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Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following three equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
For given $n>1$ I'm looking for the maximum number $k$ of colors such that there exists a $k$-coloring of the edges of the complete graph on $2n$ vertices with the property that a copy of every possible $k$-colored perfect matching is present in the colored complete graph.
For example, we know that there are $(n+1)$... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following three equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT... |
From the answer for this [question][1], I understand that *affine subspace* is the same as *affine subset*, however (despite the somewhat misleading question's title), it doesn't say that *affine space* is the same as *quotient space*.
Also, I found the following definition of *affine space* on [Wikipedia][2]:
> ... |
Is *affine space* the same as *quotient space*? |
Find the maximum length $m$ of the sequence $a_1 a_2 \dots a_m$ such that
(1) Each $1\le a_i\le n(\in\mathbb{N})$
(2) No $1\le i \le m-1$ such that $a_{i}=a_{i+1}$
(3) Call $(x,y)$ good if $a_i=x, a_j=y, a_k=x$ for some $1\le i<j<k\le m$.
If $(x,y)$ is good, $(y,x)$ is not good.
I found a proof using induc... |
Do you know modular arithmetic? This problem is much clearer to understand in that setting (it becomes obvious why the exponent $m$ is being taken as odd), while it seems totally opaque and strange as a plain induction problem.
Update: since you know modular arithmetic, then in order to show
$1^m + 2^m + \cdots... |
For example, consider the sequence $2,7,3,5$. The sums of the segments of this sequence are as follows, and they are all unique:
$$2, 2+7, 2+7+3, 2+7+3+5, 7, 7+3, 7+3+5, 3, 3+5, 5$$
Can we generate such a sequence using the first $n$ prime numbers for any given $n$?
Here's my attempt:
Initially, consider the ca... |
Is it possible to construct a sequence using the first $n$ prime numbers such that each segment has a unique sum? |
I read in some lecture notes the following definition of contraction rate:
Definition (Posterior rate of contraction) The posterior distribution $\Pi_n\left(\cdot \mid X^{(n)}\right)$ is said to contract at rate $\epsilon_n \rightarrow 0$ at $\theta_0 \in \Theta$ if $\Pi_n\left(\theta: d\left(\theta, \theta_0\right)... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following three equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following three equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT... |
I am looking at the Tor functor for $R$-modules $M,N$. If, for example, $M$ or $N$ is flat then $\text{Tor}_i^R(M,N) = 0$ for all $i > 0$ so this functor can be seen as a "measure" of flatness (and it makes sense that a few criteria for flatness can be stated in terms of Tor, as in Eisenbud)
But I was wondering... |
Does $\text{Tor}_i^R(M,N) = 0$ imply $\text{Tor}_j^R(M,N) = 0$ for $j > i$? |
Let $V$ be an $n$-dimensional vector space, and let $M\subset V\oplus V$ be an $n$-dimensional subspace. It is known that I can identify $M$ (non-uniquely) with a frame, i.e., with an injective linear operator
$$\left( \begin{array}{c} X \\ Y \end{array} \right):V\rightarrow V\oplus V$$ whose range is $M$ (here, $X,Y$... |
Does the frame of a direct sum of subspaces split somehow? |
When the boat's speed was $45$ km/h, its engine suddenly stopped.
$7$ minutes after the engine stopped, the boat's speed was $12$ km/h.
It is known that on the boat, in the opposite direction to its movement, the frictional force of the water acts, which is proportional to the speed of the boat.
Find the law... |
My book states that it's not always true that:
$$E(\sum_{i=1}^{\infty} X_i) = \sum_{i=1}^{\infty} E(X_i)$$ and what makes it not true in general is this equality:
$$E(\sum_{i=1}^{\infty} X_i) = E(\lim_{n\to\infty}(\sum_{i=1}^{n}X_i)) \stackrel{?}{=} \lim_{n\to\infty}E(\sum_{i=1}^{n}X_i)$$
Two special c... |
>In right triangle $ABC$, we have $\angle ACB=90^{\circ}$, $AC=2$, and $BC=3$. Medians $AD$ and $BE$ are drawn to sides $BC$ and $AC$, respectively. $AD$ and $BE$ intersect at point $F$. Find the area of $\triangle ABF$.
