Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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What is the most efficient method to evaluate this indefinite integral? $$\int x^5 e^x\,\mathrm{d}x$$
Is there another, more efficient way to solve this integral that is not integration by parts?
| You're question asked if there is a more efficient way than integration by parts to solve the indefinite integral $\int x^5e^x dx$, and other users have provided good answers.
But in case you tacitly assumed that the answer to your question would pretty much be the same for definite integrals like $\int_a^b x^n e^{kx} ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Free module finite generated with a submodule not finite generated. Anyone knowns an example of a R-module finite genereted with a submodule not finite generated?
I find the following example: Taking the set of function $f:[0,1]\rightarrow\mathbb{R}$ seen as module of it self. This is finite generated. If we take the s... | Let $R$ be the ring of function $f:[0,1]\to\mathbb R$. Let $M$ be the $R$-module of functions $f$ such that $f(x)=0$ for all $x\in[0,1]$ except for a finite numbers of points.
Assume on contrary that $M$ is finitely generated with generators $g_1,\ldots,g_n$.
For each $a\in [0,1]$ let
$\chi_a(x)=
\begin{cases}
1&x=a,\\... | {
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"url": "https://math.stackexchange.com/questions/832433",
"timestamp": "2023-03-29T00:00:00",
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How Find the diophantine equation $x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}$
let $x_{1},x_{2},\cdots,x_{n}$ such
$$n\ge 3,x_{1}\le x_{2}\le\cdots x_{n}$$
$$x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}$$
let the number of ordered pairs of postive integers $(x_{1},x_{2},\cdots,x_{n})$ is $f(n)... | Observing that, calling $\sigma_i(x_1,\dots,x_n)$ the $i$-th symmetric polynomial then $x_1+\dots+x_n=\sigma_1(x_1,\dots,x_n)$and $x_1x_2\cdots x_n=\sigma_n(x_1,\dots,x_n)$; then looking at the symmetric polynomials properties maybe you can get some informations on what you want.
| {
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Follow up on subspectra: is the restriction a subalgebra This is a question that came up after thinking about one of my previous questions here.
My question is: If we consider the algebra $A$ of continuous linear operators $u: X \to X$ where $X$ is some Banach space can the algebra $A|_C = \{u|_C: u \in A\}$ where $C$ ... | I don't think you can expect anything like that if you just require $C$ to be a subset. Not even when it is a subspace, because you need $$ (ab)|_C=a|_C\,b|_C.$$ This requires $C$ to be an invariant subspace for all operators, which would never happen.
This of course doesn't preclude the possible existence of some oth... | {
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How to check does polygon with given sides' length exist? I have polygon with $n$ angles. Then I have got $n$ values, which mean this polygon's sides' length. I have to check does this polygon exist (means - could be drawn with given sides' length). Is there any overall formula to check that? (like e.g. $a+b\ge c$, $... | No, there's no overall formula. There are some weak conditions (the sum of any $n-1$ sides' lengths must be greater than the length of the remaining side, for instance), but this is merely necessary for the existence of any polygon with those side-lengths, not one that has your desired angles. Is it sufficient? I'm not... | {
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What is the difference between 1-dim.harmonic oscillator and 2-dim. harmonic oscillator? I ask myself what exactly is meant with "2-dimensional harmonic oscillator".
I only know the situation of a bob hanging on a bar... is that 1-dimensional or 2-dimensional?
| The difference is the number of spatial dimensions in which the oscillator is allowed to oscillate.
The 1D oscillator has a potential function,
$$ V(x) = kx^2, $$
where the 2D oscillator has the potential function,
$$ V(x,y) = k (x^2 + y^2) .$$
In the context of classical mechanics the differential equations for $x$ a... | {
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Given a rational, how to find the integers whose quotient it is? I haven't found an answer to this anywhere.
Excluding brute force, given a rational $q$ in its decimal form ($1.47$, for example), is there a good algorithm to find integers $m$ and $n$ such that $\frac m n = q$?
Thank you.
| If $q$ is in decimal form well yes, there's the whole "if $q$ is not periodic then it is $q$ without . divided by $10^n$, if it's periodic do this and that" (if you meant this comment and I'll write it all)
| {
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How to solve: $\frac{3·x-5}{8·x-2}<6$ I'm trying to solve $\frac{3x-5}{8x-2}<6$ ?
I'm not sure which first step to take. I mean if I multiply both sides by $8x-2$ then I'm not sure if the sign would switch, as this could be positive or negative depending on $x$.
| Hint
You can't multiply by $8x-2$ without discuss on its sign. The best way to answer the question is:
$$\frac{3x-5}{8x-2}<6\iff\frac{3x-5}{8x-2}-6=\frac{-45x+7}{8x-2}<0$$
and now draw a signs table for this quotient.
Edit The sign table is
so the answer is $$\left(-\infty,\frac7{45}\right)\cup \left(\frac14,+\infty\r... | {
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Why is it safe to approximate $2\pi r$ with regular polygons? Considering this question: Is value of $\pi = 4$?
I can intuitively see that when the number of sides of a regular polygon inscribed in a circle increases, its perimeter gets closer to the perimeter of the circle. This is the way Archimedes approximated $\pi... | The reason why we know the Archimedean approximation works when we also know the 'troll' (rectilinear) approximation doesn't is because the Archimedean approach approximates not just the position of the curve but also its direction.
The rectilinear example shows that some information above and beyond just the position ... | {
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What are non-tagential limits? I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions being non-tangential limits on ${\mathbb R}$ of bounded analytic functions on the upper half... | If $x_n\to x$, we say that $(x_n)$ is a non-tangential sequence in the upper half plane if $\inf_{n > 0}\text{Im}(x_n-x)/\text{Re}(x_n-x) > 0$. So $f(x_n)$ converges non-tangentially to $y$ if $f(x_n)$ converges to $y$, and $(x_n)$ is a non-tangential sequence.
So $x_n = (a+ib)/n$ for $b>0$ is non-tangential, but $x_n... | {
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Covariant derivative in cylindrical coordinates I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57).
Equation (48) shows that the covariant derivative is:
$$A_{j;k} = \frac{1}{g_{kk}}\frac{\partial A_j}{\partial x_k} - ... | The thing you forgot is the scale factor $\frac{1}{r}$ given in equation (14). See Scale Factor in Mathworld.
| {
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Matching gender expectation
If there are x men and y women and we pair them randomly (do not consider gender). What are the expected number of the pair man-man, man-woman, woman-woman respectively?
(Assume x,y large so that even or odd in total is neglectible, I just want an approximation)
Note: I am confusing. Why c... | $$(x+y)^2 = x^2 + 2 x y + y^2$$
Consequently, we expect $\frac{x^2}{(x+y)^2}$ male-male assignments, $\frac{2 x y}{(x+y)^2}$ male-female assignments, and $\frac{y^2}{(x+y)^2}$ female-female assignments.
| {
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How can I prove that a square matrix is invertible if it satisfies this polynomial equation? For a 3x3 matrix $C$, it is given that
$$C^3+I=3C^2-C$$
I am then required to prove that $C$ is invertible.
I have attempted a proof, below, but I am not sure it is valid or if there is a better solution.
Attempted proof
$$C^3 ... | Subtract to get $$I=-C^3+3C^2-C$$
Then factor to get $$I=C(-C^2+3C-I)$$
Now you have $I=CD$, for $D=-C^2+3C-I$. Hence $C$ is invertible.
| {
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An "elementary" approach to complex exponents? Is there any way to extend the elementary definition of powers to the case of complex numbers?
