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I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place?
In this moment I just know that it is named the derivative and that it is the slope of the tangent line at that point. We can find that slope by finding the limit of closer and ... |
Let $\mathcal F$ be a $\sigma$-algebra and $X_n$ RV s.t. $X_n$ is independent of $\mathcal F$ for all $n$. Also let $X_n \to X$ $\mathbb{P}-$a.s.. Is $X$ independent of $\mathcal F$ now too?
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Given that R is an integral domain. I'm trying to prove that divisibility on this set constitutes a partial ordering. In particular, I have defined the relation $y \leq_{\,d} x$ on R by $y|x$. Reflexivity is, of course, very easy. Let $x \in R$ be arbitrary. Then, $x|x$ and as a result $x \leq_{\,d} x$. Anti-symmetry i... |
The first $12$ natural numbers are given. Two distinct numbers are selected. What's the probability that their sum is divisible by $3$?
This looks very easy. I know answer is $1/3$ but in spite of knowing permutations and combinations I had to find favourable cases. The numbers were few. What if we were given questio... |
Assume you have defined variables $A$, $B$ and $C$ in the text. There is also a defined function $f$ that applies on those variables. In addition you may have some variables such as $\kappa_A$, $\kappa_B$ and etc.
The question is, how to best express $f$ and $\kappa$ without repeating it for the three variables. I mea... |
There are three parameters:
$y_s=y[0]$ start value
$y_t=y[n]$ target value
$\alpha, 0>\alpha\leq1$ smoothness
Starting at $y[0]=y_s$ the sequence is developed with this recursive formula:
$y[i]=\alpha\cdot y[i-1]+(1-\alpha)\cdot y_t$ for $i>0$
It can be transformed to a non-recursive form:
$y[i]=\alpha^i\cdot y_s+(1-... |
I have 3 (not independent) events $A, B, C$ and I know everything about how any two of them correlate. For example, I know:
$$ P[A], P[B], P[C], P[A,B], P[A,C], P[B,C], P[A|B], P[A|C], P[B|C], P[B|A], ...$$
Is there any way to use this information to calculate a correlation for the three of them, i.e.
$$ P[A,B,C] \t... |
Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$.
The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. I'm interesting in knowing whether the operator has a unique and asymptotic distribution $f^*$ (think e.g. whether an at... |
I started studying about binom's and sums, How do I calculate
$$\sum_{k=0}^{33}\binom{33}{k} k$$
Note: I do know that it is $\binom{33}0\cdot0 + \binom{33}1 \cdot 1 + ... + \binom{33}{33} \cdot 33$, but how do I write it briefly?
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Find the exact value of
$$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} \mathrm {dx} $$
Where, $a$ is a positive constant
Hi, guys can give me tips to solve this ? Should we use like u substitution?
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There is an array which contains points as shown below;
[ -0.0249795, -0.00442094, -0.00397789, -0.00390947, -0.00384182, -0.0037756, -0.00371057, 0.00180882, 0.00251853, 0.00239539, 0.00244367, 0.00249255, 0.00254166, 0.00259185, 0.0116467, 0.0155782, 0.016471 ]
First of all, honestly, i don't know whether there is a... |
We have $(T_l)_l$ a sequence of bounded linear operators from $\ell^2$ to $\ell^2$.
$\bullet$ We say $(T_l)_l$ satisfies the property "A" if $\sup_{||x||_{\ell^2}=1}\sum_{l=1}^\infty||T_l(x)||^2<\infty$.
$\bullet$ And we say that it satisfies the property "B" if $\sup_{N\in\mathbb{N}}\int_0^{2\pi} ||\sum_{l=1}^{N}e^{il... |
Given a basis $U$, what conditions are needed for an orthogonal basis for it?
For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an orthogonal basis for it?
Is it enough to have a basis of dimension $2$ that's orthogonal? or are there more conditions?
EDI... |
Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$.
Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?
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$Tr(XY) = 1$ and $Tr(Y) = 1$ implies that $Tr(X) = 1$.
I tried to prove by contradiction and switch the dummy variable of $X$ and $Y$. But I don't think my approach is right and if there is any much easier proof.