<hr>
I aim to use the Shoelace Theorem to calculate the area of $\triangle ABF$, but I'm stuck ... |
Let $V$ be an $n$-dimensional vector space, and let $M\subset V\oplus V$ be an $n$-dimensional subspace. It is known that I can identify $M$ (non-uniquely) with a frame, i.e., with an injective linear operator
$$\left( \begin{array}{c} X \\ Y \end{array} \right):V\rightarrow V\oplus V$$ whose range is $M$ (here, $X,Y$... |
For this question I am asked to find the *radius and centre-point* of the
circle of curvature for the following function:
> $−7.87e^{2.65x}$
I calculate the radius correctly with the formula: $R =\frac{1}{ρ}$
which is:
R = 0.9838228573
[working out for radius][1]
However, when it comes to finding *the... |
Is there a way to define in a canonical way a $p$-adic valuation on the complex numbers or at least on the algebraic numbers, that extends the $p$-adic valuation on $\mathbb{Q}$? That is, is there a canonical embedding of $\overline{\mathbb{Q}}$ to $\overline{\mathbb{Q}_p}$?
To give a concrete example, take the poly... |
How to solve this limit with e power in both numerator and denominator? |
I have $F(G(x))=G(x)$, where $x$ is any real number, and function $F$ can be computed if value $G(x)$ is given. I have also proved that function $G$ is monotone with respect to $x$ and $G(x)\in[0,1]$.
I want to obtain a mapping $G$ (or an algorithm G).
Is this a fixed-point problem? How to solve it, either ... |
$F(G(x))=G(x)$. what is this problem? |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following three equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT... |
A and B play sudoku until A has 3 consecutive win. Find expected value of the number of matches that they played. Known that there are three cases: win, tie, lose and the probability is uniform.
I've tried using recursion approach such that:
Assume f(x) is the probability that A has 3 consecutive win in x matches. ... |
Let $V$ be an $n$-dimensional vector space, and let $M\subset V\oplus V$ be an $n$-dimensional subspace. It is known that I can identify $M$ (non-uniquely) with a frame, i.e., with an injective linear operator
$$\left( \begin{array}{c} X \\ Y \end{array} \right):V\rightarrow V\oplus V$$ whose range is $M$ (here, $X,Y$... |
I'm trying to evaluate the following integral, but I think I don't do something correctly.
$ \int\frac{\cos x}{1+x^2}\;dx\ $
I tried in 2 ways:
I defined $\gamma = \gamma_1 + \gamma_2 $, where $ \gamma_1 = [-R, R], \gamma_2 = R e^{it}, t \in [0, \pi] $.
so the integral on $\gamma_2$ approaches zero, and by res... |
I have this integral: $\int_{-\infty}^{\infty}\frac{xe^{a - x}}{(1+e^{a-x})^2}dx$
I've tried to find its: $\int_{-\infty}^{\infty}\frac{xe^{a - x}}{(1+e^{a-x})^2}dx = | a-x = t| = -\int_{-\infty}^{\infty}\frac{(a-t)e^{t}}{(1+e^{t})^2}dx$, then I used integration by parts:
Let $u = a-t$ so $du = -dt$ and let $dv = \... |
Well I give an answer [here][1] and I want to know if it's true for a special example
*Answer :*
Not an answer just some speculation about gamma function and Bernstein's polynomial which is of independent interest here but could help .
Using fallaciously the inverse function of $f(x)$:
$$f\left(x\right... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
I don't really get what's the difference between them. What does each thing physically and mathematically signify?
Aren't both things just a dot product with the del operator? |
What is gradient? What's the difference between gradient and divergence? |
Mathematically, the **gradient** is a property of a scalar function $f:\mathbb R^n\to\mathbb R$, found as
$$\mathrm{grad} (f)=\nabla f=\begin{bmatrix}\frac{\partial f}{\partial x_1}\\\frac{\partial f}{\partial x_2}\\ \frac{\partial f}{\partial x_3}\\\vdots\end{bmatrix}.$$
In physical terms you can think of it as ... |
A group of *2n* individuals consisting of *n* couples, are randomly arranged at a round table. You are required to find an upper bound for the probability that none of the couples are seated next to each other.
