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H: Indefinite Integration : $\int \frac{dx}{(x+\sqrt{(x^2-1)})^2}$
Problem :
Solve : $\int \frac{dx}{(x+\sqrt{(x^2-1)})^2}$......(i)
I tried :
Let $x =\sec\theta$ therefore , (i) will become after some simplification
$$\int \frac{\sin\theta}{(1+\sin\theta)^2}d\theta$$
but i think its wrong method of approaching pl... |
H: Determine depth of a partially filled hemisphere
Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer.
The question goes like this:
A bowl is in the shape of a hemisphere with radius 10cm. The surface of the wa... |
H: How many different numbers are composed by n repeated digits?
For example, there are 3 digits: 1, 1, 4 and they compose 3 different numbers: 114, 141, 411.
My questions is: given n repeated digits: 1 * n1, 2 * n2, 3 * n3, ..., 9 * n9, in which ni >= 0 and n1 + n2 + ... + n9 = n, how many different numbers are compo... |
H: Find the Dirichlet inverse of the identity function
I thought I got this one but now I'm having doubts. I have that if $f$ is a function such that $(f\ast\text{id})(n)=(\text{id}\ast f)(n)=\iota$ then $f$ is multiplicative since it is the inverse of a multiplicative function, and so it suffices to examine its value... |
H: Prove that $\log _5 7 < \sqrt 2.$
Prove that $\log _5 7 < \sqrt 2.$
Trial : Here $\log _5 7 < \sqrt 2 \implies 5^\sqrt 2 <7.$ But I don't know how to prove this. Please help.
AI: Observe that:
$$ \begin{align*}
\log_5 7 &= \dfrac{3}{3}\log_5 7 \\
&= \dfrac{1}{3}\log_5 7^3 \\
&= \dfrac{1}{3}\log_5 343 \\
&< \dfra... |
H: Getting a values from nodes
The goal: get horizontal values of vertical level N where level 1 is pinacle node (1).
Example: level 4 as input should produce: | 1 | 3 | 3 | 1 |
Note: the sum of two adjacent nodes of level N is the value of node of level N + 1 in between those nodes of level N (marked with circles on... |
H: Why does $\lim_{n\to \infty}\sqrt{n}(1-x^2)^n=0$ if $0<|x|\leq 1$?
I was working on constructing an approximation of the identity, and one point in my construction requires me to show that there is a sufficiently large $n$ such that given $\epsilon>0$ and some $0<\delta<1$, then $\sqrt{n}(1-x^2)^n<\epsilon$ if $\de... |
H: Does the integrability of $\log(f(x))$ imply $f(x)$ is bounded?
Let $f(x):(a,b)\rightarrow \mathbb{R}$. Suppose $\int_a^b\log{f(x)}\,\mathrm{d}x<\infty$, can we claim that $0<f(x)<M<\infty$ a.e.. Why and why not?
AI: Hint: Consider $f\colon (0,1)\to \Bbb R, x\mapsto \dfrac{1}{x}$. |
H: How to find this mass?
Let $R$ be the region in the first quadrant of the plane bounded by the lemniscates of the following equations:
$\rho^2=4\cos(2\theta)$,
$\rho^2=9\cos(2\theta)$,
$\rho^2=4\sin(2\theta)$, and
$\rho^2=9\sin(2\theta)$.
Find the mass of this portion of plane if the density at the point $P(\rho,... |
H: holomorphic on right half plane
Could any one tell me how to solve this one?
Let $f$ be holomorphic function in the right half plane, with $|f(z)|<1$ for all $z\in \{z:Re(z)>0\}$, $f(1)=0.$ Find out largest possible value of $|f(2)|.$
AI: Let $\phi(z) = \frac{1-z}{1+z}$. $\phi$ is a Möbius transformation that maps ... |
H: a codeword over $\operatorname{GF}(4)$ -> two codewords over $\operatorname{GF}(2)$ using MAGMA
A codeword $X$ over $\operatorname{GF}(4)$ is given. How can I write it as $X= A+wB$ using MAGMA? where $A$ and $B$ are over $\operatorname{GF}(2)$ and $w^2 + w =1$.
Is there an easy way, or do I have to write some for l... |
H: Lie algebra isomorphism between $\mathfrak{sl}(2,{\bf C})$ and $\mathfrak{so}(3,\Bbb C)$
I think that this is an exercise. I can not find a solution.
