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 |
|---|---|---|---|---|
4,167,747 | <p>I need to prove the following statement: If <span class="math-container">$1+ \alpha = \alpha$</span>, <span class="math-container">$\alpha$</span> is an infinite ordinal.</p>
<p>I am trying to use Bernstein's Theorem(CBS) to show if <span class="math-container">$1+ \alpha \leq \alpha$</span>, i.e., there is an injec... | Asaf Karagila | 622 | <p>Using Cantor–Bernstein is not the right tool here. You're not trying to prove that cardinalities are equal, but rather than the ordinals are equal. For this you need more than a bijection. You need an order isomorphism.</p>
<p>One way to simplify this is to remember that <span class="math-container">$1+\alpha=1+\ome... |
125,610 | <p>I have question about sets. I need to prove that: $$X \cap (Y - Z) = (X \cap Y) - (X \cap Z)$$</p>
<p>Now, I tried to prove that from both sides of the equation but had no luck.</p>
<p>For example, I tried to do something like this: $$X \cap (Y - Z) = X \cap (Y \cap Z')$$ but now I don't know how to continue.</p>
... | Rudy the Reindeer | 5,798 | <p>In general, to show equality of sets $A = B$ you show $A \subset B$ and $B \subset A$.</p>
<p>To show $X \cap (Y - Z) \subset (X \cap Y) - (X \cap Z)$ assume $a \in X \cap (Y -Z)$. Then $a \in X$ and $a \in Y - Z \subset Y$. Hence $a \in X \cap Y$. Also, $a \notin Z$ hence $a \notin X \cap Z$ and hence $x \in (X \c... |
374,105 | <p>Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.</p>
| tessellation | 71,044 | <p>If you know about ideal polygon then this is easy because every ideal polygon has angle sum zero.</p>
|
16,802 | <p>In an attempt to squeeze more plots and controls into the limited space for a demo UI, I am trying to remove any extra white spaces I see.</p>
<p>I am not sure what options to use to reduce the amount of space between the ticks labels and the actual text that represent the labels on the axes.</p>
<p>Here is a smal... | Emilio Pisanty | 1,000 | <p>This can also be achieved by</p>
<ul>
<li>encasing the graphic inside a <code>Show</code>,</li>
<li>setting the outer <code>Show</code>'s <code>PlotRangeClipping</code> to <code>False</code>, and</li>
<li>adding the labels as <code>Text</code> commands inside an <a href="http://reference.wolfram.com/language/ref/Ep... |
1,821,411 | <p>$f:[a,b]\rightarrow R$ that is integrable on [a,b]</p>
<p>So we need to prove:</p>
<p>$$\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$$</p>
<p>1.) So we'll use a property of definite integrals: (homogeny I think it's called?)</p>
<p>$$\int_{-b}^{-a}f(-x)dx=-1\int_{-b}^{-a}f(x)dx$$</p>
<p>2.) Great, now using the fun... | Alex M. | 164,025 | <p>Your first step is mistaken: it seems that you mistake $\int _a ^b (-f) (x) \ \Bbb d x$ for $\int _a ^b f (-x) \ \Bbb d x$; these two are completely different, and the homogeneity property applies only to the first formula, not to the second.</p>
<p>Just use the substitution $y = -x$, this will solve the problem in... |
863,561 | <p>A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere which describes the exiting radiance is supposed to be equal to π. Is there a way I can prove that the surface of the ... | johannesvalks | 155,865 | <p>Given your question, I think you need to evaluate</p>
<p>$$
\int_0^{\pi/2} d\theta \int_0^{2\pi} d\phi \sin(\theta) \cos(\theta)
$$</p>
<blockquote class="spoiler">
<p> $$= \pi \int_0^{\pi/2} \sin(2\theta) d\theta = \pi$$</p>
</blockquote>
|
863,561 | <p>A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere which describes the exiting radiance is supposed to be equal to π. Is there a way I can prove that the surface of the ... | Jacques MALAPRADE | 163,723 | <p>Based on the p.571 and 566 in Stroud's 'Engineering Mathematics' I am setting out the answer below. The surface of revolution based on the parametric equation where in our case the rotation is around the y-axis the equation is as follows:
$$
A = \int_0^{\pi/2} 2\pi x \sqrt{(\frac{dx}{d\theta})^2 + (\frac{dy}{d\theta... |
4,374,521 | <p>In the <span class="math-container">$(x,t)$</span>- plane, the characteristic of the initial value problem <span class="math-container">$$u_t+uu_x=0$$</span> with <span class="math-container">$$u(x,0)=x,0\leq x\leq 1$$</span> are</p>
<p><span class="math-container">$1$</span>. parallel straight lines .</p>
<p><span ... | Henry Lee | 541,220 | <p>I'm just going to expand on my comment. This appears to fit the form of the <a href="https://en.wikipedia.org/wiki/Burgers%27_equation#Inviscid_Burgers%27_equation" rel="nofollow noreferrer">Inviscid Burgers' equation</a> which is:
<span class="math-container">$$\frac{\partial u }{\partial t}+u\frac{\partial u}{\par... |
3,996,090 | <p>I believe I have found the recurrence relation to be
<span class="math-container">$$B\left(n\right)=B\left(n-1\right)+2^{n-1}-1$$</span>
with Initial Condition B(0)=0 (I am a bit unsure about the initial condition though but I think it is correct)</p>
<p>Now I am trying to solve B(n) using iteration, this is what I ... | RobPratt | 683,666 | <p>Yes, <span class="math-container">$B_0=B_1=0$</span>. Let <span class="math-container">$A_n = 2^n - B_n$</span> be the number of bit strings that do <em>not</em> contain <span class="math-container">$01$</span>, so <span class="math-container">$A_0=1$</span>. For <span class="math-container">$n>1$</span>, condi... |
3,888,146 | <p>When we give a proof that the tangent is the sine to cosine ratio of an oriented angle,</p>
<p><span class="math-container">$$\bbox[5px,border:2px solid #C0A000]{\tan \alpha=\frac{\sin\alpha}{\cos \alpha}}$$</span>
with <span class="math-container">$\cos \alpha \neq 0$</span>, we take the tangent <span class="math-c... | heropup | 118,193 | <p>Since <span class="math-container">$\ell$</span> is a diameter, reflecting either <span class="math-container">$A$</span> or <span class="math-container">$B$</span> across <span class="math-container">$\ell$</span> will give a third point through which the circle passes. You can then use this as the construction.</... |
3,752,162 | <p>I already knew that normal subgroups where important because they allow for quotient space to have a group structure.
