instruction stringlengths 12 30k |
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Firstly, you need to define an inner product in $T^1(V)$. As far as I am aware this is defined by summing all of the component-wise inner products for each tensor power; but, this is only guaranteed to converge in the range of the signature thanks to the factorial decay. Ilya's work that you link doesn't define the inn... |
Your **idea can work** as follows.
We can show that for $\color{blue}{0<t<1}$:
$$\color{blue}{F_{X_{N}|N=i}(t)=1-\left ( 1+ \frac{1}{i-1} \right ) t +\frac{1}{i-1} (1-t)^{i}} \tag{1}.$$
This gives
$$\color{blue}{\mathbb E[X_{N}|N=i]}=\int_0^1 \left [1- F_{X_{N}|N=i}(t) \right ] \text d t \color{blue}{=\fr... |
I've read somewhere an MSE that we can understand the normalized valuation of a polynomial in the function field of a curve at a smooth point as the first non-vanishing coefficient (or exponent) of a corresponding laurent series, meaning that (using Silvermans notation) for the function $f \in \overline{K}(C)$ it's suf... |
I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you!
There is a common/standard treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, a... |
Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the quotient $R/I$. My question is the following: is the ring $R/I$ injective? Since $R\rightarrow R/I$ is a flat ring homo... |
Is the quotient of an infinite product of fields by the direct sum injective? |
The question [here][1] has the following set up
> Let $\{e_i\}$ be a basis of a $\mathbb{C}$-linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix of $h$ for $\{e_i\}$, that is
$$h_{ij}=h(e_i,e_j).$$
Let now $g\in GL(E)$ be a complex automorphism and consider the new basis $\{e'... |
This is question 1a in section 6.3.6 of Zorich's Analysis I.
I went about it as follows:
$
\lim_{n\to\infty}\left[ \frac{n}{(n+1)^2} + ... + \frac{n}{(2n)^2}\right] =
\lim_{n\to\infty}\left[ \frac{n}{[n(1+1/n)]^2} + ... + \frac{n}{[n(1+n/n)]^2}\right] =
\lim_{n\to\infty}\left[ \frac{n}{n^2}\frac{1}{(1+1/n)^2} ... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f$*. C... |
I've found myself in a situation where I need to prove a PBW theorem for a certain quotient of an algebra $A/I$ where $A$ is a filtered algebra given by generators and relations. Furthermore $A$ has a PBW theorem in the sense that $grA \cong U(\mathfrak{g})$ for some infinite dimensional Lie algebra $\mathfrak{g}$.(For... |
I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you!
There is a common/standard treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, a... |
Consider the Cauchy problem
$$\begin{cases}x'(t)=f(x) \\ x(0)=x_0\end{cases}$$
with $t\in \mathbb{R}$ and $f:I\subseteq\mathbb{R} \longrightarrow\mathbb{R}$.
If $f(x_0)=0$, then $x(t)=x_0$ is the solution of the equation. If $f(x_0)\neq 0$, **then $f(x(t))\neq 0 \, \forall t$ in the maximal interval of definition ... |
I am attempting to prove that the function $ f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} $ satisfies the functional equation $f(x+y) = f(x)f(y) $ for all real $ x $ and $y$, without relying on the knowledge that $ f(x) $ is the exponential function or using Taylor series expansions. I have been unable to establish the re... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f$*. C... |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
Is the polynomial $f=x^p-u$ irreducible over $\mathbb{Q}_p$? If it is reducible, what are the possible factors?
I tried Newton polygons but they are not conclusiv... |
Is it true that $i \in \mathbb{Q}(\sqrt{2},\xi)$, where $\xi = 1_{\frac{2\pi}{3}} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}$? |
Suppose that $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I / \sim$, that the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I / \sim \to \lbrace x_0 \rbrace \ti... |
X and Y have the joint mass function:
$f(x,y) = \frac{C}{(x+y+1)(x+y)(x+y-1)}$ x, y =1,2,3...
