Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is $dx$ in integration? When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.
$$\int f(x)\, dx$$
When he came to explain the meaning of the $dx$, he told us "think of it as a full stop". For whatever reason I did not raise my hand and question him ... | I once went at some length illustrating the point that for the purpose of evaluating integrals it is useful to look at $d$ as a linear operator.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "183",
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"answer_id": 2
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Extensions of Bertrand's Postulate Two questions came to mind when I was reading the proof for Bertrand's Postulate (there's always a prime between $n$ and $2n$):
(1) Can we change the proof somehow to show that: $\forall x > x_{0}$, there exists a prime $p$ $\in [x, ax]$, for some $a \in (1, 2)$?
(2) Suppose the (1) ... | I think you would enjoy the page PRIME GAPS.
My own version of the conjecture of Shanks, actually both a little stronger and a little weaker, is $$ p_{n+1} < p_n + 3 \,\left( \log p_n \right)^2, $$
for all primes $p_n \geq 2.$ This is true as high as has been checked.
Shanks conjectured that $$ \limsup \frac{p_{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Determine the equations needed to solve a problem I am trying to come up with the set of equations that will help solve the following problem, but am stuck without a starting point - I can't classify the question to look up more info.
The problem:
Divide a set of products among a set of categories such that a product d... | Let $x_{ij} = 1$ if you put product $i$ in category $j$, $0$ otherwise. You need
$\sum_i x_{ij} \ge m_j$ for each $j$, where $m_j$ is the minimum for category $j$,
and $\sum_j x_{ij} = 1$ for each $i$, and each $x_{ij} \in \{0,1\}$. The last requirement takes it out of the realm of linear algebra. However, look up ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proving a inequality about sup
Possible Duplicate:
How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
I want to prove that $\sup\{a+b\}\le\sup{a}+\sup{b}$ and my approach is that I claim $\sup a+ \sup b= \sup\{\sup a + \sup b\}$ and since $\sup a +\sup b \ge a+b$ the inequality is proved.... | Perhaps this is what you are looking for. Consider
$$
\sup_{x\in X}(a(x)+b(x))=\color{#C00000}{\sup_{{x\in X\atop y\in X}\atop x=y}(a(x)+b(y))\le\sup_{x\in X\atop y\in X}(a(x)+b(y))}=\sup_{x\in X}a(x)+\sup_{x\in X}b(x)
$$
The red inequality is true because the $\sup$ on the left is taken over a smaller set than the $\s... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve an nth degree polynomial equation The typical approach of solving a quadratic equation is to solve for the roots
$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$
Here, the degree of x is given to be 2
However, I was wondering on how to solve an equation if the degree of x is given to be n.
For example, consider this... | If the equation's all roots are real and negative, The range bound answer for one of a root is between $\displaystyle -\frac{k}{z}$ and $\displaystyle -n \frac{k}{z}$, where $k$ is constant, $z$ is coefficient of $x$ and $n$ is the highest power of $x$. And the coefficient of $x^n$ must be $1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question: Find all values of real number a such that $ \lim_{x\to1}\frac{ax^2+a^2x-2}{x^3-3x+2} $ exists. Thanks in advance for looking at my question.
I was tackling this limits problem using this method, but I can't seem to find any error with my work.
Question:
Find all values of real number a such that
$$
\li... | Since the denominator's limit is 0, the numerator cannot have a nonzero limit if the limit of the quotient is to be defined. The only hope is that the numerator's limit is also 0, and that after analyzing the indeterminate form, it does have a limit.
So, it must be the case that $\lim_{x\to1} ax^2+a^2x-2=0$, and conseq... | {
"language": "en",
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Numerical optimization with nonlinear equality constraints A problem that often comes up is minimizing a function $f(x_1,\ldots,x_n)$ under a constraint $g(x_1\ldots,x_n)=0$. In general this problem is very hard. When $f$ is convex and $g$ is affine, there are well known algorithms to solve this. In many cases however,... | If you have a single equality constraint you might try to rewrite your constraint $g(x_1,...,x_n)$ as:
$x_i = h(x_1,...x_{i-1},x_{i+1},...x_n)$
and then substitute for the $i$th variable in your objective function and solve the problem as an unconstraint optimiation problem.
| {
"language": "en",
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Statements in Euclidean geometry that appear to be true but aren't I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some examples of statements that may appear to be true (perhap... | Here is one example that is quite similar in nature to the statement in the question about the center of the circumcircle lying inside a triangle, but the dubious part ("lie inside") is somewhat better disguised. I report it only because I just found it in Wikipedia, with a literature reference.
The incenter (that is,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inducing a well-defined function on a set What does it mean to say that $f$ induces a well-defined function on the set $X$?
I'm confused about what the term induce means here, and what role the set
$X$ has.
| It means that a function is such that we can define a(nother) well defined function on some set $\,X\,$ that'll depend, in some definite way, on the original function.
For example: if $\,f:G\to H\,$ is a group homomorphism and there's some group $\,N\leq \ker f\,$ , with $\,N\triangleleft G\,$ , then $\,f\,$ induces a ... | {
"language": "en",
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Is it possible to take the absolute value of both sides of an equation? I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$.
Since the quadratic is greater than 0, I assume that there are no real solutions since
$y = 3x^2+bx+7$, and $3x^2+bx+7 > 0$, $y > 0$
since $y>0$ ... | What you've written is an inequality, not an equation. If you have an equation, say $a=b$, you can conclude that $|a|=|b|$.
But notice that $3>-5$, although $|3|\not>|-5|$.
If $3x^2+bx+7>0$ for every value of $x$, then the quadratic equation $3x^2+bx+7=0$ has no solutions that are real numbers. THat implies that the ... | {
"language": "en",
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Error in proof of self-adjointness of 1D Laplacian I have successfully checked self-adjointness of simple and classic differential operator - 1D Laplacian
$$D = \frac {d^2}{dx^2}: L_2(0,\infty) \rightarrow L_2(0,\infty)$$
defined on
$$\{f(x) | f'' \in L_2(0,\infty), f(0) = 0\},$$
open an article and see the first Exa... | The operator in Pavlov's article is not the same as yours. His has a domain of functions "vanishing near the origin", i.e. on a neighborhood of 0. For your operator, functions in the domain need only vanish at the origin. So there is no error; your operator is self-adjoint and his is not.
Regarding your last paragra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Example about hyperbolicity. $\def\abs#1{\left|#1\right|}$I would like to understand this example:
*
*Why is the following set a hyperbolic manifold?
$X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w < \abs{\exp(1/z)}\}$
It's an examples given in the book Hyperbolic Manifolds and Holomorphic Mappings: An Intro... | $\mathbb{CP}^2$ is a natural complex manifold where the chart are given by the maps :
$\begin{array}{lclc} \varphi_i: & U_i:=\{[z_0:z_1:z_2]\in \mathbb{CP}^2 \ | \ z_i\neq 0\} & \longrightarrow & \mathbb{C}^2 \\ & {[z_0:z_1:z_2]} & \longmapsto & (\dfrac{z_j}{z_0},\dfrac{z_k}{z_0}) \end{array}$
where $j,k\neq i$.
Howe... | {
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Prove that $\zeta (4)\le 1.1$ Prove the following inequality
$$\zeta (4)\le 1.1$$
I saw on the site some proofs for $\zeta(4)$ that use Fourier or Euler's way for computing its precise value, and that's fine and I can use it. Still, I wonder if there is a simpler way around for proving
this inequality. Thanks!
| $$\zeta(4) < \sum_{n=1}^{6} \frac{1}{n^{4}} + \int_{6}^{\infty} \frac{dx}{x^4} < 1.1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/201151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of a complex function How to find the limit of such a complex function?
$$
\lim_{z\rightarrow \infty} \frac{z \left| z \right| - 3 \Im z + i}{z \left| z \right|^2 +2z - 3i}.
$$
|
Consider moduli and use the triangular inequality.
