Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is there a minimum number of vertices that a graph must have to be of a given genus? It seems intuitive that the higher the genus of a graph the more vertices and edges it would have to have. Is it known if the minimum number of vertices must be larger for graphs of higher genus?
| There is at least an upper bound on the genus in terms of the number of vertices and edges in the graph. Let $\gamma(G)$ be the genus of a graph $G=(V,E)$ and $\beta(G)=|E|-|V|+1$ be its Betti number. Then the following holds:
$$\gamma(G)\leq\left\lceil\frac{\beta(G)}{2}\right\rceil$$
For a reference on how to prove th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Simplest way to get the lower bound $\pi > 3.14$ Inspired from this answer and my comment to it, I seek alternative ways to establish $\pi>3.14$. The goal is to achieve simpler/easy to understand approaches as well as to minimize the calculations involved. The method in my comment is based on Ramanujan's series $$\frac... | Do you want a solution that is a compromise between calculation efficiency and ease of understanding? This story might help.
Many, many years ago in school, I was introduced to Fortran and had limited access to a computer. It was physically large but very low power by today's standards. I knew the famous $\frac{\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
"answer_count": 11,
"answer_id": 9
} |
Center of mass of triangle created by parabola and two lines The parabola $y^2 = 3x$ is given.
Two perpendicular straight lines are drawn from the origin point, intersecting the parabola in points P and Q.
Find equation (in cartesian form) of the set of centers of mass of all triangles OPQ (where O is the origin point... | Step 1: calculate points $P$ and $Q$. You already have the equation for those lines, make them intersect with the parabola. Assume $a>0$. Note that $x>0$
$$ax=\sqrt{3x}$$ yields $$x=\frac{3}{a^2}$$ and $$y=\frac{3}{a}$$
The second equation is $y=-x/a$. You square it and get $$x=3a^2$$ and $$y=-3a$$
Step 2: The center o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Proof explanation for Order of Subgroups, by Fraleigh Theorem: $G$ a cyclic group, $H < G$, $H = \langle a^s \rangle$ then $|H|=\frac{G}{\operatorname{gcd}(s,n)}.$
Proof:
Let $|H|=| \langle a^s \rangle |=m$ for $m$ smallest such that $(a^s)^m=e$
So $(a^s)^m=e$ if and only if $n|ms$.
How do we find $m$ smallest satisfy... | The proof in Fraleigh uses Bezout's identity to show that $n∣ms$ iff $\frac{n}{d}\mid m\frac{s}{d}$ where $d=\gcd(s,n)$, so that $m=\frac{n}{d}$ is the smallest such $m$.
To see this another way, and perhaps see why the $\gcd$ is relevant, clearly we want the smallest $m$ such that $n\mid ms$. Now, $s$ contains some o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Shifting of Line Segment
$L_1$ and $L_2$ are skew lines in $3$-dimensional space.
$A_1 C \mathbin{\bot} L_1,L_2$. If line segment $AB$ is shifted parallelly, such that $A$ moves along $L_1$ and reaches $A_1$ while $B$ reaches $B_1$, then prove that $B_1 C \mathbin\bot A_1 C$.
It seems really int... | Here's a vector solution, but coordinate-free, and really simple . . .
The advantages? It works in $n$ dimensions, for all $n \ge 2$, and requires minimal visualization.
From $A_1 C \mathbin{\bot} L_1,L_2$, we get
$$(\mathbf{A_1}-\mathbf{C})\cdot(\mathbf{A}-\mathbf{A_1})=0\tag{eq1}$$
$$(\mathbf{A_1}-\mathbf{C})\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving limits without limit theorems I have to prove the following limit without using any limit theorems. I can only do so by using the Archimedean Proprety and the definition of a limit.
I have to prove:
$$\lim_{n\to\infty} \frac{n^2 - 2}{3n^2 + 1} = \frac13$$
| I'll help get you started. You want to show that N>n => |(n^2 - 2)/(3n^2 + 1) -1/3| < epsilon. (Definition of convergence of a sequence).
First, get a common denominator.
|-7/(9n^2+3)|< epsilon
Then you're gonna want to solve for n in terms of epsilon and set N equal to the value you get for n.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Is a certain inverse image under an equivariant map a submanifold? Suppose $G$ is a Lie group and $K$ a maximal compact subgroup of $G$, and suppose $G$ acts smoothly and properly on a manifold $X$.
Question 1: Suppose we are given a $G$-equivariant smooth map $f:X\rightarrow G/K$. Is $S:=f^{-1}(eK)$ necessarily a sub... | Suppose $f:X\to G/K$ is smooth and $G$-equivariant, with $G$ a Lie group and $K$ a closed subgroup. Then, in particular, $f$ must be a submersion: if $p\in X$ and $g\in G$, then $f(gp)=gf(p)$, so $f(\exp(t\xi)p)=\exp(t\xi)f(p)$.
Thus, $f^{-1}(gK)$ is a submanifold for every $gK\in G/K$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that MNPQ and ABCD have the same centroid. If ABCD is a quadrilateral and M,N,P,Q are exterior points such that ABM, BNC, CPD, DQA are equilateral triangles. Show that MNPQ and ABCD have the same centroid.
I tried to solve it by vectors but I don't know how can I use that the triangles are equilateral.
| Let $\varepsilon = e^{-i{\pi\over 3}} = \cos {\pi\over 3} -
i\cdot \sin{\pi\over 3}$ and let $G$, $G'$ be gravity centers of $ABCD$
and $MNPQ$, so
$$ G ={1\over 4}(A+B+C+D)$$ and $$G'= {1\over 4}(M+N+P+Q)$$
Now since we get the vector $\vec{AM}$ with rotation of $\vec{AB}$ around $A$ for angle $-{\pi\over 3}$ we have (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Natural density and Poincare's recurrence Let $(X,\mathcal{B},\mu,T)$ be a dynamical system and let $A \in \mathcal{B}$ such that $\mu(A)>0$ and $\forall x \in X$ we define $$L_x=\{n \in \Bbb{N}|T^nx \in A\}$$
Prove that $\mu(\{x:\bar{d}(L_x)>0\})>0$ where $$\bar{d}(L_x)=\limsup_n \frac{|L_x \cap \{1,2...n\}|}{n}$$.
My... | I am also assuming $\mu$ is a probability measure. For a measurable set $A$ with positive $\mu$-measure, define $A_0$ to be the set of points $x\in A$ such that $\overline{d}(L_x)=0$. Let $\chi_{A_0}(.)$ be the characteristic function over the set $A_0$ and for any $x\in X$, consider
$$
F^+(x) = \limsup_{n\to +\infty} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How do I show that formal logarithm is the inverse of the formal exponential? Let $A$ be a unital commutative and associative $\mathbb{Q}$-algebra.
Define $exp(f):=\sum_{n=0}^\infty \frac{f^n}{n!}$ for each $f\in XA[[X]]$.
Define $log(f):=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} (f-1)^n$ for each $f\in 1+XA[[X]]$.
Hence,... | Here is one possible argument. One way to define the exponential $\exp(x)$ as a formal power series is that it is the unique formal power series $f(x)$ (over any commutative $\mathbb{Q}$-algebra) satisfying $f(0) = 1$ and
$$f'(x) = f(x).$$
Repeatedly differentiating this identity easily gives $[x^n] \exp(x) = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Prove $1+2+3+...+n<=n^2$ for all n∈N
Prove $1+2+3+...+n<=n^2$ for all n∈N
This is what I have so far. Not sure what to do now.
| One potential stumbling block is that your "in other words" isn't quite complete. What "$P(k)$ implies $P(k+1)$" really says is that if
$$
1+2+\cdots+k \leq k^2
$$
then
$$
1+2+\cdots+k+(k+1) \leq (k+1)^2
$$
In other words (!), you use the premise that $1+2+\cdots+k \leq k^2$ as the starting point. If you add $k+1$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Annoying primes first of all, I'm no mathematician at all. I was just playing with prime numbers and ended up with this list:
2 = 2¹
3 = 3¹
5 = 5¹
7 = 3¹ + 2²
11 = 2¹ + 3²
13 = 13¹
17 = 5¹ + 2² + 2³
19 = 2¹ + 3² + 2³
23 = 11¹ + 2² + 2³
29 = 17¹ + 2² + 2³
31 = 19¹ + 2² + 2³
37 = 37¹
41 = 5¹ + 3² + 3³
43 = 7¹ + 3² + 3³
4... | I can only find 6 annoying primes: 2, 3, 5, 13, 37, and 61.
You gave these and three other examples, but they don’t hold:
$$29 = 17 + 2^2 + 2^3$$
$$67 = 7 + 2^2 + 2^3 + 2^4 + 2^5$$
$$97 = 13 + 3^2 + 3^3 + 2^4 + 2^5$$
By general density arguments one would expect:
Conjecture: There are only finitely many annoying primes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 1,
"answer_id": 0
} |
Let $p>3$ be a prime number. Prove that for every $a$, $1$ $\lt$ $a$ $\lt$ $p-1$ Let $p>3$ be a prime number. Prove that for every $a$, $1$ $\lt$ $a$ $\lt$ $p-1$, there is a unique b $\neq$ a , $1$ $\lt$ b $\lt$ $p-1$ such that $ab$ $\equiv$ (1 mod p)
I started off with a proof of contradiction where suppose b is not u... | Let $P= \{0, 1, \dots, p-1\}$ and consider $\mu: P \to P$ given by $x \mapsto ax \bmod p$. Then, $\mu$ is injective (*) and so is surjective, because $P$ is finite. Therefore, $\mu$ is bijection and there is a unique $b \in P$ such that $1=\mu(b)=ab \bmod p$.
