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**The question**
The natural numbers $a,b,c$ are considered non-zero and distinct two by two, so that $a^3+b^3+c^3=\frac{a^2b^2}{c}+\frac{a ^2c^2}{b}+\frac{b^2c^2}{a}$
a) It is possible that one of the numbers is equal to the gemometric average of the other two?
b) Show that $a+b+c$ divides at least one of the... |
Elementary example where a neighborhood basis exists but no countable neighborhood basis exists? |
Imagine we have a function f(x,y). For simplicity, let's say:
$$f(x, y) = \begin{cases}
c & \text{if } 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1 \\
0 & \text{otherwise}
\end{cases}$$
With so much imagination, think this is the function:
[![enter image description here][1]][1]
I know how to integrate with ... |
how to integrate in the diagonal? |
We all agree that the provided answer is incorrect for the reasons you state and others. Then the fun starts. Here is how I see it:
First, choose the leaders, eight out of 16 students C(16,8) ways.
Then assign a partner for each leader from the remaining 8 students 8! ways.
Finally, choose team A from the ... |
Consider the family of functions $f_n$ and $(g_n)_{n\in\mathbb{N}}$, where for each $n$ in $\mathbb{N}$:
- $f_n: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f_n(x) = x^n \cos(x)$,
- $g_n: \mathbb{R} \rightarrow \mathbb{R}$ defined by $g_n(x) = x^n \sin(x)$.
Additionally, let there be a function $f: \mathbb{R}... |
In the derivation of the Bernoulli-Seki formula the first step is unfolding the exponential generating function of $S_n(k)=\sum_{a=1}^k a^n$
as
$$
\sum_{n \geq 0} S_n(k) \frac{T^n}{n !}=\sum_{a=1}^k \sum_{n \geq 0} \frac{(a T)^n}{n !}=\sum_{a=1}^k e^{a T}=e^T \frac{e^{k T}-1}{e^T-1}
$$
This part is clear.
H... |
I sew in wiki the following theorem
[![enter image description here][1]][1]
Does anyone know a textbook including it? Thanks in advance.
[1]: https://i.stack.imgur.com/Zc64f.png |
> The quartic equation $2x^4 - 3x^3 + kx^2 - 16 = 0$ has roots $\alpha, \beta, \gamma, \delta$. Given that $\alpha \beta = 5$ find the value of $k$
So I have deduced that $\gamma \delta = \frac{-8}{5}$ using product of roots $=-8$ but I am not sure how to proceed from here.
I have written out Vieta's formulae, b... |
> Suppose $e$ and $\bar e$ are two frames for a vector bundle $E \to M$ over $U \subset M$ such that $\bar e = eg$ for some $g: U \to \text{GL}(r,\Bbb R)$. Then $$\nabla \bar e = \nabla(e)g + edg.$$
I'm trying to understand this which is seemingly just the Leibniz rule, but I have some trouble with the definitions. ... |
There are lots of posts on MSE and the web titled "converse to CRT" but this is not the same. The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
In Chinese Remainder Theorem direction is to go from two numbers ... |
New to automata, looking for some clearification and some guidance.
I understand the concept of prefix and proper prefix, but I'm struggling with the following question: (It is translated, so forgive me for the misspelling)
For $L$ over finite letters:
$\max(L) = \{w \in L \mid L \text{ does not contain $w'$ in ... |
Is there a bijective function from $\mathbb{N}$ to $\mathbb{N} \times \mathbb{N}$ and supports additivity? Namely if $n,m \in \mathbb{N}$ then:
$$f(n)+f(m)=f(n+m)$$
If there isn't, how can one prove such a thing?
|
b) If 3 specific persons, among 6, insist on following each other, how many ways are possible?
I understand that a) is just 6!, but apparently b) is 4! 3!. Can anyone explain where they got 4! from? I thought it was 3! 3!, since 3 people are following each other, and 3 aren't.
Thank you. |
In how many ways can 6 people be lined up to get on a bus? |
I was thinking it to be true, as the inverse function theorem gives us contious differentiablity of the inverse locally at any point. With continous differntiablity being a local attribute of the function, the inverse should be continous differentibale right? |
Does being bijective and having a full rank jacobian Matrix imply being a Diffeoorphism? |
I'm trying to show that the image of the function f(z)=(z+2)^4 from the unit disk in the complex plane is not convex but i can not seem to figure out an explicit example to show this. Does anyone have any ideas? |
a) In how many ways can $6$ people be lined up to get on a bus?
b) If $3$ specific persons, among $6$, insist on following each other, how many ways are possible?
