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 |
|---|---|---|---|---|
562,707 | <p>This is a famous rudimentary problem : how to use mathematical operations (not any other temporary variable or storage) to swap two integers A and B. The most well-known way is the following:</p>
<pre><code>A = A + B
B = A - B
A = A - B
</code></pre>
<p>What are some of the alternative set of operations to achieve... | Steven Stadnicki | 785 | <p>Any <em>group operator</em> — that is, any operator that is associative and has an identity and inverse — provides a solution to this problem. Suppose that the operator is $\bigtriangleup$, and that every element $X$ has an inverse $X^{-1}$ such that $X\bigtriangleup X^{-1} = X^{-1}\bigtriangleup X = e_... |
533,399 | <p>Starting with the classical propositional logic, is there a rather canonical way to prove that $$p\wedge q=q\wedge p$$ for the commutativity of the conjunction and analogously for the other properties and connectives, other than using truth tables, visualizing with Venn diagrams akin <a href="http://en.wikipedia.org... | hmakholm left over Monica | 14,366 | <p>Once you select a particular proof system, you should be able to write down a <em>formal proof</em> of $(p\land q)\leftrightarrow (q \land p)$. How such a proof will look will vary wildly between different proof systems, though.</p>
<p>For example, in (classical or intuitionistic) sequent calulus, the formal proof ... |
1,237,595 | <p>Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with parameter 1? Thanks for help.</p>
| mathifold.org | 231,554 | <p>The function $F$ does exist, because you only have to redistribute the values going to $[1,2]$ to values going to $(0,\infty)$. But how to find $F$ explicitely? It must satisfy $F(1)=0$, $F(2)=\infty$ (so strictly speaking the function <strong>will not</strong> be $\mathbb{R}\longrightarrow \mathbb{R}$, or at least ... |
1,237,595 | <p>Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with parameter 1? Thanks for help.</p>
| zoli | 203,663 | <p>The corresponding cdf is:</p>
<p>$$G_X(x)=\begin{cases}0, & \text{ if }x<1\\
\frac{1}{7}x^3& \text{ if }1\le x<2\\
0, & \text{ otherwise. }\end{cases}.$$
Let us calculate the distribution of the random variable $G(X)$. Let it be denoted by $H$:
$$H(x)=P(G_X(X)<x)=P(X<G_X^{-1}(x))=G_X(G_X^{... |
90,480 | <p>Given two simplicial topological spaces $X_{\bullet}$ and $Y_{\bullet}$ (i.e. a simplicial object in Top) and a continuous map between their geometric realizations $f \colon \lvert X_{\bullet} \rvert \to \lvert Y_{\bullet} \rvert$. Is $f$ homotopic to $\lvert \varphi_{\bullet} \rvert$ for a map $\varphi_{\bullet}$ o... | Thomas Nikolaus | 11,002 | <p>The answer is no. For an arbitrary simplicial space $X_\bullet$ we can consider $|X_\bullet|$ as a constant simplicial space, lets call this $Y_\bullet$. Then there is clearly the identity $|X_\bullet| \to |Y_\bullet|$, but there is in general no nontrivial map $X_\bullet \to Y_\bullet$ (take e.g. BG for a top. grou... |
36,568 | <p>To do Algebraic K-theory, we need a technical condition that a ring $R$ satisfies $R^m=R^n$ if and only if $m=n$. I know some counterexamples for a ring $R$ satisfies $R=R^2$. </p>
<p>Are there any some example that $R\neq R^3$ but $R^2 = R^4$ or something like that?</p>
<p>(c.f. if $R^2=R^4$, then we need that $R... | Seamus | 6,701 | <p>Rings that satisfy the condition
$R^n \cong R^m \iff n=m$ are said to have <em>invariant basis number</em> or the <em>invariant basis property</em>. P. M. Cohn has constructed examples of rings which fail to have this property, even giving examples of (non commutative) integral domains for which e.g. $R^3\cong R$ b... |
2,909,022 | <p>I don't understand how to get from the first to the second step here and get $1/3$ in front.</p>
<p>In the second step $g(x)$ substitutes $x^3 + 1$.</p>
<p>\begin{align*}
\int_0^2 \frac{x^2}{x^3 + 1} \,\mathrm{d}x
&= \frac{1}{3} \int_{0}^{2} \frac{1}{g(x)} g'(x) \,\mathrm{d}x
= \frac{1}{3} \int_{1}^{... | MPW | 113,214 | <p>Looking at the denominator, you define $g(x)=x^3+1$. This means that $g'(x)=3x^2$.</p>
<p>Your goal is to manipulate the integrand so that it is in the form $g'(x)/g(x)$, that is, $3x^2/(x^3+1)$. As it stands, the factor $3$ you need is missing. You can't just throw it in, because that changes the integrand. But yo... |
440,082 | <blockquote>
<p><span class="math-container">$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$</span>
Note: <span class="math-container">$\mu$</span> here is an extremely small constant.</p>
</blockquote>
<p>I have tried:</p>
<ol>
<li>Estimating the integral by Tayl... | AccidentalFourierTransform | 106,114 | <p>It is more or less straightforward to write down an asymptotic expansion around <span class="math-container">$\mu\to0$</span>,
<span class="math-container">$$
I(\mu)\sim \frac{2 \mu ^2}{15}-\frac{\mu ^4}{9}+\frac{\pi \mu ^5}{15}+\frac{1}{450} \mu ^6 (60 \log (\mu )+60 \gamma -67)+\cdots
$$</span>
where <span class=... |
937,064 | <p>The title pretty much says it all:</p>
<p>If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?</p>
<p><em>Edit:</em> Let me attempt to be a little more precise:</p>
<p>Suppose you have a proposition. Furthermore, suppose that assuming the proposition is false... | Mark Bennet | 2,906 | <p>It depends on the statement. Some statements e.g.</p>
<blockquote>
<p>This statement is false.</p>
</blockquote>
<p>lead to a contradiction whether you assume them true or false, so don't have an assignable truth value.</p>
<p>You also need to know or prove that your statement has a truth value (i.e. is either ... |
2,989,494 | <p>I am trying to derive properties of natural log and exponential just from the derivative properties.</p>
<p>Let <span class="math-container">$f : (0,\infty) \to \mathbb{R}$</span> and <span class="math-container">$g : \mathbb{R} \to \mathbb{R}$</span>.
Without knowing or stating that <span class="math-container">$f... | Derek Elkins left SE | 305,738 | <p>Do a case analysis on either <span class="math-container">$x\geq y$</span> or <span class="math-container">$y > x$</span>. In the <span class="math-container">$x \geq y$</span> case, <span class="math-container">$x+x \geq x+y$</span> therefore <span class="math-container">$2x\geq 100$</span> so <span class="math-... |
221,053 | <p>Given two uncorrelated random variables $X,Y$ with the same variance $\sigma^2 $ I need to compute $\rho= \frac{COV(X,Y)}{\sigma(X)\sigma(Y)}$ between $X+Y$ and $2X+2Y$. I know it should be a number between $-1$ and $1$ and I don't understand how come I get $4$. </p>
<p>Here's what I did:</p>
<p>$COV(X+Y,2X+2Y)... | André Nicolas | 6,312 | <p>The reason things went wrong is probably due to an unfortunate choice of notation, the use of $X$ and $Y$ with two different meanings. </p>
<p>We want the correlation coefficient $\rho(U,V)$, where $U=X+Y$ and $V=2(X+Y)$. So we need to divide $\text{Cov}(U,V)$ by the product of the standard deviations of $U$ and of... |
1,076,292 | <p>I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$</p>
<p>Is there any direct formula to compute.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Set the values of $x,y$ to form two linear simultaneous equation for the unknowns $A,B$</p>
<p>Solve for $A,B$ in terms of $C, x_1,y_1,x_2,y_2$</p>
|
3,738,508 | <p>If <span class="math-container">$G$</span> is order <span class="math-container">$p^2q$</span>, where <span class="math-container">$p$</span>, <span class="math-container">$q$</span> are primes, prove that either a Sylow <span class="math-container">$p$</span>-subgroup or a Sylow <span class="math-container">$q$</sp... | Angina Seng | 436,618 | <p>If we defined the Frobenius map by <span class="math-container">$F(a)=a^5$</span>, then <span class="math-container">$a$</span> lies in <span class="math-container">$\Bbb F_{5^k}$</span>
iff <span class="math-container">$F^k(a)=a$</span>.</p>
<p>Here let <span class="math-container">$a$</span> be a root of <span cla... |
2,339,974 | <p>I know that there is a theorem that states that If $(G, *)$ and $(H, •)$ are groups, $e_G$ (identity of $G$) and $e_H$ (identity of $H$). Let $f: G\to H$ be a homomorphism. Then </p>
<ol>
<li>$f(e_G) = e_H$.</li>
</ol>
<p>I don't know how to use this, or begin my proof or should I use kernel for this problem?</p>... | Lukas Heger | 348,926 | <p>Let $f:G \to H$ be a homomorphism of groups, where $|G|=20$ and $|L|=17$.