We can define Lie bracket multiplication on $\mathbb{C}^3$ : $$ x\wedge y $$ where $x=(x_1,x_2, x_3)$, $y=
(y_1,y_2,y_3)$, and $\wedge $ is the wedge product we know.... |
H: How to determine the no. of integral partitions into $k$ parts?
I wanted to know, if I was to partition $500$ into positive $k$ integers, not necessarily distinct under the following constraints
1.k is +ve.
2.all k parts need not be distinct.
3.the first integer subtracted from the last integer in the k parts is sm... |
H: How to justify $\frac14(n^2(n+1)^2)+\frac14(4(n+1)^3) = \frac14(n+1)^2(n^2+4(n+1))$?
How does my lecturer go from :
$$ \frac {n^2(n+1)^2} {4} + \frac {4(n+1)^3} {4} \text{ to } \frac {(n+1)^2} {4} \times [n^2+4(n+1)] $$
I can understand that $$ \frac {n^2(n+1)^2} {4} = \frac {(n+1)^2} {4}\times n^2 $$
But I'm not ... |
H: $\log(0,05)$ is minus, but $\log(0,04999\ldots)$ is plus?
How is this calculated, and why is this?
We're calculating fixed-rate mortgage, with following formular:
$$
n = 1-\frac{\log(\frac{L\cdot x}{y})}{\log(1+x)}
$$
Where: $L$ is the loan size, $x$ is the interest, $y$ is the amount to pay back per month, and $n$... |
H: adding infinitely many equations side by side in a recurrence relation
we are given that $x+\beta y_{n+1}=k_n+y_n$ for all $n\in\mathbb{N}\cup\{0\}$, where $\beta\in(0,1)$, $y_0=0$, and $k_n$ is 6 whenever $n$ is even and 4 whenever odd. Being the naive mathematician I do the following
$x+\beta y_1=6$
$\beta x+\bet... |
H: Hilbert's finiteness theorem over arbitrary fields; reductive groups
As a generalization of the finiteness result Hilbert proved in his 1890 paper, one usually formulates the following nowadays:
Let $G\to\operatorname{GL}(V)$ be a rational representation of a linearly reductive group $G$ on a finite-dimensional $k... |
H: Open and closed set on a given metric
I'm stuck on this one, could you please give me a tip or two:
Let $d$ be a metric in $X$ such that $d : X^2 \rightarrow \mathbb{R} : d(x, y) = d_{1}(x,y) + d_{2}(x,y)$ for $x,y \in X$. Let $d_{1}$ be a discrete metric. Show that in the metric space $(X,d)$ for any $x\in X$ the... |
H: Tangent map $F_{*}: T_{p}(M) \to T_{F(p)}(N)$ are linear tranformations.
How to show that tangent map $F_{*}: T_{p}(M) \to T_{F(p)}(N)$ are linear tranformations?
I know that it suffices to show that
$$F_{*}(ax_{u}+bx_{v})=aF_{*}(x_{u})+bF_{*}(x_{v})$$
where $x$ is a patch in $M$.
By definition,
$$F_{*}(ax_{u}+bx_{... |
H: Function Projection: Orthogonal Polynomials
I am currently reading a paper called "Numerical Quadrature" by Timothy J. Giese (2008) which describes the numerical quadrature technique in detail. At one point (just before equation 19) it states that:
$"\dots$, we begin by projecting $f(x)$ into a set of orthogonal po... |
H: Trying to understand the standard deviation's formula
I have just started learning standard deviation and I'm trying to understand the formula.
$$ s = \sqrt{ \frac{\sum(x-\bar x)^2}{n-1} } $$
Can anyone explain to me the square (and square root) part? If they are using the square & square root to prevent having neg... |
H: Check for directional derivative and show that $f(1,1,1) > f(0,0,0)$ when given the partial derivatives
I'm kinda lost in this exercise
Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be of class $C^1$ and $\forall x=(x_1,x_2,x_3)\in\mathbb R^{3}$
$$\frac{\partial f}{\partial x_{1}}(x)=x_{2}, \frac{\partial f}{\pa... |
H: Expected value: Showing $[\Bbb E(X)-\Bbb E(Y)]^2 \geq 2 \cdot \Bbb{Cov}(X,Y)$
The original question is to show that for any Random variables $X,Y$ and $0\leq p \leq 1$
$$p\Bbb V(X)+(1-p)\Bbb V(Y) + p(1-p)[\Bbb E(X)-\Bbb E(Y)]^2 \geq p^2 \Bbb V(X)+(1-p)^2 \Bbb V(Y) +2p(1-p) \Bbb{Cov}(X,Y)$$
I tried to focus on showi... |
H: Asymptotic behaviour of $1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$
I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of
$$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$$
p... |
H: Summing a series - Calculus 1.
