instruction stringlengths 12 30k |
|---|
Let $\sum_2=\{s=(s_0s_1s_2)|s_j=0\ \text{or} \ 1\}$, which is a sequence space of $0$ and $1$'s.
We make $\sum_2$ into a metric space as follows, For two sequences $s=(000\dots)$ and $r=(1010\dots)$, then the distance between them is
$$d[s,r]=\sum_{i=0}^\infty \frac{1}{2^{2i}}=\frac{1}{1-\frac14}=\frac43$$
B... |
Why is this function almost Lipschitz? |
According to [Wikipedia][1], if the Laplace transform
$$\mathcal{L}\{f\}(s)=\lim_{b\to\infty}\int_0^b e^{-st}f(t)\,dt$$
converges for some $s_0\in\mathbb{R}$, then it automatically converges for all $s\in\mathbb{R}$ with $s>s_0$. (I'm considering the Laplace transform on the real line.) How can I prove this? If
$$\l... |
Let $\sum_2=\{s=(s_0s_1s_2)|s_j=0\ \text{or} \ 1\}$, which is a sequence space of $0$ and $1$'s.
We make $\sum_2$ into a metric space as follows: for two sequences $s=(000\dots)$ and $r=(1010\dots)$, then the distance between them is
$$d[s,r]=\sum_{i=0}^\infty \frac{1}{2^{2i}}=\frac{1}{1-\frac14}=\frac43$$
s... |
Given a function $g(\boldsymbol{x})$,
$g(\boldsymbol{x})=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $\boldsymbol{x} \in \mathbb{R}^{n}$, and $f_{i}$ is a $L_{i}$-smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and cove... |
Given a function $g(\boldsymbol{x})$,
$g(\boldsymbol{x})=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $\boldsymbol{x} \in \mathbb{R}^{n}$, and $f_{i}$ is a $L_{i}$-smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and cove... |
Given a function $g(\boldsymbol{x})$,
$g(\boldsymbol{x})=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $\boldsymbol{x} \in \mathbb{R}^{n}$, and $f_{i}$ is a $L_{i}$-smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and cove... |
you are watering a garden. The height h (in feet) of water spraying from the garden hose can be modeled by h(x)=−0.1x^2+0.7x+3, where x is the horizontal distance (in feet) from where you are standing. You raise the hose so that the water hits the ground 1 foot farther from where you are standing. Write a function that... |
Why is this the correct solution? |
I have a proof that if $A$ and $B$ are in the Event Space ($F$) then $A-B\in F$. However, I don't think it's very elegant. Do people have another proof that is perhaps more elegant?
Here is the proof:
$A,B \in F $
$\implies A \cup B \in F$ ... (given)
$\implies (S-A)\cup B \in F$ ... ($X\in F\implies S-A \in ... |
Better proof that if $A$ and $B$ are in the Event Space ($F$) then $A-B\in F$? |
You are watering a garden. The height $h$ (in feet) of water spraying from the garden hose can be modeled by $h(x)=−0.1x^2+0.7x+3$, where $x$ is the horizontal distance (in feet) from where you are standing. You raise the hose so that the water hits the ground $1$ foot farther from where you are standing. Write a funct... |
Why is my answer not correct? |
Let $G$ be an open set in $R^{2}$ (the whole plane),$F$ be a closed set in $R^{2}$. Is it possible that$G-Q^{2}=F-Q^{2}$ in nontivial situations ? ($Q^{2}$ is the set of all rational points and $G-Q^{2}=F-Q^{2}$ can't be empty or $R^{2}-Q^{2}$. |
Given a standard one-dimensional Brownian motion $B$, we define the hitting time as
\begin{align*}
\tau_a:=\inf\{t\geq0 \, : \, B_t = a\},
\end{align*}
with $a \in \mathbb{R}$. From the continuity of the Brownian motion, we have that
\begin{align*}
B_{\tau_a}= a.
\end{align*}
Let $\tau_a^{-1}$ the right inverse... |
The question specifies that you **raise** the hose. This implies a vertical dilation or translation.
Of course, there are multiple transformations which achieve the required result, but in the *context* of the problem I believe the writer of the problem was after a vertical translation.
In a related matter - I ha... |
I have a proof that if $A$ and $B$ are in the Event Space ($F$) then $A-B\in F$. However, I don't think it's very elegant. Do people have another proof that is perhaps more elegant?
