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 |
|---|---|---|---|---|
3,794,158 | <p>I am trying to prove that: Let <span class="math-container">$(M,d)$</span> an metric space and <span class="math-container">$(x_n)$</span>,<span class="math-container">$(y_n)$</span> sequences in <span class="math-container">$M$</span> such that <span class="math-container">$d(x_n,y_n) \leq \frac{1}{n}$</span> <spa... | José Carlos Santos | 446,262 | <p>Your approach would, at most, prove that <strong>if</strong> the sequence <span class="math-container">$(y_n)_{n\in\Bbb N}$</span> converges, <strong>then</strong> its limit must be <span class="math-container">$L$</span>. But it does not prove that the sequence must converge.</p>
<p>Given <span class="math-containe... |
971,617 | <p>Suppose $p,q$ are two distinct prime numbers, $q \geq 3$ and $p \not\equiv 1 \pmod q$. Then I have the following problem: Prove that there is no integer $x \in \mathbb{Z}$ such that $1+x+x^2+...+x^{q-1} \equiv 0 \pmod p$. </p>
<p>It is obvious that $x$ cannot be $0 \pmod p$, and I also found that when $p$ is even, ... | Timbuc | 118,527 | <p>Doing arithmetic modulo $\;p\;$ all the time:</p>
<p>$$0=1+x+x^2+\ldots+x^{q-1}=\frac{x^q-1}{x-1}\implies x^q=1$$</p>
<p>But we <em>also</em> have $\;x^{p-1}=1\;$ since clearly $\;x\neq 0\;$ .</p>
<p>Get now your contradiction.</p>
|
1,005,330 | <p>The theorem states (quoted from my book)</p>
<p>"If $x_0$ is any solution of a consistent linear system Ax=b, and if S = {$v_1, v_2, ..., v_k$} is a basis for the null space of A, then every solution of Ax=b can be expressed in the form</p>
<p>$x = x_0 + c_1v_1 + c_2v_2+...+c_kv_k$</p>
<p>Conversely, for all choi... | M. Vinay | 152,030 | <p>Your impression is indeed correct.</p>
<p>If $x_0$ is a solution, then $Ax_0 = b$.</p>
<p>As $S$ is a basis of the null space of $A$, any linear combination of the vectors of $S$ is also in the null space of $A$, so $A(c_1x_1 + \cdots + c_k x_k) = 0$.</p>
<p>Then, for $x = x_0 + c_1x_1 + \cdots c_k x_k$, we have:... |
1,005,330 | <p>The theorem states (quoted from my book)</p>
<p>"If $x_0$ is any solution of a consistent linear system Ax=b, and if S = {$v_1, v_2, ..., v_k$} is a basis for the null space of A, then every solution of Ax=b can be expressed in the form</p>
<p>$x = x_0 + c_1v_1 + c_2v_2+...+c_kv_k$</p>
<p>Conversely, for all choi... | Eoin | 163,691 | <p>It is saying that for any choice of scalars, and any specific solution to the equation $Ax_0=b$ we have $Ax=b$. In other words, $x$ is a solution to the equation as well, for any choice of scalars.</p>
<p>This is equivalent to saying $Ax=b$ iff $Ax_0=b$. The proof of which can be done by letting $c=c_1v_1+...+c_nv_... |
1,005,330 | <p>The theorem states (quoted from my book)</p>
<p>"If $x_0$ is any solution of a consistent linear system Ax=b, and if S = {$v_1, v_2, ..., v_k$} is a basis for the null space of A, then every solution of Ax=b can be expressed in the form</p>
<p>$x = x_0 + c_1v_1 + c_2v_2+...+c_kv_k$</p>
<p>Conversely, for all choi... | Timbuc | 118,527 | <p>Your interpretation is, I think, correct, though I can't understand what does "...for which $\;x_o\;$ is consistent" can possibly mean within this context.</p>
<p>This is one of the facts that, at least in my case, made get in love even more with mathematics while in university: if we have a non-homogeneous system ... |
192,334 | <p>I want to partition string into longest substrings that each contain only specific characters, beginning from left to right with no overlaps, always choosing the longest one possible at current position. In my example only substrings that contain only characters <code>d,f,g</code> or <code>d,e,h</code> or <code>a,b,... | Alexey Popkov | 280 | <blockquote>
<p>But <code>Alternatives</code> should be from definition something that is independent on arguments order. </p>
</blockquote>
<p><code>StringCases</code> is based on the PCRE regular expression engine for which <a href="https://mathematica.stackexchange.com/a/108399/280">it isn't true</a>: a regex eng... |
3,873,882 | <p>(Follow-up question to <a href="https://math.stackexchange.com/questions/3873041/how-to-do-proofs-by-induction-with-2-variables">How to do proofs by induction with 2 variables?</a>)</p>
<p>Suppose you want to prove that <span class="math-container">$P(x,y,z)$</span> is true for all <span class="math-container">$x,y,... | TheSilverDoe | 594,484 | <p>You should not expand the power : as you said, you want to find <span class="math-container">$k$</span> such that
<span class="math-container">$$(k-10)^3=1$$</span></p>
<p>which is equivalent to
<span class="math-container">$$k-10=1$$</span></p>
<p>so
<span class="math-container">$$k=11$$</span></p>
|
2,690,433 | <p>If $V$ is a vector space that has closure properties and satisfies the axioms and $S$ is a subset of $V$, why wouldn't $S$ always have closure under addition and scalar multiplication (which are required to show $S$ is a subspace) because since $S$ is a subset of $V$, doesn't that mean $S$ would have the same proper... | celtschk | 34,930 | <p>Take the trivial example of $V=\mathbb R$ as vector space over $\mathbb R$. Take $S$ to be any set other than $\{0\}$ and $\mathbb R$ itself. Then $S$ is <em>not</em> a subspace of $V$.</p>
<p><strong>Proof:</strong></p>
<p>If $S$ does not contain $0$, it does not contain a neutral element of addition (this especi... |
3,506,482 | <p>In my A-level textbook, it states that if there is a stationary point at <span class="math-container">$x=a$</span> and <span class="math-container">$f''(a)>0$</span> then the point is a local minimum because "the gradient is increasing from a negative value to a positive value, so the stationary point is a minimu... | twentyeightknots | 739,641 | <p>For <span class="math-container">$a$</span> to be a stationary point, <span class="math-container">$f'(a)=0$</span>. </p>
<p>The second derivative of the function represents the gradient of the gradient, and therefore can be used to find whether the <em>gradient</em> is increasing or decreasing. </p>
<p>If <span c... |
3,506,482 | <p>In my A-level textbook, it states that if there is a stationary point at <span class="math-container">$x=a$</span> and <span class="math-container">$f''(a)>0$</span> then the point is a local minimum because "the gradient is increasing from a negative value to a positive value, so the stationary point is a minimu... | Allawonder | 145,126 | <p>I think this is a question that shows you're engaging with your material. That's good.</p>
<p>The last thing you need to understand why this works is the notion of <em>continuity.</em> Forget for the moment about second derivatives and think instead about any function <span class="math-container">$g(x)$</span> cont... |
2,249,707 | <blockquote>
<p>$$\int f(x)\sin x \cos x dx = \log(f(x)){1\over 2( b^2 - a^2)}+C$$</p>
</blockquote>
<hr>
<p>On differentiating, I get,</p>
<p>$$f(x)\sin x\cos x = {f^\prime(x)\over f(x)}{1\over 2( b^2 - a^2)}$$ </p>
<p>$$\sin 2x (b^2 - a^2) = {f^\prime( x)\over (f(x))^2} $$</p>
<p>On integrating, </p>
<p>$${... | mickep | 97,236 | <p>The solutions are OK (contrary to what I first wrote). After integrating, you can check that you have
$$
\frac{1}{f}=\frac{(b^2-a^2)\cos 2x}{2}
$$
and, according to the solution in the book,
$$
\frac{1}{f}=a^2\sin^2x+b^2\cos^2x.
