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H: Prove $I(Z_1, Z_2) = 0$.
Define $\mathbb{R}P^k$ as the quotient of $S^k$ by the antipodal map, with
smooth structure defined so that the projection $p: S^k \to \mathbb{R}P^k$ is a local diffeomorphism. Suppose that $k$ is odd and $Z_1, Z_2 \subset \mathbb{R}P^k$ are compact submanifolds of positive dimension for... |
H: Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?
This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that $[x+f(h)]^2+1=[x+g(h)]^2$ and this would get rid ... |
H: Proof or Counterexample on the Convergence of a Series
So one of my professors proposed a problem to me and it has stumped me for some time now. Here's how it goes:
Suppose you have a sequence $a_n$ of real numbers such that $$\lim_{n\to\infty} a_{n} = 0$$ and suppose the sequence of partial sums $s_n$ is bounded. ... |
H: Bill calculation (very simple math question)
I hope this question isn't too simple! If it is, let me know if I should post it elsewhere, thank you. (Also, I didn't know what to tag it with, sorry!)
My wife and I live with my brother-in-law Bob, and, for 30% of the time, also live with Bob's three children.
Our last... |
H: show that $x_n$ converges to root of alpha
I solved (a), (b) but
it's hard to show that c) is true.
(By intuition, since $x_1$ is larger root of $\alpha$, and all $x_n$ other than $x_1$ is $\lt$ $x_1$, I think all $x_n$ is in the interval (root of $\alpha$, $x_1$ )
I want to use theorem that if the sequence is mon... |
H: Ratio test: Finding $\lim \frac{2^n}{n^{100}}$
$$\lim \frac{2^n}{n^{100}}$$ as n goes to infinity of course.
I know that the form os $\frac{a_{n+1}}{a_n}$
$$\frac {\frac{2^{n+1}}{(n+1)^{100}}}{\frac{2^n}{n^{100}}}$$
$$\frac{2n^{100}}{(n+1)^{100}}$$
I am not clever enough to evaluate that limit. To me it looks like ... |
H: Example of a variety that is not toric
My question is simple, but I haven't seen it to be addressed anywhere:
What would be a simple example of an affine variety that is not a
toric variety?
Toric varieties (the ones I have studied) are constructed by a fan using "gluing". (For some examples, see Example 2.2 in... |
H: Automorphisms of $\mathbb Z_p[x]$
I am trying to find all automorphisms of $\mathbb Z_p[x]$ (polynomials with coefficients from $\mathbb Z_p$ where $p$ is prime).
I know that automorphisms of $\mathbb Z[x]$ are $x\to x$ and $x\to -x$, but now when coefficients are in $\mathbb Z_p$, I am not entirely sure.
AI: As... |
H: If $x_{n+1}/x_n\to x$ and $x_n\nearrow+\infty$ then $\frac{x_1+\cdots+x_{n+1}}{x_1+\cdots+x_n} \to x $
So, here we go again, the sequence $x_n$ is increasing and $x_n\to\infty$ as $n\to\infty$,
and also, $\lim\limits_{n\to\infty}\dfrac{x_{n+1}}{x_n}= x$ which is a real non zero number,
Prove that :
$$\lim\limits_{... |
H: Real Period of an Elliptic Curve
Trying to work out what the real period of an elliptic curve is as seen in the Birch Swinnerton-Dyer conjecture.
From what I've read, given an elliptic curve E over the rationals, one can associate to it a value $\displaystyle \Omega_{E} = \int_{E(\mathbb{R})}|\omega|$ where $\omega... |
H: Subgroups of order $125$ in $S_{15}$
I know symmetric group $S_{15}$ contains a copy of $C_5 \times C_5 \times C_5$ given by generators
$a=(1,2,3,4,5)$
$b=(6,7,8,9,10)$
$ c=(11,12,13,14,15)$
so $\langle a,b,c \rangle \cong C_5 \times C_5 \times C_5$. I am supposed to show every subgroup of $S_{15}$ of order $5^3=1... |
H: Projecting a vector onto a line: Question in article
At the top of page 3 in the article found here, the author claims that the projection of a vector $\mathbf{x_i}$ onto the line $\mathbf{w}$, denoted by $y_i$, is given by
$$
y_i = \mathbf{w}^{T} \mathbf{x_i}
$$
I'm not sure how to interpret this. Usually isn't a... |
H: How many solutions are possible to the equation $a^x-b^y=c$?
