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With
$g(x) = {\large\int} \ \sin(a x) \cos(b x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(a x) \cos(b x)}{\frac{1}{f(x)} \ + \ c} dx$
that may contain $g(x)$ and possibly $g'(x)$?
I have a hunch that the antiderivative of $sin(x)/(e^{-x}+a)$,
https://www.wolframa... |
I am currently working through [*The International Winter School on Gravity and Light*][1] lecture series and have had some trouble understanding the following claim made in lecture $9$ at the [22:30][2] minute mark.
Dr. Schuller says that asking
> "$(1)$ Can gravity be encoded in the curvature of space, such t... |
How is an autoparallel equation related to Newton's second axiom? |
The first point you should think about is ... when the limit is a number (not $\infty$) so must be a $(x-3)$ in numerator. so $$\lim _{x\to 3}\frac{3x^2+ax+b}{x^2-2x-3}=\frac{5}{2}\\\lim _{x\to 3}\frac{3x^2+ax+b}{(x-3)(x+1)}=\frac{5}{2}\\\lim _{x\to 3}\frac{(x-3)(something)}{(x-3)(x+1)}=\frac{5}{2}
$$
now you can di... |
Consider a 2-manifold $(M,g)$ isometric to $(\mathbb{R}^2,\bar{g})$, where $\bar{g}$ is the Euclidean metric, and let $F:\mathbb{R}^2 \to M$ be an isometry. Let $K_0 \subset \mathbb{R}^2$ and $K \subset M$ be two compact subsets (actually, I consider geodesic triangles, if that matters), such that $K = F(K_0).$
I co... |
I am currently working on these two models and I don't understand the connection between them and hyperbolic space. In case of spherical geometry one can imagine everything well as a 2 dimensional sphere in 3 dimensional space and all the major facts on parallel lines make sense. But in the case of hyperbolic geometry ... |
How is the Poincare disc model or the upper half plane model a representation of hyperbolic space? |
> First why there are (−1)! ways to seat the women? Shouldn't it be
> !?
In circular permutations, when seats are indistinguishable, the total number of permutations is $(n-1)!$. That's because each unique permutation in the indistinguishable case gives $n$ unique permutations in the distinguishable case - by rotat... |
The probleme des menages consists of the following:
How many ways can $n$ married couples si at a round table in such a way that there is one man between every two women and no man is seated next to his wife?
The solution on Titu Andreescu's book A path to combinatorics for undergraduates, is by Inclusion-Exclusi... |
Suppose $M$ is a manifold of dimension $n$.Let $p\in M$ and let $\psi:U\subset \mathbb R^n\to M$ be a local parametrization on $M$ near $p$.So,$\psi:U\to \psi(U)$ where $\psi(U)$ is open in $M$ and $p\in \psi(U)$.Let,the coordinates on $U$ be given by $(x_1,x_2,...,x_n)$.Now consider $\hat p\in U$ which maps to $p$.The... |
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: *How much information does $\tau(n)$ contain about $n$*?
The answer is obviously: *not very much*. There are infinitely many numbers which have the same value for $\tau$. However, if there were an *infinit... |
Let $p: \mathbb{F} \to \mathbb{F}$ be a polynomial of degree of at most $m$, i.e. $p \in \mathcal{P}_m(\mathbb{F})$ such that $p(r) = 0$ for some $r \in \mathbb{F}$. Is it true that in that case $p$ can always be written as
$$p(z) = b_1(z- r) + b_2(z-r)^2 + \cdots + b_m(z-r)^m$$
for some $b_i \in \mathbb{F}$, i.e. $... |
Suppose $M$ is a compact Riemannian manifold and let $d$ be the induced distance function on $M$. Let $\mu$ be a probability measure on $M$ with continuous density. The Fr$\acute{\mathrm{e}}$chet function is defined as
$$
F_{\mu}(x) = \int_{M} d^2(x,y) \,\mathrm{d}\mu(y)
$$
I want to ask whether $F_{\mu}(x)$ i... |
I have a set of points $\left\{ \left(−\frac{6}{4}, 0\right), \left(−\frac{4}{4}, −\frac{\sqrt{6}}{3}\right), \left(−\frac{3}{4}, −1\right), \left(−\frac{2}{4}, −\frac{\sqrt{6}}{3}\right), \left(\frac{0}{4}, 0\right), \left(\frac{2}{4}, \frac{\sqrt{6}}{3}\right), \left(\frac{3}{4}, 1\right), \left(\frac{4}{4}, \frac{\s... |
Is it possible to adjust the sine function to pass through specific points? |
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... |
Random variable ξ has a uniform distribution on [0, 1], and η is its independent binomial random variable Bin(n, p). Find the density of the random
variable ξ + η.
