Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $? So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its elements.
But how do we go a step further in under... | For any set of points of $\mathbb R^2$ we can draw the set in 2-D by visualizing $\mathbb R^2$ as a sheet of paper and printing a pixel black if it belongs to the set or leaving it white if it doesn't belong (keep extra toner handy, as every page is infinite in size and there are uncountably many sheets). Print every s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Evaluating a double integral from zero to infinity How do I evaluate this integral? I don't understand at which point the limit notation should set in? And my method yields $0$ in the end. The integral is:
$$
\int_0^{\infty} \int_0^{\infty} c\,x\,y\,e^{-(x+y)} \;\mathrm{d}y\;\mathrm{d}x
$$
| You can separate $x$ and $y$ and integrate successively with respect to each variable since $$cxye^{-(x+y)}=c(xe^{-x})(ye^{-y})$$ Now $$\int_{0}^{\infty}xe^{-x}dx$$ is equal to the expectation of an exponentially distributed random variable with parameter $1$. Therefore it is also equal to $1$. Thus $$
\int_0^{\infty} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Proving of the multiplication theorem for Bernoulli polynomial How the expression below can be proven:
$$B_n(mx) = m^{n−1} \sum\limits_{k=0}^{m-1}B_n\left(x+\frac{k}{m}\right)$$
Where $B_n(x)$ is Bernoulli polynomial
I know it is already proved by Joseph Ludwig Raabe, but I don`t know how exactly.
| By induction over $n$.
If $n=1$ we have from definition of $B_1(x)=x-1/2$,
\begin{eqnarray*}
B_1(mx) = mx - \frac{1}{2},
\end{eqnarray*}
On the other hand, the right hand side with $n=1$ is
\begin{eqnarray*}
\sum_{k=0}^{m-1} B_1 \left ( x +\frac{k}{m} \right )
&=& \sum_{k=0}^{m-1} \left ( x+ \frac{k}{m} - \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 1
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How do I compute the kernel of this map? How do I compute $\ker{(\mathbb{Z} \otimes A \longrightarrow \mathbb{Q} \otimes A)}$ where this map comes from the short exact sequence:
$0 \rightarrow Tor(\mathbb{Q}/\mathbb{Z}, A) \rightarrow \mathbb{Z} \otimes A \rightarrow \mathbb{Q} \otimes A \rightarrow \mathbb{Q}/\mathbb{... | The map from $\mathbb{Z} \otimes A \to \mathbb{Q} \otimes A$ is induced by the canonical inclusion, so is defined on generators by $n\otimes a \mapsto \frac{n}{1}\otimes a$ (for any $n \in \mathbb{Z}, a \in A$).
First, the torsion subgroup of $A$ is contained in the kernel.
Recall that there is a canonical identificat... | {
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"url": "https://math.stackexchange.com/questions/1013712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do the $n\times n$ matrices over a division ring $D$ form a free $D$-module? Let $D$ be a division ring. Then the set of $n\times n$ matrices over $D$ is free as a
$D$-module.
I think this is wrong because they are linearly dependent, right?
But what does the given of $D$ as a division ring changes ?
| It's easy to see that $M_n(R)$ is isomorphic to the direct sum of $n^2$ copies of $R$ as an $R$ module for any ring $R$.
It's clearly free and has a basis consisting of the "unit matrices" $E_{ij}$ that are $1$ on the $i,j$ entry and zero elsewhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving various relations are partial orders I am given these relations, in which I have to prove or disprove each and every one.
a. The relation $\trianglelefteq$ defined on ℕ by a $\trianglelefteq$ b if a ≤ b²
b. The relation $\preceq$ defined on ℤ by m $\preceq$ n if m ≤ n + 5.
c. The relation $\ll$ on the set of cont... | The first proof is not correct. $a\leq b^2$ and $b \leq c^2$ does not imply $a \leq c^2$. Consider $a=15, b=4, c=3$.
The second proof is also incorrect because again transitivity fails. Consider $m=10,n=5,p=0$.
Your problem is that you state, for example, that "if m ≤ n + 5, and n ≤ p + 5, then m ≤ p + 5" but do not gi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it true that an equivalent 'absolute value' is an absolute value? I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by $|x|'=|x|^r$ for some $r> 0$ also an absolute value? How do I ... | The triangle inequality holds if $r\leq 1$, but may not hold if $r>1$. To see that it may not hold if $r>1$, you may use $\mathbb{Q}$ with the standard absolute value as an example. If $r<1$, then note that for $x$ and $y$ positive with $y\leq x$ we have that
for some $c$ with $x\leq c\leq x+y$
$$(x+y)^r-x^r=ryc^{r-1}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $n! = O(n^{n})$ As the title says, how would you show that $n! = O(n^{n})$? I'm not really understanding how one "shows" the Big O notation of a function mathematically (at least when you're dealing with things that aren't polynomials).
| Note that $1\cdot 2\cdot \ldots \cdot n\le n\cdot n\cdot \ldots \cdot n$, so we even have $n!\le n^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear Indepence Question involving an arbitrary vector Suppose $u, v, w$ are linearly independent in $V$ and $x \in V$.
Prove $u + x, v + x, w + x$ are always linearly independent.
I'm stuck on this problem because I don't know what to do with $x$. Any hints or help would be appreciated.
| You're stuck because the result is false as it is stated now. Let ${\bf x} = -{\bf u}$, or the opposite of any of the other two vectors. If $\{{\bf u},{\bf v}, {\bf w}, {\bf x}\}$ is linearly independent, for example, it seems true, and follows straight from definition. Maybe there is some other hypothesis we can add ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine whether there is linear independence or not Determine if the following set of vectors is linearly independent:
$$S = \{e^x, e^{2x}\}$$ of continuous functions.
So my problem isn't in not knowing what linearly independence is, it is setting up a linear combination of this form:
$$a_1e^x + a_2e^{2x} = ????$$
W... | your matrix will be in a 2x2 form where row 1 is ae^x + be^2x = 0 and row 2 = e^x + 2e^2x =0, you find the determinant by factoring out e^xe^(2x) multiplied by your simplified matrix with (1,1) and (1,2) as rows 1 and 2. Hence determinant is e^xe^(2x)* (2-1)= e^xe^(2x)(1) which is not zero, meaning you have a set of li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Sum of 6 cards being multiple of 6 I pick 6 cards from a set of 13 (ace-king). If ace = 1 and jack,queen,king = 10 what is the probability of the sum of the cards being a multiple of 6?
Tried so far:
I split the numbers into sets with values:
6n, 6n+1, 6n+2, 6n+3
like so:
{6}{1,7}{2,8}{3,9}{4,10,j,q,k}{5}
and then gro... | Your combinations are $4\cdot 4+1\cdot 0+1\cdot 2, 2\cdot 1+2\cdot2+2\cdot3, 4 \cdot 4+2\cdot 1,+$ something that doesn't make sense because there are three places you choose from $1$ and only $0,5$ qualify. The left number is the number of cards of that value $\pmod 6$ . You could also have $3\cdot 4+1 \cdot 5+1 \cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many distinct binary bit strings of length fifteen are there? I know this is a simple question but i'm not sure of which combinatorial selection equation to use.
How many distinct binary bit strings of length fifteen are there?
Using a simple example, would someone be able to explain the difference between
*
*or... | With binary strings of length $15$ you can't have without repetition. There are only two characters, so by the third character you will have repetition. If the string is not ordered you only care how many $0$s and $1$s there are. The number of $0$s can range from $0$ to $15$-how many is that? Then the number of $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014555",
"timestamp": "2023-03-29T00:00:00",
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Are epimorphisms in the category of magmas surjective? The question says it all, but let me recall the definitions.
*
*A magma $(X, \cdot)$ is a set $X$ with a binary operation $\cdot \colon X \times X \to X$ (without any further assumptions like associativity).
