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I have seen two motivations of decomposing an $L^2(\mathbb{T}^n)$ function into a Fourier series with basis functions $e^{2\pi inx}$ with $n$ being integers. The first one is the intuitive frequency representation of a signal; the second one is that the trigonometrical functions are eigenfunctions of the Laplacian, so ... |
*Hexagons* are made of 6 embedded rhombuses. The Rhombic dodecahedron
Has twice the number of rhombuses as the hexagon has yet those rhombuses are its exterior boundary. The exterior boundary of a hexagon is a flat line.
So, might there be an expansion sequence for these hidden-rhombus-like geometries and their re... |
Am I onto something regarding hidden rhombus-like structures hiding within N-dimensional honeycombs? |
How do I prove this using inequalities? I know that since $4x^2 +9$ is $\Omega (x^2)$ it follows that $4x^2 +9$ is $\Omega (x log x)$ since Big Omega is a lower bound. But how do I prove it using inequalities and choose a $n_0$ and c? I don't really know how to get from $4x^2 +9$ to $\Omega (x^2)$, or $\Omega (x log x)... |
Bruce and Walls in their paper *On the classification of cubic surfaces* state at the end the "final observation" that a number of distinct lines on cubic surface with $k$ isolated du Val singularities is given by formula
$$
\binom{8-c}{2} + k - 1
$$
where $c$ is the codimension of space parameterising surfaces of... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_... |
A solution obtained when O,A,B are not collinear, even if derived on that basis, may be due to some inherent precession ? in the curve.
A circle radius $a$ is drawn through A,B, centred on their perpendicular bisector. Let
$$OB=r, OA= r- 2 a \sin \psi, \psi= \psi(s), r'(s)= \cos \psi $$
Constant Harmonic me... |
Why $\min (s,t) =\int_0^1(t-u)_{+}^0(s-u)_{+}^0 du$?
Background of the question is:
An elementary illustration of the integral representation theorem arises from the kernel
$$
\begin{aligned}
K(s, t) & =\min (s, t) \\
& =\int_0^1(t-u)_{+}^0(s-u)_{+}^0 d u
\end{aligned}
$$
defined on $[0,1] \times[0,1]... |
Say you have a list of $n$ random integers in the range $[1, m]$. The binomial PMF can find the probability of exactly $k$ occurrences of a number with $p = \frac{1}{m}$
$$ {n \choose k} p^k (1-p)^{n-k} $$
Now say the list is constructed in the following way: the $i$th element in the list is a random number in $[1,... |
A solution obtained when O,A,B are not collinear, even if derived on that basis, may be due to some inherent precession ? in the curve.
A circle radius $a$ is drawn through A,B, centred on their perpendicular bisector. Let
$$OB=r, OA= r- 2 a \sin \psi, \psi= \psi(s), r'(s)= \cos \psi $$
Constant Harmonic me... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_... |
Say you have a list of $n$ random integers in the range $[1, m]$. The binomial PMF can find the probability of exactly $k$ occurrences of a number with $p = \frac{1}{m}$
$$ {n \choose k} p^k (1-p)^{n-k} $$
Now say the list is constructed in the following way: the $i$th element in the list is a random number in $[1,... |
Let $B$ be a Brownian motion. How can $\mathbb{E}\left[B_t^n\right]$ and $\mathbb{E}\left\lvert[B_t\right\rvert^n]$ with $n\in\mathbb{N}$ and $t\geq 0$ be computed? |
I am attempting to derive the quadratic formula by completing the square on the generic generic expression:
>$$ax^2+bc+c=0$$
I'm struggling with the right hand side of the equation which, for the step I'm on I know should be $\frac{b^2-4ac}{4a^2}$. However, I arrive at $\frac{b^2a-4a^2c}{4a^3}$
## Here's my wo... |
When $z_{1,2} \in \mathbb{R}$ then we get the modified Bessel function $I_0(\sqrt{{z_1}^2+{z_2}^2})$. What would be the solution for complex $z_i$? |
What is the integral: $\int_{-\pi}^{\pi} \frac{d\theta}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}$? |
**HINT**: Suppose you were given the corresponding *series* and asked to determine whether it converges. What test do you immediately apply? |
What is the integral: $\int_{-\pi}^{\pi} \frac{1}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}\, d\theta$? |
The question is as follows
Find the domain of the function
$$f(x) = \frac{1}{1 + 2 \sin(x)}$$
since $\sin(x) \neq \frac{-1}{2}$ therefore the general expression for this should be as follows
$$x \neq n\pi + (-1)^{n-1}\frac{\pi}{6}$$
This equation i derived my self and seems about right however the answer g... |
What is the integral $\int_{-\pi}^{\pi} \frac{1}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}\, d\theta$?
