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The game setup is as follows: there is a bag with 6 matching pairs of cards (two 1s, two 2s, two 3s, two 4s, two 5s, two 6s).
We randomly draw one card at a time, but put matching cards aside as soon as they appear in our hand. The game ends and we lose if we ever hold three cards, no two of which match. What is th... |
So, to first answer your main question:
> can we, for all ordered pairs $(x, y)$ find a monadic predicate, call it $\def\Fst{{\rm First}}\Fst$ such that $\Fst((x, y))=x$?
And the answer is, yes, absolutely. If someone put forth some notion of ordered pairs, and there was _not_ some way of extracting the $x$ from... |
I'm supposed to use Rouche's theorem to solve this problem, but I'm pretty sure it's not possible. Can anyone confirm this? I want to determine how many zeros $e^z-z$ has on $B_1(0)$. The obvious set up is to take $f(z)=e^z$ and $g(z)=-z$, but Rouche's theorem can't be applied on $\partial B_1(0)$, as $e^{-1}<1$.
W... |
Let $(f_n)_{n \geq 1}$ be a sequence of measurable functions from $(\Omega, \mathcal F)$ to a metric space $(X, d)$. Suppose that $f$ is such that
$$ \lim_{n \geq +\infty} d(f_n(\omega),f(\omega)) = 0, \ \forall \omega \in \Omega. $$
Is the function $f$ $\mathcal F$-measurable?
What I know: if the space $X$ i... |
Limit of measurable functions in a general metric space is measurable? |
Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol.
$$\begin{array}{r}\dagger\,\color{violet}0,&\bullet\,\color{brown}1,&*\,\color{red}3,&27,&531,\\
1\,035,&4\,635,&6\,363,&11\,475,&19\,683,\\
4\,131,&80\,955,&*\,\color{red}{266\,4... |
Our requirement is to use the intermediate value theorem to prove that there must be some root between an interval $ (a,b) $.
We must first show that the function is continuous on $ (a,b) $. But that might be better left for later.
I suppose you may just need a reminder about how to pick $ a$ and $b$ to show the ... |
How about proving that the provided expression satisfies the recursive formula?
\begin{align}
\mathbb E_t\left[\int_t^\infty \left(u(c_s)-\beta\mathbb E_s\left[\int_s^\infty e^{-\beta(w-s)}u(c_w)dw\right]\right)ds\right]
\end {align} |
Let $F$ be a field and $p \in F[x]$ be monic and irreducible.
Let $X=\begin{bmatrix}
C(p) & A \\
0 & B
\end{bmatrix}$ with minimal polynomial $p$. Then $X$ is similar to $C(p) \oplus D$ for some matrix $D$.
Question: Can the matrix of similarity be taken to be block upper triangular conformal to $X$? |
How about proving that the provided expression satisfies the recursive formula?
\begin{align}
\mathbb E_t\left[\int_t^\infty \left(u(c_s)-\beta\mathbb E_s\left[\int_s^\infty e^{-\beta(w-s)}u(c_w)dw\right]\right)ds\right]&=\mathbb E_t\int_t^\infty \left(u(c_s) - \int_s^\infty \beta e^{-\beta(w-s)} \mathbb E_s\left[u... |
Let $F$ be a field and $p \in F[x]$ be monic and irreducible.
Let $C(p)$ be the companion matrix of $p$.
Let $X=\begin{bmatrix}
C(p) & A \\
0 & B
\end{bmatrix}$ with minimal polynomial $p$. Then $X$ is similar to $C(p) \oplus D$ for some matrix $D$.
Question: Can the matrix of similarity be taken to be block up... |
How about proving that the provided expression satisfies the recursive formula?
\begin{align}
\mathbb E_t\left[\int_t^\infty \left(u(c_s)-\beta\mathbb E_s\left[\int_s^\infty e^{-\beta(w-s)}u(c_w)dw\right]\right)ds\right]&=\mathbb E_t\int_t^\infty \left(u(c_s) - \int_s^\infty \beta e^{-\beta(w-s)} \mathbb E_s\left[u... |
Let $0<k\in \mathbb{Z}$, $F$ be a field, and $p \in F[x]$ be monic and irreducible.