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So i want to prove that
$$x^2e^x=1$$ has at least one solution for $$x\in\mathbb{R}$$
I am kinda lost and would appreciate any help. This is suppose to be solved using basic calculus but i am not sure what to use.
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I have a little problem with quadratic programing problem:
${\bf v}^T \Sigma {\bf v} \rightarrow min $,
and constrains are
$ {\bf v}^T \mu = \mu_*, {\bf v}^T {\bf 1 }= 1, 0 \leq {\bf v}. $
Where v is vector and $\Sigma$ is matrix.
I know that Lagrange function should be:
$ L( {\bf v }, \bar{\tau}, \bar{\gamma}, \... |
Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and $\text{diam}(D_n)\to0$ as $n\to\infty$, then $\bigcap_{n=1}^\infty D_n$ consists of precisely one point.
Give examples with $X=\math... |
Let $\{B_t:t\geq 0\}$ be a standard Brownian Motion and let $\{\mathcal{F}_t\}_{t\geq 0}$ be the natural filtration associated to Brownian Motion (that is, $\mathcal{F}_t=\sigma(B_s:0\leq s\leq t)$). Fix $T>0$. Let $u:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a measurable function such that $u\in L_{a,T}^2$, that is:
... |
I have a $64$-card deck, with $4$ colours - red, green, blue, yellow, $4$ numbers - $1,2,3,4$ and $4$ letters A,B,C,D. So an example card could be yellow2D.
I was attempting to calculate the probability of $4$ of a kind, i.e. $4$ green cards, $4$ $3$'s, etc... having been dealt $k$ cards.
But I got all stuck!
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I am ultimately trying to prove, for an Exercise in Burton's Elementary Number Theory, that $x^4 - y^4 = 2z^2$ has no solution in the positive integers.
I can establish that if there is a solution, the solution with the smallest value of x has gcd(x,y)=1
I see that $x^4 - y^4 = (x^2 + y^2)(x^2 - y^2) = (x^2 + y^2)(x +... |
Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the $\mathbb{Z}$, explicitly we do the following. Let
$$ \cdots\to P_2\to P_1 \to P_0 \to \mathbb{Z} $$
be the standard resolution. ... |
In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic manipulation rather than much more stronger visual (intuition) method.
Is it a good notion to rely on visualization and intu... |
Given $$\dfrac{d^2y}{dx^2}-(3-2i)y=0,\quad y(0)=1,\quad y(x\rightarrow \infty)\rightarrow 0$$ then what is $y(\pi)$ ?
The answer given is $-e^{-\pi}$.
But I cannot understand how its solution can be pure real.
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If $$A+B+C=\pi$$, prove that
$$\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}=4\cos\frac{\pi-A}{4}.\cos\frac{\pi-B}{4}.\cos\frac{\pi-C}{4}$$.
My solution:
Here
$$A+B+C=\pi$$
$$A+B=\pi-C$$
Taking 'sine' on both sides gives
$$\sin(A+B)=\sin C$$.
Now,
$$L.H.S=\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}$$
Which formu... |
How to prove that not all norms are equivalent in an infinite-dimensional vector space?
In particular, I would like to prove that for a space $X$ of continuous real-valued functions defined on interval $[0,1]$, every two norms $\|\ .\|_p$ ($p \in [1, \infty]$) are not equivalent.
|
"Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even, and at most 3 roots if $n$ is odd?"
To do this, I assume I must show that there are certain values of $x$, given by some functions of $k$ and $l$ such that $f_{1}(k,l)<0, f_{2}(k,l)>0, f_{3}(k,l)<0$, but I'm unsure of how t... |
I am trying this question:
We have a bag with 10 blue jacks and 20 red jacks. We pick 3 jacks from the bag at random and with replacement. We are told that at least one jack is red.
Compute the expected values of blue jacks and red jacks, again being told that we have at least one red jack
My attempt so far:
I am tryi... |
Michael lives $10$ km away from where I live. Ahmed lives $5$ km away and Susan lives $7$ km away from where I live. Arun is farther away than Ahmed but closer than Susan from where I live. From the information provided here, what is one possible distance (in km) at which I live from Arun’s place?
