Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Circle bisecting the circumference of another circle If the circle $x^2+y^2+4x+22y+l=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-m=0$,then $l+m$ is equal to
(A)$\ 60$
(B)$\ 50$
(C)$\ 46$
(D)$\ 40$
I don't know the condition when one circle intersects the circumference of other circle. So could not solve t... | Circle $C_1: x^2+y^2+4x+22y+l=0$ has its center $(-2, -11)$ & a radius $\sqrt{(-2)^2+(-11)^2-l}=\sqrt{125-l}$
Similarly, circle $C_2: x^2+y^2-2x+8y-m=0$ has its center $(1, -4)$ & a radius $\sqrt{(1)^2+(-4)^2-(-m)}=\sqrt{m+17}$
Now, solving the equations of circles $C_1$ & $C_2$ by substituting the value of $(x^2+y^... | {
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A basic root numbers question If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
| Notice, we have $$\sqrt{x^2+5} - \sqrt{x^2-3} = 2\tag 1$$ let $$\sqrt{x^2+5} + \sqrt{x^2-3} = y\tag 2$$ Now, multiplying both (1) & (2), we get $$(\sqrt{x^2+5} - \sqrt{x^2-3} )(\sqrt{x^2+5} + \sqrt{x^2-3})=2y$$ $$(\sqrt{x^2+5})^2-(\sqrt{x^2-3})^2=2y$$ $$x^2+5-(x^2-3)=2y$$ $$8=2y\implies y=4$$ Hence, $$\bbox[5px, borde... | {
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find the complex number $z^4$
Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$.
Find $c+d$.
I got that the distance is:
$$|z^3|\cdot|z^2 - (1 + 2i)| = 125|z^2 - (1 + 2i)|$$
So I need to maximize the distance... | Let $\theta$ denote the argument of $z$, then the arguments of $(1+2i)z^3$ and $z^5$ are respectively $3\theta +\arctan(2)$ and $5\theta$. Now imagine that these points lie on concentric circles, so in order to maximize the distance, the difference between the arguments must be $\pi$ (or $-\pi$ it won't matter):
$$5\t... | {
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Dedekind Construction Of Real Numbers If we define Dedekind-real numbers as Dedekind cuts, i.e. $\sqrt 2 = \{\text{rationals less than }\sqrt2\} \cup \{\text{rationals more than } \sqrt2\}$, can we define addition and multiplication of these real numbers as follows:
These real numbers $\mathbb R$ are a complete lattic... | The idea of Dedikind cuts is that one can define $\mathbb{R}$ and its arithmetic operations and order in an elementary manner, directly in terms of $\mathbb{Q}$ and its the arithmetic operations and order. One doesn't need the concepts of analysis or topology to do this. Of course, after the real numbers and their oper... | {
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Sorting triangles by hypotenuse length I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time consuming operation. As I understand, I can easily omit square root for sorting purpose... | No.
Consider $x_1-x_2=4$, $y_1-y_2=4$ then $(x_1-x_2)^2+(y_1-y_2)^2=32$ and $|x_1-x_2|+|y_1-y_2|=8$. But $x_1-x_2=6$, $y_1-y_2=1$ then $(x_1-x_2)^2+(y_1-y_2)^2=37$ and $|x_1-x_2|+|y_1-y_2|=7$. So this would give the wrong comparison.
| {
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Trying to make sense of this proof in Hatcher So I'm trying to understand this proof in Hatcher's Algebraic Topology.
Lemma: The composition $\Delta_n(X)\xrightarrow{\partial_n} \Delta_{n-1}(X)\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)$ is zero
where here $\partial_n$ is a boundary homomorphism $$\partial_n(\sigma_{\a... | When taking the second boundary map, one can not remove the $ i $th term since it was already removed by the first boundary map. So $ j \neq i $ in the sum, and so it is convenient to split up the sum as
$$ \sum_{\substack{ j \\ j \neq i } } \cdot = \sum_{\substack{ j \\ j < i } } \cdot + \sum_{\substack{ j \\ j > i }... | {
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Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true? I would like to know what are the mathematical reasons for which Bernhard Riemann believed that his hypothesis is true, and I would like to know if those mathematical reasons were cited in his original paper.
My question H... | Harold Edwards provides some informed speculation on this question in his book. I'm not home right now but will come back and summarize his thoughts in a couple days.
And by the way his book is a gem. It includes a translation of the original paper, and one of the more enjoyable and satisfying couple months of my "mat... | {
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What is the variance of the volumes of particles? According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger particles hinder the movements of the smaller ones. Thus, it is importa... | I'm going to use units of hundredths of a centimeter, then convert back to centimeters.
The diameter of a given particle is modeled as a random variable $$D \sim \operatorname{Uniform}(1,5);$$ namely, $$f_D(x) = \begin{cases} \frac{1}{4}, & 1 \le x \le 5 \\ 0, & \text{otherwise}. \end{cases}$$ If $$V = \frac{4}{3}\pi ... | {
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determination of probability Recently I have faced a problem related to probability.I have tried it by bayseian theorem but failed.Here is the problem:
A doctor knows that pneumonia causes a fever 95% of the time. She knows that if a person is selected
randomly from the pneumonia. 1 in 100 people suffer from fever. Yo... | There are many ambiguities in the question itself and the question can be interpreted in two ways.
*
*At first it says that "pneumonia causes a fever 95% of the time" and later it is being contradictory/(false in terms of stats) by saying " if a person is selected randomly from the pneumonia. 1 in 100 people suffer... | {
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Convexity under diffeomorphisms Let $K \subset \mathbb{R}^n$ be a compact convex subset with non-empty interior, and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism. Then is it true that $f[K]$ is convex?
| I'm afraid that in general it's not true. In $\mathbb{R}^2$ you can construct a diffeomorphism $(x, y) \mapsto (x + hy^2, y), \; h > 0$ and apply it to unit square (or unit circle). The "top part" of this figure will be stretched much stronger than parts are near to the $Ox$ axis, which will lead to fail of convexity. ... | {
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If $g$ is a permutation, then what does $g(12)$ mean? In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5):
Prove that exactly half of the $n!$ permutations in $S_n$ are even.
(Hint: Show that if $g$ is an even permutation, then $g(12)$ is odd. Try to use this to def... | It is the composition of the permutation $g$ and the transposition $(1\ 2)$. Since $g$ is even and all transpositions are odd, it is elementary that $g (1\ 2) = g \circ (1\ 2)$ is odd, as the product of an even and odd permutation is odd.
(There is a space between $1$ and $2$; this is not the number $12$.)
Edit: To als... | {
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Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$ Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Prove that $$S = x \sin^2 A + y \sin^2 B + z \sin^2 C \leq \dfrac{\left(yz+zx+xy\right)^2}{4xyz}$$
and find when it holds equality
| Update
This is an intricate problem, since not only the value of $S_{\max}$ is changing with the values of the parameters $x$, $y$, $z$, but also the global extremal situation.
Claim: If $0<2x<y\leq z$ then $S_{\max}=y+z$.
Proof: If one of the angles is $>{\pi\over2}$, e.g., $\alpha\leq\beta<{\pi\over2}<\gamma$, it is... | {
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Sizes of Quotient Rings of DVRs with Finite Residue Field If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are finite. My question is:
Can we say anything about the sizes o... | We have a short exact sequence: $$0\to\mathfrak m/\mathfrak m^2\to R/\mathfrak m^2\to R/\mathfrak m \to 0.$$ Since $\dim_{R/\mathfrak m}\mathfrak m/\mathfrak m^2=1$ we get $\#(\mathfrak m/\mathfrak m^2)=\#(R/\mathfrak{m})$. It follows that $\#(R/\mathfrak m^2)=\#(R/\mathfrak{m})^2$.
