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 |
|---|---|---|---|---|
3,042,433 | <blockquote>
<p>Rosanne drops a ball from a height of 400 ft. Find the ball's
average height and its average velocity between the time it is
dropped and the time it strikes the ground.</p>
</blockquote>
<p>My trial...</p>
<p>So, I tried to use average value theorem for integrals. I took acceleration as positive... | user | 505,767 | <p>Assuming <span class="math-container">$v_0=0$</span>, we have that</p>
<ul>
<li><p><span class="math-container">$h(t)=400-\frac12gt^2 \implies t_{max}=\sqrt{\frac{800}{g}}=20\sqrt{\frac{2}{g}}$</span> time to strike the ground</p></li>
<li><p><span class="math-container">$v(t)=gt \implies v_{max}=20\sqrt{2g} \impli... |
4,134,365 | <p>This is a question in a book that I am studying, and I have attempted to answer it but got it wrong.<br />
There were two parts that I wrong.<br />
The question is about five digit numbers where the digits are 1, 2, 3, or 4.</p>
<p>The first part asked how many numbers were such that the sum of the digits was even. ... | Ritam_Dasgupta | 925,091 | <p>For the first part you are correct, both expressions evaluate to <span class="math-container">$512$</span>. For the second part, you are clearly overcounting. Suppose you choose the last <span class="math-container">$3$</span> digits to be even, and then you freely set the first two. There would be a case where all ... |
4,134,365 | <p>This is a question in a book that I am studying, and I have attempted to answer it but got it wrong.<br />
There were two parts that I wrong.<br />
The question is about five digit numbers where the digits are 1, 2, 3, or 4.</p>
<p>The first part asked how many numbers were such that the sum of the digits was even. ... | Parcly Taxel | 357,390 | <p>Your answer for the first part is numerically the same, just arrived at by a more tedious method. For the second part your approach overcounts numbers; with <span class="math-container">$22424$</span> for example, you would count the processes "pick positions <span class="math-container">$1,2,3$</span> first&qu... |
1,258,198 | <p>Suppose $a > 1$. I want to compare
$$\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$ and $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx$$</p>
<p>My instinct suggests that after a certain value of $a$, $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx < e^{-a}\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$</p>
<p>bu... | Ben Grossmann | 81,360 | <p>Note that for $a>0$ and $x \geq 1$, we have
$$
e^{-2ax} = e^{-ax}e^{-ax} \leq e^{-a}e^{-ax}
$$
Thus, we have
$$
\int_1^\infty \frac{e^{-2ax}}{1+x^2} \leq e^{-a}\int_1^\infty \frac{e^{-ax}}{1+x^2}
$$
Perhaps this will suffice for your purposes.</p>
|
1,559,946 | <p>Can anyone solve this?</p>
<p>Find the sum of the series $1 + \frac{1}{2} +\frac{1}{3} + \frac{1}{4} + \frac{1}{5}+ \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \cdots,$ where the denominators are of the form $(2^i) (3^j)(5^k)$?</p>
<p>The test came with the next answer choices:</p>
<p>a) $\frac{7}{2}$</p>
<p>b) $... | Jack D'Aurizio | 44,121 | <p>We have an <a href="https://en.wikipedia.org/wiki/Euler_product" rel="nofollow">Euler product</a>:
$$ \prod_{p\leq 5}\left(1-\frac{1}{p}\right)^{-1} = \prod_{p\leq 5}\left(1+\frac{1}{p}+\frac{1}{p^2}+\ldots \right) = \sum_{n\in A}\frac{1}{n} $$
where $A$ is the set of positive integers whose prime divisors are $\leq... |
78,617 | <p>I'm sorry I'm French so the subject may not be properly translated, but here's my try:</p>
<p>A goat lives in a rectangular place. She's tied to the point P. The length of the row is 8 meters. The problem is that she can eat flowers: it's the shaded area. The farmer doesn't want the goat to eat the flowers, so he h... | Ross Millikan | 1,827 | <p>This assumes that the area to be strengthened is some of the border between the shaded area and the white area and no new fence may be built. The left extreme is 5 meters from the lower left corner of the hut. So it is 3 meters to the left of the left wall of the hut. The right extreme is 3 meters from the upper ... |
78,617 | <p>I'm sorry I'm French so the subject may not be properly translated, but here's my try:</p>
<p>A goat lives in a rectangular place. She's tied to the point P. The length of the row is 8 meters. The problem is that she can eat flowers: it's the shaded area. The farmer doesn't want the goat to eat the flowers, so he h... | Charles | 1,778 | <p>Place an $\varepsilon$-cm fence $\varepsilon$ cm to the left of P starting at the hut and extending downward. Then run $50+\varepsilon$ cm of fence from the bottom of the first fence toward the rightmost side of the pen. The goat's shortest path to the flowers is then $4.5+\sqrt{9+\varepsilon^2}+\sqrt{0.25+\varepsi... |
93,383 | <blockquote>
<p>A hotel can accommodate 50 customers, experiences show that $0.1$ of those who make a reservation will not show up. Suppose that the hotel accepts 55 reservations. Calculate the probability that the hotel will be able to accommodate all of the customers that show up. </p>
</blockquote>
<p>I only ... | Community | -1 | <p>You need to make use of the binomial distribution here. It is not hard to evaluate the answer using binomial distribution itself.</p>
<p>The probability that the hotel will be able to accommodate all of the customers that show up is the probability that at-most $50$ customers show up. Hence, $$\mathbb{P}(\text{at-m... |
561,648 | <p>this is the limit to evaluate: </p>
<p>$$\eqalign{
& \mathop {\lim }\limits_{n \to \infty } \root n \of {{a_1}^n + {a_2}^n + ...{a_k}^n} = \max \{ {a_1}...{a_k}\} \cr
& {a_1}...{a_k} \ge 0 \cr} $$</p>
<p>As far as I understand, $a_1..a_k$ is finite. right?<br>
Suppose it is, I'm clueless about the... | Haha | 94,689 | <p>Another solution is this one:</p>
<p>$T$ is self-adjoint iff $\langle Tx,x\rangle \in \mathbb{R}$ for every $x\in \mathcal{H}$.</p>
<p>Let $x\in \mathcal{H}$. Then $0=\langle Tx,x \rangle-\langle T^{*}x,x \rangle = \langle Tx,x \rangle-\overline{\langle Tx,x \rangle}=2i\operatorname{Im}\langle Tx,x\rangle \iff \l... |
450,228 | <p>Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ?</p>
<p>I want a series expansion such that $\zeta(s)=\sum_{n}^{\infty} f(n,s)$ where the $f(n,s)$ are standard functions without irrational constants.</p>
| mick | 39,261 | <p>Not really an answer but perhaps close.</p>
<p>A simple solution without the conditions that $f(n,s)$ are standard functions though is relatively easily obtained by : </p>
<p>$$\zeta^*(s)=\frac 1{1-2^{1-s}}\sum_{n=1}^\infty \frac {(-1)^{n-1}}{n^s}$$</p>
<p>$$\zeta^{**}(s)=\sum_{n=1}^\infty \frac {1}{n^{s+\frac{1}... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Alex B. | 3,212 | <p>Let me try as elementary as is humanly possible:</p>
<p>$X=\{p,q_1,q_2\}$, consisting of only three elements! The open sets are $U_0=\{p\}$, $U_1=\{p,q_1\}$, $U_2=\{p,q_2\}$, and of course the empty set and $X$. Define $\mathscr{F}(U)=\mathscr{G}(U)=\mathbb{Z}$ on all non-empty open sets $U$. Now, the trick is goin... |
3,550,162 | <p>if a complex number is prime in Gauss integers, does it follow that its complex conjugate is also prime?</p>
<p>I know in general if a “regular” number divides <span class="math-container">$a+bi$</span>, it also divided <span class="math-container">$a-bi$</span> but can’t show the same for all cause integers. Irred... | mjw | 655,367 | <p>Yes, if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> satisfy the conditions, then both <span class="math-container">$x+iy$</span> and <span class="math-container">$x-iy$</span> are Gaussian prime.</p>
<p>Please see <a href="http://mathworld.wolfram.com/GaussianPrime.html" rel=... |
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | David K | 139,123 | <p>If you recognize that $x^2 + 4x + 3 = (x+2)^2 - 1,$
then it can quickly be seen that the graph of that function is an
upward-opening parabola with its minimum at $x=-2$, where the value of the
function is $-1$.
