Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Simple solution to combinations with limited repetition Suppose that you are going to prepare a fruit salad with oranges, apples, and bananas. The salad should consist of 10 pieces. No fruit should be used more than 5 times.
Without the constraint, the solution would be
with $n=10$ and $k=3$, I believe.
However, the p... | To be found are the number of tuples $(o,a,b)$ with $o+a+b=10$ under constraint:$$(o,a,b)\in\{0,1,2,3,4,5\}^3$$
Setting $o'=5-o$, $a'=5-a$ and $c'=5-c$ this comes to the same as finding the number of tuples $(o',a',b')$ with $o'+a'+b'=5$ under constraint:$$(o',a',b')\in\{0,1,2,3,4,5\}^3$$
But here the constraints can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove or Disprove $2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx) $ converge uniformly to $x$ on $(-\pi,\pi)$ I want to prove $2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx) $ converges pointwise and uniformly to $x$ on $[-\pi,\pi]$. I know $\sum_{k=1}^{\infty}\frac{(-1)^n}{n}$ converge by alternating series t... |
NOTE: The original question that the OP asked was
"Prove $2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx) $ converge pointwise and uniformly to $x$ on $[0,2\pi]$ using elementary analysis"**
Let $a_n(x)=(-1)^{n-1}\sin(nx)$ and $b_n(x)=\frac1n$. Obviously, $b_n(x)\to 0$ monotonically and uniformly as $n\to\infty... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a,b$ are irrational numbers, Is $K=[a,b] \cap \mathbb Q$ closed in $\mathbb Q$? Suppose $a,b \in \mathbb R-\mathbb Q , a <b$. Consider $K=[a,b] \bigcap \mathbb Q$.
Now, $[a,b]$ is closed in the metric superspace of $\mathbb Q$ i.e in $\mathbb R$. Thus, $K=[a,b] \bigcap Q$ is closed in $\mathbb Q$
But, $K=[a,b] \big... | A subset $C$ of the metric space $(X,d)$ is closed if every sequence $(x_n\in C)$ which converges towards $c$, $c\in C\subset X$.
Here $X=\mathbb{Q}$,, $C=[a,b]\cap\mathbb{Q}$, if you take $a=\sqrt2, b=\sqrt5$ and a sequence of rationals $x_n\in C, limx_n=\sqrt5$, $\sqrt5$ is not in $X$.
What you have is the fact that ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does fundamental theorem of calculus not work for this integral $\int_0^{2\pi}\frac{dx}{(3+\cos x)(2+\cos x)}$? $$\int\frac{dx}{(3+\cos x)(2+\cos x)}= \frac{2\arctan(\frac{\tan(\frac x2)}{\sqrt3})}{\sqrt3} - \frac{\arctan(\frac{\tan(\frac x2)}{\sqrt2})}{\sqrt2} + C $$
This is the antiderivative . By the FTC :
$$\i... | Let $f(x)=\frac{1}{(3+\cos x)(2+\cos x)}$
and $F(x)$ be its anti-derivative.
Pertaining to this very question,what if we tried to make a substitution $\cos x=u$.
Since we would have to change the limits accordingly we see that the limit comes out to be
$\displaystyle\int_1^1 g(u) \mathrm du=0$, where the function $g$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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$\Psi: V_1\longrightarrow V_2$ such that $\Psi(F)=\Phi \circ F$. Let $V_1=Lin(\mathbb{R_{<=2}[x]},\mathbb{R_{<=2}[x]})$ and $V_2=Lin(\mathbb{R_{<=2}[x]},\mathbb{R})$ two vector spaces and let
$$\Phi \in V_2, \qquad \Phi(p(x))=p'(1).$$
Consider
$$\Psi: V_1\longrightarrow V_2\qquad \Psi(F)=\Phi \circ F.$$
How can I fin... | Consider the base $\mathcal B =\{x^2,x,1\}$ of $\mathbb R_{\leq 2}[x]$ and consider the coordinates isomorphism:
$$
\mathbb R_{\leq 2}[x] \rightarrow \mathbb R^3,\qquad a_2x^2+a_1x+a_0\mapsto (a_2,a_1,a_0)^T
$$
By this isomorphism an element of $\text{Lin}(\mathbb R_{\leq 2}[x],\mathbb R_{\leq 2}[x])$ is just a $3\time... | {
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Finding the integral $\int\frac{e^{x}}{e^{2x}+1}$ This question has been puzzling me for a bit and I'd like an explanation for what I'm doing wrong as my answer doesn't coincide with the correct one.
Let's say we're asked to find:
$$\int \frac{e^{x}}{e^{2x}+1}\mathrm{d}x$$
The way I chose to solve this was factor out a... | One of the best thing you can do with indefinite integrals is take the derivative of your result and check if it is the integrand function:
$$\frac{\text{d}}{\text{d}x} \left[-\arctan \left (\frac{1}{e^x}\right)+c\right]=\frac{\text{d}}{\text{d}x} \left[-\arctan \left (e^{-x}\right)+c\right]=-\frac{1}{1+e^{-2x}}({-e^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3648726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove union/intersect of two powersets is/is not equal to powerset of their union? I am supposed to show whether or not the union/intersection of two powersets is equal to the powerset of the union/intersection, respectively.
I found these two links that have both the proofs, and say that the union is not always equal... | The powerset of intersection is equal to the intersection of powersets, i.e. $P(A\cap B)=P(A)\cap P(B)$, can be proven as follow:
1) $P(A \cap B) \subseteq P(A) \cap P(B)$
*
*$\forall E \in P(A \cap B)$
*$[\forall x \in E(x \in A \cap B \equiv x \in A \land x \in B)]$
*$(E \subseteq A) \land (E \subseteq B)$
*$(E... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $|f(y)−f(x)−f′(x)(y−x)|≤ε|y−x|, ∀x,y∈[0,1], |x−y|<δ$. Suppose f is a differentiable function and f′ is continuous on [a,b],
prove that ∀ε > 0, there is δ > 0 such that
$|f(y)−f(x)−f′(x)(y−x)|≤ε|y−x|, ∀x,y∈[0,1], |x−y|<δ$.
Since f is differentiable and f' is continuous, then by using the definitions of both, I ca... | Let $x \in [a, b]$ then since $f(t)$ is differentiable at $x$ it follows that
$\lim_{y \to x} \frac{f(y)-f(x)}{y-x} = f^\prime(x)$
Translating this to $\epsilon, \delta$ we get that given $\epsilon > 0$ there exists a $\delta>0$ such that $|\frac{f(y)-f(x)}{y-x} - f^\prime(x)| < \epsilon$ if $|y - x|< \delta$. Which ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof for Uniform Convergence for $\{f_n\}$
Suppose $\{f_n\}$ is an equicontinuous sequence of functions defined on $[0,1]$ and $\{f_n(r)\}$ converges $∀r ∈ \mathbb{Q} \cap [0, 1]$. Prove that $\{f_n\}$ converges uniformly on
$[0, 1]$.
Since I know that $\mathbb{Q} \cap [0, 1]$ is not compact, I am a bit stuck on ... | This is actually a simple case of applying the Arzela-Ascoli Propagation Theorem which states:
Point-wise convergence of an equicontinuous sequence of functions on a dense subset of the domain propagates to uniform
convergence on the whole domain.
The rational numbers, $\mathbb{Q}$, are dense in the interval $[0,1]\sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3649221",
"timestamp": "2023-03-29T00:00:00",
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One inequality about the average integral: $\bar{f}_A:=\frac{1}{\mu(A)} \int_A f \,d\mu$ For a measurable function $f$, if we consider the average integral:
$$\bar{f}_A:=\frac{1}{\mu(A)}\int_A f \, d\mu$$
where $\mu$ is Lebesgue measure.
\begin{equation}
\begin{split}
\int_A (f-\bar{f}_{A})^2\,d\mu
... | $f-\overline {f}_B =(f-\overline {f}_A )+ (\overline {f}_A -\overline {f}_B )$. Square both sides and integrate over $A$. Note that the cross term vanishes: $2\int_A (f-\overline {f}_A )(\overline {f}_A -\overline {f}_B )=2(\overline {f}_A -\overline {f}_B ) \int_A (f-\overline {f}_A )$ and $\int_A(f-\overline {f}_A ) ... | {
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Finding / Proving the Order of Dihedral and Symmetric Subgroups Lately I've been trying to follow along with this YouTube lecture series called "Visual Group Theory" by Clemson's Mathew Macauley and have had some issues grasping the concept of a group's "order"..
I (think I) understand how it works when referring to r... | You started with a slight error. There's a difference between the order of a group, and the order of an element. To illustrate, a favorite theorem of mine is Cauchy's theorem.
It states that if $p$ is a prime dividing the order of the group, then the group has an element of order $p$.
But getting back to your que... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3649556",
"timestamp": "2023-03-29T00:00:00",
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Mathematical Operation (PEMDAS AND BODMAS) I am learning Python and came across PEMDAS.
Python uses PEMDAS to solve mathematical equations.
