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Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. The regular triangle inequality states that the sum of any two sides is greater than the third side. But what happens if we take the sum of the reciprocal of any... |
$x$ and $y$ are real numbers (edited: thanks to @mcd's answer, now the "positive" requirement is removed) constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, if there exists solution(s) to $a\cdot f(x)+b\cdot f(y)=z$ where $z$ is ... |
I'm reading this book:
https://doc.lagout.org/science/0_Computer%20Science/2_Algorithms/The%20Art%20of%20Computer%20Programming%20(vol.%201_%20Fundamental%20Algorithms)%20(3rd%20ed.)%20%5BKnuth%201997-07-17%5D.pdf
And the first exercise is the following:
Let a be any positive number. For all positive integers ... |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
a) $Y$ is the Hausdorff space.\
b) $A=\{(y_1,y_2)\in Y\times Y:y_1<y_2\}$ is open in product topology $Y\times Y$.\
c) $B=\{x\in X:f(x)\leq g(x)\}$ is closed in $X$.\
d) Th... |
How to Integrate the following?
$$\int_{2}^{343}\{x-\lfloor{x}\rfloor\}^2dx = \int_{2}^{3}x^2dx + \int_{3}^{4}x^2dx+\cdots+\int_{342}^{343}x^2dx$$
This is what I did, How can I proceed further with this? Any help is appreciated. |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
a) $Y$ is the Hausdorff space.\
b) $A=\{(y_1,y_2)\in Y\times Y:y_1<y_2\}$ is open in product topology $Y\times Y$.\
c) $B=\{x\in X:f(x)\leq g(x)\}$ is closed in $X$.\
d) Th... |
y = {1,2,3,4,5,6}
My first try is $(1/6)*(y/6)^{n-1}$ , but this doesn’t work, should I consider the order?
How to find the formula for pmf and cdf? |
$x$ and $y$ are real numbers (edited: thanks to @mcd's answer, now the "positive" requirement is removed) constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, if there exists solution(s) to $a\cdot f(x)+b\cdot f(y)=z$ where $z$ is ... |
Let $I$ be an ideal in an integral domain $R$, and define the ideals $I_a = (I, a)$ where $a \in R-I$, and $J = \{r \in R: rI_a \subseteq I\}$. It is also known that $I_a$ and $J$ are principal ideals, so $I_a = (\alpha)$ and $J = (\beta)$. We then have $I_aJ = (\alpha\beta)$. To show set containment, we let $x \in I_a... |
Is there any formula known?
There is the following asympotic formula for unlabelled trees:
$$t(n) \sim C \alpha^n n^{-\frac{5}{2}}$$
With $t(n)$ the number of unlabelled non-isomorphic trees on $n$ vertices.
Is there a similar formula known for forests?
I actually just wanna know if the number of forests is $F(... |
How many non-isomorphic, non-labelled, forests are there (asymptotically) on $n$ vertices? |
$x$ and $y$ are real numbers (edited: thanks to @mcd's answer, now the "positive" requirement is removed) constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, if there exists solution(s) to $a\cdot f(x)+b\cdot f(y)=z$ where $z$ is ... |
Let $U=(0,\infty)\times (0,2\pi]$
Define a function $F:U \rightarrow \mathbb R^2$ by
$$F(r,\theta)= (r\cos(\theta),r\sin(\theta))$$
Here all $4$ partial derivative of component function are exist and continuous. So $F$ is continuously differentiable with non zero Jacobian $r$ at $(r,\theta)$. So By invers... |
I've looked at some highly composite numbers and realized that a lot of them are almost primes, i.e. differ only by $1$ to the next closest prime. Here's a short list (I made) of highly composite numbers and whether they are almost prime:
$$ 1 \text{, true $(+1)$} \\
2 \text{, true $(+1)$} \\
4 \text{, true $(\pm 1)... |
Are there infinitely many primes that are a highly composite number $\pm 1$? |
Let $a$,$b$ $\in$ $\Bbb N$ such that the equations
$$x^2+ax+b$$
$$x^2+bx+a$$
have all of their roots as integers. Find all $a$ and $b$.
**My Approach**
I tried using Vieta's Relations but the process got too much complicated and I gave up.
Any help? |
Is it possible (using ZFC or any other axiom system which is known to be as consistent as ZF) to prove the existence of an ultrafilter $\mathcal F$ on the set of prime numbers s.t. there is an embedding $\mathbb R\hookrightarrow \tilde{\mathbb F}=\prod_p \mathbb F_p/\mathcal F$, but without this field $\tilde{\mathbb F... |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
a) $Y$ is the Hausdorff space.\
b) $A=\{(y_1,y_2)\in Y\times Y:y_1<y_2\}$ is open in product topology $Y\times Y$.\
c) $B=\{x\in X:f(x)\leq g(x)\}$ is closed in $X$.\
d) Th... |
This is pretty clear geometrically. Why do you need an algebraic proof?
