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Assume
$$
F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, F\left( x \right) := \sum_{n=-\infty}^{+\infty}{\left( -1 \right) ^nx^{n^2}},
$$
I have to prove
$$
\left( G\left( x \right) \right) ^2+\left( H\left( x \right)... |
How to prove the equality of power series below? |
Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th [Jacobi polinomial][1]. An old result of Bateman shows that $$
\left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\right)}{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(1\right)}=\frac{\Gamma\left(\alpha+\mu+1\right)}{\Gamma\left(... |
This is a question on something on something more general, but for now I'd like to keep in simple. Consider a module homomorphism $\phi:\mathbb{Z}^n\to\mathbb{Z}$, where $n$ is a positive integer. Here I will consider $\mathbb{Z}^n$ as a (left) $\mathbb{Z}$-module. My question is: what are all the types of (module) hom... |
Let $n\geq 2$ be an integer, and consider $H=$\langle (1 \ 2 \ \cdots \ n), (a \b)\rangle\subset S_n.$
It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.
**Question.** What is the order of $H$ in general ?
For the moment, I think I have managed to prove the following facts:
Let $d=gcd(n, b-... |
What is the order of the following subgroup $\langle (1 \ 2 \ \cdots \ n), (a \b)\rangle$ of $S_n$? |
In Discrete math and its app book from Kenneth H. Rosen says that
> What is the symmetric closure of the relation $R=\{(a,b) | a >b\}$ on the set of positive integers ?
The answer of book following:
> $R \cup R^{-1}=\{(a,b) | a >b\} \cup \{(b,a) | a >b\}= \{(a,b) | a \neq b\}$
I think that it should have b... |
I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry".
In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he doesn't prove it. And he says we can find elliptic results in "Gilbarg, D., and N.S. Trudinger, Elliptic Partial Differential Equatio... |
A fair coin is tossed $n\geq 2$ times. Player 1 wins if there are more HH than HT in the sequence, Player 2 wins if there are more HT than HH. Who has the higher probability of winning ?
Modelling the tosses with iid $X_i$, the number of $HH$ is $S_n = \sum_{i=1}^{n-1} 1_{X_i=H,X_{i+1}=H}$ and the number of HT is $R... |
Find generating function of $\frac{5-3x}{2-3x+x^2}$. My attempt: $\frac{5-3x}{2-3x+x^2} = \frac{1}{2} \cdot (5-3x) \cdot \frac{1}{1-x/2} \cdot \frac{1}{1-x}$. Here you use geometric series, but I don't know, how to find the coefficient before $x^n$ in this form. Any idea?
|
Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$
It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.
**Question.** What is the order of $H$ in general ?
For the moment, I think I have managed to prove the following facts:
Let $d=\gcd(n, b... |
What is the order of the following subgroup $\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle$ of $S_n$? |
I'm showing that the coefficients of the Fourier series of a $2\pi$-periodic function $f$ can be written as
$$a_k = \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \left(f(x) - f\left(x - \frac{\pi}{k}\right)\right)\cos(kx) dx$$
and
$$b_k = \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \left(f(x) - f\left(x - \frac{\pi}{k}\right)... |
Why the integral in $[-\pi-\frac{\pi}{k}, \pi-\frac{\pi}{k}]$ equals the integral in $[-\pi, \pi]$? |
Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$
It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.
**Question.** What is the order of $H$ in general ?
For the moment, I think I have managed to prove the following facts:
Let $d=\gcd(n, b... |
I'm learning Radon-Nikodym theorem and I have some doubts now. Let $\mu(B(x,r))=r^2$ for each $x\in\mathbb{R}$ and $r>0$, then $\mu$ is a radon measure on $\mathbb{R}$ and $\mu<<\mathscr{H}^1$. By Radon-Nikodym theorem, we have $\mu(E)=\int_E f d\mathscr{H}^1$ for some $f$. We can calculate that $f(x)=\lim_{r\to 0}\fra... |
Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$
It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.
