Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do you prove the left and right side of an identity of a set are equal? I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right identity of S, how can I prove those two are equal... | Well this is the proof.
$$ L = L * R = R$$
No need to use associativity.
For an elaboration.
A left identity is an element $L$ satisfying $L * g = g$ for every $g \in G$. Hence $L * R = R$.
A right identity is an element satisfying $g * R = g \;\; \forall g \in G$. Hence, $L * R = L$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What makes a number representable? The set of real numbers contains element which can be represented (there exists a way to write them down on paper). These numbers include:
*
*Integer numbers, such as $-8$, $20$, $32412651$
*Rational numbers, such as $\frac{7}{41}$, $-\frac{14}{3}$
*Algebraic numbers
*Any other ... | The numbers you are describe in your list are called computable numbers meaning they can be computed to arbitrary precision. Equivalently, a number $x$ is computable if it is decidable, given rational $a$ and $b$, if $x\in (a,b)$.
The other numbers are said to be noncomputable. It is worth noting, however, that some no... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that an operator $T$ on a Hilbert space is compact Let $H$ be a Hilbert space, $T:H \to H$ be a bounded linear operator and $T^{*}$ be the Hilbert Adjoint operator of $T$. Show that $T$ is compact if and only if $T^{*}T$ is compact.
My attempt:
Suppose first that $T$ is compact. The Hilbert Adjoint operator of ... | Suppose that $f = \text{w}-\lim_{n \to \infty} f_n$. So we have $\lim_{n \to \infty} \| T^* T (f_n-f)\|=0$ because $T^*T$ is compact. Also, we know that sequence $\{f_n-f\}_{n=1}^{\infty}$ is bounded, so we have
\begin{align*}\lim_{n \to \infty} \|T(f_n-f)\|^2 = \lim_{n \to \infty} \langle T^*T (f_n-f),f_n-f\rangle \l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve logarithmic equation $\log_{\frac{x}{5}}(x^2-8x+16)\geq 0$ Find $x$ from logarithmic equation:
$$\log_{\frac{x}{5}}(x^2-8x+16)\geq 0 $$
This is how I tried:
$$x^2-8x+16>0$$
$$ (x-4)^2>0 \implies x \not = 4$$
then
$$\log_{\frac{x}{5}}(x^2-8x+16)\geq \log_{\frac{x}{5}}(\frac{x}{5})^0 $$
because of base $\frac{x}{... | Given $$\displaystyle \log_{\frac{x}{5}}(x^2-8x+16)\geq 0\;,$$ Here function is defined when $\displaystyle \frac{x}{5}>0$ and $\displaystyle \frac{x}{5}\neq 1$
and $(x-4)^2>0$. So we get $x>0$ and $x\neq 5$ and $x\neq 4$
If $$\displaystyle \; \bullet\; \frac{x}{5}>1\Rightarrow x>5\;,$$ Then $$\displaystyle \log_{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1422858",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Implicit solution of ODE to explicit or approximate explicit function Working with the following ODE and implicit solution but need an explicit solution for J:
The ODE with$J_c$ and $G$ as constants is:
$$-\frac{1}{J^2}\frac{dJ}{dt} = G(J-J_c)$$
The implicit solution given by Field et al. (1995) is:
$$Gt = \frac{1}{J_... | Based on contribution from JJacquelin the problem needs a numerical approach to solve. No solution based on standard functions (possibly non-standard) is likely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Another messy integral: $I=\int \frac{\sqrt{2-x-x^2}}{x^2}\ dx$ I found the following question in a practice book of integration:-
$Q.$ Evaluate $$I=\int \frac{\sqrt{2-x-x^2}}{x^2}\ dx$$ For this I substituted $t^2=\frac {2-x-x^2}{x^2}\implies x^2=\frac{2-x}{1+t^2}\implies 2t\ dt=\left(-\frac4{x^3}+\frac 1{x^2}\right)\... | Let $$\displaystyle I = \int \frac{\sqrt{2-x-x^2}}{x^2}dx = \int \sqrt{2-x-x^2}\cdot \frac{1}{x^2}dx\;, $$ Now Using Integration by parts
$$\displaystyle I = -\frac{\sqrt{2-x-x^2}}{x}-\int\frac{1+2x}{2\sqrt{2-x-x^2}}\cdot \frac{1}{x}dx $$
So $$\displaystyle I = -\frac{\sqrt{2-x-x^2}}{x}-\underbrace{\int\frac{1}{\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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Is this a sound demonstration of Euler's identity? Richard Feynman referred to Euler's Identity, $e^{i\pi} + 1 = 0$ as a "jewel."
I'm trying to demonstrate this jewel without recourse to a Taylor series.
Given $z = cos\theta + i sin\theta\; |\;|z| = 1$,
$$\frac{dz}{d\theta}= -sin\theta + icos\theta =i(isin\theta+cos\... | This is not a proof of Euler's discovery, but of something much weaker.
What Euler noticed is that there is a connection between the exponential function and the trigonometric functions. Your proof only shows that solutions to $F'(t)=iF(t)$ (for real $t$) are equivalent to uniform circular motion in the complex pla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Plot a single point on number line in interval notation For example, I want to plot the solution set $\{3\}\cup (2, \infty$). How do I represent 3 as a single point?
| If you needed to represent $\{2\}\cup(4,\infty)$, then you could use a filled-in dot for individual points, so you might draw something like the following:
As Alex G. mentioned, since $3$ is in $(2,\infty)$, $\{3\}\cup(2,\infty)=(2,\infty)$ so you wouldn't have to do this for the case you mentioned in your original qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423243",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Proof that bisecting a line segment with straightedge is impossible This is the proof I read from here. I will quote it fully:
The answer is NO. To see why, consider a line L in the plane P, and
two marked points A, B on it. It is desired to construct the midpoint
M of the segment AB using the straightedge. Suppos... | It depends on your "initial position". Given A, B, and possibly other points on the line AB (but not M) we cannot obtain other points on AB unless we are given at least 2 points C,D not on AB. Now suppose M lies on the line CD. If we have a projection ,that is, a perspective view P' of P, the recipe "Take the intersec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to calculate the probability of drawing numbers under certain constraints We have numbers $1,2,3,\dots, n$, where $n$ is an integer, and we select $d$ distinct numbers from them, $i_1,i_2,\dots,i_d$. What is the probability that there exist at least two numbers whose difference is less than $k$, i.e $|i_{h}-i_{j}|<... | I think the best approach is to enumerate the number of ways to choose $d$ numbers so that the difference between any two is at least $k$. That is, we aim to enumerate the number of sets of the form $\{i_1,i_2,\ldots,i_d\}$ such that $|i_j-i_{j-1}|\geq k$ for all $2\leq j\leq d$.
One way to think about this type of pr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can we prove that $\frac{a}{b }\times\frac{c}{d} =\frac{ac}{bd}$ I am slowly reading calculus by michael spivak and it is one of the problems in first chapter.
however I cant prove it please help me with it...
| To see that equality holds, we will write the elements $\frac1b$ and $\frac1d$ as $b^{-1}$ and $d^{-1}$, respectively. Thus
\begin{align*}
\frac ab\cdot\frac cd&=(a\cdot b^{-1})\cdot(c\cdot d^{-1})\\
&=(a\cdot c)\cdot(b^{-1}\cdot d^{-1})\\
&=(a\cdot c)\cdot(b\cdot d)^{-1}\\
&=\frac{ac}{bd}.
\end{align*}
The second line... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isomorphic or Equal in Vect? In the category of vector spaces, how many elements are there in the isomorphism class of 1-dimensional vector spaces? Secondly, is the polynomial algebra generated by the symbol $x$ equal to the polynomial algebra generated by the symbol $y$, or merely isomorphic to it.
| The first question can have no answer. Vector spaces over a given field (even $1$-dimensioanl vector spaces) are not a set, unless you consider a small category. It is for the same reason as the set of all sets does not exist.
For the second question, the answer is ‘yes: two polynomial algebras in one indeterminate ove... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Laurent series of $e^{z+1/z}$ What is the Laurent series of $e^{z+1/z}$?
