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The entire problem is like physics-math. Generally we have n cables whose signals are desynchronized and we have m receivers. We know what are waves of voltage at receivers and distances between receivers. We also know that the signals on cables are diffrent only be frequency, phase and amplitude. We want to know the p... |
Given that f: Z+ —> R, prove that f(2x)=2f(x) is injective.
I’m new to analysis and can’t understand how to approach this problem. Should I find the f(x) first? |
I'm trying to find a general unit formula for $\mathbb{Q}(\sqrt{2},\sqrt{3}) = \lbrace a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \ \vert \ a,b,c,d\in\mathbb{Q}\rbrace$ and then plug in my specific unit $1+\sqrt{2}+\sqrt{3}+\sqrt{6}$ to prove is it a unit:
$$a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} = (a+b\sqrt{2})+(c+d\sqrt{2})\sqrt{3}... |
In Donald Marshall's notes on complex analysis he defines a region, $\Omega$ as simply connected if $S^2 \setminus \pi(\Omega)$ is connected where $\pi$ is the steregraphic projection. This seems to include regions like $\Omega = \\{z : \Re{z} > 1\\}$ since $S^2 \setminus \pi(\Omega)$ is the just lower hemisphere which... |
Are these two different definitions of simply connected? |
In Donald Marshall's notes on complex analysis he defines a region, $\Omega$ as simply connected if $S^2 \setminus \pi(\Omega)$ is connected where $\pi$ is the steregraphic projection. This seems to include regions like $\Omega = \{z : \Re{z} > 1\}$ since $S^2 \setminus \pi(\Omega)$ is the just lower hemisphere which i... |
Consider $A ∈ \mathbb{R}^{m \times n}$ and $W ∈ \mathbb{R}^{m \times m}$ such that $n < m$ and $A^TA$ is invertible. If we know that $ \operatorname{rank}W = w \ge n $ what can we say about $\operatorname{rank}(A^TWA)$? My intuition is $ n + w - m \le \operatorname{rank}(A^TWA)$ but I can't prove it. I think using $ \o... |
If $A^TA$ is invertible and $W$ is square what is $\operatorname{rank}A^TWA$? |
The entire problem is like physics-math. Generally we have n cables whose signals are desynchronized and we have m receivers. We know what are waves of voltage at receivers and distances between receivers. We also know that the signals on cables are diffrent only be frequency, phase and amplitude. We want to know the p... |
In Donald Marshall's notes on complex analysis he defines a region, $\Omega$ as simply connected if $S^2 \setminus \pi(\Omega)$ is connected where $\pi$ is the steregraphic projection. This seems to include regions like $\Omega = \{z : \|z \| > 1\}$ since $S^2 \setminus \pi(\Omega)$ is the just lower hemisphere which i... |
In WolframAlpha, you can get the Hessian determinant for a multivariable function $f(x_1,x_2,\ldots)$ with the syntax
Hessian matrix (or determinant) f(x1, x2, ...)
Both matrix and determinant are included in either flavor of input. For your use case, one would resort to
Hessian matrix norm({x, y, z})
... |
If $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x) = 2f(x)$. Is it injective?
I’m new in analysis and can’t understand how to approach this problem. Should I find an analytic expression for $f(x)$ first? |
When $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x)=2f(x)$, then is it injective? |
I'm trying to find a general unit/inverse formula for $\mathbb{Q}(\sqrt{2},\sqrt{3}) = \lbrace a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \ \vert \ a,b,c,d\in\mathbb{Q}\rbrace$ and then plug in my specific unit $1+\sqrt{2}+\sqrt{3}+\sqrt{6}$ to prove is it a unit/inverse:
$$a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} = (a+b\sqrt{2})+(c+d\... |
Let $f:X \rightarrow S$ be a smooth proper family of K3 surfaces and let $X_t=f^{-1}(t)$. Let's suppose $S$ is a disk in $\mathbb{C}^n$.
