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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\mid f(g) - h\cdot f(g) = f(h) - g \cdot f(h),\ \ \ \forall g,... |
How do I solve this Diophantine equation, $a^{2n}+b^2=c^{2n}$, where $n>1$ & $a,b,c\ne0$?
I tried it by applying the Pythagorean triplets generating formula, but unable to find any integer solutions & nor I'm able to prove that there are no integer solutions except $a,b,c=0$. Thanks in advance for any help or guidan... |
How do I solve this Diophantine equation? |
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... |
Given the following differential equation:
\begin{equation}
\Delta f - \lambda^2((\Delta f)^2 - (\partial_i \partial_j f)^2) = g,
\end{equation}
where there is an implicit sum over i,j and $g$ is a function which is constant non-zero inside an ellipsoid and outside zero. Is there a way how to extract analytical... |
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... |
Suppose we have $x_1$ lines on a plane parallel in one direction ,$x_2$ lines parallel in another direction, $\cdots$,$x_{10}$ lines parallel in another direction. If $x_i$=$i$ for $i$ $\in${$1,2,3,4,5,6,7,8,9,10$} then what is the maximum possible number of intersections of these lines?
$My$ $approach$ - I tried t... |
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... |
Maximum number of intersections possible for the group of parallel lines? |
I have to prove that if 3 divides the product of two integers then it divides one of the integers. At this point in my instructors notes, we do not have Euclid's lemma, nor the concept of primes or gcd, nor modular arithmetic. Basically just elementary algebra. One thing we do have is that every integer is of the form ... |
I've been studying statistics for a while now, and honestly, I've been struggling to gain an intuition for both the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) up to this point. Now, I just came across [this][1] paragraph which describes both theorems as first and second-order asymptotic expansions. ... |
I have to prove that if 3 divides the product of two integers then it divides one of the integers. At this point in my instructors notes, we do not have Euclid's lemma, nor the concept of primes or gcd, nor modular arithmetic. Basically just elementary algebra. One thing we do have is that every integer is of the form ... |
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... |
I managed to find the PDF of $X^3$ given $X$~$N(0,1)$:
$Y=X^3$ then for any $y$, $P(0<X^3<y)= \phi (\sqrt[3]{y}) - \phi (0)$ and then derivate that to find $f_Y(y)$. But that didn't give me anything. I know that to find the mean of a function of a variable, in this case I will need to calculate the integral on the P... |
Let $a\in(0,\infty)$. Let $(X_t)_{t\in[0,a]}$ be a standard Brownian motion and $(W_s)_{s\in[0,\infty)}$ be a standard Brownian motion such that $W_s$ is independent of $X_t$ for all $s \geqslant 0$ and $t \in [0, a]$. Define a new process $(Y_t)_{t∈[0,\infty)}$ as follows.
\begin{equation}
Y_{t} =
\begin{ca... |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
Consider an $n \times n$ grid graph $G$. Define the following quantity
\begin{equation}
m(G) = \text{max}\bigg\{\frac{|E|_{H'}}{|V|_{H'}},~ H' \subseteq G, ~~|V|_{H'} > 0 \bigg\},
\end{equation}
where $|V|_{H'}$ and $|E|_{H'}$ are the number of vertices and edges of the graph $H'$ respectively. I am trying to c... |
Let $a\in(0,\infty)$. Let $(X_t)_{t\in[0,a]}$ be a standard Brownian motion and $(W_s)_{s\in[0,\infty)}$ be a standard Brownian motion such that $W_s$ is independent of $X_t$ for all $s \geqslant 0$ and $t \in [0, a]$. Define a new process $(Y_t)_{t∈[0,\infty)}$ as follows.