If $a,b,c\in \mathbb Z$ are known and $a>b>1,(a,b)=1$, how many integer solutions are possible to the equation $$a^x-b^y=c~?\tag1$$
Can $(1)$ has more than $4$ integer solutions ?
AI: There are at most two solutions in positive integers $x$ and $y$, due ... |
H: Basis of a $n \times n$ matrix which trace is zero
For a matrix $2 \times 2$ it's easy. But for$n \times n$, I don't understand. Someone can help me?
AI: Recall that there are $n^2$ total entries in the matrix. The matrix (which we call $A$) is subject to the condition that if
$$A = [a_{i, j}]_{1 \leq i, j \leq n}... |
H: divisibility test
let $n=a_m 10^m+a_{m-1}10^{m-1}+\dots+a_2 10^2+a_1 10+a_0$, where $a_k$'s are integer and $0\le a_k\le 9$, $k=0,1,2,\dots,m$, be the decimal representation of $n$
let $S=a_0+\dots+a_m$, $T=a_0-a_1\dots+(-1)^ma_m$
then could any one tell me how and why on the basis of divisibility of $S$ and $T$ by... |
H: An inequality problem with complex numbers
Knowing that $p$ and $q$ are complex numbers, $|p| < 1$ and $|q|<1$ show that $|\frac{p - q}{1 - q\bar{p}}| < 1$.
I've tried to write:
$p=x + yi$ and $q=a+bi$ which led me to $x^2 + y^2 + a^2 + b^2 < 1 + (x^2 + y^2)(a^2 + b^2)$ after some algebra, but I've not been able to... |
H: Supremum of a sequence
I encountered this question in a grad-level exam. i hope somebody could help.
we have to choose one correct option.
Let $a_n = \sin (\pi/n).$ For the sequence $a_1, a_2,...$ the supremum is:
$a)$ $0$ & it is attained
$b)$ $0$ & it is not attained
$c)$ $1$ & it is attained
$d)$ $1$ & it is not... |
H: Compute the value of $\sum\limits_{k=1}^{100}\left|\bigcup_{i=1}^k A_i\right|$
Earlier today I was given the following problem by a friend:
Define the sets $A_i$ as $A_1=\{1,2,3\},A_2=\{2,3,4\},\dots,A_k=\{k,k+1,k+2\}$. Given that $\bigcup\limits_{n=1}^kA_n=A_1\cup A_2\cup\dots\cup A_k$, compute the value of $\su... |
H: Don't know the equation in an SMBC comic.
The comic is here. The equation is: $$ L\left(q\,, \dot{q}\,,t\right) = \mbox{everything}$$
I have tried searching for it, but haven't found it. So I am asking here.
AI: It's from Lagrangian mechanics. $q$ is a generalized coordinate and $\dot{q}$ is its derivative with r... |
H: Surjectivity of composite functions when 3 functions are involved
Suppose that we have three $\mathbb{Z} \rightarrow \mathbb{Z}$ functions such as $ \ f$, $g$ and $h$. How should $f$ and $h$ be so that $f \circ g \circ h$ can be onto (surjective) given that $g$ is a one to one (injective) function?
AI: There’s not ... |
H: Are functions which say $X$ onto $Y$ mean the function is surjective?
Throughout my reading I've encountered theorems which use certain wording, which is unclear at times. For example, consider the following corollary taken out of Intro to Topology by Mendelson,
Let $X$ and $Y$ be topological spaces, let $f:X\to Y$... |
H: First Order Logic Problem
I have the following problem I'm trying to understand/solve using first order logic.
Predicates:
Set(S), which states that S is a set, and
x ∈ S, which states that x is an element of S,
Using first order logic, I need to write :
For any x and y, there is a set containing just the elements... |
H: Question on set theory and first order logic
I need help with this problem on set theory.
For any sets $A$ and $B$, consider the set $S$ defined below:
$$S = \{ x \mid \neg (x ∈ A \to x ∈ B) \}$$
I need to write an expression for $S$ in terms of $A$ and $B$ using the standard set operators (union, intersection, etc... |
H: denseness of rational numbers in $Q$
Firstly, Let's consider the denseness of rational numbers in real numbers.
Between two arbitrary real numbers, there is at least one rational number. And we say that $Q$ is dense in $R$.
One equivalent definition: one arbitrary real number $a$, there must exist one rational numb... |
H: Remainder division
What is the remainder when $35^{245}$ is divided by $41$?
I know this is a basic question to ask but I seems to forgot the method, tried to search for it online but to no avail.