Can you continue from here?
| {
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Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$ I have faced difficulties while trying to prove that
$$\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$$
I don't have any clue how can I start to work with it. Any hint will be helpful.
| I am coming to this party about three years too late, but the answers posted here, and some here as well: enter link description here, may leave readers with the false impression that the vector Laplacian acting on a vector field (appearing on the right-hand side) always takes the form
$$\begin{align}
\nabla^2 \vec{A} ... | {
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Finding the work from $(0,0)\to(1,1)$ of $\vec F(x,y)=xy^2\hat i+yx^2\hat j$
I need to find the work from $(0,0)\to(1,0)\to(1,1)$ of the following vector field:$\vec F(x,y)=xy^2\hat i+yx^2\hat j$
My attempt:
$$\oint_{c}\vec F d\vec r=\int_{(0,0)\to (1,0)}\bigg(xy^2\; dx +yx^2 \; dy\bigg)+\int_{(1,1)\nwarrow(1,0)}\bi... | METHOD 1:
We can simplify the problem by noting that $\nabla \times \vec F=0$. Therefore, $\vec F$ is therefore conservative on any connected domain. By Stokes' Theorem, the integral of $\vec F$ over any contour $C$ that bounds a connected domain $S$ is
$$\oint_C \vec F\cdot d\vec \ell =\int_S \nabla \times \vec F\c... | {
"language": "en",
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Prove that if $\forall A \in \mathcal F (B\subseteq A)$ then $B \subseteq \bigcap \mathcal F $ This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $.
My approach so far:
Suppose ... | Basically yes, but you should try to be a bit more methodical to obtain a 100% correct proof. Here are the steps to take (compare with what you've actually done).
1) Suppose $x\in B$. The aim is to show that $x\in \bigcap \mathcal F$.
2) To show $x$ is in an intersection, it suffices to show that $x\in A$ for all $A\... | {
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Area of a triangle with sides $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$,$\sqrt{z^2+x^2}$ Sides of a triangle ABC are $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$ and $\sqrt{z^2+x^2}$ where x,y,z are non-zero real numbers,then area of triangle ABC is
(A)$\frac{1}{2}\sqrt{x^2y^2+y^2z^2+z^2x^2}$
(B)$\frac{1}{2}(x^2+y^2+z^2)$
(C)$\frac{1}{2}... | hint: Let $a = \sqrt{x^2+y^2}, b = \sqrt{y^2+z^2}, c = \sqrt{z^2+x^2} \to a^2 = x^2+y^2, b^2 = y^2+z^2, c^2= z^2+x^2 $. Use this and Cosine Law to find $\cos^2 A$, then $\sin^2 A$, and use $S^2 = \dfrac{b^2c^2\sin^2 A}{4}$, to find $S^2$ and then take square-root to get back $S$.
| {
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Can functions within a matrix adjust its size? I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix:
\begin{bmatrix}G\begin{pmatrix}1\\0\\0\end{pmatrix}&G\begin{pmatrix}0\\1\\0\end{pmatrix}&G\begin{pma... | That is mostly a question of notation and convention. You can certainly choose to define that in your work, the notation
$$ A = \begin{bmatrix}X & Y & Z\end{bmatrix} $$
where $X$, $Y$ and $Z$ are column matrices of the same height, will mean that $A$ is the $3\times n$ matrix that has those three columns. As a matter o... | {
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Geodesic connectivity implies geodesic convexity? Assume a subset $C$ of a Riemannian manifold $M$ is "geodesically-connected", that is: given any two points in $C$, there is a geodesic contained within $C$ that joins those two points.
Is it true that $C$ must be geodesically convex?
(given any two points in $C$, there... | The answer is negative.
Take the sphere $S^2$ with the round metric, and cut out the part which is souther than some latitude line which is close to the south pole. (The threshold line also stays out).
We are left with an open Riemannian submanifold $M$.
Any two points in $M$ will be connected via a geodesic. For some... | {
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integrating product of PDF and CDF I am trying to show that the following integral:
$$
\int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!}
$$
Where $F$ is the cumulative distribution function of some continuous random variable X, and $f$ is the probability density function. Not quite sure how this conclusion was reached?
| Clearly $F'(x)=f(x)$ (wherever CDF is continuous).
Substitute $F(x)=t$. Then $dt=f(x)dx$.
$$\int_{-\infty}^a F(x)~f(x)~dx \rightarrow \int_{0}^{F(a)} t~dt= \frac{(F(a))^2}{2}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Iterative calculation of $\log x$ Suppose one is given an initial approximation of $\log x$, $y_0$, so that:
$$y_0 = \log x + \epsilon \approx \log x$$
Here, all that is known about $x$ is that $x>1$. Is there a general method of improving that estimation using only addition & multiplication, i.e. without exponentiatio... | Instead of solving $y - \ln(x) = 0$ for $y$ you can solve $g(y) = e^y - x = 0$
Given initial approximation $y_0 \approx \ln(x)$ you can try to solve $y$ using Newton's medhod:
$$y_{n + 1} = y_n - \frac{g(y_n)}{g'(y_n)}= y_n - \frac{e^{y_n} - x}{e^{y_n}} = y_n - 1 +\frac{x}{e^{y_n}}$$
When exponentiation isn't allowed y... | {
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Minimum value of reciprocal squares I am bit stuck at a question. The question is : given: $x + y = 1$, $x$ and $y$ both are positive numbers. What will be the minimum value of: $$\left(x + \frac{1}{x}\right)^2 + \left(y+\frac{1}{y}\right) ^2$$
I know placing $x = y$ will give the right solution. Is there any other ... | $(x+1+y/x)^2+ (y+1+x/y)^2 >= 1/2 (x+1+y/x+y+1+x/y)^2 >= 1/2(2+1+x/y+y/x)^2 >=25/2 $
Equality holds when $x/y=y/x$ or $x=y$
| {
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"timestamp": "2023-03-29T00:00:00",
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Name for continuous maps satisfying $\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U)$ I have recently come across particularly kind of continuous maps $f
\colon X \to Y$ between topological spaces with the property that
$$
\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U),
$$
for all $U \in \wp(X)$.
If $X$ is ... | Look at the fibers. Your condition implies that $f^{-1}(y) ⊆ cl(\{x\})$ for every $x ∈ f^{-1}(y)$, i.e. each fiber is indiscrete. So, such mapping is injective even if $X$ is $T_0$. Conversely, if each fiber is indiscrete, then the closure contains the “fiber-closure”.
Every continuous map is a unique composition of a ... | {
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Whether the set $A$ , $ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges to } +\infty \}$ is connected Let $f: \mathbb R \rightarrow \mathbb R$ be continuous function and $A\subset \mathbb R$ be defined by $$ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where ... | As has been sort of said, $x\sin(x)$ is a counterexample to (B) and to (C).
But (A) is true. (And this is really awful notation, using "A" for a certain set and also for a condition that set may or may not satisfy).
It's enough to show that if $a,c\in A$ and $a<b<c$ then $b\in A$. Say $x_n\to\infty$, $f(x_n)\to a$, $z_... | {
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Permutations of the elements of $\mathbb Z_p$ Let $p$ be prime. Describe all permutations $\sigma$ of the elements of $\mathbb Z_p$, having the property that $\{\sigma(i)-i: i\in\mathbb Z_p\}=\mathbb Z_p$
(Added by Robert Lewis in an attempt to provide background, motivation, and other context for this engaging proble... | These are known as complete mappings of the finite field $\mathbb{F}_{p}$. I think you will find that it is quite a difficult problem to describe all such permutations.
Here is a good overview:
Niederreiter, Harald; Robinson, Karl H., Complete mappings of finite fields, J. Aust. Math. Soc., Ser. A 33, 197-212 (1982). Z... | {
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Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$
Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$.
I know that maybe by using Characteristic function of $S_n$ I will be able to get the d.f. of $S_n$ using Inversion Theorem but I d... | Note that the vector $(X_1,X_2,...,X_n)$ has density $f(x_1,x_2,...,x_n)=1$ on the unit hypercube. So $\mathbf{P}(S_n\leq 1)$ is just the volume of the region bounded by the axis planes and the plane $x_1+x_2+\cdots+x_n=1$ and equals $1/{n!}$.
| {
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Limit of Fraction $$\lim_{x \to \infty} \frac{(1 + x)^{x/(1 + x)}\cos^{4}x}{e^{x}}$$
Attempt: I've tried evaluating the limits of the terms individually using the property of limits. Also, $y=1/x$ subsituition hasn't helped me, any help will be appreciated.
| Hint: For positive $x$, the top is $\ge 0$ and $\le 1+x$.