\begin{equation}
Y_{t} =
\begin{ca... |
Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote
the i'th term of the lower central series of $G$. Let $\displaystyle\frac{G}{\gamma_3(G)}\cong M(p^n), n\geq 3$,
where $M(p^n)$ denotes the modular $p$-group with the following presentation
$$\langle a, b| a^{p^{n-1}}=b^{p}=1, [a,b]=a^{... |
Consider an $n \times n$ grid graph $G$. Define the following quantity
\begin{equation}
m(G) = \text{max}\bigg\{\frac{|E|_{H'}}{|V|_{H'}},~ H' \subseteq G, ~~|V|_{H'} > 0 \bigg\},
\end{equation}
where $|V|_{H'}$ and $|E|_{H'}$ are the number of vertices and edges of the graph $H'$ respectively. I am trying to c... |
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... |
Is this formula for calculating prime numbers suitable for $n^2$ digits? $f(x)=|\sin(πx/P_1 )|+|\sin(πx/P_2 )|+⋯+|\sin(πx/P_n )|$ |
Consider the set of all permutations of $n$ numbers $\mathfrak{S}(n)$. Each permutation can be seen as a total ordering relation of $n$ elements $a_1,...,a_n$, such that
\begin{equation}
a_{\pi(1)}<a_{\pi(2)}<...<a_{\pi(n)}.
\end{equation}
Specific subsets of the set of permutations can be used instead to identify ... |
Throughout studying a paper about finite $p$-groups,
I have the following question
Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote
the i'th term of the lower central series of $G$. Let $\displaystyle\frac{G}{\gamma_3(G)}\cong M(p^n), n\geq 3$,
where $M(p^n)$ denotes the modular $... |
Let $\{a_n\}$ be a sequence of real positive numbers. We wish to prove that $$ \lim \inf \frac{a_{n+1}}{a_n} \leq \lim \inf a_n^{1/n} \leq \lim \sup a_n^{1/n} \leq \lim \sup \frac{a_{n+1}}{a_n}$$
The answer to this [post][1] shows that $\lim \sup a_n^{1/n} \leq \lim \sup \frac{a_{n+1}}{a_n}$, and it mentions there ... |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
Is there a differential equation:
$$\ddot x(t)=f(t, x(t)),\ f\in C^1(\mathbb{R}^2), \forall t\in \mathbb{R}\ f(t,0)=0;$$
that for any solution x(t) the following is true:
$$\forall \epsilon>0\ \exists \delta>0\ (|x(0)|<\delta \implies \forall t\ge0\ |x(t)|<\epsilon)\ ? $$
I guess that there is no such an equatio... |
I'm learning about intersection homology, and I'm trying to write a proof of this statement:
>Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to Y$ such that for all stratum $S$ of Y, there is a strata $S^f$ of Y such that $f(S)\subset S^f$ and $codim_Y(S^f)\leq codim_X... |
Let $\Omega\subset\mathbb{R}^n$ be nonempty, open and bounded with $C^1$ boundary. Let $p\in[1,n)$. Let $g\in C(\mathbb{R})$ satisfy $$|g(y)|\leq C(1+|y|^q)$$
for some $C<\infty$ and some $q$ with $1\leq q<p^*$. Let $(u_k)_k$ be a sequence with $u_k\in W^{1,p}(\Omega)$ and $$\int_{\Omega}g(u_k(x))dx=0 \text{ for ever... |
Why is the relative interior of a set defined in terms of its affine (vs convex) hull? |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
I am reading this paper "[Choosability and fractional chromatic numbers][1]"
It concerns the fractional chromatic numbers of a graph $G$: let $\mathcal{S}(G)$ be the collection of independent sets in $G$. A fractional coloring is a function $f:\mathcal{S}(G)\rightarrow\mathbb{R}_{\ge 0}$ such that $\sum_{S\in\mathca... |
Why every vertex is in $q$ sets? |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
Let $\Omega\subset\mathbb{R}^n$ be nonempty, open and bounded with $C^1$ boundary. Let $p\in[1,n)$. Let $g\in C(\mathbb{R})$ satisfy $$|g(y)|\leq C(1+|y|^q)$$
for some $C<\infty$ and some $q$ with $1\leq q<p^*$. Let $(u_k)_k$ be a sequence with $u_k\in W^{1,p}(\Omega)$ and $$\int_{\Omega}g(u_k(x))dx=0 \text{ for ever... |