But in lower classes like 5th or 6th we were taught BODMAS.
I got confused and then made an equation to check which method gives me the correct answer.
My equation was
100-2⁵×8÷2+... | In both PEMDAS and BODMAS, there is no particular preference for multiplication or division, either can be done first and answer will be same. In general, an expression like $\frac{abcd}{ghij}$ can be evaluated by multiplying or dividing number in any order whatsoever and you will get the same result.
For example, chec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3649733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Subgraphs of a bipartite graph $H$ as the intersection of two copies of $H$ in $K_{n,n}$. Let $H$ be a bipartite graph. Is it necessarily the case that every subgraph of $H$ can appear as the intersection of two copies of $H$ in $K_{n,n}$ for large enough $n$? (Here $K_{n,n}$ is the complete bipartite graph with $n$ ve... | The answer is negative for $H=C_4$ and its subgraph $G$ which equals $H$ without an edge. Indeed, if a copy of $G$ is an intersection of two copies of $H$ then both copies share the same four vertices and it is easy to check that the intersection of the copies of $H$ cannot be a copy of $G$.
On the other hand, the an... | {
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"url": "https://math.stackexchange.com/questions/3649895",
"timestamp": "2023-03-29T00:00:00",
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Prove that $R^{n}\setminus R^{k} \simeq S^{n} \setminus S^{k} \simeq S^{n-k-1} $ $\simeq$ is homotopy equivalence space
homotopy equivalence:
Two topological spaces X and Y are homotopy equivalent if there exist continuous maps $f:X\rightarrow Y$ and $g:Y \rightarrow X$, such that the composition f degreesg is homotop... | Your question is imprecise because you do not specify how $\mathbb R^k$ is regarded a subspace of $\mathbb R^n$ (similarly for $S^k$ and $S^n$). But certainly you identity
$\mathbb R^k$ with $\{(x_1,\ldots,x_n) \in \mathbb R^n \mid x_{k+1} = \ldots = x_n = 0 \}$ and $S^k$ with $\{(x_1,\ldots,x_{n+1}) \in S^n \mid x_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3650017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Geometric Interpretation of Matrix Additiom Is there a geometric meaning to matrix addition similar to how matrix multiplication acts as a linear transformation? I'm really curious thanks!
| As you note, each matrix corresponds to a linear transformation. Given a matrix $A$, the linear transformation that $A$ corresponds to is defined via matrix-vector mulitplication. In particular, if $A$ has size $m \times n$, then the corresponding transformation $T:\Bbb R^n \to \Bbb R^m$ is given by $T_A(x) = Ax$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3650199",
"timestamp": "2023-03-29T00:00:00",
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Show that $f$ has exactly one zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}. Let $f(z) = z + g(z)$ where $g$ is holomorphic. Suppose that
$|\operatorname{Im} g(z)| < 1$ for $z ∈ [−1 − i, 1 − i]∪[−1 + i, 1 + i]$ and $|\operatorname{Re} g(z)| < 1$ for $z ∈ [−1 − i, −1 + i] ∪ [1 − i, 1 + i]$.
Show that ... | For this problem the following stronger version of Rouche works (sometimes it is called the symmetric Rouche and is expressed as $|f-g| <|f|+|g|, z \in K$):
If $\Omega$ is the interior domain of a Jordan curve $K$ and $f(z)+\lambda h(z) \ne 0, \lambda \ge 0, h(z) \ne 0, z \in K$ then $f,h$ have the same number of zero... | {
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Why do we need to determine the definiteness of the Hessian to decide what a critical point is? In univariate calculus, if we know that $f'(c)=0$, we can determine if the function $f$ has a minimum at $c$ by checking that $f''(c) > 0$. The multivariate analogue of the second derivative is the Hessian matrix. I now lear... | 1) For example, the function $f(x,y) = x^2 + 4 x y + y^2$ has all entries of the Hessian matrix $> 0$, but the critical point $(0,0)$ is a saddle (e.g. $f(t,-t) < 0$ for $t \ne 0$).
2) A smooth function of $n$ variables is convex in an open set $R$ iff its Hessian is positive semidefinite there.
3) A real symmetric mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3650650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Variance calculation bivariate probability distribution I understand the following statement is true
$$V(XY)=E((XY)^2)-(E(XY))^2$$
In my question X and Y are independent.
The solution recommends
$$ V(XY)=E((X)^2)E((Y)^2)-(E(XY))^2 $$
and I curious to understand why this approach fails
$$V(XY)=E(XY)E(XY)-(E(XY))^2$$
See... |
The solution recommends
V(XY)=E((X)^2)E((Y)^2)-(E(XY))^2
Because $X$ and $Y$ are independent, so too are $X^2$ and $Y^2$. The expectation of a product of independent random variables is the product of their expectations.
You can further say $\mathsf V(XY)=\mathsf E(X^2)~\mathsf E(Y^2)-(\mathsf E(X))^2~(\mathsf E(Y))^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3650830",
"timestamp": "2023-03-29T00:00:00",
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Continuous Random Variable Transformations vs Discrete My Textbook, Introduction to Mathematical Statistics, has the following example of finding the pdf of a transformation of a continuous random variable:
Let $X$ be a random variable with pdf $f_X(x)=2x$ for $0 < x < 1$, zero elsewhere, and cdf $F_X(x)=x^2$. Let $Y ... | Because the pdf is an unsigned derivative†, we must apply the chain rule for derivation.
$$\begin{align}f_Y(y) &=\begin{vmatrix}\dfrac{\mathrm d F_Y(y)}{\mathrm d y}\end{vmatrix}\\[1ex] &=\begin{vmatrix}\dfrac{\mathrm d F_X(g^{-1}(y))}{\mathrm d y}\end{vmatrix}\\[1ex] &=\begin{vmatrix}\dfrac{\mathrm d F_X(g^{-1}(y))}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3650976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area of the shaded region of the intersection between two triangles Given figure below
Find the area of the shaded region.
The only thing I found is DCO triangle is congruent with ABO triangle
$$\frac{AB}{CD}=\frac{AO}{CO}\\
\frac{9}{CD}=\frac{AO}{17}$$
I don't think this lead anywhere. Any clue what to do? Thanks.
| I will assume $17$ is from $A$ to the point of intersection of $DB$ and $AC$ which you call $O$. The triangles $ABO$ and $DCO$ are similar so that $AO=BO=x$ and $\angle ADB=\angle ACB=\theta$. Using the law of cosines you get the following equations which you can easily solve for $x$ (without solving for $\theta$)
$$\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3651328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The outside of a $180$-sheet roll of toilet paper is covered by two sheets; the inner cylinder, by one. What's wrong with how I counted the layers?
Puzzle: A roll of toilet paper has 180 sheets on it. The outside is covered with exactly two sheets. The inside around the cardboard cylinder is covered by exactly one. Qu... | Consider the cross-section of the roll.
It might be easier to think of it as a continuous spool of paper that's later going to be divided lengthwise equally into sheets.
The circumference of the cross section at a radial distance $r$ from the centre is $2\pi r$. The thickness of one layer is $t$. The circumference one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3651513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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real number and decimal expansions For any real number $x$ we define its decimal expansion as $N\cdot x_1x_2x_2\cdots$ where $N=\lfloor x\rfloor$ and
$$x_i=\left\lfloor 10^i \left(x- \left(N+\sum_{j=1}^{i-1}\frac{x_j}{10^j}\right) \right)\right\rfloor.$$
Now I have two questions regarding this definition:
*
*Why wi... | *
*WLOG, $N=0$ (you can rescale $x$), and
$$0\le(x-0.)<1$$ starts the induction.
Then $$0\le10^n(x-0.x_1x_2\cdots x_n)<1\implies0\le10^{n+1}(x-0.x_1x_2\cdots x_n)<10$$ so that taking the floor, the next digit is one of $0,1,\cdots 9$. And in turn
$$0\le10^{n+1}(x-0.x_1x_2\cdots x_nx_{n+1})<1$$ because this is the frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3651649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$g$ not continuous in $(0,0)$, differentiable in every direction AND $|D_vg(x)| \leq |v|$ I have found plenty of simliar questions to mine, but in this case there is one more condition that needs to be satisfied, this is the problem:
"Find a function $g:\mathbb{R}^2 \rightarrow\mathbb{R}$, so that all directional deri... | Suppose $x,y\in \mathbb R^2.$ Define $f:\mathbb R\to \mathbb R^2$ to be the function $f(t)= g(x+t(y-x)).$ Then $f'(t) = D_{y-x}g(x+t(y-x))$ for all $t.$ By the MVT, $f(1)-f(0) = f'(c)$ for some $c\in (0,1).$ It follows that
$$|g(y)-g(x)|=|f(1)-f(0)| = |f'(c)| = |D_{y-x}g(x+c(y-x))|\le |y-x|.$$
Thus $g$ is Lipschitz on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3651816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Square root of the max is the max of the square root? I apologize if the question seems so obvious, but I don't have a strong base on maths nor I know the tools to prove this simple statement.
