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H: contradicting identity theorem?
the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$
Let $f(z) = \sin z + \cos z$,
$g(z) = \cos z$
then $(f-g)(0) = 0$
$\implies$ by the above theore... |
H: Solving recurrence relation, $a_n=6a_{n-1} - 5a_{n-2} + 1$
I'm trying to solve this recurrence relation:
$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$
I calculated generator function as:
$$
A = \frac{31x - 24x^2}{... |
H: Isomorphism $\left(\mathbb{C}^{n}\setminus\{0\}\right)/\mathbb{Z}$ with $S^{1} \times S^{2n-1}$
I have to prove that there is this isomorphism:
$$\frac{\mathbb{C}^{n}{\setminus\{0\}}}{ \mathbb{Z}} \simeq S^{1} \times S^{2n-1},$$
where there is this equivalence relation in the left side:
$(w_1, \dots, w_n) \sim (z_1... |
H: Quick way to solve computational congruences
The specific problem at hand is $$34x \equiv 60 \bmod{98}$$
I reduced to get $$17x \equiv 30 \bmod{49}$$
and from this I have
$$17x \equiv 30 \bmod{7}$$
which is easy to solve and yields $x \equiv 3 \bmod{7}$. How can I use this fact to help me solve the more complex eq... |
H: Expected value of function of negative binomial
With $X$ representing the total number of trials, and m the fixed number of successes. The pdf is then
$f(x|p)=$${x-1}\choose{m-1}$$p^m (1-p)^{x-m} \ \ \ \ x \ge m$
As a step in something else I'm trying to find $E[\frac{1}{X-1}]$.
Here's what I've got up to now:
$... |
H: Expected value of $|H-T|$ in $n$ coin flips
Let $H_n$ be the number of heads in $n$ coin flips.
Let $T_n$ be the number of tails in $n$ coins flips.
Is there a good way to calculate $E_n = E[|H-T|]$ that isn't brute force computation, i.e. directly evaluating $$E[|H-T|] = \frac{1}{2^n} \sum {n \choose r} |n-2r|.... |
H: Partitions of $n$: proving $p(n+2)+ p(n) \geq 2p(n+1)$
For $n \geq 2$ give an alternative description of $p(n) - p(n-1)$ as the number of partitions of $n$ which have a certain property.
I have done that part, it is fine. I have not included it here as I am not accustomed with LaTeX but can put it up if wanted. B... |
H: A question regarding random variables
Consider the average $Y$ of $n$ independent random variables, each uniformly distributed on $[a,b]$. Does that mean $Y= \sum_{i=1}^{n}Y_{i}$ or $Y= \dfrac{\sum_{i=1}^{n}Y_{i}}{n}$?
Thank you for your time.
AI: $\text{The sample $\color{red}{\text{mean}}/\text{average}$ always ... |
H: Notation minimum of a column vector
I'd like to know the notation to express the minimum of a column vector.
Is this notation correct?
\begin{equation}
\min
\left[\matrix{
\left|b_{n}-b_{n+1}\right| \cr
\left|\left(b_{n}+360\right)-b_{n+1}\right| \cr
\left|b_{n}-\left(b_{n+1}+360\right)\right| \cr
}\right]... |
H: The last 2 digits of $7^{7^{7^7}}$
What is the calculation way to find out the last $2$ digits of $7^{7^{7^7}}$? WolframAlpha shows $...43$.
AI: $7^2=49=50-1$
$\implies 7^4=(50-1)^2=50^2-2\cdot50\cdot1+1\equiv1\pmod {100}$
Alternatively, $100=4\cdot25, 7^2\equiv1\pmod 4$
and $7^2\equiv-1\pmod{25}\implies 7^4\equi... |
H: Lie subalgebra, Lie subgroup and membership
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$.
We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t \in \mathbb{R} $ .