When $z_{1,2} \in \mathbb{R}$ then we get the modified Bessel function $I_0(\sqrt{z_1^2+z_2^2})$. What would be the solution for complex $z_i$? |
What is the integral $\int_{-\pi}^{\pi} \frac{1}{2\pi}\exp{(z_1 \cos\theta + z_2 \sin\theta)}\, d\theta$? |
## Problem
> Let $M\in\mathbb{M}^{loc}_C$, i.e. $M$ is an (a.s.) continuous local martingale with $M_0=0$. Show that $M$ is constant on an interval $[a,b]$ with $0\leq a < b$ if $\langle M\rangle$ is constant on $[a,b]$.
## My Attempt
Let $\left\langle M\right\rangle$ be constant on $[a,b]$, i.e. $\forall t\in... |
Below I am trying to make the asked points clear, computations do not stay in focus. First of all a picture:
[![mse 43881764][1]][1]
The problem starts somehow with the middle, making the least number of words in a random language count for exposing the problem. This is not the way we think, and not the way we st... |
How to find the approximate value of $f(x) = \sum_{n = 1}^{\infty} \frac{n}{n^3 + x}$ as a function of $x$ when $x$ goes to $\infty$? I want a closed form function $g(x)$ s.t. $f(x) \sim g(x) $ for $x \to \infty$.
I have tried to integrate $f(x)$:
$$ \int f(x) dx= \sum_{n = 1}^{\infty} n\int\frac{1}{n^3 + x} dx = \... |
How to approximate \sum n / (n^3 + x) for x approaching \infty? |
Are analytic functions with highest convergence radius at a point analytic everywhere? |
Determine the the points where the function $\sin^n x \sin nx$ is maximum or minimum.
Differentiating,we get that the critical points turn out to be $\frac{k\pi}{n+1}$. But from this how can the points at minima or maxima be differentiated when we derived only one form as critical points? Does that mean the function... |
The permanent of a square matrix is known to have the following properties -
1) The permanent of the identity matrix is one.
2) For any permutation matrices $P$, $Q$, and matrix $A$, the permanent of $A$ equals the permanent of $PAQ$.
Is the permanent the only function of a square matrix that has these two pro... |
In a ΜAΘ contest from 1991, I found this problem in my problem book. I know how to solve problems like this, and I know how to solve it if the problem tells me to find the digits in $2^{44}$, but $5^{44}$ makes me think about the problem like this:
$${5^{44} = \frac{10^{44}}{2^{44}}}$$
Given this, I don't think ... |
I want to show that a statement with several quantifiers, e.g., "$f(a, b) <= 0$ for all $a\in [0, 3]$ and all $b\in [-\infty, -1]$", is equivalent to another statement with several quantifiers, e.g., "$g(a, b) <= 0$ for all $a \in[-10, 15]$ and $b < 0$", where $f$ and $g$ are two functions from $\mathbb{R}^2 \rightarr... |
While exploring possible applications for this *[new trick][1]*, I stumbled upon an entire family of integrals that "always" yield $a\pi$, where $a$ is a real number. The following integral captivated me greatly:
>$$\boxed{\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}} (x^2 - x) \, dx = \color{red}{\phi}... |
What is meant by "decimal fraction" ? Why have rationals been referred to as periodic and irrationals as non periodic? |
Does anyone know how to fill the details of [Lemma 2.3.4](https://www.google.com/books/edition/Field_Arithmetic/FNkMaZrvpvsC?hl=en&gbpv=1&pg=PA27) from the book "Field Arithmetic" by M. D. Fried and M. Jarden. The statement of the lemma goes as follows:
Let $v$ be a discrete valuation of a field $E$, $h\in O_v[X]$ a... |
I tried to learn commutative algebra and I found a problem to do the following:
Consider an integral domain $R$ and an algebraically closed field $M$. Let $\varphi_0$ be a homomorphism from $R$ to $M$, and let $F$ be a field that contains $R$. In this context, we can explore two possibilities for extending $\varphi_... |
<ul>1. First, what is $z^w$ when $z$ and $w$ are complex numbers. It is not trivial at all to define a power in complex field. It is multivalued function, i.e. you will not receive a unique value $z^w$ for given $z,w$. In fact, even $i^i$ with $i$ is the imaginary unit is not really well-define, it can be $e^{-\frac{\p... |