The modulus of the numerator is at most $|z|^2+3|z|+1$ because $|\Im z|\leqslant|z|$ and $|\mathrm i|=1$. The modulus of the denominator is at least $|z|^3-2|z|-3$ because $|\mathrm i|=1$. Hence the limit of the ratio is $0$ when $|z|\to\infty$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Affine Subspace Confusion I'm having some trouble deciphering the wording of a problem.
I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$.
It's the next part where I can't figure out w... | By definition the set $L$ in your question consists of all the points on a line. So you may think of $L$ as a line (or the line that passes through the two points $v_1$ and $v_2$).
Hence if you are considering the two points ($v_1$, $v_2$) giving you the line $L$, then a subset $X$ containing all lines (the one line) t... | {
"language": "en",
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Recommended book on modeling/differential equations I am soon attending a undergrad course named differential equations and modeling. I have dealt with differential equations before, but in that course just learned a bunch of methods for solving them. Is there any cool books with more 'modeling' view of this subject? L... | Note: this list is different if you meant partial differential equations.
*
*A First Course in Differential Equations, Modeling, and Simulation Carlos A. Smith, Scott W. Campbell
*Differential Equations: A Modeling Approach, Frank R. Giordano, Maurice Weir
*Differential Equations And Boundary Value Problems: Compu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/201353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Open Measurable Sets Containing All Rational Numbers So I am trying to figure out a proof for the following statement, but I'm not really sure how to go about it. The statement is: "Show that for every $\epsilon>0$, there exists an open set G in $\mathbb{R}$ which contains all of the rational numbers but $m(G)<\epsilo... | Hint: if you order the rationals, you can put an interval around each successive one and take the union for your set. If the intervals decrease in length quickly enough....
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the following limit $\lim_{x\to 0}\frac{\sqrt[3]{1+x}-1}{x}$ and $\lim_{x\to 0}\frac{\cos 3x-\cos x}{x^2}$
Find the following limits
$$\lim_{x\to 0}\frac{\sqrt[3]{1+x}-1}{x}$$
Any hints/solutions how to approach this? I tried many ways, rationalization, taking out x, etc. But I still can't rid myself of the sing... | Like N.S. said, looking this limit as derivative is a way to solve.
You could also do $u=x+1$ to simplify your expression and consider $f(u)=u^{1/3}$.
$$u=x+1\rightarrow \lim_{u\rightarrow 1} \frac{u^{1/3}-1}{u-1}=\lim_{u\rightarrow 1} \frac{f(u)-f(1)}{u-1}=f'(1)$$
But $f'(u) = \frac{1}{3}u^{-2/3}$, then $f'(1) = \frac... | {
"language": "en",
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Uniform convergence of $f_n\rightarrow f$ and limit of zeroes to $f_n$ I'm having some doubts on a homework question:
Let $f_n\rightarrow f$ uniformly on compact subsets of an open connected set $\Omega \subset \mathbb{C}$, where $f_n$ is analytic, and $f$ is not identically equal to zero.
(a) Show that if $f(w)=0$ th... | Take a small circle around $w$.
Then by Rouché's theorem $f_n$ has a zero $z_n$ inside the circle for $n$ large enough (and maybe several if $w$ is a multiple zero of $f$).
Now shrink the circle and repeat: you will obtain the convergent sequence $(z_n)$.
By the way, this sketch of proof shows why we must assume th... | {
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Pathologies in module theory Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.
If one replaces the underlying field by a ring, an... | I am surprised that it is not mentioned here-
Example of a free module M which has bases having different cardinalities.
Let $V$ be a vector space of countably infinite dimension over a division ring $D$. Let $R=End_D(V)$. We know that $R$ is free over $R$ with basis $\{1\}$. We claim that given a positive integer $n$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
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How to show a quasi-compact, Hausdorff space be totally disconnected? This is from Atiyah-Macdonald. I was asked to show if every prime ideal of $A$ is maximal, then $A/R$ is absolutely flat, Spec($A$) is a $T_{1}$ space,further Spec($A$) is Hausdorff. The author then asked me to show Spec($A$) is totally disconnected.... | Okay, so first, there is a distinction made between quasicompactness and compactness. A topological space $X$ is quasicompact if every open cover of $X$ has a finite subcover. The topological space $X$ is said to be compact if it is quasicompact and Hausdorff.
We know the $Spec(A)$ is quasicompact for any ring $A$. If ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/201661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is it called when a function is not continuous but still can have a derivative? Consider the following function (I think it has a name, but I don't remember it):
$$
f(x) = \cases{-1 & $x < 0$ \\
0 & $x = 0$ \\
1 & $x > 0$}
$$
$f'(x)$ is zero everywhere except at $x=0$, where $f$ is not continuous. But suppose we i... | Interestingly, you can assign a derivative of the function $\operatorname{abs}$ at $0$ by using the following definition:
$$\frac{\mathrm df(x)}{\mathrm dx}=\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}.$$
Thus, taking the limit,
$$\operatorname{abs}'(x)=\frac{\mathrm d}{\mathrm dx}|x|=\operatorname{sgn}(x)=\cases{-1, & $x < 0;... | {
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Probability problem: cars on the road I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and ... | There are already two answers that show that under a certain interpretation of the question the answer is the $N$-th harmonic number. This can be seen more directly by noting that the $k$-th car is the "leader" of a group iff it is the slowest of the first $k$ cars, which occurs with probability $1/k$. Thus the expecte... | {
"language": "en",
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What is the probability of the box? Your box of cereal may be a contest winner! It's rattling, which 100% of winning boxes do. Of course 1% of all boxes rattle and only one box in a million is a winner. What is the probability that your box is a winner?
| The correct solution would be $0.0001$ ($1/10000$), wouldn't it? It's late, but it seems to me that Drew Christianson miscalculated and dedocu mixed $p(A)$ and $p(B)$ - correct me please, if I'm wrong.
| {
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"timestamp": "2023-03-29T00:00:00",
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Proof that a perfect set is uncountable There is something I don't understand about the proof that perfect sets are uncountable. The same proof is present in Rudin's Principles of Mathematical Analysis.
Do we assume that our construction of $U_n$ must contain all points of $S$? What if we are only collecting evenly-ind... | There is an alternative proof, using what is a consequence of Baire's Theorem:
THM Let $(M,d)$ be a complete metric space with no isolated points. Then $(M,d)$ is uncountable.
PROOF Assume $M$ is countable, and let $\{x_1,x_2,x_3,\dots\}$ be an enumeration of $M$. Since each singleton is closed, each $X_i=X\smallsetm... | {
"language": "en",
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Please explain how this ratio is being calculated A,B and C are partners of a company. A receives $\frac{x}{y}$ of profit. B and C share the remaining profit equally among them.
A's income increases by $I_a$ if overall profit increases from P% to Q%. How much A had invested in their company.
I know the answer: $\frac{I... | Let $A$ be the amount that Alicia has invested in the company. Let $\frac{x}{y}$ be the fraction of the company that she owns. So if $V$ is the total value of the company, then $A=\frac{x}{y}V$.
The old percentage profit was $P$. So the old profit was $\frac{P}{100}V$.
Alicia got the fraction $\frac{x}{y}$ of this, so ... | {
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Taylor's Tangent Approximation This is my question,
A function of 2 variable is given by,
$f(x,y) = e^{2x-3y}$
How to find tangent approximation to $f(0.244, 1.273)$ near $(0,0)?$
I need some guidance for this question.