*A morphism between two magmas $X$ and $Y$ is a map $f... | Yes. Suppose $f:X\to Y$ is not surjective, and let $Z$ be the complement of $f(X)$ in $Y$. Then the disjoint union $f(X)\cup Z\cup Z'$ of $f(X)$ and two copies of $Z$ can be given a magma structure so that the two obvious embeddings of $Y$ (as $f(X)\cup Z$ and $f(X)\cup Z'$) are inclusions of submagmas, and have the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A function is convex and concave, show that it has the form $f(x)=ax+b$ A function is convex and concave, it is called affine function. That is the function:
$$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$
Force $y=0$(suppose $0$ is in the domain of $f(x)$), we obtain:
$$f(tx)-f(0)=tf(x)-tf(0)=t[f(x)-f(0)]$$
So
$... | First show it's homogenous. That is, $F(ax) = aF(x)$ for any $a\in\mathbb{R},x\in\mathbb{R}^n$. If $a \in (0,1)$, we have:
$$
F(ax) = F(ax + (1-a)0) = aF(x) + (1-a)F(0)=aF(x)
$$
If $a\ge 1$, he have:
$$
F(x) = F((1/a)ax + (1-1/a)0) = (1/a)F(ax) + (1-1/a)F(0)=F(ax)/a
$$
If $a<0$, we use the above cases combined with the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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How can we use $(im(A))^\perp = \ker(A^T)$ to prove $rank(A)=rank(A^T)$? Why is it that these two statements are essentially equivalent? $(im(A))^\perp$ represents all vectors orthogonal to the $im(A)$. Yet I'm not sure what this being equal to $\ker(A^T)$ exactly means, nor how it relates to the rank of $A$ and $A^T$.... | There are a few things we need to prove the equivalence here:
*
*the rank of a matrix is the dimension of its image
$\DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\im}{im}
\DeclareMathOperator{\null}{null}$
*The rank nullity theorem: $\rank(A) + \null(A) = \rank(A^T) + \null(A^T) = n$, where $\null(A)$ me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1014915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$? Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by $f$ acting on the matrix elements.
My question is, what is the mo... | Here are some standard examples: Consider $a$, $b$ integers $>0$ so that $b \mid a \ $. We have the canonical surjective morphism of rings
$$\mathbb{Z}/(a) \to \mathbb{Z}/(b)$$
and the induced morphism of groups
$$GL_n(\mathbb{Z}/(a)) \to GL_n(\mathbb{Z}/(b))$$
is surjective.
Even for $n=1$ this is a good exercise. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $(A\setminus B) \cup (A\setminus C) = B \Leftrightarrow A=B \wedge (B \cap C) = \emptyset$ I believe there are 3 parts to this.
1) $(A\setminus B) \cup (A\setminus C) = B \Rightarrow A=B $
2) $(A\setminus B) \cup (A\setminus C) = B \Rightarrow (B \cap C) = \emptyset$
3) $A=B \wedge (B \cap C) = \emptyset ... | Hint: Let $(A-B)\cup (A-C)=B$ hold true and assume that there is $x \in B \cap C$. This means in particular that $x\in B$ and $x \in C$. Hence $$x \notin A-B$$ and for the same reason $$x\notin A-C$$ Therefore $$x\notin (A-B)\cup (A-C)$$ but on the other hand $x\in B$ which a contradiction to the assumption that $$(A-B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Linear Algebra quadratic forms diagonalization I have a question that reads:
Diagonalize the quadratic form $A(x,x) = 2x^2 - 1/2 y^2 -2xy - 4xz$ by completing the squares, and find the change of basis matrix and the new basis in which A will be diagonalized.
HINT: The change of basis matrix is the inverse of the chang... | Note: My answer below is more than a bit disorganized; if any one cares to clean it up a bit, I welcome the effort.
It seems that diagonalize in this context would mean something like writing
$$
2x_1^2 - (1/2) x_1^2 -2x_1x_2 - 4x_1x_3 =
A(x,x) = x^TU^T DUx = (Ux)^TD(Ux)
$$
for some diagonal matrix $D$ and invertible m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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System of equations, limit points This is a worked out example in my book, but I am having a little trouble understanding it:
Consider the system of equations:
$$x'=y+x(1-x^2-y^2)$$
$$y'=-x+y(1-x^2-y^2)$$
The orbits and limit sets of this example can be easily determined by using polar coordinates. (My question: what i... | When you see an $x^2 + y^2$ in a problem that is probably going to be tractable, one good first thing to try is to see how the problem looks transformed to polar coordinates. (Even in a problem that comes about in real life, where you don't know the solution will be possible to obtain, this is a good first shot.)
When... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability of coins and boys example is unsolvable I have a couple of problems that are giving a headache.
P1)
A coin is tossed three times. Let $A = \{\text{Three head occurs}\}$, $B = \{\text{At least one head occurs}\}$
Find $P(A\cup B)$.
Now $P(A) = 1/8$ , $P(B) = 1-1/8 = 7/8$ so
$$P(A \cup B) = P(A)+P(B)-P(A\cap... | You calculated $P(A)$ and $P(B)$ correctly. But you cannot in general calculate $P(A\cup B)$ as $P(A) + P(B)$, because there might be events where both $A$ and $B$ have occurred, and $P(A) + P(B) $ counts those events twice, once as part of $P(A)$ and once as part of $P(B)$.
Suppose, for example, that the cafeteria se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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convergence of the derivatives I am trying to solve the question:
Let $u_n$ a sequence converging uniformly to $u$ where $u_n\in C^3(\Omega)$ for each $n$ and $\Omega$ is a subset limited of $\mathbb{R^n}$. Suppose $u_n=0$ on
$\partial\Omega$. To show that
$$
\displaystyle\lim_{n\rightarrow0}\int_{\Omega}|\nabla u_n|^... | You can suppose, without loss of generality, that $u \equiv 0$ (if not, take $v_n = u_n - u$, and prove the statement for $v_n$).
Then, you want to show that $\lim_{n\to\infty}\int_\Omega |\nabla u_n|^2 = 0$.
Use integration by parts:
$$\int_\Omega |\nabla u_n|^2 = \int_{\partial \Omega} u_n (\nabla u_n \cdot \hat n) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015465",
"timestamp": "2023-03-29T00:00:00",
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Determine the Number of Integer Solutions $x_1 + x_2 + x_3 + x_4 = 32$ with restrictions The Question
My Problem
Part a is straight forward, just $C(35,32)$. I'm having a little difficulty with the restrictions and understanding what they mean. $x_1 > 0$ means we shouldn't have any solutions of the form $(0 + 32)$? If... | For part (b), $x_i > 0$ for all $i$, let $y_i = x_i - 1$. Then the problem is equivalent to problem (a), only with a sum of 28 instead of 32 -- the answer is $C(31, 28) == C(31,3)$.
For part (c) let $y_i = x_i-5$ for $i\in \left\{ 1,2 \right\} $ and $y_i = x_i-7 $ otherwise. Then you have problem (a) again, with a s... | {
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Proving any odd number is a factor of $2^n -1$ for some $n$ I'm struggling with a proof of the following. I feel like it should be a one-liner or something simple but I'm just not grasping the idea:
Suppose that m is an odd natural number. Prove that there is a natural number $n$ such that $m$ divides $2^n -1$
Any help... | $$(2k+1) \mid 2^{\phi(2k+1)} -1 $$
by http://en.wikipedia.org/wiki/Euler%27s_theorem
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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implications, equivalence, disjunction I might not be very clear with this but i hope someone gets it
Prove that $f : X→Y$ is surjective then and only then when $g_1, g_2$ which $Y → Z$ we have $g_1 \circ f= g_2 \circ f \Rightarrow g_1 = g_2$
it would be very helpful for me if someone could break this down cause i don'... | *
*Assume $f$ is surjective, and that $g_1 \circ f = g_2 \circ f$. Want to show $g_1 = g_2$
Assume $g_1 \neq g_2$. Then, there is some $y \in Y$ so that $g_1(y) = z_1 \neq z_2 = g_2(y)$. Since $f$ is surjective, there is an $x \in X$ so that $f(x) = y$. What does this tell you about $g_1(f(x))$ and $g_2(f(x))$?
*
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inclusion-exclusion principle (?) in a counting problem A group of pre-school children is drawing pictures ( one child is making one picture ) using 12-colours pencil set. Given that
(i) each pupil employed 5 or more different colours to make his drawing;
(ii) there was no identical combination of colours in the differ... | HINT: Each color appears in at most $20$ drawings, and there are $12$ colors, so there are at most $12\cdot20=240$ drawings. However, each drawing uses at least $5$ colors, so ... ? See if you can finish it from here; I’ve left the conclusion in the spoiler-protected block below.