Clearly for values of $x$ far from $-2$, the function is positive,
those parts of the graph of the functio... |
3,301,881 | <p>I'm reading the proof that <span class="math-container">$L_XY=[X, Y]$</span> on page 225 <a href="https://books.google.com/books?id=xQsTJJGsgs4C&q=225#v=snippet&q=225&f=false" rel="nofollow noreferrer">in this book</a> and I believe it is not quite correct to due an error in equation 20.6. The author wri... | merle | 299,107 | <p>I also stumbled across this error, and worked out the right derivative. I'll share the details of the computation, they might save someone some time.</p>
<p>We want to know the value of <span class="math-container">$\frac{\partial}{\partial t}|_{t=0} \left[ \frac{\partial{\phi_i}}{\partial x^j}(-t, \phi_t(p))\right... |
3,225,176 | <p>Prove that the number A is not primary</p>
<p>Such that : </p>
<p><span class="math-container">$A=\frac{2^{4n+2}+1}{5}$</span> </p>
<p><span class="math-container">$n≥2$</span></p>
<p>n=2 then <span class="math-container">$A=205$</span></p>
<p>Please I need some ideas to approach it</p>
| marty cohen | 13,079 | <p>The somewhat well-known identity
<span class="math-container">$4n^4+1
=(2 n^2 - 2 n + 1) (2 n^2 + 2 n + 1)
$</span>
gives</p>
<p><span class="math-container">$\begin{array}\\
2^{4n+2}+1
&=4(2^n)^4+1\\
&=(2\cdot 2^{2n}-2\cdot 2^n+1)(2\cdot 2^{2n}+2\cdot 2^n+1)\\
&=(2^{2n+1}-2^{n+1}+1)(2^{2n+1}+2^{n+1}+1)... |
2,803,069 | <p>Prove:
$$|z_1+z_2|\ge\frac{1}{2}(|z_1|+|z_2|)*|\frac{z_1}{|z_1|}+\frac{z_2}{|z_2|}|$$</p>
<p>Except inserting $(a+bi)$ instead of $z$ (which I think will lead me to a dead end), I really don't have a good idea how to confront this exercise, any tips or hints?</p>
<p>I'm not student yet and I just started learning ... | Jonathaniui | 420,908 | <p>The difference is that Arrow's theorem says that if you want to have two desirable conditions in your voting system (unanimity and independence of irrelevant alternatives) then necessarily your voting system is a dictatorship. It gives consequences for having "nice" properties. In this case if you do not want a dict... |
4,558,977 | <p>I have a circle. I know the radius (800) and I know the point coordinates (0, -800) under the circle. I double the point and move this one to the right. And a second point now has coordinates (500, -800). I have to define y (z coordinates according to my screenshot) value like a point is located on the circle and de... | cigar | 1,070,376 | <p><strong>Hint</strong></p>
<p>You can do two charts by gluing together two cylinders. But they won't be simply connected, because cylinders have the homotopy type of the circle.</p>
<p>In fact, if you take a simply connected piece out of <span class="math-container">$T^2$</span>, what's left won't be simply connec... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | kcrisman | 1,608 | <p>This is not so geometric, but if you have students who can actually do some difference quotients (numerically only, I mean) then just having them do a table where they estimate the derivative (say, $(f(3.001)-f(3))/(3.001-3)$) for various values, and then compare with the actual value of $e^x$, they might get surpri... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Dan Fox | 672 | <p>Geometrically the relation $f^{\prime} = f$ means that the slope at $(x, f(x))$ of the graph of $f$ is equal to $f(x)$. We look for a function $f$ that has this property. Consider $g(x) = 1 + x$. Its graph is $(x, 1 + x)$ and the slope is $1$ at every point. Hence for $x$ very small this function almost has the desi... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Community | -1 | <p>Take the graph of an exponential function such as $2^x$. If, for example, you shift it to the right by 3 units and then stretch it vertically by a factor of 8, you get the same graph back again. But the vertical stretch also increases the slope of the tangent line by a factor of 8. This holds for any shift, and ther... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | abel | 3,441 | <p>here is a way i teach about the natural exponential function. we look at the slope of the graphs $y = 2^x$ and $y = 3^x$ at $x = 0, y = 1$ by evaluating the average rate of change(slope of the secant line) of these functions on $[0, h]$ for small values of $h.$ conclude that there is a number $e$ with the property t... |
2,571,843 | <p>Let $A = \mathbb{R}\setminus\mathbb{Q}$. Then it can be shown that $A + A = \mathbb{R}$, for example by using the fact that $A$ is $G_{\delta}$. Let $q\in\mathbb{Q}$. This means that $q = r_1+r_2$ where $r_1,r_2$ are irrational numbers. But this is not too surprising, as every rational can be written $q = \left(\fra... | zwim | 399,263 | <p>You are engaging in a slippery terrain.</p>
<p><em>When you write $r_1=q_1+r$, do you imply that $r_1$ is the sum of a rational $q_1$ and a "rational free" irrational $r$ ? We shall see that this concept is not so simple.</em></p>
<p><br/>
In fact since $(\mathbb Q,+)$ being a normal subgroup of $(\mathbb R,+)$ we... |
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | samerivertwice | 334,732 | <p>Your proposed solution assumes A and B are disjoint in all possible cases. The question appears to intend A and B are not necessarily disjoint, and requires them to be disjoint for success.</p>
<p>Also you don't state whether it's equally likely to choose e.g. 3 or 1 elements since the natural scenario in the real... |
1,776,177 | <p>Given the definite integral:</p>
<p>$$\int_{1}^{2}\left(x\sqrt{x+3}\right)\text{d}x$$</p>
<p>We can make the Power Substitution:
$$\begin{align}
u^2=&&x+3 \\
2u\text{d}u=&&\text {d}x
\end{align}$$</p>
<p>We get the following: (without the limits)</p>
<p>$$\int{\left(\left(u^2-3\right)\times u\tim... | Eric Towers | 123,905 | <p>The problem you have is that the map $u \mapsto x$ is not one-to-one (injective). This is discussed with some complexity <a href="https://math.stackexchange.com/questions/623053/in-the-change-of-variables-theorem-must-%CF%95-be-globally-injective">here</a>. There are two intervals in $u$ that are mapped to the sam... |
175,661 | <blockquote>
<p>Prove that the series $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $, $x\in\mathbb{R}^{n} $, does not converge uniformly on the unit ball $\left\{ x\in\mathbb{R}^{n}\mid\left\Vert x\right\Vert <1\right\} $. </p>
</blockquote>
<p>I am not sure how to show this. What I got to is that the given s... | gifty | 36,492 | <p>The statement is true:</p>
<p>Take any be an open cover $\mathcal U$ of $f[K]$. Then, by continuity of $f$, the set $f^{-1}[U]$ is open in $K$ for any $U\in\mathcal U$. Thus, $\{f^{-1}[U]: U\in\mathcal U\}$ is an open cover of $K$.</p>
<p>By using that assumption that $K$ is compact, there is a finite sub-cover $\... |
1,137,930 | <p>Please, help me to understand the mathematics behind the following formula of CPI. Why do we calculate CPI the way it's done on the pic? The formula reminds me the expected value from stochastic, but do we have a random value here? </p>