Now use Squeezing.
| {
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Continued Fraction Counting Problem The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him.
There is more than fifty houses on that side of the street, but not so many as fiv... | If it is house number $x$ in a street of $y$ houses, we have
$$\frac{x(x-1)}{2}+x+\frac{x(x-1)}{2}=\frac{y(y+1)}{2}$$
which simplifes to
$$(2y+1)^2-8x^2=1\ .$$
This can be solved by computing the continued fraction
$$\sqrt8=2+\frac{1}{1+{}}\frac{1}{4+}\frac{1}{1+{}}\frac{1}{4+\cdots}\ .$$
The table of convergents is
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1382629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$ I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that
$$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$
I started following the path of $2$ on the right:
$$2\to3\to4\to5\to \ ?$$
Where does $5$ go? I should stop here, right? ... | This is another conjugation problem in disguise:
$(2\ 3)(3\ 4)(4\ 5)(4\ 3)(3\ 2) = (2\ 3)[(3\ 4)(4\ 5)(3\ 4)^{-1}](3\ 2)$
$= (2\ 3)(3\ 5)(2\ 3)^{-1}$ (since $(3\ 4)$ takes $4 \to 3$ and fixes $5$)
$= (2\ 5)$ (since $(2\ 3)$ takes $3 \to 2$ and fixes $5$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1382683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Consider the function f(x)=sin(x) in the interval x=[π/4,7π/4]. The number and location(s) of the local minima of this function are? This is MCQ of a competitive exam(GATE), Answer is (d) given by GATE , and from other sources ,explanation is (b) somewhere and (d) somewhere , I am going with (b) as minimum at $270$, I ... | The local minima is at $x=\frac{3\pi}{2}$
This is very obvious from the graph of $f(x)=\sin{x}$
On a second look at the graph below, I believe $x=\frac{\pi}{4}$ is also a local minimum. This is because it is lesser than all other values within its locality.
Thus we have two local minima: $x=\frac{\pi}{4}, \frac{3\pi}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1382765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Does $A\cap B =\varnothing \Rightarrow B\subseteq \overline{A}$? How to prove $A\cap B =\varnothing \Rightarrow B\subseteq \overline{A}$?
If I going by definitions, there is no $x$ s.t $x\in A$ and $x\in B$.
But, what do we can tell about $\overline{A}$? What i'm missing?
| Let's assume $A\cap B =\varnothing$ (start hypothesis)
Let $x \in B$
Since $A$ and $B$ are disjoint (start hypothesis), then $x \notin A$
By definition of $\overline A$, since $x \notin A$ then $x \in \overline A ~~~~(= \Omega - A)$
Therefore $B\subseteq \overline A$, because for all $x \in B$, we have $x \in \over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1382863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Given vector $\vec x = \left\{ x_i\right\}_{i=1}^n$ find an algebraic expression for $\vec y = \left\{ x^2_i\right\}_{i=1}^n$ Given vector
$$\vec x = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},$$
How can we write out vector
$$\vec y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} := \begin{bmatrix} x^2_... | $\begin{pmatrix}
x_1 & 0 & 0\\
0 & x_2 & 0\\
0 & 0 & x_3\\
\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}x_1^2\\x_2²\\x_3^2\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1382955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Use stirlings approximation to prove inequality. I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't prove it to myself but I think it is true from checking values with Math... | For
$\left(\frac{e}{2}\right)^{3-2/p} \frac{1}{(p-1/2)^{2p-1}}
\le 1
$,
$\begin{array}\\
(p-1/2)^{2p-1}
&=p^{2p-1}(1-1/(2p))^{2p-1}\\
&\approx p^{2p-1}(1/e)(1-1/(2p))^{-1}
\quad\text{since }(1-1/(2p))^{2p} \approx 1/e\\
\end{array}
$
so
$\begin{align*}\\
\left(\frac{e}{2}\right)^{3-2/p} \frac{1}{(p-1/2)^{2p-1}}
&\app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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What is the solution of $a^b=a+b$ in terms of $a$?
Let $a, b$ be real numbers. Solve
$$a^b=a+b$$
for $a$.
If there isn't a solution with $a, b$ real, maybe $a, b$ should be complex.
But no matter how hard I try, this is proving to be very difficult to
do. Would anyway be kind enough to show me the solution to thi... | I will use Lambert-W function defined as the following:
$$W(x)e^{W(x)}=x$$
Your equation:
$$a^b=a+b$$
Multiply it by $a^a$ to make the exponent more "friendly" and use some tricks
$$a^{a+b}=(a+b)a^a$$
$$\frac{1}{a+b}a^{a+b}=a^a$$
$$(a+b)a^{-(a+b)}=a^{-a}$$
$$-(a+b)a^{-(a+b)}=-a^{-a}$$
$$-\ln(a)(a+b)e^{-\ln(a)(a+b)}=-\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$
Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\... | $$\langle f,f \rangle=\iint_Df(z)\times\overline{f(z)}dxdy=\iint_D |f(z)|^2dxdy \leqslant 3\pi$$
Power series of $f(z)$ about $z=0$:
$$\sum_{n=0}^{\infty}a_nz^n=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^n$$
then you can easily prove (using polar coordinates, edit: or more easily by defining a base using $z^n$ then usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Math Subject GRE 1268 Question 55 If $a$ and $b$ are positive numbers, what is the value of $\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. | The integral being considered is, and is evaluated as, the following.
\begin{align}
I &= \int_{0}^{\infty} \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx \\
&= \int_{0}^{\infty} \frac{dx}{1 + e^{bx}} - \int_{0}^{\infty} \frac{dx}{1 + e^{ax}} \\
&= \left( \frac{1}{b} - \frac{1}{a} \right) \, \int_{1}^{\infty} \frac{dt}{t(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 7,
"answer_id": 4
} |
How can the axiom of choice be called "axiom" if it is false in Cohen's model? From what I know, Cohen constructed a model that satisfies $ ZF\neg C $.
But if such a model exists, how can AC be an axiom? Wouldn't it be a contradiction to the existence of the model?
Only explanation I can think of, is that this model re... | The existence of a model of a statement does not mean that statement is "true" (whatever that means; see below). For example, the Poincare disk is a model of Euclid's first four postulates plus the negation of the parallel postulate; this does not mean that the parallel postulate is "false."
What having a model of a se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Derivative of sum of powers For fixed $n \geq 1$ and $p \in [0,1]$, is there a nice expression for the derivative of
$\sum_{k=0}^n p^k (1-p)^{n-k}$
with respect to p?
| I believe you can use the fact that $\displaystyle \dfrac{\text{d}}{\text{d}p} \sum \left( \dots \right) = \sum \dfrac{\text{d}}{\text{d}p} \left( \dots \right) $. $$ \begin{aligned} \dfrac{\text{d}}{\text{d}p} \sum_{k=0}^{n} p^k (1-p)^{n-k} & = \sum_{k=0}^{n} \dfrac{\text{d}}{\text{d}p} \left( p^k (1-p)^{n-k} \right) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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$z^3=w^3 \implies z=w$? I've reached this in another problem I have to solve: $z,w \in \Bbb {C}$. $z^3=w^3 \implies z=w$?
I've scratched my head quite a bit, but I completely forgot how to do this, I don't know if this is correct:
$$
z^3=|z^3|e^{3ix}=|w^3|e^{3iy}
$$
I know the absolute values are equal, so I get $3ix... | We assume $z \ne 0 \ne w$; it is obvious that in the other circumstance $z = w = 0$ is the only possible solution.
