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Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$.
If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get some properties about $\sum_{m=1}^\infty \phi_{m}z^m$ and $\sum_{m=1}^\infty \psi_{-m}z^-m$ from this equation $P(z)\p... |
Is it true that a field automorphism must fix the elements of any subfield?
Ie, since $\mathbb{Q}\subset\mathbb{R}$, all automorphisms from R to itself must preserve the map from $I:\mathbb{Q}\to\mathbb{Q}$?
note: I know that there is only one automorphism of $\mathbb{R}$ --- I am more curious to see if the reas... |
> Is there a general procedure for converting an axiomatic proof using normal Detachment (Modus Ponens) into a proof using CD?
Yes, straightforwardly. Use the same reasons in the same lines, and refill all the formulas with [most general unifiers](https://en.wikipedia.org/wiki/Unification_(computer_science)). Since ... |
First, let me say that I'm impressed by your maturity and wisdom. It's not easy to recognize your own limitations, accept them, and adapt. Most people have to learn the hard way, by living through a few decades of struggle and frustration. Some people actually enjoy struggle and frustration, though. Your choice.
I h... |
> Is there a general procedure for converting an axiomatic proof using normal Detachment (Modus Ponens) into a proof using CD?
Yes, straightforwardly. First note that something like "[(line) 2, r:=p]" is not a dedicated reason of the proof system, so lines like that have to be merged with the line they reference.
... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
> Is there a general procedure for converting an axiomatic proof using normal Detachment (Modus Ponens) into a proof using CD?
Yes, straightforwardly. First note that something like "[(line) 2, r:=p]" is not a dedicated reason of the proof system, so lines like that have to be merged with the line they reference.
... |
> Is there a general procedure for converting an axiomatic proof using normal Detachment (Modus Ponens) into a proof using CD?
Yes, straightforwardly. First note that something like "[(line) 2, r:=p]" is not a dedicated reason of the proof system, so lines like that have to be merged with the line they reference.
... |
>What is the formal process behind finding a Cayley permutation representation.
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 ex... |
Let $\sqrt{6\sum_{k=1}^\infty{\frac{1}{k^2}}}$ be Euler's approximation of $\pi$; $\lim_{n\rightarrow\infty}\frac{2}{g_n}$ Gauss approximation of $\pi$; and $2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}...$ Viète's approximation of $\pi$ where:
- $\{g_n\}_{n=1}^\inf... |
How to calculate the ratio of convergence for Euler's, Gauss' and Viète's approximation of $\pi$? |
Let $n \geq 0$. Consider a pushout square in the category of simplicial sets as follows:
$$\begin{array} ~
\Lambda^{n+2}_{i} & \stackrel{}{\longrightarrow} & X \\
\downarrow{} & & \downarrow{f} \\
\partial \Delta^{n+2} & \stackrel{}{\longrightarrow} & Y,
\end{array}
$$
where $\Lambda^{n+2}_{i}$ denotes 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 ... |
Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ 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 $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic? |
I'm learning that if $n \times n$ positive definite matrix A A's then $\forall v$ $v^TAv>0 \ $,
but this requires A is symmetric(then A have orthonormal eigenvectors)
so I'm asking:
1.if A is not symmetric, but still have n eigenvectors with n positive eigenvalues, would $v^TAv>0 \ $ forall
$v$
2.if A hav... |
I'm trying to find eigenvector that dotproduct < 0,
but without so luck,try many different matrixs in wolfram alpha,
then I find out, that if two vectors $v_1$ and $v_2$'s dotproduct < 0,
I can always negate one of them, since $-v_1$ would also be eigenvector
of a matrix.
so my question is:
1.for the wolfram a... |
Can two eigenvector's dotproduct be < 0? |
I am currently reading about the Eilenberg-Watts theorem. I got the original work from Charles Watts "Intrinsic Characterizations of Some Additive Functors" and theorem 1 is what I am interested in.
Alternatively the proof is stated [here][1] and the important part [here in Lemma 1][2].
