instruction stringlengths 12 30k |
|---|
My apologies if this is a silly question. I feel like I'm misunderstanding something basic and just need a quick clarification.
I have an eigenvalue problem stated in terms of the inner product. More specifically, I have an operator $O: X \to Y$, where $X$ and $Y$ are Hilbert spaces. I would like to find the eigenv... |
How to compute eigen values and eigen vectors for eigen value problem in terms of inner product? |
Let $R$ be a ring with $1$ and $M$ be a right $R$-module. A homomorphism $f:P\to M$ is said to be a *projective cover* of $M$ if $P$ is projective and $f$ is an epimorphism and $\ker f \ll P$. Recall that $N$ is a *small* submodule of $K$ (written $N \ll K$) if $K=N+L \Longrightarrow K=L$.
I need to prove that the ... |
Suppose $A$, $B$, $C$, and $D$ are compact and injective linear operators in $L_2[0,1]$ and that $AB^{*}=CD^{*}$. Is it possible that both $D^{-1}B$ and $C^{-1}A$ are unbounded? |
Can both of these linear operators be unbounded? |
Below is a problem I did. I believe my answer is right but I would like somebody to check it for me.
> **Problem:** Show that the Black-Scholes formula for call and put options satisfy put-call parity equation. The put-call parity equation is:
$$ c + Ke^{-rT} = p + S_0 $$
We have:
\begin{align*}
c &= S_0 N(d... |
How can I visualise the meaning of isotopy that appears while defining _Teichmuller_ space?
Can you suggest a picture where two maps are not isotopic? I want more clarification about isotopyic maps. Isotopy is sometimes referred to as homotopy via homeomorphism. I need geometric visualisation. |
The [Iwasawa decomposition][1] of a real symplectic matrix $S \in \text{Sp}(2n,\mathbb R)$ gives a convenient decomposition $S=KAN$ where
$$K\in \text{Sp}(2n,\mathbb R) \cup \text{SO}(2n,\mathbb R)\\
A \in \mathcal{A}=\{D(\vec{\kappa}): \vec{\kappa}=(\kappa_{1},\kappa_{2}\dots,\kappa_{1}^{-1},\dots,\kappa^{-1}_{n})\... |
$\newcommand{\spec}{\operatorname{Spec}}\newcommand{\O}{\mathcal{O}}$Liu claims the following exercise:
> Let $S$ be a locally Noetherian scheme. Let $X,Y$ be $S$-schemes of finite type and $s\in S$.
>
> Then all morphisms of $S$-schemes $\phi:X\times_S\spec\O_{S,s}\to Y\times_S\spec\O_{S,s}$ arise as the base cha... |
Why can we suppose a scheme is locally local affine, as an improvement to locally affine? |
**Question:** Let $\tau_i\sim \text{Exp}(\lambda_0)$ iid and $\gamma_i\sim \text{Exp}(\lambda_1)$ iid and independent of each other and set $N_t=max\{k\geq 0: \sum_{i=1}^k{(\tau_i+\gamma_i)}\leq t\}$. What is $\mathbb{E}(N_t)$?
**My attempt:**(inspired by section 3 of [this][1] paper) Let $(\sigma_t)_{t\geq 0}$ be a... |
Can anyone explain this statement?
"Firstly, let’s define the input space as
X (sensory observations) and the label space as Y (semantic categories). The data distribution is represented
by the joint distribution P(X, Y ) over the space X ×Y.
Distribution shift can occur in either the marginal distribution P(X), ... |
Why does a shift in the probability distribution over a label space naturally trigger a shift in the distribution over input space? |
The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line.
In space (3 dimensional solid geometry), can a similar augmentation be made? Can we augment space, adding a plane at infinity, such that parallel planes ... |
Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity? |
My understanding of the definition of a statistical model is a probability space $(\Omega,\Sigma,p)$, a random variable $X:\Omega\to\mathbb{R}$, and a collection of probability measures $\mu_{\theta}$ on $\mathbb{R}$, where $\theta$ is a parameter.