find $V = X-Y$
if $z>=0$
$P(X-Y = z) = P(X=k+z, Y=k)$
$=C\sum_{k=1}^{\infty}\frac{1}{(x+y+1)(x+y)(x+y-1)}$
$=\frac{1}{2}C \{\sum_{k=1}^{\infty}\frac{1}{(2k +z-1)(2k+z) - (2k+z)(2k+z+1)} \}$
$=\frac{1}{2}C... |
Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine
$$\frac{d}{dz}E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$
$X$ and $Y(X)$ have no density functions and no joint density.I thought about changing integral and derivative to get $$E[e^X(A-Y(X))\frac{d}{dz}\chi_... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
I am trying to do polynomial interpolation algorithmically with Newton's divided differences. I am able to find the divided differences $f[x_0, x_1, ... , x_n]$ okay, but I am struggling to find the coefficients of the Newton basis polynomial algorithmically:
$$
n_k(x) = \prod_{j=0}^{k-1}(x - x_j)
$$
Where the ... |
Suppose that $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I /\! \sim$, that the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I /\! \sim\; \to \lbrace x_0 \rbra... |
Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of
$$F(z)=E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$
$X$ and $Y(X)$ have no density functions and no joint density.I thought about changing integral and derivative to get $$E[e^X(A-Y(X))\frac{... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
I'm currently studying multiple linear regression on my own, and I've come here with three questions.
**Question 1** https://i.stack.imgur.com/0SZTs.jpg
I understand that the function is a quadratic function, but as you all know, to find the minimum value by finding the point where the derivative is 0, it must be... |
Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{H}/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{H}$ and the fundamental group $\pi_1(\Sigma,x_0)$ ($x_0\in\Sigma$) is isomorphic to $\Gamma$. Fix a point $\tilde{x_0}\in\math... |
From [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5Mw&ptn=3&ver=2&hsh=3&fclid=15624c3d-15b7-699a-3f5e-5f941464685c&psq=influencbof+conjugacy+classes&u=... |
Suppose that a CW complex $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I /\! \sim$, that the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I /\! \sim\; \to \lbr... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(\rho)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $... |
I've found myself in a situation where I need to prove a PBW theorem for a certain quotient of an algebra $A/I$ where $A$ is a filtered algebra given by generators and relations. Furthermore $A$ has a PBW theorem in the sense that $grA \cong U(\mathfrak{g})$ for some infinite dimensional Lie algebra $\mathfrak{g}$.(For... |
Suppose that a CW complex $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I /\! \sim$, the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I /\! \sim\; \to \lbrace x... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(\rho)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $... |
Given a graph G with N nodes we need to find if cycle of length 100 exists.
I've been told that this problem is NP-Complete and can be reduced to the Hamiltonian Cycle problem. I believe this is just P.
My first thought was just to perform a DFS on each node with a max depth of 101 nodes and check if the last nod... |
Find a length 100 cycle in an undirected graph is NP Complete? |
From [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5Mw&ptn=3&ver=2&hsh=3&fclid=15624c3d-15b7-699a-3f5e-5f941464685c&psq=influencbof+conjugacy+classes&u=... |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then at least two vertices i... |
I'm working in "Operator Theoretic aspects of Ergodic theory" by Eisner, Farkas Haase and Nagel. The question is as follows
> Consider the Hilbert space $H := \ell^2(\mathbb{N})$ and a sequence $(a_n)_{n\in\mathbb{N}}$ in $\mathbb{C}$ with $\sup_{n\in\mathbb{N}}\lvert a_n \rvert\leq 1$. Define $T: H\to H$ by $T(x_n)_... |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
I am wondering how the polynomial $f=x^p-u$ factors over $\mathbb{Q}_p$. I managed to prove (without using that $u$ is a unit) that f is irreducible if and only if i... |
The Claim:
---
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
>[T]he existence of natural numbers and the fact that given a natural number $n$, there is always a successor $(n+1)$, do not imply the existence of an infinite set. You need an extra... |
What are the number of ways of dividing $n$ balls into $2n$ bins, such that each bin contains two balls?
My working was as follows:
First select $n$ balls in $\binom{2n}{n}$ ways, then put one ball into each bin, and arrange it in a total of $n!$ possible ways, then, put the rest of the $n$ balls into the $n$ b... |
Suppose that a path-connected CW complex $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I /\! \sim$, the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I /\! \sim\... |
What are the number of ways of dividing $2n$ balls into $n$ bins, such that each bin contains two balls?