The proof in Watts says the... |
Based on the fact that:
1. There are 120 non-cyclical permutations of {1,2,3,4,5,6}
2. Each permutation could be represented by a hexagon with 6 triangles of different colours; e.g.,
[![enter image description here][1]][1]
Is it possible to write a formula or algorithm (function) to arrange (in a tessellati... |
I've been working on solving some equations in the moduli and conjugate of complex numbers and it always seems to take me awfully long to arrive at an answer.
For example, solve $1+2|z|^2 = |z^2+1|^2+2|z+1|^2$. My first thought was to put $z=x+iy$ which didn't work out quite well since I got coupled quartic equation... |
Let $V$ be a finite-dimensional complex inner product space and suppose that there is an operator (a matrix) $A$ on $V$ that satisfies the following anti-commutation relations:
$$AA + AA = 0$$
$$A^{*}A + AA^{*} = I,$$
where $I$ is the identity matrix and $A^{*}$ is the adjoint of $A$. In this case, one can show that... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
This post is a call back to one of my old questions that got no response. That is, why multiply an extra t (but not anything else) when we get repeated roots from either the characteristic polynomial of the linear ODE or from getting repeated roots from the homogeneous kernel and the non-homogeneous kernel? The reasoni... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
I'm trying to find eigenvector that dotproduct < 0,
but without so luck,try many different matrixs in wolfram alpha,
then I find out, that if two vectors $v_1$ and $v_2$'s dotproduct < 0,
I can always negate one of them, since $-v_1$ would also be eigenvector
of a matrix.
so my question is:
1.for the wolfram a... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
I will assume that even though your language is propositional, your *logic* $L$ is non-classical. Why? Because the deduction theorem fails as per the condition of your question.
In general, if you do not have the deduction theorem, you cannot say that $\Gamma\cup\{\neg\phi\}$ is consistent iff $\Gamma\not\vdash_L\ph... |
At 54 yo, differential equations are still a wall for me, I have difficulties to go through.
One video for beginner helped me, explaining me that
- in common equations, we are searching for the values of a variable $x$, for solutions
- in differential equations, for the values of a function $f(x)$
But t... |
Beginner question on differential equations: they look useful to search for a function for solution of an equation, but why are derivatives mandatory? |
I'm trying to find eigenvector that dotproduct < 0,
but without so luck,try many different matrixs in wolfram alpha,
then I find out, that if two vectors $v_1$ and $v_2$'s dotproduct < 0,
I can always negate one of them, since $-v_1$ would also be eigenvector
of a matrix.
so my question is:
1.for the wolfram a... |
$$\int \ln(\cos x)\,dx$$ In this integral can I substitute $\cos x$ by $e^t$? by proceeding this way i am using byparts but is looping back to the point from which i have started
|
I'm trying to find eigenvector that dotproduct < 0, but without so luck,try many different matrixs in wolfram alpha, then I find out, that if two vectors $v_1$
and $v_2$
's dotproduct < 0, I can always negate one of them, since −$v_1$
would also be eigenvector of a matrix. so my question is:
1.for the wolfram a... |
Can two eigenvector's dotproduct be < 0? |
I am reading the "Probability & Statistics for Engineers & Scientists" 9th edition and encounter a proof of theorem 4.4 where it said:
$$\mu_{X}=\sum_{x}xf(x,y),\ \ \mu_{Y}=\sum_{y}yf(x,y)$$ where $f(x,y)$ is the joint probability distribution of the random variables X and Y . But as far as I'm aware, the mean of a ra... |
Definition of mean of a random variable in joint probability distribution? |
I will assume that even though your language is propositional, your *logic* $L$ is non-classical. Why? Because the deduction theorem fails as per the condition of your question.
In general, if you do not have the deduction theorem, you cannot say that $\Gamma\cup\{\neg\phi\}$ is consistent iff $\Gamma\not\vdash_L\ph... |
More precisely,let's say A is n*n matrix
1.If all n eigenvalues and eigenvectors are known,
does it uniquely determine A?
2.let's say A have r ( r<n) n eigenvectors, and
all r eigenvalues and eigenvectors are known
, does it uniquely determine A? (the question is
different from question 1 since A has less tha... |
Would all eigenvalues and eigenvectors of Matrix A uniquely determine A? |
Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$.