This defines a map $\iota: P \to P$ given by $a \mapsto b$ s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Mean duration of the renewal intervals $\textbf{Q}$: Let $\{X_n; n = 0, 1,...\}$ be a two-state Markov chain with the transition probability matrix
$$\begin{bmatrix}
1-a & a\\
b & 1-b
\end{bmatrix}$$
where state $0$ represents an operating state of some system, while state 1 represents a repair state. We assume that th... | Let $S_0 = \inf\{n>0:X_n=0,X_{n-1}=1\}$ and $S_{k+1} = \inf\{n>S_k:X_n=0,X_{n-1}=1\}$ for $k\geqslant 1$. Then $\{S_n:n\geqslant0\}$ is a renewal process. Let $X\sim\mathrm{Geo}(a)$ and $Y\sim\mathrm{Geo}(b)$ distribution, then $S_{k+1}-S_k\stackrel d=X+Y$, so
\begin{align}
\mathbb P(S_{k-1}-S_k=m) &= \mathbb P(X+Y=m)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How to apply Kuratowski's theorem here
I'm unsure which subgraph to find and which of $K_{3,3}$ or $K_5$ I should compare it to. First this looks like $K_6$. If it was $K_6$, I could simply remove one of the vertices to get $K_5$ and I would be done.
However, there is an extra vertex $I$ in the centre. I thought of re... | $A$ is joined to $C,E,F$, also $D$ is joined to $C,E,F$, and $B$ is joined to $C$ and $F$, and to $E$ via $I$, so there's also a $K_{3,3}$ there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove $1+{x\over 2}-{x^2 \over 8}<\sqrt{1+x}$ for all $x>0$? How to prove $1+(1/2)x-(1/8)x^2<\sqrt{1+x}$ for $x>0$ by Taylor Expansions up to $n$ order or Mean Value Theorem? I tried to apply MVT on $\sqrt{1+x}$ and get $\displaystyle \sqrt{1+x}=1+\frac{1}{2\sqrt{1+\xi}}x$ for $\xi\in (0,x)$. How to do next?
| Let $f(x)=\sqrt {1+x}\;.$ For $x>0$ we have $$f(x)=f(0)+xf'(0)+x^2f''(y)/2$$ where $0<y<x.$ We have $f(0)=1$ and $f'(0)=1/2.$ We have $$f''(y)=-\frac {1}{4(1+y)^{3/2}}>-\frac {1}{4} .$$ So $\sqrt {1+x}=f(x)=1+x/2-x^2f''(y)/2>1+x/2-x^2/8.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Elegant proof of combinatorial statement: at least $l-k+1$ children who were in larger teams There are $n$ children, who play a game where in round $1$ they spilt up in $k$ teams (every team non-empty, pairwise disjoint). After a while it gets boring, so they form new teams to play round $2$; this time, they split up i... | Say a child in a team of size $m$ takes up $1/m$ of a team. Say child $i$ takes up $x_i$ of a team in round $1$ and $y_i$ of a team in round $2$. Now we need to show there are at least $l-k+1$ values of $i$ for which $y_i>x_i$.
But $\sum_ix_i=k$ and $\sum_iy_i=l$, so $\sum_i(y_i-x_i)=l-k$. Each term in this sum is stri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Solution to second order non constant differential equation without first derivative Cheers!
So I am studying the book ‘Introduction to quantum mechanics’ by David J. Griffiths for my introductory course of quantum mechanics. On page 51 at the bottom it introduces the differential expression:
\begin{equation}
\Phi’’ = ... | For solving $y''-(\xi^2-K)y=0$ we have solution
$$y=Ae^{\sqrt{\xi^2-K}}+Be^{-\sqrt{\xi^2-K}}~~~~~,~~~~~\xi^2-K>0$$
with $\xi>>K$
$$\sqrt{\xi^2-K}=\sqrt{-K}\sqrt{1-\dfrac{\xi^2}{K}}=\sqrt{-K}\left(1-\dfrac{\xi^2}{K}\right)^\frac12\approx\sqrt{-K}\left(1-\frac12\dfrac{\xi^2}{K}\right)\approx\sqrt{-K}\left(-\frac12\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solving, a Square with broken diagonal: $AB=x$? I am reading a geometry lecture note and it was given this figure below.
Where ABCD is square of size $AB=x$. $BE =12$, $EF=3$ and $FD=9$.
Please help me to find the value of $x$.
I do not know how to start. So far I know that $BD =x\sqrt 2.$
| The hint:
Use the following.
$$\measuredangle DFB=90^{\circ}+\arctan4,$$
$$DF=\sqrt{153},$$
$$FB=9$$ and
$$DB=x\sqrt2.$$
I got $x=15$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Choosing $x$, $y$, $z$ parts in a pumping lemma $w$ string I want to proof that $L = \left\{u0v \mid u, v \in \{0, 1\}^* \land \#_1(u) = \#_0(v) \right\} $ is not regular. But my understanding of the pumping lemma is somehow not bulletproof, so I'm not sure if I'm right in what I'm doing.
I chose $w$ to be $1^k00^k = ... | Inside any substring of length $>k$ you can find a pumpable substring. So taking that substring inside the $0$ part is OK. You should try to understand the pumping lemma, rather than just applying it, the idea is simple: if you go $k+1$ times to $k$ places, you have to go to the same place at some point (like bar hoppi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is the average of two endpoint's instantaneous rates of change always the same as the average rate of change of the interval on a parabola? Is the average of two endpoint's instantaneous rates of change always the same as the average rate of change of the interval on a parabola? If not, which additional circumstances w... | I'll give you a hint, but you need to know some calculus to understand the problem itself.
Recall that average value of a function $f$ on the interval $[a,b]$ is: $$\frac{f(b)-f(a)}{b-a}$$
And the average of the two endpoints instantaneous rate of change is given by:
$$\frac{f'(b) + f'(a)}{2}$$
Where $f'(x)$ is the der... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If 9 points are chosen from within the rectangle, explain why two of the points must be at most 18cm away from each other. I've been revising for an exam and the following question has stumped me. I thought it might involve pythagorean theorem but after trying it out it doesn't seem to...
"A rectangle has width 6 cm a... | Divide the rectangle into squares $3cm\times 3 cm$, there are $2 \times 4 = 8$ of them. Two of the points will land in the same square, so the distance between them is at most the diagonal of the square $= 3\sqrt{2} cm = 4.24\ldots cm$.
Obs: Maybe the problem stated $\sqrt{18} cm$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that if $f$ is increasing on $[a, b]$ and satisfies the intermediate value property, then $f$ is continuous on $[a, b]$ I know this question has been asked before but I feel like my approach to solving the problem is "different" (EDIT: turned out to be different because it's wrong!)
Since $[a,b]$ is closed and bou... | Often it is better just to go back to the $\epsilon$-$\delta$ definitions. I will show that $f$ is right continuous first. Left continuity should be easy enough for you to show yourself. Let $\epsilon > 0$. For a fixed $x \in [a,b)$, let $\epsilon' = \min\{\epsilon, f(b)-f(x)\}$, then
$$f(x) < f(x)+\epsilon' \leq f(b)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Why does $1+\sqrt{6-2\sqrt{5}}=\sqrt{5}$? Actually, I've already found a clumsy proof of my statement, but I want to know if there's something deeper going on here. Specifically, Mathematica's "Simplify" function doesn't seem to know how to reduce this, which caused me hours of woe before I realized that Mathematica wa... | $$\sqrt{6-2\sqrt{5}}=\sqrt{5}-1$$
$$6-2\sqrt{5}=\left(\sqrt{5}-1\right)^2$$
$$6-2\sqrt{5}=6-2\sqrt{5}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Factored $z^4 + 4$ into $(z^2 - 2i)(z^2 + 2i)$. How to factor complex polynomials such as $z^2 - 2i$ and $z^2 + 2i$? I was wondering how to factor complex polynomials such as $z^2 - 2i$ and $z^2 + 2i$?
I originally had $z^4 + 4$, which I factored to $(z^2 - 2i)(z^2 + 2i)$ by substituting $X = z^2$ and using the quadra... | You can also go brute force, it sometimes has advantages when the polar form is not sympathetic.
$z^2=(x+iy)^2=2i\iff x^2+2ixy-y^2=2i\iff\begin{cases}x^2-y^2=0\\xy=1\end{cases}\iff z=\pm(1+i)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
About Riesz's lemma Read about Riesz's lemma and its proof as given in the Functional Analysis text (2nd ed) by Taylor.
In the proof it's said that $X$ is the normed linear space. $X_0$ is its closed and proper subspace.
$x_1\in X-X_0$.
$x_0,x\in X_0$.
$h=||x_1-x_0||^{-1}$.