$3.00$
$4.99$
$6.02$... |
Solve for $\theta$ $[0°<\theta<180°]$
$$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$
My solution is here:
$$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$
After using the transformation formula, I got
$$\sin3\theta\cos\theta=\cos3\theta\cos\theta.$$
I could not proceed from here.
|
Given
$$g(x)=\int_{0}^{x} (x-t)e^{t}dt$$ find out $g''(x)$
I thought of using Lebnitz theorem to differentiate it but using Lebnitz I get this $g'(x)=1\cdot (x-x)e^{x}=0$ I don't know how to find the second order derivative of this function.
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I have read that there are two 2-dimensional algebras over the complex numbers, but I wanted to see what they are and how they are formed. I understand how to show that there are three 2-dimensional algebras over the reals and am trying to alter that but I don't know if what I'm doing is right!
I'm quite new to this ar... |
$$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad(n\in I) \\ \text{where $\lfloor\cdot\rfloor$ is the greatest integer function.}$$
This is what I did:
Since $[x] = x - \{x\}$ we get our limit equal to
$$\lim_{n\to\infty}\left\{\sqrt{n^2+n+1}\right\}$$
Moving the limit i... |
I asked someone about this problem:
Let $V,W$ be vector spaces with bases $(\alpha_i)_{i\in I}, (\beta_j)_{j \in J}$ respectively. Define $f_{ij}(\alpha_k) = \delta_{ik}\beta_j$. Show that $(f_{ij})_{i\in I, j \in J}$ is the basis of $Hom(V,W)$.
The solution presented was something like: take some formula from nowher... |
The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to?
I think the question can be solved by writing the series in reverse order but I am not able to solve further.
|
$\Omega=\{H,T\}^\mathbb{N}$,
so that a typical point $\omega$ of $\Omega$ is a sequence
$\omega=(\omega_1,\omega_2,\dots), \omega_n \in \{H,T\}.$
Let $\mathcal{A}$ be the set of all maps $\alpha : \mathbb{N} \to \mathbb{N}$ such that $\alpha(1)\lt \alpha(2)\lt \dots.$
For $\alpha \in \mathcal{A}$, let
$$F_{\alpha} ... |
Given a matrix:
$A = \begin{pmatrix} -\epsilon & tf_1 \\ tf_2 & -\epsilon \end{pmatrix}$
Compute the eigenvectors. I can easily find the eigenvalues to be $\lambda = -\epsilon \pm t\sqrt{f_1 f_2}$. Maple tells me that the eigenvectors are
$\begin{pmatrix} f_1 /\sqrt{f_1 f_2} \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -f_... |
I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm :
$K:\mathbb{R}_+\times\mathbb{R}_+$ with $K(x,y)=\log(1+xy)$
I've tried a lot of different things (using series, writing it as an integral), ... |
Could somebody help me prove that there are a infinite number of natural numbers for which their sum of digits exceeds the number of divisors?
If $S(n)$ denoted the sum of digits, and $\sigma_k(n)$ was the divisor function, how can I prove that there are infinite such $n$ where $S(n) > \sigma_0(n)$.
Note that there are... |
W is closed and convex subset of Hilbert space $H$. Suppose $x\in H$, $x_0\in W$. I am trying to proof, that $d(x, W) = ||x-x_0||$ iff $\Re(<x-x_0, y-x_0>)\le 0$ for all $y\in W$.
Could anyone help me with that?
($\Re$ is the real part)
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Let $f:\{z\in\mathbb{C}:|z|>1\}\rightarrow\mathbb{C},f(z)=\frac{1}{z}.$ Now my question is how many entire functions $g$ are there such that $f=g$ for $|z|>1$?According to me there is no such entire function because if there is so then it must be equal to $\frac{1}{z}$ itself. Am I right? Please help. Thanks.
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Let $V$ and $W$ be Brownian motions such that $\mathbb{E}W_tV_t=\rho t$. Let $$R_t=\sup_{u \le t} V_u \mbox{ and } Z_t=\sup_{u \le t}W_u .$$
Show that $$\mathbb{E}R_tZ_t=tf(\rho) .$$
Can you find $f(\rho)$ ?