If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get some properties about $\sum_{m=1}^\infty \phi_{m}z^m$ and $\sum_{m=1}^\infty \psi_{-m}z^-m$ from this equation $P(z)\p... |
When i want to check if f(x,y) is continous in (0,0) i basically have to look at the limes:
$\lim_{(x,y)\to (0,0)} \frac{3x^2-5y^2sin(x)}{|x|+|y|}$
if we come to (0,0) along the axis we have the following:
x-axis:
$\lim_{(0,y)\to (0,0)} \frac{3*0^2-5y^2sin(0)}{|0|+|y|} = 0$
y-axis:
$\lim_{(x,0)\to (0,0)... |
$f(x,y)=\frac{3x^2-5y^2sin(x)}{|x|+|y|}$ continous in (0,0)? |
How to construct an algebraic function that is not rational? |
When i want to check if $f(x,y)$ is continous in $(0,0)$ i basically have to look at the limes:
$\lim_{(x,y)\to (0,0)} \frac{3x^2-5y^2\sin(x)}{|x|+|y|}$
if we come to $(0,0)$ along the axis we have the following:
$x$-axis:
$$\lim_{(0,y)\to (0,0)} \frac{3\cdot 0^2-5y^2\sin(0)}{|0|+|y|} = 0$$
$y$-axis:
$$... |
$f(x,y)=\frac{3x^2-5y^2\sin(x)}{|x|+|y|}$ continous in (0,0)? |
When I want to check if $f(x,y)$ is continous at $(0,0)$ I basically have to look at the limit:
$$\lim_{(x,y)\to (0,0)} \frac{3x^2-5y^2\sin(x)}{|x|+|y|}$$
If we approach $(0,0)$ along the axes we have the following:
$x$-axis:
$$\lim_{(0,y)\to (0,0)} \frac{3\cdot 0^2-5y^2\sin(0)}{|0|+|y|} = 0$$
$y$-axis:
... |
Is $f(x,y)=\frac{3x^2-5y^2\sin(x)}{|x|+|y|}$ continous at $(0,0)$? |
When I want to check if $f(x,y)$ is continous at $(0,0)$ I basically have to look at the limit:
$f(0,0)=0$ for $(x,y)=(0,0)$ is also defined
$$\lim_{(x,y)\to (0,0)} \frac{3x^2-5y^2\sin(x)}{|x|+|y|}$$
If we approach $(0,0)$ along the axes we have the following:
$x$-axis:
$$\lim_{(0,y)\to (0,0)} \frac{... |
I'm trying to find eigenvector that dotproduct < 0, but without so luck,try many different matrixs in wolfram alpha, then I find out, that if two vectors $v_1$
and $v_2$
's dotproduct < 0, I can always negate one of them, since −$v_1$
would also be eigenvector of a matrix. so my question is:
1.for the wolfram a... |
Is $$f(x,y)=\begin{cases}\frac{3x^2-5y^2\sin(x)}{|x|+|y|} & \text{ if }\, (x,y)\neq (0,0),\\ 0 & \text{ if }\, (x,y)=(0,0).\end{cases}$$ continous at $(0,0)$?
When I want to check if $f(x,y)$ is continous at $(0,0)$, I basically have to look at the limit:
$$\lim_{(x,y)\to (0,0)} \frac{3x^2-5y^2\sin(x)}{|x|+|y... |
I am looking for references in discrete differential geometry for a concept I've been interested in.
It is very common to approximate smooth surfaces using discrete triangulations. I am interested in the opposite problem. I start off with some finite collection of edges that can form triangles, and I want to know wh... |
I'm trying to find eigenvector that dotproduct < 0, but without so luck,try many different matrixs in wolfram alpha, then I find out, that if two vectors $v_1$
and $v_2$
's dotproduct < 0, I can always negate one of them, since −$v_1$
would also be eigenvector of a matrix. so my question is:
1.for the wolfram a... |
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
**Direct problem** - If we know the PDF $p_X$, then the PDF $p_Y$ is for... |
> One video for beginner helped me, explaining me that
> - in common equations, we are searching for the values of a variable $x$, for solutions
> - in differential equations, for the values of a function $f(x)$
That statement is true, but misleading. Here is a more accurate statement:
* An "ordinary" e... |
For a [nonnegative random variable][1]
$$E[X]=\int_0^\infty P(X >x)dx$$
This [video][2] on a question from IIT JAM 2023 extends it to
$$E[X^2]=\int_0^\infty 2x P(X >x)dx$$
How was the result was arrived at?