By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots a}_{n\;\text{factors}}.$$ (Meaning I am not interested in the power series or "compoun... | So here's a good place to start
$$e^{i\theta}$$
Is interpreted as the complex number that is formed if you form a circle of radius 1 in the complex number field. And starting from the point 1 + 0i you move along the circle for an angle $\theta$ to a new number in the complex number field:
$$\sin(\theta) + \cos(\theta)i... | {
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"timestamp": "2023-03-29T00:00:00",
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Where's my error on finding all the solutions of a linear congruence? I'm supposed to find all solutions of each of the linear congruence.
9x ≡ 5 (mod 25)
I know there are other posts on the site about this, but I don't really follow.
Here's what I did:
I used the Euclidean Algorithm to find the gcd, which was 1 and ... | $$\text{Hint: Use euclidean algorithm}$$
$$\text{Find $a,b$ such that $9a+25b=1$, which can be done since gcd(9,25)=1}$$
$$\text{You'll finish with: $1 \equiv 9 \cdot a \mod (25) \Rightarrow x \equiv 9x \cdot a (\mod 25)$. Remember $9x \equiv 5 (\mod 25)$}$$
| {
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Find the norm of functional Consider the functional from $l_2$.
$$
x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}.
$$
What is the norm of the functional?
| It should not be very difficult to rewrite your functional to the form
$$x\mapsto \sum x_n y_n$$
where $y=y_n\in\ell_2$.
Then the norm of this functional is precisely $\|y\|_2$, i.e., it is the same as the $\ell_2$-norm of the sequence $y$.
To see this, just notice that you have the functional of the form
$$f(x)=\langl... | {
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"timestamp": "2023-03-29T00:00:00",
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Is the derivative of a bump function still a bump function? My question is rather simple : are derivatives of bump functions still bump functions ?
For example, for a bump function $u\in D(\mathbb{R}^d)$, that is,
$$u \in C^\infty|\;\text{supp}\;u\subseteq K : \mathbb{R}^d \to \mathbb{R}$$
Is this always true for all d... | Yes, because, obviously, $\operatorname{supp}\partial u/\partial x_j\subset\operatorname{supp}u$.
| {
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An Integral inequality involving vanishing boundary data Suppose that $f$ is twice differentiable on $[0,1]$ with $f(0)=f(1)=0$. Also $f$ is not identically zero. Show that $$|f(x)|\leq \frac{1}{4}\int_0^1 |f''(x)|dx,\ \forall\ x\in [0,1].$$
Thank @FisiaiLusia, I am sorry that on my computer "Mathematics Stack Exchange... | We can assume that there exists point $c\in (0,1) $ such that $$ \sup_{v\in [0,1] } |f(v)|=f(c) .$$
By Lagrange Theorem we have that there exist points $u_1 \in (0,c) $ and $u_2 \in (c, 1)$ such that $$ f(c) =f(c) -f(0) =cf' (u_1 )$$ $$f(c) =f(c) -f(1) =(c-1)f' (u_2 )$$
so we have $$\int_0^1 |f''(s)|ds \geq \int_{u_1}^... | {
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Exponent of polynomials (of matrices)
$A$ is a matrix over $\mathbb R$ (reals).
Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$
I tried using the sigma writing but got stuck
(I wrote $f(x)=\sum a_ix^i$, $g(x)=\sum bix^i$ and then started to develop the left part of ... | You should know that for square matrices $A,B$ the following holds:
$AB=BA \implies e^Ae^B=e^{A+B}$.
Can you prove that if $P,Q$ are polynomials and $A$ a square matrix, $P(A)Q(A)=Q(A)P(A)$ ?
| {
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Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT
Prove for all $x\in\mathbb R$:
$$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$
Mclauren expansion:
$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$
$$e^{-x}=1+-x+\frac{x^2}{2}-\frac{x^3}{3!}+S_4(x)$$
Adding both together:... | An alternative approach:
$$ \ln\cosh x = \int_0^x \tanh t\,dt \le \int_0^x t\,dt = \frac{x^2}{2} $$
This uses the fact that $\tanh x\le x$ for $x\ge 0$ (and the reverse for $x\le 0$), which can be proved by computing the second derivative of $\tanh$, concluding that it's convex on $(\infty,0]$ and concave on $[0,\infty... | {
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Homology of product of topological space and sphere is direct sum of homologies.
Show that for $i > n \in\mathbb{N}$: $$H_{i}\left(X \times \mathbb{S}^{n}\right) \simeq H_{i}\left(X\right) \oplus H_{i - n}\left(X\right).$$
My first idea motivated by $n=0$ case (which is obvious) was to try induction but I cannot see ... | Ok so here is what I've got so far:
Lemma 1. $$H_i(X\times S^n) \simeq H_i(X \times \{s\})\oplus H_i(X\times S^n,X\times\{s\})$$
Let's write seuquence for pair $(X \times S^n, X \times \{s\})$:
$$H_{i + 1}(X \times S^n, X \times \{s\})\to H_i(X \times \{s\}) \to H_i(X \times S^n) \to H_i(X \times S^n, X \times \{s\}... | {
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Prove $u_{n}$ is decreasing $$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?
In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.
| If there is a limit, it will be defined by $$L=\frac{1}{3-L}$$ which reduces to $L^2-3L+1=0$. You need to solve this quadratic and discard any root greater than $2$ since this is the starting value and that you proved that the terms are decreasing.
| {
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Find the number of real solutions to the equation $ x^7=x+1$. The equation $x^7 = x+1 $ has:
a) no real solution
b) no positive real solution
c) a real solution in the interval (0,2)
d) a real solution but not within (0,2)
Which is the correct answer and why ? How do i find the answer to such questions?
| Consider the 2 equations:
$$y = x^7 \tag{A}$$
$$y = x + 1 \tag{B}$$
What does equation (A) look like? It has an intersection point at $(0, 0)$, grows incredibly fast before $x=-1$ and after $x=1$.
Equation (B) grows so much slower that it can only intersection (A) at one point. Since $1^7 < 1 + 1 < 2^7$, the intersec... | {
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Geometrically, what is the span of vectors? Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?
| Assuming it makes sense that the span of a single vector is a line, we can imagine the two vectors in 3-space. Because the span of each vector lies within the space of each of them, we can draw the two lines that are in the direction of these two vectors:
*
*if the two lines are equal, then this is all of the span.... | {
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Probabilistic game Suppose a rich person offers you $\$1000$ and says that you can participate in $1000$ rounds of this game:
In each round a coin is flipped and you get a $50$% return on the portion of your money that you risked if it lands on heads, or get a $40$% loss if it lands on tails. For example if you choose ... | The expected value of your game is positive ($E[X]=0.5\alpha-0.4\alpha\ge 0$, where $\alpha$ is the amount of money bet) so the more you bet, the more you'll earn.
Moreover, betting all your money doesn't prevent you playing the $1000$ games since the result only affect a percentage of your bet, so the best strategy ... | {
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How many Sylow-$ 3$ subgroup does $G$ have? Let $G$ be a noncyclic group of order $21$. How many sylow-$3$ subgroup does G have?
The possible orders of Sylow $3$ subgroups is $1, 7$. But how to check the exact number?
| If it is $1$, $G$ must be cyclic as Sylow-$7$ subgroup is uniqe so it must be $7$.
Notice that $n_7$ must be equal to $1$, so it has a uniqe sylow-$7$ subgroup.