Problem: Define the distance function between $(x_1, y_1)$ and $(x_2, y_2)$ for two
points (where $x_i
, y_i$ are real numbers) $d((x_1, y_1), (x_2, y_2))$ in the plane to be
$$|y_1 - y_2| \text{ if } x_1=x_2; \quad 1 + |y_1- y_2| \text{ if } x_1 \ne x_2.$$
$d$ is indeed a metric, and $(X, d)$ is a Locally compac... |
The subring of Q[x] which contains all rational polynomials with constant term integer sits in between Z[x] and Q[x] both of which is Noetherian, but this subring is not so although it has properties very much similar to UFD. Does this has any particular interest and any generalization? |
While exploring possible applications for this *[new trick][1]*, I stumbled upon an entire family of integrals that "always" yield $a\pi^n$, where $a$ is an algebraic number and $n$ is a natural number. The following integral captivated me greatly:
>$$\boxed{\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}}... |
In a ΜAΘ contest from 1991, I found this problem in my problem book. I know how to solve problems like this, and I know how to solve it if the problem tells me to find the digits in $2^{44}$, but $5^{44}$ makes me think about the problem like this:
$${5^{44} = \frac{10^{44}}{2^{44}}}$$
Given this, I don't think ... |
I have a question about a small part of the proof of the open mapping theorem. I've boxed it in red and posted the proof that I am looking at below. This is from Axler's Measure, Integration & Real Analysis. How is he applying $\frac{r}{\Vert h\Vert} h$ to get 6.82? This is the only part of this proof I'm not understan... |
As per some sources fractions have been defined as a quotient of two numbers whereas some sources restrict the numerator to be a whole number and denominator to be a positive integer.
What is the exact definition? |
What is the exact mathematical definition of a fraction? Can all real numbers be expressed as fractions? |
A solution obtained when O,A,B are not always collinear, even if derived on that basis. Could be due to some inherent precession? in the spiral.
A circle radius $a$ is drawn through A,B, centred on their perpendicular bisector. Let
$$OB=r, OA= r- 2 a \sin \psi, \psi= \psi(s), r'(s)= \cos \psi $$
Constant Ha... |
Let me begin by a remark concerning the geometric tangent space $\mathbb R^n_a$ of $\mathbb R^n$ at $a$. Lee defines it as
$$\mathbb R^n_a = \{ a \} \times \mathbb R^n $$
with the vector space structure canonically induced from $\mathbb R^n$, i.e. the unique vector space structure making the canonical bijection $\phi... |
This morning I found that [Carnot's theorem][1] for the conic transversal of a triangle can be generalized to any polygon, and the generalization can be proven by applying mathematical induction to Carnot's theorem itself. For example, when applying the generalization to a quadrilateral, the equation becomes as shown i... |
Is this generalization of Carnot's theorem known? |
I have the following problem:
Suppose an unbiased coin is tossed 10 times. Let D be the random
variable that denotes the number of heads minus the number of tails.
What is the variance of D?
My solution is: Let X be the number of heads and Y be the number of tails, so $X~Bin(10,\frac{1}{2})$ and $Y~Bin(10,\frac{1... |
A friend of mine showed me an SAT question today in which one of the choices is:
> No genetic variations that were common to those finches that used technique Q were not common to the finches that did not use technique Q.
I just learned a bit about logic so I tried breaking down the statement into its logical fo... |
How to translate this statement into $A \land B$? |
The integral ∫sin(x²)dx doesn't have an elementary solution.But i Don't understand why?
Can anyone please explain it. |
It is a common knowledge that a function is $c$-strongly convex if $f - c\|x\|^2_2$ is convex.