My working was as follows:
First select $n$ balls in $\binom{2n}{n}$ ways, then put one ball into each bin, and arrange it in a total of $n!$ possible ways, then, put the rest of the $n$ balls into the $n$ b... |
This is part of Exercise 7.1.9 from Vershynin's book "High Dimensional Probability":
> Suppose $X_1(t),\dots,X_N(t)$ are $N$ independent, mean zero random processes indexed by points $t\in T$. Let $\varepsilon_1,\dots,\varepsilon_N$ be independent symmetric Bernoulli random variables (Rademacher variables). Prove th... |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
What are the number of ways of dividing $2n$ distinct balls into $n$ identical bins, such that each bin contains two balls?
My working was as follows:
First select $n$ balls in $\binom{2n}{n}$ ways, then put one ball into each bin, and arrange it in a total of $n!$ possible ways, then, put the rest of the $n$ b... |
Suppose that a path-connected CW complex $X$ is homotopy equivalent to its own suspension $SX$. Is it true that $X$ must be contractible?
I have tried to show that since $X \times I \simeq SX = X \times I /\! \sim$, the projection $X \times I \to X \times \lbrace 0 \rbrace$ induces a projection $X \times I /\! \sim\... |
What are the number of ways of dividing $2n$ distinct balls into $n$ identical bins, such that each bin contains two balls?
My working was as follows:
First select $n$ balls in $\binom{2n}{n}$ ways, then put one ball into each bin, and arrange it in a total of $n!$ possible ways, then, put the rest of the $n$ b... |
Already answered here: [1]. Using this answer to share some additional facts.
Let $n$ be some even natural number.
Let $S(n)$ be the set of all schedules for $n$ teams.
Let $f(n)$ denote the number of ways to schedule $n$ teams. $f(n) = |S(n)|$.
### Conjecture: A Solution Exists
For all $n \geq 2$, there... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
I’m including the link to my geogebra website
>[here](https://www.geogebra.org/m/fhge9wjr),
in which $AGH$ is a triangle, $B$ is the center of its inscribed circle, and $D$ and $E$ are the intersections of that circle with $[AG]$ & $[AF]$ respectively. I couldn’t figure out how to prove that the perpendicular bis... |
Do we really know that we are near a local min when using gradient descent or [Nelder-Mead][1] method?
Suppose that
- I have a function $f$ that is analytic on the open ball $B = B(x, r)\subset R^n$,
- if the coordinates of $x$ are rational, then $f(x)$ and $\nabla f(x)$ are rational,
- I can evaluate $f$ and $... |
Since the bins are identical, the question essentially reduces to - in how many ways can $2n$ distinct balls be divided into $n$ pairs?
Let's line up the balls in a row. The first two balls make one pair, the next two another pair, and so on. Consider any one arrangement. There are $n$ pairs and they can be arranged... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have an example of holomorphic line bundle over $\mathbb P^n$,
> Example 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ is an open covering of $\mathb... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have an example of holomorphic line bundle over $\mathbb P^n$,
> Example 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of $\mathb... |
What is the number of ways of dividing $2n$ distinct balls into $n$ identical bins, such that each bin contains two balls?
My working was as follows:
First select $n$ balls in $\binom{2n}{n}$ ways, then put one ball into each bin, and arrange it in a total of $n!$ possible ways, then, put the rest of the $n$ ba... |
What is the number of ways of dividing $2n$ distinct balls into $n$ identical bins, such that each bin contains two balls? |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
There is not necessarily any "middle" of a four-torus, nor is there necessarily any "hole" in any place you might call the "middle".
When we try to visualize space in cases where the space is not Euclidean, there's a strong inclination to try to do this by embedding the space in a Euclidean space of higher dimension... |
As noted in the comments, the structure ($V_\omega$, $\in$) [satifies all the axioms][1] of ZFC except for the axiom of existence of an inductive set, and moreover every object of it is finite under all (usual?) definitions (, since choice holds), so it follows that some axiom form of infinity is in fact necessary to p... |
In an exam there are $20$ questions, one can obtain either $-1,0$ or $4$ in each question based on marking scheme. How many different totals can he obtained in the test?
One thing I observed different that even without repetition there is repetition as different permutation can obtain same number.
I tried several... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
I want to show that the adjoint of the identity $i: L^2(\Omega) \rightarrow H_0^s(\Omega)$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ is defined by measurable functional calculus. We assume $\Omega$ to be open and bounded. Any ideas to support me?
What I know is tha... |
Adjoint of the identity is the fractional Laplacian? |
I want to show that the adjoint of the identity $i: L^2(\Omega) \rightarrow H_0^s(\Omega)$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ is defined by measurable functional calculus. We assume $\Omega$ to be open and bounded. Any ideas to support me?