**Solution:**
This is a combinatorial problem. Let's denote the total number of ways to arrange *2n* indi... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
I encountered very similar and related question. That is whether the following conditions are equivalent to each other,
(1) $f$ is $L$-smooth ($\|f(x) - f(y)\| \leq L \| x-y \|$)
(2) $|D_f(x,y)| \leq \frac{L}{2}\|x-y\|^2$
(3) $\|\nabla^2 f\| \leq L$.
Without otherwise stated, we consider Euclidean norm fo... |
Is the $u$ substitution needed? Consider the following:
$\sqrt{\frac{a}{x}}-\sqrt{\frac{x}{a}}=\frac{a^{2}-1}{a}$
Now, re-write the RHS as $a-\frac{1}{a}$ and square both sides (this is the part that I'm not totally sure is valid) to give:
$\frac{a}{x}+\frac{x}{a}-1=a^{2}+\frac{1}{a^{2}}-1$
$\frac{a^{... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
Let $X\to \operatorname{Spec } K$ be a projective integral variety. Let $L|K$ be a field extension and consider an $L$ point $x\colon \operatorname{Spec} L\to X$ in $X$ (morphism over $K$).
Under which conditions on the extension $L|K$, the morphism $x \colon \operatorname{Spec} L\to X$ is projective or quasi-projec... |
Mathematically, the **gradient** is a property of a scalar function $f:\mathbb R^n\to\mathbb R$, found as
$$\mathrm{grad} (f)=\nabla f=\begin{bmatrix}\frac{\partial f}{\partial x^1}\\\frac{\partial f}{\partial x^2}\\ \frac{\partial f}{\partial x^3}\\\vdots\end{bmatrix}.$$
In physical terms you can think of it as ... |
> Show that for any real random variable $X$ and any proper open subset $U$ of $\mathbb{R}$, we have $$\mathbb P(X \in U)=\sup\limits \left \{\mathbb P(X \in K): K \subseteq U \text { is compact} \right \}.$$
**My Attempt** $:$ Let $U \subseteq \mathbb R$ be open. Let $$\alpha : = \sup\limits \left \{\mathbb P (X \... |
If I have an arbitrary vector $A = (a,b,c,0)$ how can I find a transformation matrix $M$ such that $M \times A = (0,1,0,0)$?
We can assume $A$ has a magnitude of $1$ if it helps simplify the derivation process.
The trivial case $A = (0,1,0,0)$ would cause $M$ to be the identity matrix.
If $A = (0,-1,0,0)$ then... |
Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this part. I managed to repeat most of the steps, but the "this shows" (highlighted with **!!!**) sentence confuses me. How was ... |
May this [paper][1] answer your question?
*NB: there are typos in the last Corollary.*
[1]: https://ijpam.uniud.it/online_issue/202044/71%20RamiAlahmad.pdf |
I'm trying to evaluate the following integral, but I think I don't do something correctly.
$ \int_{-\infty}^\infty\frac{\cos x}{1+x^2}\;dx\ $
I tried in 2 ways:
I defined $\gamma = \gamma_1 + \gamma_2 $, where $ \gamma_1 = [-R, R], \gamma_2 = R e^{it}, t \in [0, \pi] $, and letting R aproach infinity.
