Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
discrete math Quantifiers? In the following exercises, V(x,y) means "x + y = 2xy", where x and y are integers. Determine
the truth value of the statement: ∀x∃y¬V(x,y)
what this says is for every x, there exists an y such that negation of V? what is the negation v?
and also for this question, U(x,y) means "2x + 3y = xy"... | The negation of V is $x + y \neq 2xy$. $\forall x \exists y¬V(x,y)$ means that whatever number $x$ you are given, you can find another (non necessarily distinct) number $y$ for which $x + y \neq 2xy$. Do you think this is true or false? This is true, think of a simple way of proving it!
For $∃x∀yU(x,y)$, mean that ther... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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A compact Hausdorff space is metrizable if it is locally metrizable
A space $X$ is locally metrizable if each point $x$ of $X$ has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if it is locally metrizable.
Hint: Show that $X$ is a finite union of open... | For every $x\in X$, there exists a neighborhood $U_x$ which is metrizable. These neighborhoods cover $X$, i.e., $X=\bigcup_x U_x$. Now use the definition of compactness to reduce this to a finite union, $X=U_1\cup\ldots\cup U_n$. Each of these sets is metrizable, so pick metrics which are defined locally on each $U_i$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that the union of the limit points of sets is equal to the limit points of the union of the sets My instructor gave us an exercise to do, to show that this equality holds:
$$\bigcup_{k=1}^m A_k'=\left(\bigcup_{k=1}^m A_k\right)^{\!\prime}.$$
My thoughts on approaching this question is by contradiction, suppose ... | It’s very easy to show that $A_\ell'\subseteq\left(\bigcup_{k=1}^mA_k\right)'$ for each $\ell\in\{1,\ldots,m\}$ and hence that
$$\bigcup_{k=1}^mA_k'\subseteq\left(\bigcup_{k=1}^mA_k\right)'\;;$$
the harder part is showing that
$$\left(\bigcup_{k=1}^mA_k\right)'\subseteq\bigcup_{k=1}^mA_k'\;.\tag{1}$$
Proof by contradic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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A nice number is an integer ending in 3 or 7 when written out in decimal. Prove that every nice number has a prime factor that is also a nice numbers. My teacher just asked me a question like this but i do not know how to start and work it out at all. Can someone help me out with that?
| Hint: The number is odd, so all of its prime factors are odd. Could they be all nasty (end in $1$ or $5$ or $9$? Examine products of nasty numbers. You will find they are all nasty.
Remark: Note that a nice number can have some nasty prime factors. For example, $77$ has the nasty prime factor $11$, but it also has the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/551073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How find this sum $\sum_{n=1}^{\infty}n\sum_{k=2^{n-1}}^{2^n-1}\frac{1}{k(2k+1)(2k+2)}$ Find the sum
$$\sum_{n=1}^{\infty}n\sum_{k=2^{n - 1}}^{2^{n}\ -\ 1}\dfrac{1}{k(2k+1)(2k+2)}$$
My try:
note
$$\dfrac{1}{k(2k+1)(2k+2)}=\dfrac{2}{(2k)(2k+1)(2k+2)}=\left(\dfrac{1}{(2k)(2k+1)}-\dfrac{1}{(2k+1)(2k+2)}\right)$$
Then I ca... | Are you really sure about the outer $n$? If not, Wolfram Alpha gives a sum that telescopes nicely
$$\sum_{k=2^{n-1}}^{2^n-1} \frac{1}{k (2 k+1) (2 k+2)}=-2^{-n-1}-\psi^{(0)}(2^{n-1})+\psi^{(0)}(2^n)-\psi^{(0)}\left(\frac{1}{2}+2^n\right) +\psi^{(0)}\left(\frac{1}{2}+2^{n-1}\right)$$
Update:
It is not hard to prove user... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
} |
Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3? So, do you guys have any ideas? Sorry if this might seem like dumb question, but I have asked everyone I know and we haven't got a clue.
| $x \mod 3=\{0,1,2\}$ for $x\in \mathbb{N}$.
In the other words, if you divide any natural number by $3$, the remainder will be $0$ or $1$ or $2$.
*
*$x \mod 3=0 \Rightarrow (2x^3 + x) \mod 3 = 0$
*$x \mod 3=1 \Rightarrow (2x^3 + x) \mod 3 = 0$
*$x \mod 3=2 \Rightarrow (2x^3 + x) \mod 3 = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/551148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 4
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Limit of $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$ What is the limit of this sequence $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$?
Where $a$ is a constant and $n \to \infty$.
If answered with proofs, it will be best.
| Let $f(x) = 1+x + x^2+x^3+ \cdots$. Then the radius of convergence of $f$ is $1$, and inside this disc we have $f(x) = \frac{1}{1-x}$, and $f'(x) = 1+2x+3x^2+\cdots = \frac{1}{(1-x)^2}$.
Suppose $|x|<1$, then we have $xf'(x) = x+2x^2+3x^3+\cdots = \frac{x}{(1-x)^2}$.
If we choose $|a| >1$, then letting $x = \frac{1}{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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is this conjecture true or false? I want to know if this conjecture istrue or false
$$\Large e^{\frac{ \ln x}{x}} \notin \mathbb{Z} $$
for every $x \in \mathbb{R} \setminus \{1,-1,0\} $
| You should ask this only for $x>0$, as the expression is not well defined otherwise. You can rule out the case $x\in(0,1)$ easily, since it implies $\ln(x)/x<0$. Now find the maximum of $\ln(x)/x$ on $(1,\infty)$, and conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/551337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calculate area of this shape?
I was trying to solve a complicated problem then I came accros to this complicated problem. I believe that there is enough information to calculate the area. Can you help me to find a general formula for the area of this shape, in terms of $x,\alpha,\beta$?
I forgot to write on the... | Set up a coordinate system with its original at $B$. Then, in this coordinate system, we have:
$$
A = (x_0, y_0) = \frac{1}{\sqrt2}(x,x)
$$
$$
B = (x_1, y_1) = (0,0)
$$
$$
C = (x_2,y_2) = (-\cos\beta, \sin\beta)
$$
$$
D = (x_3,y_3) = \frac{1}{\sqrt2}(x,x) + (-\cos\alpha, \sin\alpha)
$$
Then apply the area formula from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/551522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Convert a WFF to Clausal Form I'm given the following question:
Convert the following WFF into clausal form:
\begin{equation*}
\forall(X)(q(X)\to(\exists(Y)(\neg(p(X,Y)\vee r(X,Y))\to h(X,Y))\wedge f(X)))
\end{equation*}
This is what I've gotten so far, but I'm not confident that I'm in the proper form at the end.
... | To write $\neg q(X)\lor\Big(\big(p(X,g(X))\lor r(X,g(X))\lor h(X,g(X))\big)\land f(X)\Big)$ in clausal form, you first need to put it in conjunctive normal form. To do so, you can distribute the first disjunction over the paranthesis to yield:
$$\big(\neg q(X)\lor f(X)\big)\land\big(\neg q(X)\lor p(X,g(X))\lor r(X,g(X)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/551594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Asymptotic relationship demonstration I have to demonstrate that if
$$
\begin{split}
f_1(n) &= \Theta(g_1(n)) \\
f_2(n) &= \Theta(g_2(n)) \\
\end{split}
$$
then
$$
f_1(n) + f_2(n) = \Theta(\max\{g_1(n),g_2(n)\})
$$
Actually I have already proved that $$f_1(n)+f_2(n) = O(\max\{g_1(n),g_2(n)\}).$$
My problem is $$f_1(n)+... | Note that if $f = \Theta(g)$ then $g = \Theta(f)$ and since you already proved the $O()$ bound, you can conclude that $f_1 + f_2 = O(g_1 + g_2)$ and by a symmetric argument that $g_1 + g_2 = O(f_1 + f_2)$ which is equivalent to the $\Omega()$ bound you seek.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Can somebody help me to understand this? (Baire Category Theorem)
Theorem $\mathbf{6.11}$ (Baire Category Theorem) Every residual subset of $\Bbb R$ is dense in $\Bbb R$.