For a given function $f(x)$, is it true that
$$ \left(\max |f(x)|^2 \right)^{\frac{1}{2}} = \max |f(x)|.$$
I wonder if it shoul... | It's a big assumption that $\max |f(x)|$ or $\max(|f(x)|^2)$ exist but if one or the other does, they both do $(\max(|f(x)|^2))^{\frac 12} = \max |f(x)|$.
Suppose $\max |f(x)|$ exist. That means there is $a\in \mathbb R$ so that for every $y\in \mathbb R$ we have $|f(y)| \le |f(a)|$ and $\max|f(x)| = |f(a)|$.
If $|f(y... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that for sets $A,B,C$, if $C \subseteq B$, then $(A\setminus B)\cap C = \varnothing$. I just need the proof of this. How does one prove that given $A, B, C$, if $C\subseteq B$, then $(A\setminus B)\cap C$ is equal to an empty set.
| Assume that $(A \setminus B) \cap C \neq \varnothing$. Let $x \in (A \setminus B) \cap C$. Then $x\in (A \setminus B)$, and hence, not in B. However, $x\in C$ which is a subset of $ B$. So $x\in B$. That's a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Subgroup inavariant subextension of Galois extension Let $L/K$ be a Galois extension. Let $a$ be a generator of normal basis $ga$. Let $H$ be any subgroup of G. I need to prove that $M/K$ ($H$-invariant subextension) is generated by $x=\sum_{h\in H}ha$. The only thing i had managed to find out is that one of bases of $... | Lets check H-fixed elements in $K[G] \cong L/K$ (isomorphism as $K[G]$-modules is given by normal basis). It would be elements looking like $k\sum_{h\in Hg}h$ because only cosets of $H$ are stable under $H$ multiplications on the left and $H$ acts transitively on such cosets. Subspace $K$-linearly generated by such ele... | {
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"answer_id": 2
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Is this a rigourous and correct way to evaluate a limit including an $n^{th}$ degree derivative? I'm a student in Physics' first year, I have an introductory knowledge on real analysis, and I'm not sure about how to solve the following limit:
$$\left.e^{2x}\frac{d^n}{dx^n} e^{-x^2}\right\rvert_{-\infty}^\infty$$
What I... | The expression can be written
$$\frac{P(x)}{e^{(x-1)^2}}$$ or $$\frac{Q(x-1)}{e^{(x-1)^2}}$$
where $P,Q$ are polynomials.
Whatever their degree $d$, $$e^{(x-1)^2}=\Omega(|x-1|^{d+1}).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3652361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Evaluating : $\int \frac{\sec x-\tan x}{\sqrt{\sin^2x-\sin x}} \mathrm{d}x$ As a part of a bigger question, I was asked to evaluate the integral :
$$\int \frac{\sec x-\tan x}{\sqrt{\sin^2x-\sin x}} \mathrm{d}x$$
Here's what I tried:
(Please bear with me, it gets quite lengthy)
$$\int \frac{\sec x-\tan x}{\sqrt{\sin^... | Letting $\sin x=\sec^2 \theta$, we have
$$
\begin{aligned}I&=\int \frac{2 \sec ^2 \theta \tan \theta d \theta}{\left(1+\sec ^2 \theta\right) \sec \theta \tan \theta}\\&= 2 \int \frac{d(\sin \theta)}{2-\sin ^2 \theta}\\&= \frac{1}{\sqrt{2}} \ln \left|\frac{\sqrt{2}+\sin \theta}{\sqrt{2}-\sin \theta}\right|+C\\&= \frac 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3652486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Calculate the perimeter of a circle with a continuously increasing radius I have a circle (if that would even be the correct name for this shape), with a radius function equal to $R=\frac 2\theta + 1$, where $\theta$ is the angle in radians. The domain is between $\theta = 0.25$ and $\theta = 2\pi$.
How do I calculate ... | There are three main ways to describe a curve
*
*by a cartesian equation $y=f(x)$,
*by parametric equation(s) $x=x(t), \ y=y(t)$,
*by a polar equation $r=r(\theta)$. We are here in this third case with
$$r(\theta)=\frac{2}{\theta}+1$$
Each "way" has its own formulas for the computation of areas, lengths, etc. whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3652603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Non equivalent colourings of regular hexagon( Brualdi Chapter-14 , Exercise -32) I have a question in this exercise of Richard Brualdi's Introductory Combinatorics.
Exercise is -> Determine the number of non equivalent colourings of corners of regular hexagon with colours red, white and blue.
Now, Taking motivation f... | The dihedral group of the hexagon is $\rho^0,\rho^1,\rho^2,\rho^3,\rho^4,\rho^5,\tau_1,\tau_2,\tau_3,\sigma_4,\sigma_5,\sigma_6$.
The $\rho^i$ are the rotations, the $\tau_i$ are the reflections through axes which pass though the vertices of the hexagon and the $\sigma_i$ are the reflections which do not pass through t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Prove that $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n}\zeta(n) = \gamma$ How do you prove that
$$\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n}\zeta(n) = \gamma$$
where $\gamma$ is the Euler-Macheroni constant? This series kind of appeared in one of the questions I asked earlier; you just need to do some rearranging to get to this... | We can't write
$$\sum_{k=1}^{\infty} \left(\frac{1}{k} - \ln\left(\frac{k+1}{k}\right)\right)=\sum_{k=1}^\infty\frac1k-\sum_{k=1}^\infty\ln\left(\frac{k+1}{k}\right)$$
because both of these two series are divergent. To fix this issue, we use the limit
$$\sum_{k=1}^{\infty} \left(\frac{1}{k} - \ln\left(\frac{k+1}{k}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Is Hamiltonian path NL? NL is what can be solved by a non-deterministic Turing machine in logspace.
Could you non-deterministically "guess" the correct Hamiltonian path in logspace, keeping track of the current vertex (log(n) bits) and a count of how many vertices you've visited (log(n) bits)?
Does that mean finding a ... | This describes a non-deterministic walk, not a path.
You may end up counting the same vertex more than once.
Given a start configuration of (v, 0) (meaning you're at vertex v and you've traversed 0 edges), when you're at configuration (w, n-1) (meaning now you're at vertex w and you've traversed n-1 edges), all you kno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Exponent of primes in factorial n Let $n$ be a natural number and $V2,V3$ and $V5$ denote the exponent of $2,3$ and $5$ in $n!$ respectively. Then it true that $(2^{V2})^2(3^{V3})^2(5^{V5})^2>n!$. I have verified it by calculator. How do I prove it mathematically?
| This is not true, for example $59!=2^{54}\cdot 3^{27} \cdot 5^{13}\cdot d$ where $d$ is coprime to $2,3$ and $5$ (see factorize 59!), yet
$$
(2^{54}\cdot 3^{27} \cdot 5^{13})^2 \not > 59!,
$$
see is (2^54 * 3^27 * 5^13)^2 > 59!.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove ${n+2 \choose 3}=1\cdot n + 2 \cdot (n - 1) + \ldots + n \cdot 1$? I saw this problem as an exercise in Combinatorial Identities :-
Prove that $${n+2 \choose 3}=1\cdot n + 2 \cdot (n - 1) + \ldots + n \cdot 1\,.$$
After giving some time to this, I think that it is quite similar to the identity :-
${n \ch... | I suggest proving it combinatorially. $\binom{n+2}3$ is the number of $3$-element subsets of the set $[n+2]=\{1,2,\ldots,n+2\}$. We can classify those sets by their middle elements: let $\mathscr{A}_k$ be the family of all $3$-element subsets of $[n+2]$ of the form $\{j,k,\ell\}$, where $j<k<\ell$; clearly
$$\binom{n+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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convergency of a function series I've got the following function series:
$$ f(x) = \sum_{n = 0}^{\infty} \frac{e^{nx}-1}{2^ne^{nx}}$$
Is it punctually convergent and uniformly convergent?
What I have done is:
I've chosen the series $\frac{1}{2^n}$ which is bigger than the series given in the problem, and because of th... | I believe this can be put into the form of a geometric series s.t.
$$f(x) = \sum_{n = 0}^{\infty} \bigg(\frac{1}{2}\bigg)^n - \sum_{n = 0}^{\infty} \bigg(\frac{1}{2e^x}\bigg)^n $$
The first term can be evaluated without thought via the equation for the geometric series: $\sum_\limits{n =0}^{\infty} ar^k = \frac{a}{1-r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3653902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The quotient of an integral extension is also an integral extension I would like to prove the following fact:
Suppose that $K$ is a field and that $A$ is a ring and an integral extension of $K$. Given a prime ideal $\mathcal{P} \subseteq A$, then the quotient $A / \mathcal{P}$ is also an integral extension of $K$.
This... | I think the approach may be too elementary. You want to show that the map $K\to A\to A/\mathcal{P}$ is injective. Morphisms from a field to another ring are always injective. Indeed, let $f:K\to B$ be just about any ring morphism and assume that $f(a)=f(b)$ and $a\ne b$, hence $t:=a-b$ is invertible and so you get the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are strings of length $n^2$ sparse? Let L be the language of strings of length $m$, where $m$ is a perfect square. (So strings of length $1, 4, 9, 16, 25, \dots$ are accepted, other lengths are not.)