$$
The difference should be a constant (the integrating constant). And indeed,
$$
\frac{... |
213,872 | <p>I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that</p>
<blockquote>
<p>The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$. </p>
</blockquote>
<p>Also... | Lord_Farin | 43,351 | <p>The fundamentally nice properties of half-open intervals are that:</p>
<ul>
<li>They are closed under arbitrary intersections</li>
<li>For two half-open intervals $I_1, I_2$, their difference $I_1 \setminus I_2$ is a union of half-open intervals (a trivial union for $\Bbb R$, but not so in $\Bbb R^n$, in general)</... |
3,041,656 | <p>I need some help in a proof:
Prove that for any integer <span class="math-container">$n>6$</span> can be written as a sum of two co-prime integers <span class="math-container">$a,b$</span> s.t. <span class="math-container">$\gcd(a,b)=1$</span>.</p>
<p>I tried to go around with "Dirichlet's theorem on arithmetic ... | RandomMathDude | 626,706 | <p>Just to provide an answer synthesized out of the comments already posted, your best (read as easiest) approach to this kind of problem is to toy around with general patterns until something either clicks and you can write a clever proof or until you accidentally exhaust all possible cases.</p>
<p>In this particular... |
428,841 | <p>Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$</p>
<p>a) Show that $x_{n} < x_{n+1}$</p>
<p>b) Show that $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$</p>
<p>Hint : Square $x_{n+1}$ and factor a 2 out of the square root</p>
<p>c) Hence Show that $x_{n}$ is bounded above by 2. Deduce that $\lim\limits_{n... | Hagen von Eitzen | 39,174 | <p>a) Note that $0<u<v$ implies $0<\sqrt u<\sqrt v$. This allows you to show the claim by starting from $0<n<n+\sqrt {n+1}$ and walking your way to the outer $\sqrt{}$.</p>
<p>b) Follow the hint</p>
<p>c) By induction: $0<x_1<2$ and $0<x_n<2$ implies $1+\sqrt 2 x_n<1+2\sqrt 2<4$</p... |
3,985,447 | <p>Which of these is a possible solution for
<span class="math-container">$$\cos^2(x)+\sin^2(x)-1=0$$</span>
in the interval <span class="math-container">$x\in[0,2\pi]$</span></p>
<p>a. <span class="math-container">$x=\frac{\pi}{4}$</span><br />
b. <span class="math-container">$x=\pi$</span><br />
c. <span class="math-... | Aryan | 866,404 | <p>All of the above.<br />
The equation you wrote is basically a n identity and is true for all <span class="math-container">$x\in[0,2\pi]$</span></p>
|
1,832,177 | <p><em>(see edits below with attempts made in the meanwhile after posting the question)</em></p>
<h1>Problem</h1>
<p>I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a variable <span class="math-container">$x \in [0,1]$</span>, a function <span class="math-contain... | user10333 | 458,535 | <p>Look for this substitution (kx-x)/(2kx-k-1) x from 0 to 1 k from -1 to 1
Then you can scale the argument if you will need</p>
<p><a href="https://dinodini.wordpress.com/2010/04/05/normalized-tunable-sigmoid-functions/" rel="nofollow noreferrer">https://dinodini.wordpress.com/2010/04/05/normalized-tunable-sigmoid-fu... |
4,198,263 | <p><a href="https://www.cliffsnotes.com/study-guides/algebra/linear-algebra/real-euclidean-vector-spaces/projection-onto-a-subspace" rel="nofollow noreferrer">https://www.cliffsnotes.com/study-guides/algebra/linear-algebra/real-euclidean-vector-spaces/projection-onto-a-subspace</a></p>
<p>I am following this example he... | Gerry Myerson | 8,269 | <p>When it says, "projections ... onto the individual basis vectors," that's a little bit sloppy; what it actually means is, projections onto <em>the subspaces generated by</em> the individual basis vectors. One projects onto a subspace, not onto a vector.</p>
<p>Now, you ask about subspaces that don't have a... |
642,631 | <p>What is $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$?</p>
<p>On the one hand, we have $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}(i,\sqrt{2})]\cdot[\mathbb{Q}(i,\sqrt{2}):\mathbb{Q}(i)]\cdot[\mathbb{Q}(i):\mathbb{Q}]=2^3=8.$</p>
<p>On the other hand, the minimum polynomial in $\mathbb{Q}[x]$ containing $i,\sqrt... | egreg | 62,967 | <p>You're correctly applying the degree theorem, but the equality
$$
[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}(i,\sqrt{2})]=2
$$
is not obvious. I'd use a different chain of subfields, recalling that $\mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}(\sqrt{2}+\sqrt{3})$, which has degree $4$ over $\mathbb{Q}$.</p>
<p>It is i... |
3,130,939 | <p>Suppose the following function with pi notation, with the pi denoting the iterated product, multiplying from <span class="math-container">$i = 0$</span> to <span class="math-container">$i = n$</span>:</p>
<p><span class="math-container">$$\prod_{i=0}^n \ln(y_i^{x - 1})$$</span></p>
<p>That is, the natural logarith... | Tom Chen | 117,529 | <p>We don't even need a product rule. We have
<span class="math-container">\begin{align*}
g(x) = \prod_{i=0}^{n}\ln(y_i^{x-1}) = (x-1)^{n+1}\prod_{i=1}^{n}\ln(y_i)
\end{align*}</span>
And so
<span class="math-container">\begin{align*}
\frac{d}{dx} g(x) = (n+1)(x-1)^{n}\prod_{i=0}^{n}\ln(y_i)
\end{align*}</span></p>
|
203,456 | <p>Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.</p>
| Bill Dubuque | 242 | <p>${\bf Hint}\quad\begin{array}{cccccc} &\rm x^{\,I} &\rm C\quad\\
& \ \nearrow & \\
\rm A\!\!\!\! & & \downarrow \rm x^{\,J} \\
& \nwarrow & \\
&\rm x^K &\rm B\quad\ \
\end{array}\rm\ \Rightarrow\ \ IJK\, =\, 1$</p>
|
627,444 | <p>I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, but they seem to be way to general notions as to admit a really interesting theory.</p>
<p>Thus, I'm wondering whet... | Pavel Čoupek | 82,867 | <p><a href="http://en.wikipedia.org/wiki/Racks_and_quandles">Quandles</a> arise (allegedly) quite naturally in knot theory. They are also connected to group theory, since the conjugation operation on a group gives rise to a quandle.</p>
|
500,632 | <p>Find all such lines that are tangent to the following curves:</p>
<p>$$y=x^2$$ and $$y=-x^2+2x-2$$</p>
<p>I have been pounding my head against the wall on this. I used the derivatives and assumed that their derivatives must be equal at those tangent point but could not figure out the equations. An explanation will... | Rocco Dalto | 559,298 | <p><img src="https://i.stack.imgur.com/oBQQX.png" alt="enter image description here"></p>
<p>Let $f(x) = x^2$ and $g(x) = -x^2 + 2x - 2 \implies \dfrac{d}{dx}(f(x))|_{x = a} = 2a$ and $\dfrac{d}{dx}(g(x))|_{x = b} = -2b + 2$</p>
<p>$2a = -2b + 2 \implies a = 1 - b \implies$</p>
<p>$A: (1 - b, (1 - b)^2), B: (b, -b... |
105,190 | <p>Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/conjectured about the next term?</p>