So the i... |
There is the following analague of natural density on the positive real numbers:
Suppose that $A\subseteq (0,\infty)$ is Lebesgue-measurable, then one can look at the limit (if it exists) $\frac{\lambda(A\cap(0,n))}{n}$ for $n\rightarrow\infty$, where $\lambda$ is the usual Lebesgue measure on $(0,\infty)$. My questio... |
That means if a one parameter differentiable group $\Phi(x,t):\mathbb{R}^d\times \mathbb{R}\to\mathbb{R}^d$ satisfies $$\Phi(\Phi(x_0,t_1),t_2)=\Phi(x_0,t_1+t_2),\Phi(x_0,0)=x_0$$
holds for $\forall x_0\in\mathbb{R}^d,t_1,t_2\in\mathbb{R}.$ Then is there exist a suitable function $F$ on $\mathbb{R}^d$ s.t. $\dot{\Phi(... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
Is $\textit{affine space}$ the same as $\textit{quotient space}$? |
During my research I stumbled upon this system over $\mathbb{Z}_{2^n}$. Let $k=2^{n-1}-1$ then the system looks like
\begin{equation*}
\begin{cases}
x_1+2^{n-1}=d_k +x_k \\
x_2+2^{n-1}=d_1+x_1 \\
x_3+2^{n-1}=d_2+x_2 \\ \tag{1}
\dots \\
x_k+2^{n-1}=d_{k-1}+x_{k-1} \\
x_i \notin \{x_j,x_j + 2^{n-1}\}... |
>How to rigorously prove that $\sum_{n=1}^ \infty \frac{1}{4n-1} - \frac{1}{4n} =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}$$
$$f_N'(x) = \sum_{n=1}^ N x^{4n-2}- x^{4n-1}= x^{4}\left(\frac{1-x}{x^2} \right)\frac{x^{4N+4}-1 }{x^4-1}$$
I nee... |
How to rigorously prove that $\sum\limits_{n=1}^ \infty \frac{1}{4n-1} - \frac{1}{4n} =\frac{\ln(64)- \pi}{8}$? |
I'm performing a close critical study of David Nelson's *Penguin Dictionary of Mathematics* (4th ed., 2008).
Under the entry **hypergeometric differential equation**, it suggests the form:
$$x (1 - x) \dfrac {\mathrm d^2 \phi} {\mathrm d x^2} + [c - (a + b - 1) x] \dfrac {\mathrm d \phi} {\mathrm d x} - a b \phi = ... |
Form of Hypergeometric Differential Equation - possible mistake? |
Suppose I have some tensor, for concreteness I will consider a rank 4 tensor with components $R_{abcd}$ where the indices run over 0,1,2,3 (my question arises in a physics context). Under what conditions is the existence of another tensor, call it $G_{abcd}$, which satisfies $G_{ab}^{~~~mn}G_{cdmn}=R_{abcd}$ guaranteed... |
>How to rigorously prove that $\sum_{n=1}^ \infty \frac{1}{4n-1} - \frac{1}{4n} =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}$$
$$f_N'(x) = \sum_{n=1}^ N x^{4n-2}- x^{4n-1}= x^{4}\left(\frac{1-x}{x^2} \right)\frac{x^{4N+4}-1 }{x^4-1}$$
I nee... |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
I have the equation $x^2+xy+y^2=z^2$ to solve it in natural numbers.
The discriminant of it $D=4z^2-3b^2$ and must be perfect square.
I wrote Python program to get solutions for $1<x<100$ by enumeration.
def Solution():
A=[]
nMaximum=100
for a in range(1,nMaximum):
... |
How to find multiple solutions for 3 variable, 2 degree Diophantine equation? |
>How to rigorously prove that $\sum_{n=1}^ \infty \frac{1}{4n-1} - \frac{1}{4n} =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}$$
$$f_N'(x) = \sum_{n=1}^ N x^{4n-2}- x^{4n-1}= x^{4}\left(\frac{1-x}{x^2} \right)\frac{x^{4N+4}-1 }{x^4-1}$$
I nee... |
## Setup ##
So... I kinda handled most of my proof but I need help with some of the stuff I just kinda went with until it worked out. The problem relates to medicine and its decay in the body. We are given that the medicine will release over a period of $b$ hours and another dose is given at time $T$.
### Known Val... |
I'm trying to evaluate the following integral, but I think I don't do something correctly.
I'm interested where my try failed, compared to other ways.
$ \int_{-\infty}^\infty\frac{\cos x}{1+x^2}\;dx\ $
I tried in 2 ways:
I defined $\gamma = \gamma_1 + \gamma_2 $, where $ \gamma_1 = [-R, R], \gamma_2 = R e^{i... |
we have $(X,F,\mu)$ be the measure space and let $M_{\mu}=\{p:X\to R \text{ measurable }: p>0 \mu -a.e and \int p d\mu =1\}$.
For each positive density $p\in M_{\mu}$ we define the Orlicz space $L^{\Phi}(p.\mu)$.
Then we have $B_{p}=\{u\in L^{\Phi_1}(p.\mu): E_{p}[u]=0\}$ where $E_{p}$ is the expectation.