$\mathbf{6.4.5}$ Suppose that $\bigcup_{n=1}^\infty A_n$ contains some interval $(c,d)$. Show that there is a set, say $A_{n_0}$, and a subinterval ... | To apply the Baire category theorem (BCT), I assume that it is intended that the sets $\{A_n\}_{n=1}^{\infty}$ are closed subsets of $\mathbb{R}$. Now, let $c<a<b<d$ and consider $[a,b]$, which is a closed subset of a complete metric space, and is therefore a complete metric space itself. So, by the BCT it can not be t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/551760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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deducibility from peano axioms I have to solve the following problem:
Using $\exists$ Introduction prove that PA$\vdash x\leq y \wedge y\leq z \longrightarrow x\leq z$: I used that if $x\leq y$ then $\ \exists \ r\ x+r=y$ and in the same way $\exists t \ y
+t=z$, but in logic term I don't know how to use these equalit... | See Wikipedia Peano Axioms.
The point of this answer is to show that you can define an ordering of the natural numbers before you define addition.
Proposition 1: If $n$ is any number except $0$, there exist an $m$ such that
$n = S(m)$
Moreover, if the successor of both $m$ and $m^{'}$ is equal to $n$, then $m = m^{'}$... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof that a limit point compact metric space is compact. If $(X,d)$ is limit point compact, show it is compact.
I have found multiple proofs of this that first show that limit point compact implies sequential compact, which in turn implies compactness. I was wondering if there is a direct proof showing that limit poin... | Pedantic comments are in parentheses.
WLOG, take $C = \{1\} \cup \{1 + \frac{1}{n}\}$. Let $\scr{U}$ be an open cover of $C$. Then $\exists U_d \in \scr{U}$ such that $1 \in U_d$.
Since $U_d$ is open then $1$ is an interior point of $U_d$. Therefore $\exists \epsilon > 0$ such that $N_{\epsilon}(1) \in U_d$.
Note that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/551921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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prove a statement (complements of unions) I want to prove this statement:
$$(A_1 \cup A_2)^c = {A_1}^c \cup {A_2}^c$$
where the $c$ means the complement.
Any help would be greatly appreciated.
| This is false in the general case. For example, if $A_1$ is the set of integers ($\{\ldots,-2,-1,0,1,2,\ldots\}$) and $A_2$ is the set of positive real numbers, and the universe is the set of all real numbers, then $(A_1 \cup A_2)^c$ doesn't contain $-1$ or $\frac12$, but $A_1^c \cup A_2^c$ contains them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/551994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Determine homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $y = 2x - 6$
Determine the homogeneous transformation matrix for reflection about the line
$y = mx + b$, or specifically $ y = 2x - 6$.
I use $mx - y +b =0$: $\text{slope} = m$, $\tan(\theta)= m$
intersection w... | Note that any matrix $\mathbf A\cdot \vec{0}=\vec{0}$, so there is no matrix that can flip over $y=2x-6$ as it must map $\vec 0 \to (4 \frac{4}{5},-2\frac{2}{5})$. You might want to do something about that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/552129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$f=g$ almost everywhere $\Rightarrow |f|=|g|$ almost everywhere? Suppose $(X, \mathcal{M}, \mu)$ is a measure space. Assume $f: X\to\overline{\mathbb{R}}$ and $g=X\to\overline{\mathbb{R}}$ are measurable maps. Here $\overline{\mathbb{R}}$ denotes the set of extended real numbers. My question is:
If $f=g$ almost everyw... | Note that since $f$ and $g$ are measurable, then so are $|f|$ and $|g|$ by continuity of $|\cdot|$, and hence, $h=|f|-|g|$ is measurable. Noting that $$F=\left(h^{-1}\bigl(\{0\}\bigr)\right)^c,$$ we readily have $F\in\mathcal M$. By nonnegativity and monotonicity of measure, since $F\subseteq E$ and $\mu(E)=0,$ then $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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90 people with ten friends in the group. Prove its possible to have each person invite 3 people such that each knows at least two others
A high school has 90 alumni, each of whom has ten friends among the other alumni. Prove that each alumni can invite three people for lunch so that each of the four people at the lunc... | Pick any vertex v. It is connected to 10 vertices. Keep these 11 vertices aside. We are left with 79 vertices on the other side.
Realize that the central case is when we have five disjoint pairs(that is, each pair consists of distinct vertices) of edge connected vertices among the 10 vertices v is connected to. Thus w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why are these equations equal? I have racked my brain to death trying to understand how these two equations are equal:
$$\frac{1}{1-q} = 1 + q + q^2 + q^3 + \cdots$$
as found in http://www.math.dartmouth.edu/archive/m68f03/public_html/partitions.pdf
From what I understand if I substitute $5$ for $q$ the answer:
$$\frac... | $S=1+q+q^2+\dots$
For any $k\in \mathbf{N}$,
Let $S_k=1+q+q^2+\dots+q^k$,
Then, $qS_k=q+q^2+q^3+\dots+q^{k+1}$.
So, $(1-q)S_k=S_k-qS_k=1+q+q^2+\dots+q^k-(q+q^2+q^3+\dots+q^{k+1})=1-q^{k+1}$
If $|q|<1$, then $q^k \longrightarrow 0$ as $k \longrightarrow \infty$, so,
$$S=\lim_{k \to \infty}S_k=\lim_{k \to \infty}\frac{1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$ Let $p$ be an odd prime number. Prove that $$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \left( \frac{3 \cdot 4}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \... | Let $a^\ast$ be the inverse of $a$ modulo $p$. Then
$$\left(\frac{a(a+1)}{p}\right)=\left(\frac{a(a+aa^\ast)}{p}\right)=\left(\frac{a^2(1+a^\ast)}{p}\right)=\left(\frac{1+a^\ast}{p}\right).$$
As $a$ ranges from $1$ to $p-2$, the number $1+a^\ast$ ranges, modulo $p$, through the integers from $2$ to $p-1$. But the su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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How does the mathematical form of nCr ensure that the result is a whole number The mathematical form for nCr is
(n!)/(r!(n-r)!)
How does this form ensure that nCr is indeed a whole number. Is there a mathematical proof?
| The nice thing is that it is the combinatorial argument that $\frac{n!}{r!(n-r)!}$ counts something (i.e., the number of ways to choose $r$ items from $n$ unordered items) that proves (in the most rigorous sense) that it is a natural number. What are proof is needed?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/552590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating integral with standard normal distribution.
I have a problem to solving this,
Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate.
However, at the bottom I found that Integral from zero to infinity of 1 goes to infinity and ... | You have managed to state a closed form for $F_x(x)$.
This is not in fact possible for a normal distribution, so you have an error in your integral about a third of the way down: the integral of $\displaystyle e^{-\frac12 t^2}$ is not $\displaystyle -\frac{1}{t} e^{-\frac12 t^2}$
There is an easier solution, but it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$ If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if :
$$(tr(A))^n\geq n^n \det(A)$$
What i have tried is :
As $A\in M_{n\times n} (\mathbb{R})$ a positive definite sym... | This is really a Calculus problem! Indeed, let us look for the maximum of $h(x_1,\dots,x_n)=x_1^2\cdots x_n^2$ on the sphere $x_1^2+\cdots+x_n^2=1$ (a compact set, hence the maximum exists). First note that if some $x_i=0$, then $h(x)=0$ which is obviously the minimum. Hence we look for a conditioned critical point wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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$F:G\to B$ is an isomorphism between groups implies $F^{-1}$ is an isomorphism $F$ is an isomorphism from group $G$ onto group $B$. Prove $F^{-1}$ is an isomorphism from $B$ onto $G$.
I do not know which direction to go in.
| Because $F$ is a bijection, $F^{-1}$ is a bijection as well. Now we know
$$F(a\cdot b)=F(a)\cdot F(b)$$
This means that
$$F^{-1}(a\cdot b)=F^{-1}(F^{-1}(F(a\cdot b))) = F^{-1}(F^{-1}(F(a)))\cdot F^{-1}(F^{-1}(F(b))) = F^{-1}(a)\cdot F^{-1}(b)$$
Thus $F^{-1}$ is an isomorphism from $B$ to $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/552867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
$\frac{\prod_{i=1}^n (1+x_i)-1}{\prod_{i=1}^n (1+x_i/\delta)-1} \stackrel{\text{?}}{\le} \frac{(1+x_n)^n-1}{(1+x_n/\delta)^n-1} $ . Let $x_1 \le x_2 \le \cdots \le x_n$. Let $\delta>1$ be some positive real numbers. I assume that $0\le x_i <1$, for $i=1,\ldots,n$ and $x_n >0$.
