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I am having trouble understanding the proof of
**Theorem** If $F : M \to N$ is an embedding, then $F (M)$ is an embedded
submanifold of $N$
Proof. Let $p \in M$. Since $F$ is an immersion, by Theorem 1 (see below) there are charts $(U,\varphi)$ around $p$ and $(V,\psi )$ around $F(p)$ such that $F(U) \subseteq V... |
I am having trouble understanding the proof of
**Theorem** If $F : M \to N$ is an embedding, then $F (M)$ is an embedded
submanifold of $N$
Proof. Let $p \in M$. Since $F$ is an immersion, by Theorem 1 (see below) there are charts $(U,\varphi)$ around $p$ and $(V,\psi )$ around $F(p)$ such that $F(U) \subseteq V... |
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \infty \frac{2^n}{2^n n\ln(2)(\ln(n \ln(2)))^2}=\sum_{n=3}^\infty \frac{1}{n \ln(2)(\ln(n \ln(2)))^2} $$
and
$$\sum_... |
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \infty \frac{2^n}{2^n n\ln(2)(\ln(n \ln(2)))^2}=\sum_{n=3}^\infty \frac{1}{n \ln(2)(\ln(n \ln(2)))^2} $$
and
$$\sum_... |
I think this is a bad question, because it doesn't really matter if there is a name for sets with this specific property, as the property isn't sufficiently interesting to warrant a name being described to it.
If you wanted to be wordy about it, you could just say, "$\mathbb{N}\setminus A$ is less than a geometric s... |
I think this is a bad question, because it doesn't really matter if there is a name for sets with this specific property, as the property isn't sufficiently interesting to warrant a name being described to it.
I also don't see how the sets you describe are "refined notions of natural density."
*Looking for* names... |
Let $R$ be a positive integer, $\mathcal{X}$ be the sample space and $x \in \mathcal{X}$ be an event of the sample space; $P(x)$ denotes the probability of occurrence of event $x$. The problem is to upper bound the following expression:
$$
\sum_{x\in \mathcal{X}}x(P(x))^{\frac{1}{R}},
$$
in terms of $$\sum_{x\in ... |
I'm reading Jean-Pierre Serre's 1970 "Cours d'arithmetique". I'm having trouble reading the beginning of his chapter 2 devoted for example to $\mathbb Z_3$, $3-$adic numbers that I discover and that interest me because of [questions][1] I have about the ring sequence
$$(\mathbb Z/2\mathbb Z,\mathbb Z/6\mathbb Z,\mathb... |
The surface integral $\int_S (z^2 + y^2 + x^2) \, dS$ over the cube $S$? |
The surface integral $\iint_S (z^2 + y^2 + x^2) \, dS$ over the cube $S$? |
Given 9 real numbers $x_1, x_2, ... , x_9\in [-1,1]$ such that $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$.
I have tried ordering the numbers from smallest to largest and then dividing the set of integers $\{x_1, x_2, ... , x_9 \}$ into two subsets of only negative numbers and only pos... |
I am a post graduate student who is currently studying some optimization for quadratic form. From the class lecture, I know that we can always turn any quadratic functions into theirs corresponding canonical form using some changes of variable.
For example: $f\left( {x,y} \right) = x \times y$ can be rewritten as $... |
Existence of canonical from for cubic and quartic form? |
I was solving a problem from old exams and got stuck here. I'd appreciate the help.
We have the three variables p, q, and r. There are 8 valuations of the variables. If F is a propositional logic formula containing p, q, and r, and we construct a truth table for F, the table will thus have 8 rows.
Provide a form... |
I'm looking for the inverse Laplace transform of
$$F(s) = \frac{1}{s + e^{-s\tau}}$$
where τ is a positive real parameter.
I am trying to use general inverse formula of Laplace transformation to solve it. But then, I need to find the singularities of F(s), that is, $ s + e^{-s\tau} = 0$.
Transform the euqation... |
There is a standard result in measure/integration theory which I just cannot seem to obtain.