Now, since
$z^3 = w^3 \Leftrightarrow z^3 - w^3 = 0, \tag{1}$
then
$(z - w)(z^2 + zw + w^2) = z^3 - w^3 = 0; \tag{2}$
we see that if
$z^2 + zw + w^2 \ne 0, \tag{3}$
then
$z = w; \tag{4}$
furthermore, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Angle between two parabolas I'm a little confused about a problem that asks me to find the angle between the two parabolas $$y^2=2px-p^2$$ and $$y^2=p^2-2px$$ at their intersection. I used implicit differentiation to find the slopes $$y'=\frac{-p}{y}$$ and $$y'=\frac{p}{y}$$ Using the formula for the angle between two... | First, let's find the point of intersection. This occurs when
$$p^2-2px_0=2px_0-p^2\implies x_0=p/2\implies y=0$$
At the point of intersection, $y=0$ and thus $yy'=\pm p$ implies that $y'$ is undefined and therefore the tangent to both parabolas is a vertical line in the $x-y$ plane defined be the equation $x=p/2$.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Existence of solutions of the equation with a limit. Let f be continuous function on [0,1] and
$$\lim_{x→0} \frac{f(x + \frac13) + f(x + \frac23)}{x}=1$$
Prove that exist $x_{0}\in[0,1]$ which satisfies equation $f(x_{0})=0$
I suppouse that the numerator should approach $0$ which would implicate that for
x near $0$ $f... | First, prove that the numerator should be zero. You can do this by contradiction. Prove that if $$\lim_{x\to 0} f(x+\frac13) + f(x+\frac23)\neq 0$$
then the original limit cannot exist.
Now, you can use continuity to show that $f(\frac23) + f(\frac13)=0$, and then use a well known theorem to finish your proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Determining if function odd or even This exercise on the Khan Academy requires you to determine whether the following function is odd or even
f(x) = $-5x^5 - 2x - 2x^3$
To answer the question, the instructor goes through the following process
*
*what is f(-x)
*f(-x) = $-5(-x)^5 - 2(-x) - 2(-x)^3$
*f(-x) = $5(x)^5 ... | Let's follow the simplest process
Notice, $$f(x)=-5x^5-2x-2x^3$$ $$f(-x)=-5(-x)^5-2(-x)-2(-x)^3$$ $$ =5x^5+2x+2x^3$$ $$\implies f(x)+f(-x)=-5x^5-2x-2x^3+5x^5+2x+2x^3=0$$
Hence, $f(x)$ is odd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Can someone help me give a proof for this? I know there are theorems about integrals of odd and even functions, but i kept wondering about integrals that share symmetry around an axis $x=c$. I've been trying to give a proof for this but can't seem to get around it; could someone help me prove/disprove this?
$$
\large \... | Two initial remarks:
*
*For clarity, you should not use the same letter in the limits of integragion and as the integration variable itself.
*I assume your hypothesis is that, for all $x$, $f(c-x)=f(c+x)$.
Let us prove that
$$
\large \int_{c-x}^{c+x}f(t)dt=2\int_{c}^{c+x}f(t)dt
$$
Proof:
First, let us consider:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"Mean value like" problem. Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad \frac{f(b')-f(a')}{b'-a'}=f'(c').$$ My first tries were connected with mean value because we can find such $c,c'$ but we do... | Consider the function
$$
g(x)=f(x)-{f(b)-f(a)\over b-a}(x-b).
$$
It is easy to prove that $g(a)=g(b)$ and $g'( c)=0$. Moreover, if
$\displaystyle{g(x_2)-g(x_1)\over x_2-x_1}=g'(\xi)$ then also
$\displaystyle{f(x_2)-f(x_1)\over x_2-x_1}=f'(\xi)$. So we can prove the theorem for $g$ and the result will hold for $f$ too.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Can a space $X$ be homeomorphic to its twofold product with itself, $X \times X$? Let $X$ be a topological space of infinite cardinality. Is it possible for any $X$ to be homeomorphic to $X\times X$ $?$
For example, $\mathbb R$ is not homeomorphic to $\mathbb R^{2}$, and $S^{1}$ is not homeomorp... | At this level of generality you can make $X=X \times X$ happen quite easily. Take a discrete space of any infinite cardinality, for instance. Or topologize $X=A^B$ by whatever means and compare $X \times X = A^{B \sqcup B}$; under various mild assumptions on $B$ those spaces would be homeomorphic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Proving an Inequality using a Different Method Is there another way to prove that: If $a,b\geq 0$ and $x,y>0$
$$\frac{a^2}{x} + \frac{b^2}{y} \ge \frac{(a+b)^2}{x+y}$$
using a different method than clearing denominators and reducing to $(ay-bx)^2 \ge 0$?
| One method of solving any inequality is to determine points where the associated equality is true or where the functions involved are not continuous. That is because, as long as $a$ and $b$ are continuous, we can only go from "$a> b$" to "$a< b$", or vice-versa, by going through "$a= b$".
However here, the simplest wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Sums of Fourth Powers While fooling around on my calculator I found:
$$7^4 + 8^4 + (7 + 8)^4 = 2 * 13^4$$
$$11^4 + 24^4 + (11 + 24)^4 = 2 * 31^4$$
I'm intrigued but I can't explain why these two equations are true. Are these coincidences or is there a formula/theorem explaining them?
| You have a disguised version of triangles with integer sides and one $120^\circ$ angle. These are
$$ 3,5,7 $$
$$ 7,8,13 $$
$$ 5,16,19$$
$$ 11,24, 31, $$
which solve
$$ a^2 + ab + b^2 = c^2. $$
Square both sides and then double both sides and you get your identities.
These can be generated by a coprime pair of number $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m. Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m.
I attempted to solve by letting them both equal each other, but it didn't work. How do I do this question?
Th... | So the two functions are $y=-\frac{1}{4}x^2$ and $y=mx+m^2$.
Setting them equal, we have $-\frac{1}{4}x^2=mx+m^2$
which rearranges to $0=x^2+4mx+4m^2$
then you can use the quadratic formula (or simply complete the square) to get the result. I suspect that this is where you are supposed to use the discriminant, since in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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sum of the residues of all the isolated simgualrities Prove that, for $n \geq 3$, the sum of the residues of all the isolated singularities of
$$\frac{z^n}{1+z+z^2+\cdots+z^{n-1}}$$
is 0
Can someone show me how to do this problem. Thank you.
| Let
$$
F(z)=\frac{z^n}{1+z+z^2+\ldots+z^{n-1}}=\frac{P(z)}{Q(z)}.
$$
Since $Q(1)=n\ne 0$, then, for every $z\ne 1$ we have
$$
Q(z)=\frac{1-z^n}{1-z},
$$
and $F$ can be redefined as
$$
F(z)=\begin{cases}
\frac{(z-1)z^n}{z^n-1} &\mbox{ for } z\ne 1\\
\frac1n &\mbox{ for } z=1
\end{cases}
$$
Therefore, the set of isolated... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Calculate 3D Vector out of two angles and vector length What is the easiest way to calculate vector coordinates in 3D given 2 angles vector length?
Input:
*
*Angle between X and Y axis: $$\alpha \in [0, 360).$$
*Angle between Y and Z axis: $$\beta\in [0, 360).$$
*Scalar length.
Expected output:
X, Y, Z coordinat... | You have presented a point in spherical coordinate system, which needs to be converted to Cartesian. The line joining the origin to the point will be the vector. Let the scalar length be $r$. If the $\beta$ angle is measured from $Y$ axis towards $Z$ axis and $\alpha$ from $X$ axis towards $Y$:
$$x = r \cos \beta sin \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1385137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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does simply connectedness require connectedness? My question consists of two parts.
$1)$
suppose domain $D=\{(x,y)\in\mathbb R^2~|~xy>0\}$ is given. Now that is first quadrant and third quadrant with exclusion of $x$ and $y$ axis. We can easily see that $D$ is not connected, since there is a discontinuity at origin. Bu... | According to the Wikipedia, that definition of simply connectedness excludes all the non path connected spaces.
The injectivity of the closed curves is not mentioned and not required.
Your second example thus becomes simply connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1385232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Find $\lim\limits_{x\to 0}\frac{\sqrt{2(2-x)}(1-\sqrt{1-x^2})}{\sqrt{1-x}(2-\sqrt{4-x^2})}$ I use L'Hospitals rule, but can't get the correct limit.
