qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
757,702 | <p>I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as <span class="math-container">$P = Q\cdot L + R$</span></p>
<p>I am unable to solve the following:</p>
<blockquote>
<p>Show that <span class="mat... | enigne | 143,559 | <p>If $n=0$, then it is trivial.</p>
<p>For $n>0$, we have $$(x+1)^{2n+1}+x^{n+2}\\=(x^2+2x+1)(x+1)^{2n-1}+x^{n+2}\\=(x^2+x+1)(x+1)^{2n-1}+x(x+1)^{2n-1}+x^{n+2}\\=(x^2+x+1)(x+1)^{2n-1}+x((x+1)^{2n-1}+x^{n+1})$$.
By using induction, suppose $(x+1)^{2n-1}+x^{n+1}$ can be divided by $x^2+x+1$, then, $(x+1)^{2n+1}+x^{... |
1,705,656 | <p>I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it.</p>
<p>$$ g(x)=3x+4 , \quad f(g(x)) = \cos\left(x^2\right)$$</p>
<p>So far I've managed to get to the point where I have... | Roman83 | 309,360 | <p>$$y=g(x)=3x+4$$
$$x=\frac{y-4}{3} \Rightarrow g^{-1}(x)=\frac{x-4}{3}$$
$$f=f \circ (g \circ g^{-1})=(f \circ g) \circ g^{-1} =\cos \left(\frac{x-4}{3} \right)^2 $$</p>
|
3,007,785 | <p><span class="math-container">$\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$</span> I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, there is very difficult calculations. I have the answer <span class="math-container... | Community | -1 | <p>It is not so "very awful" as you say, if you tame the computation.</p>
<p>For the second term of the numerator, you can work this out as</p>
<p><span class="math-container">$$y^2=1+2x+2x^2,$$</span></p>
<p><span class="math-container">$$2yy'=2+4x,$$</span></p>
<p><span class="math-container">$$2y'^2+2yy''=4,$$</... |
3,007,785 | <p><span class="math-container">$\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$</span> I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, there is very difficult calculations. I have the answer <span class="math-container... | trancelocation | 467,003 | <p><span class="math-container">$$\begin{eqnarray*} \frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}
& = & \underbrace{\frac{1}{e^x+\sqrt{1+2x+2x^2}}}_{\mbox{harmless}}\cdot \frac{e^{2x}-(1+2x+2x^2)}{x+\tan (x)-\sin (2x)} \\
& \stackrel{\mbox{Taylor}}{=} & \frac{1}{e^x+\sqrt{1+2x+2x^2}}\cdot \frac{\fr... |
1,952,337 | <p>I have consulted some answers on SE already, such as these two:</p>
<p><a href="https://math.stackexchange.com/questions/473831/if-am-i-then-a-is-diagonalizable">If $A^m=I$ then A is Diagonalizable</a></p>
<p><a href="https://math.stackexchange.com/questions/507199/show-that-if-an-i-then-a-is-diagonalizable">Show ... | 211792 | 211,792 | <p>To show that $A$ is diagonalizable, it's sufficient to show that its minimal polynomial has distinct roots. Since $A$ satisfies $A^m-I$, its minimal polynomial must divide $\lambda^m-1$, which has distinct roots. So the minimal polynomial has distinct roots, meaning that $A$ is diagonalizable.</p>
<p>For a proof ... |
1,952,337 | <p>I have consulted some answers on SE already, such as these two:</p>
<p><a href="https://math.stackexchange.com/questions/473831/if-am-i-then-a-is-diagonalizable">If $A^m=I$ then A is Diagonalizable</a></p>
<p><a href="https://math.stackexchange.com/questions/507199/show-that-if-an-i-then-a-is-diagonalizable">Show ... | Martin Argerami | 22,857 | <p>Here is a different argument that does not use characteristic or minimal polynomials. </p>
<p>We have that $A$ is similar to its Jordan form, i.e., $A=SJS^{-1}$. Then
$$ J^m=(S^{-1}JS)^m=A^m=I. $$</p>
<p>Now, to have $J^m=I$, we need that each Jordan block $J_k(\lambda)$ satisfies $J_k(\lambda)^m=I_k$. This can o... |
4,351,725 | <p>This is what the solution says:</p>
<p>Since each string of 4 digits are independent, having 2018 in a string has probability of <span class="math-container">$(1/10)^4$</span></p>
<p>By geometric distribution, expected value of digits to obtain 2018 in a string would be <span class="math-container">$10^4$</span></p>... | gnasher729 | 137,175 | <p>The digits of 2^n are pseudo-random (hard to predict, behaving like random except they are of course deterministic) except for the first and last few digits. So if you calculate values 2^n until you produced 10,000 digits, there is a good chance that four consecutive digits equal 2018, or any other four digit sequen... |
118,486 | <p>I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.</p>
<p>Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ generates two operators $pre: \wp(B) \to \wp(A)$ and $post: \wp(A) \to \wp(B)$ where $pre$ maps a set $X \subseteq ... | Qiaochu Yuan | 290 | <p>You should think of the preimage as taking the pullback of $\mathbb{F}_2$-valued functions. For any topological space $X$, the space of continuous functions $X \to \mathbb{F}_2$ may be identified with the Boolean algebra / ring of clopen subsets of $X$, the pullback of such a function is another such function, and t... |
3,780,781 | <p>How can I shift selected column of a matrix (column wise) by matrix multiplication?</p>
<p>Let's say I have a matrix like this:
<span class="math-container">$$A=\begin{pmatrix}1 & 0 & 0\\\ 1 & 0 & 1 \\\ 1&1&1\end{pmatrix}$$</span></p>
<p>and I want to shift the first column as well as the thi... | A learner | 737,804 | <p><strong>Hints</strong> : as <span class="math-container">$A$</span> is invertible, just find <span class="math-container">$A^{-1}$</span>.</p>
<p>Then , <span class="math-container">$S=A^{-1} B $</span></p>
<p>Can you continue ?</p>
|
3,780,781 | <p>How can I shift selected column of a matrix (column wise) by matrix multiplication?</p>
<p>Let's say I have a matrix like this:
<span class="math-container">$$A=\begin{pmatrix}1 & 0 & 0\\\ 1 & 0 & 1 \\\ 1&1&1\end{pmatrix}$$</span></p>
<p>and I want to shift the first column as well as the thi... | user | 505,767 | <p>We can operate by left multiplication which corresponds to operation on the rows, notably in this case we exchange first and second rows by the permutation matrix</p>
<p><span class="math-container">$$S_1 = \begin{pmatrix}0 & 1 & 0\\\ 1 & 0 & 0 \\\ 0&0&1\end{pmatrix}$$</span></p>
<p>then we e... |
4,244,983 | <p>In maths it is assumed that every statement is either true (t) or false (f). When proving theorems such as: For all vector spaces <span class="math-container">$V$</span> over a field <span class="math-container">$K$</span> and <span class="math-container">$\lambda \in K$</span> it holds that <span class="math-contai... | Léreau | 351,999 | <p>It seems to me that, so far you have encountered the (metaphorically speaking) coin that represents <em>logic and proofs</em> but you haven't seen or been shown yet, that it really has two individual sides.</p>
<p>There is as a deductive and a semantic side of it.</p>
<p>On the <strong>deductive side</strong>, one s... |
21,182 | <p>In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.</p>
<p>Does anyone have ... | Miguel | 5,316 | <p>I would say the "correct conditions" for a statement are discovered by trying to prove it and seeing what you need to be true in order for the statement to follow.</p>
<p>Proofs are generally discovered in the opposite direction than they are written because proofs are written for elegance and conciseness, not for ... |
21,182 | <p>In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.</p>
<p>Does anyone have ... | cheater | 5,251 | <p>My first intuition is always to talk to somebody. This can help, since the speech center is a different part of the brain, works differently, taps different parts of the brain. In that way, even speaking out loud what you want to talk about can supply you with a surge of inspiration. Don't forget that written words ... |