I'm learning Calculus 1 at the collage,
and the semester's end is close, which bring with it the exams period.
So I pretty much understand all the topics,
except for a series summing.
I don't know why, but I just don't get it. I Googled a lot, but nothing. so I hope to find my luck he... |
H: Quick method /Birds eye view to determine the value
Is there any way to guess the answer without doing elaborating calculations?
AI: Corrected after a comment of André Nicolas:
We have $4^5=1024>1016$. This shows that $y$ can only be $1$, $2$ or $3$. Now if $y=3$, then since $3^2\times 3^5>1016$, the only possible... |
H: Finding the number of each type of fruit bought
A man has $\$100$. If we have that:
$1$ apple costs $\$1$
$1$ oranges cost $\$0.05$
$1$ watermelon costs $\$5$
and the man buys exactly $100$ pieces of fruit with exactly $\$100$ (He must buy at least 1 fruit from each category), then how many of each fruit did he ... |
H: Please the Inequality in the proof of The Isoperimetric Inequality
From A proof of the Isoperimetric Inequality, can you please explain the starred inequality $$A + \pi r^2 = \int_{\gamma} x\,dy + \int_C -y\,dx = \int^l_0 x(s)y_s(s)\,ds - \int^l_0 \overline{y}(s)x_s(s)\,ds = $$
$$ = \int^l_0 ( x(s)y_s(s) - \overlin... |
H: Prove $f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$
I need to show that $$f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$$
where $X/{\sim}$ is a quotient topology and $\pi$ is the quotient map.
I understand the pr... |
H: Application of Cauchy theorem to prove normality of a subgroup
Let $G$ be a group $o(G)=pq$, where $p,q$ are both distinct prime numbers. Let $H<G$ be a subgroup of $G$ and $o(H)=p$. I want to show that $H$ is normal in $G$. My argument goes as follows. First we observe that based on Cauchy theorem for groups there... |
H: How do I show that an angle is a certain value in a triangle with two sides given?
In the following example I am told that angle x is 60° and that I have to prove it is (without a calculator). What is the simplest way of showing that it is true?
AI: $12-5\sqrt{3}=\sqrt{3}*(4\sqrt{3}-5)$ |
H: Joining finite sequences
How do I describe the joining of two finite sequences in mathematical notation? For example, suppose the following:
$$
A=(a_i)_{i=1,2}=(4,2)\\
B=(b_i)_{i=1,2}=(9,5)\\
C=(c_i)_{i=1,...,4}=(4,2,9,5)
$$
Sequence $C$ can be considered sequence $A$ with sequence $B$ attached to the end. How do I... |
H: How to write two for loops in math notation?
I have a vector of numbers that looks like this:
[1, 2, 3, 4, 5]
For every number the vector, I would like to multiply each number by every other number and find the sum:
1*1 + 1*2 + 1*3 + 1*4 + 1*5 + 2*1 + 2*2 + 2*3 + 2*4 + 2*5 + ... + 5*5
How can I write this in math... |
H: Finding all the integer solutions for :$y^2=x^6+17$
Assume that $x,y$ are integers .How to find the solutions for:
$$y^2=x^6+17$$
AI: $y^2-x^6=17\Rightarrow (y-x^3)(y+x^3)=17$.And we know $17$ is a prime.
So the only possibility is
$y+x^3=17$ and $y-x^3=1$ .
or $y+x^3=-17$ and $y-x^3=-1$
or $y+x^3=1$ and $y-x^3=1... |
H: Contractive mapping on compact space
A contractive mapping on $M$ is a function $f$ from the metric space $(M,d)$ into itself satisfying $$d(f(x),f(y))<d(x,y)$$ whenever $x,y\in M$ with $x\ne y$. Prove that if $f$ is a contractive mapping on a compact metric space $M$, there exists a unique point $x\in M$ with $f(... |
H: Principal ideal and free module
Let $R$ be a commutative ring and $I$ be an ideal of $R$.
Is it true that $I$ is a principal ideal if and only if $I$ is a free $R$-module?
