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In lecture we were told to think about the following:
> **Kummer's Lemma:** Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ is real and $g$ is an integer.
This theorem says that the units of $\mathbb{Z}[\zeta]$ can be thougth as the... |
Here is something I've found on the internet
$$\begin{aligned}
f-\int f&=1\\
\left(1-\int\right)f&=1\\
f&=\left(\frac1{1-\int}\right)1\\
&=\left(1+\int+\int\int+\dots\right)1\\
&=1+\int1+\int\int1+\dots\\
&= 1+x+\frac{x^2}2+\dots\\
&= e^x
\end{aligned}$$
At first I interpreted this as a joke, but on closer in... |
Given a graph $G$ and a maximum independent set (MIS) $I$, can we comment on the size of a maximum independent set of the residual graph $G\setminus I$? Note that $G\setminus I$ denotes the induced graph obtained from $G$ after deleting the set of vertices in $I$.
A trivial upper bound is: $|\text{MIS}(G\setminus ... |
> Is there a simpler way?
Yes. Rather than multiplying, the method is to nest the integrals.
You have that:
$\qquad\begin{align}\mathsf E(X_{1}\mid X_{1}{\,=\,}\max\{X_1,X_2\}) &=\dfrac{\int_0^1\int_0^{x_1} x_1\,\mathrm d x_2\,\mathrm d x_1}{\int_0^1\int_0^{x_1} \mathrm d x_2\,\mathrm d x_1}\\[1ex]&=\dfrac ... |
In lecture we were told to think about the following:
> **Kummer's Lemma:** Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ is real and $g$ is an integer.
This theorem says that the units of $\mathbb{Z}[\zeta]$ can be thougth as the... |
I have been trying to become more rigorous in general, and have been trying to practice with elementary set theory. I came across the idea of existential instantiation, in which we can suppose we have an arbitrary object which satisfies a particular property. In relation to set theory, I believe this means if we have a... |
Limit in Invoking Existential Instantiation? |
The title's question is not entirely accurate, as I had to cut down the number of characters to be under the maximum number of characters allowed. Here is my real question:
Let $M$ be a connected manifold with dimension $\geq 2$ and $p_1,\dots,p_k$ and $q_1,\dots,q_k$ be two collections of distinct points. Show that... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
> Is there a simpler way?
Yes. Rather than multiplying, the method is to nest the integrals.
You have that:
$\qquad\begin{align}\mathsf E(X_{1}\mid X_{1}{\,=\,}\max\{X_1,X_2\}) &=\dfrac{\int_0^1\int_0^{x_1} x_1\,\mathrm d x_2\,\mathrm d x_1}{\int_0^1\int_0^{x_1} \mathrm d x_2\,\mathrm d x_1}\\[1ex]&=\dfrac ... |
Does there exist a natural number $n$ for which there is a supremum of a set $$A = \{a\in\mathbb{Q}^+ | a^3+a\leq n^2\}$$ in the set of rational numbers?
Since $f(a) = a^3+a$ is increasing function for $a\in \mathbb{Q}^+$, set $A$ has upper bound $n^2$, so we need to solve $a^3+a=n^2$? Is there such $n$? |
What is this sequence called? [0,0,1,0,2,0,2,2,1...] |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
The problem presented is an augmented ver of matching pennies so P1 will get +2 for matching with P2 and -2 for not matching essentially.
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1 |
One of the questions regarding this game is about finding the Maxmi... |
Game Theory: Maxmin Value for matching pennies game?? |
The problem presented is an augmented ver of matching pennies so P1 gets doubled payoffs.
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1)|
One of the questions regarding this game is about finding the Maxmin value for P1. I was a bit lost on the explanat... |
The problem presented is an augmented ver of matching pennies so P1 gets doubled payoffs for picking heads (we are given this matrix so assume this is correct).
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1)|
One of the questions regarding this game is ... |
> Approximate the large positive root of the equation
> $$e^{e^x} = 10^{10} x^{10}e^{10^{10} x^{10}}$$
>
> (attributed to G.H. Hardy)
This is a great problem which I can find no verifiable source or discussion of online. Is there a source where Hardy actually asks this question? How did Hardy (or others) approa... |
The problem presented is an augmented ver of matching pennies so P1 gets doubled payoffs for picking heads (we are given this matrix so assume this is correct).
