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Covariant representable functors preserve monomorphisms. What about contravariant representable functors? |
> Prove the nonzero polynomial $$T(\theta)=a_0+\sum\limits_{k=1}^{n}(a_k\cos
k\theta+b_k\sin k\theta)$$ has no more than $2n$ distinct roots on
> $[0,2\pi).$
My attempt:\
From $\cos k\theta=\dfrac{e^{ik\theta}+e^{-ik\theta}}{2}$ and $\sin k\theta=\dfrac{e^{ik\theta}-e^{-ik\theta}}{2i}$, $T$ can be rewritten a... |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
*Given a finite set of points in Euclidian plane such that any line passing through two of the points passes through at least one more of them, all points have to be collinear.*
Proof
Let n be... |
Let $\theta\in[0,2\pi)$ be a root of $T$, so that $T(\theta)=0$. Then you have found that $p(e^{i\theta})=0$ for some (nonzero?) polynomial $p$ of degree at most $2n$. This means $e^{i\theta}$ is a root of $p$. Of course the map
$$[0,2\pi)\ \longrightarrow\ \Bbb{C}:\ \theta\ \longmapsto\ e^{i\theta},$$
is injective. ... |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
*(1) Given a finite set of points in Euclidian plane such that any line passing through two of the points passes through at least one more of them, all points have to be collinear.*
Proof
Let ... |
Well, despite using programs this "appears" simple (the solutions are numerically easy at sight), I am not able to solve this system of two equations by hand, except for the trivial solution $(0, 0)$. Can you help me perhaps?
$$\begin{cases} 3y^3 + 6xy^2 + 3x^2y - 3y + 6x^3 - 7x = 0 \\ 2y^3 - 3xy^2 + 2x^2y - y - 3x... |
Let's consider the following definition (I'm not an expert, so let me know if this is not the usual one!).
> *Let $I\subset \mathbb{R}$ be an open interval. A function $f:I\to \mathbb{R}$ is said to be **algebraic** on $I$ if there is a nonzero polynomial $p\in \mathbb{R}[x,y]$ such that $p(x,f(x)) = 0$, for every $... |
If $f$ and $g$ are algebraic functions, is $x\mapsto (f(x),g(x))$ contained in an algebraic curve? |
It is often claimed that the following equation holds for the [modular discriminant](https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Modular_discriminant) $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the [Dedekind eta ($\eta$) function](https://en.wikipedia.org/wiki/Dedekind_eta_function) ... |
This is a follow-up question or variant question to https://math.stackexchange.com/questions/4873818/finding-1-periodic-h-omegass-so-that-lim-omega-rightarrow-2-pi-n.
Let $n$ be a nonzero integer. I am wondering now if it is possible to find $1$-periodic functions $h_{\omega}(s)$ such that
$$ \lim_{\omega\rightar... |
Well, despite using programs this "appears" simple (the solutions are numerically easy at sight), I am not able to solve this system of two equations by hand, except for the trivial solution $(0, 0)$. Can you help me perhaps?
$$\begin{cases} 3y^3 + 6xy^2 + 3x^2y - 3y + 6x^3 - 7x = 0 \\ 2y^3 - 3xy^2 + 2x^2y - y - 3x... |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
Well, despite using programs this "appears" simple (the solutions are numerically easy at sight), I am not able to solve this system of two equations by hand, except for the trivial solution $(0, 0)$. Can you help me perhaps?
$$\begin{cases} 3y^3 + 6xy^2 + 3x^2y - 3y + 6x^3 - 7x = 0 \\ 2y^3 - 3xy^2 + 2x^2y - y - 3x... |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
We know, a character $\chi$ mod $q$ is induced by a primitive character $\chi^{*}$ mod $q^{*}$. I have the following questions.
1) If $\chi$ is non principal, can $\chi^{*}$ be principal?
2) Let $\chi$ be a character mod $q$. Can we say $\chi$ is principal iff conductor of $\chi=1?$
Any help or hint would be app... |
We know, a character $\chi$ mod $q$ is induced by a primitive character $\chi^{*}$ mod $q^{*}$. I have the following questions.
1) If $\chi$ is non principal, can $\chi^{*}$ be principal?