Am i suppose to do the linear approximation or quadratic approximation?
Need some explanation for th... | More precise approximation we obtain if represent $f(x,\,y)$ as $$f(x,\,y)=e^{2x-3y}=e^3e^{2x-3y-3}=e^3e^{2x-3(y-1)}.$$ Then apply formula for tangent approximation to function $g(x,\,y)=e^{2x-3(y-1)}$
with $a=0; \,b=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/202046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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The drying water melon puzzle I couldn't find an explanation to this problem that I could understand.
A watermelon consist of 99% water and that water measures 2 litre. After a day in the sun the water melon dries up and now consist of 98% water. How much water is left in the water melon?
I know the answer is ~1 li... | At the beginning the solid material is $1\%$ of the total which is a trifle (to be neglected) more than $1\%$ of $99\%$ of the total, or $1\%$ of $2000\ {\rm cm}^3$. Therefore the solid material has volume $\sim20\ {\rm cm}^3$.
After one day in the sun these $20\ {\rm cm}^3$ solid material are still the same, but now t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/202095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Direct construction of Lebesgue measure I have seen two books for measure theory, viz, Rudin's, and Lieb and Loss, "Analysis".
Both use some kind of Riesz representation theorem machinery to construct Lebesgue measure.
Is there a more "direct" construction, and if so, what is a source?
| The most popular way is constructing it using the Caratheodory extension theorem, from Lebesgue outer measure. This approach is not very intuitive, but is a very powerful and general way for constructing measures.
An even more direct construction and essentially the one developed by Lebesgue himself defines Lebesgue me... | {
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"url": "https://math.stackexchange.com/questions/202145",
"timestamp": "2023-03-29T00:00:00",
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Zero polynomial
Possible Duplicate:
Polynomial of degree $-\infty$?
Today in Abstract Algebra my instructor briefly mentioned that sometimes the zero polynomial is defined to have degree $-\infty$. What contexts have caused this to become convention?
| Persistance.
You want formulas to make sense also when abusively applying them to cases involving the zero polynomial.
For example, we have $\deg(f\cdot g)=\deg f +\deg g$ and $\deg (f+g)\le \max\{\deg f, \deg g\}$. Therefore we assign a symbolic value - and be it only for mnemonic purposes - of $-\infty$ as the degree... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding error bounds for hermite interpolation I am unsure how to find the error bounds for Hermite interpolation. I have some kind of idea but I have a feeling that I am going wrong somewhere.
$f(x)=3xe^x-e^{2x}$ with my x-values being 1 and 1.05
My hermite interpolating polynomial is:
$H(x)=.7657893864+1.5313578773(x... | I think that should be $(x-1)(x-1.05)$ instead of $(x-1)^2(x-1.5)$
| {
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Why are Darboux integrals called Riemann integrals? As far as I have seen, the majority of modern introductory real analysis texts introduce Darboux integrals, not Riemann integrals. Indeed, many do not even mention Riemann integrals as they are actually defined (with Riemann sums as opposed to Darboux sums). However, ... | There are other examples, such as "An introduction to Real Analysis" by Wade. I don't know the history of these definitions at all. Once the dust settles over partitions, we have just one concept of integral left. The term "Riemann integral" is entrenched in so much of the literature that not using it isn't an option. ... | {
"language": "en",
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Measuring orderedness I've found this a frustrating topic to Google, and might have an entire field dedicated to it that I'm unaware of.
Given an permutation of consecutive integers, I would like a "score" (real [0:1]) that evaluates how in-order it is.
Clearly I could count the number of misplaced integers wrt the ord... | One book which treats metrics on permutations (that is, metrics on the symmetric group) is Persi Diaconis:"Group representations in probability and statistics"
which it is possible to download from here:
Link
| {
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Retraction of the Möbius strip to its boundary Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Möbius strip.
I've tried to find a contradiction using $r_*$ homomorphism between the fundamental groups, but they are both $\mathbb{Z... | If $\alpha\in\pi_1(\partial M)$ is a generator, its image $i_*(\alpha)\in\pi_1(M)$ under the inclusion $i:\partial M\to M$ is the square of an element of $\pi_1(M)$, so that if $r:M\to\partial M$ is a retraction, $\alpha=r_*i_*(\alpha)$ is also the square of an element of $\pi_1(\partial M)$. This is not so.
(For all t... | {
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"source": "stackexchange",
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How to find sum of quadratic I got this quadratic function from physics that I need to find the sum of each term, up to whatever point. Written thusly:
$$ \sum_{n=1}^{t}4.945n^2$$
And is there someway to quickly figure this out? Or links to tutorials
| There is the standard formula
$$\sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}.$$
It can be proved by a pretty routine induction.
| {
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Is there a largest "nested" prime number? There are some prime numbers which I will call "nested primes" that have a curious property: if the $n$ digit prime number $p$ is written out in base 10 notation $p=d_1d_2...d_n$, then the nested sequence formed by deleting the last digit one at a time consists entirely of prim... | From the comments in OEIS A024770
Primes in which repeatedly deleting the least significant digit gives a prime at every step until a single digit prime remains. The sequence ends at $a(83) = 73939133.$
| {
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"source": "stackexchange",
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Showing a vertical tangent exists at a given function. I want to apologise in advance for not having this in latex or some sort of neat code, I would be more than happy to learn how though.
Anyway, for the function $y=4(x-1)^{2/5}$ I see there appears to be a vertical tangent at $x=1$, but how can I know for certain th... | Yes, you would check if $f'(1)$ tends to $+\infty$ or $-\infty$
$$
\frac{d}{dx}4(x-1)^{2/5}=\frac{8}{5}(x-1)^{-3/5}\\
$$
| {
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What are Aleph numbers intuitively? I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
| The cardinals are the following ones:
$$0,1,2,3,4,5,6,\dots,\aleph_0,\aleph_1,\aleph_2, \aleph_3,\dots,\aleph_\omega,\aleph_{\omega+1},\dots,\aleph_{\omega2},\aleph_{\omega2+1},\dots $$
Where $\aleph_0$ is the first infinite cardinal (the cardinality of each infinite countable set), so $\aleph_0\notin\mathbb N$, is not... | {
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Solve a simultaneous equation. How do we solve $|b-y|=b+y-2\;and\;|b+y|=b+2$? I have tried to square them and factorize them but got confused by and and or conditions.
| $2\min (b,y)=b+y-|b-y|=2$ so that $\min (b,y)=1$. This implies that $b$ and $y$ are both positive so that $b+y=b+2$. Hence $y=2$ and $b=1$.
| {
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Locus perpendicular to a plane in $\mathcal{R}^4$ I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me..
In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the locus of Q points such that line PQ is perpendicular to $\pi$.
$$\pi:\beg... | It all seems good.
But, you jumped one step: how did you find your $v_1$ and $v_2$ spanner vectors of $\pi$?
| {
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Need a hint: show that a subset $E \subset \mathbb{R}$ with no limit points is at most countable. I'm stuck on the following real-analysis problem and could use a hint:
Consider $\mathbb{R}$ with the standard metric. Let $E \subset \mathbb{R}$ be a subset which has no limit points. Show that $E$ is at most countable.
... | proof by contradiction.
suppose E is uncountable. consider the set E intersection with [n,n+1] for integers n.
then there exist at least one n such that the above type of set is uncountable. clearly that set is bounded in real nos. then by Bolzano Weierstrass's theorem the above set has a limit point in R.
a contradic... | {
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What is the number of combinations of the solutions to $a+b+c=7$ in $\mathbb{N}$? My professor gave me this problem:
Find the number of combinations of the integer solutions to the equation $a+b+c=7$ using combinatorics.
Thank you.