**Question.** What is the order of $H$ in general ?
For the moment, I think I have managed to prove the following facts:
Let $d=\gcd(n, b... |
It's given that $$I_n=\int_0^{\frac{\pi}{2}}\frac{\sin^2nx}{\sin x}dx,$$and I need to prove that $$I_n\leq1+\ln\sqrt{2n-1}.$$
I have got that $$I_n-I_{n-1}=\int_0^{\frac{\pi}{2}}\frac{\sin^2nx}{\sin x}dx-\int_0^{\frac{\pi}{2}}\frac{\sin^2(n-1)x}{\sin x}dx\\=\int_0^{\frac{\pi}{2}}\frac{[\sin nx-\sin(n-1)x][\sin nx+\sin... |
Given that $I_n=\int_0^{\frac{\pi}{2}}\frac{\sin^2nx}{\sin x}dx$,how to prove that $I_n\leq 1+\ln\sqrt{2n-1}$? |
I was told that the retraction of S^2 into a point is a retract but not a deformation retract , could someone explain this to me please?
If I understood this wrongly, could someone give me an example of a retract that is not a deformation retract?
|
I can see that the map $f(z) = z^k$ is an homeomorphism for neighbourhoods around non-zero points not including zero. So I need to check only around neighbourhoods of zero.
Can anyone help me out? |
> Find generating function of $\frac{5-3x}{2-3x+x^2}$.
My attempt: $\frac{5-3x}{2-3x+x^2} = \frac{1}{2} \cdot (5-3x) \cdot \frac{1}{1-x/2} \cdot \frac{1}{1-x}$. Here you use geometric series, but I don't know, how to find the coefficient before $x^n$ in this form.
Any idea?
|
$f$ is integrable on $[a,b]$, $\int_a^bf(x)^2>0$. Prove that there is a point of continuity $x$ on $[a,b]$ that $f(x) \neq 0$. I provided the following: almost every point of $[a,b]$ is a continuity point. $F(x) = \int_a^bf(x)^2>0$ is differentiable in every continuity point and in such point $F'(x) = f^2(x)$. If $F(x)... |
> Why do we take it to be A implies B and not, say, A and B, or, other
> truth-functions?
We want to rule out the possibility that both $A$ is true and $B$ is false $\neg (A \land \neg B)$ which is often used to define logical implication:
$~~~~~~~A\implies B~~\equiv \neg(A \land \neg B)$
From this definition... |
In fact, LP has used many different tools from mathematics; for example, see [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
LP and its well-known extensions, such as convex optimization and integer programming, are also used in different branches of math... |
> Find generating function of $\frac{5-3x}{2-3x+x^2}$.
My attempt: $\frac{5-3x}{2-3x+x^2} = \frac{1}{2} \cdot (5-3x) \cdot \frac{1}{1-x/2} \cdot \frac{1}{1-x}$. Here you use geometric series, but I don't know, how to find the coefficient before $x^n$ in this form.
Any ideas?
|
A fair coin is tossed $n\geq 2$ times. Player 1 wins if there are more HH than HT in the sequence, Player 2 wins if there are more HT than HH. Who has the higher probability of winning ?
Modelling the tosses with iid $X_i$, the number of $HH$ is $S_n = \sum_{i=1}^{n-1} 1_{X_i=H,X_{i+1}=H}$ and the number of HT is $R... |
For some time now, I have been able to use the Conditional Proof rule in Natural Deduction. We assume A, derive B within the scope of the assumption and discharge the assumption to get that A implies B. This is, I believe, the syntactic approach to the proof, pure rule-based. However, what is the motivation behind this... |
I was told that the retraction of $S^2$ into a point is a retract but not a deformation retract , could someone explain this to me please?
If I understood this wrongly, could someone give me an example of a retract that is not a deformation retract?
|
I heard the following:
> For a smooth submanifold $M \subseteq \mathbb{R}^n$,
> given a local trivialization one can easily find a basis for
> the tangent space.
I want to know how.
