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H: prove equation has no rational root.
Prove
For all $n>1$, equation $\sum _{k=1}^n \frac{x^k}{k!}+1=0$ has no rational root.
I'm not sure whether there are two questions,for without brace after Sigma.
My thought is to prove it is not reducible on rational field.
AI: Multiplying by $n!$ we can make all coefficient... |
H: Using Maclaurin series with solving a multi-variable limits
I need to determine wheter there's a limit where $(x,y)=(0,0)$ of the next function:
$$\lim_{(x,y)\to(0,0)}\frac{e^{x(y+1)}-x-1}{\sqrt{x^2+y^2}}$$
In order to simplify the expression can I use maclaurin series on $e^{x(y+1)}?$ If so it's equal to $e^{x(y+1... |
H: Find the value which satisfies these equations
$$C = 0.6Y +50$$
$$I=10$$
We want to find the $Y$ for which $Y=C+I$. This is a question in a chapter about discrete dynamic models, so we have to use an appropriate method.
I tried to rewrite it and ended up with $y = 1\dfrac{2}{3} C - 73 \dfrac{1}{3}$, but that was a ... |
H: The derivative of a linear transformation
Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}, x \in \mathbb{R}^n$. Define $F: \mathbb{R}^n \times \mathrm{Mat}_{m\times n} \rightarrow \mathrm{Mat}_{m\times n}$ by $F(x,A) = df_x + A$.... |
H: How to show that space is complete?
Let $N_\alpha=\{(x_n)_{n=1}^\infty\mid \sum_{j=1}^n |x_j|\leq Mn^\alpha\}$, where $\alpha\in R$. Show that $N_\alpha$ is Banach space with the norm $\|(x_n)_{n=1}^\infty\|=\sup_{n\in N} n^{-\alpha} \sum_{j=1}^n|x_j|$.
It is is easy to check properties of norm, but I don't know h... |
H: Equivalent condition for Differentiability
Is it possible to prove that
$f:(a,b)\to\mathbb R$ is differentiable at $c \in (a,b)$
iff
there exists a real number $f'(c)$ such that for all $x \in (a,b)$, $f(x) = f(c) + f'(c)(x-c) + e(x)$
for some function $e:(a,b) \to\mathbb R$ s.t $e(x)\to 0$ as $x\to c$
without a... |
H: Sum of infinitely many integrals
I know that $\int_0^\infty \frac{1}{x^2+1} \, dx=\dfrac{\pi}{2}$. What if I integrate $f(x)=\dfrac{1}{x^2+1}$ in infinite disjoint integrals not of the same length, like
$$\int_{x_{1}}^{x_{2}} f(x)\,dx+\int_{x_3}^{x_4} f(x)\,dx+\cdots +\int_{x_{2k+1}}^{x_{2k+2}} f(x)\,dx+\cdots$$ Is... |
H: Need help with Cantor-Bernstein-Schroeder Proof at ProofWiki
This concerns Proof 6 of the CBST theorem at ProofWiki.
I am stuck on the line beginning "Similarly, let $g' = $"
The 2nd equality on this line is not immediately obvious to me. How do you prove $A-X = g(B -f(X))$?
AI: A few lines up from your quote, we s... |
H: Show that there can be no trail in $G$ that contains all edges.
Let $G=(V,E)$ be a graph. Assume that $G$ contains at least three vertices with an odd degree. Show that there can be no trail in $G$ that contains all edges.
First I checked the definition of a trail; that is a walk that does not contain any edge mor... |
H: Determinant of a matrix
Having some problems with a determinant of a 4x4 matrix M.
$
M =
\left( {\begin{array}{cc}
1 & 2 & 3 &-1 \\
0 & 1 & 2 & 2 \\
1 &1 &0 &0 \\
3&1&2&0
\end{array} } \right)
$
Went along and developed it according to the 4th column. So I end up with two matrixes A and B.
$
A = -1 \cdot det
\left... |
H: Universal covering space of connected open subset of $\mathbb R^n$
Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
AI: No. For example, let $n=3$ and $U=\Bbb{R}^3\setminus \{(0,0,0)\}$. Then the universal cover of $U$ is $U$ itself (since it's simply conn... |
H: Is $\mathbb{R}^2$ boundaryless?
I just have a quick question, as stated in the title.
Is $\mathbb{R}^2$ boundary-less?
