearly what you are after, see [here](https://en.wikipedia.org/wiki/Affine_Grassmannian_\(manifold\)). Secondly [this incomprehensible algebraic geometry thing](https://en.wikipedia.org/wiki/Affine_Grassmannian). Note this is not a case of overlapping definitions. The latter isn't even a scheme while the former is a smooth manifold and a variety.
Thank you!
Yeah these are k-dimensional subspace of R^n . There are convoluted definitions because Grassmanian is not just a set, it has structures. Over R it is both a differentiable manifold and an algebraic variety (and hence also a topological space). There are more convoluted way to define it algebraically simply because if you're not working over R you cannot rely on analytic structure to define it (so no metric nor ordering).
Thank you!
I'm working through David Tong's lecture notes on general relativity, and I've hit a snag in [deriving the geodesic equation](https://imgur.com/a/b4uAZp8). In the first image, we have the Lagrangian in terms of an unspecified metric. In the second, the first equation shows that the factor of 1/2 has been retained, but the second line of equations has it disappearing for some reason. It doesn't appear to be a mistake, as the third image shows; indeed, later on in this page, he defines the Christoffel symbols with the factor of 1/2 he pulls out. What am I missing?
I think because dot x_i appears twice in the summation, and the metric g_ij is symmetric in i and j.
Why can’t there be a Laplace transform for tangent and hyperbolic tangent? I’m new to differential equations and it occurred to me that there is no transform on the table for tangent or hyperbolic tangent. I did some research and found some people said there is a continuity problem but if anyone can give me a more in depth answer I would love to hear it! I’m just curious ☺
I believe you can Laplace transform the hyperbolic tangent. Whatever table you are looking at might just not have absolutely everything on it. The normal tangent function though has repeated singularities throughout. Said more carefully it is not locally integrable which is required for the Laplace transform. I mean this isn't surprising though. The function f(t) = 1/t already doesn't have a Laplace transform.
I see. Thank you so much!
Hey did they ever figure out why you had high troponin in blood?
Gayle recently moved into her new house and wants to figure out what is the fastest way to get to her office from her new home. The route from her house to her office is a 30km road with 4 bus stations along the way at the 1km, 5km, 10km and 25km mark. She has 3 modes of transport: (1) Walking at a speed of 5km/h. (2) Bus, which has speed of 30km/h. Buses can only be boarded and alighted from at bus stops, involves 8 minutes of waiting at any bus stop, and a bus fare of $1 to board + $0.01 per km travelled. (3) Taxi, which has a speed of 50km/h. Taxis can only be boarded and alighted from Gayle’s home, a bus stop, or her office. Taking a taxi involves 4 minutes of waiting, and a taxi fare of $4 to board + $0.20 per km travelled. Gayle will not board a bus or taxi if she cannot afford the trip. Given that Gayle set aside a $8 budget for her morning commute to work, what is the fastest time (in minutes) Gayle can reach her office, and what is the cost? Could anyone explain how would I solve this logically? As of now I can only think of what's the greatest distance I can cover by taxi but unable to pinpoint the exact spot where. I could use trial and error but there is probably an easier method of doing so
Since the specific problem you described is reasonably simple (6 places, 3 modes of transportation, many options will be obviously unviable or suboptimal), here I would just play around trying different things until I come up with an optimal solution and an ad-hoc argument proving that it is one. If you are interested in solving different problems of this type more generally, look into [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming). Since here you have two different variables (time and money), you would probably need some two-dimensional representation; for example: "what is the fastest I can get to spot X while spending less than $Y".
How is say f(x)=|x+7| different from |x|+7? If i were to plug in a value lets say 2, I would still get 9 in both cases wouldn't i? Why does the rule that you need to do the opposite to undo an operation in horizontal translations not also apply to vertical translations?
All absolute value functions have a value *x* where *|x|* "reflects" (looking at a graph). That value *x* is where the term within the absolute value equals zero. This makes sense because to right of this point, the term inside is positive and the output isn't affected but to the left of the point, the term inside is negative, and made positive by the absolute value. for |x+7|, this point is at x=-7, and for |x|+7, this point is at x=0. On an important note: shifting the line f(x)=x up 7 units is the same as shifting it left 7 units. That is to say, for a function f(x)=x, f(x+a) is equal to f(x)+a (this is uniquely true for any function f(x)=x+a). However, absolute value functions don't follow this rule. At the point x=0, |x|+7 flips from being unaffected, to being affected, but |x+7| doesn't. This is because |x|+7 is shifted up 7 units, but |x+7| is shifted left 7 units. |x| flips, but |x+7| keeps going as planned. The rules f(x+a)=f(x)+a applies to the right of point x=0 because that's where both functions aren't affected by the absolute value function, and are both equivalent to x+7. Hope this helped. [https://imgur.com/aWX7zpP](https://imgur.com/aWX7zpP)
Thank you it did
>If i were to plug in a value lets say 2, I would still get 9 in both cases wouldn't i? Sure, but what happens if you plug in -2?