UPDATE
Positive solutions
| Is this a hoax?
Perhaps I should put that differently. What institution are you studying at?
Was that the whole question? Or was there a part two asking the same thing but =702 or some other rather bigger number?
I ask these questions because the question as posed is absolutely trivial. The only tricky point is decidin... | {
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Determinant with unknown parameter. I'm given 4 vectors: $u_1, u_2, u_3$ and $u_4$. I'm going to type them in as points, because it will be easier to read, but think as them as column vectors.
$$u_1 =( 5, λ, λ, λ), \hspace{10pt} u_2 =( λ, 5, λ, λ), \hspace{10pt} u_3 =( λ, λ, 5, λ), \hspace{10pt}u_4 =( λ, λ, λ, 5)$$
Th... | If $\lambda=0$, the vectors are clearly linearly independent.
If $\lambda\ne0$, we can divide through by $\lambda$ without affecting whether the determinant vanishes; this yields
$$
\pmatrix{\frac5\lambda&1&1&1\\1&\frac5\lambda&1&1\\1&1&\frac5\lambda&1\\1&1&1&\frac5\lambda}\;.
$$
Thus the values of $\lambda$ for which ... | {
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Curves that intersect all exponential functions precisely once? Let $f(x)$ be a function on the positive real line. Suppose that for all nonnegative reals $A$, $Ae^{x}$ intersects $f(x)$ exactly once. Is there a simple description of the set of functions $f$ which satisfy this property?
| I presume that by a function on the positive real line you mean a function from the positive real line to the positive real line.
The condition is equivalent to $f(x)\mathrm e^{-x}$ taking every value $A$ exactly once. If $f$ is continuous, this is equivalent to $f(x)\mathrm e^{-x}$ either increasing monotonically with... | {
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Question regarding what appears to be an identity This is an MCQ we were posed in school recently (I hope you don't mind elementary stuff):
What is $(x-a)(x-b)(x-c)...(x-z)$ ?
Options:
$0$
$1$
$2$
$(x^n)-(abcdef...z)$
| Hint $\ $ What is $\rm\ (24-1)(24-2)(24-3)\cdots (24-26)\ $ ?
And what is $\rm\,(x_{24}\!-x_1)(x_{24}\!-x_2)(x_{24}\!-x_3)\cdots (x_{24}\!-x_{26})\ $ ?
| {
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Finding the maximum height a ball can be dropped from and still collide with a ball thrown below it I don't necessarily need a specific answer, but I could use a hint, direction, or maybe some reading material.
The question states:
A rubber ball is shot straight up from the ground with speed $V(0)$. Simultaneously, a ... | First, doublecheck your answer. Dimensionally, it doesn't make sense. $gh/v_0$ doesn't have dimensions of length.
Next, you find that they collide at some height $d$, and the problem requires that $d\ge 0$. This inequality is solved for certain values of $h$.
| {
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"question_score": "2",
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Where is the difference between the union and sum of sets? My book writes:
Definition. A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1 \cap W_2=\{0\}$ and $W_1 + W_2 = V$. We denote that $V$ is the direct sum of $W_1$ and $W_2$ by writing $V=W_1\op... | Take this example to clarify the difference: $$V=\mathbb{R}^{2}$$ $$W_{1}=sp_{\mathbb{R}}\{(1,0)\}=\{(a,0)|a\in\mathbb{R}\}$$ $$W_{2}=sp_{\mathbb{R}}\{(0,1)\}=\{(0,b)|b\in\mathbb{R}\}$$
Then,
$$W_{1}+W_{2}=\{w_{1}+w_{2}|w_{i}\in W_{i}\}=\{(a,0)+(0,b)|a,b\in\mathbb{R}\}=\{(a,b)|a,b\in\mathbb{R}\}$$
but,
$$W_{1}\cup W_{2... | {
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Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$ Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
| Use $\sqrt{1\pm i\sqrt 3}=\sqrt{2}e^{\pm i\pi/6}$ (EDIT we are picking the principal branch here) to get
$$
\sqrt{2}\left( e^{i\pi/6}+e^{-i\pi/6}\right)=2\sqrt{2}\cos(\pi /6)=2\sqrt{2}\frac{\sqrt{3}}2=\sqrt{6}
$$
| {
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"source": "stackexchange",
"question_score": "15",
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Is this function Lipschitz? Let $f:X \rightarrow \mathbb R$ be a Lipschitz function on a metric space $X$ and $K<M$ be some constants.
Is it such a function $g:X\rightarrow \mathbb R$ Lipschitz:
$$
g(x)=f(x) \textrm{ if } \ K \leq f(x) \leq M,
$$
$$
g(x)=K \textrm { if } \ f(x)<K,
$$
$$
g(x)=M \textrm{ if } \ f(x)>M... | You can write your function as $\min(\max(f(x), K), M)$. Note that the composition of Lipschitz functions is Lipschitz. So you just have to show that the functions $\min(x,K)$ and $\max(x,M)$ are Lipschitz on ${\mathbb R}$, which is not hard.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Writing "$\nabla f$" or "$\operatorname{grad} f$" When hand-writing the gradient of $f$ as "$\nabla f$" or "grad $f$", is it necessary to indicate that it is a vector using the usual vector markings (cap, arrow, wavy line, etc.)?
| It should be considered obligatory to write, for example $\vec{a}$ or $\mathbf{a}$, when you're writing in a context in which vectors and vector-valued functions are generally written that way. But that is not always done. The style should be consistent throughout the document.
However, notice that the $f$ in $\nabl... | {
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Stuck with solving a polynomial I am doing a problem for homework that says:
Suppose $s(x)=3x^3-2$. Write the expression $\frac{s(2+x)-s(2)}{x}$ as a sum of terms, each of which is a constant times power of $x$.
I was able to do the following work for this problem:
$\frac{3(2+x)^3-3(2)^3-2}{x}$
$\frac{3(x^3+6x^2+12x+8)... | You went astray at the first step, when you got $$\frac{3(2+x)^3-3(2)^3-2}{x}\;;$$ in fact
$$s(2+x)-s(2)=\Big(3(2+x)^3-2\Big)-\Big(3\cdot2^3-2\Big)=3(2+x)^3-3\cdot2^3\;.$$
Can you straighten the rest out from there? When you do it correctly, there will be no constant term in the numerator.
| {
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Proving that $x^3 +1=15x$ has at most three solutions. in the interval [-4,4]. I need someone to check my work. Thanks! This is a 2 mark homework question by the way. I am not sure why am I using such a long way to prove it. Is there a way to shorten it or is there a shorter, more intuitive method?
Proving that $x^3 +1... | Degree 3 polynomials have exactly 3 roots, some of which could be complex. If it had more than 3 solutions in your interval, you get a contradiction with the fundamental theorem of algebra.
| {
"language": "en",
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How many bits are in factorial? I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values.
In other words I need integers A and B so that A <= Log(N!) <= B
| Expanding on joriki's answer, taking more terms from the approximation,
$$ \log (n!) = n(\log n - \log e) + \frac{1}{2}\log n + \log \sqrt{2\pi} + \frac{\log e}{C_ n}, \quad 12n < C_n < 12n+1. $$
The number of binary digits is equal to $\lceil \log n! \rceil$, and for most $n$, I expect that the slight uncertainty in $... | {
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Solving for x with exponents (algebra) So I am trying to help a friend do her homework and I am a bit stuck.
$$8x+3 = 3x^2$$
I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for $x$ in this situation.
Could someone be so kind as to break down the steps in solving... | This is a quadratic equation: the highest power of the unknown is $2$. Rearrange it to bring everything to one side of the equation:
$$3x^2-8x-3=0\;.$$
If you can easily factor the resulting expression, you can take a shortcut, but otherwise you either complete the square or use the quadratic formula.