Thank you very much. :-)
AI: That depends on whether you consider $\mathbb{R}^2 \subseteq \mathbb{R}^2$ or $\mathbb{R}^2 \subseteq \mathbb{R}^n$ for $n \geq 3$. In the former case, there are no po... |
H: Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Club challenge problem. I don't think it's possible to do with only hi... |
H: Hartshorne Lemma(II 6.1)
Hi I am wondering what Hartshorne means by "an open affine subset on which $f$ is regular". This is part of the first sentence in the proof of lemma 6.1 in chapter two of the book "Algebraic Geometry by Hartshorne". I Tried looking up regular in the index, but this just refers me to regula... |
H: Is there any good reason why a protractor starts from right to left, unlike a scale, which starts from left to right?
While studying through the number system, i notice that positive side is from 0 to +ve infinity. The direction is left to right.
However, this is opposite in case of angles. The sort of curved numbe... |
H: Area of Questionably Generated Manifold
I might not possess the language to ask this question, but I'm going to try anyway.
Consider a path c(t) : $\mathbb{R}\rightarrow \mathbb{R}^n$. Let c'(t) denote the tangent vector of the path c(t). Can we "stitch" together the tangent vectors given by evaluating c'(t) at e... |
H: What is the defining characteristic of a quadratic function?
I'm helping a high school student prepare for an exam, and I'm unsure how to answer this...
Why is $x^3+2x^2$ not quadratic? I thought anything that had a power of 2 was quadratic.
AI: A quadratic must be a polynomial and it must be of degree $2$.
The ... |
H: Integrating a form and using Gauss' theorem.
Given the 2-form
$$
\varphi = \frac{1}{(x^2+y^2+z^2)^{3/2}}\left( x\,dy\wedge dz +y\,dz\wedge dx + z\,dx\wedge dy\right) \ .
$$
(a) Compute the exterior derivative $\textbf{d}\varphi$ of $\varphi$.
(b) Compute the integral of $\varphi$ over the unit sphere oriented ... |
H: Application of uniform boundedness principle
Let $(a_n)$ be a sequence in $\mathbb{K}$ such that for each $(x_n) \in c_0$ also $(a_nx_n) \in c_0$.
Derive from the uniform boundedness principle that $(a_n) \in l^\infty$.
I see that the idea is to find a family $\{T_i\}_{i\in I}$ of linear, bounded operators that is ... |
H: Showing that $\{x\in\mathbb R^n: \|x\|=\pi\}\cup\{0\}$ is not connected
I do have problems with connected sets so I got the following exercise:
$X:=\{x\in\mathbb{R}^n: \|x\|=\pi\}\cup\{0\}\subset\mathbb{R}^n$. Why is $X$ not connected?
My attempt: I have to find disjoint open sets $U,V\ne\emptyset$ such that $U\c... |
H: Selection Sort Algorithm Analysis
While performing algorithm analysis on the following C code-snippet,
void selection_sort(int s[], int n)
{
int i, j, min;
for (i = 0; i < n; i++)
{
min = i;
for (j = i + 1; j < n; j++)
if (s[j] < s[min])
min = j;
swap(... |
H: '$R$-rational points,' where $R$ is an arbitrary ring
On page 49 of Liu's Algebraic Geometry and Arithmetic Curves, we find Example 3.32. In it, he shows that if $k$ is a field and $X=k[T_1,\dots,T_n]/I$ is an affine scheme over $k$, the sections $X(k)$ (the $k$-rational points of $X$) are in bijection with the alg... |
H: Is the product of covering maps a covering map?
I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times X_2 \rightarrow Y_1 \times Y_2$ a covering map?
AI: Yes. This is a ... |
H: Showing that $\int_0^{\pi/3}\frac{1}{1-\sin x}\,\mathrm dx=1+\sqrt{3}$
Show that $$\int_0^{\pi/3}\frac{1}{1-\sin x}\,\mathrm dx=1+\sqrt{3}$$
Using the substitution $t=\tan\frac{1}{2}x$
$\frac{\mathrm dt}{\mathrm dx}=\frac{1}{2}\sec^2\frac{1}{2}x$
$\mathrm dx=2\cos^2\frac{1}{2}x\,\mathrm dt$
$=(2-2\sin^2\frac{1}{2... |
H: Why this object is a sheaf?
I whould like to know why $ \ \mathcal{C} : U \to \mathcal{C} ( U , \mathbb{R} ) $ is a sheaf ? $ U $ is an open set of $ E $ a $ \mathbb{R} $ - vector space which has a finite dimension.