Where can I get references for such results? Books or structured references (rather than single links pr... |
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$. My question is about the "inverse" problem described below.
**Direct prob... |
I'm learning about intersection homology, and I'm trying to write a proof of the following statement:
>Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to Y$ such that for all stratum $S$ of Y, there is a strata $S^f$ of Y such that $f(S)\subset S^f$ and $codim_Y(S^f)\le... |
Well, this is embarassing.
To be honest, during my PhD, I haven't really bothered too much regarding the topology of CW complexes.
Back then, I understood the first few pages of Hatcher's book (the construction is easy to follow, also the examples given there), and didn't really care much about it.
When I revised m... |
I have a function $f$ which depends on two random variables $a$ and $y$: $f(x,y)$.
I want to calculate the variance of $f$ w.r.t. $x$ and $y$, i.e. $\mathrm{Var}(f(x,y))$.
I am wondering if I can expand it similar to the definition of variance for a single random variable, i.e. if $\mathrm{Var}(f(x,y)) = E_{x,y... |
I got homework at the University and I think I solved this, but I am not sure, that my solution is right. Can somebody check it? Thank you!
Suppose the person $A$ tells the truth with a probability of $0.75$ and the person $B$ with a probability of $0.8$. Both persons act independently of each other. Determine the p... |
Let $A$ be the $n\times n$ matrix with entries $A_{ij}$ and consider the enlarged $2n\times 2n$ matrix
$$\tilde A=\begin{pmatrix}A_{11} & 0 & A_{12} & 0 & \dots & A_{1n} & 0 \\ A_{11} & 0 & A_{12} & 0 & \dots & A_{1n} & 0 \\ A_{21} & 0 & A_{22} & 0 & \dots & A_{2n} & 0 \\ A_{21} & 0 & A_{22} & 0 & \dots & A_{2n} & 0 \... |
Is there a general way to invert (solve for $u$) something like this?
$$\sum_{ij}R_{ijk}a_iu_j = -x_k$$
With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices. |
Do you know [Eisenstein's theorem](https://en.wikipedia.org/wiki/Eisenstein's_theorem)? It states that if you have any algebraic power series
$$f(z) = \sum_n a_n z^n$$
with $a_n \in \mathbb Q$, then there is a number $A \in \mathbb N$ such that $a_n A^n \in \mathbb N$ for all $n$. In other word, the power series
$$g... |
For $n\geq 1,$ is there a minimal $s$ (depending on $n$) such that $H^s(\mathbb{R}^N)$ is continuously embedded in $L^1(\mathbb{R}^n)?$ I researched several Sobolev embeddings books but I could not find such result or a counterexample. Thank you. |
Does convergence in distribution implies imply this inequity? |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
**Additional context**
The program has shown that the number of elements of... |
Let $A$ be the $n\times n$ matrix with entries $A_{ij}$ and consider the enlarged $2n\times 2n$ matrix
$$\tilde A=\begin{pmatrix}A_{11} & 0 & A_{12} & 0 & \dots & A_{1n} & 0 \\ 0 & A_{11} & 0 & A_{12} & \dots & 0 & A_{1n} \\ A_{21} & 0 & A_{22} & 0 & \dots & A_{2n} & 0 \\ 0 & A_{21} & 0 & A_{22} & \dots & 0 & A_{2n} \... |
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 ... |
In my mechanics we had a differential equation arise from a spring oscillator via its energy ($x^2_{max}$ is some constant maximum length) :
$$\begin{align}\frac{1}{2}kx^2_{max}&=\frac{1}{2}mv^2+\frac{1}{2}kx^2\\\frac{k}{m}x^2_{max}&=\dot{x}^2+\frac{k}{m}x^2\\\end{align}$$
So we have $\dot{x}^2+\frac{k}{m}x^2-\fr... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
With some intermediate derivation results I found the two integrals are exactly the same. But why?
\begin{align}
I=\int_0^\infty\frac{\sin x}{x + x^2}\mathrm dx=\int_0^\infty\frac{\pi-2\tan^{-1}x}{2 e^x}\mathrm dx
\end{align}
Both are equal to
\begin{align}
I&=\operatorname{Si}(1)\cos(1)-\operatorname{Ci}(1)\si... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
I came across this question in Pure Mathematics 2 in The Binomial Theorem section.