Does the following expression hold?
$$ \fr... | By putting:
$$A=\sum_{i=1}^n\log(1+x_i),\quad A_\delta=\sum_{i=1}^n\log(1+x_i/\delta),$$
$$B=\sum_{i=1}^n\log(1+x_n),\quad B_\delta=\sum_{i=1}^n\log(1+x_n/\delta),$$
the inequality is equivalent to:
$$e^{A+B_\delta}+e^{A_\delta}+e^{B}\leq e^{B+A_\delta}+e^{A}+e^{B_\delta}.$$
Provided that $A+B_\delta\leq B+A_\delta$, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/552955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Integral of a positive function is positive? Question:
Let $f:[a.b]\to \Bbb R \in R[a,b]$ s.t. $f(x)>0 \
\forall x \in \Bbb R.$ Is $\int _a^b f(x)\,dx>0$ ?
What We thought:
We know how to prove it for weak inequality, for strong inequality - no clue :-)
| If f is continuous, we can argue like this, Let $m$ be the infemum of the function on$[a,b]$. show that $m \gt 0$ and $\int _a^b f(x)\,dx \gt m(b-a) \gt 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/553031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Is math built on assumptions? I just came across this statement when I was lecturing a student on math and strictly speaking I used:
Assuming that the value of $x$ equals <something>, ...
One of my students just rose and asked me:
Why do we assume so much in math? Is math really built on assumptions?
I couldn't an... | Mathematical and ordinary language and reasoning are the same in this respect.
If there is a general question of whether and how human thought relies on hypotheticals, that's an interesting subject to explore, but the only reason it appears to be happening more often in mathematics is that the rules of mathematical sp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?
Let the sequence $\{a_n\}$ be defined as $a_1 = \sqrt 2$ and $a_{n+1} = \sqrt {2+a_n}$.
Show that $a_n \le$ 2 for all $n$, $a_n$ is monotone increasing, and find the limit of $a_n$.
I've been working on this problem ... | Hints:
1) Given $a_n$ is monotone increasing and bounded above, what can we say about the convergence of the sequence?
2) Assume the limit is $L$, then it must follow that $L = \sqrt{2 + L}$ (why?). Now you can solve for a positive root.
| {
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Where am I going wrong on this probability question? A person has to travel from A to D changing buses at stops B and C en-route. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time,... | You are assuming that the waiting time is a natural number. A more reasonable assumption is that the waiting time is uniformly distributed over the interval $[0,8]$ You can think of the two waiting times as choosing a point in the square $[0,8] \times [0,8]$ and the chance of exceeding a total of $13$ minutes is the ... | {
"language": "en",
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"source": "stackexchange",
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finite dimensional range implies compact operator Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in X$ is a bounded sequence, there exist a subsequence $x_{n_k}$ such tha... | You require the boundedness of $T$. Fact is that boundedness + finite rank of $T$ gives compactness of $T$. This you can find in any good book in Functional Analysis. I would recommend J.B Conway
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/553313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Homework - set theory infinite union A question from my homework I'm having trouble understanding.
We are given:
$A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$
$A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity
The questions are:
1) show that $A\times A \subseteq A$
2) I... | HINT: For (2), note that not all the elements of $A$ are ordered pairs.
Also, let's write $A(2)$, but to make it simpler let's call $A(1)=X$. Then $A(2)=X\cup(X\times X)=\{\varnothing\}\cup\{\langle\varnothing,\varnothing\rangle\}=\{\varnothing,\langle\varnothing,\varnothing\rangle\}$.
Not very difficult, $A(3)$ will ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Question on Matrix Transform Operations Given $e = Y - XB$, where
$
e = \begin{bmatrix}
e_1 \\
\vdots \\
e_n \\
\end{bmatrix}
$,
$
Y= \begin{bmatrix}
y_1 \\
\vdots \\
y_n \\
\end{bmatrix}
$,
$
X= \begin{bmatrix}
1 & x_1 & x^2_1 \\
\vdots & \vdots & \vdots \\
1 & x_n & x^2_n \\
\end{bmatrix}
$, and
$
... | The transpose is contravariant, meaning $(AB)^T=B^TA^T$ is to be used. So, we have
$$Y^TY - B^TX^TY - Y^TXB +B^TX^TXB\,.$$
Now, $Y^TXB$ is a $1\times 1$ matrix, nonetheless, is the scalar product of vectors $Y$ and $XB$, but then it is symmetric as matrix, so we have
$$Y^TXB=(Y^TXB)^T=B^TX^TY\,.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cutting up a circle to make a square We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if we don't allow overlapping?
More precisely: For N pieces that together will fit ins... | Ok not my greatest piece of work ever, but here's a suggestion in 2 parts:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/553571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
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The irrationality of the square root of 2 Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
| This question has been asked before. Please search Math.SE for an answer to your question before asking.
Regardless, yes, there are lots of ways to prove the irrationality of $\sqrt{2}$. Here are some pertinent resources:
*
*https://mathoverflow.net/questions/32011/direct-proof-of-irrationality/32017#32017 (That's a... | {
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Computing Coefficients for Generalized Combinatorial Sets I'm new to Combinatorics and am wondering if there is a well-known generalization to the binomial coefficients in the following sense:
The binomial coefficients, $n \choose d$, can be considered as the number of ways in which $d$ objects can be chosen from amon... | Ok, so you want to buy $d$ items in a store with $n$ elements, but where item $i$ only has a stock of $s_i$. What you are looking for is the coefficient of $x^d$ in the product
$(1+x^1+ x^2+ \dots+ x^{s_1})(1+x^1+ x^2+ \dots+ x^{s_2})\dots (1+x^1+ x^2+ \dots+ x^{s_n})=$
$\dfrac{(1-x^{s_1+1})(1-x^{s_2+1})\dots (1-x^{s_n... | {
"language": "en",
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"source": "stackexchange",
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Showing Inequality using Gauss Function
If $\alpha, \beta\in \Bbb{R}$ and $m, n\in \Bbb{N}$ show that the inequality
$[(m+n)\alpha]+[(m+n)\beta] \ge [m\alpha]+[m\beta]+[n\alpha+n\beta]$
holds iff m=n
I thought that we have to find a counter-example such that when $m\neq n$, there always exists certain (m, n) which ma... | You can proceed as follows.
Lemma 1. The inequality is true if $m=n$.
Proof of lemma 1. Replacing $m$ with $n$, the inequality becomes
$$
[2n\alpha]+[2n\beta] \geq [n\alpha]+[n\beta]+[n(\alpha+\beta)] \tag{1}
$$
Let us put $x=n\alpha-[n\alpha],y=n\beta-[n\beta]$. Then $x,y$ are in $[0,1)$, $[2n\alpha]=2[n\alpha]+[2x], ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How find this $\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i}{n}\right)^n$ How find this $$\lim_{n\to\infty}\sum_{i=1}^{n}\left(\dfrac{i}{n}\right)^n$$
I think this answer is $\dfrac{e}{e-1}$
and I think this problem have more nice methods,Thank you
| For each fixed $x$, Bernoulli's inequality
$$ (1 + h)^{\alpha} \geq 1 + \alpha h,$$
which holds for $h \geq -1$ and $\alpha \geq 1$, shows that whenever $x \leq n+1$ we have
$$ \left( 1 - \frac{x}{n+1} \right)^{n+1}
= \left\{ \left( 1 - \frac{x}{n+1} \right)^{(n+1)/n} \right\}^{n}
\geq \left( 1 - \frac{x}{n} \right)^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/553895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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measurability of metric space valued functions Let's say that we have a measure space $(X, \Sigma)$ and a metric space $(Y, d)$ with its Borel sigma algebra. If $f_n: X\rightarrow Y$ is an arbitrary sequence of measurable functions, then I already know that if $f$ is a pointwise everywhere limit, then $f$ is measurabl... | Not in general. Consider the functions $f_n:X\to X\times [0,1]$ defined by $f_n(x)=(x,1/n)$. This sequence converges everywhere to $x\mapsto (x,0)$. Now I pull a trick: define $Y=(X\times [0,1])\setminus (E\times \{0\})$ where $E\subset X $ is a nonmeasurable set. The functions $f_n:X\to Y$, defined as above, converge ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Judicious guess for the solution of differential equation $y''-2y'+5y=2(\cos t)^2 e^t$ I want to find the solutions of the differential equation: $y''-2y'+5y=2(\cos t)^2 e^t$.