Let $C(p^k)$ be the companion matrix of $p^k$.
Let $X=\begin{bmatrix}
C(p^k) & A \\
0 & B
\end{bmatrix}$ with minimal polynomial $p^k$. Then $X$ is similar to $C(p^k) \oplus D$ for some matrix $D$.
Question: Can the matrix of si... |
Using the axioms of $ZF$, you ensure that from a set or multiple sets, you can also create a set. However, the question that arose in my mind is whether all subsets of this set that was created are also sets. In other words, is there a proof that every class $A$, where $A\subset C$and $C$ is a set, is also a set? |
Is every subset of a set also a set? |
Imagine a Pendulum with String Length L, and Angle θ, which forms arclength S (Will be used later in the derivation.)
First V = $\sqrt{2gh} $ (Height is measured straight down) KNOWN TO BE A CORRECT FORMULA
(From Here, Assume that nothing else is proven to be correct.)
Arclength X = $Lθ $
Time (Peri... |
Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $a_1, a_2, b_1$ and $b_2$ nonzero elements of $\mathcal{A}$. Then it is not difficult to see that
>$\vert \vert a_1b_1+a_2b_2 \vert \vert^2 \leq (\vert \vert a_1 \vert \vert^2 +\vert \vert a_2 \vert \vert^2 ) (\vert \vert b_1 \vert \vert^2 +\vert \vert b_2 \vert \vert^2... |
Does there exist a countable ordinal $\alpha$ that satisfies both of the following two properties?
1. There exists an integer $n$ such that $\alpha$ is $\Sigma_n$-definable in the language of first-order ZFC (by a formula in the Lévy hierarchy) over $L_{\omega_1}$;
2. There does not exist an integer $m$ such th... |
I have a question regarding the definition of 'Markov process' given by Shreeve, book II (continuous models).
It says (see below) this: "Assume that for all $0 \leq s \leq t \leq T$ and for every nonnegative Borel-measurable function $f$, there is another Borel-measurable function $g$ such that $\mathbb E [ f(X(t))... |
We consider a complex Hadamard matrix $H$ with entries in $\{\pm 1, \pm i\}$. I have seen some readings that often write $H = A + iB$ for matrices $A$ and $B$ with entries in $\{0,\pm 1\}$ where $A \circ B = 0$ ($\circ$ indicating the Hadamard product). I've never see this formulation of Hadamard matrices before, how m... |
Using the axioms of $ZF$, you ensure that from a set or multiple sets, you can also create a set. However, the question that arose in my mind is whether all subsets of this set that was created are also sets. In other words, is there a proof that every class $A$, where $A\subset C$and $C$ is a set, is also a set?
If... |
No, there is no such ordinal (or even set).
*Actually we need to be a bit careful; what does it mean to talk about the definability of an ordinal in the context of arithmetic? Below, I'm assuming that by "$\alpha$ is $\Sigma^1_n$" we mean "the set of reals coding relations on $\mathbb{N}$ which are isomorphic to $\a... |
The problem is\
Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies
\begin{cases}
\Delta u(x,y,z)=0\\
\lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\
\lim\limits_{z\to -\infty}\sup\limits_{x,y} |u(x,y,z)|=0
\end{cases}
Show that\
(i)For given $n\in \mathbb{Z}^... |
I am watching a logarithm lecture from 3Blue1Brown (great for math dummies like me!)
Here is the video for context:
https://youtu.be/4PDoT7jtxmw?t=1306
The step that I did not follow is when he equates the LHS and RHS of the equation below.
Maybe someone can fill me in on the general rule that allows us to clai... |
Using the axioms of $ZF$, you ensure that from a set or multiple sets, you can also create a set. However, the question that arose in my mind is whether all subsets of this set that was created are also sets. In other words, is there a proof that every class $A$, where $A\subset C$and $C$ is a set, is also a set?