<p>On a related note, the BSD conjecture predicts the value of the first non-vanishi... | Andreas Holmstrom | 349 | <p>There is a related conjecture for the L-function attached to certain Galois representations, due to Colmez and stated in the beautiful survey paper <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf" rel="nofollow">Periods</a> by Kontsevich and Zagier (see section 3.6). This survey paper refers to an Annals paper ... |
237,708 | <p>Does the series </p>
<p>$$\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$$</p>
<p>converge?</p>
| André Nicolas | 6,312 | <p>Warning, ugly calculation ahead. </p>
<p>We compare with the harmonic series. We will use L'Hospital's Rule, so change the variable to $x$. We look at the behaviour of
$$\frac{1-(\log x)^{-1/(x+1)}}{\frac{1}{x\log x}}$$
for large $x$.</p>
<p>The derivative of the bottom is $-\dfrac{1+\log x}{x^2\log^2 x}$.</p>
<p... |
292,651 | <blockquote>
<p>Does an integer $9<n<100$ exist such that the last 2 digits of $n^2$ is $n$? If yes, how to find them? If no, prove it.</p>
</blockquote>
<p>This problem puzzled me for a day, but I'm not making much progress. Please help. Thanks.</p>
| Math Gems | 75,092 | <p>More generally, instead of $4,25,$ let $\rm\,p, q\,$ be coprime prime powers. By $\rm\,n,n\!-\!1\,$ coprime</p>
<p>$$\rm pq\,|\,n(n\!-\!1)\ \Rightarrow\ p\,|\,n\ \ or\ \ p\,|\,n\!-\!1\ \ \ and\ \ \ q\,|\,n\ \ or\ \ q\,|\,n\!-\!1$$</p>
<p>This yields $4$ possibilities. Write $\rm\: n \equiv (a,b)\,\ (mod\ p,q)\:$ ... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| Carsten S | 90,962 | <p>When you write $i=\sqrt{-1}$ then this is something that is sometimes useful and sensible, but really has to be done with care. All that really says is that $i^2=-1$, and of course $(-i)^2=-1$ holds as well. So correctly your calculation only yields
$$(i^3)^2=i^2\cdot i^2\cdot i^2=(-1)(-1)(-1)=-1=i^2,$$
which is tru... |
455,230 | <p>I found this proposition and don't see exactly as to why it is true and even more so, why the converse is false:</p>
<p>Proposition 1. The equivalence between the proposition $z \in D$ and the proposition $(\exists x \in D)x = z$ is provable from the definitory equations of the existential quantifier and of the equ... | Hill | 88,216 | <p>The simple modules for $F[x]$ are $F[x]/(p)$ where $p=p(x)$ is irreducible. These do not occur as submodules of $F[x]$.</p>
|
3,242,921 | <p>Prove that the equation<span class="math-container">$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$</span>
does not admit <span class="math-container">$$x=-2$$</span> as a triple root.</p>
| Wuestenfux | 417,848 | <p>You mean the <span class="math-container">${\Bbb F}_2$</span>-vector space <span class="math-container">${\Bbb F}_2^n$</span> of dimension <span class="math-container">$n$</span>? Then each <span class="math-container">${\Bbb F}_2$</span>-basis of <span class="math-container">${\Bbb F}_2^n$</span> has <span class="m... |
60,259 | <p>The independence of theorems in some propositional calculus systems seems well studied. For example, if we just have the rules of detachment, substitution, and replacement, and every theorem of this axiom set {((p->q)->((q->r)->(p->r))), ((~p->p)->p), (p->(~p->q))}=<strong>X</strong> as our system <strong>X'</stron... | Carl Mummert | 630 | <p>We could say that $A$ is independent of a set of axioms $B$ if $A$ is not an admissible rule over $B$, or if $A$ is not a derivable rule over $B$. For definitions, see <a href="http://en.wikipedia.org/wiki/Rule_of_inference#Admissibility_and_derivability" rel="nofollow">the Wikipedia article</a> on rules of inferenc... |
2,621,871 | <p>My lecture notes say that</p>
<blockquote>
<p>A topological space is an ordered pair <span class="math-container">$(X, \tau)$</span>, where <span class="math-container">$X$</span> is a set and <span class="math-container">$\tau$</span> is a collection of subsets of <span class="math-container">$X$</span> that satisf... | Kaj Hansen | 138,538 | <p>The definition you've highlighted in your box is the more general definition. You can show that the open sets in a metric space satisfy the criteria for the general definition of "open set" (e.g. it isn't too hard to see that the union of open sets in a metric space is again an open set since unions will preserve t... |
2,103,706 | <p>I tried to prove this by induction.</p>
<p>Base case $n=1$, $5$ vertices. I just drew a pentagon, which has $5$ vertices of degree $2$</p>
<p>Then I assume for $n=k$,$4k+1$ vertices, there is at least one vertex with degree $2n$. The number of edges for this graph is $\dfrac{(4k+1)(4k)}{4}=(4k+1)(k)$</p>
<p>Then ... | bof | 111,012 | <p>Let $G$ be a self-complementary graph of order $4n+1$ and let $\sigma:G\to\overline G$ be an isomorphism. Since $G$ and $\overline G$ have the same vertex set $V,$ $\sigma$ is a permutation of $V,$ an anti-automorphism of $G,$ mapping edges of $G$ to edges of $\overline G$ and vice versa.</p>
<p>Consider the cycle ... |
2,856,373 | <blockquote>
<p>If <span class="math-container">$z_{1},z_{2}$</span> are two complex numbers and <span class="math-container">$c>0.$</span> Then prove that</p>
<p><span class="math-container">$\displaystyle |z_{1}+z_{2}|^2\leq (1+c)|z_{1}|^2+\bigg(1+\frac{1}{c}\bigg)|z_{2}|^2$</span></p>
</blockquote>
<p>Try: put <s... | Batominovski | 72,152 | <p>By the AM-GM Inequality,
$$c|z_1|^2+\frac{1}{c}|z_2|^2\geq 2|z_1||z_2|\,.$$
Thus, $$(1+c)|z_1|^2+\left(1+\frac{1}{c}\right)|z_2|^2\geq \big(|z_1|+|z_2|\big)^2\geq |z_1+z_2|^2\,,$$
where the last inequality follows from the Triangle Inequality. Note that the inequality $$(1+c)|z_1|^2+\left(1+\frac{1}{c}\right)|z_2|^... |
3,481,661 | <p>Let <span class="math-container">$X$</span>,<span class="math-container">$Y$</span> be Banach spaces. And <span class="math-container">$T:X\rightarrow Y$</span> a continuous operator when <span class="math-container">$X$</span> is endowed with the weak topology and <span class="math-container">$Y$</span> with the no... | Kavi Rama Murthy | 142,385 | <p>By continuity at <span class="math-container">$0$</span> there exist <span class="math-container">$n,w_1,w_2,...,w_n$</span> and <span class="math-container">$r_1,r_2,...,r_n$</span> such that <span class="math-container">$|w_i(x)| <r_i, 1\leq i \leq n$</span> implies <span class="math-container">$\|Tx\|<1$</s... |
2,872,492 | <p>My work starts with a supposition of $N$, so that for $n > N$ we have $\vert b \vert ^n < \epsilon$.</p>
<p>Since $0 < \vert b \vert < 1$, we see the logarithm with base $\vert b \vert$ is a decrescent function meaning it will invert the inequality once taken.
$$\vert b \vert ^n < \epsilon $$
$$n >... | janmarqz | 74,166 | <p>If $0<|b|<1$ then $|b|=\frac{1}{1+A}$ for some $A>0$.
