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I want to understand the semantics of rights bit shifts `x>>k` in two's complement properly, in particular why do right bit shifts of size $k$ approximately divide $x$ by $2^k$ (and ideally making this "approx" precise and also handling the arithmetic left bit shift too).
So the representation of a number in two co... |
Consider the diagonal operator A{$x_n$} = {$a_nx_n$}. The operator A is defined on the sequence space $l_p$, where 1<p<∞. 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 $a_m$=$a_k$, m≠k. Then the... |
Given a psd matrix $H=[A, B; B^\top, C]$. I can find bounds for singular values and the norm for $A, B, C$. How can I construct a bound for the condition number of $H$? I have tried using interlacing theorem but it doesn't work. |
I've been trying to find an analytical expression (if it exists) of the followimg function:
$$
f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{n^n}x^{2n}
$$
written in form of elemental functions. I've tried to write a differential equation for the series given that:
$$
f'(x) = -2x\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^n}x... |
What is the distributional derivative of the function
$$
f(x) =
\begin{cases}
1 & \text{if } x \in \mathbb Q \\
-1 & \text{if } x \in \mathbb R \setminus \mathbb Q
\end{cases}
$$
Can we compute it explicitly? Does it have any 'good properties'? For example, is it a measure? |
I shall outline what is basically going on in the way I am familiar with, because, I suppose, once the main idea has been grasped, the rest could be worked out according to the specific details of a particular exposition.
Our intention is to build an increasing sequence of consistent set of sentences in an expanded ... |
Let $F = \Phi (X^x_T)$. $X^x_T$ behaves as a risky asset such that $dX^x_t=rX^x_tdt+\sigma X^x_tdW_t$ with $W_t$ standard Brownian Motion; $X_0^x=x$. $\Phi$ is continuous with linear growth and $P(x) = e^{−rT}\mathbb{E}[F]$. Prove that $P \in C^2(\mathbb{R}_+;\mathbb{R}) $.
I've already proved that $P \in C^1(\math... |
For posts looking for feedback or verification of a proposed solution. "Is my proof correct?" is too broad or missing context. Instead, the question must identify precisely which step in the proof is in doubt, and why so. This should not be the only tag for a question, and should not be used to circumvent site polici... |
What is the distributional derivative of the function
$$
f(x) =
\begin{cases}
1 & \text{if } x \in \mathbb Q \\
-1 & \text{if } x \in \mathbb R \setminus \mathbb Q
\end{cases}
$$
Can we compute it explicitly? Does it have any 'good properties'? For example, is it a measure?
Here are my thoughts/questions.
... |
Let us assume $A' \subset A$ and $f\colon A \to A'$ is bijective (so $A$ is an infinite set according to Dedekind‘s definition). Let us further assume $h: A \to B$ is bijective. I want to prove that $B$ is also an infinite set.
There is a proof that goes like this:
We assume further $g = h \circ f \circ h^{-1} \c... |
If $G$ is square free such that $|G|=p_1\cdots p_n$, where $p_n>p_{n-1}>\cdots$. Then we can use the N/C Theorem, along with some induction, to show that the Sylow $p_n$-subgroup of $G$, $P_n$, is normal in $G$.
Suppose $G$ isn't square free, and let's assume $G$ is solvable. Suppose $p_k$ is the largest prime di... |
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. ... |
Let S={a--z, 0--9} be the set of characters(lower case a through z) and the digits 0 through 9
I need to form a password of length 8 in which two of them must be digits. How many possible passwords can I create?
Here is what I came up with but im not positive it is correct.
p2 = 8C2 *10^2 * 26^6
p3 = 8... |
There are two generators: $a$ and $b$. It is known that $a^4 = b^4 = (ab)^{15} = (ab^2)^6 = (ab^3)^9 = 1$.
Is it possible and how to express relations $(ab^2)^6 = (ab^3)^9 = 1$ through relations $a^4 = b^4 = (ab)^{15} = 1$?
