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Darboux Theorem: If $f$ is differentiable on $[a,b]$ and $\eta$ between $f'(a)$ and $f'(b)$, then exists $c\in[a,b]$ such that $f'(c)=\eta$.
The common counterexample about $f'$ at $c$ not necessarily continuous is $f(x)=x^2\sin\frac{1}{x}(\mathbb{R}-\{0\}); 0(0)$, $\eta=0$ and $f'(0)=0$ where $f'$ is not continuou... |
Functional derivatives are a cornerstone in many engineering and physics concepts. For instance, engineering students typically encounter this concept in analytical mechanics and optimization theory.
As an engineering student, I've found the concept of functional derivatives somewhat confusing. This confusion mainly... |
I saw the statement in the question from the book Moonshine Beyond the Monster.
We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation.
I can't understand why the author said quosi-periodicity is an example of projective representation:
$\quad$ Let $\theta... |
How does Jacobi form lives projectively on the torus $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$? |
I found how to demonstrate the Taylor Series with integral remainder.
$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}(x-a)^k \Biggr] + \int_a^x \frac{f^{(n +1)}(t)}{n!}(x-t)^n dt $$
But I do you pass to this formula to the formula with the $o(x^n)$ term ?
$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}... |
I have a task to find two polynomials u(x) and v(x) such that x^m * u(x) + (1-x)^n * v(x) = 1, how can i do that?
I understand that we can consider x = 0 and get v(0) = 0, similarly for x = 1 we get u(1) = 1, and then we could consider the formal derivatives of these polynomials according to the same principle, but ... |
Why doesn't substiuting here work? |
I have a task to find two polynomials $u(x)$ and $v(x)$ such that $$x^m * u(x) + (1-x)^n * v(x) = 1$$, how can i do that?
I understand that we can consider $x = 0$ and get $$v(0) = 0$$, similarly for $x = 1$ we get $$u(1) = 1$$, and then we could consider the formal derivatives of these polynomials according to the ... |
We don't really need the Axiom of Regularity in ZFC. Having it is a matter of convenience.
We've known since the 1930s that if the variant of Zermelo-Fraenkel set theory that lacks the axiom of regularity is consistent, then so is Zermelo-Fraenkel set theory itself. The argument is essentially what you allude to in ... |
Most of the axioms of $\mathsf{ZFC}$ are "weak" in the sense that they only tell you what sets definitely exist, not that those are the only sets or that some sets don't exist. The exceptions are extensionality, foundation, and choice. In the case of foundation, it explicitly concerns set that don't exist, so indeed yo... |
After reading [this][1] post, I realized that an [earlier][2] post on this problem may be wrongly answered and replied to.
So we need to find the derivative of the given function:
\begin{equation}
f(x)= \begin{cases}1 \ \ \ \ \ x\in[0,1] \\
0 \ \ \ \ \ \ x\not\in[0,1]
\end{cases}
\end{equa... |
I have written the following algorithm pseudocode to depth-first traverse an undirected connected graph $G = (V, E)$:
```
DFS(G, v):
T = (V' = {v}, E' = empty, v)
s = [] // stack with push and pop
visited = [] // array with append and size
all_visited = [] // array
for u in {u | uv in E}:
s.push(u... |
I have a task to find two polynomials $u(x)$ and $v(x)$ such that $$x^m u(x) + (1-x)^n v(x) = 1$$, how can I do that?
I understand that we can consider $x = 0$ and get $$v(0) = 0$$, similarly for $x = 1$ we get $$u(1) = 1$$, and then we could consider the formal derivatives of these polynomials according to the sam... |
The typical definition of expectation requires a **probability space** and a **random variable**
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\mathrm{X}:\Omega\to\mathsf{X}$ be a random variable. The expectation of $\mathrm{X}$ is defined as... |
I know that:$$\int \cos x dx = \sin x +C$$
Substiute $x$ for $ax+b$:
$$\int \cos(ax+b) dx = \sin(ax+b) +C$$
but according to my book:
$$\int \cos(ax+b) dx = \frac{1}{a}\sin(ax+b) +C$$
Why doesn't substiuting work here?
|
Why doesn't substiuting work here? |
Show that trace class is a banach space with respect to norm $\| \|_{S_1}$ where the norm is infinite sum of eigenvalues of $(T^*T)^\frac{1}{2}$ and T is a compact operator in a separable hilbert space. I think this should follow quite easily when using the fact that space is complete if and only if $\sum_{k=1}^\infty... |
- If $X$ is positive definite, by Schur complement, the optimization problem is equivalent to
\begin{align*}
&\min_{U, V \in S^{m}} \quad \mathrm{tr}(V)\\
&\mathrm{subject \ to}\quad X \succeq U^2, \quad U \succ 0, \quad V \succeq XU^{-1}X.
