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I tried something like that:
For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get;
$lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (Should i look it in two cases when z=c and z=x ? since for $x>c$ wlog $z\in[c,x]$ ) After that the limit gets the form;
$l... |
Your approach is correct.
For problem 1, the $y$-axis is a characteristic curve, so you should expect things to go wrong when you try to assign values to $v$ there.
For problem 2, you can in principle solve the ODE through multiplication by the integrating factor $e^{3s}$, but then you will need to calculate
$$
... |
Suppose we have the recurrence
$T(n) = T(\frac{n}{2}) + d$ if $n = 2^j$ and where is some integer greater than $0$ (i.e n is even). I know that this recurrence is $\Theta(lg(n))$, and I want to prove it with two inductive substitution proofs, one for $O(lg(n))$ and one for $\Omega(lg(n))$. If both proofs are succe... |
Does my proof that the recurrence $T(n) = T(\frac{n}{2}) + d = \Theta(lgn)$work? |
Assume we have:
\begin{cases}
a_{n} = 2^n\left(1 - b_{\left\lfloor \frac n2\right\rfloor}\right)\quad(\forall n\ge0) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}}\quad(\forall n\ge1) \\
b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$... |
This is from my uni's algebra textbook.
Given vector $u=(1_5,3_5,4_5)$ and basis $B=\left \{ (2_5,3_5,4_5), (1_5,2_5,3_5), (3_5,3_5, 2_5) \right \}$, find the coordinates of the vector $u$ in the basis $B$.
I get that from the definition, the coordinates of $u$ are $(\alpha,\beta, \gamma )$ where $\alpha,\beta, \... |
$f(x)=\frac{x}{2}-\frac{sin(200πx)}{400π}$
I'm not a mathematician. I'm making a heater which uses this formula to calculate the watt of given x. The problem is I need inverse of this function between (0 and 0.01) to use in PID algorithm. I couldn't get inverse of it so I asked Wolfram Alpha which told me that it d... |
I tried something like that:
For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get;
$lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (Should i look it in two cases when z=c and z=x ? since for $x>c$ wlog $z\in[c,x]$ ) After that the limit gets the form;
$l... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to definition of solvable groups.
>Equivalent definition: $G$ is obtained from the group $\{1\}$ by a finite number of extensions with abelian kernels.
In the same section, the following claim is given, as... |
I tried something like that:
For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get;
$lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (Should i look it in two cases when z=c and z=x ? since for $x>c$ wlog $z\in[c,x]$ ) After that the limit gets the form;
$l... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
>Equivalent definition: $G$ is obtained from the group $\{1\}$ by a finite number of extensions with abelian kernels.
In the same section, the following claim is given, ... |
I tried something like that:
For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get;
$lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (Should i look it in two cases when z=c and z=x ? since for $x>c$ wlog $z\in[c,x]$ ) After that the limit gets the form;
$l... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
a_{0} = 1, b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\fr... |
I'm learning about the ideal class group of a number field, and am trying a few exercises where I calculate $\mathbb{Q}(\sqrt{d})$ for $d \in \mathbb{Z}$ for various $d$.
