All Questions
1,645,358
questions
1
vote
0
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48
views
Derive a regular conditional distribution of $\beta_0 + \beta_1 X + \epsilon$ given $X$, when $\epsilon \sim N(0, \sigma^2)$
Given two random variables $X $ and $\epsilon$, where $\epsilon \sim N(0,\sigma)$, I would like to derive a conditional distribution of $\beta_0 + \beta_1 X + \epsilon$ given $X$. If it is necessary ...
- 996
1
vote
0
answers
93
views
If $G$ is a abelian with $|G|=p^n$ and $a$ is an element of maximum order in $G$, then $G$ is of the form $\langle a\rangle \times K.$
Prove that if $G$ is an abelian group of prime-power order $p^n$ and let $a$ be an element of maximum order in $G$, then $G$ can be written in the form $\langle a\rangle \times K.$
Our teacher told ...
- 11
3
votes
2
answers
647
views
Is it true that if $f$ is surjective from $A$ to $B$ then there is an injective function $g$ from $B$ to $A$? [duplicate]
I think the answer is yes because otherwise $f$ wouldn't be a function.
Is this correct? And how would the formal proof go?
- 363
0
votes
0
answers
25
views
Prove the existence of a regular tetrahedron with some costraints on the center of mass
"Prove there exists in $\mathbb{R}^{3}$ a regular tetrahedron of real side $d$ such that if one puts in its vertices four strictly positive masses $m_1>m_2>m_3>m_4$ the center of mass is in $(...
1
vote
1
answer
139
views
INTEGRATION-type problem-Find the value of $\int_0^8f(t)dt$
If the function $f:[0,8] → R$ is differentiable then for $0<\alpha$, $\beta<2$ , $\int_0^8f(t)dt$ is equal to
(A) $3[\alpha^3f(\alpha^2)+\beta^3f(\beta^2)]$
(B) $3[\alpha^3f(\alpha^2)+\beta^...
- 8,810
1
vote
1
answer
66
views
comparing averages versus comparing the averages of percentage
Let’s say there are three rooms with 40 identical boxes in each room. Each box contains some number of cards inside. The number of cards in each box is different. It may be as many as 60 cards in one ...
- 818
2
votes
1
answer
130
views
Quaternion algebra: $a \equiv 3$ or $5\mod 8$ implies $(a,2)_{\mathbb{Q}}$ is a division ring.
Question: let $a\in\mathbb{Z}$ such that $a\equiv 3$ or $5\mod 8$. Proof that $(a,2)_{\mathbb{Q}}$ is a division ring.
Definition: a quaternion algebra over a field $F$ is a ring that is a $4$-...
- 824
1
vote
1
answer
166
views
Solution to non-homogenous second-order differential equation from Navier-Stokes equation.
I was using the simplified Navier-Stokes equations to find the velocity profile within a cylindrical pipe, where I got the equation as:
$ \alpha = \frac{\partial V^2}{\partial r^2}+\frac{1}{r}\frac{\...
- 55
2
votes
1
answer
154
views
Does there exist an initial arrangement of 10 black squares such that all the squares will ultimately be black?
Let there be a $12×12$ table of white squares. We draw $10$ squares in black. If a white square has $2$ black neighbours, then we draw it in black. We say that $2$ squares are neighbours if they have ...
- 587
0
votes
1
answer
25
views
One More Ordinary Differential Equation
While working on some variations of a physics problem (how does the normal force vary in a circular loop with a constant attrition coefficient with regards to the angle ) , I arrived at the ...
- 796
2
votes
1
answer
84
views
Calculus II: Limit exercise
I'm currently studying for my Calc exam and I came across this exercise.
The problem is to find the values for $a,b,c \in \mathbb{R}$ so that the following limit exists:
$$ \lim_{(x,y) \rightarrow (...
- 111
-1
votes
2
answers
72
views
Solve the equation $3x^2+5x \equiv -1357 \pmod{7919}$
Solve the modular equation $3x^2+5x \equiv -1357 \pmod{7919}$.
Is there anybody help me with this ? I have thought several times but not to find out the solution. Thank you.