I know that the Betti/Hodge numbers are constant, so the direct image $R^2f_*\mathbb{Z}$ is locally free and I can study it as a vector bundle.
1. What are the fibers of this vec... |
In WolframAlpha, you can get the Hessian determinant for a multivariable function $f(x_1,x_2,\ldots)$ with the syntax
Hessian matrix (or determinant) f(x1, x2, ...)
This usually also works if you have additional parameters, in which case one should specify which variables to use in the component derivatives, ... |
I'm having trouble understanding rencontres numbers.
The numerical values shown of the wiki page:
https://en.wikipedia.org/wiki/Rencontres_numbers
1. Looking at n = 3:
2. k = 3 = 1 I think I understand this because there is only one way to give all three items to the correct person
3. k = 2 = 0 because if... |
In WolframAlpha, you can get the Hessian determinant for a multivariable function $f(x_1,x_2,\ldots)$ with the syntax
Hessian matrix (or determinant) f(x1, x2, ...)
This usually also works if you have additional parameters, in which case one should specify which variables to use in the component derivatives, ... |
Each bunny is individually ready for the big mission with probability p. If I have n bunnies and need k for the mission, find the expected number of k-large squads I can form with bunnies ready for the mission.
I said that for binomial distribution we have P(X=x) = $$\binom{n}{x}$$$$p^{x}(1-p)^{n-x}$$.
So then E(... |
Each bunny is individually ready for the big mission with probability p. If I have n bunnies and need k for the mission, find the expected number of k-large squads I can form with bunnies ready for the mission.
I said that for binomial distribution we have $$P(X=x) = \binom{n}{x}p^{x}(1-p)^{n-x}$$
So then E(X) = ... |
I have encountered here that sometimes people write $dx/df$. For example, [here][1], the author of the answer writes down the chain rule as
$$\frac{dg}{df}=\frac{dg}{dx}\frac{dx}{df},$$
which seems to be an abuse of notation. And $dx/df$ is found to be just $\frac{1}{df/dx}$.
[Here][2], I learned that when someone... |
I would start by splitting the initial velocity vector into the horizontal and vertical directions. The initial velocity of $44$ ft/s has an angle of $45^\circ$, so...
- Initial horizontal velocity = $44*cos(45)\approx31.113$ ft/s
- Initial vertical velocity = $44*sin(45)\approx31.113$ ft/s
Now that we have th... |
Can I determine the angles of a quadrilateral if I know the lengths of the sides and the difference between the diagonals? |
I'm having trouble understanding rencontres numbers, $D_{n,k}$.
The numerical values shown of the wiki page:
https://en.wikipedia.org/wiki/Rencontres_numbers
**Looking at n = 3:**
1. $D_{3,3} = 1$ I think I understand this because there is only one way to give all three items to the correct person
3. $D... |
Since the domain of $f$ is $\Bbb{Z}^+$, defining the function $f$ means choosing real numbers to represent the values $f(1), f(2), f(3)$, etc. Try to start picking some values. What can you choose for $f(1)$? Maybe you set $f(1)=5$. What about $f(2)$? Well, according to your rule, $f(2) = f(2\cdot 1) = 2 \cdot f(1) = ... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Each bunny is individually ready for the big mission with probability p. If I have n bunnies and need k for the mission, find the expected number of k-large squads I can form with bunnies ready for the mission.
I said that for binomial distribution we have $$P(X=x) = \binom{n}{x}p^{x}(1-p)^{n-x}$$
So then $E(X) =... |
I am learning alone and I have just start studiying about variety and below a question. I want to be sure that my answer and so my understanding is correct.
------------------------
### Question:
Let $ \lambda \in \mathbb{R}$ and let $S = \left \{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2 = \lambda... |
Apologize for naive question, but I got mixed answers regarding this one.
The inverse of $f$ exists if and only if $f$ is bijective. Horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one).