**The first part is simple and goes like this:**
>If a clock with a seconds pendulum registers *x seconds* too few per day, what is the time of
one beat of the pendulum?
Since there are $24\times60\times60$ *seconds*... |
In reaction diffusion based pattern formation (or other types of pattern formation too, really), it seems that the absence of an inversion symmetry (i.e., if the field $u(x,t)$ is a stable solution/steady state, then $-u(x,t)$ is not) leads to hexagonal pattern formation (rather than stripes or squares). This can be se... |
In my mechanics class we had a differential equation arise from a spring oscillator via its energy ($x^2_{max}$ is some constant maximum length) :
$$\begin{align}\frac{1}{2}kx^2_{max}&=\frac{1}{2}mv^2+\frac{1}{2}kx^2\\\frac{k}{m}x^2_{max}&=\dot{x}^2+\frac{k}{m}x^2\\\end{align}$$
So we have $\dot{x}^2+\frac{k}{m}x... |
I'm trying to find eigenvector that dotproduct < 0, but without so luck,try many different matrixs in wolfram alpha, then I find out, that if two vectors $v_1$
and $v_2$
's dotproduct < 0, I can always negate one of them, since −$v_1$
would also be eigenvector of a matrix. so my question is:
1.for the wolfram a... |
Can eigenvector sets always to change to dotproduct between any two of them >= 0? |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
I am reading Gortz's Algebraic geometry book, Vol.2 , p.207 and some qestion arises. I think that I am begginer of cohomology theory of schemes so please understand.
Let $f:X\to Y$ be a morphism of ringed spaces, where $Y=(*,A)$ possibly be a scheme such that the underliying space is consists of a single point a... |
I am considering a family of lattices $\Lambda_n$ in $\mathbb{C}$ generated by $\langle 1,\tau_n\rangle$ for $n\geq 2$ with
$$\tau_n = -\left(\frac{n^2}{4}+1\right)^{-1} \left(\frac{n^2}{2}+1-\frac{a}{2}i\right)$$
which comes from the complex values $\omega_{1;n}=1+\frac{n}{2}i$ and $\omega_{2;n}=-1+n i$ and $\tau_n... |
I am trying to solve the DE
$$\ddot{x} + x = \frac{-3A^3}{4}\sin(t + \phi) + \frac{A^3}{4} \sin(3(t + \phi)) $$
And here is my attempt and my question is at the bottom:
First we have the homogenous solution for $\ddot{x} + x =0$ which looks like
$$x_h = c_1 \cos(t) + c_2 \sin(t), c_1,c_2 \in \mathbb{C}$$
... |
I come form General Relativity background and have a question about differential geometry.
Assume a pseudo-Riemannian manifold $M$ equipped with a metric tensor field $g$ inducing inner product on the tangent bundle. Assume a codimension 1 submanifold in $M$ called $\Sigma$ with unit normal field $n$ and pullback of... |
**Main question as posted by the OP**
> How many different arrangements can be formed from the letters PEPPER?
**Main Doubts:**
1. `6!`
permutations of the letters when the repeated letters are distinguishable from each other
2. And that for each of these permutations, there are `(3!)(2!)`
permutations ... |
I'm trying to find eigenvector that dotproduct < 0, but without so luck,try many different matrixs in wolfram alpha, then I find out, that if two vectors $v_1$
and $v_2$
's dotproduct < 0, I can always negate one of them, since −$v_1$
would also be eigenvector of a matrix. so my question is:
1.for the wolfram a... |
How to construct a algebraic irrational function $f(x)$ such that $$f(x)= \sum_{i=1}^{\infty}a_i x^i$$ with $a_1, a_2,\dots, a_i,\dots \in \mathbb{N}$.
Reference is appreciated.