I had used $$a_k= \frac{1}{2\pi i}\int_c \frac{f(z)}{z^{k+1}}\,dz $$
for a curve $c$ in which we can use $e^z$ as an analytic func. and expanded the $e^{1/z}$ series expansion.
| $$
\begin{align}
e^ze^{\frac1z}
&=\sum_{k=0}^\infty\frac{z^k}{k!}\sum_{j=0}^\infty\frac1{j!z^j}\\
&=\sum_{k=-\infty}^\infty z^k\color{#0000F0}{\sum_{j=0}^\infty\frac1{(k+j)!j!}}\\
&=\sum_{k=-\infty}^\infty\color{#0000F0}{I_{|k|}(2)}\,z^k
\end{align}
$$
where it is convention that $\frac1{k!}=0$ when $k<0$.
$I_n$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1424105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Examples of monotone functions where "number" of points of discontinuity is infinite We know that if $f:D(\subseteq \mathbb{R})\to\mathbb{R}$ be a monotone function and if $A$ be the set of points of discontinuity of $F$ then $\left\lvert A \right\rvert$ is countable. Where $\left\lvert A \right\rvert$ denotes the card... | *
*Take the graphic of any function you like, which is both continuous and monotonous over a finite interval of your choice $[a,~b]$.
*Map said interval to $[0,1]$ using $x\mapsto\dfrac{x-a}{b-a}$.
*Divide the new interval $[0,1]$ into an infinite number of subintervals of the form $\bigg(\dfrac1{n+1},\dfrac1n\bigg]... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $n^{(n+1)} > (n+1)^{n}$ when $n\geq 3$ Firstly, for $n=3$,
$$3^4 > 4^3$$.
Secondly, $(n+1)^{(n+2)} > (n+2)^{(n+1)}$
Now I'm stuck.
| The inequality at stake is equivalent to $n > \left( 1+\frac{1}{n}\right)^n$.
Taking log yields $\frac{\ln n}{n} >\ln \left(1+\frac{1}{n} \right)$.
This is true since the concavity of $\log$ implies the stronger $\frac{1}{n} >\ln \left(1+\frac{1}{n} \right)$.
Using this argument, your inequality can be refined to $(n+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Set theory and functions (1) Prove or disprove: For any sets $X,Y,Z$ and any maps $f:X \to Y$ and $g:Y \to Z$, if $f$ is injective and $g$ is surjective, then $g \circ f$ is surjective.
So i proved previously that $f$ is injective if $g \circ f$ is injective, but then i also proved that if $g \circ f$ is surjective the... | As you said in your question we have indeed:
*
*$g\circ f$ injective implies that $f$ is injective.
*$g\circ f$ surjective implies that $g$ is surjective.
Then under the condition that $f$ is injective and $g$ is surjective you seem to conclude that $g\circ f$ is bijective ("I'm saying it's true, because $g \circ... | {
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Geometric Interpretation of $|z_1-z_2|\ge ||z_1|-|z_2||$ I have to give a geometric argument that, given two complex numbers $z_1, z_2$, the following inequality holds
$$|z_1-z_2|\ge ||z_1|-|z_2||$$
I know every complex number has a nonnegative modulus, and this becomes a problem if $|z_1|\lt |z_2|$, and it contradicts... | If you want to proof it using the triangle inequality:
Apply the triangle inequality to get $|z_1-z_2|+|z_2| \geq |z_1|$ if $z_2\geq z_1$.
Apply it again to get $|z_2-z_1|+|z_1| \geq |z_2|$ if $z_1\geq z_2$.
This gives together with $|x|=|-x|$ and $|x|=x$ if $x \in \mathbb R_{\geq 0}$ the desired inequality.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Complex Analysis Geometric Series The question is:
Let $n$ be a positive integer, let $h$ be a positive integer not divisible by $n$, and let
$$ w = \cos\left(\frac{2\pi}{n}\right) + i \sin\left(\frac{2\pi}{n}\right) $$
Show that
$$ 1 + w^h + w^{2h} + w^{3h} + \dots + w^{(n-1)h} = 0$$
I believe I do something with a g... | Note that
$$w = \cos\left(\frac{2\pi}{n}\right) + i \sin\left(\frac{2\pi}{n}\right) = e^{\frac{2\pi i}{n}}.$$
Use the geometric series
$$\sum_{k=0}^{n-1} z^k = \frac{z^n-1}{z-1}.$$
Your case delivers
$$\sum_{k=0}^{n-1} \left( e^{\frac{2\pi i}{n}} \right)^{kh}=\sum_{k=0}^{n-1} \left( \left( e^{\frac{2\pi i}{n}} \right)^... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving Exponential Inequality I would like to know the range of $n$ where this condition is true. Can somebody help me with it. The equation is as follows
$$n^{100}\le2^{n^2}$$
| Both functions in your inequality are continuous, so we find the correct intervals by solving for the equality $n^{100}=2^{n^2}$ first.
Let $x=n^2$. Then
$$x^{50}=2^x$$
$$x=2^{x/50}=e^{x(\ln 2)/50}$$
$$xe^{x(-\ln 2)/50}=1$$
$$x\frac{-\ln 2}{50}\cdot e^{x(-\ln 2)/50}=-\frac{\ln 2}{50}$$
$$x\frac{-\ln 2}{50}=W\left(-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1424685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Signing a Derivative of an Expectation Let $$Z=E[F(x)]=\int_{-\infty}^{\infty} F(x) \frac{1}{y} \phi\left(\frac{x-a}{y}\right)\,{\rm d}x$$
where $F(x)$ is a convex function of $x$, $\phi$ is the standard normal PDF and $a$ is some (finite) constant.
I want to show that $$\frac{\partial Z}{\partial y}>0$$
Here's my p... | With $u=(x-a)/y$, we have
$$
\int_{-\infty}^{\infty} F(x) \frac{1}{y} \phi\left(\frac{x-a}{y}\right)\,\mathrm dx
=
\int_{-\infty}^{\infty} F(yu+a)\phi(u)\,\mathrm du\;,
$$
and
$$
\frac\partial{\partial y}\int_{-\infty}^{\infty} F(yu+a)\phi(u)\,\mathrm du
=\int_{-\infty}^{\infty}F'(yu+a)u\phi(u)\,\mathrm du\;.
$$
Now... | {
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"timestamp": "2023-03-29T00:00:00",
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Not affine, projective, geometrically connected, geometrically reduced, nor geometrically regular... Is there a field $k$ and a regular integral $k$-variety $X$ that is neither affine, projective, geometrically connected, geometrically reduced, nor geometrically regular?
| This is more of a comment on Kevin's answer, but it's too long. I think the exposition can be made clearer with less technicalities (I don't even follow the second isomorphism) and complications. The point is that affine/projective are easy to get rid of, so we focus on the latter three. If we want to make a regular th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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Is it possible for $(900q^2+ap^2)/(3q^2+b^2p^2)$ to be an integer? The original problem is: "Find all possible pairs of positive integers $(a, b)$
$$k = \dfrac{a^3+300^2}{a^2b^2+300}\tag1$$
such that $k$ is an integer."
I've tried so many different ways. Now this question comes up from one of them. Let,
$$\left(\dfra... | While your $(3)$ can be derived from $(2)$, if it is an integer does not guarantee that $(1)$ will be an integer as well. Note that it has the four variables $a,b,p,q$ that can "integerize" your expression, while you're stuck with only $a,b$ for $(1)$.
(For example, $a,b,p,q =3,2,9,5$ makes $(3)$ an integer, but $a,b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1424978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to calculate log10 x without using log? Is it possible to calculate $\log_{10} x$ without using $\log_{10}$? I'm interested because I'm working with a framework that has some simple functions, but log is not one of them.
The specific platform is capable of doing addition, subtraction, multiplication and ... | Let x = 1545 and proceed as follows:
*
*Divide x by base 10 until x becomes smaller than base 10.
1545/10 = 154.5
154.5 / 10 = 15.45
15.45 / 10 = 1.545
As we divide 3 times the value of x by 10, the whole digit of our logarithm will be 3.
*Raise x to the tenth power:
1.545 ^ 10 = 77.4969905 ...
*Repeat the proced... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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Find the value of K in a specific case of a cartesian plane I have a linear equation of a line in a cartesian plane $r:= \{(x,y) \in \mathbb{R}^2 \mid kx-(k+1)y+k-1=0, \,\, k \in \mathbb{R}\}$
and I have to find the value of k so that the line intersects the x axis in a x positive point.
