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>How to rigorously evaluate $L:=\lim\limits_{n \to \infty } n \int_0^\frac{\pi}{2}1- \sqrt[n]{\sin(t)}dt$?
I tried to use the beat function since $$B(x,y)= 2\int_0^\frac{\pi}{2}\sin^{2x-1}(t)\cos^{2y-1}(t)dt$$
$$\int_0^\frac{\pi}{2}\sqrt[n]{\sin(t)}dt= 0.5 B\left(\frac{1}{2n} + \frac{1}{2},\frac{1}{2}\right)=... |
How to rigorously evaluate $\lim\limits_{n \to \infty } n \int_0^\frac{\pi}{2}1- \sqrt[n]{\sin(t)}dt$? |
Is the starting value $k$ ensure convergence for almost all polynomials? |
Sobolev space on the unit sphere is usually defined with the help of spherical harmonics, but is there any reference where the equivalence to the classical fractional sobolev space definition is shown? And preferably also the construction of the classical fractional sobolev space.
I would greatly appreciate it, thanks... |
I have this question and it seems like I am stuck somewhere. Could someone help me with this? It involves parametric curves with the parameter of the curve in terms of $\pi$.
>Find the cartesian equation of the curve, sketch the curve and indicate with an arrow the direction in which the curve is traced as the param... |
Let $f:\mathbb{R}^n_+\mapsto\mathbb{R}$ be concave and homogeneous of degree $\lambda\in(-\infty,0)\cup(0,1)$. Does the inequality $\lambda f(x)>0$ necessarily hold for all $x>0$? I suspect that this is the case, because I can't find a counter example. I know that for $\lambda=1$ the answer is NO as can be seen from th... |
I am trying to understand if there is a way we can define the $S_n$ group, just like we do for $D_n$. We know that $D_n$ is defined as $D_n = <r,s | s^2 = e, r^n = e, srs = r^{-1}>$.
Can we do a similar definition for $S_n$>?
I know that a transposition $(a,b)$ and a n-cycle (1,2,3...,n) generate $S_n$ if and only... |
Definition of Symmetric Group $\S_n$ like $D_n$? |
I am reading the paper [1]" P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." I am confused with the statement (iii) of the Theorem 7 which says "For every minimal non-empty-valued mapping $F : Z \rightarrow X$ where Z is an $\alpha$-fa... |
Is the statement of Theorem 7 is correct? |
I am trying to understand if there is a way we can define the $S_n$ group, just like we do for $D_n$. We know that $D_n$ is defined as $D_n =\langle r,s\mid s^2 = e, r^n = e, srs = r^{-1}\rangle$.
Can we do a similar definition for $S_n$?
I know that a transposition $(a,b)$ and a $n$-cycle $(1,2,3...,n)$ generate ... |
Presentation of the symmetric roup $S_n$ like $D_n$? |
What are the uses of defining the derivative as $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$? |
I was just messing with numbers when I created this formula.
But to understand the formula, you have to understand this: Multiplication is repeated addition, as $8*6$ is more or less $8+8+8+8+8+8$, or vice-versa. In the same way, exponents are repeated multiplication, as $5^3$, can be written as $5*5*5$, which can ... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ with $n>\max\{ q_1,q_2,\ldots,q_k\},\ $ such that $a m^i \equiv q_{i}
\pmod n\ \forall\ i\in \{1,2,\ld... |
Uses of the symmetric derivative $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$? |
I am trying to understand the conditions for time series $\{y_t\}$ to be stationary, i.e.:
- $E(y_t)=\mu$ is constant for all $t >0$ and
- $Var(y_t) = V$ is constant for all $t >0.$
I came across a [youtube video by ritvikmath](https://www.youtube.com/watch?v=oY-j2Wof51c&ab_channel=ritvikmath) explaining this conc... |
Having trouble understanding this proof from Pugh's Real Mathematical Analysis 2nd Edition (p. 288).
Theorem. A function $f:U → \mathbb{R}^m$ is differentiable at $p ∈ U$ if and only if
each of its components $f_i$ is differentiable at $p$. Furthermore, the derivative of its $i^{th}$
component is the $i^{th}$ com... |
A 2-dimensional simplex is just a triangle, and a 3-dimensional simplex is just a tetrahedron… therefore for convenience, I simply use the term triangle and tetrahedron in the following words.