But I was told that normal subgroups are also important in particular because they are the only subgroups that can occur as kernels of goup homomorphisms. Why is this property a big deal in algebra?<... | Andrea Mori | 688 | <p>Let <span class="math-container">$G$</span> be a group. Since the normal subgroups of <span class="math-container">$G$</span> coincide, as a set, with the subgroups that appear as kernels of homomorphisms with domain <span class="math-container">$G$</span>, the normal subgroups are exactly the subgroups of <span cla... |
571,941 | <p>I know that $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\frac { 1 }{ n } =\ln(2) }$ .</p>
<p>How about the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 } } \frac { 1 }{ \sqrt { n } }$ </p>
<p>To what number does it converge?</p>
| 1233dfv | 102,540 | <p>Let $a_1$ be the number of combinatorics problems solved on the first day, $a_2$ be the total number of combinatorics problems solved on the first and second days, and so on. The sequence of numbers $a_1,a_2,...,a_{365}$ is an increasing sequence since each term of the sequence is larger than the one that precedes i... |
2,507,828 | <p>$$\int_C (1+\cosh(y),x\sinh(y))d\vec{s}$$
Where C is a curve that goes from $(0,0)$ to $(1,1)$</p>
<p>I am not sure on how to proceed, I can find </p>
<p>$\vec{F}=(1+\cosh(y),x\sinh(y))$</p>
<p>$\vec{f}:\nabla f=\vec{F}$</p>
<p>$\vec f=(x+x\cosh(y), x\cosh(y))$</p>
| Doug M | 317,162 | <p>Fundamental theorem of line integrals: if $f = \nabla F$ and $c$ is a contour from $a$ to $b$</p>
<p>$\int_c f \cdot dr = F(b) - F(a)$</p>
<p>The line integral does not depend on the path. It only depends on the endpoints.</p>
<p>$f = \nabla (x+x \cosh y)$</p>
<p>$\int_c f\cdot dr = 1 + \cosh 1$</p>
|
2,507,828 | <p>$$\int_C (1+\cosh(y),x\sinh(y))d\vec{s}$$
Where C is a curve that goes from $(0,0)$ to $(1,1)$</p>
<p>I am not sure on how to proceed, I can find </p>
<p>$\vec{F}=(1+\cosh(y),x\sinh(y))$</p>
<p>$\vec{f}:\nabla f=\vec{F}$</p>
<p>$\vec f=(x+x\cosh(y), x\cosh(y))$</p>
| operatorerror | 210,391 | <p>Hint: You are in search of a function $f$ with the property that
$$
\frac{\partial f}{\partial x}=1+\cosh y\implies f(x,y)=x+x\cosh y+g(y)
$$
furthermore, you would like
$$
\frac{\partial f}{\partial y}=x\sinh y+g'(y)=x\sinh y\implies g(y)=c
$$
and
$$
f(x,y)=x+x\cosh y+c
$$
try evaluating the potential $f$ at the... |
2,361,602 | <p>The "Heine–Cantor theorem" states: If $f : M → N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous.</p>
<p>I do not doubt its validity, of course, just trying to understand <strong>why</strong> it is valid.</p>
<p>If we, say, take the function: $y = x^4$.
It ... | A. Thomas Yerger | 112,357 | <p>Compactness is a finiteness property. It says that given any infinite collection of data associated to open sets in a topological space, you can in fact deal with only finitely many.</p>
<p>The great thing about a finite number of objects, is that you can compare them. Unlike with infinite sets, which only have wel... |
4,421,529 | <p><strong>Question:</strong> Let <span class="math-container">$n > 0$</span>. How can I find a function <span class="math-container">$f:\mathbb{N}\rightarrow\mathbb{R}^+$</span> such that
<span class="math-container">$$
\lim_{n\to\infty} \frac{f(n)^2}{n} \log \left(\frac{f(n)}{n}\right) = L
$$</span>
with <span cla... | ajr | 266,348 | <p>Take <span class="math-container">$f(n) = n+1$</span>. Then
<span class="math-container">\begin{align*}
\lim\limits_{n\to\infty} \frac{f(n)^2}{n}\log\bigg(\frac{f(n)}{n}\bigg) = \lim\limits_{n\to\infty} \bigg(n + 2 + \frac 1n\bigg)\log\bigg(1 + \frac 1n\bigg) = \lim\limits_{n\to\infty} \frac{n + 2 + \frac 1n}{n}\cdo... |
3,084,479 | <p><span class="math-container">$h\in \mathbb{R}$</span>, because we have defined the Trigonometric Functions only on <span class="math-container">$\mathbb{R}$</span> so far.</p>
<p>I have a look at <span class="math-container">$e^{ih}=\sum_{k=0}^{\infty}\frac{(ih)^k}{k!}=1+ih-\frac{h^2}{2}+....$</span> </p>
<p><stro... | Jam | 161,490 | <p>Hint: Use Euler's formula and split the limit into well known trigonometric limits.</p>
|
188,492 | <p>$A$ is an $n\times n$ matrix (not symmetric). If $\rho(A)$, spectral radius of $A$, is less than or equal to 1, can we say that $x^TAx\leq x^Tx$? </p>
<p>In another word,</p>
<p>if $\rho(A)\leq 1$, then $\frac{1}{2}\rho(A+A^T)\leq 1$?</p>
| Harald Hanche-Olsen | 23,290 | <p>The second part of the question is easier to answer, with the counterexample $$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}.$$</p>
|
4,350,450 | <p>For me, <span class="math-container">$\Bbb N$</span> includes <span class="math-container">$0$</span>. I am referencing, yet again, <a href="https://www.math.uni-leipzig.de/%7Eeisner/book-EFHN.pdf" rel="nofollow noreferrer">this</a> text, exercise <span class="math-container">$19$</span>, page <span class="math-cont... | José Carlos Santos | 446,262 | <p>This is not true in general. Suppose that <span class="math-container">$f$</span> is constant (you always have <span class="math-container">$f(x)=c$</span>), that each <span class="math-container">$A_n$</span> is non-empty and that <span class="math-container">$\bigcap_{n\in\Bbb N}A_n=\emptyset$</span>. Then<span cl... |
1,937,826 | <p>Ok, this seems obvious to me, but how would one prove it?</p>
<p>Let $<f(t),g(t)>$ and $<h(t),p(t)>$ be parametrized arcs in the cartesian plot. If $f,g,h,p$ are all continous and the arcs don't intersect, then there will be a line between the two that will be the shortest distance. Prove this line is n... | cjackal | 44,643 | <p>If you assume that the two curves are defined over an open interval, and there is a shortest segment connecting the two curves, then yes; just differentiate the squared length of the joining segment by one of the two parameters.</p>
<hr>
<p>Okay, let me be more explicit.</p>
<p>I assumes that the two smooth curve... |
1,392,257 | <p><strong>The definition of a conjugate element</strong> </p>
<p>We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$</p>
<p>Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = \big<a,b: a^4 =1,b^2 = a^2, b^{-1}ab = a^{-1} \big>$$</p>
<p>Now the elements of $Q_8$ a... | Erik Rijcken | 261,145 | <p>No, this does not hold: take any abelian group $G$, then $ab=ba$ for all $a,b\in G$, so $b^{-1}ab = a$ for all $a,b\in G$, so $a^G=\{a\}$ for all $a\in G$. So if $G$ contains an element of order different from $2$, it does not satisfy that $a,a^{-1}\in a^G$ for all $a$.</p>
<p>For a concrete example, take $G = \lan... |
61,106 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be Poisson random variables with means <span class="math-container">$\lambda$</span> and <span class="math-container">$1$</span>, respectively. The difference of <span class="math-container">$X$</span> and <span class="math-cont... | Adrien Hardy | 15,517 | <p>By simple computations : The definition of the modified Bessel function of the first kind yields
$$
I_k(\lambda)=\sum_{n\geq 0}\frac{1}{n!(n+k)!}\left(\frac{\lambda}{2}\right)^{2n+k}
$$
so that we get (the sums transpositions are clearly allowed)
$$F(\lambda)=e^{-\lambda-1}\sum_{k\geq 1}\sum_{n\geq 0}\frac{\lambda^... |
2,011,003 | <p>I stumbled upon this logic question in a math class recently. </p>
<p>My teacher told us that a statement that is not tested/is empty is true. For example, that if I stated that: "if the team A wins the game, I am gonna buy you a coke", and then team B goes on and wins the game, the statement would be true, indepen... | amWhy | 9,003 | <p>Do you understand that the conditional $p\rightarrow q$ is true whenever $p$ is false, or whenever $q$ is true?</p>
<p>I think the best way of representing the truth of a condition $p\rightarrow q$ is knowing that $p\rightarrow q$ IS TRUE, UNLESS both $p$ is true, and $q$ is false.</p>
<p>With two variables, $p, q... |
789,458 | <p>If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire <em>truly random</em> number generator?</p>
| qwr | 122,489 | <p>The definition of random is <em>unpredictable</em>. $\pi$'s digits have been calculated to billions of digits and they are all public information. You might be able to use a random range of the digits, but then you would need a separate method to select that range.</p>
|
1,095,621 | <p>I am looking for a way to integrate $$\int \sqrt{x^2-4}\ dx $$ using trigonometric substitutions. </p>
<p>All my attempts so far lead to complicated solutions that were uncomputable.</p>
| APGreaves | 191,763 | <p>For a trigonometric substitution, $ x = 2\sec \theta $ will work if you can integrate certain other trig functions.</p>
|
1,095,621 | <p>I am looking for a way to integrate $$\int \sqrt{x^2-4}\ dx $$ using trigonometric substitutions. </p>
<p>All my attempts so far lead to complicated solutions that were uncomputable.</p>
| Barry Cipra | 86,747 | <p>Following up on Aaron Maroja's (second) hint, note that</p>
<p>$$\sec\theta\tan^2\theta\ d\theta={\sin^2\theta\ d\theta\over\cos^3\theta}={\sin^2\theta\cos\theta\ d\theta\over\cos^4\theta}={s^2\ ds\over(1-s^2)^2}$$</p>
<p>where $s=\sin\theta$. Some tedious partial fractions can wrap things up:</p>
<p>$${s^2\over... |
2,912,152 | <p>I know there is already a question about resolving a quadrilateral from three sides and two angles, but I want to ask about a special case. Firstly, two of the sides are known to be of equal size. Secondly, I'm only interested in the area, not in the remaining angles or lengths. Can anyone suggest a simple formul... | Daniel Schepler | 337,888 | <p>One point of view which might be useful would be the <a href="https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence" rel="nofollow noreferrer">Curry-Howard correspondence</a>: this point of view interprets (formal) proofs as being programs in a certain typed lambda calculus. (And the more typical inform... |
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | Community | -1 | <p>To answer this question, we must first ask ourselves "what is a variable?" What do I mean when I say that "$x$ is a real number-valued variable"?</p>
<p>I'm going to try and describe one useful approach.</p>
<p>We might think of $x$ as being a placeholder for an unknown but specific number. Or maybe a notation for... |
3,267,499 | <p>Let <span class="math-container">$k$</span> be a field.
