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Prove that:
$n^n (\frac{n+1}{2})^{2n}$
Greater than or equal to $(\frac{n+1}{2})^3$
Greater than or equal to $(n!)^3$
Here $n\in \mathbb{N}$
How to prove this? It seems too complicated. Please suggest as how to initiate the solution?
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Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if $rs\in I \space \forall r\in R,s\in I$ part...
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This question arose through a response to this post.
For which integers $N>1$ does the fraction $\frac 1N$ appear in the Egyptian Fraction expansion of $\frac {N-1}{N}$?
To specify: As such expansions are not unique, I should say which one I refer to. Here we consider the expansion obtained through the greedy algorithm... |
Find:
$$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$
The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So if we could apply Lebesgue's monotone convergence theorem, we have:
\begin{align}
\lim_{n \rightarrow \infty} \int_0^n \le... |
Given a concrete mathematical statement, such as BSD conjecture(https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture), do we know if it is provable?
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Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$
It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
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Problem:
Show that $a_n$ is convergent sequence and find a limit of $a_n$.
$$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$
I tried to look at this as normal limit problem so I wrote this:
$$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\lim_{n \to \infty}(\frac{1}{\sqrt{1+1}})=\frac{1}{2}$$
But I didn... |
let $S: V \to V$ linear transformation in a inner product space of $\mathbb{C}$.
Prove that $S$ is normal iff $$\|S(x)\| = \|S^*(x)\|$$
That's what I have done so far:
if $S$ is normal than $$SS^* = S^*S \implies \|S(X)\|^2 = (S(x),S(x)) = (x, S^*S(x)) = (x,SS^*(x)) = (S^*(x), S^*(x)) = \|S^*(x)\|^2$$
Thats for the fir... |
Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of $\pi/m$ between their perpendicular hyperplanes $H_\alpha$ and $H_\beta$, where $m \geq 2$. I need to show that $s_\alp... |
Suppose $x$ and $y$ are some integers satisfying $$x^2-16=y^3.$$ I'm trying to show that $x+4$ and $x-4$ are both perfect cubes.
I know that the greatest common divisor of $x+4$ and $x-4$ must divide $8$, but I don't know where to go from there. Would anyone be able to help?
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Let $G$ be a group whose representation is $G = \left\langle x , y \mid x^5 = y^2 = e , x^2y = yx \right\rangle$ . Then $G$ is isomorphic to
a) $\mathbb Z_5$
b) $\mathbb Z_2$
c) $\mathbb Z_{10}$
d) $\mathbb Z_{30}$
I think order of $G$ is 10, but how to show that $G$ is cyclic.
any help would be appreciated, Thank you... |
Let $E \subset \mathbb{C}$. Show that the sets of isolated point of
$E$ is a countable set.
That question is related to this question. However, my question somewhat different.
Define $Acc(E)^c$ the set of isolated points of $E$. Then there exists an open set in $V \subset \mathbb{C}$ for each point $z \in Acc(E)^c$... |
Assume that $\sqrt 2$ exists, and let $a=\sqrt 2$. Prove that there exists a number $c > 0$ such that for all integers $q,p$, and $q\neq0$ we have $$\lvert q\alpha - p \rvert \gt \frac cq $$
Note: The same $c$ should work for all $q,p$. Try rationalizing $q\alpha - q$, i.e. take the product $(q\alpha - p)(q\alpha + p)$... |
What does it mean for two triangles to be equally oriented?
I have heard this term a lot but I haven't seen a definition of it. I know that in $3$-space two triangles are considered to be equally oriented if we can slide one on top of the other with the vertices aligning if they are similar or congruent. In $2$-space... |
To do this problem, I wrote this matrix in RREF form and found that $V_3$ is $-1V_1 + 2V_2$. This demonstrates that these planes are a basis for $ \mathbb R^2$. However, I am not sure to extend that conclusion to show $ \mathbb R^2$ is a subspace of $ \mathbb R^3$. Any ideas how to prove this?
I think I need to demon... |
Chinese remainder theorem dictates that there is a unique solution if the congruence have coprime modulus.
However, what if they are not coprime, and you can't simplify further?