My steps so far:
1) η is a discrete random variable, ξ is absolutely continuous random variable
let's do the following: $$F_{\xi + \eta}(t) = P(\... |
- $(\Omega, \mathcal{F}, \mathbb{P})$ probability space
- $(E, \mathcal{B}(E))$ a measurable space where $B(E)$ is the Borel sigma algebra and $E$ is a Banach space
- $X:\Omega\to E$ random variable with distribution $\mathbb{P}^X$ on $(E, \mathcal{B}(E))$
I know that one can write the probability of an event a... |
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I would like a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has (at least) two distincts roots in an algebraic closure $\overline{k}$ then $P(X,Y)$ has (at least) two distincts roots in ... |
Let $x_1 , x_2 , x_3 ~ N(0,1)$ iid.
$Y = \frac{x_1 + x_2 * x_3}{\sqrt{1+ x_3^2}}$
Does Y and $X_3$ jointly gaussian ?
I already showed that $E[Y|X3]=0 $ and $Var[Y|X3]=0$
And Y|X3 is gaussian... But how can I proceed?
Maybe there is a mistake in the question that make it better? |
- $(\Omega, \mathcal{F}, \mathbb{P})$ probability space
- $(E, \mathcal{B}(E))$ a measurable space where $B(E)$ is the Borel sigma algebra and $E$ is a Banach space
- $X:\Omega\to E$ random variable with distribution $\mathbb{P}^X$ on $(E, \mathcal{B}(E))$
I know that one can write the probability of an event a... |
Let $m^*$ be an outer measure and $\Delta$ denote the symmetric difference.
- Define a semi-metric $d: \mathscr{P}(\mathbb{R}) \times \mathscr{P}(\mathbb{R})\to [0, \infty]$ by $d(A, B)=m^* (A \Delta B)$.
- Denote by $\mathscr{E}$ the set of all elementary subset of $\mathbb{R}$ where an elementary subset of $\math... |
Interpreting the closure of the set of all elementary subsets of $\mathbb{R}$ with respect to the semi-metric $d$? |
Let $f(x) = 3/64(2x - 2)$ whenever $x \in [1, 5]$, zero otherwise, be the PDF of a random var. I know the expected value of $X$ is $2.75$. I was asked to find its variance. For the sake of practice I used two methods. First, I calculated $E(X^2)$ and used the fact that $V(X) = E(X^2) - E(X)^2$. This gives
\begin{ali... |
> Let $f$ be holomorphic on $\mathbb{D}$, continuous on
> $\overline{\mathbb{D}}$ s.t $|f(z)|=1$ whenever $|z|=1$. Prove that if
> $f(z)\neq 0$ for all $z\in\mathbb{D}$ then $f$ is constant on
> $\mathbb{D}$.
I already proved the following result:
Let $f$ be non-constant, holomorphic on $\mathbb{D}$, continu... |
I want to show that for every nonnegative integers $n$ and $b$ we have $\alpha \geq 0$ and $0 \leq \gamma < 2^{b+1}$ such that $n = \alpha\cdot 2^{b+1} + \gamma$.
So let $n\geq 0$ be an integer.
If $n$ is a multiple of $2^{b+1}$, then $\alpha = \frac{n}{2^{b+1}}$ and $\gamma = 0$.
But if it is not, we have $\alp... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
When a sequence satisfies a linear recurrence relation? |
Let P be a convex polygon in the plane, and let P’ be an enlarged version of P, dilated by a scale factor of 2. Show that seven copies of P can completely cover P’.
