Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Negate the following statements formally so that **no negation symbol remains:** Negate the following statements formally so that no negation symbol remains:
(i) $ \ \ \forall y \exists x (y>0 \to x \leq 0 ) \ $
(ii) $ \ \forall x \forall y \exists z(x <z \leq y ) \ $
Answer:
My approach is as follows:
(i)
The negati... | Your answer to (i) is wrong. The correct negation of $A\rightarrow B$ is $A\wedge \neg B$ so in your case, it should be
$\exists y \forall x (y>0) \wedge (x>0)$
which is not equivalent to the answer you have of $\exists y \forall x (x>0 \rightarrow y\leq 0)$. Your sentence is equivalent to $\exists y \forall x ( y\leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Is Scrabble's method of determining turn order fair? At least the way my family plays, turn order is determined by drawing tiles and seeing who has the letter closest to A (blanks taking precedence). If I recall correctly, there are 100 unevenly distributed letters.
Intuitively, it seems like it would be unfair, though... | Look at a toy problem. Suppose there are two players and three tiles: $A_{1}$, $A_{2}$, and $B$. The subscripts on the $A$,s indicate there are two $A$ tiles. Look at all the possible outcomes: the first letter is what the first player draws, the second letter is what the second player draws and the third letter is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 6,
"answer_id": 1
} |
Hamiltonian from Lagrangian $L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2}$ I'm doing the first exercises with the Lagrangians and Hamiltonians.
Let:
$$L= \frac{m}{2}(\dot{r}^2+r^2\dot{\theta^2})+ \frac{k\cos(\theta)}{r^2}$$
$$p_1=m\dot{r}$$
$$p_2=mr^2\dot{\theta}$$
$$H=\frac{p^2}{2m}-\frac{k\... | The answer given by Harry49 is correct but I think that you should also understand the more general approach to the problem. It is not always true that the Hamiltonian $H$ is equal to $T+V$; there are some features that the Lagrangian must obey and one should be careful before stating this equality. In the case above t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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solution verification for biased dice I have the following problem for which I'm not sure my solution is correct:
A dice is constructed in such a way that 1 dot occurs twice more often than the rest of the points. The probabilities for the rest of the dots are mutually equal. The dice is thrown 2 times.
Calculate the p... | I don't think the term $\frac57\frac57\frac{10}{49}$ is right. $\frac57\frac57$ is the probability that both are greater than $1$, but if that happens the probability that $i<j$ is quite a bit bigger than $\frac{10}{49}$.
The right way to approach this is that the probability of the second die showing more dots is equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to expand formula for covariance matrix? I have seen an expression for the covariance of not of just two random variables but the whole covariance matrix $\mathbf{K}$.
$$\mathbf{K} = E\bigg((\mathbf{x} - E(\mathbf{x}))(\mathbf{x} - E(\mathbf{x}))^{T}\bigg) $$ where $\mathbf{x}$ is a vector of random variables $(\ma... | The covariance between, say $x_j, x_k$ is given by
\begin{align}
\sigma_{jk} &= \text{Cov}(x_j, x_k) \\
&= \mathbb{E}[(x_j-\mu_j)(x_k-\mu_k)]\\
&= \mathbb{E}(x_j x_k)-\mu_j \mu_k \\
\end{align}
Clearly, when $k=j$ we obtain the variance.
$$\sigma_{jj} = \mathbb{E}[(x_j-\mu_j)^2]$$
For $k$-vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Combinatorics: n people visit k exhibitions Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?
The question is as follows :
n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities... | Arranging $n$ stars and $k-1$ bars is a way of modeling this situation. Each arrangement of those $n+k-1$ symbols gives us a set of numbers of people to go to each exhibition. However, the result of that counting problem is:
$$\binom{n+k-1}{k-1}=\frac{(n+k-1)!}{n!(k-1)!}$$
Do we have a condition that each exhibition ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2410118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that two summations are equivalent
Show that
$$
\sum_{i=1}^n \left(x_i - \bar{x}\right) \left(y_i - \bar{y}\right)
= \left(\sum_{i=1}^n x_i y_i\right) - n \bar{x} \bar{y}.
$$
My work is attached:
.
I'm stuck on what I should do next.
Any guidance in the right direction would be great!
| You are correct. Recall that by definition $\overline{x}=\frac{1}{n}\sum_{i=1}^nx_i$ and $\overline{y}=\frac{1}{n}\sum_{i=1}^ny_i$.
Therefore by linearity,
$$\sum_{i=1}^n\overline{y}x_i=\overline{y}\sum_{i=1}^nx_i=n\overline{x}\overline{y},\quad
\sum_{i=1}^n\overline{x}y_i=\overline{x}\sum_{i=1}^ny_i=n\overline{x}\ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2410235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proving that $\{x\in\Bbb{R}\mid 1+x+x^2 = 0\} = \varnothing$ without the quadratic formula and without calculus I'm asked to prove that $\{x\in\Bbb{R}\mid 1+x+x^2 = 0\} = \varnothing$ in an algebra textbook.
The formula for the real roots of a second degree polynomial is not introduced yet. And the book is written wit... | $x=0$ is not a root, so divide by $x \ne 0$ and write the equation as:
$$
x+\frac{1}{x} = -1
$$
This requires $x$ to be negative for the LHS to be negative, but then $y=-x$ is positive and $\displaystyle y+\frac{1}{y} \ge 2$ by AM-GM, so $\displaystyle x+\frac{1}{x} \le -2 \lt -1\,$, therefore there are no real solutio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 14,
"answer_id": 12
} |
A tricky integration problem Given: $$f(x)=\int_{x}^{0}\frac{\cos(xt)}t\, dt.$$ What is $f'(x)?$
| This is an improper integral, as $t$ approaches $0$, the numerator approaches $1$ while the denominator approaches $0$.
You must check that the limit doesn't diverge!
$$
f(x)=\lim_{c\rightarrow 0^{\pm}}\int_x^c\frac{\cos(xt)}{t}dt
$$
where you use $\pm$ on $0$ depending on the sign of $x$.
Therefore, near $0$, you're i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2410445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Baby Rudin Theorem 2.27c I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:
If $X$ is a metric space and $E\subset X$, then $\overline{E}\subset F$ for every closed set $F\subset X$ such that $E\subset F$. Note: $\overline{E}$ denotes the closure of $E$; in other wo... | If $x$ is a limit point of $E$ then $x = \lim x_n$ for some sequence $x_n \in E \setminus \{x\}$. If $E \subseteq F$ then $x_n \in F \setminus \{x\}$ so we can also say that $x$ is a limit point of $F$. Therefore
$$ E' \subseteq F' \subseteq F. $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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net convergence implies bounded? I just saw the following theorem:
Theorem Let $\alpha:[a,b] \to \mathbb{R}$ be a mapping. If the Riemann-Stieltjes integral $$I(f) := \int_a^b f(t) \, d\alpha(t)$$ exists for all continuous functions $f:[a,b] \to \mathbb{R}$, then $\alpha$ is of bounded variation.
in this answer.
but ... | You can take limits along one sequence of partitions with the norms (i.e. the maximum of the lengths of subintervals) tending to zero. There is no need to use nets here since you are not proving the existence of limits of Riemann Steiltje sums.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving an nonlinear ODE $$y'y'' = ky^2$$
I need a closed form expresion, if not atleast an almost closed form expression such as an inverse of an integral characterization. What could be the properties of its solutions?
| Multiply both sides by $y'$ and integrate to get $y'^3=ky^3+C$. This integrates again to give a very ugly hypergeometric function (according to Wolfram alpha). So just hope your boundary conditions make C=0 to give An aedonist's solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2410825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Sum of nonzero squares modulo p It is easy to prove that for prime $p$ every element of $\mathbb{Z} / p \mathbb{Z}$ can be written as the sum of two squares. An elementary proof is given here: Sum of two squares modulo p
How can we show that, provided further $p \geq 7$, any nonzero element of $\mathbb{Z}/ p \mathbb{Z}... | Adapting Mikhail Ivanov's argument to a slightly different but AFAICT equivalent question to fit here. Some of the elements appeared also in my answer to that question.
Every non-zero element of $\Bbb{Z}/p\Bbb{Z}$ is either a square or a non-square
If $a=b^2$ is a non-zero square, then, as $p>5$ we have
$$
a=b^2=(3b/5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2410920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Complex structure through two involutions The standard algebraic definition of a complex structure $I$ is $I^2=-1$. On real pairs $(a,b)$ it is represented like $(a,b)\mapsto (-b,a)$. But what if we have not had the negative pairs of reals $-(a,b)=(-a,-b)\quad\Leftrightarrow\quad I^2=-1$ and had only the single real i... | I'm not sure exactly what you're asking. On the surface, it looks like you're excited about the factorization $\begin{bmatrix}-1&0\\0&1\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=\begin{bmatrix}-b\\a\end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Is the span of any 3 n-dimensional, linearly independent vectors, $\mathbb R^3$? I am not sure if the span of a set of any 3 (n-dimensional) vectors that are linearly independent is $\mathbb R^3$.