However, can we replace $\|x\|_2$ with any norm $\|x\|_p$? I strongly suspect this holds, but from looking at several references online (see p 26 of [this][1]), it might also be true that this condition ONLY holds for $p ... |
$$\lim_{\lim_{x\to0}}\dfrac{(1+\sin^2x)^{1/3}-(1-2\tan x)^{1/4}}{\sin x+\tan^2x}=\lim_{x\to0}\dfrac{(1+\sin^2x)^{1/3}-1}{\sin x+\tan^2x}-\dfrac{(1-2\tan x)^{1/4}-1}{\sin x+\tan^2x}$$
Now rationalize the numerator:
$$\lim_{x\to0}\dfrac{(1+\sin^2x)^{1/3}-1}{\sin x+\tan^2x}=\lim_{x\to0}\dfrac{(1+\sin^2x)-1}{(\sin x+... |
The integral $\int\sin(x^2)\mathop{\mathrm{d}x}$ doesn't have an elementary solution. But I don't understand why?
Can anyone please explain it. |
**Does there exist two binary operations + and ∗ on {0,1} such that + and ∗ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality?**<p>
(Partial) answers:<p>
<ol>
<li> No, ... |
I am looking for all functions f(x) (Z→R)
that satisfy
f(x) +f(y) +f (xy+1) =f(x+y) +f(1) +f(xy)?
Thanks! |
How do I find f(x) (Z→R) that satisfy f(x) +f(y) +f (xy+1) =f(x+y) +f(1) +f(xy)? |
Given the primal resource allocation:
$$
\min_{x} \text{ costs} = \sum_{i} c_i x_i
$$
$$
\sum_{i} x_i = D \\ (p)
$$
$$
x_i \leq \text{CAP}_i \quad \forall i \text{ (}\lambda_i\text{)} \\
$$
$$
x_i \geq 0 \quad \forall i \\
$$
Now the solution of the dual is also given:
$$
\max_{(p, \lambda_... |
**Does there exist two binary operations + and ∗ on {0,1} such that + and ∗ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality?**<p>
(Partial) answers:<p>
<ol>
<li> No, ... |
The following text is taken from the a book about linear programming that I'm reading:
> A graphical illustration is useful for understanding the notions and procedures of linear programming, but as a computational method it is worthless. **Sometimes it may even be misleading, since objects in high dimension may beh... |
How do I prove this using inequalities? I know that since $4x^2 +9= \Omega (x^2)$, it follows that $4x^2 +9$ is $\Omega (x \log x)$ since Big Omega is a lower bound. But how do I prove it using inequalities and choose a $n_0$ and c? I don't really know how to get from $4x^2 +9$ to $\Omega (x^2)$, or $\Omega (x \log x)$ |
To prove the derivative of $e^x$ my math book did this
$$\begin{align}{d(e^x)\over dx}&=\lim_{h\to0} \frac{e^{x+h}-e^x}{h} \tag{1}
\\ &= \lim_{h\to0}e^x \cdot\frac{e^h-1}{h} \tag{2}
\\ &= e^x \cdot \lim_{h\to0} {\left(1+\frac{h}{1!}+\frac{h^2}{2!}...\right)-1\over h} \tag{3}
\\ &=e^x\end{align}$$
----------
B... |
How to approximate $\sum_{n = 1}^{\infty} \frac{n}{n^3 + x}$ for $x$ approaching $\infty$? |
**Does there exist two binary operations + and ∗ on {0,1} such that + and ∗ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality?**<p>
(Partial) answers:<p>
<ol>
<li> No, ... |
Define a sparse matrix set where a matrix and its inverse have the same sparse structure. Is such a set forming a certain group?
For example, $\begin{bmatrix}a & b\\0 & c \end{bmatrix}^{-1}=\begin{bmatrix}1/a & -b/(ac)\\0 & 1/c \end{bmatrix}$ has the same sparse structure with $\begin{bmatrix}a & b\\0 & c \end{bmatr... |
Recently found an exercise for high school students as follows:
Let $A,B$ be $2\times 2$ matrices with real entries and suppose $A^2=B^2$, $\operatorname{tr}(A)\neq0$,$\operatorname{tr}(B)\neq0$. Show that $AB=BA$.
Since it only ask the case for $2\times 2$ matrix and hence we can simply bash out all the equation... |
I'm studying general topology and a question has come to my mind.
We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.