Derivative of numerator in function is
$$\frac{-3x^2+4x-\sqrt{1-x^2}+1}{\sqrt{(4-2x)(1-x^2)}}$$
and derivative of denominator is
$$\frac{-3x^2+2x-2\sqrt{4-x^2}+4}{2\sqrt{(... | using Bernoulli $$x \to 0 \\ {\color{Red}{(1+ax)^n \approx 1+anx} } \\\sqrt{1-x^2} = (1-x^2)^{\frac{1}{2}} \approx 1-\frac{1}{2}x^2 \\ \sqrt{4-x^2}=\sqrt{4(1-\frac{x^2}{4}})=2(1-\frac{x^2}{4})^{\frac{1}{2}} \approx 2(1-\frac{1}{2} \frac{x^2}{4})=2-\frac{x^2}{4} $$ so by putting them in limit :
$$\lim_{x \to 0} \frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1385320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
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last two digits of $14^{5532}$? This is a exam question, something related to network security, I have no clue how to solve this!
Last two digits of $7^4$ and $3^{20}$ is $01$, what is the last two digits of $14^{5532}$?
| ${\rm mod}\,\ \color{#c00}{25}\!:\, \ 14\equiv 8^{\large 2}\Rightarrow\, 14^{\large 10}\equiv \overbrace{8^{\large 20}\equiv 1}^{\rm\large Euler\ \phi}\,\Rightarrow\, \color{#0a0}{14^{\large 1530}}\equiv\color{#c00}{\bf 1}$
${\rm mod}\ 100\!:\,\ 14^{\large 2}\, \color{#0a0}{14^{\large 1530}} \equiv 14^{\large 2} (\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1385396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
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How many bit strings of length 10 contains... I have a problem on my home work for applied discrete math
How many bit strings of length 10 contain
A) exactly 4 1s
the answer in the book is 210
I solve it
$$C(10,4) = \frac{10!}{4!(10-4)!} = 210
$$
for the last 3 though I can't even get close. Did I even do part A right... | For part A it will be simply $C(10,4)=210$
For part B it will be $C(10,0)+C(10,1)+C(10,3)+C(10,4)=386$ because at most means maximum which we actually start from zero till that particular number
For part C it will be simply $C(10,4)+C(10,5)+C(10,6)+C(10,7)+C(10,8)+C(10,9)+C(10,10)=848$
For part D it will be simply $C(1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Radical under Radical expression how to find the sum of $\sqrt{\frac54 + \sqrt{\frac32}} + \sqrt{\frac54 - \sqrt{\frac32}} $ ? Is there a method to solve these kind of equations ?
| Why does squaring work here? Well it obviously deals with part of the square root. But there are two other things going on.
The first is that $$\left(\sqrt{a+\sqrt b}\right)^2+\left(\sqrt{a-\sqrt b}\right)^2=a+\sqrt b+a-\sqrt b=2a$$
And this means that the inner square roots disappear in part of the square.
The second ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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If the product of two matrices, $A$ and $B$ is zero matrix, prove that matrices $A$ and $B$ don't have to be zero matrices I can give an example where product of two non-zero matrices is zero matrix,
$$
A=
\begin{bmatrix}
3 & 6 \\
2 & 4 \\
\end{bmatrix}
$$
$$
B=
\begin{bmatrix}... | whilst exhibiting a single value $a$ for which $P(a)$ is true is sufficient to establish the existential claim $\exists x\cdot P(x)$, i sympathize with OP's feeling that a single case doesn't give us much grasp of the why's and wherefores of the presence of zero divisors in matrix rings. there is, after all, considerab... | {
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"url": "https://math.stackexchange.com/questions/1385713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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Poisson Conditional Probability - Very lost! A discrete random variable is said to have a Poisson distribution if its possible values are the non-negative integers and if, for any non-negative integer $k$,
$$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$$
where $\lambda>0$. It turns out that $E(X)=\lambda$.
Minitab has a cal... | The distribution of the conditional probability is $$f_{X\mid W}(x\mid w)f_W(w)=f_{X,W}(x,w).$$
The probability that $X=5$ and $W=10$ is the same as the probability that $X=5$ and $Y=5$, so the joint probability on the right is the same as $f_{X,Y}(x,y)$, where $Y=W-X$. (Explicitly, this is a change of coordinates, but... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Distribution of sum of random variables Let $X_1, X_2, . . .$ be independent exponential random variables with mean $1/\mu$ and let $N$ be
a discrete random variable with
$P(N = k) = (1 − p)p^{k-1}$ for $k = 1, 2, . . . $
where $0 ≤ p < 1$ (i.e. $N$ is a shifted geometric random variable). Show that $S$ defined as $S ... | I would calculate the characteristic function :
$$E[ e^{itS} ] = E\left[ \prod_{i=1^N} e^{itX_1} \right]$$
$$= \sum_{k=1}^\infty (1-p)p^k \prod_{i=1}^k \int_{\mathbb{R}^+} e^{itx} \mu e^{-\mu x} dx$$
$$= \sum_{k=1}^\infty (1-p)p^{k-1} \prod_{i=1}^k \frac{\mu}{\mu-it}$$
$$ =\frac{(1-p)}{p}\sum_{k=1}^\infty \left( \frac... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Are there infinitely many pythagorean triples? I believe these questions are all asking different things, but:
*
*Are there infinitely many (integer) solutions to the pythagorean theorem?
*Is every positive integer part of a solution to the pythagorean theorem?
Also, is there a difference in multiplying the pytha... | That should be the easiest one:
Let n be a factor of every Integer in a pythagorean triple:
n* 3sq + n* 4sq = n* 5sq
There are infinitely many positive integers and so also infinitely many n´s. Now we just need to show that the pythagorean triples are all correct with the factor n:
n* 3sq + n* 4sq = n* 5sq (fac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Obtaining a $3$-dimensional simple random walk from a $d$-dimensional simple random walk with $d>3$. Suppose $S_n$ is a $d$-dimensional random walk with $d>3$. Let $T_n=(S_n^{(1)},S_n^{(2)},S_n^{(3)})$, that is, we obtain $T_n$ by looking only at the first three coordinates of $S_n$. It is clear $T_n$ isn't always a si... | Here's the proof for a simple random walk, which can be generalized further. Hopefully it's clear that $T_{N(k)}$ as a function of $k$ changes exactly one coordinate for each $k$ and moreover $|T_{N(k+1)}-T_{N(k)}|_\infty=1$. More importantly, $T_{N(k+1)}|$ is a Markov process. So it suffices to verify that each step i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Eigenvalues of a quasi-circulant matrix The following matrix cropped up in a model I am building of a dynamical system:
$$A=
\begin{bmatrix} 1 - \alpha & \alpha/2 & 0 & 0 &\cdots & 0 & 0 & \alpha/2\\
\alpha/2 & 1-\alpha & \alpha/2 & 0 &\cdots & 0 & 0 & 0\\
0 & \alpha/2 & 1-\alpha & \alpha/2 &\cdots & 0 & 0 & 0\\
\vdots... | Oddly enough, spectral graph theory has the analytic solution for this problem.
Let $L$ be the graph Laplacian for the path graph. Then $L$ is tri-diagonal with $[1,2,2,...,2,1]$ on the diagonal and $-1$'s on the super and sub diagonals. Then the matrix $A'$ is given by $$A'=I-\frac{\alpha}2 L.$$ Now since every vecto... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
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Phase portrait of a $2 \times 2$ system of linear, autonomous differential eqns. with a zero eigenvalue Let $\mathbf{Y} = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$ and $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $a, b, c, d \in \mathbb{R}$. Now, consider the system
$$
\frac{\operatorname d \mathb... | If you let $t' = k_2 e^{\lambda_2 t}$, then $(1)$ becomes
$$Y = k_1 V_1 + t' V_2.$$ This is the equation for a line through $k_1V_1$ with direction $V_2$ ie lines parallel to $V_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$ The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$ contains $N$ integers. Find the value of $10N$.
I tried to find the mini... | Put $\sqrt{1+\cos x}$ +$\sqrt{1-\cos x} = A$
$A^2 = 2\pm 2 \sin x ,\quad A^2 - 2 =\pm 2 \sin x$
$ -2\leq A^2 - 2\leq 2,\quad -2\leq A\leq2$
So $f(x) = \frac{5 - A^2}{A}$ or $\frac{A^2 + 1}{A}$
Find the minimum and maximum of $f(x)$ in the two conditions with $-2\leq A\leq 2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Difference between these two logical expression I am trying to solve the following problem:
Let S(x) be the predicate “x is a student,” F(x) the predicate “x is a
faculty member,” and A(x, y) the predicate “x has asked y a question,”
where the domain consists of all people associated with your school.