Then it's said that
$$h^{-1}x+x_0\in X_0$$
H... | Since $h^{-1}$ is just some scalar, $h^{-1}x+x_0$ is just a linear combination of $x$ and $x_0$. Since $x,x_0\in X_0$ and $X_0$ is a linear subspace, this means $h^{-1}x+x_0\in X_0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The sum of the cubes of three consecutive integers is divisible by $9$ The question is in the title and my try comes here:
Let $n\in\mathbb{Z}$, then we have to look at
$$
(n-1)^3+n^3+(n+1)^3=3n(n^2+2)
$$
so if we can show that $n(n^2+2)$ is divisible by $3$ we are done.
So I divide it up into three cases
*
*$n\equi... | Yes, your way is correct. An alternative method (which I leave you to determine whether it is more elegant or not):
$$(n+1)^3 +n^3 +(n-1)^3 = 3n(n^2+2) = 3n(n^2-1+3)\\=3n((n-1)(n+1)+3)=3(n-1)n(n+1)+9n$$
Of the consecutive numbers $(n-1), n, (n+1)$ one is divisible by $3$, which leaves that the term is divisible by 9.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Miscellenous binomial expansion Prove That the sum of the series: $\sum_{r=0}^{10} (-1)^{r}$ $\binom{10}{r}$ [$\frac{1}{2^{r}}+
\frac{3^{r}}{2^{2r}}+ \frac{7^{r}}{2^{3r}} +\frac{15^{r}}{2^{4r}}$ ....upto m terms =$\frac{2^{10m}-1}{2^{10m}(2^{10}-1)}$
I tried to expand this term but getting more and more complicated
| Hint:
$$\sum_{r=0}^u\binom ur(-1)^r\left(\dfrac{2^n-1}{2^n}\right)^r=\sum_{r=0}^u\binom ur\left(\dfrac{1-2^n}{2^n}\right)^r=\left(1+\dfrac{1-2^n}{2^n}\right)^u=?$$
Here $n=1,2,\cdots,m$ and $u=10$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What's the difference between "function" and "relation" in logic? What's the difference between "function" and "relation" in logic?
Why do we cosider them apart?
I think every relation expresses a function...
| In FOL (first-order logic) a function symbol has terms (i.e. "names") as input and output.
For every pair $n,m$ of natural numbers, $+(n,m)$ (usually writteh: $n+m$) denotes a number.
Predicate (or relation) symbols have terms as input and produce sentences.
For every pair $n,m$ of natural numbers, $<(n,m)$ (usually wr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Proof that the number $e$ exists based on the following definition My book defines the number $e$ as follows (Simmons' Calculus With Analytic Geometry, 2nd edition, pg. 265):
$e$ is the number for which $\lim_{h\to0}\frac{e^h-1}{h}=1$
However, it does not provide a proof that such a number exists. How can we prove th... | As you have stated that your book defines $e$ as the number for which
$$\lim_\limits{h\to 0}\frac{e^h-1}{h}=1$$
I must say that, the definition is not mathematically sound and circular in nature and also seems confusing as to whether it uses the exponential function to define $e$.
However, what I think it intended to s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Determine if $\int_{0}^{1}\sqrt{x}\ln{x} \ dx$ is convergent I'm so utterly lost when it comes to these improper integrals. Everytime i feel like I got the hang of it, I always get the wrong answer.
I can clearly see that the function is not defined for $x=0$. So I need to figure out what happens there by examining the... |
I thought it would be instructive to present a way forward that relies on elementary, pre-calculus tools only. To that end we proceed.
To show that $\sqrt x\log(x) \to 0$ as $x\to 0$, we simply exploit the inequality
$$\frac{x-1}{x}\le\log(x)\le x-1 \tag1$$
which I showed using only the limit definition of the exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Proof that $ 2\sqrt{x_1x_2} \le x_1+x_2 $ We have
$$ x_1, x_2,...,x_n \ge 0 $$
and $$ P(n): x_1x_2....x_n \le \left(\frac{x_1+...+x_n}{n}\right)^{n} $$
I have to prove that P(2) is valid.
$$x_1 x_2 \le \left(\frac{x_1+x_2}{2}\right)^{2} $$
I don't know how to achieve this, this is what I tried so far:
$$ \sqrt{x_1x_2}... | The important parts about the proof are already given in the answers up there. I want to share something different, geometrical actually (not that it works for a proof though) Consider a circle $OAB$ (the center is $T$) with a diameter $|AB|$ now inside the circle choose a point $P$ ($P\in|AB|$)such that $|AP|=x_1$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Electrical Potential Here is my question
A proton (mass $\text {m}$, charge $+e$) and an alpha particle (mass
$4\text {m}$, charge $+2e$) approach one another with the same initial
speed $\text {v}$ from an initially large distance. How close will
these two particles get to one another before turning around?
Th... | Read up on elastic collision. When particles collide, total momentum and total kinetic energy are both conserved. Easy equations to work with. You will have two more equations to work with from this setup.
I think you will find your scenario in this article where they have worked on it. It is not as complex as it looks... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Algebraic and geometric multiplicities of eigenvalues of a $3 \times 3$ matrix "Let A = \begin{pmatrix}-1&-1&-2\\ \:2&2&1\\ \:6&2&6\end{pmatrix}
Find the characteristic polynomial of A.
Find the distinct eigenvalues of A and their respective algebraic and geometric multiplicities."
So I calculated the value of the cha... | The numbers you are quoting are the algebraic multiplicity; i.e., $\lambda$ has algebraic multiplicity $n$ when the factor $(x - \lambda)$ appears exactly $n$ times in the factored characteristic polynomial.
The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of the eigenspace corresponding to $\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Prove that $\sqrt{3} \notin \mathbb{Q}(\sqrt2)$ I'm trying to prove that $\sqrt{3} \notin \mathbb{Q}(\sqrt2)$
Suposse that $\sqrt{3}=a+b\sqrt{2}$
$\begin{align*}
\sqrt{3}&=a+b\sqrt{2}\\
3&=(a+b\sqrt{2})^2\\
3&=a^2+2\sqrt{2}ab+b^2\\
(3-a^2-12b^2)^2&=(2\sqrt{2}ab)^2\\
9-6a^2-12b^2+4a^2b^2+a^4+4b^4&=8a^2b^2
\end{align*}$
... | Note that $1$ and $\sqrt{2}$ is a basis for $\mathbb{Q}(\sqrt{2}).$
Hence from
$$3=a^2+b^2+2\sqrt{2}ab$$
We have $a^2+b^2=3$ and $2ab=0$.
From there, you should be able to see a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
why does $2^{-n}n^{1000} $ converge by the limit comparison test? the series $\displaystyle\sum\limits_{n=1}^{\infty} 2^{-n}n^{1000}$ converges by the comparison test.
$2^{-n}\leqslant \displaystyle\frac{c}{n^{1002}}$
$\displaystyle\sum\limits_{n=1}^{\infty} 2^{-n}n^{1000} \leqslant \displaystyle\sum\limits_{n=1}^{\... | Because for large $n$, $2^{-n}$ decreases rapidly as compared to $\frac{1}{n^p}$ for any $p>0$. This you can prove by using Binomial theorem. So there exists $n_0$ and $c$ such that $$2^{-n}<\frac{c}{n^p}, ~~\text{for } n>n_0.$$
Now if $a_n\leq b_n$, and if $\sum b_n$ is divergent, by Comparison test nothing can be de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For any additive function, $f: (\mathbb{Q}, +)\to (\mathbb{Q}, +)$, why does $f(nx) = nf(x)$? For the homomorphism, $f: (\mathbb{Q}, +)\to (\mathbb{Q}, +)$, where
$\mathbb{Q}$ is the set of rational numbers.
Show that for all $x\in\mathbb{Q}$ and for all $n\in\mathbb{Z}$, we have $f(nx) = nf(x)$
I can't figure out how ... | We have that $f(nx)=f(\underbrace{x+x+\cdots +x}_{n\:\text{times}})=\underbrace{f(x)+f(x)+\cdots+f(x)}_{n\:\text{times}}=nf(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving difference quotient is equal to derivative at 0 everyone, thankfully I was able to figure out parts 1 and parts 3. However, I am stuck on part 2. The conditions somewhat confuse me. All I so far was (first steps are similar to part 1):
Start with:
$f(a_n)=f(0)+a_n(f'(0)+c_n $
$f(b_n)=f(0)+b_n(f'(0))+d_n $... | Note that $$0\leq\left|\frac{d_n-c_n} {b_n-a_n} \right|\leq\frac{|d_n|+|c_n|} {b_n-a_n} =\left(\frac{|d_n|} {b_n} +\frac{|c_n|} {b_n} \right) \frac{b_n} {b_n-a_n} \leq \left(\frac{|d_n|} {b_n} +\frac{|c_n|} {a_n} \right) M$$ and now the last expression tends to $(0+0)M=0$ so that by squeeze theorem $|d_{n} - c_n|/(b_n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counting techniques to find all nonisomorphic graphs with six vertices, all having degree $2.$ First of all, the graphs considered are undirected & parallel edges and loops are allowed.