So first I should maybe formalize the question:
Suppose $M \subseteq \mathbb{R}^n$ is a smooth $k$-dimensional submanifold and... |
Is it possible to prove the measurability of the following map
$\Phi_n \colon (C(\mathbb{R}^d), \sigma(\mathcal{C})) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, $n \in \mathbb{N}$, defined by
$$ \Phi_n(f) := \int_{[-n,n]^d} |f(x)|^2 \, dx, \quad f \in C(\mathbb{R}^d), $$
where $\mathcal{C}$ is the collection of fin... |
Is this form useful for to detecting prime numbers, or can it be discarded?
$f(x)= |sin\frac{\pi x}{P_1} |+|sin\frac{\pi x}{P_2} |+⋯+|sin\frac{\pi x}{P_n} | $
Here is an example:
$P_1=2; P_2=3$
$f(x)=|sin\frac{\pi x}{2} |+|sin\frac{\pi x}{3}|$
[enter image description here][1]
The result is tw... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
where $\operatorn... |
What is $\sqrt{-1}$? circular reasoning defining $i$. |
I have a universal first order universal theory T such that:
-T is universal over a finite language
-T has a model completion T*
-T* is countable, has quantifier elimination (QE) and is complete.
Now considering a theory T' extending T, suppose T' has QE. Can we say that T' is (at most) countable? If so, ar... |
I have a universal first order universal theory T such that:
-T is universal over a finite language
-T has a model completion T ⊆ T*
-T* is countable, has quantifier elimination (QE) and is complete.
Now considering a theory T ⊆ T', suppose T' has QE. Can we say that T' is (at most) countable? If so, are th... |
Let $G=\langle a,b\rangle$ be a finite $p$-group such that $a^p=b^p=1$. Is there any result about the size of the set of $p$-elements $\Omega(G):=\{g\in G\mid g^p=1\}$? In particular, I'm interested in the fraction $|\Omega(G)|/|G|$.
Edit: I was exactly wondering about if the fraction $|Ω(G)|/|G|$
could be done ar... |
There are $n$ teams in a sports tournament, and each team has to play every other team, and all teams have to play every weekend over $n-1$ weekends.
For example, rugby six nations $n=6$ pretty much follows this pattern.
Hoe many ways are there to uniquely order the matches?
Note: we are not counting home vs ... |
I was reading this part in Allen Hatcher book :
[![enter image description here][1]][1]
But I did not understand it, can someone explain it to me with drawing please?
[1]: https://i.stack.imgur.com/mFrNB.png |
There is a number $x$ containing $1999$ digits and is divisible by $9$; if we add all digits of $x$ we get $y$ and as same adding digits of $y$ we get $z$. Find the probability that $z$ contains two digits.
----------
*My Attempt*
let $y=9k \;; k\in \mathbb N$, $$y_{max}=9\times1999=17991 \qquad \qquad y_... |
**By induction:**
For $n=1,2$, $I_1=1,I_2=\frac{4}{3}<1+\ln\sqrt{3}$ is easy to check.
Suppose $I_n<1+\ln\sqrt{2n-1}$,
then
$$I_{n+1}=I_n+\frac1{2n+1}<1+\ln\sqrt{2n-1}+\frac1{2n+1}.$$
We need to prove
$$1+\ln\sqrt{2n-1}+\frac1{2n+1}<1+\ln\sqrt{2n+1},$$
which is equivalent to
$$\frac1{2n+1}<\frac{1}{2}\ln\frac{... |
> Suppose function $F$ is twice-differentiable, L-lipschitz smooth and $\mu$-strongly convex. Let $\boldsymbol{x}(\xi), \boldsymbol{y}(\xi) \in \mathbb{R}^d$, for a random variable $\xi$. Denote $\boldsymbol{H}=\int_0^1 \nabla^2 F(\boldsymbol{y}+s(\boldsymbol{x}-\boldsymbol{y})) \mathrm{d} s$. Then, prove that
$$
\m... |
1 -((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2 -(¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3 -(¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-groups ins... |
Let $M \subseteq \mathbb{R}^n$ be a smooth $k$-dimensional submanifold and let $A \subseteq \mathbb{R}^n$ be an open subset with $M \subseteq A$.
Show that if $\psi$ is a diffeomorphism from $A \rightarrow B \subseteq \mathbb{R}^n$, then the image $\psi(M)$ is a (smooth) submanifold.