If... |
I am using the formal definition of conditional expectations: $E[X|\mathscr{F}]$ is any RV $Y$ such that $Y\in\mathscr{F}$ and $\int_AXdP=\int_AYdP$ for all $A\in\mathscr{F}$.
Suppose that $\xi_1,\xi_2,\dots$ are iid RV with mean $p$, and they are independent of RV $N$. How do I show rigorously by definition that $E... |
Let us say that a $\Sigma_m^1$-formula $\phi$ defines a countable ordinal $\alpha$ if it defines a type-$1$ object (i.e. a real) $x$ that encodes a well-ordering of $\mathbb{N}$ of order type $\alpha$ (assuming that there is a fixed way to interpret a real as a well-ordering of $\mathbb{N}$).
Does there exist a ... |
Suppose $E_1$ and $E_2$ are two vector bundles over a base $B$ such that $E_1 \oplus E_2 \cong \epsilon$, i.e., the sum is trivial. After taking intersections, we get a cover $U_i$ of $B$ such that $E_1$ and $E_2$ are trivial over the $U_i$, giving transition functions $f_{ij}$ and $g_{ij}$. Thus, possible transition f... |
Conditional expectation of random summation - How to show $E[\sum_{i=1}^{N}\xi_i|\sigma(N)]=pN$? |
How is $\log _{10}(e)=\left[\log _e(10)\right]^{-1}$? |
In the article you linked ["On quadratic variation of martingales"][1], they specifically mention that they use predictable simple process in their proof. In particular to use Doob’s maximal inequality, they needed their stochastic integral to be a martingale.
[1]: https://www.academia.edu/81870538/On_quadrati... |
I've been using the following method to derive/remember the logarithm base conversion formula: If I want to convert $\log_a(x)$ to an expression in base $b$, I say,
$$\begin{align*}a^{\log_a(x)}&=x \\[0.3cm]
\log_b(a^{\log_a(x)})&=\log_b(x) \\[0.3cm]
\log_a(x)\log_b(a)&=\log_b(x)\\[0.3cm]
\log_a(x)&=\frac{\log_b(... |
In the article you linked ["On quadratic variation of martingales"][1], they specifically mention that they use predictable simple process in their proof. In particular to use Doob’s maximal inequality, they needed their stochastic integral to be a martingale.
Another reason predictability is handy is because it giv... |
In the article you linked ["On quadratic variation of martingales"][1], they specifically mention that they use predictable simple process in their proof. In particular to use Doob’s maximal inequality, they needed their stochastic integral to be a martingale.
Another reason predictability is handy is because it giv... |
The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$
I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since:
$\vec{BA}=<0,-2,2>$ and $\vec{CA}=<-1,0,2>$ with a cross product $<4,2,2>$
hence $4x+2y+2z=4 \cdot (1) + 2 \cdot (0) +2 \cdot (0) =4 $ So $$2x+y+z=2$$
The question asked... |
The game setup is as follows: there is a bag with 6 matching pairs of cards (two 1s, two 2s, two 3s, two 4s, two 5s, two 6s).
We randomly draw one card at a time, but put matching cards aside as soon as they appear in our hand. The game ends and we lose if we ever hold three cards, no two of which match. What is th... |
I'm considering functions of the form $\cos{(n\theta)}+\sin{(n\theta)}$ and have noticed that when $n=4k+1$ for positive integers $k$ the expression seems to be divisible by $\cos{\theta}+\sin{\theta}$ (also works in the trivial case when $k=0$) and when $n=4k-1$ the expression is divisible by $\cos{\theta}-\sin{\theta... |
Is this true (Trigonometry) and if so, can it be proved by induction? |
This question was already asked [here](https://math.stackexchange.com/questions/3883269/given-f-is-analytic-in-d-and-satisfy-fz-1-as-z-1-then-prove-that-f-is-ra), however I do not understand any of the solutions provided and I can’t find anything else online.
