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 |
|---|---|---|---|---|
878,914 | <p>I'm having a problem to solve this limit.</p>
<p>$$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$$</p>
<p>$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$ =
$\lim_{x \to \pi/4} \frac{\frac{\sin x}{\cos x}-1}{\sin x-\cos x}$=
$\lim_{x \to \pi/4} \frac{\frac{\sin x-\cos x}{\cos x}}{\sin x-\cos x}$=
$\lim_{x ... | Jam | 161,490 | <p>Rearrange the fraction into something simpler. Express $\tan(x)$ in terms of $\sin(x)$ and $\cos(x)$.
$$\frac{\tan(x)-1}{\sin(x)-\cos(x)}=\frac{\frac{\sin(x)}{\cos(x)}-1}{\sin(x)-\cos(x)}$$
You've done this and the next step in your own work.
$$=\frac{\frac{\sin(x)}{\cos(x)}-\frac{\cos(x)}{\cos(x)}}{\sin(x)-\cos(x)}... |
1,259,961 | <p>I was looking at $$\int_0^1\left(\frac{1}{\sqrt{s}}\right)^2\ ds.$$</p>
<p>So in calculus, I would evaluate $\ln(1) - \ln(0)$ as the answer. What I don't get and I don't remember why is the answer $\infty$?</p>
<p>I know $\ln(1)=0$ and $\ln(0)=-\infty$</p>
<p>Wouldn't the answer be $-\infty$? Any explanation woul... | Jamie Lannister | 234,745 | <p>$ln(1) = 0, ln(0) \rightarrow -\infty$</p>
<p>$ln(1) - ln(0) \rightarrow 0 - -\infty = \infty$</p>
|
112,263 | <p>When Mathematica outputs nested lists, it just gloms one record right after the next record and it wraps around in the space available, for example:</p>
<pre><code>In[10]:= FactorInteger[FromDigits["1000000000000101",Range[2,10]]]
Out[10]= {{{13,1},{2521,1}},{{11,1},{251,1},{5197,1}},{{3,2},{229,1},{520981,1}},{{47... | Tyler Durden | 10,067 | <p>Based on the comments I developed the following answer. Not completely simple, but not too onerous to do:</p>
<pre><code>In[38]:= TableForm[Part[FactorInteger[FromDigits["1000000000000101",Range[2,10]]],All,All,1]]
Out[38]//TableForm=
13 2521
11 251 5197
3 229 520981
4751 ... |
991,377 | <p>I have to evalute this integral.
$\displaystyle\iint\limits_{D}(2+x^2y^3 - y^2\sin x)\,dA$
$$D=\left \{ (x, y):\left | x \right |+\left | y \right | \leq 1\right \}$$</p>
<p>At first, I evaluated simply by putting $-1\leq x\leq 1, -1\leq y\leq 1$, thus making
$$
\int_{-1}^{1}\int_{-1}^{1}(2+x^2y^3 - y^2\sin x)\,... | Blue | 409 | <p>Consider $A$'s relationship with various circles.</p>
<p>Since $B^\prime$ and $C^\prime$ are points of tangency of lines through $A$ with the incircle, we have
$$|AB^\prime| = |AC^\prime| \qquad (\star)$$
Further, by the <a href="http://en.wikipedia.org/wiki/Power_of_a_point#Theorems" rel="nofollow">"Power of a Poi... |
2,526,865 | <p>I came across this question while preparing for an interview.</p>
<p>You draw cards from a $52$-card deck until you get first Ace. After each card drawn, you discard three cards from the deck. What's the expected sum of cards until you get the first Ace? </p>
<p>Note</p>
<ol>
<li><p>J, Q, K have point value 11, 1... | Fimpellizzeri | 173,410 | <p>At each step, four cards are removed from the deck, so a deck is exhausted in $13$ steps.
Let's call that a 'round'.</p>
<p>The game you propose can be restated as follows.
At each round, we draw $13$ ordered cards from the deck, and add up the values of the cards that came before the first ace in our $13$ drawn ca... |
3,848,517 | <p>I have a conjecture in my mind regarding Arithmetic Progressions, but I can't seem to prove it. I am quite sure that the conjecture is true though.</p>
<p>The conjecture is this: suppose you have an AP (arithmetic progression):
<span class="math-container">$$a[n] = a[1] + (n-1)d$$</span>
Now, suppose our AP satisfie... | Especially Lime | 341,019 | <p>No, for example consider the functions <span class="math-container">$f_n:\mathbb R\to \mathbb R$</span> where <span class="math-container">$f_n=0$</span> outside the region <span class="math-container">$[\sum_{m<n}a_n,\sum_{m\leq n}a_n]$</span> and on that region rises linearly from <span class="math-container">$... |
437,053 | <p>I'm struggling with this nonhomogeneous second order differential equation</p>
<p><span class="math-container">$$y'' - 2y = 2\tan^3x$$</span></p>
<p>I assumed that the form of the solution would be <span class="math-container">$A\tan^3x$</span> where A was some constant, but this results in a mess when solving. The ... | Pedro | 23,350 | <p>Have you used that $$|a^3|=\frac{|a|}{\gcd(|a|,3)}= |a|\text{ ? }$$</p>
<p>Suppose now that $xa^3=a^3x$. Note that $(a^3)^2=a$ Then $$xa^3a^3=a^3xa^3$$</p>
<p>Can you finish?</p>
<p><strong>SPOILER</strong> </p>
<blockquote class="spoiler">
<p>$$xa=xa^6=xa^3a^3=a^3xa^3=a^3a^3x=a^6=ax$$</p>
</blockquote>
<p>... |
2,409,312 | <p>In <a href="https://math.stackexchange.com/a/1999967/272831">this previous answer</a>, MV showed that for $n\in\Bbb N$,</p>
<p>$$\int\frac1{1+x^n}~dx=C-\frac1n\sum_{k=1}^n\left(\frac12 x_{kr}\log(x^2-2x_{kr}x+1)-x_{ki}\arctan\left(\frac{x-x_{kr}}{x_{ki}}\right)\right)$$</p>
<p>where</p>
<p>$$x_{kr}=\cos \left(\fr... | H. H. Rugh | 355,946 | <p>Let $\gamma=\exp(2\pi i/a)$.
The polynomial $Q(x)=x^a+1$ has $a$ simple roots $z_k=\gamma^{k+1/2}$,
$0\leq k<a$. Since $z_k^a=-1$, we have $Q'(z_k)=a z_k^{a-1}=-a z_k^{-1}$, so
$$ \frac{1}{Q(x)} =
\sum_{k=0}^{a-1} \frac{1}{Q'(z_k)} \frac{1}{x-z_k} =
-\frac{1}{a} \sum_{k=0}^{a-1} \frac{z_k}{x-z_k} $$</p>... |
3,275,966 | <p>During the drawing lottery falls six balls with numbers from <span class="math-container">$1$</span> to <span class="math-container">$36$</span>. The player buys the ticket and writes in it the numbers of six balls, which in his opinion will fall out during the drawing lottery. The player wants to buy several lotter... | Ed Pegg | 11,955 | <p>Best known is <a href="https://ljcr.dmgordon.org/cover/show_cover.php?v=36&k=6&t=2" rel="nofollow noreferrer">47 tickets</a>.<br>
The lower bound would be 42 tickets, but no-one has found a solution with fewer than 47 tickets. </p>
<p>A table of similar results is at <a href="https://ljcr.dmgordon.org/cove... |
591,207 | <p>Show a proof that $Z(G),$ the center of $G,$ is a normal subgroup of $G$ and every subgroup of the center is normal. I know that $Z(G)$ is a normal subgroup and it is abelian, but how would I show that every subgroup of $Z(G)$ is also normal?</p>
| amWhy | 9,003 | <p>Recall that the center of a group $Z(G)$ consists of all elements that commute with every element in $G$. That's pretty much all you need, here:</p>
<p>If $H<Z(G),$ and $h \in H,$ then $h \in Z(G),$ and so for every $g \in G,\;$ $g^{-1}hg =g^{-1}gh = eh = h$ ($h \in Z(B)$, so $h$ commutes with $g$.)</p>
<p>Sinc... |
100,459 | <p>Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$</p>
<blockquote>
<p>Is there any already existing evidence, like papers or proofs or something that can debunk this?</p>
</blockquote>
<p>As far as I k... | Community | -1 | <p>First, your condition seems a bit strange to me. Since it seems to me it would imply that that the absolute value of $\zeta(1/2 + it)$ is bounded which would contradict the lower bound on the moments you recall. </p>
<p>Yet, second, here is one (unconditional) result on $|\zeta(1/2 + it)|$ that gives some informati... |
100,459 | <p>Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$</p>
<blockquote>
<p>Is there any already existing evidence, like papers or proofs or something that can debunk this?</p>
</blockquote>
<p>As far as I k... | juan | 7,402 | <p>Read Theorem 8.12 in Titchmarsh:</p>
<p>For $\frac12 \le \sigma <1$ take $0<\alpha <1-\sigma$. Then the inequality
$|\zeta(\sigma+it)| > \exp(\log^\alpha t)$ is satisfied for indefinitely
large values of $t$.</p>
<p>Therefore $f(t)=\exp(\log^\alpha t)$ does not satisfy your assertion for this $\sigm... |
825,531 | <p>How many positive integers $n$ are there, such that both $2n$ and $3n$ are perfect squares? I tried to use modular arithmetic, but I'm stuck.</p>
| Karolis Juodelė | 30,701 | <p>If you factor a perfect square, you'll see that every prime has an even power. If $2n$ has an even power of $2$, then $n$ has an odd power. Now consider the power of $2$ in $3n$. Since $2$ and $3$ are (co)prime, the power of $2$ in $3n$ is the power of $2$ in $n$, thus odd. Therefore, $3n$ can't be a perfect square.... |
2,988,987 | <blockquote>
<p>For example, we have <span class="math-container">$f(x)=\frac{1}{x^2-1}$</span></p>
</blockquote>
<p>Would the domain be <span class="math-container">$$\mathcal D(f)=\{x\in\mathbb{R}\mid x\neq(1,-1)\}$$</span> or rather
<span class="math-container">$$\mathcal D(f)=\{x\in\mathbb{R}\mid x\neq \{1,-1\}\... | epi163sqrt | 132,007 | <blockquote>
<p><strong>Hint:</strong> The flaws in OP's proposals have already been identified in the comment section. Note, it is often convenient to write</p>
<p><span class="math-container">$$\mathcal D(f)=\mathbb{R}\setminus\{-1,1\}$$</span></p>
</blockquote>
|
1,546,599 | <p>What method should I use for this limit?
