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I'm trying to understand the notation I've found in a paper which doesn't seem to be defined within the text. I can't quite recreate it in MathJax but it looks something like:
$$\propto \Bbb{1}(\overset{\Delta}{\pi}(f^a_t), u^a_t)$$
In the paper the 1 is blackboard bold style. I'm not sure what the significance of th... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
To be proved: for all integers $a,b,x,y$, if $ax + by = \gcd(a, b)$ then $\gcd(x,y)=1$.
Suppose that $ax+by = \gcd(a,b)$. **I first want to prove that every common divisor of $x$ and $y$ divides $\gcd(a,b)$.**
Let $n$ be such a common divisor. We have: $n$ divides $x$ and $n$ divides $y$, implying that $x = kn$ a... |
I was asked to determine whether the following statement is true:
> If every function $F : P \to P$ is a homomorphism from $(P, \leq)$ to $(P, \leq)$, with $\leq$ an arbitrary order, then $|P| = 1$.
It is straightforward to observe that $|P| \not> 1$. However, $|P| = 0$ seems to satisfy the first part of the pred... |
Is $\emptyset : \emptyset \to \emptyset$ an isomorphism from $(\emptyset, \leq)$ to $(\emptyset, \leq)$? |
As I was surfing the Mathematics side of Instagram (as usual), I came across this integral:
$$I=\int_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$$
It encouraged me to embark on a very satisfying journey to try and evaluate it. The result turned out very nice, of course. I've found that $$I=\frac{1}{24}\ln^{4}2-\frac... |
Is there an easier way to evaluate the integral $I=\int_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$? |
Edit: Added examples
I am confused about how free variables work with linear independence in vector space bases.
I am using Linear Algebra and its Applications Fifth Edition. It states -
"The columns of a matrix A are linearly independent if and only if Ax=0 has only the trivial solution."
and "The homoge... |
How can a basis have free variables if it also has to be linearly independent? UPDATED: With examples |
I'm trying to study if the following function has continuous partial derivatives at the origin:
$$f(x, y) = \frac{x^4y^3}{x^8 + y^4} \qquad \quad (x, y) \neq (0, 0)$$
(and $f(x, y) = 0$ for $(x, y) \neq (0, 0)$).
I proved $f$ is continuous at the origin, and I also proved its partial derivatives exist at the origi... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$).
Here $f$ is not a path-length parametrization.
Suppose also that
$T(t)$ - the unit tangent at $t$ ... |
Some context: I am reading Strang's paper, *A Framework for Equilibrium Equations*. In it, he gives a simple example with beams and rods. I'm trying to reproduce his arguments for 2D linear elasticity. In the paper, he considers conservation equations of the form
\begin{equation}
A^Tw = f
\end{equation}
where here... |
Formal adjoint of the symmetric gradient operator; is there a good interpretation? |
Edit: Added examples
I am confused about how free variables work with linear independence in vector space bases.
I am using Linear Algebra and its Applications Fifth Edition. It states -
"The columns of a matrix A are linearly independent if and only if Ax=0 has only the trivial solution."
and "The homoge... |
Could someone please tell me what results are known about the zeros of L function, $L(s,\chi)$ on the imaginary axis, where $\chi$ is a character mod $q$?
In particular, what is known about zeta function?
I guess there are none because we know $L(1+it)$ is non zero from the euler product and the functional equation ... |
The top line is $y=x$ and consists of the primes and four because $\rho(p)=p$ if $p$ is a prime number, and $\rho(4)=4$.
The second to top line is $y=\frac{x}{2}+2$ and consists of all $2p$ where $p\in{P}$ and $p\neq2$, and the number 8. This is because $\rho(2p)=\rho(p)+\rho(2)=p+2$ and $\rho(8)=\rho(4)+\rho(2)=4+2$.... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$).
Here $f$ is not a path-length parametrization.
Suppose also that
$T(t)$ - the unit tangent at $t$ ... |
There’s a bit of multilinear-algebraic folklore knocking around in my head that I’d like to cite and use but that I can’t track down for the life of me:
> Let $n > 2$ be an integer, and let $a,b \in M_n(\mathbb{R})$. If $\bigwedge^2 a = \bigwedge^2 b$, then $a=\pm b$.
