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Something like this is explored in [Hopkins: Algebraic Topology and Modular Forms](https://arxiv.org/pdf/math/0212397.pdf) (very similar looking piece at PDF page 14, internal page 296). Skimming (so take with a grain of salt), it looks like only "$c_4$" (a.k.a. $E_4$) and "$c_6$" ($E_6$) are used and (if there are an... |
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a matrix of size $m\times n$, $a_{ij}\in \mathbb{N},~b\in \mathbb{N}$.
Let $S_{b'}$ be the set of solutions of this system... |
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a matrix of size $m\times n$, $a_{ij}\in \mathbb{N},~b\in \mathbb{N}$.
Let $S_{b'}$ be the set of solutions of this system... |
Numbers have certain forms. Like $3k + 1$ applies to 1,4,7 so on. I'm interested in how to characterize what numbers are not of a particular form. For example, $3k$ and $3k + 2$ numbers are not $3k + 1$.
In addition, it would be helpful if I could set a limit on a variable's value. As an arbitrary expression (I don't... |
When is a number not of a form below a certain input? x*y + y + x +1 with x < c |
If $(a,b,c)$ are the sides of a triangle and $x \ge 1$, what is the probability that $a+b > cx$? |
To get the principal answer to a complex number $(a+bi)$ raised to another complex number $(c+di)$ I understand you can get this by first determining $r=\sqrt{a^2+b^2}$ and $\theta = \arctan(b/a)$ and then use those by raising $e^{(\ln(r)*c-d*\theta)+(\ln(r)*d+c*\theta)}$. But I find that does not provide the correct ... |
Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., y_n$ respectively.Let
$$
D_{i,j} = \begin{cases}
x_i-y_j, & x_i\ge y_j \\\\
1+x_i-y_j, & x_i<y_j
\end{cases}
$... |
To get the principal answer to a complex number $(a+bi)$ raised to another complex number $(c+di)$ I understand you can get this by first determining $r=\sqrt{a^2+b^2}$ and $\theta = \arctan(b/a)$ and then use those by raising $e^{(\ln(r)*c-d*\theta)+(\ln(r)*d+c*\theta)}$. But I find that does not provide the correct ... |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
Taking partial derivative of SSD, wrt the parameters $a, b$? |
Let $u(x,t)$ be a square integrable function defined over $]0,a[ \times ]0,T[$. \
I want to simplify these two expressions : \
$$\int _0 ^a \frac{\partial ^2 u(x,t)}{\partial t ^2 }.u(x,t) \, dx$$ and
$$\int _0 ^a \frac{\partial u(x,t)}{\partial t}.u(x,t) \, dx$$
I wanted to use Leibniz integral rule to move the... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to understand some statement.
First, I propose associated question.
**Q.** Let $A \subseteq B$ be a subring with prime ideals $\mathfrak{p} \subseteq A$ and $\mathfrak{q} \subseteq B$ such that $\mathfrak{p} = \mathfrak{q} \cap A$ ... |
Yes. The proof is not really accurate. For any $y' \in \mathbb{A}^{n}_{\kappa(y)}$ lying over $y\in \mathbb{A}^{n}_k$, it is not in general $\kappa(y') =\kappa(y)$, so we cannot apply the reduced case directly. But we can choose suitable $y'$ such that $\kappa(y') =\kappa(y)$. This is as follows.
Again let $f: X \... |
Yes. The proof is not really accurate. For any $y' \in \mathbb{A}^{n}_{\kappa(y)}$ lying over $y\in \mathbb{A}^{n}_k$, it is not in general $\kappa(y') =\kappa(y)$, so we cannot apply the reduced case directly. But we can choose suitable $y'$ such that $\kappa(y') =\kappa(y)$. This is as follows.
Again let $f: X \... |
Let us denote the original graph $G_O$ and the target graph $G_T$.
I am wondering if there exists an algorithm that uses the minimum number of edge deletions and additions to make $G_0$ isomorphic to $G_T$.
I have been searching for a bit and found graph matching as the closest thing, but it only checks if two gr... |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
Someone can clarify a doubt risen solving a limit, please.