Aren't the $\mu_{\theta}$ supposed to be related to the random varia... |
The excercise wants us to show that for CW complexes $X$ and $Y$ with $0$-cells $x_0$ and $y_0$, we have a homeomorphism between the two spaces
$$X*Y/(X*\{y_0\} \cup \{x_0\}*Y) \cong S(X \wedge Y)/S(\{x_0\} \wedge \{y_0\}).$$
I was reading the proof in this post [here][1] as well as the [solution manual][2]. Both beg... |
My apologies if this is a silly question. I feel like I'm misunderstanding something basic and just need a quick clarification.
I have an eigenvalue problem stated in terms of the inner product. More specifically, I have an operator $O: X \to Y$, where $X$ and $Y$ are Hilbert spaces. I would like to find the eigenv... |
If $X_1,...,X_n$ is a random sample on a probability space $(\Omega, p)$, show that $\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator for $E(X_1)$.
It seems an obvious computation, \begin{align*}
E(\frac{1}{n}\sum_{i=1}^nX_i)&=\frac{1}{n}\sum_{i=1}^nE(X_i)=\frac{1}{n}\cdot n E(X_1)=E(X_1)
\end{align*}
Since ... |
I will follow **Nagell**'s algorithm, as described for instance in the excellent Handbook of Ian Connell, §1.4, Cubic to Weierstraß, page 115 and following, a link found quickly is
[Ian Connell, Elliptic Curve Handbook, 1999][1]
----------
$(0)$
We start with the given curve, and let us set $Z=1$ (and pass ... |
Let $H_A, H_B, H_C$ be arbitrary Hilbert spaces.
Let $ρ_{AB} ∈ D(H_A \otimes H_B)$ such that $ρ_{A}$ is pure. Prove that $ρ_{AB} = ρ_{A} \otimes ρ_{B}$
( Hint: One way could be to prove it before for when $ρ_{AB}$ is pure and then use a purification to reduce to this case)
I haven't been able to go very far with t... |
Which pairs can go on infinitely? |
Here: https://en.wikipedia.org/wiki/Imaginary_hyperelliptic_curve#The_divisor_and_the_Jacobian it says:
"*Actually, Bézout's theorem states that a straight line and a hyperelliptic curve of genus $2$ intersect in $5$ points. So, a straight line through two point lying on
$C$ does not have a unique third intersect... |
In (first-order) logic, I understand that there are two notions of the truth of a sentence $\phi$ in a theory $T$:
1. Syntactic truth: $T\vdash \phi$ if $\phi$ is provable from $T$,
2. Semantic truth: $T\models \phi$ if $\phi$ is true in every model $M$ of $T$.
Naturally, both the notion that "$\phi$ is provable... |
Can you tell me if my solution is so far correct and help me complete the second case?
**The statement of the problem :** Find all functions $f: \mathbb R \rightarrow \mathbb R$ with the following propriety :
$$f(xf(y) + f(x+y)) = y(f(x)+1) + f(x) , \forall x,y \mathbb R$$
**My approach** : For $x = y = 0$ we g... |
> Henry recorded the number of students present everyday for a total of $40$ days. Also there are a total of $29$ students. Here is the frequency table:
![enter image description here][1]
Also it is given that,on exactly half of the days, No more than one student was absent. Find the values of $x,y$.
My try: O... |
I'm reading the proof of the cone theorem from Koll\'ar-Mori and I find step 3 to be very unclear. In particular below is a simple proof of this step which must be wrong somehow. I am wondering where I went wrong.