2) Let $\chi$ be a character mod $q$. Can we say $\chi$ is principal iff conductor of $\chi=1?$
3) If $\chi$ be the principal... |
This is a nice discovery! But why does it work?
Let $n^2$ and $(n+1)^2$ be two consecutive squares.
If you expand $(n+1)^2$, you will get $n^2+2n+1$.
$(n^2+2n+1)-(n^2)=2n+1$.
This means the difference between consecutive squares is always the odd numbers.
Now, we can say that $((n+2)^2-(n+1)^2)-((n+1)^2-n^2)=(2(n+... |
I was watching this video https://www.youtube.com/watch?v=HO2TNOsrg6c&ab_channel=COSMOSLEARNING and in the proof, he applies Taylor's theorem for the interval $V_\delta(x_0)$, since $f$ and its derivatives $f', f'', ... f^{(n)}$ exist and are continuous in $V_\delta(x_0)$. But what I don't understand is how Taylor's th... |
> $(2X-8)(x-1)≥0$
>
> $2x-8\geq0 \vee x-1\geq0$
An error occurs here. In writing $\lor$, you are saying "or," but this is not the case for this. In general, $ab \ge 0$ only if both are positive, both are negative, or at least one is zero. Hence, you would instead write
$$\Big( 2x - 8 \ge 0 \text{ and } x-1 \ge... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
>Solvable groups. One says that G is solvable if there exists a sequence
$$\{1\}=G_0\subset G_1\subset\cdots\subset G_n=G$$
of subgroups of $G$, with $G_{i-1}$ normal in ... |
**Problem :**
Let $l$ be a line which passes two points : $(1,0,0), (1,1,1)$.
And $S $ be a surface which generated by revolving line $l$ around $z$-axis.
Find a volume enclosed by surface $S$ and two planes : $z=0, z=1$.
---
**My Attempt**
Parametric equation of $l$ can be obtained easily :
$$l(t)... |
Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a polynomial time algorithm?
Notice that the assumption is that $L$ and $K$ are isomorphic simply as fields. |
Say if I have the non-linear system of equations $\dot x=….$ and $\dot y=….$.
I work out the equilibrium points, eigenvalues, and then the corresponding eigenvectors for the real eigenvalues. Say if the eigenvectors are $(\matrix{2\\1})$, $(\matrix{0\\1})$ and $(\matrix{—2\\1})$.
Does this mean that the straight ... |
I have a simple logic equivalence expression. I'm thinking there must be a name for this principal of OR inversion. Can anyone help put a name to this idea? Link to expression calc below.
$$(p\lor q)\iff \neg(\neg p\land\neg q)$$
https://www.emathhelp.net/en/calculators/discrete-mathematics/truth-table-calculat... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
>Solvable groups. One says that G is solvable if there exists a sequence
$$\{1\}=G_0\subset G_1\subset\cdots\subset G_n=G$$
of subgroups of $G$, with $G_{i-1}$ normal in ... |
In my case, $R = {\mathbb{C}[x, y]}/{y^2 - x^3 + x}$, and M is the ideal generated by {x,y} over R.
Since M is projective I know such N exists, but how do I find it?
|
Given a projective R-module M, how does one find another R-module N such that $M \oplus N$ is a free R-module? |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
>Solvable groups. One says that G is solvable if there exists a sequence
$$\{1\}=G_0\subset G_1\subset\cdots\subset G_n=G$$
of subgroups of $G$, with $G_{i-1}$ normal in ... |
My friends and I have used an Elo-Ranking system to track our relative strength in the social deduction game "Secret Hitler". So far, the formula we used was a very basic adaptation of the original Elo-formula that uses a lot of assumptions and thus gives unfair results.<br />
First, I will describe the relevant featu... |
I'm very confused about the question below, which I couldn't figure out for days.
In Example 5 the author is teaching us proving
$$\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{1}{n!}) = e$$
with equation
$$x_n = (1+\frac{1}{n})^n$$
$$y_n = 1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{... |
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module? In my case,
$R = {\mathbb{C}[x, y]}/{(y^2 - x^3 + x)}$,
and $M$ is the ideal generated by $\{x, y\}$ over $R$.
Since $M$ is projective I know such $N$ exists, but how do I find it?
|
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module? |
Say if I have the non-linear system of equations $\dot x = \dots$ and $\dot y = \dots$.
