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 |
|---|---|---|---|---|
17,914 | <p>How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?</p>
| GA316 | 72,257 | <p>This is irrelevent here, still it is 'relevent'. The cardinality of set of all continuous function from $\mathbb{R}$ to $\mathbb{R}$ $(C(\mathbb{R},\mathbb{R}))$ is $2 ^ \mathbb{N_0} = \mathfrak{c}$ because any such function is determined by its value on rationals. hence #$(C(\mathbb{R},\mathbb{R}))$ = # $\mathbb{R}... |
17,914 | <p>How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?</p>
| Community | -1 | <p>This answer is based on, but differs slightly from, user Asaf Karaglia's above. </p>
<hr>
<p>First, observe that by definition, $\{\text{all real functions of real variable}\}:= \{f: \; f: \mathbb{R}\to\mathbb{R}\} := \mathbb{R}^\mathbb{R}$.</p>
<p>The question is about $|\{\text{all real functions of real varia... |
3,313,216 | <p>How to prove that vectors are parallel iff their unit vectors are equal?</p>
<p><span class="math-container">$$\mathbf{u} \parallel \mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}$$</span></p>
<p>A vector can be written as a scalar multiple of its magnitude and unit vector in its direction: <span class="math-c... | J.T | 528,854 | <p>If <span class="math-container">$\mathbf{u} = \lambda \mathbf{v}$</span> for <span class="math-container">$\lambda > 0$</span> then you can similarly find <span class="math-container">$||\mathbf{u}||$</span> in terms of <span class="math-container">$\lambda$</span> and <span class="math-container">$\mathbf{v}$</s... |
3,289,305 | <p>I was trying to the following theorem:</p>
<blockquote>
<p>let <span class="math-container">$G=\textrm{GL}(n,\mathbb{R})$</span> and <span class="math-container">$N=\{A\in G: \, \det(A)>0\}$</span>. Prove that <span class="math-container">$G/N \cong \mathbb{Z}_{2}$</span> .</p>
</blockquote>
<p>In the solutio... | James | 506,916 | <p>Consider the composition of homomorphisms
<span class="math-container">\begin{align}
GL(n,\mathbb{R}) \rightarrow \mathbb{R}^* \rightarrow \mathbb{Z}_2
\end{align}</span></p>
<p>where the first is the determinant, the second sending an element of <span class="math-container">$\mathbb{R^*} = \mathbb{R} \setminus 0$<... |
3,289,305 | <p>I was trying to the following theorem:</p>
<blockquote>
<p>let <span class="math-container">$G=\textrm{GL}(n,\mathbb{R})$</span> and <span class="math-container">$N=\{A\in G: \, \det(A)>0\}$</span>. Prove that <span class="math-container">$G/N \cong \mathbb{Z}_{2}$</span> .</p>
</blockquote>
<p>In the solutio... | Belgi | 21,335 | <p>Hint: Think of it that way, we identify all matrices with positive determinant so we think of them as one element (formally thay map to the same element in the quotient group). What would be the second element? For this recall that <span class="math-container">$G$</span> can't have matrices with zero determinant.
Th... |
666,806 | <p>The question I have is mostly on stability analysis but the problem is:<br><br><br>
Consider a nonlinear pendulum. Using a linearized stability analysis, show that the inverted position is unstable. What is the exponential behavior of the angle in the neighborhood of this unstable equilibrium position.<br><br></p>
... | yoknapatawpha | 108,381 | <p>The phrase "inverted position" means that the pendulum is pointing straight up. </p>
<p>Intuitively, we know that this is unstable since any perturbation away from perfectly vertical will cause the pendulum to swing downwards. To show this, you'll compute the linearization of the system when the pendulum is pointin... |
2,469,745 | <p>I have seen this question on this site so I know this is a duplicate. I do not understand all the explanations on the other questions and they are years old.</p>
<blockquote>
<p>Let $G$ and $H$ be groups. Suppose $J$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$. Show that $f(x,y)=(Jx,Ky)$ is a ... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$2+4+6+8+...+2x=2^{x(x+1)}$$ and $$\left(\frac{1}{4}\right)^{-36}=2^{72}$$ and you will have $$2^{2^{x(x+1)}}=2^{72}$$</p>
|
2,469,745 | <p>I have seen this question on this site so I know this is a duplicate. I do not understand all the explanations on the other questions and they are years old.</p>
<blockquote>
<p>Let $G$ and $H$ be groups. Suppose $J$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$. Show that $f(x,y)=(Jx,Ky)$ is a ... | user577215664 | 475,762 | <p>$( 0.25)^{-36}=( \frac 1 4) ^{-36}=(2^{-2})^{-36}=(2^{36})^2$</p>
<p>$2^22^42^6....2^{2x}=(2^12^22^32^4...2^x)^2$</p>
<p>So we must have:</p>
<p>$(2^{36})^2=(2^12^22^32^4...2^x)^2$</p>
<p>Or simply:</p>
<p>$2^{36}=2^12^22^32^4...2^x$</p>
<p>$1+2+3+ ....x=36$</p>
<p>$\frac {(x+1)x} 2=36$</p>
<p>$x^2+x=72$</p>... |
3,708,200 | <blockquote>
<p>Evaluation of <span class="math-container">$$\int_{C}xydx+(x+y)dy$$</span> aling the curve <span class="math-container">$y=x^2$</span> from <span class="math-container">$(-2,4)$</span> to <span class="math-container">$(1,1)$</span></p>
</blockquote>
<p>What i try </p>
<p>Let <span class="math-conta... | Ninad Munshi | 698,724 | <p>We could also directly plug in the functions</p>
<p><span class="math-container">$$\int_C xy\:dx + (x+y)\:dy = \int_{-2}^1 x^3\:dx + \int_4^0 -\sqrt{y}+y\:dy + \int_0^1 \sqrt{y} + y \: dy= -\frac{21}{4}$$</span></p>
<p>with no extra parametrization work necessary.</p>
|
1,212,815 | <p>What is the right algorithm for testing whether the graph is "weakly connected"?</p>
<p>The theory says:</p>
<blockquote>
<p>Oriented graph $G=(V,E)$ is weakly connected graph if and only if for every two vertices $u,v \in V$ exists a directed path from $u$ to $v$ or directed path from $v$ to $u$.</p>
</blockquo... | Jack D'Aurizio | 44,121 | <p><span class="math-container">$$\begin{eqnarray*}
F_z(z) = \mathbb{P}[X^2+Y^2\leq R^2]&=&\frac{1}{2\pi}\iint_{x^2+y^2\leq R^2}e^{-\frac{x^2+y^2}{2}}\,dx\,dy \\
&=& \frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{R}\rho e^{-\rho^2/2}\,d\rho\,d\theta = 1 - e^{-R^2/2}
\end{eqnarray*}$$</span></p>
<p>hence the dis... |
200,723 | <p>Trying to get the modulus of the five numbers immediately before a prime, added together in there factorial form; I'll call this operation $S(p)$. For example,</p>
<p>$$S(p) = ((p-1)! + (p-2)! + (p-3)! + (p-4)! + (p-5)!) \bmod p$$</p>
<p>$$S(5) = (4!+3!+2!+1!+0!) \bmod 5$$</p>
<p>$$S(5) = 4$$</p>
<p>However, I h... | Ross Millikan | 1,827 | <p>You could use <a href="http://en.wikipedia.org/wiki/Wilson%27s_theorem" rel="nofollow">Wilson's theorem</a>, $(p-1)!=-1 \pmod p$ when (and only when) $p$ is prime. This makes your expression $-1+\frac {-1}{p-2} + \ldots \pmod p$ so you just need four inverses $\pmod p$</p>
<p>Multiplicative inverses $\pmod p$ are ... |
144,644 | <p>I am a beginner in Mathematica, </p>
<p>Suppose I want to write a series of five vectors $p_1,...,p_5$ in terms of an arbitrary basis spanned by $\left\{p,n\right\}$ with each coefficient in such a decomposition a function of scalar variables. </p>
<p>e.g $$p_i = f_i p + g_i n,$$
with $f_i$ and $g_i$ labelling the... | SPPearce | 23,105 | <p>I'm not sure why it has stopped evaluating between versions, but your code could be significantly improved.</p>
<p>Evaluating <code>f[0.1]</code> gives a list containing a value, rather than just a value. If you take only the first (and only in this case) solution from <code>NDSolve</code> the issue will disappear:... |
29,619 | <p>From the book,<br>
Suppose $p \equiv 1 \pmod{4}$, then by law of quadratic reciprocity, we have:
$$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) $$
Next, if $p \equiv 2 \pmod{3}$, then $p \equiv 5 \pmod{12}$
Hence, $$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1$$</p>
<p>How do they get those Leg... | Douglas Zare | 8,345 | <p>If $p\equiv 2 \mod 3$ then $p$ is not a square mod $3$. That is the justification for the second line. </p>
<p>As you correctly calculated, if $p$ is $5 \mod 12$ (and $q$ is odd) then $(-1)^{\frac {p-1}{2} \cdot \frac {q-1}{2}}$ is $(-1)^{\text{even}} = 1$. </p>
<p>You have an incorrect statement of the main case... |
3,304,195 | <p>I have this statement:</p>
<blockquote>
<p>The luggage weight of a commercial aircraft follows normal distribution <span class="math-container">$W(Weight) \sim N(20kg,4kg)$</span>
.If the limit of The total luggage load of an aircraft carrying <span class="math-container">$100$</span>
passengers is <span clas... | Steve Kass | 60,500 | <p>What you need to find is the probability that the total weight of <span class="math-container">$100$</span> pieces of luggage, each of which independently has weight from a normal distribution with mean <span class="math-container">$20$</span> and standard deviation <span class="math-container">$4$</span>, exceeds <... |
774,043 | <p>How many natural numbers less than ${10^8}$ are there,whose sum of digits equals ${7}$?</p>
<p>My Try: I used multinomial theorem to solve it and I am getting an answer of 1716. I want to know whether I am correct or not. Please help me as I have no way other than this to check my answer. Thank you! :))</p>
| evil999man | 102,285 | <p>You seek the number of solutions of </p>
<p>$$a_!+a_2...a_8=7$$</p>
<p>We have all of them as whole numbers.(We need not worry about the trivial case $0$). Also, you need not worry about their upper limit of $9$ for obvious reasons.</p>
<p>Now define $A_i=a_i+1$</p>
<p>Now you have : </p>
<p>$$\sum A_i=7+8=15$$... |
470,427 | <p>I am trying to understand the topology on $\{0,1\}^X$, where $X$ is uncountable. The topology on $\{0,1\}$ is the discrete and I am using the product topology on $\{0,1\}^X$.