Completing the sq... | {
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Why does this equation have different number of answers? I have a simple equation:
$$\frac{x}{x-3} - \frac{2}{x-1} = \frac{4}{x^2-4x+3}$$
By looking at it, one can easily see that $x \not= 1$ because that would cause $\frac{2}{x-1} $ to become $\frac{2}{0}$, which is illegal.
However, if you do some magic with it. Firs... | If $\dfrac AB = 0$ then $A=0\cdot B$. But you can't say that if $A=0\cdot B$ then $\dfrac AB=0$ unless you know that $B\ne 0$. So if $A$ and $B$ are complicated expressions that can be solved for $x$, there may be values of $x$ that make $B$ equal to $0$, and if they also make $A$ equal to $0$, then they are solution... | {
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Cycling Digits puzzle I'm trying to answer the following:
"I have in mind a number which, when you remove the units digit and place it at the front, gives the same result as multiplying the original number by $2$. Am I telling the truth?"
I think the answer to that is no. It's easy to prove that it's false for numbers ... | If you follow your argument but let $N=a+10b$ where $a$ is a single digit but let $b$ have $n$ digits, then $2N=10^na+b$ and you get $b=\frac {10^n-2}{19}a$ If $n=17$, this is integral. Then $a$ has to be at least $2$ to make $b$ have $17$ digits. The smallest solution is $105263157894736842$
Another way to get ther... | {
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Find arc length of a circle using a hyperbolic metric
Given the hyperbolic metric $ds^2=\frac{dx^2+dy^2}{x^2}$ on the half plane $x > 0$, find the length of the arc of the circle $x^2+y^2=1$ from $(\cos\alpha,\sin\alpha)$ to $(\cos \beta, \sin\beta)$
I found that $ds^2=\displaystyle\frac{d\theta^2}{\cos^2\theta}$ but... | The circle $x^2 + y^2 = 1$ can be parametrised by $(\cos \theta, \sin \theta)$. If $x(\theta) = \cos \theta$ and $y(\theta) = \sin \theta$ then
$$ds^2 = \frac{dx^2+dy^2}{x^2} = \frac{(\sin^2\theta+\cos^2\theta) \, d\theta^2}{\cos^2\theta} = \sec^2\theta \, d\theta^2.$$
The arc-length that you are interested in is given... | {
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sets and functions proof help I started to get excited about mathematical analysis. So I bought a mathematical analysis book and started to study. But because of the reason that, book do not have solutions I do not have an idea where to start and how to prove the following:
Let $A_t$, $t \in T$, be a family of sets, an... | Let's look at the case there $|T| = 1$, i.e. that the family of sets has only one element.
The $X\setminus A$ is trivially $X \setminus A$.
What about when $|T| = 2$?
Then $X \setminus (A_1 \cup A_2)$ is the set of $x \in X$ such that $x \not \in A_1, A_2$.
$X \setminus A_1$ is the set of $x \in X$ such that $x\not \i... | {
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Polynomials and factoring in $\mathbb{Z}[x]$
Show that any polynomial $p(x) \in \mathbb{Q}[x]$ can be written as $p(x) = tq(x)$ where $t \in \mathbb{Q}$ and $q(x) \in \mathbb{Z}[x]$ is primitive.
I started my proof by defining $p(x)$ as $(\frac{q}{r})_n x^n + \dots + (\frac{q}{r})_0$. Then I defined $t \in \mathbb{Q... | You presumably brought the coefficients of $p(x)$ to some common denominator $r$, where $r$ is an integer. This can certainly be done.
So now the coefficients in the numerator are integers, say $b_n$ down to $b_0$. Let $d$ be the gcd of all of these, and let $c_i=b_i/d$. Then $p(x)$ is $\frac{d}{r}$ times the primitive... | {
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How does this prove AM-GM? Here is an extract from A less than B. The author claims to imply the AM-GM inequality with this reasoning, but I can't see how. So far the author has covered AM-GM, convexity, the smoothing principle and Jensen's inequality.
"Theorem 4: Let $f$ be a twice-differentiable function on an open i... | Notice that $\ln{x}$ is defined only for $x>0$, which means that its second-order derivative is always negative, which means the function is concave.
Let $f(x)=-\ln{x}$. We know that $f$ is convex. Thus, writing the Jensen's inequality for $f$ with the weights $t_{i}=\frac{1}{n}$ ($i=1,2,...,n$), we get:
$$-\ln\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Holomorphic functions as sums Are there any holomorphic functions on a connected domain in $\mathbb C$ that can not be written as a sum of two univalent (holomorphic and injective) functions? What about as a sum of finitely many univalent functions? Or even infinitely many?
| There is a growth obstruction for finite sum representation. Indeed, a theorem of Prawitz (1927) says that every univalent function on the unit disk belongs to the Hardy space $H^p$ for all $p<1/2$. Consequently, $f(z)=(1-z)^{-q}$ is not a finite sum of univalent functions when $q>2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/204428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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Generating Series for the set of all compositions which have an even number of parts.
I'm having trouble showing that the generating series for all compositions which have an even number of parts.
I'm given that each part congruent to 1 mod 5 is equal to:$$\frac{1-2x^5+x^{10}}{1-x^2-2x^5+x^{10}}$$
If you could help ... | Here's another method. First, you should convince yourself that the generating function for compositions with $k$ parts is given by
$$(x + x^2 + x^3 + \ldots)^k.$$ This is because choosing a composition $k_1 + k_2 + \ldots + k_m$ corresponds to choosing $x^{k_1}$ in the first factor, $x^{k_2}$ in the second, and so o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Chain Rule of Partial Derivatives If $f$ is a differentiable function defined $f: \mathbb{R}^2 \to \mathbb{R}$ and $f(2,1) = 3$ and $\nabla f(2,1) = (4,3)$, find $\nabla G(1,1)$ with $G(x,y) := x^2yf(x^2 +y^2, xy)$.
I wrote the $G_x$ as $2xyf(x^2 +y^2, xy) + x^2yf'(x^2 + y^2, xy)f_x(x^2 + y^2, xy)$ but I don't know wha... | Denote
$$g(x,y)=x^{2}y$$
Then by product rule
$$\nabla G=f\nabla g+g\nabla f$$
$$\nabla g=2xy\boldsymbol{i}+x^{2}\boldsymbol{j}$$
Now write
$$\begin{cases}
x^{2}+y^{2} & =2\\
xy & =1
\end{cases}$$
Multiply the second equation by 2, add and subtract from the first one obtaining respectively
$$\left(x+y\right)^{2}=4$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Complex analysis integration question Let $f(z) = A_0 + A_1z + A_2z^2 + \ldots + A_nz^n$ be a complex polynomial of degree $n > 0$.
Show that $\frac{1}{2\pi i} \int\limits_{|z|=R} \! z^{n-1} |f(z)|^2 dz = A_0 \bar{A_n}R^{2n}$.
| Let $\Gamma = \{z: |z| = R\}$. Recall that
$$ \int_{\Gamma} z^k \, dz = \begin{cases} 0 & k \neq -1 \\ 2\pi i & k = -1 \end{cases}$$
Now, when we multiply out $|f(z)|^2$ in terms of $z$ and $\overline{z}$, we are ultimately evaluating an integral of the following form:
$$ \frac{1}{2\pi i} \int_{\Gamma} \sum_j B_j z^{p_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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a geometry problem Let $DA$ be the normal on the plane of the triangle $ABC$ and $E \in (DA)$. Let's notate with $M,N,P,Q$ the proiections of the point $A$ to the lines $BD$, $CD$, $BE$, respectively $CE$.
