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I did a problem in competitive programming, the problem is split a number into $x$ and $y$ (with $x \leq y$), and make a new number by $(x - 1)(y + 1)$, do it until a new number different zero, and count how many different number can have by this split way.
It's easy for me but, when i draft, a new number always lower... |
Is there a setup in which is the Woodbury formula actually useful? |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
I did a problem in competitive programming, the problem is to split a number into $x$ and $y$ ($x \leq y$) and make a new number by $(x - 1)(y + 1)$, do it until a new number different zero, and count how many different numbers can have by this split way.
It's easy for me but, when I draft, a new number is always lowe... |
While trying to get an expression for [the number of deals from a generalised bridge deck with nobody being void in any suit](https://math.stackexchange.com/q/3443423/357390) I encountered the following subproblem.
From a generalised bridge deck with $r$ ranks instead of just $13$ deal four hands of $r$ cards each. ... |
I have the equation
$$f(x) = \arccos(x) + (C \times (1 - x^2) - x) \times \sqrt{1 - x^2}$$ where the parameter $C$ is a real number. When $-\frac{2}{3} ≤ C$, the function has one real zero at $x = 1$. When $C$ drops below $-\frac{2}{3}$, a second zero appears, starting at $x = 1$ and decreasing towards $x = -1$ as $C... |
Can the zero of $f(x)=\arccos(x)+(C \times (1 - x^2) - x) \times \sqrt{1 - x^2}$ be calculated explicitly, or is it transcendental? |
Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$.
How can I calculate $E[e^{\lambda t-\mu L(t)}]$? |
Can anyone help me with this, please?
In this context:
"Show the following property of the tangent to an ellipse: the tangent to an ellipse at a point
M coincides with the exterior bisector of the angle of the ray vectors of M."
What are the ray vectors of a point? |
Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$.
How can I calculate $E[e^{\lambda t-\mu Z(t)}]$? |
What is $r^{-1}(B)$ if $r$ is a rule and $B$ is an expression in a proof system? |
I am reading a machine learning [paper][1], and this paragraph below doesn't quite make sense. How is x+y (mod p−1) and x*y (mod p) equivalent? Suppose p = 5 and x = 3 and y = 4, then clearly 7 mod 4 =3 isn't equal to 12 mod 5 = 2. I'm also not sure how the line "every nonzero residue modulo a prime can be represented ... |
While trying to get an expression for [the number of deals from a generalised bridge deck with nobody being void in any suit](https://math.stackexchange.com/q/3443423/357390) I encountered the following subproblem.
From a generalised bridge deck with $r$ ranks instead of just $13$ deal four hands of $r$ cards each. ... |
I'm required to determine analytically the range of the following function:
[$f(x,y) = 2 + \sqrt{9-x^2-y^2}$][1]
[1]: https://i.stack.imgur.com/Q0igU.png
The domain for this function is $\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 ≤ 9\}$. I've found that the range for a very similar function (which is $f(x,y) = \... |
$xy \overset{?}{>} (x - 1)(y + 1)$
$xy \overset{?}{>} xy + x - y - 1$
$x \leq y \implies y = x + k, k \geq 0, k \in R$
So ...
$xy \overset{?}{>} xy + x - (x + k) - 1$
$xy \overset{?}{>} xy - (k + 1)$ |
Let $E,F$ be a normed space, and
$T:D(T) (\subset E) \rightarrow F$
be a densely defined unbounded linear operator.
By unbounedness,
for all $x \in E$, there is a sequence $(x_n) \subset D(T)$ such that
$x_n \to x$ and $||T(x_n)|| \to \infty$.
On the other hand, the follwing hold?
There is a $x \in E$ suc... |
I am looking for a function that must have the following requirements:
1. $f(1) = f(-1) = 0$
2. $f(x) > 0, \forall x \in (-1,1)$
3. $f$ is differentiable.