What I know is tha... |
I want to show that the adjoint of the identity $i: H_0^s(\Omega)\rightarrow L^2(\Omega), x\mapsto x$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ is defined by measurable functional calculus. We assume $\Omega$ to be open and bounded. Any ideas to support me?
What I ... |
As noted in the comments, the structure ($V_\omega$, $\in$) [satifies all the axioms][1] of ZFC except for the axiom of existence of an inductive set, and moreover every object of it is finite under all (usual?) definitions (since choice holds), so it follows that some axiom form of infinity is in fact necessary to pro... |
I want to show that the adjoint of the identity $i: H_0^s(\Omega)\rightarrow L^2(\Omega), x\mapsto x$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ is defined by measurable functional calculus. We assume $\Omega$ to be open and bounded. Any ideas to support me?
What I ... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have an example of holomorphic line bundle over $\mathbb P^n$,
> Example 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of $\mathb... |
I am concerned that I don't fully understand the Axiom of Union and how to use it **rigorously**.
Exercise 3.5.1(iii) from Terence Tao's Analysis I provides an opportunity to use it:
> (iii) Show that regardless of the definition of ordered pair, the
> Cartesian product $X \times Y$ of any two sets $X,Y$ is agai... |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f(x)$*.... |
[![enter image description here][1]][1]
$AGF$ is a triangle, $B$ is the center of its inscribed circle, and $D$ and $E$ are the intersections of that circle with $[AG]$ & $[AF]$ respectively. I couldn’t figure out how to prove that the perpendicular bisector of $[AD]$, the perpendicular bisector of $[AE]$, and the an... |
**The importance of $\lambda(n)$ dividing $n$**: The unit group $(\mathbb{Z}/n)^{\times}$ is a $\mathbb{Z}$-module and $(\lambda(n))$ is by definition, the annihilator of the module. In general, there is no canonical way to turn to raise elements of the unit group to an exponent in $(\mathbb{Z}/n)$. Of course, you can ... |
I want to show that the adjoint of the identity $i: H_0^s(\Omega)\rightarrow L^2(\Omega), x\mapsto x$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ is defined by measurable functional calculus. We assume $\Omega$ to be open and bounded. Any ideas to support me?
What I ... |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
**********************
*Intro__________*
On a [previous question](https://math.stackexchange.com/q/4838311/909869) I found that the following approximation could be used:
$$\ln\left(1+e^x\right)\approx \frac{x}{... |
Given two vectors $\mathbf{a}(t) = (r(t), s(t), -v(t), -u(t))$ and $\mathbf{b}(t) = (p(t), q(t), w(t), z(t))$, along with accompanying conditions:
1. $r(t) + s(t) - v(t) - u(t) \leq 0$
2. $r(t)p'(t) + s(t)q'(t) -v(t)w'(t) - u(t)z'(t) \leq 0$ (Dot product of $\mathbf{a}(t)$ and derivative of $\mathbf{b}(t)$)
... |
The Claim:
---
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
>[T]he existence of natural numbers and the fact that given a natural number $n$, there is always a successor $(n+1)$, do not imply the existence of an infinite set. You need an extra... |
Motivation for the problem [here][1]:
Is it possible to evaluate
$$
\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}?
$$
Or numerical value? I attempted to use Wolfram Alpha, but I didn't receive any results.
To simplify the integral let's denote $u = \tanh... |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
I am wondering how the polynomial $f=x^p-u$ factors over $\mathbb{Q}_p$. I managed to prove (without using that $u$ is a unit) that f is irreducible if and only if i... |
There is a committee of 11 people where 3 people are from Group A, 6 from Group B, and 2 from Group C. The first part is to find the number of ways they can sit, if people in the same group are indistinguishable. Now the second part is to find the probability that the first and last people in the row are from different... |
The question [here][1] has the following set up
> Let $\{e_i\}$ be a basis of a $\mathbb{C}$-linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix of $h$ for $\{e_i\}$, that is
$$h_{ij}=h(e_i,e_j).$$
Let now $g\in GL(E)$ be a complex automorphism and consider the new basis $\{e'... |
My question originates from a short proof in Strang's LA book that shows when
$A\pmb{x}=\pmb{b}$
every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in the column space of $A$. The proof is discussed [in this thread](https://math.stackexchange.com/questions/3585142/ho... |
I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an arbirtrary norm), at a point at the boundary $\partial B$. Recalling that the normal cone to a convex set $C$ at a point $x$ in... |
Let $S$ be a set of axioms, meaning, just any old set of sentences in some logical system, and let $A$ be an element of $S$. There are two definitions of the property "$A$ is redundant with respect to $S$". The first definition is, $A$ is in the deductive closure of $S - \{A\}$. The second definition is, the deductive ... |
Is there a non-monotonic logic where these two definitions of redundance are not equivalent? |
There is a committee of 11 people where 3 people are from Group A, 6 from Group B, and 2 from Group C. The first part is to find the number of ways they can sit, if people in the same group are indistinguishable. Now the second part is to find the probability that the first and last people in the row are from different... |
>whether the Fourier series of $f$
converges to $f$, or, more generally, whether f is determined by its Fourier series.