My question is, who are the basic open sets? From my understanding of the definition of product topology, basic sets should either contain fin... | Martin Sleziak | 8,297 | <p>I'll quote the beginning of the example from the document you have linked to.</p>
<blockquote>
<p>Example 3: Let $X$ be an uncountable set and let $\{0,1\}$ have the discrete topology. Consider $\mathcal P(X) = \{0,1\}^X$ with the product topology. Let $\mathcal A\subseteq \mathcal P(X)$ be the collection of all ... |
2,168,125 | <p>From an exercise list:</p>
<blockquote>
<p>Let $V$ be a inner product space over $\mathbb{C}$ and $T\in \mathcal{L}(V)$ a normal operator such that $T^2=-I$. Prove that T preserves the inner product, i.e. $\langle Tu,Tv\rangle = \langle u,v \rangle, \forall u,v\in V$.</p>
</blockquote>
<p>I found a bunch of equa... | Marc Bogaerts | 118,955 | <p>The intersection of the eigenspaces corresponding to different eigenvalues is always $\{0\}$. Indeed if $x \in E_m$, the eigenspace of $m$ and $x \in E_n$ then we have $Ax = mx$ and $Ax = nx$ so $(m-n)x = 0$ so, since $m \neq n$, $x$ must be $0$. Now let $d_i$ be the dimension of the eigenspace of eigenvalue $m_i$ ... |
2,168,125 | <p>From an exercise list:</p>
<blockquote>
<p>Let $V$ be a inner product space over $\mathbb{C}$ and $T\in \mathcal{L}(V)$ a normal operator such that $T^2=-I$. Prove that T preserves the inner product, i.e. $\langle Tu,Tv\rangle = \langle u,v \rangle, \forall u,v\in V$.</p>
</blockquote>
<p>I found a bunch of equa... | Mr. Butter | 1,093,556 | <p>Remember that, dim(U+V) = dim(U) + dim(V) - dim(U ∩ V)</p>
<p>If a matrix A has 'n' distinct Eigenvalues, then the matrix A has 'n' Linearly Independent Eigen Vectors.</p>
<p>Let the eigenspace formed by λi is Ei.</p>
<p>A vector x can not belong to two different Eigenspaces. And thus making the intersection of the ... |
2,650,913 | <p>I was looking at the solid of revolution generated by revolving $\cos(x)$ about the $x$-axis on the interval $[0, 2\pi]$, and I noticed that when the volume of the solid was approximated with $3$ or more cylinders via the disk method the approximation would equal the true volume. To prove this, I deduced that it suf... | marwalix | 441 | <p>One has</p>
<p>$$\cos^2{x}={1+\cos{2x}\over 2}$$</p>
<p>So the sum rewrites as</p>
<p>$$\sum_{i=1}^k\cos^2{2i\pi\over k}={k\over 2}+{1\over 2}\sum_{i=1}^k\cos{4i\pi\over k}$$</p>
<p>relabel the index so there is no confusion with $i=\sqrt{-1}$</p>
<p>$$\sum_{m=1}^k\cos{4m\pi\over k}=\sum_{m=1}^k\mathfrak{R}(e^{... |
2,650,913 | <p>I was looking at the solid of revolution generated by revolving $\cos(x)$ about the $x$-axis on the interval $[0, 2\pi]$, and I noticed that when the volume of the solid was approximated with $3$ or more cylinders via the disk method the approximation would equal the true volume. To prove this, I deduced that it suf... | MrYouMath | 262,304 | <p>Hint: As already pointed out by other users the given result seems to be false without further conditions. In order to determine the value of the sum, you could use Euler's Identity</p>
<p>$$\cos x = \dfrac{\exp(ix)+\exp(-ix)}{2}.$$</p>
<p>For your problem $x = \dfrac{2\pi}{k}$. Together with the finite geometric ... |
2,340,235 | <blockquote>
<p>Which way I can determine whether the polynomial $x^5 + 5x^2 +1$ is irreducible over $\mathbb Q$ or not?</p>
</blockquote>
<p>Mod $p$ Irreducibility Test and Eisenstein's criterion not applicable here.</p>
<p>Which way I should proceed?</p>
| Lukas Heger | 348,926 | <p>$7^5+5\cdot 7^2+1 = 17053$ which is a prime number. Thus the polynomial is irreducible by <a href="https://en.wikipedia.org/wiki/Cohn%27s_irreducibility_criterion" rel="nofollow noreferrer">Cohn's irreducibility criterion</a>.</p>
|
1,673,967 | <blockquote>
<p>Show algebraically that the Joukowski transformation maps the unit circle, $|z| = 1$, to the straight line segment, $-2 \le u \le 2$ and $v = 0$. </p>
</blockquote>
<p>Other information given is that $u+iv = f(x+iy)$ where $u=u(x,y)$ and $v=v(x,y)$. </p>
<p>For example if $w = z^2$, then $u+iv = (x+... | Zev Chonoles | 264 | <p>A simple counterexample is the ring $\mathbb{Z}$, and the $\mathbb{Z}$-modules $M=\mathbb{Z}/2\mathbb{Z}$ and $N=\mathbb{Z}/3\mathbb{Z}$. There is no non-trivial $\mathbb{Z}$-module homomorphism from $M$ to $N$, because there is no element of $N$ that has order $2$. (One could replace $2$ and $3$ with any relatively... |
3,152,144 | <p>Find the value of <span class="math-container">$x$</span> where <span class="math-container">$f(x)$</span> attains its minimum. (Hint: you will need the Chain Rule.)</p>
<p><span class="math-container">$$f(x) = \int_{-10}^{x^2+2x} e^{t^2}\,dt. $$</span></p>
<p>I'm a little confused by this. I thought this would be... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>Let <span class="math-container">$$\int e^{t^2}dt=F(t)+C$$</span>therefore <span class="math-container">$$f(x)=F(x^2+2x)-F(-10)$$</span>and therefore <span class="math-container">$$f'(x)={d\over dx}F(x^2+2x)$$</span>Now, applying Chain Rule leads to ...</p>
|
521,928 | <p>I would appreciate help showing $e^{D}(f(x)) = f(x+1)$</p>
<p>Where $D$ is the linear operator $D: \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ where (in the context where this statement arose) $x \in \mathbb{N}$; </p>
<p>$f(x) \mapsto \frac{d}{dx} f(x)$</p>
<p>By the Taylor series expansion $e^{D} = \sum_{n=0}^{\inf... | Michael Hardy | 11,667 | <p>Use the binomial theorem.
$$
e^D x^n = \sum_{k=0}^\infty \frac{D^k}{k!} x^n.
