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H: How can I resolve this definte integral?
$$ \int (t^2-3)^3 t dt$$
$$ \int_a^b (t^2-3)^3 t dt$$
if $$ a = -1$$
$$ b=1$$
then
$$ \frac 1 2\int (t^2-1)^3 2t dt$$
$$ \frac 1 2 \frac {u^{3+1}}{3+1}$$
$$ \frac 1 8 u^{4}$$
$$ \frac 1 8 (t^2-1)^{4}$$
$$ \frac 1 8 [(1^2-1)-(-1^2-1]$$
$$ \frac 1 8 [(0)-(0)]=0$$
or
$$ \int (... |
H: Prove that the class of well-founded sets is a proper class
Does anyone have an "elementary" proof of the following claim:
If $A$ is a class such that
$$(*)\qquad\forall x(x\subseteq A\to x\in A),$$ then $A$ is a proper class, i.e. $\forall y\ y\ne A$.
The reason that the title refers to well-founded sets is th... |
H: Finding root of equation
This question was asked in one of the enterance test of mathematics in India which is
For the equation $1+2x+x^{3}+4x^{5}=0$, which of the following is true?
(A) It does not possess any real root
(B) It possesses exactly one real root
(C) It possesses exactly two real roots
(D) It possesses... |
H: Connected Metric Space Exercise
Let $E$ be a connected metric space, in which the distance is not bounded. Show that in
$E$ every sphere is nonempty.
AI: I’m assuming that by sphere you mean a set of the form
$$S(x,r)=\{y\in E:d(x,y)=r\}\;,$$
the sphere of radius $r$ centred at $x$.
HINT: Let $S(x,r)$ be any spher... |
H: Find triangel-area with Cavalieri's principle
A triangle is given by $A=(0,0), B=(5,1)$ and $C=(2,4)$. I already know $\lambda^2(\Delta ABC)=9$. Now I want to compute the area by using Cavalieri's principle.
I know how to start when I have to use the principle for volumes, but I don't know how to start here. I thou... |
H: How prove this $\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+y}}\le\frac{2}{\sqrt{1+\sqrt{xy}}}$
Let $x,y>0$ and $xy\le 1$. Show that
$$\dfrac{1}{\sqrt{1+x}}+\dfrac{1}{\sqrt{1+y}}\le\dfrac{2}{\sqrt{1+\sqrt{xy}}}.$$
This inequality have same follow methods?
I saw this.
Let $x,y>0, xy\le 1$.
$$\dfrac{1}{1+x}+\dfrac{1}{1+y... |
H: Derivative of $-e^y = 0$?
I stumbled upon this on wolfram alpha and still wonder why $-e^x$ equals $0$ (third step).
AI: It is assuming that $x$ and $y$ are independent variables and as you're differentiating with respect to $y$, that term vanishes since it doesn't contain any $y$'s |
H: Not able to solve $({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$
I'm not able to solve $$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$
If you put values of $p$ (like $\frac{1}{2}$ or 2) back in the equation it doesn't satisfy! So please check your values also.
What I fou... |
H: Why do we restrict the range of the inverse trig functions?
I understand why we restrict the domain, but why do we restrict the range? Why do we necessarily care so much for the inverse trig relations to be functions? Thanks!
AI: Basically since it is easier and simpler to work with functions than it is to work wit... |
H: Why is $e^x$ the only nontrivial function with a repeating derivative?
Why is $e^x$ the only nontrivial function with a repeating derivative, i.e. is its own derivative?
It says so in the Wikipedia article about $e$. Is there a proof of this that I (a calculus AB student) could understand? Thanks!
AI: Technically ... |
H: What is the probability of a multidimensional rectangle?
Assume given a probability measure $P$ on $(\mathbb{R}^p,\mathcal{B}_p)$, where $\mathcal{B}_p$ denotes the $p$-dimensional Borel-$\sigma$-algebra. Let $F$ denote the $p$-dimensional CDF for $P$, given by $F:\mathbb{R}^p\to[0,1]$ with
$$
F(x_1,\ldots,x_p) = P... |
H: Proving something is not differentiable
I am looking for confirmation so that I can be sure I understand what is being asked here. I need to show that the following function $f(x,y)$ is not differentiable at $(0,0)$ but that $g(x,y)=yf(x,y)$ is:
$$
f(x,y) = \left \{
\begin{array}{ll}
{\frac {x^2 y} {x^2 + y^2} } & ... |
H: Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus
Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b),
c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or
langrange multiplier): $$74 - 37\sqrt 2 \le a+b+6(c+d) \le 74
+37\sqrt 2$$
The original problem... |
H: How to show orthonormal basis?