Let $g \in G$ be any element, then the order of $g$ divides $20$, thus $g^{20}=e_G$, so after applying $f$ we get $f(g)^{20}=f(e_G)=e_L$, so the order of $f(g)$ divides $20$ as well. Now as $f(g)$ is an element of $L$, the order of $f(g)$ also... |
4,185,100 | <p>In parallelogram <span class="math-container">$ABCD$</span> <span class="math-container">$CE=ED$</span>. <span class="math-container">$O$</span> is the intersection of <span class="math-container">$AE$</span> and the bisector of <span class="math-container">$\angle ABC$</span>. Given <span class="math-container">$AB... | Overdrowsed | 849,029 | <p>I have extended <span class="math-container">$AE$</span> and <span class="math-container">$BC$</span>, with their intersection point being <span class="math-container">$K$</span>, as shown below</p>
<p><a href="https://i.stack.imgur.com/9Nqk3.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9Nqk3.j... |
4,185,100 | <p>In parallelogram <span class="math-container">$ABCD$</span> <span class="math-container">$CE=ED$</span>. <span class="math-container">$O$</span> is the intersection of <span class="math-container">$AE$</span> and the bisector of <span class="math-container">$\angle ABC$</span>. Given <span class="math-container">$AB... | Mick | 42,351 | <p>Note: This is essentially what you have done. I found that out after I have the solution written up.</p>
<p>Let [ABO] = X, [the required] = Y, [CEK] = Z.</p>
<p><a href="https://i.stack.imgur.com/rc7qU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rc7qU.png" alt="enter image description here" />... |
351,226 | <p>I am trying to brush up on my regular grammar knowledge to prepare for an interview, and I just am not able to solve this problem at all. This is NOT for homework, it is merely me trying to solve this.</p>
<p>I want to give a regular grammar for the language of the finite automaton whose screen shot is below, pleas... | MJD | 25,554 | <p>To convert an automaton to a regular grammar is easy. You have one symbol for each state of the automaton, in this case $A, B,$ and $C$. Each symbol has one production for each transition that the automaton has out of the corresponding state. For example, when you have a transition $A\stackrel{0}{\to} B$ as you d... |
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Ilya Nikokoshev | 65 | <p>What you're asking is (or will be after a slight change of question) essentially a form of "what's the smallest number that cannot be written in words" paradox.</p>
|
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Harrison Brown | 382 | <p>Kenny's point in the comments is a good one as well, and it's also worth keeping in mind that the number of atoms in the observable universe is at most a few orders of magnitude more than 10^80, so writing down all the numbers with 100 or fewer digits is a hopeless task. (Certainly we haven't checked them all for pr... |
4,214,474 | <p>Consider <span class="math-container">$\mathbb{R}^\omega$</span> (countably infinite product of <span class="math-container">$\mathbb{R}$</span>) with the uniform metric.</p>
<p>Let <span class="math-container">$A$</span> be the set of infinite bounded sequences of <span class="math-container">$\mathbb{R}$</span>, i... | Lukas Betz | 238,388 | <p>For your unbounded <span class="math-container">$x$</span> you are only using the fact that there is an <span class="math-container">$M$</span> such that <span class="math-container">$|x_n| > M$</span> for infinitely many <span class="math-container">$n$</span>. This is to weak since it is true for any <span clas... |
1,470,760 | <p>Okay so $A=0.2, B=0.5$ and the probability that both $A$ and $B$ occur is equal to $0.12$.</p>
<p>What is $P((A \cap B) \cup A^c)$?</p>
<p>What I basically did was $0.12 \times 0.5+0.5+0.2-0.12 = 1.2$. </p>
<p>Am I doing it right?</p>
| barak manos | 131,263 | <p>HINT:</p>
<p><a href="https://i.stack.imgur.com/od1UI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/od1UI.png" alt="enter image description here"></a></p>
<p>You need the yellow part plus the blue part.</p>
<p>BTW, probability is a value between $0$ and $1$ <strong>by definition</strong>, so ... |
3,439,223 | <blockquote>
<p>Given the metric space <span class="math-container">$(X,d)$</span>:</p>
<p>If <span class="math-container">$M\subset N \subset X$</span>, <span class="math-container">$M\neq 0$</span>, we have <span class="math-container">$\text{diam}(M)\leq
\text{diam}(N)$</span></p>
</blockquote>
<p>Negating ... | José Carlos Santos | 446,262 | <p>The number <span class="math-container">$\operatorname{diam}M$</span> is the supremum of a set <span class="math-container">$S_M$</span> of numbers and the number <span class="math-container">$\operatorname{diam}N$</span> is the supremum of a set <span class="math-container">$S_N$</span> of numbers. Since <span clas... |
1,620,686 | <p>Prove that this works for all $x$ and and only some $y$
$$\sqrt{(x-1)^2-(y+2)^2}=0.$$</p>
<p>This is as far as I got so far</p>
<p>Difference of squares:</p>
<p>$\sqrt{(x-1-y-2)(x-1+y+2)}=0$<br>
$\sqrt{x-y-3}\sqrt{x+y+1}=0$</p>
<p>Therefore $x-y-3=0 \implies y=x-3$ </p>
<p>$x+y+1=0$ and $y=-1-x$</p>
<p>I ju... | BrianO | 277,043 | <p>What you need to show is the following:</p>
<blockquote>
<p>For every real number $x$, there exists a real number $y$ such that
$$\sqrt{(x-1)^2-(y+2)^2}=0,\tag{Eqn}$$</p>
</blockquote>
<p>An equivalent way of saying "there exists a real $y$" is to say "for some real $y$". Saying "for some <em>of</em> $y$" bri... |
2,483,794 | <p>I'm trying to figure out the equality $$\frac{1}{y(1-y)}=\frac{1}{y-1}-\frac{1}{y}$$</p>
<p>I have tried but keep ending up with RHS $\frac{1}{y(y-1)}$.</p>
<p>Any help would be appreciated.</p>
| Michael Rozenberg | 190,319 | <p>$$\frac{1}{y(y-1)}=\frac{y-(y-1)}{y(y-1)}=\frac{y}{y(y-1)}-\frac{y-1}{y(y-1)}=\frac{1}{y-1}-\frac{1}{y}$$</p>
|
2,118,761 | <p>How can I show that there are an infinite number of primes by using the Fundamental Theorem of Arithmetic?</p>
| marty cohen | 13,079 | <p>The standard way
is by assuming that
there are only a finite number of primes
and deducing that
if all terms of the form
$\prod_{p \in P} p^{a(p)}$
are counted,
there are not enough of them.</p>
<p>I don't remember the details,
but it might go
something like this:</p>
<p>The number of integers
of the form
$\prod_{... |
342,306 | <p>An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.</p>
<p>A critical point of such an embedding is the least ordinal $\alpha$ s... | Paul McKenney | 53,995 | <p>No. If $\kappa$ is the critical point of a full elementary embedding $j : V\to N$, with $N$ transitive, then $\kappa$ is <a href="http://en.wikipedia.org/wiki/Measurable_cardinal" rel="nofollow">measurable</a>. Yet not all large cardinals are measurable; see, for instance, <a href="http://en.wikipedia.org/wiki/Wea... |
342,306 | <p>An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.</p>
<p>A critical point of such an embedding is the least ordinal $\alpha$ s... | zyx | 14,120 | <p>There is an online PDF of lecture slides by Woodin on the "Omega Conjecture" in which he axiomatizes the type of formulas that are large cardinals. I do not know how exhaustive his formulation is. See the references under</p>