PS In this case, the generators are rotations of adjacent faces of the 2x2 Rubik’s Cube. H... |
For any module with vector set V and scalar set C, does there exist a set X such that there exists a module isomorphism between V and (X → C)? |
Given a diagonalizable matrix $A \in \mathbb{R^{N \times N}}$, we can decompose $A$ in following terms:
1. $A = V \Lambda_{C} V^{-1}$ where $V, \Lambda_{C} \in \mathbb{C^{N \times N}}$
2. $A = Q \Lambda_{R} Q^{-1}$ where $Q, \Lambda_{R} \in \mathbb{R^{N \times N}}$
In a matrix form, we can see the following rela... |
Let us consider the function
$$
f(x) =
\begin{cases}
1 & \text{if } x \in \mathbb Q \\
-1 & \text{if } x \in \mathbb R \setminus \mathbb Q
\end{cases}
$$
I have a main question about the distributional derivative of $f$. Let me split it into four thoughts/subquestions:
1. This function is not Riemann in... |
Let us consider the function
$$
f(x) =
\begin{cases}
1 & \text{if } x \in \mathbb Q \\
-1 & \text{if } x \in \mathbb R \setminus \mathbb Q
\end{cases}
$$
I have a main question about the distributional derivative of $f$. Let me split it into four thoughts/subquestions:
1. This function is not Riemann in... |
I'm having trouble understanding Edward/Pierre's argument showing that the distance function is subanalytic. Specifically, "Remarks 3.11" in Semianalytic and subanalytic sets (https://link.springer.com/article/10.1007/BF02699126). Below is a direct copy of "Remarks 3.11".
$\textit{Remarks}\; \mathbf{3.11.}$ Let $X$ ... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time. I have documented my research and implementation in C#: it can be found on [Git... |
In a recent Veritasium video discussing Euclid's Elements, Alex Kontorovich comments that Euclid's definitions of primitive objects (e.g. "A point is that which has no part.") are absurd and lead to an infinite regression of definitions. He goes on to say:
> You shouldn't have definitions; you should have undefined te... |
Is there a term for the idea that mathematical objects are defined by their relationships? |
I am struggling dealing with $o$ notation. Here $k, n$ are all natural numbers, why do we have $o\Big(\sum_{k=0}^{n-1}\frac{(k\log\log k)^{1/2}}{k+1}\Big)=o((n\log\log n)^{1/2})$ as $n\rightarrow\infty$? And another one with $O$ notation, why does $\sum_{k=0}^{n-1}\frac{X_k}{k+1}=O ((n\log\log n)^{1/2})$ as $n\rightarr... |
Currently taking a class on computability theory and the constant function described as follows:
> $C^n_a(x_1,...,x_n) = a$
First of all, why do we have a function of this sort to begin with? Can't we just refer to the constant itself as it is? Also why is $x_1, ..., x_n$ passed to it if they're not used? |
Why do we have a constant function in primitive recursive functions? |
Currently taking a class on computability theory and started learning about primitive recursive functions. The constant function described as follows:
> $C^n_a(x_1,...,x_n) = a$
First of all, why do we have a function of this sort to begin with? Can't we just refer to the constant itself as it is? Also why is $x_... |
Why do we have a constant function as part of primitive recursive functions? |
For a linear combination over $v_1, \ldots, v_n$ with coefficients $a_1, \ldots, a_n$, if $a_i = f(v_i) \in \mathbb{R}$ for some function $f$, would $\sum_{i=1}^n a_iv_i$ still be a linear combination? |
Can the coefficients of a linear combination be functions of the variables? |
Is the following second-order statement true or false? (Assume $P$ is a binary predicate statement which is always defined for two members of $S\neq\emptyset$.)
$$\forall P\left( x,y \right)\forall S \neq \emptyset \Bigl[ \left[ \exists \left( a,b \right)\! \in\! S^2\, P \left( a,b \right) \middle] \equiv \middle[ ... |
Suppose a subcategory $C$ of topological spaces has all coproducts. Is that coproduct always equal to the disjoint sum topology?
Notes:
* If $C$ happens to be a coreflective subcategory of a category $D$, then I believe all colimits in $C$ are those of $D$. This is because the inclusion functor is left adjoint, a... |
Is the coproduct in a subcategory of topological spaces with coproducts always the disjoint sum? |
Let $ (M, g) $ be a Riemannian manifold and $ \omega $ a connection on the tangent bundle of $ M $ at a given point.
Then the curvature form is the $\mathfrak{g} $-valued 2-form $ \Omega \in \Omega^2(M) $ where $ \mathfrak{g} $ is the Lie algebra of the structure group in the tangent bundle $ TM $.