<span class="math-container">$k[x,y]$</span> is a UFD by the following known argument taken from <a href="https://en.wikipedia.org/wiki/Unique_factorization_domain" rel="nofollow noreferrer">wikipedia</a>:
"If <span class="math-container">$R$</span> is a UFD, t... | Bernard | 202,857 | <ol>
<li>The ring of Laurent polynomials <span class="math-container">$R=k[x,x^{-1}]=\bigl(k[x]\bigr)_{x}$</span>, and a ring of fractions of a U.F.D. is a U.F.D., so <span class="math-container">$k[x,x^{-1},y]=R[y]$</span> is a U.F.D.</li>
<li>You can find the irreducible elements of <span class="math-container">$k[x,... |
3,090,448 | <p>I have the following question to complete.</p>
<p>Let <span class="math-container">$X$</span> be an inner product space. Let <span class="math-container">$(e_{j})_{j\geq1}$</span> be an orthonormal sequence in <span class="math-container">$X$</span>. Show that,
<span class="math-container">\begin{align}
\sum_{j=1}^... | jmerry | 619,637 | <p>The Cauchy-Schwarz inequality should just have <span class="math-container">$\|x\|\cdot\|y\|$</span> on the right hand side; you've got the statement of it mixed up.</p>
<p>In my post, the notation <span class="math-container">$(u|e_j)$</span> denotes the component of <span class="math-container">$u$</span> with re... |
116,037 | <p>I would warmly appreciate it if someone could tell me whether the following question has an affirmative answer. I am new to the field of commutative algebra, so I am simply trying to fill in some (huge) gaps. Thanks!</p>
<p>Let $ (R,{\frak{m}}) $ be a Noetherian local (commutative unital) ring. Let $ I $ be an idea... | Steven Sam | 321 | <p>The resolution that you are asking for exists for a special class of ideals, namely the perfect codimension 2 ideals. Here perfect means that a finite resolution exists, and codimension 2 means roughly (in any reasonable situation of classical algebraic geometry, at least) that $\dim R/I = \dim R - 2$.</p>
<p>The H... |
222,555 | <p>I would like to find a simple equivalent of:</p>
<p>$$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $$</p>
<p>We have:</p>
<p>$$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$$</p>
<p>So $$ u_{n} \rightarrow 0$$</p>
<p>Clearly:</p>
<p>$$ u_{n} \sim \frac{1}{n!} \int_{\sin(1)}^1 (... | Julián Aguirre | 4,791 | <p>This is not a complete answer, but an improved inequality. From
$$
\arcsin x\le \frac{\pi}{2}\,x
$$
you get
$$
u_n\le\frac{1}{(n+1)!}\Bigr(\frac{\pi}{2}\Bigl)^n.
$$</p>
|
1,299,266 | <p>How many zeros are there in the number $50!$?</p>
<p>My attempt:</p>
<p>The zeros in every number come from the 10s that make up the number. The 10s are, in turn, made up of 2s and 5s.</p>
<p>So: $\frac{50}{5*2} = 5$ zeros?</p>
| Khosrotash | 104,171 | <p>number of zeros in $n!$ equal to number of $5$ in $n!$ =$$\sum_{k=1}^{\infty}\left \lfloor \frac{n}{5^k} \right \rfloor\\=\left \lfloor \frac{50}{5} \right \rfloor+\left \lfloor \frac{50}{5^2} \right \rfloor+\left \lfloor \frac{50}{5^3} \right \rfloor+...=10+2+0=12$$it means 12 zero in front of 50!</p>
|
311,153 | <p>Im trying to resolve the next definite integral:
$$\int_{1-x^2}^{1+x^2}{\ln(t^2)\ dt}$$
Im not sure if I can use the Barrow's theorem, I think I have to use the fundamental theorem of integral calculus, but im not sure. How can I solve it?</p>
| Mikasa | 8,581 | <p>If you set $$f(x)=\int_{1-x^2}^{1+x^2}{\ln(t^2)dt}$$ then according to F.T. we get $$f'(x)=4x\ln(1-x^4)$$ You can use the integral by parts firstly to solve the above OE. It takes time to be evaluated so I personally prefer the @experimentX's point of view.</p>
|
1,553,391 | <p>Let $E$ be a measurable set of finite measure and $1\leq p_1 < p_2 \leq \infty$ . Then $L^{p_2} (E) \subseteq L^{p_1} (E)$ Furthermore $||f||_{p_1} \leq c \cdot ||f||_{p_2}$ for all $f$ in $L^{p_2}(E)$ where $c =[m(E)]^{\frac{p_2-p_1}{p_1p_2}} $ if $p_2<\infty$ and $c=[m(E)]^{\frac{1}{p1}}$ if $p_2 =\infty$ </... | Robert Israel | 8,508 | <p>It's easy. If $f \in L^\infty(E)$, then $ \int_E |f|^{p} \le \int_E \|f\|_\infty^p = \|f\|_\infty^p m(E)$.</p>
|
71,608 | <p>Consider the following question:</p>
<p>Is there a family $\mathcal{F}$ of subsets of $\aleph_\omega$ that satisfies the following properties?</p>
<p>(1) $|\mathcal{F}|=\aleph_\omega$</p>
<p>(2) For all $A\in \mathcal{F}$, $|A|<\aleph_\omega$</p>
<p>(3) For all $B\subset \aleph_\omega$, if $|B|<\aleph_\om... | Santi Spadaro | 11,647 | <p>This question has been already answered thoroughly. I just wanted to address the OP's comment "I am not sure if there is anything special about $\aleph_\omega$".</p>
<p>Actually, there is nothing special about $\aleph_\omega$ other than the fact that it's a singular cardinal. Let $\kappa$ be a cardinal and let $S(\... |
127,493 | <p>How many number less than $k$ contain the digit $3$?