Here is the proof:
$A,B \in F $
$\implies A \cup B \in F$ ... ($X_1,X_2,\in F \implies X_1\cup X_2\in F$)
$\implies (S-A)\cup B ... |
Combinatorial interpretation for Vandermonde's identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$? |
I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq \mathbf R^d$, we have two Monge problem:
$$
\min_{T\in L(P_1,Q... |
The answers from @Belgi and @joriki are erroneous because they only find the shortest walk (i.e it can go through an edge more than once): The shortest path in the bipartite graph can pass through both $(i, j')$ and $(i', j)$ in $E'$, which result in $(i, j)$ taken twice in $G$ (with notations from @Belgi answer).
T... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_... |
> Let $F:\mathbb{R}^2\to\mathbb{R}, F(x,y)=\sin |x^2+y|$.
>
> a) Prove that $F(x,y)$ is not continuously differentiable at $(0,0)$.
>
> b) Is $F$ differentiable at $(0,0)$?
I can do part b):\
If $F$ is differentiable at $(0,0)$ then $\dfrac{\partial F}{\partial y}(0,0)$ must exist. However,
$$
\dfrac{\pa... |
My program spits out links as a crossing diagram. In fact, it *does* have *some* topological data (not only graphic) since the links are constructed as sums of tangles, and instead of drawing line A to line B, I could "follow" this lines and see if at the end some Dowker code results (but my brain already hurts now). M... |
Let M be a matrix with 20 rows and 21 columns, containing the following elements
$M[i][j]=i*i$ if $i=j$
$M[i][j]=min(i,j)$ if $i≠j$
for $1\leqslant i\leqslant 20, 1\leqslant j\leqslant 21$.
What is the sum of all elements in the matrix?
**My approach:**
I drew a $4\times 5$ matrix with the specified ele... |
Consider the system $\dot x = f(x)$ with $f(0)=0$, f continuously differentiable and
$\frac{\partial f}{\partial x}(x)^TP+P\frac{\partial f}{\partial x}(x)\preceq-I$ for all x and some $P=P^T\succ0$.
How to verify that f(x) can be written as $f(x)=\int_0^1\frac{\partial f}{\partial x}(\sigma x)xd\sigma$, and use t... |
Let M be a matrix with 20 rows and 21 columns, containing the following elements
$M[i][j]=i*i$ if $i=j$
$M[i][j]=min(i,j)$ if $i≠j$
for $1\leqslant i\leqslant 20, 1\leqslant j\leqslant 21$.
What is the sum of all elements in the matrix?
**My approach:**
I drew a $4\times 5$ matrix with the specified ele... |
Let M be a matrix with 20 rows and 21 columns, containing the following elements
$M[i][j]=i*i$ if $i=j$
$M[i][j]=min(i,j)$ if $i≠j$
for $1\leqslant i\leqslant 20, 1\leqslant j\leqslant 21$.
What is the sum of all elements in the matrix?
**My approach:**
I drew a $4\times 5$ matrix with the specified e... |
Let M be a matrix with 20 rows and 21 columns, containing the following elements
$M[i][j]=i*i$ if $i=j$
$M[i][j]=min(i,j)$ if $i≠j$
for $1\leqslant i\leqslant 20, 1\leqslant j\leqslant 21$.
What is the sum of all elements in the matrix?
**My approach:**
I drew a $4\times 5$ matrix with the specified e... |
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
(a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
(b) $\sum_{k=1}^{n} (-1)^{k-1} a_k \binom{n}{k} = \sum_{k=1}^{n} a_k$.
I trie... |
I solved the following equation yesterday for fun and I have the exact result:
$\sin(x)^{\sin(x)}=2$ — $x=\frac{\pi}{2}-i\ln (\exp^{W(\ln(2))}\pm{\sqrt{\exp^{2W(\ln(2))}-1}}) +2k\pi; k\in\mathbb{Z}$
: $i$ is the imaginary unit and $W$ is the Lambert W function
But how can I solve $\sin(x)^{\cos(x)}=2$ now?
Thank... |
How do I solve the following equation exactly: $\sin(x)^{\cos(x)}=2$? |
This is an old question, but I would like to provide another angle which touches directly the statement used in the Wikipedia article and whose gist has been hinted at it in the other answers:
> -finiteness can be compared in this respect to separability of topological spaces
In a separable topological space a c... |
Let $M$ be a smooth manifold. Denote $C_M^\infty$ to the sheaf of real-valued functions on $M$. For a smooth vector bundle $\pi:E\to M$, denote $\Sigma_E$ to the sheaf of smooth sections of $\pi$ (it is a $C_M^\infty$-module). I am wondering the following:
> If $E\to F\to G$ is an exact sequence of smooth vector bun... |
For a local ring $R$, we have a bijection $\mathbf P_R^n(R) \cong (R-\left\lbrace 0\right\rbrace)/R^\times$. This allows us to refer to points $\text{Spec} R \to \mathbf P_R^n$ using homogeneous coordinates. I know the same is not true for general rings $R$.