The question is found in Donald E. Marshall’s *Complex An... |
Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $a_1, a_2, b_1$ and $b_2$ nonzero elements of $\mathcal{A}$. Then it is not difficult to see that
>$\vert \vert a_1b_1+a_2b_2 \vert \vert^2 \leq (\vert \vert a_1 \vert \vert^2 +\vert \vert a_2 \vert \vert^2 ) (\vert \vert b_1 \vert \vert^2 +\vert \vert b_2 \vert \vert^2... |
I am aware of [Ralston's paper][1] on optimal (minimum truncation error) Runge-Kutta methods for orders up to 4 with the minimum number of stages for those orders. I have copied the Butcher tableaux of the methods he derived below:
**Order 1** (this one is trivial since Euler's method is the unique order-1 RK method... |
Higher-order optimal Runge-Kutta methods? |
This is Lemma 2.4 in Pete Clark's [notes](http://alpha.math.uga.edu/~pete/integral.pdf) on commutative algebra.
Given two posets $(X,\le)$ and $(Y,\le)$, an *(antitone) Galois connection* between $X$ and $Y$ is a pair of maps $F:X\to Y$ and $G:Y\to X$ such that $F$ and $G$ are antitone (= order-reversing) and for al... |
I believe that $\log \circ f \circ \exp$ should be concave if $f$ is positive, increasing and concave. My intuitive reasoning is that since it is true (in a degenerate sense) for $f(x) = x$, a fortiori it must be true if $f$ is strictly concave.
However, I'm not sure how to prove this. Any idea or counter-example? |
Is $\log \circ f \circ \exp$ concave when $f$ is concave, positive and increasing? |
The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through computational geometry, I stumbled upon a paper addressing the calculation of the 3D Winding Number for a closed triangular mesh.... |
Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms? |
I do not think any of the current answers actually address the question. So, let me add my two cents.
The keyword is **Commutativity**. The field of complex numbers $\mathbb{C}$ has this property over multiplication. The coefficients $\{c_{n-1}, c_{n-2}, \cdots, c_1\}$ are [elementary symmetric polynomials](https://... |
This question was already asked [here](https://math.stackexchange.com/questions/3883269/given-f-is-analytic-in-d-and-satisfy-fz-1-as-z-1-then-prove-that-f-is-ra), I wouldn’t ask it again however I do not think that the question gets the point across that it’s trying to get across, and I can’t find anything else related... |
I want to solve the partial differential equation $$2x(y + z) dx + (2yz - x^2 + y^2 - z^2) dy + (2yz - x^2 - y^2 + z^2) dz = 0$$
Here is the step that I followed :
By putting $u=y+z, v=y-z$, then we have :
$$4xudx+(u^2-v^2-2x^2)du+2uvdv=0 $$
For $X=(4xu, u^2-v^2-2x^2,2uv)$ then $X rot(X)=0$, hence the condition o... |
I have a doubt in the proof of Gagliardo-Nirenberg-Sobolev inequality that is found in Evans' book "Partial Differential Equations" (Theorem 1, pages 277-279).
In this part of the proof, $ \gamma>1 $ is a constant and $ u\in C^1(\mathbb{R}^n) $ is a function with compact support (i.e., $ u\in C^1_c(\mathbb{R}^n) $).... |
I do not think any of the current answers actually address the question. So, let me add my two cents.
The keyword is **Commutativity**. The field of complex numbers $\mathbb{C}$ has this property over multiplication. The coefficients $\{c_{n-1}, c_{n-2}, \cdots, c_0\}$ are [elementary symmetric polynomials](https://... |
The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$
I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since:
$\vec{BA}= \langle 0,-2,2 \rangle$ and $\vec{CA}= \langle -1,0,2 \rangle$ with a cross product $\langle 4,2,2 \rangle$
hence $4x+2y+2z=4 \cdot (1) + 2 \cdot (0) +2 \cdot (0... |
This question was already asked [here](https://math.stackexchange.com/questions/3883269/given-f-is-analytic-in-d-and-satisfy-fz-1-as-z-1-then-prove-that-f-is-ra), I wouldn’t ask it again however I do not think that the question gets the point across that it’s trying to get across, and I can’t find anything else related... |
Why does a *compact-open topology* define a *prebase* of *continuous functions*?