Does this property generalize to modules over noncommutative rings? |
Is there a reference in English for the proof that Boolean algebras admit quantifier elimination? I'm interested in how quantifier elimination can be performed. However, the result of Tarski is not complete and the paper of Ershov is in Russian. Moreover, the book of Chang and Keisler uses Ehrenfeucht Fraissé games ins... |
I've been studying statistics for a while now, and honestly, I've been struggling to gain an intuition for both the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) up to this point. Now, I just came across [this][1] paragraph which describes both theorems as first and second-order asymptotic expansions. ... |
>Let $R$ be arbitrary and let $(e_1,e_2,...,e_n)$ be a basis for $R^{(n)}$. Show that $(f_1,...,f_m)$, $f_j=\sum_{j^\prime=1}^na_{j^\prime j}e_j$ is a basis for $R^{(m)}$ if and only if there exists an $n\times m$ matrix $B$ such that $AB=I_m$, $BA=I_n$ where $A=(a_{ij})$, $I_m$ is the usual $m\times m$ unit matrix and... |
For any group $G$ of order $n\in\mathbb N$, I think that the key point of the "formal process" is the red-typed isomorphism:
$$G\stackrel{Cayley}{\hookrightarrow}S_G\color{red}{\cong}S_n$$
That isomorphism is non-canonical, as it is explicitly given by:
$$l_a\mapsto(i\mapsto f^{-1}(l_a(f(i)))) \tag1$$
where $l_a$ i... |
Just came across this question. I'm the author of both the OEIS entry [A238363][1] and the pdf [Goin’ with the Flow: Logarithm of the Derivative][2], cited by the OP, posted at my blog Shadows of Simplicity along with an erratum. Numerous posts on the Stirling polynomials of the first, $ST1_n(x) = x(x-1)\cdots(x-n+1$),... |
$$ \dot{y}(x)+y(x)=u(x) $$
$$ y_c(x) = ce^{-x} + \int_{0}^{x} u(t) e^{(t-x)} dt $$
Show that if u is periodic with a period of T, then there exists exactly one solution to the differential equation with the period T. Where here $y_c(0) = c$ holds. The use of the second equation is hinted.
I am stuck trying to ... |
Let $\overline{M}=M \cup \partial M$ be a manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective elliptic partial differential operator of order $2$ on $\overline{M}$.
For $k\geq 0$, let $u \in H^{m+k}$, the $L^2$-Sobolev space... |
I am interested in a sequence of linear programs of the following type, over $x\in\mathbb{R}^k$:
$$
\min c_n^T x\\
x\geq 0\\
Ax = b
$$
for $x\in\mathbb{R}^k$, $n\geq 1$ and some choice of $A, b$ that remain fixed. The vector $c_n$ is given by
$$ c_n = c_1\circ c_1\circ\cdots\circ c_1 \hspace{1cm}(n\text{ times})... |
In Appendix 5A of Andrew Granville's [Number Theory Revealed: A Masterclass](https://bookstore.ams.org/mbk-127), he presents a proof of the Sylvester-Schur Theorem.
I am unclear on the last step of the following inductive proof.
Exercise 5.10.1 is used to show that for all $n\ge k \ge 1$: $$\left(1 + \frac{1}{n+k... |
For any group $G$ of order $n\in\mathbb N$, I think that the key point of the "formal process" is the red-typed isomorphism:
$$G\stackrel{Cayley}{\hookrightarrow}S_G\color{red}{\cong}S_n$$
That isomorphism is non-canonical, as it is explicitly given by:
$$l_a\mapsto(i\mapsto f^{-1}(l_a(f(i)))) \tag1$$
where $l_a$ i... |
I am interested in a sequence of linear programs of the following type, over $x\in\mathbb{R}^k$:
$$
\min c_n^T x\\
x\geq 0\\
Ax = b
$$
for $x\in\mathbb{R}^k$, $n\geq 1$ and some choice of $A, b$ that remain fixed. The vector $c_n$ is given by
$$ c_n = c_1\circ c_1\circ\cdots\circ c_1 \hspace{1cm}(n\text{ times})... |
In ∫ln(cosx)x can i substitute cosx by *e^t*?