My approach:
Since $M$ is a su... |
There Is a somewhat misconception i can't understand about P=NP. For example, the Hilbert's tenth problem says that we can't decide , given a diophantine Equation, if we can solve It or not. But we can clearly check that a given solution Is Indeed a solution in polinomial time. Doesn't that disprove the P=NP conjecture... |
$f$ is integrable on $[a,b]$, $\int_a^bf(x)^2>0$. Prove that there is a point of continuity $x$ on $[a,b]$ that $f(x) \neq 0$. I provided the following: almost every point of $[a,b]$ is a continuity point. $F(x) = \int_a^bf(x)^2>0$ is differentiable in every continuity point and in such point $F'(x) = f^2(x)$. If $F(x)... |
There is a number $x$ containing $1999$ digits and is divisible by $9$; if we add all digits of $x$ we get $y$ and as same adding digits of $y$ we get $z$. Find the probability that $z$ contains two digits.
----------
*My Attempt*
let $y=9k \;; k\in \mathbb N$, $$y_{max}=9\times1999=17991 \qquad \qquad y_... |
1 -((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2 -(¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3 -(¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-groups ins... |
I was reading this part in Allen Hatcher book :
[![enter image description here][1]][1]
But I did not understand it, can someone explain it to me with drawing please?
**Edit:**
I need a picture for the last 3 lines.
[1]: https://i.stack.imgur.com/mFrNB.png |
Why is the condition that $f$ requires only simple poles in Mittag-Leffler's Expansion theorem necessary? |
this question is from a interview a friend had and I was curious how to solve it.
The question is:
>I have two random variables, $a$ and $b$.
$a=rand(1,100)$
$b=rand(a,100)$
The function rand generates a number within the range of the numbers in the parentheses, with equal probability for each number.
How is ... |
Consider the following linear matrix inequality (LMI) feasibility problem
$$
\begin{aligned}
\textrm{find} \quad & X \\
\textrm{s.t.} \quad & X \succ 0 \\
\quad & F(X) \prec Q
\end{aligned} \tag{1}
$$
where $F$ is linear in $X$ and $Q \succeq 0$. I'm trying to solve this feasibility problem using an LMI solver,... |
How can I convert a strict (affine) LMI feasibility problem into a non-strict one? |
Consider the following system of equations:
$$\begin{cases}
1 + 6(x+y) + 8xy = 0 \\ y = 6x^2 - 8 xy^2
\end{cases}
$$
If for example from the first one I solve for $x$, and then I substitute into the second one I get a fourth-degree equation, even a bit painful since it's also complete.
I checked with some pro... |
How to know if there are real solutions or not? |
Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$
It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.
**Question.** What is the order of $H$ in general ?
For the moment, I think I have managed to prove the following facts:
Let $d=\gcd(n, b... |
Hint:
Observe that $\dfrac{2x+y+\ \ y}2=?,\dfrac{2x+y-\ \ y}2=?$
$$\dfrac{\sin(2x+y)}{\sin y}=\dfrac75$$
Apply [Componendo & Dividendo][1]
$$\dfrac{7+5}{7-5}=\dfrac{\sin(2x+y)+\sin y}{\sin(2x+y)-\sin y}$$
Apply [Prosthaphaeresis][2] Formulas
[1]: https://brilliant.org/wiki/componendo-and-dividend... |
this question is from a interview a friend had and I was curious how to solve it.
The question is:
>I have two random variables, $a$ and $b$.
$a=rand(1,100)$
$b=rand(a,100)$
The rand function generates a number within the range of the numbers in the parentheses, with equal probability for each number.
... |
Let $F(n,p)=\sum_{m=p}^{\infty} \frac{(n+m)!}{n!\ m!\ 2^m}$. What is the upper bound for $F(n,p)$, where $p\geq n+1$ and $n\geq 0$? Does there exist a tight bound where the value of $p$ could be easily estimated under a constraint $F(n,p)\leq 2^{-k}$, where $k>1$?
Note that $F(n,p)=\frac{(n+p)!}{n!\ p!}\left(\frac{1... |
Is this form useful for to detecting prime numbers, or can it be discarded?
