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H: Volume and surface area of a sphere
The volume of a spherical balloon increases by $1cm^3$ every second. What is the rate of growth of the radius when the surface area of the balloon is $100cm^2$
The surface area of a sphere is $4\pi r^2$, and its volume is $\dfrac{4}{3}\pi r^3$.
The answer sheet states that $\... |
H: Confustion with Fourier Transform of harmonic functions
Note: this is homework.
Say we assume a time-varying momentum is of the form
$p(t) = \Re \{p(w) e^{-iwt} \}$
Now we would like to know the solution of the following equation, in frequency space:
$ \tfrac{d}{dt} p(t) = -\tfrac{1}{\tau}p(t) - eE(t) $
The part wh... |
H: Sum the following series, where $e^y < 1$
Sum the following series: $$ 1 + e^y + e^{2y} + e^{3y} + .. $$ where $e^y < 1$
So $e^y$ is less than one, that means $y$ is a negative number, which means that each term will be contributing less. But I find this to be a very vague question. Can you actually say that the su... |
H: Solving nested summation $\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}p(A_i)p(B_j) $
I am having trouble solving the following nested summation:
$$\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}p(A_i)p(B_j) $$
where $p(A_i) = \frac{1}{n}$, and $n$ is a constant (length of an array). Same goes for $p(B_i) = \frac{1}{n}$.
I tried rewriting i... |
H: Find $2$ unit vectors orthogonal to $(2,1)$
I understand that in order to find an orthogonal vector to $(2,1)$ I will solve this :
$\langle(x,y),(2,1)\rangle = 0 $
but I don't understand how this is related to $2$ unit vectors.
AI: As you did:
$$2x+y=1\iff y=1-2x\,,\,\,\text{so for example}\;\;u:=\binom{\;\;1}{-1}
... |
H: Odd and even function properties...
Does it mean when the function is even it's in 100% cases y-axis symmetric, and when it's odd it's in 100% cases origin symmetric?
AI: Yes, that is the geometric interpretation of even or oddness. Also, if a function has no lines of symmetry in the plane, then it cannot be even ... |
H: Show: $C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)$.
Show that
$$
C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)~~~~~~~\forall0<\lambda\leq 1.
$$
Hello, I have some problems to show these inclusions! In order to get some help, I wrot... |
H: See if "7<4 implies 7 is ..." Is the following conclusion valid?
For my homework I need to see if the following conclusion is correct.
$$
7<4 \implies 7\ \text{is not a prime number}\\
\lnot(7<4)\\
-----------------\\
\text{7 is prime number}\\
$$
To tell you the truth, I have no idea how to start this, letalone ho... |
H: Automorphisms of order $2$ of the multiplicative group of a field
Let $k$ be any field, finite or infinite, (even though I'm more interested in the infinite case) of $\operatorname{Char}(k)\neq2$. Let $\varphi$ be an automorphism of order $2$ of the multiplicative group $k^\times$ of $k$ ($\varphi$ is not assumed t... |
H: Help with an implication of a really basic question in algebraic geometry
I'm starting to study algebraic geometry and I'm trying to prove this implication:
For a homogeneous ideal $\mathfrak a\subset S$, show that $Z(\mathfrak
a)=\emptyset \implies \sqrt {\mathfrak a}=\text{either}\ S \ \text{or
the ideal}\ S_+... |
H: How is $\sqrt{\frac{\sqrt{3}+2}{4}} = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$
How is $\sqrt{\frac{\sqrt{3}+2}{4}} = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$?
(Prove by using algebraic manipulation not by calculation)
I've tried to come up with something myself but I can't find a solution, I must be missing something.
AI: Well, not... |
H: Converting Vector Valued Function
I am having trouble turning this vector function into something like a $y=mx +b$ equation.
$$r(t) = 2cos(t)^3 \hat{i} + 2sin(t)^3 \hat{j}$$
Normally I would say $x = 2cos(t)^3$ and $y=2sin(t)^3$ and either rearrange and solve x for t or use trigonometric identities to simplify thin... |
H: Right adjoint of forgetful functor from Top
How to prove this?
The forgetful functor $U:\mathbf{Top}\to\mathbf{Set}$ has a right adjoint, namely the functor $\mathbf{Set}\to\mathbf{Top}$ which equips a set with the indiscrete topology and left adjoint which equips a set with the discrete topology.