$$
\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}
$$</p>
<p>I tried ratio test but I ended with the ugly answer
$$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}}
$$
which would go to 1? Which means we cannot use ratio test. I do not know how else I could find this limit.</p>
| sfp | 285,220 | <p>You could try to use the Stirling formel which states that for large n
$$n! \sim \sqrt{2\pi n} (\frac{n}{e})^n$$</p>
<p>So you would get :
$$\frac{n^{n-1}}{n!} \sim n^{n-1}(\frac{e}{n})^n (2\pi n)^{-\frac{1}{2}}$$
$$ \sim \frac{1}{\sqrt{2\pi}n^{\frac{3}{2}}} e^n \rightarrow \infty$$
Thus the sequence diverges.<... |
2,253,768 | <p>I am currently working on a small optimization problem in which I need to find an optimal number of servers for an objective function that incorporates the Erlang Loss formula. To this end, I have been searching for an expression for the first order difference of the Erlang Loss formula with respect to the number of... | Jesko Hüttenhain | 11,653 | <p>Consider the morphism $f:R[x]\to R[x]/(M,x)$ and the inclusion $g:R\to R[x]$. Then,
$$
\ker(f\circ g) = g^{-1}(\ker(f))= \ker(f)\cap R=(M,x)\cap R=M
$$
and as you explained, this gives you $R/M\cong R[x]/(M,x)$.</p>
|
608,875 | <p>solve this equation
$$\sqrt{\sqrt{3}-\sqrt{\sqrt{3}+x}}=x$$</p>
<p>My try: since
$$\sqrt{3}-x^2=\sqrt{\sqrt{3}+x}$$
then
$$(x^2-\sqrt{3})^2=x+\sqrt{3}$$</p>
| DeepSea | 101,504 | <p>Let $y = \sqrt{\sqrt3 + x}$, then the equation becomes: $\sqrt3 - y = x^2$, and $y^2 = \sqrt3 + x$.<br>
So:
$$\begin{align*}
y^2 - x & = y + x^2 \\
\Rightarrow (y + x)(y - x - 1) & = 0
\end{align*}$$
And we can go from here.</p>
|
3,689,051 | <p>Let <span class="math-container">$(X,d_{X})$</span> be a metric space, and let <span class="math-container">$(Y,d_{Y})$</span> be another metric space. Let <span class="math-container">$f:X\to Y$</span> be a function. Then the following two statements are logically equivalent:</p>
<p>(a) <span class="math-container... | marwalix | 441 | <p>Assume b)</p>
<p>Let <span class="math-container">$\epsilon\gt 0$</span> and <span class="math-container">$x\in X$</span>. The ball <span class="math-container">$B(f(x),\epsilon)\subset Y$</span> is an open subset. Its reciprocal image <span class="math-container">$f^{-1}\left(B(f(x),\epsilon\right)$</span> is open... |
2,841,640 | <p>What is a vector space? I can see two different formulations, and between them there is one difference: commutativity. </p>
<blockquote>
<p><strong>DEFINITION 1</strong> (See <a href="https://proofwiki.org/wiki/Definition:Vector_Space" rel="noreferrer">here</a>)</p>
<p>Let $(F, +_F, \times_F)$ be a division ... | Bernard | 202,857 | <p>In $Bourbaki$, a field $F$ is <em>not</em> necessarily commutative, and they simply define left (resp. right) $F$-vector spaces as left (resp. right) $F$-modules. </p>
<p>Ref. N. Bourbaki, <em>Algebra</em>, ch.I, <em>Algebraic Structures</em>, §9 and ch. II, <em>Linear Algebra</em>, §1, n°1.</p>
|
2,841,640 | <p>What is a vector space? I can see two different formulations, and between them there is one difference: commutativity. </p>
<blockquote>
<p><strong>DEFINITION 1</strong> (See <a href="https://proofwiki.org/wiki/Definition:Vector_Space" rel="noreferrer">here</a>)</p>
<p>Let $(F, +_F, \times_F)$ be a division ... | Community | -1 | <p>Usually, a vector space is an abelian group with a scalar multiplication with elements that come from a field.</p>
<p>It is true that most linear algebra keeps holding true if you drop the commutativity of the field (we are left with a division ring then), so that might be why the first definition calls it a vector... |
2,563,300 | <p>If we put the Cartesian coordinates of a point in a 2 dimensional locus' equation then we get zero as the value if the point lies on the locus.