Additionally, I would like it to be parametrizable to control the steepness with which it rises and falls around $x = -1$ and $x = 1$
The usual logisti... |
**"Direct" problem** - Assume that a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$; let's also assume $M\leq N$.
Formally, if we know... |
IEEE 754 floating point numbers contain the concept of `` (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return `` if either argument is ``). That is, `` is an [absorbing element](https://en.wikipedia.org/wiki/Absorbing_element). However, from an algebraic point of view, FP numbers are very u... |
Using basic calculus, it's easy to see that we have a maximum at $ x = \pi/6$, but is there a way to prove this geometrically? I can get the $\cos x \sin x$ to appear but I'm stuck here; how should I proceed with maximizing the sum of length of the base + hypotenuse ($\cos x \sin x + \cos x$)? Any hints would be apprec... |
Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x $$
where $A$ is an $n \times n$ **symmetric** matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ whit $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.$$
T... |
I am implementing a fast Manhattan Frame estimation method based on this [paper](https://onlinelibrary.wiley.com/doi/abs/10.1002/rob.22064).
A crucial step to this algorithm is computing the real, symmetric $\mathbf{M}$ matrix parameterized by 15 variables as shown in the following screen shot
[![enter image des... |
For a real and symmetric matrix M, if I take $M_1 = |M|$, is the relation $M_1 = M_1^T$ true? |
I have 120 people and groups of 6 people and i want to know the probability that I am in the same group as a specific person (call him x)?
I’m little confused about the usage of the binomial and how to count the number of combination in which I am with this person.Thanks in advice |
if $X$ be a topological space and $f\in C_c(X)$, then we know that $$f\bigg[\Big(supp(f)\Big)^c\bigg]=0 $$
>is this true that if $f\in C_c(X)$, then $f\bigg[\partial\Big(supp(f)\Big)\bigg]=0$?
if $x\in \partial \Big(supp(f)\Big) $, since $$\partial \Big(supp(f)\Big)\subseteq\overline{ \Big(supp(f)\Big)^c ... |
Let's say we have $Z_3[X]/(7x²+1)$, such that we take the quotient of the polynomial ring $Z_3[X]$ with that specific ideal. Is this still a ring, and would it make sense to ask what the result of $(x²+3x³)*(13+7x)$ is in it?
What's the procedure for it? |
Is this quotient still a ring? |
Let $E,F$ be a normed space, and
$T:D(T) (\subset E) \rightarrow F$
be a densely defined unbounded linear operator.
By unbounedness,
for all $x \in E$, there is a sequence $(x_n) \subset D(T)$ such that
$x_n \to x$ and $||T(x_n)|| \to \infty$.
On the other hand, the follwing hold?
If $D(T) \neq E$,
there... |
Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x $$
where $A$ is an $n \times n$ **symmetric** matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ whit $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.$$
T... |
Let $E,F$ be a normed space, and
$T:D(T) (\subset E) \rightarrow F$
be a densely defined unbounded linear operator.
By unbounedness,
for all $x \in E$, there is a sequence $(x_n) \subset D(T)$ such that
$x_n \to x$ and $||T(x_n)|| \to \infty$.
On the other hand, the follwing hold?
If $D(T) \neq E$,
there... |
We can define a parametrized curve in ${ℝ}^2$ as a function $r:I→{ℝ}^2$ where $I$ is an interval of $ℝ$ and we can write $r(t)=(x(t),y(t))$.
Does a parametric curve whose graph is the same as a closed disk in ${ℝ}^2$ described by the equation : $x^{2}+y^{2}\le r^{2}$ exist?
The question might be stupid but i am tryin... |
Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x $$
where $A$ is an $n \times n$ **symmetric** matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ with $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.$$
T... |
I am wondering what coordinate ring $\mathbb{C}[t]/I$ for $I=\langle t^2(3-t^2)\rangle$ looks like. For any cases with more than one variable, I can somewhat imagine the structure by using the fact that the coordinate ring encodes the structure of the zero set over the ideal $I$.