However, I am wondering about the case where we have an in... |
> -finiteness can be compared in this respect to separability of topological spaces
In a separable topological space a countable dense set can be used to approximate any element of the topological space, in the sense that any environment around any element contains a member of the countable dense set. This also all... |
Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ evolves as
$$ X_t = X_0 \exp\left[\left(\mu-\frac{\sigma^2}{2}\right)t + \sigma W_t\right] $$
where $W_t$ is a Wiener ... |
Assume that the non-compact $n$-dimensional Riemannian manifold $M$ satisfies a curvature-dimension $\mathrm{CD}(K,N)$, $K \le 0$, with respect to a measure $\mu = e^{-V} d\mathrm{Vol}$; i.e.
\begin{equation*}
\mathrm{Ric} + \mathrm{Hess} V - \frac{\nabla V \otimes \nabla V}{N - n} \ge Kg.
\end{equation*}
I wonder ... |
I solved the following equation yesterday for fun and I have the exact result:
$\sin(x)^{\sin(x)}=2$ when $x=\frac{\pi}{2}-i\ln (\exp^{W(\ln(2))}\pm{\sqrt{\exp^{2W(\ln(2))}-1}}) +2k\pi; k\in\mathbb{Z}$
: $i$ is the imaginary unit and $W$ is the Lambert W function
But how can I solve $\sin(x)^{\cos(x)}=2$ now?
Th... |
I wonder if there is any efficient way to determine whether the hash values of $h(x)=3x\ mod\ 2^{64}$ are uniformly distributed for $x$ being uniformly distributed in $[0,2^{512}-1]$?
I try to approach this by definition: for $y\in[0,2^{64}-1]$, $P(h(x)=y)=P(x=\frac y3+k\cdot\frac{2^{64}}3)$ for a set of intergers $... |
Let $\mathcal{X}$ be a collection of extended metric spaces. Let $\Pi\mathcal{X} = \Pi_{X \in \mathcal{X}} X$. For $1 \leq p \leq \infty$, define the following extended metric on $\Pi\mathcal{X}$:
$d_p: \Pi\mathcal{X} \times \Pi\mathcal{X} \to [0, \infty], d_p((x_X)_{X \in \mathcal{X}}, (y_X)_{X \in \mathcal{X}}) = ... |
Define a function: $\mathbb N \times \mathbb N \to \mathbb Q$:
$$f((a,b)) = a + \frac{1}{b}$$
choose $(a,b) \in \mathbb{N}, \ (c,d) \in \mathbb{N}$
workings:
$$
f((a,b)) = a + \frac{1}{b} \\
f((b,c)) = c + \frac{1}{d} \\
$$
Suppose $f((a,b)) = f((c,d))$:
$$
a + \frac{1}{b} = c + \frac{1}{d} \\
d(... |
Proving that $D_1$ and $D_2$ are open is not so hard, for instance by noting that for any $a \in \mathbb{C}$ the map $\mathbb{C} \to \mathbb{R} : z \mapsto |z^2 - a|$ is continuous. But I don't think you struggle with this?
For $D_1$ your intuition is correct! To prove it is not connected it suffices of course to f... |
I solved the following equation yesterday for fun and I have the exact result:
$$\sin(x)^{\sin(x)}=2 \iff x=\frac{\pi}{2}-i\ln (\exp^{W(\ln(2))}\pm{\sqrt{\exp^{2W(\ln(2))}-1}}) +2k\pi,$$
for $k\in\mathbb{Z}$, where $i$ is the imaginary unit and $W$ is the Lambert $W$ function.
But how can I solve $\sin(x)^{\cos(x... |
Consider for the short exact sequence (s.e.s)
\begin{align*}
0 \to \mathbf{Z} \xrightarrow{\cdot 2} \mathbf{Z} \xrightarrow{\pi} \mathbf{Z}/2\mathbf{Z} \to 0
\end{align*}
the covariant left exact functor $\text{Hom}(\mathbf{Z}/2\mathbf{Z}, -): \text{Ab} \to \text{Ab}$. I want to exhibit the long exact sequence asso... |
In orthogonal cartesian coordinates $c\cdot c=|c|^2$ is the square of length of the vector $c$.