If $f \colon X \to \mathbb{C}$ is measurable ($X$ is any measurable space), there exist simple measurable functions $\phi_k \colon X \to \mathbb{C}$ such that $\phi_k \to f$ pointwise and all $|\phi_k| \le |f|$. This is fine... |
A random walk S with absorbing barriers at $0$ and $N$, at each step the particle can move right,left or stay put with probabilities (p+q+r=1). Let $W$ be the event that the particle is absorbed at $0$ rather than $N$ , and let $p_k=\mathbb{P}(W|S_0=k)$. Show that if the particle starts at $k$ where $0 < k < N$, the co... |
Recall that a ring $R$ is called right (left) Noetherian if every right (left) ideal $I$ of $R$ is a finitely generated $R$-module, i.e., there exists $x_1,\ldots,x_m \in I$ such that $I=x_1R+\ldots+x_mR$ (or $I=Rx_1+\ldots+Rx_m$).
Suppose that $R$ is finitely generated as an additive group, i.e., $R=\mathbb{Z}x_1+... |
Suppose that $A\subseteq\mathbb{N}$. Then one can look at the function $f(n)=|\{0,...,n\}\setminus A|$ ($|S|$ denoting the cardinality of $S$ ). I am interested in the case when $f(n)\leq Cn^\alpha$ with $C>0$ and $\alpha<1$. In this case $A$ obviously has natural densitiy $1$. My question is: Is there already a name f... |
In machine learning there is one fundamental step which is the feedforward step of neural networks. Basically given a matrix of weights W, a input vector x, a bias vector and a g function likes tanh the algorithm is defined as
g(Wx+b)
Nothing very complex, however things get more complex if input is not a vect... |
I have $F(G(x))=G(x)$, where $x$ is any real number, and function $F$ can be computed if value $G(x)$ is given. I have also proved that function $G$ is monotone with respect to $x$ and $G(x)\in[0,1]$.
I want to obtain a mapping $G$ (or an algorithm G).
Is this a fixed-point problem? How to solve it, either ... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT,
$$
... |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
The relation between two functions $\beta$ and $\alpha$ is: $\beta(\alpha(x))=\alpha(x)$. what is this problem? |
Is $\sin\left(\frac{1}{x}\right)$
periodic? with what period time? |
Is $\sin\left(\frac{1}{x}\right)$ periodic?with what period time? |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
This is an exercise in the book by *B. Daya Reddy*.
For $f \in L^2(0, 1)$, let $u_f$ be the solution of the ODE: $u'' + u' - 2u = f$, $u(0) = u(1) = 0$. Define the functional $\ell$ by
$$
\langle \ell, f \rangle = \int_0^1 u_f(x) dx \ \forall f \in L^2(0, 1)
$$
Show that $\ell$ is a bounded linear functiona... |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
Suppose $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,y,z,w$ that:
If all the eigen values of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB+A+I)<0$ and $\d... |
Is Infinite Product of Hausdorff spaces Hausdorff with product topology? |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
> Find all $x\in \mathbb R$ such that $16sin^{3}x -14cos^{3}x = \sqrt[3]{sinxcos^{8}x + 7cos^{9}x}$
It's a tough one question I've found. I've tried with this way.
$16tan^{3}x -14 = \sqrt[3]{tanx + 7 }$
By inspection $tanx=1$ is one of the answer.
but According to [WA][1], $tanx$ is not equal to 1!!!!
... |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
I have $\beta(\alpha(x))=\alpha(x)$, where $x$ is any real number, and function $\beta$ can be computed if value $\alpha(x)$ is given. I have also proved that function $\alpha$ is monotone with respect to $x$ and $\alpha(x)\in[0,1]$.
I want to obtain a mapping $\alpha$ (or an algorithm).
Is this a fixed-poi... |
> Find all $x\in \mathbb R$ such that $16\sin^{3}x -14\cos^{3}x = \sqrt[3]{\sin x\cos^{8}x + 7\cos^{9}x}$
It's a tough one question I've found. I've tried with this way.
