instruction stringlengths 12 30k |
|---|
Question:
In general, $Z_t := \exp(\int_0^t \phi_s dBs - \frac{1}{2}\int_0^t\phi_s^2ds)$ is the exponential martingale. Is $Z_t := \exp(-\int_0^t \phi_s dBs - \frac{1}{2}\int_0^t\phi_s^2ds)$ also an exponential martingale?
Origin:
Let $(B_t)_{t \geq 0}$ be abrownian motion starting at $x>0$. Let $T_a := \inf\... |
Consider a discrete stochastic process $\{X_t\}_{t \in T}$ with the following properties. Each $t \in T$ has a value $v(t) \in \mathbb{R}_{+}$ and the value is added to the overall value conditioned on some event $\mathbb{1}_t = 1$; that is, $X_t = v(t) \cdot \mathbb{1}_t$. In addition, for every $t \in T$ the expected... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defin... |
Given a circle $C$ of radius $1$ around the origin, I want to determine the locus of the centers of all circles that touch $C$ and the $x$-axis. This is the red curve in the following [Desmos plot](https://www.desmos.com/calculator/e8upuzdtke), where the blue circle touches $C$ and the $x$-axis:
[![enter image des... |
Why is the partial derivative of the generalized momentum with respect to generalized position equal to zero? |
The radius of a blue circle is $y$. The distance between the centers is $1-y$ (do you see why?). Therefore, $x^2 + y^2 = (1-y)^2$. |
What you are asking for is the [expected value][1]
of the number of balls in a crate. It's essentially an average of all the possibilities weighted by their probabilities.
The expected value is calculated by multiplying the outcomes by the probability of each outcome.
For example, consider finding the expected v... |
>In the café, 4 people are having lunch, whose names are $v_1$, $v_2$, $v_3$, and $v_4$. Some of them know each other. The number of acquaintances of person $v_i$ (who are in the café) is denoted by $d(v_i)$. Prove that among the numbers $d(v_1)$, $d(v_2)$, $d(v_3)$, and $d(v_4)$, there are equal ones.
I guess I hav... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
By the Binomial Theorem, when $y<1$:
$(1+x)^y=1+yx+\dfrac{y(y-1)}{2!}x^2+\dfrac{y(y-1)(y-2)}{3!}...$
The coefficient of $y$ is:
$x+\dfrac{(-1)}{2!}x^2+\dfrac{(-1)(-2)}{3!}x^3+\dfrac{(-1)(-2)(-3)}{4!}x^4...$
I don't understand how the coefficient of $y$ came about except for the second term, which is obvious... |
Let $f \in End (\mathbb{R}^{3})$ such that
1. $f$ is diagonalizable and it has only two distinct eigenvalues.
2. $f(U) = V$, where $U = \{(x,y,z): x-2y-z = 0\}$ and $V = span ((1,0,1), (-1,1,1))$
3. 1 is an eigenvalue of $f$ and there is a vector in $U$ associated to that eigenvalue.
4. (1,0,-1) is an eigenve... |
Brainstorm: If $a+c \le w \le b+d$ then $w$ is "some proportion" between $a+c$ and $b+d$. Pick $x,y$ so that the are the same proportion between $a$ and $b$ and $c$ and $d$ respectively. How?
Let $r = \frac {w- (a+c)}{(b+d)-(a+c)}$. Thats the "some proportion". Note: $0 \le r \le 1$.
Let $x = a + r\times (b-a... |
How to prove this trigonometrical Identities? |
From page 5 of https://mast.queensu.ca/~kani/papers/numgenl.pdf :
$\text{Then } \psi \text{ is automatically an anti-isometry with respect to the } e_N \text{-pairings on } E \text{ and on } E':$
$$e_N(\psi(x), \psi(y)) = e_N(x, y)^{-1}, \forall x, y \in J_E[N],$$
$\text{and this condition is equivalent to the ass... |
I've just come across this question on Isaac Physics and I have no idea where to begin. The question is as follows:
You are serving on a jury in the Crown Court. The defendant has been accused of a serious crime, however the **only** evidence is that their DNA is a perfect match to the perpetrator's DNA found at the... |
The Question :
In the first step we randomly choose an integer out of the values ${1,2,...,10}$. Let X be the chosen integer.
In the second step we randomly choose an integer out of the values ${1,2,...,X}$. Let Y be the chosen integer.