The projections of those three vertices of a cube will always form a triangle unless two of them are collinear in the $xy$-plane. Your "looking upward" hypothesis prevents that since if two of the projections were collinear the third vertex woul... |
I'm trying to understand the notion of Cartier divisors and I am having trouble understanding the connection with hypersurfaces in the projective space $\mathbb{P}^n_k$, where $k$ is an algebraically closed field.
So the definition of Cartier divisors says that such a divisor is defined by an open cover and for each o... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time. I have documented my research and implementation in C#: it can be found on [Git... |
I will try to write my proof of the Riemann lebesgues Lemma, correct me if something is wrong.
If $f\in L_{1}(R)$ then its fourier transform $f^{*}$ maps to $C_{0}(R)$, a space of continuous functions that converge to zero in infinity, that is $\lim_{|\zeta|\to\infty} |f^{*}(\zeta)|=0.$
Proof: Since compactly su... |
Consider a finite state irreducible Markov Chain $(Z_n)$ on the trajectory space $(\Omega, \mathcal{A}, \mathbb{P})$ induced by $(X,P)$ where $X$ is the state space and $P$ a stochastic matrix. For any event $A\in\mathcal{A}$, we have $\mathbb{P}(A)=\sum_{j\in X}\pi(j)\mathbb{P}(A|Z_0=j)$ where $\pi$ is an initial dist... |
I have been studying transition kernels recently from "Probability and Stochastics" by Erhan Cinlar.
The book shows that for any transition kernel from $(E, \mathcal{E})$ to $(F, \mathcal{F})$, and for any $x \in E$, a measure on $(F, \mathcal{F})$ is defined which will be henceforth be signed by $K(x)$.
Kernels ca... |
Why is s-finiteness required for a transformation by a transition kernel be well defined? |
According to Wikipedia, at https://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms, for integer $m$ and large real $x$,
\begin{aligned}I_{m }(x)={\frac {1}{{\sqrt {2\pi x}}{\sqrt[{4}]{1+{\frac {m^{2}}{x^{2}}}}}}}\exp \left(-m \operatorname {arsinh} \left({\frac {m}{x}}\right)+x{\sqrt {1+{\frac {m^{2}}{x^{2}}... |
In the representation theory class, we proved that any unitary representation from group $G$ (assuming that $G$ is locally compact topological group equipped with Haar measure) acting on $GL(V)$ where $V$ is $n$ dimensional $\mathbb{C}-$vector space is semisimple (i.e. reducible into direct sum of irreducible subrepres... |
This is pretty clear geometrically. Why do you need an algebraic proof?
The projections of those three vertices of a cube will always form a triangle unless two of them are collinear in the $xy$-plane. Your "looking upward" hypothesis prevents that since if two of the projections were collinear the third vertex woul... |
I was looking at a number of different proofs of the Cauchy-Schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$).
All of them used the idea of $\lVert x-sy\rVert$ where $s$ was selected in particular fashion which in the real case, $s$ would be the value that minimized the function $f(s) ... |
Proof of Cauchy-Schwarz inequality : why select $s$ so that so that $\lVert x-sy\rVert$ would be minimized? |
Consider a surface of revolution $S$ with constant positive curvature and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with conjugate points $p,q$ anchored on $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm{dist}(p,q)=\sqrt{3}$.
>What is $\rho_{\mathrm{max}}=\mathrm{max} \lbrace \mathrm{vol}... |
This refers to the question (and answer from here: https://math.stackexchange.com/questions/3239120/show-that-a-vector-space-v-can-only-admit-a-dual-if-it-is-finite-dimensional)
I don't know the appropriate procedure here, whether I should ask in the previous thread or a separate question but anyways:
The proof i... |
This refers to the question (and answer from here: https://math.stackexchange.com/questions/3239120/show-that-a-vector-space-v-can-only-admit-a-dual-if-it-is-finite-dimensional)
I don't know the appropriate procedure here, whether I should ask in the previous thread or a separate question but anyways:
The proof i... |
[I [crossposted](https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/Non-symmetric.20monoidal.20monads) this question on the category theory Zulip and got answers there, so I can now answer my own question; see there for more examples.]