<p><a href="http://en.wikipedia.org/wiki/Omega_conjecture" rel="nofollow">http://en.wikip... |
140,358 | <p>Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are homeomorphic. But I wonder how independent $X$ and $Y$ could be ? For example is there any forced relation between their car... | Community | -1 | <p>Let $X,Y$ be arbitrary sets (of arbitrary cardinals) armed with topologies $\tau_1 = \lbrace \emptyset, X\rbrace$ and $\tau_2 = \lbrace \emptyset, Y\rbrace$. Then it is clear that $C(X) \cong \Bbb{R} \cong C(Y)$. So there is no forced relation between the cardinal numbers. </p>
|
2,309,721 | <p>The problem is: Prove that $7|x^2+y^2$ only if $7|x$ and $7|y$ for $x,y∈Z$.</p>
<p>I found a theorem in my book that allows to do the following transformation:
if $a|b$ and $a|c$ -> $a|(b+c)$</p>
<p>So, can I prove it like this: $7|x^2+y^2 =>7|x^2, 7|y^2 => 7|x*x, 7|y*y => 7|x, 7|y$ ?</p>
<p>I am no... | knm | 447,358 | <p>There is a theorem that if a prime $p$ of the form $p \equiv 3\pmod4$, and $p \mid x^2+y^2$, then $p \mid x$ and $p \mid y$.</p>
<p>At your problem take $p=7$, and with the above theorem you are done.</p>
|
119,722 | <p>For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered by reverse inclusion. The empty intersection, $V$ itself, is the unique minimal element of $L(\mathcal{A})$.</p>
<p... | Michael Falk | 15,365 | <p>The intersection poset of a (not necessarily central) hyperplane arrangement is a geometric semi-lattice, as defined by Bjorner and Wachs, who show that every such poset is isomorphic to the subposet of $x \not \geq a$ of a geometric lattice and an atom a. This corresponds geometrically to putting the arrangement in... |
354,642 | <p>Show that each of the following initial-value problems has a unique solution ($0 ≤ t ≤ 1 , y(0) = 1$).</p>
<p>$$y' = \exp(t-y)$$</p>
<p><strong>Theorem 1</strong>: Suppose that $D=\{(t,y)|a≤t≤b, −∞< y<∞\}$ and that $f(t,y)$ is continuous on $D$. If $f$ satisfies a Lipschitz condition on $D$ in the variable $... | Community | -1 | <p>Consider the sequences $$x_n = \dfrac1{2n \pi} \,\,\,\, \text{ and } x_n = \dfrac1{2 n \pi + \dfrac{\pi}2}$$ Both tend to zero. What happens to $\cos(1/x)$ along these sequences? (Recall that if a limit exists it has to be unique.)</p>
|
468,784 | <p>Two disjoint sets $A$ and $B$, neither empty, are said to be <strong>mutually separated</strong> if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is called <strong>connected</strong> if it is not disconnected.</p>
<p>With the above definition of c... | Betty Mock | 89,003 | <p>If |x| < |y| then those x's are all in the disk of radius y about the origin. That is surely connected</p>
|
468,784 | <p>Two disjoint sets $A$ and $B$, neither empty, are said to be <strong>mutually separated</strong> if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is called <strong>connected</strong> if it is not disconnected.</p>
<p>With the above definition of c... | Balbichi | 24,690 | <p>Do you know straight lines are connected set in $\mathbb{R}^2$? do you know union of connected set is connected if they have one point in common?</p>
<p>Now can you see your set with above stated properties?</p>
|
2,881,020 | <p>We're trying to figure out something, and things aren't adding up. The senario is that you make $\$ 50,000$ a year. Every year you get a $15\%$ bonus of that income, which then gets added to your next year's income. So, the first year, you get $\$ 7,500$ which then makes your base $\$ 57,500$ the next year. When we ... | mzp | 287,326 | <p>I am not sure this is what you want, but here is the calculation for the bonuses in the first $3$ years which should get you going:</p>
<p><strong>1st Year:</strong></p>
<p>$$ \text{Base} = 50,000 \quad \Rightarrow \quad \text{Bonus}=0.15*50,000 = 7,500. $$</p>
<p><strong>2nd Year:</strong></p>
<p>$$ \text{Base}... |
2,881,020 | <p>We're trying to figure out something, and things aren't adding up. The senario is that you make $\$ 50,000$ a year. Every year you get a $15\%$ bonus of that income, which then gets added to your next year's income. So, the first year, you get $\$ 7,500$ which then makes your base $\$ 57,500$ the next year. When we ... | Ross Millikan | 1,827 | <p>The general rule is that in year $n$ the base is $50,000\cdot 1.15^n$ and the bonus is $7,500\cdot 1.15^n$ This increases without bound.
<a href="https://i.stack.imgur.com/0VDsp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0VDsp.png" alt="enter image description here"></a></p>
|
162,655 | <p>Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? </p>
<p>I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in L... | Yiyan | 22,880 | <p>Eguchi - Hanson metric over $T^*S^2$ is complete Ricci flat but not flat, which can be written down explicitly. In fact, $T^*S^n$ admits Calabi-Yau structure for each $n$. </p>
<p>Ref: Stenzel, Ricci flat metrics on the complexification of a compact rank one symmetric space.</p>
|
1,779,088 | <blockquote>
<p>Prove
$$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$
for every integer $k\ge0$. </p>
</blockquote>
<p>By principle of induction,</p>
<p>$$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$
$$2\sum_{i=1}^n i = n(n+1)$$
$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$
$\implies$$(... | Mark Viola | 218,419 | <p>Rather than proceed using induction, I thought it might be instructive to present a straightforward approach. </p>
<p>To that end we proceed by using the <a href="https://en.wikipedia.org/wiki/Summation_by_parts#Newton_series" rel="nofollow">Newton Series</a> for summation by parts with $f_i=i$ and $g_i=i^k$.</p>
... |
2,314,327 | <p>I have a quick question here.</p>
<p>For an exercise, I was asked to factor:</p>
<p>$$11x^2 + 14x - 2685 = 0$$</p>
<p>How do I figure this out quickly without staring at it forever? Is there a quicker mathematical way than guessing number combinations, or do I have to guess until I find the right combination of n... | Ross Millikan | 1,827 | <p>You want to know the <a href="https://en.wikipedia.org/wiki/Divisibility_rule" rel="noreferrer">divisibility rules</a> for small numbers. We know $2685$ is divisible by $5$ because of the last digit and by $3$ because of the sum of the digits. Once you find those factors, divide them out, getting $179$. The rules... |
341,648 | <p>I'm trying to understand what a tableaux ring is (it's not clear to me reading Young Tableaux by Fulton).</p>
<p>I studied what a monoid ring is on Serge Lang's Algebra, and then I read about modules, modules homomorphism. I'm trying to prove what is stated at page 121 (S. Lang, Algebra) while talking about algebra... | Did | 6,179 | <p>The integral $I$ to be computed is
$$
I=2\int_0^{1/2}\log(\sin\pi x)\mathrm dx\stackrel{x\to 1/2-x}{=}2\int_0^{1/2}\log(\cos\pi x)\mathrm dx.
$$
Summing up yields
$$
2I=2\int_0^{1/2}\log(\cos\pi x\sin\pi x)\mathrm dx=2\int_0^{1/2}\log(\sin2\pi x)\mathrm dx-2\int_0^{1/2}\log(2)\mathrm dx.
