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Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set of functions $af(x+b)$ for $a,b \in \mathbb{R}$. If that set is closed under addition, what can $f$ be?
I suspect the possible $f$'s are exponentials, sinusoids, constants and their products. (For example, the sum of any two sinu... |
I am having issues computing Riemann tensor of 2D manifold. It should be very simple, but I cannot see the mistake I am making.
Assume a 2D manifold with metric in very simple form $$\tilde{g}=\frac{1}{f(x)^2}(dx^2+dy^2).$$
Since this is evidently conformally flat (indeed all 2D manifolds are, but here one can se... |
cos2a?
2cos3a = cosa
2cos3a = cosa ⟺ 2(4cosa^3-3cosa)=cosa ⟺8cosa^3 - 6cosa = cosa
8cosa^3=7cosa`we divide everything by cosa`
8cosa^2=7 ⟺ cosa^2=7/8
cos 2a = cos^2 - sin^2
We need to find sin^2
sin^2 + cos^2 = 1 ⟺ sin^2 = 1 - 7/8 ⟺ sin^2 = 1/8
cos 2a = 7/8 - 1/8 ⟺ cos 2a = 6/8 ⟺ cos 2a = 3/... |
> But I do not know if it is allowed or how to continue with it.
You know the law of total expectation, apply it to the conditional expectation:
$$E[Y|X] = E[ E[Y|X,U]]$$ where $U=1$ if $Z \ge 0$, $U=0$ otherwise, and the outer expectation is wrt to $U$.
Now, $E[Y|X,U=0] = a |X|$ and $E[Y|X,U=1] = b |X|$ , wh... |
I want to know the behavior of the solution without solving the ordinary differential equation for the following enzyme-substrate reactions.
[![enter image description here][1]][1]
Here, [S] is the concentration of S, and the others similarly represent the concentrations of substances sandwiched between brackets. E... |
I know this question seems too easy to post it here (it is for 9thers in India) but somehow I had to skip classes at school and couldn't attend geometry and now I am stuck with a question I couldn't solve....
> In the given semicircle; ABCDE is a cycle? pentagon in which AB is diameter. Find <C + <E
[![enter image ... |
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find a closed form for
$$\sum_{n=1}^\infty f\bigg( e^{-\sqrt{\log n}} \bigg). $$
I simplified this to
$$\su... |
Let $W(t)$ a Wiener process. By the scaling propety of the Brownian motion we have that
$$
\frac{1}{\sqrt{c}} W(ct)
$$
is a Wiener process. Let $(\mathcal{F}_t)_{t \ge 0}$ the Filtration of the Wiener process and suppose $cs <t$. Then $W(cs)$ is $\mathcal{F}_t$ measurable
As $\frac{1}{\sqrt{c}} W(cs) \sim W(s)$ ... |
I have to evaluate
$$\iint_{R}\frac{\sin(x-y)}{\cos(x+y)}dxdy$$
where $R$ is bounded by $0\leq x+y\leq1,x\geq0,y\geq0$. My first thought was to change the variables to
$$u=x-y$$
$$v=x+y$$
Since $0\leq x+y\leq 1$, then it's clear
$$0\leq v\leq1$$
However, is there a way I could find the limits of integration fo... |
Well, it is wrong because, as I see it, you tried to force the recursion when you do $$x^{\overline{n}} = x^{\overline{n-1}}(x+n+k-1),$$
that is **not true**, but you can do this:
$$x^{\overline{n}} = x^{\overline{n-1}}(x+n-1)=x^{\overline{n-1}}((x\color{red}{-k})+(n\color{red}{+k}-1)).$$
Now, when you distribute ... |
Can anyone tell me exactly what is the question that is being asked in English for this equation? I am confused.
Please check the link for the video:
https://youtube.com/shorts/mGDnsbKr7IM?si=3eLrjOzLffTp3SBb
A book costs \$1 plus half its price. How much does it cost?
A. \$1
B. \$1.50
C. \$2
Is the ... |
I know this question seems too easy to post it here (it is for 9th graders in India) but somehow I had to skip classes at school and couldn't attend geometry and now I am stuck with a question I couldn't solve
> In the given semicircle; ABCDE is a cycle? pentagon in which AB is diameter. Find $\angle C + \angle E$.