$16\tan^{3}x -14 = \sqrt[3]{\tan x + 7 }$
By inspection $\tan x=1$ is one of the answer.
but According to [WA][1], $\tan x$ is not eq... |
Suppose $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,y,z,w$ that:
If all the eigen values of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB+A+I)<0$ and $\d... |
> Find all $x$in \mathbb R$ such that $16\sin^{3}x -14\cos^{3}x = \sqrt[3]{\sin x\cos^{8}x + 7\cos^{9}x}$
It's a tough one question I've found. I've tried with this way.
$16\tan^{3}x -14 = \sqrt[3]{\tan x + 7 }$
By inspection $\tan x=1$ is one of the answer.
but According to [WA][1], $\tan x$ is not eq... |
I'm trying to find an explicit upper bound bound (explicit => without big O or small o notation, and with explicitly calculated constants) for the class number of a quadratic field.
So far, I've found that It was Littlewood who first addressed the question of how large the class number $h$ of an imaginary quadratic... |
I am trying to solve the following exercise: *Given $\phi: B(\mathcal{H}) \to \mathbb{C}$ linear functional, prove that $\phi$ is continuous with respect to weak operator topology if and only if $\phi$ is continuous with respect to strong operator topology.*
However, I don't know how to approach it, and I don't have... |
I was looking at [papers about the SYK model](https://arxiv.org/abs/1711.08482) page 33 (equation 112), in which they write
$$\int\mathscr{D}\Sigma\,\mathscr{D}G~\exp\left\{-\frac{N}{2}\int\limits_{\left[0,\beta\right]^2} d\tau~d\tau^{\prime}~\Sigma(\tau,\tau^{\prime})\left(G(\tau,\tau^{\prime})~-~\sum_{i=1}^N\psi_i... |
I want to define a mathematical model (formula) to design missing information from sampling. In my problem, I have real events whose value is binary (0, 1) which might change every second. For a given sample period, $p$, I periodically checks the value at $t$, and assumes the value stays the same until the next observa... |
What are the contraharmonic and inverse contraharmonic means? Are there any inequalities that relate them to each other like the AM-GM inequality, for example? |
What are the Contraharmonic and Inverse Contraharmonic Means? |
In statistics, the hypothesis testing is used for determining whether the null hypothesis can be rejected or failed to reject. For example, we use hypothesis testing in order determine the significance of the estimator in linear regression:
$H_0: \beta_1 = 0 \ \ \ H_1 : \beta_1 \neq 0$
My question is the approach... |
Is hypothesis test an intuitionist approach? |
Consider the following logical statement.
$$ (A \land B) \implies (C \land D)$$
**Question:** Under what conditions does the following statement follow from the above?
$$ (A\implies C) \land (B \implies D)$$
---
**Context:** I realised the heart of [my previous question](https://math.stackexchange.com/qu... |
I am a high school student and there is something I want to ask about the application of digital sums.
Let's say there is a fraction "**520/7",** let **520/7=a**, so **520= a × 7**, so if we now calculate the digital sums, it would be like **7= a × 7** , it means the digital sum of a should be 1 and nothing else so ... |
When is digital sum applicable and when it isn't? |
In statistics, the hypothesis testing is used for determining whether the null hypothesis can be rejected or failed to reject. For example, we use hypothesis testing in order determine the significance of the estimator in linear regression:
$H_0: \beta_1 = 0 \ \ \ H_1 : \beta_1 \neq 0$
My question is the approach... |
I am stuyding my combinatorics syllabus and came across two claims, that are said to be generalisations of the Matrix Tree Theorem:
G = (V,E) is a complete graph without loops. U is a subset of V. The matrix Q^U(G) is the matrix retreived from the Laplacian matrix L(G) by removing the rows and columns that correspon... |
> Find all $x$ in $\mathbb R$ such that $16\sin^3(x) -14\cos^3(x) = \sqrt[3]{\sin x\cos^8(x) + 7\cos^9(x)}$
It's a tough question I've found. I've tried using
$16\tan^3(x) -14 = \sqrt[3]{\tan x + 7 }$
By inspection, $\tan x=1$ is one of the answers.
but According to [WA][1], $\tan x$ is not equal to $1... |
I was solving a problem from old exams and got stuck here. I'd appreciate the help.