I think, since any two (2 dimensional vectors) that are independent always span $\mathbb R^2$, that if the dimension of th... | Any two $n$-dimensional vector spaces (over the same field ) are isomorphic. .. It would be easy to write down an isomorphism: just send basis vectors to basis vectors. .. i.e. $\mathcal i:V\to W$ by $\mathcal i (v_i)=w_i $ where $\{v_1, \dots, v_n\} $ and $\{w_1, \dots,v_n\} $are bases and extend linearly. ..
Secon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration by parts to prove a function is constant a.e. Let $(a,b)$ be an interval on $\mathbb{R}$. Let $f \in L^1(a,b)$. Assume that $$
\int_a^b f(x)g'(x)\, dx =0
$$
for all $C^1$ functions $g$ with support compactly contained in $(a,b)$. Prove that there is a constant $c$ such that $f(x)=c$ for almost every $x \in ... | Hint: If you happen to know that
$$ \int_a^b f(x) h(x) dx = 0 , \ \ \forall h\in C_c((a,b)): \int_a^b h\; dx=1$$
implies that $f$ vanish
then you may reduce to this situation by considering the difference
of two such $h$'s. One of them will give rise to the constant $c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Function between two measurable functions is measurable Let $f$ be a function on $\mathbb{R}^n$. Assume that for any $\epsilon>0$, there exist measurable functions $g, h \in L^1(\mathbb{R}^n)$ such that $g(x) \leq f(x) \leq h(x)$ for all $x \in \mathbb{R}^n$ and $$
\int_{\mathbb{R}^n} (h(x)-g(x))\, dx < \epsilon
$$
Pro... | (1) Measurability of $f(x)$. Thanks to the hint of @Robert Israel
Let $g_n, h_n \in L^1$ be such that $h_n(x) \leq f(x) \leq g_n(x)$ for all $x$ and $\int g_n(x)-h_n(x)\, dx < 1/n$. Let $h(x) = \limsup h_n(x)$ and $g(x) = \liminf g_n(x)$, so $h(x) \leq f(x) \leq g(x)$ for all $x$, and $h, g$ are measurable. Moreover, b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Find A and B in this limit Can you find $a$ and $b$? In how many ways can I find them?
$$\lim_{x\to0} \frac{a+\cos(bx)}{x^2}=-8$$
| For
$\lim_{x\to 0}\frac{\ a+\cos bx}{x^2}=-8
$,
note that,
for small $x$,
$\cos x
\approx 1-\frac{x^2}{2}
$.
Therefore
$\dfrac{\ a+\cos bx}{x^2}
\approx
\dfrac{\ a+1-\frac{(bx)^2}{2}}{x^2}
$.
If
$a+1 \ne 0$,
then
$\dfrac{\ a+1-\frac{(bx)^2}{2}}{x^2}
\to \infty$
as $x \to 0$.
Therefore,
to have the limit exist,
we must ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Partial Fraction Decomposition with Complex Number $\frac{1}{z^2 - 2i}$. How do I decompose the fraction
$$\dfrac{1}{z^2 - 2i}$$
into partial fractions? I understand how to do partial fraction decomposition with real numbers, but I am unsure of how to do it with complex numbers. I attempted to find examples online, b... | Note that $z^2-2i=(z+\sqrt{2i})(z-\sqrt{2i})$ and $\sqrt{2i}=\sqrt{2}e^{i\pi/4}=1+i$. To simplify, let $b=1+i$, then $$\frac{1}{z^2-2i}=\frac{1}{(z+b)(z-b)}$$ From here it actually doesn't matter if you regard $b$ as real or complex, the process to find the partial fractions is the same as long as the terms are linear ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Independent odds, am I (+ friend) seeing this wrong or is there a mistake in the practice exam? I found this exercise in a practice exam:
Any student has a 90% chance of entering a University. Two students
are applying. Assuming each student’s results are independent, what is
the probability that at least one of t... | That's the standard way of doing it:
$P(\text{At Least One}) = 1 - P(\text{Both Fail}) = 1 - 0.1 \times 0.1 = 1 - 0.01 = 0.99$
And the not so standard way:
$P(1\text{ in}) + P(2\text{ in Without }1) = 0.9 + (0.9 \times 0.1) = .99$.
Or
$$\begin{align}
P(1\text{ in Without }2) + P(2\text{in Without }1) + P(1\text{ in An... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
"answer_id": 0
} |
Better way to reduce $17^{136}\bmod 21$? What I have done:
Note $17\equiv -4$ mod 21, and $(-4)^2 \equiv 5$ mod 21. So $17^{136} \equiv (-4)^{136} \equiv 5^{68}$ mod 21. Also note $5^2 \equiv 4$ mod 21 and $4^3 \equiv 1$ mod 21, so $5^{68} \equiv 4^{34} \equiv (4^3)^{11}\cdot4 \equiv 4$ mod 21. I feel this is rather co... | You could also reduce modulo each of the factors of $21$ and then use the Chinese Remainder Theorem to recover the result modulo $21$.
$$
17^{136} \cong 2^{136} \cong (2^2)^{68} \cong 1^{68} \cong 1 \pmod{3}
$$
and
$$
17^{136} \cong 3^{136} \cong 3^{6 \cdot 22 + 4} \cong (3^6)^{22} \cdot 3^4 \cong 1^{22} \cdot 3^4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Does it follow that $\sup\emptyset = 0$ if the domain is $\mathbb{R}_{\geq 0} \cup \{\infty\}$? I am currently working on a paper where I am taking $\min$s of $\max$s over sets of non-negative real numbers including positive infinity, i.e. my equations look something like $\min\max\{x_1,\ldots,x_n\}$ where $x_i \in \ma... | Yes: Every number is an upper bound of the empty set, and $0$ is the least such number in your domain.
(By contrast, $\max\emptyset$ doesn't exist by the usual definitions. If you really wanted to define $\max\emptyset$, you would have to extend the definition of $\max$ in a way that risks being misleading.)
If your co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2411919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate a limit using l'Hospital rule Evaluate $$\lim_{x\to0} \frac{e^x-x-1}{3(e^x-\frac{x^2}{2}-x-1)^{\frac{2}{3}}}$$
I tried to apply l'Hospital rule in order to get the limit to be equal to
$$\lim_{x\to0}\frac{e^x-1}{2(e^x-\frac{x^2}{2}-x-1)^{-\frac{1}{3}}(e^x-x-1)}$$
but the new denominator has an indeterminate fo... | You can use $e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+o(x^5)$
$$\qquad{\lim_{x\to0} \frac{e^x-x-1}{3(e^x-\frac{x^2}{2}-x-1)^{\frac{2}{3}}}=\\
\lim_{x\to0} \frac{(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+o(x^5))-x-1}{3((1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+o(x^5))-\frac{x^2}{2}-x-1)^{\frac{2}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Why is this deterministic variant of Miller-Rabin not working? I am using this paper as a reference.
The Miller-Rabin test, as classically formulated, is non-deterministic -- you pick a base $b$, check if your number $n$ is a $b$-strong probable prime ($b$-SPRP), and if it is, your number is probably prime (repeat unti... | If you look at Best known SPRP base sets, you can see the remark "Depending on your Miller-Rabin implementation, you may need to take a ← a mod n." and "When the witness a equals 0, the test should return that n is prime." The latter is saying we skip that test when n divides the base.
This is especially critical for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Suppose we roll a fair $6$ sided die repeatedly. Find the expected number of rolls required to see $3$ of the same number in succession.
Suppose we roll a fair six sided die repeatedly.
Find the expected number of rolls required to see $3$ of the same number in
succession
From the link below, I learned that $258$ rol... | From Did's answer here, the probability generating function $u_0(s)=\mathbb{E}(s^T)$ for
the number of trials $T$ needed to get three consecutive values the same is
$$u_0(s)={s^3\over 36-30s-5s^2}.$$ Differentiating this and setting $s=1$ in the
derivative shows that $\mathbb{E}(T)=43.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Why alternating power series can be truncated to created upper and lower bounds for functions? I often see functions that can be represented by alternating power series like this
\begin{align}
f(x) =\sum_{i=0}^\infty(-1)^i a_i x^i,
\end{align}
being upper and lower bounded by a truncated power series. For example, the ... | There is a theorem about constant alternating series $\sum_{k=0}^\infty (-1)^k c_k$ with positive $c_k$ monotonically decreasing to $0$. Such series are automatically convergent to a finite sum $s\in{\mathbb R}$. Furthermore the even partial sums $s_{2m}:=\sum_{k=0}^{2m}(-1)^k c_k$ are all larger than $s$, and the odd ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The max norm $|. |_{\infty}$ is generating a metrics
I want to prove that the max norm $\|\cdot \|_{\infty}$ generates in the
space of continuos functions $C([a,b])$ a metrics.