It is clear that in order for a set of such functions to set a *prebase*, it must cover the space of all continuous functions of their $X$ in $Y$, but it seems to me not obvious that in some arbitrary topological space there will be a u... |
Why does a compact-open topology define a prebase of continuous functions? |
Do analytic continuations exist over $\mathbb{R}$? |
I know an example for finite lebesgue measure for an open dense set, but not for zero measure.
Otherwise can we prove that no such set exists? |
Does there exist an open dense subset of $\mathbb{R}$ with Lebesgue measure zero? |
Given two operators $A $and$ B$, where $A ≤ B$ means the operator
$B − A$ is positive semidefnite.
(i) $0 ≤ A ≤ I.$
(ii) ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$
are equivalent
I am having trouble proving $(ii) \implies (i)$
If I take any non zero vector $|\phi⟩ $, then pluging in $|\phi⟩|/\|\ph... |
How can we randomly select $k$ people out of $n$ such that for person $i$, the probability they get selected is $0<p_i<1$, and $\sum p_i=k$?
I'm struggling quite a bit on this; you could go about this by creating every combination, assigning a weight for each combination, then randomly selecting a combination, but I... |
Given two operators $A $and$ B$, where $A ≤ B$ means the operator
$B − A$ is positive semidefnite.
(i) $0 ≤ A ≤ I.$
(ii) ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$
are equivalent
I am having trouble proving $(ii) \implies (i)$
If I take any non zero vector $|\phi⟩ $, then pluging in $|\phi⟩|/\|\ph... |
I'm having trouble understanding how to calculate the high order derivative of matrix, and was wondering if anyone could help me.
The function with respect to P is
f(P)=*Tr*($XPAP^TPA^TP^TX^T$)=$||XPAP^T||^2_F$
where *Tr* is the trace.
The question is how to calculate the derivative of f(P) with respect to ... |
Let $x>0$ be some variable that I am interested in analyzing in the limit $x\to\infty$.
Let $y = x+c$ for some constant $c$.
> Is $\sqrt{y} - \sqrt{x} \geq f(c) > 0$ for some function $f$ of only $c$?
This feels "obvious", but I am having quite a bit of difficulty in establishing it, so perhaps I am just Very Wr... |
Test the convergence of $$\displaystyle \sum_{n=1}^\infty\left(\frac{\log n}{n}\right)^2$$My textbook has used Cauchy's Condensation Test (CCT) to prove that the series is convergent. But I know that to apply CCT, the function $f(n)$ has to be monotonically decreasing. $\left(\frac{\log n}{n}\right)^2$ is not a monoton... |
Does the function has to be necessarily monotonically decreasing to apply Cauchy's Condensation Test? |
**1. What went wrong?** The formula
$$\text{speed}=\frac{\text{distance}}{\text{time}} $$
actually defines the **average speed**, whereas your formula $v=\sqrt{2gh}$ is the maximum instantaneous speed of the bob (when it passes through the minimum height point). This means that you cannot directly plug $v$ to the... |
Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence:
$$ D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$
As a consequence they are also close in total variation:
$$ \|p_n-p\|_{TV}^2\leq \frac{2C}{n}\quad\forall n\geq1\;.$$
Wha... |
Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence:
$$ D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$
As a consequence they are also close in total variation:
$$ \|p_n-p\|_{TV}^2\leq \frac{2C}{n}\quad\forall n\geq1\;.$$
Wha... |
I have heard that Homotopy Type Theory makes it so that isomorphic objects are “equal”. I wonder how this squares with a lot of mathematical examples from Algebra and Set Theory, where the nature of the isomorphism, or a certain class of isomorphisms, and how they interact with other morphisms, is relevant. How can you... |
Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence:
$$ D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$
As a consequence they are also close in total variation:
$$ \|p_n-p\|_{TV}^2\leq \frac{2C}{n}\quad\forall\,n\geq1\;.$$
Wh... |