I think it is related to number theory, hope someone can point me in the right direction.
AI: An elementary approach:
$... |
H: How to understand and to construct $m-1\leq n x
$x, y\in R$, and $y>x$, prove: there exist $p\in Q$, such that $x<p<y$
Proof:
Since $y>x$ is equvalent with $y-x>0$, by Archimede's property, there exists positive integer $n$, such that
$$\begin{align*}n(y-x)>1.\tag{1}\end{align*}$$
why not $n (y - x) > 0$ since $y ... |
H: Do any of these sequences converge?
I have a homework question asking which of these 5 sequences converges, but it seems to me that none of them actually converge.
The 5 sequences are:
A) $a_n = n + \frac3n$
B) $\displaystyle a_n = -1 + \frac {(-1)^n} n$
C) $\displaystyle a_n = \sin \frac {n\pi}2$
D) $\displaystyle... |
H: Using sets to determine percentage of girls not in any of the given sets
Among the girls of a college,60 % read the Bichitra,50% read the Sandhani,50% read the Pubani(Names of magazines),30% read the Bichitra and Sandhani,30% read the B and P,20% read the S and P while 10% read all three.How can we find out the per... |
H: inductive proof of geometric series
I am stuck on understanding the inductive proof of geometric series. Specifically, I don't see how $ar^{k+1}$ equates to $\dfrac {(ar^{k+2}-ar^{k+1})}{(r-1)}$.
AI: $$\frac{ar^{k+2}-ar^{k+1}}{r-1}=\frac{ar^{k+1}(r-1)}{r-1}=ar^{k+1}$$ |
H: The map $p(x)\mapsto p(x+1)$ in vector space of polynomials
I encountered this question in a previous year paper of an exam. I wish somebody could help me how to go by this question.
Choose the correct option
Let $N$ be the vector space of all real polynomials of degree at most $3$. Define
$$S:N \to N \ \text{by} \... |
H: Is such a partition available?
Consider :
$$
S = \{1,2,\cdots, N\}
$$
We want to partition $S$ into $K$ parts ($S_1\cup S_2\cup\cdots\cup S_K=S$) to satisfy these equalities :
$$
\sum_{k \in S_d} a_k = \frac1K \sum_{k \in S} a_k\quad, \qquad d=1,2,\cdots,K
$$
where $a_k$'s are positive.
Actually the problem is fin... |
H: Statistics notation
I came across this notation and am not sure what it means.
$\downarrow$ and $\uparrow$.
For example. Let $A$ be a sample space and we can divide $A$ into $n$ disjoint sets.
$A_N \subset A_{N-1}\dots A_2 \subset A_1$ be an infinite number of subsets such that $A_N \downarrow A$. Then $P(A_N)\to ... |
H: eigenproblem and characteristic equation
Eigenvalues $\lambda$ of a matrix $A$ are defined as $Ax = \lambda x$, where $x$ are the eigenvectors. The characteristic polynomial is $\det(A-\lambda I)$.
Say I have the expression $Ax = \lambda Bx$, where $B$ is a matrix as well. Does this equation have a characteristic p... |
H: Need a hint on what's wrong - polar coordinates
I'm asked to solve the following
$$ \int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy $$
I thought about using polar coordinates:
(1) $0 \le x \le \sqrt{4-y^2}$ is the upper half of a circumference with radius 2 and centered on the origin.
(2) $ 0 \le y \le 2 $ is the... |
H: Number of elements of a subset of $\mathbb F_8$
I wish somebody could help me in this. I encountered this question in a previous year paper of an exam.
Let $F$ be a field with $8$ elements and $A=\{x\in F\mid x^7=1$ and $x^k \ne 1$ for all natural numbers $k<7\}.$ we have to find the number of elements in $A$.
AI:... |
H: Prove by contradiction that every integer greater than $11$ is a sum of two composite numbers
I have thought a lot but am failing to arrive at anything encouraging.
First try: If this is to be proved by contradiction, then I start with the assumption that let $n$ be a number which is a sum of two numbers, of which... |
H: Let $A,B\subset X$, $X$ a topological space. If $A$ is connected, $B$ open and closed, and $A\cap B\neq\emptyset$ then $A\subset B$.
I'm studying Intro to Topology by Mendelson.
The problem statement is,
Let $A,B\subset X$, $X$ a topological space. If $A$ is connected, $B$ open and closed, and $A\cap B\neq\emptyset... |
H: Let $A,B$ be nonempty subsets of a topological space $X$. Prove that $A\cup B$ is disconnected if $(\bar{A}\cap B)\cup(A\cap\bar{B})=\emptyset$.
