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Hint 1: Fundamental theorem of calculus
Hint 2: Let $F(x)=\displaystyle\int^x_{-x}\dfrac{t^2e^t}{e^t+1}~dt$. What is $F(0)$?. |
So there is a million (essentially equivalent) ways to make it obvious **algebraically** that the axis of symmetry of the parabola $y=x^2+bx+c$, must be at $x=-\frac{b}{2}$, while its real roots, if they exist, are $\pm\sqrt{\left(\frac{b}{2}\right)^2-c}$ away from that axis.
It feels like there must be a way to mak... |
When we solve a system of linear equations in $n$ variables by Gauss elimination, there are two ways to write the general solution:
1. As one $n$-tuple depending on the free variables,
2. As a linear combination of specific vectors, with free variables as coefficients, to which a fixed vector is added.
For example... |
For a matrix $A$, by $||A||$, I mean the matrix-norm induced by the $\ell^2$-norm. Let $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times p}$, $C\in\mathbb{R}^{n\times m}$, $D\in\mathbb{R}^{n\times p}$ with $||A|| < 1$. Define,
$$
N_k :=
\begin{cases}
CA^{k - 1}B, &\text{if }k \geq 1\\
D, &\text{if } k = 0\\
... |
I was reading [Moysés Nussenzveig's "Basic Physics Course 1"](https://edisciplinas.usp.br/pluginfile.php/5481240/course/section/6000551/Moyses_Mecanica.pdf) when I came across this excerpt in chapter 11, about rotations and angular momentum, in section 11.2, vector representation of rotations:
> We could then think ... |
Are "infinitesimal rotations" commutative? If so, which mathematical fact allows it? |
I'm struggling to solve a problem in group cohomology theory which could seem immediate to some more expert mathematicians here. Suppose to have a non-degenerate, skew-symmetric bicharacter $b\colon G\times G\to\mathbb{T}$ on an Abelian group $G$, namely
$$b(gg',h)=b(g,h)b(g',h),\quad g,g',h\in G$$
$$b(g,hh')=b(g,h)b... |
Zarisky topologies are defined via closed-set definition; with the closed sets being algebraic varieties $V(I)$ of ideals $I\subseteq R$ of commutative rings $R$... But what are these varieties, exactly?
I can understand easily the concept of an algebraic variety in the context of polynomial rings, since its just a... |
[![enter image description here][2]][2]
I am wondering how to prove the second part of this stochastic calculus question. In the question, $\mathcal{P}$ is the set of pre-visible processes. $\mathcal{M}_c^2$ is the set of square integrable martingales.
___
**My attempt:** I used the following formula :
$$... |
How can I re-write $2^{\sqrt{\log n}}$ as $n^?$
I tried $2^{\log n^{0.5}}$ then $2^{0.5\log n}$, then $n^{0.5 \cdot 1} = \sqrt{n}$.
But it seems to be smaller than $\sqrt{n}$ in the answers sheet. |
How can I re-write $2^{\sqrt{\log n}}$ as $n^?$ |
This question is asking for an explanation of a step in the following segment of someone else's proof of a textbook exercise regarding set membership, conjunction and implication.
---
Consider the following:
$$ (x \in A \land y \in B) \implies (x \in C\land y \in D) $$
Let me check I understand the meaning.... |
So I was thinking say you have a linear differential operator such as the exponential differential one which is renown in some fields in physics:
$$e^{\mathrm D_x}\equiv\sum_{n=0}^\infty\frac{\mathrm D_x^n}{n!},\text{ where } \mathrm D_x^n\equiv\frac{\mathrm d^n}{\mathrm dx^n},$$
and you applied it to a product of fu... |
What's the product rule for the exponential differential operator? |
Hint 1: Fundamental theorem of calculus
Hint 2: Let $F(x)=\displaystyle\int^x_{-x}\dfrac{t^2e^t}{e^t+1}~dt$. What is $F(0)$?.