$$
The first integral on the... |
977,956 | <p>Can you help me solve this problem?</p>
<blockquote>
<p>Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.</p>
</blockquote>
| Community | -1 | <p>Take the terms in opposite pairs, and note the change of sign,</p>
<p>$$\sin \dfrac{2k\pi}{n}+\sin \dfrac{2\pi(n-k)}{n}=\sin \dfrac{2k\pi}{n} +\sin(2\pi-\dfrac{2k\pi}{n})=\sin \dfrac{2k\pi}{n} -\sin\dfrac{2k\pi}{n}=0.$$
In case that $n$ is even, the central term remains, but $$\sin \dfrac{2n\pi}{2n}=0.$$</p>
|
4,216,105 | <p>In the <a href="https://www.feynmanlectures.caltech.edu/I_22.html#Ch22-S5" rel="nofollow noreferrer">Algebra chapter</a> of the Feynman Lectures on Physics, Feynman introduces complex powers:</p>
<blockquote>
<p>Thus <span class="math-container">$$10^{(r+is)}=10^r10^{is}\tag{22.5}$$</span>
But <span class="math-con... | Brick | 263,389 | <p>Feynman in that whole section is using results that he knows to be true and trying to provide some intuition, but you've picked one of several things in that chapter that are not mathematically rigorous. Even just with the equations you've shown, a mathematician would want to prove that <span class="math-container"... |
43,513 | <p>Please help me with the following question.</p>
<p>Let $F:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a continuously differentiable mapping;</p>
<p>$F_{n}(x)$ be $n$-th iteration of $F(x)$, i.e. $F_{1}(x)=F(x)$, $F_{n}(x)=F(F_{n-1}(x))$;</p>
<p>$J_{n}(x)=(F_{n}(x))'$ be Jacobian matrix of $F_{n}(x)$;</p>
<p>$\lambda_{n}... | rpotrie | 5,753 | <p>In general, this may not be true. </p>
<p>EDIT (Atending OP´s objection) The important thing is that as stated, the problem is reduced to a linear algebra problem since it is possible to construct a diffeomorphism of $\mathbb{R}^n$ such that matrix $J_n(0)$ for the orbit of $0$ is the product of any sequence of inv... |
43,513 | <p>Please help me with the following question.</p>
<p>Let $F:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a continuously differentiable mapping;</p>
<p>$F_{n}(x)$ be $n$-th iteration of $F(x)$, i.e. $F_{1}(x)=F(x)$, $F_{n}(x)=F(F_{n-1}(x))$;</p>
<p>$J_{n}(x)=(F_{n}(x))'$ be Jacobian matrix of $F_{n}(x)$;</p>
<p>$\lambda_{n}... | Alexandra Korobeynikova | 10,343 | <p>But this is not a counterexample. You proposed $F(x)=Ax+b$ if $\|x\|\leq\delta$, $F(x)=x+b$ if $\|x\|>\delta$ where $\|b\|$ is sufficiently large.</p>
<p>Here $A$ is a matrix with eigenvalues $\lambda^{(1)}, \lambda^{(2)}, \ldots\lambda^{(k)}$;</p>
<p>$\mu^{(1)}, \mu^{(2)}, \ldots\mu^{(k)}$ are eigenvalues of $... |
332,170 | <p>How can I solve <span class="math-container">$$T(n) = aT(n-1) + bT(n-2)+ cn $$</span>; where <span class="math-container">$a,b,c$</span> are constants. I could not figure it out :(</p>
<p>There are T(0) = d and T(1) = e,</p>
<p>Thanks in advance.</p>
| Brian M. Scott | 12,042 | <p>Your general problem is significantly harder than the specific problem that gave rise to it. I would not use the characteristic equation at all for the specific problem. For the specific recurrence $T(n)=T(n-2)+4n$ with initial conditions $T(0)=2$ and $T(1)=3$, I’d separate it into two sequences, one corresponding t... |
332,170 | <p>How can I solve <span class="math-container">$$T(n) = aT(n-1) + bT(n-2)+ cn $$</span>; where <span class="math-container">$a,b,c$</span> are constants. I could not figure it out :(</p>
<p>There are T(0) = d and T(1) = e,</p>
<p>Thanks in advance.</p>
| André Nicolas | 6,312 | <p>We use your specific equation. Look for a <strong>particular</strong> solution of the shape $an^2+bn+c$, actually in this case $an^2+bn$ is good enough.</p>
<p>Substituting in the equation, we get $an^2+bn=a(n-2)^2+b(n-2)+4n$, Comparing coefficients, we find that $-4a+4=0$ and $4a-2b=0$. Thus $a=1$, $b=2$. So we ha... |
28,568 | <p>Recently, I answered to this problem:</p>
<blockquote>
<p>Given <span class="math-container">$a<b\in \mathbb{R}$</span>, find explicitly a bijection <span class="math-container">$f(x)$</span> from
<span class="math-container">$]a,b[$</span> to <span class="math-container">$[a,b]$</span>.</p>
</blockquote>
<... | mjqxxxx | 5,546 | <p>Define a bijection <span class="math-container">$f:(-1,1)\rightarrow[-1,1]$</span> as follows: <span class="math-container">$f(x)=2x$</span> if <span class="math-container">$|x|=2^{-k}$</span> for some <span class="math-container">$k\in\mathbb{N}$</span>; otherwise <span class="math-container">$f(x)=x$</span>.</p>
|
28,568 | <p>Recently, I answered to this problem:</p>
<blockquote>
<p>Given <span class="math-container">$a<b\in \mathbb{R}$</span>, find explicitly a bijection <span class="math-container">$f(x)$</span> from
<span class="math-container">$]a,b[$</span> to <span class="math-container">$[a,b]$</span>.</p>
</blockquote>
<... | CopyPasteIt | 432,081 | <p>For background to this 'turn-the-crank' technique see this <a href="https://math.stackexchange.com/a/3156178/432081">answer</a>.</p>
<p>Let <span class="math-container">$A = (0,1)$</span> and <span class="math-container">$B = [0,1]$</span>. Let <span class="math-container">$f: A \to B$</span> be the inclusion mappi... |
4,132,402 | <p>Can this be solved without trigonometry?</p>
<blockquote>
<p><span class="math-container">$AB$</span> is the base of an isosceles <span class="math-container">$\triangle ABC$</span>. Vertex angle <span class="math-container">$C$</span> is <span class="math-container">$50^\circ$</span>. Find the angle between the alt... | quasi | 400,434 | <p>Using trigonometry, the angle in question is approximately equal, in degrees, to
<span class="math-container">$$
10.558536057412143196227467316938626443256567512439
$$</span>
which, given the lack of an apparent repeating block, is almost certainly not a rational number.</p>
<p>
Hence you should not expect a solutio... |
947,618 | <p>For T: V2->V2</p>
<p>T maps each point with polar coordinate (r.theta) to each point with polar coordinate (r,2theta) and T maps 0 onto itself.</p>
<p>Hi,</p>
<p>I was trying to do this by letting r= square root of x^2 + y^2 and theta=arctan(y/x) </p>
<p>but I failed.</p>
<p>can anybody please explain it? </p>
| Mohamed | 33,307 | <p>$T$ is not linear since $$T(1+i)=T\left({\sqrt 2} e^{i\frac{\pi}4} \right)={\sqrt 2} e^{i\frac{\pi}2} =i {\sqrt 2}$$
and: $$T(1)+T(i)=1+(-1)=0$$</p>
|
3,601,552 | <p>A school has <span class="math-container">$500$</span> girls and <span class="math-container">$500$</span> boys. A simple random sample is obtained by selecting names from a box (with replacement) to a get a sample of <span class="math-container">$10$</span>. </p>
<p>Find the probability of someone being picked mor... | Rezha Adrian Tanuharja | 751,970 | <p>Probability of anybody being picked more than once:</p>
<p><span class="math-container">$$
\frac{\binom{10^{3}}{10}\times 10!}{\left(10^{3}\right)^{10}}
$$</span></p>
<p>Probability of a particular person being picked more than once:</p>
<p><span class="math-container">$$
1-\frac{\left(10^{3}-1\right)^{10}+10\tim... |
666,461 | <p>The function $f(x)=x+\log x$ has only one root on $(0,\infty)$ which is in $(0,1)$.</p>
<p>Using the Intermediate value theorem: $f$ is continuous on $(0,\infty)$ and $f(0)=0+\log(0)=-\infty<0$ and $f(1)=1+\log(1)=1>0$. So there exists an $x$ such $f(x)=0$.</p>
<p>But how to show that this $x$ is the only ro... | zhw. | 228,045 | <p>Note $f(1/e) = 1/e-1<0,$ $f(1) = 1>0.$ By the IVT, $f$ has a root in $(1/e,1).$ Also note $f'(x) = 1 + 1/x > 0.$ Thus $f$ is strictly increasing, and therefore the root just found is the unique root.</p>
|
4,373,055 | <p>We define a bound vector to be a quantity with a defined starting point, magnitude and direction. A free vector has no defined starting point, just magnitude and direction.</p>
<p>So what is a position vector (of a point)? It is defined relative to something else (Origin), so it has a starting point, size and direct... | Karl | 279,914 | <p>The "free"/"bound" terminology doesn't have a precise mathematical meaning, it's just an informal description of how we're thinking about a vector quantity or what we're using it to represent. The vector itself never contains information about a starting point; it's always just a direction with a... |
323,783 | <p>How do I evaluate this definite integral?