I know
$$\lim_{x\to+\infty }\sqrt{x^{2}+x}-x=\frac{1}{2}$$
$$\lim_{x\to-\infty }\sqrt{x^{2}+x}-x=-\infty $$
Solving the first one
$$\lim_{x\to+\infty }\sqrt{x^{2}+x}-x\cdot\frac{\sqrt{x^{2}+x}+x}{\sqrt{x^{2}+x}+x}$$
$$\lim_{x\to+\infty }\frac{x^{2}+x-x^... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to understand some statement.
First, I propose associated question.
**Q.** Let $A \subseteq B$ be a subring with prime ideals $\mathfrak{p} \subseteq A$ and $\mathfrak{q} \subseteq B$ such that $\mathfrak{p} = \mathfrak{q} \cap A$ ... |
This is a historic question for which I couldn't find an answer on google or on the math history books I have at hand.
The first fundamental form $I$ of a surface $S$ embedded in $\mathbb{R}^3$ is, indeed, a differential form: a choice of bilinear map $I_x\colon T_xS \times T_xS \to \mathbb{R}$ for the tangent space... |
Who named the first fundamental form? |
Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., y_n$ respectively.Let
$$
D_{i,j} = \begin{cases}
x_i-y_j, & x_i\ge y_j \\\\
1+x_i-y_j, & x_i<y_j
\end{cases}
$... |
in \(\sigma (12)\sigma ^{-1}=\left(\sigma (1)\sigma (2)\right)=(23)\), can anyone prove why the result is equal to the transposition (23)? |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
I was studying polynomial rings over commutative rings from the book Topics in Algebra by I.N Herstein.
From there what I understood was, that:
If $R$ be a commutative ring with a unit element then $R[x]$ is defined to be the set of all symbols $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ where, $a_0,a_1,...,a_n\in R$ and... |
a, b and c are 3 Independent Vectors. We can generate Orthonormal basis vectors using those 3 vectors using Gram-Schmidt method. Lets say those 3 orthogonal basis vectors generated from a, b and c are $A$, $B$ and $C$. We can generate orthonormal basis vectors by dividing those orthogonal basis vectors with their lengt... |
This question is part of the original question: https://math.stackexchange.com/questions/4875789.
For each ${n}$, let ${A_n = (a_{ij,n})_{1 \leq i,j \leq n}}$ be an ${n \times n}$ random sign matrix (i.e. the entries ${a_{ij,n}}$ of ${A_n}$ are jointly independent in ${i,j}$ and take values in ${\{-1,+1\}}$ with a p... |
in $(\sigma (12)\sigma ^{-1}=\left(\sigma (1)\sigma (2)\right)=(23))$, can anyone prove why the result is equal to the transposition (23)? |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
I have a Chi-squared RV with 2NL degrees of freedom and I am interested in the distribution of its square ($Y=X^2$). I have tried the transformation method to get the expression for the pdf of Y as:
$f_Y(y)=\frac{1}{2^{NL+1} \Gamma(NL)} y^{NL / 2-1} e^{-\sqrt{y} / 2}$
I am not sure about the type of RV that corre... |
I thought that f(x) = 1/2x + 1/2 might be a solution. However, I am also doubting it because of bijectiveness... is it necessary? Is it too obvious? |
A 1/2-Lipschitz function f from [0,1) to [0,1) that has no fixed point? |
Let $(V,Q)$ be a finite-dimensional quadratic space over either the real or complex numbers with a non-degenerate, but not necessarily positive-definite quadratic form $Q$.
We define the Spin group $Spin(V,Q)$ as $Pin(V,Q) \cap Cl_0(V,Q)$, meaning as the subset of the Clifford algebra which is generated by an even num... |
I want to solve $\mathscr{F}\{e^{ix}\}(\xi)$ in terms of distributions.
This is what I did
$$\mathscr{F}\big(e^{ix}\big)(\xi) = \int e^{ix} e^{i(\xi,x)}\text{d}x$$
But how do I get further from here?. I used the formula in Vladimirov, on p 108, but this seems to me too general as a solution.
Any help ... |
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for any function. We know that:
$$\log(1 + z) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k+1} z^{k+1} =z + \sum_{k=2}^{\infty... |
Background: I'm an engineering student with no experience with a real investment account or anything of that sort.