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1)|
One of the questions regarding this game is ... |
As far as I know, the metric matrix is used to measure the length of vectors regardless of the chosen basis used to represent them. I read in the book "[Álgebra Geométrica e Aplicações][1]" by Fernandes, Lavor & Neto that the diagonalized metric matrix is used to identify the signature of a Geometric Algebra by coun... |
What is the relation of the metric matrix with the signature of a Geometric Algebra? |
The problem presented is an augmented ver of matching pennies so P1 gets doubled payoffs for picking heads (we are given this matrix so assume this is correct).
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1)|
One of the questions regarding this game is ... |
Given $\mathcal{O}_X$-modules $\mathcal{M}$ and $\mathcal{M}$, does the sheaf $Isom$ of isomorphisms of $\mathcal{O}_X$-modules, defined as
$$U\mapsto\mathrm{Sheaf}(U\mapsto \mathrm{Isom}_{\mathcal{O}_{X|U}}(\mathcal{M}|_U, \mathcal{N}|_U))$$
exist, where $\mathrm{Sheaf}$ is the sheafification functor? Is it a s... |
What is the average of $n(100-n)$ for $n \in {1,2,...,100}$
I saw this problem on the internet without a method of solving it without brute force or a computer. The answer is 1666.5.
If you were to integrate, you would get:
$$\int_0^{100} x(100-x)dx = \frac{500000}{3}$$
which gives $\frac{5000}{3}$ for the... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
Preface
=======
Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$):
$$x=x\left(t\right)$$
$$y=y\left(t\right)$$
In addition, there is a scalar function $f\left(x,y\right)$ defined for every value of $x$ and $y$.
Giving an example to illustrate (in this example, $t$ is the cent... |
Can I find a non-trivial cubic polynomial with rational zeroes *and* rational turning points? |
A distinction is commonly made between a ball (solid) and a sphere (the boundary of a ball). This distinction is made in other dimensions as well (e.g. circle versus disc, in 2D).
From what I've seen trying to research this online, the word "cube" is usually used to refer to the soild. Is there any standard name for... |
Is there a standard name for the boundary of a cube? |
Let $(X,\omega_X)$ be a compact Kaehler manifold. Denote by $\iota:Y\rightarrow X$ the natural embedding, where $Y$ is a submanifold in $X$. Recall that we have the
De Rham-Kodaira Hodge decomposition on smooth $(p,q)$-forms,
$$\mathcal I=\bar\partial{\bar\partial}*\mathcal G+{\bar\partial}^*{\bar\partial}\mathcal ... |
I suggest a frame change.
What you care about is that the students understand the relationships among intercepts (on both axes) and critical points. The algebra required to find those points matters less. An approximate sketch can show that knowledge. So ask questions where such an approximation can provide an answe... |
For each $n\in\mathbb{N}$, consider real $n$ numbers $x_{i,n}$, $1\le i \le n$, given by
$$\sum_{i=1}^{n}x_{i,n}=0,\\-1/n \le x_{i,n} \le 1-1/n.$$
I am trying to find value of
$$\lim_{n\to \infty} \sum_{l=1}^{\lfloor n/p \rfloor}\textbf{1}_{lp\le n}x_{lp,n},$$
where $p$ denotes a prime and $\textbf{1}_{\lbra... |
The following argument is from [a column by Peter Winkler](https://doi.org/10.1145/1378727.1389570), attributed to Ehud Friedgut; it may be equivalent to Graham and Kleitman's argument, but is easier to read. It works in the general case of a graph $G$ with $n$ vertices and $m$ edges, where the edges are labeled with d... |
Source : Buton, *Elementary number theory* (7th edition)
> Show that any integer of the form $6k + 5$ is also of the form $3 j +
2$, but not conversely.
**As to the $(\implies)$ part :**
$n= 6k+5$
$\implies n = 3(2k')+5 \space $ for some $ k' \in \mathbb Z$
$\implies n = 3(2k')+3+2 $
$\implies ... |
$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$
where $r$ is a constant (not necessarily an integer)
---
I apologize if I'm not able to contribute a lot to this problem as the floor function is really weird to me. I have never seen this inside a summation and don't know what to make of this proble... |
Source : Burton, *Elementary number theory* (7th edition)
> Show that any integer of the form $6k + 5$ is also of the form $3 j +
2$, but not conversely.