E.g. If I have to solve the following 5 congruence equations
$x=1 \pmod 2$
$x=1 \pmod 3$
$x=1 \pmod 4$
$x=1 \pmod 5$
$x=1\pmod 6$
as gcd (2,3,... |
If E is a measurable and A is any subset of E, show that $|E|=|A|_i+|E-A|_e$
where |E| is the measure of of E, $|A|_i$ is the inner measure of A, and $|E-A|_e$ is the outer measure of $E-A$.
I have proved $|E|\geq|A|_i+|E-A|_e$, but I can't seem to figure out $\leq$ part of the proof.
I just need hints. Please don't ... |
I came across this quiet new to me way of defining "tensors",
That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n \times m_p} \rightarrow \mathbb{R}^{m_1 \times m_2 \times .. m_p}$ such that given matrices $V_i \in \mathbb{R}^{n \times ... |
Prove that for every $x>0$:
$$\ln x \ge \frac{x-1}{x}$$
What I did:
$$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$
$$f(1) = g(1) = 0 $$
So it's enough to prove that $$ f'(x) \ge g'(x)$$
I proved this for every $x>1$.
How can I prove this for every $0<x\le1$ ?
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Where am I going wrong in solving this inequality?
$$\frac{p-\sqrt{9p-20}}{p-5}<2$$
On cross multiplying and squaring to remove the square root,I get the inequation
$p^2-29p+120<0$
Which gives $5<p<24$.
This is the correct answer http://www.wolframalpha.com/input/?i=solve+(p-root(9p-20))%2F(p-5)%3C2
But that is wrong a... |
$$a^2+ab+b^2\ge 3(a+b-1)$$
$a,b$ are real numbers
using $AM\ge GM$
I proved that
$$a^2+b^2+ab\ge 3ab$$
$$(a^2+b^2+ab)/3\ge 3ab$$
how do I prove that $$3ab\ge 3(a+b-1)$$
if I'm able to prove the above inequality then i'll be done
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Given the random vector $(X,Y)$ with joint probability $P(0,1)=\frac{1}{18}$, $P(1,2)=\frac{3}{18}$, $P(1,4)=\frac{5}{18}$, $P(2,0)=\frac{2}{18}$, $P(2,1)=\frac{4}{18}$, $P(2,3)=\frac{3}{18}$ and $0$ otherwise, I need to find:
Domain, probability and distribution for $(Y|X=2)$
Domain and probability for $Z=X+Y$
1) ... |
I am trying to find a polynomial time reduction from the colored graph isomorphism to the regular graph isomorphism.
Doing a search on this problem, I found this article and it seems like theorem 1 is the solution I am looking for, however I don't fully understand it.
Could someone please explain how the reduction work... |
I need help with solving one of the questions the teacher gave us to prepare for an upcoming exam. I tried solving it but with no luck. Here is the question:
On one shelf there are 5 hardcover books and 6 paperbacks and on the other shelf there are 7 hardcover and 4 paperback. From the first shelf we pick two books ra... |
I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried:
$$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi i}}2\right)^{\pi}$$
$$=\frac{(e^{\pi i}+e^{-\pi i})^{\pi}}{2^{\pi}}$$
$$(2\cos(\pi))^{\pi}=S$$
$$S_1=\sum_{n=0}^{\infty}\frac{\pi!e^{\pi^... |
Define a locally lipschitz and nonnegative function $f\colon\mathbb{R}^n\to\mathbb{R}$. Let $M\in\mathbb{R}^{n\times n}$ and $\eta>0\in\mathbb{R}$. Consider the function $h\colon\mathbb{R}^n\to\mathbb{R}^n$ defined as
$$h(\mathbf{x})=
\begin{cases}
\frac{1}{\Vert M\mathbf{x}\Vert}M\mathbf{x}, &\text{if }f(\mathbf{x})\... |
Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$)
I want to prove this using the Betweenness property of $\Bbb R$, that is, $\forall x,y \in \Bbb R$, if $x\lt y$ then $\exists \omega \in \Bbb Q$ such that $x\lt \o... |
I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar multiplication will effect part c?