The connection... |
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying
$\overline{\partial_c}f=0,$
where
$$\overline{\partial_c}=\frac{1}{2}\left(\frac{\partial}{\partial t}+\frac{\operatorname{Im}q}{r}\frac{\partial}{\partial r}\right), \ \operatorname{Im... |
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying
$\overline{\partial_c}f=0,$
where
$$\overline{\partial_c}=\frac{1}{2}\left(\frac{\partial}{\partial t}+\frac{\operatorname{Im}q}{r}\frac{\partial}{\partial r}\right), \ \operatorname{Im... |
For any module with vector set V and scalar set C, must there exist a set X such that there exists a module isomorphism between V and (X → C)? |
Suppose that $M$ is an $n-$ dimensional smooth manifold and if there exists an atlas consisting of only two charts $(U,x)$ and $(V,y)$. If $U\cap V$ is connected, then $M$ is orientable.
To prove this, I must show that the transition map $x\circ y^{-1}: y(U\cap V)\to x(U\cap V)$ has a positive Jacobian determinant. ... |
Let $(\Sigma, g)$ be a compact Riemannian surface with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(\Sigma)$ associated to the same eigenvalue $\mu > 0$ of the Laplacian $\Delta_g$ and such that $\nabla u(p)$ and $\nabla v(p)$ are parallel for any $p \in \partial \Sigma$... |
Could somebody help me prove that $\sum_{k=0}^{500}\left(\dfrac{k}{1001}\right) = 0$? I think that there might be a bijection with $\sum_{k=0}^{500}\left(\dfrac{k}{501}\right)$ but I don't know how to prove this
|
Let $(M, g)$ be a compact Riemannian $3$-manifold with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(M)$ associated to the same eigenvalue $\mu > 0$ of the Laplacian $\Delta_g$ and such that $\nabla u(p)$ and $\nabla v(p)$ are parallel for any $p \in \partial \Sigma$? It ... |
I am trying the series expansion of this function :
$$f(n)=\, _2F_1\left(-\frac{1}{n},\frac{1}{n};\frac{1}{n}+1;-1\right)\text{at}\; \infty$$
This comes from : find an asymptotic series for this integral
$$I(n)=\int_{0}^{1}\sqrt[n]{1+x^n}dx$$
By using Mathematica, I could find $$I(n)=1+\frac{\pi^2}{12n^2}-\fra... |
Recall that
$$\int_{|z|=R} |dz| = 2 \pi R$$
Now, by the triangle inequality:
$$|z|^2-|a|^2 \leq |z^2-a^2| \leq |z-a||z+a|$$
Hence,
$$\int_{|z|=R} \frac{|dz|}{|z-a||z+a|} \leq \int_{|z|=R} \frac{|dz|}{|z|^2-|a|^2} = \frac{2\pi R}{|R^2-|a|^2|}$$
Still trying to figure out how to get the $\leq$ to becom... |
I have an eigenvalue problem $y'' + \lambda y=0$, $y'(-L) = y'(L)=0$. So far, I've solved the cases $\lambda=0$ and $\lambda=-p^2<0$. In the first case, I got $y_0(x) = B$ as an eigenfunction, and the second case yields a trivial solution. Similarly in the third case, I let $\lambda=p^2>0$. In this case, $y$ is of the ... |
Let $(M, g)$ be a compact Riemannian $3$-manifold with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(M)$ associated to the same eigenvalue $\mu > 0$ of the Laplacian $\Delta_g$ and such that $\nabla u(p)$ and $\nabla v(p)$ are parallel for any $p \in \partial M$? It looks... |
In the J. Tate's paper "Relations Between K2 and Galois Cohomology" Lemma 5.2, it says that given $\alpha_1,\cdots,\alpha_r\in $Br$_lF$, i.e. $\alpha_i$ is killed by $l$, where $l$ is prime and $l\neq$ char $F$ and $F$ is a global field. If there exists for each place $v$ of Fan element $x_v\in F_v^{\times}$ such that... |
I have a bit of a problem with the Extended Cauchy Schwarz Inequality, specifically at the line
"with equality if and only if b = cB^(-1)d for some constant c."
[Extended Cauchy-Schwarz Inequality][1]
As such, I'm having problems understanding the following lemma, at a similarly worded line:
"with the maxim... |
I have a series of fractions that consists consecutive odd numbers in the following fashion:
$$\frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \ ... \ = \ ?$$
My question is whether this series converges if it were to go on forever until infinity.
