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>Starting from a vertex of an unknown, finite, strongly connected directed graph, we want to 'get out' (reach the vertex of the labyrinth called 'end'). Each vertex has two exits (edge which goes from vertex in question to an other one), one exit is labeled 'a', the other exit is labeled 'b'.
>We have limitless '... |
Consider the triangle formed by joining the circumcenter, the incenter and the centroid of a a triangle (is there is already a name for this triangle in literature?). Simulations show that the line joining the incenter and the circumcenter always subtends and obtuse angle at centroid (except when the triangle is degene... |
Suppose I want to find all the interior local minimizers of the following problem: $\underset{x,y}{min}f(x,y)$ subject to $x,y>0, g(x,y)\geq 0$.
By setting the first order conditions $\frac{\partial f(x,y)}{\partial x}=0$ and $\frac{\partial f(x,y)}{\partial y}=0$, I found $n$ candidates $(x^*_1,y^*_1)$,$(x^*_2,y^*... |
How to find all interior local min for minimization under inequality constraint? Does checking the psd of hessian of objective function suffice? |
There is [numerical evidence][1] that
$$\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
>How can this be proved?
Wolfram [does not find][2] an antiderivative.
Here is the graph of $y=\frac{1}{\sqrt{1-x^2}}\arccos\lef... |
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic and affine map respectively. The claims is th... |
For any two finite subsets $A,B$, of an abelian group, is
$$|A+B|^2 |A-B|^2 \geq |A+A||A-A||B+B||B-B| \ ?$$
I’m interested in finding out if there are sumset inequalities that are sharper than the triangle inequalities and that show that some quantity related to sums and differences is smaller when the sums and ... |
In an abelian group, does there exist inequalities involving products of number of elements of sum and differences of finite subsets? |
Can you calculate the sum of this series?
$\sum_{k=0}^\infty \left(\frac{kx^k}{(k+x)!}\right) $
|
Where did the sin(kx)/(kx) term in the two-point correlation function come from? |
Can you calculate the sum of this series?
$\sum_{k=1}^\infty \left(\frac{kx^k}{(k+x)!}\right) $
|
How could I introduce a parameter to determine the value of $\int_0^1\frac{log(1+x)}{1+x^2}dx$?
|
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic and affine map respectively. The claims is th... |
Suppose you have a 26-sided die, each face is labelled from A-Z, what is the expected number of steps to observe the sequence "ABRACADABRA" for the first time?
ANS = $26^{11} + 26^4 + 26$
A common technique to handle these kind of problems is to draw a markov chain and state which correspond to observing the patt... |
Let $k$ be an algebraically closed characteristic zero field, consider the $\mathfrak{gl}_{1}$ representation
$$
\rho: \mathfrak{gl}_{1} \rightarrow k^{2}
$$
given by
$$
\rho(x) = \begin{pmatrix} \alpha & x \\\ 0 & \beta \end{pmatrix}
$$
Then this has a subrepresentation
$$
\\{ \begin{pmatrix} a \\\... |
Is this representation of a Lie algebra decomposable? |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
|−|= $\left(\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!}\right) $ .
I've shown that
|−| = $\left({2\lambda^\lambda e^{-\lambda}}\right) $
$\sum_{k=1}^\infty \left(\frac{kx^k}{(k+x)!}\right)... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
How could I introduce a parameter to determine the value of $\int_0^1\frac{\log(1+x)}{1+x^2}dx$?
|
Stabilization problem is a fundamental problem in control theory. There are many literature to achieve stabilization but fewer results are related in computational complexity. Consequently, we consider the computational complexity of stabilization problem.
Given a linear Boolean control network (BCN)
$$x(t+1)=Ax(t)... |
Faster way to find the eigenvalues of a 4x4 real matrix? |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
Where is my mistake in this integral equation? |
I'm currently learning about neural networks and stumbled upon a confusion related to the use of Stochastic Gradient Descent (SGD) in training. Specifically, I'm puzzled about the computation of the partial derivative of the cross-entropy loss with respect to the predicted probabilities. Here's where my confusion lie... |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
|−|= $\left(\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!}\right) $ .
