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 |
|---|---|---|---|---|
3,786,553 | <p>I was trying to follow the logic in a similar question (<a href="https://math.stackexchange.com/questions/643352/probability-number-comes-up-before-another">Probability number comes up before another</a>), but I can't seem to get it to work out.</p>
<p>Some craps games have a Repeater bet. You can bet on rolling ac... | peter.petrov | 116,591 | <p><span class="math-container">$d(2+x) = d(x)$</span> so there's no problem here.</p>
<p>This integral is just <span class="math-container">$\int_{0}^{2} x ~d(x)$</span></p>
<p>Do you know how to compute this one?</p>
|
876,896 | <p>What are the poles of a polynomial? Are they the same as the roots?</p>
| paw88789 | 147,810 | <p>A (nonconstant) polynomial may be considered to have a pole at infinity.</p>
|
876,896 | <p>What are the poles of a polynomial? Are they the same as the roots?</p>
| Jason Knapp | 8,454 | <p>Brief answer: a pole refers to the location of a special type of discontinuity, in fact a special sort of singularity. Polynomials on the real line or complex plane are continuous, and thus do not have any poles in the real or complex numbers. </p>
<p>However we have a very similar object in the complex case call... |
3,862,182 | <p>I encountered this question, and I am unsure how to answer it.</p>
<p>When <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x - 4$</span>, the remainder is <span class="math-container">$13$</span>, and when <span class="math-container">$P(x)$</span> is divided by <span class="ma... | Lion Heart | 809,481 | <p><span class="math-container">$P(x)=(x^2-x-12)Q(x)+ax+b$</span></p>
<p><span class="math-container">$P(4)=4a+b=13$</span></p>
<p><span class="math-container">$P(-3)=-3a+b=-1$</span></p>
<p><span class="math-container">$a=2, b=5$</span></p>
<p><span class="math-container">$P(x)=(x^2-x-12)Q(x)+2x+5$</span></p>
<p><span... |
3,106,696 | <p>I am confused on the notation used when writing down the solution of x and y in quadratic equations.
For example in <span class="math-container">$x^2+2x-15=0$</span>, do I write :</p>
<p><span class="math-container">$x=-5$</span> AND <span class="math-container">$x=3$</span></p>
<p>or is it</p>
<p><span class=... | lulu | 252,071 | <p>Let <span class="math-container">$p$</span> be a prime dividing <span class="math-container">$\gcd(f(n+1),f(n))$</span></p>
<p>We note that <span class="math-container">$$f(n+1)-f(n)=2(n+1)$$</span> so <span class="math-container">$p$</span> must divide <span class="math-container">$2(n+1)$</span></p>
<p>Since all... |
3,834,796 | <p>I found a really interesting question which is as follows:
Prove that the value of
<span class="math-container">$$\sum^{7}_{k=0}[({7\choose k}/{14\choose k})*\sum^{14}_{r=k}{r\choose k}{14\choose r}] = 6^7$$</span></p>
<p>my approach:</p>
<p>I tried to simplify the innermost sigma as well as trying to simplify by u... | cosmo5 | 818,799 | <p>Using <span class="math-container">$${14 \choose r}{r \choose k} = {14 \choose k}{14-k \choose r-k}$$</span> given reduces to</p>
<p><span class="math-container">$$
\begin{align*}
& \sum_{k=0}^7 {7 \choose k} \bigg\{\sum^{14}_{r=k} {14-k \choose r-k} \bigg\} \\
& = \sum_{k=0}^7 {7 \choose k} \{2^{14-k}\... |
3,834,796 | <p>I found a really interesting question which is as follows:
Prove that the value of
<span class="math-container">$$\sum^{7}_{k=0}[({7\choose k}/{14\choose k})*\sum^{14}_{r=k}{r\choose k}{14\choose r}] = 6^7$$</span></p>
<p>my approach:</p>
<p>I tried to simplify the innermost sigma as well as trying to simplify by u... | Elliot Yu | 165,060 | <p><strike>First off I don't think your sum is quite right. The bounds on the outer sum should be <span class="math-container">$k=0$</span> to <span class="math-container">$7$</span>, I believe, otherwise the value isn't <span class="math-container">$6^7$</span>.</strike> (Question now corrected)</p>
<p>You are on the ... |
2,705,980 | <p>I have the following problem:
\begin{cases}
y(x) =\left(\dfrac14\right)\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2 \\
y(0)=0
\end{cases}
Which can be written as:</p>
<p>$$ \pm 2\sqrt{y} = \frac{dy}{dx} $$</p>
<p>I then take the positive case and treat it as an autonomous, seperable ODE. I get $f(x)=x^2$ as my ... | Martín-Blas Pérez Pinilla | 98,199 | <p>By the <a href="https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem" rel="nofollow noreferrer">rank-nullity theorem:</a> rank = $\dim$ of image $= n\implies\dim\ker A =0$.</p>
|
2,190,551 | <p>How can I find the degrees of freedom of a $n \times n$ real orthogonal matrix?</p>
<p>I have tried to proceed by principle of induction but I fail.Please tell me the right way to proceed.</p>
<p>Thank you in advance.</p>
| jnez71 | 295,791 | <p>For an $n \times n$ matrix with column vectors $v_1 \dots v_n$, we say that it is orthogonal if,
$$
\langle v_i, v_j \rangle = 0,\ \ \ \forall (i,j)\ \Big{|}\ i\neq j
$$</p>
<p>Recalling that the inner product is commutative, the above gives us <a href="https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%A... |
3,749,892 | <p>I'll just upload a Brilliant page first. Check this out please.</p>
<p><a href="https://i.stack.imgur.com/yj9Yc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yj9Yc.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/7iPh2.png" rel="nofollow noreferrer"><img src="htt... | Angela Pretorius | 15,624 | <p>The integers which are coprime to <span class="math-container">$n$</span> form a group under multiplication modulo <span class="math-container">$n$</span>. Denote this group by <span class="math-container">$G$</span>.</p>
<p>Let <span class="math-container">$H$</span> be the group of powers of <span class="math-cont... |
3,749,892 | <p>I'll just upload a Brilliant page first. Check this out please.</p>
<p><a href="https://i.stack.imgur.com/yj9Yc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yj9Yc.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/7iPh2.png" rel="nofollow noreferrer"><img src="htt... | Pablo Carneiro Elias | 272,929 | <p>If <span class="math-container">$a$</span> and <span class="math-container">$m$</span> are coprimes, then, <span class="math-container">$m \nmid a^i$</span>.</p>
<p>Since all reminders of <span class="math-container">$m$</span> belongs to the set <span class="math-container">$\{0,...,m-1\}$</span>, by the pigeonhole... |
4,499,058 | <blockquote>
<p>Let <span class="math-container">$A:L_2[0,1]\to L_2[0,1]$</span> be defined by<span class="math-container">$$Ax(t)=\int \limits _0^1t\left (s-\frac{1}{2}\right )x(s)\,ds\quad \forall t\in [0,1].$$</span>Compute the adjoint and the norm of <span class="math-container">$A$</span></p>
</blockquote>
<p>This... | Abhijeet Vats | 426,261 | <p>Not quite. I'll tell you what you did wrong first and try to point you in the right direction. My solution will be presented later and you can use that as a reference.</p>
<p>So, your calculation of <span class="math-container">$\left|\left|Ax\right|\right|_2$</span> is incorrect. You haven't used the correct expres... |
313,025 | <p>I got two problems asking for the proof of the limit: </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$</p>
</blockquote>
<p>and, </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,... | André Nicolas | 6,312 | <p>For the first problem, use integration by parts, followed by crude estimation. </p>
<p>Let $u=\frac{1}{t}$ and $dv=te^{-t^2}\,dt$. Then $du=-\frac{1}{t^2}\,dt$ and we can take $v=-\frac{1}{2}e^{-t^2}$. Thus our integral is equal to
$$\frac{1}{x}\cdot \frac{1}{2}e^{-x^2}-\int_x^\infty \frac{1}{2t^2}e^{-t^2}\,dt.$$
M... |
313,025 | <p>I got two problems asking for the proof of the limit: </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$</p>
</blockquote>