$$
All of the terms after $k=n$ are $0$, i.e. $D^k x^n=0$ if $k>n$, so this is
\begin{align}
\sum_{k=0}^n \frac{D^k}{k!} x^n & = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!} x^{n-k} \\[6pt]
& = \sum_{k=0}^n \binom nk x^{n-k} \\[6pt]
&... |
2,724,754 | <p>I have a set of data of size $X$, say $X = 7$. I want to find all of the unique ways that the data can be grouped into two bins of a minimum size of two. For the example where $X = 7$, I have:</p>
<pre><code> Bin A: 5 4
Bin B: 2 3
</code></pre>
<p>As the possible arrangements. What I need is what the act... | P.S | 372,604 | <p>By Lebesgue <span class="math-container">$f$</span> is continuous away from a set of measure zero . So there exists a subset <span class="math-container">$A$</span> of <span class="math-container">$[a , b]$</span> with measure <span class="math-container">$0$</span> such that the restriction of <span class="math-con... |
4,549,427 | <p>I would like to know if <span class="math-container">$\forall xx=x$</span> is an axiom in axiomatic set theory like in other first order languages, or a theorem? If it is a theorem, how to prove it?</p>
<p><strong>Update:</strong>
In the first order language materials I read,the equality is one of the logical symbol... | Bram28 | 256,001 | <p>Since it is a logical theorem, it can be proven using no assumptions whatsover. Hence, there is no need to include it in any set of axioms.</p>
<p>How to prove it? That depends on the specific rules of the proof system you are working with. Here is a formal proof in Fitch:</p>
<p><a href="https://i.stack.imgur.com/... |
3,900,569 | <p><strong>Question:</strong></p>
<p>Let <span class="math-container">$\{x_{n}\}$</span> be a recursively defined sequence defined as:
<span class="math-container">$$x_{1} = \frac{1}{2}$$</span>
<span class="math-container">$$x_{n+1} = \frac{1-x_{n}}{4}$$</span></p>
<p>I want to show first that <span class="math-contai... | TheSilverDoe | 594,484 | <p>Let <span class="math-container">$f : [0,1] \rightarrow [0,1]$</span> be the function defined for all <span class="math-container">$x \in [0,1]$</span> by <span class="math-container">$$f(x)=\frac{1-x}{4}$$</span></p>
<p>First, <span class="math-container">$f$</span> is well-defined because it is decreasing over <sp... |
100,551 | <p>I need to get a matrix $\{a(x_i-x_j)\}$, where $x_i$ form a partition of an interval, $a(x)$ is a given function. I use </p>
<pre><code>In[67]:= a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Table[-10 + 0.02 (j - 1), {j, 1, 1001}];
A = Outer[a[#1 - #2] &, x, x]; // AbsoluteTiming
Out[69]= {2.99032, ... | Jacob Akkerboom | 4,330 | <p>It appears <code>Outer</code> can be reasonably compiled to C.</p>
<pre><code>cfu = Compile[
{{x, _Real, 1}}
,
Outer[Exp[-Abs[# - #2]] &, x, x]
,
CompilationTarget -> "C"
]
</code></pre>
<p>Timings</p>
<pre><code>A = cfu@x; // RepeatedTiming
B = Exp[-Abs[x - #]] & /@ x; // RepeatedTiming
A ... |
1,751,955 | <p>$ S = \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ is obviously bounded and infinite. It also looks totally disconnected to me (it is not and does not contain as its subset an interval with more than one element). But we know that a compact, totally disconnected set must be finite. Hence we know that $ S $ is not ... | Brian M. Scott | 12,042 | <p>It <em>is</em> compact. Your error is in thinking that a compact, totally disconnected space must be finite: this is false. A compact <em>discrete</em> space must be finite, but a totally disconnected space need not be discrete, as the present example shows. An even better example is the Cantor set, which is totally... |
1,436,655 | <p>I was reading my course notes and I came across this statement:</p>
<blockquote>
<p>If we are given a set of moments, we can identify the distribution that they came from.</p>
</blockquote>
<p>My question is: how do we identify the distribution when its moments are specified?</p>
| JKJK | 1,155,037 | <p>I'm about 6 years late, but I was trying to get a quick answer to this and thought others might benefit.</p>
<p>As another physicist answering, this is a well-known problem encountered in statistical field theory, particularly with regards to the <a href="https://en.wikipedia.org/wiki/Replica_trick" rel="nofollow no... |
3,517,019 | <p>Let <span class="math-container">$k_0$</span> be a field, <span class="math-container">$k$</span> its algebraic closure, and <span class="math-container">$K$</span> a field extension of <span class="math-container">$k_0$</span>.
Let <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</s... | Bib-lost | 294,785 | <p>There are many different ways to construct a field <span class="math-container">$\Omega$</span>; even if we require the field to be minimal with respect to the demanded properties, it will in general not be uniquely determined. As GreginGre says, one could for example set <span class="math-container">$L_1$</span> an... |
3,496,985 | <p>I am confused about a sample solution to a problem (please read until the end, since I am not looking for a solution to the problem itself)</p>
<p>We were asked if the vector </p>
<p><span class="math-container">$\begin{bmatrix} -3 \\ 4\\ 7 \end{bmatrix}$</span></p>
<p>can be written as a linear combination of <... | Alessio K | 702,692 | <p>We say a map ϕ : X → Y between two vector spaces X, Y is called a linear map if the
following holds:</p>
<p>ϕ(λ·<span class="math-container">$x_1$</span>+µ·<span class="math-container">$x_2$</span>) =λ·ϕ(<span class="math-container">$x_1$</span>)+µ·ϕ(<span class="math-container">$x_2$</span>) for all vectors <span ... |
2,757,687 | <p>Can you make the claim that for any ordinal, its cardinality equals it's least upper bound.</p>
<p>This is motivated by:</p>
<blockquote>
<blockquote>
<p>$\bigcup\omega+1=\omega$ and $|\omega+1|=\omega$</p>
</blockquote>
</blockquote>
<p>where $\bigcup\omega+1$ is also the $\text{sup}(\omega+1)$</p>
<p>T... | Ross Millikan | 1,827 | <p>What is $\bigcup (\omega^2) ?$</p>
|
4,493,926 | <blockquote>
<p>Suppose <span class="math-container">$p(x)=ax^n + b_1x^{n-1}+\cdots~$</span> and <span class="math-container">$g(x)=ax^n + b_2x^{n-1}+\cdots~~$</span> (basically only the leading coefficients are same).</p>
<p>I am required to find/proof: <span class="math-container">$$\lim_{x \to \infty}{p(x)^{1/n}-g(x... | Robert Z | 299,698 | <p>Hint. Assuming that <span class="math-container">$a>0$</span>, then as <span class="math-container">$x\to +\infty$</span>,
<span class="math-container">$$(ax^n + bx^{n-1}+o(x^{n-1}))^{1/n}=a^{1/n}x\left(1+\frac{b}{ax}+o(1/x)\right)^{1/n}.$$</span>
Now use the <a href="https://en.wikipedia.org/wiki/Binomial_approx... |
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | Sue VanHattum | 60 | <p>If the students can think about graphs, you can graph y=1/x. So if 1/0 = ∞, this should approach ∞ as x -> 0. It does on one side. But on the other, it approaches -∞. Since it doesn't approach ∞ from both sides, we must say it's undefined.</p>
|
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | Community | -1 | <p>I teach mostly physics, but have taught calculus a couple of times. From the physics end, I see things in almost the opposite way that you do. Here is a typical way that this plays out in my class.</p>
<p>We have a homework problem where a cable is stretched between two buildings, with a streetlight hanging from th... |
3,947,337 | <p>Be <span class="math-container">$E$</span> a normed vector space. If <span class="math-container">$A \subset E$</span> is compact and <span class="math-container">$p \in E$</span>, prove that <span class="math-container">$A_p = \{x+p:x \in A\} $</span> is also compact.</p>
<p>Hello, everyone! All right? I tried to p... | Luca.b | 627,612 | <p>Let <span class="math-container">$\{U_n\}$</span> be an open coverage of <span class="math-container">$A_p$</span>. Let <span class="math-container">$t: x \mapsto x+p$</span> be the translation map, which is a homeomorphism. So <span class="math-container">$\{t^{-1}(U_n)\}$</span> is an open coverage of <span class=... |
2,734,442 | <p>Prove in natural deduction (Negation of existential quantifier):</p>
<ul>
<li>∀x ¬P(x) ⊢ ¬∃x P(x)</li>
</ul>
<p>Inference rules:</p>
<ul>
<li>(∀−) If $Σ ⊢ ∀xA(x)$, then $Σ ⊢ A(t)$ where $t$ is any term.</li>
<li>(∀+) If $Σ ⊢ A(u)$ and $u$ does not occur in $Σ$, then $Σ ⊢ ∀xA(x)$.</li>