Let $A$ be n*n matrix with complex entries.
Prove that $AA^*=I$ iff rows of $A$ form an orthonormal basis of $C^n$.
I know since $AA^*=\langle a_i, a_j \rangle= \delta_{i,j}$ so the rows are orthonormal.
But why does it mean that they are basis?
AI: For the rows to be a basis they ... |
H: What is wrong with treating $\dfrac {dy}{dx}$ as a fraction?
If you think about the limit definition of the derivative, $dy$ represents $$\lim_{h\rightarrow 0}\dfrac {f(x+h)-f(x)}{h}$$, and $dx$ represents
$$\lim_{h\rightarrow 0}$$
. So you have a $\;\;$$\dfrac {number}{another\; number}=a fraction$, so why can't y... |
H: A question on the symbolic powers of a prime ideal
In I. Swanson's notes about primary decomposition the author wrote:
The smallest $P$-primary ideal containing $P^n$ is called the $n$th symbolic power of $P$, where $P$ here is a prime ideal of a ring $R$.
She in fact gave this result as a consequence of a theor... |
H: What are the possible ranges of a metric
Given a metric $d$ on a space $X$, what can we say about $d(X\times X)$? What possible range can $d$ have?
More precisely, consider the set $D=\{ A \subset [0,\infty) | A = d(X\times X), \textrm{d is a metric on $X$} \} $ What properties does $D$ have?
For instance, all fin... |
H: Prove that the preimage of a prime ideal is also prime.
Let $f: R \rightarrow S$ be a ring homomorphism, with $R$ and $S$ commutative and $f(1)=1$. If $P$ is a prime ideal of $S$, show that the preimage $f^{-1}(P)$ is a prime ideal of $R$.
Define $g: S \rightarrow S/P$ with kernel $s$. Let $h = g \circ f: R \righ... |
H: A question about an infinite sum
Let $\{ a_n \}, \{ b_n \}$ be a sequence of nonnegative real numbers. Assume that $\sum_{n=1}^\infty b_n $ converges(let $\sum_{n=1}^\infty b_n = C)$ and assume that $$ a_n \le a_1 \;(\forall n \in \mathbb N).$$ Then can I conclude that $$\sum_{n=1}^\infty a_n b_n \leq a_1 \sum_{n... |
H: Determine if $\sum\limits_{n=0}^{\infty}\frac{1}{2^{\sqrt{n}}}$ converges
Apparently it can be proven with a comparison but I've tried to compare it to $\frac{1}{n^{p}}$ with not results.
I've also tried comparing $\sqrt{n}$ with $\ln{n}$ but $\frac{1}{2^{\ln n}}$ diverges so that doesn't give me anything useful.
A... |
H: Given $a_1,a_2,...,a_n>0$ where $n\in\mathbb N$$, a_1+a_2+...+a_n=n$. Is this true? $a_1a_2+a_2a_3+...+a_na_1\leq n$
Given $a_1,a_2,...,a_n>0$ where $n\in\mathbb N$$, a_1+a_2+...+a_n=n$. Is this true?
$$a_1a_2+a_2a_3+...+a_na_1\leq n$$
By observing:
When $n=1$, this is trivial;
When $n=2$, $ab\leq(\frac {a+b} 2)... |
H: How to prove that some set has the quotient topology for a function?
Let $f: X \to Y$ be a continuous function between topological spaces. Let $S$ be a set and $g: Y \to S$ a function. Assume that $g \circ f$ is surjective, and that $S$ has the quotient topology for $g \circ f$. Assume that $g$ is continuous. How d... |
H: Calculate the limit of $\lim\limits_{x\to 1^-}\left(\frac{1}{1-x^2} -\frac{1}{1-x^3}\right)$
I want to calculate this limit and wonder what is the best way to calculate it.
$$\lim\limits_{x\to 1^-}\left(\frac{1}{1-x^2} -\frac{1}{1-x^3}\right)$$
I tried to do the following thing
$$\lim\limits_{x\to 1^-}\left(\frac{... |
H: Evaluate $ \int^4_1 e^ \sqrt {x}dx $
Evaluate $ \int^4_1 e^ \sqrt {x}dx $
solution:-
Here $1<x<4$
$1<\sqrt x<2$
$e<e^ \sqrt {x}<e^ 2$
$\int^4_1 $e dx$<\int^4_1 e^ \sqrt {x}dx<\int^4_1 e^ 2dx$
$3e <\int^4_1 e^ \sqrt {x}dx<3 e^ 2 $
But in this objective question
Options are
a)$e $
b)$e^2 $
c)$2e $
d)$2e^2 $
AI: Hi... |
H: What is the easiest/most efficient way to find the taylor series expansion of $e^{1-cos(x)}$ up to and including degrees of four?