I work out the equilibrium points, eigenvalues, and then the corresponding eigenvectors for the real eigenvalues. Say if the eigenvectors are
$$
\pmatrix{2\\1}, \quad \pmatrix{0\\1},
\quad\text{and}\quad
\pmatrix{-2\\1}.
... |
In my case, $R = {\mathbb{C}[x, y]}/{y^2 - x^3 + x}$, and M is the ideal generated by {x,y} over R.
Since M is projective I know such N exists, but how do I find it?
Here is my initial attempt at solving this.
Since M is generated by {x,y}, we have a function F (the identity function) from {x,y} to M, by universal... |
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module? |
I was watching [this video][1] and in the proof, he applies Taylor's theorem for the interval $V_\delta(x_0)$, since $f$ and its derivatives $f', f'', \dots, f^{(n)}$ exist and are continuous in $V_\delta(x_0)$.
But what I don't understand is how Taylor's theorem would apply if $V_\delta(x_0)$ is an open interval. ... |
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module? In my case, $R = {\mathbb{C}[x, y]}/{(y^2 - x^3 + x)}$, and $M$ is the ideal generated by $\{x, y\}$ over $R$. Since $M$ is projective I know such $N$ exists, but how do I find it?
Here is my init... |
In Evans' book Partial Differential Equations, appendix C.2 Theorem 3, Green's formulas are established: Let $u, v \in C^{2}( \overline{U})$. Then
$$ \int_U Du \cdot Dv dx = - \int_U u \Delta v dx + \int_{\partial U } \dfrac{\partial v }{\partial \nu } u dS .$$
My question is, is this formula valid requiring less a... |
What is the minimum about the function $u$ we must have to use Green's formula? |
I'm confused on the general rules for little-oh and big-oh when doing function composition. For example, in Zorich analysis it is proven that $e^x -1 = x+o(x)$ as $x \to 0$. Later zorich uses this fact to directly say that $e^{h\ln a}-1 = h \ln a+o(h \ln a).$ While directly substituting $x$ with $h \ln a$ seems intuiti... |
Let $X \subseteq \mathbb{C}^n$ be an irreducible affine variety and $P \subseteq \mathbb{C}[z_1,\ldots,z_n]$ the corresponding prime ideal. So, by definition, $P$ consists of all the polynomials which vanish on $X$. It follows that all the elements of $P^2$ vanish to (at least) second order on $X$. To be more precise, ... |
Does the square of a prime ideal contain all the polynomials vanishing to second order on its variety? |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Considering the complex exponential map $w=e^{z}$, I want to see what the image of the function would look like for the following region of the complex plane
$$\text{The disk} \quad |z| \leq \pi$$
Note that the region is represented graphically as follows
[![region][1]][1]
Initially the points (in the axis ... |
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module? In my case, $R = {\mathbb{C}[x, y]}/{(y^2 - x^3 + x)}$, and $M$ is the ideal generated by $\{x, y\}$ over $R$. Since $M$ is projective I know such $N$ exists, but how do I find it?
Here is my init... |
How to solve a method of characteristics problem with only a point boundary condition? |
Because I am not majoring in math, I wonder if there is a standard approach to prove or disprove Lipschitz continuity.
In my case, I want to prove that the Mean Squared Error (MSE) loss function for the weight \( w \) is Lipschitz continuous.
MSE is defined as:
$$
E_{in}(w) = \frac{1}{n}\sum_{i=1}^{n}(y_i - w^T... |
How to prove or disprove a function is Lipschitz continuity? |
How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$? |
Spinning things out from [a recent question](https://math.stackexchange.com/questions/4873889/), let's recall that every second countable sequentially compact space is compact, because every Lindelof countably compact space is compact.
We cannot weaken second countable to first countable; the [first uncountable ordi... |
Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact? |
For Markov Chains, there are well known formulae (e.g. fundamental matrix approach, first step analysis) which show us how to calculate the Expected Time to Absorption (e.g https://en.wikipedia.org/wiki/Absorbing_Markov_chain). That is, given a transition matrix and the current state - how many iterations (i.e. steps... |
Let $z \in \mathbb{C}$, i want to prove that $e^{z+\lambda}=e^z$ then $\lambda=2k \pi i$, I should take $z=x+iy$ s.t.