In my attempt, I do feel it should converge. Every term... |
Does the following series of fractions of consecutive odd numbers converge? If yes, where does it converge to? |
$$L=\lim_{\epsilon\to 0^+}\int_{\epsilon}^{2\epsilon}\frac{e^{(x-1)^2}}{x}dx$$
How can one approach this integral? I've tried inserting an extra variable to perform Feynman's trick but it just complicates it more.Kindly give a few hints on how to solve this. |
I have an eigenvalue problem $y'' + \lambda y=0$, $y'(-L) = y'(L)=0$. So far, I've solved the cases $\lambda=0$ and $\lambda=-p^2<0$. In the first case, I got $y_0(x) = B$ as an eigenfunction, and the second case yields a trivial solution. Similarly in the third case, I let $\lambda=p^2>0$. In this case, $y$ is of the ... |
So... I came across this fun little math / programming problem.
[![enter image description here][1]][1]
*The programming language is Java*
Now, to shortcut a few things...
* The `n1 = n0 + (n0 = n1))` line affects a Fibonacci sequence.
* The loop is flawed, because `i` never increments
* `int` in Jav... |
How come Pisano periods develop this way for$\mod 2^m$? |
In [Sets, Logic, Computation](https://slc.openlogicproject.org/), the compactness theorem is stated as follows:
> For any sentences $\Gamma$ and $A$, the following holds:
>
> 1. $\Gamma\models A$ iff there is some finite $\Gamma_0\subseteq\Gamma$ s.t. $\Gamma_0\models A$.
>
> 2. $\Gamma$ is satisfiable iff for a... |
I am reading a lecture note about tensor category by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. The link is attached here https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/pages/lecture-notes/
In the definition 1.1.1（The definition of monoidal category), a family of isomor... |
What is a "functorial isomorphism"? |
I have a series of fractions that consists consecutive odd numbers in the following fashion:
$$\frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \ ... \ = \ ?$$
My question is whether this series converges if it were to go on forever until infinity.
In my attempt, I do feel it should converge. Every term... |
I have a series of fractions that consists consecutive odd numbers in the following fashion:
$$\frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \ ... \ = \ ?$$
My question is whether this series converges if it were to go on forever until infinity.
In my attempt, I do feel it should converge. Every term... |
I found a question in the very end of [this][1] video, which essentially is the following:
> Let $n$ be some natural number. Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>- $f(A\cdot B) = f(A)\cdot f(B),\ \forall A,B \in \mathbb{... |
$x$ and $y$ are positive numbers constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, prove that there exists solution to $a\cdot f(x)+b\cdot f(y)=z$ (if any) where $z$ is a strictly positive number features that $x\geq y$.
I do... |
I am studying from sheldon ross's first course in probability. I think this would be a silly question, but anyway I have to know the answer. This is from chapter 2.
**EXAMPLE 6a Probability and a paradox**
> Suppose that we possess an infinitely large urn and an infinite collection of balls labeled ball number 1,... |
Are you asking if two coordinate charts would always give the same components for a vector at a point? If so, I think considering cartesian and polar coordinates on $\mathbb{R}^2$ should answer your question. The vector field itself is manifestly invariant but the component functions which describe the vectors at the v... |
I am reading through Krylov's *Lectures on Elliptic and Parabolic Equations in Hölder Spaces*. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to reference the relevant proofs in the Elliptic case. Starting with some definitions, we take
$$[u]_{\delta;\Omega} ... |
I came across the following asymptotic problem in my reseach, however I don't know how to answer it.
Let $C_0,C_1,C_2,C_3$ be absolute constant (do not depend on $n$), and $p(n)$ is a function of $n$ that is larger than the following function, i.e., $$\tiny p(n)>\frac{-(4+(n-2)(2C_1+C_2))+\sqrt{((4+(n-2)(2C_1+C_2)))... |
I came across the following asymptotic problem in my reseach, however I don't know how to answer it.
Let $C_0,C_1,C_2,C_3$ be absolute constant (do not depend on $n$), and $p(n)$ is a function of $n$ that is larger than the following function, i.e., $$\tiny p(n)>\frac{-(4+(n-2)(2C_1+C_2))+\sqrt{((4+(n-2)(2C_1+C_2)))... |
Let $M$ be a smooth manifold. We know that if $X:\ M\to TM$ is a smooth vector field, then for any local chart $x=(x_1,\dots,x_n):\ U\subset M\to\mathbb R^n$, there exists smooth functions $X_1,\dots,X_n:\ U\to\mathbb R $ such that
$$X_p\,=\,\sum X_i(p)\frac{\partial }{\partial x_i}\bigg|_p\ \ \forall p\in U $$
I w... |
Γ is a rectifiable simple closed curve and Ω is its interior. f is holomorphic on Ω and continuous on Ω∪Γ. Prove the integral of f along Γ is 0.