I have seen a proof of this but I don't understand wh... |
H: bifurcation value
I tried to understand how to locate the bifurcation value for the one-parameter family. From my understanding the bifurcation value is the maximum or minimum point of a parabola, so I set the differential equals to zero to find the equilibrium points. From two equilibrium point, I'll find the midp... |
H: $\int_\Omega fd\mu_n\to\int_\Omega fd\mu,\ \forall\ f\in C_0(\Omega)$ implies $\mu_n(\Omega)\to \mu(\Omega)$?
Let $\Omega\subset\mathbb{R}^N$ be a bounded open smooth domain and $C_0(\Omega)$ the set of bounded continuous functions with compact support. It is know that $C_0(\Omega)^\star =M(\Omega)$, where $M(\Omeg... |
H: How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$?
How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ?
i know that its "1", but why?
How can i calculate this?
Thank you very very much =)
AI: Substituting $t = \log x$ you obtain
$$
\lim_{x \to 1^+} \log(x)^{\log(x)} = \lim_{t \to 0^+} t^t = \lim_{t \to 0^+} ... |
H: Disc with two points identified
Is a disc $D^2$ with two points on the boundary identified, same as $D^2 \vee D^2$ ? They both have boundary $S^1 \vee S^1$. I am confused because an exercise in Hatcher seems to ask the same question about the two spaces (i.e. to prove that there doesn't exist a retraction from each... |
H: Homeomorphism between open unit ball and $\mathbb R^n$
Let $B=\{x\in\mathbb R^n : ||x||<1\}$ the open unit ball with the subapce topology of $\mathbb R^n$. I want to show that $B^n\cong\mathbb R^n$ with the map $F(x)=\tan(\frac{\pi ||x||}{2})\frac{x}{||x||}$ for $x\not=0$ and $F(0)=0$
Well $\frac{x}{||x||}$ is more... |
H: how to calculate the real probability
Say I have a 50% probability of winning a game if I play against person 1, and a 50% probability of winning a game against person 2. I will play with both people, one after another. Before the matches, what was the probability that I would win at least 1 match? It has to be som... |
H: Is $y=|x^3|$ a smooth function?
Is this a smooth function? $y=|x^3|$
The graph of this function has no sharp cuts or corners, so I think it is a smooth function but someone told me that it's not.
AI: You can check that
$$
f'(x)=3x^2\operatorname{sign}(x)\\
f''(x)=6x\operatorname{sign}(x)\\
f'''(x)=6\operatorname{si... |
H: Wrong Reasoning about the problem of breaking a stick in $2$ points and build a triangle with the $3$ parts.
For homework I was asked to solve this classical problem "If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4" and ok, it must result $1/4$.... |
H: finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ X_t\ dW^Y_t \\
dX_t &= - \kappa X_t dt + \sigma dW^... |
H: Equivalent norms and isometries
Let $X$ be a vector space, $\|\cdot\|_1$ and $\|\cdot\|_2$ two equivalent norms on $X$. Under what further assumptions can we prove there is an isometry between $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$?
In particular, are $(\mathbb{R}^2,\|\cdot\|_1)$ and $(\mathbb{R}^2,\|\cdot\|_2)$ i... |
H: Without calculating limit directly show that it is equal to zero
$$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\frac{1}{3^n}=0$$
I am not really sure what it means by "without calculating limit" and I don't really have ideas how to do it.
AI: $\left(1+\dfrac 1 n\right)^n < e \,\,$so $\left(1+\dfrac 1 n... |
H: Inconsistent System of Linear Equations
Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C ∈ F^n$ such that
the system of linear equations $AX = C$ is inconsistent.
I thought it'd be clever to use this (underlined by my pencil... |
H: Generating primes from other primes
For a natural number $n$ let $M$ be an $n$ by $n$ matrix w/$0$'s on diagonal and natural numbers off diagonal and let $p_1, p_2, \dots, p_n$ be a set of prime numbers.
