All Questions
1,620,001
questions
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Classify $\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}/\left<(1,1,2)\right>$
I am following a solution to classify $G/H$, $G=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z},H=\left<(1,1,2)\right>$, according to the theorem of finitely generated abelian groups.
It claims that ...
-1
votes
1
answer
14
views
How to prove whether the limit as $(x,y) \rightarrow (0,0)$ of $f(x,y) = \frac{x^2 y}{\sqrt{x^4 + y^2}}$ exists or doesn't, using delta epsilon
I don't get how I can cancel out the x^4 term in the denominator
0
votes
0
answers
10
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What does $a\ne b \ne c$ mean?
Does
$a\ne b \ne c$
mean
$a \neq b \land b \ne c$
or
$a \neq b \land b \ne c \land a\ne c $ ?
These are two distinct statements when 2 $\;\not\!\!\!\implies(a \ne c)$, yet I am unaware if one ...
- 370
0
votes
0
answers
17
views
Differentiability of $f(x,y)=\frac{xy}{x^2+y^2}$ at $(1,1)$
I tried to show that $f(x,y)=\frac{xy}{x^2+y^2}$ is differentiable at $(1,1)$ by doing the following but I got stuck:
$$\lim_{(h,k)\longrightarrow(0,0)}\frac{f(1+h,1+k)-f(1,1)-f_x(1,1)h-f_y(1,1)k}{\...
- 355
-2
votes
0
answers
9
views
Which is true, should i use improper integral or integral symmetry properties
∫sqrt(sin(t)^2 + cos(t)^2 - 2*cos(t) + 1) from 0 to 2pi
from one source the result is 0
SOLUTION
Your input: calculate ∫2π0(sin2(t)+cos2(t)−2cos(t)+1−−−−−−−−−−−−−−−−−−−−−−−−√)dt
First, calculate the ...
- 1
0
votes
0
answers
5
views
Is a distribution with continuous partial derivatives $C^1$?
This answer shows that a locally integrable function on a domain $\Omega$ that has continuous weak partial derivatives must be equal to a $C^{1}$ function a.e. My question is, does this hold for ...
0
votes
0
answers
8
views
Maximum possible rate of change of length-of-day?
What is the maximum possible rate of change of length-of-day (in units of minutes per day), on Earth? It is well known (and easy to observe) that it's larger near the equinoxes, and at higher ...
- 21
0
votes
1
answer
11
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Relationship of a section cut parallel to a base of a pyramid and its height proof
Is there an algebraic proof for this? I was trying to solve how this relationship was made. Thanks! Also, what is the proper name for this relationship?
- 1
-1
votes
0
answers
14
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Can we find any odd $n\in\mathbb{N}$ satisfying $n=e^2−f^2=a(g^2−h^2)=k^2−l^2$ and $e^2f^2=a^2g^2h^2+k^2l^2$ where $a,e,f,g,h,k,l (>1)\in\mathbb{N}$?
Can we find any odd natural number $n$ satisfying $n=e^2−f^2=a(g^2−h^2)=k^2−l^2$ and $e^2f^2=a^2g^2h^2+k^2l^2$ where $a,e,f,g,h,k,l~(>1)$ are natural numbers? The problem has a relation to a famous ...
0
votes
0
answers
10
views
Distance between relations
Assume I have two binary relations $R_{1}$ and $R_{2}$ on a set $S$, i.e., two subsets of $S \times S$. I want to compare these two relations in terms of similarity. Is there any distance metric that ...
- 321
-1
votes
0
answers
10
views
Finding expected value with replacement
I have balls numbered 1 to 100 in a bag. I take one out, record it, and put it back into the bag. I repeat this 1000 times. My questions are:
What is the expected value E[X] of never picking a number?...
0
votes
0
answers
8
views
``Anti-Freiman'' homomorphisms - Reference request
For two additive sets $A,B$ with respective ambient groups $G,F$, an order $2$ Freiman homomorphism from $A$ to $B$ is a map $\phi:A\to B$ with the property that,
$$a_1+a_2=a'_1+a'_2 \Longrightarrow \...