As $m$ increases, less and less strings are accepted.
Is this language sparse?
a sparse language is a formal language (... | It's not sparse:
Every time we encounter a perfect square m=k^2 we add 2^m strings to the language.
Even though we're adding less strings as n increases (perfect squares occur less and less), every time we add strings, we're adding exponential (2^m) strings.
So overall we're still adding exponential strings.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3654234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $H\leq G$ and $x^{2}\in H$ for all $x\in G$, show that $H\lhd G$ and $G/H$ is abelian. I have already showed that $H$ is normal in $G$. But I can't show that $G/H$ is abelian.
What I have tried was taking $X,Y\in G/H$ and show that $XYX^{-1}Y^{-1}=1_{G/H}$. If $X=Hg_1$ and $Y=Hg_2$ then $XYX^{-1}Y^{-1}=Hg_1g_2g_1^{... | Hint For each $x,y \in G$ you have $xyxy\in H, xx yy \in H$ and hence
$$Hxyxy=H=Hxxyy$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3654445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding Lay's Proof of Uniqueness of Reduced Echelon Form In Lay's Linear Algebra and Its applications, he gives a proof the uniqueness of the reduced echelon form.
Consider two row equivalent matrices U and V in reduced echelon form. He says that the pivot columns of U and V are precisely the nonzero columns tha... | Suppose that the $k$-th column of $U$ is a pivot column; then it is linearly independent of the columns to its left. If the $k$-th column of $V$ were not linearly independent of the columns to its left, it would be linear combination of them, and the first $k$ columns of $V$ would therefore satisfy a linear dependence ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every *-isometric isomorphism of $B(\mathcal{H})$ keep compact operators? Let $\mathcal{H}$ be a Hilbert space, $B(\mathcal{H})$ denotes the $\mathcal{C}^*$-algebra consisting of bounded linear transformation on $\mathcal{H}$ ($*$ is the adjoint). Now consider a isometric *-isomorphism of $B(\mathcal{H})$ onto it self ... | This is a corollary of the fact that every $*$-automorphism of $\mathbb B(\mathcal H)$ is inner, and the hint given by MaoWao can also be used to prove this. I'll expand on this hint a bit.
Indeed, if $\phi$ is an automorphism of $\mathbb B(\mathcal H)$, then minimal projections are mapped to minimal projections under... | {
"language": "en",
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Computing $\int_0^1\frac{1-2x}{2x^2-2x+1}\ln(x)\text{Li}_2(x)dx$ Any idea how ot approach
$$I=\int_0^1\frac{1-2x}{2x^2-2x+1}\ln(x)\text{Li}_2(x)dx\ ?$$
I came across this integral while I was trying to find a different solution for $\Re\ \text{Li}_4(1+i)$ posted here.
here is how I came across it;
using the identity
$... | $$\int_0^1\frac{\ln(x)}{1+x}\text{Li}_2\left(\frac{x}{1+x}\right)\ dx=3\text{Li}_4(2)+\text{Li}_2(2)\log^22-3\text{Li}_3(2)\log2+6\operatorname{Li}_4\left(\frac12\right)+\frac{21}4\ln2\zeta(3)-\frac{\pi^2}{8}\log^22+\frac{1}{4}\log^42-\frac{29\pi^4}{288}$$
I being known,I deduce
$$\int_0^1\frac{x\ln(1+x)}{1+x^2}\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3655021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Evaluate $\int_0^1\frac{\tan^{-1}ax}{x\sqrt{1-x^2}}\,dx$ Evaluate $$\int_0^1\frac{\tan^{-1}ax}{x\sqrt{1-x^2}}\,dx\,,$$ where $a$ being parameter. I am not able to solve this.
| We will use Taylor series of arctan, so we have :
$$f(a)=\int_0^1\frac{\arctan ax}{x\sqrt{1-x^2}}=\sum_{n=0}^\infty{\frac{(-1)^na^{2n+1}}{2n+1}}\biggl(\int_0^1\frac{x^{2n}}{\sqrt{1-x^2}}\biggr)$$
Then we use integration by substitution for calculate the given integral (by choosing $x=\sin t$) :
$$\int_0^1\frac{x^{2n}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3655194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose $A \subseteq \mathbb R$ is countable. Show $\exists x\in\Bbb R$ s. t.$A \cap (x+A) =\emptyset$. Suppose $A \subseteq \mathbb R$ is countable. Show $\exists x\in\Bbb R$ s. t. $A \cap (x+A) =\emptyset$. (Here, $x + A = \{ x + a : a ∈ A \}$.
I'm unsure how to proceed.
I thought about taking the smallest distan... | One can show something stronger: let $I$ be uncountable set, then $(f_λ)_{λ∈I}$ a sequence of $I$-many injective functions from uncountable set $X$ to itself such that $f_a(x)=f_b(x)⇒a=b$, then for every countable set $A$, there exists some $i∈I$ such $A\cap f_i[A]=\emptyset$.
Let $I=X=\Bbb R$ and $f_a(x)=a+x$ to get y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Is there another type of number more advanced than complex numbers? I am a student and I was introduced to complex numbers about a year ago. I am curious to know whether there is another type of number system more advanced than complex numbers.
So the way I was introduced to complex numbers was by being told the histo... | There are many kinds of numbers we can obviously say are as "advanced", or more advanced. The Cayley-Dickson construction lets you double the dimension as often as you like, e.g. the already mentioned quaternions are a $4$-dimensional generalization of complex numbers.
There are also some which aren't comparable to com... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral $\frac{2}{876} \int_{0}^{T} \left[ x^{-3/8} (1+b\,x)^{3/4} (1+ab\,x)^{-3} e^{(c\,x^2)}+\log{(5/4)}\right]\,\mathrm dx$ I am trying to evaluate definite integral:
$$\frac{2}{876} \int_{0}^{T} \left[ x^{-3/8} (1+b\,x)^{3/4} (1+ab\,x)^{-3} e^{(c\,x^2)}+\log{(5/4)}\right]\,\mathrm dx$$
$a$, $b$, $c$ are all positi... | "abandon hope all ye who enter here" as wrote Dante Alighieri in "The Divine Comedy"
Using your numbers, you have
$$f(x)= \frac{1}{438} \left(\frac{ (1+3777 x)^{3/4}}{x^{3/8}
\left(1+\frac{19 }{5}x\right)^3}\,e^{\frac{3 x^2}{625}}+\log \left(\frac{5}{4}\right)\right)$$ and it seems to me that you want to find the v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3655763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Difficulty understanding proof by contradiction Here's my understanding of proof by contradiction based on what I've read and I've been taught.
We show $\neg P \implies (c \land \neg c)$ is always true. This is done by assuming $\neg P$ is true. Then, we realize that $(c \land \neg c)$ is logically equivalent to $F$ so... |
If my dog is in the bathroom, I will hear him bark there.
I do not hear him bark there.
Hence he is not in the bathroom.
Nothing more profound than that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3655947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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How can I solve this probability problem? A most unusual Irish pub serves only Guinness and Harp. The owner of this pub observes that 85% of his male costumers drink Guinness as opposed to 35% of his female costumers. On any given evening, this pub owner notes that there are three times as many males as females. What i... | What is wrong with your attempt has been elaborated in the comments : you calculated only the probability that the person is Female and drinking Guinness. You have not taken into account the fact that it was known that the person was drinking Guinness. This will increase the probability significantly.
See, the person y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3656123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing a collection of maps is a projector Suppose I have a collection of maps defined as follows:
for $d_{n}:C_{n} \rightarrow C_{n-1}$ and $s_{n}: C_{n} \rightarrow C_{n+1}$ I have :
$t_{n}=1-f'_{n} -f_{n}$ , where $f_{n}=s_{n-1}d_{n}$ and $f'_{n}=d_{n+1}s_{n}$.
Furthermore I am given that $s_{n}$ is a collection o... | $\text{id} - t_n = f_n + f_n' = s_{n-1}d_n + d_{n+1}s_n$.
Therefore $s_\#$ is the required chain homotopy between $t_n$ and $\text{id}$.
The answer your second question is, no (in general). For example, let $X$ be a topological space such that $X = \text{Int}A \cup \text{Int} B$, let $\iota$ be the inclusion $C_n(A + B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3656518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Adjusting a $y = \sin(x)$ graph slightly I have a function where
$$ y = m\sin\Bigl(\frac{x^{1.1}+30}{0.7d}\Bigr)^{2.2} $$
where $m = 45$ and $d = 120$ (constants)
I would like the turning point at the top of the sin curve to peak at 0.75d (90). I would like the starting and finishing values to remain the same.