<p><img src="https://i.stack.imgur.com/djpLX.png" alt="enter image description ... | Community | -1 | <p>Taken any $a$, $b$ and $c$, we have:</p>
<p>$(a,b)$ and $(b,a)$ $\in R$ $\Rightarrow$ $a = b = 2$ because there are no other such couples in $R$. This shows $R$ to be anti-symmetric.</p>
<p>$(a,b)$ and $(b,c)$ $\in R$ $\Rightarrow$ $a = 1$, $b=c=2$ or $a=b=c=2$. In both cases: $(a,c) \in$ $R$. This is why $R$ is t... |
3,956,400 | <blockquote>
<p>Arrange in ascending order <span class="math-container">$\tan45^\circ,\tan80^\circ$</span> and <span class="math-container">$\tan100^\circ.$</span></p>
</blockquote>
<p>We know that <span class="math-container">$\tan45^\circ=1$</span> because a right triangle with angle equal to <span class="math-contai... | CHAMSI | 758,100 | <p>Doesn't answer your question, since I think you've already got the answer to it. It ads something to it though.</p>
<p>Let's evaluate the sum of the series <span class="math-container">$ \sum\limits_{n\geq 0}{\frac{\left(-1\right)^{n}}{n+1}\sum\limits_{k=0}^{n}{\frac{1}{2k+1}}} $</span>.</p>
<p>Denoting <span class=... |
820,614 | <p>In a scalene triangle,does there exist three cevians which are equal in length,where length is measured between the corresponding vertex and the intersection point of the cevian with the corresponding side? This is just a question I had in my mind. </p>
| Michael Hardy | 11,667 | <p>I'd look at cross-sections parallel to the base. The radius of the cross-section at height $z$ from the base is $r(h-z)/h$, so the area is $\pi r^2(h-z)^2/h^2$. So the infinitesimal element of volume at that height is $\pi r^2(h-z)^2\,dz/h^2$. Integrating $z$ with respect to volume, for the $z$-coordinate of the ... |
820,614 | <p>In a scalene triangle,does there exist three cevians which are equal in length,where length is measured between the corresponding vertex and the intersection point of the cevian with the corresponding side? This is just a question I had in my mind. </p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\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{\down}{\... |
3,845,520 | <blockquote>
<p><strong>The length approximately equals width. The length is three times the height. The volume of the building is about <span class="math-container">$0.009 km^3$</span>.</strong></p>
</blockquote>
<hr />
<h2><em>The answer is 100 m by 300 m by 300 m.</em></h2>
<p>This question is supposed to be solved ... | Robert Israel | 8,508 | <p>If you can break up the <span class="math-container">$13$</span>'s:
<span class="math-container">$(13-1\times 3)^{1\times3} = 1000$</span>.</p>
|
1,832,512 | <p>How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far:</p>
<ol>
<li>$y = 2x^2+8x+13$</li>
<li>$x = 2y^2+8y+13$</li>
<li>$x-13 = 2y^2+8y$</li>
<li>$x-13=y(y+8)$</li>
</ol>
<p>This is where I got stuck. To be clear, I want to write $x$ in terms of $y.$ <strong>Credit to Jenna</strong> f... | Ivo Terek | 118,056 | <p>This function does not have an inverse as it is not injective: check that $f(x-2) = f(-x-2)$ whatever $x$ is - for example, $f(1) = f(-5)$, etc.</p>
<hr>
<p>We want to solve $y=2x^2+8x+13$ for $x$. Applying the quadratic formula for $2x^2+8x+(13-y)=0$ yields $$x = \frac{-8\pm \sqrt{64-4\cdot 2 \cdot (13-y)}}{4} = ... |
3,184,088 | <p>In Munkres, Section 22 (The Quotient Topology) he says the following:
Another way of describing the quotient map is as follows: We say that a subset <span class="math-container">$C$</span> of <span class="math-container">$X$</span> is <strong>saturated</strong> (with respect to the surjective map <span class="math-c... | Lee Mosher | 26,501 | <p>The definition of saturated is correct. It should <em>not</em> be the complete inverse image of <span class="math-container">$Y$</span>, because the complete inverse image of <span class="math-container">$Y$</span> would simply be <span class="math-container">$X$</span> itself, and why go to all that trouble with a ... |
3,184,088 | <p>In Munkres, Section 22 (The Quotient Topology) he says the following:
Another way of describing the quotient map is as follows: We say that a subset <span class="math-container">$C$</span> of <span class="math-container">$X$</span> is <strong>saturated</strong> (with respect to the surjective map <span class="math-c... | Henno Brandsma | 4,280 | <p><span class="math-container">$A \subseteq X$</span> is saturated for an equivalence relation <span class="math-container">$R$</span> on <span class="math-container">$X$</span>, if <span class="math-container">$x \in A$</span> and <span class="math-container">$xRy$</span> implies <span class="math-container">$y \in A... |
720,504 | <p>Why is $\displaystyle \lim_{x \to \infty} \ x^{2/x} = 1$ since this is an indeterminate form $\infty^{0}$ and I can't see any manipulation that would suggest this result?</p>
| ZHN | 131,755 | <p>If $f(x)=x^{\frac{2}{x}}$, then $\ln f(x)=\ln x^{\frac{2}{x}}=\frac{2}{x}\ln x$, and
$$\lim_{x\rightarrow \infty}\ln f(x)=\lim_{x\rightarrow \infty}\frac{2}{x}\ln x =\lim_{x\rightarrow \infty}\frac{2\ln x}{x},$$
and the indetermination now is in the form $\frac{\infty}{\infty}$. Then, for L'Hospital rule:
$$\lim_{x\... |
581,497 | <p>Case $1$: $4$ games: Team A wins first $4$ games, team B wins none = $\binom{4}{4}\binom{4}{0}$</p>
<p>Case $2$: $5$ games: Team A wins $4$ games, team B wins one = $\binom{5}{4}\binom{5}{1}-1$...minus $1$ for the possibility of team A winning the first four.</p>
<p>Case $3$: $6$ games: Team A wins $4$ games, team... | Community | -1 | <p>Let the best of $n$ series be decided after $k$ games. This will happen if in the preceding $k-1$ games $A$ also wins $\lfloor n/2 \rfloor$ and wins the $k^{th}$ game. Hence,
$$C(k) = \dbinom{k-1}{\lfloor n/2 \rfloor}$$
Hence, the total number of ways is
$$\sum_{k=\lfloor n/2 \rfloor+1}^n C(k)$$
In your case, set $n... |
530,301 | <p>There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B so that two triangles are made. What is the minimum length of wire needed to set up this configuration? </p>
<p>I tri... | Kaster | 49,333 | <h2>Geometric method</h2>
<p>If one mirror reflects $A$ to the $A_1$, then obviously $AC+CB = A_1C+BC$. Latter is minimal when $A_1$, $B$ and $C$ are aligned. One can even find mirror image of $B$ as well, so final $l = \sqrt{(a+b)^2 + h^2}$. One can use another triangles, triangle similarities, etc, but this one is q... |
530,301 | <p>There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B so that two triangles are made. What is the minimum length of wire needed to set up this configuration? </p>
<p>I tri... | André Nicolas | 6,312 | <p>The geometric way nicely described by Kaster is of course best.</p>
<p>We describe a calculus way. Draw an upward going perpendicular $CM$ at the point $C$. Let $\theta=\angle BCM$ and $\phi=\angle ACM$.</p>