Use quanti... | The textbook answer says that the following represents:
Some student has asked every faculty member a question.
$\forall y(F(y) \to \exists x (S(x) \land A(x,y))) $
Which seems incorrect as I read this as saying for every faculty member there exists a student that has asked a question.
I'd argue that the following i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$? How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?
I tried $$\lim_{z\to0} \frac{\bar{z}^2}{z}=\lim_{\overset{x\to0}{y\to0}}\frac{(x-iy)^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}\cd... | From your calculation :
$$=\lim_{\overset{x\to0}{y\to0}}\frac{(x^2-2xyi-y^2)(x-iy)}{x^2+y^2}$$
$$=\lim_{(x,y)\to (0,0)}\frac{x^3-3xy^2}{x^2+y^2}-i\lim_{(x,y)\to (0,0)}\frac{3x^2y-y^3}{x^2+y^2}$$
From here, show that both the limits are zero by changing polar form , $x=r\cos \theta$ , $y=r\sin \theta$.
For the first lim... | {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$ Let $\ell^\star$ be the upper logarithmic density on the set of positive integers, namely
$$
\forall X\subseteq \mathbf{N}^+, \,\, \ell^\star(X)=\limsup_n \frac{1}{\ln n}\sum_{x \in X\cap [1,n]}\frac{1}{x}.
$$
Problem: Let $... | The answer is yes. More in general, if $\alpha \in [-1,\infty[$ and $\mathsf{d}^\ast(\mathfrak F; \alpha)$ is the upper $\alpha$-density (on $\mathbf N^+$) relative to a certain sequence $\mathfrak F = (F_n)_{n \ge 1}$ of nonempty subsets of $\mathbf N^+$ with suitable properties (can make this more precise if requeste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove $\begin{pmatrix} a&b\\ 2a&2b\\ \end{pmatrix} \begin{pmatrix} x\\y\\ \end{pmatrix}=\begin{pmatrix} c\\2c \\ \end{pmatrix}$ Prove that
$$
\begin{pmatrix} a & b \\ 2a & 2b \\ \end{pmatrix}
\begin{pmatrix} x \\ y \\ \end{pmatrix}
=
\begin{pmatrix} c \\ 2c \\ \end{pmatrix}
$$
has solution
$$
\begin{pmatrix} x \\ y \\ ... | The fast approach, as I see it is (in case $a \neq 0$):
you have $ax+by = c$ , just plug in $y=0+ta$, and you get
$$ax + bta = c$$
$$x= \frac ca - bt$$.
alternatively, in case you don't have the solution in advance, use same approach and you will get: $$ x = \frac ca - \frac ba y$$, so the general solution will be $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Skorokhod space with uniform norm is Banach Let $D := D([0,t])$ be the Skorokhod space of right-continuous functions with left limits taking values in $\mathbb{R}^d$. Equip $D$ with the supremum norm $||f||_\infty = \sup_{s \in [0,t]}|f(s)|$. How would one go ahead and show that this is a Banach space? I have thought a... | If $(f_n)$ is a Cauchy sequence then $f_n$ is uniformly Cauchy, so there exists $f$ such that $f_n\to f$ uniformly. You only need to show that $f\in\mathcal D$. This follows by exactly the same argument as one uses to show a uniform limit of continuous functions is continuous...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$ Problem statement: Suppose that $\mu$ is a finite measure. Prove that a measurable, non-negative function $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$.
My attempt at a so... | Convince yourself that $$f(x)-1\leq \sum_{n=1}^\infty {\bf 1}_{(f\geq n)}(x)\leq f(x)$$
for all $x\in X$, then integrate with respect to $\mu$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Help with Spivak's Calculus: Chapter 1 problem 21 I've been stuck on this problem for over a day, and the answerbook simply says "see chapter 5" for problems 20,21, and 22. But I want to complete the problem without using knowledge given later in the book, so I've been banging my head against the wall trying all sorts ... | I think I figured it out thanks to GeoffRobinson's hint, and from RJS (thanks guys!) so I figured I'd write out my own work here.
$|xy-x_0y_0| = |x(y-y_0) + y_0(x-x_0)| \leq |x(y-y_0)| + |y_0(x-x_0)|$
So in essence we're going to show that $|x(y-y_0)| + |y_0(x-x_0)| \leq \varepsilon$ which implies that $|xy-x_0y_0| \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find $f(2015)$ given the values it attains at $k=0,1,2,\cdots,1007$
Let $f$ be a polynomial of degree $1007$ such that $f(k)=2^k$ for $k=0,1,2,\cdots, 1007$. Determine $f(2015)$.
Taking $f(x)=\sum_{n=0}^{1007} a_n x^n $, (whence $a_0=1$), I tried to combine $$a_1=1-\sum_{n=2}^{1007} a_n$$ and the easily derivable $$ ... | Hint: $f(x)=\sum_{r=0}^{1007}\,\binom{x}{r}$, where $\binom{x}{r}:=\frac{x(x-1)\cdots(x-r+1)}{r!}$ for every positive integer $r$ and $\binom{x}{0}:=1$, satisfies the required condition. Show that it is the only one satisfying this property. Then, verify that $f(2015)=2^{2014}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Jackpot probablity There are two parts to a problem:
Part (a);
In a Lottery, three white balls are drawn (at random) from twenty balls numbered 1 through 20, and a blue SuperBall is drawn (at random) from ten balls numbered 21 through 30. When you buy a ticket, you select three numbers from 1-20 and one number from 21-... | 3 matching white balls are drawn.
B: 2 matching white balls are drawn.
C: The right blue ball is drawn.
Therefore $P(A\cup B \cup C )=P(A)+P(B)+P(C)-P(A \cap C)-P(B \cap C)$
$\left[\frac{3}{20} \cdot \frac{2}{19} \cdot \frac{1}{18}\right]+\left[3 \cdot \frac{3}{20} \cdot \frac{2}{19} \cdot \frac{17}{18}\right]+\left[\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
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Extending continuous function from a dense set If $X$ is a metric space and $Y$ a complete metric space. Let $A$ be a dense subset of $X$. If there is a uniformly continuous function $f$ from $A$ to $Y$, it can be uniquely extended to a uniformly continuous function $g$ from $X$ to $Y$. I was trying to think of an exam... | Define $f:\mathbb{Q}\to\mathbb{R}$ by $f(x)=0$ if $x<\sqrt{2}$ and $f(x)=1$ if $x>\sqrt{2}$. Then $f$ is continuous, but cannot be continuously extended to $\sqrt{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Proving that cows have same weight without weighing it! My friend gave me this problem and I have no clue how to go about it:
A peasant has $2n + 1$ cows. When he puts aside any of his cows,
the remaining $2n$, can be divided into two sub-flocks of $n$ cows and each having
the same total weight. How can we prove that ... | We have $2n+1$ equations in $2n+1$ unknowns. This has a unique solution. As all the cows having the same weight is a solution, it is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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Proving an inequality with given equality of the form $a^2+b^2=c$ $$\begin{Bmatrix}
a^2+b^2=2\\
c^2+d^2=4
\end{Bmatrix} \to ac+bd\le3 $$ prove that inequality
| HINT
Note that $$(a-c)^2+(b-d)^2\ge0$$so that $$a^2+c^2+b^2+d^2\ge2ac+2bd$$
This deals with the easy inequality required. Other answers give a sharper result, which is possible because it is impossible for $a=c$ and $b=d$ with the given conditions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove identity: $\frac{1+\sin\alpha-\cos\alpha}{1+\sin\alpha+\cos\alpha}=\tan\frac{\alpha}{2}$ Prove identity: $$\frac{1+\sin\alpha-\cos\alpha}{1+\sin\alpha+\cos\alpha}=\tan\frac{\alpha}{2}.$$
My work this far:
we take the left side
$$\dfrac{1+\sqrt{\frac{1-\cos2\alpha}{2}}-\sqrt{\frac{1+\cos2\alpha}{2}}}{1+\sqrt{\frac... | Notice, $$LHS=\frac{1+\sin\alpha-\cos\alpha}{1+\sin\alpha+\cos\alpha}$$
$$=\frac{1+\frac{2\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}-\frac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}}{1+\frac{2\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}+\frac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}}$$
$$=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
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Tale of a frog that jumps a fraction of what is left to cross the pond I recall from undergraduate courses in calculus and series analysis a tale of a frog that tries to jump a fraction (e.g. 1/2) of what is left for the frog to cross the pond.