I want to do this with the judicious guessing method and therefore I want to write the right part of the differential equations as the imaginary... | Hint: $(\cos t)^2 = (1 + \cos(2 t))/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/554091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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sufficient condition for a polynomial to have roots in $[0,1]$ Question is to check :
which of the following is sufficient condition for a polynomial
$f(x)=a_0 +a_1x+a_2x^2+\dots +a_nx^n\in \mathbb{R}[x] $ to have a root in $[0,1]$.
*
*$a_0 <0$ and $a_0+a_1+a_2+\dots +a_n >0$
*$a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dot... | for third case we consider polynomial
$F(x)=\frac{a_0}{1.2}x^2+\frac{a_1}{2.3}x^3+\dots + \frac{a_n}{(n+1)(n+2)}a_nx^{n+2}$
we now assume third condition i.e., $\frac{a_0}{1.2}+\frac{a_1}{2.3}+\dots+\frac{a_n}{(n+1).(n+2)} =0$
In that case, for polynomial $F(x)$ we would then have $F(0)=0$ and $F(1)=0$ (with given con... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How to find the distance between two planes? The following show you the whole question.
Find the distance d bewteen two planes
\begin{eqnarray}
\\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\
\end{eqnarray}
Find the other plane $C3\neq C1$ that has the distance d to the plane $C2$.
Ac... | The term "normal" means perpendicularity. A normal vector of a plane in three-dimensional space points in the direction perpendicular to that plane. (This vector is unique up to non-zero multiplies.)
For determining the distance between the planes $C_1$ and $C_2$ you have to understand what is the projection of a vecto... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Show that 2 surfaces are tangent in a given point Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + c^2)$ are tangent at the point $(0, ±b,c)$
To show that 2 surfaces are tan... | The respective gradients of the surfaces are locally perpendicular to them:
$$
\nabla f_1 = 2\left(\frac{x}{a^2}, \frac{y}{b^2}, -\frac{z}{c^2} \right) \\
\nabla f_2 = 2\left(x, y, z - \frac{b^2+c^2}{c}\right)
$$
At any point of common tangency, the gradients are proportional. Therefore $\nabla f_1 = \lambda \nabla f_2... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What really is an indeterminate form? We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?
Furthermore, why can’t we call $ ... | The phrase “indeterminate form” is used to mean a function that we can't compute the limit of by simply applying some general theorem.
One can easily show that, if $\lim_{x\to x_0}f(x)=a$ and $\lim_{x\to x_0}g(x)=b$, then
$$
\lim_{x\to x_0}(f(x)+g(x))=a+b
$$
when $a,b\in\mathbb{R}$. One can also extend this to the case... | {
"language": "en",
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"source": "stackexchange",
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Unitary trivialization over Riemann surfaces with boundary I am puzzled with the proof of Proposition 2.66. in the book "Introduction to Symplectic Topology" by Salamon, McDuff.
The Proposition states, that every Hermitian vector bundle $E \rightarrow \Sigma$ over a compact smooth Riemann surface $\Sigma$ with $\partia... | Perhaps you may use the riemann-cevita connection to move the indenpendent vectors at the origin to the whole disc along radical pathes, which gives you a independent vector field, hence trivalization?
| {
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Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$ I need your assistance with evaluating the integral
$$\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}dx$$
I tried manual integration by parts, but it seemed to only complicate the i... | This is not a full answer but how far I got, maybe someone can complete
$\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}=\frac{1}{\sqrt{2\,x^2}+\sqrt{2\,x^2+1}}=\frac{\sqrt{2\,x^2+1}-\sqrt{2\,x^2}}{\sqrt{2\,x^2+1}^2-\sqrt{2\,x^2}^2}=\sqrt{2\,x^2+1}-\sqrt{2\,x^2}$
So you are looking at
$$\int_0^\infty(\sqrt{\frac{2\,x^2+1}{x^2+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/554624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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Hyperplanes in finite and infinite dimension vector spaces. I know that hyperplanes of a n-dimensional vector space are sub-spaces of dimension n-1, This is in finite dimension spaces. BUT what about infinite dimension spaces what are hyperplanes? are they the same?
| I know this is an old question, but it seems to me that no one has answered it in a "correct" fashion yet, concerning "infinite" of course.
The most generalized definition I've seen is the next one:
Let H be a subspace in a vector space X. H is called hyperplane if $H \neq X$ and for every subspace V such that $ H \su... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $ I'm having trouble calculating the integral
$$\int_{- \infty}^\infty \frac{\sin x}{x+i}\,dx $$
using residue calculus. I've previously encountered expressions of the form
$$\int_{- \infty}^\infty f(x) \sin x \,dx $$
where you would consider $f(z)e^{iz}$ o... | You're on the right track. Rewrite $\sin{x}=(e^{i x}-e^{-i x})/(2 i)$, but evaluate separately. For $e^{i x}$, consider
$$\frac{1}{2 i}\oint_{C_+} dz \frac{e^{i z}}{z+i}$$
where $C_{+}$ is the semicircle of radius $R$ in the upper half plane. The integral is then zero because the pole at $z=-i$ is outside the contou... | {
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"source": "stackexchange",
"question_score": "1",
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Integration: product containing square root. How can I integrate the following expression?
I have tried using u-substitution, but I am having problems with integrating the entire expression. So far, I have the following:
Any help on this is highly appreciated!
| You have it. $u=1+e^{-kt}$ and $du=-ke^{-kt}dt$, so $dt = -\frac{du}{k}$ where $k$ is just a constant. Rewrite this and you have $\int \sqrt{u}\frac{du}{-k}$, which I bet you know how to integrate!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/554917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof: $2^{n-1}(a^n+b^n)>(a+b)^n$ If $n \in \mathbb{N}$ with $n \geq 2$ and $a,b \in \mathbb{R}$ with $a+b >0$ and $a \neq b$, then $$2^{n-1}(a^n+b^n)>(a+b)^n.$$
I tried to do it with induction. The induction basis was no problem but I got stuck in the induction step: $n \to n+1$
$2^n(a^{n+1}+b^{n+1})>(a+b)^{n+1} $
$ \... | You can write that as $$\frac{{{a^n} + {b^n}}}{2} > {\left( {\frac{{a + b}}{2}} \right)^n}$$
Think convexity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/555002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Determination of the prime ideals lying over $2$ in a quadratic order Let $K$ be a quadratic number field.
Let $R$ be an order of $K$, $D$ its discriminant.
By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module.
Let $x_1,\cdots, x_n$ be a sequence of elements of $R$.
We deno... | I realized after I posted this question that the proposition is not correct.
I will prove a corrected version of the proposition.
We need some notation.
Let $\sigma$ be the unique non-identity automorphism of $K/\mathbb{Q}$.
We denote $\sigma(\alpha)$ by $\alpha'$ for $\alpha \in R$.
We denote $\sigma(I)$ by $I'$ for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/555045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find an ordered basis $B$ for $M_{n\times n}(\mathbb{R})$ such that $[T]B$ is a diagonal matrix for $n > 2$ I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference.
"Let T be the linear operator on $M_{n\times n}(\mathbb{R})$ defined by $T(... | Hint: What matrices satisfy $A^\top =\lambda A$ for some $\lambda$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The number of integral solutions for the equation $x-y = x^2 + y^2 - xy$ Find the number of integral solutions for the equation $x-y = x^2 + y^2 - xy$ and the equation of type $x+y = x^2 + y^2 - xy$
| Added: The approach below is ugly: It would be most comfortable to delete.
We look at your second equation. Look first at the case $x\ge 0$, $y\ge 0$. We have $x^2+y^2-xy=(x-y)^2+xy$. Thus $x^2+y^2-xy\ge xy$. So if the equation is to hold, we need $xy\le x+y$.