[... |
If $(X, \Vert \cdot \Vert_1)$ and $(X, \Vert \cdot \Vert_2)$ are Banach and $\Vert \cdot \Vert_1 \leq C \Vert \cdot \Vert_2 $ then the norms are equivalent by the open mapping theorem. I can't find an example where $(X, \Vert \cdot \Vert_2)$ is not Banach and the norms are not equivalent, can you help me?
I've tried... |
Let's say we have three lines written in their standard form:
\begin{align}
a_1x + b_1y + c_1 = 0 && (l_1) \\
a_2x + b_2y + c_2 = 0 && (l_2) \\
a_3x + b_3y + c_3 = 0 && (l_3)
\end{align}
Then if we consider the matrix
$$
M =
\begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{bma... |
Is there a visual proof for why this property of a matrix is true? |
Let's say we have three equations written in their standard form:
\begin{align}
a_1x + b_1y + c_1 = 0 && (l_1) \\
a_2x + b_2y + c_2 = 0 && (l_2) \\
a_3x + b_3y + c_3 = 0 && (l_3)
\end{align}
Then if we consider the matrix
$$
M =
\begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end... |
I am studying sheaf cohomology of complex manifolds and, while reading some proof about Dolbeault cohomology, I realized that there is a $\bar{\partial}$-Poincarè Lemma which gives us the local exactness of the complex we use when talking about Dolbeault cohomology and so allows us to use sheaf cohomology.
It's not ... |
Let $k\in\mathbb{N}$, $\alpha\in\mathbb{R}$, and $a,b\in\mathbb{R}$ such that $a,b>0$. Then
\begin{equation}\label{alpha-n=1-k-odd}\tag{q1}
\int_0^\infty\frac{\sin^{2k-1}t}{t}\sin(\alpha t)\operatorname{d} t
=\frac{(-1)^{k-1}}{2^{2k}}\sum_{q=0}^{2k-1}(-1)^q\binom{2k-1}{q}\ln\biggl|\frac{2q-2k+1-\alpha}{2q-2k+1+\alph... |
Define $$f_n(x_1,x_2,x_3,x_4)=\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}+\sum_{s_i\neq s_j, \forall i\neq j,s_i\in[0,n]} a_{n,s_1,s_2,s_3,s_4} x_1^{s_1}x_2^{s_2}x_3^{s_3}x_4^{s_4} $$ where $x_i\in (0,1)$ $\forall i$ and $$a_{n,s_1,s_2,s_3,s_4} =(-1)^{s_1+s_2+s_3+s_4}\binom{n}{s_1}\binom{n... |
I am trying to simulate a duffing oscillator. I am not sure how to read the phase diagrams I am obtaining. In the image, we can see at the top the phase diagram before the bifurcations; it has an elliptical shape and it's centered around 0,0. In the bottom picture, we can observe how now the phase is around -0.9 (corre... |
Let $K$ be a commutative ring and $R$ a $K$-algebra. Consider the $K$-algebra $R^e=R\otimes_K R^{op}$. The category of $R-R$-bimodules is a monoidal category with the tensor product $\otimes_R$ of $R$-bimodules and unit $R$. After identifying the category of $R-R$-bimodules with the category $R^e\text{-Mod}$ of left $R... |
Given a continuous (if need be $C^\infty$) function $\Phi(\boldsymbol{x})$, mapping $\mathbb{R}^3\to\mathbb{R}$, that has $n$ minima and obtains $\Phi\to0$ for $|\boldsymbol{x}|\to\infty$, is there a theorem about the number $m$ of maxima and/or saddles, i.e. points (apart from the minima) where $\boldsymbol{\nabla}\Ph... |
I am trying to simulate a duffing oscillator. I am not sure how to read the phase diagrams I am obtaining. In the image, we can see at the top the phase diagram before the bifurcations; it has an elliptical shape and it's centred around $0,0$. In the bottom picture, we can observe how now the phase is around $-0.9$ (co... |
Let's say we have three equations written in their standard form:
\begin{align}
a_1x + b_1y + c_1 = 0 && (l_1) \\
a_2x + b_2y + c_2 = 0 && (l_2) \\
a_3x + b_3y + c_3 = 0 && (l_3)
\end{align}
Then if we consider the matrix
$$
M =
\begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end... |
Let $(M, g)$ be a Riemannian manifold. I say $\lambda\in\mathbb R^\times$ is a scale factor of $M$ if $(M, g)$ is isometric to $(M, \lambda g)$. Composing isometries multiplies the scale factor, so there's subgroup $G=\{\lambda:\lambda\text{ is a scale factor of } M\}\leq\mathbb R^\times$ associated to this manifold wh... |
Which groups are achievable as achievable as the scale factors of a manifold? |
I have the polynomial $f = x^3 - y^2$, considered as a convergent power series $f \in \mathbb C\{x,y\}$. Is there a strategy to compute the group $G$ of right-equivalences (coordinate transformations) under which $f$ is fixed? Given $\varphi \in G$, set $\xi = \varphi(x), \zeta = \varphi(y)$, so that
$\xi^3 - \zeta^2 ... |
[![enter image description here][1]][1]
I have a blue input image (figure 1) with known width and height and known points A and B. I have a second red input image (figure 2) with known points A and B, which is the base image. The goal is to overlay the blue image over the red base image so that the points A and B ar... |
How can I calculate a transformation so that two reference points in two images are matching? |
My teacher defined an angle as the measure of the size of rotation of a ray around the vertex. But recently I stumbled across the Wikipedia article on angles, which defines an angle as the figure formed by two rays with a common endpoint, the vertex. So which definition is correct? Are both of them correct, perhaps?