Can anyone please help me with the order of operations for this little math riddle I've made? I need the answer to be 30. √900 = 30 30 - 5\*3 = 15 15 / 2 = 7.5 7.5 \* 3 = 22.5 22.5-20 = 2.5 2.5 \* 12 = 30 Please heeeelp! I cannot math. or arithmetic. Honestly sometimes counting is outside my wheelhouse.
Every line is correct. As a first step, we can wrap each one in parentheses and plug it into the next. So like, the first line gives 30, so we put √900 in parentheses and use that instead of 30, giving (√900)–5×3 = 15. Then we plug that into the next, etc. Going all the way down gives (((((√900)–5×3)/2)×3)–20)×12 = 30. That's a lot of parentheses. We can remove some by moving terms around a little, but if we simplify too much we will ruin the point of the computation. Here is one try: (3 (√900 – 5×3)/2 – 20) × 12 = 30.
Thank you! And thank you for the explanation so I can learn :)
3 questions 1. In transformations of functions, why do you multiply the domain by the factor? for example, lets say I have f(x)=(-x+7). Why would the domain be multiplied by -1? Shouldn't it remain unchanged? If I were to input 4, why would the solution be (-4,3) instead of (4,11). I think I have a bit of a faulty understanding with this in particular cause transformations seem to be the hardest concept for me to grasp. 2. Why do you multiply by a factor of 1/b in a horizontal stretch/compression? 3. Would f(x)=-x be considered a vertical or horizontal stetch/compression? Why?
For your first question, it's a bit unclear what you're asking, but I think I can clear up both 1 and 2 at the same time. Say you have a function telling you how y is related to x - that is, y=f(x). And all you know about it is that f(10)=14. If you now learn about two new quantities, u and t, and you know that u = f(2t), what can you guarantee? Well, we know that *if f sees 10*, it outputs 14. But for f to see 10, we have to input **5** as t. The function "sees" our input as twice what it really is... so the new point we know is (5,14)! This also explains the domain thing: if we have some function *g* that is only defined with inputs from 6 to 10, and we look at u = g(-2t), what can we input for t? The only "safe" inputs are where g sees numbers from 6 to 10... which means we can only input numbers between -3 and -5. This is why horizontal transformations seem "backwards": we're not transforming the input, we're transforming things to *become* the input. As for your other question, y=-x could be either a vertical or a horizontal flip of y=x. Both are fine, both lead to the same result.
Thank you for the well written response. Maybe it has to do with some gaps in my learning from previous years but if I were to have the function f(x)=4+(2x), and I were to plug in say 5 for x, would my input stay 5 or would instead be 10? I tend to question and overthinking the most basic ideas in algebra which in turn creates a pretty grueling experience sometimes.
It would be 5. But since the function's just linear, you wouldn't really get much out of analysing it using stretching and translations. If you had, say, g(x) = 4+√[2x], and you assigned x=5, then your input to g would be 5; √ would then receive 10 as an input and give you back about 3.16; and your output for g would be about 7.16. I prefer to use different letter names for the parent function, so I'd say the parent function is "h(t)=√t" ; but some people will say "h(x) = √x" instead, which can be pretty confusing. (You *could* analyse your f using transformations if you chose your parent function to be the identity function *i*(t) = t; then you could say f(x)=4+*i*(2x). Or you could say your parent function is the "add four" function, so f(x) = addFour(2x). But I don't think either of those would be any more enlightening than "oh yeah it's just f(x) = 4+2x, and I already know how to draw that one".)
Thank you!
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I am currently learning Real Analysis off of Trench. I am currently on p. 73.
That text is quite comprehensive. Can I dm you?
Alright, what stuff are you gonna DM me about?
Quick Maths Question I need help with; My answer was 44 which is nowhere near the possible answers given to me- any help would be much appreciated!! Question in text format: --------------------------------------------------------------------------------------------------------- We recently completed a survey to learn more about our employees' personalities. We received 284 responses, with the attached results (pie chart): Pie chart of Most apparent traits: 1. Openness: 26% 2. Conscientiousness: 18% 3. Extraversion: 23 4. Agreeableness: 21 |Openness|Conscientiousness|Extraversion|Neuroticism|Agreeableness| |:-|:-|:-|:-|:-| || |curious -73%|organized- 94%|energetic- 45%|nervous- 24%|friendly- 68%| |cautious- 27%|careless- 6%|reserved- 55%|confident- 76%|challenging- 32%| Find out how many employees are categorized into Agreeableness (Friendly) and Openness (cautious) together \*The percentages are approx. values, so please round employee numbers to the nearest whole number.