**As to the $(\implies)$ part :**
$n= 6k+5$
$\implies n = 3(2k')+5 \space $ for some $ k' \in \mathbb Z$
$\implies n = 3(2k')+3+2 $
$\implies... |
When we allocate $N:=k\cdot Z$ balls, where $\frac{N}{2}$ are red balls and $\frac{N}{2}$ are black balls, into $k$ groups, the number of red balls in each group follows a multivariate hypergeomtric distribution. Then, how to estimate or get a tight upper bound on its entropy?
I could get some trivial uppder bounds ... |
How to estimate or get an upper bound on the entropy of multivariate hypergeometric distribution? |
Let $R = \mathbb Z[X,Y]$. Construct an exact sequence of $R$-modules
$$0 \to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0,$$
where $f(g(X,Y)) = g(0,0)$ for all $g \in R$. Here $\mathbb Z$ is viewed as an $R$-module via $X\cdot 1 = Y\cdot 1 = 0$.
The only choice of $f$ is to actually *define* $f(g(x,y)) = ... |
I am trying to understand how to lower an index of a (1,2)-tensor $A(v,\alpha,\beta)$. As far as I understand A eats a vector and two covectors. The coefficients are given by $A_{jk}^{i}$.
I want to lower $i$,i.e. the "vector slot". To my understanding by lowering $i$ i get a tensor that eats three covectors, say ... |
Given a full rank matrix $W \in \mathbb{R}^{n\times n}$, and matrices $X, Y \in \mathbb{R}^{n \times k}$, $k < n$, I want to find matrices $A, B \in \mathbb{R}^{n \times k}$ which optimize:
$$
\min\limits_{A, B}\;\lVert W - (BX^T + YA^T + BA^T)\rVert_F
$$
That is, we're trying to approximate $W$ by $BX^T + YA^T... |
Decide whether the following subset of $\mathbb{R}$ is a ring, field, or neither:
S = {${a + b\sqrt[3]{2} + c\sqrt[3]{4} | a,b,c \in \mathbb{Q} } $ }
I tried approaching by considering the polynomial $f(x) = x^3 - 2$ $\in \mathbb{Q}[x] $ and observing that a root is $x = \sqrt[3]{2}$. Then, if there exists a set ... |
Let $\mathbb{F}_q$ be a finite field. Let $\widetilde M$ be the subgroup of $GL(n, \mathbb{F}_q)$ consisting of block diagonal matrices such that all diagonal blocks are invertible. Let $M=\widetilde M\cap SL(n,\mathbb{F}_q)$. Suppose $\sigma_1$ and $\sigma_2$ are irreducible representations of $\widetilde M$ such that... |
> Let $(X_n)_{n\geq0}$ be a submartingale,
$$T_0=0\text{,}\quad S_{n+1}=\inf\{k>T_n:X_k\leq a\}\text{,}\quad T_{n+1}=\inf\{k>S_{n+1}:X_k\geq b\}\text{,}$$ and $U_n=\sup\{k:T_k\leq n\}$ the number of upcrossings of $[a,b]$ before time $n$, for $n=1,2,3,\dots$. Then
$$ \mathbf EU_n\leq\frac{1}{b-a}\mathbf E(X_n-a)^+$$... |
$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$
where $r$ is a constant (not necessarily an integer). Note that $r\ge k$ and both $r, k$ are positive.
---
I apologize if I'm not able to contribute a lot to this problem as the floor function is really weird to me. I have never seen this inside a s... |
The problem presented is an augmented ver of matching pennies so P1 gets doubled payoffs for picking heads (we are given this matrix so assume this is correct).
| | Heads | Tails |
|:---: |:------:| :----:|
| Heads| (2,-2) | (2,-2)|
| Tails| (-1,1) | (1,-1)|
One of the questions regarding this game is ... |
Let $\mathbb{F}_q$ be a finite field. Let $\widetilde M$ be the subgroup of $GL(n, \mathbb{F}_q)$ consisting of [block diagonal matrices][1] such that all diagonal blocks are invertible. Let $M=\widetilde M\cap SL(n,\mathbb{F}_q)$. Suppose $\sigma_1$ and $\sigma_2$ are irreducible representations of $\widetilde M$ such... |
I am trying to understand how to lower an index of a (1,2)-tensor $A(v,\alpha,\beta)$. As far as I understand A eats a vector and two covectors. The coefficients are given by $A_{jk}^{i}$.