Well, I am not sure how to properly make it. I just know that if I have two functions $f$ and $g$ then for all $x \in [a,b]$ a I can find t... | You are, I think, trying to say:
The norm $\|\cdot\|_\infty$ generates a metric on $C([a,b])$ by $$d(f,g) = \|f-g\| = \max_{x \in [a,b]} \{| f(x) -g(x) |\}$$
This is true, as any norm $\|\cdot\|$ on a linear space defines a metric in this way (norm of the difference).
We use $\|-f\| = \|f\|$ for symmetry, $\|f\| = 0 \i... | {
"language": "en",
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A question about asymptotic notations with sums. I need to prove that $$ \sum_{k=0}^{n-2017} \binom{n}{k} \binom{2k}{k} \frac{ \sqrt{k}}{2^{k}} = \Theta(3^{n})$$
I'm pretty sure it's straightforward to prove that it's $\Omega(3^{n})$ but I'm not sure how to prove the $O(3^{n})$ part. Maybe with using square root and de... | *
*For the upper bound,, note that
$$\begin{align}
\sum_{k=0}^{n-2017} \binom{n}{k} \binom{2k}{k} \frac{ \sqrt{k}}{2^{k}}
&\leq \sum_{k=0}^{n} \binom{n}{k} \binom{2k}{k} \frac{ \sqrt{k}}{2^{k}}
\leq \sum_{k=0}^{n} \binom{n}{k} \frac{2^{2k}}{\sqrt{k}} \frac{ \sqrt{k}}{2^{k}}
\\&= \sum_{k=0}^{n} \binom{n}{k} 2^k
= (1+2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Sylow's theorem proof Sylow's theorems state that the $p$-Sylow subgroups exist for a group G of order $p^km$, where $p$ is prime and does not divide $m$. My question is how to prove that there is at least 1 subgroup of order $p^n$ exists for every non-negative integer $n \le k$.
| Actually, there exists a normal subgroup of order $p^n$ for any group of order $p^k$, where $n\leq k$. We know that the center of a nontrivial $p$-group is nontrivial, then you can use induction to prove the fact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
Inequality for Fibonacci to find an upper bound of harmonic Fibonacci series I want to find an sharp upper bound for $$\sum_{n=1}^{\infty}\frac{1}{F_n}$$which $F_n $ is the n$th$ term of Fibonacci sequence .
I wrote a Matlab program to find an upper bound ,$\sum_{n=1}^{10^6}\frac{1}{F_n}<4$
Now my question is:(1):Is ... | Using Wolfram Mathematica answer is :
Wolfram Mathematica Code
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2412980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the Fourier Transform of a function that is periodic in an interval only? I know how to find the Fourier Series for a periodic function (Periodic for all inputs, from the definition). I also know how to find the Fourier Transform for non periodic functions. But, Which formula to use to calculate the Fourier... | Let be
$$
\mathcal F\{f(t)\}=F(\omega)=\int_{-\infty}^\infty f(t)\mathrm e^{-i\omega t}\mathrm d t=\int_{0}^Mf(t)\mathrm e^{-i\omega t}\mathrm d t
$$
Let be $f_0(t)$ the basis function in the interval $[0,\, r]$ and $M=nr$ for some $n\ge 1$. We have
$$
f(t)=\sum_{k=0}^{n-1}f_0(t-kr)
$$
and then
$$
\begin{align}
F(\omeg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Understanding SVD Notation Given any $m \times n$ matrix $M$, one can write
$$
M = U\Sigma V^T
$$
is the Singular Value Decomposition, where $U$ and $V$ are orthonormal and $\Sigma$ is a diagonal matrix.
Now, the same $M$ can be written as:
$$M = \sum_{i=1}^r u_i c_i v_i^T\,,$$ where $u_i$ is the $i$th column of $U$... | First, just for clarity: $U$ is $m\times m$, $V$ is $n\times n$, and $\Sigma$ is $m\times n$.
We have, according to the first decomposition, that for any $1\leq i\leq m$ and $1\leq j\leq n$,
$$
M_{i,j} = (U\Sigma V^T)_{ij} = \sum_{k=1}^n (U\Sigma)_{ik} (V^T)_{kj}
= \sum_{k=1}^n (U\Sigma)_{ik} V_{jk}
= \sum_{k=1}^n \sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Can every domain be exhausted by compact *connected* subsets?
Let $D \subset \Bbb R^n$ be an open connected set. I would like to exhibit an increasing sequence of compact connected subsets of $D$ converging to $D$.
For example, for a ball we might take a sequence of closed ball inside it of radius $r-\frac 1n$.
User ... | Fix a point $x_0 \in D$. For every $A$ subset of $D$ containing $x_0$ denote by $A'$ the component of $A$ containing $x_0$. Assume now we have $D_m$ an increasing sequence of open subsets of $D$ containing $x_0$ and with union $D$. Then $D_m'$ (the $x_0$ components) also cover $D$. Indeed, consider $x \in D$. There ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
} |
Why does finer mesh mean worse condition number? Suppose I am working on the finite element approximation of a problem. My understanding is that the condition number of the resulting algebraic system becomes worse when the mesh becomes finer. What is the reason for this?
| It is not universally true that the condition number becomes worse with a finer mesh, for example take $10u = f$. There's no good reason to solve this via FEM, but you can do it, and the condition number doesn't increase with mesh refinement. Less trivially, if the operator in question is bounded and strictly coercive,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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All isometric embeddings that maps $\mathbb{R}$ to an inner product vector space $\mathbb{E}$ are of the form Let $\mathbb{E}$ be a normed vector spaces with inner product.
All isometric embeddings $f:\mathbb{R} \to \mathbb{E}$ are of the form
$f(t)=u+tv$, with $u,v \in \mathbb{E}$ and $\|v\|=1$.
This was an assignm... | Let $f$ be such an isometry, $u=f(0)$, $v=f(1)-u$. As $\|f(x)-f(y)\|=|x-y|$ for all $x,y\in\Bbb R$, this implies $\|v\|=1$.
Let $t\in\Bbb R$ be arbitrary.
Let $w=f(t)-(u+tv)$. Then
$$\begin{align}t^2&=\|f(t)-f(0)\|^2\\&= \langle tv+w,tv+w\rangle\\&=t^2\|v\|^2+2t\langle v,w\rangle+\|w\|^2\\&=t^2+2t\langle v,w\rangle+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If almost-periodic function is not identically zero, then it is not in L2 I have an $\mathbb{R}_+ \to \mathbb{R}$ function $f(t)$, which is a combination of sums and products of $\sin$ and $\cos$ functions of incommensurable frequencies. Thus $f(t)$ is a quasiperiodic function, or, more generally, almost-periodic funct... | Your proof is wrong. An $L^2$ function does not necessarily have limit $0$ at $\infty$.
However, what is true is that given $\epsilon > 0$, an almost periodic function $f$ has arbitrarily large "almost periods" $\tau$ such that
$|f(t+\tau) - f(t)| < \epsilon$ for all $t \in \mathbb R$. You can use this to show that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can this "illegal geometry" problem be possible? Using 2 triangles each with base of 8 and height of 3, and 2 trapezoids with heights of 3 on top, 5 on bottom and height of 5, these four figures can create an area with 64 units squared. However, when rearranged as a rectangle with 13 x 5=65, one additional unit squ... | This is a classic illusion based on the Fibonacci number identity
$$
13 \times 5 = 1 + 8 \times 8 .
$$
The "diagonal" of the rectangle isn't one. The slopes on each segment don't agree. There's one unit of area between the "diagonals".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Evaluate $\int_{0}^{1}2^{x^2+x}\mathrm dx$
Question : Evaluate - $$\int_{0}^{1}2^{x^2+x}\mathrm dx$$
My Attempt : First I tried to evaluate the indefinite integral of $2^{x^2+x}$ in order to put the limits $0$ and $1$ later on, but couldn't integrate it. Then I checked on WA and came to know that it's elementary inte... | Hint:
Using $u=\frac{2x+1}{2}$ yields an imaginary error function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2413891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Why are Desmos and W/A not plotting the graph correctly for a rational function? I had this question to plot this rational fraction function:
$$y=\frac{x-2}{x^2-4}$$
With asymptotes at $x=2,-2$
Now, I did immediately realise that this could be simplified to:
$$y=\frac{1}{x+2}$$
But, when one immediately simplifies it i... | Let $f(x)=\frac{x-2}{x^2-4}$. Let us prove that there is an asymptote at $x = -2$:
$$\lim_{x\to -2^+}f(x)=\lim_{x\to -2^+}\frac{x-2}{x^2-4} = \lim_{x\to -2^+}\frac{1}{x+2} = +\infty,$$
$$\lim_{x\to -2^-}f(x)=\lim_{x\to -2^-}\frac{x-2}{x^2-4} = \lim_{x\to -2^-}\frac{1}{x+2} = -\infty.$$
However, there is actually no asy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Uniformly continuity of a real valued function $f$
Let $f:\mathbb{R}\to \mathbb{R}$ be a function given by $f(x)=\sum_{k=1}^\infty \frac{g(x-k)}{2^k}$
where $g:\mathbb{R}\to \mathbb{R}$ is a uniformly continuous function such that the series converges for each $x$ belongs to $\mathbb{R}$.