Prove that:
1) $$MN\cap BC \cap PQ \neq \emptyset;$$
2) the quadrilateral $MNPQ$ is inscriptible.
| For 2), consider the circumscribed circle of the $ABC$ triangle, and expand it to a sphere in 3d with the same center and radius. If we cut it by a plane orthogonal to the $ABC$ plane and containing $AB$, we get a circle with $AB$ as its diameter, so, by Thales' thm the points $M$ and $P$ will be on that circle, hence ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to approximate $y=\frac{W(e^{cx+d})}{W(e^{ax+b})}$? How to approximate
$$y=\frac{W(e^{cx+d})}{W(e^{ax+b})}$$
with (a) simple function(s)?
given $a=-1/\lambda_0$, $b=(\mu_0+\lambda_0)/\lambda_0$, $c=1/\lambda_1$, $d=(\mu_1+\lambda_1-1)/\lambda_1$ for positive $\mu_0,\lambda_0,\mu_1,\lambda_1$
where $W$ is a Lamber... | You could try to estimate $W(x)$ by using Newton-Raphson iteration, because $W(c)$ is the root of $x\exp(x)-c$:
$$x_{n+1}= x_n-\frac{f(x_n)}{f'(x_n)} = x_n-\frac{x_n \exp (x_n)-c}{\exp(x_n)(x_n+1)} = \frac{c\exp(-x_n)+x_n^2}{x_n+1}$$
and as $n$ is sufficiently large, we can get approximations for $W$ using only element... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Estimate for the product of primes less than n In this paper Erdős shows a shorter proof for one of his old results stating that $$ s(n) = \prod_{p < n} p < 4^n$$ where the product is taken over all primes less than $n$. He also remarks that using the prime number theorem one can show $$ s(n)^{\frac1n} \stackrel{n\to\i... | The reason the sum
$$ \sum_{k = 2}^{n} \frac{k}{\log k} $$
works as an estimate of the sum of all primes up to $n$ is because, roughly speaking, one on $\log N$ numbers of size around $N$ are prime. You are estimating
The sum of all primes of a given size
with the approximation
The sum of all numbers of that size, m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Trying to prove that $p$ prime divides $\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$ So I'm trying to prove that for any natural number $1\leq k<p$, that $p$ prime divides:
$$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$$
Writing these choice functions in factorial form, I obtain:
$... | Letting $k=p-1$, we find that the expression equals $p$ summands of value $1$ each.
Maybe the problem was not meant to read "for some natural" but rather "for all natural" numbers $1\le k<p$.
The statement remains true, once you observer that the sum is simply $p\choose k$.
To see this combinatorially, note that you c... | {
"language": "en",
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"source": "stackexchange",
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Finding the range of rational functions I have a problem that I cannot figure out how to do. The problem is:
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?
I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have tried to find the inverse of fu... | To find the range, we want to find all $y$ for which there exists an $x$ such that
$$ y = \frac{x+2}{x^2+5}.$$
We can solve this equation for $x$:
$$ y x^2 + 5y = x+2$$
$$ 0 = y x^2 -x + 5y-2$$
If $y \neq 0$, this is a quadratic equation in $x$, so we can solve it with the quadratic formula:
$$
x = \frac{ 1 \pm \sqrt{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Bounded sequences and lim inf Let $a_n$ and $b_n$ be bounded sequences. Prove that lim inf $a_n$ + lim inf $b_n \leq$ lim inf$(a_n + b_n)$
I have no idea where to begin.
| Start by showing that for all $n$, $$\inf_{k\geq n}a_k+\inf_{k\geq n}b_k\leq\inf_{k\geq n}(a_k+b_k),$$ then take the limit as $n\to\infty$.
| {
"language": "en",
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area of a convex quadrilateral I have a quadrilateral with sides as follows: $30, 20, 30, 15.$
I do not have any other information about the quadrilateral apart from this.
Is it possible to calculate its area?
| A quadrilateral with sides $30,20,30,15?$ two sides are equal, right? Why don't you try to draw it? Divide it into two triangles. If the two equal sides have a common edge, one of the triangles is isosceles, i.e. has equal angles. Can you find the rest of the angles and the area?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A function which is in $L^1$ but does not belong to $L^\infty$ Can someone give me an example of an $L ^1$ function which does not belong to $L^\infty$. In fact we look at $L^1(\Omega,\mathcal{F},P)$, where $(\Omega,\mathcal{F},P)$ denotes a probability space. Of course the function should be unbounded but the integral... | An example for the space $X= (0,1]$ is $\dfrac{1}{\sqrt{x}},$ which has infinite $L^{\infty}$ norm because as $x\to 0^+$ the function $1/\sqrt{x}\to \infty,$ but has $L^1$ norm of $\displaystyle \int^1_0 \frac{1}{\sqrt{x}} dx = 2.$
If you want an example for $X=\mathbb{R}$ then $1_{(0,1]} \dfrac{1}{\sqrt{x}}$ works si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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elementary negation I couldn't be sure about the negation of the statement $\exists x \in N , x < 3$
is it
$$
\lnot(\exists x \in N , x < 3) \equiv (\forall x \in N , x \geq 3)
$$
or
$$
\lnot (\exists x \in N , x < 3) \equiv (\forall x \notin N , x \geq 3)
$$
can someone help me (with an explanation)?
| just to be sure, to me your statement reads as "There exists an $x$, element of the positive integers (doesn't matter with or withour $0$), such that $x$ is less than $3$. If this is correct, then the negation is: There is no $x$ element of the positive integers such that $x$ is less than $3$. This means that all $x$ e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate the following expectation There are $K$ items indexed $X_1, X_2, \ldots, X_K$ in the pool. Person A first randomly take $K_A$ out of these $K$ items and put them back to the pool. Person B then randomly take $K_B$ out of these $K$ items. What is the expectation of items that was picked by B but not taken by A... | André's solution is the best one, of course.
But for the sheer fun of it, let's calculate the sum
\begin{equation}
E = \sum_{i=1}^{K_B} i \frac{{{K}\choose{K_A}}{{K_A}\choose{K_B - i}}{{K - K_A}\choose{i}}}{{{K}\choose{K_A}}{{K}\choose{K_B}}}
\end{equation}
First, cancel the common factor
$$E = \sum_{i} i \frac{{K_A\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this category essentially small? Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in this category is a linear map such that $\phi_W f = \phi_V f$. Note this category is th... | Jordan normal form tells you what the isomorphism classes look like, but you don't need to know this: it suffices to show that the collection of isomorphism classes with a fixed value of $\dim V$ forms a set, and this is straightforward as specifying the corresponding $\phi_V$ requires at most $(\dim V)^2$ parameters. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Heat equation with initial values $U(0,t)=U_1$, $U(L,t)=U_2$,$\forall t$. My problem is given as
Arbitrary temperatures at ends . If the ends $x=0$
and $x=L$ of the bar in the text are kept at constant
temperatures $U_1$ and $U_2$ respectively, what is the temperature
$u_1(x)$ in the bar after a long time (theoreticall... | Let $u(x,t)=X(x)T(t)$ ,
Then $X(x)T'(t)=c^2X''(x)T(t)$
$\dfrac{T'(t)}{c^2T(t)}=\dfrac{X''(x)}{X(x)}=-\dfrac{\pi^2s^2}{L^2}$
$\begin{cases}\dfrac{T'(t)}{T(t)}=-\dfrac{\pi^2c^2s^2}{L^2}\\X''(x)+\dfrac{\pi^2s^2}{L^2}X(x)=0\end{cases}$
$\begin{cases}T(t)=c_3(s)e^{-\frac{\pi^2c^2ts^2}{L^2}}\\X(x)=\begin{cases}c_1(s)\sin\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of Cauchy Riemann Equations in Polar Coordinates How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives?