If we were to extend the concept of compound interest strictly mathematically, the formula would be:
$$\text{Amount} = \text{Principal} \times \text{rate} ^ t$$
where $t$ is time in number of compou... |
I try to derive the information matrix equality for the Poisson distribution with the log-Likelihood:
$$\mathcal{L}(\lambda; x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} \left[-\lambda + x_i \log(\lambda) - \log(x_i!)\right]$$
The derivation by the Hessian matrix is clear to me and yield $I(\lambda) = \frac{n}{\lambda... |
how to calculte this multiple integrals $\iiint_G 1 dxdydz$? |
I am working on a problem in the context of molecular simulation. I consider various potentials and compute the resulting forces on each of the atoms in my simulation. I managed to do this for the bond potential and the angle potential. I am stuck however on the dihedral potential.
I have the dihedral potential give... |
I *do* believe quaternions are the most direct approach to it. That is to say: to computing the rotation that corresponds to a given point on $\mathbb{S}^3/\sim_A$. Showing that this correspondence is bijective is a different story, but you said you already understood that part.
You know that elements of $SO(3)$ are... |
I thought that $f(x) = 1/2x + 1/2$ might be a solution. However, I am also doubting it because of bijectiveness... is it necessary? Is it too obvious? |
A $1/2$-Lipschitz function $f$ from $[0,1)$ to $[0,1)$ that has no fixed point? |
I'm trying to model the process scheduling mechanism in Linux using queueing theory models.
Assuming that both the arrival and processing times of processes in the system follow a Poisson distribution of, and the system contains `c` independent logical cores, this scenario approximates an `M/M/c` model.
The chal... |
How to model the round-robin scheduling mechanism using queueing theory models? |
I'm currently reading [A Course on Rough Paths: With an Introduction to Regularity Structures][1], but I find it difficult to start on even though I've taken undergraduate mathematical analysis, linear algebra, algebra, ODE and PDE courses of the math department. I'm not familiar with the concepts of "tensor", "tensor ... |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
You wrote in a comment that you’re aware that we don’t know whether infinitely many Euclid numbers are prime, and indeed your questions might be answered without knowing this. But we also don’t know whether infinitely many Euclid numbers are composite – at least we didn’t in $1996$, see [The New Book of Prime Number Re... |
$\frac{1}{(1-x^a)(1-x^b)}=\frac{A}{(1-x)^2}+\frac{B}{(1-x)}+\sum_{r^a=1}^{ }\frac{C_r}{(1-x/r)}+\sum_{t^b=1}^{ }\frac{D_t}{(1-x/t)}$ $(t,r\neq1; A, B, C, D$ are real numbers)
How did the author know that he need to separate the case r,t=1?
I would appreciate it really much if someone can guide me to resources... |
Can someone help me with partial fraction expansion, please? |
I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups.
**My question**
Let $k$ be an algebraically closed field, $R := k[(x_{ij})_{i,j=1}^n,y]/(\det(x_{ij})y-1).$ Where $x_{ij}\in A$ is a commutative algebr... |
I know that the Monty Hall problem is old news. However, I can't get this out of my head and I am looking forward to your thoughts about it.
If the rules of the game are known from the beginning, especially that the game host will guarantee to open a door containing a goat, isn't this information extremely relevant ... |
It is a problem found in Physics. From the edge of a (in distance r to the center) disk a (mathematical) pendulum is hanging. The disk spinns with a velocity w, which also stimulates the hanging pendulum.(As can be seen in the picutre below)
[![enter image description here][1]][1]
[![enter image description here][2]]... |
I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$ with Neumann boundary conditions, where $\kappa:[0,T]\to(0,\infty)$ is linear and $W$ is a cylindrical Wiener process on $H:=L^2([0,1)^2)$. I somehow need to compute/estimate (or at least find a sharp... |
I have derived that the required number of operations in my algorithm is $Y(a)$, where $a>0$ is the concerned parameter.
However, $Y(a)$ is not a closed-form expression w.r.t. $a$.
But I find that $Y(a)\leq P(a)$, and $P(a)$ can be expressed as a closed-form w.r.t. $a$, that is $P(a)=a^2+1$.
So, can I say ... |
In $\sigma (12)\sigma ^{-1}=\left(\sigma (1)\sigma (2)\right)=(23)$, can anyone prove why the result is equal to the transposition (23)?