$ \mathcal{C} ( U , \mathbb{R} ) $ contains continous maps over $ U $, with values in $ \mathbb{R} $... |
H: Complete theories - dense linear order
There are two things I would like to prove.
DLO - Dense linear order is complete, that means that when $\psi$ is a sentence of the language $\{<\}$ then $DLO\vDash\psi$ or $DLO\vDash\neg\psi$
When $S$ is recurisvely enumerable and a complete set of axioms then $M:=\{\phi: S\v... |
H: Linear multiple-variable function
I'm reading a differential equation book but i'm stuck on his definition of linear ODE.
Supposing our equation has y as its dependent variable, t as its independent variable and y^(n) reffers to the nth derivative of y with respect to t, my book defines any ordinary differential eq... |
H: Primality test bound: square root of n
I was reading about primality test and at the wikipedia page it said that we just have to test the divisors of $n$ from $2$ to $\sqrt n$, but look at this number:
$$7551935939 = 35099 \cdot 215161$$
Since $\sqrt {7551935939} \approx 86901,875$ so basically I would only have to... |
H: First order sentences and Halting problem recursively enumerable
I am just finding searching for some examples of recurisvely enumerbale models and I do not know how to prove that the following ones satisfy this property.
Consider the set of first order sentences. The subset of the sentences which have a finite mo... |
H: Limits and Inequalities
My book claims the following: Let $f$ be a continuous function. $\lim s_n = x_0$ and since $f(s_n) < y$ for all $n$, we have $f(x_0) = \lim f(s_n) \leq y$. Can someone explain why the last inequality is $\leq$ and not just $<$?
I feel like I am missing something very obvious.
AI: You have $... |
H: Relationship between $PGL_2$ and $PSL_2$
I read somewhere that $PGL_2(\mathbb{C})=SL_2(\mathbb{C})/N$ where $N$ is the normal subgroup consisting of
$\pm \left(
\begin{matrix}
1 & 0\\
0 & 1
\end{matrix}\right)$. It is unclear to me how this follows from the fact that $PGL_2(\mathbb{C})=GL_2(\mathbb{C})/aI$ where $a... |
H: find $\frac{ax+b}{x+c}$ in partial fractions
$$y=\frac{ax+b}{x+c}$$ find a,b and c given that there are asymptotes at $x=-1$ and $y=-2$ and the curve passes through (3,0)
I know that c=1 but I dont know how to find a and b?
I thought you expressed y in partial fraction so that you end up with something like $y=Ax+B... |
H: Evaluating $\lim\limits_{x\to1}\frac{\sqrt[3]{x^3+1}}{\sqrt[3]{x+1}}$?
This is the limit:
$$\lim_{x\to1}\frac{\sqrt[\large 3]{x^3+1}}{\sqrt[\large 3]{x+1}}$$
I made the calculation and the result gave $\sqrt[\large 3]{0}$, I wonder if this is the correct result, but if not, what would be the account to the correct ... |
H: Is there such a thing as a metric space of sets over an underlying metric space?
Typically we think of a metric as a notion of distance between elements of set subject to the following constraints...
$$
d(x, y) ≥ 0,\quad d(x, x) = 0,\quad d(x, y) = d(y, x),\quad d(x, z) ≤ d(x, y) + d(y, z)
$$
I want to know i... |
H: Least Upper Bounds based problem
I am having difficulty with this problem from chapter 8 in Spivak's Calculus, any help is appreciated.
(a) Suppose that $y - x > 1$. Prove that there is an integer $k$ such that
$x < k < y$. Hint: Let $l$ be the largest integer satisfying $l \le x$, and
consider $l + 1$.
(b) S... |
H: Are compositions of the Fourier sine and cosine transforms commutative?
That is to say, is it true or false that
$$\mathcal{F}_c(\mathcal{F}_s(f(x)))(\xi)\equiv\mathcal{F}_s(\mathcal{F}_c(f(x)))(\xi),$$
and if they are not then are there any conditions on $f$ for which they might be?