Then show that $f$ is un... | If $g$ is uniformly continuous in $\mathbb{R}$ then given $\epsilon>0$ there exist $\delta>0$ such that if $|x-y|<\delta$ then $|g(x)-g(y)|<\epsilon$.
Hence, if $|x-y|<\delta$, then $|(x-k)-(y-k)|=|x-y|<\delta$ and
$$|f(x)-f(y)|\leq \sum_{k=1}^\infty \frac{|g(x-k)-g(y-k)|}{2^k}\leq
\sum_{k=1}^\infty \frac{\epsilon}{2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Factors and primes I wanted to know if anyone could explain how to work out this question.
$n$ is a (natural) number. 100 is the $LCM$ of 20 and $n$. Work out two different possible values for $n$.
| $100$ is the LCM of $20$ and $n$. Hence, $100$ must be a multiple of $n$, so we only need to look at divisors of $100$ as possible values of $n$.
Furthermore, divisors of $20$ will lead to $20$ as LCM of $n$ and $20$ and not $100$, so they can be ruled out.
We are left with $n = 25$, $n = 50$ and $n = 100$.
Edit: I wou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $\lim\limits_{n\rightarrow \infty} |\frac{a_n}{b_n}|$ Suppose that $\lim\limits_{n\rightarrow \infty} |\frac{a_{n+1}}{a_n}| = \frac{1}{\alpha}$, $\lim\limits_{n\rightarrow \infty} |\frac{b_{n+1}}{b_n}| = \frac{1}{\beta}$ and $\alpha > \beta$. Does it implies that $\lim\limits_{n\rightarrow \infty} |\frac{a_n}{b_n}... | We show that $|b_n/a_n|\to 0$ if and only if $\alpha<\beta$.
If. Indeed, to be meaningful, it means that $\alpha,\beta>0$. Fix $\varepsilon>0$, hence there exists $n_0=n_0(\varepsilon)>0$ such that
$$
|a_{n+1}|\ge \left(\frac{1}{\alpha}-\varepsilon\right)|a_n| \,\,\text{ and }\,\,|b_{n+1}|\le \left(\frac{1}{\beta}+\var... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Rolle's theorem with a supplementary condition I have to find the types of roots (i.e real or complex) of the equation $$ 11^x + 13^x+ 17^x -19^x = 0 \dots (1) $$
If $$ f(x) = 11^x + 13^x+ 17^x -19^x = 0 $$ , then obviously $ f'(x)= 0 $ has a 0 solution, and indeed every derivative of $f(x)$ has a 0 solution.
In this... |
if all the conditions of Rolle's Theorem are satisfied for a function $g(x) $ in $[a,b]$, and in addition if $g'(c)=0$ ,then is it necessary that $c$ lies between $a$ and $b$ ?
No, not at all. Consider for example, $g(x)=\sin(x)$ with $a=0$, $b=\pi$, and $c=\frac32\pi$.
Or (if you mean "strictly between $a$ and $b$"... | {
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"timestamp": "2023-03-29T00:00:00",
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Probability that no letter is in alphabetical order Given a random string of distinct letters, find the probability that none of the letters are in order. A letter is in order when every letter preceding it is of lower alphabetical value and every letter after it is higher.
having trouble with the combinatorial approac... | Let's replace letters with numbers.
Without loosing in generality we can assume that they are the number $1,2,\cdots,n$.
We call $P(n)$ the sought number of permutations in which none of the numbers are ordered, according to your definition.
Then we shall have
$$ \bbox[lightyellow] {
\eqalign{
& P(n) = {\rm N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Matrix of the perpendicular Projection I am bit doubtful about my reasoning so I hope there's someone that will correct me if I'm wrong on the steps 1 to 3.
Given is:
$W = span(v_1, v_2)$
with:
$v_1 = \begin{bmatrix} 1\\0\\1\\0 \end{bmatrix}, v_2 = \begin{bmatrix} 1\\-1\\1\\-1 \end{bmatrix}$
Steps:
i)
I use the Gr... | Yes, your reasoning is correct for iii). The columns of the matrix are the images of the canonical basis, so the first column is
$$\langle e_1, b_1 \rangle b_1 + \langle e_1, b_2 \rangle b_2 = \frac{1}{\sqrt{2}} b_1$$
and similarly for $e_2, e_3, e_4$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Disjoint curves in a smooth manifold. I already asked the question here but it was in a too specific way, so I asked it again here for more visibility. How do we prove the following statement :
Let $M$ be a smooth connected manifold without boundary of dim $\geq 2$ and $(x_1, y_1,... , x_n,y_n )$ be $2n$ distinct poin... | I would recommend try proving this by induction. Proving that a connect manifold is path connected by a smooth path would be a very useful lemma.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Determine whether $\sum\limits_{n=1}^{\infty} (-1)^{n-1}(\frac{n}{n^2+1})$ is absolutely convergent, conditionally convergent, or divergent.
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
$$\sum_{n=1}^{\infty} (-1)^{n-1}\left(\frac{n}{n^2+1}\right)$$
Here's my work:
$b_... | This definitely converges by the alternating series test. The AST asks that the unsigned terms decrease and have a limit of 0. In your case, the terms $\frac{n}{n^2+1}$ do exactly that, so it converges.
Now, which flavor of convergence?
If you take absolute values, the resulting series $\sum_n \frac{n}{n^2+1}$ dive... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Suppose that 3 men and 3 women will be randomly seated at a round table having 6 evenly-spaced seats Suppose that $3$ men and $3$ women will be randomly seated at a round table having $6$ evenly-spaced seats (with each seat being directly opposite one other seat).
(a) What is the probability that all $3$ men will be se... | All of your answers are correct. I will assume that only the relative order of the people matters, that the men are named Barney, David, and Fred, and that the women are named Ashley, Carly, and Emma.
What is the probability that all three men will be seated next to each other?
We seat Ashley. The remaining people... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2414941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$n \times n$ positive matrix with $a_{ij} a_{ji} = 1$ has an eigenvalue not less than $n$
$A$ is a real $n \times n$ matrix with positive elements $\{a_{ij}\}$. For all pairs $(i, j), a_{ij} a_{ji}=1$. Prove that $A$ has an eigenvalue not less than $n$.
| Proof 1. By Perron-Frobenius theorem, $Av=\rho(A)v$ for some positive eigenvector $v$. Let $D=\operatorname{diag}(v)$ (the diagonal matrix whose diagonal is $v$), $e=(1,\ldots,1)^T$ and $B=D^{-1}AD$. Then $Be=\rho(A)e$. Since $B$ is also a positive matrix with $b_{ij}b_{ji}=1$ for all $i,j$, and $2\le b+\frac1b$ for ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2415056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluate limit of $\lim_{x \to \infty}\frac{x^x}{\left(x+2\right)^x}$ $\lim_{x \to \infty}\dfrac{x^x}{\left(x+2\right)^x}$
I tried using Taylor and L'H and wasn't able to land on an answer.
Any help would be appreciated!
| HINT: write $$\frac{1}{\left(\left(1+\frac{2}{x}\right)^{2x}\right)^{1/2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2415146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Are essential isolated singularities preserved under non-zero holomorphic functions?
Question. In univariate complex analysis, are essential isolated singularities preserved under non-zero holomorphic functions?
For example, if we've already proved that $e^{1/z}$ has an essential singularity at $0$, can we deduce tha... | Let $f\colon D\longrightarrow\mathbb C$ be an analytic function and suppose that it has an essential singularity at some point $z_0$. Let $g$ be a non-constant entire function. Then $g\circ f$ also has an essential singularity at $z_0$. This is so because, by the Casorati-Weierstrass, if $U$ is a neighborhood of $z_0$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do we calculate the chances of getting n fixed points in a permutation? I understand that attaining a derangement for n = 4 or more objects has a probability of about 1/e or about 37 %. But what about permutations with precisely n fixed points? For example, if I have exactly ten cards (numbered 1 to 10), shuffle th... | Denote by $F(n,k)$ the number of permutations of $n$ elements with exactly $k$ fixed points. It satisfies the following relations.
*
*$F(n,n)=1$ given by the identity permutation.
