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 |
|---|---|---|---|---|
1,671,933 | <p>In the question <a href="https://math.stackexchange.com/questions/1502484/the-application-of-nimbers-to-nim-strategy">on nimbers</a>, the original poster asks for the meaning of Nimber <a href="https://en.wikipedia.org/wiki/Nimber#Multiplication" rel="nofollow noreferrer">multiplication</a> in the context of imparti... | Théophile | 26,091 | <p>I don't think there's an intuitive way to understand nimber multiplication. Also note that it is distinct from repeated addition, so it isn't as simple as replacing stones with copies of piles. Multiplication is defined recursively as
$$ab = \text{mex}\left(\{a'b+ab'+a'b': a'<a, b'<b\}\right)$$
where $\text{me... |
979,656 | <p>Finding prime solutions to $100q+80 = p^3 + q^2$</p>
<p>Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?</p>
| André Nicolas | 6,312 | <p><strong>Outline:</strong> One can do it crudely. Rewrite the equation as $-p^3=q^2-100q-80$, and note that the smallest value of $q^2-100q-80$ is $-2580$. </p>
|
979,656 | <p>Finding prime solutions to $100q+80 = p^3 + q^2$</p>
<p>Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>solving your equation for $q$ we obtain
$q=50\pm\sqrt{2580-p^3}$
we get
$p\le 13$
therefore $p\in \{2,3,5,7,11,13\}$</p>
|
106,853 | <p>I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. What textbook(s) should I read to take me from the basics to practical use of the theory?</p>
| Artes | 21,946 | <p>If you are looking for practical solving of differential equations I recommend :</p>
<ul>
<li><p>Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A.Orszag.</p></li>
<li><p>Methods of Mathematical Physics R.Courant and D. Hilbert two volumes.</p></li>
</ul>
<p>The second pos... |
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | SaeidAli | 371,048 | <p>Note that the corresponding space created by a vector(non-zero) is equivalent to a line passing through zero. Now since your assumption does not involve being closed under addition you only need to consider at least two lines which pass through zero.</p>
|
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | Bernard | 202,857 | <p>Use the general definition of a <em>real</em> exponent:
$$x^{\tfrac1x}\overset{\text{def}}{=}\mathrm e^{\tfrac{\ln x}{x}}.$$
Hence
$$\Bigl(x^{\tfrac1x}\Bigr)'=\mathrm e^{\tfrac{\ln x}{x}}\cdot\frac{\cfrac1x\cdot x-\ln x}{x^2}=x^{\tfrac1x}\cdot\frac{1-\ln x}{x^2}.$$</p>
|
332,130 | <p>Here is my proof, I would appreciate it if someone could critique it for me:</p>
<p>To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, then $A⊆B$," and, If $A⊆B$, then $\mathcal{P}(A)⊆ \mathcal{P}(B)$) are true.</p>
<hr>
<p>Contrapositive of the ... | MarnixKlooster ReinstateMonica | 11,994 | <p>I know you were asking for a critique of your original proof. But since I find text-based proofs (like yours, and the one from another answer) more difficult to read than symbolic ones, let me answer by presenting the way in which I would write a proof for this.</p>
<p><em>Proof.</em> For all sets A and B,</p>
<p... |
165,888 | <p>I want to return the final value from the Table.For example something like Table[n-1] value.</p>
| J. M.'s persistent exhaustion | 50 | <p>This is slightly faster and certainly more compact than the uncompiled implementation in Henrik's answer:</p>
<pre><code>makeKMat[k1_?SquareMatrixQ, k2_?SquareMatrixQ] := Module[{m = Length[k1], n = Length[k2]},
KroneckerProduct[k1, IdentityMatrix[n, SparseArray]] +
KroneckerProduct[IdentityMatrix[m, Spars... |
3,025,375 | <p>What is<span class="math-container">$$\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)?$$</span>So it is<span class="math-container">$$\lim_{n→∞}\frac{n^3(\sqrt{n^2+\sqrt{n^4+1}})^2-(n\sqrt{2})^2}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=\lim_{n→∞}\frac{n^3(n^2+\sqrt{n^4+1}-2n^2)}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}.$$</span... | egreg | 62,967 | <p>Formally substitute <span class="math-container">$n=1/t$</span>; if the function you get has a limit for <span class="math-container">$t\to0^+$</span>, then it is the same as the limit you are looking for. So consider
<span class="math-container">$$
\lim_{t\to0^+}\frac{1}{t^3}\left(
\sqrt{\frac{1}{t^2}+\sqrt{\frac... |
2,786,610 | <p>I have to calculate $\int\operatorname{arccot}(\cot(x))\ dx.$ If I had to find the derivative it would be easy with the chain rule. How can I do this?</p>
| Will Jagy | 10,400 | <p>If you write down $(A-I)^2$ you see that any vector with the final entry nonzero is outside the kernel, so you can pick
$$
p_4 =
\left(
\begin{array}{c}
0 \\
0 \\
0 \\
1
\end{array}
\right)
$$
followed by $p_3 = (A-I) p_4 \;, \;$ $p_2 = (A-I) p_3 \;, \;$ which is automatically an eigenvector because $(A-I)p_2 = (A... |
1,633,846 | <p>I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be a polynomial with real coefficients and a positive leading coefficient.</p>
<p>Let <span class="math-container">$a \geq 0$</span>. Then,... | Aops Vol. 2 | 706,407 | <p>A more elegant proof for the upper bound (the same logic can follow with the lower) is to notice that if the quotient <span class="math-container">$q(x)$</span> has only positive coefficients all numbers greater than <span class="math-container">$a$</span> will have a positive output (and thus never cross the x-axis... |
3,699,640 | <p>How to show <span class="math-container">$x^2=20$</span> has no solution in <span class="math-container">$2$</span>-adic ring of integers <span class="math-container">$\mathbb{Z}_2$</span> ?</p>
<p>What is the general criterion fora solution of <span class="math-container">$x^2=a$</span> in <span class="math-contai... | Merosity | 741,168 | <p>I want to show a more elementary approach by directly grinding out algebra steps just for the fun of showing how Hensel's lemma and the ultrametric inequality can be used. Let's begin by supposing there is an <span class="math-container">$x \in \mathbb{Z}_2$</span> such that <span class="math-container">$x^2=20$</sp... |
1,944,630 | <p>whether every positive definite matrix has to be symmetric? If not, what will be the example?</p>
| Martin Argerami | 22,857 | <p>If the definition is simply $x^TAx\geq0$ for all $x $, this can fail over the reals:
$$
A=\begin{bmatrix}1&1\\0&1\end {bmatrix} $$ satisfies $x^TAx\geq0$ for all $x\in\mathbb R^2$.</p>
<p>For complex matrices, the condition $x^*Ax\geq0$ for all $x\in\mathbb C^n $ implies that $A $ is selfadjoint. </p>
|
2,558,619 | <p>Compute
$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt
$$
In closed form, where $\delta(t)$ is the <a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">Dirac Delta function </a>.</p>
<p>My attempt: </p>
<p>$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|}... | robjohn | 13,854 | <p>Note that
$$
\begin{align}
\sum_{n=0}^\infty n^2x^n
&=x^2\sum_{n=0}^\infty n(n-1)x^{n-2}+x\sum_{n=0}^\infty nx^{n-1}\\
&=\frac{2x^2}{(1-x)^3}+\frac{x}{(1-x)^2}\\[6pt]
&=\frac{x+x^2}{(1-x)^3}
\end{align}
$$
Therefore,
$$
\begin{align}
2\pi^2\sum_{n=0}^\infty n^2e^{-\pi n}
&=2\pi^2\frac{e^{-\pi}+e^{-2\... |
3,707,132 | <p>I believe my proof of this simple fact is fine, but after a few false starts, I was hoping that someone could look this over. In particular, I am interested in whether there is an alternate proof.</p>
<blockquote>
<p>For a real number <span class="math-container">$a$</span> and non-empty subset of reals <span cla... | Sebastian Monnet | 572,592 | <p>What you've done looks correct to me, but I think we can rework it more concisely using exactly the strategy that you suggest at the end of the question. Note that both suprema exist since the sets are non-empty.</p>
<p><strong>First direction:</strong> Let <span class="math-container">$\lambda \in a + B$</span>. T... |
4,183,734 | <p>I just started trigonometry, and I came across this problem. I know that this problem would be very simple with a calculator, but without one, I'm lost. How would you determine if <span class="math-container">$\sin 4$</span> is bigger than <span class="math-container">$\sin 3$</span> in negativity or positivity? How... | Lalit Tolani | 913,165 | <p>@Ritam_Dasgupta perfectly answered your question.</p>
<p>I will help you with other question i.e. which out of <span class="math-container">$\sin 3$</span> and <span class="math-container">$\sin 4$</span> is greater in magnitude</p>
<p>For this I will assume <span class="math-container">$\pi\approx3.14$</span></p>
<... |
39,184 | <p>Hi everybody,</p>
<p>Reading about Geometry Processing, I have realized that people in this area are very interested in regular vertices(degree=6) rather than irregular ones.</p>
<p>Can anybody give me reasons why this is an important property?