So far I have never thought about t... |
Let $E(p):=\overbrace{(n-2)(\sqrt{2}-2)}^Ap^2+\overbrace{(3n-2)}^Bp\overbrace{-n}^C$, $n\geq 3$
Since $A:=(n-2)(\sqrt{2}-2)$ is negative, the quadratic function $E$ opens downwards, thus when $p<p^*$ where $p^*=\frac{-(3n-2)+\sqrt{(3n-2)^2+4(\sqrt{2}-2)n(n-2)}}{2(n-2)(\sqrt{2}-2)}$ is the left-hand zero of $E(p)=0$
... |
I am not sure about the following derivative:
$ \frac{d\textbf{b}^T e \Omega \textbf{r} \textbf{r}^T \Omega^T \textbf{e}^T \textbf{b}}{d \textbf{r}} = ?$
where $\textbf{b} \in \mathbb{R^N}$, $\textbf{e} \in \mathbb{R^{N \times L}}$, $\Omega \in \mathbb{R^{L \times K}}$, $\textbf{r} \in \mathbb{R^{K}}$.
My gues... |
Let $E(p):=\overbrace{(n-2)(\sqrt{2}-2)}^Ap^2+\overbrace{(3n-2)}^Bp\overbrace{-n}^C$, $n\geq 3$
Since $A:=(n-2)(\sqrt{2}-2)$ is negative, the quadratic function $E$ opens downwards, thus when $p<p^*$ where $p^*=\frac{-(3n-2)+\sqrt{(3n-2)^2+4(\sqrt{2}-2)n(n-2)}}{2(n-2)(\sqrt{2}-2)}$ is the left-hand zero of $E(p)=0$
... |
I am implementing a fast Manhattan Frame estimation method based on this [paper](https://eprints.whiterose.ac.uk/183320/6/A%20Fast%20Manhattan%20Frame%20Estimation%20Method%20based%20on%20Normal%20Vectors.pdf).
A crucial step to this algorithm is computing the real, symmetric $\mathbf{M}$ matrix parameterized by 15... |
The test involves a partial sum of a diagonal whose second element is a prime. Because of the symmetry of Pascal Triangle each prime has two diagonals parallel to the sides of the triangle. These two diagonals meet at the central binomial coefficient (cbc). The number of elements of each diagonal up to the cbc is alway... |
Given a function $f\colon X \to Y$, $X$ is called the "domain", $Y$ is called the "codomain" and the range of $f$ is defined to be the set, $\{ y \in Y \mid \exists x \in X \colon y=f(x) \}$
We can extend this concept to binary relations. We call a set "graph(of a binary relation)" $R$ a set whose only elements are ... |
[![Problem][1]][1]
I don’t really understand what it means by suppressing some terms. Can you provide me with an example?
[1]: https://i.stack.imgur.com/2ZjZc.jpg |
What does the question mean by suppressing some terms? |
I am not sure about the following derivative:
$ \frac{d\textbf{b}^T \textbf{E} \Omega \textbf{r} \textbf{r}^T \Omega^T \textbf{e}^T \textbf{b}}{d \textbf{r}} = ?$
where $\textbf{b} \in \mathbb{R^N}$, $\textbf{E} \in \mathbb{R^{N \times L}}$, $\Omega \in \mathbb{R^{L \times K}}$, $\textbf{r} \in \mathbb{R^{K}}$. F... |
The series $$\sum_{n=1}^\infty \frac{5n+\cos n}{3+2n^3}\tag 1$$ converges because $$\frac{5n+\cos n}{3+2n^3}\sim_0 \frac 52 \cdot \frac1{n^2} $$
Infact
$$\frac{5n+\cos n}{3+2n^3}=\frac{n\left(5+\frac{\cos n}n\right)}{n^3\left(\frac3{n^3}+2\right)}\sim_\infty\frac 1{n^2}\cdot \frac 52$$
We have
$$\frac{\frac{5n+... |
Let $F$ be a field and $M_n (F)$ the ring of $n\times n$ matrices with entries in $F$ and $n>1$. I know that that $F[x]$ is a PID. Trivially, $M_n (F)$ is not commutative and not a field. My question is that is every left ideal of $M_n (F)[x]$ principal? |
I struggle to understand arguments that make use of $\text{d}x$-like notation (with the exception of rigorously defined differential forms, yet everything below is elementary enough so that forms are not needed). Ahlfors gives such an argument in his book _Complex Analysis_ to prove the **Result** below, and I was wond... |
Let $h \in \mathbb{R}^n\backslash \{0\}$ and $r \in \mathbb{R}$. A sure that $M$ is an affine subspace of $\mathbb{R}^n$ with $H(h, r) \cap M=\phi$. I tried to use Banach separation theorem to show that there is some hyperplane in $\mathbb{R}^n$, which strongly separates $H(h, r)$ and $M$, but it did not work. How can ... |
Let $X$ and $Y$ be complex manifolds and let $\pi \colon X \to Y$ be holomorphic proper surjective submersion. Therefore, $\pi$ is a fiber bundle whose fibers are complex manifolds.