**Edit.** Let's me provide the exact solution below in case you need.
>! Hint 1: We apply fundamental theorem of calculus on $F$. That is, $$F'(x)=\dfrac{x^2e^x}{e^x+1}\dfrac{dx}{dx}-\d... |
I have a circle like so
![circle with a given radius r, with angle \theta to the y-axis][1]
Given a rotation **θ** and a radius **r**, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees.
For example, I have a radius **r** of 12 and a rotation **θ** of... |
I'm a beginner to number theory, and in a text book, right after proving the fundamental theorem of arithmetic, the following problem is stated:
Let $H_m$ be the subset of real numbers, which can be written in the form of $x + y\sqrt{m}$, where $x$ and $y$ are integers and $m$ isn't a square number.
Show that besid... |
Zariski topologies are defined via closed-set definition; with the closed sets being algebraic varieties $V(I)$ of ideals $I\subseteq R$ of commutative rings $R$... But what are these varieties, exactly?
I can understand easily the concept of an algebraic variety in the context of polynomial rings, since its just a... |
Suppose that $f(x)=\sum_{k=1}^n a_k e^{ik b_k}$ is an exponential sum with frequencies $b_k \in \mathbb R$ and coefficients $a_k \in \mathbb C$. Further, let $S(\alpha, m)$ be the set
$$
S(\alpha, m) = \{ \alpha l : l = -m, \dots, m-1, m \} \subset \mathbb R, \quad \alpha >0, \quad m \in \mathbb N.
$$
Assuming that... |
I'm a beginner to number theory, and in a text book, right after proving the fundamental theorem of arithmetic, the following problem is stated:
Let $H_m$ be the subset of real numbers, which can be written in the form of $x + y\sqrt{m}$, where $x$ and $y$ are integers and $m$ isn't a square number.
Show that besid... |
So I was thinking say you have a linear differential operator such as the exponential differential one which is renown in some fields in physics:
$$e^{\mathrm D_x}\equiv\sum_{n=0}^\infty\frac{\mathrm D_x^n}{n!},\text{ where } \mathrm D_x^n\equiv\frac{\mathrm d^n}{\mathrm dx^n},$$
and you applied it to a product of fu... |
**Proposition 6.7** Let $X, Y \in \mathcal{P}, M \in \mathcal{M}_{\mathrm{c}}^2$ and $S \leq T$ be stopping times. Then
$$
\begin{aligned}
& \mathbb{E}\left[\left((X \cdot M)_{T \wedge t}-(X \cdot M)_{S \wedge t}\right)\left((Y \cdot M)_{T \wedge t}-(Y \cdot M)_{S \wedge t}\right) \mid \mathcal{F}_S\right]=
\math... |
I'm a retired computer engineer and I'm studying a bit of abstract algebra these days.
I have done some research on the permutation groups of puzzles I've had a copy of the full partition of the 2x2x2 Rubik's cube for a while, and I think I can say it has an interesting structure, this partition.
It's embedded in... |
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$.
Let $S = \{i\leq n: X_i\in B_n\}$.
I want to show that
$$
\frac{1}{|S|}\sum_{i\in S... |
This question is asking for an explanation of a step in the following segment of someone else's proof of a textbook exercise regarding set membership, conjunction and implication.
---
Consider the following:
$$ (x \in A \land y \in B) \implies (x \in C\land y \in D) $$
Let me check I understand the meaning.... |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
For any two finite subsets $A,B$, of an abelian group, is $|A+B|^2 |A-B|^2 \geq |A+A||A-A||B+B||B-B|$? |
How Should I Solve this Problem on 1-Form with Compact Support? |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
**Note to the moderators: This question has been solved, and is indeed a valid question. Please post a comment explaning what needs to be elaborated on. Thank you very much! - random0620**
**Proposition 6.7** Let $X, Y \in \mathcal{P}, M \in \mathcal{M}_{\mathrm{c}}^2$ and $S \leq T$ be stopping times. Then
$$
\be... |
What physics phenomenom does the Laplace equation describes?