2,495,176 | <p>For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?</p>
<p>Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?</p>
| Olimjon | 300,977 | <p>Your answer makes sense.</p>
<p>Minimum 4 digit number is $1000$</p>
<p>Maximum 4 digit number is $9999$</p>
<p>$$max = 3n = 9999$$</p>
<p>$$n_{max}=3333$$</p>
<p>$$min=\frac{n}{3}=1000$$</p>
<p>$$n_{min}=3000$$</p>
<p>Keep in mind that n must be divisible by 3. So, the answer would be: $$\frac{3333-3000}{3}+... |
2,495,176 | <p>For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?</p>
<p>Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?</p>
| Piyush Agrawal | 497,053 | <p>112 values is the number of positive values whose n/3 and n*3 both are 4-digit numbers.</p>
|
2,299,678 | <p>Question:</p>
<p>Assume $x, y$ are elements of a field $F$. Prove that if $xy = 0$, then $x = 0$ or $y = 0$.</p>
<p>My thinking:</p>
<p>I am not sure how to prove this. <strong>I can only use basic field axioms.</strong> Should I assume that both x and y are not equal to 0 and then prove by contradiction or shoul... | baharampuri | 50,080 | <p>In a field every nonzero element has a multiplicative inverse. If $x \neq 0$ there is an inverse let us say $t$, and $xy =0$ so $t(xy)=t0=0$ or $(tx)y=0$ or $1 \times y=0$ which shows $y=0$. Similarly if $y$ is not zero we can show $x$ is zero using the multiplicative inverse of $y$.</p>
|
101,078 | <p>While reading through several articles concerned with mathematical constants, I kept on finding things like this:</p>
<blockquote>
<p>The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. </p>
<p>The <strong>high-water marks</strong> are ... | Jonas Kibelbek | 1,461 | <p>The term comes from the common practice of marking bridges or walls with the water level and date during big floods. So, you can look at these high-water marks to see that in 1985, the river flooded to 5 feet above its current level. If you come back a decade later, you may find that a new high-water mark has surp... |
220,946 | <p>I am a newcomer to Mathematica and I am trying to solve this differential equation:</p>
<pre><code>s = NDSolve[{y'[x] == -4/3*y[x]/x + 4 a/(3 x^2) + 4/(9 x^3) a b -
8/(9 x^3) c, y[0.01] == 20}, y, {x, 0.01, 10}]
</code></pre>
<p>I am getting the error message "Encountered non-numerical value for a derivativ... | kglr | 125 | <p><strong>Update:</strong> Inspired by @thorimur's excellent answer, an alternative way to get a maximal matching:</p>
<pre><code>ClearAll[swapRows]
swapRows = #[[SparseArray`MaximalBipartiteMatching[# /.
Except[List, _Symbol] -> 1][[All, 1]]]] &;
</code></pre>
<p><strong><em>Examples:</em></strong></p>
... |
220,946 | <p>I am a newcomer to Mathematica and I am trying to solve this differential equation:</p>
<pre><code>s = NDSolve[{y'[x] == -4/3*y[x]/x + 4 a/(3 x^2) + 4/(9 x^3) a b -
8/(9 x^3) c, y[0.01] == 20}, y, {x, 0.01, 10}]
</code></pre>
<p>I am getting the error message "Encountered non-numerical value for a derivativ... | thorimur | 63,584 | <p>The question can be rephrased as:</p>
<blockquote>
<p>Can we assign an index to each row such that 1) the row contains a nonzero entry at that index and 2) no other row is assigned that index?</p>
</blockquote>
<p>Each row has a set of what we might call <em>relevant indices</em>, that is, the set of indices at ... |
1,573,341 | <blockquote>
<p>Prove that the roots of the equation $z^n=(z+3)^n$ are collinear</p>
</blockquote>
<p>Taking modulus on both sides, $$\vert z-0\vert =\vert z-3\vert$$</p>
<p>This represents the perpendicular bisector of line joining $x=0$ and $x=3$</p>
<p>That was easy. But I tried to solve it using algebra:</p>
... | Community | -1 | <p>$\left(\dfrac{z+3}{z}\right)=(1)^n= e^{\dfrac{2\pi i k}{n}}$ where k=0,1,2.....(n-1)</p>
<p>$\left(1+\dfrac{3}{z}\right)=(1)^n= \alpha^k$ where $\alpha=e^{\dfrac{2\pi i}{n}}$</p>
<p>$\implies z=\dfrac{3}{\alpha^k-1} ;k\neq0$</p>
<p>roots of your equation will be $z=\dfrac{3}{\alpha-1},\dfrac{3}{\alpha^2-1},......... |
4,360,054 | <p><span class="math-container">$\textbf{Question}$</span>: Show that there exist an uncountable subset <span class="math-container">$X$</span> of <span class="math-container">$\mathbb{R}^{n}$</span> with property that every subset of <span class="math-container">$X$</span> with <span class="math-container">$n$</span> ... | Bart Michels | 43,288 | <p>You can consider all such subsets of <span class="math-container">$\mathbb R^n$</span> (not necessarily uncountable). They form a poset under inclusion and it is easy to see that every chain has an upper bound (their union).
By Zorn's lemma, there exists a maximal subset <span class="math-container">$S$</span> of <s... |
3,552,695 | <p>The question is to find all real numbers solutions to the system of equations: </p>
<ul>
<li><span class="math-container">$y=\Large\frac{4x^2}{4x^2+1}$</span>,</li>
<li><span class="math-container">$z=\Large\frac{4y^2}{4y^2+1}$</span>,</li>
<li><span class="math-container">$x=\Large\frac{4z^2}{4z^2+1}$</span>, </... | Daniyar Aubekerov | 741,637 | <p>We can note that <span class="math-container">$4a^2+1 \ge 4a \ \ \forall a \in R$</span>. Also, since <span class="math-container">$\frac{4a^2}{4a^2+1} \ge 0$</span>, then we know that <span class="math-container">$x,y,z \ge 0$</span></p>
<p>Therefore <span class="math-container">$y=\frac{4x^2}{4x^2+1} \le \frac{4x... |
3,552,695 | <p>The question is to find all real numbers solutions to the system of equations: </p>
<ul>
<li><span class="math-container">$y=\Large\frac{4x^2}{4x^2+1}$</span>,</li>
<li><span class="math-container">$z=\Large\frac{4y^2}{4y^2+1}$</span>,</li>
<li><span class="math-container">$x=\Large\frac{4z^2}{4z^2+1}$</span>, </... | jacky | 14,096 | <p><span class="math-container">$\bullet\; $</span> Clearly <span class="math-container">$x=y=z=0$</span> are the solution of system of eqn</p>
<p><span class="math-container">$\bullet\; $</span> If <span class="math-container">$x,y,z\neq 0\; $</span> Then</p>
<ul>
<li><span class="math-container">$\displaystyle y=\f... |
4,026,795 | <blockquote>
<p>Let <span class="math-container">$V=\{(x,y,z)\in\mathbb{R^3}: x^2+y^2\le z, z\le x+2\}$</span></p>
<p>Then the volume of V is:</p>
<p>(A) Vol(V) = <span class="math-container">$\frac{75}{8}\pi$</span></p>
<p>(B) Vol(V) = <span class="math-container">$\frac{81}{32}\pi$</span></p>
<p>(C) Vol(V) = <span cl... | John Wayland Bales | 246,513 | <p>If we let <span class="math-container">$G^\prime(x)=g(x)$</span> then we have <span class="math-container">$f(x)=G(1)-G(x)$</span>. Then <span class="math-container">$f^\prime(x)=-G^\prime(x)=-g(x)$</span>.</p>
<p>We know that <span class="math-container">$f$</span> will have a minimum value at the value <span class... |
622,883 | <p>I'm finding maximum and minimum of a function $f(x,y,z)=x^2+y^2+z^2$ subject to $g(x,y,z)=x^3+y^3-z^3=3$.</p>
<p>By the method of Lagrange multiplier, $\bigtriangledown f=\lambda \bigtriangledown g$ and $g=3$ give critical points. So I tried to solve these equalities, i.e.</p>
<p>$\quad 2x=3\lambda x^2,\quad 2y=3\... | Eric Thoma | 35,667 | <p>For an arbitrary set of natural numbers, there is no shorter way to count the number of even numbers other than considering each number individually.</p>
<p>I will assume that you are considering a set of natural numbers forming an arithmetic progression. The sequence starts with $a_1$ and $a_{k+1} = a_{k} + d$. Th... |