We know that an arbitrary (Euclidean) triangle is either right or oblique, and an oblique triangle is either acute or obtuse.... |
What is the substantial generalization of the “acuteness” to 3-simplex? |
What this formula from econometrics textbook calculates? |
Let $f\in C^{2}[0,1]$ such that $f(0)=0,f(1/2)=f(1)=1$. Show that
$$\int_{0}^{1}[f''(x)]^2dx\ge 12.$$
My idea: try to find a function $f(x)$ such that the equality holds and use Cauchy-Schwarz. So I tried some "good" functions to achieve the minimum value.
1. Assume $f(x)=ax^3+bx^2+cx$ is a polynomial. Then ... |
How to prove this integral inequality $\int_{0}^{1}[f''(x)]^2dx\ge 12$? |
How to calculate Gal$(F(\mu_{p^\infty})/F(\mu_p))$ for a number field $F$? |
Let $f\in C^{2}[0,1]$ such that $f(0)=0,f(1/2)=f(1)=1$. Show that
$$\int_{0}^{1}[f''(x)]^2dx\ge 12.$$
My idea: try to find a function $f(x)$ such that the equality holds and use Cauchy-Schwarz. So I tried some "good" functions to achieve the minimum value.
1. Assume $f(x)=ax^3+bx^2+cx$ is a polynomial. Then ... |
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of... |
doesn't the independency phenomenon make a case for non-classical logic? |
What does this formula (from an econometrics textbook) calculate? |
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of... |
I'm looking at a problem I believe is combinatorial.
Find **all** possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \in \mathbb{N}$$
$$\sum a_k \cdot x_k = B, x_k \le l_k, x_k \in \mathbb{N}$$
$\mathbf{a}$, $\mathbf{l}$ (limit... |
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of... |
i have been working with generating functions and i am fed up with always doing the partial fraction decomposition. i have found the following trick that can (hopefully) skip
pfd.
$$
\frac{z^j}{(1-z)^k} \iff \sum_{n\geq 0} \binom{n+k-1-j}{k-1} \cdot z^n
$$
i would like to prove it, however i seem to lack the sk... |
It is well known that handles of indices k and (k+1) may be cancelled if the attaching sphere of (k+1)-handle *intersect* the belt sphere of k-handle *at exaclty one point*. Is it also necessary condition for the cancelling? In Rourke and Sanderson book I found that for dimension $n\geq 6$ and for sufficiently small ... |
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of... |
I have the following basic question regarding continuous functions.
Is the following statement correct?
> Given functions
>
> \begin{align*} h & : A \times C \to B \times D ,\\ f &: A \to B ,\\ g &: C
\to D , \end{align*}
>
> if $h$ and $f$ are continuous, then $g$ is continuous.
I know that if $f$ and... |
I'm trying to calculate this integral using residue theorem:
$$
\int_{-\infty}^{\infty}\text{d}\omega\frac{f(\omega)}{(\omega+a+i\epsilon)(\omega-a-i\epsilon)}
$$
where $f(\omega)$ is bounded in the whole complex plane and $f(a+i\epsilon)\neq f(-a-i\epsilon)$.
It seems to me that the integrand is proportional to $... |
I have an integral which I have used contour integration to calculate it. But it seems that I have some contradiction here? |
Let $A$,$B$ be two positive semidefinite symmetric matrices on $\mathbb{R}^d$ with the same eigenvectors. Let $P$ be an orthogonal projection. Does it hold that
$$
\|APB\| \leq \|AB\|,
$$
where $\|\cdot\|$ denotes the operator norm. If $P$ was on the left or right we would directly have $\|ABP\| \leq \|AB\|\|P... |
I have the following basic question regarding continuous functions.
Is the following statement correct?
> Given functions
>
> \begin{align*} h & : A \times C \to B \times D ,\\ f &: A \to B ,\\ g &: C
\to D , \end{align*}
>
> with $h := (f, g)$, if $h$ and $f$ are continuous, then $g$ is continuous.
I ... |
I'm looking at a problem I believe is combinatorial.