I'd like to check my work. Unfortunately, I can't seem to find a list anywhere of ideal class groups, nor a calculator to do it. I can only find l... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
Let be
https://i.imgur.com/hufWpv4.jpg
a quiver. We have that $\mathbb{K}Qe_2=span \{e_2,a'',a''' \}$, so $V_1=e_1\mathbb{K}Qe_2=\{0 \}$, $V_2=e_2\mathbb{K}Qe_2=span \{e_2 \}$, $V_3=a''\mathbb{K}Qe_2=span \{a'' \}$,$V_4=a''\mathbb{K}Qe_2=span \{a''' \}$, so a representation is this:
https://i.imgur.com/OPzfk... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
Assuming the elipse is just an circle being scratch on one aix, And the change in the perimeter is the change in all Vectors length. We may say that the elipse is an circle being linearly transform to an elipse and then the change in the perimeter is the basically change in all vectors length eich we can calculate by d... |
It's not true that $ \int _ { \bigcup _ i A _ i } f = \sum _ i \int _ { A _ i } f $ unless the $ A _ i $ are all disjoint (or at least have intersections of measure zero). And even then, I'm not sure how you would prove that integrals decompose like this without already knowing at least the $ \Longrightarrow $ directi... |
Assuming the elipse is just an circle being scratch on one aix, And the change in the perimeter is the change in all Vectors length. We may say that the elipse is an circle being linearly transform to an elipse and then the change in the perimeter is the basically change in all vectors length wich we can calculate by d... |
Today I've stumbled across this integral: $$\int_{0}^{\pi/2} (\frac{sin5x}{sinx})^2 \,dx $$
I was on a time limit and my intuition told me: $$\int_{0}^{\pi/2} (\frac{sin5x}{sinx})^2 \,dx = \int_{0}^{\pi/2}5dx= 5x|_0^{\pi/2}=\frac{5\pi}{2}$$
Which turned out to be the right answer. I'd like to know what is the co... |
There exists a generating function for the Chebyshev polynomials in the following form:
$$\sum\limits_{n=1}^{\infty}T_{n}(x) \frac{t^n}{n} = \ln\left( \frac{1}{\sqrt{ 1 - 2tx + t^2 }}\right)$$
**Question:** can one find a similar closed form for $\sum\limits_{n=1}^{\infty}T_{2n}(x) \frac{t^n}{n}$ ? |
I am going through a maths book and unable to understand the following:
Given that x and y are rational numbers, then if the equation $2 \sqrt{3} - 3 = (x + y) \sqrt{3} - 2 \sqrt{3xy}$ is true, then we must have $x + y = 2$ and $2 \sqrt{3xy} = 3$.
I don't understand this. We cannot reach such a conclusion with eq... |
Assuming the elipse is just an circle being scratch on one aix, And the change in the perimeter is the change in all Vectors length. We may say that the elipse is an circle being linearly transform to an elipse and then the change in the perimeter is the basically change in all vectors length wich we can calculate by d... |
The Question:
---
>In $\operatorname{SL}(2,q)$, does there always exist at least four elements of distinct order?
Thoughts:
---
I have strong reasons to suspect that this is true.
The order of $\operatorname{SL}(2,q)$ is $(q-1)q(q+1)$, so is a multiple six. Cauchy's theorem implies elements $g,h$ of order... |
Does there exist at least four elements in $\operatorname{SL}(2,q)$ of different order? |
Assuming the elipse is just an circle being scratch on one aix, And the change in the perimeter is the change in all Vectors length. We may say that the elipse is an circle being linearly transform to an elipse and then the change in the perimeter is the basically change in all vectors length wich we can calculate by d... |
Today I've stumbled across this integral: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $$
I was on a time limit and my intuition told me: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx = \int_{0}^{\pi/2}5dx= 5x\Big|_0^{\pi/2}=\frac{5\pi}{2}$$
Which turned out to be the right answer.... |
Assuming the elipse is just an circle being stretch on one aix, And the perimeter of shape equals to the sum of all Vectors length. We may say that the elipse is an circle being linearly transform to an elipse and then the change in the perimeter is the basically change in all vectors length which we can calculate by d... |
Let $R$ an equivalence relation over a set $A$. For $a \in A$ we define $a / R = \{ b \in A: (a, b) \in R\}$ and call this set the equivalence class of $A$. Furthermore, we define $A / R = \{a / R : a \in A\}$ the set of all equivalence classes for the equivalence relation $R$ over $A$.
As a last definition, if $f : ... |
How do I show that the synthesis equation of the Fourier Transform equals the original function?
[![enter image description here][1]][1]
[![enter image description here][2]][2]
I want to expand equation 2 using equation 1 and show that the integral indeed equals the original function. I am fine with some handwav... |
How do I show that the synthesis equation of the Fourier Transform equals the original function? |
In short: angle are dimensionless (m/m for rad and m²/m² for sr), and have no base units in SI, but this creates problems. The status of angle units has never been clear for the BIPM, the bureau in charge of the SI system, but as making angle units true base units creates more problems than it solves, there is a status... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
>Equivalent definition: $G$ is obtained from the group $\{1\}$ by a finite number of extensions with abelian kernels.
In the same section, the following claim is given, ... |
**In short**:
Angles are dimensionless quantities (e.g. m/m for rad and m²/m² for sr). They have no base units in SI, but this creates problems. The status of angle units has never been clear for the BIPM, the bureau in charge of the SI system, but as making angle units true base units creates more problems than it ... |
I want to solve the following optimization problem:
Let $S \in \mathbb{R}^{p \times p}$ be a symmetric matrix, and fix $n \in \mathbb{R}$ such that $n$ is significantly less than $p$ and the rank of $S$ is at most $n$. Now, I want to find a positive-semidefinite real matrix $R \in \mathbb{R}^{p \times p}$ that minim... |
I'm very confused about the question below, which I couldn't figure out for days.