So each drawing is counted at least $... | {
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"timestamp": "2023-03-29T00:00:00",
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Show $n^{\frac{1}{n}}$ is decreasing for $n \ge 3$ How to show $\displaystyle n^{\frac{1}{n}}$ is decreasing for $n \ge 3$ ?
| Late answer but $n>e$ so $$n^{1/n} > e^{1/n}>1 + 1/n = \frac{n+1}{n} \implies n^{1+1/n}>n+1 \implies n^{1/n} > (n+1)^{1/n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1015958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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A quick question on the subring generated by a finite set Let $R$ be a commutative unital ring and let $r_1,\ldots,r_n \in R$.
Let $S$ be the unital subring of $R$ generated by $r_1,\ldots,r_n$.
Let $\varphi:\mathbb{Z}[X_1,\ldots,X_n]\to R$ be the unique homomorphism sending $1\mapsto 1_R$ and each $X_i\mapsto r_i$.
Am... | Yes. It is clear that $\text{im}(\phi) \subseteq S$. On the other hand, $\text{im}(\phi)$ is a subring of $R$ that contains all the $r_i$'s, and since $S$by definition is the smallest subring of $R$ containing the $r_i$'s, we must have $S \subseteq \text{im}(\phi)$. Hence, $S = \text{im}(\phi)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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limits question with radicals, rationalizing Find the limit value
Here's what I did (Above)
I think I can rationalize the numerator to solve it,
but I'm having trouble rationalizing numerator, when I'm usually rationalizing the denominator.
How do I rationalize the numerator? (If I'm on the right track for solution)
| $$\begin{align}\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}&=
\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}\cdot\frac{\sqrt{x+2}+\sqrt2}{\sqrt{x+2}+\sqrt2}\\
&=\lim_{x\to0}\frac{x+2-2}{x(\sqrt{x+2}+\sqrt2)}\\
&=\lim_{x\to0}\frac{x}{x(\sqrt{x+2}+\sqrt2)}\\
&=\lim_{x\to0}\frac{1}{\sqrt{x+2}+\sqrt2}\\
&=\frac{1}{2\sqrt2}\\
\end{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
complex eigenvalues and invariant spaces I am currently reading Guillemin and Pollack's Differential Topology, and the following claim is made without proof:
Given a linear isomorphism $E: \mathbb{R}^k \to \mathbb{R}^k$, with $k>2$ and such that $E$ can be represented by a matrix with real entries, $E$ has a one- or t... | I think here is another way. A has eigenvalue $\lambda$ and eigenvector $\textbf{v}\in \mathbb{C}^n$. We can say $\lambda = a+bi$ and $v_j=x_j+iy_j \text{ for }j=1,2,...,n\Rightarrow \textbf{v}={\textbf{x}}+i\textbf{y} \text{ where }\textbf{x,y}\in \mathbb{R}^n$ . So now we have: $$(1)\quad A\textbf{v}=\lambda \textbf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Study the character of the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$ Discuss the character of the series
$$\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$$
where $z\in \mathbb C$ and $|z|=\frac{1}{4}$.
Any suggestions please?
Thank you very much
| Note that
$$
\left|n^{-2z}\right|^2=\left|n^{-2\Re (z)-2 i \Im (z)}\right|^2=\left(n^{-2 \Re(z)}\right)^2=n^{-4\Re(z)}
$$
for $n\ge 1$, so the sum is
$$
\sum_{n=1}^{\infty}n^{-4\Re(z)}=\zeta\left(4\Re(z)\right),
$$
with convergence only for $4\Re(z) > 1$. If we are given that $\left|z\right|=1/4$, clearly $\Re(z) \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
turning a map into a fibration In Allen Hatcher's book Spectral Sequence page 29 Example 1.18,
What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a fibration. How can we turn it into a fibration?
| Given a map $f:A\to B$, let $B^I$ denote the set of continuous functions $I \to B$ endowed with the compact open topology. Define $E_f=\{(a,\gamma)\in A \times B^I | \gamma(0)=f(a)\}$
One can show that the map $p:E_f \to B$, $(a,\gamma)\mapsto \gamma(1)$ is a fibration. We can view $A$ as a subspace of $E_f$ consisting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
A hint is given that for $0 \le \alpha \le \beta$ there exist a number $\xi \in (\beta, \alpha + \beta)$ s.t. $$(\alpha + \beta)^p - \beta^p = p \xi^{p-1}... | We want to show
$$1\le \left(\frac{|x|}{|x|+|y|}\right)^p+\left(\frac{|y|}{|x|+|y|}\right)^p,$$
or
$$1\le a^p+b^p\tag{1}$$
for $a,b\in [0,1]$, $a+b=1$, and $0< p\le1$. But (1) is clear since
$$a\le a^p,b\le b^p.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Hypergeometric function integral representation How to prove the following relation?
$$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$
where $_2{F}_1(.,.;.;.)$ is the hypergeometric function, $m\in\mathbb{R}^+$ , and $K \in\mathbb{N}$.
| Refer to the integral definition of $ _2F_1(a,b;c;z)$ in the general case :
$$ \, _2{F}_1(a,b;c;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)} \int_0^{\infty} t^{-b+c-1} \, (t+1)^{a-c} \, (t-z+1)^{-a} \, dt $$
In the particular case : $a=K$ ; $b=K$ ; $c=K+1$ ; $z=1-m$ ; $t=x$
and with $\Gamma(c-b)=\Gamma(1)=1$ leads... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Asymptotic relation between specific binomial coefficient and exponential function I need to determine the asymptotic relationship between the functions:
$$f_1(n)={n\choose{\lfloor{n\over{2}}\rfloor}}, f_2(n)=7^{\sqrt{n}}$$
(I'm going to just assume $n$ is always even.)
I've convinced myself that $f_2(n) = o\left(f_1(n... | Express combinations number via factorials, and then use Stirling's Approximation for factorials. Then you'll be able to prove what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How find this maximum of this $(1-x)(1-y)(10-8x)(10-8y)$
let $x,y\in (0,1)$, and such
$$(1+x)(1+y)=81(1-x)(1-y)$$
Prove
$$(1-x)(1-y)(10-8x)(10-8y)\le\dfrac{9}{16}$$
I ask $\dfrac{9}{16}$ is best constant?
PS:I don't like Lagrange Multipliers,becasue this is Hight students problem.
My idea:
$$(1-x)(1-y)(10-8x)... | Let $P=xy$, $S=x+y$. Rewrite both the condition and the other formula in terms of $P$ and $S$. Use the condition to eliminate $P$, and maximize as a function of $S$. Also check the edges of the square.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Alternative ways to evaluate $\int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$ In the following link here I found the integral & the evaluation of
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$
I'll also include a simpler version together with the question: is it possible to find some easy
ways of computing both i... | By series expansion
$$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^2}{x}\,dx=\sum_{k,n\geq 1}\frac{1}{(nk)^2}\int^1_0x^{n+k-1}\,dx =\sum_{k,n\geq 1}\frac{1}{(nk)^2(n+k)}$$
By some manipulations
$$\sum_{k\geq 1}\frac{1}{k^3}\sum_{n\geq 1}\frac{k}{n^2(n+k)}= \sum_{k\geq 1}\frac{1}{k^3}\sum_{n\geq 1}\frac{1}{n^2}-\sum_{k\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1016935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
Cauchy-Schwarz inequality with zero angle?