I vaguely remember seeing this problem online, but I can’t find the source.
I noticed that seven circles of radius 1 can cover a circle of radius 2. ... |
I have a set of points $\left\{ \left(−\frac{6}{4}, 0\right), \left(−\frac{4}{4}, −\frac{\sqrt{6}}{3}\right), \left(−\frac{3}{4}, −1\right), \left(−\frac{2}{4}, −\frac{\sqrt{6}}{3}\right), \left(\frac{0}{4}, 0\right), \left(\frac{2}{4}, \frac{\sqrt{6}}{3}\right), \left(\frac{3}{4}, 1\right), \left(\frac{4}{4}, \frac{\s... |
Let $m^*$ be an outer measure and $\Delta$ denote the symmetric difference.
- Define a semi-metric $d: \mathscr{P}(\mathbb{R}) \times \mathscr{P}(\mathbb{R})\to [0, \infty]$ by $d(A, B)=m^* (A \Delta B)$.
- Denote by $\mathscr{E}$ the set of all elementary subset of $\mathbb{R}$ where an elementary subset of $\math... |
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
$$u(x)\Big|_{\partial \Omega}=g(x)$$
for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I found that in Gilbarg and Tr... |
What precisely do you mean by "a variable with infinite mean will always produce a finite value"?
An ordinary real-valued random variable, by definition, can only take real values: it is a measurable function from a probability space $(\Omega, \mathcal{F}, P)$ to $\mathbb R$. And real numbers are finite. So any r... |
Assume given is a smooth manifold $M$ and smooth vector bundles $A_1\subseteq A_2 \subseteq A_3$ such that $A_i$ is a subbundle of $A_{i+1}$ (embedded submanifold and a vector bundle). Is it generally true that $$\frac{A_3}{A_1} \cong \frac{A_3}{A_2} \bigoplus \frac{A_2}{A_1}?$$
It is easy to construct a homomorphis... |
What about piecewise-defined functions? If those work for you, you can use the periodic version of [interpolation by cubic splines][1]. That would most likely satisfy the condition you want on the second derivative, as the second derivative of cubic splines is continuous and is piecewise linear.
[1]: https://en.... |
The Problem says that :-
>If $\large \mathrm{N}$ is any positive integer, and $\delta_{n}=\large \int_{_{0}}^{^{N}} {{u} \choose {n}}{{N-u} \choose {N-n}} \textit{ du }$, then the value of
$\large \sum_{n=0}^{N}\delta_{n} = \\ \ \\ \ \\ (A). 0 \\ (B). N \\(C). N^2 \\ (D). \frac{1}{N} $ .
I was thinking about ... |
I've been pondering how to intuitively visualize the commutative property of multiplication when dealing with more than three numbers. For two numbers, we can easily conceptualize this with the area of a rectangle, and for three, the volume of a cube serves as a great illustration. However, when we move to four or more... |
[Ref: Cassela and Berger Statistical Inference][1]
I solved for parts a and b. I got the $\tilde{\theta}_{MoM}$ to be $\frac{2\bar{x}}{3}$.
Also, I got $\hat{\theta}_{MLE}$ to be $\frac{X_n}{2}$.
For the c part, I am trying to find the MSE for both MoM and MLE estimators. I used the formula $MSE(\theta) = Var(... |
Let $X_1, X_2, X_3$ be a random sample of size three from a $uniform(θ, 2θ)$ distribution, where $θ > 0$
I solved to get $\tilde{\theta}_{MoM}$ to be $\frac{2\bar{x}}{3}$.
Also, I got $\hat{\theta}_{MLE}$ to be $\frac{X_n}{2}$.
For the c part, I am trying to find the MSE for both MoM and MLE estimators. I used... |
What about piecewise-defined functions? If those work for you, you can use the periodic version of [interpolation by cubic splines][1]. That would most likely satisfy the condition you want on the second derivative, as the second derivative of cubic splines is continuous and is piecewise linear.