The setup for this step is that we have the closed cone of curves $\overline{NE}$ and we define a close... |
This is quite a hacky solution, but you can nest "`Limit`", `Sum`, and `SeriesCoefficient` to produce what you want, although WA is very particular about the syntax ([link](https://www.wolframalpha.com/input?i=Sum%5BSeriesCoefficient%5BSqrt%5B100%2Bx%5D%2C+%7Bx%2C+0%2C+n%7D%5D+*+x%5En%2C+%7Bn%2C+0%2C+2%7D%5D+as+x-%3E2.... |
I have a question regarding the measurability of a function in the following paper:
https://statistics.berkeley.edu/sites/default/files/tech-reports/638.pdf
Let $\phi$ be a surrogate loss function and $(X,Y)$ a joint random variable where $X\in \mathcal{X}$ and $Y\in\{-1,+1\}$. Define the risk for a measurable f... |
The excercise wants us to show that for CW complexes $X$ and $Y$ with $0$-cells $x_0$ and $y_0$, we have a homeomorphism between the two spaces
$$X*Y/(X*\{y_0\} \cup \{x_0\}*Y) \cong S(X \wedge Y)/S(\{x_0\} \wedge \{y_0\}).$$
I was reading the proof in this post [here][1] as well as the [solution found here][2]. Both... |
The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line.
In space (3 dimensional solid geometry), can a similar augmentation be made? Can we augment space, adding a plane at infinity, such that parallel planes ... |
I'm reading the proof of the cone theorem from Koll\'ar-Mori and I find step 3 to be very unclear. In particular below is a simple proof of this step which must be wrong somehow. I am wondering where I went wrong.
The setup for this step is that we have the closed cone of curves $\overline{NE}$ and we define a close... |
given that $a,b,c,d \in \mathbb{R}$ and $u\in\mathbb{R}^4$, how can I find $a,b,c,d$ such that for a fixed $x\in \mathbb{R}$
$$e_h:=\Bigg |\frac{au(x)+bu(x+h)+cu(x+2h)+du(x+3h)}{h^2}-u''(x)\Bigg |=O(h^2)$$
I've tried to use the MVT and talyor expansion of $u(x+h),u(x+2h), u(x+3h)$, but I don't know how I am sup... |
The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line.
In space (3 dimensional solid geometry), can a similar augmentation be made? Can we augment space, adding a plane at infinity, such that parallel planes ... |
I have found this tough combinatorial geometry question in some Soviet math competition circa $1972$. It looks very simple (and rather natural), yet I cannot find the proof for quite some time now.
*$2n$ points are given on the plane. Prove that there exist $n$ non-overlapping circles, each containing exactly two gi... |
I was trying to check whether $ arc sin (sin \theta) = \theta$ and $arc cos(cos \theta) = \theta $ is satisfied when I compose Taylor series expansion of these functions, i.e.:
1. For sine:
$$ x = sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - ... $$
$$ arc sin x = x + \frac{1}{2 \cdot 3} x^3 ... |
given that $a,b,c,d \in \mathbb{R}$ and $u\in\mathbb{R}^4$, how can I find $a,b,c,d$ such that for a fixed $x\in \mathbb{R}$
$$e_h:=\Bigg |\frac{au(x)+bu(x+h)+cu(x+2h)+du(x+3h)}{h^2}-u''(x)\Bigg |=O(h^2)$$
I've tried to use the MVT and talyor expansion of $u(x+h),u(x+2h), u(x+3h)$, but I don't know how I am sup... |
I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, the last method is explained at this page. https://www.dtubbenhauer.com/slides/geotop2022/4-knot-determinant.pdf
My... |
Can you tell me if my solution is so far correct and help me complete the second case?
**The statement of the problem :**
Find all functions $f: \mathbb R \to \mathbb R$ with the following property:
$$
f(xf(y) + f(x+y)) = y(f(x)+1) + f(x) , \forall x,y \in \mathbb R.
$$
**My approach** :
For $x = y = ... |
Let $U$ be a proper subspace of a finite-dimensional vector space $V$ . Find a basis for $V$ containing no element of $U$.
We have no answer key for this question which i find annoying since that is the case for many questions in uni.