\end{align*}
From $X \succeq U^2$ and $U \succ 0$, we have $
U^{-1} \... |
For $k=1,\ldots, \infty$, and $n=2,\ldots, k-1$ we define the following recurrence relation
$$ p_{1,k} = \frac{k-1}{k} p_{1,k-1}$$
$$ p_{n,k} = \frac{k-1}{k} p_{n,k-1}+\frac{1}{k^2} p_{n-1,k-1}$$
$$ p_{k,k} = \frac{1}{k^2} p_{k-1,k-1}$$
$$ p_{1,1} = 1$$
We may evaluate $p_{1,k}$ to be given by
$$ p_{1,k} = \fra... |
I have just started Measure Theory from Bartle, The Elements of Integration and Lebesgue Measure, where he defines Measurable Functions as:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/5PIog.jpg
What is the motivation for this? How is it related/equivalent to the more standard way of... |
We don't really need the Axiom of Regularity in ZFC. Having it is a matter of convenience.
We've known since the 1930s that if the variant of Zermelo-Fraenkel set theory that lacks the axiom of regularity is consistent, then so is Zermelo-Fraenkel set theory itself. The argument is essentially what you allude to in ... |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.
If possible, it would be helpful to link to proofs of the results. These can be external p... |
How can I compute the Laurent expansion of the function
$$f(z)=\frac{1}{(2z+1)(z-2)}$$
when $\frac{1}{2}<|z|<2$
with this $\frac{1}{z^2}$ term?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/uQ1QQ.png |
Are your $v_i$ vectors in your vector space? It's up to you: you can impose the rule that only finitely many coordinates are non-zero or allow any number of non-zero coordinates.
**Finitely many non-zero coordinates**
Your $e_i$ vectors form a basis but your $v_i$ vectors are not in the space.
**Any numbe... |
> Is there some kind of projection theorem, that lets us deduce closeness of random variables from closeness of their laws?
Certainly not. Two random variables can have the same law but be highly distant as random variables.
However it's certainly true that we can go the other way: If two random variables are clo... |
Let $X=\mathbb{C}[a,b]$ with the norm $\Vert x\Vert=\max_{a\le t\le b}|x(t)|$,
If we have $t_1,\cdots, t_n\in [a,b]$ and $\lambda_1,\cdots, \lambda_n\in \mathbb{C}$. If and let $f(x)=\sum_{i=1}^n\lambda_i\cdot x(t_i), ~\forall ~x\in \mathbb{C}[a,b]$.
How can I calculate the norm of $f$ ?
By $|f(x)|=\Vert\sum... |
Say that $M$ is a smooth $n$-dimensional manifold. Is it true that if we have an orientation form (that is, a smooth nowhere vanishing $n$-form) that $\Lambda^n M = \Lambda^n T^*M$ is trivial?
My thinking is that given $\Lambda^n M$ has dimension 1, perhaps we could then show that the orientation form is a basis of ... |
If $A,B\subseteq\mathbb{N}$ are disjoint decidable sets then it is clear to me that the function $f$ is defined as:
$f\left(x\right)=\begin{cases}
1 & \text{if }x\in A\\
2 & \text{if }x\in B\\
\text{undefined} & \text{otherwise}
\end{cases}$
is partial recursive.
This can be proved on base of the fact tha... |
Two functions $f(x)$ and $g(x)$ are linearly independent if and only if neither is a constant multiple of the other. That's directly from the definition of linear independence: there do not exist constants $a_1$ and $a_2$, not both $0$, such that $a_1 f(x) + a_2 g(x) = 0$ for all $x$ in the domain of the functions.