So I have $$e^{1-cos(x)}$$ and want to find the taylor series expansions up to and including the fourth degree in the form of $$c_{0} \frac{x^0}{0!} + c_{1} \frac{x^1}{1!} + c_{2} \frac{... |
H: Do the following functions exist?
While practicing for my topology exam, I stumbled upon the following question from a previous exam:
Give a proof of your answer:
(a) Is there a continuous surjective map from $\mathbb{C}$ to $\mathbb{C} - \{0 \} $?
(b) Is there a continuous surjective map from $\mathbb{R}$ to $\m... |
H: Consistency of definition of weak derivative with classical derivative
I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have
$\int f' \varphi = -\int f\varphi'$ for all $\varphi\in C^\infty$ with compact support. Is this defi... |
H: Is it true that Quadratic residue was published and discovered before Legendre symbol and Euler's Criteria?
So Is it true that Quadratic residue was published and discovered before Legendre symbol and Euler's Criteria?
Quadratic residue came in 1801 by Gauss(1).
can you put these concepts in chronological order an... |
H: question about Ito's formula
I'm currently learning about the Ito's lemma / formula
In my textbook, a direct application of the formula is to compute quantities like that :
(W is a Brownian motion)
While trying to prove these results I am finding that these computations are really not direct.
Am I missing somethin... |
H: Composition of a convex function
If $f:[a,b]\rightarrow R$ is convex function and $f'(x)\geq 0$ for all $x\in [a,b]$ and $g:U\rightarrow [a,b]$ is convex function, how to show that $f(g(u)), u\in U$ is convex function?
AI: Note that $f$ is increasing. Let $x,y\in U, \lambda\in(0,1)$. From Jensen's inequality follow... |
H: Calculate a $\infty^0$ limit using `de l'hopital` rule.
I have to calculate the following limit:
$$\lim_{x\to \infty} 2x^{1/\ln x}$$
So I tried to start:
$$\lim_{x\to \infty} 2x^{1/ \ln x} = \infty^0 $$
From here on I noticed that I have to use de l'hopital rule. but don't really know how and I need help.
If the ma... |
H: A function $f$ is increasing on the closed internal $[a,b]$
Let the function $f$ be increasing on the closed internal $[a,b]$. If $a \le f(a)$ and $f(b)\le b$, prove that:
$\exists x_0\in [a,b]$, such that $f(x_0)=x_0$.
Thanks for your help.
Note that $f$ need not be continuous.
AI: Assume otherwise. Especially, ... |
H: Extreme values of a function with conditions
What is a way to find extreme values of a function $u(x,y,z)=xy+yz+xz$ with conditions $x+y=2, y+z=1$?
AI: $$u(x,y,z)=(2-y)y+y(1-y)+(2-y)(1-y)=y(2+1-1-2)+y^2(-1-1+1)+2=2-y^2$$
Alternately, $$u(x,y,z)=(x+y)(z+y)-y^2=2-y^2$$
So the minimum is when $y=0$.
Using Lagrange mul... |
H: Contructing a $\delta$-fine tagged partition from the old ones
Let $[a,b]\subset \mathbb{R}$. A tagged partition of $[a,b]$ is a set
$D=\{(t_i,I_i)\}_{i=1}^m$ where $\{I_i\}_{i=1}^m$ is a partition of $[a,b]$ consisting of closed non-overlapping subintervals of $[a,b]$ and $t_i\in I_i$; $t_i$ is called the tag asso... |
H: For what values of $m$ the function $y=x^m\sin(x)$ have horizontal asymptote
I want to figure for what values of $m$ the function have horizontal asymptote.$$y=x^m\sin(x)$$
so what I understand from that is this that the function dont have a vertical one, so I will find vertical asymptote and I will require th... |
H: Not able to solve $\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
If $p=\frac{7}{8}$ then what should be the value of $\displaystyle\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
when $$g(x) = x \log x \quad \text{or} \quad g(x) = \frac{x}{\log x}? $$
Wondering which way to proceed?
an algebraic substituti... |
H: Is a diagonal matrix times a matrix A a linear combination of A?
Say there is a set of m $\mathbb R^n$ vectors, represented as matrix $A$ in $\mathbb R^{r*m}$; and an m-by-n diagonal matrix $D =diag(i_1, i_2,...,i_m)$.