For instance:</p>
<p>How many number contain the digit $3$ in the following list?</p>
<pre><code>Table[n, {n, 33}]
</code></pre>
<p>$\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, \
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32... | Nasser | 70 | <p>One way to find out why M can not do an indefinite integral, is to run Rubi integration package on it. Since Rubi shows step by step integration, most of the time, when Rubi gets stuck on a step, it will also be the same with Mathematica. This can point out which part of the larger integral where M was not able to ... |
1,156,907 | <p>I don't know anything about measure theory, I'm studying real analysis and this showed up in the book I'm reading as a way to characterize integrable functions. The author defined that a subset $X \subset \mathbb{R}$ has measure zero if for each $\epsilon > 0$ we can find infinitely countable open intervals $I_n$... | Hagen von Eitzen | 39,174 | <p>That's just how measures work. We start by defining that the measure of an open interval $(a,b)$ is $b-a$. Then we can attempt to define the (outer) measure of an arbitrary set $A$ as the infimum of all $\sum_{i\in J}\mu(I_i)$ where the $I_j$ are open intervals and $A\subseteq \bigcup_{i\in J}I_i$.
A few observation... |
374,209 | <p>Need to show that if $f$ is a glide reflection then there is only one line $L$
such that $f(L) = L$</p>
<p>What I know is that a glide reflection is an isometry </p>
<p>$$f(z)=a\bar{z}+b,$$ such that $|a|=1$ and $a\bar{b}+b\neq0$.</p>
<p>Now assume that two lines $L_1$ and $L_2$ such that are axes for this glide ... | rschwieb | 29,335 | <p>Here is what I arrived at.</p>
<p>We produce the axis with some vector algebra.</p>
<p>By computing $f(f(z))=z+b+a\overline{b}$, we can see the expected result that the translation that occurs along the axis is by the nonzero amount $t=(b+a\overline{b})/2$. The translation helps us by showing us the direction of t... |
1,793,854 | <p>I am messed up on solving this question. What should I do first in order to get the answer ?</p>
<p><a href="https://i.stack.imgur.com/hE4rG.png" rel="nofollow noreferrer">This is the trigonometric function</a></p>
<p>$$ \lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x} $$</p>
| AbstractSage | 296,761 | <p>Changing into cosines greatly eases the manipulation of terms.
$$
\begin{align}
\lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x}
& = \lim \limits_{x \rightarrow 0} \frac{a\sec(a+x) - a \sec(a)}{x} + \lim \limits_{x \rightarrow 0} \frac{x\sec(a+x)}{x} \\
& = A + B
\end{align}
$$
$$
\begin... |
1,738,968 | <blockquote>
<p>Let $V$ be a vector space and let $T \in \operatorname{End}(V)$. If $\operatorname{rank}(T)$ and $\operatorname{null}(T)$ are finite, prove that $\dim(V)$ is finite.</p>
</blockquote>
<p>I cannot use the Rank-Nullity Theorem as it only applies to finite
dimensional vector space and I don't know wheth... | Vishnu | 693,070 | <blockquote>
<p>This answer is preserved for those who want to understand why the sign of constants do not matter. After reading this answer please check the comments for more details from @mathlove.</p>
</blockquote>
<p>@mathlove's answer really explains the question. But, I would like to show that the "<span class... |
3,822,042 | <p>For any function <span class="math-container">$f : X \rightarrow Y$</span> and any subset A of Y, define
<span class="math-container">$$f^{-1}(A) = \{x \in X: f(x) \in A\}$$</span> Let <span class="math-container">$A^c$</span> denote the complement of A in Y. For subsets <span class="math-container">$A_1,A_2$</span>... | N. F. Taussig | 173,070 | <p>Let <span class="math-container">$b, g, r$</span> denote, respectively, the numbers on the blue, green, and red cards. Then we want to find the number of solutions of the equation
<span class="math-container">$$b + g + r = 16 \tag{1}$$</span>
subject to the restrictions <span class="math-container">$3 \leq b \leq 9... |
1,627,713 | <p>This is maybe math $101$ question:</p>
<p>Let $z_1=1+i$.</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$</p>
<p>But now if I take a look at</p>
<p>$z_2=-1-i$,</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(-1/-1)=\pi/4$ so $$z_1=\color{blue}{\sqrt ... | Mankind | 207,432 | <p>Let $a=(x_1,y_1)$ and $b=(x_2,y_2)$ be two points of $\Bbb{R}^2\setminus\Bbb{E}$. By definition of $\Bbb{E}$, at least one coordinate of both $a$ and $b$ must be irrational, so suppose for instance that $x_1$ and $y_2$ are irrational.</p>
<p>You are right that we need to find a continuous function $f\colon [0,1]\to... |
827,154 | <p>I need help with the definition of "within 1":</p>
<ul>
<li><p>If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. </p></li>
<li><p>If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$.</p></li>
<li><p>If $x = 8$ and $y = 8$, is $x$ still "within 1" of $y$?</p></li>
</ul>
<p>It's my understanding that this wou... | Community | -1 | <p>In the more general case, I would say that "$x$ is within (a distance) $d$ of $y$" means that
$$|x - y| \le d.$$
(Depending on the context, I would imagine the inequality could be strict.)</p>
|
2,028,703 | <p>I'm having this example for a simple <a href="https://en.wikipedia.org/wiki/Binary_symmetric_channel" rel="nofollow noreferrer">binary symmetric channel</a> (BSC) to bound the mutual information of $X$ and $Y$ as</p>
<p>\begin{align*}
I(X;Y) &= H(Y) - H(Y|X)\\
&= H(Y) - \sum p(x) H(Y \mid X = x) \\
&= H... | R.G. | 78,396 | <p>The reason for the validity of the equation</p>
<p>\begin{equation}
\sum p(x) H(Y \mid X = x) = \sum p(x) H(p)
\end{equation}</p>
<p>can perhaps be better seen if we denote the right-hand side by</p>
<p>\begin{equation}
\sum p(x) H_b(p)
\end{equation}</p>
<p>where $H_b(\cdot)$ is the binary entropy function ... |
1,760,242 | <p>Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a circuit in electrical engineering, but this is not very interesting, and it doesn't seem to be so real.