But $(1+A)^n\ge 1+nA$ for each $n\in \Bbb N$, so you have $$|b|^n=\frac{1}{(1+A)^n}\le\frac{1}{1+nA}$$
Now taking $n$ large enough you are going to reach $\frac{1}{1+nA}<\varepsilon$, for each $\varepsilon>0$. Hence $|b|^n$ too.</p>
|
1,957,084 | <p>Where $n$ is any positive integer.</p>
<p>I'm honestly completely at a loss at how to prove this.<br>
Tested by brute forcing it up to large numbers, and it keeps increasing, although very slowly.</p>
<p>This is actually part of a bigger problem containing harmonic numbers, but I've solved the rest, so that's why ... | carmichael561 | 314,708 | <p>Since $i\leq 2^{n+1}$ it follows that $\frac{1}{i}\geq \frac{1}{2^{n+1}}$. There are $2^n$ terms in the sum, hence
$$ \sum_{i=2^n+1}^{2^{n+1}}\frac{1}{i}\geq \frac{2^n}{2^{n+1}}=\frac{1}{2}$$</p>
|
113,725 | <p>Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.</p>
| Gerry Myerson | 8,269 | <p>You won't find a closed form, but you will find many references at <a href="http://oeis.org/A000178" rel="nofollow">http://oeis.org/A000178</a>.</p>
|
43,226 | <p>I realize the probability of the following two events are equal. I am curious: is there a reason, besides coincidence, that the probabilities are equal?</p>
<p>Suppose there are five balls in a bucket. 3 of the balls are labelled A, and 2 of the balls are labelled B. There is no way to distinguish between balls lab... | André Nicolas | 6,312 | <p>I will describe a problem that has the same flavour as yours, since it may throw some additional light on your observation.</p>
<p>A standard deck of cards is thoroughly shuffled. Someone lifts up the top $5$ cards, <em>without looking at them</em>, and looks at the sixth card. What is the probability that this s... |
3,428,995 | <p>I found this inequality on twitter and I can't seem to prove the statement.</p>
<p>Prove that for <span class="math-container">$a,b,c > 0$</span> that </p>
<p><span class="math-container">$$
\frac{a+b+c}{2} \geq \frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c}
$$</span></p>
<p>After an hour (and a crick in my ... | user | 505,767 | <p>We have that by <a href="https://artofproblemsolving.com/wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality" rel="nofollow noreferrer">HM-AM inequality</a></p>
<p><span class="math-container">$$\frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c}=\frac{1}{\frac1a+\frac1b} + \frac{1}... |
4,380,992 | <p>I'm trying to do the following exercise:</p>
<p><em>Find a non-homogeneous recurrence relation for the sequence whose general term is</em></p>
<p><span class="math-container">$$a_n = \frac{1}{2}3^n - \frac{2}{5} 7^n$$</span></p>
<p>From this expression we can obtain the roots of the characteristic polynomial <span c... | martini | 15,379 | <p>Just start with your favorite term containing <span class="math-container">$a_n$</span> and <span class="math-container">$a_{n-1}$</span>, say
<span class="math-container">$a_n - a_{n-1}$</span>, calculate the difference, here
<span class="math-container">$$ a_n - a_{n-1} = \frac 12 3^n - \frac 25 7^n - \frac 12 3^{... |
3,153,306 | <p>In other words, say I am looking for multiple X</p>
<p>let: </p>
<p>X < 1000005</p>
<p>let the fist 18 divisors of X be:
1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 25 | 32 | 40 | 50 | 64 | 80 | 100 | 125 | 160 | 200 </p>
<p>finally, I also know: X has exactly 49 divisors. </p>
<p>I will tell you what the answer is.... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\sum_{q=3}^p\frac{q^2-3}{q}
&=\int_{3^-}^{p^+}\frac{x^2-3}{x}\,\mathrm{d}\pi(x)\\
&=\left[\frac{p^2-3}{p}\pi(p)-2\right]-\int_{3^-}^{p^+}\pi(x)\left(1+\frac3{x^2}\right)\mathrm{d}x\\[3pt]
&=\frac{p^2}{2\log(p)}+\frac{p^2}{4\log(p)^2}+\frac{p^2}{4\log(p)^3}+O\... |
16,831 | <p>As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presu... | Andrés E. Caicedo | 462 | <p>(I am turning this into Community wiki, since the original version made an obvious mistake). </p>
<p>The result follows, for example, from the <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" rel="noreferrer">Stone-Weierstrass theorem</a>, once one justifies that the limit of some integrals ... |
2,463,421 | <p>The question is:</p>
<p>Nadir Airways offers three types of tickets on their Boston-New York flights. First-class tickets are \$140, second-class tickets are \$110, and stand-by tickets are \$78. If 69 passengers pay a total of $6548 for their tickets on a particular flight, how many of each type of ticket
were sol... | N. F. Taussig | 173,070 | <blockquote>
<p>How many $4$-character strings can be formed using the sixteen hexadecimal digits?</p>
</blockquote>
<p>Your answer $16^4$ is correct?</p>
<blockquote>
<p>How many $4$-digit numbers can be formed using the sixteen hexadecimal digits?</p>
</blockquote>
<p>Your answer $16^4 - 16^3$ is correct.</p>
... |
239,863 | <p>Im trying to reproduce the following graph:
<a href="https://i.stack.imgur.com/tHQX3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tHQX3.png" alt="enter image description here" /></a></p>
<p>I have the parabolas graphed, but I want to indicate the intercepts with the x axis by some point or cros... | kglr | 125 | <p>We can use a <em>function</em> as the option setting for <code>MeshStyle</code> to add labels to mesh points without having to post-process:</p>
<pre><code>Plot[{(2 x + 2) (2 x - 1), (2 x + 1) (2 x - 2)}, {x, -2, 2},
Mesh -> {{0}}, MeshFunctions -> {#2 &},
MeshStyle -> ({Directive[White, AbsolutePoi... |
8,107 | <p>Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on resear... | Tom Au | 12,506 | <p>I know something about your "business" so I'll make a minor correction, and then use a standard ROI analysis.</p>
<p>Your "widgets" will still cost 15 dollars to make if you make your research breakthrough. But they will be "superwidgets" that are three times more productive than the old ones, and therefore worth 4... |
97,062 | <p><strong>Bug introduced in 9.0 or earlier and fixed in 10.4.0</strong></p>
<hr>
<p>Why does this work?</p>
<pre><code>Solve[5 Tan[t] + 9 == 0 && 0 <= t < 2 Pi , t]
{{t -> π - ArcTan[9/5]}, {t -> 2 π - ArcTan[9/5]}}
</code></pre>
<p>But this doesn't.</p>
<pre><code>NSolve[5 Tan[t] + 9 == 0 &a... | ilian | 145 | <p>This bug has been fixed in Mathematica <a href="http://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn104.html" rel="nofollow">10.4.0</a>.</p>
<pre><code>NSolve[5 Tan[t] + 9 == 0 && 0 <= t < 2 Pi, t]
(* {{t -> 2.07789}, {t -> 5.21949}} *)
</code></pre>
|
3,339,780 | <p>The popular definition of a vector is </p>
<blockquote>
<p>A vector is an object that has both a magnitude and <strong>a</strong> direction.</p>
</blockquote>
<p>We know that zero vector has no specific <strong>single</strong> direction.</p>
<p>Then how can it be a vector?</p>
| Allawonder | 145,126 | <p>Yes, it should have <strong>a</strong> direction, but I see no place in the quoted definition where it's stated that it should have <strong>only a</strong> direction.</p>
<p>The idea is simple. This indefiniteness in direction happens only for the trivial (zero) vector. It is a vector that does nothing. And a point... |
3,012,558 | <p>Let <span class="math-container">$C \subseteq \mathbb{R}^n$</span> be a closed convex set, and <span class="math-container">$x^* \in C^c$</span> (not in <span class="math-container">$C$</span> and its closure). </p>
<p>Define the Euclidean distance from <span class="math-container">$x^*$</span> to <span class="math... | Caldera | 618,911 | <p>for <span class="math-container">$\forall z \in D$</span> and <span class="math-container">$y\in C$</span>, we have
<span class="math-container">\begin{align}
d_D(x^*) &= min_{z \in D} \|z-x^*\|_2 \\
&= min_{y\in C} \|(z_{best} -y)+ (y-x^*)\|_2 \\
&\leq min_{y \in C}\|z_{best}-y\|_2 + min_{y \in C}\|y-x... |
2,323,845 | <p>I would like to create a 4 on 4 tournament with 8 players (4 players on a team where two teams play against each other each game), where every player plays with every other player an equal number of times. A simple example of this would be if you had a 2 on 2 tournament with 4 players then:</p>
<p>12 v 34</p>
<p>... | Théophile | 26,091 | <p>You are looking for a <em>Balanced Incomplete Block Design</em>, or BIBD. A $(v,k,\lambda)$-design puts $v$ players into groups of $k$ at a time, and any two players will play in exactly $\lambda$ groups.</p>