$f(x)= |sin\frac{\pi x}{P_1} |+|sin\frac{\pi x}{P_2} |+⋯+|sin\frac{\pi x}{P_n} | $
Here is an example:
$P_1=2; P_2=3$
$f(x)=|sin\frac{\pi x}{2} |+|sin\frac{\pi x}{3}|$
[![P_1=2; P_2=3 example][1]][1]
The result is two ... |
Is there any analytical solution of $\sin mx =k \sin x$ for $x$, where $m>1$ and $k\neq 0$? I also need solutions of $\cos mx =k \cos x$ and $\tan mx =k \tan x$ for $x$. When $k=1$, the solution is $x=\pi/(1+m)$. But cannot solve for $k \neq 1$. |
Is there any analytical solution of $\sin mx =k \sin x$ for $x$, where $m>1$ and $k !=0$? |
The Vassilev invariant of degree 2, which i'll denote as $v_2$ may be computed as the sum of two multiple integrals,
$v_2=\rho_1(\gamma)+\rho_2(\gamma)$
for a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, by integrating over the configuration spaces:
$\Delta_4=\{(t_1,t_2,t_3,t_4)|0<t_1<t... |
Is there any analytical solution of $\sin mx =k \sin x$ for $x$, where $m>1$ and $k \neq 0$? |
If I have the identity
$$f(a)f(b)=f(a')f(b')$$
and additionally the constraint
$$a+b = a'+b' = const.$$
for any suitable pairs of real numbers $a,b$ and $a',b'$, would this be sufficient to conclude that
$$f(a)f(b)=f(a+b)$$
or are there further conditions required?
I know that the latter relationsh... |
The Vassilev invariant of degree 2, which i'll denote as $v_2$ may be computed as the sum of two multiple integrals,
$v_2=\rho_1(\gamma)+\rho_2(\gamma)$
for a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, by integrating over the configuration spaces:
$\Delta_4=\{(t_1,t_2,t_3,t_4)|0<t_1<t... |
COMMENT.- (Just for help you a little).
$(1)\space\space P(x)\equiv0\pmod{p^{\alpha}}\Rightarrow P(x)\equiv0\pmod p$. If $P(x)$ has degree $4$ and has more than $4$ solutions, then all the coefficients of $P(x)$ are multiples of $p$ (i.e. are $0$ modulo $p$)
$(2)$ Let $P(x_1)\equiv0\pmod p$ and put $x=x_1+pt_1$ w... |
Assume
$$
F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, H\left( x \right) := \sum_{n=-\infty}^{+\infty}{\left( -1 \right) ^nx^{n^2}},
$$
I have to prove
$$
\left( G\left( x \right) \right) ^2+\left( H\left( x \right)... |
Consider some differential equation on a square domain with $0 \leq x \leq L_1$ and $0 \leq y \leq L_2$, and consider the boundary conditions $u(x, 0) = 0$, $u(y, 0) = 0$, $u(L_1, y) = k_1$ and $u(L_2, y) = k_2$
Does this not mean that $u(L_1, L_2) = k_1 = k_2$ as it is in the corner and must satisfy both of the end... |
Am I misunderstanding boundary conditions? Should they be continuous? |
Let $\{B_j(p) | j \geq 1\}$ be a sequence of Bernoulli trials or an i.i.d. sequence of Bernoulli$(p)$ random variables, where the win probability is
$$p = P\{B_j(p) = 1\} = E[B_j(p)].$$
Now define $X$ to be the negative binomial random variable that counts the wins within all the Bernoulli$(p)$ trial winning strea... |
The problem occurs when reading [Bombardini et al., 2023, "Did US Politicians Expect the China Shock?", *American Economic Review*, Vol.1, PP174-209](https://bcf.princeton.edu/wp-content/uploads/2021/03/BLT25jan2021.pdf).