An example of a non-symmetric commutative... |
Now, it is clear (I think) that an analogy is not a proof. The thought process of that claim to say it's not logical/mathematical; it's wasting time; and the result can follow without a need to further address why might due to the reasoning as follow:
- (1) Given $\epsilon>0, \exists N\in\mathbb{N}$, then $\left\Vert ... |
*This is my own question but I assume it has been asked before; I just have not found where. I am curious of the answer but I don't know how to approach solving it. I would appreciate it if someone could help me out with tags because I don't have any experience with this area in math. Also, please let me know if I misu... |
Does an infinite amount of randomly chosen numbers within a definite interval form a connected space? |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
a) $Y$ is the Hausdorff space.\
b) $A=\{(y_1,y_2)\in Y\times Y:y_1<y_2\}$ is open in product topology $Y\times Y$.\
c) $B=\{x\in X:f(x)\leq g(x)\}$ is closed in $X$.\
d) Th... |
Yup, if you look only at the "outputs" there is no sequential order. That's why the sequence is not the same thing as its "outputs". The sequence *is* the mapping, and the domain/index set is a subset of the naturals, which does have the standard order on it. If you want to get poetical about it, "elements of the seque... |
I try to numerically solve (by finite difference method) the beam deflection equation
$$EJ\cdot\frac{\partial^4 v}{\partial x^4} = -g\cdot F\cdot\rho$$ with following boundary conditions
$$v(0)=0;\frac{\partial v(0)}{\partial x} = 0$$
$$\frac{\partial^2 v(L)}{\partial x^2} = 0;\frac{\partial^3 v(L)}{\partial x^3} = ... |
Yup, if you look only at the "outputs" there is no sequential order. That's why the sequence is not the same thing as its "outputs". The sequence *is* the mapping, and the domain/index set is a subset of the naturals, which does have the standard order on it. If you want to get poetical about it, "elements of the seque... |
According to Wikipedia, at https://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms, for large real $x$,
\begin{aligned}I_{\alpha }(x)={\frac {1}{{\sqrt {2\pi x}}{\sqrt[{4}]{1+{\frac {\alpha^{2}}{x^{2}}}}}}}\exp \left(-\alpha \operatorname {arsinh} \left({\frac {\alpha}{x}}\right)+x{\sqrt {1+{\frac {\alpha^{2... |
Let $G$ be a directed graph, and let $C$ be a circuit in $G$. Prove there exists a simple circuit $C'$ such that all of the edges of $C'$ belong to $C$.
I am new to graph theory, my idea was the following: Look at the shortest circuit in $C$. It must be a simple circuit (I don't know how to prove that) and all of it... |
Yup, if you look only at the "outputs" there is no sequential order. That's why the sequence is not the same thing as its "outputs". The sequence *is* the mapping, and the domain/index set is a subset of the naturals, which does have the standard order on it. If you want to get poetical about it, "elements of the seque... |
I'm trying to find functions $f$ over $\mathbb R$ which satisfy
$$ f\left(\frac{x+y+1}{2}\right) = \frac{f(x)+f(y)}{2} $$
for all real $x$ and $y$. One thing I immediately note is that shifting any potential function by a constant preserves the above identity, so I can assume $f(0) = 0$ and so $f(\frac{x+1}{2}) = \f... |
Can a non-constant function satisfy $f\left(\frac{x+y+1}{2}\right) = \frac{f(x)+f(y)}{2}$? |
You have the right idea. If some interior vertex of $C$ is repeated, consider the (necessarily shorter) path between two occurrences of that vertex. What can you say about that path? |
I am studying chapter 1 of Gathmann's notes about algebraic geometry and there is something that I don't understand in the following proof.
[enter image description here][1]
[1]: https://i.stack.imgur.com/3ifHF.png
My question is about the statement $\mathfrak{m} \subsetneq \mathfrak{m}^{\prime} \subsetne... |
I am studying chapter 1 of Gathmann's notes about algebraic geometry and there is something that I don't understand in the following proof.
[please click here for the proof][1]
[1]: https://i.stack.imgur.com/3ifHF.png
My question is about the statement $\mathfrak{m} \subsetneq \mathfrak{m}^{\prime} \subse... |
So I need to verify
$$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
Using Parsevals identity on $f(x)=x$ on $[0,2\pi)$. Does this mean if $a_n$ are the Fourier coefficients then
$$\sum_{n=-\infty}^\infty \vert a_n \vert^2 = \frac{1}{2\pi} \int_0^{2\pi} x^2 dx.$$
Where
$$a_n=\frac{1}{2\pi}\int_0^{2\p... |
I am trying to find a formula for number of equivalence relations on a set with n-elements however I am confused. <br/>
I have already encountered the idea of "[bell's number][1]" and "[Stirling numbers of the second kind][2]" however it seems to contradict the meaning of "Relation". <br/>
Definition: "Relation" be... |
Let $w(x_1, \dots, x_n)$ be a k-degree polynomial of $n$-variables given by
$$w(\vec x) = \sum_{\alpha : |\alpha| \leq k} c_\alpha x^\alpha$$
where $c_\alpha \in \mathbb C$ and $\text{Re}(c_\alpha) \geq 0$.