Note then, that
$$p=\sum_{i=1}^n (\pm) \prod_{j=1}^n p_i^{m_{ij}}$$
is not divisible by any of the primes $p_1,... |
H: Limit as N goes to Infinity
Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$
I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, $\frac{1}{n} \rightarrow 0$:
$$\therefore \lim_{n\rightarrow\infty}(1+0)^{n^2} = x$$
Apparent... |
H: Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$
Here is another infinite sum I need you help with:
$$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$
I was told it could be represented in terms of elementary functions and integers.
AI: Note that
$$\frac{2^{-n}}{2^{2^{-n}}-... |
H: Calculating $-\frac{4}{x^2}+\frac{1}{(x-1)^2}$
Hello everyone I have the following question.
I have the following fraction$$f(x)=-\frac{4}{x^2}+\frac{1}{(x-1)^2}$$
But how would I reduce it? I know I have to multiply the opposite numerator by denominator and I got:
\begin{gather*}
(x-1)^2(4)=4x^2-8x+4\\
1(x^2)=x^2
... |
H: Why is the free pro-c-group on an infinite set not the pro-c-completion of the free group?
The set-up is the following: $\mathfrak c$ is a collection of finite groups closed under subgroups, homomorphic images, and extensions. For any group $G$, the pro-$\mathfrak c$-completion $G(\mathfrak c)$ is defined as the to... |
H: Regular expressions, is it always true that (r+s)*=r*+s*?
I'm really confused about this, can some one please help me understand this better.
If r and s are regular expressions then is it always true that (r+s)=r+s*?
Are r and s sets and does the plus mean the union?
AI: Let $R$ be the set of words described by t... |
H: Commuting square of functors
Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective categories of presheaves with values in $\mathcal{E}$:
$$Lan_i =: i_!: [\mathcal{C}, \mathca... |
H: How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Short Version of the Question:
How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Long Version of the Question:
I'm currently attempting a Project Euler quest... |
H: Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$
I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my homework for the comparison test before the othe... |
H: Proving existence of unique polynomial satisfying some integral
I came across a problem that I was having some trouble with.
Fix a positive integer $n$. Let $f(x) \in C(R)$ be a (real valued) continuous function.
Show that there exists a unique polynomial $q(x) \in P_n(R)$, such that for every
polynomial $p(x) \in ... |
H: Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several exercises without any hints, one of th... |
H: Recursion relation for Euler numbers
I am trying to solve the following:
The Euler numbers $E_n$ are defined by the power series expansion
$$\frac{1}{\cos z}=\sum_{n=0}^\infty \frac{E_n}{n!}z^n\text{ for }|z|<\pi/2$$
(a) Show that $E_n=0$ when $n$ is an odd integer.
(b) Establish a recursion relation for the seq... |
H: Calculus II, Curve length question.
Find the length of the curve
$x= \int_0^y\sqrt{\sec ^4(3 t)-1}dt, \quad 0\le y\le 9$
A bit stumped, without the 'y' in the upper limit it'd make a lot more sense to me.
Advice or solutions with explanation would be very appreciated.
AI: $$\frac{dx}{dy} = \sqrt{\sec^4{3 y}-1}$$
Ar... |
H: Annulus Theorem
I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is
Use the Schoenflies theorem (every topological imbedding $S^1 \rightarrow \mathbb{R}^2$ is the boundary of a 2-disk) to prove the an... |
H: Arc length parameter s
Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$
Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$
for $-\frac{\pi}{2}\leq\theta\leq \frac{\pi}{2}$. Find the arclength parameter s. How would I go about doing this? Thank you in advance!
AI: Si... |
H: Perfect squares using 20 1's, 20 2's and 20 3's.
How many perfect squares can be formed using 20 1's, 20 2's and 20 3's. This is a recent exam question, which I had no clue how to solve?
There is some kind of trick here, since time allotted to solve it was just 4 minutes.
AI: Hint: The resulting number is divisi... |
H: Linearly independent subset?