Suppose that we can have a mesh with all regular vertices (genus 1) bu... | sleepless in beantown | 8,676 | <p>Nima, geometry processing is used in meshes for 3-dimensional modeling of real world structures or physical items.</p>
<p>If you try to create a regularly spaced lattice in 3-dimensions such as $\mathbb{R}^3$, a starting point would be the regions of $\mathbb{Z}^3$ which are enclosed by the object in question. Tha... |
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| Michael Hoppe | 93,935 | <p>Two vectors are parallel iff the dimension of their span is less than $2$.</p>
|
3,579,739 | <p>Its also mentioned that this question is an example of <strong>Sampling without replacement</strong></p>
<p>My question is , by the general method of how I do these kind of problems , I would assume that there are two possibilities</p>
<ol>
<li><p>You chose a defective bottle and then a non defective which will h... | JMP | 210,189 | <p>In the general case, say there are <span class="math-container">$n$</span> bottles, and you choose <span class="math-container">$k$</span> of them. There is only one flawed bottle.</p>
<p>The number of ways to pick <span class="math-container">$k$</span> bottles from <span class="math-container">$n$</span> is the b... |
4,207,823 | <p>I came across the following question in a set of lecture notes:</p>
<p>If <span class="math-container">$(G, *)$</span> is a group, when, in general, is it true that there exists <span class="math-container">$X \subset \mathbb{R}^n$</span> such that <span class="math-container">$G \cong \text{Sym}(X)$</span>? Or, les... | Jacob Manaker | 330,413 | <p>The group of isometries of <span class="math-container">$\mathbb{R}^n$</span> is a <a href="https://en.wikipedia.org/wiki/Semidirect_product" rel="nofollow noreferrer">semidirect product</a> <span class="math-container">$\mathbb{R}^n\rtimes O(\mathbb{R}^n)$</span>, where <span class="math-container">$O(\mathbb{R}^n)... |
621,438 | <p>Is there any Characterization for all measurable sets in $\mathbb{R}$?
Can I say that a set is measurable if an only if it has the property of Baire? (differs from an open set by a first category set). If the answer is no, Is there an example of a measurable set that does not have the property of Baire?</p>
<p>Than... | tomasz | 30,222 | <p>A useful characterization of measurable sets is that a set is measurable if and only if it is of the form $B\triangle N$ where $B$ is Borel (even $G_\delta$) and $N$ is null (measure zero) (this is easy to show using regularity of the measure, and is actually true for any regular Borel measure, on any space).</p>
<... |
451,001 | <p>To define the function $f(x)=|[x]|$ where $|[x]|$ is the greatest integer that is less or equal to $x$, we need to prove that indeed such an integer exists. In other words,</p>
<blockquote>
<p>$$\forall x\in \mathbb{R} \;\;\exists !\, n\in\mathbb{Z}\;\;:\;\;n\leq x<n+1.$$</p>
</blockquote>
<p>My first attempt... | Hagen von Eitzen | 39,174 | <p>By symmetry with respect to the mid-perpendicular of the side the motion is parallel to, the ball moves by an even number <span class="math-container">$2k$</span> of sides (which may be more than <span class="math-container">$\frac n2$</span>, as in the case <span class="math-container">$n=3$</span>) and the (polygo... |
451,001 | <p>To define the function $f(x)=|[x]|$ where $|[x]|$ is the greatest integer that is less or equal to $x$, we need to prove that indeed such an integer exists. In other words,</p>
<blockquote>
<p>$$\forall x\in \mathbb{R} \;\;\exists !\, n\in\mathbb{Z}\;\;:\;\;n\leq x<n+1.$$</p>
</blockquote>
<p>My first attempt... | Servaes | 30,382 | <p>Let <span class="math-container">$R$</span> be a regular <span class="math-container">$n$</span>-gon, and number the midpoints of its sides <span class="math-container">$0$</span> through <span class="math-container">$n-1$</span>.</p>
<p>Note that if a line passing through the midpoint of a side is parallel to anot... |
199,374 | <p>Consider the two table set below</p>
<pre><code>t1= {0.44, 0.62, 0.77, 0.87, 0.93, 0.96, 0.98, 1}
t2= {0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25}
</code></pre>
<p>how can I create a plot which wil plot a rectangle with a heat map normalized to the values of the table?</p>
<p>The result should look like</p>
... | Vitaliy Kaurov | 13 | <p>Function <strong>Blend</strong> actually does a great color interpolation. For simplicity assume that all your values are always in <span class="math-container">$[0,1]$</span> interval. Choose a specific <a href="http://reference.wolfram.com/language/guide/ColorSchemes.html" rel="nofollow noreferrer">Color Scheme</a... |
3,397,606 | <p>Find <span class="math-container">$[z^N]$</span> for <span class="math-container">$\frac{1}{1-z} \left(\ln \frac{1}{1-z}\right)^2$</span>. Here <em>generalized harmonic numbers</em> should be used. </p>
<p><span class="math-container">$$H^{(2)}_n = 1^2 + \frac{1}{2}^2 + \dots + \frac{1}{N}^2.$$</span></p>
<p>For n... | Jack D'Aurizio | 44,121 | <p>From
<span class="math-container">$$-\log(1-z)=\sum_{n\geq 1}\frac{z^n}{n}$$</span>
it follows that
<span class="math-container">$$\frac{-\log(1-z)}{1-z} = \sum_{n\geq 1} H_n z^n $$</span>
and by applying <span class="math-container">$\int_{0}^{x}\ldots dx$</span> to both sides
<span class="math-container">$$ \log^... |
783 | <p>In a category I have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ n = \mathrm{id}_b$ and $n \circ m = \mathrm{id}_a$?</p>
<p>I am looking for an example in which the id-part is not ... | Dennis | 1,646 | <p>The important point here, I think, is about Hom(a,a) and Hom(b,b). nm is guaranteed to be an element of the first and mn an element of the second; to be an isomorphism, both of these maps must be the identity by definition. The definition of category requires that you have at least the identity in both endomorphism ... |
61,406 | <p>Can I say this is a bad performance from the new V10 function <code>SubsetQ</code>?</p>
<p>Here are some tests comparing it to <code>Complement[l2, l1] === {}</code></p>
<pre><code>count1[data_,list_]:=Module[{r},
r=SubsetQ[#,list]&/@data;
Counts[r]
]
count2[data_,list_]:=Module[{r},
r=Complement[li... | ubpdqn | 1,997 | <p><strong>EDIT</strong></p>
<p>After comments from RunnyKine...</p>
<p>Just another approach and timing:</p>
<pre><code>subs[u_, v_] := Length@Intersection[u, v] == Length@v
</code></pre>
<p>Performance:</p>
<pre><code> BenchmarkPlot[{count1[#, list] &, count2[#, list] &,
Tally@Map[Function[x, subs[x, l... |
1,660,231 | <blockquote>
<p>Show that the function $g(x)=\sqrt{x^2+x+2}$
is defined and is continuous on $\mathbb{R}$.</p>
</blockquote>
<p>I have tried completion of square for $$x^2+x+2=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}$$
This means that range, $r\ge 7/4$ in domain $\mathbb{R}$.</p>
<p>Cannot find any more logic in... | Community | -1 | <p>To prove that $g (x)$ is defined on $\mathbf {R}$, we need to prove that for <strong>any</strong> $x_0 \in \mathbf {R}$ the value $g (x_0)$ exists. Indeed, since $x^2 + x + 2 > 0$ for any $x \in \mathbf {R}$, the expression $$g (x) = \sqrt {x^2 + x + 2}$$ is meaningful. Additionally, $g (x)$ is always $> \sqrt... |
51,732 | <p>A <em>Perron number</em> is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a
unique positive eigenvector whose eigenvalue is a Pe... | Bill Thurston | 9,062 | <p>I've gained some new perspective on this question, based partly on comments and on Hitachi Peach's answer. Instead of editing the original question, I'll write down some more thoughts as a (partial) answer in the hopes that it will inspire someone with different expertise to say more.</p>
<p>First, after Hitachi Pe... |
553,040 | <p>The question is: A card is drawn from an ordinary pack(52 cards) and a gambler bets that either a spade or an ace is going to appear. The probability of his winning are?