I want to lower $i$,i.e. the "vector slot". To my understanding by lowering $i$ i get a tensor that eats three covectors, say ... |
Let $\mathbb{F}_q$ be a finite field. Let $\widetilde M$ be the subgroup of $GL(n, \mathbb{F}_q)$ consisting of [block diagonal matrices][1] such that the main-diagonal blocks are invertible matrices. Let $M=\widetilde M\cap SL(n,\mathbb{F}_q)$. Suppose $\sigma_1$ and $\sigma_2$ are irreducible representations of $\wid... |
> Approximate the large positive root of the equation
> $$e^{e^x} = 10^{10} x^{10}e^{10^{10} x^{10}}$$
>
> (attributed to G.H. Hardy)
This is a great problem which I can find no verifiable source or discussion of online. Is there a source where Hardy actually asks this question? How did Hardy (or others) approa... |
Why are naturals < modulus equal if congruent? [uniqueness of remainder in a consecutive residue system] |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, f(x^2)-f(⌊x^2\over2⌋) is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows more and more slowly
... |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, $$f(x^2)-f(⌊x^2\over2⌋)$$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows more and more slow... |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, $$f(x^2)-f(⌊{x^2}\over2⌋)$$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows more and more sl... |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, $$f(x^2)-f(⌊{x^2}\over{2}⌋)$$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows more and more ... |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, $$f(x^2)-f(⌊({x^2}\over{2})⌋)$$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows more and mor... |
I've been experimenting with the function f(x) such that f(x) counts the number of primes from 1 to x
I found that after 10, $f(x^2) - f(\lfloor \frac{x^2}{2} \rfloor)$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows ... |
Let $\mathbb{F}_q$ be a finite field. Let $\widetilde M$ be the subgroup of $GL(n, \mathbb{F}_q)$ consisting of [block diagonal matrices][1] such that the main-diagonal blocks are invertible matrices. Let $M=\widetilde M\cap SL(n,\mathbb{F}_q)$. Suppose $\sigma_1$ and $\sigma_2$ are irreducible representations of $\wid... |
A prime number $\bar{a_1a_2\ldots a_n}$ is called a permutable prime number, if for any $\sigma\in S_n$, the number $\bar{a_{\sigma{(1)}}a_{\sigma{(2)}} \ldots a_{\sigma{(n)}}}$ is prime. For example 13 is a permutable prime number. Are there infinity many permutable prime numbers? |
Got a question regarding the following statement: a nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it.
If the above statement is not true, how about this statement: a symmetric nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how ... |
how to show that a nonsingular positive semidefinite matrix is positive definite? |
I'm trying to prove that:
> Given a sequence $(X_n)_{n \geq 1}$ of independent and identically
distributed random variables, $E(X_i^2) < +\infty$ for all $i \geq 1$, then $$\frac{2}{n(n-1)}\sum\limits_{1 \leq i < j \leq n} X_i X_j \xrightarrow[n\to +\infty]{} E(X)^2 \;\; \text{a.s}$$
My trick is to write $$\sum... |
I've been experimenting with the function π(x) such that π(x) counts the number of primes from 1 to x
I found that after 10, $π(x^2) - π(\lfloor \frac{x^2}{2} \rfloor)$ is always larger than x
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of primes grows ... |
There is a longer integral for which integration by parts would be appreciated:
>!$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg(\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\right)}_v\bigg)$$
as it has the below problem, but here is a ... |
Let $\mathcal{A}$ be an algebra over a set $E$, and $\mu$ be a pre-measure over $\mathcal{A}$. We can define an outer measure $\mu^*$ over $2^E$, and a $\sigma-$algebra $\mathcal{M}$ over $E$ (by Caratheodory theorem). In fact $\mu^*\mid_{\mathcal{M}}$ is a measure $\mu$. (cf. Real Analysis by Folland P29, 30, 31).