... |
>Let $a,b\in\mathbb{N},$ with $a≥b.$ Suppose $a^n+b^n|(ab)^n+1$ for each $n\in\mathbb{N}.$ Show that $b=1.$
Suppose not. Suppose $b>1.$ Now, assume $a≥b^2.$ Hence, $\frac{a}{b}-b≥0.$
We know that $a+b|ab+1$ and that $a+b|ab+b^2.$ Hence, $a+b|b^2-1.$ Since $b>1,$ we know that $b^2-1≠0.$ Hence, we get the inequalit... |
[![Labeled graph][1]][1]
[1]: https://i.stack.imgur.com/xoPWS.png
This is the graph I am working with and the edge labelings are given by $l(\{2n, 2n + 1\}) = a$ and $l(\{2n + 1, 2n + 2\}) = b$.
The question I'm working on guides us through to a description of the automorphism group of this graph. I so far... |
In *The Theory of Finite Groups*, by Kurzweil & Stellmacher, there is a proof of Gaschütz’s Theorem, which states the existence of a complement in $G$ for an abelian normal subgroup $K$, provided that $K$ has a complement in $U$, where $K\leqslant U\leqslant G$ and $\gcd(|K|,|G:U|)=1$.
There are too many details as ... |
I am studying Bessel modified $K_\nu$ functions, and I do not find any clear reference for the following question: is there a relation between $K_\nu(x)$ and $K_\mu(x)$ in general?
More specifically, I would like to know the relation (which can be merely bounds or asymptotics) between $K_{a+it}(x)$ and $K_{it}(x)$. |
>Let $a,b\in\mathbb{N},$ with $a≥b.$ Suppose $a^n+b^n|(ab)^n+1$ for each $n\in\mathbb{N}.$ Show that $b=1.$
Suppose not. Suppose $b>1.$ Now, assume $a≥b^2.$ Hence, $\frac{a}{b}-b≥0.$
We know that $a+b|ab+1$ and that $a+b|ab+b^2.$ Hence, $a+b|b^2-1.$ Since $b>1,$ we know that $b^2-1≠0.$ Hence, we get the inequalit... |
Young's inequality states the following:
For scalar time functions $x(t)\in\mathbb{R}$ and $y(t)\in\mathbb{R}$, the following holds:
$$ x\cdot y \leq \frac{1}{2\cdot \epsilon}\cdot x^2 + \frac{\epsilon}{2}\cdot y^2 \ \ \ (1) $$
for any $\epsilon > 0$. Consider now a constant $c>0$ and the absolute value of a f... |
I was studying about the mean residual life and theres a point where in evaluating an integral for the expression of the mrl you get (t-u)S(t) when t tends to infinite. How do we know for sure that goes to 0? |
Does the survival function decrease faster than linear? |
I read that geometric product $ab$ can be decomposed into 2 parts (symmetric and antisymmetric).
But I can't understand why symmetric part is a scalar product.
I mean following symmetric part:
$\frac{1}{2} (ab+ba) = \frac{1}{2}((a+b)^2-a^2-b^2)$
Why $ab+ba = (a+b)^2-a^2-b^2$ and $\frac{1}{2} (ab+ba) = a \cd... |
How to get scalar product from geometric product in Geometric Algebra? |
This link [here][1] shows that $\langle 3 \rangle$ is a generator for the given group by brute force method, that is trial and error. I was curious as to how to justify this using the theorem.
According to the Corollary, Generators of $\mathbb{Z}_n$:
> An integer $k$ in $\mathbb{Z}_n$ is a generator of $\mathbb... |
Why is $\langle 3 \rangle$ a generator of $(\mathbb{Z}_7^*, \cdot)$? |
I am interested in questions about the interaction between prime numbers and sequences generated by specific functions $f: \mathbb{N} \rightarrow \mathbb{N}$. I'm particularly interested in sequences like $k2^n+1$ (akin to Sierpiński numbers) but wish to extend this question to other sequences or functions. The core qu... |
I was trying to prove that $\varinjlim_{U \ni x} A(U) \cong A_{\mathfrak{p}}$ where $U$ are basic open sets containing $x = \mathfrak{p} \in \operatorname{Spec} A$, and $A(U) = A_f$ where $U = X_f$. $A$ is a commutative ring.
I know that the restriction maps from $A(X_f)$ to $A(X_g)$ are given by $a/f^k \mapsto au^k... |
Proof outline:
Restrict attention to $|M|$ where $MM^T=|M|^2$, aka the "matrix absolute value" or the "P.S.D term of the polar decomposition".