Any tips?
| Straight line in intercepts form:
$$ \dfrac{x}{(1-k)/k} +\dfrac{y}{(k-1)/k+1} = 1 $$
If k<1 positive x- intercept
If k=1 line through origin
If k>1 negative x- intercept
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1425132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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torsion formula for a parametric space curve I managed to prove
$$ k(t) = \frac{ | a^{\prime} \times a^{\prime \prime} | }{ | a^{\prime} | ^3 } $$
for a regular parametric curve $a : I \to R^3 $ where $k(t)$ stands for its curvature but I am stucked in proving
$$ \tau (t) = \frac{( a^{\prime} \times a^{\prime \prime}... | It's going to take too long to type it all out in MathJax, so I have uploaded two photos of the derivations of the formulas for curvature and torsion.enter image description here
| {
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"url": "https://math.stackexchange.com/questions/1425377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determining values of a coefficient for which a system is and isn't consistent. Given the system : \begin{array}{ccccrcc}
x & + & 2y & + & z & = & 3 \\
x & + & 3y & - & z & = & 1 \\
x & + & 2y & + & (a^2-8)z & = & a
\end{array}
Find values of $a$ such that the system has a unique solution, infinitely many solutions, or... | What strikes me immediately
is that if
$a^2-8 = 1$,
the first and last equations
have the same LHS.
Since choosing $a=3$
makes these equations
identical,
this becomes only two equations
in three unknowns,
so there are an
infinite number of solutions.
On the other hand,
if you choose
$a=-3$,
then the first and third equ... | {
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"url": "https://math.stackexchange.com/questions/1425571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can this function have two different antiderivatives? I'm currently operating with the following integral:
$$\int\frac{u'(t)}{(1-u(t))^2} dt$$
But I notice that
$$\frac{d}{dt} \frac{u(t)}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$
and
$$\frac{d}{dt} \frac{1}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$
It seems that both solutio... | I will explain the concept using the derivative since you are already pretty familiar with that. Lets define to $\int \:f\left(x\right)dx$ to be some fucntion where $\frac{d}{dx}\left(g\left(x\right)\right)=f\left(x\right)$. (Note this is not the exact definition of an anti-derivative, but an intuitive way of thinking ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1425660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Finding the $n$th derivative of trigonometric function.. My maths teacher has asked me to find the $n$th derivative of $\cos^9(x)$. He gave us a hint which are as follows:
if $t=\cos x + i\sin x$,
$1/t=\cos x - i\sin x$,
then $2\cos x=(t+1/t)$.
How am I supposed to solve this? Please help me with explanations bec... | De Moivre taught us that if $t=\cos x + i\sin x$ then $t^n = \cos(nx) + i\sin(nx)$ and $t^{-n} = \cos(nx) - i\sin(nx)$ so
$$
t^n + \frac 1 {t^n} = 2\cos(nx).
$$
Then, letting $s=1/t$, we have
\begin{align}
& (2\cos x)^9 =(t+s)^9 \\[10pt]
= {} & t^9 + 9t^8 s + 36t^7 s^2 + 84 t^6 s^3 + 126 t^5 s^4 + 126 t^4 s^5 + 84 t^3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1425755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Compact and Open subsets of $\ell^p$ Let $(a_n)_1^{\infty}$ be a sequence of positive real numbers. Consider $$ A = \{x \in \ell^p : |x_n| < a_n \ \ \forall n\}$$ $$ B = \{x \in \ell^p : |x_n| \leq a_n \ \ \forall n\} $$
I'm interested to know under what conditions imposed on $a_n$ would $A$ be an Open Set and likewise... | The notes you provided shows that $A$ is open if and only if $1\le p<\infty$ and $\inf a_n>0$. You are correct that if $p=\infty$, $B$ is compact if and only if $a_n\to0$ (this is also shown in the notes). You are also correct that if $1\le p<\infty$, $B$ is compact if and only if $(a_n)\in\ell^p$. I'll sketch a proof.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1425858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Essentially bounded function on $\mathbb{R}$ I don't really know a lot about measure (just finishing my undergrad) so I'm not really on good terms with this. So, let $L^\infty[a,b]$ denote the space of all essentially bounded functions on $[a,b]$ with the norm $\left\| f \right\|_\infty = \operatorname{ess} \sup_{x\in[... | An essentially bounded function $f\in L^{\infty}([a,b])$ is explicitly related to the idea of measure, so I don't think there's a way to understand this without measures. The definition can be stated that $f\in L^{\infty}([a,b])$ if there is a $g$ measurable on $[a,b], f=g$ except on a set of measure zero, and $g$ is b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 2
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Is there any winning strategy? 2015 and Game with marbles!!! Two players, Alex and Brad, take turns removing marbles from a jar which initially contains $2015$ marbles. Assume that on each turn the number of marbles withdrawn is a power of two. If Alex has the first turn and the player who takes the last marble wins, ... | I think this is essentially the same as what @PSPACE-Hard said, but here is my take on the winning strategy;
A winning strategy revolves all around the number $3$. This is because;
*
*Every number is either a multiple of $3$, or is $1$ ($2^0$) or $2$ ($2^1$) below a multiple of $3$.
*Also, if either player is left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to find the general formula for this recursive problem? I want to ask about recursive problem.
Given: $$a_0= 11, a_1= -13,$$ and $$a_n= -a_{n-1} +2a_{n-2}.$$
What is the general formula for $$a_n$$ ?
I've already tried to find the first terms of this series. From there, I got:
$$a_2 = 35, a_3= -61,$$ and $$a_4= 131... | Linear Recurrence Equations have typical solutions $a_n=\lambda^n$. Using this, we can compute the possible values of $\lambda$ for this equation from
$$
\lambda^n=-\lambda^{n-1}+2\lambda^{n-2}
$$
which means, assuming $\lambda\ne0$, that
$$
\lambda^2+\lambda-2=0
$$
This is the characteristic polynomial for the recurre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly one real root? For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly
one real root?
I think I should find a way to apply IMV and Rolle's theorem to $f(x) = \log(x) - cx^4$. I think I should first find a range of values ... | One may use a slightly geometric argument for this problem. Notice that $\ln(x)=cx^4$ when $f(x)=cx^4-\ln x=0$. Now, $f$ has a vertex, and the vertex occurs where $f'(x)=4cx^3-\frac{1}{x}=0$. Thus, we have the system of equations
\begin{align}
f(x)&=cx^4-\ln x=0\\
f'(x)&=4cx^3-\frac{1}{x}=0.
\end{align}
Solve the secon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 4
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Left and right multiplying of matrices I am new to matrix multiplication and trying to understand something. Suppose you have a matrix equation $ A x=b $. I know to solve for $x$ you should left multiply by the inverse of A. But what is the reason you can't solve for $b$ like this: $ A A^{-1} x=b A^{-1} $ so that $ x=... | You are asking why $AB=A'B'$ does not imply $AEB = A'EB'$. You noticed that the thing is false. So you should not ask yourself "why is false" (which is nonsense) but you should ask yourself "why I thought it could be true?".
You know that applying the same operation to the same object will held the same result. So if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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prove isomorphisms using first isomorphism theorm Using first isomorphism theorem, prove the following isomorphisms
*
*$\Bbb R/\Bbb Z\xrightarrow\sim S^1,\; $
*$\Bbb C/\Bbb R\xrightarrow\sim \Bbb R,\; $
*$\Bbb C^\times/\Bbb R^\times_+\xrightarrow\sim T,\;$
*$\text{GL}_n(\Bbb C)/\text{SL}_n(\Bbb C)\xrightarrow\sim... | for 1th. use $f(x+iy)=y$
for 3th. use $f(A)=determinan A$.
for last one use, $f(x)=sgn x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can anyone give me $k$ projection maps $P_i:V \to V$ ;$i=1,...,k$ such that $\sum_{i=1}^k Im(P_i)$ is not a direct sum but $\sum_{i=1}^k P_i=Id$. Can anyone give me $k$ projection maps $P_i:V \to V$ ;$i=1,...,k$ such that $\sum_{i=1}^k Im(P_i)$ is not a direct sum but $\sum_{i=1}^k P_i=Id$.
I think in $\Bbb R^2$ I can'... | CLARIFICATION : This answers the case of finite-dimensional vector spaces over $\mathbb R$ (or any field with zero characteristic).