... |
[![1]][1]
This is Self Test Exercise 7.8 of Sheldon's A First Course in Probability. I'm lost at what the question is asking, and illustrate this with an example.
Suppose $r = 10$. That means that there are $10$ families in the arriving plane (OK). The confusion starts from the next sentence. "A total of $n_j$ of... |
[**Updated on Nov.10, Dec.05.2023 and Feb.18.2024**] Applying a similar approach followed in these MO questions, for [Catalan's][1] and [Apery's][2] Constants, i.e. Wilf-Zeilberger proofs starting from Dougall's sum, working together with nonlinear $_2F_1$ and $_3F_2$ hypergeometric transformations, I have found four c... |
Let $P$ be a module over a conmutative unitary ring $A$ and $\mathcal{J}^r(P)$ be the module of $r$-linear forms $P \times \cdots \times P \longrightarrow A.$ If $\alpha \in \mathcal{J}^r(P)$ and $\beta \in \mathcal{J}^s(P)$ are two multilinear forms, its tensor product is, by definition:
$$
\alpha \otimes \beta (x_1... |
> But I do not know if it is allowed or how to continue with it.
You know the law of total expectation, apply it to the conditional expectation:
$$E[Y|X] = E[ E[Y|X,U]]$$ where $U=1$ if $Z \ge 0$, $U=0$ otherwise, and the outer expectation is wrt to $U$.
Now, $E[Y|X,U=1] = a |X|$ and $E[Y|X,U=0] = b |X|$ , wh... |
Let $C^{1\text{-var}}([0,T];\mathbb{R}^d)$ denote the space of continuous bounded variation paths taking values in $\mathbb{R}^d$. Similarly, let $C^{1\text{-Höl}}([0,T];\mathbb{R}^d)$ denote the space of $1$-Hölder continuous paths. Now, to me it's clear that
$$C^{1\text{-Höl}}([0,T];\mathbb{R}^d)\subset C^{1\text{-... |
I am wondering if there is a name for an algebra which is commutative up to some group action.
To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, and each of the pieces $A_n$ comes equipped with an action of the symmetric group $S_n$ such that $$ a \cdot b = \sig... |
I am studying sheaf cohomology of complex manifolds and, while reading some proof about Dolbeault cohomology, I realized that there is a $\bar{\partial}$-Poincaré Lemma which gives us the local exactness of the complex we use when talking about Dolbeault cohomology and so allows us to use sheaf cohomology.
It's not ... |
There's this proposition I encountered in a paper on Bayesian data analysis. I am trying to use it to ease some computations for a mixed effects model. It goes as follows:
$\textbf{Proposition 1.}$ Let $z_d, d = 1, \ldots, D - 1,$ be independent Gaussian random variables with different means $\mu_d$ and variances $\... |
My teacher defined an angle as the measure of the size of rotation of a ray around the vertex. But recently I stumbled across the Wikipedia article on angles, which defines an angle as the figure formed by two rays with a common endpoint, the vertex. So which definition is correct? Are both of them correct, perhaps?
... |
I'm going over the exterior covariant derivative $$d^\nabla : \Omega^k(E) \to \Omega^{k+1}(E)$$ of a vector bundle $E \to M$ and a connection $\nabla$ on $E$.