Sorry, the $\sigma = (12...n)$. |
The Boolean function $g$ of the variables $x_1 , . . . , x_5 , y_1 , . . . , y_5$ , is given by the formula:
$$
\bigwedge_{i=1}^{5} (x_i \ \vee \ y_i)
$$
**Mission:** find the dnf($g$). (the minimum number of conjuncts in the DNF of function $g$)
**Some clarifications:** The
smallest number of conjuncts in the ... |
>Show that $p(x)=x^5+x^3+x+1$ is irreducible.
**My attempt**: First, I tried to shifting Eisenstein's up to $10$, i.e., $p(x+a)$, where $a=1,\dots, 10$ using calculator. I did not continue because if I am being honest, you will get bored after a several failures. Also, please, I ask you to do not recommend rational ... |
Suppose $X$ is a scheme (possibly smooth / $\mathbb C$), $E$ is a vector bundle of rank $e$ on $X$ with corresponding projective bundle $p:\mathbb PE \to X$, and $s: X \to \mathbb PE$ is a section. I wanted to compute the normal bundle $N$ of $s(X) \subset \mathbb PE$ in terms of the short exact sequence
$$\tag{1}\lab... |
What went wrong computing the normal bundle of a section in projective bundle? |
It is a problem found in Physics. From the edge of a (in distance r to the center) disk a (mathematical) pendulum is hanging. The disk spinns with a velocity w, which also stimulates the hanging pendulum.(As can be seen in the picutre below)
[![enter image description here][1]][1]
[![enter image description here][2]]... |
Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth isotopy between two diffeomorphisms $f_0$ and $f_1$ of M in the sense that $\forall t \in I ,F(\cdot,t):M\rightarrow M $ is an embedding and that $F(\cdot,0)=f_0,F(\cdot,1)=f_1$.
My question is can we find a new isoto... |
I am studying the branching structure of a Directed Acyclic Graph and need to define a kind of set that includes all the possible reachable nodes according to the branching path.
For example, I have a graph that consists of three nodes $A$, $B$, and $C$, with directed paths $A\rightarrow B\rightarrow C$ and $A\right... |
This is a proof of a formula in a math class. There are some parts that I don't quite understand, so I would like to ask for some clarification.
In $\sigma (12)\sigma ^{-1}=\left(\sigma (1)\sigma (2)\right)=(23)$, can anyone prove why the result is equal to the transposition (23)?
Sorry, the $\sigma = (12...n)$. |
I'm trying to understand how to calculate the Hausdorff dimension of some profinite groups and came across the following paper ([here][1]) which defines the Hausdorff dimension for a closed subgroup $H \leq G$ as
$$dim(H)=\liminf_{n \rightarrow \infty} \frac{\log{ \vert H:H \cap G_n \vert }}{\log{\vert G:G_n \vert}}$... |
$$\frac{1}{(1-x^a)(1-x^b)}=\frac{A}{(1-x)^2}+\frac{B}{(1-x)}+\sum_{r^a=1}^{ }\frac{C_r}{(1-x/r)}+\sum_{t^b=1}^{ }\frac{D_t}{(1-x/t)}$$ $(t,r\neq1; A, B, C, D$ are real numbers)
How did the author know that he need to separate the case r,t=1?
I would appreciate it really much if someone can guide me to resourc... |
Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth (say $C^{\infty}$) isotopy between two diffeomorphisms $f_0$ and $f_1$ of M in the sense that $\forall t \in I ,F(\cdot,t):M\rightarrow M $ is an embedding and that $F(\cdot,0)=f_0,F(\cdot,1)=f_1$. My question is can we f... |
Let $X \subset \mathbb P^N$ be a smooth projective surface. Let $H$ be an ample divisor on $X$. Let $M_{X,H}(r;c_1,c_2)$ be the moduli space of rank r, slope stable (w.r.to $H$) vector bundles with fixed Chern classes $c_1,c_2$. Let $C$ be a smooth curve lying on $X$ such that for every $E \in M_{X,H}(r;c_1,c_2)$, $E|_... |
I was trying to writ metric tensor over a surface $M$ starting with the usual arc length, but I find some unclear steps.