<p>and, </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,... | Mhenni Benghorbal | 35,472 | <p>Let</p>
<p>$$ f(x)=\ x e^{x^2}\int_x^\infty e^{-t^2} \implies f(x)=\ x e^{x^2}g(x).$$ </p>
<p>We can see that $ f(0)=0 $ and $f(x)>0,\,\, \forall x>0$. Taking the limit as $x$ goes to infinity and using L'hobital's rule and Leibniz integral rule yields</p>
<p>$$ \lim_{ x\to \infty } xe^{x^2}g(x) = \lim _{x... |
184,772 | <p>Is there any way to write a code that has a <strong>function</strong> include <strong>Block[ ]</strong> and <strong>Do[ ]</strong> loop instead of my code?</p>
<p>Here is my code:</p>
<pre><code>(* m = Maximum members of "list" *)
list = {{12, 9, 10, 5}, {3, 7, 18, 6}, {1, 2, 3, 3},
{4, 5, 6, 2}, {1, 13, 1, 1}}... | ZaMoC | 46,583 | <p>if you want to use your code try</p>
<pre><code>F[list_] := Block[{m}, m = {};
Do[AppendTo[m, Max[list[[All, i]]]];, {i, Length[list[[1]]]}];m]
F[list]
</code></pre>
<p>otherwise you can use this function </p>
<pre><code>F[list_]:=Max/@Transpose@list
</code></pre>
|
2,750,931 | <p>I'm in the process of exploring Bra Ket notation. In it, I often find operators in the form $\lvert a\rangle\langle b\rvert$, which can be thought of as multiplying a row vector $a$ with a column vector $b$.</p>
<p>This strikes me as a construction which should probably have a name that I can research to understan... | Pietro Paparella | 414,530 | <p>It can be shown that a matrix $M$ has rank equal to one if and only if $M = ab^\top$, where $a$ and $b$ are column vectors with complex entries, so the matrices you are thinking of can be referred to as rank-one matrices.</p>
<p>The product $ab^\top$ is also known as the outer product whereas the product $a^\top b$... |
2,247,498 | <p>Imagine a circle of radius R in 3D space with a line l running threw it's center C in a direction perpendicular to the plane of the circle. Basically, like the axel of a wheel. </p>
<p>From a given point P that is not on the circle or on l, a ray extends to intersect both l and the circle. What would be the equatio... | Futurologist | 357,211 | <p>No need to rotate anything, it is a matter of very simple geometry, which if you follow through, gives you a very simple explicit algorithm for computing the points you need. </p>
<p>Assume the orientation of the lines $l$ is given by a vector $\vec{v}$. Then the circle, call it $k$, has given center $C$ and radius... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Stefan Smith | 55,689 | <p>An antiderivative of a function $f$ is <em>one</em> function $F$ whose derivative is $f$. The indefinite integral of $f$ is the set of <em>all</em> antiderivatives of $f$. If $f$ and $F$ are as described just now, the indefinite integral of $f$ has the form $\{F+c \mid c\in \mathbb{R}\}$. Usually people don't bot... |
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Bill Dubuque | 242 | <p>In the Theorem below put $\rm\:p\!=\!2,\ c=-5,\ d = 1\:$ to deduce that $\rm\:mod\ 16,\ (x\!+\!5)(x\!-\!1)\:$ has roots $\rm\:x \equiv -5,1\pmod 8,\:$ which are $\rm\:x,x\!+\!8 \equiv -5,1,3,9\pmod{16}.$ It has $\rm\,4\ (\!vs.\,2)$ roots by $\rm\:x\!+\!5 \equiv x\!-\!1\pmod{2},\:$ so both are divisible by $2$, so th... |
1,839,693 | <p>I think it must be true. Yet I have no rigorous proof for that. So, what I need to prove is that "group being simple" is invariant under isomorphism.</p>
<p>That if $G \cong H$, then either both are simple groups, or both are not simple.</p>
| Eric Wofsey | 86,856 | <p>Hint: Let $f:H\to G$ be an isomorphism and let $K\subseteq H$ be a normal subgroup. What can you say about $f(K)\subseteq G$?</p>
|
341,602 | <p>Let <span class="math-container">$L$</span> be a semisimple Lie algebra and let <span class="math-container">$(V,\varphi)$</span> be a finite-dimensional <span class="math-container">$L$</span>-module representation. Our main goal is to prove that <span class="math-container">$\varphi$</span> is completely reducible... | Jim Humphreys | 4,231 | <p>Maybe it would clarify matters if I gave a little more background, in community wiki format.</p>
<p>The basic idea of this algebraic proof goes back to a short paper by Richard Brauer (1936) in German in <em>Mathematische Zeitschrift</em>, an interesting time in his life when he had been expelled from his professor... |
112,320 | <p>What is the number of strings of length $235$ which can be made from the letters A, B, and C, such that the number of A's is always odd, the number of B's is greater than $10$ and less than $45$ and the Number of C's is always even?</p>
<p>What I can think of is </p>
<p>$$\left(\binom{235}{235} - \left\lfloor235 -... | Ross Millikan | 1,827 | <p>Hint: I would do it with a program. You have 44*2*2 allowable states-number of B's, parity of A's and parity of C's. Write the recurrence relations and note that the empty string has even C's and no B's.</p>
|
4,821 | <p>A quick bit of motivation: recently a question I answered quite a while ago ( <a href="https://math.stackexchange.com/questions/22437/combining-two-3d-rotations/178957">Combining Two 3D Rotations</a> ) picked up another (IMHO rather poor) answer. While it was downvoted by someone else and I strongly concur with the... | Rick Decker | 36,993 | <p>Consider the "symmetric" situation: would you hesitate to upvote another--good--answer after you had already provided one of your own? I certainly wouldn't. I put quotes around "symmetric" because it's not completely clear to me that the two positions are indeed equivalent. My feeling is that they are, especially si... |
4,155,900 | <p>I'd like someone to verify my sketch proof of the below exercise 3.3.7 from Abbott's, <em>Understanding Analysis</em>. If it's incorrect, could you hint/point at the correct approach to the proof. Thanks!</p>
<blockquote>
<p>Prove that the sum <span class="math-container">$C+C=\{x+y:x,y\in C\}$</span> is equal to th... | Zerox | 737,359 | <p>Note that <span class="math-container">$\{x_n\}$</span> is bounded, so there is a subsequence <span class="math-container">$\{x_{n_k}\}$</span> converging to some <span class="math-container">$x \in [0,1]$</span>. Use contradiction to show that <span class="math-container">$x \in C$</span>: If not, then <span class=... |
4,155,900 | <p>I'd like someone to verify my sketch proof of the below exercise 3.3.7 from Abbott's, <em>Understanding Analysis</em>. If it's incorrect, could you hint/point at the correct approach to the proof. Thanks!</p>
<blockquote>
<p>Prove that the sum <span class="math-container">$C+C=\{x+y:x,y\in C\}$</span> is equal to th... | dan_fulea | 550,003 | <p>Here is - in essence - the same argument, written differently. Let <span class="math-container">$C$</span> be the <a href="https://en.wikipedia.org/wiki/Cantor_set" rel="nofollow noreferrer">Cantor set</a>. Each element <span class="math-container">$a$</span> in <span class="math-container">$C$</span> can be written... |
4,393,925 | <p>Is the integral of <span class="math-container">$\tan(x)\,\mathrm{d}x$</span> equal to the negative <span class="math-container">$\ln$</span> of absolute value of <span class="math-container">$\cos(x)$</span>, the same as integral of <span class="math-container">$\tan(x)\,\mathrm{d}x$</span> equal to the <span class... | EvPlaysPiano | 1,031,001 | <p>So, when dealing with logarithms, you can use this following property: <span class="math-container">$\ln(\frac{1}x) = -\ln(x)$</span>. and this is the same when working with inverse trig functions, such as cos and sec; so, <span class="math-container">$-\ln|\cos(x)$</span>| is the same thing as <span class="math-con... |
714,000 | <p>Lets say:</p>
<p>$X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$</p>
<p>If I have to increase mean of set $X$ by $3$, each number in the set has to be increase by $3$.