Thank you!</p>
| GEdgar | 442 | <p>Your computer animation programs do 3D geometric transformations before they can draw their pictures. These involve $3 \times 3$ matrices.</p>
|
134,574 | <p>$a^{p-1} \equiv 1 \pmod p$</p>
<p>Why do Carmichael numbers prevent Fermat's Little Theorem from being a guaranteed test of primality? Fermat' Little theorem works for any $a$ such that $1≤a\lt p$, where $p$ is a prime number. Carmichael numbers only work for $a$'s coprime to $N$ (where $N$ is the modulus). Doesn't... | joriki | 6,622 | <p>In case you looked at the <a href="http://en.wikipedia.org/wiki/Carmichael_number">Wikipedia article</a> on Carmichael numbers, your question may have resulted from the sentence "Since Carmichael numbers exist, [the Fermat] primality test cannot be relied upon to prove the primality of a number". This is a bad formu... |
1,878,734 | <p>Is it true that if an isomorphism $f$ maps a cyclic group $G$ to group $H$ that $H$ must also be cyclic? It seems intuitive but until I can actually prove it I'm always a bit dubious to believe it. </p>
| Justin Benfield | 297,916 | <p>To give a direct proof: Let <span class="math-container">$y\in H$</span>, and <span class="math-container">$\phi:G\rightarrow H$</span> an isomorphism of the groups <span class="math-container">$G$</span> and <span class="math-container">$H$</span>. Consider <span class="math-container">$\phi^{-1}(y)\in G$</span>. S... |
1,346,286 | <p>Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? </p>
<p>At the very best, I know that the area $\sin x$ covers between $0$ to $\pi\over 2$ has the same magnitude bet... | Brian M. Scott | 12,042 | <p>Make the substitution $u=\pi-x$; then</p>
<p>$$\int_{\pi/2}^\pi\frac{dx}{1-\sin x}=\int_{\pi/2}^0\frac{-du}{1-\sin u}=\int_0^{\pi/2}\frac{du}{1-\sin u}\;.$$</p>
|
1,346,286 | <p>Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? </p>
<p>At the very best, I know that the area $\sin x$ covers between $0$ to $\pi\over 2$ has the same magnitude bet... | John Hunsberger | 240,638 | <p>It appears by symmetry in that sin(x) x = 0 .. pi describes an arc that will define an area that is twice that of the half arc described by sin(x) x = 0 .. pi/2 so the second integral is half the area of the first hence multiplied by two makes them equal areas.</p>
|
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Evpok | 15,102 | <p>A common convention in French is</p>
<p>$$
x∈⟦1, 50⟧
$$</p>
<p>and I am genuinely surprised to learn that it might not be common elsewhere ! In any case, $\{1, …, 50\}$ or maybe $\{1, 2, …, 50\}$ should be universal and more readable for most people.</p>
<p>For your other question, still from the French perspecti... |
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Hammerite | 23,931 | <p>One possibility is $\{i\}_{i = 1}^{50}$, by analogy with $\sum_{i = 1}^{50}(\cdots)$ and other similar notation.</p>
|
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Newton fan 01 | 560,959 | <p>Another fancy way of writing the set is this one: </p>
<p><a href="https://i.stack.imgur.com/5Ny73.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5Ny73.png" alt="enter image description here"></a></p>
<p>I got this idea when reading Hammerite's answer. However, the formulas are different. Or ... |
1,627,619 | <p>Could anyone please check my solution to the following problem?</p>
<blockquote>
<p><strong>Problem:</strong> Let $f(x, y) = (x^2 + y^2)e^{-(x^2 + y^2)}$. Find global extrema of $f$ on $M = {\mathbf R}^2$.</p>
</blockquote>
<p><strong>Proposed solution:</strong> Taking partial derivatives of $f$, we conclude tha... | Tryss | 216,059 | <p>Or you could just remark that the function is radial with value $f(x) = |x|^2 e^{-|x|^2}$. so if it has a maximum/minimum at $x_0$, it's on the whole circle of radius $|x_0|$.</p>
<p>So it suffice to study the function $h(t) = t^2 e^{-t^2}$.</p>
<p>Here you can just differentiate, $h'(t) = (2t - 2t^3) e^{-t^2}$, a... |
457,557 | <p>Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$<br>
This is how I started solving the problem, but the way I was solving it lead me to 0, which is incorrect. $$\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_{1}^{5-y}dzdx... | Mikasa | 8,581 | <p>You can also use the Cylindrical Coordinates to find the volume. Take a look at the area in which all solid is being projected on $z=0$ plane. It is really a circle with radii $3$.</p>
<p><img src="https://i.stack.imgur.com/aoosR.png" alt="enter image description here"></p>
<p>So we have the following triple integ... |
3,047,686 | <p>Prove that: <span class="math-container">$$(p - 1)! \equiv p - 1 \pmod{p(p - 1)}$$</span></p>
<p>In text it's not mentioned that <span class="math-container">$p$</span> is prime, but I checked and this doesn't hold for non-prime, so I guess <span class="math-container">$p$</span> is prime ..
I know that <span class... | Julio Trujillo Gonzalez | 272,343 | <p>1) Theorem. If <span class="math-container">$a|c$</span>, <span class="math-container">$b|c$</span> and <span class="math-container">$(a,b)=1$</span> then <span class="math-container">$ab|c$</span></p>
<p>We know that <span class="math-container">$ (p,p-1)=1$</span> and also <span class="math-container">$p-1|(p-1)!... |
2,019,070 | <p>I got this equality $(A \times B) = (A\cup B) \times(A\cup B)$. I've already shown the left-to-right inclusion, and I want to refute the other way by giving a counterexample. I want to show that given an ordered pair $(x,y) \in (A\cup B) \times(A\cup B)$, $(x,y)$ does not always belong to $(A \times B)$, thus giving... | Simply Beautiful Art | 272,831 | <p>Notice that</p>
<p>$$\frac1{n(n+1)}=\frac1n-\frac1{n+1}$$</p>
<p>This makes this a telescoping sum:</p>
<p>$$\begin{align}S&=\quad\frac1{1\times2}\ \ \ \quad+\frac1{2\times3}\ \ \ \ \ \ \ \ +\frac1{3\times4}\ \ \ +\dots+\quad\ \frac1{n(n+1)}\\&=\left(\frac11-\color{#ee8844}{\frac12}\right)+\left(\color{#e... |
2,019,070 | <p>I got this equality $(A \times B) = (A\cup B) \times(A\cup B)$. I've already shown the left-to-right inclusion, and I want to refute the other way by giving a counterexample. I want to show that given an ordered pair $(x,y) \in (A\cup B) \times(A\cup B)$, $(x,y)$ does not always belong to $(A \times B)$, thus giving... | Community | -1 | <p>By induction,</p>
<p>If $$S_n=\frac n{n+1}$$</p>
<p>then</p>
<p>$$S_{n+1}=S_n+\frac1{(n+1)(n+2)}=\frac n{n+1}+\frac1{(n+1)(n+2)}=\frac{n+1}{n+2}.$$</p>
|
1,917,790 | <p>Can anyone help me to solve this? </p>
<blockquote>
<p>Determine the value or values of $k$ such that $x + y + k = 0$ is tangent to the circle $x^2+y^2+6x+2y+6=0$.</p>
</blockquote>
<p>I don't know how to calculate the tangent.</p>
| DonAntonio | 31,254 | <p>Developed hint:</p>
<p>If the line is tangent to the circle then the line's distance to the circle's center equals the circle's radius. The circle's equation is</p>
<p>$$x^2+y^2+6x+2y+6=(x+3)^2-9+(y+1)^2-1+6\implies$$</p>
<p>$$(x+3)^2+(y+1)^2=4$$</p>
<p>Well, now use the formula for the distance of the point $\;... |
1,917,790 | <p>Can anyone help me to solve this? </p>
<blockquote>
<p>Determine the value or values of $k$ such that $x + y + k = 0$ is tangent to the circle $x^2+y^2+6x+2y+6=0$.</p>
</blockquote>
<p>I don't know how to calculate the tangent.</p>
| 5xum | 112,884 | <p><strong>Hints</strong>:</p>
<ul>
<li>A line is <em>tangent</em> to a circle if there exist <em>precisely</em> one point that is both on the straight line and on the circle</li>
<li>A point $(x,y)$ is both on the line and the circle if it satisfies <em>both</em> equations.</li>
<li>A quadratic equation has <em>exact... |
2,349,982 | <p>According to the <a href="http://www.fftw.org/fftw3_doc/1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" rel="nofollow noreferrer">FFTW Website</a>, the Fourier Sine Transform (FST) returns:</p>
<p>$$Y_k = 2 \sum_{j=0}^{N-1} X_i \sin [\pi (j+1)(k+1)/(N+1)]$$</p>
<p><a href="http://r... | spaceisdarkgreen | 397,125 | <p>You need to take the expectation value of the expression and compare to $\sigma^2$. We have $E(X_i^2) = \sigma^2+\mu^2$ and $E(X_i X_j) = \sigma^2\rho + \mu^2$ for $i\ne j.$ And then we also have $$ \left(\sum_i X_i\right)^2 = \sum_i X_i^2 + 2\sum_{i<j} X_i X_j$$ </p>
<p>Now it should be relatively straightforwa... |
4,348,969 | <p>Let <span class="math-container">$a<b\in\mathbb{R}$</span>. A sequence <span class="math-container">$P:=(p_0,\ldots,p_n)$</span> is a called a partition of <span class="math-container">$[a,b]$</span> if
<span class="math-container">$$a=p_0<\ldots<p_n=b.$$</span>
The size of <span class="math-container">$P$<... | Sammy Black | 6,509 | <p><strong>Geometric partition</strong> for integrating <span class="math-container">$f(x) = \frac{1}{x}$</span> on interval <span class="math-container">$[1, b]$</span> for defining the <strong>natural logarithm</strong> <span class="math-container">$\,\ln b$</span>:
<span class="math-container">$$
F(b) = \int_1^b \fr... |
310,462 | <p>I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete.</p>
<p>It is easy to see that a discrete space is complete because its topology can be derived from the distance <span class="math-container">$d(x,y)=1$</span> iff <span class="math-conta... | Francis Adams | 6,342 | <p>Take any countable, closed subset of a Polish space and it again will Polish. There are non-discrete examples of this, like $\{\frac{1}{n}\}_{n=1}^\infty\cup\{0\}$ as a subset of $\mathbb{R}$.</p>
|
3,179,505 | <p>Help me please , I am not able to solve this problem.I have tried in many ways to figure out such as Ration test , Integral test , Comparison test , Limit Comparison Test , Root Test but i can't find the way out . This is my first question and i'm not good at English. If there is something wrong or you are not comfo... | uniquesolution | 265,735 | <p>When considering a product of two metric spaces, say <span class="math-container">$(X_1,d_1)$</span> and <span class="math-container">$(X_2,d_2)$</span>, you want to consider metrics on the set <span class="math-container">$X_1\times X_2$</span> that are naturally related to the individual metrics <span class="math-... |
1,915,782 | <p>I'm attempting to teach myself some vector calculus before starting university next month in hope of getting my head around some of the concepts as I can foresee this being a weak topic for me.</p>
<p>I have been 'learning' from some online lecture notes related to my course. The notes talk about line integrals but... | gt6989b | 16,192 | <p>The trick mainly consists of parameterizing the curve $C$ in some parameter $t \in [0,1]$ and then you integrate
$$
\int_C f(x,y,z) = \int_0^1 f(x(t), y(t), z(t))
\sqrt{|x'(t)|^2 + |y'(t)|^2 + |z'(t)|^2}dt.