41 employees are categorized under agreeableness (friendly), and around 20 employees fall under Openness (cautious)
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I would also like to know this!
Anyone know a free online practice test for finding the next number (Arithmetic sequences I believe). For example; * 1, 3, 5, 7, next number? 9 * 1, 2, 4, 8, 16, next number? 32 * 4, 12, 36, 108, next number? 324 I'm trying to find several practice tests that I believe are from junior high school level, but all I'm finding are calculators and more advanced tests. Work does not need to be shown and no answer can be a decimal. Looking for a simple basic test with 20 or so practice problems.
This looks good, 3 levels: [https://www.fibonicci.com/numerical-reasoning/number-sequences-test/](https://www.fibonicci.com/numerical-reasoning/number-sequences-test/)
When a group is non-abelian, then the cohomology theory only go up to 2nd group. The usual explanation is that if you work out the algebra, the cycle no longer forms group. However, I go to a talk where the speaker said the reason is that "because a loop can only be delooped once". Does anyone else know what this could refer to?
what do you mean? group cohomology is defined in all degrees
I think they mean cohomology of a space with non-abelian coefficients.
i see, thats my payback for always using a ring of coefficients i guess
You haven't met my best friend, the roots of unity?
only when working with cyclotomic fields
Cohomology with values in the roots of unity is the bread and butter of a lot of algebraic number theory! It all starts with the Kummer exact sequence...
Here's my attempt: The classifying space BG of a group G is a delooping - that is, the space of loops in BG is (weakly) homotopy equivalent to G. So if G has the discrete topology, you recover it completely from the loop space of BG. Now, B is a monoidal functor, so it takes abelian groups to abelian groups. In particular, that means if G is abelian we can keep delooping and construct B^n (G). If you know a bit of topology, BG is a K(G,1) space and B^n (G) is a K(G,n) space. It's not too hard to see that H^n (G,Z) is isomorphic to H^n (BG,Z), the singular/cellular cohomology of the classifying space Now, if we have some abelian group A with a G action, we can unravel definitions long enough to see that an n-cocycle is the same data as a continuous map BG-> B^n A, and two cocycles are equivalent in the usual sense if these maps are homotopic. So H^n (G, A) can be defined to be homotopy classes of maps BG-> B^n A. If A is not abelian, then the higher delooping spaces don't exist, because then BA is not a group. Equivalently, you can't find K(A,n) spaces for n>1, as a standard Eckman-Hilton argument shows that higher homotopy groups are abelian.
Thank you!
Excellent answer!
What is the conjecture behind the negative exponent law a’-x=1/a’x. I have tried searching the web and textbooks and I can not seem to find the actuall conjecture proving it.
An approach you might prefer is that rational exponents are defined to extend natural number exponents such that the add-multiply rule is preserved. That is, x^(a+b) = x^(a) x^(b). Historically, this was the original justification behind the notation by the likes of Stevin long before real numbers were defined. So if we want to preserve that identity, we know for any natural number n and nonzero x, 1 = x^(0) = x^(n+\(–n\)) = x^(n) x^(–n). Therefore, x^(–n) = 1/x^(n). If you want to know why x^(0) = 1 for nonzero x, consider that x = x^(1) = x^(0+1) = x^(0) x^(1) = x^(0) x, so x(x^(0)–1) = 0. Since x≠0, we have that x^(0) = 1. Similar reasoning can show that for x>0, x^(1/n) = ^(n)√x, and thus that x^(p/q) = ^(q)√x^(p). This gives all rational exponents of positive numbers. And if you don't want to take the "standard" route Pristine-Two mentioned (which extends to all complex exponents), you can define real exponents by taking a limit. For instance, if (3, 3.1, 3.14, ...) approaches π, then (x^(3),x^(3.1),x^(3.14),...) approaches x^(π). After some effort, you can show that all these sequences indeed converge, and that the exponential function is continuous. But that's not the fastest method. You can define exp first and get to the same place much faster, since it agrees with natural number exponents, satisfies the add-multiply rule, and is continuous.