My question: Does there exists for every $x \in X$ a neighborhood $U \subset X$ of $x$ with a vertical holomorphic vector field $v$ on... |
Let $f_n$ be a sequence of Schwartz functions on $\mathbb{R}^N$ converging to another Schwartz function in the Schwartz space $\mathcal{S}(\mathbb{R}^N)$.
Now, let $A : \mathbb{R}^N \to \mathbb{R}^N$ be any invertible matrix. Then, functions $f_{A,n}$ and $f_A$ defined by
\begin{equation}
f_{A,n}(x):= f_n(Ax) \qqu... |
If $f_n(x) \to f(x)$ in the Schwartz space, $f_n(Ax) \to f(Ax)$ for any invertible matrix $A$? |
If from the assumption of some A it follows that A is a theorem, then under threat of inconsistency of the system ¬A is not derivable from A. Is this statement true? |
I've been studying topology and I am trying to find a relation between the compactification of a space $X$ and it's quotent space. For example, one-point compactification of the open interval $(0,1)$ can be thought of as bringing its end points to the point infinity (we consider the embedding $f(t) = (cos(2πt),sin(2πt)... |
One good strategy is to divide and conquer, while following some well-known patterns.
For example, many statements involving quantifiers follow one of the following four forms:
"All of [these] are like [that]"
"Some of [these] are like [that]"
"None of [these] are like [that]" (which is equivalent to "All o... |
I struggle to understand arguments that make use of $\text{d}x$-like notation (with the exception of rigorously defined differential forms, yet everything below is elementary enough so that forms are not needed). Ahlfors gives such an argument in his book _Complex Analysis_ to prove the **Result** below, and I was wond... |
How can $P_1 ⇔ P_2 ⇔ P_3 ⇔ \ldots ⇔ Pn$ be true when not all the $P$’s have the same truth value?
---
For example: If P1 = T P2 = T P3 = F P4 = F would this be true?
T(P1) ⇔ T(P2) ⇔ F(P3) ⇔ F(P4) =
T(P1&2) ⇔ F(P3) ⇔ F(P4) =
F(P1,2,3) ⇔ F(P4) =
True
If so how can an evaluation for a series of compound... |
How can compound biconditional statements whose individual statements don't all have the same truth values be true? |
**"Direct" problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we... |
Map between random variables: how to practically deal with the PDF "inverse" problem? |
Let $A$ be a (commutative, associative, unital) ring. Let $f$ be a non-zerodivisor in $A$, and let $I$ be a finitely-generated ideal in $A[\frac1f][T_1,\dotsc,T_r]$. Can it happen that $I\cap A[T_1,\dotsc,T_r]$ is no longer a finitely-generated ideal in $A[T_1,\dotsc,T_r]$? By the Hilbert basis theorem, this cannot hap... |
If I have two matrices, one of which is 3*2 and the second is 2*3, how can I find the determinant of the square matrix that results from multiplying them without multiplying them and getting the resulted matrix in order to get the determinant of it? |
B={[a,b): a<b} is the basis inducing the metric topology $T_B$. We want to prove that If $K\subset\mathbb{R}$ is compact w.r.t $T_B$, then $K$ is countable.