Is $D$ times $A$ a linear combination of that set of vectors? Is $0A$ the trivial linear combinat... |
H: Find push down automata and context free grammar
I have the following language:
$$
L = \{a^nb^{2n+1} \mid n \ge 0\}
$$
I must find the push down automaton and a context free grammar for the language.
For the push down I have no idea how to approach the problem.
For the context free grammar I think I know the soluti... |
H: Proving $\sum_{k=1}^m{k^n}$ is divisible by $\sum_{k=1}^m{k}$ for $ n=2013$
I got an interesting new question, it's about number theory and algebra precalculus. Here is the question:
a positive integer $n$ is called valid if $1^n+2^n+3^n+\dots+m^n$ is divisible by $1+2+3+\dots+m$ for every positive integer $m$.
P... |
H: Continuity of the function $f=1/x$
How do I show that the function $f(x)=\frac{1}{x}$ is continuous using the $\epsilon - \delta$ definition?
I have been trying for quite a while now without success.
My attempts
Suppose that $\left |x-x_0 \right| < \delta$ for some $\delta >0$ then $\left |f(x)-f(x_0)\right| =\le... |
H: Closed operator
I've got a very straightforward question : if $T : B \rightarrow B$ is a linear continuous operator and $B$ is a Banach space, is $T$ a closed operator?
This is obviously true in finite dimension, but I'm not sure what can happen in infinite dimension. Maybe it can't get "too bad" if $B$ is a Banach... |
H: A problem about mollification
The problem is :
Given $M > 0$ a constant, show that exists $\phi \in C^{\infty}(R)$ with the following properties:
i) $\phi(x) = x , \forall x \in [-M,M] $
ii) $ 0 \leq\varphi^{'}(x) \leq 1, \forall \ x $
This question arises form my question in the link
In the previous link the u... |
H: On convergence of $\prod (1 - \alpha_n)$
Suppose $\{ \alpha_n \}$ is a decreasing sequence of real numbers such
that $0 < \alpha_n < 1$ and $\alpha_n$ goes to $0$ as $n$ goes to infinity.
I was wondering if there is a known condition for $\{ \alpha_n \}$ so that
the product $\prod (1- \alpha_n)$ will not be $0$?
Th... |
H: Integrating $\ln x$ by parts
I am asked to integrate by parts $\int \ln(x) dx$. But I'm at a loss isn't there supposed to be two functions in the integral for you to be able to integrate by parts?
AI: Hint: Write $\log(x)$ as $1 \cdot \log(x)$ and use integration by parts. |
H: CDF of the distance of two random points on (0,1)
Let $Y_1 \sim U(0,1)$ and $Y_2 \sim U(0,1)$.
Let $X = |Y_1 - Y_2|$.
Now the solution for the CDF in my book looks like this:
$P(X < t) = P(|Y_1 - Y_2| < t) = P(Y_2 - t < Y_1 < Y_2 + t) = 1-(1-t)^2$
They give this result without explanation. How do they come up with ... |
H: Where I can find the Pythagorean theorem deduced from Hilbert's axioms?
Hilbert took years to make a rigorous revision and formalization of Euclidean geometry in his Foundations of Geometry. As he intended to organize only the most basic aspects of the theory, he didn't write about things like the Pythagorean Theor... |
H: Points from an affine subspace with equal distance from given points
Given vector space $\mathbb{R}^3$ with dot product defined as $x \cdot y = 2x_1y_1 + 3x_2y_2 + x_3y_3$ where $x = (x_1,x_2,x_3),y = (y_1,y_2,y_3)$ and given an affine subspace $W: x - y - z - 2 = 0$ .
I need to find all points from W which have t... |
H: MODULAR problem
What will be the remainder when 64! is divided by 71?
Do we need to solve this problem by using MOD theorem or need to expands the factorial?
AI: Hint $\displaystyle\ \ {\rm mod}\ 71\!:\,\ 64! = \frac{70!}{\color{#c00}{70}\cdots \color{#0a0}{65}}\!\!\stackrel{\rm\ Wilson_{\phantom{ I_I}}}\equiv\!\!\... |
H: Solving trigonometric identity with condition.
Problem : If $\sin\theta +\sin^2\theta +\sin^3\theta=1$ Then prove $\cos^6\theta -4\cos^4\theta +8\cos^2\theta =4$
My working :
As $\sin\theta +\sin^2\theta +\sin^3\theta=1 \Rightarrow \sin\theta +\sin^3\theta = \cos^2\theta$
Now the given equation : $\cos^6\theta -4\c... |
H: What is "every inductive set"?