*$F(n,k) = \binom{n}{k}F(n-k,0)$ given by choosing $k$ points to fix and then taking a permutation of the other elements with no fixed p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$ How do i find the When $x$ is ranging from $-1$ to $1$?
I want know why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$
I have already tried inverse-function.
thanks.
| The "co" in "cosine" stands for "complement". It means, the sine of the complementary angle. Two angles are complementary if they add to a right angle, $\pi/2$ radians. Thus:
$$
\arcsin x + \arccos x = \frac{\pi}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2415527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Differentiating an expression with f(x) in it $\frac{d}{dx}$ $\frac{x^4}{f(x)}$ at $x=1$ where $f(1)=1$ and $f'(1)=3$
I've tried simply differentiating the expression using the quotient rule, obtaining $\frac{4x^3*f(x)-x^4*f'(x)}{f(x)^2}$ But I'm not sure about where to go from here, I'm confused by the use of f(x) in ... | Since $f$ is differentiable at $1$, the composition $x\mapsto f(x)/x^4$ is differentiable at $1$ aswell. Hence by the quotient rule
$$\left .\frac{\mathrm d}{\mathrm d x }\frac{f(x)}{x^4}\right|_{x=1} = \left.\frac{4x^3 f(x)-x^4 f'(x)}{f(x)^2}\right|_{x=1}=\frac{4-3}{1}=1,$$
where we used $f(1)=1$ and $f'(1)=3$.
| {
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How can I write this in a different ways? $\vert(\bigcup_{i=0}^n Ai)\vert$ I'm trying to write this equation in a more compressed way. For me, I think it would contribute a lot to my practice to see different ways of writing the same identity.
How can I write this in a different ways?
$$\vert(\bigcup_{i=1}^n Ai)\vert =... | Two different ways:
$$ \left\lvert \bigcup_{i = 0}^n A_i \right\rvert = \sum_{\emptyset\ne S \subseteq\{0,\dots,n\}} (-1)^{|S|-1} \left\lvert \bigcap_{j \in S} A_j \right\rvert = \sum_{k = 1}^{n + 1} (-1)^{k-1} \sum_{0\le j_1 < \dots < j_k \le n} |A_{j_1} \cap \dots \cap A_{j_k}|. $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Why is p ∧ q ⇒ r true when p is true and q is false? I'm taking Intro to Logic on Coursera. One of the exercises has this:
Consider a truth assignment in which p is true, q is false, r is true.
Use this truth assignment to evaluate the following sentences.
The answer key says $p ∧ q ⇒ r$ is true, but I don't under... | If we have predicates $p$ and $q$ then $p\implies q$ is true when either $p$ is false or $q$ is true (or both). That is,
$$(p\implies q)\iff (\lnot p\lor q)$$
The implication will not hold only when $p$ is true and $q$ is false. That is, $p\implies q$ is false when $p\land \lnot q$ is true.
So in your example, $p$ is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2415924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Determining the value of required parameter for the equations to have a common root
Determine the value of $a$ such that $x^2-11x+a=0$ and $x^2-14x+2a=0$ may have a common root.
My attempt:
Let the common root be $\alpha$
On substituting $\alpha$ in both equations and then subtracting, $a = -3\alpha$
How do I contin... | If you do the calculation correctly $a=3\alpha$. So,$x^2-11*x+3\alpha=0$ is the first equation and it has root $\alpha$. So $\alpha^2-11*\alpha+3\alpha=0$, that is $\alpha(\alpha-8)=0$. So, $\alpha=0,8$. If common root $\alpha=8$ then $a=3\alpha=3*8=24$. If $\alpha=0$ then $a=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2416028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Linear algebra - changing basis and spanning the space let $|e_i\rangle$ and $|f_i\rangle$ be basis vectors, and matrix $\textbf{S}_{ij}$ that
$|e_j\rangle = \sum_j\textbf{S}_{ij}|f_i\rangle$
so that
$\textbf{a}^f=\textbf{Sa}^e$ where the superscript indicates the basis.
My textbooks says that $\textbf{S}^{-1}$ exists ... | Any kind of vectors and matrices we see in undergraduate texts, like $S=\begin{pmatrix}1&1\\0&3\end{pmatrix}$, are closely related to orthonormal vectors in Euclidean spaces. That is, we use the fact that when a basis is given, any vectors in the span of the basis can be represented by the unique linear transformation ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2416144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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If $x_{n^2}= 2x_{2n}+n^{1/n} - \sum\limits_{k=1}^{\infty}\frac{1}{2^k}$ and $\sum(x_{n+1}-x_n)$ converges, then $\lim x_n = 0$ Let $x_n$ be a real number sequence such as: $x_{n^2}= 2x_{2n}+n^{1/n} - \sum_{k=1}^{\infty}\frac{1}{2^k}$, for every $n=1,2,3...$ and the sum $\sum_{n=1}^{\infty}(x_{n+1}-x_n)$ converges. Show... | Note that:
*
*The series $\sum(x_{n+1}-x_n)$ converges and its $n$th partial sum is $x_{n+1}-x_1$, hence the sequence $(x_n)$ converges, call $\ell$ its limit.
*In the identity $x_{n^2}= 2x_{2n}+n^{1/n} - \sum\limits_{k=1}^{\infty}\frac{1}{2^k}$, the term $\sum\frac1{2^k}$ equals $1$ and $n^{1/n}\to1$ when $n\to\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing the local maximum or minimum, while the function changes sign infinitely often Please I need a hand in solving this problem:
These 3 functions' values at $0$ are all $0$ and for $x\ne0$, $$f(x)=x^4\sin\frac{1}{x}, \, g(x)=x^4\left(2+\sin\frac{1}{x}\right), \, h(x)=x^4\left(-2+\sin\frac{1}{x}\right)$$
b- Show th... | HINT: we have for 1) $$f'(x)=4x^3\sin\left(\frac{1}{x}\right)+x^4\cos\left(\frac{1}{x}\right)\cdot \left(-\frac{1}{x^2}\right)$$
and for 2)$$g'(x)=4x^3\left(2+\sin\left(\frac{1}{x}\right)\right)+x^4\cdot\cos\left(\frac{1}{x}\right)\left(-\frac{1}{x^2}\right)$$
and 3)$$h'(x)=4x^3\left(-2+\sin\left(\frac{1}{x}\right)\rig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Comparing big powers
Which of the following is the largest?
A. $1^{200}$
B. $2^{400}$
C.$4^{80}$
D. $6^{300}$
E. $10^{250}$
I'm stuck trying to solve this. Obviously A and C are wrong ($4^{80}$ is less than $2^{400}$ and 1 to any power is always 1). And cancelling $2^{200}$ from each of the remaining choices, I can a... | $$6^{300} = (6^6)^{50} \ \ ; \ \ 10^{250} = (10^5)^{50}$$
so it is enough to check what is larger between $6^6$ and $10^5$. Now,
$$6^6 = (6^3)^2 = 216^2 < 300^2 = 90000 < 100000 = 10^5$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Tangent space of closure of subgroup of Lie group I am working my way through Stillwell's Naive Lie Theory. I am looking for suggestions how to address question 8.2.4 - Show that
$\{\text{sequential tangents to }H\} = T_{\mathbf 1}\overline H$ where $H$ is an arbitrary subgroup of a matrix Lie group?
| If $X$ is a sequential tangent vector to $H$ at $\mathbf 1$, then it is also a sequential tangent vector to $\overline H$ at $\mathbf 1$ and therefore $X\in T_{\mathbf 1}\overline H$. (In Stillwell's book, this is proved right after the definition of sequential tangent vector.)
Now, let $X\in T_{\mathbf 1}\overline H$.... | {
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How do I prove, using Dirac notation, that $\langle A|A\rangle$ is a real number? Using bra-ket notation, how do I prove that the inner product of a ket with itself (ie, $\langle A | A \rangle$) is a real number?
I understand that the rule of inner products states that: $\langle B | A \rangle = \overline{\langle A | B ... | If you want/need more detail you could set $\langle A|A \rangle = x + iy$ and then use $\overline{\langle A|A \rangle} = \langle A|A \rangle$ to conclude that $-y = y$, which is true only for $y=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2416759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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The second derivative of $\log\left(\sum\limits_{i=1}^ne^{x_i}\right)$ seems negative, but I have to prove the function is convex For the second derivative I got $$\frac{\partial^2}{\partial x_k x_j}\log \left(\sum_{i=1}^{n} e^{x_i}\right)=-\frac{e^{x_k}e^{x_j}}{\left(\sum_{i=1}^{n} e^{x_i}\right)^2},$$ where $j \neq k... | There is nothing wrong. What you have only shown is not that the Hessian matrix $H$ is negative definite, but merely that the off-diagonal entries of $H$ are negative. The matrix $\pmatrix{1&-1\\ -1&1}$, for instance, has negative off-diagonal entries, but the matrix itself is positive semidefinite.