Such a counterexample does not exist. Indeed, let $V_i={\textsf{Im}}(P_i)$ and $r_i={\textsf{dim}}(V_i)={\textsf{trace}}(p_i)$ since $p_i$ is a projector. Since $\sum_{i=1}^k p_k=\textsf{I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do these 2 nested for loops differ in terms of Big Oh Loop 1:
sum $\gets 0$
for $i\gets 1$ to $n$ do
$~~~~$ for $j \gets 1$ to $i^2$ do
$~~~~~~~~~$ sum $\gets$ sum + ary$[i]$
Loop 2:
sum $\gets 0$
for $i\gets 1$ to $n^2$ do
$~~~~$ for $j \gets 1$ to $i$ do
$~~~~~~~~~$ sum $\gets$ sum + ary$[i]$... | The first one runs in
$$
\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \Theta\left(n^3\right)
$$
and the second one takes
$$
\sum_{i=1}^{n^2} i = \frac{n^2\left(n^2+1\right)}{2} = \Theta\left(n^4\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Some clarification needed on the Relation between Total Derivative and Directional Derivative I will consider here functions of several variables only.
If both directional derivative $D_{v}f(x)$ at $x$ along $v$ and total derivative $D f(x)$ at $x$ exist then
$$D_{v}f(x)=Df(x)(v).$$
Existence of total deriva... | There’s a very nice discussion of this topic on Math Insight that might be helpful. The key point for your question is that directional derivatives only “look” along straight lines, but the total derivative (also called the differential) requires you to look at all ways to approach the point. For a function on the real... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Finding global maximizers and minimizers I want to find if global maximum or minimum exists in $ f(x,y)=e-^{(x^2+y^2)}$
I found that (0,0) is the only critical point.
In the Hessian matrix $H_{(f)}(0,0)$
was negative definite and so (0,0) is a local maximizer.
There are few things I need to clarify .
1)As $(0,0)$ ... | While it is always nice to practice multi-variable extremal analysis, with partial derivatives, gradients, Hessians, etc, sometimes a much simpler argument will do. Since the function $e^{-(x^2+y^2)}$ depends only on the length of the vector $(x,y)$, we can look at the function $e^{-r^2}$. We can justify this formally ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The number of distinct partial binary operations on a finite set of n elements I am asked to show that there are exactly $(n+1)^{n^2}$ partial binary operations on a finite set of n elements.
My professor said that this can be done using a combinatoric argument, but I have failed to see how.
Things I know: There are ... | HINT: Add an $(n+1)$-st value, $\text{undefined}$, to the set of possible values of the operation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Convergence of the power series I like to determine where the following power series converges.
$$\sum_{k=1}^\infty \dfrac{x^k}{k} $$
Since the harmonic series diverges, I think that the series would converge if I make the numerator small by forcing $|x|<1$, but I cannot rigoroulsy show where the series converges.
How... | Hint. You may use the ratio test, evaluating
$$
\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}(x)}{a_n(x)}\right|
$$ with $$ a_n(x)=\frac{x^n}n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Solve $x^5 + y^5 + z^5 = 2015$ If $x, y, z$ are integer numbers, solve:
$$x^5 + y^5 + z^5 = 2015$$
A friend of mine claims there is no known solution, and, at the same time, there is no proof that there is no solution, but I do not believe him. However, I wasn't able to make much progress disproving his claim.
I tried ... | These problems are usually done allowing the variables to have mixed signs, some positive, some negative or zero.
i think I will make this an answer. The similar problem for sums of three cubes has been worked on by many people; as of the linked article, the smallest number for which there are no congruence obstruction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Looking to solve an integral of the form $\int_1^\infty (y-1)^{n-1} y^{-n} e^{(\alpha -\alpha n)\frac{(y-1) }{y}} \; dy$ Looking for a solution the following integral. With $n \geq2$, $\alpha>1$,
$$z(n,\alpha)=\frac{\left(\alpha (n-1)\right)^n}{\Gamma (n)-\Gamma (n,(n-1) \alpha )} \int_1^\infty (y-1)^{n-1} y^{-n} e^{... | Integrand behaves as $\frac{C}{y}$ as $y\rightarrow\infty$; hence the integral diverges.
Indeed,
$$ (y-1)^{n-1} y^{-n} \sim y^{-1}$$
as $y\to\infty$, and
$$ e^{(\alpha -\alpha n)\frac{(y-1) }{y}}\to e^{(\alpha -\alpha n) } $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How to explain Clairaut-Schwartz's Theorem, $f_{xy}=f_{yx}$? I am looking for a non-technical explanation of Clairaut's theorem which states that the mixed derivative of smooth functions are equal.
A geometrical, graphical, or demo that explains the theorem and its implications will be helpful. I am not looking for a p... | I don't think that there is a simple geometric or physical argument that makes this theorem intuitively obvious. In the following I try to explain what kind of information about $f$ the mixed partials do encode. That they are equal then follows from symmetry.
Consider the small square $Q_h:=[-h,h]^2$ and add up the val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Number Theory - Remainders A number is of the form $13k_1+12$ and of the form $11k_2+7$
That is $N = 13k_1 + 12 = 11k_2 + 7$
Now why must N also equal $(13 \times 11)k_3 + 51$ ?
Thanks
| Alternately, $N=13k_1+12 = 11k_2+7 \implies N-51 = 13(k_1-3)=11(k_2-4)$
Thus $N-51$ must be a multiple of both $13$ and $11$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Inverse vs Direct Limits This is probably a basic question but I haven't found anything satisfying yet.
I'm trying to understand the difference between inverse and direct limits other than the formal definition. In my mind, an inverse limit is like $\mathbb{Z}_p$ and a direct limit is like the germ of functions at a p... | Direct limits don't have to be small: any set is the direct limit of its finite subsets under inclusion, for instance. An inverse limit of some sets or groups is always a subset (subgroup) of their product, and dually a direct limit is a quotient of their disjoint union (direct sum), if that helps with intuition. I wou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Conditional expectation given an event is equivalent to conditional expectation given the sigma algebra generated by the event This problem is motivated by my self study of Cinlar's "Probability and Stochastics", it is Exercise 1.26 in chapter 4 (on conditioning).
The exercise goes as follows: Let H be an event and let... | That is really strange. How about showing the context? Anyway:
If $\mathbb{E}_\mathcal{F}(X) = \mathbb{E}(X | \mathcal{F})$
and if $\mathbb{E}_HX = \mathbb{E} [X|H]$,
then $\mathbb{E}(X | \mathcal{F}) = \mathbb{E}(X | H)1_H + \mathbb{E}(X | H^C)1_H^C$
If $\omega \in H$, then
$$\mathbb{E}(X | \mathcal{F})(\omega) = \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1427760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Surjectivity of a map from a space to itself I am wondering how to prove that a non-zero degree map from $A \to A$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. This is a map of degree $k$. How can I show that it is surjective?
Note that I... | If $f$ is not surjective it has degree zero. To see this, assume there is some $y$ not in the image of $f$, and factor $f:S^1\to S^1$ as $f=h\circ g$, with
$g:S^1\to S^1/\{y\}$, and $h:S^1 / \{y\}\to S^1$. Since $S^1/\{y\}$ is contractible, it's first homology group is the trivial group and so the induced homomorphism ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the solution space to $x_1x_4-x_2x_3=0$ in $\mathbb{R}$ a manifold? I was wondering whether the set of $(x_1,x_2,x_3,x_4)$ that solve the equation $x_1x_4-x_2x_3=0$ in $\mathbb{R}$ is a manifold. My first guess is that it is most likely not because if I define a function $J(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$ then $0$ i... | No, it is not a manifold because it is a cone with apex the origin $O$, and a cone that is not a vector subspace is never a manifold.
The fact that $dJ(0,0,0,0)=0$ indeed essentially implies that $J=0$ is not a manifold at $O$, but this is a very subtle point never explained in differential geometry books.
See here ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Worker A and worker B doing a project OK, so embarrassingly I've forgotten how to do this type of problem which I'll generalize:
$n$ workers of type $A$ can do a job in $x$ hours. $m$ workers of type $B$ can do the same job in $y$ hours. How long would it take $n_1$ workers of type $A$ and $m_1$ workers of type $B$ ... | $nx$ man-hours of type A = $ym$ man-hours of type B
thus 1 man-hour of type B = $\dfrac{nx}{ym}$ man-hours of type A.