I think the answer is $\frac{16}{52} = \frac{4}{13}$. Did I go "probably" go wrong somewhere?</p>
| Ahaan S. Rungta | 85,039 | <p>Nope, you're correct! (Converting this to an answer.)</p>
|
1,252,857 | <p>Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto?</p>
<p>According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in (-\infty,3)$ so all numbers in the codomain $(-\infty,3)$ wont have pre-images.</p>
<p>But many say $5x^3+7$ as an odd ... | Hagen von Eitzen | 39,174 | <p>You overlook that real cube roots (in contrast to square roots) are defined for all reals For example $\sqrt[3]{-8}=-2$ because $(-2)^3=-8$.</p>
|
1,252,857 | <p>Let $f\colon \mathbb R \to \mathbb R$ be defined by $f(x)= 5x^3+3$. Is it onto?</p>
<p>According to me, if $y=5x^3+3$, then $x = \sqrt[3]{(y-3)/5}$ is not an element of $\mathbb R$ for all $y \in (-\infty,3)$ so all numbers in the codomain $(-\infty,3)$ wont have pre-images.</p>
<p>But many say $5x^3+7$ as an odd ... | Daniel W. Farlow | 191,378 | <p>Simply looking at Wolfram|Alpha shows that $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=5x^3+3$ is not just onto but also one-to-one. Let's prove that $f$ is onto though, using your choice of $x$ to do this. </p>
<p><strong>Claim:</strong> The mapping $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=5x^3+3$ i... |
2,932,139 | <blockquote>
<p>How would I show that the line <span class="math-container">$A=[(x,y,z)=(0,t,t)\mid t\in\mathbb{R}]$</span> is parallel to the plane <span class="math-container">$5x-3y+3z=1$</span>?</p>
</blockquote>
<p>I know the normal vector would be <span class="math-container">$(5,-3,3)$</span>, but how would I... | Community | -1 | <p>The direction of <span class="math-container">$A$</span> is <span class="math-container">$(0,1,1)$</span>. Check if <span class="math-container">$(0,1,1)$</span> and <span class="math-container">$(5,-3,3)$</span> are perpendicular. Since the dot product is zero, they are. </p>
|
2,932,139 | <blockquote>
<p>How would I show that the line <span class="math-container">$A=[(x,y,z)=(0,t,t)\mid t\in\mathbb{R}]$</span> is parallel to the plane <span class="math-container">$5x-3y+3z=1$</span>?</p>
</blockquote>
<p>I know the normal vector would be <span class="math-container">$(5,-3,3)$</span>, but how would I... | irchans | 372,582 | <p>In the equation <span class="math-container">$5x-3y+3z=1$</span>, replace the <span class="math-container">$x$</span> by 0, the <span class="math-container">$y$</span> by <span class="math-container">$t$</span>, and the <span class="math-container">$z$</span> by <span class="math-container">$t$</span>. After replac... |
4,014,768 | <blockquote>
<p>Find the sationary points of the curve and their nature for the equation <span class="math-container">$y=e^x\cos x$</span> for <span class="math-container">$0\le x\le\pi/2$</span>.</p>
</blockquote>
<p>I derived it and got <span class="math-container">$e^x(-\sin x+\cos x)=0$</span>.</p>
<p><span class="... | Dr. Wolfgang Hintze | 198,592 | <p>Graphically we see immediately that <span class="math-container">$x=\frac {\pi}{4}$</span>. But how about deriving this analytically?</p>
<p>Consider <span class="math-container">$\cos(2x)$</span> and use the formula for the double argument <span class="math-container">$\cos(2x) = \cos(x)^2 - \sin(x)^2$</span></p>
<... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | guest | 13,762 | <p>I would just say you have a square in 2D, and a cube is the similar shape in 3D, then what is the next shape in 4D. Then show them the cube in cube view and cross like fold out. </p>
<p>Don't jump to Schlegel diagrams and the rotating pictures on Wiki so fast...they are confusing. Do like I said instead. </p>
... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Cees Timmerman | 13,523 | <p>Use arrays (aka <a href="https://www.doitpoms.ac.uk/tlplib/tensors/what_is_tensor.php" rel="nofollow noreferrer">tensors</a> in math lingo) to store the values of a place:</p>
<p>0 dimensions / zero rank (or order) tensor:</p>
<pre><code>a = 1
v = a
</code></pre>
<p>These simple values are also known as <em>scalars<... |
1,328,082 | <p>I've been trying to prove the following inequality, but until now I've had problems coming up with a solution:</p>
<p>$$
2^{mn} \ge m^n
$$</p>
<p>$m$ and $n$ can assume any natural number.</p>
<p>I wasn't able to find any counterexample that would invalidate this inequality, so I am assuming that this statement i... | Somabha Mukherjee | 21,617 | <p>Look at the function $f(x) = 2^x-x$. </p>
<p>$f'(x)=(2^x)(\log 2)- 1 > 0$ for $x \geq 1$. So $f$ is increasing. Now, $f(1) = 1$. Hence, $f(m) > 0$ for every natural number $m$. </p>
|
2,466,913 | <p>I am having trouble with a question on my homework and I have done some work but I'm not sure if I have completed all that is asked or how to take it further if needed.</p>
<p>Problem: </p>
<p>Let $\theta = \arccos(5x^3)$. Find $\frac{d\theta}{dx}$ in two ways:</p>
<p>(a) Use implicit differentiation starting fro... | quasi | 400,434 | <p>Since whenever a piece is cut into $7$ pieces, you gain $6$ pieces ($7$ new pieces minus the replaced old one), it follows that at any stage, the number of pieces remains the same mod $6$.
<p>
Since at the start, you have $7$ pieces, it follows that at any stage, the number of pieces is congruent to $1$ mod $6$.
<p>... |
3,742,707 | <p>How many different fractions can be made up of the numbers 3, 5, 7, 11, 13, 17 so that each fraction contains 2 different numbers? How many of them will be the proper fractions?</p>
| Mathematician | 804,713 | <p>The numbers of fraction which have two different numbers according to @Benjamin wang = 6C2×2!=30. Now you can find the solution of next question. To be proper fraction I can't take the value 2!