$$</p>
<p>Let's do the first one together. The parameterization is obvious $x=y=z=t$ with $t \in [0,1]... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| anon | 11,763 | <p>One way is to write the vertices as vectors $a,b,c,d$ with norm $\|\cdot\|=1$. Then $a+b+c+d=0$. But</p>
<p>$$ 0=\|a+b+c+d\|^2=4+2{4 \choose 2}\cos\theta,$$</p>
<p>so $\theta = \arccos(-1/3)$.</p>
|
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| trinetr | 822,174 | <p>Consider a sphere passing through the four vertices of the regular tetrapod with it's centre at the centre of the tetrapod.
Each set of three vertices form four congruent equilateral spherical triangles on the surface of the sphere.
For a spherical triangle ABC (unlike plane triangles),</p>
<ol>
<li>The sides a,b,c ... |
1,659,075 | <p>In linear algebra, the Rank-Nullity theorem states that given a vector space $V$ and an $n\times n$ matrix $A$,
$$\text{rank}(A) + \text{null}(A) = n$$
or that
$$\text{dim(image}(A)) + \text{dim(ker}(A)) = \text{dim}(V).$$</p>
<hr>
<p>In abstract algebra, the Orbit-Stabilizer theorem states that given a group $G$ ... | Marc Olschok | 19,950 | <p>The orbit-stabilizer formula stems from the fact that, given an action
of $G$ on $X$, the map</p>
<p>$$ G/G_x \ni gG_x \mapsto gx \in Gx $$</p>
<p>is an isomorphism of $G$-sets between the coset space $G/G_x$ and
the orbit $Gx$ of $x \in X$. Here $G$ acts on $G/G_x$ via left
multiplication.</p>
<p>The closest ana... |
1,659,075 | <p>In linear algebra, the Rank-Nullity theorem states that given a vector space $V$ and an $n\times n$ matrix $A$,
$$\text{rank}(A) + \text{null}(A) = n$$
or that
$$\text{dim(image}(A)) + \text{dim(ker}(A)) = \text{dim}(V).$$</p>
<hr>
<p>In abstract algebra, the Orbit-Stabilizer theorem states that given a group $G$ ... | Simon Burton | 360,303 | <p>The intuition behind this question is spot-on. I'm going to try to fill out some of the details to make this work.</p>
<p>The first thing to note is that a linear map $A:V\to V$ also gives a genuine group action: it is the additive group of $V$ acting on the set $V$ by addition. That is, any $v\in V$ acts on $x\in ... |
4,038,904 | <p>Your statistics teacher challenges you to write a mathematics paper, which depending on your time spent, will earn you a cash reward. You're given the paper and a candle stands on the table in front of you. The life span <span class="math-container">$X$</span>, in minutes, of the candle is a continuous random variab... | Community | -1 | <p>Primes very much behave like random numbers once you erase the basic properties and you start looking at very large numbers. So your question is if probability of a large number having a property P is <span class="math-container">$\frac{1}{\ln(n)}$</span> and we erase one digit, what is the probability of a new numb... |
1,480,871 | <p>$1^3$ + $2^3$ + $2^3$ + ... + $n^3$ = ($1 + 2 + 3 + ... + n)^2$</p>
<p>I start with $P(1)$ and get $1 = 1$.</p>
<p>Then I do it with $P(n+1)$ and I get stuck.</p>
<p>$1^3$ + $2^3$ + $2^3$ + ... + $n^3$ + $(n+1)^3$ = ($1 + 2 + 3 + ... + n +(n+1))^2$</p>
<p>then I've tried substituting values and both ways and I ... | marty cohen | 13,079 | <p>The product of
two number of the form
4k+1 is also of the form
4k+1.</p>
<p>Therefore,
a number of the form
4k+3
must have at least one factor
of the form 4k+3.</p>
<p>Also,
since the product of
two numbers of the form 4k+3
is of the form 4k+1,
a number of the form
4k+3
must have an odd number of factors
of the fo... |
1,480,871 | <p>$1^3$ + $2^3$ + $2^3$ + ... + $n^3$ = ($1 + 2 + 3 + ... + n)^2$</p>
<p>I start with $P(1)$ and get $1 = 1$.</p>
<p>Then I do it with $P(n+1)$ and I get stuck.</p>
<p>$1^3$ + $2^3$ + $2^3$ + ... + $n^3$ + $(n+1)^3$ = ($1 + 2 + 3 + ... + n +(n+1))^2$</p>
<p>then I've tried substituting values and both ways and I ... | Deepak | 151,732 | <p>Case 1: $n$ is prime, in which case, you're done.</p>
<p>Case 2: $n$ is composite. It has no even factors so all prime factors are odd.</p>
<p>An odd prime factor is either of the form: $p \equiv 1 \pmod 4$ or $p \equiv 3 \pmod 4$.</p>
<p>If <em>all</em> prime factors were of the first form, then their product wo... |
4,508,796 | <p>How to find the integral
<span class="math-container">$$\int_0^1 x\sqrt{\frac{1-x}{1+x}}dx$$</span></p>
<p>I tried by substituting <span class="math-container">$x=\cos a$</span>. But it's leading to a form <span class="math-container">$\sin2a\cdot\tan a/2$</span> which I can't integrate further.</p>
| David Quinn | 187,299 | <p>Hint..substitute <span class="math-container">$x=\cos2\theta$</span> and you get a simple trig integral…</p>
|
3,236,067 | <p>I am having some trouble understanding where some linear boundary conditions are derived from </p>
<p>The following is an extract from my lecture notes on boundary value problems for second-order Linear ODE's</p>
<blockquote>
<p>In this section we are going to consider the different situation when some condition... | Mohammad Riazi-Kermani | 514,496 | <p>The boundary conditions are found by the physical condition of the problem at the end points. </p>
<p>For example if the end points are moving according to certain rule involving velocity or if the temperature at end points are controlled according to some rules dictated by the problem.</p>
|
257,121 | <p>The question is very simple and I apologize for that, but I am not an expert of this kind of problem.
