instruction stringlengths 12 30k |
|---|
The problem occurs when reading Bombardini et al., 2023, "Did US Politicians Expect the China Shock?", *American Economic Review*, Vol.1, PP174-209.
The authors define $\xi$ to be a Gaussian term obeying $N(0,2\sigma_{\xi}^2)$, and $Y = \mathbf{1}\{a\theta+b+\delta E[S|\mathcal{I}]\geq\xi\}$. To my understand, $a\th... |
Suppose we have a sequence $(x_k)_{k \in \mathbb{N}}$ with $-e^{-1} \leq x_k \leq 0 \quad \forall k \in \mathbb{N}$ and $x_k \to 0$ as $k \to \infty$. Is the series
$$
\sum_{k=1}^{\infty} W_0 (x_k)
$$
necessarily finite? If not, what conditions would we need on $x_k$? |
Might be an odd question, but why the l2 norm of a matrix is defined the way it is defined?
Given the definition,
$$\|A\|_2 = \text{max} \frac{\|Ax\|_2}{\|x\|_2}$$
Specifically:
- why is it the **maximum** of the argument rather than the **minimum**, for example?
- Also, why do we divide by the l2 norm of... |
I am familiar with methods for solving quadratic programming problems of the form
$$\max_{x\in C} \mu^T x- 0.5x^T \Sigma x,$$
where
$$C = \{x\in \mathbb{R}^d: 1^Tx = 1, x\succeq 0\}.$$
For example the unconstrained solution is $x^*= \Sigma^{-1} \mu$ and the constrained solution can be obtained by any QP-Solver ... |
Let $F(n,p)=\sum_{m=p}^{\infty} \frac{(n+m)!}{n!\ m!\ 2^m}$. What is the upper bound for $F(n,p)$, where $p\geq n+1$ and $n\geq 0$? Does there exist a tight bound where the value of $p$ could be easily estimated under a constraint $F(n,p)\leq 2^{-k}$, where $k>1$?.
Note that $F(n,p)=\frac{(n+p)!}{n!\ p!}\left(\frac{... |
Does the initial-value problem $d y / d x=x / y$, $y(0)=0$ have a solution?
My answer of this quesstion is that equation, no have solution, because is not defined and also by the theorem of existence and unicity, that is not continous in 0.
My question is: Exist any justification or proof more formal to this que... |
I'm having trouble understanding what Hammack means with this and how he arrived at the result.
Solution: This set contains all numbers of form 7a + 3b, where a and b
are integers. Each such number 7a +3b is an integer, so A contains only
integers. But which integers? If n is any integer, then n = 7n +3(−2n), so
... |
Terence Tao's Analysis I 4th edition defines set intersection as follows:
> Given any non-empty set $I$ , and given an assignment of a set
> $A_{\alpha}$ to each $\alpha \in I$, we can define the intersection
> $\bigcap_{\alpha \in I}A_{\alpha}$ by first choosing some element
> $\beta$ of $I$ (which we can do sin... |
why is Terence Tao's definition 3.3 of set intersection so complicated? |
It is **not** recursive.
Consider $a,b$ as functions of $n$. Namely: given $n$ you have $a(n) = n$, $b(n) = −2n$.
About 7 and 3: the important fact is that $\gcd(7,3) = 1$ (why?). |
Terence Tao's Analysis I 4th edition defines set intersection as follows:
> Given any non-empty set $I$ , and given an assignment of a set
> $A_{\alpha}$ to each $\alpha \in I$, we can define the intersection
> $\bigcap_{\alpha \in I}A_{\alpha}$ by first choosing some element
> $\beta$ of $I$ (which we can do sin... |
> How can n = 7a+3b, and a = n, which is part of n, doesn't that mean the a is going to be n, so it's somehow recursive?
It is **not** recursive.
Consider $a,b$ as functions of $n$. Namely: given $n$ you have $a(n) = n$, $b(n) = −2n$.