If you could help me by explaining how to approach each part that would be lovely.Thanks... |
I feel like I understand the construction of dependent product types relatively well, it makes sense to me how the introduction and elimination rules work together to create the concept of a function whose co-domain depends on the values of the function in the domain, but given the definitions and motivation on n-cat l... |
Prove that if $K$ is a normal subgroup of $G$, $K$ has order $q$, where $q$ is prime, $G$ has order $pq$ where $p$ is a prime and less than $q$, and $Z(G) = {1}$, then the Centralizer of $K$ in $G$ is equal to $K$.
It is clear that either the order of the Centralizer is $q$ or $pq$, but I don't know how to show that th... |
Find:
$$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{\mathrm dx}{x^2 \log{(1+ \frac{x}{n})}}$$
The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ \frac{x}{n})}}{\frac{x}{n}}}$ converges pointwise to $\frac{1}{x^3}$. So if we could apply Lebesgue's Dominated Convergence Th... |
This concerns the characteristic polynomial of a matrix.
http://www.math.umn.edu/~olver/num_/lnv.pdf p. 7 (or p. 92).
every term is prescribed by a permutation π
of the rows of the
matrix
identity permutation is obtained by multiplying the diagonal entries together
What is the identity permutation?
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I don't understand the example here.
If we have that $\alpha=0.04$, $k=2$ then $[L_1,U_1]$ is a $1-\frac{0.04}{2}=0.98$ confidence interval for $\theta_1$ which is correct, however $[L_2,U_2]$ must be a $1-\frac{0.04}{2}=0.98$ confidence interval for $\theta_2$, which is incorrect as $0.98>0.97$
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I have proved this expression but I want to prove that they both are not equal.
$$(A \times B ) \cup ( C \times D ) \subset ( A \cup C ) \times ( B \cup D )$$
May be I have to prove that $( A \cup C ) \times ( B \cup D )$ is not a subset of $(A \times B ) \cup ( C \times D )$.
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An operation $*$ is defined on the set $\Bbb{Z} \times \Bbb{Z}$, ie. the set containing all pairs of integers by:
$$
(u,v) * (x,y)=(u+x,v \cdot y)
$$
What are all the invertible elements of the monoid $(\Bbb{Z} \times \Bbb{Z}, *)$?
And if $(\Bbb{Z} \times \Bbb{Z}, *)$ is not a group what is an equation that does not ha... |
How does one determine the ideals of $\mathfrak{gl}_n(C)$?
My guess is that the only ones are $(0) $ and $\mathfrak{sl}_n(C)$.
I think approaching the problem by the fact that each $\mathfrak{g}^{
(i)}$ is an ideal of $\mathfrak{g}$ would be wise.
|
I wish to prove the exercise which states that for a set $A$ containing more than half elements of a group $G$, every element of $G$ is a product of two elements of $A$.
My attempt:
By Lagrange Theorem, subgroup generated by $A$ must coincide with $G$, so every element in $G$ is a product of elements from $A$. How can ... |
This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of the problem. Recall, there are many ways to implement the primitive. Two standard approaches are:
$$\eta_i = e_i^{T}y = e... |
Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin,
$$\tau = \inf \{ j \geq 0 \, : \, X^{-1}(j) = 0 \, \, \mbox{or}\, \, X^{1}(j) = 0 \}.$$
How much is $E[\tau]$, the expectation of $\tau$?
Comment: If th... |
I have a function defined as:
$f(z)=\frac{\Gamma{(z)}+1}{z}$
Are there any $z ∈ C$ (with nonzero imaginary part) such that $f(z)∈R$?
I tried substituting in $z=a+bi$ with $b≠0$ into the product and integral definitions of the Gamma function, but with little effect.
EDIT:
$f(z)=\frac{\Gamma{(z)}+1}{z}$
$f(a+bi)=\frac... |
Let $X$ be a nonnegative integer-valued random variable depending on a parameter $L$.
Assume that $E[X] \rightarrow \infty$ as $L\rightarrow \infty$.
Does this imply that for any $k$, $P(X > k)$ is going to 1 with $L$?
|
I currently am trying to create a three layer neural network and I want to use the back propagation.