In Apostol's «Calculus I», on page 22 there is the following definition:
A set of real numbers is called an inductive set if it has the following two properties:
(a) The number 1 is in the set
(b) For every x in the set, the number x + 1 is also in the set.
Next there is a definition ... |
H: Contronominal property
Let $P$ be a set, $\leq$ a binary relation on $P$, reflexive, antisymmetric and transitive. Let $\wedge$ and $\vee$ be two binary operations, both commutative and associative and distributive one to each other. Let $0$ be the minimum element of $P$, $1$ the maximum. Suppose that for every $a\... |
H: If $\langle T(x),y \rangle=0$ then $T=T_0$ - Prove this result if the equality holds for all $x,y$ in some basis for $V$
Let $T$ be a linear operator on an inner product space $V$. If $\langle T(x),y \rangle=0$ for all $x,y \in V$ then $T=T_0$ where it means zero transformation.
Prove this result if the equality ho... |
H: Prove that $f(x)=0$ has no rational solutions
$f(x)$ $\in$ $Z[X]$ monic polynomial of degree $n$
$k,p$ $\in$ $N$
If none of the numbers $f(k), f(k+1), \ldots , f(k+p)$ is disivible by $p+1$, then $f(x)=0$ has no rational solutions.
AI: Hint $\ $ Suppose not, so $\,f(x)\,$ has a rational root $\,r.\,$ By the Rationa... |
H: Rank of a matrix with an added all-1 row
As part of a proof I have the following statement, $A$ being an $n × n$ matrix:
Let us assume that $rank(A) ≤ n − 2$. If we add an extra row consisting of all $1$s to $A$, the resulting $(n+1) × n$ matrix still has rank at most $n − 1$.
I don't understand how adding an ext... |
H: Evaluate $\sum_{k=1}^nk\cdot k!$
I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$.
But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just check it out on the internet. :)
AI: HINT: $k(k!)=(k+1-1)(k!)=(k+1)!-k!$. Now do the su... |
H: Norm-paradox of normal endomorphism
Let A be a normal endomorphism $A:V\rightarrow V$ and V is a unitary vector space. Now every normal endomorphism is unitary diagonalizable, meaning: $A=QDQ^{-1}$ for some unitary matrix $Q$ and $D$ is a digonal matrix.
Now we have if $||.||$ is an arbitrary operator norm, that $|... |
H: Am I understanding vectors and matrices properly?
So, here is my understanding of a Vector:
A vector is an ordered set of real numbers that lie in the space $R^n$ where $n$ is the size of the vector.
So if $n$ equals 4, the vector is of size 4.
I understand matrices to be a set of vectors - row vectors and col... |
H: Prime generating functions
I'm studying prime numbers at school and I've seen some functions that generate mostly prime numbers. I'm talking about : $$\text{Euler's polynomial : } n^2+n+41$$ $$\text{Legendre's polynomial : } 2n^2+29$$ $$\text{Ruby's polynomial : } 103n^2-3945n+34381$$ $$\text{Mersenne numbers : }2^... |
H: If $\frac{\cos^4\theta}{\cos^2\phi}+\frac{\sin^4\theta}{\sin^2\phi}=1$, show $\frac{\cos^4\phi}{\cos^2\theta} +\frac{\sin^4\phi}{\sin^2\theta}=1$
If $\dfrac{\cos^4\theta}{\cos^2\phi}+\dfrac{\sin^4\theta}{\sin^2\phi}=1$, prove that $\dfrac{\cos^4\phi}{\cos^2\theta} +\dfrac{\sin^4\phi}{\sin^2\theta}=1$.
Unable to m... |
H: How can I find the smallest possible of full miles to get full kilometers?
$1 \textrm{mile} = 1.609344 \textrm{km}$
I know that using $1000000$ miles I can move the decimal point and get a full number of $1609344 $km.
But how can I find the smallest amount of full miles that is equal to full kilometers and do the s... |
H: Find gradient of this implicit function
How to find a gradient of this implicit function?
$$
xz+yz^2-3xy-3=0
$$
AI: EDIT: Abhinav pointed out a mistake, which has been corrected for posterity.
To find $\frac{\partial z}{\partial x}$ by implicit differentiation means to differentiate both sides of the equation with ... |
H: Exercise over the connected component of a point $x$ in a metric space $E$
In a metric space $E$, how to prove that the connected component of a point $x\in E$ is contained in every open and closed set containing $x$.