If $u,v,w$ and $z$ are distinct elements in $R$, then
{$(1,u,u^2,u^3), (1,v,v^2,v^3), (1,w,w^2,w^3), (1,z,z^2,z^3)$} is a linearly independent subset of $R^4$.
Is that true or false?
I even can't start the first step..
AI: If they are linearly dependent, then the equation
$$
\underbra... |
H: Is it a standard to say that $a \oplus a_{\small 1}=0$ or $a \veebar a_{\small 1}=0$?
I am trying to express the following:
$a$ or $a_{\small 1}=0$ but only one of them equals zero.
so if $a=0$ then $a_{\small 1}\neq 0$ and if $a\neq 0$ then $a_{\small 1}=0$.
And I'm looking for a standard way to say that, from ... |
H: Curvature of a non-compact complete surface
Assume $\Sigma$ is a non compact, complete surface. Assume the integral $$\int_{\Sigma}K$$ is convergent, where K is the Gauss curvature of $\Sigma$. Is it always true that $$\frac{1}{2\pi}\int_{\Sigma}K$$ is an integer?
AI: No. Start with a one-nappe circular cone. Cut o... |
H: Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.
Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer.
Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$
My initial thought was to try and induct on $n$, but the negative powers annoyed be badly.
I still think that induction is the... |
H: Inverse and derivative of a function
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
AI: We find a function defined for all positive $x$, with the right properties.
There may be no general theory, so let's fool... |
H: Homomorphism from $\mathbb{Z}/n\mathbb{Z}$
Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
AI: Suppose there is a homomorphism $f: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}$. What could $f(1)$ be? Let's call it $a$. Then $f(2) ... |
H: Systems of Differential Equations and higher order Differential Equations.
I've seen how one can transform a higher order ordinary differential equation into a system of first-order differential equations, but I haven't been able to find the converse. Is it true that one can transform any system into a higher-order... |
H: Trigonometric equality $x = 99 \sin (\pi x)$
Find the number of real solutions of $\displaystyle x = 99 \sin (\pi x)$.
I am getting stuck in some trigonometric relations.
AI: There is an obvious root at $x=0$. Since $\sin(-\pi x)=-\sin(\pi x)$, there are just as many negative solutions as there are positive soluti... |
H: Schema of separation and set of all sets
The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$.
The notes I'm reading say that from this we can conclude that the set of all sets doesn't exist, by applying th... |
H: Number of real roots of $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$
Prove the following, without induction. Is it possible?
The equation $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$ has no real root if $n$ is even.
And if $n$ is odd, it has only one real root.
I also tried searching the proofs many times with search key w... |
H: power of a number in a factorial
what is the largest power of 24 in 150! ?
HINT : answer is 48
I need to know the method for solving such questions when the highest power of the number to be found is non-prime ..
AI: You have to compute the highest power of each prime factor
of the number you want
(for 24, these ... |
H: How to calculate $\sum_{i=2}^n {\frac 1{\log_2 i}}$
How to calculate
$$\sum_{i=2}^n {\frac 1{\log_2 i}}$$
AI: If you meant $$\sum_{i=2}^n \dfrac1{\log_2(i)},$$ then there is no closed form; though you can compute its asymptotic using Euler-Maclaurin formula:
$$\sum_{k=2}^{n} \dfrac1{\log_2(k)} \sim \log 2\times\in... |
H: A basic doubt on the sojourn time of a CTMC
By using the memoryless property, I find the CDF of sojourn time of a CTMC as follows :
$$F_X(t) = 1-e^{-F_X'(0)t}$$
I am slightly confused about the intuitive meaning of the term $F_X'(0)$. How this term represents the rate of leaving that state.