<p>In your case, you want an $(8,4,\lambda)$-design, and it happens that an $(8,4,3)$-design exists. This e... |
315,004 | <p>For a knot <span class="math-container">$K$</span>, let <span class="math-container">$\Sigma_K$</span> be the double cyclic branched cover of a knot. </p>
<p>By the classical work of <strong>Casson</strong> and <strong>Gordon</strong>, we know that if <span class="math-container">$K$</span> is smoothly slice, then ... | Danny Ruberman | 3,460 | <p>Here's a particularly subtle counterexample, from the work of Kirk and Livingston (Topology Vol. 38, No. 3, pp. 663--671, 1999). They show that the pretzel knots <span class="math-container">$J = P(-3,5,7,2)$</span> and <span class="math-container">$K = P(5,-3,7,2)$</span> are not concordant (even locally flat). The... |
425,663 | <p>Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that
$$ \forall x \in X, \|x\|_1 \leq K\|x\|_2 .$$
show then that these two norms are equivalent</p>
| Chris Eagle | 5,203 | <p>This is a variant of the <a href="http://en.wikipedia.org/wiki/Open_mapping_theorem_%28functional_analysis%29">open mapping theorem</a>. If we consider the identity map $i$ on $X$ as a linear mapping from $(X, \|\cdot\|_2)$ to $(X, \|\cdot\|_1)$, then your condition says that $i$ is continuous. Then by the theorem $... |
1,000,705 | <p>I have been trying to solve this problem for hours. </p>
<p>$\dfrac{9e^{2x}}{8x+3}$</p>
<p>I know $u'(x)$ will be $18e^{2x}$
and $v'(x)$ will be $8$</p>
<p>Written out, it will be $\dfrac{(8x+3)(18e^{2x})-(9e^{2x})(8)}{(8x+3)^2}$</p>
<p>I get to the part above^^ and I'm not sure what to do. I know it's probably... | Przemysław Scherwentke | 72,361 | <p>The determinant is equal (e.g. from the Rule of Sarrus) to $(0-4-4)-(0+4+4k)=4(-3-k)$. It is equal to 0 for $k=-3$ and then $A$ is not invertible.</p>
|
649,570 | <p>How do we show that there is only one solution to,$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+x}}}}$$</p>
<p>I guess it is only $x=2$.
Please help.</p>
| Igor Rivin | 109,865 | <p>Hint: raise both sides to the sixth power.</p>
|
3,206,730 | <blockquote>
<p>Let <span class="math-container">$f : (-1,1)\to (-\pi/2,\pi/2)$</span> be the function defined by <span class="math-container">$f(x)= \tan^{-1}\left(\frac{2x}{1-x^2}\right)$</span> the verify that <span class="math-container">$f$</span> is bijective</p>
</blockquote>
<p>To check objectivity I assumed... | Fred | 380,717 | <p>I guess that <span class="math-container">$f(x)= \arctan(\frac{2x}{1-x^2}).$</span> I am right ?</p>
<p>Now, if <span class="math-container">$x,y \in (-1,1)$</span> and <span class="math-container">$f(x)=f(y)$</span>, then we have to show that <span class="math-container">$x=y.$</span> We get <span class="math-cont... |
18,879 | <p>A first-order sentence is (logically) valid iff it's true in every interpretation. And it's valid iff it can be deduced from the FO axioms alone.</p>
<p>One normal case of showing that a FO sentence is true is deducing it (syntactically).</p>
<p>I guess that indirect proofs have to be interpreted more "semanticall... | Isaac | 72 | <p><a href="https://math.stackexchange.com/questions/15865/why-not-write-the-solutions-of-a-cubic-this-way/18873#18873">In my answer here</a>, I give a cubic formula that works for complex coefficients and works with the way principal roots are defined on non-real complex numbers in the vast majority of calculators and... |
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Szabolcs | 12 | <p>Everyone will have their own preferences about coding style. This is especially true for Mathematica, as most work done in this language is interactive, and until recently there was relatively little open collaboration between people that could have led to the development of standards. The existence of this site (... |
72,669 | <p>I encountered this site today <a href="https://code.google.com/p/google-styleguide/">https://code.google.com/p/google-styleguide/</a> regarding the programming style in some languages. What would be best programming practices in Mathematica, for small and large projects ?</p>
| Chris Degnen | 363 | <p>I found Roman Maeder' package template useful, as a general setup guide. (More comprehensive than any setup I actually use.) From <em>Programming in Mathematica 3rd Ed</em>. (1996), page 290.</p>
<p><a href="https://i.stack.imgur.com/y1PFk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/y1PFk.p... |
1,441,603 | <p>Solve the PDE for $u(x,y)$ $$\frac{\partial^2 u}{\partial x \, \partial y} = 0$$
I was thinking to integrated both sides in respect to $x$ first to get $$x= c(x)$$ then i will have $$c(x)-x=0$$ then i will integrate in respect to y but i think this wrong because it does not making any sense to me. </p>
| Chappers | 221,811 | <p>The functions $a(x,y)$ for which
$$ \frac{\partial}{\partial x} a(x,y) = 0 $$
everywhere are precisely those that are constant in $x$, i.e. functions of $y$ alone. If we set $a = \partial u/\partial y$, the original equation becomes after one integration
$$ \frac{\partial u}{\partial y} = A(y), $$
for $A$ an arbitra... |
1,441,603 | <p>Solve the PDE for $u(x,y)$ $$\frac{\partial^2 u}{\partial x \, \partial y} = 0$$
I was thinking to integrated both sides in respect to $x$ first to get $$x= c(x)$$ then i will have $$c(x)-x=0$$ then i will integrate in respect to y but i think this wrong because it does not making any sense to me. </p>
| coffeemath | 30,316 | <p>After one integration w.r.t. $x$ you have $u_y=k(y),$ and then integrating that w.r.t. $y$ arrive at $u=\int(k(y))+h(x).$ So it looks like you can say $u(x,y)=f(x)+g(y)$ for two single variable functions $f,g.$</p>
|
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| Jasser | 170,011 | <p><strong>HINT</strong> :</p>
<p>It is the summation of $\sum \frac {n(n+1)}2$ from 1 to n</p>
<p>which is equal to $\sum (\frac {n^2}2 + \frac n2)$ from 1 to n</p>
|
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| A. Thomas Yerger | 112,357 | <p>I thought about this problem differently than others so far. The problem is asking you to essentially sum up a bunch of sums. So by observation, it appears that $1$ appears $n$ times, $2$ appears $n-1$ times, $3, n-2$ times and so on, with only $1$ $n$ term. So instead, let's add up a sum from $1$ to $n$ which does ... |
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| hello | 185,595 | <p>Sums we know:<br>
$\sum^n_{i=1} i = 1+2+\cdots+n=\frac{n^2+n}{2}$<br>
$\sum^n_{i=1} i^2 = 1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}6$</p>
<p>Your sum is $$(1+2+3+ \cdots + n) + (1 + 2 + \cdots + (n-1)) + (1 + 2 + \cdots + (n-2)) + \cdots + (1)$$ $$= \sum^n_{k=1} \sum^k_{i=1} i$$ $$= \sum^n_{k=1} \frac{k^2+k}{2}... |
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| Akiva Weinberger | 166,353 | <p><img src="https://i.stack.imgur.com/wYvp3.png" alt="Pascal Triangle"></p>
<p>Sorry for the horrible resolution. In any case: That's Pascal's triangle. The blue is the triangular numbers. The red is the sum of the blue (can you see why?)</p>
<p>Now you can use the formula for the elements of Pascal's triangle: The ... |
47,603 | <p>Is it possible to express the functions $S(x)=x+1$ and $Pd(x)=x\dot{-}1$ in terms of the functions $f_1$, $f_2$, $f_3$ and $f_4$, where $f_1(x)=0$ if $x$ is even or $1$ if $x$ is odd, $f_2(x)=\mbox{quot}(x,2)$, $f_3(x)=2x$ and $f_4(x)=2x+1$? For example, $S(x)=f_4(f_2(x))$ if x is even. Is there a similar formula if... | Michael Renardy | 12,120 | <p>It depends what "express in terms of" means. Are the following allowed?</p>
<p>$$S(x)=x+f_1(f_4(x)),$$
$$Pd(x)=x-f_1(f_4(x)).$$
Or perhaps something like:
$$S(x)=f_2(f_4(x)+f_1(f_4(x))).$$</p>
|
144,864 | <p>This is my homework question:
Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$</p>
<p>For solving the problem, I need to calculate fourth derivative of $x^2\ln(x)$. It is $-2/x^2$ and it's maximum value will be $\infty$ between $(0,1)$ and I can't calculate $h... | Jonas Meyer | 1,424 | <p>I will expand on copper.hat's comment. Let $f(x)=x^2\ln(x)$ on $(0,1]$, and $f(0)=0$. Note that $f$ is continuous on $[0,1]$. The first derivative of $f$ is $f'(x)=x+2x\ln(x)$. The only critical point in $(0,1)$ is at $x=1/\sqrt{e}$, and $f$ is decreasing on the interval $[0,1/\sqrt{e}]$. Therefore if $0<c<... |
3,275,892 | <p>Im having problem graphing the following inequality:
<span class="math-container">$(x+y) \div (x-y) \ge 0$</span>.