I've shown that
|−| = $\left({2\lambda^\lambda e^{-\lambda}}\right) $
$\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\lamb... |
I want to prove that $5^{2n}-2^{3n}$ is divisible by 17 for all positive integers $n$. I know this can be done by induction (sketch proof shown below) but want to know if:
1. Are there any alternative proof methods that do not use induction?
2. How many different induction proof approaches are possible for a ques... |
Divisibility proof - alternative? |
I'm currently learning about neural networks and stumbled upon a confusion related to the use of Stochastic Gradient Descent (SGD) in training. Specifically, I'm puzzled about the computation of the partial derivative of the cross-entropy loss with respect to the predicted probabilities. Here's where my confusion lie... |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$ f(x^2+y^2) + f(xf(y) + f(x)f(y)) = (f(x+y))^2. $
My attempt:
Plugging in $x=y=0$ we get $f(0) + f(f(0)^2) = f(0)^2. $
Setting $x:=y$ and $y:=x$ gets us $f(xf(y)+f(x)f(y)) = f(yf(x)+f(x)f(y)).$
Setting $f(0)=a$ and plugging in $x=0$... |
I am solving for the rate of change of magnetic field strength( ($\frac{dB}{dt}$) Because I want to calculate the induced emf after this) in a current carrying coil of 100 turns(Note: Current is changing), the coil is connected to a 30kHz sine wave function generator. And I got confused while taking derivative of angle... |
How to calculate the rate of change of angle $\theta$ with respect to time, If the sin wave frequency is 30kHz? |
> Two cards are drawn from a well shuffled pack of $52$ cards. Find the
> probability that one of them is a red card and the other is a queen.
**My Attempt**
The relevant cards are $26$ red cards and $2$ black queens i.e. in total $28$ cards.
I took four cases.
**Case 1 : One non-queen red card and one red... |
How to calculate the rate of change of angle $\theta$ with respect to time, If the sine wave frequency is 30kHz? |
Your equation for $C(t)$ can be broken down into equations for $x$ and $y$ separately:
$$
x(t) = \frac{1-t^2}{1+t^2} \quad ; \quad
y(t) = \frac{2t}{1+t^2}
$$
It’s easy to check that $[x(t)]^2 + [y(t)]^2 = 1$ for all $t$. This means that every point $C(t)= (x(t),y(t))$ lies on the unit circle. Also it’s clear that ... |
My textbook is asking me to prove that, given $p_n(x,y)$ polynomial of degree $n\geq1$ in $x,y$, it is true that:
$$ lim_{|(x,y)|\rightarrow\infty} |p_n(x,y)| = +\infty $$
But if I take, for example, $p_n$ to be $x^2 - y^2$, then i get that the limit does indeed approach $+\infty$ everywhere except along the lines $y... |
Is the limit at infinity of an absolute multivariable polynomial always infinity? |
Let $f:\mathbb{R}^2\to \mathbb{R}$ be a smooth function such that for all $t$ $f_t:=f(t,\cdot)$ has a unique maximum at $x^*(t)$ (e.g. $f_t$ is a strictly concave function for all $t$).
My question is: is the function $t\mapsto x^*(t)$ smooth in general? If not, are there some reasonable conditions on $f$ that guarant... |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
{|−|}= $\left(\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!}\right) $ .
I've shown that
{|−|} = $\left({2\lambda^\lambda e^{-\lambda}}\right) $
$\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\... |
Your equation for $C(t)$ can be broken down into equations for $x$ and $y$ separately:
$$
x(t) = \frac{1-t^2}{1+t^2} \quad ; \quad
y(t) = \frac{2t}{1+t^2}
$$
It’s easy to check that $[x(t)]^2 + [y(t)]^2 = 1$ for all $t$. This means that every point $C(t)= (x(t),y(t))$ lies on the unit circle. Also it’s clear that ... |
Suppose $M$ is a smooth manifold and $A$ is a $\mathcal{C}^\infty(M)$-submodule of $\Gamma(TM)$ with the property that for all $x \in M$, the values of the vector fields in $A$ evaluated at $x$ determine a $p$-dimensional subspace of $T_x M$. Then $A$ defines a smooth distribution $P$ i.e. a smooth subbundle of $TM$ of... |
Does a distribution uniquely determine a (the) submodule of vector fields which generates it? |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
{|−|}= $\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!} $ .