There are some peculiar notions which I'm facing. I'm trying to verify that $$d^\nabla( d^\nabla \omega) = F^\nabla \wedge \omega$$ where $F^\nabla$ is the cu... |
Let $R=\mathbb{Q}+ x\mathbb{R}[x]$.
What is its field of fractions and the integral closure in its field of fractions? |
How can you show that the centrifugal force is conservative? (From the definition of being the grad of some potential) |
Let $G$ be a Lie group, $T:G\to GL(\mathbb{V})$ a representation of $G$ in a vector space $\mathbb{V}$. A $\mathbb{V}$-valued one-cocycle is a (smooth) map $S:G\to V$ satisfying the property $S(fg)=T(f)\big( S(g)\big) +S(f)$, for all $f,g\in G$. In the book I'm studying, the author says that: given a one-cocycle $S$, o... |
What is the definition of an affine $G$-module? |
[**Updated on Nov.10, Dec.05.2023 and Feb.18.2024**] Applying a similar approach followed in these MO questions, for [Catalan's][1] and [Apery's][2] Constants, i.e. Wilf-Zeilberger proofs starting from Dougall's sum, working together with nonlinear $_2F_1$ and $_3F_2$ hypergeometric transformations, I have found four c... |
Examine the convergence of $\int_0^1 \frac{\sin(1/x)}{x^p} \space dx $
$$\bigg| \frac{\sin(1/x)}{x^p}\bigg| \le \frac 1{x^p} $$
Now $$ \int _0^1 \frac {dx}{x^p} $$ is convergent for $p\ < 1$
So by comparisons test given integral is convergent for $p<1$
When $ 1\le p <2$
By the transforma... |
I encountered this problem: Consider the following function：
$$
f(x,\Delta k)=\lim_{\Delta k \to \infty} \frac{\sin ^{2}\frac{x\Delta k}{2}}{\Delta k\left ( \frac{x}{2} \right )^{2} }\equiv 2\pi\delta(x)
$$
It can be verified that $f(x)$ satisfies the following condition:
$$
\delta(x)=\begin{cases}
0 &(... |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
The proof is very much ambiguous to me. For example:
1- The claim $\int_1^x u^{-\sigma} du = (x^{1- \sigma} -1)/(1- \sigma) < x^{1- \sigma}/(1- \sigma)$ is... |
1. Regarding what constitutes the metatheory, Kunen defines it as "basic finitistic reasoning" in his *Foundations of Mathematics* (in I.7.2) and elaborates in his section III:
> We conclude with two additional questions: What exactly *is* the metatheory? Why is it beyond reproach, as we said above, for even consist... |
Here is a version of the $\bar \partial$-Poincaré lemma as formulated in [1]
> **Proposition 2.31** *Let $\alpha$ be a $C^1$-form of type $(p,q)$ with $q > 0$. If $\bar \partial \alpha = 0$, then there locally exists on $X$ a $C^1$-form $\beta$ of type $(p,q-1)$ such that $\alpha = \bar \partial \beta$*.
Now to ... |
Exercise IV.2.12 of Hungerford's *Algebra* asks to show the following:
If $F$ is a free module over a ring with identity such that $F$ has a basis of finite
cardinality $n > 1$ and another basis of cardinality $n + 1$, then $F$ has a basis of
cardinality $m$ for every $m > n\ (m \in \mathbb N)$.
This is easy... |
I want to know the behavior of the solution without solving the ordinary differential equation for the following enzyme-substrate reactions.
[![enter image description here][1]][1]
Here, $k_1,k_2,k_3>0$, and [S] is the concentration of S, and the others similarly represent the concentrations of substances sandwiche... |
Question from an old test:
>The bakery produces doughnuts with weights $W_1 \text{g}, W_2 \text{g}, W_3 \text{g}, \ldots$, where $W_i$ for each doughnut is drawn from a distribution with a finite expected value $\mu$ and variance $\sigma^2$. Let $p_n$ denote the probability that the weight of $n$ doughnuts exceeds $n ... |
I want to find a function $a: \mathbb{N} \to \mathbb{S}^2$ s.t. $\lim_{n \to \infty} a_{n}=(0, 0, 1)$, if $(x_{1}, x_{2}, x_{3}) \in \mathbb{S}^2$ then it must happen $x_{1} \to 0$, $x_{2} \to 0$ and $x_{3} \to 1$
As it is a point on the sphere naturally $x^2+y^2+z^2=1$, could the same constant function work, i.e. b... |
In one differential geometry class I'm taking, a critical point $p$ of a function $f$ on a manifold $(M,\mathcal{A})$ is defined as a point where $\forall$ coordinate chart $(U,\phi_U)\in\mathcal{A}$, $Df\circ\phi_U^{-1}(\phi_U(p))=0$. However, I don't understand why do we want to define it this way. Why can't we defin... |
Let $X$ be a completely regular topological space and $X^n$ its $n$-ary Cartesian product for positive integer $n$.