Convert to type A
so hours needed by given mix = $\dfrac{nx}{n_1 + \dfrac{nx}{ym}\cdot m_1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the sum of the following series to n terms $\frac{1}{1\cdot3}+\frac{2^2}{3\cdot5}+\frac{3^2}{5\cdot7}+\dots$ Find the sum of the following series to n terms $$\frac{1}{1\cdot3}+\frac{2^2}{3\cdot5}+\frac{3^2}{5\cdot7}+\dots$$
My attempt:
$$T_{n}=\frac{n^2}{(2n-1)(2n+1)}$$
I am unable to represent to proceed furthe... | The $n$th term is $$n^2/(4n^2-1) =$$ $$ \frac{1}{4}.\frac {(4n^2-1)+1} {4n^2-1}=$$ $$\frac{1}{4} + \frac{1}{4}.\frac {1}{4n^2-1}=$$ $$\frac{1}{4}+ \frac {1}{4}. \left(\frac {1/2}{2n-1}- \frac {1/2}{2n+1}\right).$$ Is this enough?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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What is exactly the difference between a definition and an axiom? I am wondering what the difference between a definition and an axiom.
Isn't an axiom something what we define to be true?
For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the successor ... | I'm not sure whether this answer would help but if you see the axioms and definitions listed in this page you might get to know what's the difference. Axioms acts as fundamentals while definitions are statements that include axioms to say about something.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 13,
"answer_id": 12
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Probability of two Daniels in one group my professor gave us a "fun" problem to work on at home, but I am relatively new to probability.
The question is as follows: There are four Daniels in a class of 42 students. If the class breaks into groups of three what is the probability that there is at least two Daniels in o... | The $P(E)$ you've calculated is correct.
However, for solving the original problem, you'll have to calculate the probability of the event that there are $3$ Daniels in a group, and then add probabilities of both events.
Edit:
I re-read the problem, and your calculation seems a bit off.
For event $E$,
No of Daniels $= 4... | {
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"source": "stackexchange",
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Proving basic fact about ordinals In a set of online lecture notes, I saw the following proposition.
Let $C$ be a set of ordinals. Then $\sup \left\{ \alpha +\beta:\beta\in C \right\} =\alpha +\sup C$.
How can I prove this?
| For brevity, we'll write $\beta^+ := \beta+1$ for the successor ordinal of $\beta$.
If $\sup(C) = 0$ then it's trivial.
If $\sup(C) = \beta^+$ a successor ordinal, then $\beta^+ \in C$ so we get that the LHS is $\alpha + \beta^+$ immediately.
If $\sup(C) = \lambda$ a nonzero limit, then if $\lambda \in C$, we're done a... | {
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What is $5^{-1}$ in $\mathbb Z_{11}$? I am trying to understand what this question is asking and how to solve it. I spent some time looking around the net and it seems like there are many different ways to solve this, but I'm still left confused.
What is the multiplicative inverse of $5$ in $\mathbb Z_{11}$.
Per... | 56 mod 11=1, because 56=55 (multiple of 11)+1
So the answer is 9, since $$9*5= 44+1$$
| {
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Sum of the series $\sum\limits_{n=1}^{\infty }\frac{n}{3^n}$ I want to calculate the sum: $$\sum _{n=1}^{\infty }\:\frac{n}{3^n}\:$$
so $:\:\sum_{n=1}^{\infty}\:nx^n;\:x=\frac{1}{3}\:$
$$=x\sum_{n=1}^{\infty}\:nx^{n-1}=x\sum_{n=1}^{\infty}\:n\:\left(\int\left(x^{n-1}\right)dx\right)'=x\sum_{n=1}^{\infty}\:\left(x^n\ri... | Although this does not address the specific question, I thought it might be instructive to present another approach for solving a problem of this nature. So, here we go
Let $S=\sum_{n=1}^\infty nx^n.\,\,$ Note that we could also write the sum $S$ as $S=\sum_{n=0}^\infty nx^n,\,\,$ since the first term $nx^n=0$ for $... | {
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Inequation: quadratic difference equations Given:
$$\frac{(x - 3)}{(x-4)} > \frac{(x + 4)}{(x + 3)}$$
Step 1:
$$(x + 3)(x - 3) > (x + 4)(x - 4)$$
Step2 : Solving step 1:
$$x^2 - 3^2 > x^2 - 4^2$$
*Step 3:
$ 0 > -16 + 9$ ???
As you see, I can delete the $x^2$, but there is no point in doing that.
What should be the nex... | What you did in step 1 amounts to multiply both sides by $(x-4)(x+3)$. Unfortunately, you have to reverse the inequation if this expression is negative, and leave it as is if it is positive. And as you don't know the sign of this product…
You can simplify solving this inequation writing both sides in canonical form:
$$... | {
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Monotonicity, boundaries and convergence of the sequence $ \left\{ \frac{a^n}{n!} \right\} $. everyone.
I have a doubt on the following question:
Let $ \left\{ \frac{a^n}{n!} \right\}, n \in \mathbb{N} $ be a sequence of real numbers, where $ a $ is a positive real number.
a) For what values of $ a $ is the sequence a... | If
$a_n = \frac{a^n}{n!}
$,
then
$\frac{a_{n+1}}{a_n}
=\frac{\frac{a^{n+1}}{(n+1)!}}{\frac{a^n}{n!}}
=\frac{a}{n+1}
$.
Therefore,
if
$a < n+1
$,
then $a_n$ is decreasing.
In particular,
for any positive real $a$,
$a_n$ is eventually
monotonically decreasing.
If $n+1 < a$,
then
$a_n$ is increasing.
Therefore
$a_n$ first... | {
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Find the supremum and infimum of {x $\in$ [0,1]: x $\notin$ $\mathbb Q$}. Prove why your assertions are correct Ok I am lost from this question. Does that mean $x$ can only be $0$ or $1$? And it can't be any rational?
|
Does that mean x can only be 0 or 1? And it can't be any rational?
No, it means $x$ is any irrational between $0$ and $1$.
$\{x\in [0;1]: x\notin \Bbb Q\}$ is: the set of numbers from the real interval $0$ to $1$ inclusive, such that these numbers are also not in the rationals.
Now, find the supremum of this set. F... | {
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Proving that if $n \in \mathbb{Z}$ and $n^2 − 6n + 5$ is even, then $n$ must be odd. Prove that if $n \in \mathbb{Z}$ and $n^2 − 6n + 5$ is even, then $n$ must be odd.
$p= n^2 - 6n + 55$ is even, $Q= n$ is odd
Proof: Assume on contrary $n$ is even. Then $n= 2k$ for some $k \in \mathbb{Z}$. Then, $$n^2 -6n + 5= 2k^2-6(2... | Hint: Note that $$n^2-6n=n(n-6).$$ Since $n$ and $n-6$ are both _____ or both _____, then we see that $n$ is _____ if and only if $n(n-6)$ is. Since $5$ is odd and $n(n-6)+5$ is even, then $n(n-6)$ must be _____, and so....
| {
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Is $BAB'$ (with positive definite $A$ and full-row rank$(B) = k$) itself of rank $k$? Here's the setup:
A matrix $B$ with dimensions $k \times p$ with $k \leq p$ and rank$(B) = k$.
A matrix $A$ with dimensions $p \times p$ is positive definite (not necessarily symmetric).
Question: Is the square matrix $BAB'$ (which wi... | Hint: motivate the following steps to prove your statement
$BAB^Tx=0$ $\Rightarrow$ $x^TBAB^Tx=0$ $\Rightarrow$ $y^TAy=0$ $\Rightarrow$ $y=B^Tx=0$ $\Rightarrow$ $x=0$.
| {
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Is "probability distribution function" a distribution? I can understand the definition of distribution as written in https://en.wikipedia.org/wiki/Distribution_(mathematics) On the other hand there are three different terms in the definition of probability distribution function(PDF) : https://en.wikipedia.org/wiki/Pr... | The only thing that relates them are that they are both restricted cases of measures (or at least the PDF can be interpreted as such).
A distribution in probability theory is very like a measure with the restriction that the measure of the whole space is 1 (ie $\int dp = $int p(x) dx = 1$).
The other distribution is q... | {
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Is it possible to construct a metric in $\mathbb{R}^n $ s.t. it does not induce CONVEX balls? I'm studying point set topology and looking for a counterexample of "Balls are convex".
We say set $K \subset \mathbb{R}^n $ is convex if $\forall x, y\in K$ implies $\lambda x + \left(1-\lambda y\right)\in K, \forall \lambda... | Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be any bijection, and define a metric $d'$ by$$d'(x,y)=d(f(x),f(y)),$$where $d$ is the Euclidean metric. Let $B$ denote the unit ball around zero in the Euclidean metric. Then the unit ball around zero in the new metric is$$B'=f^{-1}(B).$$The bijection $f$ can be chosen so that $f^{-... | {
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How is $0\cdot\infty= -1$? It is known that the product of slopes of two perpendicular lines is equal to $-1$ ($m_1*m_2=-1$ for $m_1$ and $m_2$ being the slopes of the perpendicular lines $l_1$ and $l_2$). The slope of $x$-axis $=0$; the slope of $y$-axis$=$ undefined (or $\infty$); $x$-axis and $y$-axis are perpendicu... |
How is $0*\infty=-1$? Is it really?