<p>Then the rope has length $f(\theta)=15\sec\theta+30\sec\phi$. We also have the relation $15\tan\theta+3... |
1,599,510 | <p>I understand this proof <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf" rel="nofollow">http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf</a> (Lemma 2.2) until the point "and hence of $-1 - m^2\mod p$ ". Why is this true, and how does the final line then follow?</p>
| TonyK | 1,508 | <p>All it's saying is that if $m_1^2 \not\equiv m_2^2 \bmod p$, then $-1-m_1^2 \not\equiv -1-m_2^2 \bmod p$. So the number of different values of $m^2 \bmod p$ is the same as the number of different values of $-1-m^2 \bmod p$.</p>
|
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | Soham | 242,402 | <p>A conceptual error commonly made-</p>
<p>$({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1 $</p>
|
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | martini | 15,379 | <p>There is this <em>false</em> induction type of thing, where the step does not work in all cases.</p>
<blockquote>
<p><strong>Theorem</strong>. <em>If there is one white horse, all horses are white.</em></p>
</blockquote>
<p><em>Proof</em>. Induction on the number $n$ of horses existing. If there is only one hors... |
229,127 | <p>Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?</p>
| Shane Kelly | 91,255 | <p>I wanted to leave this as a comment, but I don't have enough reputation points yet.</p>
<p>There is a short exact sequence for every $n$
$$ 0 \to I^{n - 1}(k) \to K_n^{MW}(k) \to K_n^M(k) \to 0 $$
This is proven independently of Morel's article by Gille, Scully, Zhong in "Milnor-Witt $K$-groups of local rings.", se... |
51,898 | <p>I owe the idea of asking this question to Max Muller and
<a href="https://mathoverflow.net/questions/26035/">his curiosity</a>.</p>
<p><em>What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which
the alternating sum</em>
$$
\sum_{n=1}^\infty\frac{(-1)^{n+[n^\alpha]}}n
$$
<em>converges</em>? Here $[... | Anthony Quas | 11,054 | <p>Let me try and give an answer for $\alpha>1$ also. This one uses some technology from a
<a href="http://www.math.uvic.ca/faculty/aquas/paper18.pdf" rel="nofollow">paper</a> of mine with Boshernitzan, Kolesnik and Wierdl.</p>
<p>I want to use exponential sums. Using the notation of that paper we take $a(n)=n+n^\... |
662,590 | <p>It is mentioned that using the interpolation inequality</p>
<p><span class="math-container">$$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$</span></p>
<p>one can deduce that the space <span class="math-container">$L^{1} \cap L^{\infty}$</span> is dense in <span class="math-contain... | J.R. | 44,389 | <p>The inequality implies $L^1\cap L^\infty\subset L^p$. </p>
<p>Density is then clear, since the space $C_0^\infty$ of compactly supported smooth functions is dense in $L^p$ for every $\infty>p\ge 1$ and $C_0^\infty\subset L^1\cap L^\infty$.</p>
|
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Mark Bennet | 2,906 | <p>Others have answered the direct question, in that multiplication is commutative and that applies also to multiplication by a reciprocal (the equivalent of division). However there is an issue here with associativity and division which I think is worth mentioning. This has to do with the order in which operations are... |
50,209 | <p>Over the past few days I have been pondering about this: I enjoy technical things (like programming and stuff) and try to find the patterns and algorithms in everything. My life is number oriented. I'll spend all day working on a programmatic problem. I'll spend however much time is needed to think of an elegant/eff... | Stephen J. Herschkorn | 13,048 | <p>Try getting yourself a private tutor. One who will be attuned enough to your needs and abilities that s/he might be able to help you personally analyze and get over your difficulties. One who loves mathematics, not just helps people pass SAT's.</p>
<p>You might start by looking at TutorsTeach.com and/or Directory... |
2,317,929 | <p>Let $X=\mathrm{Spec}A$ and $U$ be an affine open subscheme of $X$.
Is there some $a\in A$ such that $D(a)=U$?</p>
| KReiser | 21,412 | <p>This is true if <span class="math-container">$A$</span> is a noetherian UFD, but not in general - consider the complement of a point of infinite order inside an elliptic curve (<a href="https://mathoverflow.net/questions/7153/open-affine-subscheme-of-affine-scheme-which-is-not-principal">ref</a>). There is a closely... |
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | celtschk | 34,930 | <p>If "almost infinite" makes any sense in any context, it must mean "so large that the difference to infinity doesn't matter."</p>
<p>One example where this could be meaningful is if you have parallel resistors and one is so large compared to the others that it doesn't measurably affect the total resistance. Then you... |
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | Slade | 33,433 | <p>Here is a practical example. Suppose that $A$ and $B$ are two genes, each of which is found in exactly $50\%$ of the population. Suppose also that we have established that there is zero correlation between the occurrence of one gene and another.</p>
<p>So what percentage of the population has both genes? $25\%$,... |
2,289,668 | <p>I have a problem which states:</p>
<p>Let $f(x)=4x-2$ and $\epsilon > 0$.</p>
<p>I must find a $\delta>0$ s.t. $0<|x-1|<\delta$ implies $|f(x)-2|<\epsilon$.</p>
<p>How can I solve a problem such as this?</p>
| fleablood | 280,126 | <p>Two ways. Directly forward, and backwords.</p>
<p>Forward: Just keep asking yourself: If $|x-1| \le \delta$, what does that say about $4x -2$ and $2$. </p>
<p>$|x - 1| \le \delta \implies$</p>
<p>$-\delta < x - 1 < \delta \implies$</p>
<p>$1 - \delta < x < 1 + \delta \implies$</p>
<p>$4 - 4\delta... |
87,902 | <pre><code>eq1 := Abs[-3.533147671810^-6] ==
A1 Exp[-(-0.53326099689) ((μ1))^2]
eq2 := 7.2716492165 10^-4 == A2 Exp[-(0.53326099689) ((μ2))^2]
eq3 := Abs[-4.0740049497 10^-10] ==
A3 Exp[-(-8.8857611784 10 ⁻²) ((μ3))^2]
eq4 := -3.1704480355 10^-6 ==
2 (-0.53470532215)... | bbgodfrey | 1,063 | <p>This is more an extended comment, although it does answer the question. Using Belisarius' interpretation of the equations, one might think that </p>
<pre><code>Solve[{eq1, eq4}, {A1, m1}]
</code></pre>
<p>could yield a solution directly, but it returns unevaluated. However, <code>Solve</code> can yield a solutio... |
3,997,992 | <p>Taken from <a href="https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10" rel="nofollow noreferrer">https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10</a></p>
<p>Problem <br>
Given the polynomial <span class="math-container">$f(x)=x^n+a_{1}x^{n-1}+a_{2}... | Community | -1 | <p>Hint.</p>
<p>You can't solve the equation since you incorrectly equating a vector with a scalar.</p>
<p>Instead, you should use the "length" of the vector on the left.</p>
|