If $e^{z+\lambda}=e^z$, $e^{x+iy+\lambda}=e^z$ then $e^{x}e^{iy} e^{\lambda}=e^xe^{iy}$, and I can take $e^{\lambda}=1$ , but note that:
$$e^{i2k\pi} e^{ix}=e^{i2k\pi +ix}=e^{i(2k\pi +x)}= \cos(x... |
> $\mathrm{e}^{2x+y+z}=x^2+y^2-2x-4y+4$ is a rotation surface?
Why?
1. By translation, $e^{2u+v+w+4}=u^2+v^2-1$.
2. By orthogonal transform as $u=(2x+y+z)/6^{1/2}$, or $u=(2x+y)/5^{1/2}$, we do not find ...
3. Which translation and orthogonal transform should be put so that the surface becomes a rotation surf... |
$\mathrm{e}^{2x+y+z}=x^2+y^2-2x-4y+4$ is a rotation surface? |
Alternative approach (but Calculus rather than Combinatorics):
For $~n \in \Bbb{Z^+},~$ let $~G(n)~$ denote the *geometric mean* of $~\{1,2,\cdots,n\}.~$ That is, $~\displaystyle G(n) = (n!)^{1/n}.$
As $~\displaystyle n \to \infty, ~\frac{G(n)}{n} \to \frac{1}{e}.~$ Further, as $~n~$ increases, the fraction $~\... |
> If the definition above implies that the value of $\dot{G}(\check{u})$ is equal to $u$ for every $u \in B$, doesn't $\dot{G}$ become essentially the same as $\check{B}$ ?
No, $\dot G$ does not equal to $\check B$. For example, for $u\in B$ with $u\ne 1$, $\dot G(u)=u < 1 = \check B(u)$, so they are different name... |
I couldn't fit the whole question in the title, sorry! I'm trying the following bonus question from my Convex Analysis homework:
>Let $X$ be a vector space over $\mathbb R$, and let $F\subset X$ be an absorbing convex set. Given a nonempty set $\Omega$ in $X$, define
$$T_F(x;\Omega)=\inf\{t\geq 0\mid (x+tF)\cap... |
The first step is fine: you show that $e^{z+\lambda} = e^z \implies e^\lambda = 1$. I probably wouldn't split $z$ into its real and imaginary components. Instead, I would say,
$$e^{z+\lambda} = e^z \implies e^{-z} e^{z + \lambda} = e^{-z}e^z \implies e^{-z + z + \lambda} = e^0 \implies e^\lambda = 1.$$
This works jus... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to understand some statement.
First, I propose associated question.
**Q.** Let $A \subseteq B$ be a subring with prime ideals $\mathfrak{p} \subseteq A$ and $\mathfrak{q} \subseteq B$ such that $\mathfrak{p} = \mathfrak{q} \cap A$ ... |
I am reading the paper [Minimal surfaces of constant curvature in $S^n$,][1] wrote by Professor Robert Bryant in 1985. In this paper, the author introduced some operators to give a new method in order to classify the minimal surfaces of constant ($>0$, $<0$ or $=0$) curvature in $\mathbb{S}^n$.
In section 1, the aut... |
Dan Willard proved that you can have weak formal systems than can prove their own consistency, thus avoiding Gödel's second incompleteness theorem.
But although Willard's theories can avoid Gödel's second incompleteness theorem, they cannot avoid Gödel's *first* incompleteness theorem: namely, there will always be a... |
Is the Gödel sentence of Dan Willard's self-verifying theories a $Π_1$ sentence? |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to understand some statement.
First, I propose associated question.
**Q.** Let $A \subseteq B$ be a subring with prime ideals $\mathfrak{p} \subseteq A$ and $\mathfrak{q} \subseteq B$ such that $\mathfrak{p} = \mathfrak{q} \cap A$ ... |
Recently, I'm trying to learn geodesic currents on free groups through a [paper](https://arxiv.org/abs/0711.3806) by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form and $I$ be a bipartite graph where vertices are $\overline{CV}(F_N)\cup\mathbb{P}Curr... |
Thank you for reading my questions.