Note that the condition $\bigwedge^2 a = \big... |
Two bijections from a finite set to itself commute with each other when they are both powers of on bijection. They also commute with each other when their cycles do not intersect each other.
But some other pairs commute: $(ab)(cd), (ac)(bd), (ad)(bc)$ commute with each other.
Can anything of interest be said abou... |
Which permutations commute with each other? |
Is torsion of a multivariate curve defined in n-dimensional space? |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For the full context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that al... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Let $X$ and $Y$ be topological spaces. A continuous map $F:X \rightarrow Y$ is called *proper* if the preimage of any compact subset in $Y$ is a compact subset of $X$. I wish to understand the definition of a proper map because it is a key ingredient in studying when a quotient of a smooth manifold is again a smooth ma... |
I am posed with the following question:
Let $f(u, v) = (3u − v, u + 4v).$ Fix $(x_0, y_0) ∈ \mathbb{R}$. Prove that $f^{−1}$ exists and is differentiable on some nonempty open set containing $(x_0,y_0)$ and compute the derivative there.
I'm a bit confused by this. If the set containing $(x_0,y_0)$ is $V$, shouldn't... |
I'm working through Ted Shiffrin Linear algebra and when introducing the dot product he motivates it by saying that if P and Q are two points in the plane and the angle $\angle POQ$ is a right angle then the triangles $\triangle OAP$ and $\triangle OBQ$ are similar where A is the projection of $\vec{OP}$ along the x-ax... |
I am posed with the following question:
Let $f(u, v) = (3u − v, u + 4v).$ Fix $(x_0, y_0) ∈ \mathbb{R}^2$. Prove that $f^{−1}$ exists and is differentiable on some nonempty open set containing $(x_0,y_0)$ and compute the derivative there.
I'm a bit confused by this. If the set containing $(x_0,y_0)$ is $V$, shouldn... |
Fix an open, bounded $U \subset \mathbb{R}^n$ and consider a sequence $C^2(U) \cap C(\bar{U})$ of harmonic functions.
(i) Suppose that $u_n$ converges uniformly to a function $u.$ Show that u is harmonic.
(ii) Suppose instead that $u_n$ is pointwise increasing and that for some $y \in U$ that $u_n(y)$ is bounded.... |
let $K$ be a field an $A$ a commutative $K$-algebra with unit. fix an element $f(X) \in A[X]$, unequal elements $a$ and $b$ of $K \subset A$, and an integer $t > 0$.
**Question.** if $(X - a)^t \mid f(X)$ and $(X - b)^t \mid f(X)$, then does $(X - a)^t \cdot (X - b)^t \mid f(X)$ in $A[X]$?
this is "trivial" when ... |
divisibility in polynomial ring: if $(X - a)^t | f(X)$ and $(X - b)^t | f(X)$, then does $(X - a)^t \cdot (X - b)^t \mid f(X)$? |
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$$
where $l$ is the value we want from the anti... |
Is this a valid alternate definite integral riemann sum? What is it called if so? |
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$$
where $l$ is the value we want from the anti... |
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$$
where $l$ is the value we want from the anti... |
I'm working through Ted Shifrin Linear algebra and when introducing the dot product he motivates it by saying that if P and Q are two points in the plane and the angle $\angle POQ$ is a right angle then the triangles $\triangle OAP$ and $\triangle OBQ$ are similar where A is the projection of $\vec{OP}$ along the x-axi... |
The conditional operator $\Rightarrow$ can be tricky to motivate in the cases where it is True. An approach taken by Elliott Mendelson in Number Systems and the Foundations of Analysis is to examine the expression $(C\land D)\Rightarrow D$. My understanding is as follows: Because English sentences that can be paraphras... |
I am having trouble understanding the definition of cofibration as given in Bredon's "Topology and Geometry." In particular, he defines a cofibration using the following diagram:
[![enter image description here][1]][1]
I am not sure how to interpret what Brendon means by "filling in" the diagram. I assume that ... |
How to understand cofibration as a commutative diagram? |
Is $\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$ a valid definite integral riemann sum? What is it called if so? |
let $K$ be a field an $A$ a commutative $K$-algebra with unit. fix an element $f(X) \in A[X]$, unequal elements $a$ and $b$ of $K \subset A$, and an integer $t > 0$.