I can't seem to find any docume... |
H: Probability Question - Baye's Rule
Here's the question:
Bob's burglary alarm is on. If there was a burglary, the alarm goes off with probability 50%. However, on any given day, there is a 1% chance the alarm is triggered by a dog. The burglary rate for the area is 1 burglary in $10^4$ days. What is the probability ... |
H: One sided limits (1)
Does these limits exists:
$$1)\lim_{x \to 0^+}\frac{x^2+1}{x^3}$$
$$2)\lim_{x \to 0^-}\frac{x^2+1}{x^3}$$
$$3)\lim_{x \to \infty}\frac{x^2+1}{x^3}$$
$$1)\lim_{x \to 0^+}\frac{x^2+1}{x^3}= \lim_{x \to 0^+}\frac{1+\frac{1}{x^2}}{x}=\lim_{x \to 0^+}(\frac{1}{x}+\frac{1}{x^3})= " \frac {1}{0^... |
H: Some doubts about interpretation of an atomic formula in predicate calculus
I have some doubt related to the interpretation of atomics formulas in predicate calculus.
In predicate calculus a formula will be interpreted on a specific domain that is where I take the allowed values for the formula constants.
So I have... |
H: Is $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?
My professor introduced subspaces of $\mathbb{R}^n$ today and I don't think I understand them very well.
He posed this question as an example:
Is the set $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?
He said that it wasn't. Could... |
H: How to interpret "computable real numbers are not countable, and are complete"?
On page 12 of this (controversial) polemic
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
Wildberger claims that
Even the "computable real numbers" are quite misunderstood. Most mathematicians reading this paper suffer from... |
H: A question on limit
Suppose that $p_k>0, k=1,2,\dots, $ and
$$
\lim_{n\to\infty} \frac{p_{n}}{p_1+p_2+\dots+p_n}=0,\quad \lim_{n\to\infty} a_n=a.
$$
Show that
$$
\lim_{n\to\infty}\frac{p_1a_n+p_2a_{n-1}+\dots+p_na_1}{p_1+p_2+\dots+p_n}=a
$$
The hints are much appreciated. I don't want complete proof.
Thanks for y... |
H: Why is this cardinal regular?
I have the following problem in front of me.
Show that if $\kappa$ is the least cardinal such that $2^\kappa>2^{\aleph_0},$ then $\kappa$ is regular.
I've scribbled this:
Suppose $$\kappa=\coprod_{\alpha<\lambda}X_\alpha,$$ where $\lambda<\kappa$ is a cardinal, and for each $\alpha<\... |
H: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
Here is an exercise:
Suppose that $\{x_n\}$ is a sequence such that $\lim \limits_{n\to\infty}(x_n-x_{n-2})=0$.
Show that:
$$\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $$
Thanks.
AI: Hints:
Let $y_n=|x_n-x_{n-1}|$. Note that $|y_n-y_{n-1}|\le |... |
H: Show that the function $f: X \to \Bbb R$ given by $f(x) = d(x, A)$ is a continuous function.
I'm studying for my Topology exam and I am trying to brush up on my metric spaces.
Suppose $(X, d)$ is a metric space and $A$ is a proper subset of $X$. Show that the function $f: X \to \Bbb R$ given by $f(x) = d(x, A)$ is ... |
H: Convergence of $\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$
Does the series $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$$
converge absolutely, converge conditionally, or diverge?
I've tried applying the ratio test and the root test, and in both cases the limit is $1$, so I cannot conclude anything.
AI: This se... |
H: Evaluate the limit: $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$
Could someone give me a hint? I would like to continue to attempt it. The limit to evaluate that I would like a hint on is:
$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$
AI: Along the curve $x=y=z^2$, the function is ident... |
H: Show that $\int_{|z|=1}(z+1/z)^{2m+1}dz = 2\pi i {2m+1 \choose m}$
Show that
$$
\int_{|z|=1}(z+1/z)^{2m+1}dz = 2\pi i {2m+1 \choose m}
$$
, for any nonnegative integer m.
I can't solve this problem..
I tried to find singularities but failed.
$(z+1/z)$ is not familiar to me.
Is there anyone to help?
AI: Due to binom... |
H: Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
Testing with my calculator in degree mode, I have found the following to be true:
$$\tan \left(90 - \frac{1}{10^n}\right) \approx \frac{180}{\pi} \times 10^n, n \in \mathbb{N}$$
Why is this? What is the proof o... |
H: Finding in a string S if is possible to create a set with perfect cubes or perfect squares with elements of S.
You have a sequence $S[1...n]$ with $n$ digits(0 to 9) and you wanna know if its possible break then in perfect square or perfect cube. For example, if $S = 1252714481644$, then the answer is $YES$ because... |
H: Question about Group notation
Say I have elements $g$ and $h$ in a group $G$.
What does $g^h$ mean? Seeing this notation a lot but I can't find an explanation for it anywhere.