In the third step we randomly choose an integer out of the values ${1,2,..... |
From page 5 of https://mast.queensu.ca/~kani/papers/numgenl.pdf :
> Then $\psi$ is automatically an anti-isometry with respect to the $e_N$ -pairings on $E$ And on $E'$:
$$e_N(\psi(x), \psi(y)) = e_N(x, y)^{-1}, \forall x, y \in J_E[N],$$
and this condition is equivalent to the assertion that the "graph subgroup s... |
I am walking in circles with the question why wolframalpha gives me as geometric description of
$$(x-y)^2+(x-z)^2+(z-y)^2-14=0$$
and of
$$(x-y)^2+(x-z)^2+(x-u)^2+(z-y)^2+(u-y)^2+(z-u)^2-30=0$$
a glome, i.e., 4 sphere. The first is unbounded and the second is compact but their defining equations are just extensions... |
I am walking in circles with the question why wolframalpha gives me as geometric description of
$$(x-y)^2+(x-z)^2+(z-y)^2-14=0$$
as the infinite cylinder and of
$$(x-y)^2+(x-z)^2+(x-u)^2+(z-y)^2+(u-y)^2+(z-u)^2-30=0$$
a glome, i.e., 3 sphere. The first is unbounded and the second is compact but their defining equa... |
Let us imagine making a big list of all of the ways to choose two different birthdays for two pairs of people. The list looks something like this:
1. Alice, Bob on Jan 1; Charlie, Diane on Jan 2.
2. Alice, Bob on Jan 1; Charlie, Evan on Jan 2.
3. Alice, Bob on Jan 1; Diane, Evan on Jan 2.
Etc.
Now, there a... |
From page 5 of https://mast.queensu.ca/~kani/papers/numgenl.pdf :
> Then $\psi$ is automatically an anti-isometry with respect to the $e_N$ -pairings on $E$ and on $E'$:
$$e_N(\psi(x), \psi(y)) = e_N(x, y)^{-1}, \forall x, y \in J_E[N],$$
and this condition is equivalent to the assertion that the "graph subgroup s... |
In Zieschang's "Theory of Association Schemes", in section 5.2, he remarks that the composition of homomorphisms is not always a homomorphism. I've been struggling to find an example of that claim. I've found certain papers which reference that fact, but I'm unable to find any example.
What would be a simple counter... |
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
>I need asymptotic formula of $I_n$ as $n\to\infty$
Firstly we see that since $x(1-x)\leq \frac{1}{4}$, so the defini... |
I have a question about how to prove the limit using $\limsup$.
On the other day, I was asked the following problem in the exam:
> Let $m$ and $n$ be positive integers.
Calculate the following limit, where $m$ is fixed.
$$
\lim_{n \to \infty} \int_1^e x^m e^x (\log x)^n dx
$$
To this problem, I answered in t... |
Starting at 7:52 of [this video][1], the professor presents a proof of the following theorem: *for any infinite set of well-formed formulas $\Sigma$ such that $\Sigma \vDash \alpha$, where $\alpha$ is a w.f.f., there exists a finite subset $\Sigma_0 \subset \Sigma$ such that $\Sigma_0 \vDash \alpha$*.
This is how I ... |
Isn't this proof of what this professor calls the "Finite Models Theorem" flawed? |
2-weak algorhythm implies weak-algorhythm? |
I'm currently reading the paper by Miller and Piccirillo titled [Knot Traces and Concordance](https://arxiv.org/abs/1702.03974). I have some confusion about the proof of theorem 1.8. In particular the 'if' direction.
I'll restate some of the definitions of objects they use. The trace $X_K$ of a knot $K$ is defined a... |
what are $\mathcal{O}(x)$ and $\mathcal{o}(x)$? and where did they came from? |
Given independent random samples $(X_1,...,X_m)$ and $(Y_1,...,Y_n)$, respectively, from the following distributions of $X$ and $Y: X \sim \lambda_1 \exp (-\lambda_1 x) I(x >0)$, and $Y \sim \lambda_2 \exp (-\lambda_2 y) I(y >0)$. Consider the problem of testing the null hypothesis $H_0:\lambda_1=\lambda_2 $ against th... |
Is ∞ a limit point of $\mathbb R$ ? If not, how to understand Rudin's definition at the beginning of chapter $4$ (PMA)? |
I recently asked a question about the proof of theorem 1.8 in Miller and Piccirillo's paper:
https://math.stackexchange.com/questions/4638913/knot-traces-and-trace-lemma
I will continue to keep the same notation as in my previous question.