**My attempt**
Given $K$ is compact, then for $x\in K$. we can find the open cover $[x,+\infty) \cup_{n\in \mathbb{N}}(-\infty,x-\frac{1}{n} )$. By compactne... |
If $K\subset\mathbb{R}$ is compact w.r.t $T_B$, then $K$ is countable? |
If I have two matrices, one of which is $3 \times 2$ and the second is $2 \times 3$. How can I find the determinant of the square matrix that results from multiplying them *without multiplying them and getting the resulted matrix in order to get the determinant of it?* |
Let be $C$ a non empty, compact and convex set and $S:=\mathbb{R}^n\setminus C$, where $n>1$. In many occasions our professor simply skips the part where we construct a path that connects two arbitrary point of $S$. I was wondering if it is possible to construct a curve that connects two such points of $S$ by the follo... |
B={[a,b): a<b} is the basis inducing the topology $T_B$. We want to prove that If $K\subset\mathbb{R}$ is compact w.r.t $T_B$, then $K$ is countable.
**My attempt**
Given $K$ is compact, then for $x\in K$. we can find the open cover $[x,+\infty) \cup_{n\in \mathbb{N}}(-\infty,x-\frac{1}{n} )$. By compactness, thi... |
Is it *problematic* to allow for expressions such as
$\forall x\exists x (x < x)$
to count as well formed sentences? No (as others have explained). You can set up your formal first-order logical languages that way, while keeping everything working systematically in an acceptable way.
Is is *necessary* to allow fo... |
Consider the question
>Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$, $m>1$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no as we have the following counterexample
$$f(n)=\begin{cases}
1/n & \text{when $n$ is even}\\
2/n & \text{when $n$ is odd}
\... |
I just came to know that it is possible to define a _generalized base_ as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\in \mathbb N$ can be uniquely expressed as
$$m=\overline{x_n\dots x_1}^\mathbf b := x_1 + x_2\cdot b_1 + x_3\cdot b_1b... |
Let's say we have $\mathbb{Z}_3[X]\mathbin{/}(7x^2 + 1)$, such that we take the quotient of the polynomial ring $\mathbb{Z}_3[X]$ with that specific ideal.
> Is this still a ring, and does it make sense to ask what the result of $(x^2 + 3x^3) \cdot (13 + 7x)$ is in it?
What's the procedure for it? |
I'm studying multivariable calculus, and I was wondering:
Is there a way to characterize all sets in $\mathbb{R}^2$ that are continuously parametrizable? What I mean with a continuously parametrizable set $A$ is that there exists a continuous function $f:[0, 1] \rightarrow \mathbb{R}^2$ onto $A$.
I know that *cur... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} \exp \Bigg[ -2s \bigg(\log(s) - 1 \pm o_n(1) \log(n)\bigg) \Bigg]
$$
Can someone help me to find the asymptotic behavior of sum? (My problem is the term $\pm o_n(1) \log(n)$ behaviour) |
Let $f_n$ be a sequence of Schwartz functions on $\mathbb{R}^N$ converging to another Schwartz function in the Schwartz space $\mathcal{S}(\mathbb{R}^N)$.
Now, let $A : \mathbb{R}^N \to \mathbb{R}^N$ be any invertible matrix. Then, functions $f_{A,n}$ and $f_A$ defined by
\begin{equation}
f_{A,n}(x):= f_n(Ax) \qqu... |
> Let $0<\lambda_{1} \leq \ldots \leq \lambda_{n}$ be the eigenvalues of
> the symmetric and positive definite matrix $A \in \mathbb{R}^{n \times
> n}$ and $u_{1}, \ldots, u_{n} \in \mathbb{R}^{n}$ corresponding
> pairwise orthonormal eigenvectors. For $x \neq 0$ let $$
F(x):=\frac{\left(x^{\top} x\right)^{2}}{\le... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} \exp \Bigg[ -2s \bigg(\log(s) - 1 \pm o_n(1) \log(n)\bigg) \Bigg]
$$
Can someone help me to find the asymptotic behavior of sum? (My problem is the term $\pm o_n(1) \log(n)$ behaviour)
P.... |
$B=\{[a,b): a<b\}$ is the basis inducing the topology $T_B$. We want to prove that If $K\subset\mathbb{R}$ is compact w.r.t $T_B$, then $K$ is countable.