AI: In work by group theorists, this is the right action of $G$ on itself by conjugation: $$g^h = h^{-1} g h$$
This has the nice property th... |
H: Equivalence for quantifications
Considering:
(1) $(\exists x)(E(x) \land (\forall y)(E(y)\rightarrow M(x,y)))$
There are equivalences that say:
$\lnot(\forall x) A \equiv (\exists x) \lnot A$
$\lnot(\exists x) A \equiv (\forall x) \lnot A$
So, if I want to express (1) only with the "forall" quantifier I got:
... |
H: Calculating mean and standard deviation for a set
Suppose we have a set $A = \{a_1, a_2, a_3, a_4, a_5\}$ where $a_n \in
\mathbb{R}$ and a set $B = \{b_1, b_2, b_3, b_4, b_5\}$ where $b_n \in
\mathbb{R}$ and a set $C = \{ma_1 + nb_1, ma_2 + nb_2, ma_3 + nb_3, ma_4 + nb_4, ma_5 + nb_5\}$ where $m, n \in (0,1) \subs... |
H: A function continuous on all irrational points
Let $h:[0,1]\to\mathbb R$
$h(x)=\begin{cases}0&\text{if }x=1\\\frac{1}{n}& \text{otherwise if }x\in\mathbb Q,x=\frac{m}{n},\;m,n\in\mathbb N,\gcd(m,n)=1\\0&\text{otherwise if }x\in\mathbb R\setminus\mathbb Q\end{cases}$
How do you prove that $h$ is continuous on all ir... |
H: Elementary proof, convergence of a linear combination of convergent series
Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{... |
H: Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?
Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?
I just made this question up to practice determining if sets are vector subspaces or not.
From what I can tell, the zero vector, $\vec{0}=(0,0,0)$ exists, whe... |
H: Question about radical of powers of prime ideals.
Let $Q$ be an ideal of a commutative ring $A$ and $$r(Q) = \{x \in A : x^n \in Q \text{ for some } n >0 \},$$
the radical of $Q$. Suppose that $P$ is a prime ideal of $A$. How to show that $r(P^n) = P$ for all $n>0$?
It is clear that $P \subseteq r(P^n)$ for all $n>... |
H: Smallest positive $n$ divisible by $2$ and $3$, and which is an $n$th power and $m$th power, is $2^{\mathrm{lcm}(n,m)}3^{\mathrm{lcm}(n,m)}$
What is the smallest positive integer divisible both by 2 and 3 which is both a perfect square and a sixth power?
Answer: $2^6 3^6$
More generally, what is the smallest positi... |
H: joint pdf setting up the integral
Let $X$ and $Y$ have the joint pdf
$f_{x,y}(x,y)=2e^{-(x+y)}, 0<x<y, 0<y$. Find $P(Y<3X)$
my question is regarding setting up the integral: $\int_0^\infty\int_x^{3x}2e^{-x}e^{-y}dydx$
(1) i understand why the y integral is $0$ to infinity because it is given that $0<y$ however why ... |
H: Help on a proof of dimension of a vector space
The proof shows that two bases have the same number of elements, and I can't understand one step. The proof goes:
As $v_1, . . . , v_n$ is a basis of $V$, each $w_k$ can be expressed as a linear combination of the $v_j$ ; thus for each $k$ there are scalars $λ_{1k}, . ... |
H: No members of a sequence is a perfect square
Prove that no member of the sequence 11, 111, 1111, ... is a perfect square.
I noticed that the first four terms of the sequence (above) are none of them are a perfect square. But I do not know what to do afterwards.
AI: Since this particular question has been discussed ... |
H: Poisson counting process
Let $N_t$ be number of customers that arrived at shop until moment $t$. Let's say that shop opens at 9:00. $N_t$, $t\geq0$ is Poisson process with $\lambda=1$ per hour. What is the probability that, there will be at least 2 arrivals between 10:00 and 10:30?
We have time interval (10:00,10:3... |
H: Reduction formulae question.
$I_n=\int_0^\frac{1}{2}(1-2x)^ne^xdx$
Prove that for $n\ge1$
$$I_n=2nI_{n-1}-1$$
I end up (by integrating by parts) with: $I_n =e^x(1-2x)^n+2nI_{n-1}$
I am not sure how $e^x(1-2x)^n$ becomes $-1$?