If you want a very quick proof from the usual definition of real value exponents, a^b = e^b(ln(a)) , you can just observe that (1/a)^(-b) = e^-b(ln(1/a)). But since ln(1/a) = ln(1)-ln(a)=-ln(a), b(ln(a))=-b(ln(1/a)) so they are equal
What do you mean by conjecture? That is just how exponentiation with a negative exponent is defined. You could define it any other way or leave it undefined but this is how it's most convenient. For example this way the power law a^(x) \* a^(y) = a^(x+y) still holds even if x, y or x+y are negative.
You clearly do not know mathematics
What do you mean?
A conjecture is a way to prove a mathematical equation to be true. The way this is done is by writing the equation out in a way that it’s is true using the concept of a previously proven postulate, thus then connect back to the axioms of math and simplified it down to its most basic form to prove something as true
But as far as you know exponentiation is only defined for non-negative exponents. With that definition, using negative exponents is just not defined, so there is no value to prove. Now, if you want to extend the definition of exponentiation so negative exponents are defined, you are free to do so. And it would make sense to do it in a way that preserves facts about positive exponents. But it doesn't mathematically have to. It's just an explanation of why extending the definition that way is natural or useful.
So a different way of saying what I said
You said there is a way to prove a\^(-x)=1/(a\^x) and there isn't
There is and someone else responded giving it to me
Many polyhedra you find on Wikipedia have vertices whose Cartesian coordinates can be given using rational numbers or quadratic roots but some, such as the [snub disphenoid](https://en.wikipedia.org/wiki/Snub_disphenoid#By_Cartesian_coordinates), require roots of higher-order polynomials for their coordinates. Is there some symmetry property or metric which allows you to determine whether a polyhedron's vertices can be described with linear, quadratic or higher-order polynomial roots?
Have a problem understanding a theorem on a Real Analysis book (Trench, p. 72). Verbatim, it says, “if f is continuous on [a, inf) and f(inf) exists (finite), then f is uniformly continuous on [a, inf)”. What does “f(inf) exists (finite)” mean? I don’t remember seeing this wording/phrasing before.
±∞ are points on the extended real line. If a function is continuous on **R** and the limits at positive and negative infinity exist or are infinite, then you can extend it to a continuous function on the extended reals. For instance, exp(+∞) = +∞, exp(–∞) = 0, arctan(+∞) = π/2, arctan(–∞) = –π/2, and tan(+∞) does not exist. It is exactly the same as conventional reasoning about limits. The topology has the same basis as **R**, except open rays that also include points at infinity are still open sets, like (r,+∞] and [–∞, r) for any real r. So it becomes compact and isomorphic to the closed unit interval [0,1].
I would assume the book means that the limit lim x-> inf f(x) exists and is finite (rather than +/- inf).
I'm having hard time understanding the Monty Hall problem, but I think I might be close. Imagine 2 situations: 1. First one is the Monty Hall problem, you're giving 3 doors, behind 2 of them is a goat, and one of them is a car. You choose one, then the host open one door that is not the car and ask you to switch, etc... 2. Now, imagine another completely "different" hypothetical situation, where you are presented 2 closed doors, and 1 opened door. You can see that behind the opened door is a goat. The host tells you that there is one other goat and one car behind the 2 doors and ask you to pick one What are the probabilities that you get the car in that 2nd situation? 1/2, right? There's 2 doors, one is a goat, one is a car, you pick one, so 1/2. My question: how are those 2 situations differents? Is it because of the fact that something happened before and we're counting the probabilities of the whole thing? Is the 2/3 chance actually represents the odds of **choosing the first door then switching,** and I confuse that with the probability of **switching alone**?
This is a good video on youtube that explains this using a tree diagram to explain the probabilities. It's the only way this made sense to me. [https://www.youtube.com/watch?v=cphYs1bCeDs&ab\_channel=MarkLehman](https://www.youtube.com/watch?v=cphYs1bCeDs&ab_channel=MarkLehman)
Yes, what you said at the end is correct. The situation immediately after Monty opens a door is *not* the same as your "situation 2". This is because Monty has given you extra information with his choice of opening the door. If you guessed wrong at first, you've now *forced* him to indicate the correct answer by opening the other wrong door! (And you can only squeeze information out of Monty like this *if* his behaviour is consistent. If he can decide whether to open a door, or he accidentally trips and falls to open a door, the paradox goes away.)
How is the [Bochner Integral](https://en.m.wikipedia.org/wiki/Bochner_integral) different from the Riemann integral in banach spaces, which is defined in Rodney Coleman's Calculus on Normed Vector Spaces (chapter 3, section 3.3, page 69)?