In the limit, the fraction of the pond the frog travels is:
$1/2 + 1/2(1/2... | This looks like a geometric series, which is a series of the form
$$\sum_{n=0}^\infty a r^n$$
In your case $a = \frac{1}{2}$ and $r = \frac{1}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Integrate $\int_{-\infty}^{\infty} \frac{\cosh(\beta x)}{1+\cosh( \beta x )} e^{-x^2} x^2 \rm{d}x$ Integrate
$$ \int_{-\infty}^{\infty} \frac{\cosh(\beta x)}{1+\cosh( \beta x )} e^{-x^2} x^2 \rm{d}x, $$
with $\beta \in \mathbb{R}$ and $\beta > 0$.
Numerical integration shows that this integral exists, but I have been... | I have some idea, if you are willing to take into account something different by residues, and if you are willing to proceed by your own adopting some tricks in the same way I will do (this message refers to the part $2$ of what I will do in a while. I'll show you my ideas for part $1$ only).
Starting with writing $\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1387960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
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Compare growth of function and its derivative. Suppose $f:\mathbb R \rightarrow \mathbb R$, $f(x)<O(x)$ (i.e. has slower than linear growth), $\lim_{x\rightarrow \infty} f'(x)=0$. Is it possible to show that there exists a $\delta$ such that for $|x|>\delta$, $x f'(x) < f(x)$ for all functions $f$?
| This is not true. For example, take $f(x)=\frac{1}{x}\sin x$. Then $f'(x)=\frac{1}{x}\cos x - \frac{1}{x^2}\sin x \to 0$ as $x\to \infty$. However, $xf'(x) = \cos x - \frac{1}{x} \sin x \not\to 0$ while $f(x) \to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Question about Euler's formula I have a question about Euler's formula
$$e^{ix} = \cos(x)+i\sin(x)$$
I want to show
$$\sin(ax)\sin(bx) = \frac{1}{2}(\cos((a-b)x)-\cos((a+b)x))$$
and
$$ \cos(ax)\cos(bx) = \frac{1}{2}(\cos((a-b)x)+\cos((a+b)x))$$
I'm not really sure how to get started here.
Can someone help me?
| Try using the identity
$$\sin A \cos B \equiv \tfrac{1}{2}\left(\sin(A + B) + \sin(A - B)\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
$2^{nd}$ order PDE: Solution I am trying to solve the following equation:
$$\frac{\partial F}{\partial t} = \alpha^2 \, \frac{\partial^2 F}{\partial x^2}-h \, F$$
subject to these conditions:
$$F(x,0) = 0, \hspace{5mm} F(0,t) = F(L,t)=F_{0} \, e^{-ht}.$$
I know that I am suppose to simplify the equation with:
$$F(x,t)=... | The following field redefinition, similar to what the other answers have done, simplifies the problem:
$$F = (F_0 + \phi)e^{-ht} \implies \frac{d\phi}{dt} = \alpha^2\frac{d^2\phi}{dx^2}$$
with $\phi(0,t) = \phi(L,t) = 0$ and $\phi(x,0) = -F_0$. One can now proceed with your favoritte method like separation of variable... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
how did Cardano obtain three solutions for cubic? So, if I am not mistaken Complex numbers were discovered after Cardano's method. But from Cardano's Method on Wikipedia, it says to get the three solutions, we should use the root of unity. In that case, when Cardano's method was published it could only find one solutio... | The method of solving a cubic is explained very clearly in Stroud's Further Engineering Mathematics. According to Stuart Hollingdale's book "Makers of Mathematics", Cardano plagiarized Tartaglia's solution of the cubic. Paul Halmos said that he solved this problem for himself as a high school student.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Prove $\gcd(n, n + 1) = 1$ for any $n$ Let $n \in \mathbb Z$ be even. Then $n + 1$ is odd. So, $2$ doesn't divide $n + 1$. Thus there's no even number for which $\gcd(n, n+1)$ is not $1$. I am not sure how to show it for odd numbers. Is there a better way to prove the statement?
| Suppose $gcd(n,n+1)=d >0$ $$\to \left.\begin{matrix}
d|n\\
d|n+1
\end{matrix}\right\}\Rightarrow d|(n+1)-(n)\Rightarrow d|1\\\frac{1}{d} \in \mathbb{Z}\Rightarrow d=\pm1 \overset{d>0}{\rightarrow} d=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 3
} |
Average of a list of numbers, read one at a time I have a dilemma
Lets take the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
If we wanted to calculate the average of this set we would add up all the numbers in the set $(45)$ and then divide by the total number of items in the set $(9)$ and arrive at the correct average of $5$
W... | You can:
*
*Keep track of the total number of values you have seen so far, call it $n$, and the average of the numbers so far, $A$.
Then when you see a new number $x$, map
\begin{align*}
n &\mapsto n+1 \\
A &\mapsto \frac{nA + x}{n+1}
\end{align*}
Or:
*
*Keep track of the total number of values you have seen so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limits (Three Variable function). We're given :
$f(x,y,z) = \dfrac{xyz}{x^{2}+y^{2}+z^{2}}$ ,
Also , it's given that $\lim_{(x,y,z) \to (0,0,0)} f(x,y,z)$ exists.
We need to prove that $\lim_{(x,y,z) \to (0,0,0)} f(x,y,z) = 0$.
What I figured :
First I approach $(0,0,0)$ along $x$ -axis , and thus the limit becomes :... | Little observation: in general (if you do not know whether the limit exists), if the limit exists along one path it does NOT mean that it exists, so your proof for $(x,0,0)$ is not sufficient.
For the general case, therefore, I would either use the demonstration of answer $1$ , or majorize at the numerator $|xy|$ by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove the inequality - inequality involving surds Prove that for $r$ greater than or equal to 1:
$\displaystyle 2(\sqrt{r+1} - \sqrt{r}) < \frac{1}{\sqrt{r}} < 2(\sqrt{r}-\sqrt{r-1})$
Any help on this would be much appreciated.
| Let $f(x) = \sqrt x$. By mean value theorem, there exist $q\in (r-1,r)$ and $s \in (r, r+1)$ that satisfy
$$f(r) - f(r-1) = f'(q),\quad f(r+1)-f(r) = f'(s).$$
By the concavity of $f$, since $q<r<s$,
$$\begin{array}{rcl}
f'(q) >& f'(r) &> f'(s)\\
f(r)-f(r-1) >& f'(r) &> f(r+1) - f(r)\\
\sqrt{r}-\sqrt{r-1} >& \dfrac1{2\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to approach to fitting curve? I'd like to approximate fitting curve some kind of curves like below.
(1, 3.5), (2, 4.3), (3, 7.2), (4, 8) which is having 4 points.
and I heard that this solver is PINV() of matlab function.
But I don't know how to use.
Would you please let me know how to use and find a approximatio... | Problem statement
Fit the given data set with a sequence of polynomial fits:
$$
y(x) = a_{0} + a_{1} x + \dots + a_{d} x^{d}
$$
Indicate where we can solve the linear system with the normal equations, and when we must rely exclusively upon the pseudoinverse.
Linear System
The linear system for the polynomial with high... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1388811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Generic fiber of a scheme over a DVR What is (usually?) meant by the generic fiber of a scheme over a discrete valuation ring? I've seen in some talks now, could somebody give a precise definition?
Thank you very much in advance!
| If $R$ is a discrete valuation ring, then $Y=\mathrm{Spec}(R)$ has two points, often denoted $\eta$ and $s$, the generic and the special (or closed) point, corresponding to the ideal $(0)$ and the unique maximal ideal $\mathfrak{m}_R$, respectively. The names are apt, as $\{\eta\}$ is dense in $Y$, while $\{s\}$ is clo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1389883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Distinct Roots of $x^2+(a-5)x+1=3|x|$ Problem:
$$x^2+(a-5)x+1=3|x|$$ Find 3 distinct solutions to the above problem.