AI: These are the basic... |
H: Making a cube root function analytic on $\mathbb{C}\backslash [1,3]$
I am still not convinced by the post that the function$$\sqrt[3]{(z-1)(z-2)(z-3)}$$ can be defined so it is analytic on $\mathbb{C}\backslash [1,3]$. We define for each $z\in \mathbb{C}\backslash (-\infty,3]$ the function $$f(z)=\int_4^z \frac{((z... |
H: Proving an equality
Let $f(n) = n^ {\log n}$. Let $p(n)$ and $q(n) \geq n$ be polynomials. I want to show that for
$n$ sufficiently large $f (n)$ satisfies
$$p(n) < f (n) < 2^{q(n)}$$
starting from the above inequality doesn't yield any satisfying result.
AI: Since $p(n)$ is a polynomial,
$p(n) < c n^d$ where $d$ i... |
H: Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$
Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
$$x^6+x^4+x^2+1=(x^2+1)(x^4+1)$$
Let $f_n(x)=x^{2n}+x^{2n-2}+\cdot... |
H: Infinitely many primes of the form $4n+3$
I've found at least 3 other posts$^*$ regarding this theorem, but the posts don't address the issues that I have.
Below is a proof that for infinitely many primes of the form $4n+3$, there's a few questions I have in the proof which I'll mark accordingly.
Proof: Suppose the... |
H: How do you graph $x + y + z = 1$ without using graphing devices?
How can I graph $x + y + z = 1$ without using graphing devices?
I equal $z = 0$ to find the graph on the xy plane. So I got a line, $y = 1-x$
But when I equal 0 for either the $x$ or the $y,$ I get $z = 1-y$ or $z = 1-x$ , and those are two different ... |
H: Differentiation Operator a Contraction Mapping
Let $C^{\infty}[a,b]$ be the space of all infinitely differentiable functions on [a,b] with norm $$ || f || = \max _{[0,1]} | f(x) | , f \in C^{\infty}[a,b]$$
Is the differentiation operator $\frac{d}{dx}$ a contraction mapping on $C^{\infty}[a,b]$?
I'm confused. Opera... |
H: Finitely additive measure on $\mathbb R$
Suppose $\mathcal B$ is the Borel $\sigma$-algebra on $\mathbb R$. Let $\mu : \mathcal B \rightarrow [0, \infty ]$ be a finitely additive(but not necessarily countably additive), translation-invariant, ''measure'', with the property that $\mu(K) < \infty$ for every compact $... |
H: Find the greatest integer $k$ for which $1991^k$ divides $1990^{{1991}^{1992}}+1992^{{1991}^{1990}}$
Find the greatest integer $k$ for which $1991^k$ divides $$1990^{{1991}^{1992}}+1992^{{1991}^{1990}}$$
It is easy to see that $k \geq 1$ as $1990 \equiv -1$ and $1992 \equiv 1 \pmod{1991}$
Also, I thought that perha... |
H: Integral of fractional expression $\int^3_0 \frac{dx}{1+\sqrt{x+1}}$
I want to solve this integral and think about call $\sqrt{x+1} = t \rightarrow t^2 = x+1$
$$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$
Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your suggestions.
Thanks.
AI: The substitution is a good... |
H: Trace of an Inverse Matrix
I want to know if there is a way to simplify, or a closed form solution of $tr(\Sigma^{-1})$ where $\Sigma$ is a symmetric positive definite matrix.
AI: Let $A$ be symmetric positive definite matrix hence $\exists$ a diagonal matrix $D$ whose diagonal entries are nonzero and $A=P D P^{-1}... |
H: If $A$ is invertible show that $\det(A) \neq 0$
If $A \in M_{n\times n}(F)$ is not invertible then the rank of $A$ is less than $n$,
thus $\det(A) =0$.
I proved that way, but looks like too simple so I think maybe there is a trick that I missed.
Or I proved it right?
Modification:
($\Rightarrow$) If $A$ is inver... |
H: Integrate over the region bounded by two regions. Using Polar coordinates.