The notation $w(-i \nabla)$ is somewhat idiosyncratic and means
$$w(-i \vec\nabla) = \sum_{\alpha : |... |
Let $X$ be a normed vector space. We define the weak-$^\star$ topology on $X^{\star}$ to be the initial topology generated by all linear functionals $\phi_x: X^{\star}\rightarrow F$ defined by
$$
f\mapsto f(x).
$$
I was trying to give an explicit formula for the subbasis and basis of the weak-$^\star$ topology and... |
Is this proof right?
I approached this problem by showing that B(x;$\epsilon$) is a subset of the complement of the closure of V, for all x in X, and epsilon bigger than 0. We know B(x;$\epsilon$)={a $\in$ X: d(x,a) < $\epsilon$}. Let b $\in$ B(x;$\epsilon$). Let us prove that b is also in $(\bar{V})^c$. This would ... |
This is in reference to the zero-one loss example in the Duda hart book. The figure shows likelihood ratio plotted against x. In the image attached, the example states that there is set threshold if the errors are treated equally. But if we penalize classifying a state w1 as w2, more than the converse, the threshold in... |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
The map $h: X\to Y$ defined by $$h(x)=\min\{f(x),g(x)\}$$ is continuous on $X$.
PS: Assume $(X,\leq)$ is the set with linear order (if $x,y$ in $X$ then $x\leq y$ or $y\l... |
$$\Large \text{Question}$$
#### Question ####
Let's say that $y^{\left( \alpha \right)}\left( x \right)$ is the $\alpha$.th derivative of $y\left( x \right)$ with respect to $x$ ($x \in \mathbb{R} \vee x \in \mathbb{C}$ and $\alpha \in \mathbb{N}_{0}$). How to solve
\begin{align*}
y\left( y^{\left( 1 \right)}\l... |
How to solve $y(y'(y''(\cdots(y^{(n)}(x))))) = f(x)$ for $y(x)$? |
In Tom Apostol's book, he credits the proof of the divergence of the sum of reciprocal of primes to James A. Clarkson.
>**Theorem**: Let $\{p_n\}_{n\in\mathbb N}$ be the prime numbers. The infinite series $$\sum_{n=1}^\infty\frac{1}{p_n}$$ diverges.
>*Proof*. We assume the series converges and obtain a contrad... |
Using the three-point central difference formula,
$$f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}$$
to approximate the $f'(x)$ of $f(x)= e^x sin(x)$ for the interval $[0,9]$ using different step sizes ($h$) produces the following error vs $x$ [graph][1].
I wanted to ask why there are these uniform 'bumps' in the error? ... |
Why does the graph of the absolute error vs x for the three-point central difference formula have this shape? |
Imagine that I have the following system of equations:
\begin{equation}
\left\{
\begin{aligned}
x + 2 &= 3 \\
y - 5 &= 3
\end{aligned}
\right.
\end{equation}
The solution (x=1, y=8) of this equation is trivial (equations do not even depend on each other).
But imagine that to solve this system I do the f... |
What is wrong with substituting a constant in a system of equations? |
Given a system of ODE or PDE and a solution $u$ such that $$ \partial_t u = \mathcal{L}u , \quad u(t=0 ) = 0.$$
for some operator $\mathcal{L}$, one can ask the following question
> Let $u$ be the solution of the PDE with the initial condition $u(t = 0) = u_0$. Determine a norm $\lVert \, \cdot \, \rVert$ and a thr... |
I will try to write my proof of the Riemann lebesgues Lemma, correct me if something is wrong.
If $f\in L_{1}(R)$ then its fourier transform $f^{*}$ maps to $C_{0}(R)$, a space of continuous functions that converge to zero in infinity, that is $\lim_{|\zeta|\to\infty} |f^{*}(\zeta)|=0.$
Proof: Since compactly su... |
I am currently working on "my own" system of infinitesimals. They are pretty much analogous to dual numbers, except for the fact that mine are not nilpotent. My main question concerns possible extensions of the exponential function, but before that, I will give some background information on how these numbers work.