Following the discussion in the comment section of Lee Moscher's answ... |
Your proof is correct. It invokes a simple special case of CRT = Chinese Remainder Theorem when the values $\,a_1 = a_2\,$ are *constant*, say $\,a,\,$ which is equivalent to the following basic result
[UL = Universal property of LCM:](https://math.stackexchange.com/a/88411/242) $\ \rm \,\ j,k\mid n\!\!\color{#0a0... |
2-term weak algorhythm implies weak-algorhythm? |
Here's the definition of a sequence as laid out in the text:
> Let $m$ be an integer. A *sequence* $(a_n)_{n=m}^\infty$ *of rational
> numbers* is any function from the set $\{n \in \mathbf{Z} : n \geq m\}$ to $\mathbf{Q}$.
I can make sense out of this definition, but I was under the impression that a sequence h... |
I don't get the parts in blue, can you help me please?
If $B_m=\bigcup_{k=1}^m A_k$ then $\bar{B}_m=\bigcup_{k=1}^m \bar{A}_k$
By hypothesis, we have that $B_m=\bigcup_{k=1}^m A_k$, so $\color{Blue}{\bigcup_{k=1}^m A_k\subseteq \bar{B}_m}$ and $A_k\subseteq B_m$, then $\bar{A}_k\subseteq \bar{B}_m$ and so:
$\i... |
2-term weak algorithm implies weak-algorithm? |
Let $(H, \ast)$ be a group, where $H$ is included in $(0, \infty)$, which has these properties: $x \in H \Rightarrow \frac{1}{x} \in H$, $2023 \in H$, and $x \ast y = \frac{1}{x} \ast \frac{1}{y}$ for any $x, y$. Prove H is not an interval.
As far as my attempts go, I haven't been able to do much, apart from modifyi... |
Let $M$ be a smooth manifold. We denote $C^\infty(M)$ is the space of all smooth functions $f:\ M\longrightarrow\mathbb R$ and $\mathfrak X(M)$ is the space of all smooth vector fields $X:\ M\longrightarrow T(M)$. It is known that if $X$ is a smooth vector field, then for any local chart $x:\ U\subset M\to\mathbb R^n$,... |
Recently I was trying to solve this problem.
Suppose that $X$ is a set and $\mu^{\star}$ is an outer measure on $2^X$. Let $A \subseteq X$ be a set such that $\mu^{\star}(A)<+\infty$ and suppose that $C \in \Sigma_{\mu^{\star}}$ satisfies $A \subseteq C$ and $\mu^{\star}(C)=\mu^{\star}(A)$. Show that
$$
\mu^{\star... |
I have been working on a problem where I want to prove an injective immersion $f$ from a submanifold $M \subset R^n$ (actually, a level set) to $R^k,\; k < \dim M$ is a topological embedding. If I knew $M$ was connected I could do it (I believe) and so I have been twisting myself into a pretzel trying to come up with a... |
Can you embed a disconnected manifold in Euclidean space? |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time.
Goal: Find the global minimum of a univariate, nonnegative rational function... |
The law of total expectation states
$$\operatorname{E}[Y] = \operatorname{E}[\operatorname{E}[Y \mid X]]$$ but students at first tend to find this notation confusing or opaque. What is really happening is that the inner conditional expectation $\operatorname{E}[Y \mid X]$ regards $X$ as fixed, and calculates the expe... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time. I have documented my research and implementation in C#: it can be found on [Git... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial $aX^2 + bX + c$, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta = b^2 - 4 a c$ of such a polynomial is positiv... |
Isn't this is a trivial problem? If you have n points with no 3 on a line, 4 on a plane, etc... then they lie on a n-2 sphere, and the smallest spheres in higher dimensions will be those with the same centre and radius as this n-2 sphere. |
I am looking at the equation $x^4=-1$ over a generic field $k$ of characteristic $0$. If $k$ is algebraically closed, it will certainly have $4$ roots. I think that they should all be different, but I am not completly sure about this. Could someone tell me if this is the case, and explain why? If we are looking at the ... |
How many solutions can $x^4=-1$ have on a field of characteristic $0$? |
I'm not a maths person and the formulas you see on here with all the symbols is completely beyond me - I'm sorry. However I had help about 18 months ago to turn a provided formula into lua script to make the calculation for two aircraft to intercept each other. I will admit, I don't know why that formula works, and I w... |
How to tell when calculating an Intercept Point between two moving objects, when it will never intercept? |
Isn't this is a trivial problem? If you have $n$ points with no $3$ on a line, $4$ on a plane, etc... then they lie on a $n-2$ sphere, and the smallest spheres in higher dimensions will be those with the same center and radius as this $n-2$ sphere. |
I have been working on a problem where I want to prove an injective immersion $f$ from a submanifold $M \subset R^n$ (actually, a level set) to $R^k,\; k \geq \dim M$ is a topological embedding. If I knew $M$ was connected I could do it (I believe) and so I have been twisting myself into a pretzel trying to come up wit... |
I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$ with Neumann boundary conditions, where $\kappa:[0,T]\to(0,\infty)$ is linear and $W$ is a cylindrical Wiener process on $H:=L^2([0,1)^2)$. I somehow need to compute/estimate (or at least find a sharp... |
For every integer $n\geq 0$ we define
$a_n=\sum_{k=0}^{n} 3^{k}(n-k)$.