- 623
1
vote
1
answer
69
views
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
Let $G$ be a group and $c$ be a fixed element of $G$. Define a new operation $*$ on G by:
$$x*y=xdy$$
for all $x$ and $y$ in $G$, where $d$ is the inverse of $c$. Prove that $G$ is a group under ...
user732928
0
votes
2
answers
91
views
Sum of Legendre sequence $S=\sum_{x=0}^{p-1}\left(\frac{x(x+k)}{p}\right)$
Calculate the sum of Legendre sequence $S(x):=\displaystyle \sum_{x=0}^{p-1}\left(\dfrac{x(x+k)}{p}\right)$ with $p > 3$ prime number, $k \in \mathbb{N}$ and $\text{gcd}(p,k)=1$.
I have tried to ...
- 623
-1
votes
2
answers
28
views
squareroot of x^2 in limits infinity to -infinity...
$\frac{x}{a^2\sqrt{x^2+a^2}}\bigg|^{\infty}_{-\infty} = ?$
$=\Bigg[\frac{x}{a^2\sqrt{(x^2)(1+\frac{a^2}{x^2})}}\Bigg]^{\infty}_{-\infty}$
$=\Bigg[\frac{x}{a^2\sqrt{(x^2)}\sqrt{(1+\frac{a^2}{x^2})}}\...
- 951
0
votes
1
answer
272
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Example of a dense set which is not nowhere dense
I can't find an example of a dense set A in $\mathbb{R}^{2}$ which is not nowhere dense. The definition of nowhere dense set given in the exercise is as follows:
Let $\mathrm{X}$ be a topological ...
-1
votes
2
answers
44
views
calculating some limits with elementary ways
How the following limits can be computed without using Taylor series, Laurent series, or L'Hospital?
I)
$$\lim_{x\to\pi/4} \frac{\ln\left(\tan\left(x\right)\right)}{\cos\left(2x\right)}$$
II)
$$\...
- 369
0
votes
2
answers
172
views
why commutative integral with limit is important in real analysis? [closed]
why commutative integral with limit is important in real analysis ?
Why $\lim_{n\to\infty }\int f_n=\int \lim_{n\to\infty } f_n $ is important ?
- 4,629
1
vote
2
answers
98
views
Open subset Proof
Let $U\subseteq\mathbb{R}^2$ be defined as $\{$ $(x_1,x_2) \in \Bbb R^2$ $:$ $x_1>0$ $\}$. Then $U$ is open.
My proof:
Let $\textbf{x} \in U$ then $\textbf{x}= (x,y)$ where $x>0$. Let $r= x$. ...
user643073
0
votes
2
answers
162
views
Monotonicity of a function without using differentiation
I want to prove the monotonicity of the following function WITHOUT using derivatives.
f(x)=x²-4x+3, on the interval [-1,1]
let x1 be less than x2
square both sides
x1² less than x2²
add -4x to ...
- 11
0
votes
0
answers
125
views
Parabola - Converting Cartesian Coordinates to Polar Coordinates
Just a simple parabola, $ y = x^2 $ .
So, doing the maths: $r sin \theta = r^2 (cos(\theta))^2$.
Then, I have $ r = \frac{sin(\theta)}{(cos(\theta))^2}$ .
I'm interested when $ \frac{3\pi}{4} < ...
- 27
1
vote
1
answer
434
views
Geometry question with perpendicular lines and angle bisectors in triangle
For any triangle ABC
1) Draw angle bisector of angle BAC. Draw perpendicular bisector of BC. Mark the intersection point as D.
2) Draw altitudes from D to AB and AC. (Extend line as necessary).
...
- 1,853
1
vote
3
answers
2k
views
Solutions for problem 1-1 of Introduction to Algorithms, Third Edition, by CLRS
I'm working problem 1-1 of the textbook Introduction to Algorithms, Third Edition, by CLRS, and need to solve $n\log(n) = 10^6$, where the logarithm is base $2$. My first thought was to use the ...
- 4,192
0
votes
0
answers
49
views
Can we define sum over an uncountable set? [duplicate]
Is it possible by some means to define a notion of sum over the elements of an uncountable set $S$ of real numbers.I thought of something like $\sum_{\alpha \in \tau }
a_\alpha:=\sup_{T\subset S ,|T|=...