However, I read many answers that to determine whether $f$ has an inverse we sho... |
Let $X_1,X_2,...$ be i.i.d. nas $a_n$ be a sequence of positive numbers with $a_n/n$ increasing. Why the inequality $\sum^n_{i=1} E(|X_i|; |X_i|<a_i) \leq a_N + E(|X|; a_N \leq |X| < a_n)$ holds for any fixed $N$? Here $E(X;C)=\int_C XdP$.
This is the very last part of the proof of theorem 2.5.13 from Durrett, Proba... |
Why does a *compact-open topology* define a *subbase* of *continuous functions*?
It is clear that in order for a set of such functions to set a *subbase*, 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 subbase of continuous functions? |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Consider the Abel equation of the second kind,
\begin{align}
yp-y=f(x);\quad [p=y'_x]
\end{align}
for some arbitray non-zero function $f$.
Suppose $y$ were an odd function, then $p$ is even and it follows that
\begin{align}
yp-y=-f(-x)=f(x).
\end{align}
So odd $y \implies$ odd $f$. Is the opposite true? Doe... |
How to calculate the distance from $X^2$ to $\operatorname{span}(1,X)$? |
Can one show that any cardinal has a successor without using ordinals? I mean: cardinals are not defined as special ordinals, but have a naive definition, as equivalence classes of equipotent sets, and ordinals are not used in the proof. If you don't like that, this is another formulation: how does one prove the theore... |
From wolfram mathematica I got this result.
$\sum\limits_{i=0}^{k}\frac{k!x^i}{i!(k-i)!^2}$
This is $\frac{e^{i\pi k}x^k U(-k,1,-1/x)}{k!}$
Where U(a,b,z) is the confluent hypergeometric function of the second kind. I tried to plot it in Mathematica but it doesn't show anything. How can I plot this for very b... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
>Andrea has $x$ euros, Marco has $y$ euros more than Andrea but $z$ euros less than Luke. Determine the formula that expresses the total sum $s$ owned by the three friends.
Since Andrea has $x$ euros, Marco has $y$ euros more than Andrea and Luca has $z$ euros less than Marco, we can write the following equations:... |
In the context of math classes where you're told to use the horizontal line test, you're working with 2-dimensional graphs in the plane and "has an inverse" typically means you can define a function $f^{-1}$ which maps points from the image of $f$ to the domain of $f$. This is good enough for those types of classes bec... |
It is true that $$\text{int}\Bigl(\bigcap_{i\in I}A_i\Bigr)\subset \bigcap_{i\in I}\text{int}(A_i),$$ and the inclusion can be proper if the index set is infinite. Keeping our attention only on metric spaces, which are Hausdorff, if metric space $(X,d)$ has a non-isolated point $x$, the sets $A_i=\{y\in X:d(x,y)<1/i\}$... |
***Question:***
Let $E = \Bbb{R}_2[X]$ an euclidian space with a dot product $\left\langle P,Q\right\rangle \ = \int^1_0 {P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $\operatorname{span}(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector. We have:
\begin{equatio... |
Let $\mathcal{C}$ be an $\infty$-category, and let $X, Y$ be a pair of its objects. It is said that $\text{Mor}_{\mathcal{C}}(X,Y)$ has again the structure of an $\infty$-category.
What is this structure, how is it defined?
I guess the definition I am familiar with is saying that an $\infty$-category is a simplic... |
I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes:
> If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k \leq n$ are non-negative integers, then prove the following:
$$ P(N(s) = k | N(t) = n) = \binom{n}{k} \Big{(} \frac{s}... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
I have been studying a course on Stochastic Processes and recently encountered the following result with the proof listed as an exercise.
> Let $X_t$ be a Compound Poisson Process defined by the sum of iid random variables $(Y_n)$ independent of $N$, written explicitly as:
$$ X_t := Y_1 + Y_2 + \cdots + Y_{N_t}$... |
A scatter diagram of MSAT and VSAT scores of a large group of students is football shaped (homoscedastic). The correlation is positive but not 1.