No, $0$ times $\infty$ is not equal to $-1$. In fact, the product isn't even defined. It is not a question of this somehow giving a contradiction, it just isn't defined.
The rule you are referring to says that: Given two lines with slopes $m_1$ and $m_2$ (real numbers) then
the lin... | {
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Identical Zero Sets for two distinct irreducible polynomials In these notes on page 13, question number 9 asks to give example of two irreducible polynomials in $\mathbb{R}[X,Y]$ with identical zero sets. I can think of trivial examples like $x^2+y^2$ and $x^4+y^2$, both of which vanish only on the origin. Are there no... | The answer is no. This is simply Bezout's Theorem. A proof may be found in Michael Artin's Algebra.
| {
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How to find the distance from a point outside a circle to any point on a circle. I'm looking for a way to find out the distance from a point outside a circle to a point on a circle, where the point on the circle is based on radians, degrees, or both (whatever the formula works with).
With this, I know the distance fro... | The data of your problem is: $P=(a,b)$ outside point; $C=(c,d)$ center of circle; $r$ radius of circle. Let Q=(x,y) be your generic point and $D$ the searched distance. You have two equations to use:
$$(1)....( PQ)^2=(x-a)^2+(y-b)^2= D^2$$
$$(2)....(x-c)^2+(y-d)^2=r^2$$
Hence (1)-(2) gives $$D^2=2(c-a)x+2(d-b)y+a^2+b... | {
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Evaluate $\iint_{0Let $$f(x,y)=\begin{cases}xy &\text{ if } 0<x<y<1, \\ 0 &\text { otherwise. }\end{cases}$$ Evaluate the integral $\displaystyle \iint f(x,y)\,dx\,dy$.
I'm having trouble with the limits on integration.
| HINT
your inequality $0<x<y<1$ indicates you can integrate $\int_0^y fdx$ first. What are the correct limits on the outer $dy$ integral then.
| {
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Is there a simpler way to calculate correlation?
Let's consider that a variable y constructed from x
$x_i ∈ \left\{1,3,5,7,8\right\}$
$f(x_i)=2x_i+1$
$y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $
where $ε_i$ is a identically and independantly distributed random variable which follows a normal law $\mathcal{N(0,2)}$... | Hints:
Actually $\sigma^2_x=8$ so $\sigma_x \approx 2.83$.
It is $\sigma_{f(x)}$ which is about $5.66$.
You should then be able to calculate $\sigma^2_y$ (an integer) since it is $f(x)+\epsilon$, assuming the $x$ and $\epsilon$ are independent. That gives you $\sigma_y$.
The covariance $\sigma_{f(x) y}$ is equal to t... | {
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Is $\mp a$ actually different than $\pm a$? So, the way I understand $\pm a$ as a general concept is basically as follows: $\pm a$ is really just two numbers, functions, or whatever $a$ represents, but the catch is that one of the $a$'s is positive, and the other is negative.
All of that makes sense to me. Mathematicia... | On its own, $\mp a$ means the same thing as $\pm a$.
However -- and this is a big however -- you almost never see $\mp$ unless it occurs in an expression with $\pm$ being used as well. And then it means "the opposite of whatever sign $\pm$ is currently."
For example, the sum-or-difference of cube factorizations can be... | {
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Does $\sum_{n=1}^\infty \frac{1}{p_ng_n}$ diverge? I know of Euler's proof that the sum of the reciprocals of the primes diverges. But what if we multiply the primes by it's following prime gap.
In other words, is $$\sum_{n=1}^\infty \frac{1}{p_ng_n} = \infty$$ true or false?
| TRUE: We may get rid of the prime gaps by using Titu's lemma.
We have:
$$ \frac{1}{p_n g_n}+\ldots+\frac{1}{p_N g_N}\geq \frac{\left(\sum_{k=n}^{N}\frac{1}{\sqrt{p_k}}\right)^2}{p_N-p_n}\tag{1}$$
hence if $N$ is around $n^2$ and $n$ is big enough, by partial summation the RHS of $(1)$ is roughly:
$$ 4\cdot\frac{p_N+p_n... | {
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Profit Share Distribution > 100% Total? So I read an article today which asserted the following:
Canaccord’s latest estimate shows Samsung making 15 percent of profits in smartphones, with Apple making 92 percent. (The numbers add up to more than 100 because everyone else in the smartphone industry loses money, so the... | Particularly in the press, people often do silly things with percentages, but the values you quote are possible. It appears they are computing the total profits of the industry and allocating it to the various companies. This would guarantee that the percentages add to $100\%$, but then it appears that the values wer... | {
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The integer $n$ is not zero if and only if there is some prime $p>n$ such that $p-n$ is composite I recently solved a problem (it was posed in a spanish-talking forum) in which I used the following lemma.
The integer $n$ is not zero if and only if there is a prime $p>n$ such that $p-n$ is composite.
This is my proof ... | For any non-zero $n$, some arithmetic progression mod $n$ contains infinitely many primes (this doesn't require Dirichlet's theorem, only Euclid + pigeonhole). But if $n$ is a counterexample, then the stated condition forces every (sufficiently large) term of that arithmetic progression to be simultaneously prime (thi... | {
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"source": "stackexchange",
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Proof by Induction for Natural Numbers Show that if the statement
$$1 + 2 + 2^{2} + ... + 2^{n - 1} = 2^{n}$$
is assumed to be true for some $n,$ then it can be proved to be true for $n + 1.$ Is the statement true for all $n$?
Intuitively, then I don't think it holds for all $n.$
| If we assume that $1+2+\cdots+2^{n-1}=2^n$, we can easily prove that $1+2+\cdots+2^n=2^{n+1}$. For
$$1+2+\cdots+2^n=(1+2+\cdots+2^{n-1})+2^n=2^n+2^n=2^{n+1}.$$
But of course it is not true that $1+2+\cdots+2^{n-1}=2^n$. The base case $n=1$ does not hold. For then $1+2+\cdots+2^{n-1}=1\ne 2^1$.
The whole point of this... | {
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$A \in B$ vs. $A \subset B$ for proofs I have to prove a few different statements.
The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differs.
Prove that if $A \in B$ and $B \in C$ then $A \in C$.
I don't really understand ... | Take $A=\varnothing$, $B=\{\varnothing\}$ and $C=\{\{\varnothing\}\}$.
Then clearly $A\in B\wedge B\in C$, but $A\in C$ implies $\varnothing=\{\varnothing\}$ wich cannot be true.
This because $\{\varnothing\}$ has elements and $\varnothing$ has not.
We conclude that the implication is false.
| {
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Solve this exponential inequality $$
{5}^{(x+2)}+{25}^{(x+1)}>750\\
t=5^{(x+1)}\\
t^2+5t-750>0\\
t^2+5t-750=0\\
$$
$$
a=1, b=5, c= -750\\
D=35+3000=3025 \\
t_1= 25; t_2=-30
$$
$t_1=25\Longrightarrow x=1; t_2=-30$ Doesn't have a solution
So the solution is $x>1$.
Is this correct? I say is bigger than $1$ and not smaller... | $$5^{(x+2)} = 25 \cdot 5^x$$
$$25^{(x+1)} = 25 \cdot 25^x$$
$$750 = 25 \cdot 30$$
So your inequation may be expressed:
$$5^x + 25^x >30$$
As the derivative of $5^x + 25^x$ is always possitive, there is only one $x$ that makes the equality true:
$$x=1$$
And therefore, the solution of your inequation is $x>1$
| {
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Estimate for a specific series For a positive integer $m$ define
$$
a_m=\prod_{p\mid m}(1-p),
$$
where the product is taken over all prime divisors of $m$, and
$$
S_n=\sum_{m=1}^n a_n.
$$
I am interested in an estimate for $|S_n|$. Any references, hints, ideas, etc., will be appreciated.
| As Oussama Boussif notice, using the Euler product for the totient function $$\phi\left(m\right)=m\prod_{p\mid m}\left(1-\frac{1}{p}\right)
$$ we have $$a_{m}=\prod_{p\mid m}\left(1-p\right)=\left(-1\right)^{\omega\left(m\right)}\frac{\phi\left(m\right)}{m}\prod_{p\mid m}p
$$ where $\omega\left(m\right)
$ is the num... | {
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Show that there are $c,d \in (a,b), c\lt d$ such that $\frac{1}{b-a}\int_a^b f=\frac{1}{d-c}\int_c^d f$. Let $f: [a,b] \to R$ be continuous.