**Question.** if $(X - a)^t \mid f(X)$ and $(X - b)^t \mid f(X)$, then does $(X - a)^t \cdot (X - b)^t \mid f(X)$ in $A[X]$?
this is "trivial" when ... |
I am having trouble understanding the definition of cofibration as given in Bredon's "Topology and Geometry." In particular, he defines a cofibration using the following diagram:
[![enter image description here][1]][1]
I am not sure how to interpret what Bredon means by "filling in" the diagram. I assume that t... |
I'm trying to construct the "correct" (read: a good) notion of moduli space of $n$-dimensional lattices.
Here, an $n$-dimensional lattice is the $\mathbb{Z}$-span of an $\mathbb{R}$-basis of $\mathbb{R}^n$.
For this, one has to find the "correct" (read: a good) notion of isomorphism between two lattices.
The first... |
Consider the sequence of functions $\{Q_n\}_{n=1}^\infty$ defined in the proof of Rudin's theorem 7.26:
$$Q_n(x)=c_n(1-x^2)^n$$
where coefficient $c_n$ is choosen so that $\int_{-1}^1 Q_n(x)dx=1$ for each $n\in\mathbb N$. I wish to show that $\{Q_n\}$ is not equicontinuous.
Here is my attempt:
Pick $\epsilo... |
I was trying to assess the impact of non-idealities on the outcome of a classical geometric construction, performed on paper with actual compass and straightedge.
I was thinking of possible approaches, but at the same time I didn't want to start from scratch, expecting that someone must have investigated this topic ... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
> Let $n$ be an odd integer.
>
> Show that there is an odd number of odd numbers in the sequence:
> $\binom{n}{1}, \binom{n}{2}, \binom{n}{3}, ..., \binom{n}{\frac{n-1}{2}}$
I've tried doing it by separating the problem into two cases, one in which $n=4k+1$ and one in which $n=4k+3$ where $k$ is a full number... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Let $\mathbb{H}_+\subseteq \mathbb{C}$ be the upper half plane,and $\bar{\mathbb{H}}_+=\mathbb{H}_+\cup \mathbb{R}$. Then consider the following equation:
$$z=-\frac{1}{m}-m+\frac{a^2}{ca^2m+1}.$$
Show that for each fixed $a,c\in \mathbb{R}_+$ and $z\in \bar{\mathbb{H}}_+$, there is a unique $m=m(z)\in \bar{\mathbb... |
For an assignment, I have the prove the equivalence of 5 statements for a linear transformation $T: V \rightarrow V$:
a) $image(T^2) = image(T)$
b) $ker(T) = ker(T^2)$
c) $image(T) \cap ker(T) =$ \{$0$\}
d) $V = image(T) \oplus ker(T)$
e) $ker(T_{image(T)}) = $ {$0$}
So I try to assume a) is true and ... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Let $K$ be a field, $G$ a group and $KG$ the corresponding group ring.
Let $M$ be a $KG$-module, therefore it can also be viewed as a $K$-vector space.
Suppose that $M$ has finite $K$-dimension, $\dim_KM = n$.
Then, $\exists U \leq M$ a $KG$-submodule of $M$ such that $U$ is simple.
Is this result true?
|
What is the average length of longest substring occurring at more than one position in a uniformly random binary string S of length N ?
For example,
```
n answer
1 0/2
2 2/4
3 10/8
4 26/16
```
For n=4
```
0000 3
0001 2
0010 1
0011 1
0100 1
0101 2
0110 1
0111 2
1000 2
1001 1
1010 ... |
The conditional operator $\Rightarrow$ can be tricky to motivate in the cases where it is True. An approach taken by Elliott Mendelson in Number Systems and the Foundations of Analysis is to examine the expression $(C\land D)\Rightarrow D$. My understanding is as follows: Because English sentences that can be paraphras... |
> Let $n>1$ be an odd integer.