AI: The fact that $E$ is metric is really irrelevant, what matters is that it is a topological spa... |
H: Is this function "$h$" symmetric of the plane $x=y$?
$h=\left\{\begin{matrix}
f,x<y\\ g,x\geq y
\end{matrix}\right.$
$g=f(y,x)$.
Is $h$ symmetric of $x=y$? Here $g$ is the function that changes all $x$ to $y$ and changes all $y$ to $x$ in $f(x,y)$.
For example, $h=\left\{\begin{matrix}
x^2-y^2, x<y\\ y^2-x^2,x\geq... |
H: Finding indefinite integral by partial fractions
$$\displaystyle \int{dx\over{x(x^4-1)}}$$
Can this integral be calculated using the Partial Fractions method.
AI: HINT:
We need to use Partial Fraction Decomposition
Method $1:$
As $x^4-1=(x^2-1)(x^2+1)=(x-1)(x+1)(x^2+1),$
$$\text{Put }\frac1{x(x^4-1)}=\frac Ax... |
H: Proof using natural deduction
Prove that
$$\lnot r\Rightarrow \lnot p,\lnot(q\lor r),s\Rightarrow(p\lor q)\models\lnot s
$$
I'm completely stuck on this one. Only natural deduction inference rules can be used, no de morgan's law etc. The premises given all seem to be really irrelevant, and since we can't use trans... |
H: Proving Inner Product Space
Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$
Is $\langle f,g \rangle$ an inner product space?
I'm just checking the four conditions from Kreyszig pg 129.
I have a few questi... |
H: Subgroups of a cyclic group and their order.
Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states,
Let $G = \langle a \rangle$ be a cyclic group.
(i) Every subgroup $S$ of $G$ is cyclic.
(ii) If $|G|=n$, then $G$ has a unique subgroup of order $d$ for each divisor $d$ of $n$.
I under... |
H: Problem of convolution.
If we are given with a polynomial $\mathcal P$ and a compactly
supported distribution $g$. Can we prove that their convolution will
be a polynomial again?
AI: It should be somewhat easy to show the following:
Lemma
If $f:\Bbb{R}^n\rightarrow\Bbb{R}$ is $m$ times continuously differentia... |
H: Test for convergence of improper integrals $\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$ and $\int_{1}^{\infty}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$
I need to test if, integrals below, either converge or diverge:
1) $\displaystyle\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$
2) $\displaystyle\int_{1}^{\infty}\fr... |
H: My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?
Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes:
Let $R$ be a unitary commutative ring, and $I$ be an ... |
H: For given $n\times n$ matrix $A$ singular matrix, prove that $\operatorname{rank}(\operatorname{adj}A) \leq 1$
For given $n\times n$ matrix $A$ singular matrix, prove that $\operatorname{rank}(\operatorname{adj}A) \leq 1$
So from the properties of the adjugate matrix we know that
$$ A \cdot \operatorname{adj}(A) = ... |
H: Deducing Euler Equation
From Sydsaeter / Hammond (Further Mathematics for Economic Analysis, 2008, 2nd ed., p. 293):
$$ \max \int\limits_{0}^T [N(\dot{x}(t)) + \dot{x}(t)f(x(t))] e^{-rt} dt $$
where N and f are $C^1$ functions, r and T positive constants, $x(0) =
x_0$, and $x(T)=x_T$. Deduce the Euler Equation:... |
H: Sum of a countable dense set and a set of positive measure
Assume $A$ is a countable dense set in $\mathbb{R}$, and set $B$ has positive (Lebesgue) measure. Prove that $A+B=\{a+b:a\in A, b\in B\}=\mathbb{R}\backslash N$, where $N$ is a set of measure zero.
I haven't come up with a good idea.
Thanks in advance!
AI... |
H: Using the beta function
Show that $\displaystyle \int_{0}^{\frac{\pi }{2}}\cos^{n} \theta d\theta=\int_{0}^{\frac{\pi }{2}}\sin^{n} \theta d\theta=\frac{\sqrt{\pi}[\frac{(n-1)}{2}]!}{2(\frac{n}{2})!}$
AI: First substitute $u=\cos{\theta}$ in the first integral to get
$$\int_0^1 du \, (1-u^2)^{-1/2} u^n$$
Now sub $u... |
H: Finding SVD from unit eigen values
Suppose $A$ is a 2 by 2 symmetric matrix with unit eigenvectors $u_1$ and $u_2$. If its eigenvalues are $\lambda_1=3$ and $\lambda_2=-2$, what are the matrices $U,\Sigma,V^T $ in its SVD?