My attempt : By handshaking lemma, $$\text{Sum of degrees of all vertices} = 2 (\text{Number of edges})$$
Thus we get $\frac {2+2+2+2+... | Each one of those isomorphism classes can be represented as an integer partition of 6, that is, a way of writing positive numbers which add up to 6, where the order doesn't matter. For example your 3(i), 3(ii) and 3(iii) correspond to $2 + 2 + 2, 4 + 1 + 1$ and $3 + 2 + 1$ respectively. There is a well-understood the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How can I prove the convexity of $f(x)=x^{-\frac{1}{3}}$ by using the definition I'd like to prove that the function $f(x)=x^{-\frac{1}{3}}$, with $x> 0$ is convex. Actually, I already know it's convex because I have studied its derivatives but I'd like to give a more "formal" prove by using convex definition, that is:... | Hint:, As $t \mapsto t^3$ is increasing, we may cube both sides and replace $x, y$ with $a^3, b^3$ to equivalently prove:
$$ \left(\frac{\lambda}a + \frac{1-\lambda}b \right)^3 (\lambda a^3 + (1-\lambda) b^3) \geqslant 1 \tag{$\star$}$$
which follows from Hölder's inequality ...
--
P.S. Hölder's inequality for our cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Cubed exponent equation
$$\left(2 · 3^x\right)^3 + \left(9^x − 3\right)^3 = \left(9^x + 2 · 3^x − 3\right)^3$$ Solve the equation
I got the answer to the problem, in which I evaluated the whole expression (which was quite hard) which was $0$ and $1/2$, is there a way to solve this problem in a less tedious way?
| Using $t:=3^x$ and following the factorization by @Arthur ($(a+b)^3-a^3-b^3=3ab(a+b)$), the equation is equivalent to
$$t(t^2-3)(t^2+2t-3)=t(t-\sqrt 3)(t+\sqrt 3)(t-1)(t+3)=0.$$
Only the positive $t$ yield a solution and we immediately have
$$x=\frac12\text{ or }x=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
} |
Infinite sum of squares It is known that $1 + \frac{1}{4}+\frac{1}{9}+\frac{1}{16} + \cdots = \frac{\pi^2}{6}$
. Find the sum
$1 + \frac{1}{9}+\frac{1}{25} + \frac{1}{49} + \cdots$.
What method can we use to answer this? I tried expressing the 2nd equation into 2 fractions which contain the first summation but i could... | $S = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} +... $
Take 1/4 common from the terms whose denominator is even.
$S = 1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{4} ( 1 + \frac{1}{4} + \frac{1}{9} + ... ) $
$S = 1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{4} S$
$ 1 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find Matrix A by certain operations Given as such:
$4(A^T+2I)^{-1} =$ \begin{bmatrix}1&1\\-3/2&1/2\end{bmatrix}
Now to find Matrix A, would I have to inverse the matrix on the RHS to make it equal to A on the LHS and perform the opposite operations? A little stumped on the operations necessary to get the right result.
... | Hints:
Divide both sides by $4$ to get
$$(A^T+2I)^{-1}=\left[\begin{array}{cc}
1/4 & 1/4 \\
-3/8 & 1/8 \\
\end{array}\right]$$
Then take the inverse of both sides:
$$A^T+2I=\left[\begin{array}{cc}
1/4 & 1/4 \\
-3/8 & 1/8 \\
\end{array}\right]^{-1}.$$
I'll leave it to you to find the inverse.
Then subtract $2I$ from bot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many primes do I need to check to confirm that an integer $L$ is prime? I recently saw the 1998 horror movie "Cube", in which a character claims it is humanly impossible to determine, by hand without a computer, if large (in the movie 3-digit) integers are prime powers, i.e. they are divisible by exactly one prime ... |
Is my reasoning valid, and, is it possible to reduce the number of primes you'd need to check even more?
Yes, your reasoning is valid. This method of trying to deduce whether a number $N$ is prime by testing for prime factors up to $\sqrt N$ is called trial division.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 2
} |
Convergence of $\sum_{n=0}^{\infty} a_n \overline{z}^n$ based on $\sum_{n=0}^{\infty} a_n z^n$. Let $\sum_{n=0}^{\infty} a_n z^n$ be a convergent series, where $\{a_n\}_{n \in \mathbb{N}} \in \mathbb{R}$ and $z \in \mathbb{C}$. Is $\sum_{n=0}^{\infty} a_n \overline{z}^n$ a convergent series?
This question popped into m... | Let $z=\displaystyle \sum_{n=0}^{\infty} a_nz^n$. By definition we have that for every $\varepsilon >0$, there exists $N \in \mathbb{N}$ such that $$\left \Vert \sum_{n=0}^m a_nz^n-z \right \Vert <\varepsilon, \: \forall m \geq N.$$ Note that $$\left \Vert \sum_{n=0}^m a_n\overline{z^n}-\overline{z} \right \Vert=\left\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Prove for all all $x\in\mathbb{R}: \exp(x-1) \geq x$ i just tried to solve this question, which is a small part of a bigger one.
Prove for all all $x \in \mathbb{R}: \exp(x-1) \geq x$
My first attempt was to simplify:
$$ e^{x-1} \geq x $$ $$ \Leftrightarrow \ln(e^{x-1}) \geq \ln(x) $$ $$ \Leftrightarrow x-1 \geq \ln(x)... | the answer Michael gave is a good answer but if you want different way,
$$e^{x-1}=x\implies x=1$$ now because both $x$ and $e^{x-1}$ are continuous you just need to check one value of $x$ that is above $1$ and one that is less. for example:$$\text{for $x<1,\,x=0$},\,e^{0-1}={1\over e}\gt0\\\text{for $x>1,\,x=2$},\,e^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does there exist a smooth approximation of $x \bmod y$? I'm looking for a function $m(x,y)$ that smoothly approximates $x \bmod y$, and I'm assuming there would be some $n$ or $\varepsilon$ in the body of $m(x,y)$ that defines the degree of approximation such that as $n$ goes to infinity or $\varepsilon$ goes to zero, ... | One approach I found was to find a smooth floor function, and then simply apply it to the equation:
$$
x \bmod y=x-{\lfloor}\frac{x}{y}{\rfloor}y
$$
And a definition for the smooth floor function I derived is:
$$
{\lfloor}\frac{x}{y}{\rfloor}=-\frac{1}{2} + x + a ln(1/a) -\frac{i}{2 \pi} \left[ln\left(1-e^\left(-2 i \p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Number of conjugates of $S_3$ in $S_4$ How to quickly deduct there are 4 conjugates of $S_3$ in $S_4$? Since conjugate subgroups are isomorphic, we can have at least 4 conjugates of $S_3$ in $S_4$. But I'm not sure why there aren't more.
*I'm aware there are posts that address similar questions but most involve "label... | Let $G = S_4$ act on its subgroups by conjugation. The conjugates of the standard copy of $S_3$ constitute an orbit for this action. The size of the orbit is the index of the stabilizer subgroup. The stabilizer $N$ is what's called the normalizer. It is the set of $\pi \in S_4$ such that
$\pi S_3 \pi^{-1} = S_3$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I prove that $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_n$ is a homomorphism? I was wondering how one shows that $\phi(m):\mathbb{Z}\rightarrow\mathbb{Z}_n$ is a homomorphism? I know that we define $m:=qn + r$, but I don't really know how to continue from there.
| You have $m=qn + r,$ and there you should mention that $r\in\{0,1,2,\ldots,n-1\}.$
The homomorphism would be $\varphi(m) = r.$
Showing that that is a homomorphism means showing that $\varphi(m_1+m_2) = \varphi(m_1)+\varphi(m_2).$
That means if $m_1 = q_1 n + r_1$ and $m_2 = q_2 n + r_2$ and $m_1+m_2 = q_3 n + r_3$ then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If a vector space has a countable basis, can we construct it? Given that we cannot (?) exhibit a basis for a vector space of all real sequences (certainly that basis would be uncountable), but there is a countable basis for polynomials, i.e. $1,x,x^2,x^2,\ldots$, I would like to know,
If a vector space $V$ has a counta... | It is known that there is a computable vector space over $\mathbb{Q}$ with no computable basis. This also shows that we cannot prove constructively that every countable vector space over $\mathbb{Q}$ has a basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determine constant c so that g(x,y) is continuous at every point $$g(x,y)=\begin{cases} \frac{x^3+xy^2+2x^2+2y^2}{x^2+y^2} & \text{if} & (x,y) \neq (0,0) \\
c & \text{if} & (x,y) = (0,0) \end{cases}$$
Should I set the first function equal to c and then solve using polar coordinates?
| Hint. By using polar coordinates one gets, for $(x,y)\ne (0,0)$,
$$
g(x,y)=\frac{x^3+xy^2+2x^2+2y^2}{x^2+y^2}=\frac{r^3\cos^3 \theta+r^3\cos\theta \sin^2\theta +2r^2}{r^2},\quad r\ne0,
$$ that is
$$
g(x,y)=r\cos^3 \theta+r\cos\theta \sin^2\theta +2,\quad r\ne0,
$$ then this tends to $2$ as $r \to 0^+$.
Can you take it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Computing: $ \lim_{(x,y)→(0,0)}\frac{(x^5+y^5)\ln(x^2+y^2)}{(x^2+y^2)^2} $ $$
\lim_{(x,y)→(0,0)}\frac{(x^5+y^5)\ln(x^2+y^2)}{(x^2+y^2)^2}
$$
The answer is 0. Cannot seem to understand how the answer is 0.I know that the first part is 0 but I'm confused on how to deal with the natural log?