So, the answer of next question=6C2=15</p>
|
14,746 | <p>Is there a case to be made that the topic of line integrals should only involve vector fields?</p>
<p>My colleagues and our textbook take the position that line integrals should only be taught from a vector field perspective (specifically for calculating "work"). [In fact, our textbook <em>defines</em> a line integ... | Nick C | 470 | <blockquote>
<p>Please take 12-15 minutes to <em>introduce</em> your lesson</p>
</blockquote>
<p>Your intuition is correct that this is far too much for a 12-15 minute lesson. The committee knows that there is no way you could deliver a whole lesson, even an introductory one, on this topic.</p>
<p>Since you've only got... |
2,346,921 | <blockquote>
<p>Evaluate the following integral using Laplace transform
$$\int_0^\infty\frac{e^{-ax}\sin bx}{x}\,dx$$</p>
</blockquote>
<p>I obtained this partial result
$$=\int_0^\infty \frac{1}{p+1} \frac{b}{p^2+b^2}\,dp$$
and I am stuck here. I know that the final answer is $$\arctan\frac{b}{a}.$$ I would appr... | Siddhartha | 427,792 | <p>the standard form of LT is
$$\mathcal{L}f(t)=\int_{0}^{\infty}e^{-st}f(t)\,dt$$
So for the given integral you just have to calculate LT of $\frac{\sin bx}{x}$ then put $s=a$
these two properties of LT may be useful
$$\mathcal{L} \sin (bx)=\frac{b}{b^2+s^2}$$and $$\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_{s}^{... |
2,346,921 | <blockquote>
<p>Evaluate the following integral using Laplace transform
$$\int_0^\infty\frac{e^{-ax}\sin bx}{x}\,dx$$</p>
</blockquote>
<p>I obtained this partial result
$$=\int_0^\infty \frac{1}{p+1} \frac{b}{p^2+b^2}\,dp$$
and I am stuck here. I know that the final answer is $$\arctan\frac{b}{a}.$$ I would appr... | robjohn | 13,854 | <p>This doesn't use the Laplace Transform, however, it is pretty simple.</p>
<p>$$
\begin{align}
\frac{\partial}{\partial a}\int_0^\infty\frac{e^{-ax}\sin(bx)}{x}\,\mathrm{d}x
&=-\int_0^\infty e^{-ax}\sin(bx)\,\mathrm{d}x\tag{1}\\
&=-\frac1b+\frac ab\int_0^\infty e^{-ax}\cos(bx)\,\mathrm{d}x\tag{2}\\
&=-\f... |
1,422,666 | <p>I've just run into this problem, and was able to go as far, and understand the induction step up to the bolded section. The last part I found in the back of my book, italicized, I can't understand.</p>
<p>Use induction to prove: $n^2\geq 2n + 1$ for all $n\in \mathbb{N}$ and $n\geq 3$.</p>
<p>Base: $3^2\geq 6+1$.<... | Selene Routley | 10,549 | <p>By your expansion, </p>
<p>$$\left(k^2\geq 2\,k+1\right)\;\Rightarrow\; \left((k+1)^2 \geq 4\,k+2\right)\tag{1}$$</p>
<p>since, as you have found, $(k+1)^2 = k^2 + 2\,k+1 \geq 2\,(2\,k+1)$ by the predicate on the LHS of the $\Rightarrow$ in (1).</p>
<p>But, for $k\geq 1$, we also have $4\,k+2 > 2\, k+3 = 2(k+1... |
643,237 | <p>This statement is suggested as a correction to <a href="https://math.stackexchange.com/questions/643122/splitting-field-containing-nth-root">this question</a>:</p>
<p>If $K$ is the splitting field of the polynomial $P(x)=x^n-a$ over $\mathbb{Q}$, prove that $K$ contains all the $n$th roots of unity.</p>
<p>How to ... | Robert Lewis | 67,071 | <p>Let $F$ be an <em>arbitrary</em> field, $P(x) = x^n - a \in F[x]$ with $p = \text{char} \; F \not\mid n$ and $a \ne 0$, and let $K$ be the splitting field of $P(x)$ over $F$. Then $P(x)$ splits into linear factors in $K$; that is, there exist $\beta_j \in K$, $1 \le j \le n$, with</p>
<p>$P(x) = x^n - a = \prod_1^... |
444,865 | <p>Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.</p>
| individ | 128,505 | <p>I like this approach to the solution of this equation.</p>
<p>If we consider the Diophantine equation: $qX^2+Y^2=Z^2+j$ </p>
<p>If the root is a : $a=\sqrt{\frac{j}{q}}$ </p>
<p>We use the solutions of Pell's equation: $p^2-(q+1)s^2=1$ </p>
<p>Solutions can be written: </p>
<p>$X=2s(s\pm{p})L\pm{ap^2}+2aps\pm{a... |
3,176,593 | <p>I'm having a hard time understanding how to find all solutions of the form <span class="math-container">$a_n = a^{(h)}_n+a_n^{(p)}$</span></p>
<p>I show that <span class="math-container">$a_n=n2^n \to a_n=2(n-1)2^{n-1} +2^n=2^n(n-1+1)=n2^n$</span>.</p>
<p>I can show that <span class="math-container">$a_n^{(h)}$</s... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$a_m=b_m+2^m(a_0+a_1m+a_2m^2+\cdots)$</span></p>
<p><span class="math-container">$$2^n=a_n-2a_{n-1}$$</span>
<span class="math-container">$$=ba_n-2b_{n-1}+2^n(a_0+a_1n+a_2n^2+\cdots)-2^n(a_0+a_1(n-1)+a_2(n-1)^2+\cdots)$$</span></p>
<p><span class="math-container">$$=ba_n-2b_{n-1}+2... |
1,412,177 | <p>Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?</p>
<p>I'm not sure how to do this. Any solutions or hints are greatly appreciated. I know that in order for anything to be a subspace of something the zero ... | Yes | 155,328 | <p>Since $W$ is a subspace of $\mathbb{R}^{5}$ if and only if (i) $0_{\mathbb{R}^{5}} \in W$, (ii) $x,y \in W$ implies $x+y \in W$, and (iii) $c \in \mathbb{R}$ and $x \in W$ imply $cx \in W$, and since $(1,2,0,3,4) \in W$ and $(4,0,3,2,1) \in W$ but their sum is a vector having no zero components, so $W$ is not a sub... |
1,412,177 | <p>Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?</p>
<p>I'm not sure how to do this. Any solutions or hints are greatly appreciated. I know that in order for anything to be a subspace of something the zero ... | Jason | 195,308 | <p>No. For instance, let $v=(0,1,1,1,1)$ and $w=(1,0,1,1,1)$. Then $v,w\in W$ but $v+w=(1,1,2,2,2)\notin W$.</p>
|
1,344,001 | <p>I need to solve an expression of this kind (solve for $x$):</p>
<p>$e^{\pi i x} -e^{-\pi ix} = 2yi$</p>
<p>Both $x$ and $y$ are real numbers, $y$ is given. I have no clue on how to solve it analytically.</p>
<p>All I know is that I can rewrite this as:</p>
<p>$\sin(x\pi) = y$</p>
<p>so:</p>
<p>$x=\frac{\arcsin... | Rory Daulton | 161,807 | <p>$$e^{\pi ix}-e^{-\pi ix}=y\cdot 2i$$
$$\left( e^{\pi ix}-e^{-\pi ix} \right)e^{\pi ix}=(2iy)e^{\pi ix}$$
$$\left(e^{\pi ix}\right)^2-1=2iye^{\pi ix}$$
$$\left(e^{\pi ix}\right)^2-2iy\left(e^{\pi ix}\right)-1=0$$
$$e^{\pi ix}=\frac{-(-2iy)\pm\sqrt{(-2iy)^2-4(1)(-1)}}{2(1)}$$
$$=\frac{2iy\pm\sqrt{-4y^2+4}}{2}$$
$$=iy\... |
1,481,106 | <p>I am having a difficult time solving this problem. I have tried this several different ways, and I get a different result, none of which is correct, every time. I've derived an answer geometrically and cannot replicate it with a double integral.</p>
<p>Here's the problem: Use a double integral to find the area betw... | Michael Medvinsky | 269,041 | <p><a href="https://math.stackexchange.com/questions/670029/area-between-two-circles-as-a-double-integral-in-polar-coordinates">Here</a> is a similar problem solved.</p>
<p><a href="http://www.math.utah.edu/~mmedvin/Teaching/Math1321/LectureNotes/27_1321L_TripleIntegrals.pdf" rel="nofollow noreferrer">Here</a> is an e... |
2,306,122 | <p>Show $X=\{n \in \mathbb{N}: \text{n is odd and} \ n = k(k+1) \text{for some} \ k \in \mathbb{N}\}=\emptyset$</p>