---
Consider a parametric curve in $\Bbb R^n$
$$\gamma = \{ \mathbf x(t), t \in [t_0,t_1]\}$$
we approximate with small line this curve and $d\mathbf x=(dx_1,\dots,dx_n)$ is a little piece o... |
This is a very easy question I suppose. I am trying to help my niece for solving some equations, i.e. finding real roots. There was one equation which we solved, but I wasn't particularly satisfied with how we solved it, we guessed the end result, which was correct in the end. Let me state the equation:
$$(x^2-4)(x... |
Consider the model $y_t = F_{S_t} x_t + \varepsilon_{S_t}$ and $x_t = A_{S_t} x_{t-1} + \nu_{S_t}$, where $\varepsilon_{S_t}, \nu_{S_t} \sim N(0, R_{S_t})$ and $N(0, Q_{S_t})$ and $S_t$ is Markov chain with $\mathbb{P}(S_t = i | S_{t-1} = j) = p_{i,j}$.
We want to estimate the parameters of the model using EM algor... |
Can an isotopy between diffeomorphisms be through diffeomorphisms? |
In the book titled: Analysis of Straight-line data, by Forman S. Acton; it is given on page #10:
-------------------------------------------------
The classical 'least squares' procedure is most commonly derived by forming an expression for the sum of the squared vertical deviations from a general line, then dema... |
I'm trying to figure out the step from equation (19) to equation (20) in [this document][1] when $\mathcal{F} = 0$ y $\mathcal{A} = 0$. In this case, equation (19) reduces to
$$ - e^{\alpha \phi} \hat{\omega}^{z} {}{b} \wedge e^{b} = \beta e^{\beta} \partial{\mu} \phi (d x^{\mu} \wedge d x^{4} )$$
using $\hat{e}^... |
I have the following optimization
\begin{align}
\max_{\|x\|^2\le1} \|Lx - y\|
\end{align}
where $L$ is a lower triangular and $y$ is a given vector.
Does it admit a closed-form solution?
I am interested in the general case where $L$ may be not invertible and even not square, but any progress under some assump... |
I am trying to find the radon transform of the gaussian function
$$f(x,y) = e^{-(x^2 + y^2)}$$
Now, I am using the formula for radon transform as
$$
[\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \delta(x \cos(\theta) + y \sin(\theta) - \rho) dx dy
$$
How do i prove a... |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integers $a$ and $b$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For exponentiat... |
I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map at a point $x \in M$.
$f$ is defined by $x^T A x$ for a positive definite symmetric matrix, and the exponential map is given... |
I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map at a point $x \in M$. $T_x M$ is the tangent space at a point $x.$
$f$ is defined by $x^T A x$ for a positive definite symme... |
how to simplify this expression so that I get an expression only?
when $t = 1, 2,.... $
$$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \... |
I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map from $T_x M$ to $M$ at a point $x \in M$. $T_x M$ is the tangent space at a point $x.$
$f$ is defined by $x^T A x$ for a pos... |
Let $\mathscr{R}$ be a $\sigma$-ring and $\mu$ be a (positive) measure on $\mathscr{R}$. Suppose that $\{A_n\}$ is in $\mathscr{R}$ with $A_1 \supseteq A_1 \supseteq A_2 \supseteq\cdots$ and $A = \bigcap_{n=1}^{\infty} A_n$. Given those information, I need to provide a counter-example for the following equality when $\... |
I am working on a problem in the context of molecular simulation. I consider various potentials and compute the resulting forces on each of the atoms in my simulation. I managed to do this for the bond potential and the angle potential. I am stuck however on the dihedral potential.
I have the dihedral potential give... |
I study the following paper about finite $p$-groups,
Finite groups in which the non-normal subgroups have nontrivial intersection,
N. Blackburn, Journal of algebra, 3, 30-37 (1966).
In this paper the intersection
of all non-normal subgroups of $G$ is denoted by $R(G)$.