As you said $n_3\in \{1,7\}$, if $n_3=1$;
it has also normal subgroup of order $3$ which means $G=HK$ and $H\cap K=1$ and $H,K$ is normal in $G$ which means t... | {
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Find $\lim_{k \rightarrow \infty} \sup a_k$ and $\lim_{k \rightarrow \infty} \inf (a_k)$, with $a_k=(\frac{1}{k})_{k \in \mathbb{N}}$ Find $\lim_{k \rightarrow \infty} \sup a_k$ and $\lim_{k \rightarrow \infty} \inf (a_k)$, with $a_k=(\frac{1}{k})$
Per definition: $\lim_{k \rightarrow \infty} \sup(a_k) = \lim_{k \right... | We know that a bounded sequence$(x_n)_{n=1}^{\infty}$ is convergence if and only if $\lim\sup\limits_{n\to\infty} x_n=\lim\inf \limits_{n\to\infty}x_n$. Since $(a_{k})=(\frac{1}{k})_k$ converges to $0$, so that $\lim\sup\limits_{n\to\infty} x_n=\lim\inf \limits_{n\to\infty}x_n=0$
| {
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Help understanding the characteristic polynomial Let
$$A = \begin{bmatrix}
1 &2 &1 \\
2 & 2 &3 \\
1 & 1 &1
\end{bmatrix}$$
I'm calculating the characteristic polynomial by the following:
$$P(x) = -x^3 + Tr(A)x^2 + \frac{1}{2}(a_{ij}a_{ji} - a_{ii}a_{jj})x + \det(A)$$
Now when I calculate using matlab (using the ch... | See Wolfram Mathworld for your formula. It notes that Einstein summation is used in the coefficient for $x$, so you have to sum over both $i$ and $j$ to get the actual answer.
The matlab answer gives the negative of $P_3(x)$ (as these have the same roots anyway): 1 stands for $x^3$, the -4 for the negative trace $1+2+1... | {
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calculation of $(x,y,z)$ in $x+\lfloor y \rfloor -\{z\} = 2.98\;\;,\lfloor x \rfloor +\{y\}-z = 4.05\;\;,-\{x\}+y+\lfloor z \rfloor = 5.01$ The no. of real solution of the equation
$x+\lfloor y \rfloor -\{z\} = 2.98\;\;,\lfloor x \rfloor +\{y\}-z = 4.05\;\;,-\{x\}+y+\lfloor z \rfloor = 5.01$.
Here I did not understand... | I'm assuming that indeed $\{a\} = a - \lfloor a\rfloor$. Since then $0 \leqslant \{a\} < 1$ for all $a$, we have
$$-1 < \{a\} - \{b\} < 1$$
for all $a,b$. Then from the given equations we obtain, by writing $x = \lfloor x\rfloor + \{x\}$ and analogously for $y,z$, the three equations
$$\begin{gather}
\{ x\} - \{ z\} = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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About second uniqueness primary decomposition theorem I'm self-learning commutative algebra from Introduction to Commutative Algebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem for you to reference (page 54):
Let $\mathfrak a$ be a decomposa... | The proof of the first statement: Let $q_i$ an isolated prime ideal. Then {$p_i$} is a isolated set because $p_i$ is minimal in the set of the associated primes. If you take S=A-$p_i$ then $q_i=S(a)$, where $S(a)=a^{ec}$. So, $q_i$ is uniquely determined by $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/834965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How many ways to pick $X$ balls Suppose i have $3$ types of balls $A,B$ and $C$ and there are $n_a, n_b,$ and $n_c$ copies of these balls. Now i want to select $x$ balls from these $3$ types of balls $x < n_a + n_b + n_c$. Can anybody help me to arrive at the closed formula for this.
I thought, if I partition $x$ into ... | This problem can be easily solved with Principle of Inclusion Exclusion (PIE) and the Balls and Urns technique. The answer is:
$\dbinom{x+2}{2} - \dbinom{x-n_a+1}{2} - \dbinom{x-n_b+1}{2} - \dbinom{x-n_c+1}{2} + \dbinom{x-n_a-n_b}{2} + \dbinom{x-n_a-n_c}{2} + \dbinom{x-n_b-n_c}{2} - \dbinom{x-n_a-n_b-n_c-1}{2}$.
Hint: ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Integer solutions to $a^{2014} +2015\cdot b! = 2014^{2015}$
How many solutions are there for $a^{2014} +2015\cdot b! = 2014^{2015}$, with $a,b$ positive integers?
This is another contest problem that I got from my friend.
Can anybody help me find the answer? Or give me a hint to solve this problem?
Thanks
| Taking this equation mod $2015$ yields $a^{2014} \equiv -1 \pmod{2015}$.
Since $2015 = 5 \cdot 13 \cdot 31$, we get the following:
$a^{2014} \equiv -1 \pmod{5}$
$a^{2014} \equiv -1 \pmod{13}$
$a^{2014} \equiv -1 \pmod{31}$
By Fermat's Little Theorem, $a^{31} \equiv a \pmod{31}$. Hence, $a^4 \equiv a^{2014} \equiv -1 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Power series convergence radius My question is: how do I calculate the radius convergence of a power series when the series is not written like
$$\sum a_{n}x^{n}?$$
I have this series:
$$\sum\frac{x^{2n+1}}{(-3)^{n}}$$
Can I use the criterions as I was working with $x^{n}$, not $x^{2n+1}$?
I tried this:
$$k=2n+1\Right... | Since $x\ne0$, we can apply the ratio test for Absolute Convergence, which is stated precisely as so:
Let $ \sum u_{k}$ be a series with nonzero terms and suppose that
$$ p= {Lim_\xrightarrow{k\to\infty}}{\frac{|u_k+1|}{|u_k|}}$$
(a) The series converges absolutely if $p<1$.
(b) The series diverges if $p>1$ or $ p = \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Geometric proof of dot product distributive property I'm working my way through a text book for fun in order to keep my math brain fresh and came across this simple yet perplexing problem.
"Demonstrate geometrically that the dot product is distributive"
I can do this algebraically but what would a geometric proof of th... | Maybe something like this?
$$a \cdot (b+c) = a \cdot b + a \cdot c$$
"The 'projection' of $a$ onto $b+c$ is the same as the sums of the 'projections' of $a$ onto $b$ and of $a$ onto $c$." (You'll have to draw the triangle formed by $b$, $c$, and $b+c$. Also, you will need to be precise about what the 'projection' is.)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why must polynomial division be done prior to taking the limit? Suppose I wish to evaluate the following,
$$\mathop {\lim }\limits_{x \to 2} \left( {{{{x^2} - 4} \over {x - 2}}} \right)$$
If I just substitute two into $x$, it can't be done because the answer would be undefined (division by zero).
But, if I complete the... | The point is that the functions
$$\frac{x^2-4}{x-2}\quad\hbox{and}\quad x+2$$
are equal except at $x=2$, where the second is defined and the first is not. If you look closely at the definition of a limit as $x\to a$, you will see that it is carefully framed in such a way that the value of the function (if any) when $x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that if a square matrix A satisfies the equation ....then A must be invertible.
(a) Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible. What is the inverse?
(b) Show that if $p(x)$ is a polynomial with a nonzero constant term, and if $A$ is a square matrix for w... | If $A$ is not invertible, then $0$ is an eigenvalue of $A$. Thus, $p(0)$ must be an eigenvalue of $p(A) = 0_{n\times n}$. But all of the eigenvalues of $p(A) = 0_{n\times n}$ are $0$. So, we must have $p(0) = 0$. This is a contradiction since $p$ has a non-zero constant term, and so, $p(0) \neq 0$. Therefore, $A$ is in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Egg drop problem
Suppose that you have an $N$-story building and plenty of eggs. An egg breaks if it is dropped from floor $T$ or higher and does not break otherwise. Your goal is to devise a strategy to determine the value of $T$ given the following limitations on the number of eggs and tosses:
Version 0: $1$ egg,... | A very broad hint for version 4: consider that in your version-3 answer you don't have to use uniform intervals between drops of the first egg. Can you see how to use a non-uniform distribution so that the worst-case total is identical no matter where the first egg breaks?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/835582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 0
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How to simplify this equation to a specific form? How can I simplify this expression?
$$ 2(4^{n-1})-(-3)^{n-1} + 12 ( 2 (4^{n-2})-(-3)^{n-2})$$
The correct answer is $2 · 4^n − (−3)^n$
| hint:
use the fact that
$$a^n = a\cdot a^{n-1}$$
and that
$$a^n = a^2\cdot a^{n-2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the set of 2 continuous functions is closed. Let $f: \mathbb R \to \mathbb R $ and $g: \mathbb R \to \mathbb R$ be continuous functions. Show the set $ E = \{ x \in\mathbb R: f(x)=g(x)\} $ is closed.