Cauchy-Schwarz Inequality:
If $\textbf{u}$ and $\textbf{v}$ are vectors in a real inner product space $V$, then $$|\left\langle\textbf{u},\textbf{v}\right\rangle|\leq||\textbf{u}||\ ||\textbf{v}||$$
What will happen with the Cauchy-Schwarz inequality if the angle between the... | the definition of the angle $\alpha\in[0,\pi]$ between $u,v$ is:
$$
\cos\alpha = \frac{\langle u,v\rangle}{\|u\| \|v\|}
$$
so $\alpha =0 $ iff
$$
1=\frac{\langle u,v\rangle}{\|u\| \|v\|}\iff
\langle u,v\rangle = \|u\| \|v\|
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Inequality of sides of triangle If a,b,c are the sides of a triangle ABC then prove $$ a/(b+c) + b/(a+c) + c/(a+b) < 2 $$I tried to solve this by triangle inequality but i was not able to get to the solution.
| Hint: let $x=a+b-c$, $y=a+c-b$, $z=b+c-a$. Clearly
$$a=\frac{x+y}2,\ b=\frac{x+z}{2},\ c=\frac{y+z}{2},\ a+b=\frac{2x+y+z}{2},\cdots$$
hence the original inequality reads
$$\frac{x+y}{x+y+2z}+ \frac{x+z}{x+2y+z}+\frac{y+z}{2x+y+z}<2.$$
Note that
$$\frac{x+y}{x+y+2z}+ \frac{x+z}{x+2y+z}+\frac{y+z}{2x+y+z}<\frac{x+y}{x+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove: For any integer $n \geq 2$, there is an odd number P such that $2n \lt P \lt 3n$ I am in high school and had this for a homework problem. I got it wrong, but the teacher did not post the correct answer. Any help would be appreciated. It is about writing proofs.
Prove that for any integer n greater than or equ... | Write numbers between $2n$ and $3n$
$$
2n<2n+1 ,2n+1 ,2n+3 ,...,2n+(n-1)<3n
$$
if $n>2$
there is $n-1$ numbers between $2n$, $3n$
number of terms=
$$ \frac{\text{last} -\text{first}}{\text{step}}+1=\\\frac{3n -2n}{1}+1=n+1\\$$
between $2n$, $3n$ are $(n+1)-2$ terms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Probability of going from node a to node b in an undirected graph. I have a graph with n nodes. Each of which represents an activity (play, walk, sleep, etc). If I'm standing at node 1 (any), what is the probability of going from 1 to j (another node) if probability is a number between 0 and 1 (closed) between 2 nodes ... | Your question seems to be a bit ill-posed (it's not really clear what question you're trying to answer), but what you've got here is a Markov chain, and there are standard techniques for answering questions about them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are two matrices similar iff they have the same Jordan Canonical form? Are two matrices similar if and only if they have the same Jordan Canonical form?
Does the Jordan form have to have ordered eigenvalues?
For example, if $\lambda_1$ and $\lambda_2$ are eigenvalues of $A$, are $\begin{pmatrix}\lambda_1&0\\0&\lambda_2... | *
*Up to arbitrary ordering of Jordan blocks, yes
*No
*Yes
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Find a positive integer with prime factors of at most 2, 3, 5, 7 and ends in the digits 11 Does there exist a positive integer whose prime factors include at most 2, 3, 5, and 7, and ends in the digits 11? If so find the smallest positive integer. If not, show why none exists.
My professor gave us this question to thin... | Clearly, $2$ and $5$ cannot divide the number. Hence, let the number be of the form $3^a7^b$. We have
$$3^a \equiv 1,3,9,7 \pmod{20} \text{ and }7^b \equiv 1,7,9,3\pmod{20}$$
Hence,
$$3^a7^b \equiv 1,3,7,9\pmod{20}$$ However, the number ending in $11$ is $\equiv 11\pmod{20}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Direction of unit vector that maximize directional derivative Firstly, I am aware that there are quite a few question regarding with "maximizing direction derivative" already being asked. But after scanning through, I am still not able to figure out my question thus posting it here.
Let's say we have a function of a su... | Yes it will be the (unit) vector that is parallel to the derivative itself but you still have the task of finding that vector don't you?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving the expected value of the square root of X is less than the square root of the expected value of X How do I show that $E(\sqrt{X}) \leq \sqrt{E(X)}$ for a positive random variable $X$?
I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq E(X^2)E(Y^2)$, but I'm not sure how.
| $\sqrt{x}$, $x\geq 0$ is a concave function so Jensen's inequality gives us the result without further effort.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1017891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Polynomials close to idempotents in quotient ring of $\Bbb R[x_1,x_2,\dots,x_n]$ Let $S=\Bbb R[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)$.
Given $t\in \Bbb N$, what are the polynomials $p\in S$ that satisfy the relation $$p^2=tp$$ modulo $x_i$ and $x_i-1$ for all $i$ (same as evaluation at $x_i\in\{0,1\}... | Since $(x_i)$ and $(x_i-1)$ are coprime for $1\le i\le n$ we have by CRT
$$\frac{k[x_1,\cdots,x_n]}{(x_1^2-x_1,\cdots,x_n^2-x_n)}\cong\frac{\displaystyle\frac{k[x_1,\cdots,x_{n-1}]}{(x_1^2-x_1,\cdots,x_{n-1}^2-x_{n-1})}[x_n]}{(x_n^2-x_n)}$$
$$\cong\frac{\displaystyle\frac{k[x_1,\cdots,x_{n-1}]}{(x_1^2-x_1,\cdots,x_{n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification We represent an element $(x_1,y_1,...,x_n,y_n,x_{n+1},y_{n+1})\in S^{2n+1}$ as an element $(z_1,...,z_{n+1})\in \mathbb C^{n+1}$ where $z_k = x_k+iy_k$. Now I'm considering the action of $S^1$ on $S^{2n+1}$ as $e^{iθ}·(z_1,.... | $$
S^{2n+1} \subseteq \mathbb{R}^{2n+2} = \mathbb{C}^{n+1}
$$
The usual map restricts to a surjection (check this)
$$
S^{2n+1} \rightarrow \mathbb{C}P^n
$$
Now check that two points $w,v$ on the sphere produce the same 1-dimensional subspace in
$\mathbb{C}^{n+1}$ iff they differ by an element of $S^1$:
$$
w=\lambda v ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$ Is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$? I have proved it to be open,not closed,not connected but not sure about this property .How to do this?
| The way I like writing this is as follows: Consider $A\in M_{n}(\mathbb R)$ and let $\varepsilon>0$. Let $\lambda _1, \ldots, \lambda_n$ be the eigenvalues of $A$. Take $\delta$ such that $0<\delta<\frac{\varepsilon}{n^{1/2}}$ and $\delta\neq \lambda_j$ for every $j\in \{1, \ldots, n\}$. Define
$$A_\delta:=A-\delta I.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
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Space time algebra isomorphic to matrix algebra i have the following problem:
I already know that there exists representation of the Clifford Algebra of the Minkowski space $\mathcal{C}l(M,\eta)$. Here $M$ denotes the Minkowski space and $\eta$ is the metric tensor with the signatur $(+1,-1,-1,-1)$.
I know that a repr... | The gamma matrices are four $\Bbb C$-linearly independent matrices of $M_2(\Bbb C)$, a space with $\Bbb C$-dimension $4$, so the gamma matrices form a basis. Since the image of $\gamma$ contains this basis, it is the entire space.
Thus $\gamma$ is surjective and hence injective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How does does exponent property work on $\left(xe^{\frac{1}{x}}-x\right)$ How does
$\left(xe^{\frac{1}{x}}-x\right)$
become
$\frac{\left(e^{\frac{1}{x}}-1\right)}{\frac{1}{x}}$
| With more intermediate steps :
$$\left(xe^{\frac{1}{x}}-x\right)=$$
$$=x\left(e^{\frac{1}{x}}-1\right)$$
Let $x=\frac{1}{t}$ Replace $x$ by $\frac{1}{t}$
$$=\frac{1}{t}\left(e^{\frac{1}{x}}-1\right)$$
$$=\frac{\left(e^{\frac{1}{x}}-1\right)}{t}$$
$t=\frac{1}{x}$ Replace $t$ by $\frac{1}{x}$
$$=\frac{\left(e^{\frac{1}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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System of nonlinear equations that leads to cubic equation The system of equations are:
$$\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$
I can solve it through substitution but it is an arduous process to reach this cubic equation:
$$20x^3 + 56x^2 - 243x - 544 = 0$$
And I can only solve t... | The roots are all real:
$x_{1}=-\frac{14}{15}-\frac{\sqrt{4429}.cos\Big(\frac{acot(-f)}{3}\Big)} {15}$,
$x_{2}=-\frac{14}{15}+\frac{\sqrt{4429}.sin\Big(\frac{atan(f)}{3}+\frac{\pi}{3}\Big)} {15}$
$x_{3}=-\frac{14}{15}-\frac{\sqrt{4429}.sin\Big(\frac{atan(f)}{3}\Big)} {15}$
$f=\frac{192158\sqrt{222021105}}{3330316575}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 9,
"answer_id": 8
} |
question about a special case of an $n$ sided polygon Here is an interesting question that I have been thinking about for awhile now but do not know the answer to.