But how to increase mean of set $X$ by $3$, by only changing a subset of X. Is there any mathem... | Semsem | 117,040 | <p>Now the sum of the <span class="math-container">$n$</span>elements is <span class="math-container">$s$</span> and so <span class="math-container">$$s=nm$$</span> Now we want a new mean <span class="math-container">$M=m+3$</span> and the new sum <span class="math-container">$S$</span> should be <span class="math-cont... |
366,724 | <p>Suppose $D$ is an integral domain and that $\phi$ is a nonconstant function from $D$ to
the nonnegative integers such that $\phi(xy) = \phi(x)\phi(y)$. If $x$ is a unit in $D$, show that
$\phi(x) = 1$.</p>
| Math Gems | 75,092 | <p><strong>Hint</strong> $\, $ By multiplicativity, $\rm\:\phi\:$ preserves $1$ and divisibility, so it preserves divisors of $1$ (= units).</p>
<p>$\rm(1)\quad \phi\ $ preserves $\:1\!:\ $ apply $\rm\:\phi\:$ to $\rm\:1^2 = 1\ $ to deduce $\rm\ \phi(1) = 1.$</p>
<p>$\rm(2)\quad \phi\ $ preserves divisibility: $\rm\,... |
717,980 | <p>In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a
6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled simultaneously by each player on his turn, and the
first player to obtain a total of k (or more) 6’s, accumulated over all... | JSS | 136,447 | <p>It might be easier to approach this problem with generating functions.</p>
<p>When you roll a die, you have 1/6 chance to roll a 6, and a 5/6 chance to roll any other face. Let a represent rolling a 6, and let b represent rolling any other face. </p>
<p>Rolling a single die, the outcomes may be represented as $({a... |
696,869 | <p>Question:
Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible.</p>
<p>My work: Based on the section I read, I will treat I to be an identity matrix, which is a $1 \times 1$ matrix with a $1$ or as an square matrix with main diagonal is all ones and the rest is zer... | Max | 130,322 | <p>(an alternative to your approach)</p>
<p>if $0=A^{2}+2A+I=\left(A+I\right)\left(A-I\right)$you have that the set $\left\{+1,-1\right\}$ contains all the eigenvalues of $A$. thus $0$ is not an eigenvalue of $A$ and this is equivalent to being invertible.</p>
<p>Edit:</p>
<p>as LutzL has commented correctly, $A-I$ ... |
696,869 | <p>Question:
Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible.</p>
<p>My work: Based on the section I read, I will treat I to be an identity matrix, which is a $1 \times 1$ matrix with a $1$ or as an square matrix with main diagonal is all ones and the rest is zer... | nmasanta | 623,924 | <p>Here you have to show <strong><span class="math-container">$A$</span> is invertible</strong>. For this only you have to show <span class="math-container">$ det (A) \neq 0$</span>. </p>
<p>Now since <span class="math-container">$A$</span> satisfies the polynomial equation <span class="math-container">$ x^{2} + 2x +... |
4,220,518 | <p>A point is moving along the curve <span class="math-container">$y = x^2$</span> with unit speed. What is the magnitude of its acceleration at the point <span class="math-container">$(1/2, 1/4)$</span>?</p>
<p>My approach : I use the parametric equation <span class="math-container">$s(t) = (ct, c^2t^2)$</span>, then ... | David Quinn | 187,299 | <p>You can use Cartesian coordinates for this.
You have <span class="math-container">$$y=x^2\implies \dot{y}=2x\dot{x}$$</span>
Therefore at <span class="math-container">$x=\frac12$</span>, <span class="math-container">$\dot{y}=\dot{x}$</span></p>
<p>For constant speed we have <span class="math-container">$${\dot{x}}^2... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | ajotatxe | 132,456 | <p>$$\begin{align}
\binom n{m-1}+\binom nm&=\frac{n!}{(m-1)!(n-m+1)!}+\frac{n!}{m!(n-m)!}\\
&=\frac{n!m+n!(n-m+1)}{m!(n-m+1)!}\\
&=\frac{n!(n-m++1+m)}{m!(n-m+1)!}\\
&=\frac{n!(n+1)}{m!(n-m+1)!}\\
&=\frac{(n+1)!}{m!(n-m+1)!}=\binom{n+1}m
\end{align}$$</p>
|
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | Community | -1 | <p>Simpler than the simplest, simplify by $n!$, $m!$ and $(n-m+1)!$:
$$\frac{{n\choose m-1}+{n\choose m}}{n+1\choose m}=\frac{\frac{n!}{(m-1)!(n-m+1)!}+\frac{n!}{m!(n-m)!}}{\frac{(n+1)!}{m!(n-m+1)!}}=
\frac{\frac1{1.m^{-1}}+\frac{1}{1.(n-m+1)^{-1}}}{\frac{n+1}{1.1}}=1.$$</p>
|
1,517,502 | <p>Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus.</p>
<p><strong>Answer</strong>:</p>
<p>$$(x-3)^2 + (y-5)^2 = (x+2)^2 + (y-4)^2$$
or, $$10x+2y-14=0$$
or, $$5x+y-7=0$$</... | SchrodingersCat | 278,967 | <p>$$(x-3)^2 + (y-5)^2 = k\left[ (x+2)^2 + (y-4)^2 \right]$$ where $k$ is a constant</p>
<p>Now $(0,0)$ lies on the locus. <br>
Therefore $$9+25=k(4+16) \Rightarrow k=\frac{34}{20} = \frac{17}{10}$$</p>
<p>Using this value of $k$ in the equation, we get
$$(x-3)^2 + (y-5)^2 = \frac{17}{10}\left[ (x+2)^2 + (y-4)^2 \rig... |
1,517,502 | <p>Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus.</p>
<p><strong>Answer</strong>:</p>
<p>$$(x-3)^2 + (y-5)^2 = (x+2)^2 + (y-4)^2$$
or, $$10x+2y-14=0$$
or, $$5x+y-7=0$$</... | Piquito | 219,998 | <p>The same answer than Aniket. You have a cercle centered at the point $(-\frac{64}{7},\frac{18}{7})$ and radius $\frac{2\sqrt{5\cdot13\cdot17}}{7}$</p>
|
1,032,926 | <p>I'm trying to understand this proof of the following Lemma, that I found in an article on Existence of Eigenvalues and Eigenvectors, but I don't understand the following step:</p>
<p><em>Let $V$ be a finite-dimensional complex vector space, $v\in V$ and $c\gt 0$. Since for every $v\in V\setminus \{ 0 \}$ and $k\in\... | Martín-Blas Pérez Pinilla | 98,199 | <p>$$T(v) - kv =0\implies 0=\|T(v) - kv\| \ge c \|v\|\implies v=0.$$</p>
|
137,571 | <p>As the title, if I have a list:</p>
<pre><code>{"", "", "", "2$70", ""}
</code></pre>
<p>I will expect:</p>
<pre><code>{"", "", "", "2$70", "2$70"}
</code></pre>
<p>If I have</p>
<pre><code>{"", "", "", "3$71", "", "2$72", ""}
</code></pre>
<p>then:</p>
<pre><code>{"", "", "", "3$71", "3$71", "2$72", "2$72"}
... | Carl Woll | 45,431 | <p>In place modification of a list works well here:</p>
<pre><code>fc[list_] := Module[{out=list},
With[{empty = Pick[Range[2,Length@list], Rest@list,""]},
out[[empty]]=out[[empty-1]]
];
out
]
</code></pre>
<p>A comparison with Mr Wizard's fn:</p>
<pre><code>data = RandomChoice[{"","","","a","b"}... |
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| Asaf Karagila | 622 | <p>Easily.</p>
<p>Let $P=\{2,3,5,7,11,\ldots\}$ be all the prime numbers. Take for $p\in P$, any prime number let $A_p = \{p^n\mid n\in\mathbb N, n\neq 0\}$ and take $A_1 = \{n\in\mathbb N\mid n \text{ have at least two different prime divisors}\}\cup\{0\}$.</p>
<p>Every $A_i$ is disjoint of the rest, and every natur... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | Bill Dubuque | 242 | <p>Both sides are equal since they count the same set: the RHS counts naturals <span class="math-container">$\rm\:\le n\:x\:$</span>. The LHS counts them in a unique mod <span class="math-container">$\rm\ n\ $</span> representation, <span class="math-container">$\:$</span> viz. <span class="math-container">$\rm\ \: j \... |
70,143 | <p>Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?</p>
<p>Everything I find in the literature is for blow-ups along smooth invariant centers.</p>
<p>Thanks!</p>
| Sasha | 4,428 | <p>If you are interested not in the toric picture, but in the structure of the variety itself, the following approach is very useful. Note that the ideal of the pair of lines has the following resolution on $P^3$:
$$
0 \to O(-3) \to O(-2) \oplus O(-1) \to I \to 0.