AI: If $X$ is a t... |
H: does $\int ^1_0 \frac {\ln x}{1-x^2}\ \mathrm dx$ - converge?
Question:
$\int ^1_0 \frac {\ln x}{1-x^2}dx$ - converges or diverges?
What we did:
We tried to compare with $-\frac 1x$ and $-\frac 1{x-1}$ but ended up finding that these convergence tests fail. Our book says this integral diverges, but Wolfram on the o... |
H: Probability of getting Required Sum
What is the probability of when four dice rolled together once,and getting a sum of Thirteen
If we do by just calculating all possible values of sum, then it will take more time; so we can solve the above problem as Multinomial Coefficents of sum, i.e.:
$$
x_1+x_2+x_3+x_4 = 13\... |
H: Obvious inequality
Suppose $f$ and $g$ are Lebesgue integrable functions on $E$. Then:
$$ \int\limits_E | f + g | \leq \int\limits_E ( |f| + |g| ) $$
Does this follow easily? I can see we have to use the triangle inequality, but does it work even with integrals? I'm not seeing this inequality.
AI: Of course $|f+g| ... |
H: Need help with Sigma-algebra
I am confused on how to determine a Sigma-algebra.
The following partitions of a set are given:
$$
A1 = \{1,3\} $$ $$
A2 = \{2,4,6,8\} $$ $$
A3 = \{5,7,9\}$$
And Omega is $$
\Omega = \{1,2,3,4,5,6,7,8,9\}
$$
I know that at least Omega and the empty set has to be part of the Sigma-algeb... |
H: Recurrence Relation.
I was searching the internet when I came a across a question, and just couldn't solve it. I kept rearranging and substituting but kept going around in loops.
"For $n:= 1,2,3,.....,$ Let $$ I_n = \int_0^1 \frac{x^{n-1}}{2-x}.dx $$
Writing $x^n =x^{n-1}(2-(2-x))$, show that this sequence of numb... |
H: Algebra rules when multiplying matrices with unknown variables
I'm about to compute the determinant of a given matrix, this matrix however contains variables instead of actual values, so I'm a bit uncertain what to do here.
I have the 3x3 matrix A:
$$A = \left(\begin{matrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b\\ 2c &... |
H: last two digits of $9^{1500}$ (Dummit Foote -Abstract Algebra preliminaries $0.3.5$)
Question is to find last two digits of $9^{1500}$ (No Euler totient theorem please)
What i have done so far is :
$9^2\equiv 81\pmod{100}$
$9^4 \equiv 61\pmod{100}$
$9^8\equiv 21\pmod{100}$
$9^{16} \equiv 41\pmod{100}$
$9^{32} \e... |
H: Convergence of the series $ \sum_{n=1}^\infty \frac 1{n!} $ using the Cauchy convergence criterion
Study the convergence of the following series, using the Cauchy Convergence criterion:
$$ \sum_{k=1}^n \frac 1{k!} $$
Following the $ \sum_{k=n}^{n+p} \frac 1{k!} <\epsilon $, I must show that $ \frac 1{(n+1)!}+\f... |
H: How to write this conversion algorithm
I have to calculate Grade Average from the scale of 1 to 5 such that 1 being the best (100%) and 5 being the worst (0%). So we have,
1 = 100% (Best)
2 = 75%
3 = 50%
4 = 25%
5 = 0% (Worst)
For a given subject, the student scored 40 out of 100. So we have 40%.
Now how do I conv... |
H: identity for squared binomial coefficient
I was wondering if there is an identity for squaring a binomial coefficient.
I know there is one with converting it to a linear equation, but I am looking to stay at a "coefficient" level. something like:
$${n \choose k}^2={n \choose K}$$
where K is a function of k. Did no... |
H: Mapping cone not homotopy equivalent to quotient space
My question is about a "non-example" to theorem 1.6 in chapter VII in Bredon.