Using Polar Coordinates integrate over the region bounded by the two circles:
$$x^2+y^2=4$$
$$x^2+y^2=1$$
Evaluate the integral of $\int\int3x+8y^2 dx$
So what I did was said that as $x^2+y^2=4$ and $x^2+y^2=1$
That $1 \le r \le 2$. And as t... |
H: Find limits for functions of natural numbers
So I am dealing with some problems about random graphs where we find limits of functions of natural numbers. A simple example can be the limit of $\ln(n)$ as $n \rightarrow \infty$, where $n$ is the number of vertices of a graph. My concern is how do we intepret this typ... |
H: Easy question about $H_0^1$ space
I have some trouble with proper understanding of $H_0^1(0,1)$ space. Consider the following space $$H_D = \{u\in H^1(0,1): u(0) = u(1) = 0\}.$$ What can we say about the connection between $H_D$ and $H^1_0(0,1)$. Is $H_D$ in $H^1_0(0,1)$? In literature stays, that functions in $H_0... |
H: How to express the sum of a set?
Suppose I have a set of numbers. How can I express in set-theory terms the sum of the elements in that set?
AI: There is no set-theoretic notation for that since a sum of numbers (presumably real or complex numbers, but more generally could be elements in a ring) is not a set-theore... |
H: Probability that subsets intersect
Given a set $N$ I would like to calculate the probability that two arbitrarily chosen and equally likely subsets $K\subseteq N$ and $J\subseteq N$ both of fixed size intersect. Let's say $n=\#N$, $k=\#K$, and $j=\#J$ denote the number of elements of the respective sets. The probab... |
H: Uniqueness of "Punctured" Tubular Neighborhoods (?)
Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo)
$f : \mathbb{R}^{n-1} \times [0, \infty) \rightarrow \mathbb{R... |
H: $\int^1_0 \frac{xdx}{x^2+2x+1}$
I need some suggestion how to solve this integral.
$$\int^1_0 \frac{xdx}{x^2+2x+1}$$
I think about to do the following step :
$$\frac{1}{2}\int^1_0\frac{2x+2-2dx}{x^2+2x+1}$$$$ t=x^2+2x+1 \rightarrow 2x+2dx=dt$$
then the integral will be : $$\frac{1}{2}\int^1_0 \frac{-2dt}{t}$$
its a... |
H: If an integer $n$ is such that $7n$ is the form $a^2 + 3b^2$, prove that $n$ is also of that form.
If an integer $n$ is such that $7n$ is the form $a^2 + 3b^2$, prove that $n$ is also of that form.
I thought that looking at quad residues mod $7$ might??? help. But that didn't take me anywhere so apart from that I'm... |
H: Is uncountable subset of separable space separable?
I have to prove that any uncountable $B\subseteq \mathbb{R}$, where $(\mathbb{R},\epsilon^1)$ is euclidean topology and topology on B is relative, is separable. And I know it's true because every subset of separable metric space is separable.
But what if we are g... |
H: Showing that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ and $w$ iff $Re[z\bar{w}]=0$
I'm reading Beardon's Algebra and Geometry.
Suppose that $zw\neq0$. Show that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ to $w$ if and only if $Re[z\bar{w}]=0$.
From... |
H: Do Boolean rings always have a unit element?
Let $(B, +, \cdot)$ be a non-trivial ring with the property that every $x \in B$ satisfies $x \cdot x = x$. How does one prove that such a ring $(B, +, \cdot)$ must have a unit element $1_B$? (Or, in case this is not true in general, what is a counterexample?)
BTW, I'm... |
H: Can you use modulus to make 0 > 2?
I wanted to create a rock-paper-scissors game that didn't use a lot of conditionals, and I was wondering if there were any mathematical way of representing the cycle of rock-paper-scissors. So Rock beats Paper beats Scissors beats Rock, or Rock > Paper > Scissors > Rock. If you as... |
H: Trigonometry Addition Thereom
Using the expansion of
a. $\sin(+)$, prove that $\sin75°=\sqrt 6+\sqrt{24}$
b. $\sin(+)$, prove that $\tan75°=2+\sqrt 3$
Where to start? draw up triangle of sin 75? find other values? help please.