I've shown that
{|−|} = $\left({2\lambda^\lambda e^{-\lambda}}\right) $
$\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\lambda)!}\rig... |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
{|−|}= $\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!} $ .
I've shown that
{|−|} = ${2\lambda^\lambda e^{-\lambda}} $
$\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\lambda)!}\right) $
but I... |
I'm struggling with the proof of Eulerian trail in walk through combinatorics.
As you now, the theorem states that "A connected graph G has a closed Eulerian trail if and only if all vertices of G have even degree."
I'll only ask <- part. Book gives an algorithm how to construct an Eulerian trail actually.
1. S... |
Where did the sin(kx)/(kx) term in the two-point correlation function $\xi(x)$ come from? |
There are lots of quadratic recursion sequences
$$
z_{n+1}=az_n^2+bz_n+c.
$$
Inserting a linear transformation $z=\alpha x+\beta$ results in another quadratic recursion
$$
x_{n+1}=a'x_n^2+b'x_n+c'
$$
with the same qualitative properties. Now one can ask what are the most simple, most easy to compare examples i... |
Since e is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of 0. Am I doing soemthing wrong?
$$e^z = \sum_{n=0}^\infty \frac{1}{n!}z^n$$
Setting z = 0:
$$e^0 = \sum_{n=0}^\infty \frac{1}{n!}0^n = \sum_{n=0}^\infty 0 = 0$$
What have I done... |
Why does the power series form of the exponential equal 0 when evaluated at 0? |
Since e is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of 0. Am I doing something wrong?
$$e^z = \sum_{n=0}^\infty \frac{1}{n!}z^n$$
Setting z = 0:
$$e^0 = \sum_{n=0}^\infty \frac{1}{n!}0^n = \sum_{n=0}^\infty 0 = 0$$
What have I done... |
Since $e$ is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of 0. Am I doing something wrong?
$$e^z = \sum_{n=0}^\infty \frac{1}{n!}z^n$$
Setting z = 0:
$$e^0 = \sum_{n=0}^\infty \frac{1}{n!}0^n = \sum_{n=0}^\infty 0 = 0$$
What have I do... |
Since $e$ is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of 0. Am I doing something wrong?
$$e^z = \sum_{n=0}^\infty \frac{1}{n!}z^n$$
Setting $z=0$:
$$e^0 = \sum_{n=0}^\infty \frac{1}{n!}0^n = \sum_{n=0}^\infty 0 = 0$$
What have I do... |
Since $e$ is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of $0$. Am I doing something wrong?