On putting coordinates of all other points in the equation which do not lie on the locus, we get a numerical value. Does this numerical value convey any information about th... | gandalf61 | 424,513 | <p>You have four separate cases:</p>
<p>$P(A \land B) = \dfrac{1}{15}$</p>
<p>$P(A \land \lnot B) = \dfrac{4}{15}$</p>
<p>$P(\lnot A \land B) = \dfrac{2}{15}$</p>
<p>$P(\lnot A \land \lnot B) = \dfrac{8}{15}$</p>
<p>(this assumes A and B are independent).</p>
<p>To find the probability that B succeeds given that ... |
339,331 | <p>This is a question of terminology.</p>
<p>Suppose we have a line segment AB in a plane. The line segment forms three "zones" in the plane, where the "middle zone" is comprised of points for which some line perpendicular to AB passes through both that point and AB.</p>
<p>Is there a name for this "middle zone"? I... | wchargin | 15,886 | <p>If you want a mathematical term, I can't help you.</p>
<p>But, as a programmer myself, I understand the need for a memorable, intuitive method name. So, here's the first thing that comes to my mind when I see the pictorial representation: may I suggest the <strong>asteroid belt</strong>?</p>
<p><a href="https://i.... |
339,331 | <p>This is a question of terminology.</p>
<p>Suppose we have a line segment AB in a plane. The line segment forms three "zones" in the plane, where the "middle zone" is comprised of points for which some line perpendicular to AB passes through both that point and AB.</p>
<p>Is there a name for this "middle zone"? I... | Simon | 68,248 | <p>I'm not sure if there is a term for this, but the area between two parallel lines is usually called a strip. I'd call this the 'perpendicular strip'.</p>
|
33,311 | <p>I'm (very!) new to <em>Mathematica</em>, and trying to use it plot a set in $\mathbb{R}^3$. In particular, I want to plot the set</p>
<p>$$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{u^2+v^2+w^2}(vw,uw,uv) \text{ for some } (u,v,w) \in \mathbb{R}^3\setminus\{0\} \big\}. $$
This is just the image of function from $\mathbb{... | rm -rf | 5 | <p>In such cases, you can get better flexibility by switching to a <code>DynamicModule</code> and building up the GUI yourself. Then, you can pull the data generating step out of the plotting dynamic, so that the latter can be manipulated freely without regenerating the data. </p>
<pre><code>DynamicModule[{function = ... |
313,298 | <p>For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?</p>
<p>I'm using the Buchanan textbook, but I'm not understanding their proofs at all.</p>
| copper.hat | 27,978 | <p>You need to show that for any number $\epsilon>0$ you can find $N$ such that if $n\geq N$, then $|c_n - \frac{1}{2}| < \epsilon$.</p>
<p>So, pick an $\epsilon>0$ and start computing: $|c_n - \frac{1}{2}| = | \frac{n^2+10}{2 n^2} - \frac{1}{2}| = |\frac{10}{2 n^2}| = \frac{5}{n^2}$.</p>
<p>Now you need to ... |
1,923,298 | <p>I am really struggling with this question and it isn't quite making sense. Please help and if you don't mind answering quickly.</p>
<p>Reflection across $x = −1$</p>
<p>$H(−3, −1), F(2, 1), E(−1, −3)$.</p>
| Elle Najt | 54,092 | <p>I want to suggest proving it in the following way, which I think is less confusing. I'm assuming that $A$ and $B$ are subsets of a single set, in which case $A \cap B$ may not be empty, and the number of elements in $A \cup B$ may not be $m + n$.</p>
<ol>
<li>$A \cup B = (A \setminus B) \cup (B \setminus A) \cup (A... |
1,923,298 | <p>I am really struggling with this question and it isn't quite making sense. Please help and if you don't mind answering quickly.</p>
<p>Reflection across $x = −1$</p>
<p>$H(−3, −1), F(2, 1), E(−1, −3)$.</p>
| fleablood | 280,126 | <p>You have the right idea but you are confusing whether $x \in A \cup B$ or $x \in \mathbb N$. If $f| A \rightarrow \{1,...., n\}$ then you can't say $x \in \{1,...., n\}$. And if $g(x)| \{1,...., m\}\rightarrow B$ then you can't say $h(x) = m + g(x)$ because $g(x)$ is not a number.</p>
<p>Let $f:A \rightarrow \{1... |
1,019,981 | <p>I'm a high school student and I'm giving a lecture in my high school math class on ordinal numbers, and I would like to prove that the von Neumann ordinals are well-ordered by set membership.</p>
<p>The definition of von Neumann ordinal that I'm using is as follows. An ordinal is a set $A$ such that the elements of... | Stefan Mesken | 217,623 | <p>Observe that for given ordinals $A$ and $B$ the following claims hold true:</p>
<ul>
<li>If $A$ is a proper subset of $B$, then $A \in B$.</li>
<li>$A \cap B$ is an ordinal.</li>
</ul>
<p>Now, given ordinals $A \neq B$ consider $A \cap B \subseteq A,B$. If $A \cap B = A$, then $A \subseteq B$, so that either $A = ... |
1,019,981 | <p>I'm a high school student and I'm giving a lecture in my high school math class on ordinal numbers, and I would like to prove that the von Neumann ordinals are well-ordered by set membership.</p>
<p>The definition of von Neumann ordinal that I'm using is as follows. An ordinal is a set $A$ such that the elements of... | Akira | 368,425 | <p>For any ordinals $A$ and $B$, below statements hold true:</p>
<ol>
<li>$A\subsetneq B\implies A\in B$. (I presented a proof for this at <a href="https://math.stackexchange.com/questions/1983906/if-alpha-ne-beta-are-ordinals-and-alpha-subset-beta-show-alpha-in-beta/2845626#2845626">If $\alpha\ne\beta$ are ordinals a... |
2,441,793 | <p><strong>Questions.</strong> </p>
<p>(0) Is there a usual technical term in ring theory for the following kind of module?</p>
<blockquote>
<p>$M$ is a free $R$-module over a commutative ring $R$, and for each $R$-basis $B$ of $M$, each $b\in B$, and each <em>unit</em> $r$ of $R$, we have $r\cdot b=b$</p>
</blockq... | Peter Heinig | 457,525 | <p>Ad 1. You write</p>
<blockquote>
<p>But my question is when the use of ≡ or = is mandatory, and when it is permissible. </p>
</blockquote>
<p>There is no absolute answer to this, yet the use of '=' between terms like the one you give, which can be seen as a elements of the 'rational function field' $\mathbb{Q}(x... |
2,441,793 | <p><strong>Questions.</strong> </p>
<p>(0) Is there a usual technical term in ring theory for the following kind of module?</p>
<blockquote>
<p>$M$ is a free $R$-module over a commutative ring $R$, and for each $R$-basis $B$ of $M$, each $b\in B$, and each <em>unit</em> $r$ of $R$, we have $r\cdot b=b$</p>
</blockq... | Thomas | 26,188 | <p>It is absolutely fantastic that you are thinking about these issues as you start your undergraduate career. If only more students would appreciate these subtleties in mathematics. One nice thing in mathematics is that everything can be made precise.</p>
<p>A couple of remarks though:</p>
<ol>
<li><p>Often we will ... |
2,795,652 | <blockquote>
<p>Given a set of decimal digits. And given a set of primes $\mathbb{P}$, find some $p \in \mathbb{P}$ such that $p^n, n \in \mathbb{N} $ contained in itself all the digites from a given set, and it does not matter in what order.</p>
</blockquote>
| Barry Cipra | 86,747 | <p>Remark: this answer pertains to the initial version of the question. The current edit restricts the search to powers within a given set of primes.</p>
<p>Another theoretically "cheap" way to look for prime powers that contain a given set of digits is to concatenate the $n$ digits, append a $1$ to get an $(n+1)$-dig... |
85,117 | <p>Let $\mathcal{A}$ and $\mathcal{B}$ be two $2$-categories and $F : \mathcal{A} \to \mathcal{B}$ be a lax $2$-functor. Given $1$-cells $(f_{i})_{0 \leq i \leq n}$ of $\mathcal{A}$ such that the composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}$ makes sense, this data together with the structural $2$-cells of ... | Nacho Lopez | 8,482 | <p>The closest statement I know is Theorem 1.6 in Gordon-Power-Street "Coherence for tricategories." It actually deals with pseudofunctors instead of lax functors, but I believe it can easily be modified to cover lax functors. The proof is the same proof of Theorem 1.7 in Joyal-Street "Braided tensor categories."</p>
|
975,210 | <p>\begin{align}
\left| f(b)-f(a)\right|&=\left| \int_a^b \frac{df}{dx} dx\right|\\ \ \\
&\leq\left| \int_a^b \left|\frac{df}{dx}\right|\ dx\right|.
\end{align}</p>
<p>I do not understand why the second line is greater or equal than the top equation. Can anyone explain please?</p>
| Hagen von Eitzen | 39,174 | <p>Let $u$ be integrable (in your case it is $\frac{df}{dx}$).
Because $u(x)\le |u(x)|$ we have
$$\int_a^bu(x)\,\mathrm dx\le \int_a^b|u(x)|\,\mathrm dx $$
for $a\le b$.