The authors define $\xi_{it}$ to be a Gaussian term obeying $N(0,2\sigma_{\xi}^2)$ (see line 7,... |
If I have the identity
$$f(a)f(b)=f(a')f(b')$$
for a given normalized probability distribution $f$, and additionally the constraint
$$a+b = a'+b' = const.$$
for any suitable pairs of real numbers $a,b$ and $a',b'$, would this be sufficient to conclude that
$$f(a)f(b)=f(a+b)$$
or are there further cond... |
Indeed, there is an apparent paradox as can be seen on the following example (similar to yours) with two hyperbolas (featured with their asymptotes) :
$$r=\frac{p_0}{1 - e_0 \cos \varphi} \ \text{and} \ r=\frac{p_1}{1 - e_1 \cos \varphi}\tag{1}$$
with $$e_0=1.5, e_1=2, \ p_0=1, \ p_1=-0.5\tag{2}$$
[![enter ... |
Consider some differential equation on a rectangular domain with $0 \leq x \leq L_1$ and $0 \leq y \leq L_2$, and consider the boundary conditions $u(x, 0) = 0$, $u(0, y) = 0$, $u(L_1, y) = k_1$ and $u(x, L_2) = k_2$
Does this not mean that $u(L_1, L_2) = k_1 = k_2$ as it is in the corner and must satisfy both of th... |
Finding value of $$I=\int^{\pi\over2}_0 \frac{sin(nx)}{sin{x}} dx=?$$
----------
I got this question in my book. I got if $n$ is even then $I$ is $0$, by using king's rule it turns out to be $I=-I$
But I couldn't solve it for $n \in$ odd number. I tried by taking $n=1,3,$ and $5$ and simply expanding $sin3... |
Finding value of $I=\int^{\pi\over2}_0 \frac{sin(nx)}{sin{x}} dx=?$ |
Mac Lane and Moerdijk have a neat proof on p. 197 that, in a topos $\mathcal{E}$, every epimorphism $f \colon C \to B$ is a coequalizer. But this depends on the assumption that the slice category $\mathcal{E}/B$ is itself a topos.
The Elephant gets there sooner, and has a proof by p. 38 that, in a (pre)topos, every ... |
Elementary proof that, in a topos, every epimorphism is regular? |
$$f(a)f(b)=f(a+b)f(0)=f(a+b)\iff f(0)=1$$
Hence, $f(a)=c^a$ for some constant $c$ (see e.g. https://math.stackexchange.com/questions/1548249/a-function-with-a-property-fxy-fxfy?noredirect=1&lq=1).
|
I have some recorded data s(x), and need to compute the value and uncertainty in
$$S = \int_{x_{1}}^{x_{2}} s(x) dx$$
I am using the trapezium rule for the integration, and need to estimate the uncertainty in the integral due to uncertainties in x and s(x). I have uncertainty estimates for every x and s(x) value (w... |
Each point of a unit sphere is conveniently also its own normal. I am starting with a unit sphere and then multiplying each point by a scale vector to create an ellipsoid. I would like to know the normal vector at each vertex. Is there some simple math to get the normal vector given the unit sphere vector and the scale... |
Let $M$ be a compact metric space and $\mathcal C^0(M)=\{f:M\to\mathbb R; f\ \text{is continous}\}$. Let $T:\mathcal C^0(M)\to \mathcal C^0(M)$ be a bounded linear transformation.
Suppose that for every $f\in\mathcal C^0(M)$ we have that $Tf$ is $1/2$-Hölder. Does this imply that $T$ is compact?
----------
It ... |
Does improvement of regularity implies compactness? |
Is this form useful for to detecting prime numbers, or can it be discarded?
$f(x)= |sin\frac{\pi x}{P_1} |+|sin\frac{\pi x}{P_2} |+⋯+|sin\frac{\pi x}{P_n} | $
Here is an example:
$P_1=2; P_2=3$
$f(x)=|sin\frac{\pi x}{2} |+|sin\frac{\pi x}{3}|$
[![P_1=2; P_2=3 example][1]][1]
The result is two ... |
Let $\mathbb F_q$ be a finite field of $q$ elements.
Let $G$ be a group with two generators $\{a,b\}$. Let $V$ be a non-trivial representation of $G$ over $\Bbb F_q$.