I don't know whether this question is already posted or not. I tried but didn't find. May be i am too dumb to ask it. Please pardon me for using capital letters if anyone finds them unnecessary and don't comment "don't yell!" or something similar. |
Can i substitute trigonometric function by exponential one? |
In $\int\ln(\cos x)\,dx$, can I substitute $\cos x$ by $e^t$?
I don't know whether this question is already posted or not. I tried to but didn't find it. Maybe I am too dumb to ask it. |
I'm working on a general version of the problem linked [here](https://math.stackexchange.com/questions/2590059/coupling-married-people-so-that-no-couples-are-spouses).
To quote the original:
> Suppose =10 married couples are invited to a bridge party. Bridge partners are chosen
> at random, without regard to ... |
In the (x,y)-plane a function is given: $f(x,y)=x+y$.
Let A be the area that is in the xy-plane and is encapsulated by $x=0$, $x=\frac{\Pi}{4}$, $x+cos(x)$ and $x+sin(x)$.
a) Make a parametrization of A and find its jacobi-function. Then calculate the area.
In my textbook the parametrization i am to make is on... |
$$\int \ln(\cos x) dx$$ In this integral should I substitute **$\cos x$** by **$e^t$**?
|
$$\int \ln(\cos x)\,dx$$ In this integral can I substitute **$\cos x$** by **$e^t$**?
|
$$\int \ln(\cos x)\,dx$$ In this integral can I substitute $\cos x$ by $e^t$?
|
When integrating $\int \ln(\cos x)\,dx$, can I substitute $\cos x $ by $e^t$? |
***Interviewer:***
Assume $a\sim U[0,1]\,,$ and conditional on $a\,,$ $$b\sim U[a,1]\,.$$
What is the unconditional distribution of $b\,?$
***Candidate*** (sweating):
\begin{align}
&\mathbb P\big\{b\le y\big\}=\int_0^1\mathbb P\big\{b\le y\,\big|\,a=x\big\}\,dx
=\int_0^1\frac{y-x}{1-x}1_{\{x\le y\}}\,dx=\i... |
You, Ada, and Carlos are on a game show where the host will ask contestants to wager (or risk) money. Assume that you are all very smart, and that each of you will be willing to wager money if and only if there is a greater than 50% chance you will win that wager. The three of you each roll a 12-sided die (numbered 1 t... |
I've been studying statistics for a while now, and honestly, I've been struggling to gain an intuition for both the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) up to this point. Now, I just came across [this][1] paragraph which describes both theorems as first and second-order asymptotic expansions. ... |
Let $\mathbb F_q$ be a finite field of $q$ elements.
Let $G$ be a group with two generators $\{a,b\}$. Let $A$ be a non-trivial one-dimensional representation of $G$ over $\Bbb F_q$.
Consider the set of maps
\begin{equation}
\begin{aligned}
S:=\{f:G\longrightarrow A\mid f(g) - h\cdot f(g) = f(h) - g \cdot f(h),... |
You, Ada, and Carlos are on a game show where the host will ask contestants to wager (or risk) money. Assume that you are all very smart, and that each of you will be willing to wager money if and only if there is a greater than 50% chance you will win that wager. The three of you each roll a 12-sided die (numbered 1 t... |
In Wikipedia's [proof of Bertrand's Postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate), in the second lemma, it is concluded that:
$$R = R(p,{{2n}\choose{n}}) \le \log_p 2n$$
where $R(p,n)$ is the [p-adic order](https://en.wikipedia.org/wiki/P-adic_valuation) of ${2n}\choose{n}$
Later in... |
Assuming **>1** (because that is cheating :p), would it be reasonable to think there is an ᵗʰ digit of π where stopping there would yield a palindromic number **(3.14159...951413)**?
Would it be more likely that this occurs an infinite amount of times or never? |
Might there be an ᵗʰ digit of π where the sequence becomes palindromic? |
From [Wikipedia][1]:
> If $F, G$ partial functions on the natural numbers, $F(n) \simeq G(n)$ indicates that, for each $n$, either $F(n)$ and $G(n)$ are both defined and equal, or are both undefined.