- 4,575
1
vote
0
answers
11
views
Explicit example of a Fourier transform of $f \in L^2(\mathbb R) \setminus L^1(\mathbb R)$
The Fourier transform shall be defined by
$$
\hat f(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \cdot \xi x} dx
$$
The Fourier transform is well-defined for $f \in L^1(\mathbb R)$, that is, $f$ ...
- 7,375
0
votes
0
answers
31
views
Showing the function $f'(a)=f(a)f'(0)$
Let $f(x+y)=f(x)+f(y)$ for all $x$ and $y$. Show that if $f'(0)$ exists then $f'(a)$ exist and $f'(a)=f(a)f'(0)$.
I can't really seem to make much progess, I let $x=y=0$ and got $f(0)=0$ and from ...
- 1
6
votes
0
answers
28
views
Can the multiplicative product of bijective functions $\mathbb{R} \to \mathbb{R}$ be bijective?
Given two functions $f$ and $g$, which are bijective $\mathbb{R} \rightarrow \mathbb{R}$, can $h(x) = f(x)g(x)$ also be bijective on $\mathbb{R} \rightarrow \mathbb{R}$?
I can prove no such $h$ exists ...
- 234
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0
answers
8
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Brezis' exercise 8.23.4: how to prove $\|u\|_{L^p} \leq \frac{1}{k+\delta / (p p')}\|f\|_{L^p}$ in case $p \in (1, 2)$?
Let $I$ be the open interval $(0, 1)$ and $k >0$. I'm trying to solve a problem in Brezis' Functional Analysis, i.e.,
Exercise 8.23
Given $f \in L^1(I)$, prove that there exists a unique $u \in ...
- 16.7k
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answers
22
views
A tricky question on kinematics asked in a competitive exam
So I found this rather ambiguous question based on kinematics from a competitive examination,
the question with options is -
A car travels in such a way that its velocity becomes twice the previous ...
0
votes
0
answers
6
views
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
0
votes
0
answers
9
views
On the role of PDFs in the determination of impossible events
Let's consider a random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$. Is it true to say that for any $x_0:p(x_0)=0$ the event $X=x_0$ is impossible?
By ...
- 501
0
votes
0
answers
8
views
Meromorphic infinite product mapping upper half-plane to itself [duplicate]
Given 2 sequences $(a_n)_{n\in\mathbb{Z}}$, $(b_n)_{n\in\mathbb{Z}}$ with $b_n<a_n<a_{n+1}$, and $a_{-1}<0<b_1$ I'm trying to show that the function defined by
$$\theta(z)=\frac{a_0-z}{b_0-...
- 1
-2
votes
1
answer
22
views
Does the given sequence converges or diverges?
If b>1 and is a real number then does the sequence converge or diverge. I intuit that it diverges because b^n grows faster than (n!)^1/2?
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0
answers
13
views
Definition of uniforlmly positive definite matrix
Let $M\in C(\mathbb R, \mathbb R^{n^2})$. What does it mean that $M$ is uniformly positive definite?
I was not able to find a definition anywhere, but I think it could be something like
$$\exists \...
- 456
1
vote
1
answer
14
views
Why using w-domain instead of f-domain in Fourier Transform??
I saw that when we change time domain into frequency domain, we can use frequency itself (denoted as X(f)) or angular frequency domain like X(w).
I know that w = 2πf. Is there any reason there are two ...
- 45
-1
votes
0
answers
9
views
Distance, speed and time word problem. Calculate Time
A Business owner Alex, got a call for an urgent business meeting and he had to drive through a road which did not have network coverage. He immediately left in his Car driving at constant speed of 87 ...