In Example 5 the author is tearching us proving
\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{1}{n!}) = e
with equation
x_n = (1+\frac{1}{n})^n
I'm thinking x_n < e if we take e is the s... |
How to prove \lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{1}{n!}) = e in Calculus With Analytic Geometry By George Simmons? |
I'm very confused about the question below, which I couldn't figure out for days.
In Example 5 the author is teaching us proving
$$\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{1}{n!}) = e$$
with equation
$$x_n = (1+\frac{1}{n})^n$$
I'm thinking $x_n < e$ if we take $e$ is th... |
How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{1}{n!}) = e$ in Calculus With Analytic Geometry By George Simmons? |
A capable group is a group that is isomorphic to $G/Z(G)$ for some group $G$.
So, is every member of the OEIS sequence [A216594][1] (consisting of those $n$ for which there is a non-nilpotent group, but no centerless group, of order $n$) the order of a capable group?
Every even number $2n$ is the order of a capab... |
Is every member of A216594 the order of a capable group? |
I was trying to proof Milstein Algorithm so I read the book "Stochastic Numerical Methods_ An Introduction for Students and Scientists" of Toral. Here, they show all the proof in one dimension and many dimensions but just considerer white noise in "time", not in space.
I was also reading the book "Noise in Spatiall... |
Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that resembles the plus construction in $\mathsf{Top}$?
More precisely, let $G$ be a (perfect) group and $|BG|$ the classifying ... |
**In short**:
Angles are dimensionless quantities (e.g. m/m for rad and m²/m² for sr). They have no base units in SI, meaning angles have no fundamental existence (contrary to a length), they are *derived* from something else. This actually creates problems and since a long time proposals have been made to give angl... |
This is a question from Villani's Topics in Optimal Transportation. Let's define $\mu, \upsilon$ probability measures on some measure spaces $X, Y$ respectively, and $\pi$ a probability measure on the product space $X \times Y$.
In the Introduction chapter, pg. 2, there is an intuitive assertion that for a transfere... |
A capable group is a group that is isomorphic to $G/Z(G)$ for some group $G$.
So, is every member of the OEIS sequence [A216594][1] (consisting of those $n$ for which there is a non-nilpotent group, but no centerless group, of order $n$) the order of a capable group?
Every even number $2n$ is the order of a capab... |
**In short**:
Angles are dimensionless quantities (e.g. m/m for rad and m²/m² for sr). They have no base units in SI, meaning angles have no fundamental existence (contrary to a length or a time), they are *derived* from something else. This actually creates problems and since a long time proposals have been made to... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\frac{x}{2}) =... |
>Let $b_n$ be a sequence of positive real numbers such that $$\lim_{n \to \infty} n\left( \frac{b_n}{b_{n+1}}-1\right)=p > 0$$
the limit exists and it has a positive value. Then $b_n$ must approach $0$.
My attempt:
Let $\varepsilon > 0$. Then there is $n_\varepsilon > 0$ such that for all $n > n_\varepsilon$ we ... |
I'm trying to implement an inverse Z-Transform using the Fourier series based technique by Abate and Whitt:
> The Fourier-series method for inverting transforms of probability distributions.
> Numerical inversion of probability generating functions.
We approximate the integral numerically by
$$f(n) \approx \fra... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
I am new to the study of (undergraduate) convexity and I have recently come across a new definition which generalizes to the classic concept of convex function and is the following. Let be $I\subset \mathbb{R}$ an interval. We say that $f$ is a exponential type convex function if it holds that for every $x, y \in I$ an... |
We will execute a "degree matching argument". That is,
- The determinant of the matrix
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & 1 & 1 & 1 & \cdots & 1 \\ x^{2} & 2 x^{2} & 2^{2} x^{2} & \cdots & 2^{m-1} x^{2} & 1 & 2 & 2^{2} & \cdots & 2^{m-1} ... |
>Let $b_n$ be a sequence of positive real numbers such that the limit from the Raabe-Duhamel's test is positive, i.e. $$\lim_{n \to \infty} n\left( \frac{b_n}{b_{n+1}}-1\right)=p > 0$$
the limit exists and it has a positive value. Then $b_n$ must approach $0$.