Because $-u(x)\le |u(x)|$ we have
$$-\int_a^bu(x)\,\mathrm dx=\int_a^b(-u(x))\,\mathrm dx\le \int_a^b|u(x)|\,\mathrm dx $$
for $a\le b$. As $|y|=\max... |
2,197,065 | <p>Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space.</p>
<p>Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?</p>
| Akay | 359,648 | <p>As others have mentioned, there are several valid use cases for notions of average(or mean) distance. A mundane example would be:</p>
<blockquote>
<p>If you alternately run $3$ and $5$km every day beginning from Monday
through Saturday, what is the average distance you've run in the
entire week?</p>
</blockqu... |
391,020 | <p>Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly? </p>
| Asaf Karagila | 622 | <p><strong>Hint:</strong> Every element of $\Bbb{Z/nZ}$ has finite order.</p>
|
106,708 | <p><strong>My fragile attempt:</strong> Note that if $1987^k-1$ ends with 1987 zeros, that means $1987^k$ has last digit 1 (and 1986 "next" ones are zeros). For this to be satisfied, $k$ has to be in form $k=4n$, where $n\in N$. This means out number can be written in form </p>
<p>$$ [(1987^n)^2+1][1987^n+1][1987^n-1]... | André Nicolas | 6,312 | <p>The standard approach is to use the fact that if $a$ is divisible neither by $2$ nor by $5$, then
$$a^{\varphi(10^n)}\equiv 1\pmod {10^n},$$
where $\varphi$ is the Euler $\varphi$-function.</p>
<p>The approach below is much more low-tech! All we need is some comfort with the Binomial Theorem. Suppose that $b_1$ al... |
310,669 | <p>This is related to a course I'm taking in computer science theory. </p>
<p>Let $\sum$ be an alphabet. Then the set of all strings over $\sum$, denoted as $\sum^*$ has the operation of concatenation (adjoining two strings end to end). Clearly, concatenation is associative, $\sum^*$ is closed under concatenation, a... | Jim | 56,747 | <p>It is not a group, it is a <em>monoid</em>, which is essentially a group without inverses. The natural thing to do if you want a group is to just arbitrarily allow inverses. So you would consider strings whose letters are of the form $a$ or $a^{-1}$ where $a \in \Sigma$. You say that two strings are equivalent if... |
937,443 | <blockquote>
<p>Evaluate $\displaystyle \int \tan^2x\sec^2x\,dx$</p>
</blockquote>
<p>I tried several methods: </p>
<ul>
<li>First method was I changed $\tan^2x = \sec^2x-1$, and then substitute $\sec x$ to $t$, but it doesn't work.</li>
<li>Second method was to use substitute $\tan^2x = v$, $\sec x = u$. And, it ... | Anay Mehrotra | 659,080 | <p>Substituting <span class="math-container">$x = \tan^{-1}t$</span> and using <span class="math-container">$\sec^2(x)= \tan^2(x)+1$</span> we get
<span class="math-container">$$\int (1+t^2)\cdot t^2\cdot \frac{1}{1+t^2}\, dt = \int t^2dt = \frac{t^3}{3}+C.$$</span>
Since <span class="math-container">$t=\tan(x)$</span>... |
522,714 |
<p>$$
dz_t \sim O\left(\sqrt{dt}\,\right)
$$</p>
<p>$z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation on wikipedia because I thought it might be related, but I can't see the link.</p>
| rschwieb | 29,335 | <p>The idea you're having to change it to terms of $x^2$ isn't bad, but it seems a little overfancy. (Maybe I overlooked some economy about it, but I haven't seen the benefit yet.)</p>
<p>Why not just calculate it directly? (Hints follow:)</p>
<p>$x^2=3+2+2\sqrt{6}=5+2\sqrt{6}$</p>
<p>$x^4=(5+2\sqrt{6})^2=25+24+20\s... |
1,758,194 | <p>Consider the function f: {-1, +1} -> R defined by</p>
<p>$f(x)= \arcsin (\frac{1+x^2}{2x})$.</p>
<p>Due to the following two inequalities :</p>
<p>(i) $1+x^2 \geq 2x$</p>
<p>(ii)$1+x^2 \geq -2x$ , </p>
<p>the function can only be defined at $x=1$ and $x=-1$. I have learnt that the epsilon delta definition only ... | Aloizio Macedo | 59,234 | <p>The bottom line is: </p>
<blockquote>
<p>Limits are only defined on limit points. Continuity is only defined on the domain.</p>
</blockquote>
<p>If we have a point $p$ on the domain such that $p$ is a limit point, then continuity at $p$ is equivalent to the limit being the value. However, if we have a point not ... |
484,281 | <p>I recently got interested in mathematics after having slacked through it in highschool. Therefore I picked up the book "Algebra" by I.M. Gelfand and A.Shen</p>
<p>At problem 113, the reader is asked to factor $a^3-b^3.$</p>
<p>The given solution is:
$$a^3-b^3 = a^3 - a^2b + a^2b -ab^2 + ab^2 -b^3 = a^2(a-b) + ab(... | Barry Cipra | 86,747 | <p>The second equality is obtained by first grouping terms in the middle expression and then factoring the grouped terms:</p>
<p>$$\begin{align}
a^3-a^2b+a^2b-ab^2+ab^2-b^3&=(a^3-a^2b)+(a^2b-ab^2)+(ab^2-b^3)\cr
&=a^2(a-b)+ab(a-b)+b^2(a-b)\cr
\end{align}$$</p>
<p>If the OP is wondering where the middle express... |
1,422,859 | <p>$$\sqrt{1000}-30.0047 \approx \varphi $$
$$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$
Simplifying Above expression we get<br>
$$1.0000952872327798^{5050.3535}=1.1618033..... $$
Is this really true that
$$[\varphi^2-\varphi]^{5050.3535}=\varphi $$</p>
| JJacquelin | 108,514 | <p>Of course, this is an approximation. No need for more proof than those already given by Vincenzo Oliva and Claude Leibovici - kindest regards !</p>
<p>I would add that it is easy to find a lot of such approximations.</p>
<p>Some amazing ones, to be compared to :</p>
<p>$1.6180339887...\simeq\frac{1+\sqrt5}{2}=\va... |
2,532,327 | <p>Expand the following function in Legendre polynomials on the interval [-1,1] :</p>
<p>$$f(x) = |x|$$</p>
<p>The Legendre polynomials $p_n (x)$ are defined by the formula :</p>
<p>$$p_n (x) = \frac {1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^2$$</p>
<p>for $n=0,1,2,3,...$</p>
<p>My attempt :</p>
<p>we have using the fa... | Steven Yang | 675,191 | <p>We know that the Fourier-Legendre series is like
<span class="math-container">$$
f(x)=\sum_{n=0}^\infty C_n P_n(x)
$$</span></p>
<p>where
<span class="math-container">$$
C_n=\frac{2n+1}{2} \int_{-1}^{1}f(x)P_n(x)\,dx
$$</span></p>
<p>So now we are going to calculate the result of
<span class="math-container">$$... |
1,037,621 | <p>There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (and who has the black ones). </p>
<p>I thought it might be $$\large2^{\frac{20(20-1)}2}$... | Uddipan Paul | 537,089 | <p>Here we have 2 things to keep in mind:</p>
<ol>
<li>Number of ways of matching up</li>
<li>Arrangements does matter</li>
</ol>
<p>Firstly, Number of ways of matching up players:</p>
<p>1st player: 19 player options; chooses: 1; players left after choosing: 18</p>
<p>2nd player: 17 player options; chooses: 1; pla... |
1,037,621 | <p>There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (and who has the black ones). </p>
<p>I thought it might be $$\large2^{\frac{20(20-1)}2}$... | acala | 910,755 | <p>First we choose 2 from 20 to form pair 1, then choose 2 from 18 to form pair 2... until last 2 players to form pair 10: there are totally <span class="math-container">$\binom{20}{2}\binom{18}{2}\ldots\binom{2}{2}$</span> possible combinations to form 10 ordered pairs.</p>
<p>However, in this problem, order of pairs ... |
1,142,530 | <p>Find an equation of the plane.
The plane that passes through the point
(−3, 2, 1) and contains the line of intersection of the planes
x + y − z = 4
4x − y + 5z = 2</p>
<p>I know the normal to plane 1 is <1,1,-1> and the normal to plane 2 is <4,-1,5>. The cross product of these 2 would give a vector that is ... | PdotWang | 212,686 | <p>The line passes $z=0$ at $(1.2, 2.8, 0)$.</p>
<p>A vector from this point to $(-3,2,1)$ is $(4.2,0.8,-1)$.</p>
<p>This vector cross $(4,-9,-5)$ is $(13,-17,41)$, which is the normal of the plane. </p>
<p>The result is:</p>
<p>$$13(x-(-3))+(-17)(y-2)+(41)(z-1)=0$$</p>
<p>This is the scalar product of a vector in... |
2,780,186 | <blockquote>
<p>SOA Exam C Question #140
<a href="https://i.stack.imgur.com/VqAE6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VqAE6.png" alt="SOA Exam C Question #140"></a></p>
<p>Solution
<a href="https://i.stack.imgur.com/6YK4b.png" rel="nofollow noreferrer"><img src="https://i.stack... | heropup | 118,193 | <p>The expected counts in each interval are computed by taking the hypothesized probability of an observation occurring in that interval, and multiplying by the total number of observed events in your sample. So for the interval $(0, 500]$, we have $F(500) = 0.27$, so for $n = 30$, we should see $(0.27)(30) = 8.1$ eve... |
2,780,186 | <blockquote>
<p>SOA Exam C Question #140
<a href="https://i.stack.imgur.com/VqAE6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VqAE6.png" alt="SOA Exam C Question #140"></a></p>
<p>Solution
<a href="https://i.stack.imgur.com/6YK4b.png" rel="nofollow noreferrer"><img src="https://i.stack... | Michael Hardy | 11,667 | <p>The proportion less than or equal to $310$ according to the null hypothesis is $0.16$.</p>
<p>The proportion less than or equal to $500$ but more than $300$ is $0.27-0.16= 0.11.$</p>
<p>The proportion less than or equal to $2498$ but greater than $500$ is $0.55-0.27 = 0.28$</p>
<p>And so on.</p>
<p>Now multiply ... |
2,027,556 | <p>The definition I have of a tensor product of vector finite dimensional vector spaces $V,W$ over a field $F$ is as follows: Let $v_1, ..., v_m$ be a basis for $V$ and let $w_1,...,w_n$ be a basis for $W$. We define $V \otimes W$ to be the set of <strong>formal linear combinations</strong> of the mn symbols $v_i \otim... | Mattia Ghio | 360,782 | <p>I propose you this answer:
<a href="https://math.stackexchange.com/questions/1029851/understanding-the-meaning-of-formal-linear-combination-and-tensor-product">understanding the meaning of formal linear combination and tensor product</a></p>
<p>In particular:
"Regarding the tensor product, we want it to be bilinear... |
2,294,321 | <blockquote>
<p>Suppose $F(x)$ is a continuously differentiable (that is, first derivative exist and are continuous) vector field in $\mathbb{R}^3$ that satisfies the bound $$|F(x)| \leq \frac{1}{1 + |x|^3}$$
Show that $\int\int\int_{\mathbb{R}^3} \text{div} F dx = 0$.</p>
</blockquote>
<p>Attempted proof - Suppos... | Giovanni | 263,115 | <p>The third attempt seems to be on the right track, however there is no divergence and it is not entirely clear how you are using the given bound. To expand on Gio67's answer, </p>
<p>$$\Big|\int_{\mathbb{R}^3}\text{div}F\,dx\Big| = \lim_n\Big|\int_{B(0,n)}\text{div}F\,dx\Big| = \lim_n\Big|\int_{\partial B(0,n)}F \cd... |
3,741,859 | <p>I am trying to prove that, for all non-negative integers <span class="math-container">$x$</span> and all non-negative real numbers <span class="math-container">$p$</span>,
<span class="math-container">$$
\left(p-x\right)\left(x+1\right)^p+x^{p+1}\geq0.