From what I know of, its the static problems but what it actually means?
|
Quesiton:
Let's say that F(x,y) is a function with d^2/dx^2F + d^2/dy^2F >= 0 everywhere (such a function is called subharmonic). What effect does the gradient flow of F(x,y)
have on area? |
We want to find
$$I=\int_{-\pi}^\pi1+\lim_{n\to\infty}\min(\cos(x),\dots,\cos(nx))dx$$
Here is a picture of $1+\min(\cos(x),\dots,\cos(nx))$ for $n=50$:
<img src="https://i.stack.imgur.com/2aJQu.jpg" width="800">
$\displaystyle I_n= \int_{-\pi}^\pi1+\min(\cos(x),\dots,\cos(nx))dx$ has the following forms f... |
As I mentioned in the question statement, we now have the following quadratic equation in $m = \dfrac{y}{x} $
$ m^2 ( b^2 \sin^2 s - \dfrac{1}{b^2} ) + m (a b \sin(2 s) ) + (a^2 \cos^2 s - \dfrac{1}{a^2} ) = 0 $
The two solutions $m_1 , m_2$ satisfy Vieta's formulas
$ m_1 m_2 = (a^2 C^2 - \dfrac{1}{a^2}) / (b^... |
Let $X$ be a normed space, $A$ a real vectorspace. Let a map $\phi:X \setminus \{0\} \to A$ such that it satisfies the following properties
- $\phi(\lambda x) = \phi(x)$ for every $\lambda >0$
- if $\|x\|=\|y\|$, then $\phi(x+y) = \frac{1}{2} \left( \phi(x)+\phi(y) \right)$
be called a direction.
In $\mathbb{... |
im really stuck with this , is there any way?
[enter image description here][1]
[1]: https://i.stack.imgur.com/sohQz.png |
Let $X$ be a normed space, $A$ a real vectorspace. Let a map $\phi:X \setminus \{0\} \to A$ such that it satisfies the following properties
- $\phi(\lambda x) = \phi(x)$ for every $\lambda >0$
- if $\|x\|=\|y\|$, then $\phi(x+y) = \frac{1}{2} \left( \phi(x)+\phi(y) \right)$
be called a direction.
In $\mathbb{... |
For anyone who happens to have Pazy's book on hand,on page 104, it says
"Therefore T(t) can be extended to all of X by continuity.After this extension T(t) becomes a C0 semigroup on X"
I have no idea what this sentence means. Could you please elaborate on how you come to this conclusion? |
[![enter image description here][1]][1]
I tried to garph the function $f(x)=\int_0^\pi \ln |x+\cos(t)| dt$ and $f'(x)$ in desmos as above. Even $f(x)$ behaves like a constant function in $(-1,1)$ , but it seems not differentible in $(-1,1)$. Why? Thank you for your help.
[1]: https://i.stack.imgur.com/cT7Ff.jpg |
Question.
Let's say that $F(x,y)$ is a function with $$\frac{d^2}{dx^2}F + \frac{d^2}{dy^2} F \ge 0$$ everywhere (such a function is called subharmonic). What effect does the gradient flow of $F(x,y)$
have on area? |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
NOTE: I am not an actual math pro and neither am studying some kind of advanced math. I just thought that in binary, the prime numbers might have something that could make It possible to find logic in It. Could be all in my head or just because binary is a simple logic system and I could be confusing things, but I woul... |
Could this be useful to find a prime numbers formula or something? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
Suppose we have,
$$y_t = a + {\alpha}y_{t-1}+u_t$$ for $t>k$, where $k$ is a positive integer and $\alpha \in (0,1)$. And, $$y_t = b + {\alpha}y_{t-1}+u_t$$ for $t\leq k$. And assume that $a$ and $b$ are two different real constants. $u_t$ are iid with mean $0$ and a variance, $\sigma$ constant and finite.