In oblique coordinates the square of the length of $c = c * g * c$ where g is the metric tensor and $g*c$ is a covector.
If $e_x$ and $e_y$ are the basis vectors of vector $c$ then how does one calculate the new basis ... |
Let $F$ be a close set in the plane and $G$ be an open set in the plane. Denote $C$ by the set of rational points in the plane.
Prove that $F-C=G-C$ if and only if $F-C$ and $G-C$ are both empty or $R^{2}-C$.
I'm stuck in the stage that I have to prove the union of a set of rational numbers and an open set can't be c... |
Could the union of a set of rational numbers and an open set be close? |
It's a follow up of https://math.stackexchange.com/questions/4883613/does-three-iteration-are-sufficient-to-use-newtons-method
Come back on the previous question :
$x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0\tag{I}$
... |
Is the starting value $C$ ensure convergence for almost all polynomials? |
Trying to learn some basics of algebraic topology and wanted to ask clarifications about this sentence of Hatcher ( third page, just the beginning ;) ): 'It is true in general that two spaces X and Y are homotopy equivalent if and only if there exists a third space Z containing both X and Y as deformation retracts'. He... |
Define a function: $\mathbb N \times \mathbb N \to \mathbb Q$:
$$f((a,b)) = a + \frac{1}{b}$$
choose $(a,b) \in \mathbb{N}, \ (c,d) \in \mathbb{N}$
workings:
$$
f((a,b)) = a + \frac{1}{b} \\
f((b,c)) = c + \frac{1}{d} \\
$$
Suppose $f((a,b)) = f((c,d))$:
$$
a + \frac{1}{b} = c + \frac{1}{d} \\
d(... |
How to verify Slater's conditions for a convex optimisation problem under box constraints? |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined. I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{-1}\... |
What is the Asymptotic Oder of the Sum of Random Variables with non-finite Moments? |
I'm trying to solve the following definite integral:
$\int_{R-r}^{R}sin^{-1}(\frac{x^2+R^2-r^2}{2xR})dx$
I know that: $0<r<R$. But I'm stuck after that. Wolfram doesn't seem to have any ideas either. Any suggestions to procced forward? |
Assumptions for reaching the [logistic map equation][1]
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
... |
I've been trying to better understand the primitives of the function $ f(x)=1/x^3 $ defined on the interval $ R−{0} $. In my research, I attempted to find two primitives that do not differ by a constant, but I couldn't come up with a satisfactory example.
In my attempt to find two primitives that do not differ by a... |
So the question is to find the minimum value of given modulus function.
$$f(x)=|x-1|+|3x-1|+|5x-1|+|7x-1|+|9x-1|+|11x-1|$$
So my first approach was to just make a graph of it
On making the graph the minimum value was at $x=1/9$. And also rechecked the value in Wolfram alpha and it was indeed $1/9$
But now let'... |
Let me begin by a remark concerning the geometric tangent space $\mathbb R^n_a$ of $\mathbb R^n$ at $a$. Lee defines it as
$$\mathbb R^n_a = \{ a \} \times \mathbb R^n $$
with the vector space structure canonically induced from $\mathbb R^n$, i.e. the unique vector space structure making the canonical bijection $\phi... |
How to prove strong duality by Slater's condition without the assumption $\ker(A^T)=\{0\}$? |
It's a follow up of https://math.stackexchange.com/questions/4883613/does-three-iteration-are-sufficient-to-use-newtons-method
Come back on the previous question :
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0\tag{I... |
What is a "regular point" for KKT, and how does it relate to Slater's condition? |
I am trying to understand the following estimate from a book of John Roe. The setting is as follows. Suppose $\nabla$ is a connection on a vector bundle $S$ over a closed manifold $M$. We want to make a local estimate of the $L^2$-norm
$$\|\nabla s\|_0^2$$
of $\nabla s\in C^\infty(T^*M\otimes S)$. Here the notati... |
There are two major categories of the quaternions.