<li>(∃−) If $Σ, A(u) ⊢ B$ an... | Hagen von Eitzen | 39,174 | <p>Here's a semi-formal proof. Try to build your proof along that.</p>
<p>Assume $\exists x\,P(x)$. For such an $x$, we have $P(x)$. On the other hand, specialization from $\forall x\,\neg P(x)$ gives us $\neg P(x)$. So we obtain $P(x)\land \neg P(x)$. Conclude.</p>
|
3,958,304 | <p>We consider the following Itô-Integral <span class="math-container">$$Z_t := \int_0^t \exp(-\mu s) W_s ds$$</span> for <span class="math-container">$\mu\geq0$</span>. I wonder if I could calculate the limit as <span class="math-container">$t\to\infty$</span> in some convergence, but how should I start? Some ideas?</... | John Dawkins | 189,130 | <p>As shown by @fesman, <span class="math-container">$Z_t$</span> differs from the stochastic integral <span class="math-container">$\int_0^t e^{-\mu s}\,dW_s$</span> by an amount that converges a.s (and also in <span class="math-container">$L^2$</span>) to <span class="math-container">$0$</span>. It's not hard to show... |
3,237,430 | <p>Suppose there are <code>n</code> voters and <code>k</code> candidates. In how many different ways can the vote be split among the candidates?</p>
<p>To be clear, I am only concerned with the number of votes that each candidate gets, not with how individual voters vote.</p>
<p>Each voter can and must vote for only ... | Bernard | 202,857 | <p>This is indeed <em>Raabe's test</em>, and it is a consequence of <em>Kummer's rule</em>:</p>
<p>If <span class="math-container">$\dfrac{a_{n+1}}{a_n}\le 1-\dfrac cn$</span> for all <span class="math-container">$n\ge N$</span>, we have
<span class="math-container">$$\frac{a_n}{a_{n+1}}\ge \frac 1{1-\cfrac cn}=\frac n... |
1,000,349 | <p>$\def\Li{\operatorname{Li}}$
I wonder how to prove:
$$
\int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2}
$$
I'm not used to polylogarithm, so I don't know how to tackle it. So any help is highly appreciated.</p>
| David H | 55,051 | <p>The basic idea behind most polylogarithmic integrals is integration by parts. Note that we have the indefinite integral,</p>
<p>$$\int\frac{1}{(1-x)^2}\,\mathrm{d}x=\frac{1}{1-x}+\color{grey}{constant}.$$</p>
<p>And the derivative of the dilogarithm is of course:</p>
<p>$$\frac{d}{dx}\operatorname{Li}_{2}{\left(x... |
753,077 | <p>First of all, sorry for the crappy title, I have no idea how to ask this question.</p>
<p>I have a formula:
$$S = \left(\frac{T\cdot D\cdot C}{1000}+V^{\frac13}\right)^3$$</p>
<p>I now need to turn it around to find $D$, when I have $S$. I don't even know where to start. Any pointers in the right direction, or eve... | Alec | 108,039 | <p>Given $S = (\frac{TDC}{1000} + \sqrt[3]V)^3$ right? That's just my interpretation.</p>
<p>$\sqrt[3]S = \frac{TDC}{1000} + \sqrt[3]V$</p>
<p>$\sqrt[3]S - \sqrt[3]V = \frac{TDC}{1000}$</p>
<p>Next, we multiply by $\frac{1000}{TC}$</p>
<p>Which gives us $\displaystyle D = \frac{1000(\sqrt[3]S - \sqrt[3]V)}{TC}$</p>... |
2,326,259 | <p>I tried the following $$I = \langle X^2,X+1\rangle =\langle X^2,X+1,X^2+2(X+1)\rangle =\langle X^2,X+1,(X+1)^2+1 \rangle$$</p>
<p>Yet no matter how I arrange it, I cannot obtain $1$. Can someone help me out?</p>
| Bill Dubuque | 242 | <p>From your <a href="https://math.stackexchange.com/a/2326276/242">prior question</a> $(x+1,f(x)) = (1)\iff f(-1)\mid 1$ which is true for $\,f(x) = x^2.$</p>
<p><strong>Remark</strong> $ $ The other answers are essentially a special case of the proof in the prior question.</p>
<p>If you wish to gain further <em>alg... |
203,370 | <p>I'm new to Mathematica and would like to ask a question about equation systems. I have a linear equation system given in the link. I would like to obtain kappa1 in terms of psi1, kappa2 in terms of psi2, w1 in terms of psi1 and w2 in terms of psi2. Is there a way to manipulate this system ?</p>
<p><a href="https://... | Ulrich Neumann | 53,677 | <p>With</p>
<pre><code>A = Array[a, {4, 6}];
</code></pre>
<p>you can solve the euations <code>A.{\[Psi]1, \[Kappa]1, w1, \[Psi]2, \[Kappa]2, w2 }==0</code> for
<code>{\[Kappa]1, w1, \[Kappa]2, w2}</code></p>
<pre><code>Solve[A.{\[Psi]1, \[Kappa]1, w1, \[Psi]2, \[Kappa]2, w2 } == 0, {\[Kappa]1, w1, \[Kappa]2, w2} ... |
203,370 | <p>I'm new to Mathematica and would like to ask a question about equation systems. I have a linear equation system given in the link. I would like to obtain kappa1 in terms of psi1, kappa2 in terms of psi2, w1 in terms of psi1 and w2 in terms of psi2. Is there a way to manipulate this system ?</p>
<p><a href="https://... | Cesareo | 62,129 | <p>You can make a partition such as follows</p>
<pre><code>M = Table[Subscript[a, i, j], {i, 1, 4}, {j, 1, 6}]
X = {Subscript[\[Psi], 1], Subscript[\[Kappa], 1], Subscript[w, 1],Subscript[\[Psi], 2], Subscript[\[Kappa], 2], Subscript[w, 2]};
X1 = {Subscript[\[Kappa], 1], Subscript[w, 1], Subscript[\[Kappa 2],Subscri... |
3,694,658 | <p>How can I prove that?</p>
<p><span class="math-container">$1^3+2^3+\cdots+(n-1)^3<\frac{n^4}{4}$</span> </p>
| Stefan Lafon | 582,769 | <p>If you compare with the integral of <span class="math-container">$x\mapsto x^3$</span> (increasing function):
<span class="math-container">$$\sum_{k=1}^{n-1}k^3\leq \sum_{k=1}^{n-1}\int_{k}^{k+1}x^3dx=\int_1^nx^3dx=\frac{n^4-1}4<\frac {n^4} 4$$</span></p>
|
3,694,658 | <p>How can I prove that?</p>
<p><span class="math-container">$1^3+2^3+\cdots+(n-1)^3<\frac{n^4}{4}$</span> </p>
| Anas A. Ibrahim | 650,028 | <p>Well, let's seek a simple answer:
<span class="math-container">$$\sum_{k=1}^{n}(k^3)=\left(\sum_{k=1}^{n}(k)\right)^2=\frac{n^2(n+1)^2}{4}$$</span>
Thus, it'd be sufficient to prove
<span class="math-container">$$\frac{n^2(n-1)^2}{4}<\frac{n^4}{4}$$</span>
while <span class="math-container">$n-1<n$</span> impl... |
1,427,970 | <p>Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but not Burnside's lemma) that there exists a non-trivial element having a fixed point. For example let $\Omega = \{\alph... | whacka | 169,605 | <p>If a finite group $G$ acts transitively on $\Omega$ then $|\Omega|$ is a divisor of $|G|$ by the orbit-stabilizer theorem, and so if $|\Omega|=|G|-1$ then we know $(|G|-1)\mid |G|$ which implies $|G|=2$ and $|\Omega|=1$, and hence the only element of $\Omega$ must be a fixed point.</p>
<p>In any case, pick $\omega\... |
135,423 | <p>I would like to label a curve inside ListLinePlot. Let's say I have the following list:</p>
<pre><code>Table[{x, x^2}, {x, 0, 10, 0.1}]
</code></pre>
<p>What I expected is a labeled curve. The label should also be placed above the curve and in the middle. It should also be rotated with the curve like the following... | MinHsuan Peng | 1,376 | <p>In Version 11.0.1, Labeled can be used in the dataset level.</p>
<pre><code>data = Table[{x, x^2}, {x, 0, 10, 0.1}];
ListLinePlot[Labeled[data, Style[Rotate[x^2, 40 Degree], Bold, 14], {6, 42}]]
</code></pre>
<p>The third argument of Labeled in this case is a precise location.</p>
<p><a href="https://i.stack.imgu... |
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| John Douma | 69,810 | <p>Take $\{(x,y)\in \mathbb Z^2: x = -y\}$</p>
|
386 | <p>The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the... | Noah Snyder | 22 | <p>I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds. The MTC is the thing assigned to a circle by this extended TQFT.</p>
<p>There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly." The 3d TQFTs coming from modular tensor catego... |
189,603 | <p>Let's suppose that we have the following equation</p>
<pre><code>Clear["Global`*"];
m = 1/2;
V = m/Sqrt[(x - m)^2 + (y - m)^2 + (z - m)^2] + m/Sqrt[(x + m)^2 + (y + m)^2 + (z + m)^2]
+ 1/2*(x^2 + y^2);
Vx = D[V, x];
Vy = D[V, y];
Vz = D[V, z];
</code></pre>
<p>Then we can use the <code>FindRoot</code> module fo... | Cesareo | 62,129 | <p>With</p>
<pre><code>V = m/Sqrt[(x - m)^2 + (y - m)^2 + (z - m)^2] +
m/Sqrt[(x + m)^2 + (y + m)^2 + (z + m)^2] + 1/2*(x^2 + y^2);
</code></pre>
<p>The system </p>
<p><span class="math-container">$$
V_x = 0\\
V_y = 0\\
V_z = 0
$$</span></p>
<p>has notoriously three solutions which are for <span class="math-co... |
3,833,629 | <p>So I need to find <span class="math-container">$E(X_2)$</span>, when <span class="math-container">$p=1$</span> and <span class="math-container">$X_0 = 0$</span>.</p>
<p>This is how the task sounds:</p>