I'm reading Intro to Topology by Mendelson.
The problem statement is,
Let $A,B$ be nonempty subsets of a topological space $X$. Prove that $A\cup B$ is disconnected if $(\... |
H: how to show that $A_kB_k\to AB?$
Let in the space $M(n,\mathbb R)=$ set of all $n\times n$ real matrices endowned with $\| \cdot \|_2,~A_k\to A,~B_k\to B.$ Then how to show that $A_kB_k\to AB?$
AI: Hint
Use the triangular inequality with
$$A_kB_k-AB=A_k(B_k-B)+(A_k-A)B$$ |
H: Matrix expansion does not decrease norms
Given a block matrix $A = \left[ {\begin{array}{*{20}{c}}
{{A_{11}}}&{{A_{12}}}\\
{{A_{12}}}&{{A_{22}}}
\end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean norm of any sub-block $A_{ij}$, for $i,j = \left\{ {1,2} \right\}$ satisfy the inequal... |
H: How to prove two trigonometric identities
I want to show that
$${\sin}^2 \alpha + 4{\sin}^4\frac{\alpha}{2} = 4{\sin}^2 \frac{\alpha}{2}$$
and
$${\sin}^2 \alpha + 4{\cos}^4\frac{\alpha}{2} = 4{\cos}^2 \frac{\alpha}{2}$$
They should be true, as Wolfram Alpha says so. However, I want to prove them, and I have no idea... |
H: Prove that $\succeq = \bigcap L \left(\succeq \right)$ - understanding elementary order theory
Dear reader of this post,
I am working on a elementary question on order relations. I would be very glad to receive some guidance since I think understanding this question is important for making further progress with ord... |
H: Interpretation of $\epsilon$-$\delta$ limit definition
The epsilon-delta definition for limits states that (from Wikipedia) for all real $\epsilon > 0$ there exists a real $\delta > 0$ such that for all $x$ with $ 0 < |x − c | < \delta$, we have $|f(x) − L| < \epsilon$ - however, the definition of the limit requir... |
H: Expectation of a distance in a triangle
A point $x$ is uniformly distributed in an isosceles triangle with top angle $\alpha$, what is the expected distance of the point $x$ to the side opposite to angle $\alpha$.
AI: $$
f(x,y) = \frac1{ h^2 \tan \alpha}, \qquad {(y-h) \tan \alpha} \leq x \leq {(h-y) \tan \alpha} ,... |
H: What is the natural norm on these spaces?
In [1] the authors define the function spaces \begin{align*} V(\mathbb{R}^N) = \lbrace v \in \mathbb{R}^N \to \mathbb{C}: ~ &\nabla v \in L^2(\mathbb{R}^N), \\ &\Re(v) \in L^2(\mathbb{R}^N), \\ & \Im(v) \in L^4(\mathbb{R}^N), \\ &\nabla \Re(v) \in L^{4/3}(\mathbb{R}^N) \rbr... |
H: How do i show that there exists $\theta \in [0,2\pi)$ such that a given matrix is not a rotation?
Let $x \in [0,2\pi)$.
Define $A_x = \left (\begin{matrix} \cos(x) & -\sin(x) & 0&0 \\ \sin(x) & \cos (x) & 0 &0 \\ 0 & 0 & \cos(x) & -\sin(x) \\ 0& 0& \sin(x) & \cos(x) \end{matrix} \right)$.
Then $\forall x\in [0,2\pi... |
H: On the real line, prove that the set of nonzero real numbers is not a connected set.
I'm studying Intro to Topology by Mendelson.
The problem is stated in the title.
My proof is,
Let $A=\mathbb{R}-\{0\}$. Then $C(A)=\{0\}$. Moreover, let $P=(0,\infty)$ and $Q=(-\infty,0)$. Then $A\subset P\cup Q$ and $P\cap Q=\empt... |
H: Central extension
Let $K$ be a cyclic group of order 12 and $Q$ be projective special linear group $PSL(4,4)$. If $G$ is a central extension of $K$ by $Q$ ($K$ is normal in $G$), then how many choices we have for the group $G$? Actually I wonder whether $G$ can be a non-split extension of $K$ by $Q$?
AI: Apart from... |
H: Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$.
My justification for this question is as follows;
Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) \subsetneq F(\alpha)$.