> For example, why is 7 and 3 used, is there any relevance to these numbers?... |
As part of a physics project studying renormalization group flows of scalar field theories, I've come across the following quartic plane curve in the variables $m$ and $n$:
$$
36 + 16 m^4 - 108 n + 105 n^2 - 18 n^3 + n^4 - 8 m^3 (-18 + 17 n) +
m^2 (420 - 468 n + 81 n^2) + m (216 - 408 n + 234 n^2 - 34 n^3)=0.
$$... |
Let $M$ be a finite group.
Let $M^*=\text{Hom}_{conti}(M,\Bbb{Q}/\Bbb{Z})$.
Let $M'$ be an abelian group. Let $f:M\to M'$ be a homomorphism of abelian group.
Let $f^*: M'^*\to M^*$ be a map defined by $g\to f・g$.
Suppose $M'^*\cong M'$, then $f^*: M'\cong M'^*\to M^*$.
Does $\text{Im}(f)\cong \text{Ker}(f... |
Let $M, M'$ be a profinite group.
Let $M^*=\text{Hom}_{conti}(M,\Bbb{Q}/\Bbb{Z})$ be a Pontryagin dual of $M$.
Let $f:M\to M'$ be a homomorphism of abelian group.
Let $f^*: M'^*\to M^*$ be a map defined by $g\to f・g$.
Suppose $M'^*\cong M'$, then $f^*: M'\cong M'^*\to M^*$.
Does $\text{Im}(f)\cong \text{Ke... |
In reading "[PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES][1]" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ form a "branchless cycle".
For some context,
- $U$ is a set of elliptic curves, each one being a uniquely determined by ... |
Why do we get a connected 2-regular graph? |
Is there a version of the Bernstein von Mises theorem that allows us to perform a Gaussian approximation of the posterior density (assuming such a density with respect to the Lebesgue measure exists)? Most versions of the theorem I am aware of usually establish convergence in distribution or in Total Variation (i.e.
... |
Let $(Z,d)$ be a Polish space, and for $p\geq 1$, consider a metric space $(W_p,d_{W^p})$ defined by
The **Wasserstein Space** $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel probability measure on Z such that} \int_{Z}d(z_0,z)^p\mu(dz)<\infty \textrm{ for some }z_0 \} \end{align}$
and
The **Wasserstein dista... |
Geodesic Wasserstein space => the base space is also geodesic? |
I'm having trouble understanding what Hammack means with this and how he arrived at the result.
>Solution: This set contains all numbers of form 7a + 3b, where a and b
are integers. Each such number 7a +3b is an integer, so A contains only
integers. But which integers? If n is any integer, then n = 7n +3(−2n), so
... |
Let $f:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ be a proper convex lower semicontinuous function. Suppose that $f$ is bounded by below, and for simplicity that $\inf f = 0$. Set $\varphi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ as $\varphi(x):=\sqrt{f(x)}$. It is known that
\begin{align*}
\partial\varphi(x) = \frac{1... |
In reading "[PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES][1]" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ form a "branchless cycle".
For some context,
- $U$ is a set of elliptic curves, each one being a uniquely determined by ... |
Suppose that $\vec{x}\in\mathbb{R}^2$ is a vector and we want to find the minimum and the maximum of its components, using a two-layer neural network: $$\vec{y} = f_2(W_2f_1(W_1\vec{x}+b_1)+b_2), \ \ W_1,W_2 \in \mathbb{R^{2x2}},b_1,b_2\in \mathbb{R^2} \tag{1}\\ y_1 = \min(\vec{x}) = \min(x_1 , x_2) \\ y_2 = \max(\vec... |
Let $R$ be a ring. $R$ may not be reduced and commutative. Is it true in general ring that if for any $a,b,c\in R$, we have $abc=bca=cab=0$ then it implies $acb=0$. We can prove this in case of reduced rings. But what if the ring is not reduced? |
Let $X$ be a closed subspace of $\ell^2$ such that $X$ is contained in $\ell^1$. It is easy to show that the inclusion operator $J \colon X \hookrightarrow \ell^1$ is closed, hence, by the closed graph theorem $J$ is bounded.
Is it true that $X$ is automatically finite dimensional?