Consider the set of maps
\begin{equation}
\begin{aligned}
S:=\{f:G\longrightarrow A\bigg|f(g) - h\cdot f(g) = f(h) - g \cdot f(h),\ \ \ \forall g... |
Does improvement on regularity implies compactness? |
Is there any constraint on the functional $F$ for an Euler-Lagrange problem? For instance, if we take
$$f(y,y')=y'+y$$
and let $F$ be the integral of $f$ on a certain interval.
Then Euler-Lagrange equation becomes
$$\frac{\partial f}{\partial y}=1\ne0=\frac{d}{dx}\frac{\partial f}{\partial y'}$$
which is obv... |
I'm currently working on an interesting problem in function approximation thta just came to mind and am seeking insights or methodologies that might aid in approaching it. The problem is as follows:
Given a continuous function say, $f(x) = \sin(x)$, I want to find a polynomial $P(x)$ of degree $n$, where each ... |
**Background**
In the <a href="https://math.stackexchange.com/questions/170747/sum-equals-integral-identities-similar-to-int-01-fracdxxx-sum-n%20">following question</a> and references therein, a number of "Sum equals integral" identities are described.
For instance, we have $$\int_{-\infty}^{+\infty} {\rm sinc}... |
Consider a set of (not necessarily consecutive) prime numbers, $S: = \{ p_1, p_2, \ldots, p_k\}.\ $ For each integer $n,$ for each $1\leq j \leq k,$ let (the function) $u_n(p_j)$ be the greatest nonnegative integer such that ${p_j} ^ {u_n(p_j)}$ divides $n.$
Next, define the function $f_S(n):= \displaystyle\sum_{j=... |
Question about sum of indices of prime factorisation of consecutive numbers that might be solved via Chinese remainder theorem? |
Let the partial sums $s_n$ of $\sum^{n}_{j=1} a_j$ be bounded above by $A > 0$ and the sequence $b_n$ satisfy $b_1\geq b_2\geq b_3...\geq b_n\geq 0$. I'm trying to show the following:
$$ |\sum^{n}_{j = 1}a_jb_j| \leq 2Ab_1 $$
I tried breaking the summation up via "summation by parts" below and ended up with a differe... |
Consider the cutting stock problem where $L$ is the length of the stock, $m$ is the number of different roll widths, and $n$ is the total number of patterns. Why is it the case that $n \geq 2^{\sqrt{L}-1}$? |
Let X be a separable Hausdorff space such that for every $x \in X$ there exists an open neighborhood $U$ of $x$ such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$.
Show that:
(i) $X$ is locally compact
(ii)there exists a countable compact cover of $X$.
My attempt:
(i)
Suppose $x \in X$ and ... |
Is there a matrix calculus rule for this? |
Consider a set of (not necessarily consecutive) prime numbers, $S: = \{ p_1, p_2, \ldots, p_k\}.\ $ For each integer $n,$ for each $1\leq j \leq k,$ let (the function) $u_n(p_j)$ be the greatest nonnegative integer such that ${p_j} ^ {u_n(p_j)}$ divides $n.$
Next, define the function $f_S(n):= \displaystyle\sum_{j=... |
The group of orientation-preserving $H^\infty$-diffeomorphisms on $\mathbb{R}^n$ is defined as
$$\operatorname{Diff}(\mathbb{R}^n):=\{\operatorname{id}+f\,|\,f\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)\text{ and } \det(\operatorname{id}+df)>0 \}$$
where
$$H^\infty(\mathbb{R}^n,\mathbb{R}^n)=\bigcap_{q\geq0}H^q(\math... |
I have been studying semidirect product of groups these many days. I came to know that for every group $G$, there is a group $Hol(G) = G \rtimes_{f} Aut(G)$ (here, $f: Aut(G) \rightarrow Aut(G)$ is the identity map), called the holomorph of the group $G$, such that every automorphism of $G$ can be realised as an inner ... |
So what I want to find out is the path drawn by the parabola $y=x^2+bx$ minimum as b changes.
Im pretty sure it draws an upside down parabola because if I put in Desmos and slide the b it goes in the pattern if you look at it. How can find this formula and what is it called? |
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