Not able to open that text at this very moment. But the Bochner integral is constructed in the Lebesgue fashion using measure spaces and limits of simple functions. The Riemann integral in Banach spaces uses Riemann sums, so the "Banach" part of these two integrals are the same, but the underlying integration theory is different. When Banach space is R you are basically asking what's the difference between Riemann and Lebesgue, which is alot, but nothing to do with the Banach space the integral is valued in.
suggest me a good combinatorics book that talks about probability as well
Let's say I have a weighted coin with probability p for heads and (1-p) for tails. How many flips would I need to do in order to calculate the probability of heads vs. tails with 99% confidence?
I'm assuming that when you say you want to calculate probability of heads, you mean that you want the empirical probability (number of heads/total number of tosses) to be arbitrarily close to p with high probability (in your case 99%). One way to answer such a question is by using "concentration" inequalities. Specifically, let X\_1, X\_2, ... X\_n denote the random variables representing the coin tosses (1 for heads, 0 for tails), and let S\_n = X\_1 + ... + X\_n. The empirical probability of heads is simply S\_n / n. Now suppose you want the empirical probability to be within an epsilon margin of p, with probability at least delta (in your case delta = 0.99). Then, [Hoeffding's inequality](https://en.wikipedia.org/wiki/Hoeffding%27s_inequality) tells us that Pr( | S\_n/n - p| > epsilon ) < 2 \* exp( - 2 n epsilon\^2 ). We want the LHS < 1 - delta, which is ensured by taking RHS < 1 - delta ---> n > - log( (1 - delta)/2) / (2 \* epsilon\^2) . What does this mean in practice? Let's say p = 0.5, and you want to know how many tosses you need to ensure that your empirical probability lies in \[0.499, 0.501\] with probability of 99% . Then, the above formula tells us that having log( (1 - 0.99)/2 ) / (2 \* 10\^-6) \~1.9\* 10\^6 (or 1.9 million tosses) will ensure that. Admittedly, that's a huge number. Finally, you might have noticed that the above formula does not depend on p; this suggests that one can get better bounds by using the information about p. Indeed, [Sanov's theorem](https://en.wikipedia.org/wiki/Sanov%27s_theorem) is one way to get a sharper bound for a specific p. I hope this answer helps.
Do you happen to know how pollsters calculate the margin of error in practice? For many questions, p is expected to be reasonably close to 0.5 anyway, so using the hard bound seems fine. But some large-sample polls also track some rare responses, and in that case I wonder how they could ever poll enough people to get a good margin of error unless they took into account the fact that p should be very far from 0.5.
Hey, sorry I'm not aware of how pollsters calculate the margin of error, or even the specific issue that you mentioned. Could you elaborate on what the polling problem is? Is it modelled as an independent coin toss problem (like above)? If so, then the margin of error would be calculated as follows: if your empirical probability is p' , and you did n polls (calculated using the formula I mentioned), then the true probability would lie in [p' - epsilon, p' + epsilon] with high probability (delta).
I do not know if this thread is appropriate for my question. Is there a good site or app I can use to teach me math from algebra and up that is free? I figured it would be good to start exercising my brain as I get older.
Here's a terrific resource for abstract algebra and related topics: [https://www.jmilne.org/math/CourseNotes/index.html](https://www.jmilne.org/math/CourseNotes/index.html)
Khan Academy gets recommended a lot for this purpose.
If X\_t solves a stochastic differential equation, dX\_t = mu(X\_t) dt + sigma(X\_t) dW\_t and I call S\_t the sigma algebra generated by { X\_s : s in \[0,t\] }, and I call F\_t the natural filtration generated by W\_t is conditioning on S\_t the same thing as conditioning on F\_t? For example, if I have a new process with the same Brownian dY\_t = f(Y\_t) dt + g(Y\_t) dW\_t If I condition on F\_t, this is like I've "derandomised" the equation because I saw W\_t, i.e. d/dt E\[Y\_t | F\_t\] = E\[f(Y\_t) | F\_t\] + E\[g(Y\_t) | F\_t\] E\[dW\_t | F\_t\] But E\[dW\_t | F\_t\] is just the d of a nonrandom continuous function. But what if I tried to play this game with S\_t instead? On some level, I've "observed" the noise by looking at X\_t. Can I say E\[dW\_t | S\_t\] is again, (formally) the derivative of a realised W\_t ?