Why is squeeze theorem not a g... | Hint. Note that by letting $x=\rho\cos(\theta)$ and $y=\rho\sin(\theta)$, we have that as $(x,y)\to(0,0)$ then $\rho\to 0$ and
$$0\leq \left|\frac{(x^5+y^5)\ln(x^2+y^2)}{(x^2+y^2)^2}\right|\leq
\frac{(|\rho\cos\theta|^5+|\rho\sin\theta|^5)|\ln(\rho^2)|}{\rho^4}
\leq \frac{\rho^5(1+1)|\ln(\rho^2)|}{\rho^4}={4\rho|\ln(\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
simple sum inequality I have an inequality for sums that I can't proof, although I know it is true.
Let $h_{ij} = h_{ji}$ a real $n\times n$ matrix, and $h_{ijk} = - h_{ikj}$ a real $n\times n\times n$ tensor with $$\sum_{i=1}^n h_{ii} =0, \quad \sum_{j=1}^n h_{jji} = 0, \, \forall \, 1 \leq i\leq n$$
Then there shoul... | Isn't this just an application of the Cauchy-Schwarz inequality in $l^2$?
Let $a = (h_{ii})_i$, $b = (h_{iik})_i$. We have $$\bigg(\sum_{i=1}^n a_i \, b_i \bigg)^2 = (a,b)^2_{l^2} \leq \|a\|_{l^2}^2 \, \|b\|_{l^2}^2 = \sum_{i=1}^n h_{ii}^2 \sum_{i=1}^n h_{iik}^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inequality proof as a part of calculus lesson As part of a calculus lesson I was required to prove that:
(1) if $\ |x-3| < \frac{1}{2},\ $ then $\ \bigg|\displaystyle{\frac{\sin(x^2 -8x+15)}{4x-7}}\bigg| < \frac{1}{2}$
So, by using $|\sin(t)| \le |t|,$ I can prove that:
$$\bigg|\frac{\sin(x^2 -8x+15)}{4x-7}\bigg| \le \... | Because, $x^{2}-8x+15=(x-3)(x-5)\ $, you can start saying that:
First,
$$ \vert{\sin(x^{2}-8x+15)}\vert\leq {x^{2}-8x+15} $$
and then you start like this
$$ \bigg\vert\frac{\sin(x^{2}-8x+15)}{4x-7}\bigg\vert=\frac{\vert{\sin(x^{2}-8x+15)}\vert}{\vert{4x-7}\vert}\leq\frac{\vert x^{2}-8x+15\vert}{\vert{4x-7}\vert}=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Different version of chromatic number I'm wondering whether or not there is a symbol or value such as the chromatic number of a graph that asks
What is the minimum coloring of the graph such that not only adjacent vertices have different colors, but vertices adjacent to a mutual vertices have different colors.
I'm j... | I've seen this called the distance-2 coloring, where distance-k coloring asks for a vertex coloring where every two vertices of the same color are more than k steps apart. It's used in algorithms and computational results, but I'm not familiar with many of its properties.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\mathbb Z_8$ and $\mathbb Z_{24}/\langle 8\rangle$ are isomorphic.
Use the Fundamental Homomorphism Theorem to prove that $\mathbb Z_8$ and $\mathbb Z_{24}/\langle 8\rangle$ are isomorphic.
The following function is a homomorphism from $\mathbb Z_{24}$ to $\mathbb Z_8$:
$\bigl(\begin{smallmatrix}
0& 1 &... | The general idea is correct. To use the first isomorphism theorem here you want to:
*
*Find a homomorphism $\phi:\mathbb{Z}_{24} \to \mathbb{Z}_8$
*Show that $\phi$ is surjective (i.e. $\text{Im}(\phi) = \mathbb{Z}_8$)
*Show that $\ker(\phi) = \langle 8 \rangle$
Then you may conclude that $\mathbb{Z}_{24}/\langl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Computing the radioactive probability integral of a non-uniform system How do I integrate $e^{-y\alpha- x\beta- \gamma\sqrt{xy}}\,dy\,dx$, with $x$ and $y$ from $0$ to infinity, i.e.,
$$\int_0^\infty\int_0^\infty e^{-y\alpha- x\beta- \gamma\sqrt{xy}}\,dy\,dx\tag{1}$$ &
$$N\int_0^\infty\int_0^x e^{-y\alpha- x\beta- \ga... | The change of variable $(2\beta x,2\alpha y)\to(x^2,y^2)$ shows that the integral to be computed is $$\iint_{x>0,y>0}e^{-\alpha y-\beta x-\gamma\sqrt{xy}}\,dxdy=\frac1{\alpha\beta}I\left(\frac\gamma{2\sqrt{\alpha\beta}}\right)$$ where, for every $|w|<1$,
$$I(w)=\iint_{x>0,y>0}e^{-Q_w(x,y)/2}\,xydxdy$$ and $$Q_w(x,y)=x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Nth Branch of Lambert W function I have a program to calculate the primary branch of the Lambert W function, how do I calculate the other branches (based off of the first one if possible)?
Example:
$$W(\ln(2)) = 0.44443609101$$
But (using 1st branch)
$$W_1(\ln(2)) = -1.91415552885386478373 + 4.2929649070568775i$$
How ... | See https://en.wikipedia.org/wiki/Lambertw and get the first reference: Corless et. al. "On the Lambert W function" http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf. In this basic paper the branch $W_k(z)$ is computed in formula 4.20, but I guess in practice the function is calculated with iterations.
You f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the Taylor polynomial of $f(x) = \frac{1}{x}$ with induction So I am asked to find the Taylor polynomial of $f(x) = \frac{1}{x}$ about the point $a=1$ for ever n$\in{N}$, and then use induction to justify the answer.
I got the Taylor polynomial which was simple enough:
$$T_{n}(x)=\sum_{n=0}^{\infty} \frac{f^n(... | The known form of your Taylor polynomial is $P_n(x)=\sum^{n}_{k=0} a_k(x-1)^k$ where the coefficients satisfy $a_k = \frac{f^{(k)}(1)}{k!}$
You managed to find the coefficients $a_k = (-1)^k$. What the question is probably asking is to prove that this is correct using induction.
You would take the base case and show th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Auxiliary Epsilon - is it mathematically rigorous if it's larger than epsilon? Assume $a_n \rightarrow L$. Thus, applying the standard definition, $\forall \epsilon > 0, \exists N\in\mathbb{N},\, \forall n\geq N, |a_n-L| < \epsilon$.
I'm in an introductory real analysis class, and my professor will sometimes use an "au... | Your friend is correct in this case. Keep in mind that in an $\varepsilon-\delta$ proof, we are never really choosing an explicit value for $\varepsilon$; rather, we are saying that for any arbitrarily small $\varepsilon$, our sequence of $\{a_n\}$ is within $\varepsilon$ of $L$ eventually. If $\varepsilon_0=c\varepsil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Necessary and sufficient condition for $u(x,y,z)$ , $v(x,y,z)$ ,$w(x,y,z)$ are functionally dependent Show that necessary and sufficient condition for that $u(x,y,z)$ , $v(x,y,z)$ ,$w(x,y,z)$ are functionally dependent through equation $F(u,v,w)= 0$ is $\nabla u\cdot (\nabla v \times \nabla w )$.
Any clue on this?
Wh... | Hint. If $F(u(x,y,z),v(x,y,z))=0$ and $F$ is differentiable then
$$\frac{\partial F}{\partial u}\nabla u+\frac{\partial F}{\partial v}\nabla v={\bf 0}.$$
Note that $\frac{\partial F}{\partial u}$ and $\frac{\partial F}{\partial v}$ are scalars.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given that $\operatorname{E}[Y\mid X] = 1$, show that $\operatorname{Var}(XY)\geqslant\operatorname{Var}(X)$
Given that $E[Y\mid X] = 1$, show that $\operatorname{Var}(XY)\geqslant\operatorname{Var}(X)$.
So I tried to expand $\operatorname{Var}(XY) = \operatorname{E}(X^2 Y^2) - 1$ and was stuck here.
| We have $$\operatorname{Var}[XY] = \operatorname{E}[X^2 Y^2] - \operatorname{E}[XY]^2.$$
Try to tackle each of these two terms by first conditioning on $X$ and then taking expectation. For a full proof, hover below.
Write $\operatorname{E}[XY] = \operatorname{E}[\operatorname{E}[XY \mid X]] = \operatorname{E}[X \ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Showing that there are no $3$-digit primes $\overline {abc}$ such that $b^2-4ac=9$.
Problem: Show that there are no $3$-digit primes $\overline {abc}$ such that $b^2-4ac=9$.
Solution. Upon the given condition, the quadratic equation $ax^2+bx+c$ can be written as $(px+q)(rx+s)$ and we have:
$$\overline{abc}=100a+1... | We show a more general fact:
There is no $3$-digit primes $\overline {abc}$ such that $b^2-4ac=d^2$ where $d$ is a non-negative integer.