As $[F(\alpha):F]=[F(\alpha):F(\a... |
H: Limit of joint survival function as variables become perfectly correlated
Let $Y,X$ be jointly normally distributed and assume that they are highly correlated. I'm interested in knowing what happens to the survival function as the variables become perfectly correlated. Specifically, I'm interested in this instance:... |
H: Expressing a set in tabular method
Set $A=\{\,x \in\mathbb R : x^2-(a+b)x +ab=0\,\}$. We have to express it using tabular method.
At first, we factorize it, getting $(x-a)(x-b)=0$ or $x =a$ and $x=b$. But how do we know that $a$ and $b$ belongs to $\mathbb R$? Is there any proof of this fact,
or are we just assumi... |
H: Proof $\frac{1}{2}+\frac{1}{3}+....\frac{1}{n}$ is not an integer for integer $n>1$
I found a way to prove this using Chebychev's theorem, are there ways to solve it without relying on this?
AI: Hint: Pick the largest $m$ so that $2^m \leq n$.
Isolate $\frac{1}{2^m}$ and add all the other fractions. Then your sum w... |
H: The matrix notation of signum?
The following question on a notation might look trivial but I am really not sure how to deal with it.
If I have a variable $x$, I could write out:
$$x=|x|\;\text {sgn} (x)$$
a notation that helps me with an operator for the signs that could point to $-1$, $0$ or $+1$.
But then I have ... |
H: Unsure about a maths symbol
Help, help, help!
I've come across this maths symbol,
$[n(i,j)]^{0.5}$ where $n$ is a square matrix.
Does this mean that it is the $(i,j)$ element of $n^{0.5}$? or $n(i,j)^{0.5}$?
source: http://outobox.cs.umn.edu/Random_Walks_Collaborative_Recommendation_Fouss.pdf
2nd page, fifth line f... |
H: Commutative matrices and symmetric property
Assume we have two commutating matrices, [A,B]=0. Can we say that A and B are symmetric?
Regards
AI: No, we cannot say in general that $0=[A,B]=AB-BA$ implies that $A$ or $B$ are symmetric: take $A$ non-symmetric and $B$ the identity matrix, for example.
However a relate... |
H: find the inverse function
This question was on one of our tests.I couldn't solve it but I'm curious to know the answer:
Find the inverse function of:
$$f(x)=(\ln(x))^2-\ln(x).$$
I found they domain of definition and studied the characteristics of this function but couldn't know how to find it's expression.
AI: Hint... |
H: Bounded linear operator and inverse
Suppose $A$ and $B$ are two bounded invertible linear operator on Banach space $E$. Suppose for $t\in $some interval, $A+tB$ is also invertible. My question is:
Is operator $(A+tB)^{-1}$ is bounded uniformly for all $t\in$ some interval of $0$?
AI: Yes. This is true in a gene... |
H: Computation of the Wirtinger derivative of a product
Let's have a function $f = (A/2)\phi\bar{\phi}$, where $\phi=\phi(z)$ is a complex-valued scalar field. I need to obtain $df/d\phi$. If I treat the real and imaginary parts of $\phi$ as independent variables, I can write $df/d\phi^R=A\phi^R$ and $df/d\phi^I=\phi^... |
H: Cyclic Groups: Modulo operations in exponents possible?
I'm trying to follow CCat's Zero Knowledge Proof example, which was quite similar to the $\Sigma$-protocol example in my books. And whith both of them I'm struggeling.
When I try to test CCats Example:
Setup:
Cyclic Group $G$ of size $p=17$. Generator $g=3$.... |
H: If $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $. Find $f(x)$
Problem: If $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $. Find $f(x)$
Solution: $\int f(x) \sin{x} \cos{x}\,\mathrm dx = \frac {1}{2(b^2 - a^2)} \log f(x) +c $
Differenting both si... |
H: Find the sum of the digits in the number 100!
I am working on a Project Euler problem http://projecteuler.net/problem=20.
$n!$ means $n(n - 1)\dots...3.2. 1.$
For example, $10!$ $=$ $10$ $9$ $...$ $3$ $2$ $1$ $=$ $3628800$, and the sum of the
digits in the number $10!$ is $3 + 6 + 2 + 8 + 8 + 0 + 0$ $=$ $2... |
H: If $\,p\,$ is prime, is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n$?
Is $\,p^n\mathbb{Z}_p=\mathbb{Z}_p\,$ for any positive integer $n\,?$
$\mathbb{Z}_p =$ ring of $p$-adic integers, $\,p$ prime.
Thanks.