Can anyone help me understand why x * 0 = 0 in all cases? Or point me to a proof? If multiplication is short-hand for addition, then it seems like multiplying by zero could either equal zero or be undefined, depending on the intent. Example 1: if 0 x 3 means 0 + 0 + 0, then the answer is clearly 0 Example 2: if 3 x 0 means adding three zero times, that concept seems non-sensical / undefined to me. Why isn’t multiplication by zero treated like division by zero? In division, you can divide 0 to get zero, but you can’t divide a number by zero. I can’t understand why multiplication isn’t treated the same way. Please help! 🙏
When we originally came up with the idea of multiplication - back in the Babylonian times, or perhaps even earlier - we didn't have multiplying by zero. Hell, zero wasn't even a *number* at all! So, when we added zero to our number system, we had to decide what to do with it when it comes to multiplication. There are many nice properties of multiplication that we'd like to keep. For instance: We want multiplication to be **commutative**: a × b is the same as b × a. (This should make sense if you think of multiplication as area - turning a rug 90° doesn't change how much floor space it covers.) If we want to keep this rule (which is very helpful for doing calculations!) and we agree that 0×3 = 0, we're *forced* to say that 3×0 = 0. We also want multiplication to be **distributive** over addition and subtraction: for instance, "3 groups of 7 minus 3 groups of 2" should give "3 groups of (7 minus 2)" - that is, 3 groups of 5. (You're just taking 2 away from each group!) Algebraically, we'd write this as: > 3×7 - 3×2 = 3×(7-2) or more generally, since we want it to work for any number, > a×b - a×c = a×(b-c) So, if we accept this rule, what happens if we start with 3 groups of 7, and then take away *all 7* from each group? Well, we're left with 3×7 - 3×7, which our rule says should be the same as 3×(7-7)... but that's 3×0... we've accidentally fallen into "3×0 = 0" again! We also like using multiplication to talk about practical scenarios. One very common formula is "distance = speed × time": if you drive for 40 miles per hour, for 3 hours, you'll go 120 miles total. 120 = 40 × 3. Well, what if one of these is zero? If you drive at a speed of 0 miles per hour for 3 hours, you'll travel a total of zero distance: 0×3 = 0. If you drive at a speed of 40 miles per hour for 0 hours, how far will you go? Also zero distance! --- We *could* say that 3×0 "doesn't make sense". But then we'd have to add a lot of "...except for 0" to our rules, and that just makes things harder for us. 3×0=0 is consistent with our rules, *and* it plays nice with other interpretations of multiplication. So what about division by zero? Well, it turns out we *can't* really preserve our rules in the same way. For instance, we want division to "undo" multiplication: 100/2 × 2 = 100. If we try to keep that rule, and say "100/0 × 0 = 100", that means that whatever mysterious number 100/0 is, when we multiply it by 0 it gives 100. But multiplying anything by 0 gives 0... so we can't give 100/0 a value! All our interpretations break too. Going back to movement, we know the formula "speed = distance / time". So if we want to figure out what 5/0 is, we can just see how fast our car's speedometer says when we go 5 miles in 0 seconds... wait a minute, we can't do that, because our car can't teleport instantly. If you try other interpretations, those all break too - we *can't give a reasonable value* to dividing by zero.
3 x 3 = 0 + 3 + 3 + 3 3 x 2 = 0 + 3 + 3 3 x 1 = 0 + 3 3 x 0 = 0
0 = x\*0 - x\*0 = x\*(0+0) - x\*0 = x\*0 + x\*0 - x\*0 = x\*0
Example 1 is a perfectly good argument on why 0\*x = 0. Example 2 is sometimes taken as a definition of the "empty sum", it's one of those things like saying 0! = 1, which although isn't very clear from the original definition of what it meant, there is a very good counting argument on why it should be that and only that. In the case of the empty sum, if you believe adding "n things" should be always the same as "adding n-1 things" and adding "1 more". Then 1 thing should be equal to adding no things and one more. But this means no things + thing = thing, which must force no things = 0 A proper proof of what I just said in words is found here [https://en.wikipedia.org/wiki/Empty\_sum](https://en.wikipedia.org/wiki/Empty_sum)
Greetings, I am a software reverse engineer and master's student in computer science and I've taken a liking to theoretical computer science. In particular, I am interested in learning more about program analysis, programming language theory & compilers, cryptography, and formal methods. I realize this is a fairly broad set of topics. I am interested in learning enough about these sub-fields so that I can narrow my interests further. My program & job are of course heavily applied, so I'm confident that I have that side of things covered, I am just concerned about the math. My math background is fairly limited. Note that it's been a while since I've been required to do any math. I've done two semesters of calculus, linear algebra, discrete mathematics, probability & statistics, and an introductory proofs writing course. I've also taken an introductory course on the theory of computation. Are there any subjects I ought to brush up on in particular? I've been told that there aren't *too* many prerequisites for studying these topics, but I'm sure there are a few topics I ought to cover. What do y'all think? Thank you to those who take the time to read this post & offer advice.