This was on page 186 and then on page 187
~
If $\phi_n(x)$ is orthonormal on $[a, b]$ and if
$c_n =\int_a^b f(t) \overline{\phi_n(x)}dt $
we call $c_n$ the $n$th Fourier coefficient o... |
Is $f= \sum c_n \phi(x)$ in baby Rudin 8.11? |
I am pretty new to cryptography and trying to understand some mathematical aspects. Studying different types of cipher, I came across the definition of index of coincidence which is as follows:
> Index of coincidence for a ciphertext of length $n$, in which the
> letter $A$ occurs $f_A$ times, $B$ occurs $f_B$ time... |
I am pretty new to cryptography and trying to understand some mathematical aspects. Studying different types of cipher, I came across the definition of index of coincidence which is as follows:
> Index of coincidence for a ciphertext of length $n$, in which the
> letter $A$ occurs $f_A$ times, $B$ occurs $f_B$ time... |
My question originates from a short proof in Strang's LA book that shows when
$A\pmb{x}=\pmb{b}$
every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in the column space of $A$. The proof is discussed [in this thread](https://math.stackexchange.com/questions/3585142/ho... |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
I am pretty new to cryptography and trying to understand some mathematical aspects. Studying different types of cipher, I came across the definition of index of coincidence which is as follows:
> Index of coincidence for a ciphertext of length $n$, in which the
> letter $A$ occurs $f_A$ times, $B$ occurs $f_B$ time... |
I am pretty new to cryptography and trying to understand some mathematical aspects. Studying different types of cipher, I came across the definition of index of coincidence which is as follows:
> Index of coincidence for a ciphertext of length $n$, in which the
> letter $A$ occurs $f_A$ times, $B$ occurs $f_B$ time... |
> Let $\{X_n \}_{n \geq 1}$ and $\{Y_n \}_{n \geq 1}$ be independent random variables such that $X_n \stackrel d \to X$ and $Y_n \stackrel d \to Y.$ Suppose that $X$ and $Y$ are also independent. Then show that $(X_n, Y_n) \stackrel d \to (X,Y).$
In order to show that I have to use the following fact $:$
> Let $\... |
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$
Why is that so ?
Notice
$$\int_0^{2 \pi} \ln(\sin(x) + \frac{5}{4}) dx = Re \int_0^{2 \pi} \ln (\sin(x) - \frac{5}{4}) dx... |
Your **idea can work** as follows.
We can show that for $\color{blue}{0<t<1}$:
$$\color{blue}{F_{X_{N}|N=i}(t)=1-\left ( 1+ \frac{1}{i-1} \right ) (1-t) +\frac{1}{i-1} (1-t)^{i}} \tag{1}.$$
This gives
$$\color{blue}{\mathbb E[X_{N}|N=i]}=\int_0^1 \left [1- F_{X_{N}|N=i}(t) \right ] \text d t \color{blue}{... |
Can someone recommend a Calculus book that emphasizes and clarify the intuition behind the theorems to the maximum, while in the same time is fairly rigorous?
During my bachelor in computer science, we used Stewart. I didn't like it as it was far from rigor besides I don't just want a lot of numerical examples, but ... |
Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the quotient $R/I$. My question is the following: is the ring $R/I$ injective? Since $R\rightarrow R/I$ is a flat ring homo... |
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$
Why is that so ?
Notice
$$\int_0^{2 \pi} \ln(\sin(x) + \frac{5}{4}) dx = Re \int_0^{2 \pi} \ln (\sin(x) - \frac{5}{4}) dx... |
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