We have the three variables p, q, and r. There are 8 valuations of the variables. If F is a propositional logic formula containing p, q, and r, and we construct a truth table for F, the table will thus have 8 rows.
Provide a form... |
This is an exercise in the book by *B. Daya Reddy*.
For $f \in L^2(0, 1)$, let $u_f$ be the solution of the ODE: $u'' + u' - 2u = f$, $u(0) = u(1) = 0$. Define the functional $\ell$ by
$$
\langle \ell, f \rangle = \int_0^1 u_f(x) dx \ \forall f \in L^2(0, 1)
$$
Show that $\ell$ is a bounded linear functiona... |
Let $R \subseteq A \times A$ and $S \subseteq A \times A$ be two arbitary equivalence relations.
Prove or disprove that $R \cup S$ is an equivalence relation.
Reflexivity: Let $(x,x) \in R$ or $(x,x) \in S \rightarrow (x,x) \in R \cup S$
Now I still have to prove or disprove that $R \cup S$ is symmetric and tran... |
I have the following coinflip game:
We flip a coin infinite many times(Maybe countably many times/in practice we have a given/it holds for infinite).
For every pair of tails, tails twice in a row, A gets a point. While for every pair of tails-heads, tails followed by heads, B gets a point. If it is heads-heads or... |
Let $I = \int_2^\infty \frac{sin^2(x)}{x^\alpha(x^\alpha + \sin(x))}dx$. Then $I$ is divergent iff $\exists \epsilon > 0: \forall A(\epsilon) = A\ge 2 : \exists A_1, A_2 > A: |\int_{A_1}^{A_2} \frac{sin^2(x)}{x^\alpha(x^\alpha + \sin(x))}dx| \ge \epsilon$. Let $A_1 = n\pi$ and $A_2 = 2n\pi$ for some large enough $n\in\... |
> Show that for any real random variable $X$ and any proper open subset $U$ of $\mathbb{R}$, we have $$\mathbb P(X \in U)=\sup\limits \left \{\mathbb P(X \in K): K \subseteq U \text { is compact} \right \}.$$
**My Attempt** $:$ Let $U \subseteq \mathbb R$ be open. Let $$\alpha : = \sup\limits \left \{\mathbb P (X \... |
A group of *2n* individuals consisting of *n* couples, are randomly arranged at a round table. You are required to find an upper bound for the probability that none of the couples are seated next to each other.
**Solution:**
This is a combinatorial problem. Let's denote the total number of ways to arrange *2n* indi... |
This is an exercise in the book by *B. Daya Reddy*.
For $f \in L^2(0, 1)$, let $u_f$ be the solution of the ODE: $u'' + u' - 2u = f$, $u(0) = u(1) = 0$. Define the functional $\ell$ by
$$
\langle \ell, f \rangle = \int_0^1 u_f(x) dx \ \forall f \in L^2(0, 1)
$$
Show that $\ell$ is a bounded linear functiona... |
> Show that for any real random variable $X$ and any proper open subset $U$ of $\mathbb{R}$, we have $$\mathbb P(X \in U)=\sup\limits \left \{\mathbb P(X \in K): K \subseteq U \text { is compact} \right \}.$$
**My Attempt** $:$ Let $U \subseteq \mathbb R$ be open. Let $$\alpha : = \sup\limits \left \{\mathbb P (X \... |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$ I get the integral:
$$\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\frac{a}... |
I want to derive $\sin(A-B) = \sin A \cos B - \cos A \sin B$ from $\cos(A-B)=\cos A \cos B +\sin A \sin B$
$$\cos(A-B)=\cos A \cos B +\sin A \sin B$$ substitute $A$ for $A + \frac{\pi}{2}$
$$\cos(A+\frac{\pi}{2}-B)=\cos( A+\frac{\pi}{2}) \cos B +\sin( A+\frac{\pi}{2}) \sin B$$
$$\sin(A-B)=\sin(A) \cos B +\cos(A) \... |
I have the following coinflip game:
We flip a coin arbitrary many times(finite).