**My confusion**
I understand that a) is just 6!, but apparently b) is $4! \cdot 3!$. Can anyone explain where they got 4! from? I thought it was $3... |
Does being bijective and having a full rank jacobian Matrix imply being a Diffeomorphism? |
I've just begun my grad program and we were introduced to this problem in our Analysis I course: consider two uniformly continuous functions $f$ and $g$, say from $\mathbb R$ to $\mathbb R$, where $f$ is bounded. I've learned that the canonical way to disprove that $fg$ is uniformly continuous is countering the claim w... |
There are lots of posts on MSE and the web titled "converse to CRT" but this is not the same. The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
In Chinese Remainder Theorem direction is to go from two numbers ... |
In the derivation of the Bernoulli-Seki formula the first step is unfolding the exponential generating function of $S_n(k)=\sum_{a=1}^k a^n$
as
$$
\sum_{n \geq 0} S_n(k) \frac{T^n}{n !}=\sum_{a=1}^k \sum_{n \geq 0} \frac{(a T)^n}{n !}=\sum_{a=1}^k e^{a T}=e^T \frac{e^{k T}-1}{e^T-1}
$$
This part is clear.
H... |
I want to lower bound the following expression of choosing $d$ numbers from $[n]:=\{ 1,2,\dots,n \}$.
$$
\sum_{\{ s_{1},..,s_{d}\}\subset [n]} d!\frac{1}{s_{1}s_{2}\dots s_{d}}
$$
Here $s_{1},\dots,s_{d}$ are all distincts. I know I can upper bound this by $\left( 1+\frac{1}{2}+\dots+\frac{1}{n} \right)^{d} = H_{n... |
Consider $S^2$ with the two stereographic projections obtained by removing respectively the North pole and the South pole, find the gluing function obtained through the definition of Riemannian surface with the two charts given by the two projections.
We call $\varphi_1$ the stereographic projection from $S^2- \{N \... |
I'm trying to show that the image of the function $f(z)=(z+2)^4$ from the unit disk in the complex plane is not convex but i can not seem to figure out an explicit example to show this. Does anyone have any ideas? |
Thanks for the question, Burdy. Perhaps I have misunderstood the definition of Bloom filters, but here is a simple counter-example. The first two random variables $Y_0, Y_1$ are in fact deterministic, namely $P(Y_0 = 0) = 1$ and $P(Y_1 = k) = 1$, where $k$ is the number of hashing functions. For any $1 \leq \sigma \leq... |
I have the following series:
$$\sum\limits_{n=1}^\infty(e^\frac{1}{n} - e^\frac{1}{n+2})$$
I know that this series converges to $e + \sqrt{e} - 2$ (I found its sum using $S = \lim\limits_{n \to \infty} S_n$, where $S_n$ is a partial sum of $n$ elements).
But is there any way to prove that the series converges ... |
How to prove that the following series converges? |
In Tao's book *Nonlinear Dispersive Equations*, Appendix A, a Bernstein inequality has puzzled me a long time:
Suppose that $P_{\geq N}$ is smoothed out projection to region $|\xi|\geq N$, where $N\in 2^{\mathbb{Z}}$ is a dyadic number, $\hat{\varphi}(\xi)$($|\hat{\varphi}(\xi)|\leq1$) is a real valued raidally symm... |
I'm trying to solve the following integral:
Integral of: $\int\frac{1}{1+ax^2}dx$
Where $a$ is some positive constant.
We can't of course use basic U Substitution as the derivative of $1 + ax^2$ is $2ax$, which isn't found elsewhere in the expression.
I tried to use fractional decomposition to integrate this ... |
How can we integrate $\frac1{1 + ax^2}?$ |
> The quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$ has roots $\alpha, \beta, \gamma, \delta$. Given that $\alpha \beta = p$ find the value of $k$
So I have deduced that $\gamma \delta = \frac{e}{ap}$ using product of roots $=-\frac{e}{a}$ but I am not sure how to proceed from here.