Find the generating function of the sequence $(a_n)_{n=0}^\infty$ and write it down without using series.
Attempt: $G(x)=\sum_{n=0}^{\infty} (\sum_{k=0}^{n} 3^{k}(n-k))x^n=\sum_{n=0}^{\infty} (n\sum_{k=0}^{n} 3^{k}-\sum_{k=0}^{n}k3^k)x^n=\... |
I have been working on a problem where I want to prove an injective immersion $f$ from a submanifold $M \subset R^n$ (actually, a level set) to $R^k,\; k \geq \dim M$ is a topological embedding. If I knew $M$ was connected I could do it (I believe) and so I have been twisting myself into a pretzel trying to come up wit... |
If each connected component of a submanifold of Euclidean space embeds does the entire manifold embed? |
I have arrived at the point where I have a series
$$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$$
that I know should diverge (checking via python implies it might be divergent, and wolframalpha times out when trying to evaluate it, although yes, it doesn't truly mean anything).
I have tried multiple approaches and... |
How does one show that an alternating series $\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$ is unbounded/diverges? |
I have boolean function of next variables: $x_{ij}$, $ 1 \leq i \leq n$ and $1 \leq j \leq m$. And the function is equal to 1 only when we have such $i$ that $x_{ij} = 1$ for any j. Can I calculate the value of the function in less than $n*m$ questions ($x_{ij} = 0?$)? |
Let $G$ be a group. A representation on a vector space is a group homomorphism $G \to GL(V)$ for some $k$-vector space $V$. More general on could replace $V$ by an $R$-module $M$ over a commutative ring and then maybe over a non-commutative ring and get $R$-representations of $G$. Now one could also take for example in... |
More general representation theory? |
I have arrived at the point where I have a series
$$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{2n}$$
that I know should diverge (checking via python implies it might be divergent, and wolframalpha times out when trying to evaluate it, although yes, it doesn't truly mean anything).
I have tried multiple approaches and... |
I found the next problem, I know that I will use the Chebyshev's inequality but I don't understand the "question" of the problem
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/PBZdj.png
Precisely, I don't understand the phrase "...fraction of $x_1,\ldots,x_n$ included in the interval.... |
Let $R$ be a commutative Noetherian integral domain. Let $M$ be a finitely generated projective $R$-module of positive rank. Are the abelian categories $\operatorname{mod-}R$ and $\operatorname{mod-}\operatorname{End}_R(M) $ equivalent ? |
When are abelian categories $\operatorname{mod-}R$ and $\operatorname{mod-}\operatorname{End}_R(M) $ equivalent? |
For every integer $n\geq 0$ we define
$a_n=\sum_{k=0}^{n} 3^{k}(n-k)$.
Find the generating function of the sequence $(a_n)_{n=0}^\infty$ and write it down without using series.
Attempt: $G(x)=\sum_{n=0}^{\infty} (\sum_{k=0}^{n} 3^{k}(n-k))x^n=\sum_{n=0}^{\infty} (n\sum_{k=0}^{n} 3^{k}-\sum_{k=0}^{n}k3^k)x^n=\... |
The Ablowitz-Ladik system is written as $i\dot{u}_n = (1+|u_n|^2)(u_{n+1} + u_{n-1})$ and is completely integrable. To my understanding, such integrable systems always have a so-called Lax Pair, but I cannot find what it is in literature. Do discrete integrable systems have these? |
Lax pair for discrete integrable systems? |
For every integer $n\geq 0$ we define
$a_n=\sum_{k=0}^{n} 3^{k}(n-k)$.