Note that $xy-x-y=(x-1)(y-1)-1$. The only way we can have... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Twisted sheaf $\mathcal{F}(n)$. Let $\mathcal{F}$ be a sheaf on a scheme $X$ and $O_X(k)$ as usual. We define $\mathcal{F}(n) = \mathcal{F} \otimes_{O_X} O_X(n)$, I don't undertand this definition. What is this tensor product? Then, if we can try to find examples, if $n=1$ and $\mathcal{F}=\mathcal{O}_{\mathbb{A}^1_k}$... | I am sorry, but I think $O_X(n)$ can be defined when the underlying scheme is projective.
So I think the proper example can be $\mathcal{O}_{\mathbb{P}^1_k}(1)
$,and this is just tautological line bundle which is made from the dgree one parts of the oringinal graded ring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/555310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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"answer_id": 2
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What are the most overpowered theorems in mathematics?
What are the most overpowered theorems in mathematics?
By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'm looking for the biggest guns a research mathematician can wield... | there are no theorems whose conclusions contain more that their fully-spelled-out premises.
all purported examples of such just use, e.g., definitions that allow a short syntactic statement of the theorem while shoving all the actual work of it under the rug.
well, unless math is inconsistent at any rate - in which cas... | {
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Proof of natural log identities I need to prove a few of the following identities from a real analysis perspective- this means I do not have access the $\ln e^2 = 2$ type definition of the log function. I am developing the log function from the definition $log x = \int_1^x \frac1t \mathrm dt$ for $0 < x$.
I need to pro... | Let $f(x) = \ln(x)$ and let $g(x) = \ln(ax)$, where $a$ is some constant. We claim that $f$ and $g$ only differ by a constant. To see this, it suffices to prove that they have the same derivative. Indeed, by the fundamental theorem of calculus and chain rule, we have that:
\begin{align*}
g'(x) &= \frac{d}{dx} \ln(ax)
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding an Approximation I am reading the paper A Group-theoretic Approach to Fast Matrix Multiplication and there is an approximation in the paper I don't fully understand.
In the proof of Theorem 3.3. it is stated that
$$
\frac{\ln (n(n+1)/2)!)}{\ln (1!2!\dots n!)}= 2+ \frac{2-\ln 2}{\ln n}+O\left( \frac{1}{\l... | The highest order term in $\log \prod\limits_{k=1}^n k!$ is $\frac{n^2}{2}\log n$, so in the desired
$$\log \left(\frac{n(n+1)}{2}\right){\Large !} = \left(2 + \frac{2-\log 2}{\log n} + O\left(\frac{1}{\log^2 n}\right)\right)\log \prod_{k=1}^n k!,\tag{1}$$
we get on the right hand side a term $O\left(\frac{n^2}{\log n}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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is the number algebraic? Is the number $\alpha=1+\sqrt{2}+\sqrt{3}$ algebraic?
My first attempt was to try a polynomial for which $p(\alpha)=0$ for some $p(x)=a_{0}+a_{1}b_{1}+\cdots +b_{n-1}x^{n-1}$ i. e $x=1+\sqrt{2}+\sqrt{3}$ and then square it many times to get rid of the irrationals. This procedure was futile.
Se... | I can't do this in my head but I hope the pattern is obvious
$$\begin{align}
& (x - 1 - \sqrt{2} - \sqrt{3})(x - 1 - \sqrt{2} + \sqrt{3})
(x - 1 + \sqrt{2} - \sqrt{3})(x - 1 + \sqrt{2} + \sqrt{3})\\
= & x^4-4x^3-4x^2+16x-8
\end{align}$$
In general, given any two algebraic numbers $\alpha, \beta$ with minimal polynomi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/555553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Chaoticity and randomness in a time series Suppose we have a time series : $X=\{X_t,t\in T\}$. How can we check if the data $X_t$ are random or they are the result of some chaotic behaviour of a nonlinear dynamical system? Is there some test useful to prove the chaoticity of the series? Thanks.
| You can apply chaotic time series analysis to the data. There are some useful tools about this subject here. Such implementations try to find positive Lyapunov exponents from the data set based on the studies of Rosenstein et. al. ("A practical method for calculating largest Lyapunov exponents from small data sets") or... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Number of elements in a group and its subgroups (GS 2013)
Every countable group has only countably many distinct subgroups.
The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have considered some counterexample only like $(\mathbb{Z}, +)$, $(\mathbb{Q}... | The countably infinite sum $S$ of copies of the group $G=\mathbb Z/2\mathbb Z$ (cyclic group of order two) indexed by $I$ is countable. Every subset $A$ of the index set $I$ corresponds to a subgroup of $S$ consisting of elements with nonzero components only in the copies of $G$ corresponding to the subset $A\subset I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/555724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Equivalence relation class $\bar{0}$ In the set $\mathbb{Z}$ we define the following relation:
$$a\Re b \iff a\equiv \bmod2\text{ and }a\equiv \bmod3$$
1)Prove that $\Re$ is an equivalence relation. (Done)
2) Describe the equivalence class $\bar{0}$. How many different equivalence classes exist?
My thought on $\bar{0}... | Your thoughts?: exactly.
Now, $a \equiv 0 \pmod 2 $ and $a \equiv 0 \pmod 3 \implies a \equiv 0 \pmod 6$.
So, the equivalence class of $\bar{0}$ is equal to the set of all integer multiples of $6$: $$\bar{0} = \{6k\mid k\in \mathbb Z\}$$
Can you see that the equivalence classes of $\Re$ are the residue classes, modulo ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are some examples of notation that really improved mathematics? I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational conventions that simplify problem statements and ideas, (t... | So - what about fraction notation? Using this:
$$\frac{a+b}{c+d}$$
Instead of this:
$$(a+b)\div(c+d)$$
And to some extent anything else that trades vertical space for horizontal compactness, e.g.:
$$\sum^{10}_{x=1}x^2$$
instead of (for example) $\mathrm{sum}(x,1,10,x\uparrow 2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/555895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "133",
"answer_count": 28,
"answer_id": 6
} |
Proving A Trigonometric Identity- Double Angles $(\cos(2x)-\sin(2x))(\sin(2x)+\cos(2x)) = \cos(4x)$ I'm trying to prove that the left side equals the right side. I'm just stuck on which double angle formula of cosine to use.
| $$(\cos(2x)-\sin(2x))(\sin(2x)+\cos(2x))=(\cos^2(2x)-\sin^2(2x)) = \cos(4x)$$
From
$$\cos(a+b)=\cos a \cos b-\sin a\sin b$$ if $a=2x,b=2x$ then
$$\cos(4x)=\cos^22x-\sin^22x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/555934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there thoughtfully simple concepts that we cannot currently prove? I was driving and just happened to wonder if there existed some concepts that are simple to grasp, yet are not provable via current mathematical techniques. Does anyone know of concepts that fit this criteria?
I imagine the level of simple could var... | we know that $e$ is transcendental, $\pi$ is transcendental. but still we dont know that whether $e + \pi$ and $e - \pi$ is transcendental or not. we know that atleast one of them is transcendental which follows from simple calculation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/556033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Colored blocks and towers
George got a big box from his parents. In this box are colored blocks.
He has white, black, red, blue and orange blocks. These blocks are all
exactly the same size and of he has the same amount of blocks for each
color. George will build towers of $10$ blocks high. Two towers are
equal if the... | For C), we choose the $2$ positions (from the $10$) that will be occupied by colour 1. Then we choose the $2$ positions, from the remaining $8$, that are occupied by colour $2$. And so on.
For B), for every way of selecting where the whites will go, there are $4^4$ ways to fill in the rest. So you need to multiply, not... | {
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"source": "stackexchange",
"question_score": "1",
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Test for convergence/divergence of $\sum_{k=1}^\infty \frac{k^2-1}{k^3+4}$ Given the series
$$\sum_{k=1}^\infty \frac{k^2-1}{k^3+4}.$$
I need to test for convergence/divergence. I can compare this to the series $\sum_{k=1}^\infty \frac{1}{k}$, which diverges.
To use the comparison test, won't I need to show that $... | Hint: For $k\ge3$,
$$
\frac{k^2-1}{k^3+4}\ge\frac1{k+1}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/556414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculating the probability of an event occurring in a specific time period I am confused at how to approach the following question, i.e. what probability formula I am supposed to use.