Let $V$ be a subset of $X^n$ which is homeomorphic to $X^m$.
Does it imply $m\leq n$?
I expect the answer to be simple, but at the moment I am stumped to find it, as well as I don't seem to find ... |
Can an n-ary cartesian product contain a copy of m-ary Cartesian product with m>n? |
The first point you should think about is ... when the limit is a number (not $\infty$) so must be a $(x-3)$ in numerator. so $$\lim _{x\to 3}\frac{3x^2+ax+b}{x^2-2x-3}=\frac{5}{2}\\\lim _{x\to 3}\frac{3x^2+ax+b}{(x-3)(x+1)}=\frac{5}{2}\\\lim _{x\to 3}\frac{(x-3)(something)}{(x-3)(x+1)}=\frac{5}{2}
$$
now you can di... |
I have an expression in terms of the empirical covariance of $(X,Y)$, where X and Y are two variables in some data that I have. I want to be able to say what happens to this expression as a function of the covariance of $(X,Y^2)$. Anyone have any ideas?
Is this underspecified? If so, what other information do I nee... |
Covariance of $X$, $Y^2$? |
Russell's Paradox involves a test whether a set x is an element of itself.
But from a naive set theory perspective, sets have a hiearchy, per definitionem a set cannot be an element of itself. This seems to be a formalisation problem.
What axioms are meant to preclude this? |
Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities.
This doesn't make sense over $\mathbb{C}$ because it isn't an ordered field, but we do have something not present over $\mathbb{R}$ : a nontrivial automorph... |
There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or express the degree of approximation in terms of the error. There is an understanding of how to estimate the approximation e... |
I just watched a [video][1] by blackpenredpen where he solved this equation. Here are the steps he took (The process transitions from one line to another. He didn't explicitly use an implies or iff sign, so I'll leave it as such):
$x^{x^{x^{x^{x^{...}}}}} = 2$
$x^2 = 2$
$\left[
\begin{array}{l}
x = \sqrt{2}... |
$x^{x^{x^{x^{x^{...}}}}} = 2$. Why is $-\sqrt{2}$ not a solution? |
I have already read the standard part in Evan's PDE text book. However, in his book, the theorems like Sobolev embeddings only involve bounded domain, and it talks just a little about fractional Sobolev spaces. I want to look at a book that is deeper than this. I know the book written by Adams is a standard book, but t... |
Let $a(x)$, $b(x)$, $c(x)$ be smooth coefficient on $\mathbb{R}$ and $u(x)$ be the solution to the linear ODE over $\mathbb{R}$:
$$ au''+bu'+cu=-1 $$
Suppose $a(x)>1$, can we have some estimate on $|u'|$ independend of the derivatives of $a(x)$, $b(x)$, $c(x)$? |
I'm trying to understand Grigorieff's proof (Theorem 3, page 478, Intermediate Submodels and Generic Extensions in Set Theory) that symmetric submodels of $M[G]$ are exactly $(HOD(M[x]))^{M[G]}$ for $x\in M[G]$. I'm struggling to follow the proof (this is not helped by the fact that the notation is quite dated and the ... |
My question is about what I think is a mistake in Roman Kossak's book *Model theory for beginners* (which is in my opinion a good book; I at least enjoyed a lot what I've read so far).
In chapter 5, Kossak defines the **type** $\operatorname{tp}^{\mathfrak M}(\bar a)$ of a tuple $\bar a \in M^{< \omega}$ in the doma... |
What's the coefficient of the $x^{302}$ in the following expression:
$$\dfrac{1}{(1-x^2)(1-x^3)(1-x^6)}$$
I don't know where to start. Also x are in the denominator how this expression can have x to power of a positive term? |
Let $K$ be a commutative ring and $R$ a $K$-algebra. Consider the $K$-algebra $R^e=R\otimes_K R^{op}$. The category of $R-R$-bimodules is a monoidal category with the tensor product $\otimes_R$ of $R$-bimodules and unit $R$. After identifying the category of $R-R$-bimodules with the category $R^e\text{-Mod}$ of left $R... |
I'm recently learning Shannon entropy. The discrete case seems to be easy to understand and I'm trying to apply it to the continuous case.