AI: $\displaystyle I_n=\int_0^\frac{1}{2}(1-2x)^ne^xdx$
$=\displaystyle [(1-2x)^n\int e^{x... |
H: How to use the Chain rule for: $y=\cosh(a\sinh^{-1}x)$
$$y=\cosh(a\sinh^{-1}x)$$ where $a$ is a constant. What do i substitute for $u$ and $v$ to then find $\frac{du}{dx}$ and $\frac{dv}{dx}$?
I am then suposed to prove that: $$(x^2+1)\frac{d^2y}{dx^2} +x\frac{dy}{dx}-a^2y=0$$
AI: HINT:
So, $$\cosh^{-1}y=a\sinh ^{-... |
H: central limit theorem using expected value
Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in 1957. Included as part of the test were some three thousand military and civilian "observers". Now, more then fifty years later, eight cases of leukemia have... |
H: How do I prove $\frac{M_1 \times M_2}{N_1 \times N_2} \simeq \frac{M_1}{N_1} \times \frac{M_2}{N_2}$?
I seem to encounter this issue whenever a question involves quotient objects. In this case, I have modules $M_1$ and $M_2$ and subsets $N_1$ and $N_2$ thereof respectively. It is given that $N_1$ and $N_2$ are subm... |
H: How to prove this result on limit?
$$\lim_{x \to 0} \frac{({a+x^m})^{\frac{1}{n}}-({a-x^m})^{\frac{1}{n}}}{x^m}=\frac{2}{n}a^{\frac{1}{n}-1}$$
I have tried it using L'Hôpital's rule, but I don't get any way out of that.
Any Help will be appreciated. Thanks.
AI: Putting $y=\frac {x^m}a,$
$$\lim_{x \to 0} \frac{({a... |
H: Can we get the line graph of the $3D$ cube as a Cayley graph?
Given a graph $G=(V,E)$, the line graph of $G$ is a graph $\Gamma$ whose vertices are $E$ (the edges of $G$) and in $\Gamma$, two vertices $e_1,e_2$ are connected if, as edges in $G$, they share an endpoint.
Now let $G$ be the $3D$ cube graph. It has $8$... |
H: Proving that the series $\sum\limits_{n=1}^\infty a(n)$ diverges, where $a(n) = (1/2)^k$ for $2^{k-1} \le n < 2^k$
Let $a(n)=(1/2)^k$ when $2^{k-1} \le n < 2^k$ and $k$ is a natural number. How do you prove that $\sum\limits_{n=1}^\infty a(n)$ diverges?
AI: compute the sum of $a_n$s for n in such a $k$-block. you h... |
H: Orthogonal complement of orthogonal complement
Let $U$ be a subspace of $V$ (where $V$ is a vector space over $C$ or $R$). The orthogonal complement of the orthogonal complement of $U$ is not equal to $U$ in general (equal only for dim $V$ finite).
Can anyone give me a simple example when the orthogonal complement ... |
H: The Set $\{x\in \mathbb{R}^n : \|x\|\leq r\}$ is Closed
How to show that a set $A=\{x\in \mathbb{R}^n : \|x\|\leq r\}$ is closed?
May I do it showing that for $x\in A, a\in \mathbb{R}$ it holds $ax\in A$ and for $x,y\in A$ it holds $x+y \in A$?
AI: Here closed means that $\partial A$= boundary of A belongs to $A$.... |
H: Show that $x=2\ln(3x-2)$ can be written as $x=\frac{1}{3}(e^{x/2}+2)$
Show that $x=2\ln(3x-2)$ can be written as $x=\dfrac{1}{3}(e^{x/2}+2)$.
Is there a rule for this?
AI: Solve for the "other" $x$. Notice that:
$$ \begin{align*}
x &= 2\ln(3x-2) \\
\dfrac{x}{2} &= \ln(3x-2) \\
e^{x/2} &= 3x-2 \\
e^{x/2}+2 &= 3x \... |
H: Showing that two real matrices are not congruent over $\mathbb{Q}$
Maybe it is a stupid question but I will still ask it here.
How can I prove that the following matrices are not congruent over $\mathbb{Q}$?
\begin{pmatrix}
-1 & 0\\
0 & 2\\
\end{pmatrix}
\begin{pmatrix}
-1 & 0\\
0 & 1\\
\end{pmatrix}
Th... |
H: Expectation of maximum of a function whose expectation is concave
An analysis of a data structure yields a property of the form
$\qquad \mathbb{E}[ f(k) ] = H_k + H_{n-k+1} - 1$
for some natural $n$ and all $1 \leq k \leq n$. Note that the $f(k)$ are not independent.