How to do this?
Is it something with the matrix beeing symmetric?
David
AI: Since $A$ is sym... |
H: Is there a proof of this that does not use idempotents?
I am going to present a statement and a proof. The proof makes use of idempotents which makes it a little cumbersome.
Is there a proof that does not use idempotents?
(using well-known theorems is OK even if their proofs do use idempotents)
Statement:
Let ... |
H: Prove that $1, x, x^2, \dots , x^n$ are linearly independent in $C[-1,1]$
As it states in the title, I'd like to prove that $1, x, x^2, \ldots , x^n$ are linearly independent in $C[-1,1]$.
Should I use an induction argument or integrate for $x^m$ and $x^n$ with cases $m=n$ and $m \neq n$?
The inner product is $$ \l... |
H: How prove $\mathbb Q$ is close in the following metric space?
assume $(d,\mathbb R)$ be a mertic space such that $$d:\mathbb R\times \mathbb R \to [0,\infty)$$$$d(x,y)=
\begin{cases}
0, & \text{if x=y} \\
max\{|x|,|y|\}, & \text{if x$\neq$y} \\
\end{cases}$$
How prove $\mathbb Q$ is close in this metric space a... |
H: Simple number theory problem
I found this question in a textbook on number theory:
For which integer c will $\;\displaystyle{\frac{c^6 - 3}{c^2 + 2}}\;$ also be an integer?
I wonder if there is a solution which is not based on trial and error.
AI: If $(c^6 - 3)/(c^2 + 2)$ is an integer, then so is $$\frac{c^6 - 3... |
H: Proof for $\displaystyle\sum_{k=1}^n k^a$ equaling a sum of fractions
I know $\displaystyle\sum_{k=1}^n k^2$ equals $n/6+n^2/2+n^3/3$, but... why?
And I also know that $\displaystyle\sum_{k=1}^n k^3$ equals $n^2/4+n^3/2+n^4/4$, but... is there a pattern so I can easily get $\displaystyle\sum_{k=1}^n k^a$? And could... |
H: Cumulative distribution function of the generalized beta distribution.
Suppose $Z$ has a beta distribution on the interval $(0,1)$ and its probability density function is $f_Z(x)$. I know that the cumulative density function is,
$$F_Z(x) = \mathbb{P}(Z \leq x) = \int_{0}^x f_Z(u) \, du.$$
I also know that if $X = c... |
H: An operator between $\mathcal{L}(X, Y)$ and $\mathcal{L}(Y, X)$
Please, I need help with this problem.
Let $X$, $Y$ be two vector normed spaces. Let $A_0\in\mathcal{L}(X, Y)$ such that $A^{-1}_0\in\mathcal{L}(Y,X)$. Show that there's an operator $\mathcal{T}_0\in\mathcal{L}(\mathcal{L}(X, Y), \mathcal{L}(Y, X))$ s... |
H: Distribution with singularities.
I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty
\psi^{(n)}(n)$ is a distribution which has singularities of infinite
order. Here $\psi$ is a test function that belongs to $ \mathcal D(\Bbb R)$.
Thanks.
AI: I will give you the main idea. T... |
H: Wedge product $S^1 \vee S^2$
I am trying to compute $\pi_1(S^1 \vee S^2$) by Van Kampen. I know Hatcher has a solution but I need to verify if my approach is correct and rigorous. I have seen a previous post on this topic, but I am using a different decomposition of $X=S^1 \vee S^2$, so please bear with me.
Let $z$... |
H: Finding Root of an Equation with Variables Dependent on Each other
Sorry for the title. I'm sure there is better terminology. I'd be interested to here what that terminology is haha.
Here is my problem:
If x < 40, y = 0.01
If x > 40, y = 0.02
If x > 50, y = 0.03
0 = -1000 + (x-30–[x*y])*50
Solve for $x$ an... |
H: Continuity of a function $f$ in a metric space from of the continuity of $f$ in every compact subset of $E$
Let $E$, $E'$ be two metric spaces,$f$ a mapping of $E$ into $E'$. Show that if the restriction of $f$ to any compact subspace of $E$ is continuous, then $f$ is continuous in E.
AI: HINT: If $\langle x_n:n\in... |
H: Invertibility in a finite-dimensional inner product space
Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{*}$ is also invertible and $( T^{-1} )^{*} = ( T^{*} )^{-1}$.
$$ \circ \circ \circ ~ Answer ~ from ~ Below ~ \circ \ci... |
H: Find closest vector to A which is perpendicular to B
To start, I would like to apologize if the answer to my question was easily googled, I am quite new to this and googling "Find closest vector to A which is perpendicular to B" gave me no results.