$$\int_{0}^{\frac{\pi}{12}}{\sin^4x \, \cos^4x\, \operatorname{d}\!x}$$
I know this is a trig. function. </p>
| Community | -1 | <p>We have
$$\sin^4x\cos^4x=\frac{1}{16}\sin^4(2x)=\frac{1}{16}(\frac{e^{2ix}-e^{-2ix}}{2i})^4=\frac{1}{16^2}(2\cos(8x)-8\cos(4x)+6), $$
so we integrate and we find
$$\int_{0}^{\frac{\pi}{12}}{\sin^4x \, \cos^4x\, \operatorname{d}\!x}=\frac{1}{512}π-\frac{7}{2048}\sqrt{3}.$$</p>
|
945,651 | <p>Use mathematical induction to prove the following statement:</p>
<p>For all $b\in\mathbb R$, and for all $n\in\mathbb N$, $$b>-1\implies (1+b)^n \geq 1+nb$$</p>
<p>When $n=1$, the inequality still holds
$1+b \geq 1+b$.</p>
<p>For n+1$:
$$(1+b)^{n+1} \geq 1+(n+1)b$$
Here I'm not sure the best way to simplify...... | Kim Jong Un | 136,641 | <p>For the induction step, if $(1+b)^n\geq 1+bn$, then
$$
(1+b)^{n+1}-(1+b(n+1))\geq (1+b)(1+bn)-(1+b(n+1))=b^2n\geq 0.
$$
Note that the first inequality above uses both the induction hypothesis and $b>-1$.</p>
|
945,651 | <p>Use mathematical induction to prove the following statement:</p>
<p>For all $b\in\mathbb R$, and for all $n\in\mathbb N$, $$b>-1\implies (1+b)^n \geq 1+nb$$</p>
<p>When $n=1$, the inequality still holds
$1+b \geq 1+b$.</p>
<p>For n+1$:
$$(1+b)^{n+1} \geq 1+(n+1)b$$
Here I'm not sure the best way to simplify...... | Community | -1 | <p>$$(1+b)^{n+1} =(1+b)^n (1+b) \geq (1+nb)(1+b) =1+(n+1)b +nb^2 \geq 1+ (n+1)b.$$</p>
|
288,051 | <p>In enumerative combinatorics, a <i>bijective proof</i> that $|A_n| = |B_n|$ (where $A_n$ and $B_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection between $A_n$ and $B_n$.
Bijective proofs are often prized because of their beauty and because of the insight that ... | Per Alexandersson | 1,056 | <p>It is perhaps hard still to automate bijection finding, but if you have a database of statistics on A and B, you can automatically check if there is (empirically) a bijection which sends some statistic on A, to some other statistic on B. That is, you refine the bijection.</p>
<p>I have used this approach successfull... |
4,138,292 | <p>Can anybody help me understand why these terms are always reciprocals? (theta <= 45°)</p>
<p><span class="math-container">$$ x = \frac{1}{\cos \theta} + \tan{\theta} $$</span>
<span class="math-container">$$ \frac{1}{x} = \frac{1}{\cos \theta} - \tan{\theta} $$</span></p>
<p>I understand that if we multiply the... | David K | 139,123 | <p>There is a theorem (or set of theorems) of geometry called the
<a href="https://artofproblemsolving.com/wiki/index.php/Power_of_a_Point_Theorem" rel="noreferrer">Power of a Point</a>.
Note that this theorem is easily proved <a href="https://www.cut-the-knot.org/pythagoras/PPower.shtml" rel="noreferrer">without using... |
256,612 | <p>I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (which implies they may exist). I've found suggestions that recognition and untangling require exponential time. (Untangl... | Ian Agol | 1,345 | <p>Recently, Marc Lackenby discussed a new algorithm for unknot recognition in a <a href="https://www.newton.ac.uk/seminar/20170130113012301" rel="noreferrer">talk at the Newton Institute</a> (see time around 1:03). He conjectures (but indicates at the time that it is work-in-progress) that his algorithm runs in quasi-... |
95,314 | <p>To evaluate this type of limits, how can I do, considering $f$ differentiable, and $ f (x_0)> 0 $</p>
<p>$$\lim_{x\to x_0} \biggl(\frac{f(x)}{f(x_0)}\biggr)^{\frac{1}{\ln x -\ln x_0 }},\quad\quad x_0>0,$$</p>
<p>$$\lim_{x\to x_0} \frac{x_0^n f(x)-x^n f(x_0)}{x-x_0},\quad\quad n\in\mathbb{N}.$$</p>
| N. S. | 9,176 | <p>For the second limit you can also observe that if $x_0 \neq 0$ then</p>
<p>$$\lim_{x\to x_0} \frac{x_0^n f(x)-x^n f(x_0)}{x-x_0} = \lim_{x \to x_0}x^nx_0^n \frac{ \frac{f(x)}{x^n}- \frac{f(x_0)}{x_0^n}}{x-x_0}= (x_0)^{2n} (\frac{f(x)}{x^n})'(x_0) \,.$$</p>
<p>The case $x_0=0$ is trivial.</p>
|
4,236,148 | <p><span class="math-container">$R=\{(x,y):x^2=y^2\}$</span> and I have to determine whether its an equivalence relation.</p>
<p>I found that it's reflexive but for the symmetry part I got confused as <span class="math-container">$x=y$</span> is sometimes said to be symmetric others not so I don't know what to take it ... | Ivo Terek | 118,056 | <p>You can even replace taking squares with an arbitrary function. Namely, if <span class="math-container">$X$</span> is a set, <span class="math-container">$Y$</span> is a second set, and <span class="math-container">$f\colon X\to Y$</span> is a function, then define <span class="math-container">$x\sim x'$</span> iff ... |
222,093 | <p>For what value of m does equation <span class="math-container">$y^2 = x^3 + m$</span> has no integral solutions?</p>
| LieX | 46,134 | <p>LHS being a perfect square must have "digital root" $1$,$4$,$7$ or $9$. Cubes have digital root $1$,$8$ or $9$. Hence RHS doesn't have same digital roots as LHS so can't be equal.</p>
|
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | Eric Astor | 16,599 | <p>Option (A) is almost right!</p>
<p>Specifically, it is a property of limits that if $c$ is a constant,
\begin{equation*}
\lim_{x\to a}{c}=c.