What is the difference between this and $F = G$? If $F, G$ take on exactly the same values for each $n$ on which ... |
Can
$$
x^{2/3} + y^{2/3}
$$
be expressed as a fraction of two polynomials in x and y?
How can we see this easily?
(It is the curve swept by a stick sliding down a wall) |
Is $x^{2/3} + y^{2/3}$ a rational function? |
In answering [this question][1] from Hartshorne, we ultimately end up using the following diagram
$$\require{AMScd}
\begin{CD}
X \times_Z Y @>{}>> Y\\
@V{}VV @VV{g}V \\
X @>{g\ \circ f}>> Z
\end{CD}$$
and it occurred to me to look for an example where $X \times_Z Y \to Y$ is not the same as $f$. What I mean... |
I have the function
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/5aE0Y.gif
In this function, all variables x, y, and z have a value. The goal is to optimize the function, but only one variable can be increased. Based on the values of the variables, what is the way to decide which... |
This is a problem from my homework:
> Given the plant $y_p = a_pu(t)$ ($a_p\neq 0$) and the reference model $y_m = a_mr(t)$, where $r(t)$ is bounded and continuous. Design a __differentiator free__ control $u(t)$ such that all signals are bounded and $|y_p(t) - y_m(t)|\to 0$. Consider two different scenarios: (1) $|a_... |
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... |
> Show that \begin{equation}\sum_{m=1}^k\frac{1}{m}>\log k.\end{equation}
My intuition here is that the LHS looks a lot like $\int_1^k\frac{1}{x}\textrm{d}x$, and this evaluates to $\log k$. To justify why the sum is larger, note that $1/m$ is convex, and my understanding is that more generally, \begin{equation}\sum... |
Consider three, rather generic, equations in three variables $x,y,z$:
$y z+x=a$
$z x+y=b$
$x y+z=c$
Now, assuming that there is a solution to this system, is it possible to find an expression in terms of $a,b,c$ only for the sum and/or product of the solutions $x,y,z$?
By this I mean something along the ... |
System of 3 non-linear equations in 3 unknowns - sum and product of solutions without needing to actually solve? |
Say we have two algebraic sets, $X = V_{\mathbb A^2}(x+y-1)$ and $Y = V_{\mathbb A^2}(x^3+x^2y + y - 1)$.
Is it necessarily true that $\overline{X \cap Y} = \overline{X} \cap \overline{Y}$ for $\overline{X}$ indicating the closure of $X$ in $\mathbb P^2$? In general, is it true that for two algebraic sets is $\overl... |
When does $\overline{X \cap Y} = \overline{X} \cap \overline{Y}$ in $\mathbb P^2$? |
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... |
For simplicity, assume we are working with simple regression where the predictor $x\in\mathbb{R}$.
First write $y=E[y \mid x]+u$, and immediately it says $E[u|x]=0$. I understand $E(y \mid x)$ minimizes the mean square prediction error (MSPE). We call $E[y \mid x]$ the true regression.
I understand the best (sim... |
Is conditional expectation of the error of best linear predictor given $X$ is $0$ (Is it true that $y = a^*+b^*x + \eta$, where $E[\eta|x]=0$)? |
```
| Exit 1 | | Exit 4 |
| | | |
| | | |
| | | |
_____________| |_____________| |________
Exit 2 S1 s2 Exit 6
_______... |
What is the probability a person leaves via a certain exit? |
How to decompose $\mathbb{R} G$ module in its irreducible submodules, for $\mathbb{R^{15}}$? |
Here is my attempt to prove it:
Fix X a finite set so either X $= O$ or X $\approx \mathbb{N}_{k}$. Hence card(X) $= 0$ or card(X) $= k$. Fix $k \in \mathbb{N}$
Suppose Y is also a finite set so either Y $= O$ or Y $\approx \mathbb{N}_{n}$. Hence card(Y) $= 0$ or card(Y) $= n$
Let P(n) be the statement: For al... |
Suppose that $f g$ is in $L^1([a,b])$ for every $f$ in $L^2([a,b])$, I understand that this implies that $g$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it?