0
votes
0
answers
4
views
$\frac{d}{dt}(d\psi^t)_x(Z)=(d\psi^t)_x([X_H,Z])$ for a critical point $x$ of a Hamiltonian $H$
Let $W$ be a closed symplectic manifold, $H:W\to \Bbb R$ a Hamiltonian, $x\in W$ a critical point of $H$, $X_H$ the associated Hamiltonian vector field, and $\psi^t:W\to W$ the (global) flow of $X_H$. ...
- 2,100
1
vote
0
answers
21
views
Degree 1 divisor on a curve
Let $C$ be a smooth projective curve of genus $g$ over complex numbers.
Is it true that any degree one divisor is linearly equivalent to $1.p$ for some point $p$?
I know it's true for $\mathbb P^1$. ...
- 2,286
0
votes
0
answers
11
views
Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $
Let $a,b,c\ge 0: ab+bc+ca+abc=4.$ Prove that
$$\color{black}{\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. }$$
Equality holds at $a=b=...
- 661
-1
votes
0
answers
5
views
Kronecker sum of a matrix with its transpose
I know that if $A + A^T = B$, then $B$ is symmetric. How about the Kronecker sum? If $A \oplus A^T = B$, $B$ or $A$ have some particular properties?
-1
votes
0
answers
9
views
The number of integral values of a for which the equation x⁴ - (a + 2)x³ + 2ax²+ 4(a - 2)x -16= 0 has at least two positive roots; a ∈[-10, 10] is-
The number of integral values of a for which the equation
x⁴-(a + 2)x³+2ax²+ 4(a - 2)x -16= 0
has at least two positive roots; a ∈[-10, 10] is/are
0
votes
0
answers
16
views
Are there some famous conjectures related to perfect graphs?
I am reading the book "Graph Classes: A Survey", and I know that Perfect Graph Conjecture (PGC) and Strong Perfect Graph Conjecture (SPGC) once were famous conjectures about perfect graphs.
...
- 656
-1
votes
1
answer
28
views
Monotone functions on a compact interval that converge pointwise to a continuous function, show uniform convergence.
This is Exercise V.2.5 from Amann's Analysis I, page 375.
Let $(f_n)$ be a sequence of monotone functions on a compact interval $I$ which converges pointwise to a continuous function $f$. Show that $...
- 21
0
votes
0
answers
6
views
Do modal calculi work with possible worlds?
I use a natural deduction calculus for modal propositional logic, but my question eventually is about any (sound) modal calculi with/without axioms.
Just as an example take a proof like $\square$A $\...
- 9
0
votes
0
answers
47
views
Can't figure out integration by parts mistake in integrating f(x)/f'(x)
Start with $$ \int\frac{f(x)}{f'(x)} \ dx $$
Let $$ u = \frac{f(x)}{f'(x)}\ $$ $$ dv=dx $$
Then $$ du = (1-\frac{f(x)f''(x)}{(f'(x))^2})dx $$ $$ v=x $$
Integrate by parts: $$\begin{align*}
\int\frac{...
- 11
0
votes
1
answer
22
views
Prove $\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}$
For all $a,b,c\ge 0: ab+bc+ca>0$ then prove
$$\color{black}{\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}....
- 665
1
vote
1
answer
16
views
How to define the sum of cardinals in the definition of an inaccessible cardinal?
I'm reading the Wikipedia definition of an inaccessible cardinal and I'm trying to understand it.
On Wikipedia, a (strongly) inaccessible cardinal $\kappa$ is defined in the following way:
$\kappa$ ...
- 10.1k
-1
votes
1
answer
21
views
Why in the Alternating Series Test we need to consider both partial sums with even and odd number of terms?
I'm sorry, I think this is a pretty stupid question, but can not we just prove it for 1 of them? I mean, by Definition of Series Convergence :
if partial sum is convergent then the series is ...
0
votes
0
answers
10
views
Second place in the Turing Machine race
Given a deterministic Turing machine T which begins on an infinite blank strip, let its growth rate $G_T(t)$ represent the number of non-blank squares after the machine is run for $t$ time steps.