>
> Show that there is an odd number of odd numbers in the sequence:
> $\binom{n}{1}, \binom{n}{2}, \binom{n}{3}, ..., \binom{n}{\frac{n-1}{2}}$
I've tried doing it by separating the problem into two cases, one in which $n=4k+1$ and one in which $n=4k+3$ where $k$ is a full numb... |
I'm trying to construct the "correct" (read: a good) notion of moduli space of $n$-dimensional lattices.
Here, an $n$-dimensional lattice is the $\mathbb{Z}$-span of an $\mathbb{R}$-basis of $\mathbb{R}^n$.
For this, one has to find the "correct" (read: a good) notion of isomorphism between two lattices.
The first... |
I'm trying to construct the "correct" (read: a good) notion of moduli space of $n$-dimensional lattices.
Here, an $n$-dimensional lattice is the $\mathbb{Z}$-span of an $\mathbb{R}$-basis of $\mathbb{R}^n$.
For this, one has to find the "correct" (read: a good) notion of isomorphism between two lattices.
The first... |
Almost every program considers $\frac{1}{0}$ as $+\infty$ and $-\frac{1}{0}$ as $-\infty$ according to the IEEE 754 floating-point standard. Thus $\pm\infty\cdot 0=\pm\frac{1}{0}\cdot 0=\pm\frac{0}{0}=\text{NaN} $.
Some programs also consider $\log(0)$ as $-\infty$ according to:
$$ \lim_{x\to 0^+}\log(x)=-\infty $$... |
What is the Taylor series expansion of the lower incomplete gamma function $\gamma(s,-x)$?
The standard answer from wikipedia is interms of $x^k$:
$$\gamma(s,-x)=\sum_{k=0}^{\infty}\frac{(-x)^se^{x}(-x)^k}{s(s+1)\cdots(s+k)}=(-x)^se^{x}\Gamma(s)\sum_{k=0}^{\infty}\frac{(-1)^k x^k}{\Gamma(s+k+1)}\tag{1}$$
Mat... |
Let $X$ be an (infinite-dimensional) inner product space with an (countable) orthogonal system $\lbrace e_i \rbrace_{i \in \mathbb{N}}$. The field is $\mathbb{R}$ or $\mathbb{C}$. **Here X may not be a Banach space**. Suppose further that $Span(\lbrace e_i \rbrace_{i \in \mathbb{N}})$ is dense in $X$, do we still have ... |
I'm trying to construct the "correct" (read: a good) notion of moduli space of $n$-dimensional lattices.
Here, an $n$-dimensional lattice is the $\mathbb{Z}$-span of an $\mathbb{R}$-basis of $\mathbb{R}^n$.
For this, one has to find the "correct" (read: a good) notion of isomorphism between two lattices.
The first... |
Let $M \subset B(\mathcal{H})$ be a von neumann algebra and $\phi$ be a uwo continuous faithful tracial state. I understand that we obtain the GNS $(\mathcal{H}_\phi, \pi_\phi, \xi_\phi)$ with $\xi_\phi$ the cyclic vector can be shown to be separating and a tracial vector for the image of $\pi_\phi$. From the represent... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Let $X$, $Y$, $Z$ be jointly Gaussian random variables with zero mean and known variances.
What can we say about $E[XZ]$ given we know $E[XY]$ and $E[YZ]$?
We can definitely get a valid range by requiring the covariance matrix to be positive semi-definite. But is there any more structure?
AFAIK being jointl... |
My question: Is it true that every locally compact Polish space must be boundedly compact?
\
Explanation of terminology:
- Locally compact: Every point admits a compact neighborhood;
- Polish space: Complete, separable metric space;
- Boundedly compact: Every closed and bounded subset is compact.
It seems ... |
Is it true that every locally compact Polish space must be boundedly compact? |
The conditional operator $\Rightarrow$ can be tricky to motivate in the cases where it is True. An approach taken by Elliott Mendelson in Number Systems and the Foundations of Analysis is to examine the expression $(C\land D)\Rightarrow D$. My understanding is as follows: Because English sentences that can be paraphras... |
This is from my notes in class. I don't get the parts in blue. Can you explain/elaborate on that, please?
But first, I don't understand why we can define such $\varepsilon$ in a way that $B(x,\varepsilon)$ can fit into $A$. How do we determine that? If you can be explicit I'll appreciate that.