Now I am interested in the quantity
$\qquad \mat... |
H: finiteness and first order sentences
Lets consider a set of sentences $T$ and a signature $\sigma$. I proved (using compactness theorem) that when $T$ has arbitrary large models than also an infinite model. Now there are several consequences from this statement but I have problems formulating the connection to the ... |
H: prove statement about fixed point iteration
I have the following fixed point iteration:
$$ p_{n+1} = \frac{p_n^3 + 3ap_n}{3p_n^2 + a} $$
By defining $$g(x) = \frac{x^3 + 3ax}{3x^2 + a}$$ en some algebra I found that the fixed point is $x = \sqrt{a}$. So $g(\sqrt{a}) = \sqrt(a)$.
Now I need to show that this iterat... |
H: $x$ such that $\sqrt{x^y} > \sqrt{(x y)^2}$
Let $y>1$, $x>0$
Of what form are numbers $x$ such that:
$$\sqrt{x^y} > \sqrt{(x y)^2}$$
I.e: What is the solution for $x$?
EDIT:
As the first answer implied:
$$\sqrt{x^y} > \sqrt{(x y)^2} \implies \log(\sqrt{x^y}) > \log(\sqrt{(x y)^2})\implies$$
$$\log(x^y) > \log((x y)... |
H: Solving a conjecture by bruteforcing
Say we wanted to check the Beal conjecture ["If $A^x+B^y=C^z$, where $[A, B, C, x, y, z \in N] \wedge [x, y, z \gt 2] \to $ A, B and C must have a common prime factor", from the official site]. What is the most efficient way of solving it by brute-forcing?
First, define a set ... |
H: problem concerning continuity of functions
Let $f,g:X \rightarrow \mathbb R$ be both continuous functions. Prove that if $X$ is open, then the set $A=\{ x \in X \mid f(x) \neq g(x) \}$ is also open. Show that if $X$ is closed, then $A=\{x \in X \mid f(x)=g(x)\}$ is closed.
Is there anyone who can help me to solve... |
H: What is this method called - Abelization of a Group.
Today, I wanted to make a post for this question. There are some approach in which we can overcome the problem like this and this. According to my knowledge, I could solve the problem via the approach I had learned. Since I don't know what is this way called, so ... |
H: Determinant of symmetric matrix
Given the following matrix, is there a way to compute the determinant other than using laplace till there're $3\times3$ determinants?
\begin{pmatrix}
2 & 1 &1 &1&1 \\
1 & 2 & 1& 1 &1\\
1& 1 & 2 & 1 &1\\
1&1 &1 &2&1\\
1&1&1&1&-2
\end{pmatrix}
AI: You can substract the firs... |
H: A Surjective Local Smooth Diffeomorphism That is Not A Covering Map
Let $\pi:M_1\rightarrow M_2$ be a surjective $C^{\infty}$ map between two connected manifolds with $d\pi$ an isomorphism.
If $M_1$ is compact, it is seen that $|\pi^{-1}(m_2)|$ is finite, so $\pi$ is a covering. If we only have $M_2$ compact, do we... |
H: Is there software to help with group presentation
I wrote a computer program that generates group presentations.
I would like to know the sizes of the resulting groups. I know that this is undecidable.
Are there good heuristic programs that can try to compute the size of a group given by generators and relations?
... |
H: Why is big-Oh multiplicative?
If $f$ is $O(g)$ over some base, this means that $f(x) = \beta(x)g(x)$, where $\beta$ is eventually bounded. So this means that eventually, $f$ is at most $c$ times $g$, where $c$ is some constant.
But I thought $O$ was a way of expressing that two functions are eventually approximatel... |
H: Prove that a function f is continuous (1)
$f:\mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) =
\begin{cases}
x \sin(\ln(|x|))& \text{if $x\neq0$} \\
0 & \text{if $x=0$} \\
\end{cases}$$
Is $f$ continuous on $\mathbb{R}$?
I want to use the fact that 2 continuous functions:
$$f:I \rightarrow J ( \subset \mat... |
H: Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$
Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function.
As
$${\log(i^2)\over i} = \pi$$
and $${\log(x^2)\over x}=\pi$$
Does $x = i$?
AI: Hint:assume $x=a+ib$ then $${\log(x^2)\over x}=\pi\to \pi(a+ib)=\log(x^2)=\ln(|x^2|)+i\arg(x^2)=2\mathrm{... |
H: Questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.
I have some questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.