We say that a group $G$ is an $R$-gro... |
Do you have a probabilistic interpretation of this monster? If $Y$ is a random variable with density $g(y)=f'(y)$ I found that
$$h(t)=\frac{1-\int_0^tyg(y)dy}{(\phi-t)\int^1_tg(y)dy}-\frac{1-E(Y)}{\int^1_t(y-t)g(t)+(\phi-t)\int^1_tg(y)dy}$$ This suggests to check the property when $g(y)dy$ is replaced by the Dirac m... |
Let $\mathscr{R}$ be a $\sigma$-ring and $\mu$ be a (positive) measure on $\mathscr{R}$. Suppose that $\{A_n\}$ is in $\mathscr{R}$ with $A_1 \supseteq A_2 \supseteq A_3 \supseteq\cdots$ and $A = \bigcap_{n=1}^{\infty} A_n$. Given this information, I need to provide a counter-example for the following equality when $\m... |
I am trying to find the radon transform of the gaussian function
$$f(x,y) = e^{-(x^2 + y^2)}$$
Now, I am using the formula for radon transform as
$$
[\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \delta(x \cos(\theta) + y \sin(\theta) - \rho) dx dy
$$
When I am solvin... |
Firstly I am answering my own question after 2 days because I found a very cool solution for finding the best 'u' value in this approximation:
> $log_2(1+x)≈x+u$
My solution starts with finding the error value with an integral
> $f_{error}(x,u)=log_2(1+x)-(x+u)$\
$f_{totalErr}(u)=∫_{0}^{1}|f_{error}(x,u)|dx$ ... |
Suppose that $f(x)g(x)$ is in $L^1([a,b])$ for every $f(x)$ in $L^2([a,b])$, I understand that this implies that $g(x)$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it?
I have seen a more general version of this question [here][1] but its focus is slightly different and the explanation provi... |
I am really confused about the confidence interval for n=100 tosses and k=52 heads. If I estimate the probability $\hat{p}$:
$\hat{p} = \frac{52}{100} = 0.52$
$Var(\hat{p}) = pq = 0.52 * 0.48 = 0.2496$, being a bernoulli process
$\sigma = \sqrt{Var(\hat{p})} = 0.5$
$SE = \frac{\sigma}{\sqrt{n}} = \frac{0.5}... |
Confidence interval 100 coin tosses, 52 heads? |
You seem to be using some kind of [Wagner Principle][1] in your reasoning. [The Wagner Principle][2] states that if an event can either happen or not, then there is a 50-50 chance of it happening or not happening.
You are likewise saying: if you consider two sets, the first one being the set containing just the do... |
I *do* believe quaternions are the most direct approach to it. That is to say: to computing the rotation that corresponds to a given point on $\mathbb{S}^3/\sim_A$. Showing that this correspondence is bijective is a different story, but you said you already understood that part.
You know that elements of $SO(3)$ are... |
Let $f:[0,1] \to \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)|$ for all $x \in [0, 1]$. Show that $f(x) = 0$ for all $x \in [0, 1]$.
My attempt is this:
Suppose that $x \in (0, 1]$. Since $f$ is differentiable, by the mean value theorem there exists some $c \in (0, x)$ s... |
Is my proof of [If the premises are false, the conclusion is true] correct?
1. If the premises are false, the conclusion is false.
Statement 1 is false
because
I am cup. cup is an animal. i am an animal
The premise (I am cup, cup is an animal) is false, but the conclusion (I am an animal) is tru... |
Is it true that [if the premises are false, the conclusion is true]? |
Suppose we have a 2^n×2^n board. I understand the proof that you can use any rotation of L shaped trominoes and a monomino to fill the board completely, given you can mix different rotations in the same tiling.
I am stuck trying to devise a way to count the ways to tile such a board for any given n. I know the base c... |
I recently encountered a problem involving the construction of equations involving factorials or combinatorial numbers. I recall reading somewhere (although I cannot recall the reference) that this can be a tricky subject.
In a particular scenario, the author of the problem suggested that the solution would be $x=1$... |
How to set equations containing factorials? |
I am reading a [book of E. Rio][1] and I found there a theorem (*without a proof*) about coupling. Please see below.
[![enter image description here][2]][2]
**Question 1:** I am wondering what are the conditions these theorem to hold? IS my understanding correct that the only condition that $\mathcal{X} = (\Omega... |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with an infinite order generator and integer points $(-1,\pm1)$. The
[elliptic divisibility sequence](
https://en.wikipedia.org/wiki/Ellipti... |
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