My attempt:
Let $\varepsilon > 0$. Then there is $n_... |
A particle of mass $m$ moving along the $x$-axis, is acted upon by a force:
$$F\left(t\right)=\begin{cases}A\mathrm{e}^{-\gamma t}\sin\left(\omega t\right),&t\ge0;\\0,&t<0.\end{cases}$$
Initial conditions are: $x(0)=x_{0},\dot{x}(0)=\dot{x}_{0}.$ Here $\dot{x}$ is $\frac{dx}{dt}$.
Find the equation of motion $x(t)$ ... |
[This is the solution given in book][1]
[1]: https://i.stack.imgur.com/Wuvv9.png
I just want to ask that if it is written two equations of relations of l, l1, m, etc. How did he write it in that ratio or fractional form like l upon something and m upon something etc and made it equal. How did he solve that step?... |
We will execute a "degree matching argument". That is,
- The determinant of the matrix
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & 1 & 1 & 1 & \cdots & 1 \\ x^{2} & 2 x^{2} & 2^{2} x^{2} & \cdots & 2^{m-1} x^{2} & 1 & 2 & 2^{2} & \cdots & 2^{m-1} ... |
Suppose we want to evaluate
$$\int_C f(z) \, d \overline{z}$$
(I'm intentionally being vague with how $f$ and $C$ are defined because my question doesn't focus on these details - see example below for such details). Suppose that the best way to go about this is to instead evaluate the integral with respect to $z$... |
Edit: Using asymptotic to and not equal to as it keeps the meaning intact and makes sense of the relation
Would it be correct to imply from Euler's prime equality function
$x is ~ (asymptotic) to \frac {(\log x)^{\pi(x)}}{\prod \log p}$
where $\pi(x)$ is prime counting function?
$(\sum_{i=0}^x 1/i)$ ~ $(1 + ... |
Smallest triangular number with prime signature the same as [A025487][1](n), or 0 if no such number exists. (there is a similar sequence [A081978][2])
For n = 1 through n = 29, this sequence is: (I have not confirmed the n = 15 term is 0, but it seems that it is 0, i.e. it seems that there is no triangular number of... |
Here is my system of equations:
$C: \begin{cases}x^2 + y^2 +z^2 = 14\\ x^3+y^3+z^3=36 \end{cases}$
Firstly I managed to show that for all $a \in C$, the implicit function theorem applies to express $x$ and $y$ as a function of $z$. Which implies the existence of $\phi$ such that, for all $(x,y,z)$ close enough t... |
In [this paper][1] the following equation appears under Theorem 2.3.:
[![enter image description here][2]][2]
How should I interpret the $j$-indexed denominator? I am familiar with the pochhammer symbol notation, however these typically only have one parameter (on top of the integer index).
Thanks.
[1]: h... |
What does this specific notation mean? |
I'm trying to understand the OEIS sequence [A059078][1]:
> Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement.
>
> 0, 0, 0, 1, 2, 6, 12, 27, 54, ...
I'm able to recreate the exact number of such necklaces for $n=3$, $n=4$, and $n=5$, but when $n... |
![This is the solution given in book][1]
[1]: https://i.stack.imgur.com/Wuvv9.png
I just want to ask that if it is written two equations of relations of l, l1, m, etc. How did he write it in that ratio or fractional form like l upon something and m upon something etc and made it equal. How did he solve that step... |
Your system in a neighbourhood of the point $P=(1,2,3)$ defines a differentiable curve $\gamma$. By using the Implicit Function Theorem, we may find the tangent line to $\gamma$ at $P$: there exist two functions $x=\phi(z)$ and $y=\psi(z)$ such that $\phi(3)=1$ and $\psi(3)=2$ and the desired tangent line has the follo... |
Suppose we want to evaluate
$$\int_C f(z) \, d \overline{z}$$
(I'm intentionally being vague with how $f$ and $C$ are defined because my question doesn't focus on these details - see example below for such details). Suppose that the best way to go about this is to instead evaluate the integral with respect to $z$... |
Your system in a neighbourhood of the point $P=(1,2,3)$ defines a differentiable curve $\gamma$. By using the Implicit Function Theorem, we may find the tangent line to $\gamma$ at $P$: there exist two functions $x=\phi(z)$ and $y=\psi(z)$ such that $\phi(3)=1$ and $\psi(3)=2$ and the desired tangent line has the follo... |
Hmm... it's not as obvious as I thought
See, my first thought was this as a case of if $W$ is irrational and $a,b,c,d$ are rational and $a+bW = c+dW$ then $b=d$ and $a=c$.