Suppose you have a convex polygon with $n$ sides. What would be an example of such a polygon s.t. if you randomly picked three sides of the polygon, they co... | Most polygons that people draw will have all the sides about the same length, and any set of three sides can form a triangle. I think it is more interesting to ask what polygons have some set of three sides that cannot form a triangle.
One kind that has a set that cannot form a triangle is a single long side, say $10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Computing a tricky limit $\lim_{n\to\infty} \sqrt{n}\int_0^{\infty} \cos^{2n-1}(x) e^{- \pi x} \ dx$ I'm interested in some neat approaches for
$$\lim_{n\to\infty} \sqrt{n}\int_0^{\infty} \cos^{2n-1}(x) e^{- \pi x} \ dx$$
Since I suspect my approach is wrong, and I don't wanna influence you in any way, I'll add it
in a... | Since
$$\int_{0}^{+\infty}\cos(n x)\,e^{-\pi x}\,dx = \frac{\pi}{n^2+\pi^2}$$
it is sufficient to compute the Fourier cosine series of $\cos^{2n-1}x.$ We have:
$$\cos^{2n-1}x = \frac{2}{4^n}(e^{ix}+e^{-ix})^{2n-1} = \frac{1}{4^{n-1}}\sum_{k=0}^{n-1}\binom{2n-1}{k}\cos((2n-2k-1)x)$$
hence:
$$\int_{0}^{+\infty}\cos^{2n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
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Evaluation of $\int_{0}^{\infty}\frac{1}{\sqrt{x^4+x^3+x^2+x+1}}dx$ Evaluation of Integral $\displaystyle \int_{0}^{\infty}\frac{1}{\sqrt{x^4+x^3+x^2+x+1}}dx$
$\bf{My\; Try::}$ First we will convert $x^4+x^3+x^2+x+1$ into closed form, which is $\displaystyle \left(\frac{x^5-1}{x-1}\right)$
So Integral is $\displaystyle... | The answer in terms of elliptic integrals turns out to be very simple – but it requires a very sneaky trick.
As pointed out in Jack's answer, the integrals over $[0,1]$ and $[1,\infty]$ are the same, so
$$I=\int_0^\infty\frac1{\sqrt{x^4+x^3+x^2+x+1}}\,dx=2\int_0^1\frac1{\sqrt{x^4+x^3+x^2+x+1}}\,dx$$
The sneaky trick is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Disproving existence of real root in some interval for a quintic equation Disprove the statement: There is a real root of equation $\frac{1}{5}x^5+\frac{2}{3}x^3+2x=0$ on the interval (1,2).
I am not sure whether to prove by counter-example or by assuming the statement is true and then proving by contradiction.
| First step: what is the value of the polynomial for $x=1$?
Second step: what is the sign of the derivative of this polynomial on the interval $[1,2]$?
Third step: conclusion.
edit
Altenative: $x\ge 1$, therefore $x^n\ge 1$ for all $n\ge 1$. Hence $$\frac{1}{5}x^5+\frac{2}{3}x^3+2x\ge \frac{1}{5} +\frac{2}{3} +2 $$
whe... | {
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"url": "https://math.stackexchange.com/questions/1018817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Energy dissipation I've been asked to prove the following, but I don't find the way..
Let $\Omega\subset\left\{0<x_n<a\right\}$ be a subset of $\mathbb{R}^n$ such that it is bounded in the $n^{th}$ coordinate. Prove that the solution of the heat equation
\begin{equation*}
\begin{array}{lc}
u_t=\Delta u && in\ \Omega \... | The operator $\Delta$ should be selfadjoint on the spatial domain with the desired boundary conditions. That allows you to write the solution as $u=e^{t\Delta}u_{0}$. The spectrum of $\Delta$ is normally going to be non-positive. So all of the modes will decay in time unless you have a non-trivial mode with eigenvalue ... | {
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"timestamp": "2023-03-29T00:00:00",
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AM-GM inequality proof Let $a_1,...,a_n>0$. The arithmetic mean is defined by $A(a_1,...,a_n) =\frac{a_1+...+a_n}{n}$ and the geometric mean by $G(a_1,...,a_n)=\sqrt[n]{a_1\cdot ...\cdot a_n}$.
Let $S(n)$ be the statement:
$$\forall a_1,...,a_n >0: G(a_1,...,a_n) \leq A(a_1,...,a_n)$$
a) Prove $S(2)$ is true
b) Let $n\... | To $b):$
Assume $S(2)$ and $S(n)$ hold. Then,
$$\sqrt[2n]{a_1\cdots a_{2n}}=\sqrt{\sqrt[n]{a_1\cdots a_n} \sqrt[n]{a_{n+1}\cdots a_{2n}}}\underbrace{\le}_{S(2)} \frac{\sqrt[n]{a_1\cdots a_n }+\sqrt[n]{a_{n+1}\cdots a_{2n} }}{2} \\ \underbrace{\le}_{S(n)} \frac{\displaystyle \frac{a_1+\cdots +a_n }{n}+\frac{a_{n+1}+\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1018972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Laplace transform of a function divided by t Using the formula
$$\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_s^\infty F(u)~du$$
I'm trying to determine the transform with $f(t)=1-e^{-t}$.
The formula gives me
$$\mathcal{L}\left\{\frac{1-e^{-t}}{t}\right\}=\int_s^\infty \left(\frac{1}{u}-\frac{1}{u+1}\right)~du$$
whic... | This certainly converges. Write the integral as
$$\begin{align}\lim_{N \to \infty} \int_s^N du \left( \frac1{u}-\frac1{u+1} \right ) &= \lim_{N \to \infty} \left [\log{N}-\log{s} - \log{(N+1)}+\log{(s+1)}\right ]\\ &= \log{\left (1+\frac1{s}\right)} - \lim_{N\to\infty} \log{\left (1+\frac1{N}\right)}\end{align}$$
This... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How is $\ln(-1) = i\pi$? How do I derive:
$\ln(-1)=i\pi$ and
$\ln(-x)=\ln(x)+i\pi$ for $x>0$ and $x \in\mathbb R$
Thanks for any and all help!
| take the exponential to get $$e^{ln(-1)}=e^{\pi i}$$ $$-1=e^{\pi i}$$ $$e^{\pi i}+1=0$$ this is Euler's identity
| {
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Reference for principal bundles and related concepts I am looking for a good reference for fibre bundles on differential manifolds, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could anyone advise me on this? I have looked at th... | Late to the party, but: Chapter 6 in Loring Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes seems friendly enough, and well explained. I think sections §27 and §28 are what you're looking for.
| {
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Why can't a value for this definite integral be found? I was trying to find out if $\int _0^{\infty }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ converges or diverges. I split it into a sum, that is
$\int _0^{1 }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ + $\int _1^{y }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{... | Hint: Let $~t=\dfrac1{1+x^3}~$ and then recognize the expression of the beta function in the new integral.
| {
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"source": "stackexchange",
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Why $f(t)= t+ 2t^2\ sin (\frac 1t) , t\neq 0$ & $f(t)=0 , t=0$. Prove that this function is not $1-1$ in any neighbourhood of zero. Why $f(t)= t+ 2t^2 \sin (\frac 1t) , t\neq 0$ & $f(t)=0 , t=0.\,$Prove that this function is not $1-1$ in any neighbourhood of zero.