My training data is:
Input : Output
0 | 1 : 0
1 | 1 : 2
In many articles that I've read, they use
1
f(x) = ---------
1 + e-input
to calculate the output of each neuron.
But this function can only retu... |
I am trying to solve the following problem
$$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\
u(0,t)=\frac{\partial u}{\partial x}(1,t)+u(1,t)=0,&t>0,\\
u(x,0)=1,&0<x<1.\end{matrix}\right.$$
Starting as usual with $u(x,t)=X(x)T(t)$, I get that $X$ has to verify
$$\left... |
Find $a$ so that the sum of all probabilities is $1$, and hence $P(k)$ is a
well-defined formula for a probability distribution.
|
A circular conductor, with cross section given by $(x-d)^2+y^2=b^2$, i.e. radius $b$ and centered on $x=d$, has a circular core, made up of the interior of the circle $x^2+y^2=a^2$, with $b-d>a$, removed. A total current $I$ flows in the region between these two circles, with a uniform density, $J$ per unit area. By c... |
I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises.
If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $a$ and graded for 5%, so it's supposed to be pretty easy.
Let $b,c>0$, $(b+c\neq 0)$, $I\subseteq\mathbb{R}$ a space. If $f:I\to\mathbb{... |
How to integrate
$$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$
where $a>0$
The real problem is this integral
$$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) dx$$
I tried integration by parts and then the change of variable $z=x^2$ but it does not work.
|
$$\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$$
I have multiply by $\frac{\sqrt{2x^2+2}+(x+1)}{\sqrt{2x^2+2}+(x+1)}$
and got: $$\lim_{x\to 1} \frac{|x-1|(\sqrt{2x^2+2}+(x+1))}{(2x^2+2)-(x+1)^2}=\lim_{x\to 1} \frac{|x-1|(\sqrt{2x^2+2}+(x+1))}{(x^2-2x+1)}=\lim_{x\to 1} \frac{|x-1|(\sqrt{2x^2+2}+(x+1))}{(x-1)^2}=\li... |
Say we role $n$ identical, fair dice, each with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides are numbered from $1$ to $d$ with no repeating number, as you would expect. So an ordinary $d$ sided die pool.
Every dice in the outcome that shows a number equal or higher tha... |
Find the total no. of divisors of $75600$ which of the type of $4n+2$
where $n\in \mathbb{N}$ and
$75600=2^4 \cdot 3^3 \cdot 5^2 \cdot 7^1$
Now I think divisors of type $(4n+2)$ should be of type
$2^1\cdot 3^a\cdot 5^b\cdot 7^c$. Am I correct?
|
Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$
I am trying to find an expression, possibly asymptotic, for the sum of the $a_i$'s for a given $\frac{p}{q}$.
I understand that this is related to the Stern-Brocot tree. In particular, our problem is e... |
How do I find the Lagrangian derivative of:
$$\frac{D}{Dt} \left(\frac{x}{y^{a}}\right) = 0$$
where a is a constant?
|
I've seen two definitions (or axioms?) of set equality:
$a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$
$a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$
That is, two sets are considered equal, if
They contain the same sets
They are contained in the same sets
ZFC seems to inclu... |
I have that $G=S_4$ and $N = \{1, (12)(34), (13)(24), (14)(23)\}$, and thus far I have shown that N is a normal subgroup of G. I'm trying to figure out what group $G/N$ will be isomorphic to, but I can't figure it out. Any ideas on what the group is, and how I'd proceed proving that is the case?
I think the subgroup wi... |
My text book uses the linearity of the expected value to compute it. It defines a random variable $X_i$ that indicates whether the urn $i$ contains $k$ balls or not. So the asked value is $E[X_1 + X_2 + ... + X_n]$.
What is not entirely clear for me is the following. It states that $P\{X_i=1\}={r \choose k}({{1}\over{n... |
Here is the definition of the Bessel function I am starting with a definition as an integral.
$$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$
Essentially we have computed the $n$-th Fourier coefficient of a certain function:
$$ e^{-ix \sin t} = \sum e^{int} J_n(x) $$
How can I show the integral... |
I have the following series
$$\sum_{n=2}^{\infty} \frac{n}{(n-1)^2+\alpha 2^n}$$
I have to find for which $\alpha$ this series converges.