AI: One way to see this is to consider $\mathbb{Q}_p$ as the completion of $\mathbb{Q}$ with... |
H: Seperating the O.D.E $\frac{dv}{dt} = mg - \kappa v$
I am having some trouble seperating the following (autonomous) O.D.E.
$$\frac{dv}{dt} = mg - \kappa v$$
From what I understand, I have to get the $v$ to the left side while having the $\kappa$ on right side with $dt$. The solution is given by:
$$\begin{align}
\... |
H: Functional equations very like the Taylor Series
Let $g(x,y)=0$ be a closed curve, that means, any point inside that curve satisfies $g(x,y)<0$ and any point outside that curve satisfies $g(x,y)>0$.
Given a point $(a,b)$ outside the curve ($g(a,b)>0$),my question is:
is there one or more points (p,q) satisfying bot... |
H: Surjectivity $f:\mathbb Z \to \mathbb Z$, $f(x) = 5x$.
Is the function $f(x)=5x$ surjective if $f:\mathbb{Z}\to \mathbb{Z}$?
I believe it is not as $f\left(\dfrac{x}{5}\right) = x$ can be rational, not an integer. Could someone confirm this?
Thank you.
AI: A function $f: \mathbb{Z} \to \mathbb{Z}$ is surjective if ... |
H: Is $p^n\mathbb Z_p\cong \mathbb Z_p$ as additive groups?
Is it true that $p^n\mathbb{Z}_p\cong \mathbb Z_p$ as additive groups?
Here $\mathbb Z_p$ is the ring of $p$-adic integers for $p$ prime and $n$ is any positive integer.
Thanks
AI: If $R$ is any integral domain (unitary commutative ring without zero divisors)... |
H: Recovering the spatial Fourier transform from the space-time Fourier transform
This CW question is aimed at developing some intuition (grokking) about a certain formula of Fourier analysis. Any kind of explanation (physical, geometrical, analytical ...) is welcome.
If we have a function
$$\begin{array}{cc}\phi\co... |
H: If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$.
If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$. How do I find a value for $\sin\alpha$ or $\cos\alpha$ so I can use a double angle formula?
I know how to solve a problem like
"If $\cos\alpha = \frac{\sqrt... |
H: Is this proof about $\pi$ is irrational correct?
A proof is
If $\pi$ is rational, then $\pi$=$\frac{a}{b}$ where $a,b \in \mathbb{N}$
let $f(x)=x^n\left[\dfrac{(a-bx)^n}{n!}\right]$ , for $0<x<\frac{a}{b}$
$0<f(x)<\dfrac {\pi^na^n} {n!}$ , $0<\sin x<1$
thus $0<f(x)\sin x<\dfrac {\pi^na^n} {n!}$
where $n$ is big en... |
H: How to integrate $\int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } } } dx$?
I tried using gamma function but there's no way I can change the limits as : $0$ to $1$.
AI: One observation:
$$\int_{0}^{1}e^{x^2}dx=\sqrt{\int_{0}^1\int_{0}^1e^{x^2+y^2}dx \ dy}\\
\le\sqrt{\int_{0}^{\pi/2}\int_{0}^1 e^{r^2} r dr d\theta}=\sqrt{(e-1... |
H: "Question: Show that $n^5 - n$ is divisible by 30; for all natural n"
Show that $n^5 - n$ is divisible by $30;$ $\forall n\in \mathbb{N}$
I tried to solve this three-way. And all stopped at some point.
I) By induction:
testing for $0$, $1$ and $2$ It is clearly true.
As a hypothesis, we have $30|n^5-n\Rightarrow ... |
H: Kind of a silly question, but need confirmation regarding the closed unit interval $[0,1]$
I know that $[0,1]$ is a closed subset of $\mathbb{R}$, since its complement $(-\infty,1) \cup (1, \infty)$ is open in $\mathbb{R}$. Clearly, $(0,1)$ is an open subset of $\mathbb{R}$, but is it an open subset of the unit int... |
H: $\lim x_0^2 + x_1^2+...+x_n^2$ where $x_n=x_{n-1}-x_{n-1}^2$
So, we are given a sequence $x(n)$ for which $x_{n+1}= x_n-x_n^2$ , $x_0=a$, $0 \le a \le 1$
I was first requested to show that it converges and to find $\lim_{n \to \infty} x_n$. I will post my answer here for you to check if it is right :S
so, $x_n-x_... |
H: Derivative of $\dfrac{dy^2}{dx}$
I just finish learning the chain rule and am now learning Implicit Differentiation and I am wondering: why is it not possible to take the derivative of $\dfrac{dy^2}{dx}$? Why do we need to apply the chain rule and find $(\dfrac{dy^2}{dy})\cdot\dfrac{dy}{dx}$?