Given $A=\{x\in \Bbb... |
Let $E_n$ be the elliptic curve defined by $y^2=x^3-n^2x$, and let $\mathbb{F}_{p^r}$ be a finite field of $p^r$ elements, where $p$ is prime and divides $n$. How can I determine how many points the curve has on such field? I found this problem on the book "Introduction to elliptic curves and modular forms$ chapter II:... |
I need to prove that the given point $(1, -3)$ in
$$D = \{(x, y): x>0 \space \mathrm{and} \space y \leq -2 \}$$
is an interior point of $D \subseteq \mathbb{R}^2$. That is, $(1, -3) \in D^{\mathrm{o}}$. I understand that $(1, -3)$ is an interior point of $D$ if and only if there exists a positive real number $r > 0$ ... |
How do I prove that a given point in a given subset of R² is an interior point? |
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$$
where $l$ is the value we want from the anti... |
I need to prove that the given point $(1, -3)$ in
$$D = \{(x, y): x>0 \space \mathrm{and} \space y \leq -2 \}$$
is an interior point of $D \subseteq \mathbb{R}^2$. That is, $(1, -3) \in D^{\mathrm{o}}$. I understand that $(1, -3)$ is an interior point of $D$ if and only if there exists a positive real number $r > 0$ ... |
I need to prove that the given point $(1, -3)$ in
$$D = \{(x, y): x>0 \space \mathrm{and} \space y \leq -2 \}$$
is an interior point of $D \subseteq \mathbb{R}^2$. That is, $(1, -3) \in D^{\mathrm{o}}$. I understand that $(1, -3)$ is an interior point of $D$ if and only if there exists a positive real number $r > 0$ ... |
$\newcommand{\image}{\operatorname{image}}$For an assignment, I have the prove the equivalence of 5 statements for a linear transformation $T: V \rightarrow V$:
a) $\image(T^2) = \image(T)$
b) $\ker(T) = \ker(T^2)$
c) $\image(T) \cap \ker(T) =$ \{$0$\}
d) $V = \image(T) \oplus \ker(T)$
e) $\ker(T_{\image... |
I need to prove that the given point $(1, -3)$ in
$$D = \{(x, y): x>0 \space \mathrm{and} \space y \leq -2 \}$$
is an interior point of $D \subseteq \mathbb{R}^2$. That is, $(1, -3) \in D^{\mathrm{o}}$. I understand that $(1, -3)$ is an interior point of $D$ if and only if there exists a positive real number $r > 0$ ... |
A simple drawing will convince you that $r = 1$ works. Algebraically: if $(x,y) \in B_1(1,-3)$ then $|x-1| < 1$ and $|y - (-3)| < 1$ (why?). The former implies $x > 0$ and the latter implies $y < -2$ as desired. |
How do I prove that a given point in a given subset of R^2 is an interior point? |
This is really just an extended comment, but you are unlikely to get a satisfactory answer of how the logical properties of the ordinary English *if* should be understood. There is vast amount of discussion. For example, [Humberstone, The Connectives][1], is a study of the four basic connectives (*and*, *or*, *if*, *no... |
I need to show that the given point $(1, -3)$ in
$$D = \{(x, y): x>0 \space \mathrm{and} \space y \leq -2 \}$$
is an interior point of $D \subseteq \mathbb{R}^2$. That is, $(1, -3) \in D^{\mathrm{o}}$. I understand that $(1, -3)$ is an interior point of $D$ if and only if there exists a positive real number $r > 0$ s... |
Can we find a number, let's say $x$ that's not a simple fraction, and if you raise it to the power of $π$ or raise $π$ to the power of $x$, both results end up being not simple fractions too? |
So let’s write the problem out using the definition of the derivative:
$$\frac{d}{dx}b^x=\lim_{h \to 0}\frac{b^{x+h}-b^x}{h}$$
In the equation above, $b^{x+h}-b^x$ represents a small change in $y$ while $h$ on the denominator represents a small change in $x$. It’s kinda similar to elementary linear algebra.
Now, l... |
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$$
where $l$ is the value we want from the anti... |
Does there exist an irrational number $x$ such that both $x^π$ and $π^x$ are also irrational? |
A geometry problem I worked on in a REU program from years ago just came to my mind.