$$
Hence the graded algebra $\oplus I^k$ is the quotien... |
70,143 | <p>Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?</p>
<p>Everything I find in the literature is for blow-ups along smooth invariant centers.</p>
<p>Thanks!</p>
| Allen Knutson | 391 | <p>Since you already know how to blow up along either ${\mathbb P}^1$ individually, we can concentrate on what's happening nearby the intersection. Which means we can work affinely.</p>
<p>Then the polyhedron for ${\mathbb A}^3$ is the octant $({\mathbb R}_{\geq 0})^3$, and your two lines correspond to two of the thre... |
609,245 | <p>Is it always possible to extend a linearly independent set to a basis in infinite dimensional vector space?</p>
<p>I was proving with the following argument:
If S is a linearly independent set, if it spans the vector space then done else keep on adding the elements such that the resultant set is also linearly indep... | Quimey | 10,443 | <p>Let $V$ be a vector space, $S\subseteq V$ a linearly independent subset and $\mathcal{A}=\{T\subseteq V: S\subseteq T \text{and $T$ is linearly independent}\}$. It is easy to see that any chain on $\mathcal{A}$ has an upper bound on $\mathcal{A}$ (we can take the union). Then, it follows from Zorn's lemma that $\mat... |
609,245 | <p>Is it always possible to extend a linearly independent set to a basis in infinite dimensional vector space?</p>
<p>I was proving with the following argument:
If S is a linearly independent set, if it spans the vector space then done else keep on adding the elements such that the resultant set is also linearly indep... | K. Carrillo | 544,243 | <p>Let be $V$ a vector space and $W\leq V$ a subspace of $V$ with basis $Y$. If we consider the quotient space $V/W$ , by Zorn's lemma, we can obtain a basis of $V/W$, denoted $\overline{S}$.</p>
<p>If $v\in V$, $\overline{v}=\beta_1\overline{\alpha_1}+\cdots+\beta_k\overline{\alpha_k}$
where $\alpha_i\in S$, then $... |
2,698,553 | <p>Is the natural ring morphism $\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}$ an isomorphism?</p>
<p>In other words, is there a $\mathbb Z$-linear map $f:\mathbb{C}\otimes_{\mathbb{Q}}\mathbb{C}\to\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{C}$ such that
$$
f(z\otimes w)=z\otimes w
$... | Angina Seng | 436,618 | <p>If $U$ and $V$ are vector spaces over $\Bbb Q$, then $U\otimes_\Bbb Z V$
and $U\otimes_\Bbb Q V$ are always isomorphic. To see this,
$$U\otimes_\Bbb Z V\cong U\otimes_\Bbb Q\Bbb Q\otimes_\Bbb Z V
\cong U\otimes_\Bbb Q V.$$
As the last stage we use $\Bbb Q\otimes_\Bbb Z\Bbb Q\cong\Bbb Q$
and the fact that $V$ has a b... |
1,216,302 | <p>I am looking for some sequence of random variables $(X_n)$ such that </p>
<p>$$ \lim_{n \rightarrow \infty} E(X_n^2) = 0 $$</p>
<p>but such that the following <strong>almost sure</strong> convergence does <strong>NOT</strong> hold:</p>
<p>$$ \frac{S_n - E(S_n)}{n} \rightarrow 0$$</p>
<p>where the $S_n$ are the ... | D. Thomine | 20,413 | <p>Here is an algorithm which gives you such a sequence. Let us work on the probabilised space $[0,1)$ with the Lebesgue measure.</p>
<p>For all $0 \leq k < n$, let $I_{k,n} := [k/n, (k+1)/n)$. Fix $\varepsilon \in (0,1)$</p>
<p>Start from $n = 1$, $k=0$, time $N=0$.</p>
<p>If $S_N < \varepsilon N$ on $I_{k,n}... |
1,636,207 | <p>I understand the basics of Cartesian products, but I'm not sure how to handle a set inside of a set like $C = \{\{1,2\}\}$. Do I simply include the set as an element, or do I break it down?</p>
<p>If I use it as an element I think it would be something like this:</p>
<p>$$\{(1,\{1,2\}), (2,\{1,2\})\}$$</p>
<p>If... | MPW | 113,214 | <p>That's right. Just proceed as you normally would. $B$ has two elements and $C$ has one element. Pair them up (it may help to write "$A$" in place of "$\{1,2\}$" temporarily):</p>
<p>$$A\times B = \{(1,A), (2,A)\} = \{(1,\{1,2\}), (2,\{1,2\})\}$$</p>
|
3,035,228 | <p>Show that two cardioids <span class="math-container">$r=a(1+\cos\theta)$</span> and <span class="math-container">$r=a(1-\cos\theta)$</span> are at right angles.</p>
<hr>
<p><span class="math-container">$\frac{dr}{d\theta}=-a\sin\theta$</span> for the first curve and <span class="math-container">$\frac{dr}{d\theta}... | Jean Marie | 305,862 | <p><strong>Solution 1 :</strong> take a look at the following figure :</p>
<p><a href="https://i.stack.imgur.com/P2aND.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/P2aND.jpg" alt="enter image description here"></a></p>
<p>Representing the two cardioids we are working on and 2 circles described ... |
1,333,994 | <p>We have a function $f: \mathbb{R} \to \mathbb{R}$ defined as</p>
<p>$$\begin{cases} x; \ \ x \notin \mathbb{Q} \\ \frac{m}{2n+1}; \ \ x=\frac{m}{n}, m\in \mathbb{Z}, n \in \mathbb{N} \ \ \ \text{$m$ and $n$ are coprimes} \end{cases}.$$</p>
<p>Find where $f$ is continuous</p>
| user26486 | 107,671 | <p>$$\log_{3}(1/2)+\log_{1/2}(3)=\log_3(1/2)+\frac{\log_3(3)}{\log_3(1/2)}=$$</p>
<p>$$=\log_{3}(1/2)+\frac{1}{\log_3(1/2)}$$</p>
<p>$$=-\left(\left(\sqrt{-\log_{3}(1/2)}-\sqrt{-\frac{1}{\log_3(1/2)}}\right)^2+2\right)\le -2$$</p>
<p>Equality can't occur, because $\sqrt{-\log_{3}(1/2)}\neq \sqrt{-\frac{1}{\log_3(1/2... |
345,766 | <p>I'm trying to calculate this limit expression:</p>
<p>$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $$</p>
<p>Both the numerator and denominator should converge, since $0 \leq a, b \leq 1$, but I don't know if that helps. My guess would be to use L'Hopital's rule and take the ... | lab bhattacharjee | 33,337 | <p>If $ab=1,$</p>
<p>$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s}= \lim_{s \to \infty} \frac{s}{s+1}=\lim_{s \to \infty} \frac1{1+\frac1s}=1$$</p>
<p>If $ab\ne1, $</p>
<p>$$\lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s}$$</p>
<p>$$=\lim_{s \to... |
3,337,475 | <p>This is definitely the most difficult integral that I've ever seen.
Of course, I'm not able to solve this.