A friend of mine at my coaching center came up with this problem which nobody was able to solve. Unfortunately, I have been unable to contact my professor and understand how to solve this problem. Desp... | Case when $x \geq 0$:
$x^2+(a-8)x+1=0$
Then $x=\frac{(8-a)\pm \sqrt{a^2-16a+60}}{2}$
Case when $x < 0$:
$x^2+(a-2)x+1=0$
Then $x=\frac{(2-a)\pm \sqrt{a^2-4a}}{2}$
As we're only working with real solutions: for the determinants to be non-negative, we need $a \leq 6$ or $a \geq 10$ in the former, and $a \leq 0$ or $a \ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1389980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that $\lim_{z\to 0}|z|^{\sqrt 2}f(z)=0$ in $D$
Let , $f$ be analytic in $\mathbb D\setminus\{0\}$ and unbounded near $z=0$. If the function $|z|^{\sqrt 2}f(z)$ is bounded at $z=0$ then show that $$\lim_{z\to 0}|z|^{\sqrt 2}f(z)=0\text{ & }\lim_{z\to 0}|z|^{\sqrt 2/2}f(z)=\infty.$$where , $\mathbb D=\{z\in \math... | Hint: $z^2f(z)$ is analytic in some $\{0<|z|<r\},$ and from the given information, $z^2f(z) \to 0$ at $0.$ In particular $z^2f(z)$ is bounded in this domain. Hence it has a removable singularity at $0.$ Thus $z^2f(z) = \sum a_nz^n $ in this domain. Think about the first few of the coefficients $a_n.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Does other solutions exist for $29x+30y+31z = 366$? I was asked this trick question:
If $29x + 30y + 31z = 366$ then what is $x+y+z=?$
The answer is $12$ and it is said to be so because $29$ , $30$ and $31$ are respectively the number of days of months in a leap year. Therefore $x + y + z$ must be $12$, the total num... | With $x,y,z$ being reals, no, there's tons of answers. With $x,y,z$ be natural numbers, $12$ is the only answer. Proof: $11$ is too small, because even if all $11$ was in the biggest number, $11\cdot 31=341<366$, and $13$ is too big because $13\cdot 29=377>366$.
The "middle" case if you allow negative integers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Algebra question from practice GRE exam The following is a question from the GRE exam GR9367:
Let $n > 1$ be an integer. Which of the following conditions guarantee that the equation $x^n = \sum_{i=0}^{n-1} a_ix^i$ has at least one root in the interval $(0,1)$?
I. $a_0 > 0$ and $\sum_{i=0}^{n-1}a_i < 1$
II. $a_0>0$ and... | The key to this question lies in three known facts. Namely:
*
*A continuous function $f$ has a root in an open interval $]a,b[$ if $f(a)<0<f(b)$ or $f(b)<0<f(a)$
*All power series are continuous.
and
*$1>0$
So, to prove that a power series $\sum_{i=0}^\infty a_i x^i$ has a root between 0 and 1, wee need to show t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem From Zeidler's Applied Functional Analysis
Brouwer
The continuous operator $A:M \to M$ has a fixed point provided $M$ is
compact, convex, nonempty set in a finite dimensional normed space
over $\mathbb{K}$
Schaude... | Your claim is correct.
The point is that in finite dimension all continuous operators are compact, while in infinite dimension you can have continuous operators which are not compact, as the following example shows:
In $\ell_2(\mathbb{N})$ consider the operator $T(x)=(\sqrt{1 - \| x\|^2},x_1, x_2, \dots)$
defined for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$.Then find the value of $n$ If $A_1A_2A_3.....A_n$ be a regular polygon and $\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$.Then find the value of $n$(number of vertices in the regular polygon).
I know that sides of a regular polygon are equal but i could not rel... | This problem can be trivialized by Ptolemy's theorem.
First we take the l.c.m and simplify both sides of the given equation.
Let $A_1A_2=a$, $A_1A_3=b$, $A_1A_4=c$. Then we have
$$\frac{1}{a}=\frac{1}{b}+\frac{1}{c} \qquad\to\qquad b c = a b + a c \tag{1}$$
Now, note that regular polygons can always be inscribed in a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
Binomial expansion. Find the coefficient of $x$ in the expansion of $\left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6$.
I've used the way that my teacher teach me.
I've stuck in somewhere else.
$\left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6=\left(2-\frac{4}{x^3}\right)\left(x^6\left(1+\frac{2}{... | Notice, we have
$$\left(x+\frac{2}{x^2}\right)^6=^6C_0x^{6}\left(\frac{2}{x^2}\right)^{0}+^6C_1x^{5}\left(\frac{2}{x^2}\right)^{1}+^6C_2x^{4}\left(\frac{2}{x^2}\right)^{2}+^6C_3x^{3}\left(\frac{2}{x^2}\right)^{3}+^6C_4x^{2}\left(\frac{2}{x^2}\right)^{4}+^6C_5x^{1}\left(\frac{2}{x^2}\right)^{5}+^6C_6x^{0}\left(\frac{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What was Gauss' 2nd Factorization Method? Reading Jean-Luc Chabert's A History of Algorithms, I learned that Gauss, prompted by the poor state-of-the-art, designed two distinct methods for fast integer factorization.
Chabert's book discusses the first, the Method of Exclusions, and D.H. Lehmer gives a really nice expla... | Gauss's second factorization method is pretty much a take on the difference of squares method. If one finds $a^2 \equiv b^2 \ (\mathrm{mod} \ M)$, then either $\mathrm{gcd}(M,a+b)$ or $\mathrm{gcd}(M,a-b)$ is a non-trivial factor of M.
Gauss's genius idea was using quadratic forms to force a congruence of squares, by m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$\int\limits_{0}^{1}(\prod\limits_{r=1}^{n}(x+r))(\sum\limits_{k=1}^{n}\frac{1}{x+k})dx$ The value of $\int\limits_{0}^{1}(\prod\limits_{r=1}^{n}(x+r))(\sum\limits_{k=1}^{n}\frac{1}{x+k})dx$ is equal to
$(A)n\hspace{1cm}(B)n!\hspace{1cm}(C)(n+1)!\hspace{1cm}(D)n.n!$
I tried:$\int\limits_{0}^{1}(\prod\limits_{r=1}^{n}(x... | Differentiate $\prod_{k=1}^n(x+k)$ with respect to $x$, and you get the integrand. So the answer is $(n+1)!-n!=n\cdot n!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Fraction of original velocity in series of elastic collisions I need to find the answer to this question:
Three particles A, B, and C, with masses $m$, $2m$ and $3m \, \mathrm k \mathrm g$ respectively, lie at rest in that order in a straight line on a smooth horizontal table. The particle A is then projected directly... | The collisions are perfectly elastic. So you should write energy balance equation.
$$ K.E._{initial}= K.E._{final} $$
So for the $1^{st}$ collision
$$\frac{1}{2}mu^2=\frac{1}{2}mV_a^2+ \frac{1}{2}(2m)V_b^2 $$
Now we get 2 unknowns $ V_a \hspace{0.2cm} and \hspace{0.2cm} V_b $ and 2 equations namely momentum conserva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do I convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$? I would like to convince my student in high school level using a simple
mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{Z}$ which
have only one integer solution that is $x=4$ .... | Hint:
plot the graph of $y=2^x$ and $y=4x$ and shows that the only other solution is between $0$ and $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 0
} |
Solving equations with logarithmic exponent I need to solve the equation :
$\ln(x+2)+\ln(5)=\lg(2x+8)$
With the change of base formula we can turn this into:
$\ln(x+2)+\ln(5)=\frac{\ln(2x+8)}{\ln(10)}$
We can also simplify the LHS with the product rule so:
$\ln(5(x+2))=\frac{\ln(2x+8)}{\ln(10)}$
Solving the fraction g... | You can find answer by researching of graphic of functions,I don't think you can get it by algebric method
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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