AI: I suggest, you start by listing all angles of which you know exact values for $\sin$ ... |
H: Understanding big O notation
I'm not a mathematician by any stretch and I'm trying to translate some maths terms into simple maths terms. Please don't laugh, I do consider this complicated!
The equations in question are
O(n) and O(n ^ 2)
Now, I have read up on Wiki about this but it has been written (IMO) for peo... |
H: Free online mathematical software
What are the best free user-friendly alternatives to Mathematica and Maple available online?
I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, even though the calculation time there was limited to $60$ seconds.
Very basic com... |
H: Showing that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}[\exp[tX]]$
The problem is to show that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}(\exp[tX])$ given $\exp(tX)<\infty$ for $t\in \mathbb{R}$ where $X$ is a random variable.
Then to show that $\mathbb{P}[X\geq a]\leq \inf\limits_{t\in\mathbb{R}} \exp[-ta]\mat... |
H: Quadratic form positive semidefinite if limits in every direction are nonnegative?
Let
$$q(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in \mathbb{R}.$$
be a quadratic form with real coefficients.
Suppose that the limit is nonnegative in every direction. That is, for any unit vector $u$,
$$ \li... |
H: Limit of a continued fraction
Given the continued fraction:
$$f(x,N)=\left[2,3,4,...N,x\right]$$
$$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$
is it possible to find an expression for the integral:
$$g(x,N)=\int f(x,N)dx$$
as function of $N$ and $x$?
Thanks.
AI: UPDATE
Let's observe... |
H: Integral of $\int(4-2x)^\frac{1}{3}dx$
I solved this integral then I did $\frac{d}{dx}$ of $F(x)$ and saw that it's not the same, so I did something wrong in my integration process.
$$\int(4-2x)^\frac{1}{3}dx$$
What I did is $$F(x) =\frac{(4-2x)^\frac{4}{3}}{-2\times\frac{4}{3}} + C$$
Thanks!
AI: Hint:
use this fo... |
H: Binary vs. Ternary Goldbach Conjecture
Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current techniques? As a non-expert it seems to me that these conjectures should be equally har... |
H: Why do the French count so strangely?
Today I've heard a talk about division rules. The lecturer stated that base 12 has a lot of division rules and was therefore commonly used in trade.
English and German name their numbers like they count (with 11 and 12 as exception), but not French:
# | English | ... |
H: Binomial Coefficients Combinatorics
For a positive integers n, prove that
$$\displaystyle\sum\limits_{v=0}^n \frac{(2n)!}{(v!)^2 ((n-v)!)^2} = \binom{2n}{n}^2.$$
If somebody could please help me with this question, I would greatly appreciated it.
AI: A combinatorial proof. The right-hand side corresponds to choosin... |
H: Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$
I am trying to solve this integral and I need your suggestions.
I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now.
$$\int^1_0 \frac{dx}{1+e^{2x}}$$
Thanks!
AI: With the change of variable $u=e^{x}$, you get
$$
\int_{[0,1]}\frac{dx}... |
H: In how many ways can we distribute $m$ balls to $n$ baskets when $n > m$
Given $m$ balls and $n$ baskets, in how many ways can we distribute the $m$ balls to the $n$ baskets when given that $n > m$ (the number of baskets is greater than the number of balls)?
So I said, that it does not differ if $n>m$ the solutio... |
H: Trigonometry Addition Thereom With Only one exact value?
Use the expression of $\sin(A+B)$ to evaluate $\sin 195$.
Do I use one exact value like $45+150$ or $60$ or is there another way?
AI: Here you go: $\sin 195° = -\sin 15° = -\sin(45-30)°$. After that you use standard formulas. |
H: hyperbola: equation for tangent lines and normal lines
Find the equations for
(a) the tangent lines, and
(b) the normal lines,
to the hyperbola
$y^2/4 - x^2/2 = 1$ when $x = 4$.