We first note that $d\leq b\leq 9$. Since $b^2-4ac=d^2$, we have that
$$P(x):=ax^2+bx+c=a\left(x-\frac{-b+d}{2a}\right)\left(x-\frac{-b-d}{2a}\right)$$
Assume that $P(10)=\overline{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Integrate $\ln(1 + x^\frac{1}{2})$ from $0$ to $1$ This integral is from an Integration Contest. The substitution $x^2 = u$ allowed me to evaluate the integral but I keep getting $\frac{1}{2} - \ln2$ as the answer as opposed to just $\frac{1}{2}$ which was the posted answer. Any help would be appreciated.
https://www.... | An alternative way, by Feynman's trick:
$$ \mathfrak{I}=\int_{0}^{1}2x\log(1+x)\,dx=\left.\frac{d}{d\alpha}\int_{0}^{1}2x(1+x)^{\alpha}\,dx\right|_{\alpha=0^+} $$
and by writing $2x$ as $2(1+x)-2$ we get:
$$ \mathfrak{I}=2\left.\frac{d}{d\alpha}\left(\frac{2^{\alpha+2}-1}{\alpha+2}-\frac{2^{\alpha+1}-1}{\alpha+1}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to formalize this last step in my proof? Let $X_k \sim N(\xi, \sigma^2)$. Let $\xi > 0$.
Consider $\frac{1}{k}X_k$, and the corresponding partial sums. I wish to show divergence to infinity, almost surely, of the partial sums.
I manage to show that $\sum_{k=1}^n X_k/k - \sum_{k=1}^n \xi/k$ converges to some finite... | The event $A=\{\sum_{k=1}^n X_k/k \to +\infty\}$ contains the event B = $\{\exists a, \sum_{k=1}^n X_k/k - \sum_{k=1}^n \xi/k \to a\}$. You've shown $P(B)=1$, therefore $P(A)=1$.
To show $B\subseteq A$, let $\omega\in B$, so
$$\sum_{k=1}^n X_k(\omega)/k - \sum_{k=1}^n \xi/k \to a$$
Now just work deterministically to sh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluate an trigonometric limit in $0$ Find the limit
$$\lim_{x \to 0}\frac{\sin(\sqrt{x})}{x}$$
whithout using and using L'Hospital Rule
We have
$$\frac{\sin(\sqrt{x})}{x}=\frac{\sin(\sqrt{x})}{\sqrt{x}}\times\frac{1}{\sqrt{x}}
\to 1 \times (+ \infty)=+\infty$$
Is correct is approach?
| We know that $\sin x=x-\dfrac{1}{3}x^{3}+\cdots$ for $|x|<1$, by alternating series grouping, one can see that $\sin x\geq x-\dfrac{1}{3}x^{3}$ for all small $x>0$, so considering small $x>0$, we have
\begin{align*}
\frac{\sin(\sqrt{x})}{x}\geq\frac{\sqrt{x}-3^{-1}x^{3/2}}{x}=\frac{1}{\sqrt{x}}-\frac{1}{3}\sqrt{x},
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $f,g$ are continuous functions, then $fg$ is continuous? Let $X$ be a topological space and let $f:X \to \mathbb{R}$ ,$g:X \to \mathbb{R}$ be continuous functions. Show that $fg$ is continuous.
My work:
To show $fg$ is continuous at $x$ for each $x \in X$, let $y=fg(x)$.
To show if $N_y$ is a neighborhood of $y$, th... | Here is an alternative approach:
Claim 1. If $f: X\to\mathbb{R}$ and $g: X\to\mathbb{R}$ are continuous, then $f+g: X\to\mathbb{R}$ is continuous.
Proof. Given a point $x\in X$ and $\epsilon>0$, we want to show that there exists a neighbourhood $N_{x}$ of $x$ such that $(f+g)(N_{x})\subseteq B_{\epsilon}((f+g)(x))$. By... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
How to find indicial equation How can I find the indicial equation of $x(x-1)y''+3y'-2y=0$? I tried the method of Frobenius but I keep getting lost in the algebra. Is there any other way to get the indicial equation?
| $y''+\dfrac{3}{x(x-1)}y'-\dfrac{2}{x(x-1)}y=0$ then $p(x)=\dfrac{3}{x(x-1)}$ and $q(x)=-\dfrac{2}{x(x-1)}$. The equation has two regular singular points $x=0$ and $x=1$. For $x=0$ we see
$$p_0=\lim_{x\to0}xp(x)=\lim_{x\to0}\dfrac{3}{x-1}=-3$$
and
$$q_0=\lim_{x\to0}x^2q(x)=\lim_{x\to0}\dfrac{-2x}{x-1}=0$$
then the indi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int_{0}^{\pi\over 4}{\ln(\tan(x))\over \cos^{2n}(x)}\mathrm dx$ $$\int_{0}^{\pi\over 4}{\ln(\tan(x))\over \cos^{2n}(x)}\mathrm dx=F(n)\tag1$$
$n\ge1$
$F(1)=-1$
$F(2)=-{10\over 9}$
$F(3)=-{284\over 225}$
How do we evaluate the closed form for $(1)$?
$u=\tan(x)$ then $\cos^2{(x)}\mathrm du=\mathrm dx$
$$\int_... | Start with integration by parts:
$u = \ln(\tan x) \to du = \frac{1}{\tan x} \sec^2 x dx$,
$dv = (\sec^2 x)^n = \sec^2 x(1 + \tan^2 x)^{n-1}$.
Before continuing, we'll need to compute $v = \int dv$, and for this we'll use the substitution
$w = \tan x \to dw = \sec^2x dx$. Then by the binomial theorem,
\begin{align*}v &=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Convergence and non-negative supermartingales Given $ (X_n)_{n\geq0}$ is a non-negative supermartingale and $X_{\infty}$ is an almost sure limit, I want to show that $ \forall n\geq0$ $$\mathbb{E}(X_\infty \mid \mathcal{F}_n) \leq X_n$$
My approach:
First, for $(X_n)$ a supermartingale I would say we then have $$ \mat... | By conditional Fatou, $E(\liminf_k X_{n+k}\mid\mathcal F_n)\leq \liminf_k E(X_{n+k}\mid\mathcal F_n)$, hence $$E(X_\infty\mid\mathcal F_n)\leq \liminf_k E(X_{n+k}\mid\mathcal F_n)$$
Since $\forall k, E(X_{n+k}\mid\mathcal F_n)\leq X_n$, we have $\liminf_k E(X_{n+k}\mid\mathcal F_n)\leq X_n$ and the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability when given the percentage chance There is a 30% chance that a driver will have an accident in their first year of driving. From 18 people getting their license this year in June, what is the probability that less than a quarter will have an accident before June next year?
So i had a go at it. 30/100 *18= 5.... | Assuming that the events are independent, this is equal to a binomial distribution. Calculate the chance that less than $\frac14$ of the 18 people have an accident is the same as calculating if less than or equal to 4 people have an accident. We can use a table for the cumulative distribution function, which would give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How prove this definition $a\oplus b=a+b$ Define $\oplus$: if for any real numbers $a,b,c$ there have
$$\left(a\oplus b\right)\oplus c=a+b+c$$
show that
$$a\oplus b=a+b$$
| Let $k =0\oplus 0$.
Since $(0\oplus 0)\oplus c=c$ we see that $k\oplus c=c$ for each $c$, so $k$ is left neutral.
Next we see that $\oplus $ is commutative:
$$b\oplus c=(k\oplus b)\oplus c=k+b+c=k+c+b=(k\oplus c)\oplus b=c\oplus b$$
Now we see that $k=0$:
$$(a\oplus k)\oplus k=a+2k \Rightarrow a=a+2k \Rightarrow k=0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If 'a' is divisible by 'b^2' then 'a' is divisible by 'b'. It's something that I've never really thought about before but it makes sense nonetheless. Bearing in mind that 'a' and 'b' are both positive integers, what would be the best way to go about proving this statement? Which method of proof , for example, would be ... | By the definition of divisibility you have $$b^2|a\Rightarrow a=mb^2=(mb)b=nb$$ for an $m \in \mathbb{Z}$ which means that $$b|a$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
For what natural $n$ does $3^n > n^3$ hold true? Prove by induction
For what natural $n$ does $3^n > n^3$ hold true?
I figured that it holds true for all $n$ except $n = 3$. I am not sure how to prove it by induction. I proved it by $p(k) \implies p(k+1)$ but that doesn't show that $n \neq 3$
| For $n=4$ we have
$$3^n> n^3$$ is true.
Let $$3^n> n^3$$ for all $n>3$.
Thus, $$3^{n+1}=3\cdot3^n>3n^3$$ and it's enough to prove that
$$3n^3>(n+1)^3$$ or
$$\sqrt[3]3n> n+1,$$ which is true for $n>3.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
How to find the limit of series? (What should I know?) There is a couple of limits that I failed to find:
$$\lim_{n\to\infty}\frac 1 2 + \frac 1 4 + \frac 1 {8} + \cdots + \frac {1}{2^{n}}$$
and
$$\lim_{n\to\infty}1 - \frac 1 3 + \frac 1 9 - \frac 1 {27} + \cdots + \frac {(-1)^{n-1}}{3^{n-1}}$$
There is no problem to c... | They are sums of geometric progressions.