The curr... | Consider the curve $$y=\frac{(x-4)^4}{12}-\frac{kx^2}{2}+cx+d,$$ where the constants are given by $d=5-\frac{90^4}{12}$ and $$240=2×30^4-12k(120^2)+24(120c)+24d,\\2M=2d+180c-8100k,$$ with $M$ being the maximum value you want the function to attain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3656661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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If $M$ finitely generated as an $R$-module, is $M$ is finitely generated as an $S$-module, and $S$ is finitely generated as an $R$-module? Let $S$ be a commutative ring, $R$ a subring of $S$, and $M$ a non-zero $S$-module. If $M$ finitely generated as an $R$-module, do we have that $M$ is finitely generated as an $S$-m... | If $M$ is finitely generated as an $R$-module, then since $R$ is a subring of $S$ we have that $M$ is finitely generated as an $S$-module (we just happen to be able to restrict the coefficients to be only elements of $R$ if we want, which are still elements of $S$). But $S$ need not be finitely generated as an $R$-modu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3656800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Weak*-convergence to 0 on L^\infty and convergence almost everywhere I am stuck with something standard...
Let $f_n \in L^\infty(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$, $n\geq1$, be such that
$$
\sup_{n\geq1} \|f_n\|_{L^\infty(\mathbb{R}^d)}<\infty
$$
and
$$
\lim_{n\to\infty} \int_{\mathbb{R}^d}|f_n(x)-f(x)| g(x)\,dx=0, ... | I think it is not true in general.
Further more you cannot control all elements of $L^{\infty,*}$ using $L^1$ because $L^1$ it is not reflexive.
Finally you can state that if $(f_n)_n$ converges weak* only if you allow to take a subsequence of it because the unitary ball is weak* compact. ($L^{\infty,*}=L^1$).
| {
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"url": "https://math.stackexchange.com/questions/3656938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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If $x$ is real find the maximum possible value of $10^x-100^x$ According to the person who gave this question it apparently has something to do with the range of a quadratic expression. But I can't see the connection with a quadratic equation.
So I tried to solve this by finding the maxima of the expression.
But I don'... | $$F(x)=10^x-100^x=10^x(1-10^x)$$
Let $$f(a)=a(1-a)$$
$$f'(a)=1-2a$$
the maximum of $ f(a) $ is $$ f(\frac 12)=\frac 14.$$
Thus, the maximum of $ F(x) $ is $ \frac 14 $ attained for $ x$ such that
$$10^x=\frac 12 = e^{x\ln(10)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3657080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
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Product rule for matrix-valued and vector-valued functions Given $g: \mathbb{R}^n \to \mathbb{R}^{n \times m}$ and $f: \mathbb{R}^n \to \mathbb{R}^{m}$ how can we compute $\nabla (g(x)f(x))$?
| The $(i,j)$-term of $\nabla(fg)$ is given by
\begin{align}
\partial_i((g(x)f(x))_j) &= \partial_i\left(\sum_{k=1}^n g_{kj}(x)f_k(x)\right)\\
&= \sum_{k=1}^n (\partial_ig_{kj})(x)f_k(x) + \sum_{k=1}^n g_{kj}(x)(\partial_i f_k)(x)\\
&= ((\partial_ig)(x)f(x))_j + (g(x)(\partial_i f)(x))_j
\end{align}
so
\begin{align*}
\na... | {
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"answer_id": 1
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Prove that if $p$ is an odd prime such that $p\mid(x^2+1)$ for some $x\in\mathbb{Z}$, then $p\equiv 1 \pmod 4$ I've tried proving the statement using that $x^2\equiv -1\pmod p$, and someone told me that this actually just implies that $p\equiv 1\pmod 4$. But I don't see it. Can anyone help me with this problem?
| Hint: Let $p = 2k + 1$. Consider $ x^{p-1} \pmod{p}$.
Can you conclude that $k$ must be even?
We are given that $ x^2 \equiv -1 \pmod{p}$, so $ 1 \equiv x^{p-1} \equiv (-1)^{k} \pmod{p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3657303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Proving the tangent map is linear Let $M$ and $N$ be some vector spaces and define the map $f:M \rightarrow N$, which is a smooth diffeomorphism. Let $Tf$ be the induced mapping between the tangent spaces of $M$ and $N$. Then it is said that the map $$Tf: T_pM \rightarrow T_{f(p)}N$$ is a linear mapping. This is where ... | Yes, this is an abstract object but it has a specific definition. Apply Taylor's theorem $$f(x+\lambda y)=f(x)+\frac{\partial f}{\partial x}\cdot (\lambda y)+\mathcal{o}\left(|\lambda|\right)$$ So your object $$Tf:=\frac{\partial f}{\partial x}$$ Do you see why this is a linear operator?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3657602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of $\sum_{n=1}^{\infty}\frac{(-1)^{T_n+1}}{n},$ where $T_n$ is the $n$th Triangular number Consider the series $$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$ This is clearly not absolutely convergent. On the other hand, obvious choice, alter... | $$1\color{red}{+\frac12}\color{blue}{-\frac13}-\frac14+\frac15\color{cyan}{+\frac16}\color{magenta}{-\frac17}+\cdots$$
$$=1\color{red}{-\frac12+2\cdot\frac12}\color{blue}{+\frac13-2\cdot\frac13}-\frac14+\frac15\color{cyan}{-\frac16+2\cdot\frac16}\color{magenta}{+\frac17-2\cdot\frac17}+\cdots$$
$$=1-\frac12+\frac13-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3657751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Convergence of a sequence of fixed points Let $f:[0,1]\to [0,1]$ and $g:[0,1]\to [0,1]$ be two continuous functions, each having a unique fixed point $x_f$ and $x_g$. Assume $\Vert f-g\Vert_\infty<\epsilon$. Is it possible to say something about $\vert x_f-x_g\vert$, maybe under some additional hypothesis on $f$ and $g... | I'll assume $g$ is continuous.
If $x_g$ is the unique fixed point of $g$ on $[0,1]$, then for any $\delta > 0$ we have $\epsilon = \inf \{|g(x)-x|: x \in [0,1], |x - x_g| \ge \delta\} > 0$.
If $f_n \to g$ uniformly, there is $N$ such that $|f_n(x) - g(x)| < \epsilon$ for all $n > N$ and $x \in [0,1]$, and then if $|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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show this inequality $\sum_{cyc}\frac{1}{5-2xy}\le 1$ let $x,y,z\ge 0$ and such $x^2+y^2+z^2=3$ show that
$$\sum_{cyc}\dfrac{1}{5-2xy}\le 1$$
try:
$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le 2$$
and
$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le\sum_{cyc}\dfrac{(x+y)^2}{\frac{5}{3}z^2+\frac{2}{3}x^2+\frac{2}{3}y^2+(x-y)^2}\le\sum\dfrac{3(x+y... | SOS helps.
For $a^2+b^2+c^2=1$ after $x=\sqrt3a$, $y=\sqrt3b$ and $z=\sqrt3c$ we need to prove that:
$$\frac 1{5-6ab}+\frac 1{5-6bc}+\frac 1{5-6ca}\leq 1$$ or$$\sum_{cyc}\left(\frac{1}{3}-\frac{1}{5-6ab}\right)\geq0$$ or
$$\sum_{cyc}\frac{2-6ab}{5-6ab}\geq0$$ or $$\sum_{cyc}\frac{3(a-b)^{2}+2c^{2}-a^{2}-b^{2}}{5-6ab}\g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$h:[0,1] \to\mathbb{R}$ continuous, and ivt The question is as follows:
$$ \text{Supposd } h:[0,1] \rightarrow \mathbb{R} \text{ is continuous. Show that there exist } w \in [0,1] \text{ such that}
\\h(w)= \frac{w+1}{2}h(0)+\frac{2w+2}{9}h(\frac{1}{2})+\frac{w+1}{12}h(1)$$
I know that I have to use Intermediate Value T... | Let $f(x)=\frac {h(x)}{(x+1)}$ which is continuous on $[0,1]$ and $T=1/2f(0)+1/3f(1/2)+1/6f(1)$.
Then we want to prove ,$$f(w)=T \ \ \text{for some } \ w\in [0,1]$$
Let $M$ and $m$ be maximum and minimum value taken by $f(x)$.
Then note that, $m\leq T\leq M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3658583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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How can I caculate the segment length necessary to separate a certain area from a sector under a given angle? Consider the geometry in this drawing.
A circular sector with radius $r$ is divided into two parts by a segment $l$ starting from one of the ends if its arc and meeting the opposite side of the sector at an an... | It looks to me that $$A=\frac{r^2 \beta}{2}-\frac{rl}{2}\sin(\alpha-\beta)$$ thus
$$l=\frac{r^2\beta-2A}{r\sin(\alpha-\beta)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3658760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Why is $15-\sqrt{15}-\sqrt{15-\sqrt{15}}-\sqrt{15-\sqrt{15}-\sqrt{15-\sqrt{15}}}$ so close to $5$? Basically I started with the number $15$. Then I subtracted its square root to get roughly $11.127$.