<p>Let's say that <span class="math-container">$(X_n, n=0,1, \ldots)$</span> is Markovs chain where <span class="... | redroid | 411,951 | <p>The series <span class="math-container">$X_n$</span> represents the state of the Markov process at step <span class="math-container">$n$</span>, and each successive value is related by way of the matrix, <span class="math-container">$P$</span>. The relationship between terms of <span class="math-container">$X_n$</sp... |
3,833,629 | <p>So I need to find <span class="math-container">$E(X_2)$</span>, when <span class="math-container">$p=1$</span> and <span class="math-container">$X_0 = 0$</span>.</p>
<p>This is how the task sounds:</p>
<p>Let's say that <span class="math-container">$(X_n, n=0,1, \ldots)$</span> is Markovs chain where <span class="... | Brian Moehring | 694,754 | <blockquote>
<p>Why we use <span class="math-container">$X_2$</span> and <span class="math-container">$X_0$</span>, if the formula is <span class="math-container">$p_{i,j}=P(X_{n+1}=j|X_n=i)$</span> with every <span class="math-container">$i,j=0,1,\ldots,N$</span></p>
</blockquote>
<p>You want to find <span class="math... |
2,301,960 | <p>I'm quoting from the <a href="http://stacks.math.columbia.edu/tag/001L" rel="nofollow noreferrer">Stacks Project</a>:</p>
<blockquote>
<p><strong>Definition 4.3.6.</strong> A contravariant functor <span class="math-container">$C \to \text{Sets}$</span> is said to be representable if it is isomorphic to the functor o... | John Doe | 399,334 | <p>The terms would be as follows: $$\ln(1+2x^2)=\ln(1+u)=u-\frac{u^2}2+\frac{u^3}3-\mathcal{O}(u^4)=2x^2-\frac{4x^4}{2}+\frac{8x^6}{3}-\mathcal{O}(x^8)$$ So to $3^{rd}$ order in $x$, this is why they ended the series where they did. This happens because $u=O(x^2)$, so you only need to expand $u$ until order $\frac n2$ ... |
2,682,599 | <p>Several teams played a baseball tournament (as a reminder, there are no ties in baseball); each team played every other team exactly once. We say that team $A$ is dominating team $B$ if either $A$ beat $B$ heads up or if there exists a team $C$ such that $A$ beat $C$ and $C$ beat $B$. (Notice that it is entirely pos... | Jan Bohr | 317,480 | <p>Given a set $\mathscr{D}$ of test-functions (say $\mathscr{D}\subset C^0$) one defines a corresponding set of generalized-functions as the corresponding dual space, i.e.
$$
\mathscr{D}'=\{\mathscr{F}\colon\mathscr{D}\rightarrow \mathbb{R} ~\text{linear}\}.
$$
(That is only half the truth, as one usually also specifi... |
1,715,945 | <p>I need help solving/understanding this question:</p>
<p>L (x,y) : "x loves y".
Translate "there are exactly two people whom Lynn loves".
Its answer includes a variable "z". I do not get that part with the variable "z". How did it come here when it was not introduced in the question? Detailed solution is appreciated... | Adam Hughes | 58,831 | <p>Define the map</p>
<p>$$\phi: \begin{cases}\langle a_1,\ldots, a_n\rangle = F_n\to G = \langle g_1,\ldots, g_n\rangle \\
a_i\mapsto g_i & 1\le i\le n\end{cases}$$</p>
<p>Then this is the map. Since the $g_i$ generate $G$ it's surjective. That's the basic idea.</p>
|
573,637 | <p>I'm trying to solve this problem.
A = {3,14}</p>
<p>What is the number of elements in this set?</p>
<p>I am thinking about the answer is 1. Because the priority of comma's mathematical decimal function is more than math's grammar in my opinion. I inspired computer languages' mechanism.</p>
<p>If there is a wrong ... | Magdiragdag | 35,584 | <p>Priority of mathematical operators has nothing to do with this. If the language the book is written in uses . as decimal separator, then $A$ contains two elements. If it uses , as decimal separator, then it depends on how the book denotes a set; from what I've seen, that would still be a comma, but followed by white... |
42,864 | <blockquote>
<p>Given a monic polynomial <span class="math-container">$f\in\mathbb{Z}[x]$</span>, how can I determine whether there is <span class="math-container">$k\in\mathbb{Z}^+$</span> such that <span class="math-container">$f\mid x^k-1$</span>?</p>
</blockquote>
<p>For example, <span class="math-container">$x^2-x... | Arturo Magidin | 742 | <p>The polynomials <span class="math-container">$x^n-1$</span> have as roots the complex <span class="math-container">$n$</span>th roots of unity. They factor as
<span class="math-container">$$x^n - 1 = \prod_{d|n}\Phi_d(x)$$</span>
where <span class="math-container">$\Phi_d(x)$</span> is the <a href="https://en.wikipe... |
3,820,580 | <p>Given vector <span class="math-container">$\boldsymbol{x} \in \mathbb{R}^n$</span> and real symmetric matrices <span class="math-container">$\boldsymbol{A} \in \mathbb{R}^n \times \mathbb{R}^n$</span> and <span class="math-container">$\boldsymbol{B} \in \mathbb{R}^n \times \mathbb{R}^n$</span>. I am trying to find t... | Biel Roig-Solvas | 317,868 | <p>A good trick whenever you have optimization problems with two matrices where their eigenvalues play a role is to try to co-diagonalize them to simplify the problem. In the case where one of the matrices is PD and the other is PSD, you can always do that by combining a Cholesky decomposition and an eigenvalue decompo... |
2,397,249 | <p>Well, to quote from Wolfram MathWorld directly,</p>
<blockquote>
<p>Given an affine variety $V$ in the $n$-dimensional affine space $K^n$, where $K$ is an algebraically closed field, the coordinate ring of $V$ is the quotient ring $K[V] = K[x_1 , \dots , x_n] / I(V)$.</p>
</blockquote>
<p>My question is simply t... | Community | -1 | <p>A lot of algebraic geometry is developed by analogy with differential geometry.</p>
<p>In differential geometry, we speak of <em>manifolds</em> rather than varieties. The main feature of the definition of a manifold is that it is covered by <em>coordinate charts</em>: open sets $U$ together with a (smooth) homeomor... |
1,184,963 | <p>Toss two fair dice. There are $36$ outcomes in the sample space $\Omega$, each with probability $\frac{1}{36}$. Let:</p>
<ul>
<li>$A$ be the event '$4$ on first die'.</li>
<li>$B$ be the event 'sum of numbers is $7$'.</li>
<li>$C$ be the event 'sum of numbers is $8$'.</li>
</ul>
<p>It says here $A$ and $B$ are ind... | Martín-Blas Pérez Pinilla | 98,199 | <p>$A$ effects the probability of $B$... and leaves it untouched. Getting 4 on the first die excludes many cases of $A$ and $A^c$, but does so evenly, so $P(B|A)=1/6=6/36=P(B)$.</p>
|
4,039,336 | <p>I've calculated the probability for rolling a particular value or higher on both dice when rolling 2d6, as follows.</p>
<p>1 or higher: 100%
2 or higher: 69.44%
3 or higher: 44.44%
4 or higher: 25%
5 or higher: 11.11%
6 or higher: 2.778%</p>
<p>If I add a third dice to this, how does it affect the probability if I s... | Notone | 408,724 | <p>There are two kinds of "natural" affine subsets of (affine) schemes:</p>
<ol>
<li><p>Localisations: This is the one you are alluding to probably. For <span class="math-container">$R$</span> a ring and <span class="math-container">$f\in R$</span>, one has a natural inclusion <span class="math-container">$$S... |
2,176,080 | <p>'$\Leftrightarrow$' Is very much important in this question . Actually, it seems very obvious to me.</p>
<p>We say a function is differentiable at $x=a$ iff </p>
<p>$\lim_{ h\rightarrow 0 }{ \frac { f(a+h)-f(a) }{ h } } = lim_{ h\rightarrow 0 }{ \frac { f(a-h)-f(a) }{ -h } }$</p>
<p>Now, let</p>
<p>$f'(x)=g(x)... | N. S. | 9,176 | <p>The part</p>
<p>If $f(x)$ is differentiable at $x=a$ it means</p>
<p>$$f′(a^+)=f′(a^−)$$</p>
<p>is incorrect.</p>
<p>The statement
$$f′(a^+)=f′(a^−)$$
means
$$\lim_{t \to 0^+} f'(a+t)=\lim_{t \to 0^-} f'(a+t)$$
which is not the same as the limit you wrote, it is actually the double limit:
$$\lim_{t \to 0^+} \li... |
277,331 | <p>One can write functions which depend on the type of actual parameter before they are actually called. E.g.:</p>
<pre><code>Clear[f,g,DsQ];
DsQ[x_]:=MatchQ[x,{String__}];
f[i_Integer, ds_?DsQ] :=Print["called with integer i and DsQ[ds]==True"];
f[i_String, ds_?DsQ] :=Print["called with String i and Ds... | flinty | 72,682 | <p>There is an analytic solution for the minimum of your particular ellipses though it's complicated and you're better off with the numerical methods given already, but here's my working:</p>
<p>Start with the parametric form of an axis oriented ellipse where <span class="math-container">$\mathbf{s}$</span> gives the s... |
2,538,390 | <p>I don't understand the concept of circle inversion.</p>
<p>$OP \cdot OP' = k^2$</p>
<p>For example, in a circle $x^2+y^2=k^2$.