Now we ha... |
Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$, Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}.$
**My attempt:-**
$\overline{\langle y, x \rangle}=\overline{\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert y +i^k x\Vert^2}=\overline{\frac{1}{4}(\Vert ... |
During my research I stumbled upon this system over $\mathbb{Z}_{2^n}$. Let $k=2^{n-1}-1$ then the system looks like
\begin{equation*}
\begin{cases}
x_1+2^{n-1}=d_k +x_k \\
x_2+2^{n-1}=d_1+x_1 \\
x_3+2^{n-1}=d_2+x_2 \\ \tag{1}
\dots \\
x_k+2^{n-1}=d_{k-1}+x_{k-1} \\
x_i \notin \{x_j,x_j + 2^{n-1}\}... |
>How to rigorously prove that $\sum_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n}) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N (\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n})$$
$$f_N'(x) = \sum_{n=1}^ N( x^{4n-2}- x^{4n-1})= x^{4}\left(\frac{1-x}{x^2} \right)\frac{x^{4N+4}-1 }{x^4-1}$$
... |
How to rigorously prove that $\sum\limits_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n} )=\frac{\ln(64)- \pi}{8}$? |
I have the equation $x^2+xy+y^2=z^2$ to solve it in natural numbers.
The discriminant of it $D=4z^2-3y^2$ and must be perfect square.
I wrote Python program to get solutions for $1<x<100$ by enumeration.
def Solution():
A=[]
nMaximum=10**2
for x in range(1,nMaximum):
... |
How to check Pentagon axiom with induced associator of skeletal category? |
>How to rigorously prove that $\sum_{n=1}^ \infty\left( \frac{1}{4n-1} - \frac{1}{4n}\right) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f_N'(x) = \sum_{n=1}^ N( x^{4n-2}- x^{4n-1})= x^{4}\left(\frac{1-x}{x^2} \right)\frac{x^{4... |
How exactly does one detect if it is in a curved manifold? |
Relationship for the real-value A and C given the medium B? |
As I understand, in a function, the subset of the codomsin actually mapped to is called the range.
What about the domain? Is there a name for the subset of the domain actually used in a mapping? |
Disclaimer : Not a direct answer but a methodology.
As your question can be understood as "how can I attack such an issue ?", I would like here to propose two natural tools for such questions involving angle bissectors in a triangle : [**trilinear coordinates**](https://en.wikipedia.org/wiki/Trilinear_coordinates) $... |
Suppose that $A\subseteq\mathbb{N}$. Then one can look at the function $f(n)=|\{0,...,n\}\setminus A|$ ($|S|$ denoting the cardinality of $S$ ). I am interested in the case when $f(n)\leq Cn^\alpha$ with $C>0$ and $\alpha<1$. In this case $A$ obviously has natural densitiy $1$. My question is: Is there already a name f... |
Say $r$ is an Asperger radius of $n$ if $(n-r,n+r)=(p^{a},q^{b})$ with $p$ and $q$ primes and integral $a$ and $b$ such that $ab=r$. If $r$ is a divisor of a prime number then of course $\min(a,b)=1$.
The famous twin prime conjecture can be reformulated as :
$1$ is an Asperger radius of infinitely many integers.
... |
Let $X$ be a normed linear space over the field $\mathbb C$ with the norm $||.||$. Let $x,y \in X$,
Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$, Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}.$
**My attempt:-**
$\overline{\langle y, x \r... |
As I understand, in a function, the subset of the codomain actually mapped to is called the range.
What about the domain? Is there a name for the subset of the domain actually used in a mapping? |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
In [this][1] question it is proved in the answers that If $f :[0 , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to ... |
>How to rigorously prove that $\sum\limits_{n=1}^ \infty\left( \frac{1}{4n-1} - \frac{1}{4n}\right) =\frac{\ln(64)- \pi}{8}$ ?