Why? Because $a-c = (d-b)W$. The left hand side is rational so the right hand side must be rational. But $W$ is irrational and the *only* wa... |
I am self-studying some basic linear algebra. I encountered the following theorem:
> **Theorem**$\quad$ Let $A$ be an $n \times n$ matrix.
>
> 1. $A$ is diagonalizable if and only if it has eigenvectors $\mathbf{x_1},\mathbf{x_2},\dots,\mathbf{x_n}$ such that the matrix $P=\begin{pmatrix}\mathbf{x_1} & \mathbf{x_2} ... |
My professor has given us a take-home exam with a question regarding the cardinality of sequences. He defines a sequence as (verbatim): $$\text{given } A \in \{\text{sets}\}, (s \text{ is a sequence} \iff s : \mathbb{N} \to A)$$
I am supposed to evaluate the truth of the following statement: $$s, t \in \{\text{sequen... |
How do I Interpret the Cardinality of a Sequence? |
Let $(X_i)_{i \in \mathbb{N}} \sim \mathcal{N}(0,1)$, where all variables are i.i.d. For the random vector
$$\mathbf{X} = \left [ X_1, \ \ \ \ X_1 + X_2, \ \ \ \ X_2 + X_3, \ \ \ \ \cdots \ \ \ \ X_{n-2} + X_{n-1}, \ \ \ \ X_{n-1} \right ]^T,$$
how can I find $\mathbb{P}(\mathbf{X} \geq 0)?$ In other words, th... |
In *[Bilinear generating functions for orthogonal polynomials][1]* ([PDF][2]) the following equation appears under Theorem 2.3.:
[![enter image description here][3]][3]
How should I interpret the $j$-indexed denominator? I am familiar with the pochhammer symbol notation, however these typically only have one para... |
A capable group is a group that is isomorphic to $G/Z(G)$ for some group $G$.
So, is every member of the OEIS sequence [A216594][1] (consisting of those $n$ for which there is a non-nilpotent group, but no centerless group, of order $n$) the order of a capable group?
Every even number $2n$ (for $n>1$) is the orde... |
How can I simplify the following expression? |
A group $G$ is said to be capable if there is some group $H$ for which $H/Z(H)$ is isomorphic to $G$.
It is known that the only capable cyclic group is the trivial group. So, if $n$ is a cyclic number (i.e., the cyclic group is the only group of order $n$) other than $1$, then there is no group whose center has inde... |
This question arises from the following problem:
> We define the set $N=\{1,2,...,n\}$. How many different possibilities are there to form three sets $A$, $B$ and $C$ such that $A\subseteq B\subseteq C\subseteq N$?
There are two ways to find the solution to this problem:
**Strategy 1**
For each $m\in N$, t... |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y$$
I cannot pursue it because of the floating point in comp... |
Determine the domain-sets of $$ Log(e^z-e^{-z})$$
I only know that $e^z-e^{-z} > 0$ and so z must be positive. Also, $e^z = e^{x+iy} $
> The solution is $\Bbb{C}$ \ { $k \pi i | k \in \Bbb{Z}$ }
I don't understand the steps that will lead to this. Is there some other theory that I should know? Any help is ap... |
I understand that structures are means by which we know how to interpret certain languages. So for example, in the language of number theory, we might have $\displaystyle L=\left\langle 0,S,+,\cdot,E,< \right\rangle$. As I understand it, without specifying our strucutre, we do not know how to inerpret the symbols of ou... |
https://twitter.com/martinmbauer/status/1763622278128947464?t=YObGhK4ZqjAXrwPaHxB8gw&s=19
> Many know Euler's identity, but did you also know that
>
> $|e^{i\pi}| = |\pi^{ie}| = |i^{\pi e}| = 1$
While the claim about $\pi$'s powers feels intriguing to me (for I don't know any better), the claim about $i$'s pow... |
A capable group is a group that is isomorphic to $G/Z(G)$ for some group $G$.
So, is every member of the OEIS sequence [A216594][1] (consisting of those $n$ for which there is a non-nilpotent group, but no centerless group, of order $n$) the order of a capable group?
Every even number $2n$ (for $n>1$) is the orde... |
I am self-studying some basic linear algebra. I encountered the following theorem:
> **Theorem**$\quad$ Let $A$ be an $n \times n$ matrix.