My approach
A solution I found is the following:
$h=f-g$
$h(x)=f(x)-g(x)=0$
f and g are continious and h is ... | How much do you know about continuous functions? For example, do you know that for a continuous function $h$ and a closed set $X$, the set $h^{-1}(X)$ is always closed?
If you know that, the continuation is badly written, but captures the idea. First, you define the function $h = f-g$. Then, you can see that $$E=\{x\i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand? Is there any way we can prove this definite integral inequality by hand:
$$
\int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4
$$
I don't where to start even, please help. That
$\displaystyle\cos\left(1 \over x\right)\ \leq\ 1$ doesn't seem to help bec... | Hint:
$$\begin{align}
∫_1^π x \cos \left(\frac1{x}\right)dx&=∫_{1}^{1/π}\frac{\cos {u}}{u}\left(-\frac{du}{u^2}\right)\\
&=∫_{1/π}^{1}\frac{\cos {u}}{u^3}du\\
&\leq∫_{1/π}^{1}\frac{1-\frac12u^2+\frac{1}{24}u^4}{u^3}du
\end{align}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Is $M=\{\frac{1}{k}|k \in \mathbb{N}\}$ closed? Is $M=\{\frac{1}{k}|k \in \mathbb{N}\}$ closed?
I think it is closed, but I'm not sure whether my argumentation is correct.
Since $(\frac{1}{k})_{k \in \mathbb{N}}$ is convergent $\Rightarrow$ from Cauchy $ \exists k_0 \in \mathbb{N}: \forall k \geq k_0:|\frac{1}{k}-\frac... | It is not true that $M=\mathbb{R}-O$; the set $\mathbb{R}-O$ still contains all the intervals $(\frac{1}{k+1},\frac{1}{k})$ for $k\geq k_0$, and these intervals are not contained in $M$.
It's possible some of your confusion is coming from the order of the quantifiers in the definition of Cauchy - you have to fix $\vare... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/835926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Practical trigonometry question I can't figure out (Highschool Level) "Jack is on a bearing of 260 degrees from Jill. What is Jill's bearing from Jack?"
The answer is 080 degrees. I really can't figure out how. Any help is appreciated.
| Consider the following diagram:
The rule about bearings is: "Point north and go clockwise".
The bearing of Jill from Jack $(\theta)$ and the angle $\gamma$ ($=260-180)$ are alternate angles, so they're equal.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$ If $0\le\alpha\le\frac{\pi}{2}$, then which of the following is true?
A) $\sin(\cos(\alpha))<\cos(\sin(\alpha))$
B) $\sin(\cos(\alpha))\le \cos(\sin(\alpha))$ and equality holds for some $\alpha\in[0,\frac{\pi}{2}]$
C) $\sin(\cos(\alpha))>\cos(\sin(\alpha))... | Let $a=\cos{x}$, $b=\sin{x}$, and $a,b \in[0,1]$. We are now going to see which one ($\sin{a}$ or $\cos{b}$) is larger?
Noticg that $a$ and $b$ satisfy $a^2+b^2=1$, and if we regard the value $a$ and $b$ as a pair $(a,b)$ on the $(a,b)$-plane, it should be a circle with radius $1$ in the first quadrant.
If $\sin{a}=\co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 2
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finding the combination of sum of M numbers out of N I was thinking a problem of finding the number of way to add M numbers, ranged 0 to K,
to a give a desired sum. Doing some researches online, I find a way to use polynomial to achieve the goal. For example, suppose there are M numbers, each one ranged from 0 to K in... | The number of ways to select $M$ distinct values from $1,2,\dots, N$ with sum $S$ is the coefficient of $y^{M} x^S$ in the product $\prod_{j=1}^N (1+y \, x^j)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Constraint to unconstraint optimization problem by subsitution Given the following convex optimization problem
$\min_{x,p} ||x|| - p$
subject to $p > 0$
Can I change the above to an unconstrained convex optimization problem by substituting $c = \log(p)$ and minimize
$\min_{x,c} ||x|| - \exp(c)$
If this is possible ... | Yes, of course you can do this. If $\phi$ is a diffeomorphism, over the appropriate domains
$$\min_x\ f(x) = \min_y\ (f\circ \phi)(y).$$
I don't know a reference offhand unfortunately, but it is rather intuitive that this should work -- also you can see that $[J\phi]\nabla f = 0$ if and only if $\nabla f =0$ since $J\p... | {
"language": "en",
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Maximal ideal in the ring of polynomials over $\mathbb Z$
Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal.
I tried first see that $5$ and $x^2+2$ are both polynomial in that ring but how can i get that i... | If you quotient by the ideal $I$, then
$$5 \equiv 0 \pmod{I} \text{ and } x^2+2 \equiv 0 \pmod{I}.$$
This also suggests that $x^2 \equiv -2 \equiv 3 \pmod{I}$
This helps you get the idea that perhaps $x \mapsto \sqrt{3}$ and $5 \mapsto 0$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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2D Fourier Transform proof of Similarity Theorem I have to solve an exercise, but if i could use the following theorem, it would be piece of cake
Similarity Theorem
if $ \mathscr{F}\{g(x,y)\}= G( f_x,f_y)$
then
$ \mathscr{F}\{g(ax,by)\}= \frac {1} {| a \cdot b|}G( f_x /a, f_y/b) $
i just need the proof, (You can find... | $$x' = ax, y' = by$$
Jacobian is $\dfrac{1}{|ab|}$
$$\int e^{i(xf_x + yf_y)} g(ax,by) \,dx\,dy = \dfrac{1}{|ab|}\int e^{i\left(\dfrac1ax'f_x + \dfrac1by'f_y\right)} g(x',y') \,dx'\,dy' \\
= \dfrac{1}{|ab|}\int e^{i\left(x'\dfrac{f_x}{a} + y'\dfrac{f_y}{b}\right)} g(x',y') \,dx'\,dy'\\
= \dfrac{1}{|ab|} G\left(\dfrac{f_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/836419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fair die being rolled repeatedly A fair die is rolled repeatedly, and let $X$ record the number of the roll when the 1st $6$ appears. A game is played as follows. A player pays \$1 to play the game. If $X\leq 5$ , then he loses the dollar. If $6 \le X \le 10$, then he gets his dollar back plus \$1. And if $X... | The probability of failures before the first success is modeled by the geometric distribution. So you have to figure out what the expectation of the random variable with payout:
$$
\begin{matrix}
-1 & P(X \leq 5)\\
1 & P(6 \leq X \leq 10)\\
3 & P(X \geq 11)
\end{matrix}
$$
Where $X$ is geometrically distributed with p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why can't we define more elementary functions? $\newcommand{\lax}{\operatorname{lax}}$
Liouville's theorem is well known and it asserts that:
The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.
The problem I got from this is what is an elementary function? Who ... | Elementary functions are finite sums, differences, products, quotients, compositions, and $n$th roots of constants, polynomials, exponentials, logarithms, trig functions, and all of their inverse functions.
The reason they are defined this way is because someone, somewhere thought they were useful. And other people bel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/836556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
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"answer_id": 1
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Finding parametric equations for the curved path of a particle around a half-circle I have a question about parametric equations. So far I've learned how to find the parametric equations for a straight line, I know about replacing $x^2$ and $y^2$ in the equation of the unit circle, but I'm having trouble with this part... | A generic circle of radius $r$ centered at the origin is given parametrically by $\alpha(t) = \langle r\cos(t), r\sin(t) \rangle$ such that $0 \leq t \leq 2\pi$. This is pretty straightforward. Simply draw a circle centered at the origin and draw a line segment from its center to an arbitrary point on its perimeter. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A machine has $9$ switches. Each switch has $3$ positions. How many different settings are possible? A machine has $9$ switches. Each switch has $3$ positions.