I have written out Vieta... |
in Hoffman linear algebra, chapter 1.3 exercises, this is the question:
Find all solutions to the system of equation:
(1 - i)Xl - iX2 = 0
2Xl + (1-i)X2=0
in solutions to this using elemenatary row operations, this was given:
[enter image description here][1]
[1]: https://i.stack.imgur.com/ezgJf.png
... |
I am new to tropical geometry with addition is the $\max$ and multiplication the usually $+$ and I an trying to understand the translation of polynomials from the tropical setting to the classical setting and for example the abstract structure
$$a\odot x^2\oplus b\odot x\oplus c= \max\lbrace 2x+a,x+b,c\rbrace$$
is... |
in Hoffman linear algebra, chapter 1.3 exercises, this is the question:
Find all solutions to the system of equation:
(1 - i)Xl - iX2 = 0
2Xl + (1-i)X2=0
in solutions to this using elemenatary row operations, this was given:
[picture of equation][1]
[1]: https://i.stack.imgur.com/ezgJf.png
how did ... |
I have currently opted for a course titled Differential Geometry of Curves and Surfaces for which my instructor is referring to Montiel and Ros' Curves and Surfaces. Having finished Chapter 1, I am still not getting how to solve the exercises given. For example:
(18) Let $\alpha:I \rightarrow \mathbb{R}^3$ a curve p... |
How does one solve questions in Montiel's Curves and Surfaces? |
a function h: $Q \rightarrow R$, with $h(x) = 0$ for $abs(x)<sqrt(2)$ and $h(x) = 1$ for $abs(x) > sqrt(2)$ is continuous for all x in Q. (Q = rational numbers, R = real numbers).
It states in the solution that for $x_0$ in Q and $\delta =$ min{abs($x_0$ + $\sqrt(2)$),abs($x_0$ - $\sqrt(2))$} it follows that $h(x) -... |
**This is the problem in question:**
You have two identical bowls: the first one contains **3 white** balls and **4 black** balls, and the second one contains **4 white** balls and **5 black** balls. If you choose randomly a ball from the two bowls, what is the probability it is **white**?
**Let's define our even... |
I am new to tropical geometry with addition is the $\max$ and multiplication the usually $+$ and I am trying to understand the translation of polynomials from the tropical setting to the classical setting. For example the abstract structure
$$a\odot x^2\oplus b\odot x\oplus c= \max\lbrace 2x+a,x+b,c\rbrace$$
is cl... |
The wording of the problem is ambiguous. If we interpret it to mean you will be selecting the bowl first, and then a ball from that bowl, then your first approach is correct. The second one is incorrect. Let me explain.
> Because the two bowls are identical, we could just say we don't even choose between two bowls, ... |
I came across this definition while reading Jaap van Oosten's [Topos Theory lecture notes][1] (pg 29, def 1.3).
In a category with finite limits, an **equivalence relation** on
an object $X$ is a subobject $R$ of $X \times X$ for which the following statements
hold:
- The diagonal embedding $X \to X \times X$... |
I am trying to understand the intuition behind unordered sampling with replacement.
This is the problem I have: I want to distribute 4 identical balls to 2 people. Let 1 represent person 1, and let 2 represent person 2. Then, the potential distributions are 1111, 2222, 1112, 2221, and 1122.
Formulaically, we know... |
I need to find the Laplace transform of the product of a function $f(t)$ with the Unit Heaviside Step Function $H(t-c)$, i.e., $\text{L}[H(t-c)f(t)]$.
Given that
$$\text{L}[H(t)f(t-c)] = e^{-sc}F(s)\hspace{5cm} \text{EQ. 1}$$
is a well known result, I made the substitution $y=t-c$ to get
$$\text{L}[H(t-c... |
Assume that the function $f(x,y)$ is not symmetric in $x,y$.
Consider the curve $f(x,y)=0$ in $\mathbb{R}^2$. Is it always true that $f(a,b)=f(b,a)=0$ only if $a=b$?
It does appear to be true when I draw an arbitrary curve $f(x,y)$ in $\mathbb{R}^2$. However, I'm unable to prove this. I also remember seeing an ... |
Is it true that the curves $f(x,y)=0$ and $f(y,x)=0$ can only intersect on the line $x=y$? |
Following this question:https://math.stackexchange.com/questions/1563669/show-that-every-lebesgue-integrable-function-can-be-approximated-in-norm-and-alm
Let $f\in L^1(\mathbb{R})$. Show that there exists a sequence of continuous functions $\{g_n\}$ so that $g_n\to f$ in $L^1$ as $n\to \infty$.