We have $|AAA^TA^T|=|A^2|^2$ and $|AA^TAA^T|=|A|^4$. Consider $|A|^2$ vs $|A^2|$. Substituting polar decomposition of $A=|A|\cdot U$ we have $|A^2|=|A|\cdot U\cdot |A|$ ag... |
I am currently reading "Lectures on Harmonic Maps" by R. Schoen and S. T. Yau and have problems understanding one step in the proof of a Corollary on p. 13.
> **Corollary**: Suppose $N = S^2$, and $M, S^2$ are equipped with arbitrary metrics. If $u: M \to S^2$ is harmonic and $\mathrm{deg}(u) > g_M - 1$, then $u$ is... |
Why does a holomorphic degree $1$ map imply that the Jacobian is strictly positive? |
I am trying to integrate the following first-order ordinary differential equation and confirming whether it is possible such that t(r) = 0 when r = 0.
$$\int \frac{dr}{r^2((a - r)^2 + b)} = \int c dt$$
From computationally integrating the above equation, I have obtained $t(r)$ as the following.
$$t(r) = -\frac... |
Take any positive real number $a$. Is there always some natural number $n$ such that $a\cdot n \geq 1$? And, I guess more generally, is there always some natural number $n$ such that $a\cdot n\geq b$ for any $b\in\mathbb{R}$? |
Here's a proof using only basic trigonometry and algebra, no calculus or infinite series required.
We'll do a proof by contradiction. Suppose $\cos(x)$ is a polynomial on some closed interval $[a,b]$, with $a\ne b$. We'll split it into two cases, depending on whether or not $0\in [a,b]$.
**Case 1.** Suppose your... |
I am trying to integrate the following first-order ordinary differential equation and confirming whether it is possible such that $t(r) = 0$ when $r = 0$.
$$\int \frac{dr}{r^2((a - r)^2 + b)} = \int c dt$$
From computationally integrating the above equation, I have obtained $t(r)$ as the following.
$$t(r) = -\... |
Let $U \subset \mathbb{R}^d$ be a convex open set.
Suppose a continuously differentiable vector-valued function $f : U \to \mathbb{R}^d$ satisfies that its derivative $D f : U \to \mathbb{R}^{d \times d}$ is symmetric and positive semidefinite.
Is there a convex function $\varphi : U \to \mathbb{R}$ such that $\nabla... |
For a vector-valued $f,$ is there a convex $\varphi$ such that $\nabla \varphi = f?$ |
My question is on Exercise 7.3.8 of *Chaos and Nonlinear Dynamics* (2nd ed) by Strogatz:
> **7.3.8.** Recall the system $\dot{r} = r(1-r^2) + \mu r \cos \theta, \; \dot{\theta} = 1$ of Example 7.3.1. Using the computer, plot the phase portrait for various values of $\mu > 0$. Is there a critical value $\mu_c$ at whi... |
I have an expression of the form
$$
\tag{1}
A(\chi) = \int_{0}^\infty\sum_{i=0}^\infty (-1)^{i+1}\frac{(2i+1)\chi}{\sqrt{t}}\exp\left(\frac{-(2i+1)^2\chi^2}{t}\right)\mathrm{d}t.
$$
If I am not mistaken, this expression is equal to
$$
\tag{2}
A(\chi) = \frac{1}{4}\int_{0}^\infty \frac{\chi}{\sqrt{t}}\frac... |
Given $A,B\subseteq\mathbb R^n$ two disjoint, nonempty, closed, convex sets, can we always find a vector $v\in\mathbb R^n\setminus\{0\}$ such that $\langle v,a\rangle<\langle v,b\rangle$ for all $a\in A,b\in B$?
Note that this requirement is weaker than other notions of strict separation (such as the existence of $\... |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.
If possible, it would be helpful to link to proofs of the results. These can be external p... |
I'm a bit stuck on telling what the ordinal $\epsilon_0 \cdot \omega$ is.
$\epsilon_0 = \sup \{1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}, \dots \}$
so
$\epsilon_0 \omega = \sup \{\omega, \omega^2, \omega^{\omega + 1}, \omega^{\omega^{\omega} + 1, \dots \}$
My confusion is whether or not in the lim... |
What is $\epsilon_0 \cdot \omega$? |
I'm a bit stuck on telling what the ordinal $\epsilon_0 \cdot \omega$ is.