<p>My proof is as follow, please point if I have made any mistake. </p>
<p><strong>proof:</strong></p>
<p>we have $\emptyset \subseteq X$
suppose $X≠\emptyset$ pick $n \in X$</p>
<p>Then there are 2 ... | Sri-Amirthan Theivendran | 302,692 | <p>Here is a combinatorial proof that $k(k+1)$ is always even for $k\in\mathbb{N}$. The number of two element subsets of a $k+1$ element set is
$$
\binom{k+1}{2}=\frac{k(k+1)}{2}
$$
as there are $k+1$ choices for the first element of the subset and $k$ choices for the second element of the subset, but as $\{a,b\}$ and... |
2,645,406 | <p>I am reading Rudin's real and complex analysis. On page 14 we see $\underset{n} {\text{sup }} f_n$, and on page 15 we see max$\{f,g\}$. These two are obviously different. I searched some answers and found an example: </p>
<p>sup $\{f,g\} = f(x)$ wehn $f(x)\geq g(x)$ and $g(x)$ when $f(x) < g(x)$ </p>
<p>But the... | saulspatz | 235,128 | <p>"max" stands for "maximum" the greatest value in a set. "sup" stands for "supremum" or "least upper bound" of the values in a set. When the set is finite, these mean the same thing, and which one you use is a matter of state. When the set is infinite, it may not have a maximum. For example, what is $\max \{ x | ... |
2,414,721 | <p>For all perfect numbers $N$, $\sigma (N) = 2N$, where $\sigma$ is the divisor sigma function.</p>
<p>Let $s$ be a perfect number of the form $3^m 5^n 7^k$, where $m,n,k \geq 1$ are integers.</p>
<p>Then $\sigma (s)= \sigma (3^m 5^n 7^k)$</p>
<p>$ =\sigma (3^m) \sigma (5^n) \sigma (7^k)$ since $3, 5,$ and $7$ are ... | Joffan | 206,402 | <p>In order to get a perfect number, we need the ratio $\dfrac{\sigma(n)}{n}=2$, which also means that we need the prime powers in the denominator to appear in the sigma values on top. And we need exactly and only one power of $2$ to appear in that sigma value, because the denominator is odd.</p>
<p>Unrolling this int... |
1,136,486 | <p>Can anyone help me find the first digit of $2015^{2015}$?</p>
<p>It is easy to find the last digit but I have no idea for the first digit.</p>
| Luigi D. | 164,401 | <p>The units digit is of course a 5. </p>
<p>For the leading digit, see that $n^n = 10^{n\cdot log_{10}(n)} \approx 10^{2015\cdot3.304275} \approx 10^{6658.114} = 10^{6658}10^{0.114} \approx 1.3008\cdot 10^{6658}$. Therefore, the first digits of your number are 13008...</p>
<p>Check: <a href="http://www.wolframalpha.... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| lab bhattacharjee | 33,337 | <p>As $\displaystyle i^2=-1,i=-i^3\implies z^3=i=-i^3=(-i)^3\iff z^3-(-i)^3=0$</p>
<p>Now, $\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2),$</p>
<p>$\displaystyle\implies\{z-(-i)\}\{z^2+z(-i)+(-i)^2\}=\{z-(-i)\}(z^2-iz-1)$</p>
<p>If $z-(-i)=0, z=-i$</p>
<p>Else $z^2-iz-1=0\implies z=\dfrac{i\pm\sqrt{i^2-4(-1)}}2=\dfrac{i\... |
1,601,427 | <blockquote>
<p>Let $a,b,c$ be three nonnegative real numbers. Prove that $$a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$$</p>
</blockquote>
<p>It seems that the inequality $a^2+b^2+c^2 \geq ab+bc+ca$ will be of use here. If I use that then I will get $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq ab+bc+ca+3\sqrt[3]{a... | asomog | 183,714 | <p>Let $a=x^3$, $b=y^3$, $c=z^3$, then it can be rewritten as:
$$
x^6+y^6+z^6+3 x^2 y^2 z^2-2 \left(x^3 y^3+x^3 z^3+y^3 z^3\right)\geq 0
$$
Use the following notations:
$$S_{3}:=xyz\qquad S_2:=xy+yz+xz\qquad S_1=x+y+z$$
Then:
$$
x^6+y^6+z^6=S_1^6-6 S_2 S_1^4+6 S_3 S_1^3+9 S_2^2 S_1^2-12 S_2 S_3 S_1-2 S_2^3+3 S_3^2
$$
$... |
1,601,427 | <blockquote>
<p>Let $a,b,c$ be three nonnegative real numbers. Prove that $$a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$$</p>
</blockquote>
<p>It seems that the inequality $a^2+b^2+c^2 \geq ab+bc+ca$ will be of use here. If I use that then I will get $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq ab+bc+ca+3\sqrt[3]{a... | Michael Rozenberg | 190,319 | <p>Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.</p>
<p>Hence, our inequality it's $f(v^2)\geq0$, where $f(v^2)=3u^2-4v^2+w^2$.</p>
<p>Thus, $f$ is a linear function, which says that $f$ get's a minimal value for an extremal value of $v^2$,
which happens for equality case of two variables.</p>
<p>Let $b=a=x^3$ and... |
243,743 | <p>I have been studying this sequence (<a href="https://oeis.org/A266882" rel="nofollow">A266882</a> in the OEIS) and found the following pattern:</p>
<p>$13 + 17 + 19 + 23 + 29 = 101$ (101 is prime)</p>
<p>37 does not hold.</p>
<p>$223 + 227 + 229 + 233 + 239 = 1151$ (1151 is prime)</p>
<p>The same is true for 108... | aosjckajsd | 94,470 | <p>Here's a Haskell program to display the values for which the conjecture is false.</p>
<p>Within a minute, the largest value it found was the following, rather nice quintuple: $$[468883,468887,468889,468893,468899]$$</p>
<pre><code>import Math.NumberTheory.Primes
specialPrime n = ((nthPrime n) + (nthPrime (n + 3))... |
729,441 | <p>This question comes up after going over Arzela-Ascoli theorems.</p>
<p>For a set of continuous functions $\mathbb F$ from $\mathbb R$ to $\mathbb R$ that is equicontinuous. How do I show that if sup{$|f(0)|:f \in F$} $< \infty$ , then $\mathbb F$ is pointwise bounded?</p>
<p>I know I need to show that since su... | Marc van Leeuwen | 18,880 | <p>If you know that change of basis is realised by conjugating by an appropriate invertible matrix, then you can reason in terms of matrices as follows. $E_{i,j}$ is the matrix with unique nonzero entry $1$ at position $i,j$.</p>
<ul>
<li><p>The (unique) matrix $M$ of $T$ can have no nonzero off-diagonal entries: if $... |
238,809 | <p><a href="https://i.stack.imgur.com/W5ILn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W5ILn.jpg" alt="enter image description here" /></a>
How to construct a tree like <a href="https://i.stack.imgur.com/XKYl9.png" rel="nofollow noreferrer">this</a>? I was looking <code>CompleteKaryTree</code> i... | Daniel Huber | 46,318 | <p>We create the points recursively. Given the numbers of siblings in every generation by e.g. <code>ngen=ngen = {4, 3, 2, 1}</code>, in a first step we create 4 descendants. Then for every sibling we create another 3 descendants, then 2, then 1. Finally we use <code>TreePlot</code>. You may play with labels, I simply... |
238,809 | <p><a href="https://i.stack.imgur.com/W5ILn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W5ILn.jpg" alt="enter image description here" /></a>
How to construct a tree like <a href="https://i.stack.imgur.com/XKYl9.png" rel="nofollow noreferrer">this</a>? I was looking <code>CompleteKaryTree</code> i... | Szabolcs | 12 | <p>Using my package <a href="http://szhorvat.net/mathematica/IGraphM/" rel="nofollow noreferrer">IGraph/M</a>,</p>
<pre><code>Needs["IGraphM`"]
IGSymmetricTree[{4, 3, 2, 1}, GraphLayout -> "LayeredEmbedding"]
</code></pre>
<p><a href="https://i.stack.imgur.com/qyxFc.png" rel="nofollow noreferrer"... |
3,466,707 | <p>In that problem</p>