In that answer, ... |
In Hoffman Linear Algebra, Chapter 1.3 Exercises, the following question is presented:
Find all solutions to the system of equations:
$(1 - i)X_1 - iX_2 = 0$
$2X_1 + (1-i)X_2=0$
In the solutions to this (using elementary row operations), this was given:
[picture of equation][1]
[1]: https://i.stac... |
A function $h:\mathbb{Q} \rightarrow \mathbb{R}$, with $h(x) = 0$ for $|x|<\sqrt2$ and $h(x) = 1$ for $|x| > \sqrt2$ is continuous for all $x$ in $\mathbb{Q}$.
It states in the solution that for $x_0$ in $\mathbb{Q}$ and $\delta =$ min{|$x_0$ + $\sqrt2$|,|$x_0$ - $\sqrt2|$} it follows that $h(x) - h(x_0) = 0$ and t... |
I am looking for a solution $X \in \mathbb{R}^{n \times n}$ which minimizes the following quadratic energy:
$$
\textrm{min}_X || X A X^T - B||^2 + \mu ||X \Lambda X^T - \Lambda ||
$$
where $A, B \in \mathbb{R}^{n \times n}$ are P.D. and $\Lambda$ is diagonal with positive entries. I am hoping to find an analyt... |
Let's say we have a vector space $X$ and a function $p:X\to \mathbb{R}^+$ such that $p(ax)=|a|p(x)$ for all $a\in \mathbb{R}$ and $x\in X$, and such that $p(x)=0$ iff $x=0$.
Then $p$ satisfies the triangle inequality (and is therefore a norm) iff it's a convex function. This is where our connection between the tria... |
I am trying to understand the intuition behind unordered sampling with replacement.
This is the problem I have: I want to distribute $4$ identical balls to $2$ people. Let $1$ represent person $1$, and let $2$ represent person $2$. Then, the potential distributions are $1111, 2222, 1112, 2221, \text{and } 1122$.
... |
The Bolzmann-Stefan formula for the improper integral $\int_0^\infty f(x)\; dx $, proposed and used by Bolzmann (who claimed to have learned it from his teacher Stefan, see reference [Boltzmann's Probability Distribution of 1877, by
Alexander Bach][1]; see also [This math trick revolutionized physics][2] youtube vid... |
I've the simple following trigonometric equation to solve:
$$
\sin\left(\frac{x}{2}\right)=3+2\cos(x)\,.
$$
There are some different ways to solve this, but one way is giving me a different answer and it's intriguing me because I don't see the flaw in the argument. If I use the fact that:
$$
\sin\left(\frac{x}{2... |
I am new to tropical geometry with addition is the $\max$ and multiplication the usually $+$ and I am trying to understand the translation of polynomials from the tropical setting to the classical setting. For example the abstract structure
$$a\odot x^2\oplus b\odot x\oplus c= \max\lbrace 2x+a,x+b,c\rbrace$$
is cl... |
So I recently stumbled upon a maths post which caught my eye.
[Here is the link][1]
[1]: https://mathforums.com/t/very-easy-way-to-calculate-pi-via-circle-as-a-n-sided-regular-polygon.371176/
It talked about calculating pi by perceiving the circle as a n-sided regular polygon.
Eventually ending up with the ... |
How practical is this formula to calculate pi? |
(Remark: if you haven't sketched the function $h$ -- on paper, or in your head if your visual imagination is up to it -- then you absolutely should do that first. Don't try to do this sort of thing just by pushing symbols around.)
Obviously the function $h_0\,:\,\mathbb{R}\rightarrow\mathbb{R}$ with "the same defini... |
So I recently stumbled upon a maths post which caught my eye.
[Here is the link][1]
[1]: https://mathforums.com/t/very-easy-way-to-calculate-pi-via-circle-as-a-n-sided-regular-polygon.371176/
It talked about calculating pi by perceiving the circle as a $n$-sided regular polygon.
Eventually ending up with th... |
How practical is this formula to calculate $\pi$? |
The **duplication formula** for an elliptic curve over rationals:
$$y^2 = x^3 + ax^2 + bx + c$$
for the $x$-coordinate is given by:
$x$-coordinate of $2(x, y)=(x^4-2bx^2-8cx+b^2-4ac)/(4x^3+4ax^2+4bx+4c)$
Here $2(x, y)=(x, y)+(x,y)$ via the group law of addition of two points on the curve.