It is not possible with $\frac 1 {n \pi}$.
| Hint: $f$ is continuously differentiable for all $t>0$; you can check that
$$f'\Bigl(\frac{1}{n\pi}\Bigr)$$
is positive if $n$ is odd and negative if $n$ is even.
| {
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Mathematical Science Writers without PhD Is there science writers who have written on mathematics without holding a ph.d in the subject? I am aware that Robert Kanigle is one such but does there exist any other?
| Nate Silver who wrote "The Signal and the Noise" holds a BA in Economics,
The book describes methods of mathematical model-building using
probability and statistics. Silver takes a big-picture approach to
using statistical tools, combining sources of unique data (e.g.,
timing a minor league ball player's fastbal... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Parameterizing a surface The question I was asked goes like this:
The part of the hyperboloid $5x^2 − 5y^2 − z^2 = 5$ that lies in front of the yz-plane. Let x, y, and z be in terms of u and/or v. Find a parametric representation for the surface.
So, by fixing $x$ as a constant $u$, I find that the slice is an ellips... | I interpret being 'in front of the $yz$-plane' as the first coordinate being non-negative, thus the surface $S$ in question $\{(x,y,z)\in \mathbb R^3\colon x\ge 0\land 5x^2 − 5y^2 − z^2 = 5\}$.
It can easily be proved that this set equals $\left\{\left(\sqrt{1+y^2+\dfrac{z^2}5}, y,z\right)\colon y,z\in \mathbb R\right\... | {
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Rotation matrix in spherical coordinates When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of $a$ around $\vec{A}$ to the angle $\beta$?
Points and axes are not on the coo... | I think what you might be looking for is Rodrigues' Rotation Formula. Using spherical coordinates:
Your arbitrary point on the unit sphere is:
$$
\mathbf{a} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)
$$
Your arbitrary axis is represented by the unit vector:
$$
\hat{\mathbf{k}} = (\sin\Theta\cos\Phi, \sin\Th... | {
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$A\in M_2(\mathbb C)$ and $A $ is nilpotent then $A^2=0$. How to prove this? $A\in M_2(\mathbb C)$ and $A $ is nilpotent then $A^2=0$. How to prove this?
I am not getting enough hints to start.
| $A$ can be put into upper triangular form, and since $A$ is nilpotent the its eigenvalues are both zero and the upper triangular form will have zeroes on its diagonal. Squaring this matrix gives zero, and so $A^2 = 0$ also.
Note that this same argument gives that any $n$ by $n$ nilpotent matrix over ${\mathbb C}$, when... | {
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Puzzle of gold coins in the bag At the end of Probability class, our professor gave us the following puzzle:
There are 100 bags each with 100 coins, but only one of these bags has gold coins in it. The gold coin has weight of 1.01 grams and the other coins has weight of 1 gram. We are given a digital scale, but we can... | According to the given values, the gold coins have essentially the same weight (down to a single percent) as the base ones. Since gold is heavy this means that each gold coin is significantly smaller.
Forget about weighing and just take the bag whose coins are much smaller than the coins in the other bags.
| {
"language": "en",
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Correctly predicting salt and sugar in the proper sequence. We have $10$ canisters, $5$ containing sugar and $5$ containing salt. What is the probability of naming them all in the correct order? (example: salt, sugar, sugar, salt, sugar, salt, salt, ...)
It's different from predicting $10$ coins head or tails, because... | If we assume that each arrangement of $5$ salt and $5$ sugar canisters are equally likely, then any given arrangement is uniquely determined by the placement of, say, the $5$ sugar canisters among the $10$ canisters total. For example, we can describe one arrangement as $(2, 3, 5, 7, 10)$, meaning that the sugar canis... | {
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How to prove the equality of determinant? Let A be a square matrix, and B a nilpotent. (size of A and B is same)
Assume AB=BA
Show that, det(A+B)=det(A)
| Hint: Since they commute, they are simultaneously upper triangularizable. For some unitary $U$, we can write
$$
UAU^* = \pmatrix{\lambda_1&&&&*\\&\lambda_2\\&&\ddots\\&&&&\lambda_n}: = T_A\\
UBU^* = \pmatrix{0&&&&*\\&0\\&&\ddots\\&&&&0} := T_B
$$
Note that $\det(A + B) = \det(T_A + T_B)$
| {
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How many valuations of these literals satisfy this expression? considering all the possible valuations of literals A, B, C, D, E, F, G and H (256 valuations in total), how would you go about finding how many of these valuations satisfy this expression:
$$ (A\rightarrow B) \wedge (B\rightarrow C) \wedge (D\rightarrow E)... | Note that you can divide the expression in three parts:
$$
\underbrace{(A\rightarrow B) \wedge (B\rightarrow C)}_{P} \wedge \underbrace{(D\rightarrow E)}_{Q} \wedge \underbrace{(F\rightarrow G) \wedge (G\rightarrow H)}_{R}
$$
The value of $P$ depends only on $A,B,C$, the value of $Q$ depends only on $D,E$, and the valu... | {
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CD is height of right-angled triangle ABC, M and N are midpoints of CD and BD: prove AM⊥CN I was having some troubles proving this:
CD is the height that corresponds to the hypotenuse of right-angled triangle ABC. If M and N are midpoints of CD and BD, prove that AM is perpendicular to CN.
Here's the illustration (by m... | It suffices to show $\angle MAD=\angle NCD$, which means two right triangles
$ADM$ and $CDN$ are similar. This can be proved from $AD/DM=CD/DN$, which follows from $AD/CD=CD/BD$.
| {
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Formula for the angle of a line $y = mx$ as a function of $m$. I was wondering if there was a way to calculate the angle made by a line $(\space y=mx)$ in the Cartesian plane using only $m$. I used the Pythagorean theorem in this figure:
$$AO= \sqrt{AB^2+OB^2}=\sqrt{x^2+m^2x^2}=x \sqrt{1+m^2}$$
Now I know that $\alpha ... | Looks good! Alternatively, notice that:
$$
m = \frac{y}{x} = \tan \alpha
$$
So we have:
$$
\alpha = \tan^{-1}(m) = \tan^{-1}(0.5) \approx 26.57^\circ
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1020525",
"timestamp": "2023-03-29T00:00:00",
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Conditions for using The Rational Zeros Theorem Barron's SAT Math 2 and SparkNotes state that if p/q is a rational zero of P(x) with integral coefficients, then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .
Then they show an example in which p/q isn't even a zero of t... | The theorem says if $p/q$ is a rational zero of $P(x)$, then .... The way you usually use this is to narrow down the possibilities for rational zeros. In this case $9/2$ does not happen to be a zero, but the theorem doesn't rule it out.
Once you identify the possible rational zeros using the theorem, you can test eac... | {
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Is $ G \cong G/N \times N$? If G is a finite group and N is a normal subgroup in G , then can we say G $\cong$ G/N $\times$ N always? Is it true for like normal nilpotent or normal solvable or any such special classes. I couldn't help but ask for it. What if G is infinite? Please tell me if you can, the concept I am ... | I've intended to ask a similar question. Fortunately, the community has avoided a duplicate. Moreover, I've had more information to better my proposition, which is written below.
The proposition: The necessary and sufficient conditions of group $G$ and its normal subgroup $N$ (regardless of commutativity and finiteness... | {
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Traffic with Poisson distribution The number of cars that cross an intersection during any interval of length t minutes between 3:00 pm and 4:00 pm has a Poisson distribution with mean t. Let W be the time that has passed after 3:00 pm and before the first car crosses the intersection. What is the probability of W bein... | Hint: The probability that $W\ge 2$, or equivalently the probability that $W\gt 2$, is the probability that $0$ cars go through the intersection in $2$ minutes.