In your case, it on... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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convex hull of a set of points equivalent to a set I am trying to prove the following problem:
Given a set of points $S = \{(x_i,t_i)_{i = 1}^K \}$ where $x_i \in R^n, t_i >0 ,\forall i = 1,...,K$ and $Y = \{y \in R^n: y = \frac{x}{t},(x,t) \in conv(S) \}$.
I have proved that $Y \subseteq conv(x_1/t_1,...,x_K/t_K)$. B... | To give a motivation as to my answer, let me first write out the proof that $Y\subseteq conv(x_1/t_1,\ldots,x_K/t_K)$. In this case, there exist $x\in\mathbb R^n,t>0$ and non-negative numbers $\lambda_1,\ldots,\lambda_K$ such that $y=\frac xt$ and
$$
\sum\limits_{i=1}^K\lambda_ix_i=x,\qquad\sum\limits_{i=1}^K\lambda_it... | {
"language": "en",
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Why use $\forall$ instead of $\in$? So, here are two ways to say what I interpret as the same statement:
$f_i(x,y)\geq0 \hspace{0.85cm} \forall i \{0,1,2\}$
which implies that $f_0(x,y) \geq0$ and $f_1(x,y)\geq0$ and $f_2(x,y)\geq0$
but doesn't
$f_i(x,y)\geq0 \hspace{0.85cm}\{i\in\mathbb{Z}|i\in[0,2]\}$
imply the s... | Actually, in formal mathematics, the condition should come first:
$$ \forall i \in \{0,1,2\} \; f_i(x,y) \ge 0 $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to construct a strictly monotonic sequence of all rational numbers? I know that the set of all rational numbers is countable, and can be enumerated by a sequence, say $\{a_n\}$. But can we construct a monotonic $\{a_n\}_{n=1}^{\infty}$, e.g. with $a_k<a_{k+1}$? It doesn't seem plausible to me, because ... | As stated, the answer is no, because the question uses the symbol $<$ which has the implied meaning: The usual ordering of $\mathbb{Q}$ where $\frac{a}{b}<\frac{c}{d}$ iff $ad < bc$ in $\mathbb{Z}$.
But.
As mentioned in another answer, $\mathbb{Q}$ can be well-ordered, i.e. one can define a different order $\prec$ wit... | {
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"source": "stackexchange",
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Show that the limit of the resulting sequence is $\sqrt{2}$. Consider the sequence of rational numbers defined recursively by the following formula: $$(x,y) \mapsto (x^2 + 2y^2, 2xy)$$ and starting at the point $(x,y) = (2,1)$. Show that the limit of the ratio of the resulting sequence is $\sqrt{2}$.
Is there any easy ... | Define $x_n = x_{n-1}^2+2y_{n-1}^2$, $y_n =2x_{n-1}y_{n-1}$ and $x_0 = 2$, $y_0=1$. It is obvious that $x_n$ and $y_n$ are positive for all $n$.
Assume that $\frac{x_n}{y_n}$ converges and denote it's limit by $L$. Then we have $$\frac{x_n}{y_n} = \frac{x_{n-1}^2+2y_{n-1}^2}{2x_{n-1}y_{n-1}} = \frac{\frac{x_{n-1}^2}{y_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} $.
Find the radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} \ ?$
I applied Cauchy-Hadamard test and the result is coming $0$ (radius of convergence). To obtain the limit I also used Cauchy's... | If we represent the factorials everywhere as gamma-functions and Pochhammer symbols, this is actually a generalized hypergeometric function 4F3(...;;;z/3). According to wikipedia the radius of convergence is where the argument becomes 1, which means in our case, |z|=3.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $X$ and $Y$ be $N(0,1)$, show that $Z$ is $SN(\lambda)$. Let $X$ and $Y$ be $N(0,1)$ and let $Z$ be
*
*$Z = Y$ if $\lambda Y ≥ X$
*$Z = -Y$ if $\lambda Y < X$.
Show that $Z$ is $SN(\lambda)$, (skew normal distribution).
I've tried using the transformation theorem without success, and I'm realizing that I'm ... | Start with the obvious decomposition:
$$
P(Z\leq z)=P(\lambda Y\geq X)P(Z\leq z|\lambda Y\geq X)+P(\lambda Y< X)P(Z\leq z|\lambda Y< X).
$$
Now:
$$
P(Z\leq z,\lambda Y\geq X)=P(Y\leq z,\lambda Y\geq X).
$$
$$
P(Z\leq z,\lambda Y< X)=P(-Y\leq z,\lambda Y< X).
$$
Since $X$ and $Y$ are both normally distributed, denote th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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System of differential equations $dx=\frac{dy}{y+z}=\frac{dz}{x+y+z}$ This is the first time I have seen system of differential equations in this form: $$dx=\frac{dy}{y+z}=\frac{dz}{x+y+z}$$
Can you please help me solve it because I don't even know where to start?
| $$\frac{dx}{1}=\frac{dy}{y+z}=\frac{dz}{x+y+z}$$
This system looks like to be involved in solving a PDE with the method of characteristics. The PDE should be :
$$\frac{\partial z(x,y)}{\partial x}+(y+z(x,y))\frac{\partial z(x,y)}{\partial y}=x+y+z(x,y)$$
$\underline{\text{If this supposition is true}}$,
unfortunately t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2417674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$X^n= \begin{pmatrix}3&6\\ 2&4\end{pmatrix}$, How many solutions are there if n is odd? $X^n= \begin{pmatrix}3&6\\ 2&4\end{pmatrix}$, $n \in N^*$
How many solutions are there if n is odd?
From the powers of $\begin{pmatrix}3&6\\ 2&4\end{pmatrix}$ I got that $X=\begin{pmatrix}\frac{3}{\sqrt[n]{7^{n-1}}}&\frac{6}{\sqrt[n... | $\det(X^n)=(\det X)^n=\det \begin{pmatrix}3&6\\ 2&4\end{pmatrix} = 0$, so $\det X=0$, and $X$ is singular.
$X$ cannot be the zero matrix, so it has two real eigenvalues: $0$ and $a\neq 0$. $X$ is diagonalizable: there is some non-singular $P$ such that $X=P\begin{pmatrix}0&0\\ 0&a\end{pmatrix}P^{-1}$, yielding $X^n=P\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I find the minimizer of the following optimization problem? I want to know whether the following optimization problem falls under "convex optimization". The problem at hand is minimizing the following objective function, i.e.,
$$
\left\{\min_{\boldsymbol{\beta}\in \mathbb{R}^{d}} \sum_{i=1}^{n} \left(y_{i}-\bol... | This particular instance should be trivial to solve. Let $x = \beta^TA\beta$. Solve the scalar least-squares-problem that arises in $x$. Denote that solution $x^{\star}$. Let $v$ be any vector such that $x^{\star}$ and $v^TAv$ have the same sign (e.g., if positive let $v$ be an eigenvector associated to a positive eig... | {
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"timestamp": "2023-03-29T00:00:00",
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Does $\operatorname{tr} (A)=0$ imply $\operatorname{tr} (A^3)=0$? Let $A_{(2n+1)\times(2n+1)}$ be a symmetric matrix of Rank $2n$. Then does $\operatorname{tr}A=0$ imply $\operatorname{tr}A^3=0$? If not, Under what condition?
| The answer is NO. For a counterexample, let
$$A=\operatorname{diag}(1,3,-2,-2,0)$$
We have $\operatorname{Tr}(A)=0$ and $\operatorname{Tr}(A^3)=12\ne0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Minimize $ {L}_{p} $ Norm Regularized with a Linear Term (Conjugate Function of the Norm Function) The problem is given as following:
$$ \min_{x} {a}^{T} x + \lambda \left\| x \right\|_{p} $$
Namely minimizing a ${L}_{p} $ norm term regularized by a linear term.
The above form occurs repeatedly on the dual forms of con... | I'll take $\lambda = 1$ for simplicity. Let $f(x) = \| x \|_p$. Note that
\begin{equation}
\inf_x \, a^T x + \|x \|_p = - \sup_x \, \langle - a, x \rangle - \| x \|_p = - f^*(-a).
\end{equation}
The conjugate of a norm is the indicator function for the dual norm unit ball.
Moreover, the dual norm for the $p$-norm is th... | {
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Roots of polynomial and unit circle Let $$p(k)=a_0+a_1x+a_2x^2+a_3x ^3+\dots+a_kx^k$$
Is possible known just looking at the coefficients $a_0,a_1,\dots,a_k$ $(a_k\in\mathbb{R})$ if the polynomial $p(k)$ will have roots out of the unit circle for values of $k=2,3,4$?
EDIT: I'm asking about both complex and real roots a... | Firstly, to clarify - are you allowing complex roots or only real roots?