<p><span class="math-container">$$\lim\limits_{x \to \infty} \left(1-\frac{1}{x}\right)^\left(e^x\right)$$</span></p>
<p>I use <span class="math-container">$\ln$</span>, then it gave me <span class="math-container">$0\times\infty$</span> indeterminate, then I use L'Hospital Rule but I cannot re... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Let</p>
<p><span class="math-container">$$ \left(1-\dfrac{1}{x}\right)^{(e^x)}= \left[\left(1-\dfrac{1}{x}\right)^{x}\right]^{\dfrac{e^x}x}$$</span></p>
<p>and use standard limits.</p>
<p>As an alternative we can use that </p>
<p><span class="math-container">$$\left(1-\frac1x\right)... |
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | Hippalectryon | 150,347 | <p>If the sum of the rows is zero, <a href="https://math.stackexchange.com/questions/347408/prove-that-if-the-sum-of-each-row-of-a-equals-s-then-s-is-an-eigenvalue-o?rq=1">then the matrix has the eigenvalue</a> $0$. As a result its $\ker$ is of dimension $\ge1$, i.e. there is a nonzero solution to $AX=0$, hence it's no... |
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | Nitin Uniyal | 246,221 | <p><strong>Hint:</strong> For a matrix $A$ having such a property has vector $(1,1,...,1)$ in its kernel thereby giving $\dim(\ker A)>0$.</p>
|
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | user548010 | 548,010 | <p>I'm not too sure but my reasoning is that if the sum of a row is 0, then the rows of the matrix A are linearly dependent because they are a linear combination. If the rows of A are linearly dependent then the columns of A transpose is linearly dependent. Therefore matrix A is not invertible by the Invertible Matrix ... |
81,019 | <p>In my youthful naiveté (i.e. fifteen minutes ago), I was trying to prove that if $X$ was a random variable with zero mean, then $XY$ also had zero mean for any random variable $Y$. My proof used integration by parts:</p>
<p>$\int_{\Omega} x(\omega)y(\omega)f(\omega)d\omega = y(\omega) \int_{\Omega}x(\omega)f(\omeg... | Ashok | 16,921 | <p>If $X$ and $Y$ have $0$ mean then $\mathbb{E}(XY)=0$ if $X$ and $Y$ are independent; otherwise one cannot conclude anything about $\mathbb{E}(XY)$.
For example, take $X$ to be a random variable with mean $0$ and variance $1$ and $Y=X$.</p>
|
81,019 | <p>In my youthful naiveté (i.e. fifteen minutes ago), I was trying to prove that if $X$ was a random variable with zero mean, then $XY$ also had zero mean for any random variable $Y$. My proof used integration by parts:</p>
<p>$\int_{\Omega} x(\omega)y(\omega)f(\omega)d\omega = y(\omega) \int_{\Omega}x(\omega)f(\omeg... | Ilya | 5,887 | <p>The formula you used in your answer is incorrect and not only because of using integration by parts. It seems that you're mixing definitions of an expectation because sometimes one think of three definitions while there is only one. Just for the case, let us recall some theory. Maybe it seems boring - but I hope if ... |
2,053,017 | <p>The first thing I was trying to do is to substitute $a$ with something like $1 \over b$ or $1 + \frac{1}{b}$, but I got no results. Then I tried to use Squeeze theorem here, but I also got confused because, for example, $(a + 1)^{1/n}$ is not a decreasing function. </p>
| Thomas Andrews | 7,933 | <p>Let $b_n = a^{1/n}$. </p>
<p>Assume $a\neq 1$ ($a=1$ is easy.) </p>
<p>Use that $$\frac{1-b_n}{1-a} =\frac{1-b_n}{1-b_n^n}= \frac{1}{1+b_n+b_n^2+\cdots + b_n^{n-1}}$$
Use this to get a good lower and upper bound on $1-b_n$ for use with the squeeze theorem.</p>
<p>--</p>
<p>The Mean Value Theorem version is that... |
194,312 | <p>I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.</p>
| Austin Mohr | 11,245 | <p>There are many such problems (with solutions!) in the <a href="https://math.stackexchange.com/search?q=pigeonhole">Math.SE archives</a>. You can try other keywords in the search box at the top right of the page to look for more problems.</p>
|
194,312 | <p>I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.</p>
| A.Schulz | 35,875 | <p>The famous <a href="http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK" rel="nofollow">Proofs from the book</a> contains a chapter on <strong>Pigeon-hole and double counting</strong>. You can find there several cute applications of the pigeon-hole principle. </p>
|
194,312 | <p>I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.</p>
| VividD | 121,158 | <hr>
<p><em>I will divide my answer into two parts: resources from internet, and resources from this very site.</em></p>
<hr>
<h2>Some resources on the internet</h2>
<p><a href="http://www.math.lsa.umich.edu/~hderksen/ProblemSolving/PS7.pdf" rel="noreferrer">This short paper</a> contains a lot of pigeonhole princip... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | k1.M | 132,351 | <p>Suppose that $f$ is reducible, observe that $f$ is monic, hence $f$ can be written as the product of two polynomials $g$ and $h$ of degree at least one and with integer coefficients.<br>
Now observe that $f$ has no integer root, because with rational root theorem an integer root to $f$ must be a divisor of $f(0)=16$... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Adhvaitha | 228,265 | <p>Below is an explicit proof. Note that $x^4-4x^2+16 = (x^2-2)^2 + 12$, which clearly has no real root. Hence, the only possible way to reduce $x^4-4x^2+16$ over $\mathbb{Q}$ is $(x^2+ax+b)(x^2+cx+d)$. However, the roots of $x^4-4x^2+16$ are $x = \pm \sqrt{2 \pm i\sqrt{12}}$, which are all complex numbers. Since compl... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | egreg | 62,967 | <p>The associated quadratic polynomial $t^2-4t+16$ has negative discriminant, so there's no real root. Then the polynomial can be factorized over the reals as a product of degree two polynomial. You get them by a process similar to completing the square:
\begin{align}
x^4-4x^2+16
&=x^4+8x^2+16-12x^2\\
&=(x^2+4)... |
2,598 | <p>Should Wolfram Alpha Notebook questions be considered on-topic?</p>
<p>Here's an example: <a href="https://mathematica.stackexchange.com/questions/240780/calculating-double-integral-bounded-by-domain-in-wolfram-alpha-notebook">Calculating double integral bounded by domain in Wolfram Alpha Notebook</a></p>
<p>Here ar... | Szabolcs | 12 | <p>Thanks to Penelope Benenati, it came to light that Wolfram|Alpha Notebook Edition does not behave like Wolfram|Alpha. Unlike Wolfram|Alpha (which frequently interprets input differently from Mathematica), it appears to be able to understand Wolfram Language syntax.</p>
<p>Based on her screenshots, it appears to me t... |
3,366,442 | <p>I am asked to figure out when this limit exists for polynomials <span class="math-container">$f$</span> and <span class="math-container">$g$</span>, if so what the limit is, and then to prove my findings. So far I have gathered that the limit exists if and only if <span class="math-container">$\deg f\leq \deg g$</sp... | Kavi Rama Murthy | 142,385 | <p>Hint: <span class="math-container">$\frac { \sum\limits_{k=0}^{p} a_kx^{k}} {\sum\limits_{k=0}^{m} b_kx^{k}}= x^{p-m} \frac { \sum\limits_{k=0}^{p} a_kx^{k-p}} {\sum\limits_{k=0}^{m} b_kx^{k-m}}$</span> and <span class="math-container">$\frac { \sum\limits_{k=0}^{p} a_kx^{k-p}} {\sum\limits_{k=0}^{m} b_kx^{k-m}} \to... |