But if we look at a $R^2$ with basis vectors {$e_1, e_2$} and for example $U$ ... |
This is quite a hacky solution, but you can nest "`Limit`", `Sum`, and `SeriesCoefficient` to produce what you want, although WA is very particular about the syntax ([link](https://www.wolframalpha.com/input?i=Sum%5BSeriesCoefficient%5BSqrt%5B100%2Bx%5D%2C+%7Bx%2C+0%2C+n%7D%5D+*+x%5En%2C+%7Bn%2C+0%2C+2%7D%5D+as+x-%3E2.... |
I am learning about gaussian integers and I have a few questions about the following argumentation.
> What are the prime elements in $\mathbb{Z}[i]$? We remember that only
> the units $+1,-1,+i,-i$ in $\mathbb{Z}[i]$ have Norm equal to one. We
> will find the prime elements by using the norm. Let $\pi \in
> \math... |
I am learning about gaussian integers and I have a few questions about the following argumentation.
> What are the prime elements in $\mathbb{Z}[i]$? We remember that only
> the units $+1,-1,+i,-i$ in $\mathbb{Z}[i]$ have Norm equal to one. We
> will find the prime elements by using the norm. Let $\pi \in \mathbb{... |
What you "saw somewhere" is of course incorrect, as $v_{k+1},\ldots,v_n$ cannot be a basis for $V$ if you had a basis for $V$ with $n$ elements; this one just has $n-k$ elements.
Instead, let us consider the following: if $v_1,\ldots,v_n$ form a basis for $V$, and $1\lt i\leq n$, then
$$v_1+v_i, v_2+v_i,\ldots, v_{... |
I have been working on the following question:
Find a recurrence relation for the number of regions created by 'n' mutually intersecting circles on a piece of paper (no three circles have a common intersecting point).Then, show using the generating function that a_n = n^2 - n +2 for n greater of equal to 1.
I hav... |
Say you have a triangle with the following verticies on a 2D plane:
Tri = [ (0,0), (0,1), (0.5, 0.5) ]
You wish to move V3 from (0.5,0.5) to (2,2), but you can not directly interact with this point.
Is it possible to apply a scaling transformation to the triangle such that V3 will move upwards, but V1 and V2 r... |
Is it possible to translate points on a shape using a scaling transformation? |
The intersection of event has 2 formula
P(A n B) = P(A)P(B) for dependent event
P(A n B) = P(A)P(B | A) for independent event
There are several things that make me confused. My teacher said intersection events are events that can occur simultaneously and with intersection probability we can calculate the proba... |
I have been working on the following question:
Find a recurrence relation for the number of regions created by $n$ mutually intersecting circles on a piece of paper (no three circles have a common intersecting point). Then, show using the generating function that
$$a_n = n^2 - n +2$$
for $n>=1$.
I have found ... |
I'm teaching a basic measure theory course using Royden's book, and told my students that there does not exist a set function $M : \mathcal{P}(\mathbb{R}) \rightarrow [0, \infty]$ such that
$1) M(I) = \ell(I)$ where $I$ is an interval and $\ell(I)$ is the length of $I$
$2) M\left(\bigcup_{k = 1}^\infty E_... |
Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol.
$$\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... |
Why do so many solutions to $n+1\mid3^n+1$ satisfy $n\equiv27\pmod{72}$? |
Can a oblique antiprism be constructed?
Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane.
I'm baffled, though, because Google shows countless models and pictures of (right) antiprisms, but [not a single oblique antiprism](https:... |
If I want to generate two independent uniform random variables $Y_1,Y_2$ from a single uniform random variable $X$, I know I can do it by converting $X$ into its binary representation, then taking its odd decimal digits as $Y_1$ and even decimal digits as $Y_2$ based on [this answer][1].