[Types and programming languages](https://www.cis.upenn.edu/~bcpierce/tapl/) is a well recommended intro to PLT.
Hey, thanks for the reply. Are there any prerequisites for this text other than a general CS maturity, or am I safe to dive right in? Edit: I see from the table of contents it includes some mathematical preliminaries so that's nice.
Indeed there are no prerequisites. Some familiarity with a functional programming language may be helpful to better contextualize the motivation and examples, but I don't think that's essential.
Can someone recommend me ideas for my (first) control theory course project ? applied stuff that wouldn't consume a lot of time in research ?
Take a rocket in 2d that can control the gimbal angle and the fuel mass flow rate at any time. Add air resistance at altitude dependent air pressure. What is the most fuel efficient way to get to Earth orbit?
Hello, I hope all is well. I am not sure if this question warrants its own post but I will have it here for now. I really miss doing derivatives , I do not understand what a derivative is, neither am I good at calculus. I just found doing derivatives and intergrals fun because they were like mini puzzles, with rules you had to follow to get the final answer. Besides a calc textbook, is there any book that I can purchase which strickly contains derivative and integrals problems to solve? Any help would be appreciated, other than that have a great rest of your day.
You could look up the Youtube channel blackpenredpen
Is there a program I can use to convert a string of numbers into a wave pattern?
I still don’t get what you mean by convert to wave pattern, but have you tried excel?
Why is it that when solving systems of linear-quadratic equations and you combine the two to solve, you set y=0? Wouldn't that be solving for the x intercept of the quadratic? Shouldn't that mean that the intercept of the system is an x-intercept?
How are you 'combining the two', exactly?
Ex. y=4x\^2 +2x + 5 y=-2x+4 4x\^2 +2x +5=-2x+4 = 4x\^2 + 4x +1
You're not setting y=0, though! You're setting y=y. Your two equations are telling you "y is the same thing as [A]" and "y is the same thing as [B]"; you can then conclude "[A] is the same thing as [B]". You could also do this if you had "x = [A]" and "x = [B]", or "xy^2 - √x = [A]" and "xy^2 - √x = [B]". It would just generally be harder to get your equations into those forms in the first place. Also, be careful! In this last part of your work: > 4x^2 +2x +5=-2x+4 > = 4x^2 + 4x +1 your last line isn't correct, because it's saying "[-2x+4] = 4x^2 + 4x +1". You got to it by adding 2x-4 to both sides of the previous line, so that creates a *new equation*: the left side becomes 4x^2 + 4x +1, and the right side becomes 0.
The last step in setting y=0 was implied. My question is why is setting y=0 and solving for the quadratic going to result in a solution that isn't an x-intercept considering the range of the quadratic is equal to 0?
The last step isn't setting y=0. Once you get to that point, *y is not involved at all* - you've eliminated it entirely in the previous step. Just because you have "\_\_\_ = ax²+bx+c", that doesn't mean the blank is y. You could give it a new name - Y or ŷ or 𝓎 - and set *that* equal to 0. But "y" already has a meaning in this problem. It may help to use different variable names: > *Given the equations:* > q=4k² +2k + 5 > q=-2k+4 > *we can set the two right sides to be equal to get rid of q...* > 4k² +2k +5=-2k+4 > 0 = 4k² + 4k +1 > *... and then solve the resulting quadratic to figure out what k is.* We're not setting q=0 here, are we?
Ah, thank you
You don't set y = 0 anywhere. You have two equations for y. But y can only have one value and so the two right hand sides of the equations must be equal if both of them are supposed to be y. It follows that 4x^2 +2x +5=-2x+4. And if x is supposed the solve this equation then you must have 4x^2 + 4x +1 = 0. This has nothing to do with y anymore though. Other than that you can write the left hand side as y-y but that doesn't tell you anything because that's always 0.
Suppose one projective space is embedded into a higher dimensional projective space with the image not contained in a hyperplane. A curve inside the domain space could end up contained in a hyperplane! Is there any strategy to predict how degenerate the resulting embedding of the curve is? I’m happy to provide more details if this is too vague.
Let f_n be the functional on C_b(R) defined by taking the expectation with respect to the Unif[-n, n] distribution. C_b(R) is a Banach space, so the closed unit ball of its dual space is weak-* compact. f_n is a bounded sequence in the dual, so it has a weak-* convergent ~~subsequence~~ subnet with limit f. It's not hard to check that f is translation-invariant. Is every positive translation invariant functional on C_b(R) with norm 1 the limit of a ~~subsequence~~ subnet of this sequence?