If a student is at the nth percentile (say, 70th) on the VSAT and on the mth percentile (say, 60th) on the MSAT, can we say whether the student's MSAT score is greater tha... |
If x is at the 70th percentile, can we say what percentile we expect y to be compared to other values with x at the 70th percentile? |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Problem: A set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, . . . , a_n)$, where each element of the set appears exactly once and $gcd(a_i, a_{i+1}) = 1$ for $i = 1, . . . , n$ where $a_{n+1} = a_1$. For example, $(7, 8, 9, 10, 11, 14, 13, 12)$ is a coprime cycle for... |
May someone help me understand the following inductive definition of *concatenation of strings*:
We will define $\alpha\cdot\beta$ by induction on $|\beta|.$
- If $|\beta|=0\Rightarrow \beta=\epsilon,$ so then $\alpha\cdot\beta=\alpha\cdot\epsilon=\alpha$
- If $|\beta|=n+1,$ then $\beta=\gamma b,|\gamma|=n.$ T... |
The phrase "prebase" it a bit unusual, most frequently one uses the word *[subbase](https://en.wikipedia.org/wiki/Subbase)*.
Given a set $X$ and a set $\Sigma$ of subsets of $X$, there is a coarsest topology $\mathcal T(\Sigma)$ containing $\Sigma$. It is called the *topology generated by $\Sigma$*. Explicitly it ca... |
Yes, you can extend Euclidean 3-dimensional space to projective 3-space by adding a plane at infinity. A projective plane at infinity, to be precise.
The conflict you observed is due to an incorrect assumption of transitivity. Let me explain in more detail, and do so in the projective plane because that's enough to ... |
Assume that $W$ is a $n \times n$ matrix with elements $w_{ij}$, and that $\mathbf{b} \in \mathbb{R}^n$ is a column vector with elements $b_{i}$. We know that the Jacobian matrix to the affine transformation $\mathbf{F}(\mathbf{x}) = W\mathbf{x} + \mathbf{b}$ is $\mathbf{F}'(\mathbf{x}) = W$. We are now looking at the ... |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? *i.e.* when
$$
a_n= \sum_{i=0}^{n-1} f_i(n) a_{i} + g(n)
$$
where
$$
f_i(n)=\frac{P_i(n)}{Q_i(n)} r^n
$$
for $P_i, Q$ polynomials and $r \in \mathbb{C}$ and al... |
[Use](https://www.wolframalpha.com/input?i=lim%5Bsum%5B%28%CF%80%2F2%29%5En%2Fn%21+D%5B%281%2Bsin%28x%29%2Fx%29%5E-n%2C%7Bx%2Cn-1%7D%5D%2C%7Bn%2C1%2C11%7D%5D%2Cx-%3E0%5D)
> Kaplan, S. R. : The Dottie Number. Math. Mag. 80, 73-74,2007.
$$\theta=\sum_{n=0}^\infty\frac{\left(\frac\pi2\right)^{2n+1}}{(2n+1)!}\left.\f... |
This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $E$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramif... |
Yes, you can extend Euclidean 3-dimensional space to projective 3-space by adding a plane at infinity. A projective plane at infinity, to be precise.
The conflict you observed is due to an incorrect assumption of transitivity. Let me explain in more detail, and do so in the projective plane because that's enough to ... |
I have a task:
The annual rate of return in year $t$ , i.e., $(1 + i_{t})$ where $i_{0}$ represents the interest rate from $t=0$ to $t=1$, follows a log-normal distribution with an expected value of $108%$ and a standard deviation of $20%$ . The rates of return in subsequent years are independent of each other. What... |
$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. Moreover, let $Q = [0,1]^2$. Specifically, $f:Q \to \mathbf{R}$ is real-analytic and non-negative. Moreover, $f$ is bounded o... |
The following is problem 13 [here][1].
>Consider functions in $\mathcal{C}^2[0,1]$ and $a,b>0$. In this case, if we define:
>$$\lVert f \rVert:=\lVert f \rVert_\infty+ a \lVert f' \rVert_\infty +b \lVert f'' \rVert$$
>This norm makes $\mathcal{C}^2[0,1]$ into a Banach algebra if and only if $a^2\geq 2 b$
I h... |
To get this off the unanswered list and turn comments into an answer:
You seem to be using the $10$-adic numbers $\mathbb Z_{10} = \projlim \mathbb Z / \mathbb 10^k$.