Subtracting the square root of that returned roughly $7.791$, and finally after taking the square root of that, I got ro... | I don't think there is anything fundamental about this. Here's a plot of the real part for repeated applications of the process:
Mathematica:
iter = Re[N[NestList[(# - Sqrt[#]) &, 15, 15]]];
ListPlot[iter, Joined -> True,
PlotRange -> {-2, 15}]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3658882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Question on the existence of a boundary between divergence and convergence It is said that $\sum1/n^{1+\epsilon}$ will converge and $\sum1/n$ will not. There are various proofs showing that such a boundary between divergence and convergence does not, in principle, exist. This is of course evident in the above, where $\... | The question is interesting. Actually, things are not so simple. There are many (infinite) different series "between"
$$
\sum \frac{1}{n}=\infty \quad\textrm{and}\quad \sum \frac{1}{n^{1+\varepsilon}}<\infty
$$
(for any $\varepsilon>0$),
some convergent and some other divergent. For example:
$$
\sum \frac{1}{n\cdot \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that there is no operator $T \in L(\ell^2(\mathbb{N}))$ Let $S \in L(\ell^2(\mathbb{N}))$ be the unilateral shift. How do we show that if $K \in L(\ell^2(\mathbb{N}))$ is a compact operator, then there is no operator $T \in L(\ell^2(\mathbb{N}))$ such that $T^2=S^3 +K$?
| "Compact perturbation" should always make one consider "Fredholm". And this is the key here: if we consider the Fredholm index, noting that $S^3+K$ is Fredholm we would have
$$
2\operatorname{ind}(T)=\operatorname{ind}(T^2)=\operatorname{ind}(S^3+K)=\operatorname{ind}(S^3)=-3.
$$
This would require $\operatorname{ind... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3659207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Change in coordinates for a metric in a given form With a change in coordinates, transform
\begin{align}
ds^2 = -z^2dt^2 + dz^2
\end{align}
to
\begin{align}
ds^2 = -dT^2 + dX^2.
\end{align}
My attempt. It is clear that incoming null geodesics $\dot{z}<0$
have the form $z=ce^{-t}$, and outgoing null geodesics have $z=... | Look up Rindler coordinates. They're the Lorentzian analogue of polar coordinates. Set $T = z \sinh t$ and $X = z\cosh t$. So $$\begin{align*} -{\rm d}T^2 + {\rm d}X^2 &= -(\sinh t\,{\rm d}z + z\cosh t\,{\rm d}t)^2 + (\cosh t\,{\rm d}z + z\sinh t\,{\rm d}t)^2 \\ &= -\sinh^2t\,{\rm d}z^2 - 2z\cosh t\sinh t\,{\rm d}z\,{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $ If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I proceed to the results? Please help me to proceed. Thanks in advance.
| Also, we can use a Tangent Line method:
$$\sum_{cyc}(a^3-a^2)=\sum_{cyc}(a^3-a^2-\ln{a})\geq0$$ because easy to see that
$$a^3-a^2-\ln{a}\geq0:$$
$$(a^3-a^2-\ln{a})'=3a^2-2a-\frac{1}{a}=$$
$$=\frac{3a^3-3a^2+a^2-a+a-1)}{a}=\frac{(a-1)(3a^2+a+1)}{a},$$
which gives $a_{min}=1$ and we are done!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What do the "d" in SDE notation mean? An SDE is often written in the form $ dX_t=\mu dt + \sigma dW_t $.
What is the meaning of this equation in English?
If I had to construct an SDE, I would write something like $ \frac{dX_t}{dt} = \mu + \sigma \frac{dW_t}{dt} $. Why are SDEs not written in that way?
I know that the B... | An SDE is just a short way of writing the stochastic integral equation
$$X_t = X_0 + \int_0^t \mu ds + \int_0^t \sigma dW_s$$
So if we take the SDE form and integrate both sides we get:
$$\int_0^t dX_s=\int_0^t\mu ds + \int_0^t\sigma dW_s$$
With the natural equation $$\int_0^t dX_s = X_t - X_0$$ we get original stocha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3659675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simulation of a General equation I have solved a programming problem with a Equation. But can't simulate this equation briefly. Anyone can help me?
Question:
I have $n$ rubles initially. The cost of one plastic litre bottle, the cost of one glass litre bottle, and the money one can get back by returning an empty glass ... | Let $p$ and $g$ be the number of plastic and glass bottles purchased, respectively. Assume that you return all glass bottles. Consider the problem of maximizing $p+g$ subject to linear constraints:
\begin{align}
a p + (b - c) g &\le n \\
p,g &\ge 0
\end{align}
For $(n,a,b,c)=(10,11,9,8)$ the optimal solution is $(p,g)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3659881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $\sum_i A_i=I$ with $A_i$ positive imply $\{A_i\}_i$ are mutually diagonalisable? As discussed in this other question, if $A$ and $B$ are matrices such that $A+B=I$, then trivially they commute, and thus if they are both diagonalisable they are also mutually diagonalisable.
The same argument doesn't, however, appl... | Let
$$
A_1 =\frac{1}{9}
\begin{bmatrix}
3 & 2 & -1\\
2 & 3 & -1\\
-1 & -1 & 3\\
\end{bmatrix},
\quad
A_2 =\frac{1}{9}
\begin{bmatrix}
3 & -1 & 2\\
-1 & 3 & -1\\
2 & -1 & 3\\
\end{bmatrix},
\quad
A_3 =\frac{1}{9}
\begin{bmatrix}
3 & -1 & -1\\
-1 & 3 & 2\\
-1 & 2 & 3\\
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3659957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability about lifetime of 100 bulbs (exponential distribution) I have some doubts about the following problem:
I have 100 bulbs with a lifetime represented by an exponential distribution, with an expected value of 1000 hours. Find the probability that, at least one bulb, blown down after at most 500 hours.
I have c... | I assume that it means "after at most 500 hours" right? In that case your computation makes sense for one bulb. What is $\lambda$ btw?
For the second part, we may assume that the bulbs are all independent and blow down within $500$ hours with a probability of $p=0.394$. You have $100$ bulbs. What is the chance that non... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3660125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Automatic complexity of word problem Suppose $L$ is a regular language. Let’s define its automatic complexity $ac(L)$ as the minimal possible number of states of a DFA, that recognizes $L$.
Now, suppose $G$ is a finite group. $A \subset G$, $\langle A \rangle = G$. Let’s define the map $\pi: A^* \to G$ using the follo... | I think that $L(G)$ must have at least $|G|$ states, which proves that in fact $|G|$ is the smallest number possible.
Let $v,w \in A^*$ represent two distinct elements of $G$, and let $\bar{v} \in A^*$ represent the inverse of the group element represented by $v$.
Then, after reading the word $v$, the DFA accepting $L... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Need help with calculus II series I am working on a problem that my professor isn't really explaining well, so i decided to ask here.
The following is the question
$ f(x) =\sum_{n=1}^\infty \frac{\mathrm{(-1)}^{n+1}\mathrm{(x-5)}^{n}}{(n\mathrm{5}^{n})} $
I am asked to find the interval of convergence of the following
... | For the series $\sum_{n = 1}^\infty a_n$, where there is an $N$ such that $a_n \neq 0$ for all $n \geq N$, the ratio test has you calculate
$$ L = \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right| $$
If $L < 1$, the series converges absolutely. If $L = 1$ or the limit fails to exist, the test is inconcl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3660480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving a seperable ODE using substitution proof This is the question my softmore differential equations professor asked on our practice exam:
Show that the substitution $y=ux$ in the first-order differential equation
$$p(x,y)\;dx+q(x,y)\;dy=0$$
results in an ODE (in u and x) which can be solved by "separation of... | Unfortunately I did not manage to read your notes. Follow a deduction in case it can help.
Making $y = \lambda(x) x$ and considering $dy = \lambda dx+x d\lambda$ we have
$$
p(x,\lambda x)dx+q(x,\lambda x ) dy = x^qp(1,\lambda)dx + x^q q(1,\lambda)(\lambda dx+x d\lambda) = 0
$$
so we follow with
$$
p(1,\lambda)dx + q(1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3660666",
"timestamp": "2023-03-29T00:00:00",
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Branch cut of $\sqrt{z^2-1}$. I was reading something that defined the function $f(z)=\sqrt{z^2-1}$ on $\mathbb{C}\setminus [-1,1]$ where the branch cut is such that the argument of $z$ and $\sqrt{z^2-1}$ are in the same quadrant. I think I understand what this means and I think it corresponds to the usual branch of th... | Let $f(z)=\sqrt{z^2-1}$ for $z\in \mathbb{C}\setminus[-1,1]$, with the branch cut on $[-1,1]$ such that $\arg(z)$ and $\arg(\sqrt{z^2-1})$ are in the same quadrant.
Branch points of $f(z)$ are at $z=-1$ and $z=1$. Corresponding branch cuts are contours that begin at $z=-1$ and $z=1$ and end at the point at infinity.
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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SVD for image compression I want to make sure I understood the concept behind SVD for image compression.