Find the generating function of the sequence $(a_n)_{n=0}^\infty$ and write down the answer without series.
Attempt: $G(x)=\sum_{n=0}^{\infty} (\sum_{k=0}^{n} 3^{k}(n-k))x^n=\sum_{n=0}^{\infty} (n\sum_{k=0}^{n} 3^{k}-\sum_{k=0}^{n}k3^k)x^n... |
Let $F = \Phi (X^x_T)$. $X^x_T$ behaves as a risky asset such that $dX^x_T=rX^x_Tdt+\sigma X^x_TdW_T;X_0^x=x$. $\Phi$ is continuous with linear growth and $P(x) = e^{−rT}\mathbb{E}[F]$. Prove that $P \in C^2(\mathbb{R}_+;\mathbb{R}) $. I've already proved that $P \in C^1(\mathbb{R}_+;\mathbb{R}) $, but I have no idea o... |
Suppose $F_q$ is a finite field of cardinality $q$ and $V$ is a vector space over $F_q$. The finite field Kakeya problem asks for a lower bound on the cardinality of a Kakeya set in $V$, which is a set that contains a line in every direction.
Intuitively, why is this problem a hard one?
A Kakeya set is equivalent... |
Why finite field Kakeya problem is/was hard? |
This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of sequences by taking general theorems about nets and replacing the term "net" with the term "sequence". For example: A set ... |
I've been trying to understand the decomposition of the second tangent bundle, $TTQ$, of a smooth Riemannian manifold, $(Q,g)$, into so-called horizontal and vertical bundles, $TTQ = HTQ \oplus VTQ$, using the Levi-Civita connection $\nabla$. I know that such a decomposition can be applied to much more general vector b... |
I am reading through Krylov's *Lectures on Elliptic and Parabolic Equations in Hölder Spaces*. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to reference the relevant proofs in the Elliptic case. Starting with some definitions, we take
$$[u]_{\delta;\Omega} ... |
How does the mean value theorem with Hölder seminorms depend on dimension? |
I converted $\Gamma(\tfrac14)^2=16\Gamma(\tfrac54)^2$ to a double integral by definition of Gamma function. By using polar coordinates I eliminated one integral and found $12\sqrt{\pi}$ times the integral
$$\int_0^{\tfrac\pi 2}\frac{(\sin\theta\cos\theta)^{\tfrac14}}{(\cos\theta+\sin\theta)^{\tfrac52}}d\theta\tag1$$
... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time. I have documented my research and implementation in C#: it can be found on [Git... |
I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive.
Suppose to have a degree $2$ homogeneous (non-trivial) polynomial $p(\mathbf{x})=\sum\limits_{j=1}^n\sum\limits_{i=1}^ja_{ij}x_ix_j\in\mathbb{F}_2[\mathbf{... |
I have N entities that I want to compare them to; for example, A has 5 options, B has 3 options, and C has 6 options. 1 means an option exists, 0 otherwise as in the example next:
A = [1 1 0 1 1] (satisfied by 4 options out of 5)
B = [1 0 0] (satisfied with 1 option out of 3)
C = [0 0 0 0 0 1] (satisfied wit... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time. I have documented my research and implementation in C#: it can be found on [Git... |
I converted $\Gamma(\tfrac14)^2=16\Gamma(\tfrac54)^2$ to a double integral by definition of Gamma function. By using polar coordinates I eliminated one integral and found $12\sqrt{\pi}$ times the integral
$$\int_0^{\tfrac\pi 2}\frac{(\sin\theta\cos\theta)^{\tfrac14}}{(\cos\theta+\sin\theta)^{\tfrac52}}d\theta\tag1$$
... |
There are two sets of four 2D points.
There is a 1 to 1 correspondence between the two sets.
The points in both sets are close to the following nominal positions, only deviate with a random value in both direction: (-1.5, 0.0), (-0.5, 0.0), (0.5, 0.0), (-1.5, 0.0).
I would like to find a transformation (translatio... |
Given independent random samples $(X_1,...,X_m)$ and $(Y_1,...,Y_n)$, respectively, from the following distributions of $X$ and $Y: X\sim N(\mu_1, \sigma^2)$ and $Y \sim N(\mu_2,\sigma^2)$,consider the problem of testing
the null hypothesis $H_0: \mu_1=\mu_2$ against the alternative $H_1: \mu_1 \neq \mu_2$ with unknow... |
Given a commutative ring $R$, is every ring with a homomorphism from $R$ an $R$-algebra if and only if $R$ is a solid ring?