2,733,142 | <p>This question has been asked a <a href="https://math.stackexchange.com/questions/207029/a-b2-for-which-matrix-a">few</a> <a href="https://math.stackexchange.com/questions/583442/square-root-of-nilpotent-matrix">times</a>. In the former case I noticed that there was some argument trending towards using the Jordan fo... | Theo Bendit | 248,286 | <p>I'm not 100% sure if I follow, but it looks invalid. When you write $A^2 = \lambda I + N'$, where $N'$ and $A^2$ are both nilpotent, the only scalar $\lambda$ that can possibly fit the bill here is $\lambda = 0$. Why? Because adding $\lambda I$ to a matrix increases all eigenvalues by $\lambda$, and nilpotent matric... |
4,010,160 | <h2>The Problem</h2>
<p>Let <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> be a continuous i.i.d. random variables. By definition, we know that...</p>
<ol>
<li>Independence: <span class="math-container">$\mathbb P(X_1 \in A \ \cap \ X_2 \in B)=\mathbb P(X_1 \in A) \ \mathbb P(X... | Hayden | 27,496 | <p>No, consider <span class="math-container">$X = [0,1]$</span> and <span class="math-container">$Y = [0,1] \cup [2,3]$</span>, with their usual subspace topologies. You may let <span class="math-container">$A = [0,1/3] \cup [1/3,2/3]$</span> and <span class="math-container">$B = [0,1]$</span>. <span class="math-contai... |
2,716,036 | <p>In reviewing some old homework assignments, I found two problems that I really do not understand, despite the fact that I have the answers.</p>
<p>The first is: R(x, y) if y = 2^d * x for some nonnegative integer d. What I do not understand about this relation is how it can possibly be transitive (according to my n... | bitesizebo | 547,202 | <p>For your first relation, note you're allowed to use a different $d$ each time. So if $y = 2^dx$ and $x = 2^{d'} z$ then $y = 2^{d + d'} z$ so $R(y, z) $. </p>
<p>For your second relation, we need $R(x, x) $ for all $x$ in order to be reflexive, but (for example) 1 will not be related to 1 for this relation. </p>
|
1,187,706 | <p>I'd imagine this is a duplicate question, but I can't find it:</p>
<p>How many quadratic residues are there $\pmod{2^n}$. </p>
<p>I tried small $n$: $n=1: 2, n=2:2, n=3: 3, n=4: 4, n=5: $not 5: 0, 1, 4, 9, 16, 25</p>
<p>No pattern :/</p>
| Asinomás | 33,907 | <p>The multiplicative group $\bmod 2^n$ is isomorphic to $\mathbb Z_{2^{n-2}}\times \mathbb Z_2$ so we have $\mathbb Z_{2^{n-3}}$ odd quadratic residues.</p>
<p>How many even quadratic residues do we have? How many do we have of the form $d2^{2k}?$ with $d$ odd? $d$ is an odd quadratic residue $\bmod 2^{n-2k}$ so ther... |
2,900,283 | <p>I'm having trouble proving this theorem from Rudin: that if $x \neq 0$ then $\frac{1}{\frac{1}{x}} = x$. </p>
<p>Rudin seems to solve this by referring to an earlier result that $xy = xz$ for $x \neq 0$ implies that $y = z$. I haven't quite been able to grasp this approach, as we don't seem to have access to this a... | José Carlos Santos | 446,262 | <p>This is correct, except that the justification for the third equality is not appropriate. There is no “simplification” axiom. But, by definition of multiplicative inverse,$$\frac1x\cdot\frac1{\frac1x}=1,$$since, for each $z\neq0$, $z\cdot\frac1z=1$.</p>
|
2,900,283 | <p>I'm having trouble proving this theorem from Rudin: that if $x \neq 0$ then $\frac{1}{\frac{1}{x}} = x$. </p>
<p>Rudin seems to solve this by referring to an earlier result that $xy = xz$ for $x \neq 0$ implies that $y = z$. I haven't quite been able to grasp this approach, as we don't seem to have access to this a... | Community | -1 | <p>$$\frac1{\dfrac1x}=x\iff \dfrac1x\frac1{\dfrac1x}=\dfrac1xx\iff1=1.$$</p>
|
563,129 | <p>Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming that $\mathcal{X}$ coincides with the set of extreme points of $\text{conv}(\mathcal{X})$, we can write $Z = \sum_{i=1... | Alex Ravsky | 71,850 | <p>It seems the following.</p>
<p>We have no unique choice of $\alpha$. For instance, let $d=2$ and $\mathcal{X}$ be a set of vertices of a regular hexagon $H.$ Let $\mathcal{X_1}$ be a set of odd vertices of $H$ and
$\mathcal{X_2}$ be a set of even vertices of $H$. Then each point $z$ from the intersetion of triangl... |
563,129 | <p>Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming that $\mathcal{X}$ coincides with the set of extreme points of $\text{conv}(\mathcal{X})$, we can write $Z = \sum_{i=1... | Tomilov Anatoliy | 54,348 | <p>I can propose the next algorithm for picking the point from the polytope:</p>
<ol>
<li><p>Firstly, triangulate your polytope (it is simple for convex ones).</p>
</li>
<li><p>Calculate an absolute values of hypervolumes of simplexes from <strong>1.</strong> (<a href="https://math.stackexchange.com/a/828854/54348">hyp... |
2,727,237 | <p>$$
\begin{matrix}
1 & 0 & -2 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{matrix}
$$</p>
<p>I am told that the span of vectors equal $R^m$ where $m$ is the rows which has a pivot in it. So when describing the span of the above vectors, is it correct it saying that they don't span $... | Mohammad Riazi-Kermani | 514,496 | <p>The coordinates may be orthonormal but the basis vectors do not have to be orthonormal. </p>
<p>Let $v_1=(1,3)$ and $v_2= (3,2)$ then the coordinates of $v_1$ with respect to the basis $\{ v_1, v_2 \}$ is $(1,0)$.</p>
<p>Also the coordinates of $v_2$ with respect to the basis $\{ v_1, v_2 \}$ is $(0,1)$</p>
<p... |
3,542,485 | <p>Suppose my birthday fell on a Sunday. What is the maximum number of years I may have to wait till my birthday falls on a Sunday again?</p>
<p>My intuition says 11 years:</p>
<ul>
<li><p>At the earliest, the next Sunday is in 6 years time. Reason: before I can encounter another Sunday, there must be at least one le... | Toby Mak | 285,313 | <p><span class="math-container">$365$</span> leaves a remainder of <span class="math-container">$1$</span> when divided by <span class="math-container">$7$</span>. Therefore, without accounting for leap years, your next birthday (<span class="math-container">$365$</span> days after your previous birthday) will shift by... |
132,053 | <p>I have an expression $e^{x-a}(e^{bx}+c)$. I just want it multiplied throughout to get $e^{(b+1)x-a}+ce^{x-a}$. To achieve this I have tried <code>Expand</code>, <code>Simplify</code>, <code>Collect</code> (using pattern <code>Exp[q_*x]</code>). The best output I could get is</p>
<pre><code>Expand[Exp[x-a]*(Exp[b*x]... | corey979 | 22,013 | <p>Make a rule:</p>
<pre><code>Expand[Exp[x - a]*(Exp[b*x] + c)] /. Exp[a_] :> Exp[Collect[a, x]]
</code></pre>
<p><a href="https://i.stack.imgur.com/32DmV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/32DmV.png" alt="enter image description here"></a></p>
<p>See also:</p>
<ul>
<li><a href="... |
1,489,362 | <p>$$\lim_{x\to\infty} \left(\frac{3x^2}{\sqrt{4x^2+x+1}+\sqrt{x^2+x+1}}-x\right)$$</p>
<p>I need help in finding this limit.</p>
| mjo | 235,913 | <p><strong>Hint:</strong> Try putting it under the same denominator, then multiply by the denominators conjugate and see if nice stuff happens.</p>
|
2,145,098 | <p>This was part of a three part question where I was supposed to prove two sets have equal cardinality by finding bijections. I've created a bijection $f: \Bbb Z \Rightarrow 2\Bbb Z$ by $f(x)=2x$. I've created a bijection $g: (0,1) \Rightarrow (4,50)$ by $g(x)=46x+4$. I think those are both correct. My last question i... | Przemysław Scherwentke | 72,361 | <p>HINT: Find first the bijection between sets $\{1/n:n\in\mathbb{N}\}\cup\{0\}$ and $\{1/n:n\in\mathbb{N},\ n\geq2\}$</p>
|
2,071,828 | <p>Find the splitting field of $x^6-2x^4-8x^2+16$ over $\mathbb {F}_3$ and list the intermediate fields between the base camp and the splitting field.</p>
| Dietrich Burde | 83,966 | <p>Hint: Over $\mathbb{F}_3$ we have
$$
x^6-2x^4-8x^2+16=(x^2 + 2x + 2)(x^2 + x + 2)(x^2 + 1).