I would really appreciate any ... |
**Prove that for every simple graph G with maximum degree at most 3 and at least 4 vertices, k'(G) = k(G).**
I am still relatively new to the definitions of k(G) and k'(G). So far my understanding is that k(G) is the number of vertices, |s|, removed from G so that G - s is disconnected, and k'(G) is the number of ed... |
I'm going over Kobayashi's Differential Geometry of Complex Vector Bundles, and I see no mention of the famous second structural equation $$\Omega^i_j = d\omega^i_j + \sum_k \omega^i_k \wedge \omega^k_j,$$ where $\Omega = [\Omega^i_j]$ and $\omega=[\omega^i_j]$ are the connection and curvature matrices relative to a fr... |
I am reading Bollobás' Linear Analysis. Chapter 8. Theorem 6., as the title says:
> $B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$
The proof starts by saying that i) $B(X^{**}$) is $w$-closed, and ii) $B \subset B(X^{**})$ where $B$ is the $w$-closure of $B(X)$ in $X^{**}$. Then, suppose that there is $x_... |
To study the convergence properties of the series of functions:
$$\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$$
We begin with point convergence. For any fixed value of $x$, we need to check whether the sequence of functions converges. Let us denote by
$$f_n(x) = \left[\arcsin\left(\frac{2x^... |
I am working on a problem, where I have to show that the sum of functions of the form:
$ f_i(x)=\left( \left\lVert x-a_i \right\rVert - l_i \right)^2, a_i\in\mathbb{R^3}, l_i\in\mathbb{R} $
is not coercive. I have shown that a single $f$ is not coercive, but how do i show that some sum of them is also not coerci... |
Under what conditions is the sum of two non-convex functions convex? |
I am working on a problem, where I have to show that the sum of functions of the form:
$ f_i(x)=\left( \left\lVert x-a_i \right\rVert - l_i \right)^2, a_i\in\mathbb{R^3}, l_i\in\mathbb{R} $
is not coercive. I have shown that a single $f$ is not coercive, but how do i show that some sum of them is also not coerci... |
Given a set of intervals, when an element is not in said set when is that considered statistically suspicous? |
I am working on a problem, where I have to show that the sum of functions of the form:
$$ f_i(x)=\left( \left\lVert x-a_i \right\rVert - l_i \right)^2, a_i\in\mathbb{R^3}, l_i\in\mathbb{R} $$
is not coercive. I have shown that a single $f$ is not coercive, but how do I show that some sum of them is also not coer... |
Let $S(t)$ be the stock proce at time $t\geq 0$ with $S(0)=s_0$ and let $\Pi(S_t)=\max\{K-S_t,0\}=(K-S_t)_+$ the payoff of an american put with strike price $K$. How can I calculate the derivate $\frac{d}{d \psi} \mathbb E[e^{-\delta t} \Pi(S(t))\chi_{\{S(t)<\psi\}} | S(0)=s]$ for a stochastic $t$ and fixed $s$? I dont... |
Smooth complex vector bundles are just special instances of real vector bundles, so anything that applies to real vector bundles also applies to complex vector bundles.
For the second part, I'll assume you are talking about holomorphic connections on holomorphic vector bundles. Otherwise the first sentence of my post ... |
Let $X$ and $Y$ be two random variables on a probability measure space $(\Omega, \mathcal F, \mathbb P)$ such that $\mathbb E [f(X)] = \mathbb E [f(Y)]$ for all $f \in \mathcal C_b (\mathbb R)$ (bounded continuous functions). Can we conclude that $\mathbb P (X = Y) = 1$ i.e. $X = Y$ almost surely? |
I am working on a problem, where I have to show that the sum of functions of the form:
$$ f_i(x)=\left( \left\lVert x-a_i \right\rVert - l_i \right)^2, a_i\in\mathbb{R^3}, l_i\in\mathbb{R} $$
is not convex. I have shown that a single $f$ is not convex, but how do I show that some sum of them is also not convex? ... |
I'm stuck at the finish line on a pair of Exercises from Tao's *Introduction to Measure Theory:*
> **Exercise 1.6.12 (Rising Sun Inequality).** Let $f : \mathbb{R} \to \mathbb{R}$ be an absolutely integrable function, and let $f^* : \mathbb{R} \to \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function... |
What is the puntual, absolute and uniform convergence of the $\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$? |
In Section 7B of Geometry of Syzygies by Eisenbud, we have the following setup:
Let $S=k[x_0,\dots,x_r]$ and write $W=S_1$ for the space of linear forms. Let $\hat W = \mathrm{Hom}_k(W,k)$.