However, can I do it without... |
Generate two independent uniform variables from one independent uniform variable not using binary representation? |
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.
$$\begin{array}{r}\dagger\,\color{violet}0,&\bullet\,\color{brown}1,&*\,\color{red}3,&27,&531,\\
1\,035,&4\,635,&6\,363,&11\,475,&19\,683,\\
4\,131,&80\,955,&*\,\color{red}{266\,4... |
We say that an extension of rings $R \to S$ (the map is a ring homomorphism and so takes $1 \to 1$) is finitely-generated as an $R$-algebra if the image of $R$ is contained in the center of $S$ and every element of $S$ can be written as a polynomial with coefficients in the image of $R$ whose variables are from a finit... |
There are several things that make me confused. My teacher said intersection events are events that can occur simultaneously and with intersection probability we can calculate the probability of events a and b occurring simultaneously.
But what is meant by happening simultaneously ?, is that something like What is t... |
You posted this exact same question yesterday and I gave you the answer in a comment. Perhaps you didn't see it, so I'm making it an actual answer: by using the term "simultaneously", your teacher has misled you into thinking of the intersection of events in a temporal manner, but this need not be the case. Nor does it... |
I would like to decompose an $m\times n$ matrix $A$ into two matrices $U\in\mathbb{R}^{m\times n}$ and $V\in\mathbb{R}^{n\times n}$ such that $UV=A$, and the $m$ rows of $U$ each have unit magnitude. In other words, I want the rows of $U$ to lie on the unit (hyper-)sphere. Is such a decomposition possible in general? I... |
Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence:
$$ D(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$
What can I say about the difference
$$ \Big|\int f\,dp_n-\int f\,d p\Big| $$
when $f:\mathbb R^d\to\mathbb R$ is continuous, $... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
The probability of the intersection of events $A$ and $B$ has 2 formulas:
$P(A \cap B) = P(A)P(B)$ for independent events,
$P(A \cap B) = P(A)P(B \vert A)$ for dependent events.
There are several things that make me confused. My teacher said intersection events are events that can occur simultaneously and with... |
I would want to see a strengthening or a disproof of a result from an exercise from Rudin's *Real and Complex Analysis*, that is asked before at here https://math.stackexchange.com/questions/961745/construct-a-borel-set-on-r-such-that-it-intersect-every-open-interval-with-non-z and here https://math.stackexchange.com/q... |
Does there exist a subset $S$ in $[0,1]$ such that for all interval, $S$ and its complement in the interval has the same Lebesgue masure? |
Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol.
$$\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... |
The short answer: The "curved line" is not a straight line.
The longer answer: It may be useful to have an explicit situation, where we add reduced divisors. Let us consider the [Example][1] from the linked [wiki page][2].
$$
(C)\ :\qquad
y^2 =x(x+1)(x+3)(x-3)(x-5)\ .
$$
Consider the points $P,Q,R,S$, and the r... |
I would want to see a strengthening or a disproof of a result from an exercise from Rudin's *Real and Complex Analysis*, that is asked before at here https://math.stackexchange.com/questions/961745/construct-a-borel-set-on-r-such-that-it-intersect-every-open-interval-with-non-z and here https://math.stackexchange.com/q... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
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))... |
Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but could not find the proof of this lemma. Also tried to generalize the Zig-zag Lemma for homology groups to $R$-modules doing ... |
It is well-known that the following is one of the equivalent definitions of "jointly normal":
**Definition 1.** Let $(X_1,X_2)$ be a random vector. Then $X_1$ and $X_2$ are said to be *jointly normal* if and only if there exist real numbers $\mu_1, \mu_2, \alpha_1, \ldots, \alpha_k, \beta_1,\ldots, \beta_k$ and ind... |
I want to find a counterexample to verify that the following is false $$P(A \cap B | C) = P(A |C)P(A \cap C)$$
So I thought I would take $A=C$ and then
$$P(A \cap B | C)=\frac{P(C \cap B)}{P(C)}$$
and if for example $C \cap B =\emptyset$ with $C \neq \emptyset$ then
$$P(A \cap B | C)=0$$ but
$$P(A |C)P(A... |
**(Please, *read the question properly* before answering.)**
I have a question regarding the definition of 'Markov process' given by Shreeve, book II (continuous models).