We have an inclusion $i: A \to X$, with $A = \{0\} \cup \{1/n | n = 1,2,...\}$, and $X = [0,1]$. Then $X/A$ is a one-point union of an infinite sequence of circles with radii going to... |
H: Derivative of an integral? $f(y) = \frac{d}{dy} F(y) = \color{red}{\frac{1}{\sqrt{y}}}\Phi'(\sqrt{y})$
Am I right to say if I differenciate an integral, I get back the thing inside the integral?
$$\frac{d}{dx} \int f(x) \, dx = f(x)$$
Then why is it in the below question,
The last line marked by the arrow ...
$... |
H: What is the use of Euler Totient or Phi Function?
What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
AI: RSA, or public-key cryptograph... |
H: Why $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible over $\mathbb{Q}$?
I'm tring to know why $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible over $\mathbb{Q}$.
Can you help me with a proof or showing me some references, please.
AI: The usual trick is to look instead at $f(x+1) = \frac{(x+1)^p-1}{x}$. Since the binomial coef... |
H: If a function $f$ is continuous in $[a,∞)$ and finite $\lim_{x→+∞}f(x)$ exists, then it's uniformly continuous in $[a,+∞)$.
Prove that if $f$ is defined and continuous in $[a,+∞)$ and if there exists a finite limit $\lim_{x→+∞}f(x)$, then $f$ is uniformly continuous in $[a,+∞)$
I know that since there exists a fi... |
H: $T_2$ spaces and isolated points
Is there a topological Hausdorff space with an infinite number of isolated points such that any infinite set of isolated points have an infinite number of limit points !? (Of course it would be impossible if a limit point is a limit of a sequence.)
AI: Yes: $\beta\Bbb N$, the Čech-S... |
H: Calculate $10,000e^{-\int_2^{10}\left(0.05+0.01/(t+1)\right)\,dt}$
This equation is used as an example in a text book with a given answer of $\approx$ 6,617
I cannot get to this solution as somewhere along the way I must be making an error. If it is a problem with the integration, please can you point out any funda... |
H: Simplification of boolean algebra from "not s and p" to "not s"
I am trying to learn more about "Rules of Inference" and their application, but one thing always confuses me, and that is simplification "not s and p" to "not s".
I have looked at some examples:
http://www.site.uottawa.ca/~lucia/courses/2101-10/lecture... |
H: Does the sequence $1,-1,1,1,-1,1,1,1,-1,1,1,1,1,-1,1,1,1,1,1,-1,\ldots$ have a closed form?
Question : Can we represent the following sequence $\{a_n\}\ (n\ge 0)$ as a closed form?$$a_n : 1,-1,1,1,-1,1,1,1,-1,1,1,1,1,-1,1,1,1,1,1,-1,\ldots$$ Suppose that there exist ${(i+1)}$ $1_s$ between the $i_{th}$ $(-1)$ and ... |
H: Giving an explicit example of a vector that is perpendicular to $v$
Let $v\in\mathbb{R}^3$ be a unit vector. It is possible to show that there exists vectors $\{w_1,w_2\}$ such that $\{v,w_1,w_2\}$ is orthonormal by applying the Gram-Schmidt process, but can we do so continuously?
Question: Is it possbile to give ... |
H: What is the standard notation for $\arcsin$
I found a lecture notes that claims the following. Is this standard?
The notation $\overline{\text{arc}}\text{ sin }x$ is the inverse function of $\sin x$ restricted to $\left [ -\frac{\pi}{2},\frac{\pi}{2}\right ]$ and $\text{arc sin }x $ mean all those $y$ satisfying $\... |
H: Mayer-Vietoris for a cover without triple intersections
Let $M = \bigcup_i U_i$ be a cover with open sets $U_i$ such that for
for distinct $i,j,k$ we always have $U_i \cap U_j \cap U_k = \emptyset$.
I would like to show the existence of the following exact sequence
$$\rightarrow \bigoplus_{i < j} H_q(U_i \cap U_j)... |
H: Modulo operation, the remainder of division of one number by another
The equation is:
$241 \equiv_{N} 35$
I have no clue how to get value of $N$, any ideas?
AI: Hint: In order to find $N$ you need to remember that if $a = b \pmod N$ then $N \mid a-b$, in your case $N \mid 241-35$, can you take over from here?
In y... |
H: In which the real number system that sum of geometric progression involve?
I want to know about sum of geometric progression
a and r
Are they real number it integer .. Etc ?