Find **all** possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \in \mathbb{N}$$
$$\sum a_k \cdot x_k = B, x_k \le l_k, x_k \in \mathbb{N}$$
$\mathbf{a}$, $\mathbf{l}$ (limit... |
Is the statement of Theorem 7 is correct? The points of some subset which is of the second Baire category in every non-empty open subset of Z means? |
Let $A$ be a Noetherian commutative ring. Suppose that there exists a maximal ideal $\mathfrak{m}$ of $A$ such that there exists an isomorphism $A/\mathfrak{m}\cong A_{\mathfrak{m}}$ of rings where $A_{\mathfrak{m}}$ is the localization of $A$ at $\mathfrak{m}$. **Is it true that $\mathfrak{m}$ is the zero ideal? If no... |
When do we have $A/\mathfrak{m}\cong A_{\mathfrak{m}}$? |
I'm looking at a problem I believe is combinatorial.
Find **all** possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \in \mathbb{N}$$
$$\sum a_k \cdot x_k = B, x_k \le l_k, x_k \in \mathbb{N}$$
$\mathbf{a}$, $\mathbf{l}$ (limit... |
Presentation of the symmetric group $S_n$ like $D_n$? |
I am reading Rudin's "Principles of Mathematical Analysis" and I am stuck at the following theorem
> Theorem 8.14:
>
> If, for some $x$, there are constants $\delta > 0$ and $M < \infty$ such that:
> $$\left| f(x + t) - f(x) \right| \leq M|t|$$
>
> for all $t \in (-\delta, \delta)$, then
> $$\lim_{N\rightarrow... |
Numbers are represented using binary digits "bits" which are integers in the set $\{0,1\}$.
Positive integers for example, are usually encoded as a sequence of bits so that which power of 2 they encode correspond to the order which they are stored. For example the number $a$ represented by a binary vector $$[b_{N-1... |
How can I design a non-integral-power efficient range coding that is statistically "fair" over the whole range? |
I'm looking at a problem I believe is combinatorial.
Find **all** possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \in \mathbb{N}$$
$$\sum a_k \cdot x_k = B, x_k \le l_k, x_k \in \mathbb{N}$$
$\mathbf{a}$, $\mathbf{l}$ (limit... |
Doing my homework on functional analysis I faced with the following problem.
Let $\displaystyle (Uf) (s)=\int\limits_{-1}^{1}\frac{f(t)}{|s-t|^{5/6}}dt$.
Using Young inequality it can be shown that $U$ is bounded operator from $L^3(-1,1)$ to $L^r(-1,1)$ with $r<6$ and it is compact. If $r>6$ then it is easy to see ... |
I want to find the strong form of Euler-Lagrange equation of the following variational problem( $\partial\Omega$ smooth ):
$$\min_{u\in H^2(\Omega)} I[u]:=\int_\Omega (\partial_x^2u)^2 + 2(\partial_x\partial_yu)^2+(\partial_y^2u)^2 \,dx\,dy$$
I can get the weak form:
$$\forall v\in H^2(\Omega),\int_\Omega \partial... |
For a strongly regular simple undirected graph to be disconnected, does each of its component has to be complete (aka $\mu=0$)? |
You can split the integral in to two separate ones
$$
\int_0^{\frac12} (f''(x))^2 \, \mathrm{d}x+\int_{\frac12}^{1} (f''(x))^2 \, \mathrm{d}x
$$
and find the smallest function for both using calculus of variations and combine them piecewise. The reason that you have to do this is because calculus of variations only... |
I am reading the [Paper][1] " P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before the Theorem 2 of the paper, the authors of the paper mentioned "The set of points of continuity $C(f)$ is not necessarily residual in Z. It is h... |
How the set $C(f)\cap V$ is a second category in $V$? |
How to proof a sigma algebra is equal to the union of two given subsets? |
I cannot give a satisfying answer, but I had a look in Neustrev's book which convinced me that the author's exposition in not really clear.
Quote from Chapter 3:
> Here we give a detailed answer to the following fundamental question: Given an abstract $\mathbb R$-algebra $\mathcal F$, find a set (smooth manifold) $... |
I cannot give a satisfying answer, but I had a look in Neustrev's book which convinced me that the author's exposition in not really clear.
Quote from Chapter 3:
> Here we give a detailed answer to the following fundamental question: Given an abstract $\mathbb R$-algebra $\mathcal F$, find a set (smooth manifold) $... |
The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation.
\begin{equation}
d X_t = \mu X_t \cdot dt + \sigma X_t^\beta \cdot d B_t \tag{1}
\end{equation}
where $\mu,\sigma,\beta$ are positive real parameters and $B_t$ is t... |
$\newcommand{\ga}{\gamma}$Let
\begin{equation*}
L(s):=\int_0^{\pi/2} \frac{\sin^2(sx)}{\sin^2x}\,f(x)\,dx- \frac\pi2f(0)s
-\frac{f'(0)}2\ln s,
\end{equation*}
\begin{equation*}
R:=\int_{0}^{\pi/2} \frac{f(x)-f(0)-f'(0)x}{2\sin^2x}\,dx+\frac{f'(0)}2\,(1+\ln2+\ga).