I haven't found any proof of this online.
One of my ideas was writing ou... | We can derive using purely polar coordinates. Start with
\begin{align}
z(r,\theta) &=r\,\mathrm{e}^{\mathrm{i}\theta} \\
f(z) &= u(r,\theta) + \mathrm{i} v(r,\theta)
\end{align}
We define $f'(z)$ using the limit
$$ f'(z) = \lim_{z\to 0} \frac{\Delta f}{\Delta z} $$
where
\begin{align}
\Delta f &= \Delta u + \mathrm{i} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Real Analysis (Riemann Integral) Let $f(x)=c$ for all $x$ in $[a,b]$ and some real number $c$. Show by definition below that $f$ is Riemann integrable on $[a,b]$, and $\int f(x) dx = c(b-a)$.
Definition: A function $f$ is Riemann integrable on $[a,b]$ if there is a real number $R$ such that for any $\varepsilon > 0$, ... | Let $R = c (b-a)$ and let $\varepsilon > 0$.
You want to show that there is $\delta > 0$ such that for all tagged partitions $P$ with $\|P\| < \delta$ you have
$$ \left | \sum_{k=1}^n f(x_i) (t_{i+1}-t_i) - R \right | < \varepsilon$$
where $x_i \in [t_i , t_{i+1}] \subset [a,b]$ form a tagged partition of $[a,b]$. We ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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escape velocity using limits I have the formula for a rocket's escape velocity from earth, $V$ being velocity, $v$ being initial velocity, and $r$ being the distance between the rocket and the center of the earth.
$$V = \sqrt{\frac{192000}{r}+v^2-48}$$
I am trying to find the value of $v$ for which an infinite limit f... | Let $r\to\infty$. Note that $\dfrac{192000}{r}\,$ approaches $0$. Thus since
$$V=\sqrt{\frac{192000}{r}+v^2-48},$$
$V$ approaches $\sqrt{v^2-48}$. If we want $V$ to approach $0$, we want $v^2-48=0$.
(Presumably we are measuring velocity in miles per second.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/205833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find the Circular permutation with Repetition
Possible Duplicate:
In how many ways we can put $r$ distinct objects into $n$ baskets?
Need some guidance with the following problem :
There are 'n' different types of objects which needs to be placed in a circle of length 'r' , such that no two adjacent items ar... | Yes, it is.
There is a good article about combinations and variations in codeproject.
You need for "Combinations (i.e., without Repetition)" there.
Also if you familar with C# you can use simple and short solution from stackoverflow.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/205890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you find the center of a circle with a pencil and a book? Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
| The only book I have is usually under my pillow. It has become skewed over the years - no 90 degrees joy. As a consequence I failed to have success applying the (elegant) description of Patrick Li. Moreover, my book is too small anyway to connect diametrically opposing points on the circle.
Therefore I had to revert to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "101",
"answer_count": 17,
"answer_id": 1
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The Hungarian Algorithm In reading the proof of the Hungarian algorithm for the assignment problem in a weighted bigraph, I could not understand why the algorithm terminates. In the algorithm we choose a cover (namely labels for the vertices of the bigraph: $u_1\cdots u_n$, $v_1,\cdots v_n$ with $u_i+v_j\ge $weight of ... | Let $\mathrm{M}$ be a maximum matching in the equality subgraph. According to König's theorem, we can find a vertex cover $C$ such that $\mathrm{|M|}=|C|$. We Call $R$ the set $C\cap X$, and $T$ the set $C\cap Y$.
Now, either $\mathrm{M}$ is a perfect matching, which means the algorithm should stop and return our curre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Visualizing Exterior Derivative How do you visualize the exterior derivative of differential forms?
I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I imagine two vectors, constituting a parallelogram.
So I think provided I can imagine a fiel... | I usually think of differential forms as things "dual" to lines, surfaces, etc.
Here I picture forms in a 3-dimensional space. The generalization is obvious, with little care.
A foliation of the space (think of the sedimentary rocks) is always a 1-form. Not all 1-forms are foliations, but they can always be written as ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "18",
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understanding the language of a theorem The theorem is stated as follows in the book:
Let $\phi:G\rightarrow G'$ be a group homomorphism, and let
$H=Ker(\phi)$. Let $a\in G$. Then the set
$\phi^{-1}[\{\phi(a)\}] = \{x\in G | \phi(x)=\phi(a)\}$
is the left coset $aH$ of $H$, and is also the right coset $Ha$ of
$H$... | The condition is that $H$ is the kernel of a group homomorphism, not just any random subgroup.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/206138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can all Hermitian matrices $H$ be written as $H=A^* A$? All the matrices below are square, complex matrices.
1) Is it true that, for every Hermitian matrix $H$, there exists $A$, that $A^*A=H$?
2) For any $A$, does $A^*A$ always have a square root? If it's not, is there any simple presumption of $A$ that makes $A^*A$ a... | *
*This property is only true for matrices $H$ such that $x^*Hx\geq 0$ for all $x$ (non-negative definite). To see that, note that if it's the case $x^*A^*Ax=\lVert Ax\rVert^2\geq 0$. Conversely, if $x^*Hx\geq 0$ for all $x$, then we can diagonalize $H$, and each eigenvalue is non-negative (hence admits a square root)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Embeddings of bundles in projective space. Consider the projective variety $X = \mathbb{P}^2$, and the line bundle $\mathcal{O}_X(dH)$ where $H$ is a plane and $d \in \mathbb{N}$.
Let $L$ be the total space of $\mathcal{O}_X(dH)$. I know how to form this in the following way: Let $U_i$ be the standard affine opens of $... | If $E$ is a vector bundle of rank $r$ on a projective variety $X$, then the total space of $E$ is a quasi-projective variety : this is a vast generalization of what you are asking.
And yet the proof is easy: the vector bundle $E$ has a projective completion $\bar E=\mathbb P(E\oplus \theta)$ where $\theta=X\times \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Time & Work Problem 12 men do a work in 36 days.In how many days can 18 men do the same work?
Sol:This can be solved as
1 man's can do work in 12*36 days
So,1 man's 1 day work (1/12*36) days
Hence 18 men's 1 day work is (18/12*36)
So Days taken for 18 men is (12*36)/18 days.
Similarly Here is another question where i ... | 4 men need 24 days
8 women need 24 days
one man needs 96 days [4*24]
one woman needs 192 days [8*24]
so 12 men and 8 women need 6 days [192 / (96 / 12 + 192 / 8)]
==> 192 / ((4 * 24) / 12 + (8 * 24) / 8) = 6
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/206399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Are inversion and multiplicaton open? If $G$ is a topological group, are inversion $G \to G$ and multiplication $G\times G \to G$ open mappings? More concretely, I try to show that division of complex numbers
$$\{(z,w) \in \mathbb{C}^2;\; w \neq 0\} \to \mathbb{C},\; (z,w) \mapsto \tfrac{z}{w}$$
is an open mapping. I w... | Yes, open maps. The inversion $g\mapsto g^{-1}$ is in fact a homeomorphism, continuous by def. and its inverse is itself.