... |
Let $M$ be an $n \times n$ matrix, then the $(i,j)$-submatrix of $M$ is the $n-1 \times n-1$ matriz $M_{ij}$ given by removing the $i$-th line and the $j$-th column from $M$. The $(i,j)$-minor is $m_{ij} = \det M_{ij}$ and the $(i,j)$-cofactor is $c_{ij} = (-1)^{i + j}m_{ij}$
Is there a way to define these things fo... |
Let $M$ be an $n \times n$ matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, then $M_2 = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 5 & ... |
Is there a good way to express this matrix sum? |
Integration by parts does not work for this complex integral. Why? |
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted:
>!$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg(\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\right)}_v\bigg)$$
as it has the ... |
Let $M$ be an $n \times n$ matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, then $M_2 = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 5 & ... |
Let $M$ be an $n \times n$ nonnegative row-stochastic matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1/6 & 2/6 & 3/6 \\ 4/15 & 5/15 & 6/15 \\ 7/24 & 8/24 & 9/24 \end{bmatrix}$... |
I've been experimenting with the function $π(x)$ such that $π(x)$ counts the number of primes from $1$ to $x$.
I found that after $10$, $π(x^2) - π(\lfloor \frac{x^2}{2} \rfloor)$ is always larger than $x$.
However, after doing some research (see Prime Number Theorem), I'm a bit confused because the number of... |
Let $M$ be an $n \times n$ nonnegative row-stochastic matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1/6 & 2/6 & 3/6 \\ 4/15 & 5/15 & 6/15 \\ 7/24 & 8/24 & 9/24 \end{bmatrix}$... |
Let $M$ be an $n \times n$ nonnegative row-stochastic matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1/6 & 2/6 & 3/6 \\ 4/15 & 5/15 & 6/15 \\ 7/24 & 8/24 & 9/24 \end{bmatrix}$... |
Jacobi and Madden found infinitely many primitive solutions to,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$
using an elliptic curve. We will use a different approach that, like the method for,
$$a^4+b^4+c^4 = d^4$$
discussed in [this post][1], also yields **pairs** of solutions.
---
**I. The system**
Define,... |
I'm trying to define a class of integral integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property:
$$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\mathfrak{S}_\psi\{\psi(x)\}(t)$$
With $\alpha_f\in\mathcal{C}^\infty$ for any $f\in\mathcal{C}^\infty$.
Such an ... |
How can i find the kernel of this integral transform? |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are **the number of positive integers up to n+1 divisible by 2, 3 or 5.** We can check them by the following code:
```
//A2... |
In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if
$$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(d_2)$$
is the price of a call option and we want to find $\partial C/\partial t$, we never include the term $\partial... |
Let $M$ be an $n \times n$ nonnegative row-stochastic matrix, and let $M_i$ be the matrix whose $i$-th row is the same as that of $M$, and whose remaining rows are all the same as that of the identity matrix. For example, if $M = \begin{bmatrix} 1/6 & 2/6 & 3/6 \\ 4/15 & 5/15 & 6/15 \\ 7/24 & 8/24 & 9/24 \end{bmatrix}$... |
Decide whether the following subset of $\mathbb{R}$ is not a ring, or is a ring but not a field, or is a field:
S = {${a + b\sqrt[3]{2} + c\sqrt[3]{4} | a,b,c \in \mathbb{Q} } $ }
I tried approaching by considering the polynomial $f(x) = x^3 - 2$ $\in \mathbb{Q}[x] $ and observing that a root is $x = \sqrt[3]{2}$... |
I am reading a lecture note on complex analysis, and the following is in it:
>Let $p\in\mathbb{C}$ and $r>0$. Let $\gamma$ be the curve in $\mathbb{C}$ with parametrization $ z: [0,2\pi] \to \mathbb{C}$ given by
$$
z(t) := p + re^{it}
$$
If $z_0\in B(p,r)$, then
$$
\int... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are **the number of positive integers up to $n+1$ divisible by 2, 3 or 5.** We can check them by the following code:
```
//... |
What is the number of integers divisible by either 2, 3, or 5 from the integers 1 to $n+1$? |
$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$
where $r$ is a constant (not necessarily an integer). Note that $r\ge n$ and both $r, n$ are positive.