| {
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Why does the higher order derivative test work? I'm an AP Calculus BC student, so all I know about derivatives is the increasing/decreasing/ relative max/min function relation with first derivative (first derivative test), concavity (second derivative test). I showed that you can create a third derivative test to my te... | One way to understand how the test works is by looking at the Taylor Series of the function $f(x)$ centered around the critical point, $x = c$:
$$
f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2}(x-c)^2 + \cdots
$$
Note: In your question you said that the n-th derivative is non-zero. Here I'm assuming the n+1-st derivative ... | {
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If $f(1)=1\;,f(2)=3\;,f(3)=5\;,f(4)=7\;,f(5)=9$ and $f'(2)=2,$ Then sum of all digits of $f(6)$
$(1):$ If $P(x)$ is a polynomial of Degree $4$ such that $P(-1) = P(1) = 5$ and
$P(-2)=P(0)=P(2)=2\;,$Then Max. value of $P(x).$
$(2):$ If $f(x)$ is a polynomial of degree $6$ with leading Coefficient $2009.$ Suppose
furthe... | For part 2, here's one way to simplify the differentiation. We have
$$f(x)-2x+1=2009\cdot (x-1)\cdot(x-2)\cdot (x-3)\cdot (x-4)\cdot (x-5)\cdot(x-r)$$
Substituting $x + 3 \to x$,
$$f(x + 3) - 2(x+3)+1 = 2009(x+2)(x+1)(x)(x-1)(x-2)(x+3-r)$$
$$f(x + 3) -2x - 5 = 2009x(x^2-1)(x^2-4)(x + 3 -r)$$
$$f(x + 3) - 2x - 5 = 2009(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1021098",
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calculating limit for a sech function is any answer in terms of dirac delta function?
how can i calculate this expression?
$$\lim_{x\to 0} {1\over \sqrt{x}} \text{sech} \left(\dfrac{1}{x}\right)$$
| $\lim_{x\to 0} {1\over \sqrt{x}} \text{sech} \left(\dfrac{1}{x}\right)$
$=\dfrac{\frac{1}{\sqrt{x}}}{\cosh (\frac{1}{x})}$
$t=\frac{1}{x}$
$=\lim_{t\to \infty} \dfrac{\sqrt{t}}{\cosh (t)}$
Applying L'Hopital's rule:
$=\lim_{t\to \infty} \dfrac{1}{2\sqrt{t}\sinh{t}}$
$=\frac{1}{\infty}$
$=0$
| {
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Approach towards second order differential equation I have the following equation to be solved. Can anybody explain to me how I am supposed to approach this problem?
$$4 \frac{d^{2}y}{dx^{2}} + 4 \frac{dy}{dx} + y = (8x^{2} + 6x + 2)e^{-x/2}$$
edited
I am supposed to find the particular integral for the same.
| First find the general solution of the homogeneous equation $4y''+4y'+y=0$ in the usual way, using the characteristic equation $4r^2+4r+1=0$. Then use the method of undetermined coefficients to find a particular solution of the nonhomogeneous equation. In "guessing" your undetermined coefficients form for the particula... | {
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Sum of cells on infinite board is even Let $a,b,c$ be pairwise relatively prime positive integers. In an infinite checker board (infinite in all directions), each cell contains an integer. The sum of the integers in any $a\times a$ square is even, in any $b\times b$ square is even, and in any $c\times c$ square also ev... | Suppose that we put a light under each cell in this infinite grid, all of the lights starting off. We can, at will, switch all the state of all the lights in any $a\times a$, $b\times b$, or $c\times c$ square - that is, any light in this square that was on is turned off, and any light that was off is turned on. So, by... | {
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Calculating the value of $\frac{a-d}{b-c}$ If $\frac{a-b}{c-d}=2$ and $\frac{a-c}{b-d} = 3$ then determine the value of:
$$\frac{a-d}{b-c}$$
Where $a,b,c,d$ are real numbers.
Can someone please help me with this and give me a hint? I tried substitutions and solving them simultaneously but I couldn't determine this val... | plugging $c=\frac{a+3b}{4}$ and $d=-\frac{a}{4}+\frac{5b}{4}$ in the given term we get $-5$ as the given result.
| {
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Evaluating $\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx$ What starting point would you recommend me for the one below?
$$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$
EDIT
Thanks to Felix Marin, we know the integral evaluates to
$$\displaystyle{\large{\ln^{2}\left(\, 2\,\right) \over 2\pi}}$$
| $\def\Li{{\rm{Li}_2\,}}$Denote the considered integral as $I$ and set $y=\pi x$, we have
\begin{equation}
I=\frac{1}{\pi^2}\int_0^\pi \frac{y\sin y}{\cos y}\,\ln(\sin y)\,dy
\end{equation}
Perform integration by parts by taking $u=y$, we have
\begin{align}
I&=-\left.\frac{y}{2\pi^2}\int\frac{\ln\left(1-\cos^2y\right)}{... | {
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"url": "https://math.stackexchange.com/questions/1021647",
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Expected Value of product of Ito's Integral Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit?
For example:
$$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$
I only know how to compute when the upper limit r and s are the same...but don't know h... | You can split the integrals up into parts over their domain. The part where they overlap can use the usual formula, and the variables are independent on the part where they don't overlap, so those expectations are products of the expectation of the factors.
| {
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} |
Hopf bifurcation and limit cycles $$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$
$$dR/dt=0.8(-R+1.25V+1.5)$$
Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether each is subcritical or supercritical.
Setting both equal to 0 to find the equilibrium:
$0... | 1) You forgot the constants when you calculated the Jacobian matrix.
2) In general, when you want to prove theorically that a certain bifurcation occurs in a system, the way is verify that your system is under the hypothesis of the associate theorem (in this case the Hopf bifurcation theorem). In the case of local bifu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1021869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve a coupled differential equations I tried different ways to solve this differential equation but I did not succeed. These is the first couple ODEs I try to solve. I hope somebody can give me a hint.
\begin{eqnarray}
\ddot{x} + ax - b\dot{y} = c
\end{eqnarray}
\begin{eqnarray}
\ddot{y} + ay + b\dot{x} = 0
\e... | Go to the complex plane. Do "first" + i"second" equation and you'll get something elegant. Sum of equations with second multiplied by the imaginary unit:
$$(\ddot{x}+i\ddot{y})+a(x+iy)+b(-\dot{y}+i\dot{x})=c+0i$$
Use new complex variable $z=x+iy$:
$$\ddot{z}+a z +ib\dot{z}=c$$
This is a simple second order DE with know... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1021984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
plotting an fft function need to plot the folowing:
$$X(j\omega)=\cal F\{{2\over\pi\lambda}\}*\cal F\{sinc(2w_m\lambda)\}-\cal F\{{4\over\pi\lambda}\}*\cal F\{sinc(w_m\lambda)\}$$
as a plot of $|X(j\omega)|$ for $|\omega|<10\pi$
the code I've wrote is:
%% define
t = -4:1/100:4;
w= -10*pi:1/100:10*pi;
wm=3*pi;
z=t-(pi/w... | You have two arrays of different dimensions in your plot.
After running your code, I am told that mag is 1 by 1601 and w is 1 by 6000 something.
Fix 1 quick and dirty:
w = linspace(-10*pi, 10*pi, 1601);
A better solution is to define w after mag since w isn't used until plotting. Then you can do
Fix 2 (a) and (b):
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine whether the series $\sum _{n=1}^{\infty \:\:}\frac{\left(-1\right)^n}{(3n)!}$ is convergent or divergent. Determine whether the series
$\sum _{n=1}^{\infty \:\:}\frac{\left(-1\right)^n}{(3n)!}$
is convergent or divergent. If it is convergent, then how many terms of the series do we need in order to find the ... | Edit: This is an answer for the previous version, where an error was $10^{-15}$. According to a suggestion I am leaving it, as a hint, what one should do if the error is smaller than $10^{-5}$.
Near good, but in your question is $10^{-15}$. It is known, that the error is less than the firts of remaining terms, hence $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Estimations in a ordered field? My Problem: I am stuck with a proof strategy on the following:
So i have got an ordered field $ (K,+,*,<) $ given. I also have
$x,y\in K$
and $0\le y < x$
I have to proof that, for every n $\in \mathbb{N}, n \ge 2$ there is:
$ny^{n-1} < \frac{x^n-y^n}{x-y} < nx^{n-1}$
What i got so far... | Since $\;x-y>0\;$ , we have that
$$ny^{n-1}<\frac{x^n-y^n}{x-y}<nx^{n-1}\iff ny^{n-1}(x-y)<x^n-y^n<nx^{n-1}(x-y)$$
and for example (left inequality):
$$ny^{n-1}(x-y)<x^n-y^n\iff nxy^{n-1}-ny^n<x^n-y^n\iff y^n(1-n)<x^n-nxy^{n-1}$$
But
$$x^n-nxy^{n-1}>x^n-nx^n=x^n(1-n)>y^n(1-n)$$
using what you mention that "can be eas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove two subgroups have the same order given an equivalence relation. Let $X$ be the set of all subgroups of a group $G$. The following is an equivalence relation $H$~$K$ where $H=xKx^{-1}$.