742,160 | <p>Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample.</p>
<ol>
<li>For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then $A ⊆ (B ∩ C)$</li>
<li>For all sets $A$ and $B$, if $|A| \le |B|$, then $A ⊆ B$</li>
</ol>
| Ellya | 135,305 | <p>The best way to remember is to only remember one, then by elimination you know the other.</p>
<p>I choose to remember injective as follows:</p>
<p>Injections cure things, and you have one injection for one cure. I.e. one to one.</p>
|
742,160 | <p>Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample.</p>
<ol>
<li>For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then $A ⊆ (B ∩ C)$</li>
<li>For all sets $A$ and $B$, if $|A| \le |B|$, then $A ⊆ B$</li>
</ol>
| Hovercouch | 36,369 | <p>The way I remember it is that when you get a flu shot your entire body doesn't turn into a giant flu virus, because the needle is smaller than your arm is. Then you can easily remember surjection as "the other one".</p>
<p>Another one is that <strong>in</strong>-jections are <strong>in</strong>-ferior and <strong>s... |
4,532,322 | <p>I was reading <em>"Ricci flow, an introduction"</em> by B. Chow and D. Knopf and I was stuck by the equation on Page 13. As below, <span class="math-container">$x,y,z>0$</span> satisfying
<span class="math-container">\begin{equation}
\begin{array}{ll}
\dfrac{\mathrm{d}x}{\mathrm{d}t} \!\!\!\!\!&=-8+... | K.defaoite | 553,081 | <p>It is shown <a href="https://math.stackexchange.com/questions/318169/determinant-of-matrix-exponential">here</a> that
<span class="math-container">$$\det(\exp\mathbf A)=\exp(\operatorname{tr}\mathbf A)$$</span>
But <span class="math-container">$\mathbf A\in \mathfrak{su}(2)\implies \operatorname{tr}\mathbf A=0$</spa... |
4,532,322 | <p>I was reading <em>"Ricci flow, an introduction"</em> by B. Chow and D. Knopf and I was stuck by the equation on Page 13. As below, <span class="math-container">$x,y,z>0$</span> satisfying
<span class="math-container">\begin{equation}
\begin{array}{ll}
\dfrac{\mathrm{d}x}{\mathrm{d}t} \!\!\!\!\!&=-8+... | orangeskid | 168,051 | <p><a href="https://en.wikipedia.org/wiki/Lie_product_formula" rel="nofollow noreferrer">Lie product formula</a> implies that if <span class="math-container">$\exp t A$</span>, <span class="math-container">$\exp t B$</span> are in a (closed) matrix group for all <span class="math-container">$t$</span>, then so is <sp... |
4,202,451 | <p>I have been reading about the Miller-Rabin primality test. So far I think I got it except the part where the accuracy is stated.<br />
E.g from <a href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Accuracy" rel="nofollow noreferrer">wiki</a></p>
<blockquote>
<p>The error made by the primality t... | egglog | 626,167 | <p>Since induction is required, consider this:
<span class="math-container">$\displaystyle\prod_{i=1}^{n+1}\frac{n+1+i}{2i-3}=\prod_{i=1}^{n}\frac{n+i}{2i-3}\Big(\frac{(2n+1)(2n+2)}{(2n-1)(n+1)}\Big) \implies$</span><br />
<span class="math-container">$2^{n+1}(2n+1) = 2^n(2n-1) \frac{(2n+1)(2n+2)}{(2n-1)(n+1)}$</span>... |
2,008,653 | <p>According to Riemann (I think) the (exact) prime counting function is given by:
<span class="math-container">$$
\pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$</span>
with <span class="math-container">$ \operatorname{R}(z) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li... | Raymond Manzoni | 21,783 | <p>I don't know a formal proof of the identity
$$\pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}$$
either but the way this identity was obtained is pretty clear and in fact "hinted" in <a href="http://www.claymath.org/publications/riemanns-1859-manuscript" rel="nofollow noreferrer">Riema... |
3,630,482 | <p>Show that, if 13 divides <span class="math-container">$n^2$</span> + <span class="math-container">$3n$</span> + <span class="math-container">$51$</span> then 169 divides <span class="math-container">$21n^2$</span> + <span class="math-container">$89n$</span> + <span class="math-container">$44$</span></p>
<p>We have ... | gt6989b | 16,192 | <p>You region consists of some linear constraints, and you would like to solve the linear optimization problem to maximize <span class="math-container">$x$</span> subject to your constraints.</p>
<p>You can use, for example, the <a href="https://en.wikipedia.org/wiki/Simplex_algorithm" rel="nofollow noreferrer">Simple... |
1,277,827 | <p>I came across a question from Munkres:
Problem 24.11: If A is a connected subspace of X, does it follow that IntA and BdA are connected? Does the converse hold?</p>
<p>Thinking about the converse, I know that in general, if Bd A is connected, it doesn't necessarily mean that A is connected (e.g. the rationals). But... | Matt Samuel | 187,867 | <p>A discrete space with more than one point is a trivial counterexample. The boundary of any subset is empty, hence connected, yet there exist disconnected closed subsets of the space (e.g. any subset with more than one point).</p>
<p>If the space is connected however, we may prove that the result is true as follows.... |
263,718 | <p>I have 2d data in the form <code>{x,y,f(x,y)}</code> which is randomly stored. the random <code>{x,y}</code> for specific shape (square for simplification here) can be created as</p>
<pre><code>Regn = {{-\[Pi], -\[Pi]}, {-\[Pi], \[Pi]}, {\[Pi], \[Pi]}, {\[Pi], -\
\[Pi]}, {-\[Pi], -\[Pi]}};
xylist=Select[RegionMembe... | Daniel Huber | 46,318 | <p>You can first calculate an interpolation of your data. As your arguments are not on a grid, only Interpolation order of 1 can be used:</p>
<pre><code>fun = Interpolation[datxy, InterpolationOrder -> 1];
</code></pre>
<p><a href="https://i.stack.imgur.com/urQCf.png" rel="nofollow noreferrer"><img src="https://i.st... |
263,718 | <p>I have 2d data in the form <code>{x,y,f(x,y)}</code> which is randomly stored. the random <code>{x,y}</code> for specific shape (square for simplification here) can be created as</p>
<pre><code>Regn = {{-\[Pi], -\[Pi]}, {-\[Pi], \[Pi]}, {\[Pi], \[Pi]}, {\[Pi], -\
\[Pi]}, {-\[Pi], -\[Pi]}};
xylist=Select[RegionMembe... | Hugh | 12,558 | <p>Here is my suggestion for interpolation. I start by making your data.</p>
<pre><code>Regn = {{-π, -π}, {-π, π}, {π, π}, {π, -\
π}, {-π, -π}};
xylist = Select[RegionMember[Polygon@(Regn)]][
Join @@ CoordinateBoundsArray[CoordinateBounds@(Regn), Into[50]]];
datxy = Table[{xylist[[i, 1]], xylist[[i, 2]],
Sin[x... |
401,967 | <p>This question is about logical complexity of sentences in third order arithmetic. See <a href="https://en.wikipedia.org/wiki/Arithmetical_hierarchy" rel="nofollow noreferrer">Wikipedia</a> for the basic concepts.</p>