For every pair of tails, tails twice in a row, A gets a point. While for every pair of tails-heads, tails followed by heads, B gets a point. If it is heads-heads or heads-tails neither get a point. The winner is the player with the m... |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$ I get the integral:
$$\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\frac{a}... |
Let $F$ be a field and $M_2 (F)$ denotes the ring of all $2\times 2$ matrices with entries in $F$.
I was wondering if someone could help me about this claim: Why is $M_2 (F)$ a direct sum of two minimal left ideals?
Thanks in advance. |
Why is $M_2 (F)$ a direct sum of two minimal left ideals? |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$ I get the integral:
$$\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\frac{a}... |
Pauli matrices satisfy
$$
\sigma_i \sigma^{\dagger}_j = \delta_{ij} \sigma_0 + \epsilon_{ijk}\sigma_k\,.
$$
Can one construct a set of three complex-valued $2\times 2$ matrices $a_i$, $i=1,2,3$, such that
$$
a_i a^{\dagger}_j = \delta_{ij} \sigma_0 - \epsilon_{ijk}\sigma_k\,?
$$
If not, is it possible for matri... |
We have the following mod $p$ technique to detemine the Galois group of a polynomial in $\mathbb{Z}[x]$:
>Assume $f\in\mathbb{Z}[x]$ is irreducible, $p$ is a prime,
and $\bar{f}$ is the polynomial obtained by $f$ mod $p$.
(Or more precisely, $\bar{f}$ is the image of the homomorphism $\phi: \mathbb{Z}[x]\to F_p[x]... |
Is the converse of $\mod p$ method in determining Galois group true? Or is there a prime corresponding to every member of the Galois group? |
i have been self studying algebraic geometry and have always struggled to grasp how affine spaces naturally "lie" in projective Space, as in, i haven't found any convincing sketches of what that would look like in even the simplest of cases; $\mathbb{A}^1$ and $\mathbb{P}^1$. I do understand the concept of "points at i... |
Can the Square root of $(-9)$ have two answers namely $+3i$ or $-3i$?
Squaring both $3i$ and $-3i$ does give $-9$.
Why does my text book say only $3i$? |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT,
$$
... |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$
I get the integral:
$$l =\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\fr... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0,\quad \mbox{and} \quad 2a^2\ne b^2.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T... |
Is the converse of $\bmod p$ method in determining Galois group true? Or is there a prime corresponding to every member of the Galois group? |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$
I get the integral:
$$l =\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\fr... |
Hi learning Lean I came cross the following property of category equivalence:
>**Definition** An equivalence of categories cosnists of $(F, G, \eta, \epsilon)$ where $F: C \rightarrow D$ and $G: D\rightarrow C$ are functors and $\eta: 1_C \cong GF$, $\epsilon: FG\cong 1_D$ are natural isomorphisms.
and
> **Theo... |
To avoid confusion, the following somewhat cumbersome notation seems appropriate to me :
Let us write $(2:1:7)_{ten}$ instead of $217$. It means $$(\color{red}2:\color{green}1:\color{blue}7)_{ten}=\color{red}2\times ten^2+\color{green}1\times ten^1+\color{blue}7\times ten^0$$
That's base *ten*
__________________... |
So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$.
I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $
Then using the formula $$l = \int_{0}^{a}\sqrt{1+(y'(x))^2} dx$$
I get the integral:
$$l =\int_{0}^{a}\sqrt{2-2\sqrt{\frac{a}{x}}+\fr... |
Given a set $X$ and a disjoint collection $R \subseteq \operatorname{Pot}(X)$, define a relation $\sim$ on $X$ via $x \sim y \Leftrightarrow x = y \text{ or } \exists S \in R: x \in S \text{ and } y \in S$. Define $X/R = X/\sim$ as the set of equivalence classes. Also, if $X$ is equipped with a topology, equip $X/R$ wi... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0,\quad \mbox{and} \quad 2a^2\ne b^2.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T... |
To avoid confusion, the following somewhat cumbersome notation seems appropriate to me :
Let us write $(2:1:7)_{ten}$ instead of $217$. It means $$(\color{red}2:\color{green}1:\color{blue}7)_{ten}=\color{red}2\times ten^2+\color{green}1\times ten^1+\color{blue}7\times ten^0$$
That's base *ten*
__________________... |
Give an interpretation where
$$∃_{x} (\neg P(x) ∨ Q(x)) \to (∃_{x} P(x) ∧ ∀_{x} \neg Q(x))$$
is false.