---
I apologize if I'm not able to contribute a lot to this problem as the floor function is really weird to me. I have never seen this inside a s... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$.
As joriki said, they are **the number of positive integers up to $n+1$ divisible by 2, 3 or 5.** We can check them by the following code:
```
//... |
I'm trying to prove that:
> Given a sequence $(X_n)_{n \geq 1}$ of independent and identically
distributed random variables, $E(X_i^2) < +\infty$ for all $i \geq 1$, then $$\frac{2}{n(n-1)}\sum\limits_{1 \leq i < j \leq n} X_i X_j \xrightarrow[n\to +\infty]{} E(X)^2 \;\; \text{a.s}$$
My trick is to write $$\sum... |
Problem goes as follows:
Let $\mathbb{R}_{1}[x]$ be the linear space of polynomials of degree $\leq 1$.
Define the covectors $f_{1}, f_{2} \in (\mathbb{R}_{1}[x])^{*}$ (dual space), as:
$$ f_{1}(p(x)) = \int_{0}^{1} p(x) \,dx $$
$$ f_{2}(p(x)) = \int_{0}^{2} p(x) \,dx $$.
Determine the basis of $\mathbb... |
If $f(0)=f(1)=0$ and for every $x\in (0,1)$ exists $0<\delta \inf\{x,1− x\}$ with $f(x)=\frac{f(x−\delta)+f(x+\delta)}2$. How to show that $f(x)=0$? |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
$$
S_a(x) =... |
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted:
$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg(\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\right)}_v\bigg) \tag{0}$$
as... |
I'm trying to define a class of integral integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property:
$$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\mathfrak{S}_\psi\{\psi(x)\}(t)$$
With $\alpha_f\in\mathcal{C}^\infty$ for any $f\in\mathcal{C}^\infty$.
Such an ... |
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted:
$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg[\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\right)}_v\bigg] \tag{0}$$
as... |
I am trying to find the time taken by a bead to slide down the Brachistochrone curve (between 0,0 and 1,-1) under kinetic friction, but I'm getting inconsistent answers with analytical and numerical methods.
[This article on Wolfram MathWorld][1] solves the problem analytically, giving the equations:
$$
x = a[(\... |
I'm trying to define a class of integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property:
$$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\mathfrak{S}_\psi\{\psi(x)\}(t)$$
With $\alpha_f\in\mathcal{C}^\infty$ for any $f\in\mathcal{C}^\infty$.
Such an integral ... |
I was reading Lazarsfeld's Positivity in Algebraic Geometry II
In page 312,there is a remark says if $f: V \longrightarrow W$ is any proper morphism of varieties, with $W$ reduced, and if $\mathcal{F}$ is any coherent sheaf on $V$, then there is a non-empty Zariskiopen subset $U \subset W$ such that the natural maps... |
To find all $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
\begin{gather}
\frac{f(x+y)+f( x)}{2x + f( y)} = \frac{2y + f( x)}{f( x + y)+ f( y)} \notag\\
x = 1,\ y = 1 \notag\\
f( 2) + f( 1) = 2 + f( 1) \notag\\[3mm]
f( 2) = 2\tag{1}\\[3mm]
x= 1,\ y= 2 \notag\\
\frac{f( 3) + f( 1)}{4} = \frac{f( 1) + 4}{f(... |
I wanted to find the properties of a circle from 3 points that are on its circumference, I understand how the problem should be solved, but the provided code after the explanation, does not match the approach, the code is as below:
```python
def define_circle(p1, p2, p3):
"""
Returns the center and radiu... |
Given a fixed channel $p(y|x)$, and denoting the discrete input pmf $p(x)$, and the corresponding output pmf by $p(y)$, prove that:
$$
\frac{\partial I(X;Y)}{\partial p(x)} = I(X=x;Y) - \log{e}
$$
where,
$$
I(X=x;Y) = \sum_{y} p(y|x) \log{\left(\frac{p(y|x)}{p(y)}\right)}
$$
**Note$^*$: the $\log$... |
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