It says (see below) this: "Assume that for all $0 \leq s \leq t \leq T$ and for every nonnegative Borel-measurable function $f$, there is anot... |
I want to find a counterexample to verify that the following is false $$
P(A \cap B \mid C) = P(A \mid C) \, P(A \cap C)
$$
So I thought I would take $A=C$ and then
$$
P(A \cap B \mid C) = \frac{P(C \cap B)}{P(C)},
$$
and if, for example, $C \cap B = \varnothing$ with $C \neq \varnothing$, then
$$
P(A \cap B \... |
My apologies if this is a silly question. I feel like I'm misunderstanding something basic and just need a quick clarification.
I have an eigenvalue problem stated in terms of the inner product. More specifically, I have an operator $O: X \to Y$, where $X$ and $Y$ are Hilbert spaces. I would like to find the larges... |
My apologies if this is a silly question. I feel like I'm misunderstanding something basic and just need a quick clarification.
I have an eigenvalue problem stated in terms of the inner product. More specifically, I have an operator $O: X \to Y$, where $X$ and $Y$ are Hilbert spaces. I would like to find the larges... |
Lets say I have a $(N,N)$ matrix $A$ and want to quickly calculate the inverse of an updated matrix $A' = A+D$, where $D$ is a diagonal matrix with only a few non-zero entries with positions $M = \{m_i\}$
i.e
$$
D_{i,j} = \begin{cases}
d_n,& \text{if } i=j \text{ and } i,j\in M \\
0, & ... |
(for all $x, y$ in K)
> Hint: show that $f$ is injective and continuous and consider the sequence $x_j = f(x_{j−1})$, where $x_0 \notin f(K)$. Show that $||x_j - x_{j+m}|| = ||x_0 - x_m||$ and arrive at a contradiction.
I've been able to show that $f$ is injective and continuous, but I have no idea how to use th... |
Two numbers are randomly selected from bag which has numbers from $1$ to $20$ , what is the probability that one number is not twice the other ?
My turn :
1-p(one is twice the other )
= $1$ - p[(1,2) , (2,4) ,(4,8) ,(6,12) ,(8,16) ,(9,18) ,(10 ,20 )
= $1 -\frac{1}{20}\times \frac{1}{19}\times 2 \times 10$ = $... |
Two numbers are randomly selected from bag which has numbers from $1$ to $20$ , what is the probability that one number is not twice the other ?
My turn :
$$
\begin{aligned}
& 1-p(\text{one is twice the other})
\\
&\qquad= 1 - p[(1,2)\ ,\ (2,4)\ ,\ (4,8)\ ,\ (6,12)\ ,\ (8,16)\ ,\
(9,18)\ ,\ (10 ,2... |
I found this exercise:
> $f:K \rightarrow K$ where $K$ is compact and $||f(x) - f(y)|| = ||x - y||$, for all $x, y$ in K, implies $f$ is bijective. Hint: show that $f$ is injective and continuous and consider the sequence $x_j = f(x_{j−1})$, where $x_0 \notin f(K)$. Show that $||x_j - x_{j+m}|| = ||x_0 - x_m||$ and... |
Peano axioms characterize natural numbers, they use neither sums nor multiplication. Is there an axiomatic characterization for the integer numbers which uses neither sums nor multiplications?