**Update**: an instance is $$f(x)=\frac{2}{1-\sqrt{1-4x}} =\sum C_n x^n$$ with $c_n$ being the
$n_{th}$ Catalan number. Thank to Jy... |
I am considering a family of lattices $\Lambda_n$ in $\mathbb{C}$ generated by $\langle 1,\tau_n\rangle$ for $n\geq 2$ with
$$\tau_n = -\left(\frac{n^2}{4}+1\right)^{-1} \left(\frac{n^2}{2}+1-\frac{n}{2}i\right)$$
which comes from the complex values $\omega_{1;n}=1+\frac{n}{2}i$ and $\omega_{2;n}=-1+n i$ and $\tau_n... |
Is it possible to prove the existence of the set of all function from a given set $X$ to $\{0,1\}$ without using the axiom of power set in $\mathsf{ZFC}$? |
Can I always construct eigenvector sets which dotproduct between any two of them >= 0? |
I often see functions written as $f:\Bbb{R}\rightarrow\Bbb{R}$ or similarly $f:\Bbb{C}\rightarrow\Bbb{C}$.
My concern is that there are functions where $f:\Bbb{R}\rightarrow\Bbb{R}$ doesn't satisfy.
And the more specific the function gets, the more likely it doesn't map to just basic number system (Such as reals, ... |
Is identifying the range of function necessary? |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
For $n\geq 1,$ is there a minimal $s$ (depending on $n$) such that $H^s(\mathbb{R}^n)$ is continuously embedded in $L^1(\mathbb{R}^n)?$ I researched several Sobolev embeddings books but I could not find such result or a counterexample. Thank you. |
I am considering a family of lattices $\Lambda_n$ in $\mathbb{C}$ generated by $\langle 1,\tau_n\rangle$ for $n\geq 2$ with
$$\tau_n = -\left(\frac{n^2}{4}+1\right)^{-1} \left(\frac{n^2}{2}+1-\frac{n}{2}i\right)$$
which comes from the complex numbers $\omega_{1;n}=1+\frac{n}{2}i$ and $\omega_{2;n}=-1+n i$ and $\tau_... |
Here, I **first** provide a proof with required details based on the relative entropy method, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved. Indeed, without the expectation constraint, the covariance of the distribution cannot be fixed, and... |
I often see functions written as $f:\Bbb{R}\rightarrow\Bbb{R}$ or similarly $f:\Bbb{C}\rightarrow\Bbb{C}$.
My concern is that there are functions where $f:\Bbb{R}\rightarrow\Bbb{R}$ doesn't satisfy.
And the more specific the function gets, the more likely it doesn't map to just basic number system (Such as reals, ... |
Is it possible to divide a square into $n>2$ parts, such that any $m$ of these $n$ parts can be used to reconstruct the original square, wherein $1<m<n$? The parts may overlap, but cannot rotate.
For the simple case of $n=3$ and $m=2$, I found a solution: dividing the square into four quadrants and then defining the... |
In a group of children, each child has a certain number of pencils from $1$ to $n$. If the number of children who have i or more pencils is $2^{n-i}$, for $i=1,2,3, \ldots,n$ and the total number of pencils is $511$, find the maximum number of pencils with any child.
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**My attempt:-**
Number of ch... |
I am looking for references in discrete differential geometry for a concept I've been interested in.
It is very common to approximate smooth surfaces using discrete triangulations. I am interested in the opposite problem. I start off with edges of prescribed lengths, which I can use to form triangles, and I want to ... |
Consider the $\mathbb{R}^3$ vectors expressed on terms of the canonical basis $$X=\sum_{i=1}^3x_ie_i,\,Y=\sum_{i=1}^3 y_ie_i,$$ so the wedge product of vectors is $$X\wedge Y=(x_2y_3-x_3y_2)e_1-(x_1y_3-x_3y_1)e_2+(x_1y_2-x_2y_1)e_3.$$ I was wondering if there's any relantio between this wedge and the usual wedge of for... |
While trying to evaluate $\int_0^1 k^2K(k)dk$ related to elliptic integral of the [first kind][1], by integral switching method, I reached the trigonometric integral
$$\int_0^{\frac\pi 2}\frac{\theta-\cos\theta\sin\theta}{2\sin^3\theta}d\theta$$
which is evaluated to $\frac{2C+1}4$ by Wolfram Alpha. Here, $C$ is the ... |
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