AI: The formula
$$(1-r)(a+ar+ar^2+....+ar^n)=a(1-r^{n+1})$$
hold in any ring. Thus it is true in integers, rationals, reals and complex numbe... |
H: Are the numbers 1, 0, -1 necessarily in every field?
Do the numbers 1, 0, and -1 belong to every field? To me, this seems a fairly obvious conclusion of the field axioms, though I haven't seen it stated like so in any textbooks.
AI: $0$ is in $F$, because $F$ is a group w.r.t. addition, it must have an identity ele... |
H: How to calculate the integer part of the value of the following equation?
How to calculate the integer part of the value of the following equation?
$$y=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{1000000}}$$
It should be calculated in a special way, after all the equation is so ... |
H: Prove that $1/f$ is uniformly continuous on ...
I need hints for this particular question:
Prove that if a function $f$ is uniformly continuous on $A\subseteq \mathbb{R}$ and $|f(x)|\geq k>0$ for all $x\in A$, then the function $\frac{1}{f(x)}$ is also uniformly continuous on $A$.
My attempt: From the rough work g... |
H: $\int_{-\infty}^{0}xe^{x}$ diverge or converge
How would one find whether the following improper integral converge or diverge.
$\int_{-\infty}^{0}xe^{x}$
I did the following.
$t\rightarrow\infty$
$\int_{t}^{0}xe^x$
I did the integration by parts.
$u=x$
$dv=e^x$
$xe^x-\int 1e^x$
$xe^x-1xe^x$
$(0)(e^0)-e^0(0)-te^t-te... |
H: tangent planes and linear approximations and partial derivatives
I have to study tangent planes and linear approximations, there is this theorem :
THEOREM: if the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$, then $f$ is differentiable at $(a,b)$
Actually, it's foggy in my ... |
H: Some questions on convex sets.
Are all bounded closed convex sets in a metric space $(M,d)$ compact? or if not are they complete?
The positive definite matrices form a convex set (Why does a positive definite matrix defines a convex cone?), do they also form a metric space? If so what is the metric? Are they comple... |
H: History of complex numbers
I'm interested in the history of complex numbers - their origin and their subsequent development.
I'd be very interested if anyone can provide references for finding out more about this topic.
AI: The book An Imaginary Tale: The Story of $i$ by Paul Nahin is very nice, and has a fair amou... |
H: Smallest such $n \in \mathbb{N}$ that $2^{n} \equiv 1 \pmod{5\cdot 7\cdot 9\cdot 11\cdot 13}$
Can anybody give me a hint about how to find smallest such $n \in \mathbb{N}$ that $2^{n} \equiv 1 \pmod{5\cdot 7\cdot 9\cdot 11\cdot 13}$?
I thought that I will find it piece by piece with help from my friend Fermat's L... |
H: Counter examples to inscribed squares conjecture
Can the counter examples found by me qualify as a counter proof for the "inscribed squares problem" (the Toeplitz' conjecture) ?
I ask this here because the problem stands as unsolved for 100 years, and the counterexamples below appear too simple. It is hard to belie... |
H: help find my error in a statistics transformation computation
This is a low-priority question, but it has been bugging me so I thought I'd ask. In my stats homework I have the following exercise:
Exercise. Suppose $X_1$ and $X_2$ are iid observations from the pdf $f(x\mid\alpha)=\alpha x^{\alpha-1}e^{-x^\alpha}$,... |
H: How to read a contour plot?
I am taking machine learning, and I have seen a few contour plots in the course.
It seems that I can't understand how to read this plot, I have tried looking it up in Wikipedia, but I don't even understand the first example (the figure on the right).
Another example is from the course:
... |
H: Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$
Solve for $x$
$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$
solution should be by hand
AI: First it is clear that $x \geq 0$.
By AM-GM inequality we have:
$$x^{2010}+1 \geq 2 x^{1005}$$
$... |
H: Describing the ideal in polynomial ring with $n$ indeterminates
Let $R$ be a commutative ring and let $m$ be a natural number. Describe the ideal $(X_1,X_2,...,X_n)^m$ of the ring $R[X_1,X_2,...,X_n]$ of polynomials over $R$ in indeterminates $X_1,...,X_n$.