For the product, if $U,V\subseteq G$ open subsets, then the image of $(U,V)$ under multiplication is the complex product
$$U\cdot V =\{u\cdot v\mid u\in U,v\in V\} =\bigcup_{u\in U}(u\cdot V)$$
is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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how to find arrange the following functions in increasing or decreasing order? I have the following three functions
$f_1(x) = \frac{1}{4} (8-3x + \sqrt{(x-2) (5x-14)}) (1-x)$
$f_2(x) = \frac{1}{8} (12-4x + \sqrt{2} \sqrt{(5x-14)(x-3)} + \sqrt{2} \sqrt{(x-2)(x-3)} )(1-x)$
$f_3(x) = (x-1)(x-2)$
How possible can it be sh... | If you are interested as $x \to \infty$, you can just look for the highest power of $x$. For $f_1(x)$, the square root goes as $\sqrt 5x$, so the whole thing goes as $(3-\sqrt 5)x^2$. You can do similarly with the second and the third is $x^2$. This says that as $x \to \infty, f_3(x) \gt f_1(x)$
If you are intereste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove $\ell_1$ is first category in $\ell_2$ Prove that $\ell_1$ is first category in $\ell_2$.
I tried to solve this, but had no idea about the approach. Any suggestions are helpful.
Thanks in advance.
| Write $F_n:=\{x\in \ell^2,\sum_{j=1}^{+\infty}|x_j|\leq n\}$. Then $\ell^1=\bigcup_{n\geq 1}F_n$. $F_n$ is closed in $\ell^2$, as if $\{x^{(k)}\}$ is a sequence which lies in $F_n$ and converges to $x$ in $\ell^2$; we have for an integer $N$ that
$$\sum_{j=1}^N|x_j|\leq\lim_{k\to\infty}\sum_{j=0}^N|x_j^{(k)}|\leq n,$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
ZFC set theory,first order theory
Possible Duplicate:
What is the difference between Gödel's Completeness and Incompleteness Theorems?
what is the relationship between ZFC and first-order logic?
I am a bit confused by a few things that I have read recently.
I have read that ZFC is a first order theory and that any ... | As in the comments is said, the word 'complete' has 2 different meanings.
That the first order logic is complete, is meant that it is complete w.r.t the corresponding first order models, that is: a formula is valid in all models iff it has a proof (a deduction consisting of finitely many formulas, using some specific d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/206693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove $\{0,1\}^* -\{0^i 1^i\mid i \ge 0\}$ is context free? Is the only way to prove that this language is context-free to construct a Context-Free Grammar that accepts it?
If so any hints on how to get started?
| What do you think about the following? Does it work?
$$S\to M1X$$
$$S\to X0M$$
$$M\to 0M1$$
$$M\to \Lambda$$
$$X\to 1X$$
$$X\to 0X$$
$$X\to \Lambda$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/206763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What's the probability that Abe will win the dice game? Abe and Bill are playing a game. A die is rolled each turn.
If the die lands 1 or 2, then Abe wins.
If the die lands 3, 4, or 5, then Bill wins.
If the die lands 6, another turn occurs.
What's the probability that Abe will win the game?
I think that the probabili... | You are right. You can just ignore rolls of $6$ as they leave you back in the same situation. To formalize this, the chance Abe wins on turn $n$ is $\frac 13 \left(\frac 16 \right)^{n-1}$ and the chance that Bill wins on turn $n$ is $\frac 12 \left(\frac 16 \right)^{n-1}$. You can sum these if you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/206829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 2
} |
$A = B^2$ for which matrix $A$? Is it true that for any $A\in M(n,\mathbb{C})$there exist a $B\in M(n,\mathbb{C})$ such that $A = B^2$? I think this is not true (but I don't know nay example), and then is it possible to characterize such $A$?
| The Cholesky decomposition is loosely related to the concept of taking the square root of a matrix. If $A$ is a positive-definite Hermitian matrix, then $$ A = B B^{*} $$, where $B$ is lower triangular with positive diangonal elements where $B^*$ is the conjugate transpose of B.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/207029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Do Lipschitz-continuous funcions have weak derivatives on bounded open sets? Let $\Omega\in\mathbb{R}^n$ be open and bounded. I'm wondering if a function
$f\in C^{0,1}(\Omega)$ (a Lipschitz-continuous one) is also an element of $W^{1,2}(\Omega)$ (that is the space of weakly derivatives functions whose first weak deriv... | Take a look in the page 279 of this book: "Evans - Partial Differential Equation".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/207078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Asymptotic equivalence of exponents An earlier question links to a paper of Erdos in which he says that it is "well-known" that the Prime Number Theorem is equivalent to
$(\prod_{p\leq n}p)^{1/n} \to e$ as $n\to \infty.$ **
Here is my confusion.
If $~\prod_{p\leq n}p \sim e^n$ or $e^{\log \prod p}= e^{\sum \log p} \s... | $(\Pi_{ p \leq n} p)^{1/n} \rightarrow e$, as $n \rightarrow \infty$, doesn't imply $\Pi_{p \leq n} p \sim e^n$.
Example: $(2^n n)^{1/n} \rightarrow 2$, as $n\rightarrow \infty$, but $2^n n$ is not equivalent to $2^n$.
Indeed, $n^{1/n}=e^{\frac{\log n}{n}} \rightarrow 1$ because $\frac{\log n}{n} \rightarrow 0$ as $n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$L^2$-Oscillation Let $f:[0,1]\to \mathbb{R}$ be a smooth function such that the following property is satisfied.
$$\int\limits_{[0,1]}\int\limits_{[0,1]}|f(x)-f(y)|^2dxdy\leq \varepsilon.$$
What can I most say about $\max\limits_{[0,1]}f-\min\limits_{[0,1]}f$?
| If you choose $f_n(x) = x^n$, with $n$ a positive integer, a quick computation shows that $\int_{[0,1]} \int_{[0,1]} |f_n(x)-f_n(y)|^2 \, dx dy = \frac{2n^2}{(n+1)^2(2n+1)}$. Furthermore, $\max_{x\in [0,1]} f_n(x) - \min_{x\in [0,1]} f_n(x) = 1$ for all $n$.
Hence the integral can be made arbitrarily small, yet the ran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
For integers $a$ and $b \gt 0$, and $n^2$ a sum of two square integers, does this strategy find the largest integer $x | x^2 \lt n^2(a^2 + b^2)$? Here is some background information on the problem I am trying to solve. I start with the following equation:
$n^2(a^2 + b^2) = x^2 + y^2$, where $n, a, b, x, y \in \mathbb ... | Let $N = n^2(a^2+b^2)$ and consider its prime factorization.
Let $p$ be a prime divisor of $N$.
If $p\equiv 3\pmod4$, then necessarily $p^2|N$, $p|x$ and $p|y$ (and the factor comes from $p|n$).
If $p\equiv 1\pmod 4$, find $u,v$ such that $u^2+v^2=p$.
Let $k$ be the exponet of $p$ in $N$, i.e. $p^k||N$.
Then $x+iy$ mus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit of a continuous function Suppose that $f$ is a continuous and real function on $[0,\infty]$. How can we show that if $\lim_{n\rightarrow\infty}(f(na))=0$ for all $a>0$ then $\lim_{x\rightarrow+\infty} f(x)=0$?
| $\newcommand{\orb}{\operatorname{orb}}$If $f(x)\not\to 0$ as $x\to\infty$, then there is an $\epsilon>0$ such that for every $m\in\Bbb N$ there is an $x_m\ge m$ such that $|f(x_m)|\ge\epsilon$. Since $f$ is continuous, for each $m\in\Bbb N$ there is a $\delta_m>0$ such that $|f(x)|>\frac{\epsilon}2$ for all $x\in(x_m-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculating CRC code I think I may be under a misconception. When calculating the CRC code, how many bits do you append to the original message? Is it the degree of the generator polynomial (e.g. x^3+1 you append three 0s) or is it the number of digits used to represent the generator polynomial (e.g. x^3+1 gives 1001 w... | Calculating the CRC does not itself append any bits to the message. It gives you some output bits that you can afterwards chose to do with as you want.
(Sometimes what you want to do with the output is to use it to compute some bits to append to the message such that the CRC of the augmented message happens to be all z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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