Given the polynomial
$$ P(x_1,\ldots,x_{2n})=x_1^2+\ldots+x_n^2-x_{n+1}^2-\ldots-x_{2n}^2,$$
I would like to know if there are non trivial integer roots $(y_1,\ldots, y_{2n})$ such that
$$y_1+\cdots+y_{n}=y_{n+1}+\c... | Fedor Petrov | 4,312 | <p>Fix large $N$ and consider all $n$-tuples $(x_1,\dots,x_n)\in \{1,\dots,N\}^n$. There are $N^n$ such $n$-tuples, at least $N^n/n!$ tuples modulo permutations, and for them the pairs $(x_1+\dots+x_n,x_1^2+\dots+x_n^2)$ take at most $n\cdot N\cdot n\cdot N^2=n^2N^3$ possible values. Thus by pigeonhole principle some v... |
61,047 | <p>I can add the value of a slider to the right of it using the Appearance-->Labelled option, but what if I want to add text after the automatic label. How can I do that?</p>
<p>Normally I want to do this to show the units of the value. For example, if the slider label is "4.7", I might want it to read "4.7 meters".</... | kglr | 125 | <p>You can use <code>Quantity</code> to specify the initial value and domain of a control:</p>
<pre><code>Manipulate[x, {{x, Quantity[1, "Meters"], "x ="},
Quantity[Range[0, 1, .1], "Meters"],
ControlType -> Manipulator ,
Appearance -> "Labeled"}]
</code></pre>
<p><... |
1,363,213 | <p>I am given a chessboard of size $8*8$. In this chessboard there are two holes at positions $(X1,Y1)$ and $(X2,Y2)$. Now I need to find the maximum number of rooks that can be placed on this chessboard such that no rook threatens another. </p>
<p>Also no two rooks can threaten each other if there is hole between the... | Vishwajeet Agrawal | 715,222 | <p>Consider the generalisation of the problem, place the maximum number of rooks in an m X n board with some given squares cut out.
This problem can be reduced to bipartite matching with the following construction:
Two sets of vertices A and B.
Vertices in A represent columns such that no two rooks placed in different ... |
4,059,426 | <ul>
<li>In a longer derivation I ran into the following quantity:
<span class="math-container">$$
\nabla\left[\nabla\cdot\left(%
{\bf r}_{0}\,{\rm e}^{{\rm i}{\bf k} \cdot {\bf r}}\,\right)
\right]
$$</span>
( i.e., the gradient of the divergence ) where <span class="math-container">${\bf k}$</span> is a vector of con... | J.G. | 56,861 | <p><a href="https://en.wikipedia.org/wiki/Einstein_notation" rel="nofollow noreferrer">Summing over repeated indices</a>, the divergence is <span class="math-container">$r_{0i}\partial_ie^{\text{i}k_jr_j}=r_{0i}\text{i}k_ie^{\text{i}k_jr_j}=\text{i}(k\cdot r_0)e^{\text{i}k_jr_j}$</span>. Applying <span class="math-cont... |
2,704,247 | <p>I was given this question to prepare for an exam:</p>
<p><em>Show that the set of all functions $f(x)$ such that $f''(x)$ = -3 on ($-\infty$, $\infty$) is uncountable.</em></p>
<p>I know that this gives me a set of parabolas $f(x) = -\frac{3}{2}x^{2} + ax + b$, but I'm unsure of how to show this set is uncountable... | Martin Argerami | 22,857 | <p>The parabola determined by $a,b$ as in your formula <strong>is</strong> unique. This is trivially established by the fact that a degree two polynomial is determined by its values at three points (and you have uncountably many to choose from). </p>
|
2,704,247 | <p>I was given this question to prepare for an exam:</p>
<p><em>Show that the set of all functions $f(x)$ such that $f''(x)$ = -3 on ($-\infty$, $\infty$) is uncountable.</em></p>
<p>I know that this gives me a set of parabolas $f(x) = -\frac{3}{2}x^{2} + ax + b$, but I'm unsure of how to show this set is uncountable... | Eric Towers | 123,905 | <p>You could set $a = 0$, then the $y$-intercept of the parabola depends only on $b$ and gives a bijection with choices of $b$. That two choices of $b$ give distinct parabolas is straightforward -- all of these are vertical translates of each other and all these parabolas are functions. You have uncountably many dist... |
3,245,945 | <p>Suppose <span class="math-container">$A$</span> is a linearly ordered set without maximum or minimum and every closed interval is a finite set. I want to show <span class="math-container">$A$</span> is isomorphic to the set of integers with the usual order.</p>
<p>I know that if <span class="math-container">$A$</sp... | Alex Kruckman | 7,062 | <p>It follows from your hypothesis that every element of <span class="math-container">$A$</span> has a predecessor and a successor. So picking any <span class="math-container">$a\in A$</span>, there is an embedding <span class="math-container">$f\colon \mathbb{Z}\hookrightarrow A$</span> sending <span class="math-conta... |
3,776,889 | <p>I'm reading: <a href="https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence</a> and here it says that</p>
<blockquote>
<p>Given a probability space <span class="math-contai... | Mark | 470,733 | <p>I don't really see intuition here, the equivalence just follows from using the definition of convergence. For a sequence of sets <span class="math-container">$(A_n)$</span> the set <span class="math-container">$\lim \sup(A_n)=\{A_n\ \ i.o\}$</span> is the set of elements which belong to infinitely many of the sets <... |
3,776,889 | <p>I'm reading: <a href="https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence</a> and here it says that</p>
<blockquote>
<p>Given a probability space <span class="math-contai... | angryavian | 43,949 | <p><strong>Intuition</strong></p>
<p>There is not much intuition to be gleaned here. The second definition comes from "massaging" the definition of the [non-random] limit of real numbers (since for a fixed <span class="math-container">$\omega$</span>, the limit <span class="math-container">$\lim_{n \to \infty... |
3,746,402 | <p>For example, if we define <span class="math-container">$F(x)=\int^x_a f(t)dt$</span>, where <span class="math-container">$f$</span> is Riemann integrable, then <span class="math-container">$F(x)$</span> is a function. Or for a 2 variables real-valued integrable function <span class="math-container">$f(x, y)$</span>,... | Fernando | 335,205 | <p>That depends on how you consider the parameters <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, if they are fixed numbers, then <span class="math-container">$\int_a^bf$</span> is a number, but if you consider them as variables, you have a function of two variables</p>
<p><span cl... |
1,362,220 | <p>My question is regarding the validity of the following statement:</p>
<p>$$ (\forall a (\phi \implies \psi)) \equiv (\phi \implies \forall a \psi ),$$</p>
<p>provided, of course, there are no free occurrences of $a$ in $\phi$.</p>
<p>I am using the axiom system from <a href="http://rads.stackoverflow.com/amzn/cli... | Ramiro | 190,563 | <p>There is no error in your sketch. My only remark is that in step 5, I would also indicate that MP is used, $[5:4+UG+MP]$ and then from 1 and 5 you get the reversed implication. </p>
|
439,745 | <blockquote>
<p>Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$</p>
</blockquote>
<p>example1: $|x-1|+|x-2|\geq 1$</p>
<p>my solution：(substitution)</p>
<p>$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$</p>
<p>square,</p>
<p>$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$</p>
<p>so proved.</p>
<p><em>ques... | André Nicolas | 6,312 | <p>Note that the sum $F(x)=|x-1|+|x-2| +\cdots +|X-n|$ is the <strong>sum of the distances</strong> from $x$ to the points $1,2,\dots,n$. Draw the $n$ points $1,2,3,\dots,n$ on the number line, taking $n$ say $7$ or $8$. </p>
<p>Imagine that a particle $P$ starts far to the left of $1$, and travels to the right. </p... |
2,683,032 | <blockquote>
<p>Show the sum of the first $n$ positive even integers is $n^2 + n$ using strong induction.</p>