AI: Definitely not. Any proper principal ideal in a finite commutative ring is a counterexample.
On the other hand, a commutative ring is a prin... |
H: Equations on generic algebraic stuctures.
Is there a general theory that studies the solution of equations on structures with a binary operation like, for example, Magmas, Quasigroups, Semigroups, Monoids, Loops and Groups from the most general point of view?
If there isn't a single theory, then is there a spec... |
H: double comb space is not contractible
I'm trying to show that the double comb space is not contractible.
Intuitively I can see why this is true, but I can't seem to formalize a prof.
I try to do the following:
Let $D$ be the double comb space
Suppose $H:D\times I \rightarrow D$ so that $H(x,0)=x$ and $H(x,1)=x_0$ w... |
H: Let $f \colon \Bbb C \to \Bbb C$ be a complex valued function given by $f(z)=u(x,y)+iv(x,y).$
I am stuck on the following question :
MY ATTEMPT:
By Cauchy Riemann equation ,we have $u_x=v_y,u_y=-v_x.$ Now $v(x,y)=3xy^2 \implies v_x=3y^2 \implies -u_y=3y^2 \implies u=-y^3+ \phi(x) $. Now,I am not sure which way ... |
H: The Mystery of the Integrating Factor
The other day I was taught that to solve equations of the form
$$
f'+pf=q,
$$
where $f=f(t)$, $p=p(t)$ and $q=q(t)$, I need to use a function, or integrating factor, $\mu=\mu(t)$ such that
$$
\mu'=\mu p.
$$
However, how do I know that such function $\mu$ exists at all? In other... |
H: continuity and differentiability of two variables.
Consider the map $f:\mathbb{R}^2 \to \mathbb{R}^2$ defined by
$$f(x,y) = (3x-2y+y^2 , 4x+5y+y^2)$$
then which of the following is true?
$f$ is discontinuous at (0,0)
$f$ is continuous at (0,0) and all directional derivatives exists at (0,0)
$f$ is differentiable ... |
H: Derivative of a quotient
I am trying to find
$$y = \frac{x^2 + 4x + 3}{ \sqrt{x}}$$
I am reviewing this so I am suppose to do it without the quotient rule, just using what I know about 16 years of algebra and the power rule, difference rule and such. I cannot get it, or anything close to the answer.
I attempted t... |
H: Is there a term for "is mapped to by an isomorphism"?
In any context where isomorphisms are defined.
For example, if $G$ and $H$ are two isomorphic groups, then there exists an isomorphism mapping their identity elements together. That is to say, their identity elements are _____, where _____ is the word I'm lookin... |
H: prove that $ \lim_{x \to 0} \frac{e^{1/x}}{x}$ does not exist.
By Substitution of y = $\frac {1}{x}$ i have managed to show that
$\lim_{x \to 0^+}\frac{e^{1/x}}{x} = \infty $
but i can't find a way to show that
$\lim_{x \to 0^-}\frac{e^{1/x}}{x} = 0 $
I've tried L'Hospital rule but ended with
$\lim_{x \to 0^-}-\fr... |
H: Characterization of Subset Sum via Linear Programming
I have a sample subset sum problem.
Given numbers $x_1, x_2... x_N$ and a target value to sum to $x_S$
Minimize $x_S - x_1y_1 - x_2y_2 - x_3y_3 ... x_Ny_N$
such that
0 <= $y_1$ <= 1
0 <= $y_2$ <= 1 ...
0 <= $y_N$ <= 1
$x_S - x_1y_1 - x_2y_2 - x_3y_3.... - x_ny_... |
H: Finding the Exact Value
How would one go about finding the exact value of $\theta$ in the following:$\sqrt{3}\tan \theta -1 =0$? I am unsure of how to begin this question. Any help would be appreciated!