I know what the graph looks like, but I can't grasp the thought process behind solving the problem.</p>
<p>Thanks in advance!</p>
| Kavi Rama Murthy | 142,385 | <p>First consider points with <span class="math-container">$x-y>0$</span>. Here we need <span class="math-container">$x+y>0$</span> and this translates into <span class="math-container">$-x <y<x$</span>. These are points between the lines <span class="math-container">$y=x$</span> and <span class="math-conta... |
3,256,767 | <p>So I'm trying to understand a solution made by my teacher for a question that asks me to determine whether the following is true. I'm having trouble understanding where some values in the steps are coming from.</p>
<p>Like for the first part, I don't really get where n≥5 came from. My guess is getting 16n^2 + 25 to... | Michele De Pascalis | 63,903 | <p>Let's pretend for a moment that points two and three require that:</p>
<ul>
<li><span class="math-container">$\phi(e_1) = w_1$</span></li>
<li><span class="math-container">$\phi(e_2) = w_2$</span></li>
</ul>
<p>Where <span class="math-container">$e_i$</span> are the elements of the canonical basis for <span class=... |
2,673,465 | <p>Suppose that $n \in \mathbb{N}$ is composite and has a prime factor $q$. If $k \in \mathbb{Z}$ is the greatest number for which $q^k$ divides $n$, how can I show that $q^k$ does not divide ${{n}\choose{q}}$?</p>
<p>Clearly, since
$$
{{n}\choose{q}} = \frac{n!}{(n-q)!q!} = \frac{n(n-1)(n-2) \dots (n-q+1)}{q!}
$$
So ... | Davood | 477,916 | <p>If $q$ divides $n$, then one can conclude that: </p>
<blockquote>
<p>$n$ and $n-q$ are consecutive multiples of $q$, </p>
</blockquote>
<p>so there is no other multiple between $n-1$ and $n-q+1$, so none of these numbers are divisible by $q$. As you have mentioned none of these elements
$$
\left\{ n-1, n-2, \dot... |
1,081,717 | <p>I have a vector valued mapping $F:\mathbb{R}^2\rightarrow\mathbb{R^2}$, I'm wondering whether there's a sufficient condition for it to be a contraction mapping. </p>
<p>For example, if $F$ is $:\mathbb{R}\rightarrow\mathbb{R}$, and $F\in C^1$, then a sufficient condition is $F'(\cdot)<1$ in all its domain. So fo... | Robert Israel | 8,508 | <p>This is rather fishy. Convolution corresponds via Fourier transform to pointwise multiplication. You can multiply a tempered distribution by a test function
and get a tempered distribution, but in general you can't multiply two tempered distributions and get a tempered distribution. See e.g. the discussion in
Ree... |
2,048,282 | <p>I have been doing derivatives but I can't wrap my head around this question for whatever reason. Would appreciate anyone help.
$$g(x) = \tan(x)/e^x$$</p>
| MPW | 113,214 | <p><strong>Hint:</strong> Do you know the quotient rule for finding $\left(\frac{f(x)}{h(x)}\right)'$?</p>
<p>Even better, if you write $g(x) = e^{-x}\tan x$, do you know the product rule for finding $\left(f(x)\cdot h(x)\right)'$?</p>
<p>If you don't know how to find $(\tan x)'$, remember that $\tan x = \sin x / \co... |
3,066,530 | <p><span class="math-container">$$\lim_{x\to 0} \frac {(\sin(2x)-2\sin(x))^4}{(3+\cos(2x)-4\cos(x))^3}$$</span> </p>
<p>without L'Hôpital.</p>
<p>I've tried using equivalences with <span class="math-container">${(\sin(2x)-2\sin(x))^4}$</span> and arrived at <span class="math-container">$-x^{12}$</span> but I don't kn... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Your quotient can be simplified to <span class="math-container">$$8\cos\left(\frac{x}{2}\right)^4$$</span></p>
|
2,479,290 | <p>Came across this question in my textbook:</p>
<p>$f(x) = (1+2x)^{10}$. Determine $f^{(5)}(0)$ using the binomial theorem.</p>
<p>If I am correct, the author of the book want me not to use the power rule. How else do I compute this? </p>
| Daniel Schepler | 337,888 | <p>The argument is off a bit: in fact, $[\forall k \in \mathbb{N}, k < 0 \rightarrow P(k)]$ is vacuously true. This is because for any $k \in \mathbb{N}$, $k < 0$ is false, so the implication $k < 0 \rightarrow P(k)$ is true. So, if you've proven the required statement $\forall n \in \mathbb{N}, [ \forall k ... |
2,957,611 | <p>Let <span class="math-container">$A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$</span>, series of real numbers that are zero from some point forward.</p>
<p>Let <span class="math-container">$X$</span> be <span class="math-container">... | Rusio | 603,359 | <p>If <span class="math-container">$x$</span> is your random variable, then <span class="math-container">$x^2$</span> is just a 'transformation' of that random variable. </p>
<p>Remember that the expectation operator <span class="math-container">$\text{E}[ \cdot]$</span>, when applied to a random variable <span class=... |
262,003 | <p>Is there a positive integer $N$, besides 1 and 2, such that there is a permutation $a_1=1,a_2,a_3,\dots,a_N$ of $1,2,3,\dots,N$ in which for each $k>1$, $a_k=a_{k-1}\div k,a_k=a_{k-1}-k,a_k=a_{k-1}+k,\textrm{or }a_k=a_{k-1}\times k$?</p>
| N. Virgo | 46,551 | <p>This is implicit in Gerhard Paseman's answer but it should be made explicit: what you ask for is not possible, for a fairly simple reason. Consider $a_N$. There are four possibilities:
$$
(1)\qquad a_N = a_{N-1}/N
$$
but this can't be satisfied, because $a_{N-1}\le N$, so either $a_{N-1}$ doesn't divide $N$ or $a_{N... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Vitaliy Kaurov | 13 | <p>Use <code>ColorFunction</code> along a single dimension for gradient and a smart analytic curve for boundary. You can easily control type of color gradient via <code>ColorFunction</code>.</p>
<pre><code>RegionPlot[.7 x^8 + 80 y^8 < .3, {x, -2, 2}, {y, -2, 2},
Frame -> False, Axes -> False,
ColorFunctio... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Jens | 245 | <p><strong>Edit</strong></p>
<p>One can use either an image-based (hence rasterized) or a vector-based (resolution-independent) approach to get the rounded corners. I'll first discuss the vector based solution, and then add a raster-based solution. Although Mr. Wizard already posted a raster-based approach, I think it... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | David Elm | 44,449 | <p>The equation for Vitaliy Kaurov's solution isn't too confusing. It's just a generalization of the equation of an ellipse.</p>
<p>The equation for an ellipse</p>
<p>$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</p>
<p>generalizes to </p>
<p>$\frac{\mid{x}\mid^n}{a^n}+\frac{\mid y \mid ^n}{b^n}=1$</p>
<p>Just like an elli... |
2,391,624 | <p>This question pertains to Mosteller's classic book <em>Fifty Challenging Problems in Probability</em>. Specifically, this in regards to an algebraic operation Mosteller performs in the solution to the first question, entitled "The Sock Drawer."</p>
<p>Mosteller writes:</p>
<blockquote>
<p>Then we require the pro... | Eric Towers | 123,905 | <p>In a product of two (positive) things, if I replace one of them with something bigger, I get something bigger and if I replace one of them with some thing smaller I get something smaller. So, if in
$$\frac{r}{r+b} \cdot \frac{r-1}{r+b-1}=\frac{1}{2}$$
I replace $\frac{r}{r+b}$ with something smaller, say $\frac{r-... |
2,549,690 | <p>Is a direct sum of cyclic groups cyclic? I know every abelian group is a direct sum of cyclic groups of prime power orders, but I can't make use of this.</p>