$$e^z = \sum_{n=0}^\infty \frac{1}{n!}z^n$$
Setting $z=0$:
$$e^0 = \sum_{n=0}^\infty \frac{1}{n!}0^n = \sum_{n=0}^\infty 0 = 0$$
What have I ... |
In Lee's Introduction to Smooth Manifolds, Chapter 8 (page 176) it's written that given a manifold $M$ and arbitrary subset $A \subseteq M$, $X$ is said to be a smooth vector field along $A$ if for each point $p \in A$, there is a neighborhood $V$ of $p$ in $M$ and a smooth vector field $\tilde{X}$ on $V$ that agrees w... |
I have a small question about PCA, specifically in calculating the covariance matrix. I know that to calculate the covariance matrix $C$, I have to subtract the mean from the data points and form the data matrix $X$, and thus I have
$$
C = \frac{1}{n}XX^T
$$
And here comes the question, what if there are data... |
I'm very interested in the correct solution of this problem.\
So I also tried the convolution approach
$$f(x,t)=\mathcal{L}_s^{-1}\left[\exp \left(-x \sqrt{\frac{s}{k}}\right)\right](t)=\frac{k x e^{-\frac{x^2}{4 k t}}}{2 \sqrt{\pi } \sqrt{k^3 t^3}}$$
$$g(t)=\mathcal{L}_s^{-1}\left[\frac{1}{s+b}\right](t)=e^{-b ... |
I have a problem, let $S_n$ process such that $ S_n := \xi_1 + \dots + \xi_n$ and stoping time moment $\tau: \Omega \rightarrow \mathbb{N} \cup{\infty}$, where
\begin{equation*}
\xi =
\begin{cases}
1 & p \\
-1 & 1 - p
\end{cases}
\end{equation*}
I need to prove that $X_n = S_{n + \tau} - S_\tau$ has the s... |
I have a problem, let $S_n$ process such that $ S_n := \xi_1 + \dots + \xi_n$ and stoping time moment $\tau: \Omega \rightarrow \mathbb{N} \cup{\infty}$, where
\begin{equation*}
\xi =
\begin{cases}
1 & p \\
-1 & 1 - p
\end{cases}
\end{equation*}
I need to prove that $X_n = S_{n + \tau} - S_\tau$ has the s... |
I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then
{|−|}= $\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!} $ .
I've shown that
{|−|} = ${2\lambda^\lambda e^{-\lambda}} $
$\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\lambda)!}\right) $
but ... |
Let $u_1$ and $g$ be increasing strictly concave functions from $\mathbb{R}$ to $\mathbb{R}$. Let $u_2:=g\circ u_1$. If we regard $u_1$ and $u_2$ as utility functions of two players, this is saying that player $2$ has Arrow-Pratt coefficient of absolute risk aversion greater than that of player $1$.
Suppose now $X$ ... |
How would one prove that the row space and null space are orthogonal complements of each other? |
**Introduction**
I'm given this quadrature formula:
$$\int_a^b f(x)dx = h \sum_{i=0}^{n-1}\left[ f(x_i) + \alpha h f'(x_i) + \beta h^2 f''(x_i) \right] = Q_n(a,b;f)$$
where
$$h = \frac{b-a}{n}, \ x_i = a + ih, \ i=0,1,\dots,n$$
My goal is to find $\alpha$ and $\beta$ so that the quadrature formula has th... |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
proof:
Suppose that $3| x^3+2x+1$ and $x$ is a rational number, $\frac{p}{q}$, $gcd(p, q) = 1, q \ne 0$
sub $\frac{p}{q}$ into $x^3+2x+1$:
$\frac{p^3+2pq^2+1}{q^3} =3d$ for some integer d
$p^3+2pq^2+1 =3dq^3$
this means that,
$3|(p^3+2pq^2+1)$
I got stuck from this point onwards and could not fi... |
$\newcommand{bm}[1]{\mathbf{#1}}$Given the semi-orthogonal fat matrix ${\bm B} \in\mathbb R^{c \times d}$ (i.e., $c\leq d$, $\bm {BB}^\top=\bm I$), the matrix $\bm X \in {\Bbb R}^{m \times n}$, $c$ one-hot vectors $\mathbf y_1, \mathbf y_2, \dots, \mathbf y_c \in \mathbb R^c$, let the cost function $J : {\Bbb R}^{m \ti... |
## Setting
Suppose $Q$ is a compact metric space. Let $\mathcal{P}(Q)$ be the set of Borel probability measures on $Q$. This set is endowed with the topology of weak-* convergence: a sequence $\left\{ m_{N} \right\}$ of $\mathcal{P}(Q)$ converges to $m \in \mathcal{P}(Q)$ if $\forall \varphi \in C(Q)$
$$
\lim _N \in... |
I've seen the following two definitions in my slides:
1. $S \subseteq \aleph$ is semi-decidable iff there exists a partially computable function g where
$S = \{x \in \aleph\ |\ g(x)\downarrow \}$
2. $S \subseteq \aleph$ is re iff $S = \phi$ or there exists a totally computable function $f$ where
$S = \{y\ |\ \exi... |
What does it mean for a set to be a member of $\aleph$? |
I am reading Steele's *Cauchy-Schwarz Master Class*, and am wondering what "inversion preserving" refers to the following exercise from Chapter 1:
> **Exercise 1.6 (A Sum of Inversion Preserving Summands)** Suppose that $p_k>0$ for $1\leq k\leq n$ and $\sum_{k=1}^np_k=1$. Show that
$$\sum_{k=1}^n\left(p_k+\frac1{p... |
Why are the summands here "inversion preserving"? |
When I was in high school, I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
This formula has always fascinated me and I have memorized it.