I do not know of an if or only if test. However, there are a couple of things that will help in some circumstances. Divide through by $a_k$ then:
*
*$a_0$ is the product of the roots, so if it is outside the unit circle, then there must be a r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How are these three equations that are rewritten equal eachother?
This question has been driving me insane for about a day now. I cannot figure how these two are equal and I understand the laws of logarithms decently. Regardless, when I try to show equality in a calculator between these three functions expressed in th... | The fact that in the image they are appearing equal is a bit of a misrepresentation for me. Following the help of @Bernard the equation actually is not e^(log(x))*(log(x)) but more rather x^log(x) = e^(log(x) * ln(x)) due to the base e.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Infinite 'hex': is a win always achievable? In the game hex, at least one player always wins because they can form a chain of hexagons across the board. This led me to wonder, what happens if we generalise to infinitely many points?
Specifically, if every point in a unit square (including boundaries) is coloured red o... | Color $(x,y)\in[0,1]^2$ red if $x=0$ or $y=1/2\cdot\sin(1/x)$. Color everything else blue. There are no paths of either color connecting its respective edges.
Note that the red path does not "reach" the line $x=0$. See also this post and the counterexample in the answers: "Intermediate Value Theorem" for curves
| {
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If $x^y y^x z^z=c$, then find $\frac {\partial^2z}{\partial x \,\partial y}$ at $x=y=z$. If $x^y y^x z^z=c$, then find $\dfrac {\partial^2z}{\partial x \,\partial y}$ at $x=y=z$.
I tried taking $\log$ but that doesn't help.
Any hints will be appreciated. Thanks.
This is what I have tried:
| Here is the solution:
Hopefully the image is clear and helpful
| {
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Permanent generating function identity for $\exp{\mathbf{x^{T}}A\mathbf{y}}$ In this paper, there's an identity that I can't prove to my satisfaction, (there's a similar statement in here)
which is that, given a permanent of a $(\mathbf{k,l})$-replicated matrix $A$, (written $A^{(\mathbf{k,l})}$),
$$\sum_{\mathbf{k,l}... | Here is a double-counting argument. The coefficient of $\mathbf{x}^{\mathbf{k}}\mathbf{y}^{\mathbf{l}}$ on either side can be written as
$$\frac{1}{m!^2}\sum_{K,L,\pi} a_{K(1),L(\pi(1))}\dots a_{K(m),L(\pi(m))}$$
where $m=\sum k_i=\sum l_j$, and the sum is over:
*
*$K:[m]\to [n]$ with $|K^{-1}(\{i\})|=k_i$ for each ... | {
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Graph expression The manual for the LaTeX package TikZ v. 3.0.1a states (section 19.7 "Graph Operators, Color Classes, and Group Expressions", p. 280):
TikZ's graph command employs a powerful mechanism for adding edges between nodes and sets of nodes. To a graph theorist, this mechanism may be known as a graph express... | This book - Recent Trends in Algebraic Development Techniques: 19th International Workshop, WADT 2008 - defines graph expression and discusses operators in that context. Link
| {
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"timestamp": "2023-03-29T00:00:00",
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Dual space & orthogonal $F_1^{\perp}+F_2^{\perp}\subset(F_1 \cap F_2)^{\perp}$
$E$ is a finite-dimensional vector space over a field $K$ and $E^∗$ its dual.
$F_1$ and $F_2$ is a subspace of $E$
I would like to prove that $F_1^{\perp}+F_2^{\perp}\subset(F_1 \cap F_2)^{\perp}$
I asked this question 12 hours ago, but wi... | You can shorten this a lot. There are a great deal of extraneous constructions that miss the main idea. Since you originally reference a dual space, I assume you are asking about dual spaces, not orthogonal complements and have answered accordingly; be careful, as they are not the same thing.
Let $\phi\in F_1^0+F_2^0$,... | {
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Help needed in understanding Heron's Formula So i just started learning Trigonometry seriously and something doesn't feel right to me, either I'm missing something or not but.
Lets assume we have a triangle
and there are two ways to find the area.
1 is using the standard $$A = \frac{1}{2}bh$$ and by using the example ... | You shouldn't consider the areas different, because the relative difference is smaller than $0.05\%$.
But you can a priori expect an error on the height to be up to $0.5\%$ as some number is truncated to integer. Hence without deeper error calculus, you can't be conclusive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2418977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Use the quadratic formula to solve trigonometric trinomial Solve $$-\sin^2\theta + 2\cos\theta +\cos^2\theta = 0$$
Using the quadratic formula.
This is what you should get $$\theta = \cos^{-1}\biggl(\frac{-1+\sqrt{3}}{2}\biggr)$$
How do you set this up and solve?
| As @Mathmore said in the comments, substitute $\sin^2\theta=1-\cos^2\theta$ to obtain
$$-(1-\cos^2\theta)+2\cos \theta+\cos^2 \theta=0$$
$$2\cos^2\theta+2\cos \theta-1=0$$
Substitute $t=\cos \theta$. This is a quadratic equation
$$2t^2+2t-1=0$$
with solutions
$$t=\frac{1}{2}(-1\pm \sqrt 3)$$
Note however that $t=\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why can't the ratio test be used for geometric series? The ratio test says that, for $a_k\neq 0$, if
$$\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|=L$$
exists, then if $0\leq L <1$, then $\sum_k a_k$ converges. If $L>1$, it diverges.
The notes I'm reading say that it's inadmissible to use the ratio test to test fo... | The ratio test is not inadmissible for geometric series. Its hypotheses do not exclude geometric series, therefore it applies, and its proof must support this.
One common proof structure would be:
Theorem A: Geometric series converges. Proof: direct argument.
Theorem B: Ratio test with usual hypotheses. Proof: show tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Largest set in which $(x, y)\mapsto \sqrt{xe^y - ye^x}$ is defined I have to find the biggest $\mathbb{R}^2$ subset in which the function
$g(x,y) = \sqrt{xe^y - ye^x}$
is defined. In order to do it I have to study
$xe^y - ye^x \ge 0$
but I can't find out a solution. Can someone help me?
| $$g^2(x,y)=G(x,y)=xe^y-ye^x$$ is clearly well defined when $$G(x,y)=xe^y-ye^x=0$$ which is a curve formed by the union of all the diagonal of $\mathbb R^2$ (i.e. $x=y$) and the set 0f $(x,y)$ such that $x\ne y$ satisfying the relation $$\frac xy=\frac{e^x}{e^y}$$ This second set, looking at the plot of the curve is a k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof verification: $x^3 + px - q = 0$ has three real roots iff $4p^3 < -27q^2$ I'm trying to prove the problem in the title. I'm not sure if my proof is clear, or even correct. It seems rather long, so I am not sure. Any advice would be great, especially on parts I might need to expand.
Let $f(x) = x^3 + px - q = 0$... | I think your statement is wrong.
Try, $p=-3$ and $q=-2$.
We have $$x^3+px-q=x^3-3x+2=(x-1)^2(x+2),$$
which says that the equation $$x^3+px-q=0$$
has three real roots, but $4p^3<-27q^2$ gives $-108<-108$, which is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Is the set $f=\{(x^3,x):x\in\mathbb R\}$ a function from $\mathbb R$ to $\mathbb R$? I need to find out if $f$ is a function or not. That is, whether or not the first coordinate of the ordered pair occurs only once in $f$. If yes, then it is a function.
My answer is yes, $f$ is a function because $f=x^{1/3}$ has a diff... | Let $y = g(x):= x^3$, $x \in \mathbb{R}$.
Domain$_g = \mathbb{R}; $ Range$_g = \mathbb{R}$.
$g$ is injective and surjective, I.e. bijective.
Injective: $g$ is strictly monotonic.
Surjective: $g$ is continuous on $[a,b]$ ,
for any $a \in \mathbb{R}.$
$\Rightarrow :$
An inverse function $g^{-1}$ exists, continuous and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Subgroup of integers I know that the only possible subgroups of $\mathbb Z$ are of the form $m\mathbb Z$. But how can I prove that these are the only possible subgroups?
| Prove the following two intermediate results:
*
*If a subgroup of $\Bbb Z$ contains some number $m$, then it contains $m\Bbb Z$
*If a subgroup of $\Bbb Z$ contains two numbers $m,n$, then it contains $\gcd(m,n)$
Now let $k$ be the smallest positive integer in your subgroup. If there is an element $a\notin k\Bbb Z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2419875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Why Riemann distance better for PSD matrices than Euclidian distance? Can you explain to me why Riemann distance is better for positive semidefinite matrices (for example covariance matrices) than Euclidian distance?
Here is the riemannian distance:
$$
d\left(Σ_A,Σ_B\right)=\sqrt{ \sum_i{\ln^2{λ_i (Σ_A,Σ_B)}}}
$$
Wher... | I can see several advantages of Riemann distance over Euclidean distance for covariance matrix.