\end{equation*}
We have to show that
\be... |
I was just messing with numbers when I created this formula.
But to understand the formula, you have to understand this: Multiplication is repeated addition, as $8*6$ is more or less $8+8+8+8+8+8$, or vice-versa. In the same way, exponents are repeated multiplication, as $5^3$, can be written as $5*5*5$, which can ... |
I am a bit new to functional analysis and I stumbled upon this problem that confuses me.
Consider the space $X$ of bounded, non-decreasing, right-continuous functionals $F$ on $[0,1]$. Specifically, the space of cumulative distribution functions on $[0,1]$ is a subset of $X$. I have a linear functional $\varphi$ def... |
I was just messing with numbers when I created this formula.
But to understand the formula, you have to understand this: Multiplication is repeated addition, as $8*6$ is more or less $8+8+8+8+8+8$, or vice-versa. In the same way, exponents are repeated multiplication, as $5^3$, can be written as $5*5*5$, which can ... |
If I have two matrices of type $n\times m,$ is there a way to compare their singular values?
For example, if $A$ has singular values $\lambda_1,\cdots,\lambda_n$ and $B$ has singular values $\sigma_1,\ldots,\sigma_n,$ is there a way to compare $\lambda_1+\cdots+\lambda_n$ and $\sigma_1+\cdots+\sigma_n$ or $\lambda_1+... |
If I have two matrices of type $n\times m,$ is there a way to compare their singular values?
For example, if $A$ has singular values $\lambda_1,\cdots,\lambda_n,$ $\lambda_1\geqslant\lambda_2\geqslant\cdots\geqslant\lambda_n\geqslant0,$ and $B$ has singular values $\sigma_1,\ldots,\sigma_n,$ $\sigma_1\geqslant\sigma_... |
I'm studying from a Deep Learning book (Ian Goodfellow et al).
At page 256 the text explains that, considering a set of $k$ regression models, each produces an error $ϵ_i$ for every example, drawn from a zero-mean multivariate normal distribution with variances $E[ϵ_i^2]=v$ and covariances $E[ϵ_iϵ_j]=c$. Then the err... |
I am reading the Eisenbud's commutative Algebra book, proof of Theorem 13.9 and stuck at some statement.
Let me first raise a question that I think is relevant.
**Q.** Let $R$ be a normal domain with finite extension of fields $K' \subseteq K'' \subseteq K(R) $ ( $K(R)$ is the quotient field ), possibly si... |
Is there a good way to solve this problem simply?
I set $t=x+\frac{1}{x}$ but this will make the equation very complicated.
$$f\left( x+\frac{1}{x} \right)=\frac{x+x^3}{1+x^4}$$ find $f(x)$
|
For a strongly regular simple undirected graph to be disconnected, does each of its component has to be complete (aka $\mu=0$)?
My reasoning is that two vertices disconnected components have 0 common neighbours, thus that $\mu$ has to be zero. Then by $${\displaystyle (v-k-1)\mu =k(k-\lambda -1)}$$
either $k=0$ (di... |
Let a,b,c≥0. Prove that:
>a/b+b/c+c/a+3³√abc/(a+b+c)≥4
This is a problem from Samin Riasat's Olympiad Inequalities worksheet. The worksheet gives a hint as to prove and use a/b+b/c+c/a≥(a+b+c)/³√abc. I've done it, substituted it in the original inequality and now I have to prove:
>(9AM²+3GM²)/(3AM•GM)≥4
Whe... |
Can you please tell me where you found the above formula?\
Seems to be something wrong with your formula.\
You may use this formula:
$$f(\tilde{f},T,\rho,n)\approx \frac{\rho ^T}{n}\cdot \sum _{k=0}^{n-1} \Re\left(\tilde{f}\left(\rho\cdot \exp \left(\frac{i ((2 \pi ) k)}{n}\right)\right)\cdot \exp \left(\frac{i ((... |
$$\int_0^{+\infty}\frac{(\sin x)^a}{x^b}dx=?,\,\text{suppose} \ a,b \in \mathbb{Z^+},a-b\geq 0,b\geq 2$$
I know how to calculate if $a-b\equiv 0 \pmod 2$，but I don't know
what to do when $a-b \equiv 1\pmod 2$. |
Let $a,b,c\ge0$. Prove that:
$$\frac ab+\frac bc+\frac ca+\frac{3\sqrt[3]{abc}}{a+b+c}\ge4$$
This is a problem from Samin Riasat's Olympiad Inequalities worksheet. The worksheet gives a hint as to prove and use $$\frac ab+\frac bc+\frac ca\geq\frac{a+b+c}{\sqrt[3]{abc}}$$
I've done it, substituted it in the orig... |
It can be shown that when X, Y, and Z are all Banach spaces (or at least when X or Y are Banach spaces) over the number fields R or C, and when B : X×Y →Z is a bilinear function, the continuity of B for each component implies its overall continuity.