AI: Hint: $\sqrt{3}\tan \theta -1 =0\implies \sqrt{3}\tan \theta =1\implies \tan \theta=1/\sqrt{3}$
Recall from S... |
H: Need help with integration by parts
I absolutely despise integration by parts, because it never seems to work for me. Here's an example:
$$ \int 4x \sin(2x) \, \mathrm{d}x $$
What I did:
$$ \int 4x \sin(2x) \, \mathrm{d}x = -2x \cos(2x) - \int - 4\cos(2x) \, \mathrm{d}x$$
Here I'm already stuck. I know about the IL... |
H: On irreducible polynom $\frac{X^p-1}{X-1}$
Let $m(X) := \frac{X^p-1}{X-1}$ and $n(X) := m(X^p)$. I have shown that $m$ is irr. over $\mathbb Q$. Now I want to show that this is true for $n(X)$, too. I know that
$$
n(X+1)((X+1)^p-1)= (X+1)^{p^2}-1
$$
Working modulo $p$ I get that
$$
n(X+1)X^p \equiv X^{p^2} \mod p... |
H: Galois group of $\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$ over $\mathbb{Q}$
How to find the Galois group $Gal(E|Q)$, where $E=\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$? I know that $\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega)|\mathbb{Q})=D_3$, where $D_3$ is the dihedral group, $\omega^3=1$. Does it implies that
$$\... |
H: Nullity of Kernel, Range of transpose
Define the linear transformation $T$ by $T(x) = Ax$, where
$A=\left(\begin{matrix} \frac{9}{10} & \frac{3}{10}\\ \frac{3}{10} & \frac{1}{10} \end{matrix}\right)$.
Find (a) $\ker(T)$, (b) $\text{nullity}(T)$, (c) $\text{range}(T)$ and (d) $\text{rk}(T)$.
Apparantly, $\ker(T)... |
H: Roots of $x^{2}+e^{0.1x}-1$
I saw an exercise that asks to prove that $f(x):=x^{2}+e^{0.1x}-1$
have a root $r<0$.
The solution stated that $f''(x)=2+(0.1)^{2}e^{0.1x}>0$ hence there
is a maximum of two roots, since $0$ is a root and since $$f'(0)=0.1>0$$
there is a root $r<0$.
I know that a convex function have at ... |
H: Probability, combinations with repetition
A store sells n different kinds of fruits. A boy buys k fruits. Find the probability that he buys all the kinds of fruits.
Give me a hint, please.
Thank you.
AI: I assume that you mean for the store to have an unlimited number of each different kind of fruit. Are you also ... |
H: Prove that this graph doesn't exist
A group of 3141 students gather together. Some of them have 13 friends in this group, some have 33 friends, and the rest has 37 friends. Prove using graph theory that this group does not exist. Assume that if A is friends with B, then B is friends with A.
AI: This seems like home... |
H: Some questions about Goldbach's conjecture
I was thinking about the usage of Dirichlet's theorem in proving some facts about the Goldbach's conjecture.
I will start with an example. Using Dirichlet's theorem, we know that there are infinitely many primes in the form of, let's say, $6k+1$. This means there are infi... |
H: Abstract Algebra and Parallel Computing
I've recently been learning a bit about parallel computation. Two things I recently learned about are Reduce and Scan.
Where Reduce is defined as 2-Tuple of a Set of elements and a binary operation that is associative.
Scan is defined as a 2-Tuple of a set of elements and a b... |
H: Convergeance and Lebesgue Integral exercise
Can you help me formally prove that $f_n:\mathbb R \to \mathbb R$,$f_n=n\chi_{[\frac{1}{n},\frac{2}{n}]}$ $\forall n\in\mathbb N$ converges pointwise to $f\equiv 0$ and that every $f_n$ is Lebesgue integrable?
It is not a homework,I am solving exercises for my tomorrow's ... |
H: At least two participants in a meeting received the same number of phone numbers
The question is: there are $n$ participants in a meeting ($n \geq 2$). During this meeting, people exchanged phone numbers with each other. Prove that at least $2$ participants received the same number of phone numbers.
I was thinking ... |
H: Area between a curve and a horizontal line
I had 2 make 2 exercises which I found very similar but they have a crucial difference which puzzles me:
1 Calculate the area of the plane $V$, between the graph of the function $f(x) = \dfrac{8x}{x^2+4}$ and the line $y= 1\dfrac{3}{5}$.
2 Calculate the area of the plane... |
H: Do negative dimensions make sense?
Some time ago I read in a popular physics book that in M-theory, there are some "things" which can be said to have dimension $-1$.
Probably, the author was vastly exaggerating, but this left me wondering:
Are there mathematical theories which contain a notion that can be regarded ... |
H: Constrained ( Variable Length ) Permutation Calculation.
I am writing some tracking software, but I think this is pretty purely a math question. I don't need to know the math to accomplish this with my code, but I like math and I want to learn! Thus please keep in mind, I fully expect to have to look up definitio... |
H: Sum of positive i.i.d. random variables.