Prove: If $H$~$K$, then $|H|=|K|$.
I'm guessing $H$,$K$ are subgroups of a group $G$. Other than that, I'm not quite sure whe... | Check the map $f : K \rightarrow H$ such that $f(a) = xax^{-1}$. It's obviously surjective homomorphism. So if it is injective we are done. Suppose that $a \in \operatorname{Ker}(f)$, then $xax^{-1} = e \Rightarrow xa=x $ (right multiplication by $x$) $\Rightarrow a = e$ (left multiplication by $x^{-1}$). Therefore $f$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Given the points $A,B,C,D$ in a straight line $m$ and $A,E,F,G$ in a straight line $n$, how many triangles can be formed with these points? Given the points $A,B,C,D$ in a line $m$ and $A,E,F,G$ in a straight line $n$, how many triangles can be formed with these points?
I've done the following:
I've used the follow... | Here is an example,
If you choose BEGC: one line from BE, one from GC, and one from EG--> you have one more triangle. Do the same for the rest and I think you may get the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Best approximation for a normed vector space $X$ I am self-studying functional analysis. As far as I know, a best approximation of $X$ by a closed subspace $C \subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach space, or more generally, a reflexive space.
My question is: is there a ev... | Sharper results: Best Approximation by Closed Sets in Banach Spaces and On a sufficient condition for proximity by Ka Sing Lau.
Quote from the first:
The set $K$ is called proximinal (Chebyshev) if every point $x\in X$ has a (unique) best approximation from $X$. lt is easy to see that every closed convex set $X$ in a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Certain condition on an entire function implies the function is constant ? Let $f=u+iv$ be an entire function such that $v \ge 2u+1$ , then is it true that $f$ is constant ?
| Just to throw another (almost equivalent) method in, $0$ is not in the image of the function and there is $r>0$ such that the disc $D_r=\{z\ :\ |z|<r\}$ is not either.
For example, take $r=1/4$: if $|z|<1/4$, then, $z=x+iy$ with $|x|,|y|<1/4$, so
$3/4\geq|y-2v|$, hence it is impossible that $y\geq 2x+1$ when $x+iy=z\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How do I prove that a matrix is a rotation-matrix? I have to prove that this matrix is a rotation-matrix
$$\begin{pmatrix} \frac12 & 0 & \frac{\sqrt{3}}{2} \\
0 & 1 & 0 \\
\frac{\sqrt{3}}{2} & 0 & \frac12
\end{pmatrix}$$
How do I do this?
My idea is to multiplicate it with $\begin{pmatrix} x \\ y \\ z\end{pmatrix}$ an... | I think there is a minus sign missing. As it is, the determinant is not $1$. After fixing, this specific case is easy.
$$\begin{pmatrix} \frac12 & 0 & -\frac{\sqrt{3}}{2} \\
0 & 1 & 0 \\
\frac{\sqrt{3}}{2} & 0 & \frac12
\end{pmatrix} = \begin{pmatrix} \cos \frac{\pi}{3} & 0 & -\sin \frac{\pi}{3} \\ 0 & 1 & 0 \\ \sin \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
A finite semi-group with cancellation laws is a group . I don't understand how to get this:
A finite semigroup with cancellation laws is a group.
Thanks in advance for any help.
| Let $S$ be a such semigroup. For each $a\in S$, we can find a positive integer $n(a)$ such that $a^{n(a)+1}=a$. If such $n(a)$ does not exist, then $a^1,a^2,a^3,\cdots$ are all different and is contradicting that $S$ is finite.
We will argue that, for each $a,b\in G$, $a^{n(a)}=b^{n(b)}$. Since
$$aa^{n(a)}b=a^{n(a)+1}b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove $ \lim_{x\to -\infty } \frac{x+8}{x+3} = 1 $ using only the definition of a limit I need to prove this limit:
$$ \lim_{x\to -\infty } \frac{x+8}{x+3} = 1 $$
I started with:
$|\frac{x+8}{x+3}-1|< ϵ $
$|\frac{x+8-x-3}{x+3}|=|\frac{5}{x+3}|=\frac{5}{|x+3|} <ϵ$
$\frac{5}{\epsilon} < |x+3|$
How do I proceed from he... | $$\begin{align*}
\frac5\epsilon &< \left|x+3\right|\\
\frac5\epsilon &< -\left(x+3\right)\\
x &< -\frac5\epsilon-3
\end{align*}$$
Hence, given $\epsilon>0$, we can take $M= -\frac5\epsilon-3$ such that $x<M$ implies $$\left|\frac{x+8}{x+3}-1\right|<\epsilon$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Domain, range and zeros of $f(x,y)=\frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2}$ Given the following function with two variables:
\begin{equation}
\frac{\sqrt{4x-x^2-y^2}}{x^2y^2-4xy^2+3y^2}
\end{equation}
I need to find a) the domain for the above function.
Can anyone give me a hint on how to find the domain in f?
... |
Can anyone give me a hint on how to find the domain in f?
You need to find all the values for which $x^2 y^2-4xy^2+3y^2$ is not equal to zero and $4x-x^2-y^2$ is positive or zero.
zeros of the function
Well you just need to find where $\sqrt{4x-x^2-y^2}=0$, which is just where $4x-x^2-y^2=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1022924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Find Z Rotation based on X and Y Vector I have a vector $(x,y) = (x_2 - x_1, y_2 - y_1)$. I have an arrow pointing to 0 degrees.
With vector $(x, y)$, how can I find the number of degrees (0 - 360) that will be the direction the arrow points in order for it to point from $v_1 = (x_1, y_1)$ to $v_2 = (x_2, y_2)$?
| I am assuming that the arrow 'pointing to 0°' is pointing in $x$-direction of the coordinate system.
Then $\frac{y}{x} = \tan(\alpha-180°)$ where $\alpha$ is the angle you are looking for, because the vector $(x,y)$ is poiting from $v_2$ to $v_1$. That means $y/x$ is the slope of the line that goes throu $v_1$ and $v_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate limit without L'Hopital I need a some help with this.
Calculate:
$$\lim_{x\to\infty}\frac{\ln(x-1)}x$$
I know the answer is zero. But dont know how to handle the $\ln(x-1)$
| We will prove instead that $\frac{\log x}{x} \to 0$, and leave you to deduce what you need. Suppose $x \geq e$ (here $e$ is the base of the logarithm). Then
$$\frac{\log (ex)}{ex} = \frac{\log x + 1}{ex} \leq \frac{2}{e} \frac{\log x}{x}, $$
since $e \leq x$ implies $1 \leq \log x$ (since $\log x$ is monotone). Monoton... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
How to find all of generators in a finite fields How can I find all generators of a finite field?
For example in GF(2^3) and X^3 + x^2 + 1 as primitive polynomial.
I don`t want all of solutions. I need some hint and help to solve this problem.
Thanks a lot.
Ya Ali.
| the multiplicative group of $GF_{2^3}$ has prime order, so that tells you how many generators it has. also note what is the only subfield of $GF_{2^3}$ (a generator cannot lie in a subfield). and finally note that there are precisely two irreducible polynomials of degree $3$ over $GF(2)$. try computing the first seven ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How does $\sum_{n = 1}^{\infty} \frac{1}{n(n+1)}$ simplify? $\sum_{n = 1}^{\infty} \frac{1}{n(n+1)} = \frac{1}{1\cdot2} + \frac{1}{2\cdot3} + ...$
$= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + ... = 1$
My professor wrote this the other day. But I'm wondering...how does the series become $= 1 - \frac{1}{2} + \frac{1... | It is because $\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group
Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.
I have tried all s... | This is a very well-known presentation of the trivial group, to be compared with the presentation of Higman's infinite group with no finite quotient. I do not know of any easy proof.
The proof I'm going to give is due to Bernhard Neumann in An Essay on Free Products of Groups with Amalgamations (Philosophical Transacti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 0
} |
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