I tried the ratio test but I get $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n} \right|=1$ and it tells me nothing of convergence of this series.
I also tried Raabe's test and I get... |
Does anyone know any resource of algorithms for evaluating the Bessel function of the first kind and second kind under interval-valued domains? In particular, I am interested in complex interval arithmetic. The only paper I have found is:
Zakovic, S.: Evaluation of Bessel-Ricatti Functions using Interval Arithmetic. Wo... |
Show that if there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$
I am unable to proceed any further in this and any help in this regard would be greatly appreciated.
|
Show that if $n$ is a power of $2$, say $n = 2^k$, then we have the equality $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$.
The first step is to prove $O(\lg^2n)$:
$$ \lg \frac{2^k}{2^0} + \lg \frac{2^k}{2^1} + \dots + \lg \frac{2^k}{2^k} \leq \lg^2 n.
$$
My trouble is trying to figure out what sequence of the s... |
Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of $k[x_0,x_1,...,x_n]_d$ as an affine space.
Of course, every such polynomial has a well defined zero-locus on a projective space $\... |
Let $C_o= \{(x_n):x_n \in R, x_n \rightarrow 0 \}$ and $ M=\{(x_n):x_n \in C_0,~~ x_1+ x_2+...+x_{10}=0\}$ then dimension of $(\frac{C_0}{M})$ is equal to
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I do not know how to prove this, can anybody help me out with that?
Consider five vectors:
$\vec{a},\vec{b},\vec{c}, \vec{p}, \vec{q} \in \mathbb{R}^3$
then:
$$(\vec{p}\cdot\vec{q})(\vec{a}\cdot(\vec{b}\times\vec{c}))
= (\vec{p}\cdot\vec{a})(\vec{q}\cdot(\vec{b}\times\vec{c}))
+ (\vec{p}\cdot\vec{b})(\vec{q}\cdot(\vec{... |
The question is in the title, but I have no idea how to solve it, so a few hints would be appreciated, thanks.
|
I've been stuck with this question for a while:
I've learnt how to use t-distribution to estimate CIs for an unknown variance, but I'm unsure how that applies to this situation.
Any help would be greatly appreciated.
Thank you!
|
If $ax+3y=5$ and $2x+by=3$ represent the same straight line, then what does a+b equal?
I've tried this,
$ax+3y=5$ and $2x+by=3$
Multiply to equal 15 so they equal each other
$$3ax+9y=10x+5by$$
Now what do I do?
How do I find the value of $a$ and $b$?
Thanks
|
Let $x_n$ be a convergent sequence converging to $x$
Then claim $x_n^2$ converges to $x^2$
I wish to use the definition to show this is the case.
Recall $x_n \to x$ iff $\forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $\forall n \geq N \Rightarrow |x_n - x| < \epsilon$
Then $|x_n^2 - x^2| = |x_n - x||x_n + x|... |
Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.
Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} \frac{\Re(z)}{z} = \lim_{r \to 0} (\cos ^2(\theta) - i \cos(\theta)\sin(\theta)) = (\cos ^2(\theta) - i \cos(\theta)\sin(\thet... |
To take a taxi in downtown St. Louis, it will cost you $3.00$ to go a mile. After $6$ miles, it will cost $5.25$. The cost varies linearly with the distance traveled.
|
I'm having trouble with the following exercise
$$ y'' +4y - (4/e^x) = 0 $$
with the initial values:
$$ y(0) = 1 y'(0)=5$$
I used the formula $$ y'' = s^2Y(s) − s*f(0) − f'(0)$$
and got to:
$$ Y(s) = \left(\frac{4 + 5 e^x+se^x}{s^2e^x+e^x}\right) $$
But now I'm stuck. Can't find anything remotely similar in Laplace tabl... |
Assume : $F(y)=G(w)$ where $F,G$ are two real-valued functions from $R \to R$.
We want to find the function $C(w)$ such that : $F'(y)=C(w)$ and C should be built based on F and G.