EDIT:
I have already figured out the first part of the question. Let us fix $t > 0$ and define $\overline{W}_s=\... |
Prove that the pair of lines $6x^2+5xy-4y^2+7x+13y-3=0$ form a parallelogram with the pair of lines $6x^2+5xy-4y^2=0$. Find its area.
My attempt/
I factorized the second equation to get the two lines represented by it:
$$6x^2+5xy-4y^2=0$$
$$6x^2+8xy-3xy-4y^2=0$$
$$(3x+4y)(2x-y)=0$$.
No what should I should next to comp... |
Are there examples of predictable stochastic processes $X$ such that their past is independent of their future?
More formally, such that $\sigma\{X_s | s\in (0,t]\}$ is independent of $\sigma\{X_s | s\in (t,u)\}$ for all $0<t<u$?
Of course the process cannot have right continuous paths, but maybe a left-continuous whit... |
A section in a book shows jumping from
$$7-3(9-7) = 1$$
to
$$4(7)-3(9) = 1$$
I can't see how this happens. What steps have been taken?
Where did the four come from?
|
I am trying to find a smooth bijective map from the complex Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ to the Grassmannian of $2k$-dimensional subspaces of $\mathbb{R}^{2n}$, but I do not know how to construct it.
I think it is related to the
$$a+bi\mapsto\begin{bmatrix}a & -b\\b&a\end{bmatrix}$$
map ... |
While trying to understand a proof of a result in an article, I stumbled upon the product
$$\prod_{n=1}^\infty (1 - \gamma_n)$$
with $\gamma_n$ a real scalar belonging to $(0,1)$. I'm not really a mathematician, so I looked through some of the articles here about infinite products (like this, this and this), but got lo... |
Here is the picture of the page of the book, I am reading:
$$P_k: \qquad 1+3+5+\dots+(2k-1)=k^2$$
Now we want to show that this assumption implies that $P_{k+1}$ is also a true statement:
$$P_{k+1}: \qquad 1+3+5+\dots+(2k-1)+(2k+1)=(k+1)^2.$$
Since we have assumed that $P_k$ is true, we can perform operations o... |
Does anyone have a proof or reference for this often-quoted fact? How complicated is the proof?
If $K$ is a complete normed division ring, then $K$ is isomorphic to the real numbers, complex numbers, or the quaternions.
|
To begin with I know there is a similar question asked and answered, but it is not what exactly i was trying to find (or at least it was only partially answered there) the question that i refer to is: Probability of number of unique numbers in $37$ Roulette Wheel spins.
I would like it if someone could tell me how to f... |
Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has Tor-amplitude in some interval $[a,b]$, which means that $H^i(E^\bullet \otimes_R M) = 0$ for all $i \not\in [a,b]$ and for ... |
I have two disks of radii $R_1, R_2$ with distance between centers, $d < R_1 + R_2$.
How can I find the surface area common to the two disks?
Rationale:
Solar irradiation / energy input in penumbra during solar eclipse, a problem for sun-synchronous satellites. Knowing apparent size of the Sun and the Moon, and positio... |
Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$?
I know the theoretical approach: for every $t \in [0,1]$ there exist a neighbourhood $U_t$ of $c(t)$ and a local tangent field $C_t \in \mathcal X (U_t)$ that extends $\dot c$, so one computes... |
http://www.mathalino.com/reviewer/algebra/arithmetic-geometric-and-harmonic-progressions
Please go to this link and see how they tell you to find the sum of harmonic prgression. However I am sure it is wrong as
$\frac{1}{a^{-1}+b^{-1}}\neq a+b $
I would like too understand and see the correct formula. Links will do.
P... |
I have had this problem for a long time now:
Suppose we take the cube root of unity $\omega$.
$\omega ^ 2 = e ^ {i {4\pi\over3}} = {(e ^ {i 4\pi})}^{1\over3} = {(e ^ {i 2\pi})}^{1\over3} = e ^ {i {2\pi\over3}} = \omega$
Why doesn't this manipulation work? Are we not allowed to divide exponents like we do with real nu... |
Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$?