>
> 1. $A$ is diagonalizable if and only if it has eigenvectors $\mathbf{x_1},\mathbf{x_2},\dots,\mathbf{x_n}$ such that the matrix $P=\begin{pmatrix}\mathbf{x_1} & \mathbf{x_2} ... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$ \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n = \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E^s}\lambda(0) = \Gamma(s)\lambda(-s).$$
Th... |
Apply the Leibniz rule:
$$\frac{d^n}{dx^n}\frac{\ln^3 x}{1-x}=\sum_{m=0}^n\binom nm\frac{d^m}{dx^m}\ln^3(x)\frac{d^{n-m}}{dx^{n-m}}\frac1{1-x}$$
Now use the Stirling number of the first kind [$\frac{d^n}{dx^n}\ln^a(x)$ formula](https://functions.wolfram.com/ElementaryFunctions/Log/20/02/0004/) and simplify the resu... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\frac{x}{2}) =... |
Apply the Leibniz rule:
$$\frac{d^n}{dx^n}\frac{\ln^3(x)}{1-x}=\sum_{m=0}^n\binom nm\frac{d^m}{dx^m}\ln^3(x)\frac{d^{n-m}}{dx^{n-m}}\frac1{1-x}$$
Now use the Stirling number of the first kind [$\frac{d^n}{dx^n}\ln^a(x)$ formula](https://functions.wolfram.com/ElementaryFunctions/Log/20/02/0004/) and simplify the res... |
>If $g$ is $C^1$ on the unit circle $C(0,1)$. Then there is a function $f^+$ holomorphic on $B(0,1)$ and continuous on $\bar B(0,1)$, a function $f^-$ holomorphic on $\mathbb{C}\backslash\bar B(0,1)$ and continuous on $\mathbb{C}\backslash B(0,1)$, such that $g(z)=f^+(z)-f^-(z)$ on the unit circle.
This seems like i... |
Can we write every $C^1$ complex function on the unit circle as the the difference of two approriate functions? |
Apply the Leibniz rule:
$$\frac{d^n}{dx^n}\frac{\ln^3(x)}{1-x}=\sum_{m=0}^n\binom nm\frac{d^m}{dx^m}\ln^3(x)\frac{d^{n-m}}{dx^{n-m}}\frac1{1-x}$$
Now use the Stirling number of the first kind [$\frac{d^n}{dx^n}\ln^a(x)$ formula](https://functions.wolfram.com/ElementaryFunctions/Log/20/02/0004/) and simplify the res... |
I was learning about the harmonic series back in college to which the professor said "There is no known closed form for the harmonic sum", I felt that was strange given that the sum didn't look complicated at all. So, I took a pen and paper and started exploring a way to find a closed form. In the end I found this func... |
Let $k$ be an algebraically closed field, with $n\in\mathbb N\subset k$ invertible. I am trying to prove that if $\mathbb G_m=k[t,t^{-1}]$ is the multiplicative group scheme over $k$, and $\mu_n$ is the kernel of the group map $[n]:\mathbb G_m\rightarrow \mathbb G_m$ defined on all $k$ schemes $S$ by $s\in G(S)\mapsto ... |
Is there a "correct" $k$ scheme structure to put on $\coprod_{i=1}^n \operatorname{Spec}(k)$? |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
- (i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\inf... |
Here's a question I'm sturglling with:
> Show that for an increasing sequence of events $$A_1\subset A_2\subset
A_3\subset ...$$ the next equation holds
> $$P\big(\bigcup_nA_n\big)=\lim_{n\rightarrow\infty}P(A_n)$$
Here's what I have so far: let $(B_n)_{n=1}^{\infty}$ be a sequence of events such that
$$B_{1}... |
$x\mapsto(x,x,\cdots,x)$ is the correct one, i.e. using the map $\mathsf{Spec}(\Delta)$ where $\Delta:k\to k^n$. Why? Because you have a coproduct of schemes, which are all $k$-schemes; if you want a *coproduct of $k$-schemes* i.e. the coproduct as taken in the slice category over $\mathsf{Spec}(k)$, which is categoric... |
I understand that structures are means by which we know how to interpret certain languages. So for example, in the language of number theory, we might have $\displaystyle L=\left\langle 0,S,+,\cdot,E,< \right\rangle$. As I understand it, without specifying our strucutre, we do not know how to inerpret the symbols of ou... |
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