The first it's
$$\frac{\frac{1}{2}}{1-\frac{1}{2}}=1.$$
The second it's $$\frac{1}{1+\frac{1}{3}}=\frac{3}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 9,
"answer_id": 8
} |
Find the smallest possible value of an integral Say a have an integral, like this one
$$\int_{0}^1 (x-a)^2\, dx$$
and asked to find the smallest possible value of it, as a varies. How can I do this? Besides, is there a certain rule I can use to solve this type of questions?
I will be grateful for any help.
| \begin{align}
\int_{0}^{1} (x-a)^2 &= \frac{(x-a)^3}{3} \Bigg \rvert_0^1 \\
&=\frac{(1-a)^3}{3} - \frac{(-a)^3}{3}\\
&=a^2-a+\frac{1}3\\
&=\left(a-\frac 12\right)^2+\frac {1}{12} \ge \frac{1}{12}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Probability of getting 18 or more bulls eye out of 20 shots An Olympic archer can hit the bulls eye an average of 9 times out of 10. The probability of the archer scoring 18 or more bulls eye from 20 shots is what? I've tried it but my answer is wrong. the average probability would be 0.9 as 9/10. But if we divide 18/2... | This is what is known as a binomial distribution, because we are looking at “sampling” with two possible outcomes, the probability of which we say is constant between trials. There is a prescribed formula for binomial distributions:
$$X\sim\mathrm{B}(n,p) \implies \operatorname{P}(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$
We w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does the reduced matrix $\left(\begin{array}{ccc|c}0&1&0&-7\\ 0&0&1&10\end{array}\right)$ have infinitely many solutions? $$\left(\begin{array}{ccc|c}0&1&0&-7\\ 0&0&1&10\end{array}\right)$$
I thought the requirement for a matrix to have a unique solution was that when every variable is leading. It seems like both 1... | This is equivalent to the system:
$$\begin{eqnarray*}
{0x + y + 0z}&=&{-7} \\
{0x + 0y+ z}&=&{10}
\end{eqnarray*}$$
Whatever $x$ you choose will work because $0x=0$ for all $x$. If the system were:
$$\begin{eqnarray*}
{ y + 0z}&=&{-7} \\
{ 0y+ z}&=&{10}
\end{eqnarray*}$$
Then there is only one solution. It ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
Finding all possible Jordan forms of an $ 8\times 8$ matrix given the minimal polynomial
Find all possible Jordan forms of an $ 8\times 8$ matrix given that $$t^2(t-1)^3$$ is the minimal polynomial.
I don't really know where to start so all help would be appreciated
| Let $J(\alpha,k)$ be the upper Jordan block with minimal polynomial $(t-\alpha)^k$.
$$
\begin{pmatrix}
\alpha & 1 & 0 & \ldots & \ldots\\
0 & \alpha & 1 & 0 & \ldots \\
0 & 0 & \alpha & \ddots & \ddots\\
\vdots & \vdots & \ddots & \ddots & \ddots
\end{pmatrix}
$$
So, there is - up to conjugation - one possible uppe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding the pdf of a transformation of these independent random variables
Hi, I am looking at the question above. We know that $f_X(x) = \lambda^nx^{n-1}e^{-\lambda x}/\Gamma(n)$ and $f_Y(y) = \lambda e^{-\lambda y}$.
We also know that the range for each of these pdfs are $x>0$ and $y>0$.
Now, after computing we get ... | \begin{align}
\int_0^u f_{UV}(u, v) \mathop{dv}
&= \lambda^{n+1} e^{-\lambda u} \frac{1}{\Gamma(n)} \int_0^u (u-v)^{n-1} \mathop{dv}
\\
&= \lambda^{n+1} e^{-\lambda u} \frac{1}{\Gamma(n)} \left[- \frac{1}{n}(u-v)^n\right]_{v=0}^u
\\
&= \lambda^{n+1} e^{-\lambda u} \frac{1}{\Gamma(n)}\cdot \frac{1}{n} u^n.
\end{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Concise Solution for Algebra Problem Consider the following expression
$$\sum_{i=1}^n \left(\frac{p\alpha_ie^{\alpha_i\cdot x}}{1-p+pe^{\alpha_i\cdot x}}\right) = c$$
where $\alpha_i, c \in \mathbb{R}$ and $p \in (0,1).$
How do I solve for $x$ from the equation?
| Here's what I tried and it was too long to be a comment.
By integrating both sides:
$$\sum_{i=1}^n\ln(1-p+p\;e^{\alpha_i x})=cx+c_1$$
and since this must hold for $x=0$, then $c_1=0$.
Therefore
$$\prod_{i=1}^n\left(1+p(e^{\alpha_i x}-1)\right)=e^{cx}$$
Set $y:=e^x$ to convert the above equation to a nonlinear (and seem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proving if a function is continuous, its inverse is also continuous
Let $(X,d)$ and $(X,d')$ be metric spaces. Prove that $a)\implies b) $ and $b) \implies c)$:
a) if $x_n \to x$ in $(X,d')$ then $x_n \to x$ in $(X,d)$
b)For any metric space $(Y,p)$ and any continuous $f:(X,d)\to (Y,p), f:(X,d')\to(Y,p)$ is also conti... | Hint:
If b) is true then the function $(X,d')\to (X,d)$ prescribed by $x\mapsto x$ is continuous.
Let's denote this function with $g$.
Now if $f:(Y,p)\to(X,d')$ is continuous then so is $g\circ f:(Y,p)\to(X,d)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Galois groups of rational function fields extensions Consider the subfields $$ K_{1}:=\Bbb{C}\big(4x(1-x),4y(1-y)\big) $$
$$ K_{2}:=\Bbb{C}\Big(\frac{4x(1-x)(1-2y)}{(1-2xy)^{2}},\frac{4y(1-y)(1-2x)}{(1-2xy)^{2}}\Big) $$ of $ K:=\Bbb{C}(x,y) $.
I want to compute the Galois groups $ G_{1}:=Gal(K/K_{1}),G_{2}:=Gal(K/K_{2... | Looking at this over finite fields $\Bbb F_q$ with large $q$, one finds experimentally that the map $\phi : (x,y) \mapsto (u(x,y),v(x,y))$ (where $u$ and $v$ are the two generators of $K_2$) is basically $8$-to-$1$. For a choice of a pair $(u,v)$, either there is no solution to $(u,v) = \phi(x,y)$ (approximately $7/8$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Linear dependence/independence of functions $f(x) = x^2+1$, $g(x) = 1+x^3$ and $h(x) = \ln(1 + x)$. Questions :
Prove that the functions $f(x) = x^2+1$, $g(x) = 1+x^3$ and $h(x) = \ln(1 + x)$ are linearly independent on the interval $(0, 1)$. Can you use the Wronskian to do this?
I try this:
$$c_1 x f(x)+ c_2 x g(x) + ... | The Wronskian is \begin{align}W(f,g,h)(x)&=\begin{vmatrix}f(x)& g(x) & h(x)\\f'(x) & g'(x) & h'(x) \\f''(x) & g''(x)& h''(x)\end{vmatrix}=\begin{vmatrix}1+x^2& 1+x^3 & \ln{(1+x)}\\2x & 3x^2 & \frac1{1+x} \\2 & 6x & -\frac{1}{(1+x)^2}\end{vmatrix}\\[0.3cm]&=\left\{\text{terms without} \ln(x+1)\right\}+6x^2\ln(x+1)\end{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find the limit of the complex function. $$ \lim_{x\to a} \left(2- \frac{x}{a}\right)^{\left(\tan \frac{\pi x}{2a}\right)}.$$
I have simplified this limit to this extent :
$$e^{ \lim_{x\to a} \left(\left(1- \frac{x}{a}\right){\left(\tan \frac{\pi x}{2a}\right)}\right)}$$
I don't know how to simplify the limit after that... | \begin{align*}
\log\left(2-\frac{x}{a}\right)^{\tan(\pi x/2a)}&=\left(\tan\frac{\pi x}{2a}\right)\left(\log\left(2-\frac{x}{a}\right)\right)\\
&=\sin\left(\frac{\pi x}{2a}\right)\frac{\log\left(1+\left(1-\dfrac{x}{a}\right)\right)}{\cos\left(\dfrac{\pi}{2}\left(\dfrac{x}{a}-1\right)+\dfrac{\pi}{2}\right)}\\
&=\sin\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 0
} |
Minimum slope of a chord to parabola A line is drawn from $(-2,0)$ to intersect
$y^2 = 4x \,\,$ in P, Q within the first quadrant, such that
$$ \frac{1}{AP} +\frac{1}{AQ} < \frac{1}{4} $$
Find the minimum value of the line slope. A is the origin.
I had basically let the coordinates of the parabola in parametric form... | As it is stated in the current form, this problem - as shown below - has no real solutions. I suspect that the original formulation might have been slightly different, in particular with $A$ indicating the point $(-2,0)\,\,$ and maybe with a more general equation of the parabola (again see below). In this answer, I w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the domain and range of the function, $f(x,y) = \sqrt{x+y}$ and sketch the domain in the xy-plane. I have found the domain to be $y \geq -x$.
I have found the range to be $z \geq 0$, or $[ 0, \infty )$.
I'm not sure how to sketch the domain in the xy-plane. I figured it would be a straight line through $(0,0)$ wit... | This is not an answer. I just needed the attach picture feature to show the domain of the function, which is the hatched region.