I am asking if t... |
I would agree the wording is somewhat ambiguous. However, if we assume a crazy person is not writing this problem, then I think the intended meaning is that the two bowls are identical in every way except for having different amounts of different colored balls. Furthermore, I doubt the existence of two bowls is irrelev... |
I need to find the arc length from the function $$ y^{2}= - 2.6\times x$$
Will the result change if I replace the function with? $$x = \frac{y^{2}}{-2.6}$$ I also ask you to check my answer, I did it: $$\frac{y\times\sqrt{1+\frac{y^{2}}{1.69}}+\ln\mid\frac{y}{1.3}+\sqrt{1+\frac{y^{2}}{1.69}}\mid}{3}$$ |
Let $K/F$ be a normal algebraic extension and let $L = (K/F)^{\mathrm{sep}}$ be the subfield of elements of $K$ which are separable over $F$ (this is also called the separable closure of $F$ in $K$). Is $L/F$ necessarily normal?
I really do not have a clue about this question so any hints will be appreciated. |
Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a (weak) homomorphism is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in E(H) \lor f(u) = f(v)$.
I am interested in the problem of find a weak homomorphism $f: V(G) \t... |
Your question is actually the same for random walks $S_{n}=\sum I_{k}$ for Gaussian $I_{k}\in N(0,1)$. Then if we normalize we get
$$X_{n}:=\frac{S_{n}}{\sqrt{n}}\sim N(0,1).$$
*"Since each $X_t$ is identically distributed, $X_t(\omega)$ is constant for all $t$ and fixed $\omega$."*
Here ,however, the $X_{n}$ ... |
I would agree the wording is somewhat ambiguous. However, if we assume a crazy person is not writing this problem, then I think the intended meaning is that the two bowls are identical in every way except for having different amounts of different colored balls. Furthermore, I doubt the existence of two bowls is irrelev... |
Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a (weak) homomorphism is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in E(H) \lor f(u) = f(v)$.
I am interested in the problem of find a weak homomorphism $f: V(G) \t... |
This is a question about Gentzen calculus. $X$ is a set of formulas with the symbols $\{\lnot, \land\}$, and $a$ is such a formula.
On page 27 of "A Concise Introduction to Mathematical Logic" by Rautenberg, it states:
> it easily follows that $X$ is maximally consistent iff either $a \in X$ or $\lnot a \in X$, f... |
This is an olympiad exercise. Call a number $n\in\mathbb{N}$ smooth if there are $a_1,...,a_n\in\mathbb{Z}$ s.t.
$$
a_1+a_2+...+a_n=a_1\cdot a_2\cdot ... \cdot a_n=n.
$$
The aim is to find all smooth numbers.
________
So I think that it is evident that every composite number can be written as
$$
n=a_1+...+a... |
Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a (weak) homomorphism is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in E(H) \lor f(u) = f(v)$.
I am interested in the problem of find a weak homomorphism $f: V(G) \t... |
This is a German olympiad exercise. Call a number $n\in\mathbb{N}$ smooth if there are $a_1,...,a_n\in\mathbb{Z}$ s.t.
$$
a_1+a_2+...+a_n=a_1\cdot a_2\cdot ... \cdot a_n=n.
$$
The aim is to find all smooth numbers.
________
So I think that it is evident that every composite number can be written as
$$
n=a_1... |
We got 6 pieces of strings. The top ends of the ropes are randomly paired up
and tied together. The same procedure is done with the bottom ends of the
strings. What is the probability that, as a result of the process, the 6 pieces of strings will be connected in a single closed loop of string?
My initial attempt ... |
Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a (weak) homomorphism is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in E(H) \lor f(u) = f(v)$.
I am interested in the problem of find a weak homomorphism $f: V(G) \t... |
Why if $p(t)=(t+c)^m$ , then the time needed to compute the kernel is independent of m? |
Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a [(weak) homomorphism (definition 1.14)](https://scholarscompass.vcu.edu/cgi/viewcontent.cgi?article=2951&context=etd) is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in... |
This is a question about Gentzen calculus. $X$ is a set of formulas with the symbols $\{\lnot, \land\}$, and $a$ is such a formula.
On page 27 of "A Concise Introduction to Mathematical Logic" by Rautenberg, it states:
> it easily follows that $X$ is maximally consistent iff either $a \in X$ or $\lnot a \in X$, f... |
The statement with the biconditional is definitely *not* the correct definition of limit. Your instructor is just wrong.
Let me write the two statements so we can refer to them:
$$\begin{align*}
(\forall \epsilon\gt 0)(\exists \delta\gt 0)(\forall x\in\mathbb{R})(0\lt|x-a|\lt \delta\implies |f(x)-L|\lt\epsilon) ... |
**Context**
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and describes any progress on the problems. The book was published in 1995, so I don't know if some of the problems that Kirby stated a... |
This is a German olympiad exercise from 2015. Call a number $n\in\mathbb{N}$ smooth if there are $a_1,...,a_n\in\mathbb{Z}$ s.t.
$$
a_1+a_2+...+a_n=a_1\cdot a_2\cdot ... \cdot a_n=n.
$$
The aim is to find all smooth numbers.