- 3,481
3
votes
1
answer
2k
views
When does equality hold in the triangle inequality?
We consider the supremum norm $\|f\|=\sup_{x\in [a,b]} |f(x)|$ in the space $B([a,b],\mathbb C)$ of all bounded functions $f: [a,b] \rightarrow \mathbb C$. We obviously have in general that $\|f+g\|...
- 2,191
-1
votes
1
answer
29
views
Can Someone Help me solving this equation using modulus operation? [closed]
X mod(35-3*X)=0
How to find values of X?
- 512
2
votes
2
answers
75
views
What is $\frac{\partial^2(g\circ f)}{\partial x_i \partial x_j}(c)$
If $f:A\to \mathbb R^k$ and $g:B\to \mathbb R$ are two functions of class $C^2$ and their composition is well defined.
For $c \in A$ what is $$\frac{\partial^2(g\circ f)}{\partial x_i \partial x_j}(...
- 798
0
votes
1
answer
890
views
Sum of "positive" numbers
Given this definition of a positive real number: $$\forall x\in\mathbb{R},x\geq 0:\Leftrightarrow\exists y\in\mathbb{R},x=y^2$$
how to prove that the sum of two positive numbers is a positive number? ...
- 590
0
votes
1
answer
201
views
Distributive property conjunction / existential quantifier
Given the following predicate;
$(\exists x:X.P(x)\implies Q(x)) \wedge \forall y:X.P(y)$
Applying the distributive property of $\wedge$ / $\exists$
$\equiv \exists x:X.(P(x)\implies Q(x)) \wedge \...
- 3
2
votes
1
answer
625
views
Area of bounded region
$A$ is the region of the complex plane $\{z: z/4$ and $4/z'$ have real and imaginary part in $(0, 1)\}$, then $[p]$(where $p$ is the area of the region $A$ and $[.]$ denotes the greatest integer ...
- 546
1
vote
1
answer
75
views
Why is $\sum_{p\leq x} \log p=O(x)$?
I know that $$2^{2n}=(1+1)^{2n}={2n\choose 0}+\cdots+{2n\choose n}+\cdots+{2n\choose 2n}\geq {2n\choose n}\geq \prod_{n < p \leq 2n} p$$
If we let $n=2^{k-1}$ this implies that $\prod_{2^{k-1} <...
- 2,912
5
votes
4
answers
335
views
Understanding how contrapositive work
I want to understand contrapositive clearly. I'll start by saying: "If it is sunny, then there is light". The statement is true. But now consider the contrapositive: " If there is no light then it is ...
2
votes
2
answers
87
views
Prove $\{(2,0)\}$ is not extendable to a basis for $\mathbb{Z}^2$
I want to show that not every independent set in a free module is extendable to a basis.
Let $R=\mathbb{Z}$ and consider the $R$-module $M=R^2=\mathbb{Z}^2$. Then, $M$ is free of rank $2$ and $S=\{(2,...
- 2,398
2
votes
1
answer
166
views
Creating formula from algorithm
I have this algorithm which i have to convert to polynom of degree 2:
Algorithm is :
...
- 335
0
votes
0
answers
69
views
Example of Herbrand quotient hexagon
I want to construct an example of Herbrand quotient's hexagon diagram.
Let $0 \to p\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$ be a short exact sequence of $\mathbb{Z}$-modules and we ...
user695664
1
vote
0
answers
69
views
A guide on using integration by substitution.
Integration by substitution is a tricky thing to master, I'm learning it from books and at some place my book has done something like this
$$\int \csc\theta \cot\theta \; d\theta = ?$$
$$ \int \...
- 1,557
0
votes
2
answers
153
views
Prove:$\lim_{x\to \infty}\frac{\lfloor 3e^x\rfloor+2}{\lfloor 2e^x\rfloor+1}=\frac{3}{2}$
Prove that:
$\displaystyle\lim_{x\to \infty}\frac{\lfloor 3e^x\rfloor+2}{\lfloor
2e^x\rfloor+1}=\frac{3}{2}$
My attempt:
Let $f:\mathbb R\to S\subset\mathbb Q$
$$f(x)=\frac{\lfloor 3e^x\rfloor+...
- 4,575
3
votes
0
answers
75
views
Morita-invertible von Neumann algebras
I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of von ...