Now, I ... |
Finding the Supremum of $E(y_t)$? |
Suppose we have,
$$y_t = a + {\alpha}y_{t-1}+u_t$$ for $t>k$, where $k$ is a positive integer and $\alpha \in (0,1)$. And, $$y_t = b + {\alpha}y_{t-1}+u_t$$ for $t\leq k$. And assume that $a$ and $b$ are two different real constants. $u_t$ are iid with mean $0$ and a variance, $\sigma$ constant and finite. $t$ is disc... |
Suppose we have,
$$y_t = a + {\alpha}y_{t-1}+u_t$$ for $t>k$, where $k$ is a positive integer and $\alpha \in (0,1)$. And, $$y_t = b + {\alpha}y_{t-1}+u_t$$ for $t\leq k$. And assume that $a$ and $b$ are two different real constants. $u_t$ are iid with mean $0$ and a variance, $\sigma$ constant and finite. $t$ \in \ma... |
I am studying some papers in which the notation ${\cal O}_{\mathbb{CP}^1}(-1)$, ${\cal O}_{\mathbb{CP}^1}(-2)$ and ${\cal O}_{\mathbb{CP}^1}(1)$ appear. I am not familiar with that notation and while I have already found out ${\cal O}_{\mathbb{CP}^1}(-1)$ is the tautological line bundle, I haven't found what the others... |
What is the meaning of the notation ${\cal O}_{\mathbb{CP}^n}(k)$? |
Is there a way to construct larger cardinals without choice axiom? |
> Show that the matrix $$\begin{pmatrix}2 & 1& 1\\0 &3 & 1\\0 & 2&-1\end{pmatrix}$$
cannot be the adjoint of any invertible matrix with real entries.
im really stuck with this , is there any way? |
I am studying some papers in which the notation ${\cal O}_{\mathbb{CP}^1}(-1)$, ${\cal O}_{\mathbb{CP}^1}(-2)$ and ${\cal O}_{\mathbb{CP}^1}(1)$ appear. I am not familiar with that notation and while I have already found out ${\cal O}_{\mathbb{CP}^1}(-1)$ is the tautological line bundle, I haven't found what the others... |
According to power rule, ${d\over dx}(x) = 1x^0$. So, on the line $y=x$, we have ${dy\over dx}=x^0$. The gradient at any point on this line is 1. However, substituting 0 for x into the derivative of the function gives $0^0$, which is undefined. Does this mean the gradient at the point where the line passes through the ... |
Can 0^0 be defined when it expresses a gradient? |
My problem is as follows: Suppose you draw a card from a 52-card deck and see that the card is a spade. You then draw another card without replacement. What is the probability that the next card is an ace?
I tried solving this question using conditional probability, but the issue is that the Spade that I initially d... |
Suppose $AB=AC$, where $B$ and $C$ are matrices, and $A$ is invertible. Show that $B=C$. Is this true, in general, when $A$ is not invertible?
What can be deduced from the assumptions that will help to show $B=C$?
[1]: https://i.stack.imgur.com/bgGHn.png |
Suppose we have,
$$y_t = a + {\alpha}y_{t-1}+u_t$$ for $t>k$, where $k$ is a positive integer and $\alpha \in (0,1)$. And, $$y_t = b + {\alpha}y_{t-1}+u_t$$ for $t\leq k$. And assume that $a$ and $b$ are two different real constants. $u_t$ are iid with mean $0$ and a variance, $\sigma$ constant and finite. $t \in \mat... |
My problem is as follows: Suppose you draw a card from a 52-card deck and see that the card is a spade. You then draw another card without replacement. What is the probability that the next card is an ace?
I tried solving this question using conditional probability, but the issue is that the Spade that I initially d... |
There is [numerical evidence][1] that
$$\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
>How can this be proved?
Here is the graph of $y=\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)$... |
For an [orthographic map projection][1] (of radius $r$) centered at latitude $\varphi_0$, I believe the ellipse defining the line of latitude $\varphi$ is centered at $y$ coordinate $r\cos(\varphi_0)\sin(\varphi)$, with major axis radius $r\cos(\varphi)$ and minor axis radius $r\sin(\varphi_0)\cos(\varphi)$. What I'm h... |
Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given point. But which direction does it point?