Let $z=r(\cos(\theta)+i \sin(\theta))$. Then $\frac{\Re(z)}{|z|} = \frac{r \cos(\theta)}{r} = \cos(\theta) $; as the limit will not depend of $r$, so $f$ can't be extend by continuity.
Am I right?
|
Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$.
If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then $\lim_{n\to \infty} \|f_n-f\|_{L^p}=0$ is true.
The converse is easy to prove but I'm stuck at proving this implication.
If we also... |
Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true.
Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix $x\in I$ and suppose $f:I \rightarrow \mathbb{R}$ is a strictly increasing function. Let $\{x_k\}$ be a sequence contained in ... |
I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = \frac{6(x+1)(x+2)}{(x+2)(x-2)}$ ). But for this simplification $x$ must not be $-2$ or $+2$.
If I plot the function using diffe... |
If $$ax^2+2hxy+by^2+2gx+2fy+c=0,$$ represents a pair of lines, show that the square of the distance from origin to their point of intersection is $$\frac{c(a+b)-f^2-g^2}{ab-h^2}.$$
My attempts;
since the given equation represents a pair of straight lines, let each be
$$l_1x+m_1y+n_1=0$$ and
$$l_2x+m_2y+n_2=0.$$
Now co... |
I have found the formula for calculating the R of a circle, based on a circular segment, which is:
$$ R=\frac{h}2+\frac{c^2} {8h}$$
where $R$ is radius, $c$ is the length of the segment, and $h$ is the height of the segment.
What I want to do is solve this equation for $h$. e.g. h=[...].
I have been trying, but for me ... |
I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$.
$$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 $$
My work:
$$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 = \sum_{i=1}^{n} \lvert x_i - (A\mathbf x)_i\rvert = \sum_{i=1}^{n} \lvert x_i - A_i \cdot \mathbf x... |
I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables.
$\sum_{i=1}^{n}X_i \sim N(\sum_{i=1}^{n}\mu_i, \sum_{i=1}^{n}\sigma^{2}_i)$
Let's imagine I've a bag of 100 lotto balls ... |
Are they general techniques to determine the sef-adjoint extensions of
a symmetric linear differential operators in 2 variables with (real) variable
coefficients? Any suggestions for references? Typically I am interested in
studying the self-adjoint extensions of differential operators of the form:
$$
L= a(x,y) \parti... |
Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then
(i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq e^{-(3-\sqrt8)\lambda}.$$
(ii) Now, assume that $X, Y$ are not independent, then there exist $A, c >0$, both not depending on $\lambd... |
The question is to find all the subgroups of $R^*$ (non-zero reals under multiplication) of index $2$.
The index can be found out for finite groups. How to find subgroups having certain index for an infinte group?
The index is also the no. of distinct cosets(left/right). Is it the way to proceed to the answer? Please p... |
Suppose you have 4 sets, $S_{1}, S_{2}, S_{3}, S_{4}$ and you want to find out how many ways you can select a combination of A items from set 1, B number of items from set 2, C from set 3 and D from set 4.
If you're choosing one item from each set, due to the Rule of Product it's just
$|S_{1}|\times|S_{2}|\times|S_{3}|... |
Let $f$ be Riemann integrable on $[a,b]$. Show that for every $ε > 0$, there is a continuous function $g$ on $[a,b]$ such that
$$\int_a^b |f(x)−g(x)|\mathrm dx < ε. $$
|
Given that the Euler Polynomials $E_n(z)$ are defined in terms of the generating function
$$\frac{2e^{xz}}{e^x+1}=\sum_{n=0}^\infty E_n(z)\frac{x^n}{n!}$$
and that the Bernoulli Polynomials $B_n(z)$ are defined in terms of a similar generating function
$$\frac{xe^{xz}}{e^x-1}=\sum_{n=0}^\infty B_n(z)\frac{x^n}{n!}$$
I ... |
An investor purchases $1000$ worth of units in a mutual fund whose units are valued at $4.00$. The investment dealer takes 9 % front-end-load (deduction) from the gross payment.