... |
I want to show the following:
Let $X$ be a topological vector space (and Hausdorff), $f:X \rightarrow \mathbb{K}$ a linear functional. Then the following are equivalent:
1.$f$ is continuous
2.$f$ is continuous at $0$
3.$ker(f)$ is closed
From 1 $\Rightarrow$ 2 is obviuos.
So let's start with 2 $\Rig... |
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ (is that enough? Or do we need something more?) are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \mathrm{dim}(Y),$$
where $\mathrm{... |
Is there any formula known?
There is the following asympotic formula for unlabelled trees:
$$t(n) \sim C \alpha^n n^{-\frac{5}{2}}$$
With $t(n)$ the number of unlabelled non-isomorphic trees on $n$ vertices.
Is there a similar formula known for forests?
I actually just wanna know if the number of forests is $F(... |
$X \subset l^1(\mathbb{R})$ is the set of sequences which have only finitely many non-zero terms.
How do I prove $X$ is a subspace of $l^1(\mathbb{R})$?
I have no idea how to start this, sorry. |
Given $n$ positive real numbers $a_1\geq a_2,\cdots\geq a_n>0$. Assume $\sum_{i=1}^n a_i=b_1, \sum_{i=1}^n a_i^2=b_2$. I want to find an upper bound on $\frac{a_1}{a_n}$, and the condition when this upper bound is achieved. My intuition is that the upper bound is achieved either $a_1=a_2=,\cdots=a_{n-1}>a_n$ or $a_1>a_... |
As the title says, I don't know why countable unions have this ("commutative") property.
Is there any counterexample? Could be a countable union or not. |
Why does $\bigcup_{j \geq 1} A_j \cup B_j=(\bigcup_{j \geq 1} A_j) \cup (\bigcup_{j \geq 1} B_j)$? |
How can I prove that $\Vert \int _Xfd\mu \Vert \leq \int _X\Vert f\Vert d\mu $? |
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ is indeed always equal or greater than $n+2$, and grows exponentially as $n$ grows. I attach the graph of $k(n)$ for... |
We know that that given $X_1,...,X_n$ independent random variables which are distributed exponentially, their minimum is also distributed exponentially. Is the converse true? Given $X = \min\{X_1, ..., X_n\}$ distributed exponentially, and $X_1, ..., X_n$ are independent, is it true that $X_i$ is distributed exponentia... |
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ (is that enough? Or do we need something more?) are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \mathrm{Tr}^{L}(\mathrm{id}_{Y}) = \mathrm{dim}(Y),$$
where $\mathrm{... |
$X \subset l^1(\mathbb{R})$ is the set of sequences which have only finitely many non-zero terms.
How do I prove $X$ is a subspace of $l^1(\mathbb{R})$?
Is the following enough?
Let $\vec{x} = (x_n),\, \vec{y} = (y_n) \in X$
$\therefore \exists\ N \in \mathbb{N}$ s.t.
$x_k = y_k = 0\ \forall\ k \geq N$
... |
There are 19 people who are riding a rollercoaster and each time the rollercoaster runs with all 19 people they are organized differently among the 3 separate cars on the track. Each car can carry more than 19 people. We can see the coaster from far away, so we can see the number of people on each coaster car but the p... |
Let $w(x_1, \dots, x_n)$ be a $2k$-degree polynomial of $n$-variables given by
$$w(\vec x) = \sum_{\alpha : |\alpha| \leq 2k} c_\alpha x^\alpha$$
where $c_\alpha \in \mathbb C$ and $\text{Re}(c_\alpha) \geq 0$.
The notation $w(-i \nabla)$ is somewhat idiosyncratic and means
$$w(-i \vec\nabla) = \sum_{\alpha... |
I am currently reading Amann/Escher's *Analysis I* and am struggling to develop their formal development of the isomorphism between polynomials (as a subset of the set of formal power series) and polynomial functions. I have included their discussion here, and have some questions which pertain to it:
[, as written by the poster in another comment.
From the comments:
1. "The short answer is that you cannot handle this at all without an extra layer somewhere. Say, if the i... |
I read the following claim in the [book][1], P19 Eq.(3.9).
> For a smooth function $f:\mathcal{E} \rightarrow R$, where
> $\mathcal{E}$ is a linear space. $Df(x): \mathcal{E}\rightarrow R$ is
> the differential of $f$ at $x$, that is, it is the linear map defined
> by: $$Df(x)[v]=\lim\limits_{t\rightarrow 0} \fr... |
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