<p>Recall that the Continuum Hypothesis is a <span class="math-container">$\Sigma^2_1$</span> sente... | Noah Schweber | 8,133 | <p>Here are two <em>non</em>-examples, one erring in each direction:</p>
<ul>
<li><p><strong>Too simple</strong>: "The continuum is an <span class="math-container">$\aleph$</span>-fixed point," that is <span class="math-container">$\mathfrak{c}=\aleph_\mathfrak{c}$</span>. This is equivalent of course to <spa... |
4,337,320 | <p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$F:G^n \to G$</span> with the following property: If <span class="math-container">$x_1,…,x_n,h \in G$</span>, then <span class="math-container">$F(hx_1,…,hx_n)=hF(x_1,…,x_n)$</span>. Is there a name for this type of function prop... | Thomas Andrews | 7,933 | <p>The set of such functions are in one-to-one correspondence with the set of all functions <span class="math-container">$G^{n-1}\to G.$</span></p>
<p>For example, if <span class="math-container">$H:G^{n-1}\to G$</span> we can define:</p>
<p><span class="math-container">$$F(g_1,\dots,g_n)=g_nH(g_n^{-1}g_1,g_n^{-1}g_2,\... |
440,534 | <p>Given two <em>origin symmetric convex</em> polytopes <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> (that is <span class="math-container">$P_i=-P_i$</span>) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let <span class="math-contai... | Dima Pasechnik | 11,100 | <p>An extended comment:</p>
<p>A graph-theoretic reformulation might be as follows. Can a graph <span class="math-container">$\Gamma=(V,E)$</span>,
<span class="math-container">$2n=|V|$</span>, with two different antipodal structures of class size 2, be the edge graph of a polytope?</p>
<p>Here an <em>antipodal structu... |
1,740,535 | <p>If <span class="math-container">$X\sim U(0,1)$</span> and <span class="math-container">$Y\sim U(0,X)$</span> what is the density (distribution) function <span class="math-container">$f_Y(y)$</span>?</p>
<p>I know the answer and I also found it on this site (link bellow). However, I just can't get the intuition why t... | Arnaud Mégret | 322,412 | <p>The equality :</p>
<p>$\int_\mathbb{R} f_{Y|X=x}(y)f_{X}(x)dx=\displaystyle\int_\mathbb{R}\frac{1}{X}dx$
is incorrect</p>
<p>it should be :</p>
<p>$\int_\mathbb{R} f_{Y|X=x}(y)f_{X}(x)dx=\displaystyle\int_{-\infty}^y0dx+\displaystyle\int_y^1 \frac{1}{x}dx+\displaystyle\int_1^{+\infty} 0dx$ </p>
<p>as for $x \le... |
195,759 | <p>I have three wavelengths. Each of them has a different colour. I want to plot the resulting wave with colour that corresponds to the mixture of these colour intensities at every point. I tried this code:</p>
<pre><code>Plot[I1[y]+I2[y]+I3[y],{y,-0.0000001,0.0000001},ColorFunction ->colf]
colf[y_] := Blend[{{I1[y... | Kevin Ausman | 54,593 | <p>This was harder than it should have been. But I learned some stuff along the way.</p>
<p>Let's first set up the functions:</p>
<pre><code>f[l_, y_] = 1/2 + Cos[ Pi y/(l 10^-9)]/2;
Plot[{f[450, y], f[550, y], f[650, y]}, {y, -1 10^-6, 1 10^-6},
PlotStyle -> {Blue, Yellow, Red}]
</code></pre>
<p><a href="http... |
6,825 | <p>Consider four <span class="math-container">$n$</span>-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as the diameter of the smallest circle enclosing each polygon. The polygons do not overlap. Polygons are oriented so that each one has either exactly two si... | Ross Millikan | 1,827 | <p>Yes, and your answer is correct. Following your figure, assume the polygon has a side at the top and bottom. If n is even, so the polygons have a multiple of 4 sides, there will be a pair of sides at the left and right. There will be $n/2-1$ sides between them and four sets of that bounding the central region. I... |
292,176 | <p>I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int_{a}^{b} f$ for some collection $A$ of pairs $(a,b)\in[0,1]^2$. I'm looking for pointers to any material that discusses condtions on $f$... | Dirk | 9,652 | <p>Here is how you make an inverse problem of this problem: Choose a space <span class="math-container">$X$</span> for the function <span class="math-container">$f$</span> you are looking for (e.g. <span class="math-container">$L^2(0,1)$</span> to work in Hilbert spaces, but other spaces may be more suitable, depending... |
803,687 | <p>How to prove : </p>
<blockquote>
<p>$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.</p>
</blockquote>
<p>Any suggestions ?</p>
| rschwieb | 29,335 | <p>Suggestion/hint: <a href="https://en.wikipedia.org/wiki/Hopkins%E2%80%93Levitzki_theorem" rel="nofollow">Hopkins-Levitzki</a></p>
|
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | J.G. | 56,861 | <p>There are several subdisciplines in calculus, which I'll summarise simply below. But what they have in common is the study of how one quantity changes when another changes by a small amount.</p>
<p>Differential calculus computes gradients of tangents to curves or surfaces, integral calculus computes the sizes of re... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | MattPutnam | 170,998 | <p>For people familiar with the basic concept of limits, I like to describe it as the geometric application of indeterminate forms.</p>
<p>A derivative is the slope of a function computed at every point. For a line, we can take any two points and calculate the slope as $\frac{\Delta y}{\Delta x}$. For a curve, we do... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | George Frank | 30,674 | <p>Differential calculus is not about infinitesimals or limits. if we think of a smooth function as a curve, that curve has a steepness for every value of the independent variable. (For roads the corresponding term is "grade.") For good reason the value of that steepness is taken to be the slope of the line tangent to ... |
3,856,715 | <p>Let <span class="math-container">$k$</span> be a field and <span class="math-container">$A,B$</span> be two (finite-dimensional) central simple <span class="math-container">$k$</span>-algebras. We usually say that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <em>Brauer equi... | rschwieb | 29,335 | <p>If <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are Morita equivalent, then <a href="https://en.wikipedia.org/wiki/Morita_equivalence#Criteria_for_equivalence" rel="nofollow noreferrer">there exists</a> an <span class="math-container">$n$</span> and a full idempotent <span clas... |
3,856,715 | <p>Let <span class="math-container">$k$</span> be a field and <span class="math-container">$A,B$</span> be two (finite-dimensional) central simple <span class="math-container">$k$</span>-algebras. We usually say that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <em>Brauer equi... | Qiaochu Yuan | 232 | <p>All you need to know is that <span class="math-container">$M_n(R)$</span> and <span class="math-container">$R$</span> are Morita equivalent. What this says explicitly is that the category of modules over <span class="math-container">$M_n(R)$</span> and over <span class="math-container">$R$</span> are equivalent, and... |
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