If the probability of a flood is 0.12 during a year, what is the probability of two floods over the next 10 years...?
I have thought pe... | Correct me if I'm wrong, but wouldn't this just be another Poisson function? Assuming you could have more than 1 flood per year
The expected value --> E(x) = lambda10 = for 10 years = lambda1 * 10 = .12*10 = 1.2 floods in a 10 year period.
So then rerun Poisson's for this new period of 10 years with x = 2 --> P(2) = (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Subgroup of a nilpotent group Let $G$ be nilpotent and $H \le G$. Let $P_1,P_2,\ldots,P_k$ be the Sylow subgroups of $H$. Is it true that $H = P_1 P_2 \cdots P_k$?
I know that when $G$ is nilpotent, it is the direct product of its Sylow subgroups, but is that true for a subgroup of $G$ as well?
| Yes,because a subgroup of a nilpotent group is also nilpotent.
Let $1=G_0\leq G_1\leq ...\leq G_n=G$ be a central series of $G$ that means $[G_i,G]\leq G_{i-1}$.
Let $H_i=H\cap G_i$ .Then $1=H_0\leq H_1\leq....\leq H_n=H$ is a central series of $H$, because $[H_i,H]=[H\cap G_i,H]\leq H\cap [G_i,G]\leq H\cap G_{i-1}=H... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the sum of (n-1)+(n-2)+...+(n-k)? What is the sum of this series ?
$(n-1)+(n-2)+(n-3)+...+(n-k)$
$(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$
So how can we find the sum from $n-1$ to $n-k$ ?
| $$(n-1)+(n-2)\cdots(n-k)=\underbrace{n+n+\cdots +n}_{\text{$k$ copies}}-(1+2+\cdots k)=nk-\frac{k}{2}(k+1)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differentiation of $f(x_1, x_2) = \frac{x_1^2 + x_2^2}{x_2 - x_1 + 2}$ this question might sound stupid to you, but I am having problems right now to differentiate this function:
$$f(x_1, x_2) = \frac{x_1^2 + x_2^2}{x_2 - x_1 + 2}$$
I know the solution, from wolfram alpha, however I do not know how to come up with it b... | You need to use the definition of partial derivative, before getting used to sentences like "treat $x_2$ as constant". Let us do it:
$$\frac{\partial f}{\partial x_1}(x_1,x_2):=\lim_{t\rightarrow 0}\frac{f(x_1+t,x_2)-f(x_1,x_2)}{t}=\lim_{t\rightarrow 0}\frac{(x_1^2+x_2^2)(x_2-x_1+2)-(x_1^2+x_2^2)(x_2-x_1+2)+t(2x_1+t)(x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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some weak conditions about Vitali convergence theorem As we have known,the Vitali convergence theorem is stated:
Let $(X,\mathbb{M},\mu)$ be a positive measure space.If
(i)$\mu(X)<\infty$;
(ii)$\{f_n\}$ is uniformly integrable;
(iii)$f_n(x)\to f(x)~~a.e.as~~n\to\infty$;
(iv)$|f(x)|<\infty~~a.e$;
then $f\in L^1(\mu)$ a... | Here is the strongest version of Vitali's theorem (from O. Kavian, Introduction à la théorie des points critiques, Springer, 1993)
Definition. A sequence $\{f_n\}_n$ in $L^1(\Omega)$ is equi-integrable if the following condition is satisfied: for every $\varepsilon>0$ there exists a measurable set $A$ of finite measure... | {
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What is the equation for a line tangent to a circle from a point outside the circle? I need to know the equation for a line tangent to a circle and through a point outside the circle. I have found a number of solutions which involve specific numbers for the circles equation and the point outside but I need a specific s... | The key point is see that the line that is tangent to the circle and intersects your point forms a triangle with a right angle.
See
https://stackoverflow.com/questions/1351746/find-a-tangent-point-on-circle
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to prove a function has no maximum I have a function:
$$p\cdot(w+6s)^2+(1-p)\cdot(w-s)^2$$
where $p\in(0,1)$, $w>0$ and $s\geq0$ is a choice variable.
I am looking for the maximum of the function with respect to $s$, but can quickly see that the maximum is undefined as the function tends towards infinity as $s$ inc... | Suppose $s \ge 0$. You have
$p (w+6s)^2+(1-p)\cdot(w-s)^2 \ge p (w+6s)^2 \ge p s^2$, hence $\sup_s p (w+6s)^2+(1-p)\cdot(w-s)^2 = \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/557136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Prove that a set is countable Let $A\subseteq \Bbb R$, A is an infinite set of positive numbers.
Suppose, $\exists k \in
\Bbb Z$ for all finite $B\subseteq A$.
$\sum_{b\in B}b<k$
Prove $A$ is countable.
I have a hint: $A_n=\lbrace a \in A | a>\frac1n\rbrace$
I understand that $B \in \Bbb Q$ and $k$ can be the last el... | HINT: Prove that each $A_n$ must be finite, and hence that $A=\bigcup_{n\in\Bbb Z^+}A_n$ is countable. If $A_n$ is infinite, and all of its elements are bigger than $\frac1n$, then there is a finite $B\subseteq A_n$ such that ... ?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Radius and Interval of Convergence of $\sum (-1)^n \frac{ (x+2)^n }{n2^n}$ I have found the radius of convergence $R=2$ and the interval of convergence $I =[-2,2)$ for the following infinite series:
$\sum_{n=1}^\infty (-1)^n \frac{ (x+2)^n }{n2^n}$
Approach:
let
$a_n = (-1)^n \frac{ (x+2)^n }{n2^n}$
Take the ratio test... | The radius of convergence is right. The interval of convergence started out OK, you wrote that for sure we have convergence if $-2\lt x+2\lt 2$. But this should become $-4\lt x\lt 0$.
The endpoint testing was inevitably wrong, since the incorrect endpoints were being tested.
We have convergence at $x=0$ (alternating s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/557319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Choosing a branch for $\log$ when comparing $\prod_{n=1}^\infty(1+a_n)$ and $\sum_{n=1}^\infty \log{(1+a_n)}$ On Ahlfors on p. 191 he is talking about the relation between $\prod_{n=1}^\infty (1+a_n)$ and $\sum_{n=1}^\infty \log(1+a_n)$. He says:
Since the $a_n$ are complex, we must agree on a definite branch of the l... | Since a necessary condition for the product to converge is that the factors converge to $1$, when testing the convergence of $\prod\limits_{n=1}^\infty(1+a_n)$, we can assume that $a_n \to 0$, otherwise the product is trivially divergent. So for all but finitely many terms we have $\lvert a_n\rvert < 1$, and then $\ope... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Bound of power series coefficients of a growth-order-one entire function An entire function $f(z)$ satisfies
$$|f(z)| \leq A_\varepsilon e^{2\pi(M+\varepsilon)|z|}$$
for every positive $\varepsilon$. I want to show that
$$\limsup_{n \to \infty}\ [f^{(n)}(0)]^{1/n} \leq 2\pi M.$$
Alternatively, we can state the result a... | We have
$$\begin{align}
\left\lvert f^{(n)}(0)\right\rvert &= \frac{n!}{2\pi} \left\lvert\int_{\lvert z\rvert = R} \frac{f(z)}{z^{n+1}}\,dz\right\rvert\\
&\leqslant \frac{n!}{2\pi} \int_0^{2\pi} \frac{\lvert f(Re^{i\varphi})\rvert}{R^n}\,d\varphi\\
&\leqslant \frac{n! A_\varepsilon e^{2\pi(M+\varepsilon)R}}{R^n}
\end{a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$QR$ decomposition of rectangular block matrix So I am running an iterative algorithm. I have matrix $W$ of dimensions $n\times p$ which is fixed for every iteration and matrix $\sqrt{3\rho} \boldsymbol{I}$ of dimension $p\times p$ where the $\rho$ parameter changes at every iteration. And for every iteration I need to... | EDIT: in the following treat $\mathbf{W}$ as a square matrix. It is thus zero padded with extra rows of zero and singular values also and otherwise unchanged.
For simplicity I drop the $\sqrt{3\rho}$ and use $\alpha$ instead.