- 25.8k
0
votes
3
answers
1k
views
Monty Hall Problem - $4$ doors where you pick one by one
I encountered the problem of Monty Hall Problem with $4$ doors ($3$ doors $1$ goat). However in this variation you pick an initial door. Monty reveals one goat. Thereafter you either switch or stay - ...
0
votes
2
answers
59
views
Find the maximal value of $\mu^{-1}\cos\theta+\sin\theta$
This is part of a physics problem I was doing yesterday. I am supposed to find the maximum value of
$$\mu^{-1}\cos\theta+\sin\theta$$
This is is supposed to produce the result of $\sqrt{1+\mu^{-2}}$. ...
0
votes
2
answers
1k
views
Proving a bipartite graph is not planar
I have drawn the bipartite graph K3,3. And I need to prove this graph isn't planar. I used contradiction, assuming it was planar, using Eulers formula to prove it wasn't planar.
I concluded my graph ...
0
votes
0
answers
48
views
The gradient of $\| x - \alpha \nabla f(x) \|_2^2$ with respect to $x$?
The gradient of $\| x - \alpha \nabla f(x) \|_2^2$ with respect to $x$, where $x \in \mathbb{C}^n$, $\nabla f(x) \in \mathbb{C}^n$, and $\alpha \in \mathbb{R}$?
Following greg's comments, please see ...
- 661
1
vote
0
answers
225
views
The wedge product of covectors
On a textbook, the wedge product of the covectors $\Omega$ is spanned on basis vectors $dx$ using this notation. $$\Omega^i=\Omega^i_\mu dx^{\mu}$$
$i$ is the name of the covector, not an index.
$$\...
- 307
1
vote
1
answer
3k
views
Finding a Sine Wave That Goes Through a Set of Given Points
This question is coming from a thought experiment I was having by myself.
Let's say I have 3 random points on a x-y plane (with different x values). Let them be (1,3), (2,5), (5, 1). Is there a sine ...
- 11
1
vote
1
answer
201
views
finding the volume of a tethrahedron (course MIT 18.02 multivariate calculus)
The exercice of the session 48 of the course MIT 18.02 multivariate calculus, version 2010, obtainable here reads as follows
My question:
While I get the derivation and computation, what I'm ...
- 995
-1
votes
1
answer
83
views
Probability of drawing of $5$ cards from a deck
What is the probability of drawing of $5$ cards from a deck and getting
1) $3$ spades
2) at most $2$ spades
3) at least $3$ spades ?
My attempt is:
$(13C3)(49C2)/(52C5)$
$(13C0)(52C5)+(13C1)(...
- 99
1
vote
2
answers
111
views
In a convex pentagon ABCDE: $AB=AC$, $AE=AD$, $\angle CAD= \angle ABE + \angle AEB$, M is the midpoint of BE. Prove $2AM=CD$
I'd love to know how to prove this. I don't exactly know how to begin.
I can recognize the two isosceles triangles ABC and ADE.
I can also express M as $\frac{B+E}2$.
But then I'm stuck...
- 896
1
vote
1
answer
136
views
Limit of polynomial functions
Problem: Given a sequence $(f_n)_{n\in\mathbb{N}}$ of polynomial functions such that $f_n(x)=\sum_{j=0}^{n}\dfrac{j^3}{4^{j+2}}x^j$, find the largest real number $r$, such that $(f_n)_{n\in\mathbb{N}}$...
user731634
0
votes
1
answer
631
views
Calculate sum of sum using Z transform
$$S = \sum_{n=0}^{\infty} \frac{1}{3^n} \sum_{k=0}^{n} \frac{k}{2^k}$$
We know that if one signal y can be written as $$y(n) = \sum_{k=0}^{n} x(k)$$
then it's Z transform is $$ \frac{z}{z-1} X(z) $$ ...
- 1,219
2
votes
0
answers
225
views
Equivalence of two definitions of cross product for cohomology
Allen Hatcher's book gives two versions of the cross product.
The first one is:
\begin{eqnarray}
a\times b\equiv p_1^*(a)\cup p_2^*(b),
\end{eqnarray}
in which $p_1:X\times Y\rightarrow X$, $p_2:X\...
- 574