It ...
- 2,444
0
votes
0
answers
30
views
Check that $T-\lambda I$ is bounded
I have a question for Spectrum for a bounded linear operator and its adjoint on a Banach space are same.
I have to show that spectrum for a bounded linear operator and its adjoint on a Banach space ...
- 1,398
0
votes
0
answers
12
views
Additive white Gaussian noise (AWGN)
In my research work regarding wireless communication, I came across many research papers wherein AWGN is assumed to be modelled as "complex Gaussian with zero mean and unit variance".
I ...
- 101
1
vote
0
answers
28
views
Does infinite system of measurable subsets of a measurable set have a measurable intersection
Let $U$ be a finitely-measurable set, $G_1, G_2, \dots$ are its measurable subsets such that for every $i$ $\mu(G_i) \ge \alpha > 0$. Does there necessarily exist a measurable subset $\Gamma$ such ...
- 23
3
votes
1
answer
44
views
Understanding an inequality in the proof
I believe my question is on some very elementary computations (i.e., this is probably answerable even if you don't know what is the subgaussian norm $\| \cdot \|_{\psi_2}$ etc. However, do let me know ...
- 459
-1
votes
0
answers
11
views
Multiplicative bijective continuous map from complex number to complex number.
Is there any special form of this kind of map. Obviously it will take 1 to 1 and roots of unity to roots of unity. But how we can go further, how it can acts on real numbers. Any hint will be very ...
- 172
-2
votes
0
answers
40
views
How could you solve the equation: 2^x + .5 = 4 - x^2 without graphing it
How could you solve for $x$ without graphing it, assuming you have a non-graphing calculator
$$2^x+0.5=4-x^2$$
- 1
0
votes
0
answers
31
views
$(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$ [duplicate]
Prove that $(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$. Here $a, b, m, n\in\mathbb{Z}^+$, $m\leq n$ and $(a, b)=1$.
This is a questions from a number theory book that I am recently studying.
I have read ...
- 2,900
0
votes
0
answers
43
views
Fundamental axioms of logic used to interpret most of modern mathematics
so i'm going through Terrence Tao's analysis 1 and he has a clear emphasis on rigor, however its kind of a contradiction when he didn't even go over the basics of mathematical logic.
A mathematical ...
- 21
-7
votes
0
answers
30
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Advice: best not to use the next quantum "detectors" on Earth [closed]
It will be best not to use the next gen quantum "detectors" on Earth. <br>
A good example of an early quantum sensor is an avalanche photodiode (APD). APDs have been used to detect ...
-2
votes
0
answers
16
views
Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, calculate the conditional disribution of X given $\sigma(f), f(x)=x^TD^{-1}x$
Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, and follow the n-dimensional multivariate norm distribution $N(0,D)$. Let$$f(x)=x^TD^{-1}x,x\in R^{n}.$$ So how to get the ...
2
votes
0
answers
16
views
Covariance of the product of two random variables with another random variable
Let X, Y and V be three binomial random variables, that are NOT independent from each other. My objective is to find an expression for the covariance \begin{align}\operatorname{Cov}(XY,V)&\end{...
- 151
5
votes
1
answer
55
views
Number of natural numbers with an odd sum of number of divisors till that number
Find the number of natural numbers $n<2023$ such that $\sum_{i=1}^n \sigma_0(i)$ is odd, where $\sigma_0(x)$ denotes the number of divisors of $x$.
First, I know that $\sigma_0(x) = (n_1 +1)(n_2 +...
- 1,085
0
votes
0
answers
41
views
When is the power rule for limits acceptable to use?
Suppose you wish to calculate the following limit:
$$\lim_{x\to\infty}\frac{(1+x^2)^{-a}}{e^{-bx}}$$
for real numbers $a,b>0$. My instinct here is to rewrite as
$$\lim_{x\to\infty}\left(\frac{1+x^2}...
- 1,325