However, I'm curious about the scenario when X, Y, and Z are merely... |
What is the geometrical interpretation of $1 - i \cdot \text{Im}\left(\frac{1}{z}\right) = z$ in the Gaussian number plane?
I get that the first part is the real part 1 with no imaginary part so it lies on the real axis, so it basically means I have a point that is at 1 on the real axis and dependent on the value of... |
What is the geometrical interpretation of $1 - i \cdot \text{Im}\left(\frac{1}{z}\right) = z$ in the Gaussian number plane? |
I'm trying to calculate this integral using residue theorem:
$$
\int_{-\infty}^{\infty}\text{d}\omega\frac{f(\omega)}{(\omega+a+i\epsilon)(\omega-a-i\epsilon)}
$$
where $f(\omega)$ is bounded at infinity and $f(a+i\epsilon)\neq f(-a-i\epsilon)$.
It seems to me that the integrand is proportional to $\frac{1}{\omega... |
For a strongly regular simple undirected graph to be disconnected, does each of its component has to be complete (aka $\mu=0$)?
My reasoning is that two vertices disconnected components have 0 common neighbours, thus that $\mu$ has to be zero. Then by $${\displaystyle (v-k-1)\mu =k(k-\lambda -1)}$$
either $k=0$ (di... |
Does this show $\bar{Y} is a sufficient statistic? |
Does this show $\bar{Y}$ is a sufficient statistic? |
Kindly help me with the following integral :
$
I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad
$
($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also in Prudnikov,
$\int_{-1}^{+1}\, dx\, e^{ipx} P_l(x) = \sqrt{\frac{2 \pi}{p}} \, e^{\frac{i l \pi}{2}} J... |
Can you please tell me where you found the above formula?\
Seems to be something wrong with your formula.\
You may use this formula:
$$f(\tilde{f},T,\rho,n)\approx \frac{\rho ^T}{n}\cdot \sum _{k=0}^{n-1} \Re\left(\tilde{f}\left(\rho\cdot \exp \left(\frac{i ((2 \pi ) k)}{n}\right)\right)\cdot \exp \left(\frac{i ((... |
I have the following basic question regarding continuous functions.
Is the following statement correct?
> Given functions
>
> \begin{align*} h & : A \times C \to B \times D ,\\ f &: A \to B ,\\ g &: C
\to D , \end{align*}
>
> with $h := (f, g)$, both product spaces endowed with the product topology and al... |
A conical buoy weighing $B$ lbs with its vertex $A$ ft below the surface has its top sticking up $\frac{A}{3}$ ft above the surface. How much work to lower the top of the buoy to surface level??
If I add $\frac{37B}{27}$ lbs of weight to the top it causes a displacement of $\frac{A}{3}$. Work = (force)*(displacement... |
I'm trying to calculate this integral using residue theorem:
$$
lim_{\epsilon \rightarrow 0}\int_{-\infty}^{\infty}\text{d}\omega\frac{\omega^2-k^2v_F^2}{(\omega+kv+i\epsilon)(\omega-kv-i\epsilon)}\frac{1}{(\omega-v_q+i\epsilon)(\omega-v_q-i\epsilon)}
$$
It seems to me that the integrand is proportional to $\frac... |
I have the following basic question regarding continuous functions.
Is the following statement correct?