\end{equation*}
You can prove this using the definition of a limit (that is, using a $\delta$-$\epsilon$ proof).</p>
<p>So $\lim_{x\to\infty}{\frac{11}{7}}=\frac{11}{7}$, hav... |
228,036 | <p>I quote from the <a href="http://en.wikipedia.org/wiki/Von_Neumann_cardinal_assignment" rel="nofollow">Wikipedia article</a>:</p>
<p>"So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and $\omega_\alpha$ for w... | Cameron Buie | 28,900 | <p>Perhaps they're alluding to the fact that without the axiom of choice, not all infinite cardinals are alephs; with it, the finite ordinals and the $\aleph_\alpha$ take care of all the cardinals.</p>
|
1,946,438 | <p>I solved the equation $e^{e^z}=1$ and it seemed to easy so I suspect I must be missing something.</p>
<blockquote>
<p>Would someone please check my answer?</p>
</blockquote>
<p>My original answer:</p>
<p>$e^{e^z}=1$ if and only if $e^z = 2\pi i k$ for $k\in \mathbb Z$ if and only if $z=\ln(2\pi i k)$ for $k\in ... | MPW | 113,214 | <p>The problem is that your expression "$\ln 2\pi ik$" is multivalued (worse, it is undefined if $k=0$).</p>
<p>One may write $$e^{e^z}=1 \iff e^z = 2\pi i k \stackrel{k\neq 0}{\iff} z =
\begin{cases}
\ln 2\pi k + \frac{\pi i}{2}(4n+1), & k>0\\
\ln -2\pi k + \frac{\pi i}{2}(4n-1), & k<0
\end{cases}$$</p>... |
69,590 | <p>Consider the following code.</p>
<pre><code>f[a_,b_]:=x
x=a+b;
f[1,2]
(* a + b *)
</code></pre>
<p>From a certain viewpoint, one might expect it to return <code>3</code> instead of <code>a + b</code>: the symbols <code>a</code> and <code>b</code> are defined during the evaluation of <code>f</code> and <code>a+b</c... | Carl Woll | 45,431 | <p>I gave the following answer to essentially the same <a href="https://mathematica.stackexchange.com/q/184624/45431">question</a>:</p>
<pre><code>TagSetDelayed[x, lhs_, rhs_] ^:= SetDelayed @@ (Hold[lhs, rhs] /. OwnValues[x])
</code></pre>
<p>For your example:</p>
<pre><code>x = a + b;
x /: f[a_, b_] := x
</code></... |
4,151 | <p>Since it's currently summer break, and I've a bit more time than normal, I've been organizing my old notes. I seem to have an almost unwieldy amount of old handouts and tests from classes previously taught. I'm hesitant to get rid of these, as they may provide useful for some future course. Because I adjunct at a fe... | user1984 | 1,984 | <p>I find that the most efficient way to store things is to scan them in. You will hopefully find that the photocopier at your school can batch-scan and email the copies to you, or put them on a USB.</p>
<p>After that I give them very descriptive names according to a naming scheme, so that they easy to search for. By ... |
3,014,438 | <p>Find Number of Non negative integer solutions of <span class="math-container">$x+2y+5z=100$</span></p>
<p>My attempt: </p>
<p>we have <span class="math-container">$x+2y=100-5z$</span> </p>
<p>Considering the polynomial <span class="math-container">$$f(u)=(1-u)^{-1}\times (1-u^2)^{-1}$$</span></p>
<p><span class=... | Key Flex | 568,718 | <p>An alternative way.</p>
<p>Given <span class="math-container">$x+2y+5z=100$</span> and it is clear that <span class="math-container">$0\le z\le20$</span>.</p>
<p>For any possible values of <span class="math-container">$z$</span>, <span class="math-container">$x+2y=100-5z$</span>.</p>
<p>Let us take <span class="m... |
4,389,441 | <p>I'm dealing with a sample problem where I want to work out the probability of a fair coin toss landing <em>heads</em> and a fair die roll landing <em>6</em>. We are then told that <strong>at least</strong> one of those events has happened.</p>
<p>Why is the probability of this not as simple as <em>P(C)P(D) = 0.083</... | Siong Thye Goh | 306,553 | <p>Because now you have the additional information that at least one of those events has happened.</p>
<p>Hence the corresponding probability is <span class="math-container">$$\frac{P(CD)}{P(CD^c) + P(C^cD) + P(CD)}=\frac{P(CD)}{1-P(C^cD^c)}$$</span></p>
<p>That is I have excluded the possibility of <span class="math-c... |
1,512,528 | <p>As the title says, I'm looking to find all solutions to $$x^2 \equiv 4 \pmod{91}$$ and I am not exactly sure how to proceed.</p>
<p>The hint was that since 91 is not prime, the Chinese Remainder Theorem might be useful.</p>
<p>So I've started by separating into two separate congruences:
$$x^2 \equiv 4 \pmod{7}$$ $... | MrMazgari | 284,607 | <p>We begin with your system of equations: $$\begin{cases} x^2 \equiv 4 \pmod{7} \\ x^2 \equiv 4 \pmod{13} \end{cases}$$</p>
<p>Then, solving each congruence, we obtain the system: $$\begin{cases} x \equiv \pm 2 \pmod{7} \\ x \equiv \pm 2 \pmod{13} \end{cases}$$</p>
<p>We therefore have four systems of linear congrue... |
1,862,232 | <p>I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain:<br>
An integral domain $R$ is called to be a <em>Euclidean domain</em> precisely when there is a function $f: R\setminus\{0\}\rightarrow\Bbb N_0$, called degree function of $R$, such that:<br>
(i) If $a,b \in R\... | egreg | 62,967 | <p>You can't prove that units have degree $0$: if $f\colon R\setminus\{0\}\to\mathbb{N}_0$ is a degree function, also
$$
f_k\colon R\setminus\{0\}\to\mathbb{N}_0,
\qquad f_k(a)=f(a)+k
$$
is a degree function as well, for every integer $k\ge -f(1)$ (because $f(1)$ is the minimum value attained by $f$).</p>
<p>You can <... |
1,487,966 | <p>I have been looking at stereographic projections in books, online but they all seem...I don't know how else to put this, but very pedantic yet skipping the details of calculations.</p>
<p>Say, I have a problem here which asks;</p>
<blockquote>
<p>Let <span class="math-container">$n \geq 1$</span> and put <span class... | David K | 139,123 | <p>The point of finding equations with $\lambda(x)$ and individual coordinates
$x_k$ and $y_k$ is that you do not already know what all the values of all
the coordinates are. You may have been given the coordinates $(x_0, \ldots, x_n)$
but not the coordinates $(y_0, \ldots, y_n)$,
and you want to find those coordinates... |
190,948 | <p>So I have </p>
<pre><code>Emin[T_, d_] := If[T == 0, 0, 1/(-1 + E^(1/T)) - d/(-1 + E^(d/T))];
</code></pre>
<p>which I can solve wonderfully for </p>
<pre><code>Solve[Emin[T, 10] == 0.5, T, Reals]
{{T -> 0.910427}}
</code></pre>
<p>Now as you see <code>Emin[T_, d_]</code> is also dependent on d which I set t... | bill s | 1,783 | <p>Reformatting a little:</p>
<pre><code>Emin[T_, d_] := If[T == 0, 0, 1/(-1 + E^(1/T)) - d/(-1 + E^(d/T))];
Tlist = Table[{d, First[Solve[Emin[T, d] == 0.5, T, Reals]
// Values // Flatten // N]}, {d, 5, 10}] // Quiet
ListPlot[Tlist]
</code></pre>
<p><a href="https://i.stack.imgur.com/boO5k.png" rel=... |
8,023 | <p>I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for $n=1$, all Riemannian manifolds are modeled on $\mathbb R$. When $n=2$, I believe that it suffices for the scalar c... | Kevin H. Lin | 83 | <p>This is just a rephrasing of Deane's answer, but let me add one general comment. To any Riemannian metric (or pseudo-Riemannian metric) $g$ on a manifold $M$, you can associate a <a href="http://en.wikipedia.org/wiki/Levi-Civita_connection" rel="nofollow">Levi-Civita connection</a> $\nabla : T_M \to T_M \otimes \Ome... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | fleablood | 280,126 | <p>So you got to <span class="math-container">$b\sqrt 2$</span> is an integer.</p>
<p>Thus <span class="math-container">$b\sqrt 2- b$</span> is an integer because if you subtract two integers you get an integer.</p>
<p><span class="math-container">$b\sqrt 2-b = b(\sqrt 2 -1):=b^*$</span> is an integer.</p>
<p>Hopefu... |
4,383,800 | <p>I can already see that the <span class="math-container">$\lim_\limits{n\to\infty}\frac{n^{n-1}}{n!e^n}$</span> converges by graphing it on Desmos, but I have no idea how to algebraically prove that with L’Hopital’s rule or induction. Where could I even start with something like this?</p>
<p>Edit: For context, I came... | xpaul | 66,420 | <p>By the Stirling approximation <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Stirling%27s_approximation</a>,
<span class="math-container">$$ n!> \sqrt{2\pi n}\bigg(\frac{n}{e}\bigg)^n e^{\frac1{12n+1}}$$</span>
one has
<span class="math-c... |
3,730,083 | <p>If <span class="math-container">$a,b>0$</span> and <span class="math-container">$Q=\{x_1, x_2, x_3,..., x_a\}$</span> a subset of the natural numbers <span class="math-container">$1, 2, 3,..., b$</span> such that, for <span class="math-container">$x_i+x_j<b+1$</span> with <span class="math-container">$1 ≤ i ≤ ... | 5xum | 112,884 | <p>Literally from the first paragraph from the <a href="https://en.wikipedia.org/wiki/Hypercube_graph" rel="nofollow noreferrer">Wikipedia article on the hypercube</a>:</p>
<blockquote>
<p><span class="math-container">$Q_n$</span> has <span class="math-container">$2^n$</span> vertices, <span class="math-container">$2^{... |
225,866 | <p>If I define, for example,</p>
<pre><code>f[OptionsPattern[{}]] := OptionValue[a]
</code></pre>
<p>Then the output for <code>f[a -> 1]</code> is 1.</p>