I tried the following
$$\f... |
I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact sp... |
$sup(y \cdot f(x)+b(x)-1/2\lVert f(x) \rVert^2)$ convex and regular with respect to $y$? |
If $u^2 \ge -\dfrac{8}{3}$, then $u \ge -\sqrt{\dfrac{8}{3}}$.
Is this the correct convention? I was confused because initially I thought the negative sign would go inside the square root, but then that would lead to imaginary numbers. Thanks. |
**Problem**: assume $ a^2 + b^2 + c^2 = 1 $. Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$ where $B(\mathbf{0}, 1)=\left\{x^2+y^2+z^2 \leq 1\right\}$ is the unit ball in $ \mathbb{R^3}$.
**Attempt**: Assuming $ a,b,c \neq 0 $ ( I had difficu... |
$\newcommand{bm}[1]{\mathbf{#1}}$Given the semi-orthogonal fat matrix ${\bm B} \in\mathbb R^{c \times d}$ (i.e., $c\leq d$, $\bm {BB}^\top=\bm I$), the matrix $\bm X \in {\Bbb R}^{m \times n}$, $c$ one-hot vectors $\mathbf y_1, \mathbf y_2, \dots, \mathbf y_c \in \mathbb R^c$, let the cost function $J : {\Bbb R}^{m \ti... |
When I was in high school, I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
This formula has always fascinated me and I have memorized it.
I tried the following
$$\f... |
With latex:
How to show that if $y=f(x+a)$ is an even function where $f(x)=(x-6)^2sin(\omega x)$ then $a$ must be 6?
The original question was to solve for what $\omega$s are possible. And there was a step like this: since $y=f(x+a)$ is even, so is $y=(x+a-6)^2$ and $y=sin(\omega x+\omega a)$. I'm confused by thi... |
How to show that if y=f(x+a) is an even function where f(x)=(x-6)^2sin(bx) then a must be 6? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
How to solve these trigonometry system of equations? |
> Use the method of characteristics to find the general solution of the following PDE
$$
e^xu_x + u_y = xu.
$$
Show explicitly that your result is indeed a solution of the PDE.
So I believe I have found the general solution as follows:
$\frac{dy}{dx} = e^{-x},$ $\frac{du}{dx} = xue^{-x}$, $\frac{du}{dy} = xu$... |
I'm using an ellipsoid $f = x^a + |y|^b + |z|^c - 1 = 0$ to fit some data (a failure envelope), where $x \in [0, 1]$, $y \in [-1, 1]$, and $z \in [-1, 1]$ are all normalized variables and $a$, $b$, $c$ are to-be-optimized coefficients. The fitting is not good, so I decided to add some coupling terms into $f$, including... |
Consider the triangle formed by joining the circumcenter, the incenter and the centroid of a triangle (is there is already a name for this triangle in literature?). Simulations show that the line joining the incenter and the circumcenter always subtends and obtuse angle at centroid (except when the triangle is degenera... |
The question regarding which i am asking this question is as follows
Prove that that least positive value of $x$, satisfying $\tan(x) = x+1$, lies in the interval $(\frac{\pi}{4}, \frac{\pi}{2})$
The solution to this question in my text book is done using a graph, however this method becomes infeasible for more c... |
Well I give an answer [here][1] and I want to know if it's true for a special example
*Answer :*
Not an answer just some speculation about gamma function and Bernstein's polynomial which is of independent interest here but could help .