3,997,992 | <p>Taken from <a href="https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10" rel="nofollow noreferrer">https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10</a></p>
<p>Problem <br>
Given the polynomial <span class="math-container">$f(x)=x^n+a_{1}x^{n-1}+a_{2}... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">\begin{align}
A &= (1,0,0) \\
B &= (0,1,0) \\
C &= (0,0,z) \\
\end{align}</span></p>
<p><span class="math-container">\begin{align}
a &= BC &=\sqrt{1+z^2} \\
b &= AC &=\sqrt{1+z^2} \\
c &= AB &=\sqrt 2 \\
\end{align}</span></p>
<p><spa... |
3,168,381 | <p>Given <span class="math-container">$f(x) = \frac{{4x}}{\sqrt{x}-3}$</span>, what's the domain of <span class="math-container">$g(x) = \frac{{1}}{f(x)}$</span> ?</p>
<p>My textbook includes in the answers <span class="math-container">$x \neq 9$</span>, which I think is erroneous.</p>
| Community | -1 | <p>The book is right.</p>
<p>Because <span class="math-container">$g(x)$</span> is</p>
<p><span class="math-container">$$\frac1{\dfrac{4x}{\sqrt x-3}}$$</span> and <em>not</em></p>
<p><span class="math-container">$$\frac{\sqrt x-3}{4x}.$$</span></p>
<p>The domain of the first expression is <span class="math-contain... |
137,691 | <p>What is the eigenvalue/eigenvector relationship between matrix A,B and AB?</p>
| Misha | 21,684 | <p>I do not know about eigenvectors, but for the eigenvalues this is a special case of <em>Deligne-Simpson Problem</em>. It was completely solved by Crawley-Boevey about 10 years ago using quivers. For details, see my answer <a href="https://mathoverflow.net/questions/90287">here</a> and references therein. </p>
|
4,418,091 | <p>Is <span class="math-container">$\mathbb Q-\mathbb N$</span> dense in <span class="math-container">$\mathbb R$</span>? I believe that the solution is about two relatively prime integers. However, I do not know how to proceed.</p>
| Joe | 623,665 | <p>Assuming that you have proven that <span class="math-container">$\mathbb Q$</span> is dense in <span class="math-container">$\mathbb R$</span>, we know that if <span class="math-container">$x<y$</span> then there is a <span class="math-container">$q\in\mathbb Q$</span> such that <span class="math-container">$x<... |
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | Mohamed | 33,307 | <p>$\tan x =x +o(x)$ then $\tan (2x) \sim 2x$ , then : $\frac{x^3}{\tan^3 x} \sim \frac{x^3}{8x^3} \sim \frac 18$</p>
|
729,101 | <p>Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points?</p>
<p>Given 3 points:
$
P_1 = (0,4), P_2 = (2,2), P_3 = (4,7)
$</p>
<p>And some function $f(x)$, for which the following holds:
$$
P_1, P_2, P_3 \in f(x)
\\
f'(0) = f'(2) = f'(4... | Mutinifni | 138,384 | <p>Any element of the vector space $P_3$ is of the form,</p>
<p>$p(x)= a + bx +cx^2 +dx^3$</p>
<p>Substituting x = 1, we get,</p>
<p>$p(1)= a + b.1 + c.1^2 +d.1^3$</p>
<p>$\implies a + b + c + d = 0$</p>
<p>For p'(1), we have after differentiating,</p>
<p>$p'(x)= b + 2cx +3dx^2$</p>
<p>$p'(1)= b + 2c.1 +3d.1^2 =... |
407,253 | <p>An edge will have the same vertices as another edge that it is parallel to, so how can it be uniquely described? </p>
| Zev Chonoles | 264 | <p>For an undirected <a href="https://en.wikipedia.org/wiki/Multigraph#Undirected_multigraph_.28edges_without_own_identity.29" rel="nofollow">multigraph</a>, we simply have a <a href="https://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> of edges, not a set. Thus, if $V=\{a,b,c\}$, we might have
$$E=\{[(a... |
2,871,419 | <p>Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power.</p>
<p>I was wondering when the two interpretations are in fact the same. In other words, if we write $f^n(x)$ for $f(f(\dotsb f(x) \dotsb))$ and $f(x)^n$ for $f(x) \cd... | Henry | 6,460 | <p>Something non-constant and discontinuous everywhere: </p>
<p>$$f(x) = \left\{\begin{array}{ll}
x^2 & \text{if } x \text{ is an integer} \\
\lceil x \rceil & \text{if } x \text{ is rational but not an integer}\\
\lfloor x \rfloor & \text{if } x \text{ is irrational}
\end{array} \right.$$ </p>
... |
319,663 | <p>I need to determine whether this matrix is injective
\begin{pmatrix}
2 & 0 & 4\\
0 & 3 & 0\\
1 & 7 & 2
\end{pmatrix}</p>
<p>Using gaussian elimination, this is what I have done:
\begin{pmatrix}
2 & 0 & 4 &|& 0\\
0 & 3 & 0 &|& 0\\
1 & 7 & 2 &|&... | Alexander Gruber | 12,952 | <p>Look at vectors of the form $\left(\begin{array}{c}2x\\ 0\\ -x\end{array}\right)$. What does your matrix do to these vectors? What does injective mean?</p>
|
319,663 | <p>I need to determine whether this matrix is injective
\begin{pmatrix}
2 & 0 & 4\\
0 & 3 & 0\\
1 & 7 & 2
\end{pmatrix}</p>
<p>Using gaussian elimination, this is what I have done:
\begin{pmatrix}
2 & 0 & 4 &|& 0\\
0 & 3 & 0 &|& 0\\
1 & 7 & 2 &|&... | amWhy | 9,003 | <p>Compute the rank of your matrix; since the last row is all zeros, it has rank 2.</p>
<p><a href="http://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem" rel="nofollow"><strong>rank + nullity</strong></a> $ = 3 \implies$ nullity $= 1.$</p>
<p>Hence, you matrix cannot be injective. </p>
|
454,622 | <p>I am trying to solve a particular probability question. </p>
<p>I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. </p>
<p>I find the probability as: </p>
<p>P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd... | user212527 | 212,527 | <p>Its P(multiple of 5) + P(odd) - P(multiple of 5 and odd). It's not mutually exclusive. The two events can happen at the same time, just like a the fact that it can rain when it is sunny.
2/10 + 5/10 - 1/10 = 3/5</p>
|
1,831,052 | <p><a href="https://i.stack.imgur.com/SxyPL.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SxyPL.jpg" alt="enter image description here"></a></p>
<p>ADE is a straight line .
AE : AD = 3 : 2
Find coordinates of E</p>
<p>My workings </p>
<p>Let E ( X , Y) </p>
<p>Gradient AD = Gradient of AE
$ 1/... | levap | 32,262 | <p>You know that $\sum_{n=0}^{\infty} u^n = \frac{1}{1 - u}$ for $|u| < 1$. Integrating this series, we get</p>
<p>$$ -\ln(1 - u) = \sum_{n=0}^{\infty} \frac{u^{n+1}}{n+1} = \sum_{n=1}^{\infty} \frac{u^n}{n} $$</p>
<p>and this series converges also for $u = -1$ to $-\ln(2)$ by Abel's theorem.</p>
<p>Your series i... |
3,077,882 | <p>Simplify the expression
<span class="math-container">$$\sin\left(\tan^{-1}(x)\right)$$</span>
Using a triangle with an angle <span class="math-container">$\theta$</span>, opposite is x and adjacent is 1 meaning the hypo. is <span class="math-container">${\sqrt {x^2+1}}$</span> </p>
<p>Now because the problem has s... | mjqxxxx | 5,546 | <p>You're thinking about it just right. <span class="math-container">$\tan^{-1}(x)$</span> is the angle whose tangent is <span class="math-container">$x$</span>. So draw a right triangle and put <span class="math-container">$\tan^{-1}(x)$</span> in a corner angle. Since the tangent of that angle is <span class="math... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| Zain Patel | 161,779 | <p>We have that $\tan 20^{\circ} = \dfrac{\sin 20^{\circ}}{\cos 20^{\circ}} = \dfrac{\sin 20^{\circ}}{\sin 70^{\circ}}$.</p>