$$</p>
|
1,651,452 | <p>I am trying to solve a modular arithmetic system and I got to the point where I have to solve $22t \equiv 9 \pmod{7}$ for $t$. I researched on the internet and found that there are many ways to solve this, such as using linear diophantine equations, the euclidean algorithm or using inverses.</p>
<p>Can someone show... | Gregory Grant | 217,398 | <p>The way it works is that you can subtract as many $7$'s from the numbers as you want. So subtract three $7$'s from the $21$ and it simply becomes $1$. And subtract one $7$ from the $9$ and it becomes $2$. So this really just says $t\equiv 2\pmod 7$. And there's your solution. This was a particularly easy linear... |
2,046,492 | <p>Let's think of two events $1$ and $2$.</p>
<p>Both events happen randomly $n_1$/$n_2$-times during a given time $T$ and last for a time of $t_1$/$t_2$.</p>
<p><strong>What is probability $P$, that both events happen simultaneously at some moment?</strong></p>
<hr>
<hr>
<p><em>EXAMPLE 1:</em></p>
<p>$T = 60$ mi... | Thomas Andrews | 7,933 | <p>Consider the discrete version of this problem, where $T$, $t_1, t_2$ are integers under some fixed unit of time and the events always start at integer multiples of time.</p>
<p>Then you are selecting $n_1$ values $\{a_i\}$ from $0,...,T-t_1+1$ so that each adjacent pair differs by at least $t_1$, and $n_2$ values f... |
854,671 | <p>So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. </p>
<p>For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$.
So I kind of understand how to center the circle and solve this with polar coordinates. Since the circle equation is $(x-3)^2... | bof | 111,012 | <p>Look at a calendar and observe that the months of May, June, July, August, September, October, and November start on seven different days of the week. This will happen every year, because the number of days in each of those months is the same every year. Therefore one of those months will start on a Sunday, and so t... |
611,198 | <p>A corollary to the Intermediate Value Theorem is that if $f(x)$ is a continuous real-valued function on an interval $I$, then the set $f(I)$ is also an interval or a single point.</p>
<p>Is the converse true? Suppose $f(x)$ is defined on an interval $I$ and that $f(I)$ is an interval. Is $f(x)$ continuous on $I$? <... | Nick D. | 115,491 | <p>No to both! Define $f:[0,1]\rightarrow \Bbb{R}$, $f(x)=x$ for $x\in [0,\frac{1}{2})$, and $f(x)=\frac{3}{2}-x$ for $x\in[\frac{1}{2},1]$. $f$ is injective and $f([0,1])$ is an interval, but $f$ is not continuous.</p>
|
3,555,687 | <blockquote>
<p>There are 3 straight lines and two circles on a plane. They divide the plane into regions. Find the greatest possible number of regions.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/YbUAU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YbUAU.png" alt="enter image descrip... | David G. Stork | 210,401 | <p>I get 21:</p>
<p><a href="https://i.stack.imgur.com/XwOz5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XwOz5.png" alt="enter image description here"></a></p>
<p>There can be only six regions "at infinity" (as defined by the straight lines).</p>
<ul>
<li>6 "at infinity"</li>
<li>2 "local, but... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | Yanko | 426,577 | <p>Wow your approach to this question is way too difficult.</p>
<p>Let me first explain what's wrong with your approach before I give you another way to do this.</p>
<p>You wrote your term as a sum of <span class="math-container">$n$</span> terms, each converges to zero as <span class="math-container">$n$</span> goes... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | Jacob | 181,986 | <p><strong>Hint</strong> Look at <span class="math-container">$n\gt 10000$</span></p>
<blockquote class="spoiler">
<p> For <span class="math-container">$n\gt 10000$</span>, we have <span class="math-container">$0.9999 + \frac{1}{n} \leq 0.9999+ \frac{1}{10001}\lt1$</span>, so <span class="math-container">$\lim_{n\ri... |
29,443 | <p>I'm looking at a <a href="http://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29#Relation_to_thermodynamic_variables" rel="nofollow noreferrer">specific</a> derivation on wikipedia relevant to statistical mechanics and I don't understand a step.</p>
<p>$$ Z = \sum_s{e^{-\beta E_s}} $$</p>
<p>$... | Alex Becker | 8,173 | <p>The variance of a random variable $X$ is always defined as $<(X - <X>)^2>$; this is the expected square of the difference between the expected and actual values.</p>
|
556,807 | <p>What is the sum of this series ?</p>
<p>$(n-1)+(n-2)+(n-3)+...+(n-k)$ </p>
<p>$(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$</p>
<p>So how can we find the sum from $n-1$ to $n-k$ ?</p>
| Shobhit | 79,894 | <p>$$(n-1)+(n-2)\cdots(n-k)=\underbrace{n+n+\cdots +n}_{\text{$k$ copies}}-(1+2+\cdots k)=nk-\frac{k}{2}(k+1)$$</p>
|
1,841,173 | <blockquote>
<p>Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which says that any $7$-Sylow subgroup of $S_{20}$ is subset of $A_{20}$?</p>
</blockquote>
<p>I know that o... | Shyam Saurabh | 715,157 | <p>For the last option
Sn has two types of subgroups: a subgroup which contains half even and half odd permutations or it contains only even permutations.</p>
|
2,032,501 | <p>Let A and B be subsets of $R^n$ Define</p>
<p>$A+B=\{a+b\ |\ a\in\, A , b\in \,B\}$</p>
<p>Consider the sets
$W=\{(x,y) \in\,R^2\ |\ x>0 , y>0\}
\\ X=\{(x,y) \in\,R^2\ |\ x\in\,R , y=0\}
\\ Y=\{(x,y) \in\,R^2\ |\ xy=1\}
\\Z=\{(x,y) \in\,R^2\ |\ |x|\le 1,|y|\le 1\}$<br>
Which of the following statements are... | DanielWainfleet | 254,665 | <ol>
<li><p>Prove that $W+X=\{(x,y):y>0\}.$</p></li>
<li><p>For positive integer $n$ let $p_n=(-n,0)\in X$ and $q_n=(n,1/n)\in Y.$ Then $p_n+q_n=(0,1/n)$ which converges to $(0,0)$ as $n\to \infty.$ But $(0,0)\not \in X+Y$ because if $a=(x_1,0)\in X$ and $b=(x_2,1/x_2)\in Y$ then the second co-ordinate of $a+b$ is $... |
3,097,816 | <p>Show that the product of four consecutive odd integers is 16 less than a square.</p>
<p>For the first part I first did
<span class="math-container">$n=p(p+2)(p+4)(p+6)
=(p^2+6p)(p^2+6p+8).$</span>
I know that you are supposed to rearrange this to give an equation in the form <span class="math-container">$(ap^2+bp+c... | EngineerInProgress | 632,503 | <p>Let those consecutive odd integers be <span class="math-container">$2p-3 , 2p-1 , 2p+1, 2p+3$</span>. Then their product is:
<span class="math-container">$(2p-3)(2p+3)(2p-1)(2p+1)=(4p^2-9)(4p^2-1)=16p^4-40p^2+9$</span><br>