| Luke Peachey | 506,520 | <p>Just use the map $x \mapsto rx$. This is clearly a continuous bijection with continuous inverse $ x \mapsto \frac{1}{r} x$.</p>
|
425,400 | <p>Consider the following integral expression:
<span class="math-container">$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$</span>
for <span class="math-container">$\epsilon>0$</span>, <span class="math-container">$f \in L^\infty(\mathbb R)$</span>, and <sp... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\ep}{\epsilon}\newcommand{\R}{\mathbb R}$</span>Yes, this is true. Indeed, for <span class="math-container">$\ep\in(0,1/2]$</span> we have
<span class="math-container">\begin{equation}
\mathcal I\le2\|f\|_\infty^2\, J, \tag{1}\label{1}
\end{equation}</span>
where
<span c... |
886,626 | <p>I want to solve the following system of congruences:</p>
<p>$ x \equiv 1 \mod 2 $</p>
<p>$ x \equiv 2 \mod 3 $</p>
<p>$ x \equiv 3 \mod 4 $</p>
<p>$ x \equiv 4 \mod 5 $</p>
<p>$ x \equiv 5 \mod 6 $</p>
<p>$ x \equiv 0 \mod 7 $</p>
<p>I know, but do not understand why, that the first two congruences are redund... | Darth Geek | 163,930 | <p>Note that:</p>
<p>$$x\equiv 3 \mod 4 \Rightarrow x= 4k+3\Rightarrow x\equiv 1 \mod 2\\ x\equiv 5\mod 6 \Rightarrow x = 6k'+5\Rightarrow x\equiv 2\mod 3
$$</p>
|
6,637 | <p>I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots. </p>
<p>Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show that $H^p(\mathbb{R}^n - \Sigma)$ equals $\mathbb{R}$ for $p=0,n-k-1, n-1$ and $0$ for all other $p$. Here $1... | Carsten S | 1,037 | <p>Your question has been answered by Tom. But I am also not sure if you are aware that the point of the problem was to show that the cohomology of a sphere embedded in euclidean space is independent of the embedding. You seem to think only of the standard embedding. As Tom mentioned, duality is one way to prove this... |
399,948 | <p>How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or $\csc(50)$, etc.. I tried looking for how to do it but neither my textbook or any other place has a tutorial for it. I w... | wendy.krieger | 78,024 | <p>The chords of the rational angles solve a series of equations, which one can derive from an <em>iso-series</em>, in the form $T(n+1)=X.T(n)-T(n-1)$. You then solve for the unique factor in each even number, and the chords of a $p$-gon solves this. The process can be greatly accelerated, by using a <em>bignum</em... |
2,718,495 | <p>$$\lim_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n]),$$</p>
<p>where $\xi_{\big| \Bbb{N}}$ is the counting measure on $\Bbb{N}$.</p>
<p>I am looking for $A \subset \Bbb{N}$ for which $\lim_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n])$ is not defined. So I need to find $A$ so that the l... | Stefano Rando | 546,906 | <p>First of all let's put $p_n = \frac{1}{n} \xi (A \cap [1, n])$ to sinplify notations.</p>
<p>Then we define $A$ in such a way: $1 \in A$, so $p_1 = 1$, $[2, 10] \subset A^c$ (so that $A \cap [1, 10]$ only contains $1$) and we have $p_{10} = \frac{1}{10}$, next $[11, 100] \subset A$ so we have $p_{100}> \frac{9}{... |
463,190 | <p>How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$?</p>
<p>I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?</p>
| user60887 | 60,887 | <p>Hint: Suppose gcd(a,b)=1 and let $d=gcd(a+b,a^2+b^2) \implies d|(a+b) $ and $ d|(a^2+b^2)$. Let $dr=a+b$ and $ds=a^2+b^2$ where $r,s \in\mathbb{Z}$. We see that by squaring $dr=a+b$ we get $d^2r^2=a^2+2ab+b^2$. Then $d^2r^2-d=a^2+2ab+b^2-a^2-b^2=2ab$. Thus $d(dr^2-1)=2ab\implies d|2ab$ From this we break the proof i... |
463,190 | <p>How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$?</p>
<p>I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?</p>
| lab bhattacharjee | 33,337 | <p>Let positive integer $d$ divides both $a+b,a^2+b^2$</p>
<p>$\implies d$ divides $(a^2+b^2)+(a+b)(a-b)=2a^2$ </p>
<p>Similarly, i.e., $d$ divides $(a^2+b^2)-(a+b)(a-b)=2b^2$</p>
<p>$\implies d$ divides $2a^2,2b^2\implies d$ divides $(2a^2,2b^2)=2(a^2,b^2)=2(a,b)^2$</p>
|
1,781,269 | <p>What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg--</p>
<p>What is the slope of the curve <span class="math-container">$x^3 + y^3= 3axy$</span> at origin and how to find it because after following the process of implicit differentiatio... | Hagen von Eitzen | 39,174 | <p>It's as simple of that: No (simple) curve, no derivative, no slope.</p>
|
1,220,800 | <blockquote>
<p>Calculation of x real root values from $ y(x)=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1} $</p>
</blockquote>
<p>$\bf{My\; Solution::}$ Here domain of equation is $\displaystyle x\geq 1$. So squaring both sides we get</p>
<p>$\displaystyle (x+1)+(x-1)-2\sqrt{x^2-1}=(4x-1)$.</p>
<p>$\displaystyle (1-2x)^2=4(... | Vincenzo Oliva | 170,489 | <p>Also, for $x\ge1$ $$\sqrt{4x-1}+\sqrt{x-1}\ge\sqrt{4x-1+x-1}=\sqrt{5x-2}>\sqrt{2x}\ge\sqrt{x+1}. $$</p>
|
373,357 | <p>I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my <a href="https://math.stackexchange.com/questions/372747/what-are-the-uses-of-split-complex-numbers?noredirect=1">previous question</a>). I then found out using both together, we can have trouble on the pr... | Start wearing purple | 73,025 | <p>You can build numbers generated by such $\mathbf{i},\mathbf{j},\mathbf{k}$, I see no incoherency. They will form a subalgebra of complex $2\times2$ matrices, in which the role of $\mathbf{i},\mathbf{j},\mathbf{k}$ will be played by
$$ \mathbf{i}=\left(\begin{array}{cc} 0 & i \\ i & 0\end{array}\right),\qqua... |
272,173 | <p>I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature. </p>
<p>I'd appreciate multiple suggestions that explore the topic using different approaches too.</p>
<p>I'm par... | Graham Cox | 91,324 | <p>A good recent survey is <a href="http://epubs.siam.org/doi/abs/10.1137/120880173" rel="nofollow noreferrer">Geometrical Structure of Laplacian Eigenfunctions</a> by Grebenkov and Nguyen, with lots of nice pictures and over 500 references.</p>
|
3,977,081 | <p>I’m just a high school student, so I may be somewhat logically flawed in understanding this.</p>
<p>According to wikipedia, the definition of function requires an input <span class="math-container">$x$</span> with its domain <span class="math-container">$X$</span> and an output <span class="math-container">$y$</span... | Derek Luna | 567,882 | <p>I see what you are saying, but there are ways to think about this without using more than one mapping, as the other answers have pointed out.</p>
<p>But here is what I think you are getting at (which I agree with):</p>
<p><span class="math-container">$x \mapsto f(x) \mapsto ax+b$</span>. where we have a general func... |
3,977,081 | <p>I’m just a high school student, so I may be somewhat logically flawed in understanding this.</p>
<p>According to wikipedia, the definition of function requires an input <span class="math-container">$x$</span> with its domain <span class="math-container">$X$</span> and an output <span class="math-container">$y$</span... | David K | 139,123 | <p>Equality is not just a mapping. It means two things are one and the same.</p>
<p>We wrote <span class="math-container">$f(x) = y$</span> not to indicate that we are mapping <span class="math-container">$f(x)$</span> to <span class="math-container">$y$</span>;
it means that <span class="math-container">$f(x)$</span> ... |
820,878 | <p>I am not quite familiar with the concept of correlation.