Could you help me?</p>
<p><span class="math-container">$$\int { \sin { x\cos { x } \cosh { \left( \ln { \sqrt { \frac { 1 }{ 1-\sin { x } } } +\tanh ^{ -1 }{ \left( \sin x \right) +\tanh ^{ -1 }{ \left( \co... | Ninad Munshi | 698,724 | <p>Using the fact that <span class="math-container">$\cosh t = \frac{1}{\sqrt{1-\tanh^2 t}}$</span> and <span class="math-container">$\sinh t = \frac{\tanh t}{\sqrt{1-\tanh^2 t}} $</span> <span class="math-container">$$\cosh(a+b+c) = \cosh a \cosh b \cosh c + \sinh a \sinh b \cosh c + \sinh a \cosh b \sinh c + \cosh ... |
543,938 | <p>Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.</p>
| bof | 97,206 | <p>Since $p$ is prime, $\displaystyle\binom pj\equiv0\pmod p$ for $0\lt j\lt p$. </p>
<p>By Pascal's identity, for $0\lt j\lt p$ we have $$\binom{p-1}j=\binom pj-\binom{p-1}{j-1}\equiv-\binom{p-1}{j-1}\pmod p.$$Since $\displaystyle\binom{p-1}0=1$, it follows by induction that $\displaystyle\binom{p-1}j\equiv(-1)^j\pm... |
524,686 | <p>Let X be a uniform random variable on [0,1], and let $Y=\tan\left (\pi \left(x-\frac{1}{2}\right)\right)$. Calculate E(Y) if it exists. </p>
<p>After doing some research into this problem, I have discovered that Y has a Cauchy distribution (although I do not know how to prove this); therefore, E(Y) does not exist.<... | Zz'Rot | 51,747 | <p>My intuition would be, since $\tan$ is a periodic function, you can first consider a single cycle. (In this case, maybe $x\in(0,1)$.) Since you have infinite number of cycles, you can deduce that the integral diverges.</p>
|
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | wythagoras | 236,048 | <p>On the bottom you have axioms (things that are assumed to be true) and definitions. In the case of set theory, these might be the axioms of ZFC and the definitions that explain them. PA or KP might be another possibility.</p>
<p>We will need another informal system (like English) to build the lowest axioms. But Eng... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | user21820 | 21,820 | <p>Most set theories, such as ZFC, require an underlying knowledge of first-order logic formulas (as strings of symbols). This means that they require acceptance of facts of string manipulations (which is essentially equivalent to accepting arithmetic on natural numbers!) First-order logic does not require set theory, ... |
4,189,614 | <p>I am trying to answer the following question...</p>
<blockquote>
<p>Consider a wall made of brick <span class="math-container">$10$</span> centimeters thick, which separates a room in a house from the outside. The room is kept at <span class="math-container">$20$</span> degrees. Initially the outside temperature i... | Michał Miśkiewicz | 350,803 | <p>Following user89699's answer, let me phrase the same thought differently.</p>
<hr />
<p>In principle, you're right. The wall is 3-dimensional, so the 3-dimensional heat flow would be our first choice. For simplicity, let us assume that the wall stretches infinitely in directions <span class="math-container">$y$</spa... |
471,145 | <p>I'm reading Proofs from the Book, and I ran into following theorem:</p>
<p>Suppose all roots of polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$ are real. Then the roots are contained in the interval:</p>
<p>$$ - \frac{a_{n-1}}{n} \pm \frac{n-1}{n}
\sqrt{a_{n-1}^2 - \frac{2n}{n-1} a_{n-2} } $$</p>
<p>So, if ... | Daniel Fischer | 83,702 | <p>For an open set $U \subset \mathbb{C}$, there always are $f \in \mathcal{O}(U)$ which cannot be continued analytically across any part of the boundary $\partial U$.</p>
<p>That is an easy consequence of the (general) Weierstraß product theorem, which I quote here from Rudin (Real and Complex Analysis; Thm 15.11):</... |
220,736 | <p>I have reduced a problem I'm working on to something resembling a graph theory problem, and my limited intuition tells me that it's not so esoteric that only I could have ever considered it. <strong>I'm looking to see if someone knows of any related work.</strong> Here's the problem:</p>
<hr>
<p>Given a roadway ... | Igor Rivin | 11,142 | <p>The area is certainly the same for all smooth convex curves and small $\epsilon$ - your polygonal curve is a good way to see why that might be true. For large $\epsilon,$ it is not clear what the question means...</p>
|
1,072,656 | <p>I am building a website which will run on the equation specified below. I am in pre-algebra and do not have any idea how to go about this equation. my friends say it is a system of equation but I don't know how to solve those and no one I know seems to know how to do them with exponents. I was hoping that people on ... | turkeyhundt | 115,823 | <p>The first equation tells you that $x=\frac{8000}{y^5}$. Substituting that into the second equation you have $\frac{8000}{y^5}y^4-\frac{8000}{y^5}y^3=5000$ $$\frac{8000}{y}-\frac{8000}{y^2}=5000\\\frac{y^2}{1000}(\frac{8000}{y}-\frac{8000}{y^2}=5000)\\8y-8=5y^2\\5y^2-8y+8=0$$</p>
<p>But that gives imaginary roots, ... |
102,661 | <blockquote>
<p>$n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{... | yiyi | 23,662 | <p>$\displaystyle{p(n) = 1 - \left(\frac{364}{365}\right)^{C(n,2)} = 1 - \left(\frac{364}{365}\right)^{n(n-1)/2} }$</p>
<p>Sorry if my latex is not right. </p>
<p>The big trick with most prob questions is to ask what is the prob if it doesn't happen. </p>
<p>So you take 1 (total sample space) - P(not your birthday)... |
102,661 | <blockquote>
<p>$n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{... | Karatuğ Ozan Bircan | 12,686 | <p>Paul Halmos asked this question in his "automathography", <em>I Want to Be a Mathematician</em>, and solved it as follows:</p>
<blockquote>
<p>In other words, the problem amounts to this: find the smallest <span class="math-container">$n$</span> for which <span class="math-container">$$\prod_{k=0}^{n-1} \... |
1,530,406 | <p>How to multiply Roman numerals? I need an algorithm of multiplication of numbers written in Roman numbers. Help me please. </p>
| ByronGeorge | 656,816 | <p>I don't like any on-line solutions to Numeral multiplication so I made one up.</p>
<p>The only thing you need to remember when multiplying is V x V = XXV all other multiple combinations are column shifts</p>
<p>Example: XXVII multiplied by XVIII</p>
<p>XXXVII across the top is multiplied by each individual numera... |
2,327,675 | <p>Using the GPU Gems Article <a href="https://developer.nvidia.com/gpugems/GPUGems/gpugems_ch01.html" rel="nofollow noreferrer" title="Effective Water Simulation From Physical Models">Effective Water Simulation From Physical Models</a> I have implemented Gerstner Waves into UE4, I have built the function both on GPU f... | Tyrendel | 1,098,410 | <p>Solution that I use:
<span class="math-container">$$
y=\frac{a}{2}\sin\left(\lambda x-a\cos\left(\lambda x-a\cos\left(...\right)\right)\right)
$$</span>
with:</p>
<ul>
<li><span class="math-container">$a$</span>: wave effect strength</li>
<li><span class="math-container">$p$</span>: wavelength</li>
<li><span class="... |
1,903,333 | <p>Let $G$ be a group. Prove that $Z(G)$ (the center of $G$) is always nonempty.</p>
<p>Can anyone give me solution of this theoretical problem? I have just started learning group theory and I am very interested in this math branch</p>
| Probabilitytheoryapprentice | 361,972 | <p>Hint: Think of the neutral element^^</p>
|
2,992,454 | <p>Prove :</p>
<blockquote>
<p><span class="math-container">$f : (a,b) \to \mathbb{R} $</span> is convex, then <span class="math-container">$f$</span> is bounded on every closed subinterval of <span class="math-container">$(a,b)$</span></p>
</blockquote>
<p>where <span class="math-container">$f$</span> is convex if... | Community | -1 | <p>One way could be to first prove that <span class="math-container">$f$</span> is continuous on <span class="math-container">$(a,b)$</span>. Then, we know that <span class="math-container">$f$</span> is bounded on every closed subinterval of <span class="math-container">$(a,b)$</span>.</p>
<p>A proof of this assertio... |
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | Vincenzo Zaccaro | 269,380 | <p>Suppose that $ a, b $ are coprime numers. Note that $3 |b^2 $ then $3|b|$. If $3|b|$ then $3|a $, absurd!</p>
|
168,118 | <p>I'm attempting to differentiate an equation in the form</p>
<pre><code>D[sqrt((2*(((a*b*c+Pi*d*e^2+Pi*f*g^2+h*i*j+Pi*k*l^2)/(a*b*c+Pi*d*e^2+Pi*f*g^2))-1)*m)/(n^2 - o^2)/p), a]