<p>However, in my code, I have a function that must be called using the syntax <code>f[some parameters][some other parameters]</code>, and I want to add options to... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
vl = {6, 9, 10};
vars3d = Array[Through[{x, y, z}@vl[[#]]] &, 3]
(* {{x[6], y[6], z[6]}, {x[9], y[9], z[9]}, {x[10], y[10], z[10]}} *)
{x[#], y[#], z[#]} & /@ vl
(* {{x[6], y[6], z[6]}, {x[9], y[9], z[9]}, {x[10], y[10], z[10]}} *)
Table[{x[n], y[n], z[n]}, {n, vl}]
... |
4,549,898 | <p>I need some help with solving the following problem: Let <span class="math-container">$Q(n)$</span> be the number of partitions of <span class="math-container">$n$</span> into distinct parts. Show that <span class="math-container">$$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$</span> is convergent by estimating <span class="... | TravorLZH | 748,964 | <p>Let <span class="math-container">$p(n)$</span> be the unrestricted partition function. Then we have <span class="math-container">$Q(n)\le p(n)$</span>. Note that for <span class="math-container">$|z|<1$</span></p>
<p><span class="math-container">$$
1+\sum_{n\ge1}p(n)z^n=\prod_{k\ge1}(1-z^k)^{-1},
$$</span></p>
<p... |
76,600 | <p>The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective space $RP^3$. Does $RP^3$ cause a separation of space in the manifold of $SE(3)$? </p>
<p>(edit) Sorry about lack of cl... | Ryan Budney | 1,465 | <p>Okay, now I think I understand your question. This is the question I will answer:</p>
<ul>
<li>Question: Let $X$ be a connected $4$-dimensional subspace of $SE(3)$ that contains $SO(3)$. Is it possible for $X \setminus SO(3)$ to be connected? Disconnected? </li>
</ul>
<p>The answer to both questions is yes. So... |
2,489,988 | <p>A sequence of numbers is formed from the numbers $1, 2, 3, 4, 5, 6, 7$ where all $7!$ permutations are equally likely. What is the probability that anywhere in the sequence there will be, at least, five consecutive positions in which the numbers are in increasing order?</p>
<p>I approached this problem in the follo... | N. F. Taussig | 173,070 | <p><em>This solution does not differ in any essential way from that of N. Shales. I am posting this here so N. Shales can compare our approaches.</em></p>
<p>Since the sequence contains seven numbers, any block of five consecutive increasing numbers must start in the first, second, or third positions. Let $A_1$, $A_... |
256,549 | <p>Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a maximal element." </p>
<p>In proving the $\Rightarrow$ direction, he uses the following argument: Consider a nonempty subs... | user642796 | 8,348 | <p>The Axiom of Choice is the mechanism that allows you to construct this rule/function. Even though you know that to each $\gamma \in \Gamma^\prime$ there is some $\gamma^\prime \in \Gamma^\prime$ greater than it, this only means that for each $\gamma$ the collection $$A_\gamma = \{ \gamma^\prime \in \Gamma^\prime : ... |
256,549 | <p>Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a maximal element." </p>
<p>In proving the $\Rightarrow$ direction, he uses the following argument: Consider a nonempty subs... | Rudy the Reindeer | 5,798 | <p>No, "no maximal element" doesn't give you a choice function. A choice function is a function $\phi: \mathcal P(\Gamma')\setminus \{\varnothing\} \to \Gamma'$ such that $\phi(x) \in x \subset \Gamma'$ for all subsets $x$ of $\Gamma'$. </p>
<p>If $\Gamma'$ has no maximal element then for every $\gamma \in \Gamma'$ yo... |
4,224,417 | <p>I'm trying to learn a bit of Number Theory. And while I understand the definition of congruence relations modulo <span class="math-container">$n$</span> and that they are an equivalence relations, I fail to see the <em>motivation</em> for it. So what is congruence relation <span class="math-container">$\bmod n$</spa... | Claudio Buffara | 58,890 | <p>The short answer is that there are many problems in number theory whose solutions depend on looking at the remainders of numbers when they are divided by a specific number m. I can think of no simpler example to illustrate this than the problem of deciding when a given natural number is divisible by 3 or by 9 or by ... |
1,566,111 | <p>prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero</p>
<hr>
<p><strong>Defintions</strong></p>
<p>Def of prime Ideal (n)
$$ ab\in (n) \implies a\in(n) \vee b\in(n) $$
Def 1] integer n is prime if $n \neq 0,\pm 1 $ and only divisors are $\pm n,\pm 1$ </p>
<p>Def 2 of n is prime] If $n\neq0,\pm1$ ... | qwr | 122,489 | <p>Yes, from <span class="math-container">$p \mid a \iff a \in (p)$</span> we have more generally in a commutative ring with unity, <span class="math-container">$(p)$</span> is a prime ideal iff <span class="math-container">$p$</span> is prime (in the ring theory sense: <span class="math-container">$p \mid ab \implies ... |
2,130,658 | <p>How would I go about proving this mathematically? Having looked at a proof for a similar question I think it requires proof by induction. </p>
<p>It seems obvious that it would be even by thinking about the first few cases. As for $n=0$ there will be no horizontal dominoes which is even, and for $n=1$ there can onl... | samerivertwice | 334,732 | <p>If it's only $2$ high then any domino laid horizontally, and offset from the one above or below, leaves a single space above and a single below, which cannot be filled without leaving another single space.</p>
<p>Therefore any horizontal domino must be one of a pair laid one directly above the other.</p>
|
2,130,658 | <p>How would I go about proving this mathematically? Having looked at a proof for a similar question I think it requires proof by induction. </p>
<p>It seems obvious that it would be even by thinking about the first few cases. As for $n=0$ there will be no horizontal dominoes which is even, and for $n=1$ there can onl... | Batominovski | 72,152 | <p><strong>Coloring Argument</strong></p>
<p>Let's say that the $n$-by-$2$ board is colored black on one $n$-by-$1$ row and white on the other row. Every vertical domino takes a square of each color, whilst a horizontal domino can only take two squares of the same color. Since the number of black squares is the same ... |
2,628,220 | <p>Let $(a_{n})_{n \in \mathbb N_{0}}$ be a sequence in $\mathbb Z$, defined as follows:
$a_{0}:=0,
a_{1}:=2,
a_{n+1}:= 4(a_{n}-a_{n-1}) \forall n \in \mathbb N$. </p>
<p>Required to prove: $a_{n}=n2^{n} \forall n \in \mathbb N_{0}$</p>
<p>I have gone about it in the following: </p>
<p>Induction start: $n=0$ (condi... | Patrick Stevens | 259,262 | <p>Note that this can be done in an alternative way, using generating functions: $$\sum_{n=0}^{\infty} a_n x^n = 0 + 2x + 4x \sum_{n=1}^{\infty} (a_n - a_{n-1}) x^n = 2x + 4x f(x) - 4x^2 \sum_{n=0}^{\infty} a_{n} x^{n}$$</p>
<p>so $$f(x) = 2x+4x \left[f(x) - x f(x)\right]$$</p>
<p>Solving, $$f(x) = \frac{2x}{(2x-1)^2... |
1,707,132 | <blockquote>
<p>Let $X$ be a contractible space (i.e., the identity map is homotopic to the constant map). Show that $X$ is simply connected.</p>
</blockquote>
<p>Let $F$ be the homotopy between $\mathrm{id}_X$ and $x_0$, that is $F:X\times [0,1]\to X$ is a continuous map such that
$$ F(x,0)=x,\quad F(x,1)=x_0$$
fo... | Xiang Yu | 187,406 | <p>We can multiply $G$ by a homopoty $H$ to get around the obstacle. More precisely, let $H:[0,1]\times [0,1]\to X$ be the homopoty defined by
$H(s,t)=G(0,st)$. We note that $H(1,t)=G(0,t)=G(1,t)=H(1,t) $ for all $t\in[0,1]$, thus for every $t\in[0,1]$, the product $H_t*G_t*\overline{H_t}$ can be defined, where $H_t:[0... |
195,333 | <p>I'm looking to make a graph with the x-axis reversed and a frame and tick marks that are thicker than default. However, the x-axis tick marks do not maintain the specified thickness once I reverse the x-axis. I can't restore the thickness of the tick marks using TicksStyle or FrameTicksStyle. How can I get around th... | m_goldberg | 3,066 | <p>I think you have found a bug. As far as I can tell, using any scaling option (other than Automatic or None, doesn't annihilate all tick styling, but it does annihilate any attempt to change tick thickness. </p>
<p>Here is supporting evidence.</p>
<p>We start with a <code>ListLinePlot</code> in which we override th... |
1,093,396 | <p>I've been working on a problem from a foundation exam which seems totally straightforward but for some reason I've become stuck:</p>
<p>Let $f: \mathbb{ R } \rightarrow \mathbb{ R } ^n$ be a differentiable mapping with $f^\prime (t) \ne 0$ for all $t \in \mathbb{ R } $, and let $p \in \mathbb{ R } ^n$ be a point NO... | Jihad | 191,049 | <p>Take some $s$ on the curve. Take circle $C$ with center in $p$ of radius $|p-s|$. $\overline C \cap f(\mathbb{R}) - $ closed and bounded set (i.e. compact in $\mathbb{R}^n$). $t \to |p-f(t)|$ is continuous function. What do you know about continuous functions on compacts?</p>
|
394,321 | <p>I'm studying the asymptotic behavior <span class="math-container">$(n \rightarrow \infty)$</span> of the following formula, where <span class="math-container">$k$</span> is a given constant.