Since we already know the Cauchy's integral theorem for a piecewise-smooth curve, I want to use piecewise-linear curve (say, γ) to approximate Γ. But the problem is that I ... |
*Remark:* I have mostly doubt on **Part 3** but any general remark on my answer will be greatly appreciated.
## Question:
Let $a > 0$. For $n \in \mathbb{N}$, let the random variable $X_n$ on $\{0,n\}$ be defined by
$$
\begin{cases}
P(X_n=0)=1- \frac{1}{n^{a}}\\
P(X_n=n)=\frac{1}{n^{a}}.
\end{cases}
$$
Al... |
I came across the following asymptotic problem in my reseach, however I don't know how to answer it.
Let $C_0,C_1,C_2,C_3$ be absolute constant (do not depend on $n$), and $p(n)$ is a function of $n$.
We need to find $p(n)$ such that the following two conditions hold:
(1) $p(n)$ that is larger than the followi... |
I'm trying to derive the dual form for the L2 support vector machine. I thought the Lagrangian should be of the form:
$L_p(w, b, \xi, \alpha) = \frac{1}{2} \| w \|^2 + \frac{C}{2} \sum_{i=1}^m \xi_i^2 - \sum_{i=1}^m \alpha_i \left( y_i (x_i \cdot w + b) - 1 + \xi_i \right) + \sum_{i=1}^m \mu \xi_i$
because of th... |
Consider the family $$C_{\alpha}:\big({2(\alpha-\cos{t})\,\cos{t},\,2(\alpha-\cos{t})\,\sin{t}}\big)\,,\;t\in[0,2\pi]\,,\;\alpha\in\mathbb{R},$$ of closed parametric curves.
1. For which values of $\alpha$ these curves are homotopy-equivalent to curve $C_{0.5}$?
2. For those curves which are homotopy-equivalen... |
The map $\mathbb{Z}\hookrightarrow \mathbb{Z}[i]$ induces an isomorphism of fields
$$\mathbb{Z}/173 \simeq\mathbb{Z}[i]/(13+2i)$$
So we could work with representatives in $\mathbb{Z}$, which is sometime easier. By the above $ {80}\mapsto i$, and $107\mapsto 1-3i$. Note that $107$ is a primitive root $\mod 173$. ... |
How to prove $a+b+c\ge 3$ with the strange condition? |
I was playing around looking at perfect numbers and noticed some stuff but I don't know much about perfect numbers and would be interested what these patterns amount to.
Consider the divisors of some perfect numbers (2 - 7) including themselves :
(7th) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096,... |
Pattern of perfect numbers - does the pattern continue? |
I was working on an exercise in calculus, and I somehow failed to prove the following:
$$B_n:=\int_0^{\pi/2}x^2cos^{2n}xdx \leq \frac{C}{n}\int_{0}^{\pi/2}cos^{2n}xdx=:\frac{C}{n}A_n $$ for some constant $C \in \mathbb{R}$
By using integration by parts, we can obtain
$$2nB_{n-1}-2nB_n = \frac{1}{n}A_n$$
Is... |
*Remark:* I have mostly doubt on **Part 3** but any general remark on my answer will be greatly appreciated.
## Question:
Let $a > 0$. For $n \in \mathbb{N}$, let the random variable $X_n$ on $\{0,n\}$ be defined by
$$
\begin{cases}
P(X_n=0)=1- \frac{1}{n^{a}}\\
P(X_n=n)=\frac{1}{n^{a}}.
\end{cases}
$$
Al... |
So in virtually all English-language calculus classes I have seen, we define integration as the "accumulation of change". And that makes sense to me intuitively, but when I think about it, I feel like accumulation of value makes more sense. Because if we take the change in a function $f:\mathbb{R}\mapsto\mathbb{R}\text... |
Why do we call integration "accumulation of change"? |
$x$ and $y$ are positive numbers constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, if there exists solution(s) to $a\cdot f(x)+b\cdot f(y)=z$ where $z$ is a strictly positive number, prove that one of these solutions features th... |
How do I solve the eigenvalue problem $y'' + \lambda y=0$, $y'(-L) = y'(L)=0$?
My attempt:
- Case 1: $\lambda=0$. If $\lambda=0$, then the BVP becomes $y''(x)=0$, $y'(-L)=y'(L)=0$. The general solution to this differential equation is $y(x)=Ax+B$. Taking the derivative gives $y'(x)=A$, and the boundary condition i... |
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow this rule:
$a_i=1-F(i)$ while $F(i)=e^{-\lambda*i}$. $F(i)$ is th... |
In information theory, it is well known that the capacity of a channel can be achieved asymptotically using random coding method (Section 7 of [Cover & Thomas' textbook][1]).