One year later the units have a value of $5.00$ and the fund managers claim that the fund"s unit have a value has experienced a 25 % growth i... |
This question is from Munkres Topology, page 67:
Let $S_{\Omega}$ be the minimal uncountable well ordered set. Since there is no largest element in $S_{\Omega}$, every element in $S_{\Omega}$ has immediate successor. Let $s(b)$ denotes immediate successor of $s\in S_{\Omega}$. So, if $a\in S_{\Omega}$, then $A=\{s(a),... |
Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
|
Evaluate $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\: n \in \mathbb{N}$$
$$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + \int_{-\pi}^{0}\frac {\sin nx}{(1+2^x)\sin x}dx$$
Set $y = - x$ in the second integral.
$$\int_{-\pi}^{\pi}\frac {\sin nx}{(1... |
How do I prove that for $x,y\in\mathbb{R}$ we have $|x|+|y|\leq\sqrt{2(x^2+y^2)}$?
I thought that $(|x|+|y|)^2=x^2+y^2+2|x||y|\leq2(x^2+y^2)$, but I'm not sure why that holds.
|
The obtuse angle $B$ is such that $\tan B = -\frac{5}{12}$. Find the exact value of $\sin B$.
Is this possible to do without a calculator? If so, how?
|
Can anyone prove how the two equations are equal?
Thanks
$$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$
$$=\frac1{2\pi} \int_0^{2\pi} f(x) \frac{(N+\frac12)(x-t)}{\sin\frac12(x-t)}\, dx$$
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I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$:
$E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of $n$ such that $\sum_{i=1}^n r_i>1$ and $r_i$ are random numbers from uniform distribution on $[0,1]$.
I can think of seve... |
I'm currently self studying Ireland and Rosen's A Classical Introduction to Modern Number Theory and got stuck on the proof of Proposition 13.2.9. In this proof, $p$ is a prime not dividing $m$, $D, D_m$ denote the ring of algebraic integers in $\mathbb{Q}(\zeta_p,\zeta_m)=\mathbb{Q}(\zeta_{pm})$ and $\mathbb{Q}(\zeta_... |
I have an item A (Symphony Impromptu No. 1 for Frederic Chopin) and i want to know if it is more similar to another item B (Symphony Polonaise Heroique for Frederic Chopin) or an item C (Symphony 5th for Beethoven)
I have some measures, which are:
measure 1: has the same artist.
measure 2: has the same music key.
meas... |
Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is $C^2$) , then is it true that for any scalar field $f:\mathbb R^n \to \mathbb R$ , the determinant of the Jacobian of the fu... |
I'm stuck with this integral
$\int\frac {\cos^4x}{\sin^3x} dx$
which I rewrote as
$\int \csc^3x \cos^4xdx$
then after using the half angle formula twice for $\cos^4x$ I got this
$\frac 14\int \csc^3x (1+\cos(2x))(1+\cos(2x))dx$
then after solving those products I got these integrals
$\frac 14 \{\int \csc^3xdx+2\int ... |
Consider an investment with nonzero interest rate $i$. If $i_5$ is equal to $i_{10}$, show that interest is not computed using simple interest.
Answer is If $i$ is a simple interest rate, then $i_5=i_{10}$ implies $i=0$
Workings:
$i_5=\frac{A(5)}{A(4)}-1$
$i_{10}=\frac{A(10)}{A(9)}-1$
If equal, then by cancelling ($-1$... |
Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
|
I look for an advanced linear algebra (A complete book but wich deals indiferently with infinite/finite vector space). To give an idea i expect a book that (for exemple) would prove the existence of a base in any vector space by Zorn Lemma.
Thanks
PS: I found Bourbaki's book but the level is quite hard
|
Let $X$ be a smooth surface and $f\colon X\to B$ a fibration, with $B$ a smooth curve.
(Q1) Why is the normal bundle of any fiber $F$ trivial?
It is clear to me that it has to be of degree zero since $0=F^2=\deg(O_F(F))$, but why is $O_F(F)=O_F$ ?