Let's say we were to use the field $\vec{F}(x,y,z)=\langle-y,x,0\rangle$, which... |
Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given point. But which direction does it point?
Let's say we were to use the field $\vec{F}(x,y,z)=\langle-y,x,0\rangle$, which... |
Are your "imbricated torii" more or less in connection with the following kind of "flattened representation" of the Rubik's cube ?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/GvCwG.jpg |
Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given point. But which direction does it point?
Let's say we were to use the field $\vec{F}(x,y,z)=\langle-y,x,0\rangle$, which... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Theorem 7.1. Let $A\subset\mathbb{R}^m$; let $B\subset\mathbb{R}^n$. Let $$f:A\to\mathbb{R}^n\,\,\,\,\,\,\text{and}\,\,\,\,\,\,g:B\to\mathbb{R}^p,$$ with $f(A)\subset B$. Suppose $f(a)=b$. If $f$ is differentiable at $a$, and if $g$ is differentiable at $b... |
To prove $f_2\circ f_1$ is differentiable on $U$, can we weaken this assumption? |
I am reading the [paper][1] "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the paper, the authors of the paper state: "The set of points of continuity $C(f)$ is not necessarily residual in $Z$. It is however ... |
How is the set $C(f)\cap V$ of second category in $V$? |
I'm reading "Set Theory and the Continuum Problem" by Smullyan & Fitting, and on page 16 it says:
<blockquote>
Thus, we allow $\forall x$ for $x$ a set variable, but not $\forall A$ for $A$ a class variable.
</blockquote>
But right below that it states:
<blockquote>
$P_2$ Separation
$$(\forall A_1)\ldot... |
Is this 1st or 2nd order logic? |
Here is the binary operation $ *: R\times R \backslash (0,0) $ defined by $ (a,b)(c,d)=(ac-bd,ad+bc) $. My idea is that to show this is a group ($R\times R \backslash (0,0), * $), I need to show that $ * $ is well-defined and associative and then show it has an identity and inverse. I am struggling to do the first part... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/iwaf5.jpg
The slice maps are defined in the above sceenshot.
If we take normal state $\psi$ on $N$, then the slice map $L_\psi$ defined by
$L_\psi(\sum x_i\otimes y_i)=\sum \psi(y_i)x_i$
extends to a normal conditional expectatio... |
How limited is "very limited"? And why do you write the post in a command form ("Prove that...")? It is not how you normally go around asking questions in person, surely. That style of writing here is an obvious indication that this is homework.
Hint: Treat $p=2$ first, then let $p > 2$, rewrite the congruence as $... |
Question.
Let's say that $F(x,y)$ is a function with $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} \ge 0$$ everywhere (such a function is called subharmonic). What effect does the gradient flow of $F(x,y)$
have on area? |
I have confusion when I went through the canonical construction of 1-dimensional Brownian Motion. Here we take $\Omega:=C(\mathbb{R}_+,\mathbb{R})$, and we equip $\Omega$ with the smallest sigma algebra $\mathcal{C}$ such that all coordinate mappings are measurable, and $\mathbb{P}$ to be the Wiener measure.
Questio... |
> How do I show $∗$ is well-defined (and is the first part required)?
This is indeed required, but it is almost always true because it is very obvious very fast if an operation is well-defined.
In general, for a function $f : A \to B$ to be well-defined, these must hold:
1. To each value $a \in A$, the value $... |
For some reason, you have stopped reading Cunningham's "Theoretical Properties of the Network Simplex Method" *just* as it was about to answer your question. We read on:
> **Lemma 1.** Let $T, T'$ be strongly feasible trees, such that $T'$ is a successor of $T$ with entering edge $e$ and leaving edge $f$, and suppos... |
Let $A$ be a set.
The Cartesian product of $A$ by itself is $A^2=A\times A$.
Also the Cartesian product of $A$ by itself for $n$ times is $A^n=A\times \cdots \times A$.