$\epsilon_0 = \sup \{1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}, \dots} \}$
so
$\epsilon_0 \omega = \sup \{\omega, \omega^2, \omega^{\omega + 1}, \omega^{\omega^{\omega} + 1}, \dots \}$
My confusion is whether or not in the l... |
I am solving a second-order cone programming (SOCP) problem. I had to add a constraint that checks if at least one element of the decision variable vector is lower or equal to 0, i.e., I have to add the constraint
$$\min(y)\le0$$
Of course, this kind of constraint is concave and it is not compatible with SOCP.\
Is t... |
I'm a bit stuck on telling what the ordinal $\epsilon_0 \cdot \omega$ is.
$$\epsilon_0 = \sup \{1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \dots \}$$
so
$$\epsilon_0 \omega = \sup \{\omega, \omega^2, \omega^{\omega + 1}, \omega^{\omega^{\omega} + 1}, \dots \}$$
My confusion is whether or not in t... |
Let $U \subset \mathbb{R}^d$ be a convex open set.
Suppose a continuously differentiable vector-valued function $f : U \to \mathbb{R}^d$ satisfies that its derivative $D f : U \to \mathbb{R}^{d \times d}$ is symmetric and positive semidefinite.
Is there a convex function $\varphi : U \to \mathbb{R}$ such that $\nabla... |
Given a finite dimensional real vector space, we can view all the complex structures on it as a subspace of ${\rm GL}(V)$. I wonder if it is a submanifold of ${\rm GL}(V)$. And moreover, given a symplectic form $\omega$, will the $\omega$-compatible complex structure consist a submanifold?
For the first problem, I ... |
How to show that $\forall k\in(0,1), \exists L>0, \frac{1+kt}{1-kt} \leq e^{Lt}$ on [0,1]? |
Given pairs of spaces $(X,A)$ and $(Y,B)$, consider the inclusion
$$
\Delta(A\times Y) + \Delta(X \times B) \to \Delta(A\times Y \cup X\times B)
$$
Here $\Delta(A\times Y) + \Delta(X \times B)$ denotes the subcomplex of the singular chain complex $\Delta(A\times Y \cup X\times B)$ generated by those singular simpli... |
Given a finite dimensional real vector space, we can view all the complex structures on it as a subspace of ${\rm GL}(V)$. I wonder if it is a submanifold of ${\rm GL}(V)$. And moreover, given a symplectic form $\omega$, will the $\omega$-compatible (for simplicity, I discard the requirement that $\omega(v,Jv)>0$
fo... |
The following is taken from: *Module Theory An approach to linear algebra Electronic edition* by T.S.Blyth
*Backgroun:*
**Exercise 3.11**
A short exact sequence of the form
$$(f,E,g)\equiv 0\to A\xrightarrow{f}E\xrightarrow{g}B\to 0$$
is called an *extension* of $A$ by $B$.\
Given any $R$-modules
$A$ an... |
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k-$tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is,
$
y_n=
\begin{cases}
x_n&\text{if}\ 1\leq n \leq k \\
x_{n-k}&\text{if}\ k+1 \leq n \leq 2k.\\
\end{cases}
$
Does there exist a permutation $(\sigma(y_... |
Suppose a function $\eta (z)=log(\psi (z))$ where $$\psi (z)=\prod_{k=1}^{n} \left(z-z_k\right)$$ We know that $log(z)=log|z|+i(argz)$, this implies that $$log(\prod_{k=1}^{n} \left(z-z_k\right))=\sum_{k=1}^{n}log|z-z_k|+i\sum_{k=1}^{n}arg(z-z_k)=log|\psi (z)|+i\sum_{k=1}^{n}arg(z-z_k)$$
I am new to contour integratio... |
When working with multiple branch cuts, is there a way to chose the arguments so that log of a product can be opened as sum of individual logarithms? |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.
If possible, it would be helpful to link to proofs of the results. These can be external p... |
In three dimensions, use suffix notation and the summation convention to show that $$\boldsymbol{\nabla}r^n=nr^{n-2}\mathbf{x},$$ where $\mathbf{a}$ is any constant vector and $r=|\mathbf{x}|.$
My try:
$\nabla f=\frac{\partial f}{\partial x_i}\mathbf{e}_i$ and $r=\sqrt{x_i x_i}=\sqrt{x_1^2+x_2^2+x_3^2}$. Computing ... |
I just came across this exercise in Boyd & Vandenberghe.