A ring $R$ is said to be solid if the unique homomorphism $\mathbb{Z} \to R$ is an epimorphism in the category of rings, or equivalently, if $r \otimes 1=1 \otimes r \in R \otimes_{\mathbb{Z}} ... |
Is every ring with a homomorphism from $R$ an $R$-algebra if and only if $R$ is a solid ring? |
I have been working on a problem where I want to prove an injective immersion $f$ from a submanifold $M \subset R^n$ (actually, a level set) to $R^k,\; k \geq \dim M$ is a topological embedding. If I knew $M$ was connected I could do it (I believe) and so I have been twisting myself into a pretzel trying to come up wit... |
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying
$\overline{\partial_c}f=0,$
where
$$\overline{\partial_c}=\frac{1}{2}\left(\frac{\partial}{\partial t}+\frac{\operatorname{Im}q}{r}\frac{\partial}{\partial r}\right), \ \operatorname{Im... |
Consider the diagonal operator A{$x_n$} = {$a_nx_n$}. It must be proved that this operator has a cyclic vector if and only if all {$a_n$} are different.
I tried to consider a situation where there are indexes m and k, that $a_m$=$a_k$, m≠k. Then there is no cyclic vector, because if it existed, two situations would ... |
I had an argument about the natural domain of rational functions.
Consider $f(x) = \frac{P(x)}{Q(x)},$ where $P$, and $Q$ are (real) polynomials.
Is the natural domain of $f$ the set $\{ x \in \mathbb{R}: Q(x) \neq 0\}$? Or should we first simplify the fraction? For instance, let us consider $P = Q = x$, so that $$... |
Given a summable sequence $\{a_i\}_{i\in \mathbb{N}} \subseteq (0, 1)$, is there a necessary condition for a subsequence $\{a_{k_i}\}_{i\in \mathbb{N}}$ of $\{a_i\}_{i\in \mathbb{N}}$ to exist such that $\lim_i \dfrac{a_{k_{i+1}}}{a_{k_i}}$ exists?
Since $\{a_i\}_{i\in \mathbb{N}}$ is summable, the set $\left\{ \df... |
Is $\exists(a,b)\in S^2P(a,b)$ always equivalent to $\exists\{a,b\}\subseteq S\,P(a,b)$ for a binary predicate $P$ with domain $S^2;S\neq\emptyset$?Is the following second-order statement true or false? (Assume $P$ is a binary predicate statement which is always defined for two members of $S\neq\emptyset$.)
$$\fora... |
Is $\exists(a,b)\in S^2P(a,b)$ always equivalent to $\exists\{a,b\}\subseteq S\,P(a,b)$ for a binary predicate $P$ with domain $S^2;S\neq\emptyset$? |
Given a summable sequence $\{a_i\}_{i\in \mathbb{N}} \subseteq (0, 1)$, is there a necessary condition for a subsequence $\{a_{k_i}\}_{i\in \mathbb{N}}$ of $\{a_i\}_{i\in \mathbb{N}}$ to exist such that $\lim_i \dfrac{a_{k_{i+1}}}{a_{k_i}}$ exists? For instance, for any $r>1$, the sequence $\{n^{-rn}\}_{n\in \mathbb{N}... |
I want to understand the semantics of rights bit shifts `x>>k` in two's complement properly, in particular why do right bit shifts of size $k$ approximately divide $x$ by $2^k$ (and ideally making this "approx" precise and also handling the arithmetic left bit shift too).
So the representation of a number in two co... |
Why do k arithmetic right/left bit shifts divide by $2^k$/multiply by $2^k$ in two's complement, rigorously? |
Let us assume $A' \subset A$ and $f\colon A \to A‘$ is bijective (so $A$ is an infinite set according to Dedekind‘s definition). Let us further assume $h: A \to B$ is bijective. I want to prove that $B$ is also an infinite set.
There is a proof that goes like this:
We assume further $g = h \circ f \circ h^{-1} \c... |
If $F,G: I \rightarrow Z$ are closed path at $z \in Z$, where Z is a topological space. Why is $F \cdot G: I \rightarrow Z$ a closed path?
This does not appear to make sense to me considering my idea of a closed path. Which is unfortunately undefined in my textbook. Introduction to Topology by Bert Mendelson.
My... |
If $F,G: I \rightarrow Z$ are closed path at $z \in Z$, where Z is a topological space. Why is $F \cdot G: I \rightarrow Z$ a closed path? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.