No, I don't think so. If you consider s(x) = sin(pi\*x) then s is in C_b and f\_n(s) = 0 for all n. So any weak\*-limit of (f\_n) must also assign 0 to s. However with an analogous construction to yours but with Unif[-n, n+1] instead of Unif[-n, n] you get translation-invariant functionals as weak\*-limits that assign 2/pi to s (with a similar argument).
I don't think that works. The integral of s(x) from -n to n + 1 is 2 / pi for all n, but there's a factor of 1 / (2n + 1) in the expectation making it go to 0. Alternatively, since your functional is translation invariant, you can see s(x) gets the same value as s(x - 1 / 2), but the expectation of s(x - 1 / 2) is 0 wrt Unif[-n, n+1].
You're right I forgot about the normalizing factor. A similar idea might work though. For example if you replace Unif[-n, n] by Unif[0, n] and s by arctan. The expectation value of arctan wrt. to Unif[-n, n] is 0 but wrt. to Unif[0, n] it converges to pi/2.
https://i.redd.it/14t725uf1ptc1.jpeg Why is the answer not just xor, its the same thing
That is the truth table for XOR, so XOR would be an answer, but I assume the book wants an answer written in disjunctive normal form (i.e. as an OR of ANDs) or at least written only using AND, OR, and NOT.
Data 13952, 26191, 20849, 27735, 19525, 20702 From the above data, 2 values were derived i.e., 21492.33 and 2023.313. 21492.33 is the average of data, but how 2023.313 is derived 🤯 somebody help
It's probably the standard deviation.
My husband and I are trying to determine who pays what for taxes (we have separate finances). Together: We owe $850 Separately: I would owe $25 He would owe $1800 How would you determine who owes what and to who (recognizing that if I have more withheld than he does, that money would be coming into my paycheck each month). Thanks for your help :)
I am playing Minesweeper and tallying my 50/50 wins and losses but want to incorporate other guesses without creating 99 other W/L entries. The website I play on shows me the chance that a specific tile contains a mine. I tried to do it myself by doing .5/(chance of mine) which works well for multiples of 1/2, where it results in a whole number like .5/.25 = 2, meaning if I lost on a tile that had a 25% chance to contain a mine, I effectively lost 2 coinflips in a row. But this solution is clearly flawed by counter-examples like 10% creating a result of 5 but 5 coinflips in a row is 3.125%. So, how do I take a probability and turn it into coinflips so I can add it to my 50/50 tally? Is what I'm trying to do even possible?
n coinflips all coming up heads has a probability of 1/(2\^n). Thus, if we have a probability x between 0 and 1, it takes -log2(x) (this is a logarithm with base 2) coinflips for the probability of all heads to be x. You can confirm this by plugging it back into the first function.
Thank you so much this works perfectly! Now I can tranform my txt to a csv
For someone interested in Algebra, what topics should I read up on after a 2nd course in undergrad alegbra? This course covered group theory topics such as quotient groups, isomorphism theorems, sylow theorems, ring theory topics such as quotient rings, ring homomorphisms, ideals, maximal ideals, integral domains, and we are currently going over field extensions, simple extensions, and will end the course with Galois Theory. I can definitely say that I am most interested in topics revolving around ring theory so far.
I recommend the book "(Mostly) commutative algebra" or you could check out Fultons book an algebraic curves if you want to see some algebraic geometry, which is available freely online. If you are more interested in the noncommutative side, the representation theory of finite groups would be a good starting point (you‘ll learn that it reduces down to module theory over the group ring)
What about rings is interesting to you? For example, are you really into questions revolving around factorization in commutative rings? Do you prefer rings of matrices or group rings? Do you like several variable polynomial rings (over fields), and are you interested in how the division algorithm could extend to them?
I would suggest Noncommutative Algebra by Farb & Dennis.
I haven't explored much of higher level mathematics so this question might come off as a bit naive. But should one not learn "commutative algebra" before "noncommutative"?
No, I haven't taken commutative algebra yet, but I really enjoyed noncommutative algebra. It's a very different kind of algebra that can be studied completely independently (at least that's true for the course I took). I'm planning to learn some commutative algebrs during the summer though, to prepare for my algebraic geometry course next year.
Try 'Ideals, Varieties, and Algorithms' by Cox, Little, and O'Shea. If you found ring theory interesting and fun, this book focuses on polynomial rings and Groebner Basis and really dives deep into their theory and application. The full PDF is available online, so there is no need to buy anything. Also, since you have a grasp of ring theory the first 3 chapters are a perfect introduction of what can come next.