Next step is to notice that <span class="math-container">$16p^4-40p^2+9+16=16p^4-40p^2+25=(4p^... |
1,523,230 | <p>Prove that for any primitive Pythagorean triple (a, b, c), exactly one of a and b must be a multiple
of 3, and c cannot be a multiple of 3.</p>
<p><strong>My attempt:</strong></p>
<p>Let a and b be relatively prime positive integers.</p>
<p>If $a\equiv \pm1 \pmod{3}$ and $b\equiv \pm1 \pmod{3}$, </p>
<p>$c^2=a^2... | Mark Fischler | 150,362 | <p>Your first step was a good start: It proves that <em>at least one</em> of $a$ and $b$ must be a multiple of $3$. We will have to eliminate the case of <em>both</em> $a$ and $b$ being multiples of $3$, but that is easy: If $a=3p$ and $b=3q$ then
$$c^2 = a^2+b^2 = (3p)^2 + (3q)^2 = 9(p^2+q^2) = 9m$$ so $c$ is also... |
47,495 | <p>Firstly, let me divulge. I've been doing a lot of research on the summation of two coprime numbers and unfortunately have failed to come up with the properties I'm seeking; it is my hope that someone here might be of some help.</p>
<p>Let $(j, k)\in \mathbb{N}^2$ be coprime.</p>
<p>Can $\Omega(j + k)$ or $\omega(j... | Chris Wuthrich | 5,015 | <p>Obviously, $\Omega(j+k)$ is a function in $j$ and $k$, certainly it would be difficult to write it in a closed form other than $\Omega(j+k)$. Similar for $\omega(j+k)$. On the other hand neither of them can be a function in $\omega(j)$, $\omega(k)$, $\lambda(j)$, $\lambda(k)$ and $j$ only. For instance they take the... |
1,397,538 | <p>I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a <strong>reference</strong> which contains a statement of this fact. Any thoughts? Thanks.</p>
<p>Update:</p>
<p>I indeed found it in "Théorie des distributions" b... | David C. Ullrich | 248,223 | <p>Ok, you said you knew how to prove it. Others may not. And you may like this better than what you have (it's the second thing I always think of when this comes up, and <em>I</em> like it a lot better than the first thing I think of...)</p>
<p>Say $u$ is a distribution and $u'=0$. By definition $u(\phi')=0$ for any ... |
1,397,538 | <p>I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a <strong>reference</strong> which contains a statement of this fact. Any thoughts? Thanks.</p>
<p>Update:</p>
<p>I indeed found it in "Théorie des distributions" b... | Lorenzo Pompili | 884,561 | <p>I may be aware that this is very bad timing, but I wanted to point out the fact that there is an easy proof also in <span class="math-container">$\mathbb{R}^N$</span>. Source: lecture notes by Professor P. D'Ancona (in Italian). <a href="https://www1.mat.uniroma1.it/people/dancona/IstAnSup/dispense-esercizi/4-Distri... |
1,615,732 | <p>Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, so we have $\widetilde H_i(X)=0$ for all $i$. And if $\pi_2(X)=0$, we have that $X$ is $K(\pi,1)$.</p>
<p>My questio... | Andrey Ryabichev | 257,176 | <p>No, $\pi_2(X)=\mathbb Z^{119}$, is is obvious by considering Euler characteristic.</p>
<p>Denote universal cover of $X$ by $\widetilde X$. In Hatcher we see (and i don't know how to prove it) that $\pi_1(X)$ has order $120$. So $\widetilde X$ has 120 $0$-cells, 240 $1$-cells, and 240 $2$-cells; therefore $\chi(\wid... |
119,686 | <p>I have a very dumb question. Let $X = \mathbb{P}^2_k = Proj(k[x,y,z])$ where $k$ is algebraically closed. We have an invertible sheaf $\mathcal{O}(2)$ on $X$. Its space of global sections contains the elements $x^2, y^2, z^2, xy, yz, xz$. </p>
<p>It seems to me (by my calculations), however, that $\mathcal{O}(2... | Georges Elencwajg | 3,217 | <p>One should not confuse the two different statements :<br>
a) The sections $x^2,y^2,z^2$ generate the $k$-vector space $\Gamma(\mathbb P^2_k,\mathcal O(2))$, which happens to be false.<br>
b) The sections $x^2,y^2,z^2$ generate the line bundle $O(2)$, which happens to be true. </p>
<p>a) The first statement probabl... |
3,048,063 | <p>I have to prove this preposition by mathematical induction:</p>
<p><span class="math-container">$$\left(x^n+1\right)<\left(x+1\right)^n \quad \forall n\geq 2 \quad \text{and}\quad x>0,\,\, n \in \mathbb{N}$$</span></p>
<p>I started the prove with <span class="math-container">$n=2$</span>:</p>
<p><span class... | Word Shallow | 466,835 | <p>We will prove it is true with <span class="math-container">$n=k$</span>. Indeed we need to prove it is true with <span class="math-container">$n=k+1$</span>.</p>
<p>Or we need to prove <span class="math-container">$x^{k+1}+1<\left(x+1\right)^{k+1}$</span> </p>
<p>We have: <span class="math-container">$RHS=(x+1... |
4,623,748 | <p>Let <span class="math-container">$X,Y$</span> be smooth vector fields on the unit circle <span class="math-container">$M = S^1$</span> such that <span class="math-container">$[X,Y] = 0$</span>, i.e. (treating tangent vectors as derivations) <span class="math-container">$X(x)(Yf) = Y(x)(Xf)$</span> for all <span clas... | Lee Mosher | 26,501 | <p>Assuming <span class="math-container">$z^2+|z|=0$</span> it follows that
<span class="math-container">$$-z^2=|z|
$$</span>
Taking the modulus of both sides we get
<span class="math-container">$$|-z^2| = |(|z|)| \implies |z|^2=|z| \implies |z|=0 \,\,\text{or}\,\, 1
$$</span></p>
<p>Case 1: If <span class="math-contai... |
687,897 | <p>I need help with a differential equation, the trouble is I don't think it's separable and I have tried and failed to apply the method of characteristics to figure it out. z is also bound between zero and one.</p>
<p>$$
\frac{\partial u}{\partial x}+z\frac{\partial u}{\partial y}-Cz\frac{\partial u}{\partial z}=0
$$... | doraemonpaul | 30,938 | <p>Assume $C\neq0$ for the key case:</p>
<p>Follow the method in <a href="http://en.wikipedia.org/wiki/Method_of_characteristics#Example" rel="nofollow">http://en.wikipedia.org/wiki/Method_of_characteristics#Example</a>:</p>
<p>$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$</p>
<p>$\dfrac{dz}{dt}=-Cz$ , letti... |
19,478 | <p>Let $K$ and $L$ be two subfields of some field. If a variety is defined over both $K$ and $L$, does it follow that the variety can be defined over their intersection?</p>
| Pete L. Clark | 1,149 | <p>No, not in general. Assuming we start with a variety over an algebraically closed field, this would be true if the variety could be defined over its field of moduli, which would then be the unique minimal field of definition. (In particular it is true if the variety has no nontrivial automorphisms.) Note also tha... |
1,164,040 | <p>Let $A$ be a $m \times n$ matrix. Determine whether or no the set $W= \{y : Ay=0\}$ is a vector space.