So, we start off with a rectangular $m \times n$ matrix that stores all the pixel values of the image. We then compute the SVD of this matrix to get two orthogonal matrices that contain information about the rows a... | The SVD decomposes a matrix as a weighted sum of matrices which are themselves outer product of two vectors. Hence you trade $mn$ coefficients for $k(m+n)$, where $k$ is the number of weights retained.
For compression to be effective,
$$k(m+n)\ll mn$$ must hold.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3661229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Invariant subspace of $R^3$ Let $T:R^3→R^3$ be the linear operator defined by
$$T(\begin{bmatrix}a\\b\\c\end{bmatrix})=\begin{bmatrix}b+c\\2b\\a-b+c\end{bmatrix}$$
Show that $W=span(e_1,e_3)$ is a T-invariant subspace of $R^3$.
Let $\alpha={e_1,e_3} $ be ordered basis for W and $\beta={e_1,e_2,e_3}$ be ordered basis f... | It is easy to see that $T(e_1)=e_3 \in W$ and that $T(e_3)=e_1+e_3 \in W.$
This gives
$$T(W)=W,$$
hence $W$ is a $T-$ invariant subspace.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3661440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trapezium angle given, relation between sides
How do i check other options? Should i apply triangular inequality ? Also is there some way that all options are checked?
| Assume that (D) is true. Then, since you have established that (C) is true, we must have $BC=AC \implies \theta =60°$ which is clearly false.
If (A) was true, then $AD=CD\implies \angle ACD =60° \implies \angle ADC =60° \implies \angle PDC =120° \implies PCD =0°$ which again, is false.
If (B) was true, then $BC=CD \imp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are all injective arcs the images of an injective interval? Given a continous $f:[0,1] \to \mathbb{R}^2$ call it an injective arc if:
*
*$f(0) \neq f(1)$
*$f(a) = f(b) =c $ for $a<b$ implies that $f([a,b]) = {c}$.
Given such $f$ we also have an ordering on $Im(f)$ by taking the one of $[0,1]$.
Is it true that for a... | It seems that my answer to Reparameterisation of Curve as a Regular Curve (Topology) also answers your question in the affirmative. In the referenced question arbitrary paths $p : [0,1] \to X$ are considered and it is shown that there exists a reparametrization $\phi$ (which is a non-decreasing surjective continuous ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Linear independence of complex basis of vectors. I understand that for a $\mathbb{C}^n$ as a real vector space, we choose $$\left\{\pmatrix{1\\0\\\vdots\\0},\pmatrix{\mathrm i\\0\\\vdots\\0},\pmatrix{0\\1\\\vdots\\0},\pmatrix{0\\\mathrm i\\\vdots\\0},\dots,\pmatrix{0\\0\\\vdots\\1},\pmatrix{0\\0\\\vdots\\\mathrm i}\rig... | Let the $e_j$ be the standard basis elements for $\Bbb R^n$, so those of $\Bbb C^n$ are $e_j,\,ie_j$. A general linear combination thereof is $\sum_j(a_j+ib_j)e_j$ with $a_j,\,b_j\in\Bbb R$. If this vanishes, $a_k+ib_k=0\cdot e_k=0$, so $a_k=b_k=0$.
| {
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Fundamental set of solutions for ODE Let $a$ and $b$ be distinct positive integers. Prove that $(x^{a}, x^{b})$ cannot be a fundemental set of solutions of any second order ODE of the form
$y''+p(x)y'+q(x)y=0$
on the interval $(-1,1)$, where $p(x)$ and $q(x)$ are continuous functions on $(-1,1)$.
My progress: I managed... | If $0<a<b$, then $b\ge 2$ and thus $y(x)=x^b$ has values $y(0)=y'(0)=0$. However the only solution of an initial value problem for the given DE form with these initial conditions is the zero solution. As $x^b\ne 0$ in general, this gives a contradiction.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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On some special actions Let $G$ be a group and $A$ be an abelian group. Let $\beta$, $\alpha :G\rightarrow Aut(A)$ be two homomorphisms. It is well known that if there
exist $\sigma \in Aut(A)$, $\rho \in Aut(G)$ such that $(\beta \circ
\rho )(g)=\sigma \circ \alpha (g)\circ \sigma^{-1}$ for all $g\in G$, then the semi... | It looks like you've let $\beta=\alpha$. So of course this is true with $\sigma$ and $\rho$ the identity automorphisms. But this is basically a trivial statement.
Not sure whether anything can be done with it if $\sigma\ne\rho$. If there are any examples, I guess they could be called $\alpha$ -equivalent
| {
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"url": "https://math.stackexchange.com/questions/3662731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The time convergence of stochastic integral and Doob's convergence. Consider the process $$X_{t}=\int_{0}^{t}e^{-s}dW_{s},$$ where $e^{-s}$ is deterministic.
I am wondering if $\lim_{t\rightarrow\infty}X_{t}$ exists almost surely... I understand that $X_{t}$ in the case is a martingale, so we can use Doob's martingale ... | $E|X_t|^{2}=\int_0^{t} e^{-2s} ds=\frac 1 2(1-e^{-t}) <1$ for all $t$ and this implies $E|X_t|$ is bounded.
The limiting distribution is $N(0,\int_0^{\infty} e^{-2s} ds)$ i.e. $N(0, \frac 1 2) $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that three points lie on the same line
Show that the points $A(3;9), B(-2;-16)$ and $C(0.2;-5)$ lie on the same line.
We can say that three points lie on the same line if the largest segment bounded by two of these points is equal to the sum of the smaller ones. Can you show me why this is sufficient for three p... |
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
source: https://en.wikipedia.org/wiki/Triangle_inequality
Thus, if $AB+BC=AC$, then the points must be on the same line.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Basis of External Direct Sum of infinitely many vector spaces In the book Basic Algebra by Arthur Knapp, he states that External Direct Sum for infinitely many vector spaces can be defined as below:
$\bigoplus_{a \in A} V_a$ is the set of tuples $\{v_a\}$ of the Cartesian product $\prod_{a \in A} V_a$ with atmost finit... | Addition is a binary operation: it takes two vectors, and returns a vector. By induction, we can add finitely many vectors together. But we cannot add infinitely many vectors together.
So, for example, the vector $(1,1,1,1,\ldots)$ cannot be expressed as a linear combination of the vectors $\mathbf{e}_j$ (where $\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\lim\limits_{n\to\infty}\frac{n^2+n+1}{(n+1)^2}=1$ So, here's the sequence $\{x_n\}$ defined by the following formula:
$$x_n = \frac{n^2+n+1}{(n+1)^2}$$
I want to try and prove this from the definition. Let $\epsilon > 0$ be given. Then, we need an integer $N(\epsilon) > 0$ such that:
$$n > N \implies |x_n ... | Yes your proof seems to be good. (the answer practically ends here, but here is another way to prove it)
Proof. $$\frac {n^2+n+1} {(n+1)^2} = \frac {n^2+n+1} {n^2+2n+1}= 1 - \frac n {n^2+2n+1}$$.
We claim that $$\lim_{n\to+\infty}\frac n {n^2+2n+1} = 0$$, and it can be shown that for any $\epsilon \gt 0$, as long as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that whether it is a Banach space or not? We consider the Banach space of all continuous functions on $X$ such that for each $f$ in the space,
\begin{equation*}
||f||=\sup_{x\neq y}\frac{\left\vert
f(x)-f(y)\right\vert }{\left\vert x-y\right\vert }.
\end{equation*}
How I can prove that it is a Banach spac... | $m$ is not a norm on the set of the continuous function on $[0, 1]$ with $f(0)=0$, because not all continuous functions on $[0, 1]$ are Lipschitz. For example, take $f(x)=\sqrt{x}$. It's continuous, $f(0)=0$, but
$$\frac{|\sqrt{x}-\sqrt{y}|}{|x-y|}=\frac{1}{|\sqrt{x}+\sqrt{y}|}$$
And taking $x=0$, we get that
$$m(f)\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Let $k\geq2$ and $l\in\mathbb{N}$. Prove that $(k+1)^l > k^l$. As a part of another proof, I need to prove that $(k+1)^l > k^l$, for all $k$, with $k\geq2$ and $l\in\mathbb{N}$. I need some help starting the proof. Thank you.
| Another way to see this is to take $l$th roots. Then all you want to prove is that $$k+1>k,$$ which is clearly true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3663991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
} |
eigenvectors and eigenvalues for finding the matrix representiing T with respcet to $\alpha$ Let $T: \mathcal P_2(\Bbb R)\rightarrow \mathcal P_2(\Bbb R)$ be the linear transformation defined by
$T(p(x))=p(1)(x^2+x)+p(0)$
Let $\alpha={x^2, x, 1}$ be the standard ordered basis for $\mathcal P_2(\Bbb R)$. Find the matri... | Almost. Actually, $T(1)=x^2+x+1$. Therefore$$[T]_\alpha=\begin{bmatrix}1&1&1\\1&1&1\\0&0&1\end{bmatrix}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3664184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find an example of an entire function such that $f(z)=-f(iz)$ for all $z$ Is there a systematic way to find an example of an entire function $f$ such that $f(z)=-f(iz)$ for all $z$? By testing at monomials, we find that $f(z)=z^6$ is a solution. But how can we find other solutions?