<p>This list <a href="http://intmstat.com/blog/2011/06/exact-values-sin-degrees.pdf" rel="nofollow">here</a>, gives you the exact value of the sine of every integer angle between $1$ and $90$. T... |
129,294 | <p>I found the answer in this <a href="https://mathematica.stackexchange.com/questions/67306/insert-at-specific-resulting-positions/67309#67309">post</a> very interesting to do what I need, but I would like something where I could provide a <code>list to be modified</code>, a <code>list with values</code> that will be ... | LCarvalho | 37,895 | <p>You can start applying <code>Transpose</code> function:</p>
<pre><code>listaInicial = {0, 1, 2, 3, 4, 5, 6, 7};
listaModificadora = {a, b, c, d, e};
listaPosições = {2, 6, 7, 8, 13};
Fold[Insert[#, #2[[1]], #2[[2]]] &, listaInicial,
Transpose[{listaModificadora, listaPosições}]]
</code></pre>
<blockquote>
... |
129,294 | <p>I found the answer in this <a href="https://mathematica.stackexchange.com/questions/67306/insert-at-specific-resulting-positions/67309#67309">post</a> very interesting to do what I need, but I would like something where I could provide a <code>list to be modified</code>, a <code>list with values</code> that will be ... | corey979 | 22,013 | <p><a href="https://mathematica.stackexchange.com/a/129297/22013">LMC's answer</a> is much in the spirit of Mathematica. Since you don't like slots (while you should - they make lots of things easier) I provide a straightforward <code>Do</code> loop:</p>
<pre><code>lst1 = {0, 1, 2, 3, 4, 5, 6, 7};
lst2 = {a, b, c, d, ... |
2,841,102 | <p>I'm asked to find the minimum and maximum values of $f(x, y, z) = x^2+y^2+z^2$ given the constraints $x+2y+z=5$ and $x-y=6$. </p>
<p>I have successfully computed the following:
$x = \frac{57}{11}, y = \frac{-9}{11}, z= \frac{16}{11}$. </p>
<p>I was then able to obtain $f(\frac{57}{11}, \frac{-9}{11}, \frac{16}{11}... | Tamojit Maiti | 331,943 | <p>For this particular problem, it might be helpful to consider what is being actually asked. The question can be framed as follows: </p>
<p>Find the maximum and minimum radius of the sphere which contains the line of intersection of the planes $x+2y+z=5$ and $x-y=6$. </p>
<p>Now the two planes intersect at a line wi... |
4,224,043 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$G$</span> be a matchable graph, and let <span class="math-container">$u$</span> and <span class="math-container">$v$</span> be distinct vertices of <span class="math-container">$G$</span>. Show that <span class="math-container">$G - u - v$</sp... | Theo Bendit | 248,286 | <p>This is true in more general spaces too (not just finite-dimensional). We can prove it with a couple of results that are helpful more generally. As a general notational note, we denote by <span class="math-container">$B[x; r]$</span> the closed ball centred at <span class="math-container">$x$</span>, with radius <sp... |
484,589 | <p>a) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n},\frac{1}{n}) $</p>
<p>b) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n}, 1+\frac{1}{n})$</p>
<p>c) $\bigcup_{n=1}^{\infty}(-\frac{1}{n}, 2+\frac{1}{n})$</p>
<p>Could anyone please explain how to do this problems? I'm having a hard time trying to come up with the intervals for thes... | André Nicolas | 6,312 | <p>Consider the equation $x^2+ax+b=0$, where $a$ and $b$ are <strong>integers</strong>. </p>
<p>Let $r$ be a rational solution of our equation. Then $r=\frac{m}{m}$, where $m$ and $n$ are integers. Without loss of generality we may assume that $m$ and $n$ are relatively prime, and that $n\ge 1$.</p>
<p>Substitute $\... |
2,155,755 | <p>What is the difference (or connection) between the dimension of a vector space and the dimension in terms of bases?</p>
<p>For instance, when we talk about the vector space $\mathbb{R}^3$, we are talking about a 3-dimensional vector space. This vector space contains vectors with three elements: $(x_1, x_2, x_3)$.</... | pjs36 | 120,540 | <p>The vector space $\Bbb R^n$ means exactly one thing: $n$-tuples of numbers with entries in $\Bbb R$. (This can be formalized by saying an element of $\Bbb R^n$ is really a function from $\{1, 2, \ldots, n\}$ to $\Bbb R$: For example the vector $(\sqrt{2}, -3, \pi)$ is really the function $1 \mapsto \sqrt{2},\, 2\map... |
17,423 | <p>In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial <span class="math-container">$$f(x)=a_nx^n+\cdots a_1x+a_0$$</span> with <span class="math-container">$a_n\neq 0$</span> is defined to be <span class="math-container">$n$</span>. </p>
<p>But I am ... | David E Speyer | 51 | <p>I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring <span class="math-container">$k[x]$</span> than most answerers are. Here are two reasonable definitions:</p>
<ol>
<li><p><span class="math-container">$k[x]$</span> is the ring of for... |
1,402,887 | <p>I was going through some sample papers of math, and I found this question which I cant solve:</p>
<p>If $abc=1$, find $1/(1+a+b^{-1})+1/(1+b+c^{-1})+1/(1+c+a^{-1})$.</p>
<p>Please help me with this.... I have spent almost 3 hours on this question...</p>
<p>Thanks for the help.</p>
| Tim Raczkowski | 192,581 | <p>$$\frac 1{1+a+b^{-1}}=\frac b{b+ab+1}=\frac b{1+b+c^{-1}}$$</p>
<p>The last equation follows because, $ab=1/c$.</p>
<p>In a similar manner,
$$\frac 1{1+c+a^{-1}}=\frac a{a+ac+1}=\frac a{a+b^{-1}+1}=\frac{ab}{1+b+c^{1}}$$</p>
<p>So, putting this altogether gives</p>
<p>$$\begin{align}
& \frac 1{1+a+b^{-1}}+\f... |
363,335 | <p>Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding exponents), we know that $w$ is in the commutator subgroup $(F_2)_2 = [F_2,F_2]$. In fact I can algorithmically deco... | Derek Holt | 2,820 | <p>This calculation can be done with the <em>nilpotent quotient algorithm</em>, which can be applied to any group defined by a finite presentation. See <a href="http://www.mathematik.tu-darmstadt.de/~nickel/software/" rel="nofollow">http://www.mathematik.tu-darmstadt.de/~nickel/software/</a>. There is a GAP implementa... |
1,303,263 | <p>Given 2 vectors,$u=(3,5)$,$v=(s,s^2)$,in what situations do u and v parallel?$(s≠0)$</p>
<p>In order to be parallel,$u$ must be proportional to $v$,vice verse.Let $k$ be a scalar $neq 0$,then $ku=(3k,5k)=v=(s,s^2)$,which gives $3k=s$;$5k=s^2 \rightarrow 5k=9k^2 (k \neq 0) \rightarrow 5=9k→k=\frac{5}{9}$,plug $k=\fr... | yoann | 234,608 | <p>The pattern does not seem to go on: there are only 17 primes between 144 and 233.</p>
<p>More generally, using the Prime number theorem $\pi(x) \sim_{x \to \infty} \frac x {\ln x}$ (where $\pi(x)$ is and the number of primes lower or equal to $x$), and the formula for the n-th Fibonacci number $F_n \sim_{n \to \inf... |
1,926,382 | <p>Let $T:\Bbb R^n\longrightarrow\Bbb R^n$ be a linear transformation, where $n\geq 2$. For $k\leq n$,
let $E=\{v_1,v_2,\dots,v_k\}$ contained in, equal to $R^n$ and $F=\{Tv_1,Tv_2,\dots,Tv_k\}$.