$$</span>
I've been at this for a while and I'm stuck. I've trie... | Joseph Camacho | 731,433 | <p>Equivalently, we need to prove<br />
<span class="math-container">\begin{align*}
&x \cdot x^p \ge (x - p) \cdot (x - p) \cdot (x + 1)^p\\
&\Longleftrightarrow \left(1 + \frac1x\right)^p \le \frac{x}{x - p}.
\end{align*}</span></p>
<p>But<br />
<span class="math-container">\begin{align*}
\left(1 + \frac1x\rig... |
3,741,859 | <p>I am trying to prove that, for all non-negative integers <span class="math-container">$x$</span> and all non-negative real numbers <span class="math-container">$p$</span>,
<span class="math-container">$$
\left(p-x\right)\left(x+1\right)^p+x^{p+1}\geq0.
$$</span>
I've been at this for a while and I'm stuck. I've trie... | robjohn | 13,854 | <p>For <span class="math-container">$x,p\ge0$</span>,
<span class="math-container">$$
\begin{align}
\left(1-\frac1{x+1}\right)^{p+1}&\ge1-\frac{p+1}{x+1}\tag1\\
\color{#C00}{((x+1)-1)^{p+1}}&\ge\color{#090}{(x+1)^{p+1}-(p+1)(x+1)^p}\tag2\\[6pt]
\color{#090}{((p+1)-(x+1))(x+1)^p}+\color{#C00}{x^{p+1}}&\ge0\t... |
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | Did | 6,179 | <p>Your suggestion to rely on the iteration of the function $f$ defined by $f(x)=2x-\sqrt3$ to compute $\sqrt3$ is not used in practice for at least two reasons. First it presupposes one is able to compute $\sqrt3$, in which case the approximation procedure is useless. Second, $\sqrt3$ is not an attractive point of $f$... |
1,640,373 | <p>The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the metric. But if I take an arbitrary topology for a metric space, will this set coincide with the metric topology? </p>
<p... | fosho | 166,258 | <p>Yes you are right,$$K = \ker{T} = \{(a,0,b)| a,b\in \mathbb{R}\}$$</p>
<p>A basis for $K$ is then</p>
<p>$$\{(1,0,0),(0,0,1)\}$$</p>
<p>Since given $(a,0,b)$ with $a,b\in\mathbb{R}$ we have $$(a,0,b) = a\cdot(1,0,0)+b\cdot(0,0,1)$$</p>
|
777,691 | <p>I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$</p>
<p>I'm stuck.<br>
$n = a \cdot k_1$<br>
$n = b \cdot k_2$<br>
$\therefore a \cdot k_1 = b \cdot k_2$</p>
<p>EDIT: so for <a href="http://en.wikipedia.org/wiki/Fizz_buzz" rel="... | Andreas K. | 13,272 | <p>If <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are relatively prime, then <span class="math-container">$x a + y b = 1$</span> for some integers <span class="math-container">$x, y$</span> (Bezout's identity).
Then
<span class="math-container">$$
n = k_1 a \overset{\times yb}{\L... |
1,387,454 | <p>What is the sum of all <strong>non-real</strong>, <strong>complex roots</strong> of this equation -</p>
<p>$$x^5 = 1024$$</p>
<p>Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. Is there any way to determine number of real and non-real roots of an ... | Macavity | 58,320 | <p><strong>Hint</strong>: By Vieta, sum of all roots is $0$, and the only real root is $4$.</p>
<p>P.S. For a simple argument that there is only one real root, apply Descartes' rule of signs on $x^5-1024$.</p>
|
2,294,969 | <p>I can't seem to find a path to show that:</p>
<p>$$\lim_{(x,y)\to(0,0)} \frac{x^2}{x^2 + y^2 -x}$$</p>
<p>does not exist.</p>
<p>I've already tried with $\alpha(t) = (t,0)$, $\beta(t) = (0,t)$, $\gamma(t) = (t,mt)$ and with some parabolas... they all led me to the limit being $0$ but this exercise says that there... | Miguel | 259,671 | <p>We can also try with parabolas with the axes exchanged, e.g. $x=y^2$:
$$\lim_{(x,y)\to (0,0),x=y^2}\frac{x^2}{x^2+y^2-x} = \lim_{y\to 0}\frac{y^4}{y^4+y^2-y^2}=1
$$</p>
|
2,294,969 | <p>I can't seem to find a path to show that:</p>
<p>$$\lim_{(x,y)\to(0,0)} \frac{x^2}{x^2 + y^2 -x}$$</p>
<p>does not exist.</p>
<p>I've already tried with $\alpha(t) = (t,0)$, $\beta(t) = (0,t)$, $\gamma(t) = (t,mt)$ and with some parabolas... they all led me to the limit being $0$ but this exercise says that there... | zhw. | 228,045 | <p>In fact this function, let's call it $f(x,y),$ is so badly behaved near $(0,0)$ that it maps every neighborhood of $(0,0)$ onto $\mathbb R.$</p>
<p>More precisey, set $E= \{(x,y): x\in [0,1], y = \sqrt {x-x^2}\}.$ Then the natural domain of $f$ is $U=\mathbb R^2\setminus E.$</p>
<p>Claim: For every $r>0,$ $f(U\... |
2,486,095 | <p>Given an acute-angled triangle $\Delta ABC$ having it's Orthocentre at $H$ and Circumcentre at $O$. Prove that $\vec{HA} + \vec{HB} + \vec{HC} = 2\vec{HO}$</p>
<p>I realise that $\vec{HO} = \vec{BO} + \vec{HB} = \vec{AO} + \vec{HA} =\vec{CO} + \vec{HC}$ which leads to $3\vec{HO} = (\vec{HA} + \vec{HB} + \vec{HC}) +... | Community | -1 | <p>$$\begin{matrix}
& 1.2 ,& 1.42 ,& 2.8275 \\
& 1.1975 ,& 1.98525 ,& 2.89940625 \\
& 0.98578125 ,& 1.978459375 ,& 3.00441679687 \\
& 1.00949960937 ,& 2.00183332031 ,& 2.99547936035 \\
& 0.998362543945 ,& 1.99893212646 ,& 3.00068524384 \\
& 1.00056419818 ,... |
4,221,545 | <p>Let <span class="math-container">$f$</span> be a twice-differentiable function on <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$f''$</span> is continuous. Prove that
<span class="math-container">$f(x) f''(x) < 0$</span> cannot hold for all <span class="math-container">$x... | Thomas | 89,516 | <p>Sketch of a proof. Details remain to be filled. The devil is in the details of course, but the idea should work.</p>
<ol>
<li><p>Since f is continuous, it must be <span class="math-container">$f>0$</span> or <span class="math-container">$f<0$</span> everywhere (why?). wlog consider <span class="math-container"... |
832,715 | <blockquote>
<p>Suppose that the distribution of a random variable
$X$ is symmetric with respect to the point $x = 0$. If $\mathbb{E}(X^4)>0$ then $Var(X)$ and $Var(X^2)$ are both positive.</p>
</blockquote>
<p>How is that true? I am getting $Var(X)=\mathbb{E}(X^2)$ and $Var(X^2)=\mathbb{E}(X^4)-(\mathbb{E}(X... | Henry | 6,460 | <p>I do not think it is true. </p>
<p>For example let $X=k$ and $X=-k$ each have probability $\frac12$ for some $k\gt 0$.</p>
<p>Then the distribution is symmetric about $0$, i.e. $P(X \le -x) =P(X \ge x)$ for all $x$.