I understand that $y^... |
H: Maclaurin series of $f(x) = \frac{1}{1-2x}$
I am not sure if I am doing this right.
$$f(x) = \frac{1}{1-2x}$$
So first I find the first four derivatives.
$$f'(x) = \frac{2}{(1-2x)^2}$$
$$f''(x) = \frac{8}{(1-2x)^3}$$
$$f'''(x) = \frac{48}{(1-2x)^4}$$
$$f''''(x) = \frac{384}{(1-2x)^5}$$
From this I can see that a ge... |
H: $R[x]/(x^n-1)=R[G]$ as rings
Let $R$ be a commmutative ring with $1$ and $G$ finite cyclic group of order $n$.
Show that $R[x]/(x^n-1)=R[G]$ (isomorphic) as rings.
This is what I did. Suppose $G=\langle b\rangle $. Let $\psi\colon R[x] \to R[G]$ by $\psi(a_0+a_1x+\ldots+a_mx^m)=a_0e+a_1b+\ldots+a_mb^m$. Check it's... |
H: Not divisible by $2,3$ or $5$ but divisible by $7$
The question is to determine the number of positive integers up to $2000$ that are not divisible by $2,3$ or $5$ but are divisible by $7$. The answer is supposed to be $76$ but not sure how it was derived
I know that if the question was how many integers are not di... |
H: Modified Bessel Function of the First Kind: Changing the Limits of the Integral
Can anyone explain why:
$$\int_{0}^{2\pi}\exp(\beta(2r_{1}r_{2}\cos(\theta)))\,\mathrm d\theta=2\left(\int_{0}^{\pi}\exp(\beta(2r_{1}r_{2}\cos(\theta)))\,\mathrm d\theta\right)$$
where the right hand side is the modified Bessel function... |
H: Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ with radius of convergence equals to $R$. Show that for every $r<R$:
$$\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$$
I tried this:
$$\int_{0}^{2\pi}|f... |
H: show that this function is continuously differentiable ( application to Lebesgue theorem ?)
Consider $R>0$ and $u \in C_{0}^{\infty}(B(0,R))$ (this is the set of smooth functions with compact support contained in $B(0,R)$).
Let $|\cdot|$ be the Lebesgue measure in $\mathbb R^n$.
Fix $ y \in \mathbb R^n$ and define... |
H: How $\delta_1$ and $\delta_2$ for two different limits at $a$ can be read as $\delta=\text{min}(\delta_1,\delta_2)$?
I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous ... |
H: If the symmetric difference of $A$ and $B$ is contained in $A$, then $A$ contains $B$
The symmetric difference of two sets $A$ and $B$ is the set $A \vartriangle B = (A \setminus B) \cup (B\setminus A) = (A \cup B) \setminus (A \cap B)$. Prove that if $A \vartriangle B \subseteq A$ then $B \subseteq A$.
Proof. Sup... |
H: How do you determine what the coefficients are on a Taylor series expansion if the derivative is too hard to compute?
In a past lecture, we talked about how you need to expand the Taylor series of a composed function based on what its input is. For example:
$e^u$ where $\color{red}{u} = \cos x=1 - \frac{1}{2!}x^2 ... |
H: Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove that $\int_{\bf{I}}f\ge0$.
Let I be a generalized rectangle in $\Bbb R^n$ and suppose the function $f:\bf{I}\to\Bbb R$ is Riemann integrable. Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove th... |
H: Stuck on question regarding Cantor's theorem and sets
I'm trying to prove that a set of all sets does not exist, meaning that the following does not exist:
$$
D = \{ S \mid S \text{ is a set} \}
$$
I can use Cantor's Theorem and the proof of cardinality of sets which says that if $A⊆B$ then $A≤B$. But I'm stuc... |
H: How many sets give different answers?
Consider a set $S$ of integers taken from $[n]$ and a given threshold $k$. A query $x$ returns true if there exists an $s$ in $S$ such that $s-k \leq x \leq s+k$ and false otherwise.
Say that two sets are equivalent if they give the same answer to every possible query. How man... |
H: What is the maximum of the self root $f(x) = x^{1/x}$
This is a knowledge sharing question as I have answered it below. I am demonstrating how one would differentiate an expression such as $x^{1/x}$ and proving the following statement.