It is known that the stationary phase method has this form:
$$
\int_a^bg(t)e^{jf(t)dt}\approx\sum\limits_{t_0\in\Sigma}g(t_0)e^{jf(t_0)+j*\text{sign}(f''(t_0))\frac{\pi}{4}}\frac{\sqrt{2\pi}}{\sqrt{|f''(t_0)|}} \tag{1}
$$
You can check the wiki for this:
https://en.wikipedia.... |
Say you have an array of arbetrary length and unique elements [**X**]:=[0, 1, 2, ..., **x**] from which you independently poll **n** samples and put into [**N**]; and then, you trim [**N**] for uniqueness of its elements, and find [**N'**] which has **k** ≤ **n** elements.
**k** represents the number of unique eleme... |
Given an array, [X], and target count of unique samples, k₁, what is the probability as a function of number of samples, n, of [X], P(n), that k ≥ k₁? |
Let $\{a_i\}_{i\in \mathbb{N}}$ be a sequence of positive real numbers, whose series converges i.e. $\sum\limits_{i\in \mathbb{N}}a_i = a < \infty.$ Does this imply the convergence of the double series $\sum\limits_{i,j \in \mathbb{N}}a_i a_j?$
If yes, how can this be proven? If not, what are some counterexamples?
... |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
*(1) Given a finite set of points in Euclidian plane such that any line passing through two of the points passes through at least one more of them, then all points have to be collinear.*
Let n be... |
Consider a length-$m$ array of unique elements $A=\{1,2,\dots, m\}$. Independently poll $n$ samples (with replacement) and place them into array $B$ (of length $n$). Then, you de-duplicate $B$ to produce array $C$ with $k$ elements (of course, $k\le n$).
> Question: what is the probability $P(k\ge i)$ for each $i$? |
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$ ... |
I know that the number of positive divisors of $n$ can be given by :
$\tau(n)$ = $(a_1+1)(a_2+1)\ldots(a_k+1)$ where $n = p_1^{a_1}p_2^{a_2}.... p_k^{a_k}$, where $p_1, p_2... p_k$ are the prime factors of $n$ and $a_1, a_2... a_k$ are positive integers.
And the sum of the positve divisors of $n$ can be given by... |
Consider a length-$m$ array of unique elements $A=\{1,2,\dots, m\}$. Independently poll $n$ samples (with replacement, so duplicate elements are possible) and place them into array $B$. Then, you de-duplicate $B$ to produce array $C$ with $k$ unique elements (of course, $k\le n$).
> Question: what is the probability... |
Unfortunately, different writers define *critical point* differently:
1. Many/most define it to be either a point where the derivative is $0$ OR where the derivative doesn't exist (e.g. [Wikipedia][1], [Paul's Online Notes][2], [Brilliant][3]).
2. Some define it to be a point where the derivative is $0$ (e.g. [Adam... |
Are you solving for a falling object ?
Its easier to multiply both sides by $e^{\frac kmt}$ so
$$y''+y'{\frac km}=-g\hat{j}\\ \times e^{\frac kmt} \\
e^{\frac kmt}y''+e^{\frac kmt}y'{\frac km}=-ge^{\frac kmt}\hat{j}\\ (e^{\frac kmt}y')'=-ge^{\frac kmt}\hat{j}$$ no apply integral both sides
$$\int =\int \\e^{\f... |
Given a collection of sets $\{Y_\alpha: \alpha \in A \}$ for an index set $A$. For each $\alpha \in A$, let $f_\alpha: X \to Y_\alpha$ be a function. The map $f: X \to \prod_{\alpha \in A} Y_\alpha$ defined by
$$
f(x) = \prod_{\alpha \in A} f_\alpha(x)
$$
is called the `evaluation map` where $\prod$ denotes t... |
Question: Understanding how to compute $\int_{0}^{+\infty} \frac{(sint)}{(t)}dt $ via complex integration.
Let us first define $ I(r)=\int_{(|z|=r)} \frac{e^{iz}}{z}dz. $
(a) Show that $ I(r) \to 0 $ as $ r \to \infty. $
(b) Show that$ I(r) \to \pi i$ as $ r \to 0. $
(c) Consider a path $ \gamma (s,t) = [... |
Question: Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration.