I tried writing $f=u+iv$ and then dif... | *
*All the monomials $z^2,z^6,z^{10},\dots$ are solutions.
*If $g(z)$ is any entire function, then
$$
f(z) = \tfrac14\big( g(z) - g(iz)+g(-z)-g(-iz) \big) \tag{a}
$$
is also entire and satisfies $f(z)=-f(iz)$. (This is similar to how the combination $\frac12\big( h(x)-h(-x) \big)$ produces an odd function for any rea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is this an exponential distribution? I have a probability density function $f(x) = k \cdot 3e^{-3x}$, with $k\ne 0$ constant. I saw someone saying this is the exponential distribution with $\lambda = 3$. However, isn't the exponencial distribution of the form $\lambda e^{-\lambda x}$? Doesn't the multiplication by a co... | Hint: consider the equation $1 = \int_0^{\infty} k \cdot 3e^{-3x}dx$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $T_n = \{\frac{a}{n} \mid a \in \mathbb{Z}\}$, then what are $\bigcup_{n\in\mathbb{N}}T_n$ and $\bigcap_{n\in\mathbb{N}}T_n$?
For each positive integer $n$, let $T_n = \{\frac{a}{n} \mid a \in \mathbb{Z}\}$.
What are $\bigcup_{n \in \mathbb{N}} T_n$ and $\bigcap_{n \in \mathbb{N}} T_n$?
I'm pretty sure the first o... | You are correct that $\bigcup_{n\in\mathbb{N}}T_n=\mathbb{Q}$. That's simply because any $q\in\mathbb{Q}$ is of the form $q=\frac{a}{b}$ for some $a\in\mathbb{Z}$ and $b\in\mathbb{N}$ and so $q\in T_b$.
You are wrong that $\bigcap_{n\in\mathbb{N}}T_n=\mathbb{Q}$. This has no chance of happening since $\bigcap_{n\in\mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can Cavalieri's Principle be applied to a Pyramid and a Cylinder? I know that Cavalieri's Principle makes it so that if two prisms/cylinders, or two pyramids/cones have the same area at a cross section parallel to the base, and they have the same height, they also have the same volume. However, does it still apply to a... | As you point out, if we're comparing a solid with constant cross-sectional area to one whose cross sectional area vanishes as we move away from its base, then Cavalieri's Principle is inapplicable. Put another way, a pyramid/cone might have the same volume and height as a prism/cylinder, but even in such a case, they c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3664979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Irreducible component of a geometrically reduced algebraic variety This is from Qing Liu's Algebraic Geometry and Arithmetic Curves, page 131:
Why can we assume that $X$ is integral? Can we prove that an irreducible component of a geometrically reduced algebraic variety given with the reduced closed subscheme structur... | Liu's definition of an algebraic variety over a field $k$ is a scheme of finite type over $k$. In particular, such a scheme $X$ is noetherian and has finitely many irreducible components $X_1\cup\cdots\cup X_n$. Then $X_1\setminus (X_2\cup\cdots\cup X_n)$ is an open irreducible subscheme, and so we may pick an affine o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3665176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How Singular Integrals Theory is applied on Partial Differential Equations Currently I'm interested in Singular Integrals Theory (I'm a beginner). I have read that this theory has deep relations with PDE's. For that reason I would like to know if there exists some web page, guide, essay or book which explain how Singul... | Take for example Poisson’s equation: $-\Delta u = f \text{ on }\mathbb{R}^3$, where $f\in L^2(\mathbb{R}^3)$ is a compactly supported function. Then a solution to this partial differential equation is given by:
$$
u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}\,{\mathrm d}y,
$$
where $|x-y|$ is the Euclidean ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3665334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find best approximation of $\sin(\pi x)$ over $[0,1]$ with quadratic polynomial $a_0+a_1x+a_2x^2$ Use the theory of orthogonal functions to find best in the mean approximation of the function $\sin(πx)$ on the interval $[0,1]$ by a second-order polynomial
That is, find such coefficients $a_0, a_1$ and $a_2$ that,
$$\in... | \begin{align}
\int^1_0 ( & \sin(\pi x)-a_0-a_1x-a_2x^2)^2 \, dx, \\[8pt]
= \int_0^1 \Big( & \sin^2(\pi x) \,dx + a_0^2 + a_1^2 x^2 + a_2^2 x^4 \\
& {}-2a_0\sin(\pi x) - 2a_1x\sin(\pi x) - 2a_2x^2 \sin (\pi x) \\
& {} -{}2a_0a_1 x -2a_0a_2x^2 -2a_1a_2 x^2 \Big) \, dx \\[10pt]
= {} & \int_0^1 \sin ^2(\pi x)\, dx + a_0^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3665501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Show $\sqrt[3]{5}$ is not contained in any cyclotomic extension of $\mathbb{Q}$.
Find the Galois group of $x^3-5$ over $\mathbb{Q}$, then show $\sqrt[3]{5}$ is not contained in any cyclotomic extension of $\mathbb{Q}$.
My attempt:
The roots of $x^3-5$ are $\sqrt[3]{5},\zeta_3\sqrt[3]{5},\zeta_3^2\sqrt[3]{5}$. So the ... | The splitting field is correct and the reasoning seems the right path to take, but notice that in order to affirm that $\mathbb{Q}(\sqrt[3]5) \not\subset \mathbb{Q}(\zeta_{n})$ you don't have to put in play $S_{3}$ : If $\mathbb{Q}(\sqrt[3]5) \subset \mathbb{Q}(\zeta_{n})$ since Gal($\mathbb{Q}(\zeta_{n})/\mathbb{Q}$) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3665586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Multiplication of an orthogonal matrix and a skew symmetric matrix Let $A\in O(n;\mathbb{R})$ such that for every $1\leq i\leq n$, $1>a_{ii}>0$ and $a_{ii}\geq |{a_{ij}}|$ for $j\neq i$. Prove that there exists a skew symmetric matrix $B$ such that all diagonal elements of $AB$ are positive.
For $n=2$ the proof is simp... | This is not always possible. E.g. when $n=5$, it is easy to generate by computer a symmetric orthogonal matrix $A$ such that $0<a_{ii}<1$ and $a_{ii}\ge|a_{ij}|$ using the following Octave/Matlab script:
n=5;
D=diag([ones(n-1,1); -1]);
for k=1:10000
[U,S,V]=svd(2*rand(n,n)-1);
A=U*D*U';
if min(diag(A))>0 && max(d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3665947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why matrices commuting with $\small\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$ represent complex numbers? I am trying to understand which $2$ by $2$ real matrices represent complex numbers in following way.
Let $J=\begin{bmatrix} 0&1\\-1&0\end{bmatrix}$ and $A=\begin{bmatrix} a&b\\c&d\end{bmatrix}$ be any real matrix.
If $... | Let's consider $\varphi:\mathbb C\rightarrow M_2(\mathbb C)$, $\varphi(a+ib)=\pmatrix{a & b \\ -b & a}$ the standard embedding of $\mathbb C$ into the matrix ring.
Consider $Z(J)=\{A\in M_2(\mathbb C)\ | \ JA=AJ\}$ the set of the matrix commuting with $J$.
Your question is equivalent to show that $Z(J) = \varphi(\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
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Any $(x, y, z)$ can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$? Please tell me whether there any $(x, y, z)$
which can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$ ?
No process or just solve it by calculator are both fine.
Thank you.
| Consider that you search for the extremum of $$F=5x^2+2y^2+6z^2-6xy-2xz+2yz$$
$$\frac{\partial F}{\partial x}=10 x-6 y-2 z=0 \qquad \frac{\partial F}{\partial y}=-6 x+4 y+2 z=0 \qquad \frac{\partial F}{\partial z}=-2 x+2 y+12 z=0$$
The only solution is $x=y=z=0$ so the minimum value of $F$ is $0$.
Then $F$ is non-nega... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Probability zero vs impossible I understand that probability $0$ does not mean 'impossible' - because if we look for instance at a uniform distribution over $[0, 1]$ then while each of the singleton events $\{r\}$ for $0\leq r \leq 1$ has probability $0$, if we carried out the experiment then we would get exactly one o... |
However, if we look at a distribution defined by a density function which is zero on $[0,\frac{1}{2}]$ and non-zero on $(\frac{1}{2},1]$ (let's say with a continuous transition between them), then we know that for any $0\leq r\leq \frac{1}{2}$ the event $\{r\}$ is impossible, and for $\frac{1}{2}<r\leq 1$ the event is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
A polynomial function representing Logical OR Is there a way to represent logical OR function of two or more variables as a polynomial/value function.
So that the $f(x,y)>0$ if $\text{OR}(x,y)>0$.
Additional Edit: Considering $\geqslant 0$ means TRUE, and $<0$ means FALSE. For both the variables and the function. i.e ... | Yes: $OR(x,y)=1-(1-x)(1-y)$. Assuming $0$ is false and $1$ is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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