Let $X_1, X_2, \ldots, $ be i.i.d. random variables with $X_1 > 0$. Let $S_n = \sum_{m=1}^n X_m$. Can we conclude $[\sup_{n \ge 1} S_n = \infty$] almost surely?
Assuming the statement is true, by Kolmogorov's 0-1 law, $[\sup_{n \ge 1} S_n = \infty]$ has probability either $0... |
H: Calculate Exact Value of $\sin\theta, \cos\theta$ and $\tan\theta$
Having trouble getting a start on this problem, any help would be appreciated!
Given point $P = (-3,5)$ is on the terminal arm of angle $\theta$ in standard position. Calculate the exact value of $\sin\theta, \cos\theta$, and $\tan\theta$.
AI: Hints... |
H: Real integral $ \int_{-\infty}^{\infty} \frac{dx}{1+x^2} $ with the help of complex friends
I have to solve the integral $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2} $$ by doing this:
Given a rectangle that is defined by the points $ r+i, -r+i,-r-i,r-i$, $r>0$ and $\gamma_r$ is a closed positively oriented curve aro... |
H: Defining the winding number for a general curve
In the Complex Analysis text by Ahlfors, he says that we can define the winding number $n(\gamma,a)$ for any continuous, closed curve $\gamma$ which doesn't pass through the point $a$ (differentiability is not required).
Divide $\gamma$ into subarcs $\gamma_1,\dots,\g... |
H: Vector spaces and subspaces.
a) Find an abelian group V and a field F for which V is a vctor space over F in at least two different ways, that is, there are two different definitions of scalar multiplication making V a vector spae over F.
We can choose both V and F to be $\Bbb{Q}$ and define the two different sc... |
H: How to integrate an exponential raise to the inverse sine?
Find the $\space \displaystyle\int e^{\sin^{-1}x}~\mathrm dx$ .
I started by making a substitution. Let $u=\sin^{-1}x$, and so one can conclued that:
$\begin{align}1)&\mathrm du=\displaystyle\frac{1}{\sqrt{1-x^2}}\mathrm dx\\2)&x=\sin u
\end{align}$
So, t... |
H: Cardinal arithmetic questions
I have problem to solve:
Let $a,b$ and $c$ be cardinal numbers. Prove that $a+b=b$, $b \le c$ implies $a+c=c$.
And trying to prove this I got couple questions:
For infinite cardinal $c$, is $c+c=c$ always? And how to prove it? (Is $c \cdot c=c$?)
From $a+b=b$ can I conclude that $a \l... |
H: Closed subsets of $\mathbb{R}$ characterization
I remember the characterization of open subsets of $\mathbb{R}$ as a countable union of disjoint open intervals. I was thinking about whether this allows us to characterize closed subsets as a countable union of disjoint closed intervals. As we know, closed subsets ar... |
H: Derivative of $ y = (2x - 3)^4 \cdot (x^2 + x + 1)^5$
$$ y = (2x - 3)^4 \cdot (x^2 + x + 1)^5$$
I know that it should be the chain rule and product rule used together to get the answer
$$ y = \frac{dx}{dy}((2x - 3)^4) \cdot (x^2 + x + 1)^5 + \frac{dx}{dy}(x^2 + x + 1)^5 \cdot (2x - 3)^4 $$
this gives me something ... |
H: How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$
I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.
$$\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x... |
H: I know the basic definition of continuity. But here, the definition is applied for a ball.
I am studying the topology of $\Bbb R^n$ from W. R . Wade's Introduction to analysis book.
I know the basic definition of continuity. But here, the definition is applied for a "ball".
I dont understand the section which i ha... |
H: Prove that it is possible to divide integral area into two equal parts
Assuming $f$ is locally integrable on interval $<a,b>$, I'd like to show that it is always possible to divide it into two equal parts in terms of enclosed areas.
In other words, I'd like to show there exists $x \in [a,b]$ with $\int_a^x f(t)\ dt... |
H: What is the equation representing a constant elasticity of 1?
I'm reading the chapter in my textbook about the price elasticity of demand, and it was pointed out that most demand curves do not represent a constant elasticity of demand - even linear curves like $f(x)=x$ is not constant elasticity although it has con... |
H: computation of $\int_0^T \frac{\sin (t)}{t} dt$
I need approaches to solve the problem of the following type
$\int_{0}^{T} \frac{\sin (t)}{t} dt $. Is there a closed form solution available.