Then</p>
<p>a). If $E$ is linearly independent, then $F$ is linearly independent.</p>
<p>b). If $F$ is linearly independe... | Piquito | 219,998 | <p>WARNING: a) can be true or false depending on $T$ and $k\le n$. When $T$ is one-one, it is always true but if $T$ is not one-one and $E$ is linearly independent and $k\lt n$ both cases can happen, $F$ be linearly independent and linearly dependent (why?).</p>
<p>$b)$ is always true.</p>
<p>$c)$ is true only when $... |
2,674,802 | <h2>Problem</h2>
<p>Proof that $x=a$ is solution of polynomial $P(x)=(x-a)Q(x)$ when $Q(x)$ is also polynomial expression. $P(x)$ is product of two polynomials.</p>
<h2> Attempt to solve </h2>
<h1>1. Proof</h1>
<p>We can first examine the expression by writing parenthesis open.</p>
<p>$$ P(x)=xQ(x)-aQ(x) $$</p>
... | Deepak | 151,732 | <p>I would not use tautological statements like $0 = 0$ in a formal proof. Simply state that $P(a) = (a-a)\cdot Q(a) = 0\cdot Q(a) = 0$ as zero multiplied by any real or complex number (which covers all possible values that $Q(a)$ can take) is zero. $P(a) = 0$ implies that $x=a$ is a solution of $P(x) = 0$, and you're ... |
604,459 | <p>The complex <em>solid spherical harmonics</em> can be defined as</p>
<p>$$
U_n^m(\boldsymbol{r}) = r^n P_n^m(\cos{\theta}) e^{im\phi},
$$</p>
<p>where $r,\theta,\phi$ are the usual spherical coordinates of $\boldsymbol{r}=(x,y,z)$. Note that $U_n^m(\boldsymbol{r})$ is a homogeneous polynomial in $x$, $y$, and $z$ ... | user2122427 | 160,607 | <p>I've solved the problem numerically using a C++ code and found that not only to odd n's vanish, all m's not divisible by 4 vanish - and to my surprise all quadrupole moments vanish. So the first non-zero term past n=0 is n=4 with m=-4,0,+4, then n=6 with m=-4,0,+4, then n=8 with m=-8,-4,0,+4,+8, etc. </p>
<p>Just c... |
3,809,788 | <p>Are we able to obtain the following algebraically?<span class="math-container">$$\widehat{\mathbf{x}}=\underset{\mathbf{x}}{\operatorname{argmin}}\|A-B(I\otimes \mathbf{x})\|_F^2$$</span>where
<span class="math-container">$A\in \mathbb{R}^{m\times n}$</span>, <span class="math-container">$B\in \mathbb{R}^{m\times (m... | mathreadler | 213,607 | <p>The systematic approach to use here is to define the operations with <a href="https://en.wikipedia.org/wiki/Kronecker_product" rel="nofollow noreferrer">Kronecker products</a> (<span class="math-container">$\otimes$</span>) working on vectorizations (<span class="math-container">$\text{vec}$</span>).</p>
<p>First le... |
2,134,085 | <p>Can someone please explain where the <span class="math-container">$1$</span> goes in this expression?</p>
<blockquote>
<p>Find the measure of angle in radians by solving:
<span class="math-container">$$\sin^2 \theta +\cos \theta=1$$</span></p>
</blockquote>
<p>I got <span class="math-container">$$\sin^2 \thet... | heropup | 118,193 | <p>Given $$\sin^2 \theta + \cos \theta = 1,$$ and using the circular identity $$\sin^2 \theta + \cos^2 \theta = 1,$$ it follows that $$\cos^2 \theta = \cos \theta,$$ or $$(\cos \theta - 1) \cos \theta = 0.$$ Hence $$\cos \theta \in \{0, 1\},$$ which implies $$\theta \in \{\tfrac{(2k+1)}{2}\pi, 2k\pi\}, \quad k \in \ma... |
101,716 | <p><em>Disclaimer</em>: This is homework, however I am not looking for an answer. I'm only trying to understand the actual question.</p>
<p>I'm given four mutually exclusive and exhaustive events: $A$, $B$, $C$, and $D$. I'm also given $P(A)$, $P(B)$, $P(C)$ and $P(D)$. There is also some minor event $M$, for which I ... | Henry | 6,460 | <p>Yes it is asking for $P(B|M)$.</p>
<p>No it cannot be solved as $\dfrac{P(B)}{P(M)}$ since $B$ and $M$ are not simply related. For this to work you would need $B$ to be a subevent of $M$. </p>
<p>Instead you should use <a href="http://en.wikipedia.org/wiki/Conditional_probability" rel="nofollow">conditional proba... |
1,229,194 | <p>Problems with the following limits:</p>
<p>$$
1. \quad \quad \lim_{x\to0^+} e^{1/x} + \ln x \, .
$$</p>
<p>Substitutions such as $e^{1/x}=t$ and $1/x = t$ don't yield any useful results. </p>
<p>Pretty much the same with
$$
2. \quad \quad \lim_{x\to 0^+} e^{1/x} - 1/x \, ,
$$
Common denominator doesn't help much... | DeepSea | 101,504 | <p>For $2)$: $e^{1/x} - 1/x > 1 + 1/x^2 \to +\infty$, and for $1)$ $e^{1/x} + \ln x = e^{1/x} - \ln\left(1/x\right)> 1 + 1/x + 1/x^2 - \left(1/x - 1\right)> 1/x^2 \to +\infty$</p>
|
1,638,267 | <p>Can someone please point out where I am (If I am) going wrong during the solution process of the following question:</p>
<p>I have been presented with the following :</p>
<p>$$4sinh(2ln(2))-cosh(ln2)$$</p>
<p>and told by my tutor the solution is 10. however I cannot obtain this value, the steps I take are as foll... | robit | 108,500 | <p>The answer should be $6.25$.</p>
<p>\begin{align}
& 4 \sinh (2 \ln 2) - \cosh(\ln2 ) \\
=& 2 \left(e^{2\ln2}-e^{-2\ln2} \right) - \frac{e^{\ln2}+e^{-\ln2}}{2}\\
=& 2(4-0.25)-\frac{2+0.5}{2}\\
=& 7.5-1.25 = 6.25.
\end{align}</p>
|
2,012,564 | <p>We had to prove that if</p>
<p>$$\lim_{n\to\infty}(a_n\cdot b_n)=0$$ </p>
<p>Then either $\lim_{n\to\infty}a_n$ or $\lim_{n\to\infty}b_n$ HAS to be equal to $0$.</p>
<p>My hypothesis is that since</p>
<p>$$\lim_{n\to\infty}(a_n\cdot b_n)=\lim_{n\to\infty}a_n\cdot \lim_{n\to\infty}b_n$$</p>
<p>Then for $\lim_{n\... | 35T41 | 342,242 | <p>This isn't true.</p>
<p>Take:
$a_n=\left\{\begin{matrix}
1/n^2 & n \: odd \\
n & n \: even
\end{matrix}\right., \; \;
b_n=\left\{\begin{matrix}
1/n^2 & n \: even \\
n & n \: odd
\end{matrix}\right.$</p>
<p>Then $a_n\cdot b_n = 1/n$ for every $n$ but non of the sequences has a limit.</p>
|
2,049,525 | <p>Show that the expression
$$(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$$
is a perfect square and Find its square root.</p>
| King Ghidorah | 154,513 | <p>$$(x^2 - yz)^3 + (y^2 - zx)^3 + (z^2 - xy)^3 -
3 (x^2 - yz) (y^2 - zx) (z^2 - xy) =(x^3 + y^3 - 3 x y z + z^3)^2$$</p>
|
4,042,039 | <p>Let <span class="math-container">$X$</span> be a finite set with <span class="math-container">$|X|=n$</span> equipped with the dicrete topology.Show that <span class="math-container">$C(X,Y)$</span> is homeomorphic to <span class="math-container">$Y^n$</span> where <span class="math-container">$Y^n=\overbrace{Y\time... | user10354138 | 592,552 | <p>Compact-open on <span class="math-container">$C(A,B)$</span> is the usual product <span class="math-container">$B^A$</span> (*) if <span class="math-container">$A$</span> is discrete, since all maps are continuous (so they are the same underlying set) and all compacts in <span class="math-container">$A$</span> are f... |
2,285,276 | <p>A wheel is spun with the numbers $1, 2$, and $3$ appearing with equal probability of $1\over 3$ each. If the number $1$ appears, the player gets a score of $1.0$; if the number $2$ appears, the player gets a score of $2.0$, if the number $3$ appears, the player gets a score of $X$, where $X$ is a normal random varia... | Jason | 195,308 | <p>I think trying to work with a basis in $\mathcal P(\mathbb R)'$ is going to be a bad idea - bases of many infinite dimensional vector spaces, $\mathcal P(\mathbb R)'$ included, are impossible to right down. Thankfully, a basis of $\mathcal P(\mathbb R)$ <em>can</em> be written down: for each $k\in\mathbb N$ (includi... |
1,756,593 | <p>I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare it with? I tried ${1\over n}$, ${1 \over n^2 }$, but I need a bigger series which converges to prove that this one c... | choco_addicted | 310,026 | <p>Prove that $0\le \ln n \le \sqrt{n}$ first, and
$$
0\le \frac{\ln n^2}{n^2} =\frac{2\ln n}{n^2} \le \frac{2}{n\sqrt{n}}.