It is well-known that a (nonzero) versor represents a [three-dimensional rotation operator](https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Quaternions). A versor is a unit quaternion or a normalized quaternion: $$a+b{\bf i}+c{\bf j}+d{\bf k}\te... |
Real usage of “pure” quaternions in stereometry? |
Assumptions for reaching the [logistic map equation][1]
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
... |
Assume that $X\in \mathbb{R}^{n\times n}$ is a symmetric, positive semi-definite matrix and $C>0$ is a positive real number. If $X$ satisfies $$
\forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C,
$$ then is it true that $X$ is positive definite? Can we give a concrete lower bound for $\lambd... |
The rules of a lottery are as follows: You select 10 numbers between 1 and 50. On lottery night, the celebrity mathematician Richard Thomas chooses at random 6 'correct' numbers. If your 10 numbers include all 6 correct ones, you win.
How many ways are there to win the lottery?
From my understanding, the requested ... |
I solved the following equation yesterday for fun and I have the exact result (But these are just some solutions, not all):
$$\sin(x)^{\sin(x)}=2 \iff x=\frac{\pi}{2}-i\ln (\exp^{W(\ln(2))}\pm{\sqrt{\exp^{2W(\ln(2))}-1}}) +2k\pi,$$
for $k\in\mathbb{Z}$, where $i$ is the imaginary unit and $W$ is the Lambert $W$ funct... |
I believe, if we have a closed integral curve in a vector field then it is non-conservative. The idea is that say if it were conservative then we have a potential function say $\phi(x)$. Which gradient will be the vector field $\nabla\phi(\bar{x})$. We also know that the gradient tangent to the curve (integral curve) p... |
Closed integral curve in vector field implies vector field is not conservative? |
Consider a convex optimisation problem in the standard form: $\min f_0(\mathbf x)$ subject to $f_i(\mathbf x)\le 0$ and $A\mathbf x=\mathbf b$, with $f_i$ all convex functions.
When discussing Slater's condition, the [Wikipedia page][1] mentions (if I understand it correctly) that Slater's conditions state that stro... |
Why does Slater's condition require a point in the *relative* interior of the domain? |
$\require{AMScd}$It has been surprisingly difficult to prove the following fact about orthogonal factorization.
Let $\cal C$ be a category with finite limits and colimits, and with a functorial, orthogonal factorization $(\cal E,M)$. Consider a square
$$\begin{CD}
A @>>> B \\
@VVV @VVV\\
C @>>> D
\end{CD}$$ w... |
I asume that for you $sin^{-1}(\cdot) = arcsin(\cdot)$. As a starting point this could help
\begin{align}
&\int_{R-r}^{R}\arcsin\left(\frac{x^2+R^2-r^2}{2xR}\right)dx\\
&= \int_{R-r}^{R}\arcsin\left(\frac{x^2+\left(R-r\right)\left(R+r\right)}{2xR}\right)dx\\
&= \int_{R-r}^{R}\arcsin\left(\frac{x^2+\left(R-r\right... |
Let $X$ and $Y$ be random variables. I am trying to understand if we can conclude that $X$ and $Y$ are independent given that their joint MGF factorises into a product of the individual MGFs of $X$ and $Y$. I found the following [discussion][1], where the top answer states a relevant theorem and its proof:
> **Theorem... |
What can we say about the independence of random variables given that their joint moment generating function factorises? |
Can Slater's condition be used for non-convex objective functions? |
Could someone please clearly write out how we get from one line to the next here?
[![enter image description here][1]][1]
Above,
[![enter image description here][2]][2]
and
[![enter image description here][3]][3]
This is in the context of the expectation maximisation algorithm for a Gaussian mixtur... |
Where was Slater's condition first used? |
Probably not a great question but I was thinking about how to apply multivariable calculus in real life and came up with a question.
If you have a multivariable function with variables x,y,z and a single output.
Does the order of the partial derivatives in the gradient matter? As in, is the same (∂x,∂y,∂z) and(∂z,∂... |
I asume that for you $\sin^{-1}(\cdot) = \arcsin(\cdot)$. As a starting point this could help
\begin{align}
&\int_{R-r}^{R}\arcsin\left(\frac{x^2+R^2-r^2}{2xR}\right)dx\\
&= \int_{R-r}^{R}\arcsin\left(\frac{x^2+\left(R-r\right)\left(R+r\right)}{2xR}\right)dx\\
&= \int_{R-r}^{R}\arcsin\left(\frac{x^2+\left(R-r\rig... |
Assume that $X\in \mathbb{R}^{n\times n}$ is a symmetric, positive semi-definite matrix and $C>0$ is a positive real number. If $X$ satisfies $$
\forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C,
$$ then is it true that $X$ is positive definite? Can we give a concrete lower bound for $\lambd... |
**Problem:** Show that an $n \times n$ matrix with real coefficients can be put into *Reduced Row Echelon Form (RREF)* by a sequence of at most $n^2$ elementary row operations.