My attempt
$$f_N(x):= \sum_{n=1}^ N \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$
$$f_N'(x) = \sum_{n=1}^ N( x^{4n-2}- x^{4n-1})= x^{4}\left(\frac{1-x}{x^2} \right)\fr... |
I am trying to show that the polynomial $y-x^2-x^3$ is reducible in the formal power series ring $k[[x,y]]$. I am attempting the question by finding a polynomial in $k[[x,y]]$ which is the square root of $x^2+x^3$. In order to find the square root I wrote the general polynomial in $k[[x,y]]$, $$a_{00}+a_{10}x+a_{01}y+a... |
# Problem
Compute
$$
\oint_L\frac{xdy-ydx}{4x^2+y^2}
$$
where $L$ is a circle centered at $(1, 0)$ with a radius of $R > 1$, and the direction of $L$ is counterclock-wise.
# Solution
To by pass the singular point at $(0, 0)$ which is inside the circle $L$ surrounds, adding an auxiliary curve:
$$
C:4x^2+y^2 =... |
Consider a subgroup of the real number (additive subgroup) and a positive homomorphism $\tau$ from the subgroup to $R$. Positive homomorphism means it maps positive elements to positive elements. I believe that $\tau$ has to be a positive scalar multiple ie $\tau(x)=cx$ but I am not sure how to show this. This is easy... |
Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos? |
A straight line has, in the traditional sense of the term, two directions; we can mark a starting point and from it we observe two directions, or in other words, to reach any point on a straight line we can do it only in two directions.
I heard that there is a topological demonstration that it is not possible to alg... |
There are two ways to interpret the question. First, is an allocation in the core for the initial assignment of houses? Second, is the allocation in the core if every agent owns their assigned house. For the second question, just run the top trading cycle algorithm starting from the allocation (that seems to be your an... |
Consider a subgroup of the real number (additive subgroup) and a positive homomorphism $\tau$ from the subgroup to $\mathbb R$. Positive homomorphism means it maps positive elements to positive elements. I believe that $\tau$ has to be a positive scalar multiple ie $\tau(x)=cx$ but I am not sure how to show this. This... |
Let $X$ be a normed linear space over the field $\mathbb C$ with the norm $\|\cdot \|$. Let $x,y \in X$. Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$. Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}.$
**My attempt:**
\begin{align*}
\overl... |
The following is a particular case of rank theorem.
Theorem 1( Rank theorem for surjective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is surjective, then there are charts $(U, \varphi... |
We are given that a, b, and c are positive real numbers such that:
a + b + c = abc - 2
We want to prove the following inequality:
$sqrt(1/(1+a)) + sqrt(1/(1+b)) + sqrt(1/(1+c)) <= sqrt(3)$
**Can anyone provide a proof for this inequality under the condition that a, b, and c are positive?** |
Suppose that $M$ is a smooth manifold. Let $\omega$ be an $n-$ form on $M$ with compact support. Then we define $\int_M\omega$ using partitions of unity. If $M$ is covered by a single chart $h:M\to \mathbb R^n$, then we define $\int_M\omega:= \int_{\mathbb R^n} (h^{-1})^\ast \omega$, where $\ast$ denotes pullback. $\ta... |
Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\cdots,n$. Cosider the following mixed inter programm(MIP) with decision variable $c_i\in\{0,1\}$, and $X=[x_1,\cdots,x_n]\in\mathbb{R}^{l\times n}$ with $x_i\in\mathbb{R}^{l}$ for $i=1,\cdots,n$.
Ob... |
The following is a particular case of rank theorem.
Theorem 1( Rank theorem for surjective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is surjective, then there are charts $(U, \varphi... |
Let $a,b,c\in\mathbb{R}$ be three parameters satisfying
$$abc\ne 0.$$
Consider the following equations of $(u,v,w)\in\mathbb{R}^3$:
$$
(-c) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2vw,
$$
$$
(-b) \bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = T^2 +w^2 +2vT,
$$
$$
(-a)\bigg(3(v+T)^2-2vT-(u^2+v^2)\bigg) = 2uT,
$$
... |
We are given that a, b, and c are positive real numbers such that:
$a + b + c = abc - 2$
We want to prove the following inequality:
$\sqrt{\frac{1}{1+a}}+\sqrt{\frac{1}{1+b}} + \sqrt{\frac{1}{1+c}} \leq \sqrt{3}$
**Can anyone provide a proof for this inequality under the condition that a, b, and c are posit... |
The following is a particular case of rank theorem.
Theorem 1( Rank theorem for surjective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is surjective, then there are charts $(U, \varphi... |
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