____
**Remark:** On... |
Let $\epsilon>0$. I know that for almost every $c\in (0,2\pi)$ there is a constant $k>0$ such that $dist|c+l\cdot \omega, 2\pi\mathbb{Z}|\geq k|l|^{-(m+\epsilon)}$ for all $l \in \mathbb{Z}^m-\{0\}$.
How can i prove that $|e^{i(c+l\cdot \omega)}-1|\geq \tilde{k}|l|^{-(m+\epsilon)}$? |
How to prove that $|e^{i(c+l\cdot \omega)}-1|\geq \tilde{k}|l|^{-(m+\epsilon)}$ if $dist|c+l\cdot \omega, 2\pi\mathbb{Z}|\geq k|l|^{-(m+\epsilon)}$? |
Given a Banach vector space $X$, I need to consider an infinite tensor product of it. The simplest construction I found, in Guichardet, is by considering the inductive limit of the direct system $(X^{\otimes i}, \phi_{ij})_{i,j}$ defined by:
$$ \begin{array}{lccc}\phi_{i,j}:& X^{\otimes i} &\rightarrow& X^{\otimes j}\... |
Completion of an infinite tensor product of Banach spaces: can an infinite product converge to 0? |
I'm looking for a CAS that can solve the following integral:
$$\int^{\pi}_0\frac{4\sin^2\beta\cos^3\kappa}{(\sin^2\kappa + \sin^2\beta\cos^2\kappa)^2}d\kappa$$
I've tried Wolfram Alpha and it returns a very complicated result for the indefinite integral but is unable to produce a result for the definite integral.... |
What about piecewise-defined functions? If those work for you, you can use the periodic version of [interpolation by cubic splines][1]. That would most likely satisfy the condition you want on the second derivative, as the second derivative of cubic splines is continuous and is piecewise linear.
____
**Remark:** On... |
> Let $f(z)$ be holomorphic on $D(0,R)$ s.t $|f(z)|<M$ for all $|z|<R$.
> Prove that $$|f^{(n)}(z)|\leq\dfrac{MRn!}{(R-|z|)^{n+1}},\quad n\geq0
,\quad |z|<R.$$
My attempt:\
For each $z\in D(0,R)$, take $0<r<R$ s.t $\overline{D}(z,r)\subset D(0,R)$.\
Since $f$ is holomorphic on $ D(0,R)$, applying Cauchy integra... |
I saw the statement in the question from the book Moonshine Beyond the Monster.
We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation.
I can't understand why the author said quosi-periodicity is an example of projective representation:
$\quad$ Let $\theta... |
I am calculating the stream density S (= river length L divided by stream area A). The river has a length L of 671.8 +- 5.9 km and the stream area is 1860.2 +- 19.8 km^2. The stream density is thus and 0.36114396 km^-1 its error (calculated by the laws of error propagation by division) is 0.00498359 km^-1. What is the ... |
How to correctly report significant digits in measurements and their uncertainty? |
I have a challenge. This may be little tricky or even not possible but wanted to check if anyone has any thoughts on this?
PS : This question is in general and not related to only to R. May be I can say its general mathematics
I have a data
````
df
ColA ColB ColC
6 9 27
1 4 32... |
Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?
I know that if we assume that the covering space is finite then there is no such covering, but what about if the covering space has an in... |
# Set up
Given a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\}_{t\in T})$ people seem to say that the intuitive meaning of the stochastic process $\{X_t\,:\, t\in T\}$ being **adapted** to the filtration is that it is non-anticipating/can't see into the future or any variation of this... |
By Cayley's theorem, any group $G$ naturally injects into the symmetric group $\mathrm{Sym}(G)$ of its underlying set via the Cayley embedding. Let's say this embedding is *optimal* for $G$, if there is no embeding $G \hookrightarrow \mathrm{Sym}(I)$ for some smaller index set $I$.
The finite groups with this proper... |
When is the Cayley embedding for infinite groups optimal? |
The remainder of the division of the polynomial $$W(x)=2x^{4}+px^{3}+9x^{2}+qx+5$$ by the polynomial
$$P(x)=2x-2$$ is equal to $10$. The remainder of the division of $W(x)$ by the polynomial $$Q(x)=x+1$$
is $22$.