$(1)$ How many different settings are possible?
Each switch has $3$ different settings and we have $9$ total. So,
$3^9=19,683$
Now, the problem I am facing is ... | Hint for 2: How many different words can you spell with the letters AAABBBCCC?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/836736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Average number of Dyck words in a Dyck word Given a integer $n$, how many Dyck words are a substring of a Dyck word of size $n$, on average?
For example, if $n=2$, then Dyck words of size $2$ are :
*
*[ ] [ ]
*[ [ ] ]
(1) contains two strict "sub-Dyck words" : [ ] (with the first two parentheses) and [ ] (with... | So, I've coded a little Python program that computes for each $n$, the total number of "sub-Dyck words" in all Dyck words of semi-length $n$
Here is the output for $n$ ranging from 1 to 13 : 1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860.
Which is know as A002054 in oeis. And that's eve... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to show that $\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt = \pi^{1/2} \frac{ \Gamma(s-1/2)}{\Gamma(s)}$ I want to compute the integral
$$
\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt
$$
for $s \in \mathbb{C}$ such that the integral converges ($\mathrm{Re}(s) > 1/2$ I think) in terms of the Gamma function. I... | Hint: Let $x=\dfrac1{t^2+1}$ and then recognize the expression of the beta function in the new integral.
But first, using the parity of the integrand, write $\displaystyle\int_{-\infty}^\infty f(t)~dt~=~2\int_0^\infty f(t)~dt$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/836871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Low Level Books on Conjectures/Famous Problems I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses in Algebra and Real Analysis 1. I was wondering if there are any books abo... | The Goldbach Conjecture by Yuan Wang.
From the book's description:
A detailed description of a most important unsolved mathematical problem - the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements we... | {
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Two 3-cycles generate $A_5$ I want to solve the following exercise, from Dummit & Foote's Abstract Algebra
Let $x$ and $y$ be distinct 3-cycles in $S_5$ with $x \neq y^{-1} $.
Prove that if $x$ and $y$ do not fix a common element of $\{1,2,3,4,5\}$ then $\langle x,y\rangle =A_5$.
I know that using brute force one c... | Let $H = \langle x,y \rangle \leq A_{5}.$ Then $H$ has more than one Sylow $3$-subgroup, so has either $4$ or $10$ Sylow $3$-subgroups. If $H$ has $10$ Sylow $3$-subgroups, then $|H|$ is divisible by $30.$ If $|H| = 60,$ we are done. If $|H| = 30,$ then $H \lhd A_{5},$ and furthermore, since $H$ already contains $20$-e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/837042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A problem with my reasoning in a problem about combinations I was given the following problem to solve:
A committee of five students is to be chosen from six boys and five
girls. Find the number of ways in which the committee can be chosen,
if it includes at least one boy.
My method was $\binom{6}{1}\binom{10}{4}... | Your idea is to choose one boy and then four others, which might include further boys. Nice idea, but unfortunately it doesn't work: the reason why should be clear from the following choices.
*
*Choose the boy $B_1$, then four more people $G_1,G_2,G_3,B_2$.
*Choose the boy $B_2$, then four more people $G_1,G_2,G_3... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the remainder after dividing $(177 + 10^{15})^{166}$ by $1003 = 17 \cdot 59$ What is the remainder after dividing $(177 + 10^{15})^{166}$ by $1003 = 17 \cdot 59$? I have made several observations about the problem but I cant see how they lead me to an answer.
Observation one: $\phi(1003) = 16 \cdot 58 =8 \cdot ... | Write $x=177+10^{15}=177+1000^5$. Modulo $17$ we have
$$x=7+(-3)^5=-236=2\quad\Rightarrow\quad x^4=-1\quad\Rightarrow\quad x^{166}=x^6=-4\ .$$
Modulo $59$ we get
$$x=-3^5\quad\Rightarrow\quad x^{166}=3^{830}=3^{18}=-2\ .$$
Now use the Chinese Remainder Theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\frac{\sin\theta}{1-\cos\theta} \equiv \csc\theta + \cot\theta$ This must be proved using elementary trigonometric identities.
I have not been able to come to any point which seems useful enough to include in this post.
| Lets get rid of the trigonometry stuff first:
$$s=\sin(\theta),~~c=\cos(\theta),~~\csc(\theta)=\frac{1}{s},~~\cot(\theta)=\frac{c}{s}$$
Now we are solving this equation:
$$\frac{s}{1-c}=\frac{1}{s}+\frac{c}{s}$$
$$\Leftrightarrow$$
$$\frac{s}{1-c}-\frac{1}{s}-\frac{c}{s}=0$$
multiply by $(1-c)\neq 0$
$$s-\frac{1-c}{s}-... | {
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How to understand the variance formula?
How is the variance of Bernoulli distribution derived from the variance definition?
| PMF of the Bernoulli distribution is
$$
p(x)=p^x(1-p)^{1-x}\qquad;\qquad\text{for}\ x\in\{0,1\},
$$
and the $n$-moment of a discrete random variable is
$$
\text{E}[X^n]=\sum_{x\,\in\,\Omega} x^np(x).
$$
Let $X$ be a random variable that follows a Bernoulli distribution, then
\begin{align}
\text{E}[X]&=\sum_{x\in\{0,1\}... | {
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"timestamp": "2023-03-29T00:00:00",
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How can I show the complete symmetric quadratic form has no zeros? The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be
$$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + \sum_{j<k} t_j t_k.$$
For example, in one variable we have $... | Well there is the trivial solution $\tilde{t}=(t_1,t_2,\cdots ,t_n)=(0,0,\cdots ,0)$. Now write $$\displaystyle h_2(\tilde{t})=\frac{1}{2}\left(\sum_{i=1}^{n}t_i^2\right)+\frac{1}{2}\left(\sum_{i=1}^{n}t_i\right)^2\ge 0$$ where equality only occurs when everything is zero. I suppose I understood the question correctly.... | {
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Existence of "simple" irreducible polynomial of degree 12 in a finite field Assume that we have a finite field $\mathbb{F}_p$, where $p$ is prime, $p \equiv 1\ (\textrm{mod}\ 4)$ and $p \equiv 1\ (\textrm{mod}\ 3)$. I was looking for irreducible polynomial in a form $X^{12} + a$, where $a \in \mathbb{F}_p$ and it turne... | $x^{12}-a$ is reducible mod $p=12k+1$ if and only if $a=0$ or $a^k$'s order is a proper divisor of 12.
To see this, recall that a monic polynomial of degree $d$ is irreducible mod $p$ if and only if $\gcd(f(x), g_i(x)) = 1$ for all $g_i(x) = x^{p^i}-x$, $1\leq i < d$.
Fixing $i$, and working in $\mathbb{Z}_p[x^{12}-a]$... | {
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The total number of points of $\mathbb{R}$ at which $f$ attains a local extremum Let $f(x) = \vert x^2-25 \vert$ for all $x \in \mathbb{R}$. The total number of points of $\mathbb{R}$ at which $f$ attains a local extremum is
$A$. $1$
$B$. $2$
$C$. $3$
$D$. $4$
What I was thinking where $f'(x)=0$. but this only gives y... | Recall that
a function can have a relative maximum or relative minimum only at those numbers in its domain at which the derivative is undefined or is zero (these numbers are called critical points).