It is correct that in those, the sequence $a_n := 5^{(2^n)}$ converges to an element $a \in \mathbb Z_{10}$ that, in $10$-adic a.k.a. decimal expan... |
I am learning alone and I have just start studying about manifolds and below a question. I want to be sure that my answer and so my understanding is correct.
------------------------
### Question:
Let $ \lambda \in \mathbb{R}$ and let $S = \left \{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2 = \lambd... |
Let $u$ be a smooth compactly supported function on $\mathbb{C}$. The $\overline{\partial}$-Poincaré lemma is the formula
$$\frac{{\partial}}{{\partial} \overline{z}}\frac{1}{\pi} \int_{\mathbb{C}} \frac{u(w)}{w - z} dx \wedge dy = u(z)$$
where $dx \wedge dy$ is the standard Lebesgue measure on $\mathbb{C}$ ($w = x +... |
Failure of translation-invariance of an integral? |
To get this off the unanswered list and turn comments into an answer:
You seem to be using the $10$-adic numbers $\mathbb Z_{10} = \projlim \mathbb Z / \mathbb 10^k$.
It is correct that in those, the sequence $a_n := 5^{(2^n)}$ converges to an element $a \in \mathbb Z_{10}$ that, in $10$-adic a.k.a. decimal expan... |
This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramif... |
I have a task:
The annual rate of return in year $t$ , i.e., $(1 + i_{t})$ where $i_{0}$ represents the interest rate from $t=0$ to $t=1$, follows a log-normal distribution with an expected value of $108\%$ and a standard deviation of $20\%$ . The rates of return in subsequent years are independent of each other. Wh... |
In the passage by Alfhor's Complex Analysis, under Chapter 3 Compactness, states:
[![Alfhors][1]][1]
I am new to topology, but here's my understanding: If a neighborhood isn't "intersecting" or doesn't have a point belonging to $X$ we can ignore it, and if it does, we replace that neighborhood with $2\epsilon$-nei... |
In the passage by Alfhor's Complex Analysis, under Chapter 3 Compactness, states:
[![Alfhors][1]][1]
I am new to topology, but here's my understanding: If a neighborhood isn't "intersecting" or doesn't have a point belonging to $X$ we can ignore it, and if it does, we replace that neighborhood with $2\epsilon$-nei... |
I was wondering if there was a general way to find (or show that the sequence diverges)
$$ \lim_{n \to \infty} \{a_n\}$$
Where $a_n$ is of the form $\frac{n!}{c^n}$, $c$ being a non-zero natural number.
I think that I was able to find an intuitive answer following the reasoning below. However, I would like to know... |
Let $X_1,X_2,...$ be i.i.d. and $a_n$ be a sequence of positive numbers with $a_n/n$ increasing. Why the inequality $\sum^n_{i=1} E(|X_i|; |X_i|<a_i) \leq a_N + E(|X|; a_N \leq |X| < a_n)$ holds for any fixed $N$? Here $E(X;C)=\int_C XdP$.
This is the very last part of the proof of theorem 2.5.13 from Durrett, Proba... |
How is the concept of symmetry being used in this question - Equal Flip Timer? |
Question:
Brad and Chad both flip coins with probability $p$ of heads on each flip. Let $B$ be the event that it takes strictly fewer flips for Brad to get his first heads than Chad, $C$ be the event that it takes Chad strictly fewer flips than Brad to get his first heads than Brad, and $E$ be the event that it tak... |
Question:
Brad and Chad both flip coins with probability $p$ of heads on each flip. Let $B$ be the event that it takes strictly fewer flips for Brad to get his first heads than Chad, $C$ be the event that it takes Chad strictly fewer flips than Brad to get his first heads than Brad, and $E$ be the event that it tak... |
Suppose that $p:E \to B$ is a weak fibration (i.e. that (B,p,E,F) is a fiber bundle) with a section $s:B \to E$ ($p \circ s=id_B$).