What is the maximum of the function: $f(x) = x^{1/x}$?
[NOTE] Proof that the fou... |
H: Maclaurin series of $f(x) = e^x \sin x$
$$f(x) = e^x \sin x$$
I tried applying the given formula in my book but it didn't work.
The maclaurin for $e^x$ is given as $\displaystyle \sum \frac{x^n}{n!}$ and $\sin x$ $\displaystyle \sum \frac{(-1)^n x^{2n + 1}}{(2n+1)!}$
I attempted to multiply them together, failed te... |
H: Maximum matrix simplification
What is the most that a matrix can be simplified if row or column operations are both allowed? Intuitively, I am guessing that everything is 0 except the diagonal entries, which are a mix of 0's and 1's. However, I'm unable to conjecture how many of the diagonal entries are 0's or 1's.... |
H: summation of consecutive natural numbers does not end in 7,4,2,9
I calculated sum of n consecutive natural numbers
where n = 1 to 100 .What I mean is
$$\sum_{n=1}^{1}n = 1 $$
$$\sum_{n=1}^{2}n = 3 $$
$$\sum_{n=1}^{3}n = 6 $$
And I got answers and noticed that none of the summation answers ended in digits $7,4,2,9... |
H: Convert a closed-form generating function to a sequence
I need some help with an assignment question:
I must determine the sequence generated by the following generating function:
$2x^3 \over 1 - 5x ^ 2 $
In class we have only gone from the sequence to the closed form so, I am not really sure how to begin on this. ... |
H: Graphing a difficult function
Okay can anyone explain the graph of function $$\lfloor|y|\rfloor = 4 -\lfloor|x|\rfloor$$ where $|\cdot|$ denotes Absolute Value Function and $\lfloor\cdot\rfloor$ denotes the floor function (Greatest Integer Function).
This is an interesting function as i was told by my teacher tha... |
H: Evaluate a triple integral
Given $f(x,y,z) = \sqrt{1+(x^2+y^2+z^2)^{\frac{3}{2}}}$ and $D=\{(x,y,z) : x^2+y^2+z^2 \leq r^2\}$, evaluate $\int\int\int_D f(x,y,z)dxdydz$.
I've thought that spherical coordinates would be the best way to go, then $x=\rho \cos\theta \sin\varphi$, $y=\rho\sin\theta\sin\varphi$ and $z=\rh... |
H: With a fixed number of entries, is it better to play at a single sweepstake rather than many?
The type of the sweepstake is that there is only one prize ($\$100$) and at each sweepstake one and only one winner is guaranteed.
Suppose I have two entries and there are $99$ other entry in Sweepstake $1$ and $99$ other ... |
H: show that $\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$
show that
$$\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$$ for $a,b\geq 0$
I would like someone solve it using contour integrals, also I would like to see different solutions using different ways ... |
H: Probability Density Function from: $F(x)=x , \text{ for } 0\leq x\leq \frac12$
Probability Density Function from:
F(x)=begin{cases}0 & \text{if }x<0\
x & \text{if }0\leq x\leq\frac{1}{2}\
1 & \text{if }x>\frac{1}{2}
\end{cases}.
Do somebody know how to determine the p.d.f from that $F(x)$?
actually, I have tried to... |
H: Sum of squared quadratic non-residues
Can you prove that if $p$ is a prime greater than $5$, then the sum of the squares of the quadratic nonresidues modulo $p$ is divisible by $p$?
Note that I have just proved that the sum of the quadratic residues modulo $p$ is divisible by $p$ for $p$ greater than $3$.
AI: Let $... |
H: Example of Tetration in Natural Phenomena
Tetration is a natural extension of the concept of addition, multiplication, and exponentiation.
It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest hyper-operations, as they are called.
For example:
Adding the forces on an... |
H: equation to linear function
I'm doing homework and I've been given a line as the equation $7x-6y=5.$
I need to make a function $y = f(x)$ of this that corresponds to the equation.
What I know at this point is that I need to know what the $a$ and $b$ are in $y = ax + b.$
I also know of the formula to calculate $a$, ... |
H: How can they guess my number just by knowing which rows it appears in?
I've saw this "trick" many times in math club, I'm just wondering if it's real that they know ESP, or is it just a scam?
We're given 5 rows: $\newcommand{\tsf}[1]{\mathsf{\text{#1}}}$
$$\begin{array}{c|cccccccccccccccc}
\tsf{Row 1} & \sf 1 & \sf... |
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