</code></pre>
<p>in order to do an error propagation analysis. So I need to differentiate it against a, against b, against c and so on.</p>... | Bob Hanlon | 9,362 | <pre><code>assume = (And @@
Thread[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p} > 0]) && n > o;
</code></pre>
<p>Note that when you state that a variable is positive then it is automatically also real. And for <a href="http://reference.wolfram.com/language/ref/$Assumptions.html" rel="nofollow n... |
2,106,662 | <p>I'm trying to show that the barycentric coordinate of excenter of triangle ABC, where BC=a, AC=b, and AB=c, and excenter opposite vertex A is Ia, is Ia=(-a:b:c). I've gotten to the point where after a lot of ratio bashing I have that it's (ab/(b+c)):CP:BP, where P is the incenter, but I have no idea where to go from... | dxiv | 291,201 | <p>Hint: written in matrix form:</p>
<p>$$
\begin{pmatrix}
a_{n+1} \\ b_{n+1}
\end{pmatrix}
= \begin{pmatrix}
2 & -1 \\ 1 & 4
\end{pmatrix}
\begin{pmatrix}
a_{n} \\ b_{n}
\end{pmatrix}
$$</p>
<p>Let $A_n = \begin{pmatrix}
a_{n+1} \\ b_{n+1}
\end{pmatrix}
\,$, $A = \begin{pmatrix}
2 & -1 \\ 1 & 4... |
853,308 | <p>I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the rest of that column $0$ as entries.</p>
<p>$P = \begin{bmatrix} 1 & * & \cdots & * \\ 0 & \vdots &am... | Asaf Karagila | 622 | <p>Functions are only intuitive if you think about $f(x)=x^2+1$ or $f(x,y)=\ln x+e^y$ or so on. But how do you describe in an intuitive manner <strong>every</strong> function from $\Bbb R$ to $\Bbb R$? There are more than you can possibly imagine. How would you describe intuitively a function between two sets which you... |
3,460 | <p>I asked the question "<a href="https://mathoverflow.net/questions/284824/averaging-2-omegan-over-a-region">Averaging $2^{\omega(n)}$ over a region</a>" because this is a necessary step in a research paper I am writing. The answer is detailed and does exactly what I need, and it would be convenient to directly cite t... | Count Iblis | 52,954 | <p><a href="http://countiblis.blogspot.nl/2005/12/newcombs-paradox-and-conscious.html" rel="nofollow noreferrer">This blog posting</a> by me was cited in <a href="https://arxiv.org/abs/math/0608592" rel="nofollow noreferrer">this article</a> on page 12, footnote 2. </p>
|
2,108,558 | <p>We are started Linear programming problem question. Questions provided in examples are easy. And in exercise starting questions are easy to solve. As we have given conditions to form equations and solve them.</p>
<p>But this question little difficult.</p>
<p>Question - </p>
<p><strong>An aeroplane can carry a max... | DonAntonio | 31,254 | <p>A matrix over <em>any</em> field is non-diagonalizable iff its minimal polynomial is divisible by a linear factor to a power $\;\ge2\;$ , so $\;A\;$ is non diagonalizable iff</p>
<p>$$\min_A(x)=(x-a_1)^{n_1}g(x)\;,\;\;2\le n_1\in\Bbb N$$</p>
<p>and thus we can take $\;Q(x)=(x-a_1)g(x)\;$ and, of course, $\;Q(A)^2... |
2,928,849 | <p>I have a problem understanding the following:</p>
<p><span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent vatiables with
<span class="math-container">$$P(X = i) = P(Y = i) = \frac{1}{2^i}. \quad i = 1, 2, \cdots$$</span>
Now the book says<span class="math-container">$ ... | Narendra | 583,106 | <p><span class="math-container">$$P(X < i)+P(X \geq i) = 1 $$</span>
<span class="math-container">$P(X \geq i) = 1-P(X < i)$</span></p>
<p><span class="math-container">$P(X<i) = P(X=1)+P(X=2)+......P(X=i-1)$</span></p>
<p><span class="math-container">$= \dfrac{1}{2^1} +\dfrac{1}{2^2}+.....\dfrac{1}{2^{i-1}}$... |
1,303,274 | <p>Define a sequence {$\ x_n$} recursively by</p>
<p>$$ x_{n+1} =
\sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1
$$
Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.</p>
| Paolo Leonetti | 45,736 | <p>If it converges to $\ell$ then $|s_n-\ell|<\varepsilon$, once $\varepsilon>0$ is fixed, and $n$ is sufficiently large. Here $s_n$ is exactly the $n$-th partial sum.</p>
<p>But if you set, for example, $\ell=1/10$, then the above inequality cannot be satisfied even for $n$ enough large.</p>
|
3,044,318 | <p><span class="math-container">$$\frac{e^{z^2}}{z^{2n+1}}$$</span>
Am I right that limit as z approaches infinity does not exist? So its residue at infinity is equal to <span class="math-container">$c_{-1}$</span> of Laurent series. How am I supposed to get Laurent series of this function? Where is it centered? What r... | José Carlos Santos | 446,262 | <p>The residue at infinity of an analytic function <span class="math-container">$f$</span> is the residue at <span class="math-container">$0$</span> of <span class="math-container">$\frac{-1}{z^2}f\left(\frac1z\right)0$</span>. In the case of the function that you mentioned, it's the residue at <span class="math-contai... |
3,875,643 | <p>I am studying the nonlinear ordinary differential equation</p>
<p><span class="math-container">$$\frac{d^2y}{dx^2}=\frac{1}{y}-\frac{x}{y^2}\frac{dy}{dx}$$</span></p>
<p>I have entered this equation into two different math software packages, and they produce different answers.</p>
<p>software 1:</p>
<p><span class="... | user577215664 | 475,762 | <p><span class="math-container">$$\frac{d^2y}{dx^2}=\frac{1}{y}-\frac{x}{y^2}\frac{dy}{dx}$$</span>
You can check the solution just rewrite the DE as:
<span class="math-container">$$\frac{d^2y}{dx^2}=(x)'\frac{1}{y}+x \left (\frac{1}{y}\right )'$$</span>
Since <span class="math-container">$(fg)'=f'g+fg'$</span> we have... |
962,573 | <p>I have just learned Fermat's little theorem.</p>
<p>That is,</p>
<blockquote>
<p>If $p$ is a prime and $\gcd(a,p)=1$, then $a^{p-1} \equiv 1 \mod p$</p>
</blockquote>
<p>Well, there's nothing more explanation on this theorem in my book.</p>
<p>And there are exercises of this kind</p>
<blockquote>
<p>If $\gc... | davidlowryduda | 9,754 | <p>Fermat's Little Theorem gives that $a \equiv 1 \pmod 2$, $a^6 \equiv 1 \pmod 3$, and $a^6 \equiv 1 \pmod 7$ immediately.</p>
<p>If $a \equiv 1 \pmod 2$, then $a \equiv \pm 1, \pm 3 \pmod 8$. In either of these, it's immediate that $a^2 \equiv 1 \pmod 8$, and so $a^6 \equiv 1 \pmod 8$.</p>
<p>Now by the Chinese Rem... |
320,355 | <p>Show that $$\nabla\cdot (\nabla f\times \nabla h)=0,$$
where $f = f(x,y,z)$ and $h = h(x,y,z)$.</p>
<p>I have tried but I just keep getting a mess that I cannot simplify. I also need to show that </p>
<p>$$\nabla \cdot (\nabla f \times r) = 0$$</p>
<p>using the first result.</p>
<p>Thanks in advance for any help... | Sangchul Lee | 9,340 | <p>Introducing the Levi-Civita symbol, we have</p>
<p>\begin{align*}
\nabla \cdot (\nabla f \times \nabla h)
&= \epsilon^{ijk} \frac{\partial}{\partial x^{i}} \left( \frac{\partial^2 f}{\partial x^{j}} \frac{\partial h}{\partial x^{k}} \right) \\
&= \epsilon^{ijk} \left( \frac{\partial f}{\partial x^{i} \parti... |
609,770 | <p>We have an empty container and $n$ cups of water and $m$ empty cups. Suppose we want to find out how many ways we can add the cups of water to the bucket and remove them with the empty cups. You can use each cup once but the cups are unique. </p>
<p>The question: In how many ways can you perform this operation.</p... | Michael Hardy | 11,667 | <p>Setting $\varepsilon$ to something doesn't make sense. You need to take $\varepsilon$ to be given, and find a value of $\delta$ that's small enough.</p>
<p>Continuity should not say $\exists c\in(0,1]$ etc., where $c$ is in the role you put it in. Rather, continuity <b>at the point $c$</b> should be defined by wh... |
1,142,631 | <p>A sequence of probability measures $\mu_n$ is said to be tight if for each $\epsilon$ there exists a finite interval $(a,b]$ such that $\mu((a,b])>1-\epsilon$ For all $n$.</p>
<p>With this information, prove that if $\sup_n\int f$ $d\mu_n<\infty$ for a nonnegative $f$ such that $f(x)\rightarrow\infty$ as $x\r... | Davide Giraudo | 9,849 | <p>Note that for a fixed $t_0$, we have
$$\mu_n(\mathbf R\setminus [-t_0,t_0])\cdot\inf_{x:|x|\geqslant t_0 } f(x)\leqslant \int_\mathbf R f(x) \mathrm d\mu_n,$$
hence defining $M:=\sup_n\int_\mathbf R f(x) \mathrm d\mu_n$, it follows that
$$\mu_n(\mathbf R\setminus [-t_0,t_0])\cdot\inf_{x:|x|\geqslant t_0 } f(x)\le... |
3,976,572 | <p>Suppose <span class="math-container">$f(x)= x^3+2x^2+3x+3$</span> and has roots <span class="math-container">$a , b ,c$</span>.