On Line 8-9 of Page 42, it is said that $(xs-a)t=0$ for some $t\in S$ iff $xst\in \mathfrak{a}$. If $(xs-a)t=0$, ... |
H: Closed linear subset of a Hilbert space
If $H$ is a Hilbert space, and if
$$(a,b)_H=0$$
for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
AI: No, because if $B\neq H$, $B$ will have a non-zero orthogonal complement. That means precisely t... |
H: A simpler proof that if $m\mid n$ then there is a ring homomorphism from $\mathbb{Z}_n$ onto $\mathbb{Z}_m$?
Question 18.I.4 from Pinter's A Book of Abstract Algebra asks for a proof of the following, where $\mathbb{Z}_m$ and $\mathbb{Z}_n$ are treated as rings:
If $n$ is a multiple of $m$, then $\mathbb{Z}_m$ is ... |
H: Converting data to a specified range
I am trying to convert data to a particular range of 0-10.
Actual Data may vary from - 50000 - 26214400. I have broken this down in to 4 parts as follows -
50000 - 1048576 -----> 0 - 2.5
1048576 - 5242880 -----> 2.5 - 5.0
5242880 - 15728640 ----->5.0 - 7.5
15728640 - 26214400 --... |
H: Show that the following matrix is diagonalizable
My question is related to this question discussed in MSE.
$J$ be a $3\times 3$ matrix with all entries $1\,\,$. Then prove that $J$ is
diagonalizable.
Can someone explain it in terms of A.M. and G.M. (algebraic and geometric multiplicity) concept ? Thanks in adva... |
H: Question on Galois extension of field of fractions
I would like to ask if the following is true:
Suppose we have an integral domain $A$ and a group action $G$ on $A$. We consider $A^{G}$ the subring of $A$ fixed by $G$. Let $L$ be the field of fractions of $A^{G}$ and $K$ the field of fractions of $A.$ Can we saf... |
H: Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous
If $F$ is of bounded variation defined on $[a,b]$, and $F$ satisfies
$$\int_{a}^b |F'(x)|dx=T_F([a,b])$$
where $T_F([a,b])$ is the total variation. How to prove that $F$ is absolutely continuous?
My Attempt: I used the inequality, ... |
H: Notation for Multiple summation
Is there an alternate way to represent the multiple summation given below?
$\displaystyle \Large \sum_{i_k=k}^{n} \space \sum_{i_{k-1}=k-1}^{i_k} \dots \sum_{i_2=2}^{i_3} \space \sum_{i_1=1}^{i_2}$
It guess it is wrong to write it as a product of summations like $\displaystyle \large... |
H: Why is the harmonic function $ \log(x^2 + y^2) $ not the real part of any function that is analytic in $ \mathbb{C} - \{0\} $?
I would like to show that $ \log(x^2 + y^2) $ is not the real part of any analytic function in $ \mathbb{C} - \{0\} $ A similar question can be found here, but I don't think this argument i... |
H: Can we conclude that this summation is positive?
The summation is:
$$S_n=n\sum_{i=0}^n x_i^2- \left ( \sum_{i=0}^nx_i\right)^2$$
where $n>1$ and $x_1,x_2,\ldots,x_n\in \mathbb{R}$.
I'm trying to prove that if $x_i\neq x_j$ for $i\neq j$ then $S>0$.
If $n=2$, it's easy:
$$ S_2=(x_1-x_2)^2>0$$
But if $n\geq 3$ seems... |
H: Verifying the trigonometric identity $\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$
I have the following trigonometric identity
$$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$
I've been trying to verify it for almost 20 minutes but coming up... |
H: Every perfect cube is the difference of two perfect squares?
How would you prove this without induction? I know that one easy way is using Al Kharchi's principle (namely that $1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2$), but are there other ways? Thanks!
AI: If $a^2-b^2=n^3=n^2\cdot n$
we can set $a+b= n^2$ and $a-b=n$
s... |
H: Limits preserve weak inequalities
Does anyone have a quick way to show that if $f(x)$ tends to a limit A as $x$ tends to $0$, and $f(x) \geq 0$ then $A \geq 0$?
Thanks
AI: $|f(x)-A|<\epsilon \Rightarrow A-\epsilon < f(x) < A+\epsilon$
If $A<0$ then choose $\epsilon$ less than $|A|$ which forces $f(x)<0$. |
H: summary of a summary
At the printing company I work for, we have different materials that we print on; part numbers for those materials are assigned based on the unique supplier, material type, and material widths (so the same supplier could send us two different widths of the same material type, and they'd get dif... |
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