Using the SVD of the matrix $\mathbf{W}$, the problem can be transformed into one of doing Q... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Computing derivative of function between matrices
Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map $\phi(A)=AA^t$. Show that $D\phi(A)$ can be identified with the map $M_... | The derivative at $A$ is a linear map $D\phi(A)$ such that
$$
\frac{\|\phi(A+H)-\phi(A)-D\phi(A)H\|}{\|H\|}\to0\ \ \mbox{ as } H\to0.
$$
(the spirit of this is that $\phi(A+H)-\phi(A)\sim D\phi(A)H$, where one thinks of $H$ as the variable).
In our case, we have
$$
\phi(A+H)-\phi(A)=(A+H)(A+H)^T-AA^T=AH^T+HA^T+HH^T.
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/557623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$ Prove that in a triangle $ABC$, $\angle A = \angle 2B$, if and only if:
$$a^2 = b(b+c)$$
where $a, b, c$ are the sides opposite to $A, B, C$ respectively.
I attacked the problem using the Law of Sines, and tried to prove that if $\angle A$ was indeed equal to $2\... | $$a^2-b^2=bc\implies \sin^2A-\sin^2B=\sin B\sin C\text{ as }R\ne0$$
Now, $\displaystyle\sin^2A-\sin^2B=\sin(A+B)\sin(A-B)=\sin(\pi-C)\sin(A-B)=\sin C\sin(A-B)\ \ \ \ (1)$
$$\implies \sin B\sin C=\sin C\sin(A-B)$$
$$\implies \sin B=\sin(A-B)\text{ as }\sin C\ne0$$
$$\implies B=n\pi+(-1)^n(A-B)\text{ where }n\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/557704",
"timestamp": "2023-03-29T00:00:00",
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Polynomials: irreducibility $\iff$ no zeros in F. Given is the polynomial $f \in F[x]$, with $deg(f)=3$. I have to prove, that f is irreducible iff $f$ has no zeros in $F$.
"$\Rightarrow$": let's prove the contrapositive: "if $f$ has zeros in $F$, then $f$ is not irreducible."
If $f$ has zeros in $F$, it means there's ... | If $f(x)$ is a polynomial and $g(x)$ is any nonzero polynomial, you can always write $f(x)=q(x)g(x)+r(x)$ with polynomials $q(x)$, $r(x)$, $\deg r<\deg g$ (polynomial division with remainder).
Especially, if you let $g(x)=x-a$ be a linear polynomial,you obtain $f(x)=q(x)\cdot(x-a)+r$ where $r$ is constant. If $a$ happe... | {
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"source": "stackexchange",
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Integral converges/diverges Question:
$$p \in \Bbb R \int _0 ^\infty \frac {\sin x}{x^p}dx$$
For what values of p does this integral converge/diverge?
Thoughts
We've tried using Dirichlet criteria and bounding it from below, after splitting it from $[0,1]$ and $[1.\infty]$ There seems to be some kind of trick we are p... | For $p = 1$, it is convergent. You can show it by using.
$$
\int_{1}^{b}\frac{\text{sin}\;x}{x}dx = -\frac{\text{cos}\; b}{b} + \text{cos}\;1 - \int_{1}^{b}\frac{\text{cos}\;x}{x^2}dx
$$
The point is, you can show by using integral by parts.
| {
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"timestamp": "2023-03-29T00:00:00",
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What am I doing wrong? Integration and limits I need some help identifying what I'm doing wrong here..
What is the limit of $y(x)$ when $x→∞$ if $y$ is given by:
$$y(x) = 10 + \int_0^x \frac{22(y(t))^2}{1 + t^2}\,dt$$
What i've done:
1) Integrating on both sides(and using the Fundamental Theorem of Calculus):
$$\frac{d... | To add my comment above, the answer should be $\frac{-1}{22y} + \frac{1}{22\cdot 10} = \text{atan}(x)$. This reduces to $y(x) = \frac{-1}{22}\cdot (\text{atan}(x) - \frac{1}{220})^{-1}$. On a rigorous level, we know that this holds since $y(x)\geq 10\; \forall\; x$. Now $\text{lim}\; f(x) = \frac{-1}{22}\cdot (\frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/557994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is my reasoning correct here regarding uniform convergence? Let $f_n(x) = x^n$.
Is this sequence of functions uniformly convergent on the closed interval $[0, 1]$?
My reasoning
Consider $0 \le x < 1$. $f_n(x)$ converges pointwise to the zero function on this interval.
$f_n(x)$ is uniformly convergent if there exists so... | You're are certainly very close to a full (and correct!) solution. Let's first straighten a couple things out:
1) The $N$ you choose can depend on $\epsilon$. That is, uniform convergence of $f_n \to f$ means: given an $\epsilon >0$ there exists an $N \in \mathbb{N}$ such that for all $x$ in the domain of $f$ it holds... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/558076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
estimation of a parameter The question is:
$x_i = \alpha + \omega_i, $ for $i = 1, \ldots, n.$
where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, Gaussian random variable with known variance $\sigma_i^2$. Note that $\sigma_i^2$ and $\sigma_j^2$, fo... | For the unbiasedness, we have
$$
E\left(\hat{\alpha}\right)=E\left(\sum_{i=1}^nb_ix_i\right)=E\left(\sum_{i=1}^nb_i(\alpha+\omega_i)\right)=\alpha\sum_{i=1}^nb_i + E\left(\sum_{i=1}^nb_i\omega_i\right)=\alpha\sum_{i=1}^nb_i
$$
and we get that $\sum_{i=1}^nb_i=1$ as you say.
Now, what follows is to simply make this homo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/558151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find the sum of the series For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$
Find $\sum_{n = - \infty}^{\infty} f(n)$.
I do not know how to solve such problem of series. So I could not tr... | HINT:
Using Fermat's Little theorem, $$n^p-n\equiv0\pmod p$$ where $p$ is any prime and $n$ is any integer
$\displaystyle\implies n^7-n\equiv0\pmod 7\implies \frac{n^7-n}7$ is an integer
Show that $k(n)$ is integer for all integer $n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/560219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$ Prove that the function $\sqrt{x}$ is continuous on its domain $[0,\infty)$.
Proof.
Since $\sqrt{0} = 0, $ we consider the function $\sqrt{x}$ on $[a,\infty)$ where $a$ is real number and $s \neq 0.$ Let $\delta=2\sqrt{a}\epsilon.$ Then, $\forall x \in do... | Prove that $\sqrt{x}$ is continuous on its domain $[0,\infty[$
I have used the following definition and therom:
Definition 1:
A sequence of real numbers {$x_n$} is said to converge to a real number $a \in R$ if and only if for every $\epsilon>0$ there is an N $\in$ N such that
$n \geq N$ implies $|x_n-a|<\epsilon$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/560307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
} |
Defining irreducible polynomials recursively: how far can we go? Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such that $a_r\in \mathbb N$ is the smallest such that $p_r$ is... | The answer to your first question is YES (and a similar
method can probably answer positively your second
question as well). One can use the following lemma :
Lemma Let $n,g$ be positive integers. Then there is a polynomial
$Q_{n,g}\in {\mathbb Z}[X]$ of degree $\leq n$ such that
$Q_{n,g}(i)=i^g$ for any integer $i$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/560383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Show $|1-e^{ix}|^2=2(1-\cos x)$ Show $|1-e^{ix}|^2=2(1-\cos x)$
$$|1-e^{ix}||1-e^{ix}|=1-2e^{ix}+e^{2ix}=e^{ix}(e^{-ix}-2+e^{ix})=e^{ix}(2\cos x-2)=-2e^{ix}(1-\cos x)$$
Not sure how they got rid of the $-e^{ix}$ factor. Did I expand the absolute values wrong? thank you
| $$ |1-e^{ix}|^{2} = |1-\cos x - i \sin x|^{2} = (1-\cos x)^{2} + (-\sin x)^{2} = 1 - 2 \cos x + \cos^{2} x + \sin^{2} x $$
$$= 2 - 2 \cos x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/560461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Number of ways of sorting distinct elements into 4 sets This was on a test I just had. The first part says:
"A person donates nine antique clocks to four different museums. Supposing all clocks are identical and he can distribute them in any way he chooses, how many ways are there of donating the clocks"?