First, Riemann distance is scaling free. if you change unit for both $\Sigma_A$ and $\Sigma_B$, you would expect the similarity should keep unchanged. i.e., $$d(\Sigma_A, \Sigma_B) = d(k\Sigma_A, k\Sigma_B) $$, which is not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Integral and limit with geometric factor I would like to calculate the following limit $$\lim_{x \to x_0}{\frac{log\left(\frac{1}{x-x_0}\int_{x_0}^{x}{\frac{\sqrt{f(x)f(x_0)}}{f(u)}du}\right)}{\left(\int_{x_0}^{x}{\frac{1}{f(u)}du}\right)^2}}$$
where $f$ strictly positive and infinitely differentiable.
We can define th... | Let
$$
f(x)=\frac{f(x_0)}{g(x-x_0)}
$$
Then we have
$$
\begin{align}
g(x)&=\frac{f(x_0)}{f(x+x_0)}\\
g'(x)&=-\frac{f(x_0)\,f'(x+x_0)}{f(x+x_0)^2}\\
g''(x)&=f(x_0)\frac{2f'(x+x_0)^2-f(x+x_0)\,f''(x+x_0)}{f(x+x_0)^3}
\end{align}
$$
Furthermore,
$$
\begin{align}
&\lim_{x\to x_0}\frac{\log\left(\frac1{x-x_0}\int_{x_0}^x\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\lim_{n \to \infty} \frac{\sqrt {n^2 +2}}{4n+1}=\frac14$?
How to prove that $\lim_{n \to \infty} \frac{\sqrt {n^2 +2}}{4n+1}=\frac14$?
I started my proof with Suppose $\epsilon > 0$ and $m>?$ because I plan to do scratch work and fill in.
I started with our conergence definition, i.e. $\lvert a_n -... | Let $\epsilon>0$
$$\left|\frac {4\sqrt {n^2 +2} -4n-1}{16n+4}\right|\leq\frac{4n+8-4n-1}{16n+4}=\frac{7}{16n+4} \leq \frac{7}{16n}$$
We have that $\frac{7}{16n} \to 0$
Thus exists $n_0 \in \mathbb{N}$ such that $\frac{7}{16n}< \epsilon, \forall n \geq n_0$
So $\frac{1}{n}<\frac{16\epsilon}{7} \Rightarrow n> \frac{7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Interchanging limit and supremum Sequence of functions is bounded $f_k(x) \le c\ \forall\ k, \forall\ x \ge 0,\ c \in \mathbb{R}$ and decreasing $\forall\ x \ge 0$. Is it possible to show such inequality $$\lim_{x \to \infty} \sup_{k \ge 1} f_k(x) \le \sup_{k \ge 1} \lim_{x \to \infty} f_k(x) \text{ ?}$$
| Here is a counter-example: let $$f_k(x) = 1 - \left( \frac {x}{1+x}\right)^k, \quad x\ge 0$$
Then:
*
*since $g : [0,\infty) \to [0,1) : x \mapsto \frac {x}{1+x}$ is increasing, and $h : [0,1) \to [0,1) : x \mapsto x^k$ is also increasing, so is their composition $h\circ g$. Therefore $f_k : [0,\infty) \to (0,1] : x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove identity holds Just wondering if there is any other way to show that for each positive integer $n$ holds
$$2\left(\sqrt{n} - 1\right) < 1 + \frac{1}{\sqrt{2}} +\cdots+\frac{1}{\sqrt{n}} < 2\sqrt{n}$$
other than by mathematical induction~
| Note that
$$\frac{1}{k+1}<\int_k^{k+1}\frac{dx}{\sqrt{x}}<\frac{1}{k}$$
for every $k\geq0$. Adding this relations we get
$$\int_1^{n+1}\frac{dx}{\sqrt{x}}<1 + \frac{1}{\sqrt{2}} +\cdots+\frac{1}{\sqrt{n}}<\int_0^n\frac{dx}{\sqrt{x}}$$
Solving the integrals we get what we wanted.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
i was wondering how to evaluate $\lim\limits_{x \to0} x\sum 1/n$ I was wondering how to evaluate
$$\lim_{x\to 0}x\sum_{n=1}^\infty \frac 1 n$$
Edit
so that it's clear I mean
$$\lim_{x\to 0}x*\sum_{n=1}^\infty \frac 1 n$$
witch is equvalent to
$$\lim_{x\to 0}\sum_{n=1}^\infty \frac x n$$
| $$\lim_{x\to 0}x\sum_{n=1}^\infty\frac1n=\left(\sum_{n=1}^\infty\frac1n\right)\lim_{x\to 0}x$$
since the series does not depend on $x$. Hence, the above expression is undefined, as
$$\sum_{n=1}^\infty\frac1n$$
is divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
To prove that $B\subset A\rightarrow A=A\cup B$ is it necessary to consider the cases $A\cup B=\emptyset$ and $A\cup B\neq \emptyset$? For what I know, to prove $B\subset A\rightarrow A=A\cup B$ it suffices to prove that $A\subset (A\cup B)$ and $(A\cup B)\subset A$.
Prove that $A\subset (A\cup B)$ is trivial, because ... | You are right, and I see no errors. We can also observe that $A\subset B\iff \forall x\;(x\in B\implies x\in A).$ Therefore if $B\subset A$ then for all $x$ we have $$x\in A\cup B)\implies (x\in A\lor x\in B)\implies$$ $$\implies (x\in A\lor x\in A)\implies(x\in A)\implies$$ $$\implies(x\in A \lor x\in B)\implies(x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2420922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $X$ is first countable, is $f(X)$ also first countable? I'm trying to understand the proof of this theorem: $X$ is first countable and $f:X\rightarrow Y$ is a function. Then $f$ is continuous at $x\in X$ iff for every sequence $(x_n)$ which converges to $x$, $f(x_n)$ converges to $f(x)$.
Fix $x\in X$. To prove the "... | Firstly: $f[X]$ is not necessarily first countable, e.g. take $X = \mathbb{R}$ in the usual topology and let $\sim$ be the equivalence relation with classes $\{x\} , x \notin \mathbb{Z}$ and $\mathbb{Z}$ (We identify the integers to a point). Then the quotient space $Y = X/\sim$ in the quotient topology, induced by the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do I find the closed form of this integral $\int_0^2\frac{\ln x}{x^3-2x+4}dx$? How do I find the closed form of this integral:
$$I=\int_0^2\frac{\ln x}{x^3-2x+4}dx$$
First, I have a partial fraction of it:
$$\frac{1}{x^3-2x+4}=\frac{1}{(x+2)(x^2-2x+2)}=\frac{A}{x+2}+\frac{Bx+C}{x^2-2x+2}$$
$$A=\frac{1}{(x^3-2x+4)'}... | To address your question of how to handle loops when integrating by parts, let $I=\int e^x\sin x \ dx$. Both functions are transcendental. We'll try using $u_1=e^x$ and $dv_1=\sin x \ dx$. These give $du_1=e^x \ dx$ and $v_1=-\cos x$. Thus, $$I=-e^x\cos x+\int e^x\cos x \ dx.$$
Now we have another integral. We'll try b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Understanding a sentence in a Galois Theory paper about polynomials I've been trying to understand a sentence of a math for some time. I'll be straightforward:
(*) $$x^n + a_{n-1}x^{n-1} +...+a_1x + a_0 = 0$$
(...)
If we denote the roots of (*) by $x_1, x_2, ... , x_n$ so that
$$(x-x_1)(x-x_2)...(x-x_n) = x^n + a_{n-1}... | For $1$, what darij said.
Just expand $(x-x_1)(x-x_2)\cdots (x-x_n)$. For small $n$, we have:
$$(x-x_1)(x-x_2) = x^2 - (x_1 + x_2) x + x_1 x_2 \\(x-x_1)(x-x_2)(x-x_3) = x^3 - (x_1 + x_2 + x_3)x^2 + (x_1 x_2 + x_1 x_3 + x_2 x_3)x - x_1 x_2 x_3$$
and the pattern reveals itself as the author claims:
$$(x-x_1)\cdots(x-x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Matrix Linear Least Squares Problem with Diagonal Matrix Constraint How could one solve the following least-squares problem with Frobenius Norm and diagonal matrix constraint?
$$\hat{S} := \arg \min_{S} \left\| Y - XUSV^T \right\|_{F}^{2}$$
where the $S$ is a diagonal matrix and $U,V$ are column-orthogonal matrix. Is t... | The closed-form solution to this problem is
$$S = {\rm Diag}\bigg(\Big(I\odot U^TX^TXU\Big)^{-1}{\rm diag}\Big(U^TX^TYV\Big)\bigg) \\
$$
which was derived as follows.
For typing convenience, define the matrices
$$\eqalign{
S &= {\rm Diag}(s) \\
A &= XUSV^T-Y \\
}$$
Write the problem in terms of these new variables, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
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