Eisenbud writes that if $e_0, \dots, e_r$ is a dual basis to $x_0, \dots, x_r$, then $e_i^2=0$, $e_ie_j=-e_je_i$.
How is ... |
If $e_0, \dots, e_r$ is a dual basis to $x_0, \dots, x_r$, then $e_i^2=0$, $e_ie_j=-e_je_i$. How is it that $e_i^2=0$, $e_ie_j=-e_je_i$? |
Consider the unitary group $U(2^n)$ and let $G$ be the subgroup isomorphic to $U(2^{n-1})$ embedded as:
\begin{pmatrix}
U & 0\\
0 & U
\end{pmatrix}
How to describe the homogeneous space $U(2^n) / G$? Do we get a Stiefel manifold as in the case of embedding:
\begin{pmatrix}
Id & 0\\
0 & U
\end{pmatrix}
>
|
Let $K$ be a number field and $G_K$ the absolute Galois group of $K$. Let $M$ be a **finite** $G_K$-module.
The Tate dual of $M$ is defined as follows: $M^D = \text{Hom}(M, \mu(K))$, where $\mu(K)$ is the group of all roots of unity.
Let $v$ be a fixed place of $K$.
It is well-known that $H^1(G_{K_v}, M^D) \co... |
First problem:
$$
\sum f_n(x)\hbox{ convergent }\implies\lim_{n\to\infty}f_n(x) = 0\implies
$$
$$|f_n(x)| < 1\hbox{ for $n$ large enough}\implies$$
$$
\left|\arcsin\left(\frac{2x^2-1}{x}\right)\right|< 1\implies\cdots
$$
Can you continue? |
The implication usually goes the other way (see Corollary 7.2.2 of Villani's big yellow book: https://link.springer.com/book/10.1007/978-3-540-71050-9). For what you want, I am not sure it is true. Consider the two point space, $Z= \{-1,1\}$ equipped with the discrete metric (which is Polish as desired). I think its no... |
I'm sorry if this isn't the right place (or right tags) for questions as such; I don't frequent the math StackExchange.
I'm trying to solve the following equation in relation to F;
$$ K + \frac{N}{F} + \frac{1}{MF^N}= 1$$
With the following as symbolic context (but not needed for simplification):
$$M = \pro... |
I was thinking that question and I know that it's a very fool question but I couldn't convince myself. By definition $R$ is a ring with two operation addition and multiplication $+,.$
I know that $S_n , n\ge3$ is a non abelian group group under the composition $\circ$. I believe that we can't talk about ring structu... |
To study the convergence properties of the series of functions:
$$\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$$
We begin with point convergence. For any fixed value of $x$, we need to check whether the sequence of functions converges. Let us denote by
$$f_n(x) = \left[\arcsin\left(\frac{2x^... |
Let $S(t)$ be the stock price at time $t\geq 0$ with $S(0)=s_0$ and let $\Pi(S_t)=\max\{K-S_t,0\}=(K-S_t)_+$ the payoff of an american put with strike price $K$. How can I calculate the derivate $\frac{d}{d \psi} \mathbb E[e^{-\delta t} \Pi(S(t))\chi_{\{S(t)<\psi\}} | S(0)=s]$ for a stochastic $t$ and fixed $s$? I dont... |
The implication usually goes the other way (see Corollary 7.2.2 of Villani's big yellow book: https://link.springer.com/book/10.1007/978-3-540-71050-9). For what you want, I am not sure it is true. Consider the two point space, $Z= \{-1,1\}$ equipped with the discrete metric (which is Polish as desired). I think its no... |
The surfaces of a gaming dice are labeled with 1, 2, 3, 4, 5, 6. How many different gaming dice exist, if each face of the dice is marked with a different number from 1 to 6, and no two faces can have the same number (consider rotations)?