This proof involves nullspace work and another way of asking it is also proving that $W$ is the nullspace of $A$.</p>
<p>I think you can solve this with the use of the subspace theorem being that you can just pro... | Auslander | 218,333 | <p>Yes, the subspace theorem is the way to go here. You first need to show that $W$ is not the empty. This is pretty easy (the zero vector should come in handy). </p>
<p>Next, suppose that $x,y\in W$. This means that $Ax=0$ and $Ay=0$. Therefore,
$$A(x+y)=Ax+Ay=0+0=0.$$
Therefore $W$ is closed with respect to additio... |
4,502,150 | <p>There is a "direct" way to prove this equality.</p>
<p><span class="math-container">$\displaystyle \binom{1/2}{n}=\frac{2(-1)^{n-1}}{n4^n}\binom{2n-2}{n-1}$</span></p>
<p>I am trying to skip the induction. Maybe there is a rule or formula that will help me.</p>
<p>Thank you</p>
| G Cab | 317,234 | <p>another way</p>
<p>Let's rewrite part of the RHS in terms of gamma
<span class="math-container">$$
\frac{1}{n}\left( \begin{array}{c}
2n - 2 \\ n - 1 \\ \end{array} \right)
= \frac{{\Gamma \left( {2n - 1} \right)}}
{{n\Gamma \left( n \right)\Gamma \left( n \right)}}
= \frac{{\Gamma \left( {2n - 1} \right)}}{{\Ga... |
4,930 | <p>This is not an urgent question, but something I've been curious about for quite some time.</p>
<p>Consider a Boolean function in <em>n</em> inputs: the truth table for this function has 2<sup><em>n</em></sup> rows.</p>
<p>There are uses of such functions in, for example, computer graphics: the so-called ROP3s (ter... | Darsh Ranjan | 302 | <p>If you play around with this a bit, you'll probably come to the same conclusion that I did a long time ago, namely that there is indeed a <b>one true extension</b> of boolean functions $\{0,1\}^n\to \{0,1\}$ to functions $[0,1]^n\to [0,1]$. There are many simple characterizations of this extension (which tells you t... |
4,465,504 | <p>Take an invertible formal series <span class="math-container">$f\in \mathbb{Z}_p[[T]]$</span> of inverse <span class="math-container">$g\in \mathbb{Z}_p[[T]]$</span> and let <span class="math-container">$x\in \mathbb{Z}_p$</span> such that the value of <span class="math-container">$f$</span> evaluated at <span class... | Lubin | 17,760 | <p>The answer of @reuns was perfect, but telegraphic. If you prefer prolixity, try this:</p>
<p>You’ll find that when <span class="math-container">$R$</span> is a unital ring, the set of elements of <span class="math-container">$R$</span> that have inverses (i.e. reciprocals) is often denoted <span class="math-containe... |
1,745,832 | <p>It is stated in Kolmogorov & Fomin's <em>Introductory Real Analysis</em> that every normal space is Hausdorff. I cannot seem to find an explanation for this anywhere, and don't see why this is obviously true... since it is not necessarily the case in an arbitrary topological space that a singleton is closed. </... | Forever Mozart | 21,137 | <p>You must assume singletons are closed (equivalently, that the space is T$_1$). </p>
<p>Every indiscrete space is normal, but fails to be Hausdorff if it has more than one point.</p>
|
991,530 | <p>I am trying to prove the following:
For any positive real number $x>0$, there exist a positive integer N such that $x>1/N>0$. </p>
<p>What I know is:
since x is a real number, it can be expressed as the formal limit of a Cauchy sequence
$$x:=LIM (b_n)_{n=1}^\inf$$
since $x>0$, the sequence $(b_n)$ is b... | MAS | 182,585 | <p>Let x and $\epsilon$ be any positive real numbers,
there exist M such that $M\epsilon>x$. choose $\epsilon=1$, then $$\frac{1}{M}<x$$ $$x>\frac{1}{M}>0$$</p>
|
2,551,368 | <p>If I have a liquid that has a specific volume of 26.9 inches cubed per pound, and that liquid sells in the form of a gallon weighing 15.2 lbs, it would be correct to say it's total volume is 15.2*26.9, correct? So that would be 408.88 inches cubed. So if I had a volume of 1.157 inches cubed to fill with that liquid,... | Luke Peachey | 506,520 | <p>$X$ is a normed space, so it is Hausdorff. In particular, $C$ finite implies $C$ is closed.</p>
<p>$C$ dense means by definition $C = \bar C = B_X $. Thus $B_X$ is finite.</p>
<p>For any non-trivial vector-space, any subset with non-empty interior is infinite. Hence it is only possible when $X$ has dimension $0$.<... |
186,417 | <p>Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it should be an infinite series, but I can be wrong.
In case if it can't be written using pluses only, minuses are allowed... | André Nicolas | 6,312 | <p>There is no way to write it as a finite sum. For if you bring such a sum to a common denominator, that denominator will be a power of $2$. Minus signs won't help.</p>
<p>It can be expressed as the infinite "sum"
$$\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}+\cdots.$$</p>
<p>For note that if $|r|\lt 1$ th... |
186,417 | <p>Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it should be an infinite series, but I can be wrong.
In case if it can't be written using pluses only, minuses are allowed... | Michael Hardy | 11,667 | <p>Archimedes showed that if you have a finite sum in which each term is $1/4$ of the previous term, except that the last term is $1/3$ of the previous term, then the sum does not depend on the number of terms, but is just $4/3$ of the first term. In modern terminology:
$$
1+\frac 1 4 + \frac{1}{16} + \cdots + \frac{1... |
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | Jam | 161,490 | <h3>Animated visual demonstration</h3>
<p><a href="https://i.stack.imgur.com/tP1Mb.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/tP1Mb.gif" alt="enter image description here" /></a></p>
<p>As the sequence progresses, the indicator functions move from left to right over <span class="math-container">$[0,1]$</... |
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | shark | 474,426 | <p>The set on which <span class="math-container">${f_n}$</span> fails to converge to <span class="math-container">$0$</span> is exactly <span class="math-container">$[0, 1]$</span>, which has full measure <span class="math-container">$1$</span>. To see this, note that the interval <span class="math-container">$[0, 1]$<... |
1,935,320 | <p>If $a^2-b^2=2$ then what is the least possible value of: \begin{vmatrix} 1+a^2-b^2 & 2ab &-2b\\ 2ab & 1-a^2+b^2&2a\\2b&-2a&1-a^2-b^2 \end{vmatrix}</p>
<p>I tried to express the determinant as a product of two determinants but could not do so. Seeing no way out, I tried expanding it but that ... | Bolton Bailey | 165,144 | <p>An example of a matrix $A$ such that $A^3 = I$ is
$$
A =
\left(
\begin{matrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
\end{matrix}
\right)
$$</p>
|
765,738 | <p>I am trying to prove the following inequality:</p>
<p>$$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$</p>
<p>for all $a>0, b>0$.</p>
<p>Does anyone know how to prove it?</p>
<p>Thanks a lot in advance!</p>
| enigne | 143,559 | <p>The equality holds for $a=b$, so we can just consider the case for $a>b$. In this case, to show this inequality, it is equivalent to show that
$$4\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}\leq(\ln(a)-\ln(b)).$$
As $a>b>0$, we can suppose that $a=e^x$ and $b=e^y$ where $x>y,\ x,y\in\mathbf{R}$. So we h... |
7,072 | <p>I want to replace $x^{i+1}$ with $z_i$.</p>
<p>EDIT:
I have some latex equations, and wish to import them to MMa.</p>
<p>How to replace x_i (NOT $x_i$) with $x[i]$</p>
<p>But it seems the underscore "_" has a special meaning (function), can we do such substitution in MMa?</p>
<p><strong>EDIT2:</strong></p>
<p>F... | Artes | 184 | <pre><code>x^3 + 3 x^2 - 5 x /. {x^(i_Integer) -> Subscript[z, i], x -> Subscript[z, 1]}
</code></pre>
<p><img src="https://i.stack.imgur.com/HXjhQ.gif" alt="enter image description here"></p>
<p>As rcollyer suggested in the comment it can be shortened to</p>
<pre><code>x^3 + 3 x^2 - 5 x /. x^(i_Integer: 1) :&... |
870,174 | <p>If I have $6$ children and $4$ bedrooms, how many ways can I arrange the children if I want a maximum of $2$ kids per room?</p>
<p>The problem is that there are two empty slots, and these empty slots are not unique.</p>
<p>So, I assumed there are $8$ objects, $6$ kids and $2$ empties.</p>
<p>$$C_2^8 \cdot C_2^6 \... | KeithS | 23,565 | <p>Looks right, given some assumptions. Let's build it up a different way to check your work. </p>
<p>We're basically interested in two patterns of putting kids into rooms. Those are all kids paired with one empty, 2-2-2-0, and two pairs and two solo sleepers, 2-2-1-1. Any other arrangement either breaks the 2 kids ma... |
1,035,091 | <p>I try to get the limit of the following expression (should be valid for $a\geq 0$):</p>
<p>$\lim_{n\to\infty}\sum_{k=0}^n\frac{a}{ak+n}$</p>
<p>Unfortunately nothing worked. I tried to rewrite the expressiona and to use l'hospitals rule, but it didn't work.</p>
<p>Thank you for any help:)</p>
<p>regards</p>
<p>... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
2,101,894 | <p>As part of a larger proof, I need to use the fact that for a homomorphism $\phi : A \rightarrow B$, where $A$ is cyclic and $a \in A$ its generator, $\phi(a^{-n}) = \phi(a^n)^{-1}$. All I came up with is that $\phi(a^{-n}) = \phi((a^n)^{-1})$ and then I hit a wall. Is it true in general that $\phi(a^{nm}) = \phi(a^n... | positrón0802 | 334,676 | <p>Since $\phi(a^{n})\phi(a^{-n})=\phi(a^na^{-n})=\phi(1)=1$, then $\phi(a^{-n})=\phi(a^n)^{-1}$ since inverse of $\phi(a^{n})$ is unique.</p>
|
135,235 | <blockquote>
<p>if $f(3)=-2$ and $f'(3)=5$, find $g'(3)$ if,<br>
$g(x)=3x^2-5f(x)$</p>
</blockquote>
<p>the answer is -7,
I find that very hard to understand the question.