Then find the value of
<span class="math-container">$\left(\frac{a}{a+1}\right)^{3}+\left(\frac{b}{b+1}\right)^{3}+\left(\frac{c}{c+1}\right)^{3}$</span>.</p>
<p>My Approach :
I constructed... | See Hai | 646,705 | <p>We shall make use of the following well-known identity:
<span class="math-container">$$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-xz).$$</span>
Now, using Vieta's Formula, <span class="math-container">$a+b+c=-2, ab+ac=bc=3$</span>, and <span class="math-container">$abc=-3$</span>.
Thus, by direct expansion, we have ... |
55,232 | <p>I'm looking for a concise way to show this:
$$\sum_{n=1}^{\infty}\frac{n}{10^n} = \sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right)$$
With this goal in mind:
$$\sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right) =
\sum_{n=1}^{\infty}\left(\left(\frac{... | Qiaochu Yuan | 232 | <p>The RHS can be written $\displaystyle \sum_{n \ge 1, m \ge 0} \frac{1}{10^{n+m}}$. Collect all terms with the same value of $n+m$. To justify this rigorously you just need to know that the series converges absolutely, which should be clear. </p>
|
55,232 | <p>I'm looking for a concise way to show this:
$$\sum_{n=1}^{\infty}\frac{n}{10^n} = \sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right)$$
With this goal in mind:
$$\sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right) =
\sum_{n=1}^{\infty}\left(\left(\frac{... | Gerry Myerson | 8,269 | <p>Writing $x$ for $1/10$ we have $$\sum_{n=1}^{\infty}nx^n=\sum_{n=1}^{\infty}\sum_{m=1}^nx^n=\sum_{m=1}^{\infty}\sum_{n=m}^{\infty}x^n=\sum_{m=1}^{\infty}x^m\sum_{n=m}^{\infty}x^{n-m}=\sum_{m=1}^{\infty}x^m\sum_{n=0}^{\infty}x^n$$ and the last sum is your right-hand side (except that my $m$ is your $n$, and vice vers... |
4,536,320 | <p>Let <span class="math-container">$R$</span> be a ring and <span class="math-container">$A \subseteq R$</span> be finite, say <span class="math-container">$A = \{a\}$</span>. The set <span class="math-container">$$RaR = \{ras\:\: : r,s\in R\}$$</span> Why is this not necessarily closed under addition?</p>
<p>Take <sp... | N. F. Taussig | 173,070 | <p>Your formulation of the problem is not correct. Observe that if <span class="math-container">$m = 18$</span> and <span class="math-container">$a = 20$</span>, then <span class="math-container">$m + a = 38 > 20$</span>. Also, since we want <span class="math-container">$\frac{m + a}{2} \in \{1, 2, 3, \ldots, 20\}... |
520,285 | <p>I've been going through this representation theory <a href="http://math.berkeley.edu/~serganov/math252/notes1.pdf" rel="nofollow">lecture notes</a>, and I've found the ''hungry knights'' problem very interesting.
Do you know any interesting problems similar to that one?</p>
<p>To define ''similar'': problems which ... | Calvin Lin | 54,563 | <p>A similar one is a Russian Olympiad problem about 7 dwarfs sitting around a table drinking wine. Each of them have a wine cup in front of them. In turn, they split the wine in their glass into 6 equal portions and distribute it out. After a round of distribution, they found that they have the same amount of wine as ... |
587,198 | <p>I am having problems with this question, it would be wonderful if someone can help.</p>
<p>Given that $f(x)= x^2 + x - 3$</p>
<p>1) Find $f(x + h)$</p>
<p>2) Then express $f(x+h)-f(x)$ in its simplest form</p>
<p>3) Deduce $\lim\limits_{h->0}\dfrac{f(x+h)-f(x)}{h}$</p>
<p>Thanks for the help, i was stuck on ... | Tim Ratigan | 79,602 | <p>As Stefan Smith kindly pointed out, $a=b=0$ is a trivial solution in the integers. But what if $a,b\in \Bbb N$?</p>
<p>Assume $2a^2=b^2$. Then $2|b^2$, which implies (by unique prime factorization) that $2|b$. Therefore we can write $b^2=4k^2$ for some $k\in\Bbb N$. Then $2a^2=4k^2\Longrightarrow 2k^2=a^2$. Th... |
191,673 | <p>If I input:</p>
<pre><code>data = RandomVariate[ProbabilityDistribution[x/8, {x, 0, 4}], 10];
{EstimatedDistribution[data, ProbabilityDistribution[x/8, {x, 0, θ}],
ParameterEstimator -> "MaximumLikelihood"], data}
</code></pre>
<p>Mathematica returns:</p>
<pre><code>{ProbabilityDistribution[\[FormalX]/
8, ... | Michael E2 | 4,999 | <p>The log-likelihood of the OP's (non-normalized) distribution is piecewise constant, so any value for <code>θ</code> that occurs in a certain interval is "correct," at least from the point of view of maximizing the expression returned by <code>LogLikelihood[]</code>. As <a href="https://mathematica.stackexchange.com... |
2,870,956 | <p>Assume that f is a non-negative real function, and let $a>0$ be a real number.</p>
<p>Define $I_a(f)$ to be</p>
<p>$I_a(f)$=$\frac{1}{a}\int_{0}^{a} f(x) dx$ </p>
<p>We now assume that $\lim_{x\rightarrow \infty} f(x)=A$ exists.</p>
<p>Now I want to proof whether $\lim_{a\rightarrow \infty} I_a(f)=A$ is true ... | mfl | 148,513 | <p>Assume $f$ is continuous on $[0,\infty)$ and $\lim_{x\to \infty} f(x)=L.$ Then it is $\lim_a I_a(f)=L.$ Indeed, we have</p>
<p>$$\lim_{a\to \infty} \dfrac{\int_0^a f(t)dt}{a}=\lim_{a\to \infty} \dfrac{f(a)}{1}=\lim_{a\to \infty}f(a) =L,$$ where we have used the Fundamental theorem of calculus and L'Hôpital's rule.<... |
2,870,956 | <p>Assume that f is a non-negative real function, and let $a>0$ be a real number.</p>
<p>Define $I_a(f)$ to be</p>
<p>$I_a(f)$=$\frac{1}{a}\int_{0}^{a} f(x) dx$ </p>
<p>We now assume that $\lim_{x\rightarrow \infty} f(x)=A$ exists.</p>
<p>Now I want to proof whether $\lim_{a\rightarrow \infty} I_a(f)=A$ is true ... | James Yang | 481,378 | <p>Your counter-example works because f is not integrable on $(0,a)$ for some $a$. If $f$ is integrable on $(0,a)$ for every a, it's true.</p>
<p>WLOG, assume $\lim\limits_{x\to \infty} f(x) = 0$. Otherwise, since $f$ is integrable, we may show that $\frac{1}{a}\int_0^a (f(x)-A) dx \to 0$. For every $\epsilon >0$, ... |
2,665,723 | <p>Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.</p>
<p>solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$</p>
<p>$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$</p>
<p>$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k}}+\sqrt{2z+k... | Angina Seng | 436,618 | <p>What you have there is <strong>not</strong> $\sum_{n=0}^{99}n(n+1)$ but rather
$\sum_{n=1}^{50}2n(2n-1)$. Using standard formulae,
$$\sum_{n=1}^{50}2n(2n-1)
=4\sum_{n=1}^{50}n^2-2\sum_{n=1}^{50}n
=\frac23(50\times 51\times 101)-50\times51=169150.