I get 24, is my counting correct? |
The surfaces of a gaming dice are labeled with 1, 2, 3, 4, 5, 6. How many different gaming dice exist? |
$$
\sum f_n(x)\hbox{ convergent }\implies\lim_{n\to\infty}f_n(x) = 0\implies
$$
$$|f_n(x)| < 1\hbox{ for $n$ large enough}\iff$$
$$
\left|\arcsin\left(\frac{2x^2-1}{x}\right)\right| < 1\iff\left|\frac{2x^2-1}{x}\right| < \sin(1).
$$
And you **also** need $-1\le\frac{2x^2-1}{x}\le 1$, but as $\sin(1) < 1$, every ... |
In Ridgway Scott's Numerical Analysis, the following two problems appear:
[![problem 2.2][1]][1]
[![problem 2.4][2]][2]
In the first exercise, there was a clear mention of $x_0$ being sufficiently close to $\alpha$. So I knew that I should be proving local convergence, not global. However for the second exerci... |
$$
\sum f_n(x)\hbox{ convergent }\implies\lim_{n\to\infty}f_n(x) = 0\implies
$$
$$|f_n(x)| < 1\hbox{ for $n$ large enough}\iff$$
$$
\left|\arcsin\left(\frac{2x^2-1}{x}\right)\right| < 1\iff\left|\frac{2x^2-1}{x}\right| < \sin(1).
$$
And you **also** need $-1\le\frac{2x^2-1}{x}\le 1$, but as $\sin(1) < 1$, every ... |
Let $(G,\cdot)$ be a finite group that is not abelian. The question is: can I define an operation $+\colon G\times G\to G$ such that $(G,+,\cdot)$ is a ring?
Suppose you can do it and call $R$ this "ringification" of your starting group $G$. Since $(G,\cdot)$ is a group every non-identity element admit an inverse, so ... |
A ring is a set $A$ together with two binary operations, usually (but not necessarily) denoted $+$ and $\cdot$, satisfying certain properties (which I won't list).
The group $(S_3,\circ)$ is not a ring, because it is not a set together with two operations. Just like a cow is not a ring, because it is not a set toge... |
I was looking at an excellent lecture series on [Robotics][1] by Russ Tedrake, and he discusses Linear Quadratic Control (LQR) for system of nonlinear differential equations. So as he suggests, robots are just big collections of coupled pendulums that we are trying to stabilize around their fixed points.
Now, when ... |
In a [question][1] that I recently posted I defined a property of some concrete categories $(*)$, that if it holds, then:
A subobject of $X$ is uniquely determined by the underlying set (subset of $U(X)$), same for quotient object (namely in this case, determined by the equivalence relation on $U(X)$).
So now I w... |
A,B,C are events such that A and C are independent. How do I prove that A intersection B is also independent with C ? |
A,B,C are events such that A and C are independent. How do I prove that A intersection B is also independent with C? |
I just started to self-study some measure theory. I need to prove the following proposition
> **Proposition**$\quad$ *Lebesgue outer measure on $\mathbb{R}^d$ is an outer measure, and it assigns to each $d$-dimensional interval its volume.*
where
>**Definition**$\quad$ Let $X$ be a set, and let $\mathcal{P}(X)$ be... |
Let $(X,\mathcal{B})$ be a Hausdorff topological space with its Borel $\sigma$-algebra.
What are some general conditions we could impose on $X$ so that finitely supported measures (i.e. finite affine combinations of dirac measures) are weakly dense in the space $\mathcal{M}(X)$ of Borel probability measures of $X$?... |
I'm trying to show that the function
$$ f : [0,1] \to [0,1] \times [0,1] $$
$$ t=0.t_1 t_2 t_3 \dots \mapsto (0.t_1 t_3 t_5 \dots, 0.t_2 t_4 t_6 \dots ) $$
is continuous. My idea was to show that the function
$$ f_1 : [0,1] \to [0,1] $$
$$ t=0.t_1 t_2 t_3 \dots \mapsto 0.t_1 t_3 t_5 \dots $$
Is conti... |
How to prove that $f : [0,1] \to [0,1] \times [0,1]$ is continuous? |
In a [question][1] that I recently posted I defined a property of some concrete categories $(*)$, that if it holds, then:
A subobject of $X$ is uniquely determined by the underlying set (subset of $U(X)$), same for quotient object (namely in this case, determined by the equivalence relation on $U(X)$).