And $E[X]=0$, $E[X^2]=k^2 \gt 0$ and $E[X^4]=k^4 \gt 0$, and $Var(X)=k^2 \gt 0$.</p>
<p>But $Var... |
2,581,735 | <p>I'm studying Neural Networks for machine learning (by Geoffrey Hinton's course) and I have a question about learning rule for linear neuron (lecture 3).</p>
<p>Linear neuron's output is defined as: $y=\sum_{i=0}^n w_ix_i$</p>
<p>Where $w_i$ is weight connected to input #i and $x_i$ is input #i.</p>
<p>Learning pr... | Andreas | 317,854 | <p>The error in here is the quadratic error:</p>
<p>$$
E_k = (t_k - y_k)^2 = (t_k - \sum_i w_ix_i^k)^2
$$
for the training examples indexed with $k$. Actually, you can also sum over $k$, and you can take the gradient instead of a single derivative.</p>
<p>Question 1: Changing weights proportionally to the negative ... |
234,466 | <p>This is the second problem in Neukirch's Algebraic Number Theory. I did the proof but it feels a bit too slick and I feel I may be missing some subtlety, can someone check it over real quick?</p>
<p>Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta =\varepsilon\gamma ^n$, for $\alpha,\beta$ relativ... | sperners lemma | 44,154 | <p>The trouble with Gaussian integers is that unique factorization is only unique up to units. For example consider $-i \cdot (1+i)(2+i)(1+2i)^4 = -1 \cdot (1+i)(-1+2i)(2-i)^4$: The primes $2+i$ and $-1+2i$ are just associates rather than equal. Prime ideals factor out associates and free us from caring about choosing ... |
873,803 | <p>I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about (or really self-studied) for that matter, is general polynomials beyond quadratics. In particular, I've been having ... | MJD | 25,554 | <p><strong>Hint</strong>: $$x^3 + bx^2 + cx + d = (x-r_1)(x-r_2)(x-r_3)$$</p>
|
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | wendy.krieger | 78,024 | <p>After many years of semi-non-use, some useful things remain: process, existance, etc. </p>
<p>You might know for example, that calculus exists, and there are handsome volumes of pre-made calculus, to modify to the current end.</p>
<p>You might know how to read the runes in mathematical papers, and generally follo... |
1,047,489 | <p>let $f(x)$ be a function such that </p>
<p>$$f(0) = 0$$</p>
<p>$$f(1) =1$$ $$f(2) = 2$$ $$f(3) = 4$$ $$f'(x) \text{is differentiable on } \mathbb{R}$$ Prove that there is a number in the interval $(0,3)$ such
that $0 < f''(x)<1$</p>
<p>I'm really stuck.
thanks</p>
| Simon S | 21,495 | <p><em>Hint:</em> Show that a some point $x \in (0,1)$, $f'(x) = 1$. Then for some $y \in (2,3)$, $f'(y) = 2$. Now use that to deduce the result by applying the MVT again.</p>
<p><em>Added:</em></p>
<p>Having found such $x$ and $y$, there must be a $z \in (x,y)$ such that</p>
<p>$$f''(z) = \frac{f'(y) - f'(x)}{y - x... |
1,047,489 | <p>let $f(x)$ be a function such that </p>
<p>$$f(0) = 0$$</p>
<p>$$f(1) =1$$ $$f(2) = 2$$ $$f(3) = 4$$ $$f'(x) \text{is differentiable on } \mathbb{R}$$ Prove that there is a number in the interval $(0,3)$ such
that $0 < f''(x)<1$</p>
<p>I'm really stuck.
thanks</p>
| Idele | 190,802 | <p>$1=\frac{f(1)-f(0)}{1-0}=f'(\xi_1),\xi_1\in(0,1)$</p>
<p>$2=\frac{f(3)-f(2)}{3-2}=f'(\xi_2),\xi_2\in(2,3)$</p>
<p>then $\xi_2-\xi_1\in(1,3)$</p>
<p>$f''(\eta)=\frac{f'(\xi_2)-f'(\xi_1)}{\xi_2-\xi_1}=\frac{1}{\xi_2-\xi_1}\in(\frac{1}{3},1),\eta\in(\xi_1,\xi_2)\subset(0,3)$</p>
|
243,210 | <p>I have difficulty computing the $\rm mod$ for $a ={1,2,3\ldots50}$. Is there a quick way of doing this?</p>
| Dennis Gulko | 6,948 | <p>This seems fine. the only point is $v=\frac1y$, so $y=\frac1v$ and
$$u(x)v(x)-1\cdot\frac13=u(x)v(x)-u(0)v(0)=\int_0^x (u(t)v(t))'=\int_0^x u(t)q(t)dt=-\int_0^x te^{-\frac{t^2}{2}}dt$$
So
$$v(x)=e^{\frac{x^2}{2}}\left(\frac13-\int_0^x te^{-\frac{t^2}{2}}dt\right),\hspace{10pt} y(x)=\frac{1}{v(x)}$$</p>
|
25,413 | <p>Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.</p>
<p>We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant ... | guest | 20,089 | <p>Unless you're training a superstar, and college sophomore taking calc 1 is a clue here, I would not cover material that the regular teacher has made the choice not to cover. And there are always limits on time and ability, thus choices on what to cover.</p>
<p>Also, can the limit be found using lhopital? Which stu... |
2,363,390 | <p>This question arises from an unproved assumption made in a proof of L'Hôpital's Rule for Indeterminate Types of $\infty/\infty$ from a Real Analysis textbook I am using. The result is intuitively simple to understand, but I am having trouble formulating a rigorous proof based on limit properties of functions and/or ... | DanielWainfleet | 254,665 | <p>Assuming that $a$ is a lower limit point of $A$ \ $\{a\},$ the statement $\lim_{x\to a+}f(x)=\infty$ is equivalent to $$\lim_{c\to a+}\inf \{f(x): x\in (a,c)\cap A\}=\infty.$$ And the statement you wish to derive from $\lim_{x\to a+}f(x)=\infty$ is equivalent to $$\lim_{c\to a+}\sup \{f(x): x\in (a,c)\cap A\}=\inft... |
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| Robert Lewis | 67,071 | <p>Yet another way to do it:</p>
<p>Observe that</p>
<p>$M = I + J, \tag{1}$</p>
<p>where</p>
<p>$I = \begin{bmatrix} I_n & 0 \\ 0 & I_n \end{bmatrix} \tag{2}$</p>
<p>and</p>
<p>$J = iP, \tag{2}$</p>
<p>with</p>
<p>$P = \begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix}. \tag{3}$</p>
<p>Then </p>... |
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| Marc van Leeuwen | 18,880 | <p>Asking for the inverse of such a block matrix is the same as asking the inverse of a $2\times2$ matrix over the non-commutative ring $M_n(\Bbb C)$ (viewing the individual blocks as elements of that ring). This point of view is not really helpful unless some special circumstance arises; in this case those four "entri... |
201,358 | <p>I would like to be able to extract elements from a list where indices are specified by a binary mask.</p>
<p>To provide an example, I would like to have a function <code>BinaryIndex</code> doing the following:</p>
<pre><code>foo = {a, b, c}
mask = {True, False, True}
BinaryIndex[mask, foo]
(* expected output *) {a... | Henrik Schumacher | 38,178 | <p>The problem with Booleans in <em>Mathematica</em> is that they cannot be stored in packed arrays, reducing the efficiency of their processing. Suppose that <code>b</code> is already a packed vector of integers, e.g., <code>b = Developer</code>ToPackedArray[Boole[mask]]`. Then you should be able to obtain your result... |
3,403,364 | <p>Here's a little number puzzle question with strange answer:</p>
<blockquote>
<p>In an apartment complex, there is an even number of rooms. Half of the rooms have one occupant, and half have two occupants. How many roommates does the average person in the apartment have? </p>
</blockquote>
<p>My gut instinct was ... | Mees de Vries | 75,429 | <p>This is a well-known paradox known as the <a href="https://en.m.wikipedia.org/wiki/Friendship_paradox" rel="nofollow noreferrer">friendship paradox</a>. In this case, your intuition leads you to believe that living in a one-person household is as common as living in a two-person household, because there are equally ... |
342,096 | <p>$$\left( \left( 1/2\,{\frac {-\beta+ \sqrt{{\beta}^{2}-4\,\delta\,
\alpha}}{\alpha}} \right) ^{i}- \left( -1/2\,{\frac {\beta+ \sqrt{{
\beta}^{2}-4\,\delta\,\alpha}}{\alpha}} \right) ^{i} \right) \left(
\sqrt{{\beta}^{2}-4\,\delta\,\alpha} \right) ^{-1}$$</p>