**My attempt**
Given $K$ is compact, then for $x\in K$. we can find the open cover $[x,+\infty) \cup_{n\in \mathbb{N}}(-\infty,x-\frac{1}{n} )$. By compactness,... |
**Situation**
We have $10$ cards, with $7$ blacks and $3$ reds. Draw $3$ cards at random.
**Exercise**
(1) What is the probability that at least one of the cards is black if we have sampling with replacement?
(2) What is the probability that at least one of the cards is black if we have sampling without rep... |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
Your ‘P1 =/= P2 =/= P3 =/= Pn’ is a little unclear, but from the title of your post, it is clear that your question is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the c... |
> Let $0<\lambda_{1} \leq \ldots \leq \lambda_{n}$ be the eigenvalues of
> the symmetric and positive definite matrix $A \in \mathbb{R}^{n \times
> n}$ and $u_{1}, \ldots, u_{n} \in \mathbb{R}^{n}$ corresponding
> pairwise orthonormal eigenvectors. For $x \neq 0$ let
$$
F(x):=\frac{\left(x^{\top} x\right)^{2}}{\... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} n^ {-\Bigg[ 2s \bigg(\frac{\log(s)}{\log(n)} - \frac{1}{{\log(n)}} \pm \frac {o_n(1)}{\log(n)}\bigg) +(1 \pm o_n(1)) 2s + \frac{\alpha}{2} \Big( \log{\big( \frac{a}{b}\big)} \Big)\Bigg]}
$$
... |
I know and can show that
\begin{equation*}
{}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\end{equation*}
From here, I think I get:
\begin{equation}
\begin{split}
{}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\
& = \frac sz\left[\fr... |
Do you need to enforce the condition $\max f(x)=1$? Otherwise, any function of the form
$$f(x)=\sum_{j=1}^{n}c_j (1-x)^{a_j}(1+x)^{b_j}\tag{1}$$
with $a_j, b_j, c_j > 0$ will have the desired property. Moreover, by adjusting the values of $n$, $a_j$'s, $b_j$'s, $c_j$'s, functions of the form $\text{(1)}$ can appr... |
**Context :**
We perform measurements of a set of parameters $(p_1,p_2,...)$ to retrieve the experimental mean of each parameters and the covariance matrix between them. The means of the parameters are then combined to construct a response $\epsilon = f(p_1,p_2,...)$. The question is, how can we propagate the uncert... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} n^ {- 2s \Bigg[\frac{\log(s)-1}{\log(n)} \pm o_n(1) + 1 \pm o_n(1) + \frac{\alpha}{4s} \Big( \log{\big( \frac{a}{b}\big)} \Big) \Bigg]}
$$
Can someone help me to find the asymptotic behavio... |
I know and can show that
\begin{equation*}
{}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\end{equation*}
From here, I think I get:
\begin{equation*}
\begin{split}
{}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\
& = \frac sz\left[\f... |
I am implementing a fast Manhattan Frame estimation method based on this [paper](https://eprints.whiterose.ac.uk/183320/6/A%20Fast%20Manhattan%20Frame%20Estimation%20Method%20based%20on%20Normal%20Vectors.pdf).