Hope it helps
$$...$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Examining sequence convergence with parameter $a$? For $a\in R$, let $x_1=a$ and $x_{n+1}$=$-\frac{6x_n^2+4x_n}{x_n^2-2x_n+4}$. Examine the convergence of the sequence ${(x_n)}_{n=1}^{\infty}$ for different values of $a$. Also find $\lim_{n\to\infty}x_n$ whenever it exist.
I know that if the sequence converges to $x$ t... | The only possible real limit is
$0$,
since all the other roots of
$x
=-\frac{6x^2+4x}{x^2-2x+4}
$
are complex.
Around zero,
it can't be monotone
since
$x_{n+1}
=-\frac{x_n(6x_n+4)}{x_n^2-2x_n+4}
$
has the opposite sign
of $x_n$.
What is needed,
imho,
is to write
$x_{n+2} = \text{some function}(x_n)$.
This would probabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Intersection of Radical With Semisimple Subalgebra Suppose $\mathfrak{g}$ is a Lie algebra with radical $\text{Rad }\mathfrak{g}$ and let $\mathfrak{a}\subseteq\mathfrak{g}$ be a semisimple subalgebra. Is it necssarily the case that $\text{Rad }\mathfrak{g}\cap\mathfrak{a}=\{0\}$?
I am trying to prove that maximal semi... | Yes. $\text{Rad }\mathfrak{g}\cap\mathfrak{a}$ is an ideal of $\mathfrak{a}$, because $\mathfrak{a}$ is a subalgebra and $\text{Rad }\mathfrak{g}$ is an ideal of $\mathfrak{g}$. Also, $\text{Rad }\mathfrak{g}\cap\mathfrak{a}$ is solvable, because it is a subalgebra of the solvable $\text{Rad }\mathfrak{g}$. But since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum of positive divisors if and only if perfect square let n be a positive odd integer, prove that the sum of the positive divisors of n is odd if and only if n is a perfect square.
I know that based on the prime factorization theory that every integer n can be written as the product of primes, if their sum is odd that... | I don't really understand your argument: What does "there are equal pairs of even and odd divisors" mean, especially given that there are no even divisors of $n$? You really need to be precise about the objects you're considering, and careful about how it's written.
Here's an approach that you might find useful. For e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Derivative of l2 norm with chain rule If $X$ is a $n$ by $d$ matrix, $\alpha$ is a $n$ by $1$ vector, let $f(\alpha) = \left\Vert X^\top\alpha \right\Vert_2^2$, what is $\frac{df}{d\alpha}$
| ${df(\alpha)\over d\alpha}={d\|X^T\alpha\|_2^2\over d\alpha}={d\|X^T\alpha\|_2^2\over dX^T\alpha}{dX^T\alpha\over d\alpha}=2\alpha^TXX^T$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving a first order differential equation in terms of Lambert W-function I am having great difficulty solving the following equation
$$\ \frac{ax}{(bx^2 + c)} = \frac{dx}{dt} $$
I have re-edited the question. Any help is appreciated. Thank you.
| This was for the first edit of the post.
Welcome to the world of Lambert function !
The solution of equation $$\ x^2 + \log(x) - c = 0$$ is given by
$$x=\pm\frac{\sqrt{W\left(2 e^{2 c}\right)}}{\sqrt{2}}$$ but only the positive root must be kept because of $\log(x).
The Wikipedia page gives series espansions fot he ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
if A is turing-recognizable, and A is mapping reducible to complement of A, A is decidable Here $<$ denotes mapping reducibility.
Show that if $A$ is Turing-recognizable and $A < A'$, then $A$ is decidable.
How can I prove this? I'm not sure I quite understand how $A < A'$ is possible.
| I learned about these same ideas using different notation and terms. While the ideas are the same, my notation might be off. Here is what I got.
By $A \leq_m A'$ then for some computable (halts on every input) function $f:N \rightarrow N$ and any $n\in N$ we have.
$$ n \in A \Longleftrightarrow f(n) \in A'$$
equiva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve summation of $\sum_{j=0}^{n-2}2^j (n-j)$ Question
While Solving a recursive equation , i am stuck at this summation and unable to move forward.Summation is
$$\sum_{j=0}^{n-2}2^j (n-j)$$
My Approach
$$\sum_{j=0}^{n-2}2^j (n-j) = \sum_{j=0}^{n-2}2^j \times n-\sum_{j=0}^{n-2}
2^{j} \times j$$
$$=n \times (2^{n... | Hint
Consider $$\sum_{i=0}^p i x^i=x\sum_{i=0}^p i x^{i-1}=x\left(\sum_{i=0}^p x^i \right)'$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Is the extension of injective bounded linear transformation a injective linear transformation Let $X$ and $Y$ be two Banach spaces over $\mathbb{C}$
Let $V$ be a dense subspace of $X$
Let $T : V \to Y$ be a bounded linear operator
We know that $T$ can be uniquely extended to a bounded linear transformation $S : X \to Y... | No. Using a Hamel basis of an infinite dimensional Banach space $Y$ one find a discontinuous linear functional $f$ on $Y$. Then $\|y\|' = \|y\|+|f(y)|$ is a strictly finer norm on $Y$ and the identity $(Y,\|\cdot\|') \to (Y\|\cdot\|)$ extends to the completion $X$ of $(Y,\|\cdot\|')$. This extension isn't injective be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show the limit of a sequence of real numbers Define $(a_n)_{n=1}^\infty$ as a sequence of real numbers and let a be real number. Show that
$$\lim_{n\rightarrow\infty}(a_n)=a<=>\forall\epsilon\in ]0, 10^{-6}]∃N≥10^{-6}\forall n≥N: |a_n − a|<27\epsilon$$
Where n,N are natural numbers.
Im not sure if Ive understood it cor... | I think it's more like : $\lim\limits_{n\to\infty}(a_n) = a \iff \forall\varepsilon \in \ ]0,10^{-6}]\ \exists N \geq \fbox{$10^6$} \ \forall n\geq N : |a_n - a| < 27\varepsilon$ ?
$\lim\limits_{n\to\infty}(a_n) = a$ is :
$$\forall\varepsilon > 0\ \exists N\ \forall n\geq N : |a_n - a| < \varepsilon$$
is equivalent to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the $L^2$ norm of $\theta_\chi(z)^2$ in $\mathbb{H}/\Gamma_0(4)$ I'd like to learn about $L^2$ norms on Hyperbolic space. My Automorphic forms textbook says this function is a "cusp form" so it's in $L^2(\mathbb{H}/\Gamma_0(4))$:
$$ \theta_\chi(z) = \sum_{n \in \mathbb{Z}} \chi(n) \, e^{-\pi n^2 \, z} $$
This is ... | The standard, but non-obvious, way to compute an $L^2$ norm of a modular form $f$ is as the residue at $s=1$ of the Rankin-Selberg $L$-function (or Dirichlet series) obtained as an integral $\int_{\Gamma\\\mathfrak H} E_s\cdot |f|^2$. This is because the residue at $s=1$ of that Eisenstein series is a constant (basical... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
expectancy value of count of prior values in array that are bigger Let A be an Array of the length n. A is filled randomly with distinct numbers ($\forall (i,j) < n: i \neq j \implies A[i] \neq A[j] $).
What is the total expected value (amount) of pairs (indexed i,j) with $i < j \land A[i] > A[j]$ in an array of the le... | You are counting what are called "inversions of permutations".
For each $i$ and $j$ with $i < j$, the probability that $A[i] > A[j]$ is $1/2$ (since it's equally likely for $A[i]$ or $A[j]$ to be the larger of the two).
There are $n(n-1)/2$ pairs of indices $i$ and $j$, and so the expectation you want is $n(n-1)/4$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this NP complete? I have to say that I am newbie in graph theory, so bear with me.
Here is my question:
Assume that you are given:
1. A directed graph G
2. A positive integer K
3. A source node S in graph G.
find the minimum number of edges to be added to graph G so that there is simple path of length K from nod... | Set cover reduces to this for $K = 2$, so yes it is NP-hard.
The set cover problem can be formulated as this: given a graph $G$ with disjoint vertex sets $X$ and $Y$ and arcs going from $X$ to $Y$, choose a subset $X' \subseteq X$ of minimum cardinality such that for every element $y \in Y$, there is an arc going from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Frobenius method series solution Q: $x^2y^{''}-(x^2+2)y=0$ [1]
Solving using frobenius
$y=\sum_{n=0}^{\infty}a_nx^{x+r}$ [2]
$ y'=\sum_{n=0}^{\infty}(n+r)a_nx^{x+r-1}$ [3]
$ y''=\sum_{n=0}^{\infty}(n+r)(n+r-1)a_nx^{x+r-2}$ [4]
inserting [2,4] into [1]
$x^2 \sum_{n=0}^{\infty}(n+r)(n+r-1)a_nx^{x+r-2}-(x^2+2)\sum_{n=... | Your approach is fine. There is just a small mistake which causes the problem. In the following we use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ in a series.
Let's consider again the series equation
\begin{align*}
\sum_{k=0}^\infty[(k+1)(k+2)a_k-2a_k]x^k-\sum_{k=2}^\infty a_{k-2}x^k=0\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.