Show that there are $c,d \in (a,b), c\lt d$ such that $\frac{1}{b-a}\int_a^b f=\frac{1}{d-c}\int_c^d f$.
This is my solution. I tried to work from backwards. Letting $F(x)=\int_a... | To get some intuition, consider the case $F(b) = 0$ (here using your notation). Assume the maximum value of $F$ occurs at $x_0 \in (a,b),$ with $F(x_0) > 0.$ (Good to draw a picture.) Then by the IVT every value between $0$ and $F(x_0)$ will be taken by $F$ in both of the intervals $(a,x_0)$ and $(x_0,b).$ So for each ... | {
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Can someone show by vector means that any inscribed angle in a semicircle is a right angle Could someone explain how to prove any angle inscribed in a semicircle is a right angle using vectors. I understand that the dot product of two vectors is 0 is they are perpendicular but I don't know how to show this in a semicir... | If the semicirce has radius $a$ you can represents the two vectors as the difference between the coordinates of the points $(-a,0)$ and $(a,0)$ with respect to a generic point $(a \cos \theta, a \sin \theta)$ :
$$
\vec v_1=(a\cos \theta -a; a \sin \theta)^T \quad and \quad \vec v_2=(a\cos \theta +a, a \sin \theta)^T
$... | {
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if $\alpha$ is an ordinal is it true that ${\aleph _{\alpha +1}}^{\aleph _{\alpha}}=\aleph _{\alpha +1}$? If we denote the following cardinals: $\beta _0=\aleph _0$, $\beta _k=2^{\beta _{k-1}}$
then I know that ${\beta _{k+1}}^{\beta _k}=\beta _{k+1}$
but, is it true that for some ordinal $\alpha$,
${\aleph _{\alpha+... | No, that is not necessarily true. In particular, whenever the continuum hypothesis fails so we have $\aleph_1 < 2^{\aleph_0}$, it will still be the case that
$$ \aleph_1^{\aleph_0} \ge 2^{\aleph_0} $$
by simple inclusion.
This doesn't answer whether it is consistent with ZFC that $\aleph_{\alpha+1}{}^{\aleph_\alpha} \... | {
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Let $f: A \times B \rightarrow C$ be continuous and closed under product of closed subsets of A and B, is $f$ closed? Assume product topology on $A \times B$. To make clear th title:
$f$ is a countinuous map such that if $R \subset A$ and $S \subset B$ are closed sets, then $f(R \times S) \subset C$ is a closed set of ... | Not in general.
In this context let $f:\mathbb R^2\rightarrow\mathbb R$ be prescribed by $\langle x,y\rangle\mapsto x$, let $\mathbb R$ be equipped with its usual topology, and $\mathbb R^2$ with the product topology.
Then $f$ is continuous and $f(R\times S)=R$ for $R,X\subseteq\mathbb R$, so the conditions mentioned i... | {
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Deriving the Hessian from the limit definition of the derivative Could someone possibly help me understand how I can derive the Hessian matrix of a twice-differentiable function $f$ defined on $\mathbb{R}^n$ using the limit definition of the second derivative. Namely, how does: $\lim_{h -> 0}\frac{\nabla f(x+h) - \nabl... | Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable function.
Then, at a point $p$, the derivative $Df\big|_p: \mathbb{R}^n \to \mathbb{R}$ can be computed by (but is not defined by)
$$
Df\big|_p(v) = \lim_{h \to 0} \frac{f(p+hv)-f(p)}{h}
$$
If $f$ is differentiable then $Df\big|_p$ is a linear function from $\mat... | {
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Prove that any integer that is both square and cube is congruent modulo 36 to 0,1,9,28 This is from Burton Revised Edition, 4.2.10(e) - I found a copy of this old edition for 50 cents.
Prove that if an integer $a$ is both a square and a cube then $a \equiv 0,1,9, \textrm{ or } 28 (\textrm{ mod}\ 36)$
An outline of ... | What you did is correct, but yes, a lot of the work (especially the computer check) could have been avoided.
Firstly, if $a$ is both a square and a cube, then it is a sixth power.
This is because, for any prime $p$, $p$ divides $a$ an even number of times (since it is a square), and a multiple of 3 number of times (sin... | {
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} |
Why are the following functions not equivalent I'm trying to find
$$\lim_{x\to-\infty}\sqrt{x^2 + 2x} - \sqrt{x^2 - 2x}.$$
However, I kept getting $2$ instead of $-2$, so I graphed the function to see what was going on. See the picture below.
http://puu.sh/k7Y3f/684aeed22b.png
I found the issue in the steps I took, a... | If you do not like negative numbers, well, just apply a change of variable bringing them into positive numbers:
$$\begin{eqnarray*} \lim_{x\to -\infty}\left(\sqrt{x^2+2x}-\sqrt{x^2-2x}\right)&=&\lim_{z\to +\infty}\left(\sqrt{z^2-2z}-\sqrt{z^2+2z}\right)\\&=&\lim_{z\to +\infty}\frac{-4z}{\sqrt{z^2+2z}+\sqrt{z^2-2z}}=\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Regression Model with (Y,X) non-random? In regression, we assume that $(X,Y)$ are random variables following some certain distribution. How would the problem change if we do not assume $(X,Y)$ are randoms. Why can we just have $Y=f(X,\epsilon)$, where $(X,Y)$ are non-random, and $\epsilon$ is a random quantity??
| Regression has nothing to do with randomness. Regression means fitting some parametrized function or curve to some points.
That means to set values for the parameters and come up with some metric that describes the function or curve to fit better or worse than other functions or curves derived from other parameters. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate the whole area encosed by the curve $y^2=x^4(a-x^2),a>0$ Calculate the whole area encosed by the curve $y^2=x^4(a-x^2),a>0$.
I could not plot this curve,so could not find the area.I tried wolframalpha also.Here $a$ is not specified.Required area is $\frac{\pi a^2}{4}$.Please help me.
| $\qquad\qquad$
The figure corresponds to the the case $a=1$.
The desired area is
$${\frak A}=2\int_{-\sqrt a}^{\sqrt a}x^2\sqrt{a-x^2}dx$$
Now, the change of variables $ x=\sqrt{a}\cos(t/2)$ yields
$${\frak A}=a^2\int_{ 0}^{2\pi}\cos^2\frac{t}{2}\sin^2\frac{t}{2}dt=\frac{a^2}{4}
\int_{ 0}^{2\pi}\sin^2tdt=\frac{\pi a^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is it useful to write a vector as a finite (or infinite) linear combination of basis vectors? I'm working on a project in an applied mathematics course and a professor asked me a basic question:
What is useful about writing any element of a vector space in terms of a possibly infinite linear combination of basis ve... | Sometimes there is a linear transformation that you want to understand. If you can find a basis such that the linear transformation has a simple effect on the basis vectors (like it simply scales them, for example) then this helps a lot to understand what the linear transformation does to an arbitrary vector (which can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
What is the intersective curve between sphere and a right cone? I am confused this picture : What the curve is?
I think that the curve is not circle and not the ellipse too,
What is the intersective curve?
| Consider a plane $\alpha$ parallel to the plane of the red-green coordinate lines. Let this plane contain the center of the red sphere. The brownish right cone in question (if it is a circular cone) intersects $\alpha$ in an ellipse $\mathscr E$. The curve, $\mathscr C$, whose shape we are interested in is the invers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
inequality between matrix norms A is a $n\times k $ matrix.
I have to show that $\|A\|_2\leq \sqrt{\|A\|_1\cdot \|A\|_\infty}$.
I know that $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| $ for every $\| \cdot \|$ submultiplicative matrix norm, but I don't know how to conclude.