So now I w... |
This doesn't just apply to Earth (I first discovered it when calculating the area of the Celestial Sphere), but to any perfect sphere.
Think of it this way: The area contained between 80°N and 90°N is far less than the area contained between 0°N and 10°N, despite both being bands 10° wide.
***Anyway, I've found ... |
I've found that the area between ≈30° north and south contains exactly half of Earth's surface area. Is there an exact number out there? |
In Section 7B of Geometry of Syzygies by Eisenbud, we have the following setup:
>Let $S=k[x_0,\dots,x_r]$ and write $W=S_1$ for the space of linear forms. Let $\hat W = \mathrm{Hom}_k(W,k)$.
>Eisenbud writes that if $e_0, \dots, e_r$ is a dual basis to $x_0, \dots, x_r$, then $e_i^2=0$, $e_ie_j=-e_je_i$.
I k... |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
----
The following is an example of a set family T.
T has 36 elements.
... |
Task: In how many ways can we color the edges of a tetrahedron with black and red colors?
I have a problem because I computed the cycle index, considering the rotations of the tetrahedron based on its edges, and obtained: $Z(G,x_1,x_2,x_3)=\frac{1}{12}(x_1^6+x_3^2+3x_1^2x_2^2)$ or $Z(G,2,2,2)=12$, so the answer is ... |
In how many ways can we color the edges of a tetrahedron with black and red colors? |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
The program has shown that the number of elements of T can exist up to 36.
Ca... |
Task: In how many ways can we color the edges of a tetrahedron with black and red colors?
I have a problem because I computed the cycle index, considering the rotations of the tetrahedron based on its edges, and obtained: $Z(G,x_1,x_2,x_3)=\frac{1}{12}(x_1^6+8x_3^2+3x_1^2x_2^2)$ or $Z(G,2,2,2)=12$, so the answer is... |
What are the rings $A$ that appear as quotients $B/I$ of subrings$B \subseteq F$ of fields $F$ and for $I \subseteq B$ ideals? For each $A$, give an explicit formula for a ring $B$ and a field $F$.
For any ring homomorphism $\varphi: B \to F$ for some ring $B$ and field $F$, we know that $B/\ker \varphi \cong \textr... |
I'm studying about the non-homogeneous wave equation $u_{tt}(x,t)-c^2u_{xx}(x,t)=F(x,t)$. After the change of variables $\xi=x+ct,~\eta=x-ct$, this takes the form
$$u_{\xi\eta}\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right)=\dfrac{-1}{4c^2}F\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right),$$
I want to inte... |
after drawing N times from distribution P (over a large but discrete set of things), we have K unique things.
If we draw N+M times, how many unique things will we have, in expectation?
make assumptions on the kinds of distribution of P as needed to solve the problem, for me P is a large language model, and the "... |
number of additional unique samples from a distribution? |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
The program has shown that the number of elements of T can exist up to 36.
Ca... |
There are 20 balls in an urn labeled from 1 to 20. You randomly pick 10 balls out of this urn. What is the expected maximum value of the 10 balls you picked out?
I saw answers for this question on the forum, but I am confused why my answer is not correct. I approached this problem by determining the density function... |
I have read through a book about first order logic.
It was interesting. However, when I read through undergraduate math texts it’s unclear about what system they are working in. They don’t specify anything at all about axioms or logical axioms.
I’m so confused now. What is the point of learning about first order... |
How does first order logic influence everyday mathematics? |
Indeed, there is an apparent paradox as we can see on this example (like yours) with two hyperbolas, (with their asymptotes) having four points of intersection :
[![enter image description here][1]][1]
$$r=\frac{p_0}{1 - e_0 \cos \varphi} \ \text{and} \ r=\frac{p_1}{1 - e_1 \cos \varphi}\tag{1}$$
with $$e_0... |
Suppose we have a linear system $\dot{x} = A x$ where $x \in \mathbb{R}^n$ such that all the trajectories of the system are bounded and bounded away from the origin (i.e., none of them converge to the origin). This I believe implies that all the eigenvalues of $A$ have 0 real parts and that $A$ is diagonalizable. I als... |
Suppose we have a linear system $\dot{x} = A x$, where $x \in \mathbb{R}^n$ and $A$ is a constant matrix, such that all the trajectories of the system are bounded and bounded away from the origin (i.e., none of them converge to the origin). This I believe implies that all the eigenvalues of $A$ have 0 real parts and th... |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
The program has shown that the number of elements of T can exist up to 36.