However, in coding theory, it is only shown that we can apply random coding method to get a code that achieves Gilbert-Varshamov bound (I will... |
I'm having trouble understanding Edward Bierstone and Pierre D. Milman's argument showing that the distance function is subanalytic, specifically "Remarks 3.11" in "[Semianalytic and subanalytic sets](https://link.springer.com/article/10.1007/BF02699126)" Publications Mathématiques de l'Institut des Hautes Études Scien... |
Why is distance function subanalytic? |
I was playing around looking at perfect numbers and noticed some stuff but I don't know much about perfect numbers and would be interested what these patterns amount to.
Consider the divisors of some perfect numbers (2 - 7) including themselves :
(7th) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096,... |
I was playing around looking at perfect numbers and noticed some stuff but I don't know much about perfect numbers and would be interested what these patterns amount to.
Consider the divisors of some perfect numbers (2 - 7) including themselves :
(7th) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096,... |
Pattern of perfect numbers - why? |
I am trying to understand how a joint distribution is formed when two regular conditional distributions are involved.
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let there be the three random variables $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$, $Y:(\Omega, \mathcal{A})\r... |
How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function? |
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow this rule:
$a_i=1-F(i)$ while $F(i)=e^{-\lambda*i}$. $F(i)$ is th... |
Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The elements $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $ab=0$ when $a\in Rx$ and $b\in xR$? |
Over dampened Pendulum System:
$$
mL^{2}\ddot{\theta } +b\dot{\theta } +mgL\sin \theta =\Gamma
$$
First order approximation:
$$
b\dot{}+mgL\sin{}=Γ
$$
Nondimensionalize, diving through by mgL:
$$
\frac{b\dot{\theta }}{mgL} =\frac{\Gamma }{mgL} -\sin \theta
$$
Let $$\tau =\frac{mgL}{b} t$$ and $$\g... |
$f : [0,1] \rightarrow [0,1]$ is a continuous open mapping. Show that it has finite number of maxima and $\max(f(x)) = 1$.
It is well-known that if $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and open than it's monotonic (https://math.stackexchange.com/questions/3754965/every-continuous-open-mapping-mathbbr-to... |
$x$ and $y$ are real numbers (edited: thanks to @mcd's answer, now the "positive" requirement is removed) constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly increasing, positive and continuous function $f(\cdot)$, if there exists solution(s) to $a\cdot f(x)+b\cdot f(y)=z$ where $z$ is ... |
I was attempting a true/false question from a linear algebra textbook and one of the questions was
> Suppose that the characteristic polynomial of $A$ is $p(λ) = λ^3(λ − 2)(λ + 3)^2$ . Then the null space of $A$ can be at most $2$–dimensional.
The answer was given as false with the explanation being that the mul... |
Over dampened Pendulum System:
$$
mL^{2}\ddot{\theta } +b\dot{\theta } +mgL\sin \theta =\Gamma
$$
First order approximation:
$$
b\dot{}+mgL\sin{}=Γ
$$
Nondimensionalize, diving through by mgL:
$$
\frac{b\dot{\theta }}{mgL} =\frac{\Gamma }{mgL} -\sin \theta
$$
Let $$\tau =\frac{mgL}{b} t$$ and $$\g... |
I'm trying to solve the evolution problem below using the standard Galerkin method:
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
y=0, \ \text{on}\ \partial \Omega, \ \ \ y(0)=y_0 \in L^2(\Omega)
\end{cases}$$
In Theorem 1.2, page 102 of [Optimal Con... |
There's something I can't quite wrap my head around, and I am not satisfied with my professor's "that's just defined that way" answer. So suppose we have:
f1(x) = x - 5
f2(x) = 1 / (x - 5)
f3(x) = 1
By simple algebra, we can see that
f4(x) = f1(x) * f2(x) == f3(x)
What I mean is the rational function w... |
While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-categories. Here is one such situation where they seem to be tacitly appearing in Haugseng's [An allegedly somewhat friendly ... |
> Let $K$ be an ordered field containing the field of rationals
> $\mathbb{Q}$. Prove that $\mathbb{Q}$ is dense in $K$ if and only if
> every element of $K$ is the limit of a Cauchy sequence in
> $\mathbb{Q}$.
Definition: $\mathbb{Q}$ is dense in $K$ if for each pair $x,y$ in $K$, $x<y$, there exists $z\in\mathb... |
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