Best regards. |
Lets say I have a $(N,N)$ matrix $A$ and want to quickly calculate the inverse of an updated matrix $A' = A+D$, where $D$ is a diagonal matrix with only a few non-zero entries with positions $M = \{m_i\}$
i.e
$$
D_{i,j} = \begin{cases}
d_n,& \text{if } i=j \text{ and } i,j\in M \\
0, & ... |
We first split the integration interval into two.
$$
\int_{-1}^1 x \sqrt{x} \tan ^{-1} \sqrt{x} d x
= \underbrace{\int_0^1 x \sqrt{x} \tan ^{-1} \sqrt{x} d x}_{\frac{\pi}{10}+\frac{1}{10}-\frac{1}{5} \ln (2)} + \underbrace{ \int_{-1}^0 x \sqrt{x} \tan ^{-1} \sqrt{x} d x}_{J}
$$
For the integral $J$, we let $x\... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
I have been working on the following question:
Find a recurrence relation for the number of regions created by $n$ mutually intersecting circles on a piece of paper (no three circles have a common intersecting point). Then, show using the generating function that
$$a_n = n^2 - n +2$$
for $n \ge 1$.
I have fou... |
I can provide a partial and unfinished answer, when one of the transformations, say $T$, is injective. Then there exists a linear transformation $T^{-1}:T(X)\to X$ such that $T^{-1}Tx=x$ for each $x\in X$. Now let $V$ be any subspace of $X$. Then $T(V)=S(V)$ iff $S(V)\subset T(X)$ and $T^{-1}S(V)=V$. That is we have to... |
Peano axioms characterize natural numbers, they use neither sums nor multiplication. Is there an axiomatic characterization for the integer numbers which uses neither sums nor multiplications? I mean, is there a non-algebraic characterization of the integer numbers?
Best regards. |
I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality which confuses me for a while: \
Let $P=\{\lambda \vdash n, \lambda \neq (1^n), (n)\}$ be the set of partition of the ... |
Given some $n$, is it possible to select $k_i$ such that the following is true?
$$
3^n2^{k_0} + 3^n2^{k_1}+ 3^n2^{k_{2}} + 3^n2^{k_3} + ... + 3^n2^{k_i} = 2^m - 1
$$
for $n=1$ it's trivial, for $n=2$ it's
$$
9 + 9 \times 2 + 9 * 4 = 64 - 1
$$
for $n=3$
$$
k_i = [0, 2,3,5,6, 7, 8, 10, 13]
$$
and $n=4$:
... |
3^x2^y summed over y = 2^k-1? |
Suppose $g_{n}(\cdot)$ defined on $[0,1]$ converges in distribution to a continuous Gaussian process. Let $U_{1},...,U_{n}$ be i.i.d. random variables following $\text{Unif}[0,1]$. Allow $g_{n}$ to be correlated with the $U_{i}$s; for instance, $g_{n}(x)=\frac{1}{\sqrt{n}}\sum_{i}1(U_{i}\leq x)$. Is the following sta... |
In Shreeve's book on finance in continuous time, he "defines" the following. He says on p. 99:
> In general, to compute the first-order variation of a function up to
> time $T$, we first choose a *partition* $\Pi= \{t_0, t_1, \ldots,
t_n\}$ of $[0,T]$, which is a set of times $0=t_0 < t_1 < \cdots < t_n = T$. T... |
In his book "Probability theory", Klenke uses the following definition of transition kernel:
[![enter image description here][1]][1]
and if in $ii)$ the measure is a probability measure for all $\omega_1$ then $K$ is called a stochastic kernel.
Later on he defines regular conditional distribution of $Y$ given ... |
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 strait down) KNOWN TO BE A CORRECT FORMULA (From Here, Assume that nothing else is proven to be correct.)
Arclength X = $Lθ $
Time (Period ) = ... |
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... |
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... |
How to prove that $(x-3)^{\frac {1} {2}}=\frac{10}{x-5}$ has at least one real root?
I know that you have to let $f(x)={(x-3)^{\frac {1} {2}}}$ and $g(x)=\frac{10}{x-5}$
then you let $h(x)=f(x)-g(x)=0$
I know I must use the intermediate value theorem but I cant find the $x$ intervals (need 2) of this graph w... |
How to prove that $(x-3)^{\frac {1} {2}}=\frac{10}{x-5}$ has at least one real root? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.