I can see the ambiguity of this d... |
Given the primal resource allocation:
$$
\min_{x} \text{ costs} = \sum_{i} c_i x_i
$$
$$
\sum_{i} x_i = D \\ (p)
$$
$$
x_i \leq \text{CAP}_i \quad \forall i \text{ (}\lambda_i\text{)} \\
$$
$$
x_i \geq 0 \quad \forall i \\
$$
Now the solution of the dual is also given:
$$
\max_{(p, \lambda_... |
Here's an exercise from Thomas Calculus, I need to do an implicit differentiation:
$$
x^3=\frac{2x-y}{x+3y}
$$
If I [enter this](https://www.wolframalpha.com/input?i=derivative+of+%28x%5E3%29%3D%282x-y%29%2F%28x%2B3y%29) in Wolfram Alpha, I get:
$$
y'(x) =-\frac{3 x^4 + 18 x^3 y + 27 x^2 y^2 - 7 y}{7 x}
$$... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given $T(x)\in R $ ... |
If the equation $\frac{x^3+4}{{(x+1)}^3}=c,\forall x \in R-{\{-1\}}$ possess two roots, then what are the values of $c$ ?
> Attempt:
c=1 may be a solution if the quadratic equation thus formed has distinct real roots, i.e. $$ x^3+4={(x+1)}^3 $$ which gives $$ x^2+x-1=0$$ which has 2 differet solutions $(\neq 1)$.
H... |
(This may have been asked elsewhere but I cannot find it.)
Take the set of positive integers $\{1,2,...\frac{n}{2} (n+1)\}$ and put them into $n$ subsets as follows:
$S_{1}=\{1\}$
$S_{2}=\{2,3\}$
$S_{3}=\{4,5,6\}$
And so on so that $S_{n}$ contains exactly $n$ elements.
What is the sum of the elemen... |
Can $\frac{a^n + b^n}{a+b} = c^n$ have a solution? (CONT) |
It's a follow up of https://math.stackexchange.com/questions/4883613/does-three-iteration-are-sufficient-to-use-newtons-method
Come back on the previous question :
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0\tag{I... |
If the equation $\frac{x^3+4}{{(x+1)}^3}=c,\forall x \in R-{\{-1\}}$ possess two roots, then what are the values of $c$ ?
> Attempt:
c=1 may be a solution if the quadratic equation thus formed has distinct real roots, i.e. $$ x^3+4={(x+1)}^3 $$ which gives $$ x^2+x-1=0$$ which has 2 differet solutions $(\neq -1)$.
... |
Given the matrix A defined as A =
\begin{bmatrix}
2 & -4 & 4 & -2 \\
6 & a-2 & 7 & a-6 \\
-1 & 2-b & 8 & -b+1 \\
\end{bmatrix}
where (a) and (b) are real constants, I am interested in finding the rank and nullity of (A). Applying the normal Reduced Row Echelon Form (RREF) method, I obtained a matrix R equivalen... |
If the equation $\frac{x^3+4}{{(x+1)}^3}=c,\forall x \in R\backslash{\{-1\}}$ possess two roots, then what are the values of $c$ ?
> Attempt:
c=1 may be a solution if the quadratic equation thus formed has distinct real roots, i.e. $$ x^3+4={(x+1)}^3 $$ which gives $$ x^2+x-1=0$$ which has 2 different solutions $(... |
If the equation $\frac{x^3+4}{{(x+1)}^3}=c,\forall x \in \Bbb R\backslash{\{-1\}}$ possess two roots, then what are the values of $c$ ?
> Attempt:
$c=1$ may be a solution if the quadratic equation thus formed has distinct real roots, i.e. $$ x^3+4={(x+1)}^3 $$ which gives $$ x^2+x-1=0$$ which has $2$ different sol... |
**ORIGINAL QUESTION (UPDATED):**
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \righ... |
May this [paper][1] answer your question?
*NB: there are typos in the last Corollary.*
[1]: https://%20https://ijpam.uniud.it/online_issue/202044/71%20RamiAlahmad.pdf |
$\textbf{Proposition:}$ Let $U\subseteq \mathbb{R}^n$ be open and $f:U\to\mathbb{R}^k$ be a totally-differentiable function. Then the derivative map $$df:U\to \mathbb{R}^{k\times n}:x\mapsto df(x)$$ is continuous under the operator norm $||A|| = \sup_{|v|\leq1}\{||Av||\}$.
$\textbf{Proof 1 (incorrect)}$: Since $f$ i... |
The statement is a bit unclear. I think that you are saying that for a fixed rational $\alpha$, we have $f(\alpha)$ and $g(\alpha)$ are both rational. Then what possible values of $k$ are there (depending possibly on $\alpha$)?