I know that, if $K[X,Y]$ is the ring the generators of ide... |
H: 4D to 3D projection
Im trying to calculate the position of 4D point in 3D world. I started with 2D and tried to extend it to the 3D and then to 4D. Firstly, I found out that its easy to calculate the projected position of 2D point on the line.
Whoops, there should be () in the first equation: x/(a+y)
Now I figure... |
H: Simplifying this Further
$$2x(x^2-3)^{10} + 20x(x^2+3)(x^2-3)^9$$
I would like to double check my answer (if anyone can double check this please)
Please simplify the top
AI: $$2x(x^2-3)^{10} + 20x(x^2+3)(x^2-3)^9=2x(x^2-3)^9(x^2-3+10(x^2+3))=2x(x^2-3)^9(11x^2-27)$$ |
H: Are the solutions to the equation $f(n \cdot x)=x$ always expressible in closed form?
Are the solutions to the equation:
$$f(n \cdot x)=x$$
always expressible in closed form?
$$n=1,2,3,4,5,...$$
AI: Depends what you mean by closed form, but if you mean in terms of elementary functions, then no. Consider, for instan... |
H: How do I find the supremum and infimum of this set
$$A=\left\{\dfrac{mn}{1+m+n} \mid m,n \in \mathbb N \right\}$$
I'm relatively new to this whole infimum and supremum proving. I've tried for a long time to prove the infimum of this set (which I believe to be 1/3). Can someone please help me and provide a proof for... |
H: Prove that $\sqrt{2n^2+2n+3} $ is irrational
I have proven this by cases on $n$.
I would like to see a neater proof. One similar to the proof of the fact that $\sqrt{2}$ is irrational.
AI: Hint: $2n^2 + 2n + 3 = 2n(n+1) + 3 \equiv 3 \pmod 4$. |
H: Definition of Vector Space
What is the meaning of objects in a vector space?
Definition: A vector space is a nonempty set V of objects, called vectors, on
which are defined two operations, called addition and multiplication
by scalars, subject to the ten axioms...
Can entries of a vector be anything other th... |
H: free groups , a question of listing elements and drawing multiplication table
I'm requested to list the elements and draw the multiplication table for the group $\langle a, b : |a| = 2 = |b|\rangle$ without any more details. But hence this group is infinite isn't it ?
while listing the elements i found $ = \{ a , ... |
H: Integral convergent calculating
I have to calculate if this integral is convergent for any "$a$":
$$ \int_2^\infty \frac{1}{x\log^a(x)} dx $$
(I made sure to not make any mistake when writing the integral from paper into this forums!)
I really don't what is meant by this..
AI: Changing variable to $t=\log x$, one h... |
H: How do I find the area in this question?
Find the area bounded by
$$ f(x) = x + 6 $$
$$ g(x) = x^3 $$
$$ h(x) = -\frac{x}{2} $$
Edit Fixed simple error on h(x)
I already drew the grew, although it is very hard to really tell where they seem to intersect.
It looks like on the left side of the graph f(x) intersects ... |
H: Convergence of average of sums
Let $a_1,a_2,\ldots$ be a sequence of real numbers, and define $s_n=a_1+\ldots+a_n$ for all $n$. Define $t_n=\dfrac{s_1+s_2+\ldots+s_n}{n}$. There is a theorem that if $\{s_n\}$ converges, then $\{t_n\}$ converges (to the same limit, I believe).
Does the converse hold? That is, if $\{... |
H: Evenness of Fourier coefficients
Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is even if and only if $F(n)=F(-n)$ for all $n$.
Suppose $f$ is even. Then I have to prove $\int_{-\pi}^\pi f(x)e^... |
H: Find $\vec u$ such that $D_\vec uf(x^2+y^2)$ is maximum, minimum, and zero at $(1,2)$
I understand how to find the maximum and minimum, but I'm having trouble understanding how to find it when it's zero. My professor said that I need to set $\phi=\pi/2$ in $D_uf(x,y)=\|\nabla f\|\cos{\phi}$. But $\cos{\pi/2}=0$, so... |
H: showing existence and uniqueness of solution of $y'(t)=\frac1{1+|y(t)|}$
Given \begin{align*}
y'(t)&=\frac1{1+|y(t)|},&y(0)=y_0&&\textrm{for }t\in[a,b]
\end{align*}
I want to show that this IVP has a unique solution
My attempt: We get $f(t,x)=\frac1{1+|x|}$. If $f$ is continuous there exists a solution on $[a,b]$ ... |
H: A question on limit of functions
Let $\{f_n\}$ be a sequence of continuous functions such that$ f_n \to f$ uniformly on $\mathbb R$. Suppose that $x_n\to x_0$. Prove that $\lim_{n\to\infty} f_n(x_n)=f(x_0)$.