> Fix nonempty toplogical spaces $A, B, C, D$ with $A \times C$ and $B \times D$ endowed with the product topology. Also, fix functions
>
> \begin{align*} h & : A \times C \to B \times D ,\\ f &: A \to B ,\\... |
I was going through the proof of Fundamental theorem of algebra using 'Liouvilles theorem'. **I have two queries/doubt in a proof**.
1. In the proof we assume that, $p(z) =0$ has no roots in $\mathbb{C}$ and then defined $f(z) =\frac{1}{p(z)}$ for all $z\in\mathbb{C}$. From this, **doesn't it directly follow that**... |
You can split the integral in to two separate ones
$$
\int_0^{\frac12} (f''(x))^2 \, \mathrm{d}x+\int_{\frac12}^{1} (f''(x))^2 \, \mathrm{d}x
$$
and find the smallest function for both using calculus of variations and combine them piecewise. The reason that you have to do this is because calculus of variations only... |
I've got this theorem right here. It involves two arbitrary lines $AB$ and $CD$ and two transversals $AC$ and $BD$, intersecting **in between these lines**.
The theorem (or lemma, I don't know) in question states that $\alpha+\beta = \varphi + \theta$.
[![The drawing of the theorem / lemma][1]][1]
The simplest... |
Yoni Rozenshein (https://math.stackexchange.com/users/36650/yoni-rozenshein), Currently, what is the largest publicly known prime number such that all prime numbers less than it are known?, URL (version: 2013-03-14): https://math.stackexchange.com/q/330221
"Given current computing abilities, I'd guess your prime is so... |
I have the following partitioning condition,
$$\text{p(n|parts in N)}=\text{p(n|distinct parts in M)}, \text{for n}\ge1$$
Suppose I construct a set covered by the theorem $N=(1,3,6)$, for $$ \begin{align} n &= 1 \space 2 \space 3 \space 4 \space 5 \space 6 \\ p(n|\text{parts in (1,3,6)}) &= 1 \space 1 \space 2 \spa... |
What do parts in partition refer to? |
I am not sure if the complete answer is known, but the interior and the exterior of the image $E$ of the exponential map are known:
1. The interior of $E$ consists of all matrices in $SL(n,\mathbb R)$ which have no negative eigenvalues.
2. The exterior of $E$ consists of all matrices in $A\in SL(n,\mathbb R)$ wh... |
The finite-dimensional representations of $\frak{sl}_2$ can be constructed explicitly as partial differentiation of polynomials in two variables. Does a similar explicit construction exist for the finite-dimensional representations of $\frak{sl}_3$? |
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$.
Let $S = \{i\leq n: X_i\in B_n\}$.
I want to show that
$$
\frac{1}{|S|}\sum_{i\in S... |
For region $\Omega$ with smooth boundary, $f$ a vector field smooth enough, if $$\int_{\partial\Omega} \vec{f}\cdot\nabla v=0,\forall v\in H^2(\Omega),$$
then what can we obtain from it about $f$?
This is actually from <https://math.stackexchange.com/questions/4889518/the-boundary-condition-of-strong-form-of-a-4th-... |
I have the following partitioning condition,
$$\text{p(n|parts in N)}=\text{p(n|distinct parts in M)}, \text{for n}\ge1$$
*Where N is any set of integers such that no element of N is a power of two times an element of N, and M is the set containing all elements of N together with all their multiples of powers of tw... |
Calculate: $I = \int_{\substack{x_1 \geq 0, \ldots, x_n \geq 0 \\ , x_1+\cdots+x_n \leq 2}} \sqrt[n]{x_1 \cdots x_n\left(x_1+\cdots+x_n\right)} d x_1 \cdots d x_n$
Attempt: Perform the substitution $ x_i = 2y_i $, the jacobian will be $ 2^n $ and so $I=2^{n+1+\frac{1}{n}} \cdot \int_{\substack{x_1 \geq 0, \ldots, ... |
>In a given grocery store, apples have an average weight of $194$g and standard deviation of $40$g. Suppose that their weights are independent and follow a normal distribution.
If a customer requests $1$ kg of apples and given that the employer chooses randomly each apple and continues until the total weight is the... |
$f(x,y)$ is a real polynomial such that in the equation $f(x,y)=0$ we can express $x$ in $y$ with composition of polynomial functions and square roots, and can express $y$ in $x$ with composition of polynomial functions and square roots.
The curve $f(x,y)=0$ on $\Bbb R^2$ has $n$ connected components.
Is it true ... |
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