That is, we know only that
\begin{align}
\alpha^2 - k \alpha &= a \in \mathbb{Q} \\... |
In order to prove the completeness theorem we obviously need a framework such as ZFC (I'm aware that ZFC isn't the only possibility) so that we can talk about a language $\mathcal{L}$ and also about models of $\mathcal{L}$.
Now the completeness theorem makes perfect sense to me in so far as the language which we st... |
How are these two conditions equivalent? |
You are missing what the Author is trying to convey.
(1) A graphical illustration is **useful** for understanding the notions and procedures of linear programming, but as a computational method it is **worthless**.
(2) Sometimes it **may** (not "will") even be misleading, since objects in high dimension **may*... |
There are two major categories of the "quaternions".
It is well-known that a (nonzero) versor represents a [three-dimensional rotation operator](https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Quaternions). A versor is a unit quaternion or a normalized quaternion: $$a+b{\bf i}+c{\bf j}+d{\bf k}\... |
When I first learnt calculus I was so surprised that there is a meaningful mathematical operator $D$ that $D(x^n)= n x^{n-1}$ this seem to be a very random thing to multiply the exponent with the function and take one from the exponent this was very strange for me when I first learnt about derivatives I mean if I was ... |
Is there a linear operator $T$ such that $T(x^n) =f(n)x^{n-1}$? |
I have $$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\ 0 & |x|>n .
\end{cases}$$
and I know that the result is
$$
\sup_n|f_n(x)|=\begin{cases} \frac{1}{n} & |x|\in (n-1,n] \\ 0 & x=0 \end{cases}
$$
but how can I arrive to this ? Can you please do... |
how can I find supremum in n? |
Let me begin by a remark concerning the geometric tangent space $\mathbb R^n_a$ of $\mathbb R^n$ at $a$. Lee defines it as
$$\mathbb R^n_a = \{ a \} \times \mathbb R^n $$
with the vector space structure canonically induced from $\mathbb R^n$, i.e. the unique vector space structure making the canonical bijection $\phi... |
When I first learnt calculus I was so surprised that there is a meaningful mathematical operator $D$ that $D(x^n)= n x^{n-1}$ this seem to be a very random thing to multiply the exponent with the function and take one from the exponent this was very strange for me when I first learnt about derivatives I mean if I was ... |
I have $$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\ 0 & |x|>n .
\end{cases}$$
and I know that the result is
$$
\sup_n|f_n(x)|=\begin{cases} \frac{1}{x} & |x|\in (n-1,n] \\ 0 & x=0 \end{cases}
$$
but how can I arrive to this ? Can you please do... |
I'm studying general topology and a question has come to my mind.
We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.
I can see the ambiguity of this d... |
I have $$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\ 0 & |x|>n .
\end{cases}$$
and I know that the result is
$$
\sup_n|f_n(x)|=\begin{cases} \frac{1}{x} & |x|\in (m-1,m],m=1,2,...\\ 0 & x=0 \end{cases}
$$
but how can I arrive to this ? Can you ... |
I have $$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\ 0 & |x|>n .
\end{cases}$$
and I know that the result is
$$
\sup_n|f_n(x)|=\begin{cases} \frac{1}{m} & |x|\in (m-1,m],m=1,2,...\\ 0 & x=0 \end{cases}
$$
but how can I arrive to this ? Can you ... |
>"But how do we know that any model of ZFC exists?"
We don't. The usual way out is to assume that such a base model exists, and refer to it as "the intended model/interpretation". There are many *questions* here and at MO about this; you could consult for example this MO question: [Looking for a source for intende... |
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
b) $\sum_{k=1}^{n} (-1)^{k-1} a_k \binom{n}{k} = \sum_{k=1}^{n} a_k$.
I tried ta... |
In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that sigma(x) = {z:|z|≤||x||}
But the proof is given only for one side inclusion..that is.,
The spectrum of an element x in a Complex Banach Algebra is a subset of the closed disk with centre 0 and radius ||x||.
Is the reverse i... |
Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous.
**Attempt:**
Note
$$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$
As $C_c(\Bbb{R}) \subset L^1(\Bbb{R})$ is dense, there exists $g_n \in C_c(\Bbb{R}$) such that
$$\... |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
- I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{... |
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