Ca... |
Consider the set of equivalence classes of measurable functions on the interval [0,1] and the metric $\rho(f,g) = \int_{0}^{1} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|} \, d\mu$. It must be proved that $\rho(f_n,f) → 0$ if and only if $f_n$ → f almost everywhere in [0,1].
To prove that convergence by measure implies converg... |
I came upon the following specific property for a normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:
If a sequence $x_n$ in $X$ satisfies $\displaystyle \lim_{n\to\infty}(\|x_n+y\|-\|x_n\|)=\|y\|$ for all $y\in X$, then $\displaystyle \lim_{n\to\infty}x_n=0$.
This... |
Let $M_k \in \mathbb{R}^{n\times n}$ and $k \in \mathbb{N}$. Suppose that $M_kx_n$ converges to $M_k x$ in $L^2([0,1], \mathbb{R}^n)$ for $n \to \infty$ and $M_k x_n \in H_0^k([0,1], \mathbb{R}^n)$, $k \in \mathbb{N}$. The space $H_0^k([0,1], \mathbb{R}^n)$ denotes the Sobolev space $W_0^{k,2}([0,1], \mathbb{R}^n)$ end... |
> I know that my approach is basically assuming sampling with replacement although in the question it is implied as without. However,I thought this shouldn't matter due to linearity of expectation?
I am not sure how you connected this problem with linearity of expectation. But the expected maximum would certainly be... |
I am just confused on how this can happen as if the functions deviate at any point would there not be an infinte amount of deviations in between?
i.e say at x1 and x3 they have the same value but at x2 they have a different value, would there not be an infinte amount of different points then when you draw the functi... |
I am trying to make sense of the Cantor-Schroeder-Bernstein Theorem and it's proof. It was a theorem that was stated without proof in a real analysis book but I can't resist the urge to ask how to prove it. The theorem is stated in the first section on basic set theory (Volume 1 Book found here; https://www.jirka.org/r... |
Is this use of an inverse function correct? |
Why are these vectors linearly independent? |
Let $p$ be prime, $p \equiv 1 \pmod{3}$. I wish to show that there are $(p-1)/3$ (non-zero) cubic residues (mod $p$), for which I am having some difficulty.
There is a response [here][1] by userabc, which I am not certain how to follow. I can easily prove that $-3 \in Q_{p}$ and show that $x^3 - 1 \equiv 0 \pmod{p}... |
When $x, y \not\equiv 0 \bmod p$, we have $x^3 \equiv y^3 \bmod p$ if and only if $(x/y)^3 \equiv 1 \bmod p$, so $x \equiv yz \bmod p$ where $z^3 \equiv 1 \bmod p$. Thus the cubing function on nonzero numbers mod $p$ is $k$-to-$1$ where $k$ is the number of solutions to $z^3 \equiv 1 \bmod p$. This is similar to *squa... |
suppose an euclidean space $ \mathbb{R}^{n} $ with a quadratic form $ q: \mathbb{R}^{n} \rightarrow \mathbb{R} $ such that
$$ \forall X \in \mathbb{R}^n, \, \ q(X) = \sum_{i=1}^p X_i^2 - \sum_{j=1}^q X_j^2 $$
with $ p + q = n $
then this space is the pseudo-euclidean space $ \mathbb{R}^{p,q} $ of signatur... |
suppose an euclidean space $ \mathbb{R}^{n} $ with a quadratic form $ Q: \mathbb{R}^{n} \rightarrow \mathbb{R} $ such that
$$ \forall X \in \mathbb{R}^n, \, \ Q(X) = \sum_{i=1}^p X_i^2 - \sum_{j=1}^q X_j^2 $$
with $ p + q = n $
then this space is the pseudo-euclidean space $ \mathbb{R}^{p,q} $ of signatur... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.