thanks </p>
| Community | -1 | <p><strong>Hint</strong>:</p>
<ul>
<li><p>$(\alpha+\beta)\ '=\alpha\ '+\beta\ '$ </p></li>
<li><p>$f$ is differentiable at $x=3$ as $f\ '(3)$ exists.</p></li>
<li><p>$f(3)$ is not a relevant piece of information here...</p></li>
</ul>
|
135,235 | <blockquote>
<p>if $f(3)=-2$ and $f'(3)=5$, find $g'(3)$ if,<br>
$g(x)=3x^2-5f(x)$</p>
</blockquote>
<p>the answer is -7,
I find that very hard to understand the question.
thanks </p>
| J. J. | 3,776 | <p>The solution to the problem comes from the fact that differentiation is a linear operator. This means that $(cf(x) + dg(x))' = cf'(x) + dg'(x)$ where $c$ and $d$ are constants. Assuming we know this and the differentiation rule for powers ($(x^n)' = n x^{n-1}$) we can continue by differentiating the equation
$$g(x) ... |
1,865,868 | <p>the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was a white ball while we know that at the 3rd was a black ball".</p>
<p>What I did is I build a conditional probability... | Rodrigo de Azevedo | 339,790 | <p>This is an instance of the <strong>least-norm</strong> problem</p>
<p><span class="math-container">$$\begin{array}{ll} \text{minimize} & \| {\bf x} \|_2^2\\ \text{subject to} & {\bf A} {\bf x} = {\bf b} \end{array}$$</span></p>
<p>As <span class="math-container">$2 \times 3$</span> matrix <span class="math-c... |
191,307 | <p>How can I calculate the perimeter of an ellipse? What is the general method of finding out the perimeter of any closed curve?</p>
| Mark | 24,958 | <p>I do not know if that's what you wanted, but the only general method is to calculate the length of the curve.
If we have a ellipse equation:</p>
<p>$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</p>
<p>with parametric representation:</p>
<p>$x=a \cos t, \ \ y=b \sin t, \ \ \ t\in [0,2\pi]$</p>
<p>the length of the curve is... |
191,307 | <p>How can I calculate the perimeter of an ellipse? What is the general method of finding out the perimeter of any closed curve?</p>
| SN77 | 37,814 | <p>For any ellipse, its perimeter is given by
$p=2πa(1-(\frac{1}{2})^2ε^2-{(\frac{1.3}{2.4})}^2\frac{ε^4}{3}-\cdots)$</p>
|
216,748 | <ol>
<li><p>SO far i showed that if A matrix is left invertible (L) then in Ax = b, x has at most 1 solution.
<strong>I got that LAx = x = Lb, so x = Lb</strong></p></li>
<li><p>for right inverse (R) of A, in Ax = b, x has at least one solution.
<strong>I got that x = Rb</strong>. </p></li>
</ol>
<p>In the book it sa... | N. S. | 9,176 | <p>Yes you are right. In 1., you SOLVED the equation, and thus you PROVED that $x=Lb$.</p>
<p>In 2., you can see easily that $x=Rb$ works, but this only proves that this is one solution. It could be the only one, or there could be others....</p>
<p><strong>PS</strong> I don't know how you got the $x=Rb$, it is easy t... |
3,845,144 | <p>Let's say a number <span class="math-container">$n$</span> is insertable if for every digit <span class="math-container">$d$</span>, if we insert <span class="math-container">$d$</span> between any two digits of <span class="math-container">$n$</span>, then the obtained number is a multiple of <span class="math-cont... | Evaristo | 813,919 | <p>With <span class="math-container">$\{e_1, e_2, e_3\}$</span> as your base, check what happens when you operate on them
<span class="math-container">$$Ae_1 = e_3 \implies A^ne_1 = A^{n-1}e_3$$</span>
<span class="math-container">$$Ae_2 = e_2+e_3 \implies A^ne_2 = A^{n-1}(e_2+e_3) = A^{n-1}e_3 + A^{n-1}e_2$$</span>
<s... |
120,910 | <p>$f(x) = 2\sin x \hspace{10pt}(0 \leq x \leq \pi)$<br>
$g(x) = -\sin x \hspace{10pt}(0 \leq x \leq \pi)$</p>
<p>Rectangle ABCD is enclosed between the above functions' graphs (its edges are parallel to the axes).</p>
<p>How would I go about finding the maximum perimeter of ABCD?</p>
<p>I'm really clueless about th... | André Nicolas | 6,312 | <p>As a minor matter of strategy, after <em>drawing the picture</em>, I would move the $y$-axis to get symmetry. Then the picture becomes the picture of $2\cos x$ and $-\cos x$. If the upper right-hand corner of the rectangle in the new picture is $(x,2\cos x)$, then the perimeter is $4x+6\cos x$. </p>
<p>We have $0\... |
265,509 | <p>Given a (possibly disjoint) region, defined by a discrete set of points, how can I use <code>ListContourPlot[]</code> together with <code>Mesh</code> to highlight a specific area of the plot? For instance, how can I mesh the region where the points are smaller than a certain value?</p>
<p>Here I construct a minimal ... | Bob Hanlon | 9,362 | <pre><code>data = Table[Exp[x^2 - y^2], {x, -1, 1, .01}, {y, -1, 1, .01}];
Show[
ListContourPlot[data,
Contours -> {1},
Mesh -> 25,
MeshFunctions -> {#1 + #2 &},
MeshStyle -> Directive[Thick, #[[1]]],
RegionFunction -> #[[2]]] & /@
{{Red,
Function[{x, y, f}, f < 1]... |
3,264,935 | <p>For a given natural number <span class="math-container">$k$</span>, I'm going to call a subset <span class="math-container">$T$</span> of the plane <span class="math-container">$\Bbb{R}^2$</span> a <span class="math-container">$k$</span>-traversal if, for any <span class="math-container">$x \in \Bbb{R}$</span>,
<spa... | Vincent | 101,420 | <p>I have the feeling that this (or a variation) would work, but maybe I am overlooking something. It is definitely the union of two things, but they are not 1-traversals.</p>
<p>Take the union of the v-shape <span class="math-container">$y = |1/2(x-1)|$</span> and upside-down v-shape <span class="math-container">$y = ... |
77,092 | <p>I am wondering if anyone could please post the solution to the following differential equation for the function $f(x)$:</p>
<p>$$\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$$</p>
<p>Thanks!</p>
| anon | 11,763 | <p>$\rm\bf Start$: Multiply through by $f\,''/f$ and integrate with respect to $x$:
$$\frac{f\,''}{f\,'}=\frac{f\,'}{f} \implies \ln (f\,')=\ln f+C.$$
Now exponentiate and solve another differential equation similarly...</p>
<hr>
<p>$\rm\bf Finish$:</p>
<blockquote class="spoiler">
<p> ... |
129,072 | <p>We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with weight $\mu$. I didn't find the corresponding definition for representations of finite dimensional algebras (for example... | Qiaochu Yuan | 232 | <p>Sure. If $A$ is a finite-dimensional algebra over a field $k$ and $V$ a finite-dimensional $A$-module (so we have a morphism $\rho : A \to \text{End}_k(V)$), we can define a character $\chi_V(a) = \text{tr}(\rho(a)) : A \to k$ just as in the case of finite groups (where $A = k[G]$). Characters are additive in short ... |