A crucial step to this algorithm is computing the real, symmetric $\mathbf{M}$ matrix parameterized by 15... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} n^{ \bigg[ 2s \Big(1 - \frac{\log(s)-1}{\log (n)} + 2 \big(1 \pm o_n(1)\big) \Big) + \frac{\alpha}{2} \log{\big( \frac{a}{b} \big)} \bigg]}
$$
Can someone help me to find the asymptotic beha... |
I know and can show that
\begin{equation*}
{}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\end{equation*}
From here, I think I get:
\begin{equation*}
\begin{split}
{}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\
& = \frac sz\left[\f... |
**"Direct" problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we... |
It is given, that the following system of ODEs has a non-trivial solution with a constant y axis
\begin{alignat*}{4}
\frac{dx}{dt} = x-e^{-t}y \\
\frac{dy}{dt} = 2e^tx-y \\
\end{alignat*}
Find the general solution of this system.
Any ideas?
Thank you
|
Consider the diagonal operator $A\{x_n\}$ = $\{a_nx_n\}$. The operator A is defined on the sequence space $\ell_p$, where $1<p<\infty$. 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... |
> Is it still a ring?
Yes, for an ideal $I\subseteq R$ we have that $R/I$ is the quotient ring.
> Does it make sense to ask what the result of $(x^2 + 3x^3) \cdot (13 + 7x)$ is?
Yes. In this ring $7x^2+1=0$ and $3=0$, so $x^2=-1$. This means that every polynomial is on the form $$a+bx$$ where $a,b\in\Bbb Z_3$... |
Let $r = \sqrt{x^2 + y^2 + z^2}$. From the fact that $\nabla^2 r^{-1} = -4\pi \delta^{(3)}(\vec{r})$, is it correct to say that
$$
\frac{\partial^2}{\partial x^2}(r^{-1}) = \frac{3x^2 - r^2}{r^5} - \frac{4\pi}{3} \delta^{(3)}(\vec{r}) \\
\frac{\partial^2}{\partial x \partial y}(r^{-1}) = \frac{3xy}{r^5}
$$
The q... |
*"Is this your homework, Larry?"*
<h2>I. Look at the scatter plot</h2>
And find a relation such that $y_i\approx f(x_i), i=1,...,n $
<h2>II. "You might want to watch out that front window Larry".</h2>
Set a family of functions $F$ (ex: linear fonctions) and a cost function$L$ such that:
$\sum_{i=1}^{n}L(... |
I am implementing a fast Manhattan Frame estimation method based on this [paper](https://eprints.whiterose.ac.uk/183320/6/A%20Fast%20Manhattan%20Frame%20Estimation%20Method%20based%20on%20Normal%20Vectors.pdf).
A crucial step to this algorithm is computing the real, symmetric $\mathbf{M}$ matrix parameterized by 15... |
This is my variance equation when the numbers are drawn from a normal distribution $$2/(n_{in}+1/n_{out})$$
The answer for the uniform distribution with the same variance is
$$6/(n_{in}+1/n_{out}),-6/(n_{in}+1/n_{out})$$
How exactly does the 6 come into play is it from the integral of the uniform distribution formu... |
How do I go from a normal distribution with a variance to uniform distribution with the same variance? |
As mentioned in the comments, the quotient of a space $X$ by a subspace $Y$ is the space whose points are equivalence classes such that all of $Y$ is a single class (a point in the quotient), and every point in $X \setminus Y$ is a singleton class. So, we often speak of *collapsing $Y$ to a point in $X$,* thinking of t... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
Take a standard deck of cards, with 52 cards, 26 red and 26 black. A run is a maximum contiguous block of cards, which has the same color.
So,
(,,,,...,,) has 52 runs.
(,,,...,,,,,...,) has 2 runs.
What is the expected number of runs, $E_{run}$ in a shuffled deck of cards?
We know that the answer to this ques... |
How can we maximize the expected number of runs in a shuffled deck of cards? |
This is my variance equation when the numbers are drawn from a normal distribution $$2/(n_{in}+1/n_{out})$$
The answer for the uniform distribution with the same variance is
$$6/(n_{in}+1/n_{out}),-6/(n_{in}+1/n_{out})$$
How exactly does the 6 come into play is it from the integral of the uniform distribution formu... |
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