Let me know if I should format it better! This question seems so straightforward that I don't know what d... |
H: Which volume formula do I use for this problem?
Find the volume generated by revolving about the $x$-axis the region bounded by the curves:
$$f(x)= x^2 ,\ g(x) = 2-x^2.$$
I drew the graph and if we are revolving around the $x$-axis it looks like we are going to have some sort of gap inside part of the shape (washer... |
H: A book/text in Stochastic Differential Equations
Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in this area. So, I want a book that can introduce me in this field with... |
H: If a function is integrable, then it is bounded
Probably a simple question, but I wonder about the following.
I know that if a function $f : \mathbb{R} \rightarrow \mathbb{R} $ is (Riemann)integrable, then it is bounded. I wonder if I can generalize this to functions on $\mathbb{R}^3$ (now for an ingral over a volu... |
H: Evenness of Fourier coefficients implies even function
Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$Assume that the Fourier coefficients determine an $L^1$ function.
How can we prove, without using Fourier transform, ... |
H: the table at the end of Theoretical Computer Science Cheat Sheet
Theoretical Computer Science Cheat Sheet, created by Steve Seiden, is a hodgepodge of well-known mathematical theorems and notions.
I can understand (or guess at least) many of them, but I'm not sure about this 10-by-10 table at the end of the documen... |
H: Linear Differential Equation $y'''−3y′+2y=\cos t+e^t$
I'm trying to find the solution to this non-homogenous third-order linear differential equation.
I know the solution is supposed to be:
$$c_1e^t+c_2te^t+c_3^{-2t}+\frac{e^tt^2}{6}-\frac{\sin(t)}{5}+\frac{\cos(t)}{10}$$
So far I've solved the left side of the equ... |
H: Did I correctly set up this volume problem?
The question:
Find the volume of the solid generated by revolving about the y-axis the region bounded by:
$$y = x^2$$
$$x-axis$$
$$x = 2$$
First, I believe we want to do this problem in terms of y. If this is correct, we would then set it up like the following:
$$ V = \in... |
H: Finding a marginal PDF of a joint probability distribution
I understand the idea of how to do it, but I'm currently getting a constant as my marginal PDF, which doesn't make sense.
The overall distribution is as follows:
$f(x,y) = 5ye^{-xy}$ for $0 < x, 0.2 < y < 0.4$
I'm trying to find the probability that $0 < x... |
H: How to get the equation of tangent plane when the point is an unknown?
Problem:
Solution:
I don't understand how they got the equation in the black box.
Here is my attempt:
$$\overrightarrow { \nabla } F({ x }_{ 0 },{ y }_{ 0 },{ z }_{ 0 })=(2x-2y)\widehat { i } +(6y-2x)\widehat { j } +8z\widehat { k } \\ \overr... |
H: Evaluating $ \lim_{x\rightarrow \infty}e^{-x } + 2\cos(3x)$
Find the limit or prove that it does not exist by $\varepsilon-\delta$ approach:
$$ \lim_{x\rightarrow \infty}e^{-x } + 2\cos(3x)$$
Note:I found this question when I was doing exercise from the book Calculus:Early Transcendentals. The book just need me to... |
H: Properties of Natural Logarithm I need help finding the Derivative
$y=\ln(x)^2$
I am not sure why the answer would be $\frac{2\ln(x)}{x}$
I used this property "power rule" "$\ln(x^n) = n\ln(x)$
So i got $2\ln(x) $
the derivative of that using the constant multiplier rule i got
$\frac{2}{x}$
can I use the other cha... |
H: Does $m(E)>0$ imply that $E$ must contain a nondegenerate interval?
Does $m(E)>0$ imply that $E$ must contain a nondegenerate interval?