In the book *Gaussian Processes for Machine learning* how does one convince oneself that the highlighted text is true? Or how does one prove it?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1CNV3.png |
I am currently reading through Katok on fuchsian groups chapter 2.1. There the author introduces a classification of PSL(2, R) by the trace of the tranformation. It then says that hyperbolas are in one case and ellipses in the other case invariant under these maps. I don't see why this is the case and I also don't see ... |
What is the intuition behind the classification of möbius transformation in hyperbolic, elliptic and parabolic transformations? |
I am trying to evaluate the integral
$$F(x,y) = \int_0^1 du_1\, \int_0^{1-u_1} du_2\, \frac{\log f(x,y|u_1,u_2)}{f(x,y|u_1,u_2)}\,, \tag{1}$$
with
$$f(x,y|u_1,u_2) := u_1(1-u_1)+y\, u_2(1-u_2) + (x-y-1)u_1 u_2\,, \tag{2}$$
where $0<x<1, 0<y<1$.
My question is simple: **is it possible to evaluate this integ... |
Is it possible to evaluate this integral? If not, is it possible to determine whether the result is an elliptic function or not? |
Consider the triangular hat function:
\begin{equation}
\varphi(x) =
\begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases}
\end{equation}
It is well known that $\varphi$ is the probability distribution of a sum $Y = X_1 + X_2 - 1$ where each $X_i$ is a uniformly distr... |
I do not think any of the current answers actually address the question. So, let me add my two cents.
The keyword is **Commutativity**. It kills the information of the order of multiplication: $$ab=ba.$$ The field of complex numbers $\mathbb{C}$ has this property over multiplication. The coefficients $\{c_{n-1}, c_{... |
It would be simple if you see it as cutting a circle. How to cut a circle to 3 pieces randomly? Just choose 3 points randomly! Then the three pieces should have equal expectation of length.
Due to symmetry, we can fix one point at zero, so we just need to choose the other two points. If the problem is continuous, th... |
Consider the triangular hat function:
\begin{equation}
\varphi(x) =
\begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases}
\end{equation}
It is well known that $\varphi$ is the pdf $Y = X_1 + X_2 - 1,$ where each $X_i$ is a uniformly distributed random variable on $[0... |
The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$
I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since:
$\vec{BA}= \langle 0,-2,2 \rangle$ and $\vec{CA}= \langle -1,0,2 \rangle$ with a cross product $\langle 4,2,2 \rangle$
hence $4x+2y+2z=4 \cdot (1) + 2 \cdot (0) +2 \cdo... |
There are n! permutations that can be inserted into an initially empty binary tree, but only Catalan(n) different binary trees. A possible method (surely very ineffective, but I'm an old FORTRAN guy, meaning that I automatically hate classes and recursion) to weed out duplicates would be the following: If a left child ... |
From my lecture notes I have that:
>The Lie algebra is introduced as a set of matrices $\theta$ that gave rise to elements $M$ of the corresponding matrix group through $M=e^{i\theta}$. However, just like a group can be thought of as an abstract set of instead of matrices, we can also give an abstract definition of th... |
I guess this is technically an answer (?).
Conrad's answer [here](https://math.stackexchange.com/questions/3883269/given-f-is-analytic-in-d-and-satisfy-fz-1-as-z-1-then-prove-that-f-is-ra) is probably what you were looking for; you simply needed to know what the word "Blaschke product" means. Given a finite collect... |
I have read somewhere that we can identify the group $\mathbb{Z}_n= \lbrace 0,...,n-1\rbrace $ with the interval $[- \frac{n}{2},\frac{n}{2})$ but there was no justification for this, does anyone have an argument which prove that this is true? |
It would be simple if you see it as cutting a circle. How to cut a circle to 3 pieces randomly? Just choose 3 points randomly! Then the three pieces should have equal expectation of length.