What are the values of the parameters $p$ and $q$? |
What are the values of the parameters p and q? |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
You could also say that:
f=differentiable => f=continuous.
Since f=differentiable => $\exists$f '(x) = $\lim_{h\to 0}$$\frac{f(x+h)-f(x)}{h}$ => f '(x) = lim$_n$$\frac{f(x+1/n)-f(x)}{1/n}$.
Then from Heine's deffinition (?)* $\forall$(x$_n$)$_n$$\in$$\mathbb{R}$ and $\epsilon$>0: x$_n$ $\rightarrow$ x$_0$ $\Righ... |
The remainder of the division of the polynomial $$W(x)=2x^{4}+px^{3}+9x^{2}+qx+5$$ by the polynomial
$$P(x)=2x-2$$ is equal to $10$. The remainder of the division of $W(x)$ by the polynomial $$Q(x)=x+1$$
is $22$.
What are the values of the parameters $p$ and $q$?
After deriving the equation $$-p-q=6$$ by using... |
I was reading an explanation for a solution to an Olympiad problem as follows:
Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$. Prove that there exists a triangle with vertices in $M$ and with area at most equal with ... |
you can rewrite as below and solve the system of equation $p,q$
you may know ,if the remainder of the division of the polynomial $X(x)$ by the polynomial $P(x)$ is $10 $,it can be write as $$W(x)=P(x)R_{emainder}(x)+10$$so put $2x-2=0 \to x=1 $and you will have $$W(1)=(2x-2)R_{emainder}(x)+10|_{x=1} \\ \to W(1)=0... |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true un the following cases:... |
This is probably false but can't find a good counterexample.
Let $X$ be a compact space and consider the algebra $C(X)$ of all scalar-valued continuous functions on $X$ endowed with the supremum norm. Let $G$ be an additive subgroup of $C(X)$ that is discrete. (This implies that there is $\delta > 0$ so that $\|f-g\... |
Let $A(x) \in \mathbb{R}^{n\times n}$ be a matrix depending on a vector $x \in \mathbb{R}^{n}$. What can we say about the derivative of $f(x) = (\tau A(x)- I)^{-1}x_0$ with respect to $\tau > 0$ and $x_0 \in \mathbb{R}^{n}$? Are there any simplifications?
For the context of the problem, I am looking for a $\tau = \ta... |
I need to show that there exists a tournament of n vertices $n\ge3$ that has more than $\frac{n!}{2^{n-1}}$ Hamiltonian paths
Can I try to show that with induction?
I'm not so sure how to start with $n=3$
Can anyone give a hint?
Thanks
I have tried something else but not sure it's right
I have said that I n... |
Given a discrete-time LTI system as $$x_{i+1} = A x_{i} + B u_{i}$$ and suppose the feedback law $u_{i} = \mathcal{K}(x_{i})$ assymtotically stabilizes the closed-loop system. Now, consider the modified (i.e. delayed) input law $u_{i} = \tilde{\mathcal{K}}(x_{i-I})$ for some $I \gg 1$ (e.g. 10). Given an initial condit... |
Can you solve for g(x, t) from the following equation?: f(x) = $\int_{-\infty}^{\infty} g(x, t) dt $ or just simplify the integral using g(x, t). (we know f(x), but don't g(x, t), f and g are continuous)? |
I need to show that there exists a tournament of n vertices $n\ge3$ that has more than $\frac{n!}{2^{n-1}}$ Hamiltonian paths
Can I try to show that with induction?
I'm not so sure how to start with $n=3$
Can anyone give a hint?
Thanks
I have tried something else but not sure it's right
I have said that I n... |
> It seems tempting to conclude that the last term goes to zero as n goes to infinity. But I am not sure about it. Am I right or did I miss something?
You're right, this can be done directly and we only need a little bit more work to control the right term. The following inclusion of events is easy to check:
$$
\{... |
If
$g(x) = {\large\int} \ \sin(a x) \cos(b x) f(x) \ dx$
is known, is there a general form for
${\large\int} \ \frac{\sin(a x) \cos(b x)}{\frac{1}{f(x)} \ + \ c} dx$
that may contain $g(x)$ and possibly $g'(x)$?