Suppose a channel has discrete inputs $X$ and outputs $Y$ per unit time, with a joint probability function $p(x,y): X\times Y \to [0,1]$, where $p(x,y)$ is the probability that $... |
I am wondering if there is a name for an algebra which is commutative up to some group action.
To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, and each of the pieces $A_n$ comes equipped with an action of the symmetric group $S_n$ such that $$ a \cdot b = \sig... |
The lemniscate functions $\text{sl}$ and $\text{cl}$ are the solutions to the differential equation $$ (y')^2+y^4=1$$ with $\ y(0)=0, \ y'(0)=1$ $\ $ or $\ $ $y(0)=1, \ y'(0)=0.$
Using the integral definitions for the functions $$z = \int\limits_0^{\text{sl} z} \frac{\text{dt}}{\sqrt{1-t^4}}=\int\limits_{\text{cl... |
**Premise**. I was given the following definition of beth-numbers: <br>
$$ \beth_\alpha := \begin{cases}
\beth_0 = \aleph_0 \\
\beth_{\alpha + 1} = 2^{\beth_\alpha} \\
\beth_{\lambda} = \bigcup_{\alpha < \lambda} \beth_\alpha &\text{$\lambda$ limit}
\end{cases}$$
besides, my teacher defined cofinality both as... |
Proof, there are no holomorphic functions $g, h:\mathbb{C}\to\mathbb{C}$, such that $g(\mathbb{C})\setminus h(\mathbb{C})=\{ z\in \mathbb{C}: |z|\geq 1\}$
Can I assume, that $h(\mathbb{C})$ must be the unit disc, so by that h must be constant, therefore there are no such functions? Or that it must be bounded somehow?... |
I am trying to match the excel FT (Fourier Transform) of a Gaussian to the analytical FT.
My Column A goes from 0 to 4095, column B goes from -2048 to 2047.
Column C is: C1=(1/200*sqrt(2 pi())) exp(-0.5*(B1/200)^2), which is a Gaussian.
Using Excel's Fourier Analysis pack, I FT column C and put the output in C... |
As an application of the chain rule, we have for $f:\mathbb R^n\to \mathbb R^m$ which is of class $C^1$ the following:
$$\frac{d}{dt} f(x_1(t),\dots,x_n(t))= \sum_{i=1}^n \partial_i f(x_1(t),\dots,x_n(t))\frac{dx_i}{dt}(t). $$
Where $\partial_j$ is the partial derivative w.r.t. the $j$-$th$ variable.
Note the multip... |
I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book:
If $N(\pi) \neq 3$, the cubic residue character of $\alpha$ modulo $\pi$ is given by
$(a)$ $ (\frac{\alpha}{\pi})_3 = 0$ if $\p... |
From Wikipedia, the volume of a small ball in an $n$-dimensionl Riemannian manifold M satisfies
$$\frac{Vol(B(\epsilon, p)\subset M)}{Vol(B(\epsilon, 0)\subset \mathbb{R}^n)} = 1 - \frac{S_p}{6(n+2)}\epsilon^2 + O(\epsilon^3)$$
where $p\in M$ and $S_p$ is the scalar curvature of $M$ at $p$. My question is: what d... |
Let us consider the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and for any $g \in L^2(\mathbb{T},\mathbb{C})$, define $P_N : L^2(\mathbb{T},\mathbb{C}) \to L^2(\mathbb{T},\mathbb{C})$ as
\begin{equation}
[P_Ng](x)=\sum_{k=-N}^N \langle g, e_k\rangle_{L^2} e_k(x)
\end{equation}
where $e_k(x):=e^{2\pi ikx}$. That is,... |
I am trying to match the excel FT (Fourier Transform) of a Gaussian to the analytical FT.
My Column A goes from 0 to 4095, column B goes from -2048 to 2047.
Column C is: C1=(1/200*sqrt(2 pi())) exp(-0.5*(B1/200)^2), which is a Gaussian.
Using Excel's Fourier Analysis pack, I FT column C and put the output in C... |
Let $x_n=\frac{1}{n}$ for $n=1,2,\dots$ Then $x_n$ is not summable but $x_n^r$ is summable for every $r>1$.