The Pearson's correlation coefficient is defined as:</p>
<p><span class="math-container">$$\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}$$</span></p>
<p>which makes use of Mean and Sta... | Henry | 6,460 | <p>Correlation makes sense in any case where the two standard deviations are finite and not zero</p>
<p>As a non-normal distribution example, if you have two independent Poisson distributed random variables, <span class="math-container">$X$</span> with mean <span class="math-container">$\lambda$</span> and <span class=... |
1,653,416 | <p>We know that:
<a href="https://www.youtube.com/watch?v=w-I6XTVZXww" rel="nofollow">https://www.youtube.com/watch?v=w-I6XTVZXww</a>
$$S=1+2+3+4+\cdots = -\frac{1}{12}$$</p>
<p>So multiplying each terms in the left hand side by $2$ gives:
$$2S =2+4+6+8+\cdots = -\frac{1}{6}$$
This is the sum of the even numbers</p>
... | Jeyekomon | 29,060 | <p>Lets try the same in a more general way: Instead of $-\frac{1}{12}$ I will use $C\in\mathbb{R}$.</p>
<p>So our infinite sum:
$$1+2+3+4+\cdots = C$$</p>
<p>Using the same technique you get these two equalities:</p>
<p>$$
2+4+6+8+\cdots = 2C
\\
1+3+5+7+\cdots = 2C
$$</p>
<p>And when you add them together:</p>
<p>... |
1,818,313 | <p>IF $a,b,c$ are distinct reals how many roots does $(x-a)^3+(y-b)^3+(z-c)^3=0$ have?</p>
<p>Clearly ,$x=a,y=b,z=c$ is a solution. But are there any possibilities?</p>
| 5xum | 112,884 | <p>Another solution is $x=a+1, y=b, z=c-1$.</p>
<p>One more is $x=a+2$, $y=b$, $z=c-2$.</p>
<p>From here, you should be able to produce many more solutions.</p>
|
1,818,313 | <p>IF $a,b,c$ are distinct reals how many roots does $(x-a)^3+(y-b)^3+(z-c)^3=0$ have?</p>
<p>Clearly ,$x=a,y=b,z=c$ is a solution. But are there any possibilities?</p>
| robjohn | 13,854 | <p>For any $x$ and $y$, there is one $z$ that satisfies the equation. Thus, there is a two-dimensional sheet of solutions in $\mathbb{R}^3$.
$$
z=c+\sqrt[\large3]{(a-x)^3+(b-y)^3}
$$</p>
|
1,206,195 | <p>I am trying to find the maximum of $x^{1/x}$. I don't know how to find the derivative of this. I have plugged in some numbers and found that $e^{1/e}$ seems to be the maximum at around 1.44466786. I don't know if this is the maximum, and I would like an explanation of why it is/what the maximum is. essentially, how ... | Cookie | 111,793 | <p>Let $y=x^{1/x}$. If you add natural log to both sides, then you have $\ln y=\ln x^{1/x}$, or more importantly, $$\ln y = \frac 1x \ln x$$
You can now differentiate implicitly both sides to get (use product rule on RHS) $$\frac 1y \frac{dy}{dx}=-\frac 1{x^2}\ln x+\frac 1{x^2}$$
Multiply both sides by $y$ to get $\fra... |
504,431 | <p>I'm the teaching assistant for a first semester calculus course, and the professor has given the students the following problem:</p>
<blockquote>
<p>Find the points on the curve $xy=\sin(x+y)$ that have a vertical tangent line.</p>
</blockquote>
<p>Here is a picture of the curve:</p>
<p><img src="https://i.stac... | Felix Marin | 85,343 | <p>Let's consider $x$ as a function of $y$. We have to find points
$\left(x, y\right)$ where $x' = 0$. That yields the equation
$x = \cos\left(x + y\right)$. Then, we have a system of two equations:
$$
\left\{%
\begin{array}{rcl}
xy & = & \sin\left(x + y\right)
\\
x & = & \cos\left(x + y\right)
\end{arr... |
1,299,127 | <p>Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.</p>
<p>Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}</p>
<p>Also $|z| = 3$ is given the ... | Nikita Evseev | 23,566 | <p>Write integral in the form the function in the form $\frac{e^{zt}}{(z-2i)(z+2i)}$. Then you should split the contour in two parts such that interior of each part contain only one point $2i$ or $-2i$.</p>
<p>Then apply Cauchy integral formula for each contour. </p>
|
96,110 | <p><span class="math-container">$A = \begin{pmatrix}
0 & 1 &1 \\
1 & 0 &1 \\
1& 1 &0
\end{pmatrix} $</span></p>
<p>The matrix <span class="math-container">$(A+I)$</span> has rank <span class="math-container">$1$</span> , so <span class="math-container">$-1$</span> is an eigenvalue with an al... | Robert S. Barnes | 9,381 | <p>This is basically the same as Patrick's answer, just worded differently. </p>
<p>First, we don't need to know what A is, other than the fact that it's $3\times 3$. </p>
<p>Assume $rank(A+I)=1$. Then $-1$ is an eigenvalue of $A$ because $(A+I)=(A-(-1)I)$. By the rank-nullity theorem we know that $rank(A)+nullit... |
136,453 | <p>For every $k\in\mathbb{N}$, let
$$
x_k=\sum_{n=1}^{\infty}\frac{1}{n^2}\left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k}.
$$
Calculate the limit $\displaystyle\lim_{k\rightarrow\infty}x_k$.</p>
| tomcuchta | 1,796 | <p>One reason is that prime numbers are the basis of <a href="http://en.wikipedia.org/wiki/RSA_%28algorithm%29">RSA cryptography</a>, which is based entirely on using large prime numbers in clever ways. Studying the primes directly can change how secure we believe the RSA cryptographic algorithm to be. Currently we bel... |
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