My problem:
I am a procedural generation programmer looking for a w... |
H: Convergent Series in a dual space
I don't know how solve this problem.
Please I need help.
Let $X =\mathcal{C}[0,1]$ with the uniform norm and let
$\{p_j\}_{j\in\mathbb{N}}$, $\{q_j\}_{j\in\mathbb{N}}\subseteq X$ such
that the series $\sum\limits_{j=1}^{\infty}p_j(s)q_j(t)$ uniformly
converge to a continuou... |
H: Scheffe’s Theorem
I saw a statement of Scheffe’s Theorem as follows ([1, p84]):
... we need a simple result called Scheffe’s Theorem. Suppose we have probability densities $f_n$ , $1 \le n \le \infty$, and $f_n \to f_\infty$ pointwise as $n \to \infty$. Then for all Borel sets $B$
$$
\left| \int_B f_n (x)dx - \i... |
H: Fold, Gather, Cut
Here's a mathematical puzzle I've been thinking about. Let's say you have a strip of fabric, of length $N$ units ($N$ being an integer), which has regular markings on it every 1 unit along its length. Your task is to cut the fabric into $N$ lengths of 1 unit each, but to do so using the fewest o... |
H: Rudin Theorem 2.47 - Connected Sets in $\mathbb{R}$
I need help with the proof of the converse, as given by Rudin in Principles of Mathematical Analysis, to the following theorem:
Theorem 2.47: A subset $E$ of the real line $\mathbb{R}^1$ is connected if and only if it has the following property: If $x \in E$, $y ... |
H: Inner Product Spaces : $N(T^{\star}\circ T) = N(T)$ (A PROOF)
Let $T$ be a linear operator on an inner product space. I really just want a hint as to how prove that $N(T^{\dagger}\circ T) = N(T)$, where "$^\dagger$" stands for the conjugate transpose.
Just as an aside, how should I read to myself the following sym... |
H: variational question
Let $\Omega$ a bounded domain, connexe and regular, and let $f \in L^2(\Omega).$ Let the variational problem: Find $u \in H^1(\Omega)$ such
$$\int_{\Omega} \nabla u \nabla v dx + (\int_{\Omega} u dx)(\int_{\Omega} v dx) = \int_{\Omega} f v dx, \forall v \in H^1(\Omega)$$
1- Prouve that this va... |
H: Calculate double integral of function over triangle
Find the limits for integrals $\int\int f(x,y) \,dy \, dx$ and $\int\int f(x,y)\,dx\,dy$ and compute the integral over the region, based on the function $f(x,y) = 3x^2y$.
Region = triangle inside the lines $x=0$, $y=1$, $y=2x$.
To find what the limits of my inne... |
H: rock, paper, scissors, well
Everyone knows rock, paper, scissors. Now a long time ago, when I was a child, someone claimed to me that there was not only those three, but also as fourth option the well. The well wins against rock and scissors (because both fall into it) but loses against paper (because the paper cov... |
H: Show $|X\times X| =$ cardinality of set of all functions $2\subseteq \omega \to X.$
Show that the Cartesian product of $X\times X$ has the same cardinality of the set of all functions from the set $2 \subseteq \omega$ to the set $X.$
I wonder what strategy should work for this problem.
For example, how can I find ... |
H: Zeta function and probability
I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function)
But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the probability that $n$ numbers chossen at random are relatively prime. But why? Can you give a... |
H: Classifying groups of order 12.
I was trying to classify groups of order 12 and I ended up with 5 different groups:
$\bullet$ $\Bbb{Z}_{12}$
$\bullet$ $\Bbb{Z}_2 \times \Bbb{Z}_6$
$\bullet$ $(\Bbb{Z}_2 \times \Bbb{Z}_2) \rtimes_{\alpha} \Bbb{Z}_3$ where $\alpha: (1,1) \rightarrow \bar{-1}$
$\bullet$ $(\Bbb{Z}_2 \t... |
H: Simple projector problem
Please, consider this ("sub")problem:
Let $S$ a two-dimensional subspace of a Hilbert $H$ and let $Q\in\mathcal{L}(S,S)$, $Q\neq 0$ and $Q\neq I$, such that $Q^2 = Q$. Show that $\mbox{Im}(Q)\oplus\mbox{Im}(I-Q)$ and there's $p,q,r,s\in S$, no zero, such that $\langle p,q\rangle = \langle... |
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