Notice that $f'(x) = \begin{cases} -2x & x \in (-5, 5) \\ 2x & x \in (-\infty, -5) \cup (5, \infty) \\ \text{DNE} & x =... | {
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Prove the identity... $$\frac{\cos{2x}-\sin{4x}-\cos{6x}}{\cos{2x}+\sin{4x}-\cos{6x}}=\tan{(x-15^{\circ})}cot{(x+15^{\circ})}$$
So, here's what I've done so far, but don't know what do do next:
$$\frac{\cos{2x}-2\sin{2x}\cos{2x}-\cos{6x}}{\cos{2x}+2\sin{2x}\cos{2x}-\cos{6x}}=$$
$$\frac{\cos{2x}-4\sin{x}\cos{x}\cos{2x}-... | Using Prosthaphaeresis Formula, $$\frac{\cos2x-\cos6x}{\sin4x}=\frac{2\sin4x\sin2x}{2\sin4x}=\frac{\sin2x}{\dfrac12}$$
$$\implies \frac{\cos2x-\cos6x}{\sin4x}=\frac{\sin2x}{\sin30^\circ}$$
Now apply Componendo and dividendo and again apply Prosthaphaeresis formulae $$\sin C\pm\sin D$$
| {
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Proving that $\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$. Prove:
$$\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$$
Thanks a lot!!
I tried:
With Euler's identity I can get $\sin x= \dfrac{e^{ix} - e^{-ix}}{2i}$ and the cosine too. But i'm lost trying to... | Given that
$$ e^{ik\theta} = \cos(k\theta) + \textbf{i} \sin(k\theta) $$
Then we get
$$ \sum_{k=1}^n e^{ik\theta} = \sum_{k=1}^n \Big( \cos(k\theta) + \textbf{i} \sin(k\theta) \Big) $$
which gives
$$ \sum_{k=1}^n e^{ik\theta} = \sum_{k=1}^n \cos(k\theta) + \textbf{i} \sum_{k=1}^n \sin(k\theta) $$
| {
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Proving by calculation that $\arg(-2) = \pi$ The fact that it is true, seems very obvious, if one draws the complex number $z = (-2 + 0i)$ on the complex plane. The angle is certainly 180 degrees, or pi radians.
But how can this be proven by calculation? Using $\arg(z)=\arctan(b/a)$ or even the "extended" version $\arg... | More complicated than $\arctan(y/x)$, yet valid for all $z\ne0$ is
$$
\arg(x+iy)=\left\{\begin{array}{cl}
2\arctan\left(\frac{y}{x+\sqrt{x^2+y^2}}\right)&\text{if }y\ne0\text{ or }x\gt0\\[6pt]
\pi&\text{otherwise}
\end{array}\right.
$$
which is based on the identity
$$
\tan(\theta/2)=\frac{\sin(\theta)}{1+\cos(\theta)}... | {
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Application of Christoffel symbol in differential geometry When self-studying differential geometry, I find my book involves some clumsy, troublesome calculation about Christoffel symbol when proving theorem, which in fact doesn't have the symbols. I wonder if the symbol is actually useful when for doing calculation st... | It is possible and highly recommended to learn how to do the calculations without using Christoffel symbols, in a coordinate free manner. Personally, I like the abstract index notation, for instance.
On the other hand, a good understanding of the coordinate calculations is very helpful when one attempts to read the leg... | {
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Is triangle congruence SAS an axiom? I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms.
Thank you for your help!
| The answer to this question is a bit complicated. It's not a straight yes or no.
Euclid claims to prove side-angle-side congruence in his Proposition 4, Book 1. He does this by "applying" one triangle to the other. Essentially, this means he moves one triangle until it coincides with the other.
Euclid's proof is unive... | {
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Connectedness of $\mathbb R$ Let $\mathbb R$ denote the real number space, and let sets A and B be closed and nonempty such that $\mathbb {R} \subset A \cup B $, why is it true that due to the connectedness of $ \mathbb{R} $, $A \cap B \neq \emptyset $?
| Your assumptiom is $\mathbb{R} \subset A \cap B$.
But $\mathbb{R} \neq \emptyset$ so $\emptyset \neq \mathbb{R} \subset A \cap B$ imply $A \cap B \neq \emptyset$. Note that we didn't use connectivity of $\mathbb{R}$.
| {
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Pullback in morphism of exact sequences Suppose we have the following morphism of short exact sequences in $R$-Mod:
$$\begin{matrix}0\to&L&\stackrel{f'} \to& M'&\stackrel{g'}\to &N' & \to 0\\
&\;||&&\downarrow\rlap{\scriptsize\alpha'}&&\;\downarrow\rlap{\scriptsize\alpha}\\
0\to &L&\stackrel{f}\to& M&\stackrel{g}\to& N... | If $(m,n') \in M \times_N N'$, choose a lift $m' \in M'$ of $n'$ and consider the image $\tilde{m} \in M$. Then $\tilde{m},m$ have the same image in $N$, hence there is a unique $l \in L$ such that $m=\tilde{m}+f(l)$. Then $m' + f'(l)$ is a the unique element of $M'$ which maps to $m$ and to $n'$.
| {
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Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ been stuck with this question for the last few hours, any help would be appreciated.
$$
{\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,}
\left(\,n - k\,\right)^{n}}
$$
what I did:
$\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n=n!\... | First, ask yourself the following question: $\quad\displaystyle\sum_{k=0}^na^k{n\choose k}x^{n-k}~=~?\quad$ Hint: See binomial theorem.
Then apply the following two steps repeatedly: $(1).$ Differentiate both sides with respect to x, and
$(2).$ Multiply both sides with x. Notice how, after each two steps, $\bigg(x\dfr... | {
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When the approximation $\pi\simeq 3.14$ is NOT sufficent It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example of such a purpose, accessible to a middle school student.
Thank... | The approximation required for any number depends on the purpose, as you can see. It is a good approximation to take $\pi$ being approximately equal to 3.14.
So, generally in mechanical engineering we can assume this approximation to be true because we take into account the factor safety which serves the purpose causin... | {
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$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus... | First of all it is clear that if $v \in H_1 \cap H_2$, then $v = T(v) = -v$ and so we must have $v = 0$. It follows that $H_1 \cap H_2 = 0$.
Now let $v \in V$, by definition of $T$, $T^2(v)= T(T(v))=v$ which implies that $T(v) \in H_1$, furthermore by linearity of $T$ holds $T(v-T(v)) = T(v)-T^2(v) = -(v-T(v))$ which ... | {
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Confusion Regarding the Axiom of Countable Choice My current understanding of the Axiom of Countable Choice is that the following example needs it in order to work:
Let $X$ be a countable family of finite sets. Then there exists a choice function $f$ choosing for each $x$ a bijection between $x$ and a natural number $... | Yes, we need the axiom of countable choice for choosing such bijections.
If we can choose a bijection for each finite set, then their union is countable, since we can map it into $\omega\times\omega$ in the obvious way.
An example of a countable set of finite sets, even pairs, whose union is not countable, if so, is an... | {
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Find interval with function that solves ODE $y'(x)=1+(y(x))^2$ Let $g\in C^1(\mathbb{R})$ with $g'\gt 0$ and $g(0)=0$.
Show that for the differential equation $$\begin{cases}y'(x) & = \dfrac{1}{g'(y(x))} \\[8pt] y(0) & = 0 \\\end{cases}$$ there exists exactly one non-empty open interval $I\subseteq\mathbb{R}$ and an $... | Hint: as $g'\neq 0$, using the chain rule the equation is equivalent to
$$
g(y(x)) = g(y(0)),
\\
y(0) = 0
$$
| {
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very strange phenomenon $f(x,y)=x^4-6x^2y^2+y^4$ integral goes wild I am going over my lecture's notes in preparation for exam and I saw something a bit strange I would like someone to explain how it is possible.