849,025 | <p>Say I have 100 people each with a height, weight, and age. I make a regression that predicts age based on height and weight. Now, I would like to update that model when I meet someone new. I don't want to just re-process 101 people though--I want to take the model that I already have and incorporate the new person i... | Claude Leibovici | 82,404 | <p>Let us suppose that you fit a model $Z = a +b X+cY$ based on $N$ data points $(X_i,Y_i,Z_i)$. The so-called normal equations are $$\sum _{i=1}^N Z_i= N a + b\sum _{i=1}^N X_i+ c\sum _{i=1}^N Y_i$$ $$\sum _{i=1}^N X_iZ_i= a\sum _{i=1}^N X_i + b\sum _{i=1}^N X_i^2+ c\sum _{i=1}^N X_iY_i$$ $$\sum _{i=1}^N Y_iZ_i= a\s... |
1,668,487 | <p>$$2^{-1} \equiv 6\mod{11}$$</p>
<p>Sorry for very strange question. I want to understand on which algorithm there is a computation of this expression. Similarly interested in why this expression is equal to two?</p>
<p>$$6^{-1} \equiv 2\mod11$$</p>
| N. F. Taussig | 173,070 | <p>The notation $2^{-1} \pmod{11}$ means the <em>multiplicative inverse</em> of $2$ modulo $11$, that is, a number $x$ satisfying the equation $2x \equiv 1 \pmod{11}$. </p>
<p>Since $2 \cdot 6 \equiv 12 \equiv 1 \pmod{11}$, $2^{-1} \equiv 6 \pmod{11}$. The same calculation shows that $6^{-1} \equiv 2 \pmod{11}$. </... |
2,442,705 | <p>Like the title says, I'm wondering if there's a non-brute-force way to determine $x$ such that
$$45\le x<200,$$
$$x\bmod5\equiv0,$$
$$x\bmod8\equiv1,$$
$$x\bmod12\equiv1$$
I know I can simply enumerate all integers in $[45, 200)$ to get the answer (145), but I'm wondering if there's a more elegant solution.</p>
| paw88789 | 147,810 | <p>From the last two congruences, $x\equiv 1 \pmod{24}$</p>
<p>Then $x\equiv 1 \equiv 25 \equiv 49 \equiv...$</p>
<p>Just keep going until you find something congruent to $0 \pmod{5}$ that is within your range of interest.</p>
<p>Or you could note that $25$ works, and then so will $25 + 120k$ for any integer $k$. <... |
2,442,705 | <p>Like the title says, I'm wondering if there's a non-brute-force way to determine $x$ such that
$$45\le x<200,$$
$$x\bmod5\equiv0,$$
$$x\bmod8\equiv1,$$
$$x\bmod12\equiv1$$
I know I can simply enumerate all integers in $[45, 200)$ to get the answer (145), but I'm wondering if there's a more elegant solution.</p>
| Jihoon Kang | 452,346 | <p>Hint: Use the Chinese Remainder Theorem. In the case for modular arithmetic, this theorem is as follows:</p>
<p><strong>Theorem</strong> Suppose you have a system of congruences: $$\begin{cases} x \equiv a_1 \ \mbox{mod} \ b_1 \\ \vdots \ \ \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ x\equiv a_n \ \mbox{m... |
2,294,585 | <p>I can't figure out why $((p → q) ∧ (r → ¬q)) → (p ∧ r)$ isn’t a tautology. </p>
<p>I tried solving it like this: </p>
<p>$$((p ∧ ¬p) ∨ (r ∧ q)) ∨ (p ∧ r)$$
resulting in $(T) ∨ (p ∧ r)$ in the end that should result in $T$. What am I doing wrong?</p>
| User4407 | 81,495 | <p>It looks like you swapped a $p$ for $q$.</p>
<p>You have:</p>
<p>$$((p ∧ ¬p) ∨ (r ∧ q)) ∨ (p ∧ r)$$</p>
<p>When it should be </p>
<p>$$((p ∧ ¬q) ∨ (r ∧ q)) ∨ (p ∧ r)$$</p>
<p>It also looks like you made the mistake of thinking that $p ∧ ¬p$ is true. That would be the case in classical logic if it were OR rather... |
3,911,297 | <p>Chapter 12 - Problem 26)</p>
<blockquote>
<p>Suppose that <span class="math-container">$f(x) > 0$</span> for all <span class="math-container">$x$</span>, and that <span class="math-container">$f$</span> is decreasing. Prove that there is a <em>continuous</em> decreasing function <span class="math-container">$g$</... | Donald Splutterwit | 404,247 | <p>Your solution is fine. You could do it in one line ...
<span class="math-container">\begin{eqnarray*}
a=\frac{(a^{12})^3}{(a^7)^5}.
\end{eqnarray*}</span></p>
|
1,276,177 | <p>Four fair 6-sided dice are rolled. The sum of the numbers shown on the dice is 8. What is the probability that 2's were rolled on all four dice?</p>
<hr>
<p>The answer should be 1/(# of ways 4 numbers sum to 8)
However, I can't find a way, other than listing all possibilities, to find the denominator. </p>
| Michael Hardy | 11,667 | <p>Possibly brute force is nearly the best available option here.
\begin{align}
2+2+2+2 & & & 1 \text{ way} \\[6pt]
1+2+2+3 & & & 12\text{ ways} \\[6pt]
1+1+2+4 & & & 12\text{ ways} \\
1+1+3+3 & & & 6\text{ ways} \\[6pt]
1+1+1+5 & & & 4\text{ ways}
\end{align}... |
4,013,638 | <p>a) <span class="math-container">$F(x)>0$</span> for every <span class="math-container">$x>0$</span>.<br>
b) <span class="math-container">$F(x)>F(e)$</span> for every <span class="math-container">$x>1$</span>.<br>
c) <span class="math-container">$x=e^{-2}$</span> is an inflection point. <br>
d) None of th... | Bernard | 202,857 | <p>If you apply the <em>First fundamental theorem</em> of integral calculus, you have
<span class="math-container">$$F'(x)=x^{\sqrt x},\enspace \text{hence }\quad F''(x)=x^{\sqrt x}\biggl(\frac{\ln x}{2\sqrt x}+\frac{\sqrt x}x\biggr)=x^{\sqrt x}\,\frac{\ln x+2}{2\sqrt x}$$</span>
Therefore, <span class="math-container"... |
428,291 | <p>What is the solution of the following system?</p>
<p>$$
\begin{align}
a \cdot e-b \cdot d & =\alpha \\
a \cdot f-c \cdot d & =\beta \\
b \cdot f-c \cdot e & =\gamma
\end{align}
$$</p>
<p>Where the unknowns $a,b,c,d,e,f$ are real numbers and $\alpha, \beta, \gamma$ are fixed real numbers.</p>
<p>I trie... | Glen O | 67,842 | <p>Hint: the system can actually be interpreted as
$$
\begin{pmatrix}a\\b\\c\end{pmatrix}\times \begin{pmatrix}d\\e\\f\end{pmatrix}= \begin{pmatrix}\gamma\\-\beta\\\alpha\end{pmatrix}
$$
where $\times$ is the vector cross product.</p>
|
86,657 | <p>Throughout my upbringing, I encountered the following annotations on Gauss's diary in several so-called accounts of the history of mathematics:</p>
<blockquote>
<p>"... A few of the entries indicate that the diary was a strictly private affair of its author's (sic). Thus for July 10, 1796, there is the entry</p>
... | Barry Cipra | 15,837 | <p>I think part of the answer may be found by consulting Volume <em>X</em> of Gauss's <em>Werke</em>. "REV. GALEN" doesn't actually appear in the <em>Tagebuch</em> itself, a facsimile of which appears following page 482. It was jotted down by Gauss elsewhere, as explained on page 539, in the commentary (which runs fo... |
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