For the second question I will assume that $B=\Bbb{P}^1$ and that $X$ is... |
$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$
$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$
$x=\frac{4}{3}\sin\theta$
$dx=\frac{4}{3}\cos\theta d\theta$
$$\int \frac{3\left(\frac{16}{9}-\frac{16}{9}\sin^2\theta\right)^{\frac{3}{2}}}{x^6}d\theta=\int \f... |
I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then
\begin{equation*}
\begin{array}{ll}
x'(s)=2p\\y'(s)=-2q\\u'(s)=2p^2-2q^2=16u\\ p'(s)=8p\\q'(s)=8q
\end{array}
\end{equation*}
so
\begin{equation*}
\begin{array}{ll}
p(... |
An orbiting satellite has 3 panels of solar cells, all of which must be activate to provide an adequate power output. The panels function independently of one another. The chance that single panel will fail during the mission is 0.02. What is the probability that there will be adequate power output during the entire m... |
Why do you have to factor out $-1$ here?
$$\frac{2000}{(10-h)(10+h)}$$$$=\frac{A}{10-h}+\frac{B}{10+h}$$
Decomposing this finds A annd B to be 100, which is wrong.
Symbolab and Wolfram Alpha factor out $-1$ from $(10+h)(10-h)$ before separation. I can't see why...
|
Two circles intersect at $P$ and $Q$.Through $P$ two lines $APB$ and $CPD$ are drawn to intersect circles at $A,B,C,D$. $AC$ and $DB$ when produced meet at $O$. How do I prove that $OAQB$ is cyclic quadrilateral. Please if possible first give me image so that I can also try it, I can't draw its figure properly.
|
The set $S$ of real numbers of the form $m/10^n$, $m,n$ integers and $n$ greater than or equal to $0$, is a dense subset of $\mathbb R$ or not??
I know dense means closure of $S$ in $\mathbb R$ is $\mathbb R$ then it will be dense. How to prove or disprove I have no idea.
|
I've been trying to complete this question but have been struggling to see how to approach it.
Any help would be greatly appreciated.
Is there a standard way of approaching and answering questions like this? I've thought about Chi square and T- distributions but their definitions don't seem to be helping me. I know ... |
Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ the following way: $\tilde{f}((x,o_x)):=(f(x),o(d_xf(b)))$ where $b$ is some basis for $T_xM$ from the equivalence class ... |
I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$.
I have found answers for the case of a fair dice (i.e. $p_i={1 \over 6}$) here and here
For large $N$ I could apply the central limit theorem and use a ... |
I'm looking for a word to describe a function where every output is guaranteed to have come from exactly one contiguous range of input values.
For example, a monotonic function has this property, but that is too strong: the relative ordering is unimportant.
However, this is a looser guarantee than that the function be ... |
How do i evaluate this limit as $n \to \infty$:
$$\lim_{n\to\infty}\left(8n- \frac{1}{n}\right)^{\frac{(-1)^n}{n^2}}$$
Thanks
|
Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing?
I simplified it to $\frac{\sqrt{n^2+1}}{n}$, and I tried $a_{n+1}-a_n$ and $\frac{a_{n+1}}{a_n}$, but neither seem to work, how should I proceed?
|
We have two vehicles
A: old truck that does 17 mpg (13.84 l/100km)
B: old car that does 47 mpg (5.00 l/100km)
We are looking to replace one of these vehicles with a new one (of the same size)
C: new truck 19 mpg (12.38l/100km)
D: new car 61 mpg (3.85l/100km)
So which vehicle should we replace to save fuel? Th... |
The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$
and extends to the whole of $\mathbb{C}$ by analytic continuation. It is known that all the complex zeros of $\zeta(s)$ satisfy $0<\Re(s) <1$.
But... |
Suppose I have three finite sets $A, B, C$. I want to find a function $f$ such that
$|A \cap B \cap C| = f(|A|, |B|, |C|, |A \cap B|, |A \cap C|, |B \cap C|)$
Does such a function exist? The only reasonable solution I get involves the cardinal of the union $|A \cup B \cup C|$, which I don't want at all. If there is no... |
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