Due to symmetry, we can fix one point at zero, so we just need to choose the other two points. If the problem is continuous, th... |
Consider the triangular hat function:
\begin{equation}
\varphi(x) =
\begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases}
\end{equation}
It is well known that $\varphi$ is the pdf $Y = X_1 + X_2 - 1,$ where each $X_i$ is a uniformly distributed random variable on $[0... |
Consider the triangular hat function:
\begin{equation}
\varphi(x) =
\begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases}
\end{equation}
It is well known that $\varphi$ is the pdf
$$Y = X_1 + X_2 - 1,$$
where each $X_i$ is a uniformly distributed random variable ... |
It would be simple if you see it as cutting a circle. How to cut a circle to 3 pieces randomly? Just choose 3 points randomly! Then the three pieces should have equal expectation of length.
Due to symmetry, we can fix one point at zero, so we just need to choose the other two points. If the problem is continuous, th... |
Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$:
$$
-\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega
$$
with Neumann boundary conditions in $\partial\Omega$. If $f\in L^2(\Omega)$ and we assume that we have a solution $u\in W^{1,2}(\Omega)$, we can multiply the eq... |
Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$:
$$
-\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega
$$
with Neumann boundary conditions in $\partial\Omega$. If $f\in L^2(\Omega)$ and we assume that we have a solution $u\in W^{1,2}(\Omega)$, we can multiply the eq... |
Why does $\sum_{n=1}^{\infty} \frac{1}{n}$ not converge and yet $\sum_{n=1}^{\infty} \frac{1}{n^2}$ does converge? Looking for an intuition |
Consider the triangular hat function:
\begin{equation}
\varphi(x) =
\begin{cases} 1 - |x|, & \text{if } x \in [-1, 1], \\ 0, & \text{otherwise.} \end{cases}
\end{equation}
It is well known that $\varphi$ is the pdf
$$Y = X_1 + X_2 - 1,$$
where each $X_i$ is a uniformly distributed random variable ... |
Evaluate the value of:
$$(i)\lim_{n \to ∞} n^2 \left(\int_0^{\pi/2} (\sin^n(x)+\cos^n(x))^{1/n} dx-\sqrt2 \right)$$
$$(ii)\lim_{n \to ∞} n^2 \left(\int_0^1 (x^n+(1-x)^n)^{1/n} dx-3/4 \right)$$
Both of these limits have a similar form and it's easy to see that it is a $0$ times $∞$ form; it is easy to evaluate bo... |
I think you are looking at it from the wrong way. Instead of looking for intuition into why a series converges or not, try to think about what in your current intuition makes the results you are looking for feel... strange.
In my experience, the most common intuitive thinking people have can be summarized in "Well, ... |
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... |
Let $N\subset G$ be a nontrivial subgroup of finite group $G$ , is there a "retraction" onto $N$ , i.e. a homomorphism $\varphi : G \rightarrow N$ s.t. $ Im(\varphi) = N $ , $\varphi |_N = id_N $ . Its obviously yes when $N = G$ or $ N = \{e\}$ , if no other cases possible (I could not find them) how to proof ? What ab... |
Retraction onto subgroup? |
I have come across a lemma which states:
Let $\mu$ be a finite content on an algebra $\mathcal{A}$.
$\mu$ is a pre-measure if and only if $\mu$ is continuous on the empty set, i.e., for every sequence $(A_i)_{i \geq 1}$ from $\mathcal{A}$ with $A_i \downarrow \emptyset$, we have $\mu(A_i) \rightarrow 0$ as $i \r... |
We know that for derivability, continuity is a necessary condition and equality of left derivative and right derivative is a sufficient condition. Since equality of left derivative and right derivative is a sufficient condition, so if it is true, then it implies that the function is derivable. But I think that it is al... |
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