I have a hunch that the antiderivative of $sin(x)/(e^{-x}+a)$,
https://www.wolfram... |
For each ${n}$, let ${(V_n,E_n)}$ be an Erdös-Renyi graph on ${n}$ vertices with parameter ${1/2}$ (we do not require the graphs to be independent of each other).
If ${|T_n|}$ is the number of triangles in ${(V_n,E_n)}$, (i.e. the set of unordered triples ${\{i,j,k\}}$ in ${V_n}$ such that ${\{i,j\}, \{i,k\}, \{j,k\}... |
I want to check the convexity of the following multivariate function:
$$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+\hat{b}\frac{s\lambda}{Q}+K\frac{\lambda}{Q} +c\lambda,$$
where $Q>0$, $s \ge 0$ and $h, b, \hat{b}, c, \lambda, K > 0$ and are known.
I want to know/prove that this function is convex.
Calcul... |
Thanks, everybody! I've glanced over your proof in order to not give myself any revealing details and just get the gist of it. As far as I know now, this is the proof:
Firstly, to prove the $\Leftarrow$ implication, we assume that A$\cup$B is connected. This means that there exist V,W in $\tau$ which fulfil the foll... |
$$I=\dfrac1{4a^2}\int\limits_0^1\int\limits_0^1\int\limits_0^1\dfrac{x\ln(x)}{\sqrt{yz}(1+yx)(1+zx)}\text dx\text dy\text dz$$
$$=\dfrac1{4a^2}\int\limits_0^1\int\limits_0^1\int\limits_0^1\dfrac{x\ln(x)}{\sqrt{yz}(1+yx)(1+zx)}\text dy\text dz\text dx=\bigg|y=v^2, z=w^2\bigg|$$
$$=\dfrac1{2a^2}\int\limits_0^1\int\limi... |
I want to check the convexity of the following multivariate function:
$$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+\hat{b}\frac{s\lambda}{Q}+K\frac{\lambda}{Q} +c\lambda,$$
where $Q>0$, $s \ge 0$ and $h, b, \hat{b}, c, \lambda, K > 0$ and are known.
I want to know/prove that this function is convex.
Calcul... |
The Hadamard transform is well known in the information theory and signal processing communities and can be viewed in some sense as a discrete step version of the Discrete Fourier Transform.
There does also exist a kind of a recursive analogue to the FFT factorization of the Fourier transform >
$$H_{m+1} = \begi... |
Does there exist a hadamard style transform over bits of digital numbers? Perhaps possible to be interpreted as $\mathbb Z^2$? |
I am interested in undirected, planar, connected graphs where every edge is in a 3-cycle. If there are 4 vertices then, up to isomorphism, there is only one I believe. That is (a, b), (a, c), (b, c), (b, d), (c, d). How many are there for vertices? |
How many different undirected planar connected graphs are there where every edge is in a 3-cycle? |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true un the following cases:... |
I am interested in undirected, planar, connected graphs where every edge is in a 3-cycle. If there are 4 vertices then, up to isomorphism, there is only one I believe. That is (a, b), (a, c), (b, c), (b, d), (c, d). How many are there for 5 vertices? |
How many different undirected, planar, connected graphs are there where every edge is in a 3-cycle? |
When we say "Every bounded function on [a,b] is Riemann Integrable on [a,b]", then does it mean that [a,b] is the domain of the function or any closed interval? I think that it is true considering the fact that [a,b] is the domain of f. But upon discussion, it turns out to be false when we consider Dirichlet function:-... |
Given $f,g$ two continuous functions, let's suppose that we know $f(x)$ but we don't know $g(x,t)$. Can you solve for $g(x, t)$ from the following equation $$f(x)=\int_{-\infty}^{\infty} g(x, t) dt$$ or just simplify the integral using $g(x, t)$? |
Let $f(x) = \frac{1}{x}$ be a function to and from the reals. Is $f$ bounded?