________
So I think that it is evident that every composite number can be written as
... |
This is a somewhat technical question about a line in Sharpe's book **Differential Geometry: Cartan's Generalization of Klein's Erlangen Program** in the proof of the structure theorem, Theorem 8.3 We have assumed that the leaf space of a foliation $M/\sim$ is Hausdorff. Because of this, each leaf $\mathcal{L}$ is clos... |
Is this derivation of inverse Laplace transform solid? |
Is this derivation of the inverse Laplace transform solid? |
If $ a^2 + b^2=1,c^2+d^2=1,ac+bd=0 $ , find $ab+cd $? |
**Definitions:**
Given a category $\mathcal{C}$, and a [double category][1] $\mathbb{D}$, i.e. a category [internal][2] to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$, call $\mathbb{D}$ a **“double category over $\mathcal{C}$”** if “$\mathcal{D}_0$ is $\mathcal{C}$”, ... |
Are (commutative) squares in some sense universal among edge-symmetric double categories? |
**Motivation:**
The “minimal information guaranteed to be associated” to a 2-cell in a general double category is its source horizontal morphism and target horizontal morphism. But in an edge-symmetric double category, the minimal information guaranteed to be associated to a 2-cell is not only the source and target ho... |
If you count oriented cycles, the number of such cycles for $n=2,3,4,\ldots$ is
$1, 6, 768, 3888000, \ldots$
If you count non-oriented cycles, ie edge-injective embeddings of the cyclic graph on $n(n-1)$ vertices onto $K_n$, the number of such cycles is
$1, 4, 384, 1944264, \ldots$
Neither sequence is in th... |
I have encountered here that sometimes people write $dx/df$. For example, [here][1], the author of the answer writes down the chain rule as
$$\frac{dg}{df}=\frac{dg}{dx}\frac{dx}{df},$$
which seems to be an abuse of notation. And $dx/df$ is found to be just $\frac{1}{df/dx}$.
[Here][2], I learned that when someone... |
Show that if $f_1, f_2$ are homotopy morphisms of complexes of abelian groups $C_n, D_n$, and $g_1, g_2$ are homotopy morphisms of complexes of abelian groups $D_n, E_n$, then $g_1 \circ f_1$ and $g_2 \circ f_2$ are homotopic.
We have that $f_1 - f_2 = \delta^D h_n + h_{n-1} \delta^C$ and $g_1 - g_2 = \delta^E h'_n + ... |
Consider the loss function:
$$
\text{L} = \sum_{u=0}^{M-1}(X_{u}-(10-\sqrt{\frac{\sum_{i=0}^{N - 1} (Z_{u,i} - Y_{i})^2}{N}}))^2
$$
$$\text{where:} \\ Z = \mathbb{R}^{m{\times}n} \text{ matrix} $$
$$ X = \mathbb{R}^{m{\times}1} \text{ matrix} $$
$$ Y = \mathbb{R}^{1{\times}n} \text{ matrix} $$
$$0<=Z,X,Y<=10... |
As we all know,
$$
\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)=1
$$
And I encountered many questions where I used this standard limit and it gave the right answer. However in the following questions I am unable to get the right answer by this.
I actually want to solve this limit.
$$
\lim _{x \right... |
Let $\Sigma$ be the fan defined by $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\star\}$, where $\sigma_1=\operatorname{cone}(e_1)$, $\sigma_2=\operatorname{cone}(e_2)$, $\sigma_3=-\sigma_1$, $\sigma_4=-\sigma_2$, and $\star=\operatorname{cone}((0,0))$. We are working on a two dimensional lattice $N$ extended to the real vec... |
What Toric Variety does this fan correspond to? |
**Context**
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and describes any progress on the problems. The book was published in 1995, so I don't know if some of the problems that Kirby stated a... |
In the appendix to the SEP Proof Theory [article][1], Michael Rathjen makes reference to a topology on infinite derivations for which a variant of cut-elimination is a continuous function. The referenced paper by Mints is not very explicit. Another common referenced paper is by Kreisel, Mints, and Simpson ([paper][2]) ... |
What's the natural metric/topology on infinite derivations? |
Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
The character of the permutation representation... |
Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$? |
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