$$</p>
|
125,503 | <p>Completeness Properties of $\mathbb{R}$: Least Upper Bound Property, Monotone Convergence Theorem, Nested Intervals Theorem, Bolzano Weierstrass Theorem, Cauchy Criterion.</p>
<p>Archimedean Property: $\forall x\in \mathbb{R}\forall \epsilon >0\exists n\in \mathbb{N}:n\epsilon >x$</p>
<p>I can show that LUB ... | Elchanan Solomon | 647 | <p>For Bolzano-Weierstrass $\to$ Archimedean Property: Take $\epsilon > 0$, and consider the sequence $\{n\epsilon: n \in \mathbb{N}\}$. If the Archimedean Property fails, then this sequence is bounded, so that it has a convergent subsequence. However, it is easy to see that this sequence does not have a convergent ... |
131,741 | <p>Take the following example <code>Dataset</code>:</p>
<pre><code>data = Table[Association["a" -> i, "b" -> i^2, "c" -> i^3], {i, 4}] // Dataset
</code></pre>
<p><img src="https://i.stack.imgur.com/PZSgO.png" alt="Mathematica graphics"></p>
<p>Picking out two of the three columns is done this way:</p>
<pr... | yode | 21,532 | <p>Since nobody mentioned it, I will give a version with <code>KeyTake</code></p>
<pre><code>data[All /* KeyTake[{"a", "b"}], {"a" -> f, "b" -> h}]
</code></pre>
<p><img src="https://i.stack.imgur.com/WtVGN.png" alt=""> </p>
<p>Of course we can also do</p>
<pre><code>data[All /* Map[Take[#, 2] &], {"a" -&... |
2,963,324 | <p>I want to prove <span class="math-container">$a \equiv b\;(\text{mod} \;n)$</span> is an equivalence relation then would it be ok to write,</p>
<p>Reflexive as, for all <span class="math-container">$a$</span>, <span class="math-container">$a \equiv a\;(\text{mod} \;n)$</span></p>
<p>Symmetric as, <span class="math... | José Carlos Santos | 446,262 | <p>No, it is not ok. You just <em>claimed</em> that all three assertions are true without a trace of evidence. Therefore, you proved nothing.</p>
|
2,498,123 | <blockquote>
<p>Given a $2 \times 2$ matrix $B$ that satisfies $B^2=3B-2I$, find the eigenvalues of $B$.</p>
</blockquote>
<p>My attempt: </p>
<p>Let $v$ be an eigenvector for B, and $\lambda$ it's corresponding eigenvalue. Also, let $T$ be the linear transformation (not that this is exactly necessary for the quest... | amsmath | 487,169 | <p>Define $p(x) = (x-1)(x-2)$. You have $p(B) = 0$. Hence, the minimal polynomial of $B$ divides $p$. So, your eigenvalues are in $\{1,2\}$.</p>
|
3,118 | <p>Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:</p>
<p>If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.</p>
| Bill Dubuque | 242 | <blockquote>
<p>If <span class="math-container">$\rm\,\ a\mid m,\ a\!+\!1\mid m\ \,$</span> then it follows that <span class="math-container">$\rm\ \, \color{#90f}{a(a\!+\!1)\mid m}$</span></p>
</blockquote>
<p><span class="math-container">${\bf Proof}\rm\quad\displaystyle \frac{m}{a},\; \frac{m}{a+1}\in\mathbb{Z} \ \... |
805,341 | <p>Need a bit of help with this question. </p>
<p>We're given two invertible square $n\times n$ matrices $A$ and $B$ with entries in the reals.</p>
<p>We have to show that $AB$ is also invertible and then express $(AB)^{-1}$ in terms of $A$ and $B$. </p>
<p>I've managed to get the first part out. </p>
<p>Since $A$ ... | Andreas Caranti | 58,401 | <p>Assuming you mean $(AB)^{-1}$, there's a well-known simile that can help you finding the answer.</p>
<p>Think of $A$ and $B$ representing actions, like $A$ means putting on socks, $B$ means putting on shoes.</p>
<p>In the morning you do $AB$, socks first, then shoes.</p>
<p>In the evening you need to undo this, t... |
4,317 | <p>For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate</p>
<p>$$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}\qquad x>1.$$</p>
<p>The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{... | Qiaochu Yuan | 232 | <p>The general closed form is</p>
<p>$$\displaystyle \sum_{k=1}^{\infty} k^n x^k = \frac{1}{(1 - x)^{n+1}} \left( \sum_{m=0}^{n} A(n, m) x^{m+1} \right)$$</p>
<p>where $A(n, m)$ are the <a href="http://en.wikipedia.org/wiki/Eulerian_number">Eulerian numbers</a>. When I have time I will edit with a few more details. ... |
4,317 | <p>For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate</p>
<p>$$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}\qquad x>1.$$</p>
<p>The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{... | Arin Chaudhuri | 404 | <p>According to <a href="http://mathworld.wolfram.com/GeometricDistribution.html" rel="nofollow">mathworld</a> The $n^{th}$ moment of the geometric distribution with parameter $p$, which is $ \sum p (1-p)^n n^k $ can be expressed in terms of the polylogarithm function.</p>
|
2,413,368 | <p>I am to show that if $ w = z + \frac{c}{z} $ and $ |z| = 1 $, then $w$ is an ellipse, and I must find its equation.</p>
<p>Previously, I have solved transformation questions by finding the modulus of the transformation in either the form $ w = f(z) $ or $ z = f(w) $. However, I think the part stumping me here is th... | Donald Splutterwit | 404,247 | <p>$z$ is on the unit circle; let $z=e^{i \theta}$ so
\begin{eqnarray*}
w= (1+c) \cos( \theta) +i (1-c) \sin(\theta)
\end{eqnarray*}
which gives $ x= (1+c) \cos( \theta) , y= (1-c) \sin(\theta)$ and considered in cartesian coordinates
\begin{eqnarray*}
\frac{x^2}{ (1+c)^2} + \frac{y^2}{ (1-c)^2} =1 .
\end{eqnarray*}... |
1,439,920 | <blockquote>
<p>So, the question is:<br>
Calculate the probability that 10 dice give more than 2 6s.</p>
</blockquote>
<p>I've calculated that the probability for throwing 3 6s is 1/216.</p>
<p>And by that logic: 1/216 + 1/216 + .. + 1/216 = 10/216.</p>
<p>But I've been told that this isn't the proper way set it... | chandu1729 | 64,736 | <p>It's clear that $x=0$ is one of the roots. Hence, if we prove there are atleast 2 zeros to $ f(x) := x^4-1102x^3-2015$, we are done.</p>
<p>Observe, $f(0) < 0$ and $f(-2) > 0 $, so from Intermediate Value Theorem there exists at least one root between $-2$ and $0$. </p>
<p>Now, lets say there is exactly one ... |
1,439,920 | <blockquote>
<p>So, the question is:<br>
Calculate the probability that 10 dice give more than 2 6s.</p>
</blockquote>
<p>I've calculated that the probability for throwing 3 6s is 1/216.</p>
<p>And by that logic: 1/216 + 1/216 + .. + 1/216 = 10/216.</p>
<p>But I've been told that this isn't the proper way set it... | NoChance | 15,180 | <p>You can use <a href="http://Descartes'%20rule%20of%20signs" rel="nofollow">Descartes' rule of signs</a> to tell you the number of real roots as long as you are not interested in the value of each.</p>
<p>First observe that $x=0$ is a root for:</p>
<p>$f(x)=x^5 − 1102x^4 − 2015x$</p>
<p>Second, count positive ... |
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