My question is: Is there a sequence of positive numbers $x_n$ that is summable but $x_n^r$ is not summable for every $0<r<1$?
|
Is there a barely summable sequence in terms of power? |
Good evening,
We were playing with a friend on Desmos, and we came to $x \longmapsto \arctan\left(\exp\left(-\displaystyle\frac{1}{\sqrt{1-x^2}}\right) \right)$. Here is the graph :
[![][1]][1]
[1]: https://i.stack.imgur.com/zzhbq.png
And we have :
$$\int_{-1}^1 \arctan\left(\exp\left(-\displaystyle\... |
Let $p:E \to B$ be a covering map and let $h:E \to E$ be a homeomorphism such that $p=p \circ h$. In this case, it is known that if $E$ is path connected then $h$ is uniquely determined by the image of a point. In Dieck's *Algebraic Topology* page 65 it is claimed that this same property holds if $E$ is connected, but ... |
I am studying topology and I have some doubt about the following. I show what I have tried. Any help would be really appreciated!
Let $ X $ be the quotient space of the Euclidean Plane (with usual topology) w.r.t. the equivalence generated by "$x$ is equivalent to $y$ IFF they are both not the origin and they lie on... |
> So from the start: if we take a Platonist perspective (which I was taught as the most pedagogically effective philosophy to have when learning set theory) then we assume sets in some way or another exist along with the intuitive properties like membership.
We might do that, or we might e.g. take a more neutral for... |
I am currently learning multivariate function calculus and trying to figure out the relationship between differentiability, continuity, and directional derivatives. My main question is, if a directional derivative exists in all directions at a point $(x_0, y_0)$ and satisfies the equation
$$D_{\vec{u}}f(x_0, y_0) =... |
> So from the start: if we take a Platonist perspective (which I was taught as the most pedagogically effective philosophy to have when learning set theory) then we assume sets in some way or another exist along with the intuitive properties like membership.
We might do that, or we might e.g. take a more neutral for... |
I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book:
If $N(\pi) \neq 3$, the cubic residue character of $\alpha$ modulo $\pi$ is given by
$(a)$ $ (\frac{\alpha}{\pi})_3 = 0$ if $\p... |
I am currently learning multivariate function calculus and trying to figure out the relationship between differentiability, continuity, and directional derivatives. My main question is, if a directional derivative exists in all directions at a point $(x_0, y_0)$ and satisfies the equation
$$D_{\vec{u}}f(x_0, y_0) =... |
It's a known fact (see for example Bartoszynski and Judah: Set Theory, On the Structure of the real Line) that for a filter $F$ on $\omega$ the following conditions are equivalent:
$F$ is Lebesgue measurable.
$\exists$ a family $\{A_n: n\in \omega \}$ such that
- $A_n$ consists of finitely many finite subset... |
Let $(M,g)$ be a Riemannian manifold of dimension $n$ and let $g_{ij}$ be the matrix of the components of $g$ in coordinates. Since $g_{ij}$ is symmetric and positive-definite, it can diagonalized by $g_{ij} = PDP^{T}$, with $D = \text{diag}(\lambda_1, \ldots, \lambda_n)$ diagonal and $P$ orthogonal. For $s \in \mathbb... |
I was wondering: is there a sequence of positive numbers $(a_n)$ such that $\sum a_n<\infty$ but for every $0<p<1$, $\sum a_n^p=\infty$?
None of the convergent series I was able to come up with fulfilled this property. But it seems to me that there must be one. Any suggestions? |
Let $k$ be a field and $R = k[X_1, \ldots, X_n]$ and $I = \langle X_1, \ldots, X_n\rangle$. I am trying to prove that the quotient ring $\frac{R}{I^r}$, $r\in\mathbb{N}$ is local, i.e., has only one maximal ideal. Here we define $I^r$ as the ideal generated by the products $f_1\ldots f_r$ with each $f_i\in I$.
By th... |
Is this quotient of a polynmial ring local? |
I'm not sure how to approach this, as my understanding would be to use (maximal ideal of R, maximal ideal of C) but not sure how to find these, I know for Z max ideals are mZ where m is prime but not really sure how to find max ideals for other uncoutable sets? |
What are some of the maximal ideals in ℝ×ℂ? |
Is this quotient of a polynomial ring local? |
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