$$
$\sum_{n=1}^{\infty}\frac{2}{n\sqrt{n}}$ converges.</p>
|
1,640,110 | <p>Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix
with real entries such that its determinant is zero, but if one changes any single
entry one gets a matrix with non-zero determinant.</p>
<p>I think we may be able to construct such matrices.</p>
| user1551 | 1,551 | <p>If you want a constructive proof, the following example works over any field. Consider the <span class="math-container">$n\times n$</span> matrix
<span class="math-container">$$
A=\pmatrix{I_{n-1}&-\mathbf1_{n-1}\\ -\mathbf1_{n-1}^T&n-1},
$$</span>
where <span class="math-container">$\mathbf1_{n-1}$</span> i... |
1,662,218 | <p>Among the following, which is closest in value to $\sqrt{0.016}$?</p>
<p>A. $0.4$</p>
<p>B. $0.04$</p>
<p>C. $0.2$</p>
<p>D. $0.02$</p>
<p><strong>E. $0.13$</strong></p>
<p><strong>My Approach:</strong></p>
<p>$(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = \frac{2}{5\cdot\sqrt{10}} = \frac{\sq... | dbanet | 220,258 | <p>This absolutely is the worst way to solve this problem, but I though it might be useful to you, or at least interesting (I bet ability to evaluate things numerically on paper <em>is</em> something interesting).</p>
<p>From \begin{align}\sqrt{0.016}&=\sqrt{16\cdot10^{-3}}=4\sqrt{10^{-1}10^{-2}}=4\cdot10^{-1}\sqr... |
2,077,883 | <blockquote>
<p>$f$ and $g$ are functions of real variables, strictly increasing and strictly decreasing respectively on $\Bbb R$ (both surjections), I'm asked to prove that there exists at most one solution to the equation $f(x)=g(x)$.</p>
</blockquote>
<p><strong>My attempt:</strong> Suppose $f(x)=g(x)=m,$ for som... | user64066 | 64,066 | <p>Hint: If there were two different points, either $f$ increasing or $g$ decreasing would be wrong. </p>
|
55,679 | <p>Let $G$ be a Lie group and let $\xi.\eta$ be left invariant vector fields. We can now construct right invariant vector fields $X_\xi$ and $X_\eta$ by defining $X_\xi(e)=\xi(e)$ and $X_\eta(e)=\eta(e)$. For $GL_n$, it is true that $[X_\xi,X_\eta]=X_{[\eta,\xi]}$. Is it true for any Lie group?</p>
| gggg gggg | 16,792 | <p>The above statement ist true for GL(n). As Deane pointed out, this neceassrily implies that the statement must be true for any subgoup of GL(n) that is a Lie group as well, i.e., any subgriup of GL(n).
It can be shown, however, that every Lie-Group can be considered as subgroups of some GL(n), where one uses represe... |
2,838,938 | <p>Given:</p>
<ul>
<li>$\theta$ (a negative angle)</li>
<li>$v_0$ (initial velocity)</li>
<li>$y_0$ (initial height)</li>
<li>$g$ (acceleration of gravity)</li>
</ul>
<p>I want to find the range of a projectile (ignoring wind resistance)</p>
<p>Hours of searching have given no useful results. Those that I thought we... | Nosrati | 108,128 | <p>With horizental velocity $v_x=v_0\cos\theta$ and vertical velocity $v_y=-v_0\sin\theta-gt$ (here $\theta>0$) and integration we see
$$x=\int_0^tv_0\cos\theta\,dt=v_0\cos\theta\,t\,\,\,,\,\,\,y=\int_0^t v_y\,dy=-v_0\sin\theta\,t-\frac12gt^2$$
Eliminating $t$ between them gives
$$y=-\tan\theta\,x-\frac{g}{2v_0^2\co... |
2,838,938 | <p>Given:</p>
<ul>
<li>$\theta$ (a negative angle)</li>
<li>$v_0$ (initial velocity)</li>
<li>$y_0$ (initial height)</li>
<li>$g$ (acceleration of gravity)</li>
</ul>
<p>I want to find the range of a projectile (ignoring wind resistance)</p>
<p>Hours of searching have given no useful results. Those that I thought we... | Phil H | 554,494 | <p>$V_y = 100 \sin(-12.5) = -21.6440$ m/s</p>
<p>$V_x = 100 \cos(-12.5) = 97.6296$ m/s</p>
<p>Solving for the time of flight t</p>
<p>$S = V_y\cdot t + \frac{1}{2}\cdot g\cdot t^2$</p>
<p>$-1.65 = -21.6440t + \frac{1}{2}\cdot -9.80665\cdot t^2$</p>
<p>$4.90333t^2 + 21.6440t - 1.65 = 0$</p>
<p>$t = \frac{-21.6440+... |
3,738,622 | <p>I need change the summation order in the double sum
<span class="math-container">$$
S_{m,n}=\sum_{j=0}^m \sum_{k=0}^n a_{j,k} x^{j-k} B_{m+n-j-k},
$$</span>
to separate <span class="math-container">$B$</span> and get somethink like to
<span class="math-container">$$
S_{m,n}=\sum_{s=0}^{m+n} \left( \sum_{k=0}^* *... | Community | -1 | <p>If you look at the <span class="math-container">$(j,k)$</span> plane, the domain of the summation is an axis-aligned rectangle. Then the locus of constant <span class="math-container">$s=m+n-j-k$</span> is a pencil of parallel line segments slanted by <span class="math-container">$-45°$</span>, which meet the rectan... |
2,437,635 | <p>It is well know that $$\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$$
i.e.
$$(1+2+3+\cdots+n)(1+2+3+\cdots+n)=1^3+2^3+3^3+\cdots+n^3$$
and this is usually proven by showing that the closed form for the sum of cubes is $\frac 14 n^2(n+1)^2$ which can be written as $\left(\frac 12 n(n+1)\right)^2$, and then noticin... | Misha Lavrov | 383,078 | <p>The graphical proof can be turned into a manipulation of the sum, but at some point you do still have to use the formula for the sum $1 + 2 + \dots +k$.</p>
<p>\begin{align*}
\left(\sum_{r=1}^n r\right)^2 &= \left(\sum_{i=1}^n i\right)\left(\sum_{j=1}^n j\right) \\
&= \sum_{i=1}^n \sum_{j=1}^n ij \\
&= ... |
2,437,635 | <p>It is well know that $$\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$$
i.e.
$$(1+2+3+\cdots+n)(1+2+3+\cdots+n)=1^3+2^3+3^3+\cdots+n^3$$
and this is usually proven by showing that the closed form for the sum of cubes is $\frac 14 n^2(n+1)^2$ which can be written as $\left(\frac 12 n(n+1)\right)^2$, and then noticin... | Juan Alfaro | 490,928 | <p>proof without words
<a href="https://i.stack.imgur.com/h1IvO.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/h1IvO.jpg" alt="enter image description here"></a></p>
<p>from: (<a href="https://upload.wikimedia.org/wikipedia/commons/2/26/Nicomachus_theorem_3D.svg" rel="noreferrer">https://upload.wikimedia.or... |
3,782,170 | <p>This is probably just a minor notational issue, but I am unsure whether I should write <span class="math-container">$z=a+bi$</span> or <span class="math-container">$z=a+ib$</span> when denoting complex numbers. Though the former notation seems more common, Euler's identity tends to be written as
<span class="math-co... | Filippo | 572,841 | <p><span class="math-container">$bi=b\cdot i=i\cdot b=ib$</span>, so there is no difference. But it's more common to write <span class="math-container">$a+ib$</span>.</p>
<p>Since the sum is commutative, you can also write <span class="math-container">$bi+a$</span> or <span class="math-container">$ib+a$</span>, it's al... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | user642796 | 8,348 | <p>Are there infinite sets of axioms? Yes!</p>
<p>As a trivial (and somewhat meaningless) example, consider the first-order language whose only non-logical symbols are the constant symbol $0$ and the unary function symbol $S$. Then the following would comprise an infinite set of axioms:</p>
<ul>
<li>$0 \neq S(0)$;<... |
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | Community | -1 | <p>An example of an infinite set of axioms used for practical use is the foundation of non-standard analysis.</p>
<p>Create a first-order language whose symbols include:</p>
<ul>
<li>Every element of $\mathbb{R}$</li>
<li>Every element of $\mathcal{P}(\mathbb{R})$</li>
<li>Every element of $\mathcal{P}(\mathcal{P}(\m... |
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