I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,s(b_0)) \xrightarrow {p_*} \pi_n(B,b_0) \to 0$ is exact for every $n \geq1$.
Only thing that I don't... |
Suppose that $p:E \to B$ is a weak fibration (i.e. that (B,p,E,F) is a fiber bundle) with a section $s:B \to E$ ($p \circ s=id_B$).
I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,s(b_0)) \xrightarrow {p_*} \pi_n(B,b_0) \to 0$ is exact for every $n \geq1$.
Only thing that I don't... |
> Let $X$ be a set and $\left(X, \mathcal{T}_{\mathrm{f}}\right)$ be the topological space with the finite complement topology $\mathcal{T}_{\mathrm{f}}$.
>Show: For any subset $Y \subseteq X$, the subspace topology on $Y$ inherited from $\left(X, \mathcal{T}_{\mathrm{f}}\right)$ is the same as the finite complement t... |
Suppose that $p:E \to B$ is a weak fibration (i.e. that (B,p,E,F) is a fiber bundle) with a section $s:B \to E$ ($p \circ s=id_B$). Also here $F=p^{-1}(b_0)$.
I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,s(b_0)) \xrightarrow {p_*} \pi_n(B,b_0) \to 0$ is exact for every $n \geq1$.... |
I am reading a Teichmueller theory book and trying to understand elliptic curves as examples of Riemann surfaces.
Consider the elliptic curve
\\[X = \\{[z : w : y] \in \mathbb C \mathrm P^2 \mid w^2y = z (z - y) (z - \lambda y)\\} \\]
where $\lambda \in \mathbb C$ is a complex number not equal to $0$ or $1$.
I am t... |
$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. Moreover, let $Q = [0,1]^2$. Specifically, $f:Q \to \mathbf{R}$ is real-analytic and non-negative. Moreover, $f$ is bounded o... |
Let $A$ be a $\sigma$-weakly closed $*$-subalgebra of the $W^*$-algebra $\prod_{i\in I}^{\ell^\infty} M_{n_i}(\mathbb{C})$. I believe that we must have $A\cong \prod_{j\in J}^{\ell^\infty} M_{m_j}(\mathbb{C})$ for certain $(m_j)_{j\in J}$.
Does anyone have an idea how we can show this? |
Let $\Omega$ be an open subset of $\mathbb{R}^n$.
If $\mu$ is a Radon measure on $\mathbb{R}^n$, then we denote with $T_\mu$ the distribution defined as follow:
$$<T_\mu, \phi>=\int_\Omega \phi d\mu \quad \forall \phi \in C^\infty_c(\Omega).$$
In the same way, if $f\in L^1_{loc}(\Omega)$, then we denote with $T_f$... |
Suppose that $p:E \to B$ is a weak fibration with a section $s:B \to E$ ($p \circ s=id_B$). Also here $F=p^{-1}(b_0)$.
I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,s(b_0)) \xrightarrow {p_*} \pi_n(B,b_0) \to 0$ is exact for every $n \geq1$.
Only thing that I don't seem to get ... |
I've been trying to develop an intuition for stochastic processes and would like some help clearing up the following situation.
Suppose $\{X_t\}_{t \mathbb{R}}$ is a real valued stochastic process defined on the probability space $(\Omega, \mathcal{F}, P)$. Suppose all the $X_t$ are identically distributed for each... |
Are paths in a stochastic process $\{X_t\}_{t \mathbb{R}}$ constant if $X_t$ is identically distributed for all $t$? |
Let $\mathcal{B}$ be the Borel $\sigma$-field on $\mathbb{R}$. We denote the reflection of a set B to be $B^-:=\{x\in \mathbb{R}: -x\in B\}$. Let $\mathcal{B}^s$ be the collection of symmetric Borel sets, i.e., $ \mathcal{B}^s:=\{B\cup B^-: B\in \mathcal{B}\}$.