[enter image description here][1]
[1]: https://i.stack.imgur.com/6DbfR.png
I'm not pretty sure what's the most basic manner I could have used for solving this exercise. On top of this, I can't really make sense of the requirement.
Therefore, could ... |
Are your $v_i$ vectors in your vector space? It's up to you: you can impose the rule that only finitely many coordinates are non-zero or allow any number of non-zero coordinates.
**Finitely many non-zero coordinates**
Your $e_i$ vectors form a basis but your $v_i$ vectors are not in the space.
This space ... |
How to show that $\forall k\in(0,1), \exists L>0, \frac{1+kt}{1-kt} \leq e^{Lt}$ on $[0,1]$?
This may be a very simple question but I cannnot see the trick.
I tried differentiating but it led nowhere, or maybe I am doing it wrong.
|
How to show that $\forall k\in(0,1), \exists L>0, \frac{1+kt}{1-kt} \leq e^{Lt}$ on $[0,1]$? |
I just came across this exercise in Boyd & Vandenberghe.
> Consider an Ellipsoid defined as
$$
\mathcal E = \{x \mid (x - x_c)^\top P^{-1} (x - x_c) \leq 1\}
= \{x_c + Au \mid \|u\|_2 \leq 1\},
$$
with $P$ positive definite and $AA^\top = P$.
>
> Show that both definitions are equivalent.
[(image link)]... |
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k$-tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is,
$$
y_n=
\begin{cases}
x_n&\text{if}\ 1\leq n \leq k \\
x_{n-k}&\text{if}\ k+1 \leq n \leq 2k.\\
\end{cases}
$$
Does there exist a permutation $(\sigma(y_... |
Using two copies of a decreasing $k$-tuple, can you form a convex decreasing $k$-tuple greater than the original sequence? |
Let's say that you calculate the basis of the kernel and it spans one vector. Then let's assume the rank of the matrix is two. Normally, you would choose the columns corresponding to the pivot points of the matrix as the basis of the image. However, could they just be two linearly independent vectors that are also line... |
This is a question about possible alternative formulations of a compactness.
Suppose $R$ is an open set, and $S$ is a compact subset of $R$. Suppose we have an arbitrary open cover $R = \bigcup_{i\in I} T_i$. Then, as the cover of $R$ also covers $S$ can we, for each such cover construct a finite open cover of $R$ a... |
I'm taking a course in which the ROC curve is specified by plotting points on an XY plane such that x is the false positive rate and y is the true positive rate at a certain threshold in binary classification. Then these points are joined into rectangles and the area under each is calculated, but It's said that if the ... |
Why do you take (1-the area under the ROC curve) when the area is less than .5? |
I am given the definition of the Lebesgue Measure:
Lebesgue Measure $\mathcal P$ on the Borel $\sigma$-algebra $\mathcal B[0,1]$:
$\mathcal P[[a,b]] = b - a$ for all $0 \leq a \leq b \leq 1$
The definition states the following: $\mathcal P[[a,a]] = a - a = 0$. Therefore $\mathcal P[(a,b)] = \mathcal P[[a,b]] = b... |
Perhaps surprisingly, the situation is a bit complicated!
Let's start with a pair of results, one positive and one negative, from the same paper. [Scott showed](https://math.stackexchange.com/a/437873/28111) that $\mathsf{Z-Ext}$ is equiconsistent with $\mathsf{Z}$, but $\mathsf{ZF-Ext}$ is in fact equiconsistent wi... |
Fitting a closed curve of a given length within given dimensions.
I'm trying to fit a given length of a closed curve within given dimensions.
**Example:** Curve **length=941mm**, **width=3mm**, and **height=5mm** and it must fit within the dimensions **x=250mm** **y=210mm**
I was thinking of creating a shape ... |
This is a question about possible alternative formulations of a compactness.
Suppose $R$ is an open set, and $S$ is a compact subset of $R$. Suppose we have an arbitrary open cover $R = \bigcup_{i\in I} T_i$. Then, as the cover of $R$ also covers $S$ can we, for each such cover construct a finite open cover of $R$ a... |
Find all $y<0$ such that the matrix defined for $y<0$ $$\begin{pmatrix}-3&2&y\\ 0&1&-\frac{1}{y}\\ y&0&y\end{pmatrix}$$ has rank $2$.