AI: The integral
$$
\int_0^T\frac{\sin(t)}{t}\,\mathrm{d}t
$$
is called the Sine Integral. It has no closed form in terms of... |
H: What does an expression $[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}...$ mean?
I came across the function that describes number of partitions of $n$ (I mean partitions like $5=4+1=3+2=3+1+1$ and so on.
There was defined a Cartesian product: $$\{0,1,1+1,1+1+1,...\}\times\{0,2,2+2,2+2+2,...\}\times\{0,3,3+3,.... |
H: Galois group of a reducible polynomial over a arbitrary field
How to proceed to determine the Galois group of a reducible polynomial over a field $F$. As an example I tried to compute the Galois group of $f(X)=X^4+4\in\mathbb{Q}[X]$; one can check that $f(X)=(X^2-2X+2)(X^2+2X+2)$ and the polynomials $h(X)=X^2-2X+2$... |
H: What is convection-dominated pde problems?
Can you explain for me what is convection-dominated problems? Definition and examples if possible.
Why don't we can apply standard discretization methods (finite difference, finite element, finite volume methods) for convection-dominated equation?
What is the importance o... |
H: Nitpicky Sylow Subgroup Question
Would we call the trivial subgroup of a finite group $G$ a Sylow-$p$ subgroup if $p \nmid |G|$? Or do we just only look at Sylow-$p$ subgroups as being at least the size $p$ (knowing that a Sylow-$p$ subgroup is a subgroup of $G$ with order $p^k$ where $k$ is the largest power of $p... |
H: Is this fact about matrices and linear systems true?
Let $A$ be a $m$-by-$n$ matrix and $B=A^TA$. If the columns of $A$ are linearly independent, then $Bx=0$ has a unique solution.
If is true, can you help me prove it? If is false, could you give a counterexample?
Thanks.
AI: Yes. If $Bx=0$, then $x^T B x = (Ax)^T ... |
H: prove that $\ln (x^2+ cos^2x)$ is uniformly continuous.
I know that the direction of the proof is to show that the derivative of the function is bounded, and hence meets the Lipschitz condition. So I differentiated it:
$$f'(x) = \frac{2x-\sin2x}{x^2 + \cos^2x}$$
But I am stuck here. I can't manage to show that it i... |
H: Is a function $f:X\to Y$ continuous if and only if its graph on each connnected component of $X$ is connected?
I was thinking about this question today. Is the following true:
Let $X$ be a topological space with connected components $\{C_i\}_{i\in I}$. Let $Y$ be a topological space and let $f:X\rightarrow Y$ be ... |
H: How to understand $\delta$ and $\varepsilon$ in real analysis?
I'm a bit confused as to how to interpret $\delta$ and $\varepsilon$ mean in real analysis. My textbook gives an example demonstrating that $\frac{1}{x^2}$ is not uniformly continuous on $(0,1)$.
Definition of not uniformly continuous: $(\exists \vareps... |
H: prove $A \sim B\implies 2^A \sim 2^B$.
I want to prove that if $A \sim B$ then $2^A \sim 2^B$.
$A\sim B$: There is a bijection from $A$ to $B$
thanks.
AI: HINT: Suppose that $f:A\to B$ is a bijection, and consider the map
$$F:2^A\to 2^B:X\mapsto f[X]=\{f(a):a\in X\}\;.$$ |
H: Related rate volume increasing
I am trying to find out how fast the water level is rising in a tub that is filling at a rate of $.7 \text{ft}^3 / \text{min}$
I am not too sure how to do this, I am given that the base of the bathtub is $18 \text{ft}^2$ and that it is rectangular.
I know that $$b * h = v$$ but I am n... |
H: Why it has Trace?
The operator tr : $T_1^1 (V ) \rightarrow R$ is just the trace of $F$ when it is considered as an endomorphism of $V$ . Since the trace of an endomorphism is basis-independent.