Look at the function $f(x,y) = x^4-6x^2y^2+y^4$
if we convert it to polar coordinates, we will get $f(r,\th... | The integral is unbounded when integrated over the entire plane of $\mathbb R^2$, what you are doing by changing the variables is analogous to changing the order of the terms in an non-converging summation, you can get more than one "answer" but in fact none of them are valid.
| {
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Proving polynomial limit theorems I am pretty confused on this math question. It is a two-parter but I'm not sure what part a is asking me, perhaps someone on StackExchange could help.
The question reads as follows:
(a) If p is a polynomial, prove, using limit theorems, that
$$\lim_{x\to a}p\left(x\right) = p\left(a\ri... | Since any polynomial $p$ is continuous on $\Bbb R$ then we have
$$\lim_{x\to a}p(x)=p(a)$$
so to find the limit just evaluate the polynomial on $a$.
| {
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Probability that a card drawn is King on condition that the card is a Heart
From a standard deck of 52 cards, what is the probability that a randomly drawn card is a King, on condition that the card drawn is a Heart?
I used the conditional probability formula and got:
Probability that the card is a King AND a Heart: ... | We want to work out $P(king|heart)$.
$P(heart)=1/4$
$P(king \& heart)=1/52$
Conditional probability formula: $P(A|B)=P(A \& B)/P(B)$.
So substituting into this formula we get:
$P(king | heart) = P(king \& heart) / P(heart) = (1/52)/(1/4) = 4/52 = 1/13$ as required.
So yes, you are correct.
| {
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Help undersanding meromorphics Herglotz functions A meromorphic function $f$ is called meromorphic herglotz function if $\mathrm{Im}(z)>0$ implies $\mathrm{Im}(f(z))>0$
I need to prove that all the poles and zeros of $f$ are in $\mathbb{R}$. Morover, each pole and zero is simple and the poles and zeros alterante.
Ther... | If a meromorphic function $f$ has a pole or zero at $z_{0}$, then, for a unique non-zero integer $n$,
$$
f(z) = (z-z_{0})^{n}g(z)
$$
where $g$ is holomorphic near $z_{0}$ with $g(z_{0})\ne 0$. Then
$$
f(z) = (z-z_{0})^{n}g(z_{0})+(z-z_{0})^{n+1}\left[\frac{g(z)-g(z_{0})}{z-z_{0}}\right].
$$
By choosing $z=re... | {
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Evaluating trigonometric functions How would I evaluate cot 30 + cot 60? I know that cot 30 = 1/tan30 and cot60 = 1/tan60.
The answer must have a rational denominator where relevant.
I have tried adding them like normal fractions after evaluating, but got the incorrect answer. Any relevant online reading material would... | Well tan(a)=sin(a)/cos(a) So, you want cos(30)/sin(30) + cos(60)/sin(60) = $\frac{\sqrt{3}/2}{1/2} + \frac{1/2}{\sqrt{3}/2} = \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}}+\frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4 * \sqrt{3}}{3}$
| {
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Why exponential function on p-adic numbers is meaningless? In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaningless? Thank you very much.
| It’s not the exponential function for $p$-adic numbers that’s meaningless; rather it’s the act of multiplying the real number $\pi$ by a nonrational $p$-adic number that’s meaningless. There’s no way of multiplying a real times a $p$-adic unless one of them is rational.
On the other hand, there is a $p$-adic exponentia... | {
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How to evaluate the following indefinite integral? $\int\frac{1}{x(x^2-1)}dx.$ I need the step by step solution of this integral
please help me!
I can't solve it!
$$\int\frac{1}{x(x^2-1)}dx.$$
| 1/{x*(x^2-1)}
=x/{x^2*(x^2-1)}
If we substitute:
x^2=z
By differentiating both sides
2x dx = dz
x dx= dz/2
Now if we solve the integral
(1/2)log{(x^2-1)/x^2}+C
| {
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If $B$ the inverse matrix of $A^2$ show that the inverse of $A$ is $AB$ How do I continue from $A(AB) = (BA)A = I$ and how can we justify it if we don't know that $AB=BA$?
EDIT: Also, how can we prove that if $AB=I$ then $ BA = I$? This is seperate from the question above but I felt it didn't need a new post.
| $A(AB)=(AA)B=A^2 B=I$ and $(BA)A=B(AA)=BA^2=I$
| {
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Can the value of $(-9!)$ be found I saw this question on an fb page and I couldn't solve it.
Question:
What is the value of $(-9!)$?
a)$362800$
b)$-362800$
c) Can not be calculated
The first options seems to be incorrect,which leaves $c$ but I can't justify it.Does it have something to do with gamma function which asks... | We have the following property: $n!=\dfrac{(n+1)!}{n+1}$ . Hence, $0!=\dfrac{1!}1=1;~(-1)!=\dfrac{0!}{0_\pm}=\pm\infty$, etc.
| {
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Is there any way to solve this problem without having to do it by hand? I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement. Is there any way to group the relations below, or to find values of y in t... | As stated, the problem has a very simple answer. If $x$ is even, relation [1] is satisfied if you let $y=0$ and $k=x/2$. If $x$ is odd, none of the relations can be satisfied, since they all imply that $x$ is even -- i.e., they can be rewritten as saying $x=2k(10y+r)$ with $r=1$, $3$, $7$, or $9$.
| {
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"answer_id": 0
} |
Operations with complex numbers to give real numbers If:
*
*$|z|=|w|=1$
*$1 + zw \neq 0$
Then $\dfrac{z+w}{1+zw}$ is real. How can prove that.
| Let $z=e^{ia}$ and $w=e^{ib}$ for some $a,b \in [0,2\pi)$. This takes care of condition 1. Then $1+zw \neq 0$ means $1+e^{i(a+b)} \neq 0$, which is the same as saying $a+b$ is not an odd multiple of $\pi$.
Now consider
\begin{align*}
\dfrac{z+w}{1+zw} & = \frac{e^{ia}+e^{ib}}{1+e^{i(a+b)}}
\end{align*}
Rationalize the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integrate $\int_0^1 \ln(x)\ln(b-x)\,\mathrm{d}x$, for $b>1$? Let $b>1$. What's the analytical expression for the following integral?
$$\int_0^1 \ln(x)\ln(b-x)\,\mathrm{d}x$$
Mathematica returns the following answer:
$$2-\frac{\pi^{2}}{3}b+\left(b-1\right)\ln\left(b-1\right)-b\ln b+\mathrm{i}b\pi\ln b+\frac{1}{2}b\ln^{2... | The following is an evaluation in terms of $ \displaystyle \text{Li}_{2} \left(\frac{1}{b} \right)$, which is real-valued for $b > 1$.
$$\begin{align} \int_{0}^{1} \log(x) \log(b-x) \ dx &= \log(b) \int_{0}^{1} \log(x) + \int_{0}^{1}\log(x) \log \left(1- \frac{x}{b} \right) \ dx \\ &= - \log(b) - \int_{0}^{1} \log(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Local minimum implies local convexity? Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$.
It typically looks like
What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such that $f$ is convex over $(a-\epsilon,a+\epsilon)$ ?
The motivation for this question is int... | No in the continuous case, a part requiring in practice convexity. Yes in the smooth case.
In general local minima have nothing to do with convexity:
The function $\sqrt{|x|}$ has a local minimum in $0$ but it is not convex
The function $e^x$ is striclty convex everywhere but has no minimum.
On the other hand, as point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Correlation coefficient calculation Why do we remove of the mean of the data while calculating the correlation coefficient value of bivariate data in statistics? DotProduct/ProductOfLengthOfVectors should always give anyway a coefficient that is between -1 and 1. What does removal of the mean achieve?
| Suppose we have finite samples $\{x_1,x_2,\ldots,x_n\}$ and $\{y_1,y_2,\ldots,y_n\}$ from two distributions with sample means:
$$\bar{X}=\frac1{n}\sum_{i=1}^{n}x_i, \\\ \bar{Y}=\frac1{n}\sum_{i=1}^{n}y_i,$$
and sample variances
$$S_X^2=\frac1{n}\sum_{i=1}^{n}(x_i-\bar{X})^2, \\\ S_Y^2=\frac1{n}\sum_{i=1}^{n}(y_i-\bar{Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Analytic continuation of a real function I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique.