If I set a general point $P(x,y)$, why is its image $P'(\frac{xk^2}{x^2+y^2}, \frac{yk^2}{x^2+y^2})$?</p>
<p>Also, why does a line become a circle through O?</p>
<p>Sorry for my English,... | user | 505,767 | <p>the main relation $OP \cdot OP' = k^2$ has a clear and nice geometrical meaning by means of Euclid theorem</p>
<p><a href="https://i.stack.imgur.com/7mZYO.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7mZYO.jpg" alt="enter image description here"></a></p>
|
2,619,478 | <p>I am working on a problem connected to shallow water waves.
I have a vector:</p>
<p>$U = \begin{bmatrix} h \\
h \cdot v_1\\
h \cdot v_2\end{bmatrix}$</p>
<p>and a function</p>
<p>$f(U) = \begin{bmatrix} h \cdot v_1 \\
h \cdot v_1^2 + 0.5\cdot gh^2\\
h \cdot v_1 \cdot v_2\end{bmatrix}$</p>
<p>I now want to calcu... | Christian Blatter | 1,303 | <p>In the final analysis your $f$ is a function $f:\>{\mathbb R}^3\to{\mathbb R}^3$ taking the variables $h$, $v_1$, $v_2$ as input and producing three scalar values
$$u:=hv_1,\quad v:=hv_1^2+{g\over2}h^2,\quad w:=hv_1v_2$$
as output, whereby $g$ seems to be some constant. By definition the Jacobian of $f$ is the ma... |
1,596,297 | <p>I must prove that $7^n-1$ $(n \in \mathbb{N})$ is divisible by $6$.</p>
<p>My "inductive step" is as follows:</p>
<p>$7^{n+1}-1 = 7\times 7^n-1 = (6+1)\times 7^n-1 = 6\times 7^n+7^n-1$</p>
<p>So now, $6\times7^n$ is divisible by 6, that's obvious. But what about the other part, the $7^n-1$ ? How do we know that i... | pancini | 252,495 | <p>Perhaps a cleaner way to write it:</p>
<p>Step 1: we see that $6|7-1$</p>
<p>Step 2: assume that $6|7^k-1$ where $k\in\Bbb{N}$</p>
<p>Step 3: then $7^{k+1}-1=7\cdot 7^k-1=6\cdot 7^k+7^k-1$. Since $6|7^k-1$ and $6|6\cdot 7^k$, we have $6|7^{k+1}-1$.</p>
<p>Thus $6|7^n-1$ for all $n\in\Bbb{N}$.</p>
|
1,596,297 | <p>I must prove that $7^n-1$ $(n \in \mathbb{N})$ is divisible by $6$.</p>
<p>My "inductive step" is as follows:</p>
<p>$7^{n+1}-1 = 7\times 7^n-1 = (6+1)\times 7^n-1 = 6\times 7^n+7^n-1$</p>
<p>So now, $6\times7^n$ is divisible by 6, that's obvious. But what about the other part, the $7^n-1$ ? How do we know that i... | Stella Biderman | 123,230 | <p>As noted in the comments, you are assuming, as your inductive hypothesis, that $6|7^n-1$. An alternative proof is to factor the expression as $$7^n-1=(7-1)(1+7^2+\cdots+7^{n-1})$$</p>
<p>The edits indicate a confusion about what induction is and how it works. There is a good explanation here: <a href="https://math.... |
1,596,297 | <p>I must prove that $7^n-1$ $(n \in \mathbb{N})$ is divisible by $6$.</p>
<p>My "inductive step" is as follows:</p>
<p>$7^{n+1}-1 = 7\times 7^n-1 = (6+1)\times 7^n-1 = 6\times 7^n+7^n-1$</p>
<p>So now, $6\times7^n$ is divisible by 6, that's obvious. But what about the other part, the $7^n-1$ ? How do we know that i... | Daniel R. Collins | 266,243 | <blockquote>
<p>According to some resources it is a complete proof, however, it's not
clear for me why. Could someone please explain?</p>
</blockquote>
<p>It is <em>not</em> a complete proof, because in your writeup (as the question currently appears) you didn't include the "base case". What your "inductive step" ... |
2,262,467 | <p>Hi guys i have two questions i´m struggling with.</p>
<p>1)The mean duration of education for a population is 12 years and
the standard deviation is 2 years. What is the maximum probability that a randomly selected individual will have had less than
9 or more than 15 years of education?</p>
<p>2)Limit Theorem: A f... | Blue | 409 | <p>Let's begin with this:</p>
<blockquote>
<p>If $P$ is a point on a parabola with focus $F$, and if $R$ is the intersection of the tangent at $P$ with the "vertex tangent", then $\overleftrightarrow{FR} \perp \overleftrightarrow{PR}$. Conversely, if $R$ is on the vertex tangent, then the line perpendicular to $\ove... |
313,388 | <p>Computing without Taylor series or l'Hôpital's rule </p>
<p>$$\lim_{n\to\infty}\prod_{k=1}^{n}\cos \frac{k}{n\sqrt{n}}$$</p>
<p>What options would I have here? Thanks!</p>
| Hagen von Eitzen | 39,174 | <p>The following tries to use only "basic" inequalities, avoiding L'Hopital and Taylor. </p>
<p>The inequality
$$\tag1 e^x\ge 1+x\qquad x\in\mathbb R$$
should be well-known and after taking logarithms immediately leads to
$$\tag2 \ln(1+x)\le x\qquad x>-1$$
and after taking reciprocals
$$\tag3 e^{-x}\le \frac1{1+x}... |
507,109 | <p>If we have two homeomorphisms $f:A\to X$ and $g:B\to Y$, then is it true that $f\times g:A\times B\to X\times Y$ defined by $(f\times g)(a,b)=(f(a),g(b))$ is again a homeomorphism?</p>
<p>I think the answer is yes; </p>
<p>It's clearly a bijection. Intuitively it seems to be continuous but I don't know how to show... | Quique Ruiz | 69,937 | <p>A categorical argument would be like this: Consider a category $C$ with products. (In this case, $\mathbf{Top}$, the category with objects the topological spaces and arrows the continuous functions, has products: the product topology has the <a href="http://en.wikipedia.org/wiki/Product_%28category_theory%29" rel="n... |
2,182,371 | <p>This is the most complicated differential equation I've encountered today:</p>
<p>$$y''(x)-\frac{y'^2}{y}+ \frac{\epsilon y(x)}{x^2}=0$$</p>
<p>It is not Euler-Cauchy, I guess the ultimate approach is to do it with variable substitution, but it is hard to make a guess on what to substitute by purely looking at the... | Aritro Pathak | 238,447 | <p>This equation gives $\frac{y y'' -(y')^{2}}{y^{2}}+\frac{\epsilon}{x^{2}}=0$, and thus $(\frac{y'}{y})' +\frac{\epsilon}{x^{2}}=0$. Integrating this, you have $(\frac{y'}{y})-\frac{\epsilon}{x}=C$ where $C$ is some constant. Now this is a standard linear equation, $y' - (\frac{\epsilon}{x}+C)y=0$.</p>
|
1,600,332 | <p>I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$</p>
<blockquote>
<p>Do you know if there are other integer solutions to
$$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$
besides the trivial solutions $a=b=c=d=0$ and $a=b=c=d=2$?</p>
</blockquote>
| David Altizio | 302,136 | <p>Here's a solution when $a$, $b$, $c$, and $d$ are positive; I'm not sure how to tackle the more general problem though (although something tells me it's probably similar).</p>
<p>Note that substituting $b=cd-a$ into the second equation yields $$a(cd-a)=c+d\implies a^2-cad+c+d=0.$$ By the Quadratic Formula on $a$, w... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| zoli | 203,663 | <p>Divide both sides of $(n+1)^{n-1}<n^n$ by $n^{n-1}$. We have now $$\frac{(1+1/n)^n}{1+1/n}<n.$$ The left side tends to $e$ from below. $e$ is less than 3 and the inequality holds for $n=2$.</p>
|
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| Samrat Mukhopadhyay | 83,973 | <p><strong>Proof By Induction:</strong> The hypothesis is true for $n=2$. Let it be true for $n=k$. Then $$(k+1)^{k-1}<k^k\\\implies \left(1+\frac{1}{k}\right)^{k}<{k+1}$$Now, $$\frac{(k+2)^k}{(k+1)^{k+1}}=\left(1+\frac{1}{k+1}\right)^{k}\frac{1}{k+1}<\left(1+\frac{1}{k}\right)^{k}\frac{1}{k+1}<1$$</p>
<p... |
1,509,917 | <p>Suppose I have an arithmetic progression, e.g. 3,7,11,15,19,....