Show that $\mathcal{B}^s$ is the smallest $\sigma$-fie... |
I am interested in functions $\phi:\mathbb R^d\to\mathbb R^d$ with the following property. For all $x,y\in\mathbb R^d, (\phi(x)-\phi(y))^\top (x-y)\geq 0$.
For instance, if $\phi$ is the gradient (or a subgradients) of a convex function, it satisfies this property, but this property is more general.
Examples of s... |
As we can see we want to change the order of integration of the integral:\
\
$$
I\ = \int \int \int \ f(x,y,z) \ dy\ dx\ dz
$$
\
We want to change it into the next order:\
\
$$
I\ = \int \int \int \ f(x,y,z) \ dz\ dy\ dx
$$
\
I will use algebra to write iteration of the integral with the changed integration... |
$ \int_0^1 \ln\left(\frac{1}{x}\right)^n $ dx
i am able to understand that Integral needs to be broken in two parts so that convergence can be checked at x=1 and x=0 separately.
for the first part if n<0 then its proper integral
similarly for the second half n>0 then its proper integral
i need to check for t... |
Let $X_1,X_2,...$ be i.i.d. with $E|X_i|=\infty$ and $a_n$ be a sequence of positive numbers with $a_n/n$ increasing. Why the inequality $\sum^n_{i=1} E(|X_i|; |X_i|<a_i) \leq a_N + E(|X|; a_N \leq |X| < a_n)$ holds for any fixed $N$? Here $E(X;C)=\int_C XdP$.
This is the very last part of the proof of theorem 2.5.1... |
Suppose $D_1$ and $D_2$ are two n-dimensional closed disks (topologically embedded) in the sphere $S^n$ ($n>=1$). Is there a homeomorphism $f:S^n \rightarrow S^n$ such that $f(D_1)=D_2$? Maybe I should ask whether all the embeddings of $D^n$ in $S^n$ are ambient isotopic?
When $n=1$, a disk is just a closed connect... |
Do homeomorphisms of disks in a sphere extend? |
$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. Moreover, let $Q = [0,1]^2$. Specifically, $f:Q \to \mathbf{R}$ is real-analytic and non-negative. Moreover, $f$ is bounded o... |
Let $A$ be an $n \times n$ matrix that is (weakly) [diagonally dominant][1], i.e., $|A_{ii}| \ge \sum_{j\ne i}|A_{ij}|$ holds for all $i \in [n]$. Furthermore, suppose that every row of $A$ is non-zero. What is the lowest possible value of $\mathrm{rank}(A)$ as a function of $n$?
It is easy to see that the answer is... |
Minimum rank of a (weakly) diagonally dominant matrix with non-zero rows? |
The title pretty much sums it up. For a 2-manifold with metric $\gamma_{AB}$, a vector field $Y^A$ is said to be a conformal Killing vector field if $\nabla_A Y_B + \nabla_B Y_A = \nabla_C Y^C \gamma_{AB}$. In the case of a two-sphere with the round metric, this means $\partial_{\bar{z}} Y^z = \partial_z Y^{\bar{z}} = ... |
Why are there only six global conformal Killing vector fields on the two-sphere with the round metric? |
We can show that the dual bundle $E^*$ with fibers $(E^*)_x = (E_x)^*$ for all $x \in B$ have transition functions $g_{\alpha\beta} = (f_{\alpha\beta})^{-1} : U_\alpha \cap U_\beta \to GL(r,\mathbb R)$.
However, I can't seem to show that the $g_{\alpha\beta}$ satisfy the cocycle conditions, that is $g_{\alpha\alpha}... |
My question is about the integration technique - trigonometric substitution $x = \sec{\theta},$ with $x \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2},\pi)$, in accordance with how the inverse secant function is usually defined.
If $0<\theta<\frac{\pi}{2}$, then $\sqrt{x^2-1} = \tan{\theta}$, and in the last step where ... |
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