Now i am wondering if this is true for real functions. I mean, if $f: \mathbb{R} \to \mathbb{R}$, when is it true... | I think that most mathematicians would say that the functions you mention are restrictions to the real line of functions more naturally defined on the complex plane in the first place. So--yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/840192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 1
} |
Can every positive real be written as the sum of a subsequence of dot dot dot I answered this thing Infinite sum of prime reciprocals and now wonder what happens if we do not have such a strong condition as Bertrand's postulate. i have been fiddling with this, not sure either way.
Given a sequence $a_1 > a_2 > a_3 \cdo... | Let $x$ be the desired real number, and let $i_1$ be the smallest positive integer such that $x > a_{i_1}$ (we know such an integer exists because $a_i \to 0$). Now let $i_2$ be the smallest positive integer greater than $i_1$ such that $x - a_{i_1} > a_{i_2}$. Continuing in this way, we obtain a subsequence $(a_{i_j})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Probability Help (die problem) A die is rolled 20 times. How many different sequences
a) each number 1-6 is rolled exactly three times
My Answer: (20 choose 6)*(3 choose 1)
b) each number 1-6 are each rolled exactly once in the first six rolls?
My Answer: (20 choose 6)*(6 choose 1)
c) each number rolled is at least as... | Each number 1-6 is rolled exactly three times
No sequence of twenty rolls can be created out of 18 numbers.
Each number 1-6 are each rolled exactly once in the first six rolls
Consider just the first six rolls. The first roll can be anything. The second roll can be any of the five remaining numbers, the third roll c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
A simple probability question about two types of cards There is a hidden box that contains two different types of cards--1 Card A and 1 Card B. Card A has both sides of the card red while Card B has one side red and the other blue. If you randomly picked a card and saw a red face, what is the probability that this card... | If we assume there are the same number of type 'A' cards as type 'B' cards then if you pull a card at random and look at one side only there are 4 equally likely results
Red or Red from card A.
Red or Blue from card B.
By observing a red face we can eliminate one of these options so we now have only 3 equally likely re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Need explanation for simple differential equation I can't figure out this really simple linear equation:
$$x'=x$$
I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating both sides but it doesn't seem to work. I know this is shameful noob questi... | $$x'=x \Rightarrow \frac{dx}{dt} \Rightarrow \frac{1}{x} dx = 1 \ dt \Rightarrow \int \frac{1}{x} dx = \int 1 \ dt \Rightarrow \ln(x)= t + C \Rightarrow x(t)=e^te^c.$$
$\cdot \ \text{Let A}=e^c \ \text{then} \ x(t) = Ae^t$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/840522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
} |
Probability Of a 4 sided die A fair $4$-sided die is rolled twice and we assume that all sixteen
possible outcomes are equally likely. Let $X$ and $Y$ be the result of the $1^{\large\text{st}}$ and the
$2^{\large\text{nd}}$ roll, respectively. We wish to determine the conditional probability $P(A|B)$
where
$A = \max(X,... | min(X,Y)=2. So 5 outcomes left: (2,2) (2,3) (2,4) (3,2) (4,2)
Q: What values can m take? | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
What's the complexity of expanding a general polynomial? Suppose I have a polynomial in the form $(a_1 x_1+ a_2 x_2+...+ a_m x_m)^n$, where $x_1,...,x_m$ are the independent variables. I want to expand it to the form of sum of products. What is the complexity,i.e. the big O notation?
| If I understand your question good.
If you consider your polynomial $X=a_1x_1+\cdots+a_mx_m$, then you want to calculate the complexity of the operation $X^n$. As far as I know, it depends on the method used and the best algorithm to do $X^n$ is exponentiation by squaring which is given, for example, heretime complexi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear equation: $(A^\top A+B^\top B + D)x=c$ where $A,B$ are structured sparse and $D$ is diagonal. Updated: the goal is to solve $(A^\top A+B^\top B + D)x=c$. Maybe it is not necessary to compute $(A^\top A+B^\top B + D)^{-1}$.
Denote $e=(1,1,\ldots,1)^\top\in\mathbb{R}^n$ and
$$A=\begin{bmatrix}
e & & & \\
& e... | Let $C=A^TA+B^TB+D$. You can write it as
$$
C=D+EE^T, \quad E=[A^T,B^T].
$$
The inverse can be then using the Woodbury formula written as
$$
\begin{split}
C^{-1}&=D^{-1}-D^{-1}E(I+E^TD^{1}E)^{-1}E^TD^{-1}\\
&=D^{-1}-D^{-1}[A^T,B^T]\left\{I+\begin{bmatrix}A\\B\end{bmatrix}D^{-1}[A^T,B^T]\right\}^{-1}\begin{bmatrix}A\\B\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limit with L'Hospital with infinite indeterminate formats I'm trying to find the limit:
$$\large \lim_{x\to0}(\sin x)^x$$
Whst I did was apply L'Hospital Rule:
$$\large \text{let }y =(\sin x)^x\implies
\ln y=x\ln\sin x$$
$$\large
\lim_{x\to0}\ln y
=
\lim_{x\to0} x\ln\sin x
=
\lim_{x\to0}\frac x{\frac1{\ln\sin x}}
=
\l... | HINT:
$$\lim_{x\to0^+}\frac{\ln\sin x}{\dfrac1x}=\lim_{x\to0^+}\frac{\dfrac{\cos x}{\sin x}}{-\dfrac1{x^2}}=-\lim_{x\to0}x\cdot \lim_{x\to0}\cos x\cdot\frac1{\lim_{x\to0}\dfrac{\sin x}x}$$
Hope you can take it home from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/840915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Prove no existing a smooth function satisfying ... related to Morse Theory i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ and $f(x_1)=f(x_2) = 0$. (This part is easy).
... | As required by "This is much healthier", I post a new thread to express my opinions as related to "user126154"'s answer, which is great, however there is something that I can't convince myself.
First of all, the compactness condition in the proof of "user126154" can be relaxed as proposed in Richard Palais: Topology Vo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Showing $A \subset B \iff A\cap B=A$ Showing
$A \subset B \iff A\cap B=A$
How would I show this?
My proof
Assume
i. $A \cap B \subset A$
ii.$A \subset A \cap B$
Let $x$ be any element.
Assume $x \in A \cap B$. Then $x \in A$ and $ x \in B$. By hypthesis $x \in A \rightarrow x \in B$ Thus $x \in A$
ii.
Let $ x \in A$ B... | What you have written seems correct, but a bit confusing.
I might write the proof like this
Proof
We want to prove that
$$
A \subset B \iff A\cap B=A
$$
Assume that the right hand side is true. That is, assume that $A\cap B = A$. We want to show that $A\subseteq B$. Let $a\in A$. Since $A = A\cap B$, we have $x\in A$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/841024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
$f(x)=\log x\Leftrightarrow f^{-1}(x)=e^x$. Why $e=2.73\cdots$?
$$f(x)=\log x\Leftrightarrow f^{-1}(x)=e^x.$$
Ok, $\log x$ is defined as the function $f(\cdot)$ such that: $f'(x)=\dfrac{1}{x}$. How to get, from this, the inverse of it $f^{-1}(x)$? And why $e=2.73\cdots$?
| The other answer has not answered your question, “Why is $e$ equal to $2.718281828\cdots$?”.
Let’s form the number $E=\lim_n(1+\frac1n)^n$, and evaluate it knowing the continuity of the log function and what its derivative is. Of course this number $E$ is computable, even if slowly, directly by hand. And if you take $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/841120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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