This feels simple, but proving this somehow eludes me. I thought about proceeding by Induction - for a given $(n+1) \times (n+1)$ matrix, c... |
In the urn we have a white, b red and c black balls. We draw with replacement. Calculate the expected number of white balls, which are drawn until we drew black ball for the k-th time.
My solution:
The expected number of draws till be draw black ball for the k-th time is:
$$(\frac{a+b+c}{c})^k$$. In the sequence o... |
Corrected a typo. In the computation of $~f(n),~$ the first factor should be $~\displaystyle \binom{U}{n},~$ rather than $~\displaystyle \binom{B}{n}.~$
-----
There may be a more elegant way of attacking this problem than the one that I will use. I considered
inclusion-exclusion, but found it to be messy. So... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/pZgAd.png
I have no clue how to proceed. It seems counter-intuitive! How to prove this exactly? |
Why these two definitions on Independence of two variables equivalent? |
How can we calculate the Euler-lagrange equations? |
I'm trying to find a reference for the following result:
**Theorem:** Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $k\geq 0$. Then $C_0^{k}(\Omega)$ it is (not) a (semi-)Montel space.
I tried to find in the classical references, Trèves, Horváth, but without success. The best result I found, it is the the... |
Is the space $C_0^{m}(\Omega)$ a Montel space? |
I'm calculating an expectation of the form $$\mathbb E\left[1\over aX+b\right]$$where $X\sim\chi^2_1(0)$ obeys an central chi squared distribution with 1 degree of freedom. The integral formula is $$\frac{1}{\sqrt{2\pi}}\int_0^\infty{x^{-1/2}e^{-x/2}\over ax+b}dx$$and I can not solve this integral. Also I tried the [in... |
Corrected a typo. In the computation of $~f(n),~$ the first factor should be $~\displaystyle \binom{U}{n},~$ rather than $~\displaystyle \binom{B}{n}.~$
-----
There may be a more elegant way of attacking this problem than the one that I will use. I considered
inclusion-exclusion, but found it to be messy. So... |
Let $(M,g)$ be a Riemannian manifold. Let $p\in M$. Is it possible to find a chart $\varphi: p\in U \to \mathbb R^d$ such that the geodesic distance $d_M$ on $U$ is close to the euclidean distance $d_E$ on $\phi_i(U_i)$?
More precisely, if I take $\phi_i = \exp^{-1}_p: \{x\in M: d_M(x,p)\le \delta\} \to \{v\in \math... |
Define a function: $\mathbb N \times \mathbb N \to \mathbb Q$:
$$f((a,b)) = a + \frac{1}{b}.$$
Choose $(a,b) \in \mathbb{N}, \ (c,d) \in \mathbb{N}$
workings:
$$
f((a,b)) = a + \frac{1}{b} \\
f((c,d)) = c + \frac{1}{d} \\
$$
Suppose that $f((a,b)) = f((c,d))$:
$$
a + \frac{1}{b} = c + \frac{1}{d} ... |
As you note on MO, [algebras](https://en.wikipedia.org/wiki/Algebra_representation), as well as [groups](https://en.wikipedia.org/wiki/Group_representation), have representations. One way to view Simon's claim is that we are taking a [$*$-representation](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Se... |
I'm faced with the problem
min x in R^2: (1/2) x' M x
subject to a'x = b, c'x = 1, x >=0
where c = [1, 1, ..., 1] and a = [a_1, a_2, ..., a_n]', where a_j are in R, b is in R aswell, and M is a positive definite real symmetric matrix.
My first idea was setting up the Lagrangian
L(x, y) = (1/2) x' M x + y... |
I'm faced with the problem
min x in R^2: (1/2) x' M x
subject to a'x = b, c'x = 1, x >=0
where c = [1, 1, ..., 1] and a = [a_1, a_2, ..., a_n]', where a_j are in R, b is in R aswell, and M is a positive definite real symmetric matrix.
My first idea was setting up the Lagrangian
L(x, y) = (1/2) x' M x + y... |
I'm currently studying a proof regarding the minimal polynomial's degree, which states that it's less than or equal to $( n^2)$, where $( n )$ is the dimension of the vector space on which the operator acts. In the proof, it's mentioned:
*"The dimension of the transformations acting on $( V )$ is $( n^2 )$"*
I'm ... |
Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the plane $z = 1$ using spherical Coordinates. How one may set the limits of integration of the spherical coordinates?
Now... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.