I am very confused with this statement because to my understanding, $T_1^1$ is $V^* \otimes V$. How can it be a matrix ... |
H: $\mathbb{A}^2\backslash\{(0,0)\}$ is not affine variety
In our lecture notes we have this example, with the proof why $X = \Bbb{A}^2\setminus \{(0,0)\}$ is not an affine variety:
Let $i:X\hookrightarrow \mathbb{A}^2$ be an inclusion map. We show, that any regular function on $X$ extends uniquely to a regular funct... |
H: Find the inverse Laplace transformation of $\dfrac{s+1}{(s^2 + 1)(s^2 +4s+13)}$
My question is : find the function $f(t)$ that has the following Laplace transform
$$ F(s) = \frac{s+1}{(s^2 + 1)(s^2 +4s+13)} $$
thanks
AI: I am going to evaluate this using residues. If you have no idea of what these are, then I wil... |
H: can all triangle numbers that are squares be expressed as sum of squares
I'm not sure if this is just a subset of Which integers can be expressed as a sum of squares of two coprime integers? which in turn points to http://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity, but if so, I'm not seeing it.
... |
H: Why are projective spaces over a ring of different dimensions non-isomorphic?
Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true that $\mathbb P_A^n\not\cong \mathbb P... |
H: Let $f$ be analytic on $D=\{z\in\mathbb C:|z|<1\}.$ Then $g(z)=\overline{f(\bar z)}$ is analytic on $D.$
Let $f$ be analytic on $D=\{z\in\mathbb C:|z|<1\}.$ Then $g(z)=\overline{f(\bar z)}$ is analytic on $D$.
My attempt: Let $f$ be analytic on $D,~f(x,y)=u(x,y)+iv(x,y)$ where $u,v:D\to\mathbb{R}$ are the real an... |
H: Shortest Distance From a Point to a Plane
My professor did this question (Finding the shortest distance from a point to a plane) in class, but before doing the example he showed us a formula for calculating it, but didn't really explain how he got the formula. I was hoping someone could tell me how or why the formu... |
H: $360\times 60$ nautical mile is not equal with $6400$ km of earth radius
One degree on a great circle of earth equals to $60$ Nautical Miles. Hence:
$\ 360 \times 60 = 21,600$ M (Nautical Mile)
$\ 21,600M \times 1852$ Meters $= 40,003,200$ Meters $= 40,003.2$ Kilometers
While the radius of earth is $6400$ Kilometer... |
H: Understanding equivalent definitions of left cosets
I understand the standard definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows:
Let $H\leq G$. Then a left coset of $H$ is a nonempty set $S\subset G$ such that $x^{-1}y \in H$ for any $x,y\in S$... |
H: Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$
Trial: Here I proceed in this way $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge \dfrac{2bc}{b+c... |
H: What does the notation min max mean?
Min clearly means minimum and max maximum, but I am confused about a question that says "With $x, y, z$ being positive numbers, let $xyz=1$, use the AM-GM inequality to show that min max $[x+y,$ $x+z,$ $y+z]=2$ What does this mean? (I am not looking for the answer this particula... |
H: Is every almost complex structure tame up to sign?
Let $V$ be a real vector space equipped with a symplectic form $\omega : V\times V \to \mathbb{R}$. An almost complex structure $J$ is compatible with $\omega$ if $\omega(J(u), J(v)) = \omega(u, v)$ for all $u, v \in V$. In addition, $J$ is said to tame $\omega$ if... |
H: Related rates with a cone
I am trying to figure out the rate the water level increases in a conical tank that is 3 m height, 2 m radius at top and water flows in at $2\text{m} ^3 / \text{minute}$
I know that
$$(1/3) \pi r^2 h = V$$
$$4 \pi = V$$
or at 2 seconds the volume of the water is $8.37758$
so now I have
$$... |
H: Continuous function from non-compact space onto compact space
Give an example of metric spaces $M_1$ and $M_2$ and a continuous function $f$ from $M_1$ onto $M_2$ such that $M_2$ is compact, but $M_1$ is not compact.
So there must exist a sequence $x_1,x_2,\ldots$ with no convergent subsequence in $M_1$, but any ... |
H: Proving a lemma - show the span of a union of subsets is still in the span
This is part of proving a larger theorem but I suspect my prof has a typo in here (I emailed him about it to be sure)
The lemma is written as follows:
Let $V$ be a vector space. Let {$z, x_1, x_2, ..., x_n$} be a subset of $V$. Show that if... |
H: The definition of tensor
$(E_1, \ldots, E_n)$ are basis for $V$. But this statement makes no sense to me. How could it define $F$ with $F^{j_1 \ldots j_l}_{i_1 \ldots i_k} E_{j_1} \ldots$, but then define $F^{j_1 \ldots j_l}_{i_1 \ldots i_k}$ with $F$? Totally lost...
AI: One presumably has that $\{\varphi^i\}$ is ... |
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