I want to prove using the definition. Let $M > 0$ be given. Then there exists a $\delta > 0$ such that:
If $|x-0| < \delta$ implies $|f(x)|=|\frac{1}{x}|=\frac{1}{|x|}>\frac{1}{\delta} = M$ whenever $\delta = \frac{1}{M}$
So is it c... |
Let $f(x) = \frac{1}{x}$ be a function to and from the reals. Is $f$ bounded?
I want to prove using the definition. Let $M > 0$ be given. Then there exists a $\delta > 0$ such that:
If $0<|x-0| < \delta$ implies $|f(x)|=|\frac{1}{x}|=\frac{1}{|x|}>\frac{1}{\delta} = M$ whenever $\delta = \frac{1}{M}$
So is it... |
Why does there exist such $V$? You've not tried to justify this, which of course would be one of the most important steps, and don't seem to have used assumptions on $X$. You've also not linked that to actually obtaining, say, $F_{1000}$ from $F_{999}$. For help with justifying the claim $\mathfrak{a}_n\subsetneq\mathf... |
The Hadamard transform is well known in the information theory and signal processing communities and can be viewed in some sense as a discrete step version of the Discrete Fourier Transform.
There does also exist a kind of a recursive analogue to the FFT factorization of the Fourier transform >
$$H_{m+1} = \begi... |
The system of ODEs
$$\dot{u} = bu - v + av(u^2 + v^2)$$
$$\dot{v} = u + bv + av(u^2 + v^2)$$
can be written in polar coordinates as
$$\dot{r} = br + ar^3$$
$$\dot{\phi} = 1$$
I know that in euclidean coordiantes I can calcualte the jacobian and evaluate it at the equilibrium (0,0) to get
$$ \begin{pm... |
If we have a $2$-state model (i.e. the simplest non-trivial example) in a hidden markov model, and some generated observation-data $\mathcal{O}$ from the algorithm for generating observations, is it always the case that the maximum of a given family $\Theta = \{\theta_i\}_{i \in I}$ of parameters, where $I$ is some ind... |
In special relativity, the underlying manifold is the vector space $\mathbb R^4$ equipped with the Minkowski metric. The position of a (point) particle located at $p\in\mathbb R^4$ is the point $p$ itself, at least with respect to the standard basis. A reference frame moving with constant speed $v$ is then another basi... |
It is known that if $n$ is prime, then $a^{n-1} \equiv 1 \mod n$ for all $2 \le a \le n-1$. Reading this backwards, we get: if we can find some $a$ between $2$ and $n-1$ such that $a^{n-1} \not\equiv 1 \mod n$, then surely $n$ is not prime. If not, then $n$ is prime.
Therefore, you take the number $n$ to be tested a... |
I'm trying to make a Tarski World for this structure:
$$\forall x (Triangle(x) \iff \exists y (Square(y) \land AboveOf(x,y)))$$
I think that it means the following:
- Element is a triangle if and only if there is a square below it. So every triangle must have squares below it.
- There is a square that is b... |
What does $\forall x (Triangle(x) \iff \exists y (Square(y) \land AboveOf(x,y)))$ imply? |
When we say "Every bounded function on $[a,b]$ is Riemann Integrable on $[a,b]$", then does it mean that $[a,b]$ is the domain of the function or any closed interval? I think that it is true considering the fact that $[a,b]$ is the domain of $f$. But upon discussion, it turns out to be false when we consider Dirichlet ... |
If the sample space is a subset of the real line, random variables $X$ and $Y$ are equal in distribution (denoted $X \overset{d}{=} Y$) if they have the same distribution functions:
$\mathbb{P}(X\leq x)= \mathbb{P}(Y\leq x)$ for every $x$. Also, we know to be equal in distribution, random variables need not be define... |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true un the following cases:... |
In what situations would using means other than arithmetic/geometric/harmonic make sense? |
Given a function $f\colon \mathbb R \to \mathbb R$ and that
$$
\lim_{t\to \infty}\int_{-t}^tf(x) \ dx=1,
$$
does the integral
$$
\int_{-\infty}^{\infty}f(x) \ dx
$$
necessarily converge? I've been trying to come up with a counterexample rather than a proof, but to no avail yet. |
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