<p>What exactly is the connection between the above ... | Cocopuffs | 32,943 | <p>If $f(X) = \sum_{k=0}^{\infty} F_k X^k$ as a formal power series, where $F_k$ is the $k$-th Fibonacci number, i.e. $$F_0 = 0, F_1 = 1, F_k = F_{k-1} + F_{k-2} \; (k \ge 2),$$ then $$f(X) = X + \sum_{k=0}^{\infty} F_{k+2} X^{k+2} = X + \sum_{k=0}^{\infty} F_k X^{k+2} + \sum_{k=0}^{\infty} F_k X^{k+1}$$ $$= X + X^2 f(... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| pratik | 184,865 | <p><strong>In data analysis</strong>, the eigenvectors of a covariance (or correlation matrix) are usually calculated. </p>
<hr>
<p>Eigenvectors are the set of basis functions that are the most efficient set to describe data variability. They are also the coordinate system that the covariance matrix becomes diagonal ... |
1,693,630 | <p><a href="https://i.stack.imgur.com/TVeGv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TVeGv.jpg" alt="enter image description here"></a></p>
<p>This is my attempt at finding $\frac{d^2y}{dx^2}$. Can some one point out where I'm going wrong here?</p>
| Shuri2060 | 243,059 | <p>Following your title which disagrees with the working on your paper...</p>
<p>$$x=t^2-12t$$
$$y=t^2-1$$</p>
<p>$$\frac{dx}{dt}=2t-12$$
$$\frac{dy}{dt}=2t$$</p>
<p>$$\frac{dy}{dx}=\frac{dy}{dt}\times\frac{dt}{dx}=\frac{2t}{2t-12}=\frac{t}{t-6}$$</p>
<p>But if the title is a typo then:</p>
<p>$$x=t^3-12t$$
$$y=t^... |
2,325,421 | <blockquote>
<p>If $f$ is a linear function such that $f(1, 2) = 0$ and $f(2, 3) = 1$, then what is $f(x, y)$?</p>
</blockquote>
<p>Any help is well received.</p>
| Saketh Malyala | 250,220 | <p>A linear function is of the form $f(x,y)=ax+by$.</p>
<p>We have $f(1,2)=a+2b=0$</p>
<p>We have $f(2,3)=2a+3b=1$.</p>
<p>We let $2a+4b=0$.</p>
<p>Therefore, $b=-1$, and $a=2$.</p>
<p>So $\boxed{f(x,y)=2x-y}$.</p>
<hr>
<p>Update:</p>
<p>A linear function is of the form $f(x,y)=ax+by+c$</p>
<p>We have $f(1,2)=... |
2,325,421 | <blockquote>
<p>If $f$ is a linear function such that $f(1, 2) = 0$ and $f(2, 3) = 1$, then what is $f(x, y)$?</p>
</blockquote>
<p>Any help is well received.</p>
| hamam_Abdallah | 369,188 | <p>$$f(( x,y))=f (x (1,0)+y (0,1)) $$
$$=xf ((1,0))+yf ((0,1)) $$</p>
<p>so you need find $f ((1,0)) $ and $f ((0,1)) $ using</p>
<p>$f ((1,2))=0$ and $f ((2,3))=1$</p>
<p>$$2f (1,2)-f (2,3)=f (0,1)=-1$$</p>
<p>$$f (2,3)-f (1,2)=f (1,0)+f (0,1)=1$$
$$\implies f (1,0)=2$$</p>
<p>finally</p>
<blockquote>
<p>$$f (... |
885,129 | <p>I want to speed up the convergence of a series involving rational expressions the expression is $$\sum _{x=1}^{\infty }\left( -1\right) ^{x}\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1}$$
If I have not misunderstood anything the error in the infinite sum is at most the absolute value of the last neglected term. The formula ... | Simply Beautiful Art | 272,831 | <p>This implementation of Euler's acceleration method is apparently known as <a href="https://en.wikipedia.org/wiki/Van_Wijngaarden_transformation" rel="nofollow noreferrer">Van Wijngaarden's transformation</a>.</p>
<p>Euler's acceleration method, as mentioned in the comments and in other answers, can speed up the conv... |
4,079,711 | <p>Calculate <span class="math-container">$$\iint_D (x^2+y)\mathrm dx\mathrm dy$$</span> where <span class="math-container">$D = \{(x,y)\mid -2 \le x \le 4,\ 5x-1 \le y \le 5x+3\}$</span> by definition.</p>
<hr />
<p>Plotting the set <span class="math-container">$D$</span>, we notice that it is a parallelogram. I tried... | Paul Frost | 349,785 | <p>Leinster says that the product topology is designed so that</p>
<blockquote>
<p>A function <span class="math-container">$f : A \to X \times Y$</span> is continuous if and only if the two coordinate functions <span class="math-container">$f_1 : A \to X$</span> and <span class="math-container">$f_2 : A \to Y$</span> a... |
1,472,916 | <p>I am doing it in the following way. Is it correct?</p>
<p>Set $S = \{1,\pi,\pi^2,\pi^3,...,\pi^n\}$ is LI over $\mathbb Q$</p>
<p>Suppose
$a_0\times 1 + a_1\times\pi + ... a_n\times \pi^n = 0$ where all the a_i's are not $0$.</p>
<p>Then $\pi$ is a root of $a_0 + a_1x + ... + a_nx^n = 0$
which is imposible since... | P Vanchinathan | 28,915 | <p>A much simpler argument would be to use the fact that real numbers form an uncountably infinite set, whereas rationals form a countable set. </p>
<p>Check that a vector space of countable dimension over the rationals would still be a countable set. Hence the set of real numbers as a vector space over the rationals... |
30,305 | <p>I want to call <code>Range[]</code> with its arguments depending on a condition. Say we have </p>
<pre><code>checklength = {5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}
</code></pre>
<p>I then want to call <code>Range[]</code> 13 times (the length of <code>checklength</code>) and do <code>Range[5]</code> when <code>... | bill s | 1,783 | <p>Another straightforward approach is to just build the data using a <code>Table</code>. No <code>If</code> decisions required.</p>
<pre><code>Table[Range[checklength[[i]] - 4, checklength[[i]]], {i, 1, Length[checklength]}]
</code></pre>
|
926,069 | <p>Say that two $m\times n$ matrices, where $m,n\ge 2$, are <em>related</em> if one can be obtained from the other after a finite number of steps, where at each step we add any real number to all elements of any one row or column. For example, $\left(\begin{array}{cc}
0 & 0\\0 & 0
\end{array}\right)$
and
$\left... | Venus | 146,687 | <p>Note that
$$\sum_{k=1}^\infty\frac{\sin kx}{k}=\Im\sum_{k=1}^\infty\frac{e^{ikx}}{k}=-\Im\ln\left(1-e^{ix}\right)$$
where we use Taylor series of $\ln(1-x)$. Now, we use <a href="http://en.wikipedia.org/wiki/Complex_logarithm#Definition_of_principal_value" rel="nofollow">the principal value of complex logarithm</a>.... |
1,842,989 | <p>This question might be silly, but while teaching a tutorial as a TA, I suddenly had the need to bring up a tautological statement in first order logic that involved the quantifiers $\forall\exists\forall\exists$ in this order. That is, a tautology of the form $\forall x\exists y\forall z\exists w $ $\phi(x,y,z,w)$ f... | hmakholm left over Monica | 14,366 | <p>There's always $(y=x)\land (w=z)$ of course. But this has the disadvantage that $w$ doesn't need to depend on $x$, though. The best I can think of where this is not the case would be something like</p>
<p>$$(y\ne x \lor z=x) \land (z=x \to w=y) \land (z\ne x \to w=x)$$
In a structure with three elements, $w$ will h... |
128,015 | <p>For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make $\frac{1}{x^m}$ into a distribution for $m>1.$ In the complex plane, the function $\frac{1}{z^m}$ is locally integrab... | jbc | 26,013 | <p>It is folklore that any meromorphic function on the real line can be regarded as a distribution in a natural way. One defines $x^{-n}$ to be the $n$-th derivative of the locally integrable function $\log |x|$, with appropriate coefficient. In order to treat the general case, one uses the principle of recollement d... |
1,691,981 | <p>There is a symbolic notation for the set of all eigenvalues
$$\operatorname{spec} \varphi = \lbrace \lambda \in K \mid \lambda \textrm{ is an eigenvalue} \rbrace$$
There is also a notation for the eigenspace
$$V_\lambda = \lbrace \alpha \in V \mid \varphi(\alpha) = \lambda \alpha \rbrace$$
Is there any standard nota... | Robert Israel | 8,508 | <p>I would just write </p>
<blockquote>
<p>Let $v \in V_\lambda \backslash \{0\}$ for some $\lambda \in \text{spec}\; \phi$</p>
</blockquote>