Is it correct to call $A^2$ is the square of the set $A$ and to call $A^n$ as the set $A$ raised to the $n$th power? |
Is it correct to call $A^2$ as "the square of the set $A$" and to call $A^n$ as "the set $A$ raised to the $n$th power"? |
if A, B, A-B, and C are positive-definite matrices, is (ACA-BCB) positive-definite matrix? |
if A, B, A-B, and C are positive-definite matrices, is (ACA-BCB) positive definite? |
Suppose we have data $x\sim \mathcal{D}$ and $\mathcal D$ is unknown. $g$ and $g'$ are two bounded functions. Note $G=\mathbb E (g(x))$ and $G'=\mathbb E (g'(x))$. We also have the Monte-Carlo empirical estimation with $m$ samples: $\hat G = \Sigma_i \frac 1 m g(x_i), \hat {G'} = \Sigma_i \frac 1 m g'(x_i)$.
Is the... |
We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's center. I've learnt that in the parametric form, we take
$$x= h + a\sec t$$ and $$ y = k + b\tan t $$
These values satisfy the given equation. But so does $$x= h + a\csc t$$ and $$y=k+b\cot t$$
Th... |
Suppose we have data $x\sim \mathcal{D}$ and $\mathcal D$ is unknown. $g$ and $g'$ are two bounded functions. Note $G=\mathbb E (g(x))$ and $G'=\mathbb E (g'(x))$. We also have the Monte-Carlo empirical estimation with $m$ samples: $\hat G = \Sigma_i \frac 1 m g(x_i), \hat {G'} = \Sigma_i \frac 1 m g'(x_i)$.
Is the... |
I've recently noticed that the definition of the integral in my measure theory course seems to be quite different from the usual definition, in particular the one another one of my courses uses. What really confuses me though, is that it seems to have much stronger properties than it should have, going by other texts. ... |
There are a lot of easy examples a differentiable $f$ st $f(x)\ne 0$ and for $a \in \mathbb{Q}, \ f(a) \not \in \mathbb{Q}$ for example $\pi x , e^x, etc$ but the question is: Is the converse true ? i.e Is there a differentiable $f$ on $\mathbb{R}$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$?
Of course ... |
Is there a differentiable $f$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$? |
There are a lot of easy examples a differentiable $f$ st $f(x)$ and for $a \in \mathbb{Q}, \ f(a) \not \in \mathbb{Q}$ for example $\pi x , e^x, etc$ but the question is: Is the converse true ? i.e Is there a differentiable $f$ st $f'\ne 0$ on $\mathbb{R}$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$?
Of... |
Is there a differentiable $f$ st $f' \ne 0$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$? |
There are many easy examples for differentiable $f$ st $f'(x) \ne 0$ and for $a \in \mathbb{Q}, \ f(a) \not \in \mathbb{Q}$ for example $\pi x , e^x,$ etc, but the question is: Is the converse true ? i.e Is there a differentiable $f$ st $f'\ne 0$ on $\mathbb{R}$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$?... |
> Is the number of intervals in the solution set of $\text{sin} \frac{1}{x}>0, 0<x<1$
> countably finite or uncountably infinite?
Use a change of variable. Let $y=\frac{1}{x}$:
$$\text{sin } y>0\implies y \in {(1,\pi) \cup (2\pi,3\pi) \cup ...}$$
Changing back to the original variable:
$$x \in {(1,\frac{1}{\pi... |
I'm posting a question because I was curious about something while studying linear algebra.
As we all know, x = 1 is a point in a one-dimensional coordinate system. I understand this part.
But why does x = 1 being represented as a line in two dimensions?
Is it simply defined that way?
[ is defined as follows:
> For any real number $x$, define $f(x)$ as the distance between $x$ and the integer nearest to $x$,
> or $f(x) = \min(x - \lfloor x \rfloor, \lceil x \rceil - x)$.
>The statement of the conjecture is that for ... |
Is there any material on this simpler version of Littlewood's Conjecture? |
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