It seems ok, we just need a $0$ row, which could be the last row for example, if $y = 0$. However, this would mess up the condition in the second row, where $y≠0$. Im not sure what to... |
Let $\mathbf{H}$ be an infinite dimensional, separable Hilbert space. Moreover, let $\{A_i\}_{i=1}^N$ be a set of linearly independent, invertible operators that act on $\mathbf{H}$. In other words, $A_i:\mathbf{H}\to \mathbf{H}$ for $i=1,\ldots, N$.
$\textbf{Question:}$ Are there any general conditions on $\{A_i\}_... |
If you are interested in how these rules are derived algebraically, that's how you can do it using a few division laws: [Proof][1]
**Why do these rules always work?** Because all numbers (with a few very specific exceptions) behave the same way and obey the same laws. Why? I don't know. I think that's just how it is... |
How in general can we find a functional relationship between two functions
$$ u(x,y) ~, v(x,y)~$$
when Jacobian
$$\begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}$$
vanishes? i.e., what pde/ode can be used to find the dependence or link equation?
How does it change for three functions and three v... |
If you are interested in how these rules are derived algebraically, that's how you can do it using a few division laws: [Proof][1]
**Why do these rules always work?** Because all numbers (with a few very specific exceptions) behave the same way and obey the same laws. Why? I don't know. I think that's just how it is... |
Question: What are sufficient conditions on pairs $(X,A)$ and $(Y,B)$ so that $(A\times Y,X\times B)$ is an excisive couple?
In more detail: Given pairs of spaces $(X,A)$ and $(Y,B)$, consider the inclusion
$$
\Delta(A\times Y) + \Delta(X \times B) \to \Delta(A\times Y \cup X\times B)
$$
Here $\Delta(A\times Y) ... |
(I do not understand well where the [Stone-Von Neumann Theorem][1] enters the issue, maybe the right theorem is the [Stone Theorem][2], not the SvN one?)
Suppose that a strongly continuous one-parameter group of unitaries $\mathbb{C} \ni z \mapsto U_z$ exists in the complex Hilbert space $\cal H$ with the proper... |
In addition to Isaacs' Theorem that Arturo mentioned, below is a concrete example of the application of this theorem. It appeared in the paper *On Commutators in Groups*, by L-C. Kappe, R. F. Morse, Groups St Andrews $2005$, Vol. 2, p.541. However, their calculation in Example 4.5 is wrong and we present the improved o... |
Say I have two normal distributions that I'd like to compare in some writing: distribution A with mean = 12, standard deviation of 0.9; and distribution B with mean = 10, standard deviation of 1.1. I would like to describe how far apart these distributions are from each other in a modestly accurate but simple sense. T... |
given the Crank-Nicolson method for $f:(t,z)\mapsto az$
>$$
\begin{equation}\text{given } z^0 \\ z^{n+1}=z^n+\frac{\Delta t}{2}f(t^n,z^n)+\frac{\Delta t}{2}f(t^{n+1},z^{n+1})
\end{equation}$$
I want to show that the method is defined if, and only if $a \leq 0$. Any idea?
|
This is a question about possible alternative formulations of a compactness.
Suppose $R$ is an open set, and $S$ is a compact subset of $R$. Suppose we have an arbitrary open cover $R = \bigcup_{i\in I} T_i$. Then, as the cover of $R$ also covers $S$ can we, for each such cover construct a finite open cover of $R$ a... |
I am interested in alternative solutions to the following question:
> Prove that $\int_{0}^{\infty}x^{-\ln x}dx=\sqrt{\pi\sqrt{e}}$
I first used $\int_{0}^{\infty}x^{-\ln x}dx=\int_{0}^{\infty}e^{-\ln^{2}x}dx$
Then I used a u-sub to get:
$$\frac{1}{2}\int_{0}^{1}\left(e^{\sqrt{-\ln u}}+e^{-\sqrt{-\ln u}}\right)... |
Is there a sequence, defined in terms of integers and the basic arithmetic operations such that $\lim\limits_{n\rightarrow\infty} s_n = \pi$? |
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