Let's define a constant called "chord-arc constant" for any planar polygon as the largest ratio between 2 points' Euclidean distance and the shorter arc distance over all pairs of points on the polygon's edges. So essentially we are... |
$B_r(1, -3) = \{ (x, y) : (x-1)^2 + (y+3)^2 < r^2 \}$
....
Okay. If $(x,y) \in B_r(1,-3)$ then
$(x-1)^2 + (y+3)^2 < r^2$.
But we know $(x-1)^2, (y+3)^2 > 0$
So $(x-1)^2 \le (x-1)^2 + (y+3)^2 < r$ and $(y+3)^2 \le (x-1)^2 + (y+3)^2 < r$
And so $|x-1| < r$ and $|y+3| < r$.
So $-r < x-1 < r$ and $1-... |
A geometry problem I worked on in a REU program from years ago just came to my mind.
Let's define a constant called "chord-arc constant" for any planar polygon as the largest ratio between 2 points' Euclidean distance and the shorter arc distance over all pairs of points on the polygon's edges. So essentially we are... |
Following Akiva's advice, I'm posting this as a full answer, but I'll leave the details to you:
If for any irrational, atleast one of $x^\pi$ or $π^x$ is rational, then the set of irrationals injects in a union of two countable sets: Which ones ?
Why is this absurd ? |
What conditions are needed to have $F(f,g) \in L^\infty$ for $f, g \in L^\infty$? |
(a) implies (e): Since $\operatorname{image}(T)=\operatorname{image}(T^2)$, $T|_{\operatorname{image}(T)}$ is an isomorphism. So its kernel is zero.
(e) implies (c): Suppose there is a vector $v\in\operatorname{image}(T)\cap\ker(T)$. Then $Tv=T|_{\operatorname{image}(T)}v=0$ implies $v=0$ (because $T|_{\operatorname... |
Positive integral on every interval implies positive almost everywhere if continuous? |
Let $X$ be a random variable with known distribution $N(\mu, \sigma)$. Let $Y$ be a count variable, i.e., it is a discrete random variable with known finite expectation and finite variance, which takes integer values in the range $[0, n]$, for finite $n$. Let their population correlation be $\rho > 0$.
What in gener... |
Under what conditions does a Normal random variable have the same conditional distribution at every value of a discrete variable? |
(a) implies (e): Since $\operatorname{image}(T)=\operatorname{image}(T^2)$, $T|_{\operatorname{image}(T)}$ is an isomorphism. So its kernel is zero.
(e) implies (c): Suppose there is a vector $v\in\operatorname{image}(T)\cap\ker(T)$. Then $Tv=T|_{\operatorname{image}(T)}v=0$ implies $v=0$ (because $T|_{\operatorname... |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits can be settle to the following:
- The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on subsequent digi... |
I am reading the book *Measure and Integration: An introduction to Real Analysis* by Richard L.Wheeden, Antoni Zygmund, second edition. In chapter 5, the authors give a definition of a distribution function (p. 97):
>It turns out that there is a remarkably simple and useful representation of Lebesgue integrals over su... |
> Even when when we try to map an element in domain with an element in codomain there will be one element left in codomain which will not have a preimage.
You just wrote this statement as if it's an obvious incontrovertible fact. Can you *prove* this statement? Which element of the domain, exactly are we talking abo... |
Let
$$
f: \mathbb{R}^m \longrightarrow \mathbb{C}
$$
be a smooth, complex-valued function.
How can I write the Taylor's formula with integral remainder of it?
I guess it has the form
$$
f(x)- f(0) = x\cdot \nabla f(0)+ x\ \mbox{Hess}f(0)\ x^T + \sum_{|\alpha|=3,\alpha\in\mathbb{N}^m} \frac{x^\alpha}{\alp... |
This is a follow up to https://math.stackexchange.com/questions/4516445/if-sum-limits-n-1-inftye-n-xn-0-always-implies-sum-limits-n-1-i.
For $x \in \mathbb R$ we define the property $P(x)$
as follows: For every sequence $(a_n)$ with
1. $a_1 = x$ and
2. for every sequence $(e_n) \subset \{-1, 0,1\}$ with $\sum_{... |
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