</blockquote>
<p>I can't solve the above problem using strong induction. It will be very helpful if I can get the solution. Thanks in advance.</p>
| owen88 | 12,981 | <p>In theory I think the two descriptions you give are the same, and that perhaps it is your understanding of conditional probability that is causing the confusion.</p>
<p>In your scenario a), whilst you consider pairs $P(w_i \,| \, w_{i-1})$ as "bigrams", i.e. pairs, once you condition on the specific choice of $w_{i... |
1,694,495 | <p>Graphically, I am searching for something like this:</p>
<p><a href="https://i.stack.imgur.com/Rskpk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rskpk.png" alt="enter image description here"></a></p>
<p>The only additional requirement would be that the elements are defined by a closed formul... | barak manos | 131,263 | <p>How about $a_n=2^{n\cdot\lfloor{n/1000000}\rfloor}$?</p>
|
3,484,136 | <blockquote>
<p>Show that <span class="math-container">$n^2+n$</span> is even for all <span class="math-container">$n\in\mathbb{N}$</span> by contradiction.</p>
</blockquote>
<p>My attempt: assume that <span class="math-container">$n^2+n$</span> is odd, then <span class="math-container">$n^2+n=2k+1$</span> for all <... | CyclotomicField | 464,974 | <p>You can't choose <span class="math-container">$k$</span> here; you have to show it's true for all of them. I would prove it this way. Assume <span class="math-container">$n^2+n$</span> is odd. This means <span class="math-container">$n(n+1)$</span> is odd, but that's impossible, because a number and its successor ca... |
2,559,560 | <blockquote>
<p>Show that there are two distinct positive integers such that: $1394|2^a-2^b$</p>
</blockquote>
<p>I'm sure pigeon hole principle applies here,but don't recognize holes.Another problem statement is: show that there are two positive integers $a,b$ such that: $$2^a\equiv 2^b\pmod {1394}$$<br>
Of course... | Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>Consider $2^i \pmod{1394}$ for $1 \leq i \leq \color{blue}{1395}$</p>
|
1,250,132 | <p>Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks.</p>
<blockquote>
<p>$$\frac{-1}{x^2}+2x=0 \implies 2x^3-1=0$$</p>
</blockquote>
| trocho | 207,836 | <p>Multiply by $x^2$ both sides of the equation.</p>
|
1,250,132 | <p>Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks.</p>
<blockquote>
<p>$$\frac{-1}{x^2}+2x=0 \implies 2x^3-1=0$$</p>
</blockquote>
| Daniel W. Farlow | 191,378 | <p>You need to keep one important thing in mind: what must be true of $x$ for $\frac{-1}{x^2}+2x=0$ to make any sense? We must have that $x\neq 0$. Bear this in mind before multiplying through:
\begin{array}{rcl}
\frac{-1}{x^2}+2x &=&0\\[0.5em]
x^2\cdot\left(\frac{-1}{x^2}+2x\right)&=&x^2\cdot0\\[0.5em]... |
1,036,636 | <p>The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.)</p>
<blockquote>
<p>If $f$ is continuous on an interval $I$ and $x_0$ is the <strong>only</strong> relative (local) extremum, then $x_0$ is actua... | SPK.z | 171,119 | <p>Not sure if this is what you mean, but I'll give it a go.</p>
<p>If you consider the extrema to be the minima, you can say that an absolute minimum is always a relative minimum (because if it's not even a relative minimum, how can it be an absolute minimum?). That means that only relative minima are candidates for ... |
1,534,981 | <p>I am trying to verify my procedure to find if the extrema is correct for a function
$u \left( x,y \right) ={x}^{2}-{y}^{2}$ on the set $\left( \mathop{\rm D}~~=~\left\{\left(x ,y \right)
\in \mathbb{R}^{2}{\it} | x ^{2}+y ^{2}
\le 1\right\} \right)
$</p>
<p>By the closed interval method, to find the absolute maximu... | Dr. Sonnhard Graubner | 175,066 | <p>the critical values on $D$ are the solution of the System
$$2x=0$$
$$2y=0$$
for the others we consider
$$F(x,y)=x^2-y^2+\lambda(x^2+y^2-1)$$
and you must solve
$$2x+2x\lambda=0$$
$$-2y+\lambda2y=0$$
$$x^2+y^2=1$$</p>
|
135,911 | <p>Liouville's theorem gives such a proof for antiderivatives of functions like <span class="math-container">$e^x/x$</span> or <span class="math-container">$e^{x^2}$</span>, and differential Galois theory extends that to Bessel functions, say. But what tools exist for implicit functions like Lambert's W?</p>
| IV_ | 94,085 | <p>In [Ritt 1948] p. 53 - 56, the method of J. Liouville is given for Kepler's equation. The same method can be applied to functions <span class="math-container">$f$</span> with <span class="math-container">$f(z)=A(z,e^z)$</span> (<span class="math-container">$A$</span> an algebraic function of two complex variables wi... |
4,050,336 | <p>I am familiar with the Negative Binomial distribution <span class="math-container">$NB(p, k)$</span>, which gives the number of failures before <span class="math-container">$k$</span> successes occur in a Bernoulli process with parameter <span class="math-container">$p$</span>. I am wondering, however, if there the ... | Kenny Lau | 328,173 | <p>Let <span class="math-container">$a+bi\sqrt2$</span> be a root of <span class="math-container">$x^2+1$</span>, so <span class="math-container">$(a+bi\sqrt2)^2 = -1$</span>, i.e. <span class="math-container">$(a^2-2b^2)+2abi\sqrt2 = -1$</span>.</p>
<p>Equate coefficients and derive a contradiction.</p>
|
164,309 | <p>I came across this recurrence function:</p>
<blockquote>
<p>$$F(n) = a \times F(n-1) + b$$</p>
</blockquote>
<p>where $F(0) =1$. We have to solve for $F(n) \pmod {m}$</p>
<p>But for very large $n$, solving it with computer is also taking time. Is there anyway to simplify this. I think the values will be repeate... | Community | -1 | <p>$F(n) = T(n) + c \implies T(n) + c = aT(n-1) + ac + b$.</p>
<p>Choosing $c = b/(1-a)$, we get that $T(n) = a T(n-1) \implies T(n) = a^n T(0)$.</p>
<p>Hence, $$F(n) = T(n) + c = a^n T(0) + b/(1-a) = a^n (F(0) - b/(1-a)) + b/(1-a)\\ \implies F(n) = a^n F(0) +b \dfrac{1-a^n}{1-a} = a^n + b \dfrac{1-a^n}{1-a}$$</p>
<... |
1,159,155 | <p>If we define <strong>open</strong> as: A set $O⊆R$ is open if for all points $a∈O$ there exists an
$\epsilon$-neighborhood $V_\epsilon(a)⊆O$.</p>
<p>Where $V_\epsilon(a) = \{x \in \mathbb{R}: | x - a | < \epsilon\}$
Now consider some open interval: </p>
<p>$(c,d) = \{x \in \mathbb{R} : c<x<d \}$</p>
<p>T... | Ken | 169,838 | <p>If you allowed $\epsilon$ to be $0$, then every set is open since the $0$-neighborhood of a point is just itself. For this reason, $\epsilon$ is understood to be positive.</p>
|
36,735 | <p>In Peter J. Cameron's book "Permutation Groups" I found the following quote</p>
<blockquote>
<p>It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely).</p>
</blockquote>
<p>Indeed, one... | Benoît Kloeckner | 4,961 | <p>If you have an algorithm that produces uniform and independent samples from a set of object, you can estimate the total number of objects as follows. First, construct a subset of the objects to be counted, if possible quite large, in a way such that you know the size of the subset and you can check easily if a given... |
36,735 | <p>In Peter J. Cameron's book "Permutation Groups" I found the following quote</p>
<blockquote>
<p>It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely).</p>
</blockquote>
<p>Indeed, one... | Brendan McKay | 9,025 | <p>Suppose there are two finite sets $A$, $B$ and you have some relation on $A\times B$. If you can randomly sample from $A$ and $B$, you can estimate the average number $a$ of elements of $A$ related to each element of $B$, and the average number $b$ of elements of $B$ related to each element of $A$. Then $a/b=|A|/|B... |
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