Let us first define $ I(r)=\int_{(|z|=r)} \frac{e^{iz}}{z}dz. $
(a) Show that $I(r) \to 0 $ as $ r \to \infty. $
(b) Show that $I(r) \to \pi i$ as $ r \to 0. $
(c) Consider a path $ \gamma (s,t) = [s,r... |
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 $\mathfrak{p} = \mathfrak{q} \cap A$ ... |
Going about this problem should I sub in 3 before finding the derivative or get the general form and then substitute x.
g'(x) = x'f(x) - xf'(x) / (x^3+5)^2
Here derivative of x would be 1.
Instead if I did g'(3) the derivative of 3 would be 0, eliminating the x'f(x) part. Which is correct? |
My attempt:
The group $R^*$ means the set of real numbers with zero omitted, forms a group under multiplication.
1 is the identity.
And we can't find such a & b such that, $|a| = ∞$, $|b| = ∞$ and $ |ab| = 2$.
Am I right? If not, does there exist such a & b ? Any hint or suggestion ? |
I saw this problem:
> Find $\lim_{n \to \infty }\prod_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1+x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ with radius of c... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1+x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
Given the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x)=\frac{xf(x)}{x^3+5}$, my task is to find derivative of function g at $x=3$ where $f(3) = -1,f'(3) = 4$. Should I substitute in the value $3$ before finding the derivative or get the general form and then substitute in the value of $x$?
|
You know about $f(3)=-1,f'(3)=4$and $g(x)=\frac {xf(x)}{x^3+5}\to g'(3)=?$
$$g'(x)=\\\frac {(xf(x))'\times (x^3+5)-(x^3+5)'\times(xf(x))}{(x^3+5)^2}=\\\frac {(1f(x)+xf'(x))\times (x^3+5)-(3x^2+0)\times(xf(x))}{(x^3+5)^2}$$then put $x=3$ and you willl have
$$g'(3)=\\\frac {(1f(3)+3f'(3))\times (3^3+5)-(3(3)^2+0)\ti... |
Let $G$ be an abelian group.
Let $\widehat{G}$ be a profinite completion of $G$.
Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are **finite index** subgroups of $G$ and $N\subset M$.
Let $G_1$ and $G_2$ be groups and suppose there is an injecti... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
Given a finite set of points in Euclidian plane such that any line passing through two of the points shall pass through at least one more of them, then all points have to be collinear.
Base case... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
**Is there an integer $n$ such that $2n^2-1$ has two distinct positive divisors congruent modulo $2n$?**
**We prove that the answer to this question is "NO".**
Suppose such an integer $n$ exists: $\exists p, t \in \mathbb{N}^*$ such that $p$ and $p+2tn$ are divisors of $2n^2-1$.
Let $d = p \wedge (p+2tn)$.
We... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
The space of finite signed Radon measures $M(\mathbb{R}^n)$ with total variation norm can be identified as either the dual space of continuous functions that vanishes at infinity $C_0(\mathbb{R}^n)$ or continuous function with bounded supported $C_c(\mathbb{R}^n)$ in the uniform norm. However, do they induce the same w... |
Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of
0. There is a weakly inaccessible cardinal.
1. There is a Mahlo cardinal.
2. There is a weakly compact cardinal.
3. There is a measurable cardinal.
4. There is a strongly compact cardinal.
5. There is a supercompa... |
If there is a large cardinal, can GCH also hold? |
Consider the language F = {$a^i$$b^j$$c^k$ | i, j, k ≥ 0 and if i = 1 then j = k}.
1. Show that F is not regular.
2. Show that F acts like a regular
language in the pumping lemma. In other words, give a pumping length
p and demonstrate that F satisfies the three conditions of the
pumping lemma for... |
I know the chromatic symmetric function simplifies to the chromatic polynomial when 1's and 0's are subbed in for the x's. I was wondering if one could easily find the chromatic symmetric of a graph if given the chromatic polynomial of the graph?
Thanks in advance |
Formula for the chromatic symmetric function of a graph in terms of the graph's chromatic polynomial? |
Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of
0. There is a weakly inaccessible cardinal.
1. There is a Mahlo cardinal.
2. There is a weakly compact cardinal.
3. There is a measurable cardinal.
4. There is a strongly compact cardinal.
5. There is a supercompa... |
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