How does someone even begin with questions like this? I have interpreted it in my head and I kind of get it in a sense. But seems like the only thing I know is that since it is an implication, the only way it wil... |
Why won't Wolfram Alpha compute my Limit? |
I know that the dimension of the manifold formed by **n-dimensional symmetric matrices** is n(n+1)/2.But the manifold formed by **n-dimensional positive definite symmetric matrices** is of what dimension?Thanks! |
The manifold formed by n-dimensional positive definite symmetric matrices is of what dimension? |
I have the values of a polynomial $p(x)$ defined on the Galois field mod $p$ (with $p$ prime) at the points zero to around two million. I need an algorithm to find the coefficients of the original polynomial $p(x)$ in time $O(n \log (n))$, as $O(n^2)$ is way too slow for this application. In Chapter 3 of Pan's *Structu... |
I'm reading Jean-Pierre Serre's 1970 "Cours d'arithmetique". I'm having trouble reading the beginning of his chapter 2 devoted for example to $\mathbb Z_3$, $3-$adic numbers that I discover and that interest me because of [questions][1] I have about the ring sequence
$$(\mathbb Z/2\mathbb Z,\mathbb Z/6\mathbb Z,\mathb... |
## Setting
Suppose $Q\subset \mathbb{R}$ is a compact metric space. Let $\mathcal{P}(Q)$ be the set of Borel probability measures on $Q$. This set is endowed with the topology of weak-* convergence: a sequence $\left\{ m_{N} \right\}$ of $\mathcal{P}(Q)$ converges to $m \in \mathcal{P}(Q)$ if $\forall \varphi \in C(Q)... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0,\quad \mbox{and} \quad 2a^2\ne b^2.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T... |
Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n} $$
This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n} $ and 'evaluating' that power series at $z=2$. I am interested in a divergent renormalization for this expression.
For many such divergent seri... |
How to find the divergent renormalization of $\sum_{n=1}^{\infty} \frac{2^n}{n}$? |
I know that the dimension of the manifold formed by **n-dimensional symmetric matrices** is $\dfrac {n(n+1)}2$.But the manifold formed by **n-dimensional positive definite symmetric matrices** is of what dimension?Thanks! |
I'm studying general topology and a question has come to my mind.
Given a sequence x (i.e. map from the set N of natural numbers) in a topological space and given a point p of the latter, we say that *p is a special point for x* if the x-preimage of every neighbourhood of p is cofinal in the canonical order of N (th... |
The original question :
> Find all $x$ in $\mathbb R$ such that $16\sin^3(x) -14\cos^3(x) = \sqrt[3]{\sin x\cos^8(x) + 7\cos^9(x)}$
It's a tough question I've found. I've tried using
$16\tan^3(x) -14 = \sqrt[3]{\tan x + 7 }$
By inspection, $\tan x=1$ is one of the answers.
but According to [WA][1], $\ta... |
**Setup**
Let there be a board looking like a rectangular table. A piece is placed at any square of the board. Two players play a game. They move the piece in turns. The piece can only be moved to an adjacent square (no diagonal moves). The piece can’t be moved to a square that it has already visited. A player who c... |
**the variable (X,Y) uniformly distributed over D:{(x,y)∈R^2:|x|+|y|<=1}, A=X-Y,B=X+Y
are A and B independent**?
I tried to prove F(ab)=F(a)F(b), i tried to find the joint function of AB trough changing variables. x=(A-B)/2 and Y=(A+B)/2 and got that the joint function is F(ab)=c/2 how do I continue from here? how ... |
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