<p>This is convenient since anything useful you can say about $v$ is likely to involve the eigenvalue $\lambda$. If for some reason you're determined not to identify the par... |
394,294 | <p>I would like to know the asymptotics of the following sequences of integrals:
<span class="math-container">$$ I_n = \int _0
^{+ \infty}
e^{-t} \left ( \dfrac{t}{1 + t} \right )^n
\ dt
$$</span></p>
<p>I have tried using Laplace method ou saddle node method, but I hav... | Carlo Beenakker | 11,260 | <p>The integral
<span class="math-container">$$I_n=\int_0^\infty e^{f(n,t)}\,dt,\;\;f(n,t)=n\ln t-n\ln(1+t)-t,$$</span>
has a saddle point at <span class="math-container">$t^*$</span> where <span class="math-container">$\partial f(n,t)/\partial t=0$</span>,
<span class="math-container">$$t^\ast=-\tfrac{1}{2}+\tfrac{1}{... |
4,290,651 | <p>I understand the standard proof that there exists no surjection <span class="math-container">$f: X \to \mathcal{P}(X)$</span>, but I'm not able to tell whether it deals with the case that <span class="math-container">$X = \emptyset$</span> or whether I need to rule this out separately.</p>
<p>If I want to prove that... | Eric Wofsey | 86,856 | <p>The empty set needs no special treatment in any of these arguments. For any set <span class="math-container">$X$</span>, you can define a function <span class="math-container">$X\to\mathcal{P}(X)$</span> by <span class="math-container">$x\mapsto\{x\}$</span>. If <span class="math-container">$X$</span> is empty, th... |
3,072,995 | <p>The only thing I know with this equation is <span class="math-container">$y=\frac{x^2+1}{x+1}=x+1-\frac{2x}{x+1}$</span>.</p>
<p>Maybe it can be solved by using inequality.</p>
| Michael Rozenberg | 190,319 | <p>Does not exist. </p>
<p>Try <span class="math-container">$x\rightarrow-1^-$</span>.</p>
<p>For <span class="math-container">$x>-1$</span> by AM-GM we obtain:
<span class="math-container">$$\frac{x^2+1}{x+1}=\frac{x^2-1+2}{x+1}=x-1+\frac{2}{x+1}=$$</span>
<span class="math-container">$$=x+1+\frac{2}{x+1}-2\geq2... |
3,319,122 | <p>This is from Tao's Analysis I: </p>
<p><a href="https://i.stack.imgur.com/DYQxE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DYQxE.png" alt="enter image description here"></a></p>
<p>So far I managed to show (inductively) that these sets do exist for for every <span class="math-container">$\m... | mathsdiscussion.com | 694,428 | <p><span class="math-container">$$
f(x)=\frac{x}{e^x-1} $$</span>
Find its first derivative it is zero at x=0 only and function is not defined at and x= 0 is not the vertical asymptot ; as x approach towards 0 , y approaches towards 1 ;
f'(x)<0 for all x in its domain hence decreasing function.</p>
|
3,398,645 | <p>I have a doubt about value of <span class="math-container">$e^{z}$</span> at <span class="math-container">$\infty$</span> in one of my book they are mentioning that as <span class="math-container">$\lim_{z \to \infty} e^z \to \infty $</span></p>
<p>But in another book they are saying it doesn't exist.I am confused... | Nitin Uniyal | 246,221 | <p>Unless you are working on extended complex plane i.e. <span class="math-container">$\mathbb C\cup$</span> {<span class="math-container">$\infty$</span>}, you can use the substitution <span class="math-container">$z=\frac{1}{w}$</span> and investigate the behaviour at <span class="math-container">$w=0$</span>.</p>
<... |
4,362,221 | <p>Show that for all <span class="math-container">$z$</span>, <span class="math-container">$\overline{e^z} = e^\bar{z}$</span></p>
<p>I'm a little stuck with this one.</p>
<p>First defined the following to help solve it:
<span class="math-container">$$
z = a + bi
$$</span>
then plugging that in to the question gives
<s... | D_S | 28,556 | <p>You can also do this using the power series. Let</p>
<p><span class="math-container">$$e^z = \lim\limits_{N \to \infty} \sum\limits_{n=0}^N \frac{z^n}{n!}$$</span></p>
<p>and let <span class="math-container">$\sigma(z) = \overline{z}$</span> denote complex conjugation.
Since complex conjugation is continuous, it co... |
556,423 | <blockquote>
<p>Determine using multiplication/division of power series (and not via WolframAlpha!) the first three terms in the Maclaurin series for $y=\sec x$.</p>
</blockquote>
<p>I tried to do it for $\tan(x)$ but then got kind of stuck. For our homework we have to do it for the $\sec(x)$. It is kind of tricky. ... | Mohamed | 33,307 | <p>$\sec(x)=\frac{1}{\cos x}$. The three first terms are $1,x^2$ and $x^4$. Then we write: $$\cos x=1-\frac{x^2}2 + \frac{x^4}{24} + o(x^4)$$
Putting $$u=-\frac{x^2}2 + \frac{x^4}{24}=-\frac{x^2}{2}\left(1-\frac{x^2}{12}\right)$$ we have:
$$\sec(x)=(1+u)^{-1} =\operatorname{Tronc}_4 (1 -u +u^2-u^3+u^4)=\operatorname{... |
493,600 | <p>I am presented with the following task:</p>
<p>Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather straight-forward way, but I am worried that there's more to the task:</p>
<p>First of all, both functions is a vari... | 2'5 9'2 | 11,123 | <p>It's worth noting that $|x^4|$ and $|x|^4$ equal $x^4$, but no matter. I'll assume that any simplifications like that are off-limits throughout.</p>
<p>One way to express the derivative of $|x|$ is $\frac{|x|}{x}$. So if we applied the chain rule to $|x^4|$ we have $\frac{|x^4|}{x^4}\cdot4x^3$, which is undefined a... |
355,438 | <p>I want to find the geometric locus of point $M$ such that $|MA|^2 |MB|^2=a^2$ where $|AB|=2a$, Solving algebraic equation is not hard but I can't figure out the shape of this curve. Can anybody help?</p>
| Zoltán Kovács | 427,255 | <p>Use GeoGebra's <code>LocusEquation</code> command <a href="https://i.stack.imgur.com/G239M.png" rel="nofollow noreferrer">to create an implicit locus curve</a>. Then you can drag the free points and check how the curve is dynamically changing.</p>
|
3,836,059 | <p>The following question is a last year's Statistics exam question I tried to solve (without any luck). Any help would be grateful. Thanks in advance.</p>
<p>An Atomic Energy Agency is worried that a particular nuclear plant has leaked radio-active material. They do <span class="math-container">$5$</span> independent ... | tommik | 791,458 | <blockquote>
<p>Show it has a monotone LR</p>
</blockquote>
<p>Let's set <span class="math-container">$\theta_1 < \theta_2$</span></p>
<p>The Likelihood Ratio (LR) is the following</p>
<p><span class="math-container">$$\frac{L(\theta_1;\mathbf{x})}{L(\theta_2;\mathbf{x})}=\frac{e^{-n\theta_1}\theta_1^{\Sigma x}}{e^{... |
3,263,795 | <p>Let <span class="math-container">$A = (a_{ij})$</span> be an invertible <span class="math-container">$n\times n$</span> matrix. I wonder how to prove that <span class="math-container">$A$</span> is a product of elementary matrices. I suspect that we need to transform it into the identity matrix by using elementary r... | John Omielan | 602,049 | <p>You are correct with your idea stated in the comment, i.e., the line bisects the chord of <span class="math-container">$(0,0)$</span> to <span class="math-container">$(a,b)$</span>, except you didn't mention the line is perpendicular to this chord. This is because the equal length (i.e., the radius) lines from the c... |
739,301 | <p>Show that a Dirac delta measure on a topological space is a Radon measure.</p>
<p>Show that the sum of two Radon measures is also a Radon measure.
Please help me.</p>
| William | 68,668 | <p>you also can apply the Reisz representation theorem. $\delta(f)=f(x_0)$ is a positive linear functional, then it must be a Radon measure. Second problem is obvious.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.