This question was set as an exercise in the book *An Introduction to Lebesgue Integration and Fourier Series* by H. Wilcox and D. Myer... |
I'm a non-mathematician trying to learn geometric algebra by working my way (slowly) through "[Imaginary Numbers are not Real][1]" by Gull, Lasenby and Doran [*Found. Phys. 23(9), 1175-1201 (1993)*].
So far so good and I can prove Euler's formula with GA in two dimensions - but I'm struggling to prove the following ... |
I was asked to find the contour integral $dz/z$ of a triangle with vertices:
1-2i
-2+i
1+i
I am not sure how to begin this but my professor said that it would be:
$\int_{-1}^{2} \frac{-i}{1-ti} \, dt$
But this doesn't make sense to me. why would z be equal to 1-ti ? I can see he may be working fro... |
If $D$ is the Chern connection on a holomorphic hermitian vector bundle $(E,h)$ and $e$ is a holomorphic frame, then $$\vartheta=\partial h \cdot h^{-1}$$ where $h_{ij}=h(e_i,e_j)$. Here $\vartheta$ is the connection matrix and $\partial h$ means the del-derivative of the matrix $h$.
I was going through the followin... |
In fact, LP has used many different tools from mathematics; for example, see [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
LP has been a a main driver for the study of challenging problems such as $d$-step conjecture for polyhedrons (see this Acta Mathema... |
Consider three, rather generic, equations in three variables $x,y,z$:
$y z+x=a$
$z x+y=b$
$x y+z=c$
Now, assuming that there is a solution to this system, is it possible to find an expression in terms of $a,b,c$ only for the sum and/or product of the solutions $x,y,z$?
By this I mean something along the ... |
A Pierpont number is of the form $ p=2^a.3^b+1$. (I used a>0)
Trying to explore them, I did a very simple loop to test for primality:
loop on g over primes
- if $g^{p-1}\mod p \neq 1$, then p is not prime (Fermat's little theorem)
- else if order of g is p-1, then p is prime (can be easily checked as p-1 has ... |
In fact, LP has used many different tools from mathematics; for example, see [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
LP **has been a driver for the study of challenging math problems** such as $d$-step conjecture for polyhedrons (see this Acta Mathe... |
I've been struggling for a bit on this math problem from my homework.
$$
\int_{0}^{1}\int_{1}^{3}(8x^3-36x^2y^2)dydx
$$
From what I understand and have attempted, we'll integrate with respect to x first, so something like:
$$
(2x^4-12x^3y^2)\vert_{1}^{3}=160-312y^2
$$
and then another integration, this ... |
The formula satisfied by all formulas that are not self-applicable, and only by them, is self-applicable if and only if it is not self-applicable.
Because, if it is self-applicable, it is inconsistent with its definition, because it should be satisfied only by formulas that are not self-applicable.
And if it is n... |
Is it possible to express self-application in the Peano-Russell system? |
Note that if $X \sim \operatorname{Uniform}(0,1)$, then
$$-\log X \sim \operatorname{Exponential}(1).$$
The proof of this is left as an exercise.
Next, consider
$$\log (XY^Z) = \log X + Z \log Y.$$
What is the distribution of $Z \log Y$ when $Y, Z \sim \operatorname{Uniform}(0,1)$ and are independent? Edit your... |
I am having trouble understanding how to compute this integral $$\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} \, dx \, dy$$
My idea is to consider the whole plane xy plane instead of the half of the positive quadrant. Then the integral would look like this. $$\frac{1}{8} \int_{-\infty}^{\infty} \int... |
> Two chemicals in solution react together to form a compound: one unit of compound is formed from $a$ units of chemical $A$ and $b$ units of chemical $B$, with $a + b = 1$. Assume the concentration is low enough that the rate of reaction is directly proportional to the amount of chemical $A$ present, and directly pr... |
How to solve an ODE where the rate is directly proportional to two amounts? |
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