Thanks so much.
|
Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, where $G_{w_{i}}$ denotes the point stabiliser of $w_{i}$. I would like to show that in this case the two actions of $G$ ar... |
Let $G$ be a group, $H$ a subgroup of $G$, and $N:=\cap_{x\in G} \ \ x^{-1}Hx$. Prove, that $N$ is normal subgroup in $G$.
I did this:
Let $g\in G$.
Whether $g(xhx^{-1})g^{-1}\in N$?
Take $f=gx$.
$g(xhx^{-1})g^{-1}=(gx)h(x^{-1}g^{-1})=(gx)h(xg)^{-1}=fhf^{-1}\in N$, because $f\in G$, when $f\in G$ and $gx\in G$
So, $g(... |
Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow GL_2(\mathbb{F}_p)$. Prove that this representation is not the direct sum of irreducible representations.
As I approach this proof ... |
If $M_t$ is a martingale, is this process a martingale too ?
$X_t=\int_0^tY_sdM_s$ where $Y_t$ is some process that makes $\int_0^tY_sdM_s$ defined
If not, what about the case $Y_t=M_t$ ?
|
I am currently reading Rudin's RCA, and I have some questions about a particular definition he uses in chapter 2:
The following passage is taken from Rudin's RCA, page 47, section 2.15:
"A measure $\mu$ defined on the $\sigma$-algebra of all Borel sets in a locally compact Hausdorff space $X$ is called a $\textit{Borel... |
I have the following integration to solve.
$$f(k) = \int_0^{\pi/2} \sin^2\theta \sqrt{1-k^2\sin^2 \theta}d\theta,\quad0<k<1$$
assuming $\sin\theta = t$ which results $d\theta = \frac{dt}{\sqrt{1-t^2}}$
and when $\theta = 0, t=0$ and $\theta=\frac{\pi}{2},t=1$
so above equation can be rewritten as,
$$f(k) = \int_0^1{t^... |
I know the logic of proof sequence as I solved many proof problems, I now have one that has been taken my attention for a couple of days and as easy as it may look, I don't seem able to simplify the first hypothesis of the problem even I tried many ways.
So,the question is, How can I simplify the first hypothesis?
Give... |
Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists.
In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$.
Prove that $\lim\limits_{x\to\infty} f'(x)=0$
I think Rolle's theorem might help here, but I don't know how to use it.
Any hints are most welcome.
Thanks!
|
In my textbook they write the following inequality: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+...+\frac{1}{2^{k-1}}+...+\frac{1}{2^{n-1}}$
They say that they derive this inequality by viewing the expression $\left(1 + \frac{1}{n}\right)^n$ As a geometric series. But I am not sure how they come to that concl... |
Let $h = $"Peter is handsome", $c = $"Peter is clever", $o = $"Peter is optimistic". Rewrite the following in symbolic forms using $ \sim , \land, \lor $.
-Peter is neither handsome, clever nor optimistic.
answer: $\sim(h \lor c \lor o)$
Is this correct?
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I want to show $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision and can't quite figure out how to choose my subspaces.
For $Z \subset A \subset X$, excision gives the following isomorphism
$$ H_i(X -Z, A-Z) \cong H_i(X,A)$$
If $X=M$ is a manifold, I need $M-Z = \mathbb{R}^n$, so some ... |
I have come across two instances of "Coleman map"
Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in $\mathbb{Q}_p(\mu_{p^\infty})$ with Galois group $\Gamma = 1+p\mathbb{Z}_p \cong \mathbb{Z}_p$. Let $k_n$ be the $n$-th layer i... |
I have a linear algebra midterm tomorrow. While it's highly unlikely a question of this type shows up, I really wanted to understand this because I am curious since I've spent so long without coming up with an answer.
Let $a, b, c, d \in \mathbb{R}$. Prove that the set $$\bigg\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\big... |
We all knonw that the expansion of $e^x$ is $$1+x+x^2/2+...$$.
But what if I want to find more approximate expansion of $e^x$. I try that
$$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and substituting the values $x=1,2,3$ to get these constants. And I get more approximate results by solving 3 equations. Is ... |
is there any kind of general rules to follow or algorithm for trying to simplify something in conjunctive normal form?