$E\subset\mathbb{R}.$ $m$ refers to Lebesgue measure. $I$ refers to a nondegenerate interval.
AI: No, try any fat Cantor set. $ $ |
H: Limit of e with imaginary number
Important part:
$\lim\limits_{x\to\infty} e^{-ix} - e^{-x} $
This is suppose to approximate to "$1$" but the way I see it we have $0 - 0$ ...
AI: Ok, so the limit of a complex sequence (function) exists iff the limit of its real and imaginary parts exists:
$$e^{-ix}-e^{-x}=\cos x-e^... |
H: Wolfram-Alpha's choice of $k$ in a complex logarithm
I'm puzzling on this complex integral:
$$ \int \frac{2ie^{it}}{2e^{it} - 1}dt = \log(2e^{it} -1)$$
The numerator is the derivative of the divisor, so the primitive is the log of the divisor.
When you ask Wolfram-Alpha to compute the integral over the range $0..2\... |
H: Condition on Fourier coefficients for real-valued function
Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is real-valued if and only if $F(n)=\overline{F(-n)}$ for all $n$.
We have $$\overline{F... |
H: How can the length of a normal vector matter?
Problem:
Solution:
Graph:
For part a, I understand mathematically why the value of c matters. What I don't understand is how it can possibly matter intuitively.
I get that $\overrightarrow { \nabla } F(x_{ 0 },y_{ 0 },z_{ 0 })=(0,c,0)$ and therefore, c can't be any ... |
H: Integration for Fourier coefficients of $x$
To compute the Fourier coefficients of $x$, I was trying to integrate $$\int_{-\pi}^\pi xe^{-inx}dx$$ How to integrate this? (I already did it, and I'm posting my answer just to keep the records and possibly help others in the future.)
AI: So I used integration by parts. ... |
H: Matrix representation of Automata
Is anyone know if there is any tutorial for the matrix representation of automata?? I am taking a theoritical computer science in this semester and the professor uses the matrix in his lecture. I gonna have test next week so I have to study for it. I tried to find any tutorial on g... |
H: Proof with mathematical induction that $ (\frac{n}{n+1})^2 + (\frac{n+1}{n+2})^2 + ... + (\frac{2n - 1}{2n})^2 \le n - 0.7 $
Proof with mathematical induction.
I have the following induction problem:
$ (\frac{n}{n+1})^2 + (\frac{n+1}{n+2})^2 + ... + (\frac{2n - 1}{2n})^2 \le n - 0.7 $
This property applies to all $... |
H: Triangle bisection on non-right angle triangle with known angle and two sides
I'm creating 3D road intersections and to create the corner points I detect when the edge vectors of the road intersect. It's extremely accurate.
However, I would like to pre-calculate the corner positions instead. The interior angles of ... |
H: The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule
Let's define
$$
e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R
$$
and
$$
\frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f... |
H: the value of $\int_{0}^{2\pi}f(e^{it})\cos t dt$?
$f$ be analytic function then what is the value of $\int_{0}^{2\pi}f(e^{it})\cos t dt$?
AI: Let $z=e^{it}$ the the integral becomes $$\int_{|z|=1}f(z)({z+z^{-1}\over 2}){dz\over iz}=0={1\over 2i}\int_{|z|=1} f(z)dz+{1\over 2i}\int_{|z|=1}{f(z)\over z^2}dz=\pi f'(0)$... |
H: Using DeMorgan's Laws to complement a function
Using DeMorgan's Law, write an expression for the complement of $F$ if:
$F(x,y,z) = x(y' + z)$.
$F=x'+(y'+x)'$
$F(x,y,z) = xy + x'z + yz'$
$F=(xy)'(x'z)'(yz')'$
$F(w,x,y,z) = xyz' (y'z + x)' + (w'yz + x' )$.