Any idea?
| $\|A\|_{2}^2\leq trac(A^H\cdot A)\leq{\Vert A^H\cdot A\Vert_1\leq \| A\|_1\|A\| _\infty}$.
for more information see $D.46$ and $D.52$ of abstract harmonic analysis hewitt&ross pages 706 and 709
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
evaluate the Limit $$ \lim \limits_{z \to 0} z\sin(1/z^2)$$
Anyone can help me with this question? Not sure how to solve this. I tried to bring z to denominator but don know how to continue.
| A slight variation: Let y= 1/z. As z goes to 0, y goes to infinity so this limit becomes $\lim_{y\to\infty} \frac{sin(y^2)}{y}$. Now, as juantheron and Thomas said, sin(y) is always between -1 and 1:
$$-\frac{1}{y}< \frac{sin(y^2)}{y}< \frac{1}{y}$$
and both ends go to 0 as y goes to infinity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find the general integral of $ px(z-2y^2)=(z-qy)(z-y^2-2x^3).$ $ p=\frac{\partial z}{\partial x} $ and $ q=\frac{\partial z}{\partial y} $
Find the general integral of the linear PDE $ px(z-2y^2)=(z-qy)(z-y^2-2x^3). $
My attempt to solve this is as follows:
$ p=\frac{\partial z}{\partial x} $ and $ q=\frac{\partial z}{... | Your calculus is correct.
A first family of characteristic curves comes from $\frac{dx}{x(z-2y^2)}=\frac{dy}{y(z-y^2-2x^3)}$ which solution is $z=\frac{1}{c_1}y=c'_1y$
$$\frac{z}{y}=c'_1$$
A second family of characteristic curves comes from
$$\frac{dx}{x(c'_1y-2y^2)}=\frac{dy}{y(c'_1y-y^2-2x^3)}$$
The solution of this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Probability of getting more Heads between two gamblers I'm trying to solve this probability question I thought of. If two gamblers are playing a coin toss game and Gambler A has $(n+2)$ and B has $n$ fair coins. What is the probability that A will have more heads than B if both flip all their coins?
I tried to solve i... | You were right for using symmetry. Suppose A and B both have n coins. The probability of A winning would be 0.5. Now suppose A has $(n+2)$ coins and flips $n$ coins first then flips the other 2 coins. The probability is 0.5 for A to win with $n$ coins, and adding the 2 coins, the probability of at least one of them bei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Multivariable function limit How to approach this: $\lim\limits_{(x,y)\to(0,0)}\frac{x^2y}{x^2+y}$?
Been able to grind $\lim\limits_{(x,y)\to(0,0)}\frac{x^2y}{x^2+y^2}$, it is in(link) finnish, but formulas and idea should be selfevident. However $\dot +y$ instead of $\dots+y^2$ in divisor confuses me.
Or am I thinking... | $f(x,y)=\frac{x^2y}{x^2+y}$
Suppose the limit exist and it is finite. Then, let's take $\epsilon > 0$. There is $\delta > 0$ so that $|\frac{x^2y}{x^2+y} - L|<\epsilon$ for all $x,y$ so that $x^2 + y^2 < \delta ^2$.
Let's consider $x_n= \frac{1}{\sqrt n}, y_n= -\frac{1}{n + 1}$. There is N so that $x_n^2 + y_n^2<\delt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Find a metric space that cannot be embedded in the Hilbert cube. This is homework and I'm not looking for an answer.
I just finished an exercise that asked to prove that every metric compact space can be embedded in the Hilbert cube. Knowing this I can see that I have to find a non-compact metric space to start, but I ... | HINT: It isn’t really compactness that matters here: it’s second countability. You want a metric space that isn’t second countable; equivalently, you want one that isn’t separable. There are some really simple non-separable metric spaces. A further hint is in the spoiler-protected block below.
Try discrete spaces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How do we reach the answer to the following recursive problem? $\large{a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}, a_1 = 1}$
Let the sequence $\large{\left< a_n \right>}$ be defined as above for all positive integers n. Evaluate $\large{\left \lfloor a_{2015} \right \rfloor}$.
I wrote a C++ program to solve the problem bu... | If we let $x_n=a_n/\sqrt n$ then we have the recursive rule for $x_n$:
$$
x_{n+1}={x_n\over\sqrt{n(n+1)}}+{1\over x_n}\sqrt{n\over n+1}.
$$
It is easy to prove that $\lim_{n\to\infty}x_n=1$, so that $a_n/\sqrt{2015}$ should be very close to $1$. In other words we may conclude that
$$
\lfloor a_{2015} \rfloor = \lfloor ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why is this not allowed to solve a differential equation? How come you can't just integrate like this?$$y'=2y+x \implies y=2yx+\frac{x^2}{2}+C$$
| Because $y$ is a function of $x$. Writing $y = f(x)$, then the first term on the right you are integrating is
$$\int 2y \ dx = \int 2f(x) \ dx$$
That term is equal to $2yx$ if and only if $f$ is a constant function. In general, such an assumption is dangerous as it could be wrong.
It is wrong in this case: if $f$ is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
In how many ways can a group of 5 boys and 5 girls be seated in a row of 10 seats? In how many ways can a group of 5 boys and 5 girls be seated in a row of 10 seats?
Still having some confusion with the difference between combinations and permutations. I have tried this problem using both combination and permutation wi... | Ladies first: choose $5$ fixed places for the girls: $\binom{10}{5}$ - this (number of combinations) does not account for the order of the girls, whom you can permute in $5!$ ways. There are $5$ remaining places for the boys: Again, we can permute them in $5!$ ways. So the number is
$$\binom{10}{5}\cdot 5!\cdot 5!=\ldo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The value of $\gcd(2^n-1, 2^m+1)$ for $m < n$ I've seen this fact stated (or alluded to) in various places, but never proved:
Let $n$ be a positive integer, let $m \in \{1,2,...,n-1\}$. Then
$$\gcd(2^n-1, 2^m+1) =
\begin{cases}
1 & \text{if $n/\gcd(m,n)$ is odd} \\
2^{\gcd(m,n)}+1 & \text{if $n/\gcd(m,n)$... | Since $2^{2m}-1 = (2^m-1)(2^m+1)$, we have that
\begin{equation}
\gcd(2^n-1,2^m+1) {\large\mid} \gcd(2^n-1,2^{2m}-1) = 2^{\gcd(2m,n)}-1
\end{equation}
(Some proofs of the equality can be found
here, among other places.)
Case 1
If $n/\gcd(m,n)$ is odd, then $\gcd(2m,n) = \gcd(m,n)$, and so $\gcd(2^n-1,2^m+1)$ divides... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1433014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Whether to use AND or OR in describing an inequality My understanding is that the following:
$5 < x < 10$ is read as "x is greater than 5 AND less than 10,"
whereas the solution to $| x + 2 | > 4$,
which is $x > 2, x < -6$, should be read as "x is greater than 2 OR less than -6"
Am I using this correctly? I'm wonder... | Your first example is definitely correct. The chained inequality $5 < x < 10$ formally has the same meaning as 5 < x & x < 10, that is, "5 is less than x and x is less than 10", equivalent to "x is greater than 5 and less than 10", or in interval notation $(5, 10)$.
The second example is ambiguous, in that the piece o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1433134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Simple limit of a sequence
Need to solve this very simple limit $$ \lim _{x\to \infty
\:}\left(\sqrt[3]{3x^2+4x+1}-\sqrt[3]{3x^2+9x+2}\right) $$
I know how to solve these limits: by using
$a−b= \frac{a^3−b^3}{a^2+ab+b^2}$. The problem is that the standard way (not by using L'Hospital's rule) to solve this limit - ver... | Can you help with O-symbols? It's all right here?
$$f(x) = \sqrt[3]{3x^2}\left(1 + \frac{4}{9x} + O\left(\frac{1}{x^2}\right) - 1 - \frac{1}{x} -O \left(\frac{1}{x^2}\right)\right)= \sqrt[3]{3x^2} \left(\frac{-5}{9x} + \frac{1}{18x^2} \right). $$
Hence
$$\lim _{x\to \infty }\sqrt[3]{3x^2} \left(\frac{-5}{9x} + \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1433216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Discontinuity of Dirichlet function Define $$f(x)=
\begin{cases}
1, & \text{if }x\in\mathbb{Q}, \\
0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}.
\end{cases}$$Then $f$ has a discontinuity of the second kind at every point $x$, since neither $f(x+)$ nor $f(x-)$ exists.
Proof: We'll consider only for $f(x+)$.
Case 1. ... | Yes, your proof is correct.
To summarize, the key point is to "construct", or show the existence of, if one wishes, two decreasing or increasing convergent sequences $(x_n)$ and $(y_n)$ such that
they both converge to the same given point,
but $(f(x_n))$ and $(f(y_n))$ have different limits.
You implicitly use the foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1433308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
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