<span class="math-container">$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$</span></p>
<p>I'm trying to do a series expans... | Adi Dani | 12,848 | <p>HINT </p>
<p>$$\frac{y}{(x-y)(y-1)} = \frac{Ax+By+C}{x-y} + \frac{Dx+Ey+F}{y-1}$$</p>
|
4,231,509 | <p>I'm trying to prove that the group <span class="math-container">$(\mathbb{R}^*, \cdot)$</span> is not cyclic (similar to [1]). My efforts until now culminated into the following sentence:</p>
<blockquote>
<p>If <span class="math-container">$(\mathbb{R}^*,\cdot)$</span> is cyclic, then <span class="math-container">$\... | Shaun | 104,041 | <p><strong>Hint:</strong> If <span class="math-container">$G$</span> is a cyclic group and <span class="math-container">$H\cong G$</span>, then <span class="math-container">$H$</span> is also cyclic. Now use the fact that <span class="math-container">$(\Bbb R, +)$</span> is not cyclic.</p>
<p>Also, every group satisfie... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| Daniel Akech Thiong | 169,316 | <p>Let $A$ be $m \times n$ and $B$ be $n \times p$ both with entries in the field $F$. Consider the maps: $L_{A}: F^{n} \rightarrow F^{m}: L_{A}(X) = AX$ and $L_{B}: F^{p} \rightarrow F^{n}: L_{B}(X) = BX$. </p>
<p>Then $L_{A} \circ L_{B}: F^{p} \rightarrow F^{m}$ satisfies $Y \in \text{Image } L_{A} \circ L_{B} \imp... |
637,819 | <p>$$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$</p>
<p>Since the variable $A$ is bounded by universal quantifier, it is regarded as bounded variable, according to the rules, the variable is free to change to other letters while the meaning statement remains unchanged. But,the abov... | user121173 | 121,173 | <p>The reason for this is that when you take the conjunction of two statements, its truth depends on the truth of each of the two components, and you don't look any further than that.</p>
<p>For example, the statements</p>
<p>$$\forall x \geq 5 \quad 2x \geq 10$$
and<br>
$$\forall x \leq - 3 \quad x^2 \geq 9$$
are bo... |
85,470 | <p>We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.</p>
<p>We have 4 people in our office - each with a partner that will be at our Christmas meal.</p>
<p>Steve,
Christine,
Mark,
Mary,... | Ilmari Karonen | 9,602 | <p>If each of you just draws a random name out of the hat, the probability that nobody gets their own or their partner's name is</p>
<p>$$\frac{4752}{40320} = \frac{33}{280} \approx 11.8\%.$$</p>
<p>Thus, the expected number of times you'll need to repeat the process before getting "a solution that works" is $280/33 ... |
126,052 | <p>I have no doubt that the following observation is quite well known. Let $\varphi:[0,1]\to [0,1]$ be a continuous map. Assume that the iterates $\varphi^n$ converge pointwise to some continuous map $\varphi_\infty$. Then the convergence is in fact uniform. However, I was unable to locate a reference. Does anybody kno... | Sergei Ivanov | 4,354 | <p>I don't know a reference but maybe the following proof is shorter than yours.</p>
<p>By continuity, $\varphi\circ\varphi_\infty=\varphi_\infty$. Hence $\varphi$ is identity on the set $I:=\varphi_\infty([0,1])$. Hence $I$ is the set of fixed points of $\varphi$. And it is compact and connected. Then there are two c... |
3,510,233 | <blockquote>
<p>If <span class="math-container">$\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$</span>, then find the value of <span class="math-container">$x$</span>.</p>
</blockquote>
<p>Please solve this question by using <span class="math-container">$\cos\left(\dfrac\pi2 - \theta\... | Quanto | 686,284 | <p>The growth is the fastest when the slope of <span class="math-container">$P(t)$</span> is the largest. Evaluate <span class="math-container">$
P'(t) = \frac{ke^{-t/2}}{\left(1000 e^{-t/2}+1\right)^2}$</span>, with <span class="math-container">$k=25mm$</span>. Then, cast it in the form</p>
<p><span class="math-conta... |
3,459,532 | <p>I have a pretty straightforward linear programming problem here:</p>
<p><span class="math-container">$$ maximize \hskip 5mm -x_1 + 2x_2 -3x_3 $$</span></p>
<p>subject to</p>
<p><span class="math-container">$$ 5x_1 - 6x_2 - 2x_3 \leq 2 $$</span>
<span class="math-container">$$ 5x_1 - 2x_3 = 6 $$</span>
<span class... | Anurag A | 68,092 | <p>Use the fact that <span class="math-container">$u^Tv=u \cdot v$</span>, thus <span class="math-container">$u^Tu=\|u\|^2$</span> and note the following: </p>
<ul>
<li><span class="math-container">$a_ia_i^T$</span> is a rank one matrix.</li>
<li><span class="math-container">$a_i^Ta_j=0$</span> for <span class="math-... |
3,459,532 | <p>I have a pretty straightforward linear programming problem here:</p>
<p><span class="math-container">$$ maximize \hskip 5mm -x_1 + 2x_2 -3x_3 $$</span></p>
<p>subject to</p>
<p><span class="math-container">$$ 5x_1 - 6x_2 - 2x_3 \leq 2 $$</span>
<span class="math-container">$$ 5x_1 - 2x_3 = 6 $$</span>
<span class... | user1551 | 1,551 | <p>Extend <span class="math-container">$\{a_1,a_2\}$</span> to an orthonormal basis <span class="math-container">$\{a_1,a_2,\ldots,a_d\}$</span> of <span class="math-container">$\mathbb R^d$</span>. Then
<span class="math-container">$$
Aa_i=(I-a_1a_1^T-a_2a_2^T)a_i=
\begin{cases}
0,&i=1,2,\\
a_i,&i\ge3.
\end{ca... |
3,510,156 | <p>This is a duplicate question of <a href="https://math.stackexchange.com/questions/2088815/find-integers-solutions-of-x27-y5#">Find integers solutions of $x^2+7=y^5$</a>, however there was no full answer. The solutions <span class="math-container">$(\pm5, 2)$</span> and <span class="math-container">$(\pm 181, 8)$</sp... | Community | -1 | <p>Consider the general case
<span class="math-container">$$x^2+7=y^m \tag{1}$$</span>
(Integers <span class="math-container">$(x, y, m), \, m \geq 3)$</span>
Let
<span class="math-container">$$\rho = (1+\sqrt{-7})/2$$</span></p>
<p>Then as you are aware <span class="math-container">$(1, \rho)$</span> is a basis for ... |
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