This is choos... | Every clock goes to one of four museums, so there are $4$ choices for the first clock, $4$ for the second,... for a total of $4^9$ in all by the rule of product for combinatorics.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/560565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Adding one Random Real I'm studying Kanamori's book The Higher Infinite and now I'm stuck.
I want to prove that forcing with Borel sets of positive measure adds one random real.
Let me state the theorem:
Let $M$ be a countable transitive model for ZFC.
$\mathcal{B} = \{A \in Bor(\omega^\omega): \mu(A) > 0\}$ and $p \l... | Let $A_k=\{f\in\omega^\omega: f(n)=k\}$. Fix $C\in \mathcal B^M,$ note that $\bigcup_{k\in\omega}A_k=\omega^\omega$ thus $C=C\cap (\bigcup_{k\in\omega}A_k)=\bigcup_{k\in\omega}(C\cap A_k)$. It follows that there is a $k$ so that $C\cap A_k$ has positive measure. Hence $C\cap A_k\leq C$ and satisfies $(*)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/560635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find the maximum and minimum values of $A \cos t + B \sin t$ Let $A$ and $B$ be constants. Find the maximum and minimum values of $A \cos t + B \sin t$.
I differentiated the function and found the solution to it as follows:
$f'(x)= B \cos t - A \sin t$
$B \cos t - A \sin t = 0 $
$t = \cot^{-1}(\frac{A}{B})+\pi n$
Howe... | $A\cos t+ B \sin t = \sqrt{A^2+B^2} ( \frac{A}{\sqrt{A^2+B^2}} \cos t + \frac{B}{\sqrt{A^2+B^2}} \sin t)$. Choose $\theta$ such that $e^{i \theta} = \frac{A}{\sqrt{A^2+B^2}} + i\frac{B}{\sqrt{A^2+B^2}} $. Then
$A\cos t+ B \sin t = \sqrt{A^2+B^2} ( \cos \theta \cos t + \sin \theta \sin t) = \sqrt{A^2+B^2} \cos(\theta-t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/560711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Holomorphic function having finitely many zeros in the open unit disc Suppose $f$ is continuous on the closed unit disc $\overline{\mathbb{D}}$ and is holomorphic on the open unit disc $\mathbb{D}$. Must $f$ have finitely many zeros in $\mathbb{D}$? I know that this is true if $f$ is holomorphic in $\overline{\mathbb{D... | Any accumulation point of the zeros must be on the boundary, of course. You can show in this case that the series centered at 0 has radius of convergence 1. For a specific example, I think the following works:
For $|z| \leq 1$, $$f(z) = (z-1)\prod_{n=1}^\infty\frac{z-1+1/n^2}{1-(1-1/n^2)z}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/560784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to solve this mathematically This is a question given in my computer science class. We are given a global variable $5$. Then we are to use keyboard event handlers to do the following:
On event keydown double the variable and on event keyup subtract $3$ from this variable.
The question is after $12$ presses of any k... | Limits are way more power than necessary.
Assuming $x_0$ is really $5$, we're looking at the recurrence
\begin{gather*}
x_0=5 \\[0.3ex]
x_{n+1}=2x_n-3.
\end{gather*}
The steady state is $x=2x-3\implies x=3$, suggesting we write
$$x_{n+1}-3=2(x_n-3),$$ which implies
giving $x_n-3=(x_0-3) 2^n$, or
$$x_n=3+(5-2)\cdot 2^n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/560828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to divide a circle with two perpendicular chords to minimize (and maximize) the following expression Consider a circle with two perpendicular chords, dividing the circle into four regions $X, Y, Z, W$(labeled):
What is the maximum and minimum possible value of
$$\frac{A(X) + A(Z)}{A(W) + A(Y)}$$
where $A(I)$ deno... | When the considered quotient is not constant it has a maximum value $\mu>1$, and the minimum value is then ${1\over \mu}$. I claim that
$$\mu={\pi+2\over\pi-2}\doteq4.504\ ,\tag{1}$$
as conjectured by MvG.
Proof. I'm referring to the above figure. Since the sum of the four areas is constant the quotient in question i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/560929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Complex numbers problem I have to solve
where n is equal to n=80996.
| $$\frac{\sqrt3+5i}{4+2\sqrt3i}=\frac{\sqrt3+5i}{4+2\sqrt3i}\cdot\frac{4-2\sqrt3i}{4-2\sqrt3i}=\frac{14\sqrt3+14i}{28}=\frac{\sqrt3+i}2=\cos\frac{\pi}6+i\sin\frac{\pi}6;$$$$\left(\frac{\sqrt3+5i}{4+2\sqrt3i}\right)^n=\left(\cos\frac{\pi}6+i\sin\frac{\pi}6\right)^n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/561013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
If $3c$ is a perfect square then why must $c$ be of form $3x^2$ for some $x$? ( $x,c$ integers) For example $3c=36$,
then $36=3*4^2$ so $x=4$.
| Primes (such as $3$) have the property that if they divide a product, they divide (at least) one of the factors: If $p\mid ab$ then $p\mid a$ or $p\mid b$.
Here we take $p=3$ and from the assumption $3c=m^2=m\cdot m$ conclude that $3$ divides one of the two factors on the right, that is $3\mid m$ at any rate. So $m=3x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/561116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Simple Algebra Equation I have a simple part of a question to solve. The problem is my answer is different to the solution in my textbook.
The equation is: $$\frac{5v}{6} = \frac{(\frac{1}{2}a+b+\frac{1}{2} c)v}{a+b+c}$$
I am supposed to get $$\frac{2}{3}(a+b+c) = b$$
But I simply get: $$b=2a +2c$$
I get my answer by ... | $b=2a+2c$
Adding $2b$ on both sides gives
$b + 2b =2a+2b+2c$
Or better
$3b =2a+2b+2c$
And if you throw the $3$ to the other side you get:
$b =\frac{2}{3}(a+b+c)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/561210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding abundant numbers from 1 to 10 million using a sum my task is to implement algorithm in C of finding abudant numbers from 1 to 10 million. Fistly I don't really understand mathematics.
There is several ways how to do it, but efficient and fast (for that BIG input 10 mil) might be by summing - NOT dividing, NOT m... | The following naive sieve approach takes only 30 seconds on my modest PC.
#define abLimit 10000000
int i, j, *xp;
xp = (int*) calloc(abLimit, 4);
for (i=2; i<abLimit, i++)
for (j=i*2; j<abLimit; j+= i)
xp[j] += i;
for (i=2; i<abLimit; i++)
if (xp[i] > i)
printf("%d is abundant\n", i);
free(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/561328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Does $f_n(x) = \chi_{(n, n+1)}$ converge uniformly on $\mathbb{R}$ Does $f_n(x) = \chi_{(n, n+1)}$ converge uniformly to the zero function on $\mathbb{R}$?
It seems to me that it doesn't.
Say $\exists$ $N$ such that for all $n \ge N$
$$|\chi_{(n, n+1)} - 0|< \epsilon$$ with $\epsilon > 0$
i.e. $|\chi_{(n, n+1)}|< \epsi... | You can prove this by the definition.
For $\epsilon_0=\frac12$, for any $N\in \mathbb{N}$, choose $n_0=N+1>N $ and $x_0=N+\frac32$, then $|f_{n_0}(x_0)|=|\chi_{(N+1, N+2)}(N+\frac32)|=1>\epsilon_0$. This means that $f_n$ does not uniformly converges on $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/561430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\int_{0}^{1}{\frac{\sin{x}}{x}\mathrm dx}$ converges As title says, I need to show that the following integral converges, and I can honestly say I don't really have an idea of where to start. I tried evaluating it using integration by parts, but that only left me with an $I = I$ situation.
$$\int \limits_{0... | Notice that, for all $0 < x < 1$, we have
\begin{eqnarray*}
\left|\int_0^1 \frac{\sin x}{x} \, \operatorname{d}\!x\right| &\le& \int_0^1 \left|\frac{\sin x}{x} \right| \operatorname{d}\!x \\ \\
&\le& \int_0^1 1 \, \operatorname{d}\!x \\ \\
&\le& 1
\end{eqnarray*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/561547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
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