Specifically, I'm trying:
(P or Q) and P and (Q or R) and (P or notP or R) and (notQ or R)
I got that the 4th one is always true because it has complementary literals so
(P or Q) and P and (Q or R) ... |
I have a question about this.
Here is the def of homotopy used by Conway, in case you guys need it.
Let $G$ be an open set in $\mathbb{C}$ and let $\gamma$ be a closed smooth rectifiable curve in $G$ such that $\gamma$ is homotopic to a constant curve,$\gamma_o$ ; furthermore let $ \Gamma(s,t): [0,1]\times[0,1] \to ... |
I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral,
\begin{equation}
c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx.
\end{equation}
I'm not sure if it is possible an analytical solution, does anybody know? Please note that's on an... |
This could be a very simple question to answer, but I'm unsure how to prove this.
If you have a diagonalisable matrix $A$, prove that $\chi_A(A)$ is the zero matrix.
(where $\chi_A(x)$ is the characteristic polynomial of $A$).
I can show easily that (assuming $A$ is an $n \times n$ matrix) that $\chi_A(A)v = 0$ for e... |
How to simplify the following equation:
$$\sin(2\arccos(x))$$
I am thinking about:
$$\arccos(x) = t$$
Then we have:
$$\sin(2t) = 2\sin(t)\cos(t)$$
But then how to proceed?
|
for example, let's say: 0xE5
assume the system is 8 -bit
in decimal it's = 229
and in Binary it's = 1110 0101
the Two's Complement rules said: sign-bit, which's the most left, indicates a negative value if it's "1", or if it's "0" the value is positive.
so, the value is positive, but the Binary representation indicates... |
I understand that the basis of a subspace defined by this equation requires you to find a combination of $x_1,x_2,x_3$ that satisfy this equation [so $(-1,0,2)$ for example]. But how do you know how many vectors are in the basis of this subspace? Couldn't a subspace of $\mathbf{R}^3$ be a line or a plane in $\mathbf{R... |
Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$
My Attempt
Am I correct in saying that $F$ is a conservative vector field therefore path independent. Using this property I can calculate the integral by parameterising the line segment from $(1,0)$ t... |
Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$.
Typically to combinatorially prove something we need to show that the LHS indeed counts the same thing as the RHS. I am having trouble doing so. One thing I noticed is that on the LHS, we are choosing from a total of $(n... |
I need to figure out how to express the NOT operator in terms of the operators AND, XOR, XNOR.
I need to show that this set is functionally complete, which I'm trying to do by showing that I can represent NOT and OR, a known complete set. But I have no idea where to begin when it comes to representing a unary operator ... |
Crossposed on Quantitative Finance SE
I was able to solve this problem and find the arbitrage but only after spending a long time on it and trying out different possibilites, is there a method or technique that can help me find the arbitrage faster and in a more efficient way rather than just trying out different poss... |
This question is from one of the steps of the Proof of Proposition B.1 in Appendix B of P. H. Rabinowitz's "Minimax Methods in Critical Point Theory with Applications to Differential Equations."
Let $u \in L^r(\Omega), r,s \ge 1,$ and $a_1,a_2 \ge 0$. How do you prove the following inequality? (We can choose $a_3$ to b... |
Let's call $A\subseteq\mathbb{N}$ dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$
Is the intersection of two dense sets dense again? Or does the intersection of two dense sets at least satisfy the weaker statement $$\text{lim sup}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1?$$
|
I have a set of two PDEs:
$$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$
$$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$
These can be combined into a wave equation of the form:
$$\partial_{\tau}^{2}\theta=\partial_{\eta}^{2}\left[\theta+\alpha\partial_{\tau}\theta\right]
$$
which with the... |
I'm working through the topology problem set in baby Rudin and I'm wondering if a neighborhood $N_r(p)$ of a point $p$ in a metric space automatically contains $p$, or just $\forall q|d(p,q)<r$, but not necessarily $p$.
My intuition says that because $d(p,p)=0$ and $0<r$, $p$ should automatically be in any neighborhood... |
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