Using fallaciously the inverse function of $f(x)$:
$$f\left(x\right... |
A new formula for $\pi$ using cheated Bernstein's polynomials with composition and Gamma function? |
**Problem**: Assume $ a^2 + b^2 + c^2 = 1 $. Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$ where $B(\mathbf{0}, 1)=\left\{x^2+y^2+z^2 \leq 1\right\}$ is the unit ball in $ \mathbb{R^3}$.
**Attempt**: Assuming $ a,b,c \neq 0 $ ( I had difficu... |
I believe this is a Cauchy distribution, but I am having a hard time identifying the parameters. Could someone help me to identify the value of gamma and x0 if this is indeed a Cauchy distribution?
f(x) = 2/(pi * sqrt(3))(1 + (x^2/3))^-2 |
I was doing some problems in Complex Analysis… And I came across this.
**Let $f: \mathbb{C} \to \mathbb{C}$ be entire. Then for any bounded set $B$, f ($B$) is bounded.**
Now I know that if an entire function is constant then the above statement is necessarily true.
What about non-constant entire function?
... |
>The life of a semiconductor laser at a constant power is
normally distributed with a mean of $7000$ hours and a standard
deviation of $600$ hours.
>(d) - A product contains three lasers, and the product fails if
any of the lasers fails. Assume that the lasers fail independently. What should the mean life equa... |
$$\int\frac{1}{y}\frac{1}{\ln y}$$
$$\int\frac{1}{y}\frac{1}{\ln^2y}$$
I know that i should use some substitution, but i don't understand how.Is there any other way than substitution?
I've tried to understand the integral-calculator.com output, but no luck.
Can anyone explain to me how to solve these inte... |
How should these types of integrals be solved? $\int\frac{1}{y}\frac{1}{\ln y}$ |
I believe this is a Cauchy distribution, but I am having a hard time identifying the parameters. Could someone help me to identify the value of $\gamma$ and $x_0$ if this is indeed a Cauchy distribution?
$$f(x) = \frac{2}{\pi\sqrt{3}}\frac{1}{(1 + x^{2/3})^2}$$ |
The set of real functions has the structure of a vector space, and all vector spaces have basis. What is the basis of this space then? My first thought was maybe using taylor series but still there are a lot of functions that can not be expressed through taylor series and a linear combination can not have infinite elem... |
I'm having some problems solving this integral:
$$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$
where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that:
$$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{+\infty} \frac{\sin... |
proof:
Suppose that $3| x^3+2x+1$ and $x$ is a rational number, $\frac{p}{q}$, $gcd(p, q) = 1, q \ne 0$
sub $\frac{p}{q}$ into $x^3+2x+1$:
$\frac{p^3+2pq^2+q^3}{q^3} =3d$ for some integer d
$p^3+2pq^2+q^3 =3dq^3$
this means that,
$3|(p^3+2pq^2+q^3)$
I got stuck from this point onwards and could ... |
I'm having some problems solving this integral:
$$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$
where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that:
$$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{+\infty} \frac{\sin... |
The [Foster graph](https://en.wikipedia.org/wiki/Foster_graph) is a distance-transitive symmetric bipartite cubic graph on 90 vertices (in fact, the only such graph).
I have seen this graph described as the incidence structure of the unique flag-transitive triple cover of the generalized quadrangle associated with ... |
Is there an elementary construction of the Foster graph or its associated geometry? |
I believe this is a Cauchy distribution, but I am having a hard time identifying the parameters. Could someone help me to identify the value of $\gamma$ and $x_0$ if this is indeed a Cauchy distribution?
$$f(x) = \frac{2}{\pi\sqrt{3}}\frac{1}{(1 + x^{2}/\sqrt3)^2}$$ |
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