The sums of the first n elements form tuples of increasing size.
E.g. the sums are:</p>
<ul>
<li>S(n=1)=3</li>
<li>S(n=2)=10</li>
<li>S(n=3)=21</li>
<li>S(n=4)=36</li>
<li>S(n=5)=55</li>
</ul>
<p>So the tuples are:</p>
<ul>
<li>T(1)=[3;10)</li>
<li... | Asinomás | 33,907 | <p>If you have the arithmetic progression $a,a+d,a+2d\dots$ then the sum of the first $n$ terms is given by Gaussian summation.</p>
<p>It is:</p>
<p>$\frac{(2a+(d-1)n)(n)}{2}=\frac{2an+(d-1)n^2}{2}$</p>
<p>Now, you want to to find the maximum $n$ so that $\frac{2an+(d-1)n^2}{2}<x$. I suggest you use binary search... |
4,122,288 | <p>I have <span class="math-container">$$7^x\bmod {29} = 23 $$</span>
It is possible to get <span class="math-container">$x$</span> by trying out different numbers but that will not be possible if <span class="math-container">$x$</span> is actually big.</p>
<p>Are there any other solutions for this equation?</p>
<p>Kin... | fleablood | 280,126 | <p>I don't think there is anything but trial and error.</p>
<p>But look. <span class="math-container">$7^2 \equiv 23 \equiv -6 \pmod {29}$</span></p>
<p><span class="math-container">$7^{2x} \equiv 36 \equiv 7\pmod {29}$</span>.</p>
<p>So <span class="math-container">$7^{2x-1} \equiv 1 \pmod {29}$</span>.</p>
<p>We kno... |
3,260,648 | <p>It's an example in Thomas's Calculus</p>
<p>Q. Prove that <span class="math-container">$\lim_{x \to 2} f(x)=4$</span></p>
<p>if
<span class="math-container">$f(x)=x^2, x\neq 2,
f(x)=1, x=2$</span></p>
<p>In the book they have proved it for <span class="math-container">$\varepsilon>4$</span> even when </p>
<... | friedvir | 681,752 | <p>i believe that its supposed to be that 0<ε<4 (smaller than 4, but still positive), and not the way you wrote it, but the general idea is to show that when ε is arbitray small-positive value, you can choose it to be smaller as you want, and the value the function f(x)=x² returns will be closer and closer to 4, ... |
1,431,287 | <p>Dear StackExchange users,</p>
<p>I have a little question ... I just don't have a clue how that works</p>
<p>I have the following differential equation</p>
<p>$$\frac{dy}{dx}=xy^2-2\frac{y}{x}-\frac{1}{x^3}$$</p>
<p>My book says that one can substitute $(r,s) = (x^2y,\ln{|x|})$ to get</p>
<p>$$\frac{ds}{dr}=\fr... | MrYouMath | 262,304 | <p>I am adding an alternative method that might be useful to other users.</p>
<p>First we note that $y=y(r,s)$ and $x=x(r,s)$, hence</p>
<p>$$\dfrac{dy}{dx}=\dfrac{dy(r,s)}{dx(r,s)}.$$</p>
<p>Now, we apply the total derivative for $dy(r,s)$ and $dx(r,s)$.</p>
<p>$$\dfrac{dy(r,s)}{dx(r,s)}=\dfrac{\dfrac{\partial y}{... |
2,692 | <p>The setting is undergraduate students in Computer Science, a course in Discrete Mathematics (first proof-oriented course they take, they had a mostly computation oriented first course in calculus).</p>
<p>As examples I need nice proofs using contradiction. The typical proof that $\sqrt{2}$ is irrational is standard... | ncr | 1,537 | <p>Here's one that I use in an analogous course I teach: A lossless compression algorithm that makes some files smaller must make some (other) files larger.</p>
|
313,470 | <p>Consider the set <span class="math-container">$\mathbf{N}:=\left\{1,2,....,N \right\}$</span> and let <span class="math-container">$$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\rvert=1 \right\}$$</span>
be the set of all subsets of... | Suvrit | 8,430 | <blockquote>
<p><strong>Claim.</strong> <span class="math-container">$\lambda_\min(A_N) \le 4\lambda_\min(B_N)$</span>.</p>
</blockquote>
<p><em>Proof.</em>
Let <span class="math-container">$C_N:=\bigl[\tfrac{1}{|M_i||M_j|}\bigr]$</span>. Then, <span class="math-container">$B_N = A_N \circ C_N$</span>, where <span clas... |
20,773 | <p><strong>Background</strong></p>
<p>The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.</p>
<p>Personally, I think the definition of a function is extremely abs... | Community | -1 | <p>I think "function" is one of those notions that can be presented in different ways to people at different ages and who have different levels of ambition in math. This is similar to the notion of a "set," which I was taught in school ca. age 5 or 6 in an age-appropriate way, but then learned about... |
20,773 | <p><strong>Background</strong></p>
<p>The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.</p>
<p>Personally, I think the definition of a function is extremely abs... | user52817 | 1,680 | <p>Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.</p>
<p><a href="https://i.stack.imgur.com/QcDlR.jpg" rel="noref... |
20,773 | <p><strong>Background</strong></p>
<p>The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for middle school students.</p>
<p>Personally, I think the definition of a function is extremely abs... | Sam Watkins | 3,287 | <p>We shouldn't need to teach functions to 15 year olds, because ideally they should have already learned programming since primary school, including mathematical and general functions and inverse functions. Programming, including demos, games and robotics, is the best motivator to learn math in my opinion.</p>
|
2,936,329 | <p>Let <span class="math-container">$(X,d)$</span> be a metric space. If <span class="math-container">$a_n$</span> is a sequence in <span class="math-container">$X$</span> and <span class="math-container">$a\in X$</span> such that <span class="math-container">$\frac{d(a_n,a)}{1+d(a_n,a)} \to 0$</span>, then <span class... | Nosrati | 108,128 | <p>With <span class="math-container">$\epsilon<\dfrac12$</span>
<span class="math-container">$$\frac{d(a_n,a)}{1+d(a_n,a)} =|\frac{d(a_n,a)}{1+d(a_n,a)} - 0| < \epsilon$$</span>
then
<span class="math-container">$$d(a_n,a)<\dfrac{\epsilon}{1-\epsilon}<\dfrac{\epsilon}{2}$$</span></p>
|
2,936,329 | <p>Let <span class="math-container">$(X,d)$</span> be a metric space. If <span class="math-container">$a_n$</span> is a sequence in <span class="math-container">$X$</span> and <span class="math-container">$a\in X$</span> such that <span class="math-container">$\frac{d(a_n,a)}{1+d(a_n,a)} \to 0$</span>, then <span class... | Jonas Lenz | 450,140 | <p>I would go a more general approach to show what is behind this convergence property. In fact, one can show that if $(X,d)$ is a metric space that then $\frac{d}{1+d}$ is an equivalent metric, in fact a bounded one.