AI: If $x=4, \frac{y^2}4=1+\frac{4^2}2=9\implies y=\pm6$
Using Article 305 of this, the tangent of $$\frac{y^2}4-\frac{x^2}2=1$$
at $(h,k)... |
H: Scale free property of Pareto distribution
I am trying to show that the Pareto distribution is scale free, defined as:
p(bx) = g(b)p(x)
I get to this stage:
x dp/dx = p'(1)/p(1) p(x)
I have a rough solution which simply jumps to the next step:
ln(p(x)) = p(1)/p'(1) ln(x) + ln(p(1))
Can anyone explain?
AI: From the ... |
H: How to find the minimum of $x+y^2+z^3$?
let $x,y,z>0$, and $x+3y+z=9$, find the minimum of
$$x+y^2+z^3$$
I think this problem is very interesting. I have found this
when $$x=\dfrac{9}{2}-\dfrac{1}{\sqrt{3}},y=\dfrac{3}{2},z=\dfrac{1}{\sqrt{3}}$$
I belive this inequality have $AM-GM$ methods,becasue I have see thi... |
H: How to show $T$ is diagonalizable?
Let $T\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be linear with distinct eigenvalues $\lambda_1, \lambda_2, \lambda_3$. Show that $T$ is diagonalizable.
It seems as if this is a very simple question but I'm a bit confused about how to start!
AI: You need to prove that $\math... |
H: On the no trivial $3$-tuples $(p, q, \alpha) \in \mathbb{N}^3$ such that $\sum_{k = 1}^{n}k^p =\Big [\sum_{k=1}^{n}k^q\Big]^\alpha $.
It is well known that $\sum_{k = 1}^{n}k^3 =\Big [\sum_{k=1}^{n}k^1\Big]^2$. My question is very simple.
There are $3$-tuples $(p, q, \alpha) \in
\mathbb{N}\times\mathbb{N}\time... |
H: Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP
I am sorting some easy questions for the students in Group Theory I. One of them is:
Is $(\mathbb Z_{14},+)$ isomorphic to a subgroup of $(\mathbb Z_{35},+)$? What about $(\mathbb Z_{56},+)$?
I know the first claim i... |
H: The Subspace $M=\{f\in C[0,1]:f(0)=0\}$
Let $C[0,1]$ with the supremum norm. It's easy to see that $M=\{f\in C[0,1]:f(0)=0\}$ is a closed subspace and so $C[0,1]/M$ is a Banach space. But I'm having trouble in finding a Banach space isometric to $C[0,1]/M$.
AI: For a given $a\in[0,1]$ consider linear bounded funct... |
H: $\mathbb{Q}/\mathbb{Z}$ has cyclic subgroup of every positive integer $n$?
I would like to know whether $(\mathbb{Q}/\mathbb{Z},+)$ has
$1$. Cyclic subgroup of every positive integer $n$?
$2$. Yes, unique one.
$3$. Yes, but not necessarily unique one.
$4$. Does not have cyclic subgroup of every positive integer $n$... |
H: $\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when
$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when,
$1.|z|\le 3$
$2. |z|<\sqrt{3}$
$3.|z-1|<\sqrt{3}$
$4.|z-1|\le \sqrt{3}$
The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number $$C=\lim\s... |
H: Prove using an example that there is no plane on $\mathbb{R}^3$ that contains every group of 4 points
Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere):
The exercise is as follows:
i) Find the equation of the plane of $\mathbb{R}^3$ that contains t... |
H: Sequence of the ratio of two successive terms of a sequence
If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit:
$$
\lim_{n}\frac{a_{n+1}}{a_n}
$$
exists?
When does the limit is strictly greater than $0$ (it can be zer... |
H: Relation between continuity and connection between topologies
If $X$ is a set and $\tau_1,\tau_2$ two topologies on $X$. What does it mean to put the continuity of the identic map on $X$ (i.e $id_X(x)=x\forall x\in X$) in a relation to the comparability fo two topologies (in our case $\tau_1,\tau_2$)
AI: If the ide... |
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