$F=[(xyz')'+(y'z+x)](w'yz+x')'$
My answers are underneath ... |
H: Solution of $y''+4y=\cos^2t$
I'm trying to find the solution, which is supposed to be
$y(t)=c_2\sin(2t)+c_1\cos(2t)+\frac{t\sin(2t)}{8}+\frac{\cos^2(t)}{4}$,
but I'm doing something wrong along the way and I can't figure out what I did. I was hoping someone would be able to find my mistake for me. I'll show what... |
H: Complex integral of an exponent divided by a linear ($\int \frac{e^u}{u-1}$)
Here is the question I'm working on:
Evaluate the following integral:
$$ \oint_{|z+1|=1} \frac{\sin \frac{\pi z}{4}}{z^2-1}dz$$
I've tried along the following line. Substitute $sin(z) = \frac{e^z-e^{-z}}{2i}$:
$$ \frac{1}{2i} \int \fra... |
H: Problem with permutations
The problem says:
We have strings formed by two letters, followed by two digits and then followed by three letters. In each group repetitions are not allowed, but the last group of three letters can contain up to one of those used in the first group. If the number of letters available is ... |
H: Prove that $n^9 \equiv n \pmod{30}$ if $(30,n) > 1$
I'm looking at the following number theory problem:
Prove that $n^9 \equiv n \pmod{30}$ for all positive integers $n$ if $(30,n) > 1$.
It is easy to show that $n^9 \equiv n \pmod{30}$ if $(30,n)=1$ by using Euler's Theorem. Is there any way to prove the above easi... |
H: Examples of complex functions with infinitely many complex zeros
What are some examples of complex functions with infinitely many complex zeros? There are no particular restrictions on the functions I am just curious and having a hard time finding examples. Also what can be said about a complex function with infini... |
H: Rotation Matrix inverse using Gauss-Jordan elimination
I'd like to calculate the inverse of a rotation matrix, let take the simplest case which is a $2$ x $2$ rotation matrix:
$R =\begin{bmatrix} \cos \theta & -\sin \theta \\[0.3em] \sin \theta & \cos \theta \end{bmatrix}$
I know that the inverse is the following
$... |
H: An example where $E[X_1 X_2] - E[X_1]E[X_2] = 0$ for functions $X_1$ and $X_2$
I computed $E[X_1 X_2] - E[X_1]E[X_2]$ using $X_1 = 3x+1$ and $X_2 = 2x+5$ the following way:
$E[(3x+1)(2x+5)] - E[3x+1]E[2x+5] = E[6x^2 + 17x + 5] - (3E[x]+1)(2E[x]+5) = 6(E[x^2] - E^2[x]) = 6\sigma^2$
When I was done I asked myself wh... |
H: Should I be able to prove Law of Cosines, Half Angle formula, etc?
This is more of a general question then the title suggests, but the laws in the title are what I'm currently studying. I can read the proofs of both and understand them after a while, but I could never produce such a proof unless I committed it to m... |
H: Real projective $n$ space
We define $\sim$ on $\mathbf{R}^n - \{0\}$ by $x \sim y$ if $x = \lambda y$ for some $\lambda \in \mathbf{R}$. We define projective $n$ space by $X = (\mathbf{R}^n - \{0\})/{\sim}$.
I am having trouble showing that $X$ is an $(n-1)$-dimensional topological manifold. The definition of a top... |
H: fundamental complex integral( in Conway's book)
I am reading Conway's Functions of One Complex Variable. I have some trouble in doing some exercise.
Find $\int_\gamma(z^2-1)^{-1}dz$, where $\gamma$ is a path. $\gamma(t)=1+e^{it}$ for $0\leq t\leq 2\pi$.
Find $\int_\gamma(z^2-1)^{-1}dz$, where $\gamma$ is a path. $... |
H: Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$
After a week in my Abstract Algebra class, the professor proposed this as a problem. I'm not entirely sure where to begin. $ V = \{ e, \tau, \tau_1, \tau_2 \}$, so I'm not sure exactly what is meant by $\text{Aut} (V)$. Is it simply s... |
H: How do I get an answer of $14$ using simpsons rule for $\frac{152e}{180n^4}<.0001$
I must have the algebra wrong somewhere but here is the original equation:
$$\frac{152e}{180n^4}<.0001$$
If I then multiply like this:
$$152e<.0001(180)n^4$$
Which then gives:
$$152e < .0018n^4$$
And then dividing:
$$\frac{152e}{.00... |
H: What's the probability that nine people were born in the same two months (but not the same month)?
Find the probability that nine people were born in the same two months (but not
all in the same month).
No clue how to approach this. I was thinking well you have to choose 8 out of the 9 people and then match them up... |
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