As equivalent metrics induce the same topology one can conclude the desired convergence.</p>
|
3,819,639 | <p>Integral: <span class="math-container">$J=\int_0^1 \frac{x}{1+x^8}dx$</span></p>
<p>Consider the following assertions:</p>
<p><span class="math-container">$I:J> \frac{1}{4}$</span> and <span class="math-container">$II:J< \frac{\pi}{8}$</span></p>
<p>A. Both are true</p>
<p>B. Only <span class="math-container">... | Ty. | 760,219 | <p>A different approach is to recognize <span class="math-container">$\frac{x}{1+x^8}$</span> as the sum of an infinite geometric series with <span class="math-container">$r=-x^8$</span> and <span class="math-container">$a_0=x$</span>, which converges given the bounds of the integral.
<span class="math-container">\begi... |
15,351 | <p>For example, in MATLAB, a panel is available where one can see straightaway which variables are used and their dimension sizes. Is such a feature available in <em>Mathematica</em>? I really find it hard to scroll up and down to see where things are in <em>Mathematica</em>; I just want to see at a glance what's been ... | Jens | 245 | <p>You could do this:</p>
<pre><code>Names["Global`*"]
</code></pre>
<p>It looks for symbols in the <code>Global</code> context which is where "global variables" are defined. The <code>*</code> is a wildcart which you can modify to narrow down the search. Look at the docs for <a href="http://reference.wolfram.com/mat... |
15,351 | <p>For example, in MATLAB, a panel is available where one can see straightaway which variables are used and their dimension sizes. Is such a feature available in <em>Mathematica</em>? I really find it hard to scroll up and down to see where things are in <em>Mathematica</em>; I just want to see at a glance what's been ... | Sjoerd C. de Vries | 57 | <p>Below is something posted on <a href="http://forums.wolfram.com/mathgroup/archive/2010/Aug/msg00401.html" rel="noreferrer">Mathgroup by Jason McKenzie Alexander</a>. I made a few tiny changes and corresponded about this with Jason for a short while. He sent me his final version, which I post here with his permission... |
1,025,880 | <p>Why </p>
<blockquote>
<p>$(cf)' = c(f)'$</p>
</blockquote>
<p>but not </p>
<blockquote>
<p>$(cf)' = (c)' (f)' = 0 f = 0$</p>
</blockquote>
<p>?</p>
| k170 | 161,538 | <p>There many different ways to show this.
$$ (cf)'=\frac{d}{dx}[cf(x)] =\lim_{h\to 0}\frac{cf(x+h)-cf(x)}{h}=c \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=c \frac{d}{dx}[f(x)] =cf'$$</p>
|
90,987 | <p>I have coin, and want to get 2 heads exactly. I will throw it until this condition is met. </p>
<p>What is expected number of tries for this condition? </p>
<p>I know that it would be $$\sum\limits_{n=2}^\infty P(X=n)n=0.5^n \cdot n\cdot(n-1)$$
however I don't have an idea how to solve that sum because we didn't l... | Gerry Myerson | 8,269 | <p>The probability that it will take exactly $n$ throws is the probability of getting exactly one head in the first $n-1$ throws, times the probability of getting heads on the $n$th throw, so it's $((n-1)/2^{n-1})(1/2)$, which is $(n-1)/2^n$. So the expected number of throws is $\sum_{n=2}^{\infty}n(n-1)/2^n$. Series s... |
250,426 | <p>I want to factorize any quadratic expressions into two complex-valued linear expressions.</p>
<p>My effort below</p>
<pre><code>a := 1;(*needed*)
p := 2;(*needed*)
q := 3;(*needed*)
f[x_] := a (x - p)^2 + q;(*needed*)
AA := Coefficient[f[x], x^2];
BB := Coefficient[f[x], x];
CC := f[0];
DD = BB^2 - 4 AA CC;
EE = Tim... | Alexei Boulbitch | 788 | <p>There are already several nice solutions published here. Let me put also my five cents.
Let</p>
<pre><code>expr = (-I x + a + b*I)*(I x + a - b*I)
</code></pre>
<p>be our expression.</p>
<p>I often use the function to take the desired factor out of the parentheses:</p>
<pre><code>factor[expr_, fact_, funExpr_ : Expa... |
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | Batominovski | 72,152 | <p>Your statement has a generalization. There is a work by Erdos and Selfridge stating that the product of at least two consecutive natural numbers is never a power. Here is it: <a href="https://projecteuclid.org/euclid.ijm/1256050816" rel="nofollow noreferrer">https://projecteuclid.org/euclid.ijm/1256050816</a>.</p>... |
3,682,277 | <p>By integral test I found its converges to <span class="math-container">$\frac\pi4$</span> but thats the only thing I can find :( Hope somebody can give me a clue about how can I handle this question. </p>
<p>Show that
<span class="math-container">$$\frac{\pi}{4}\leq\sum_{n=1}^{\infty}\frac{1}{n^2+1}\leq\frac{1}{2}+... | Anas A. Ibrahim | 650,028 | <p>Well:
<span class="math-container">$$\int_{1}^{\infty}\frac{dx}{x^2+1}<\int_{1}^{\infty}\frac{dx}{{\lfloor x \rfloor}^2+1}=\sum_{n=1}^{\infty}\frac{1}{n^2+1}<\int_{1}^{2}\frac{dx}{{\lfloor x \rfloor}^2+1}+\int_{2}^{\infty}\frac{dx}{(x-1)^2+1}$$</span>
<span class="math-container">$$\implies\frac{\pi}{4}<\su... |
926,581 | <p>I find the <a href="https://en.wikipedia.org/wiki/Surreal_number" rel="nofollow noreferrer">surreal numbers</a> very interesting. I have tried my best to work through John Conway's <em>On Numbers and Games</em> and teach myself from some excellent <a href="http://www.tondering.dk/claus/sur16.pdf" rel="nofollow noref... | nombre | 246,859 | <p>I think you could say that $1 - \varepsilon$ is $0.999...$ in the sense that $\forall n \in \mathbb{N}, \left\lfloor 10^{n+1}(1-\varepsilon) \right\rfloor - 10\left\lfloor 10^n(1-\varepsilon) \right\rfloor = 9$ and $1 - \varepsilon = (+-+++...)$ is the simplest surreal satisfying this. </p>
<p>$1-\varepsilon = \{0;... |
926,581 | <p>I find the <a href="https://en.wikipedia.org/wiki/Surreal_number" rel="nofollow noreferrer">surreal numbers</a> very interesting. I have tried my best to work through John Conway's <em>On Numbers and Games</em> and teach myself from some excellent <a href="http://www.tondering.dk/claus/sur16.pdf" rel="nofollow noref... | Zemyla | 155,741 | <p>I honestly wouldn't say that, because if <span class="math-container">$0.\bar{9} < 1$</span> in the surreals, then by that token <span class="math-container">$0.\bar{3} < \frac{1}{3}$</span>, and <span class="math-container">$3.14159\dots < \pi$</span>, and in general decimal notation becomes useless for de... |
3,540,956 | <p>Let G be a finite group, H a maximal proper subgroup of G and K a subgroup of H. Is the normalizer of K in G, <span class="math-container">$N_GK$</span>, a subgroup of H.
Now <span class="math-container">$N_GK$</span> is certainly contained in some maximal subgroup, maybe more than one, but why is it contained in H... | aryan bansal | 698,119 | <p>Let the number representing sum of ages of 100 youngest members = <span class="math-container">$n_1$</span>.</p>
<p>Sum of next 100 hundred (101- 200= <span class="math-container">$n_2$</span></p>
<p>Sum of next hundred (201-300) =<span class="math-container">$n_3$</span></p>
<p>Sum of oldest membera = <span clas... |
138,801 | <p>ImageSize works fine as an option for Show when used in a call.</p>
<pre><code>Show[image,ImageSize->100]
</code></pre>
<p><a href="https://i.stack.imgur.com/rFJEK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rFJEK.png" alt="enter image description here"></a></p>
<p>But SetOptions does no... | userrandrand | 86,543 | <p>I had a similar issue. I wanted to prevent the function in PrePrint from applying when I asked it not to. I did not use any systematic conditions where the function would be automatically suppressed (like if the expression to be evaluated is a plot or an integer).</p>
<p>My solution was to use an inert global variab... |
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