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 |
|---|---|---|---|---|
307,311 | <p>My question is related to Arithmetic progression. I would solve it myself but I need some help understanding the question.</p>
<p>Q.The second, 31st and the last terms of an A.P are $$ \frac{31}{4},\frac{1}{2} $$ and $$ \frac{-13}{2} $$ respectively. Find the number of terms.</p>
<p>How can I use the $$ Tn=a+(n-1)... | Community | -1 | <p>$a+d=\frac{31}{4}$, $a+30d=\frac12$, and $a+(l-1)d=-\frac{13}{2}$. Find $l$.</p>
|
307,311 | <p>My question is related to Arithmetic progression. I would solve it myself but I need some help understanding the question.</p>
<p>Q.The second, 31st and the last terms of an A.P are $$ \frac{31}{4},\frac{1}{2} $$ and $$ \frac{-13}{2} $$ respectively. Find the number of terms.</p>
<p>How can I use the $$ Tn=a+(n-1)... | André Nicolas | 6,312 | <p>The algebraic approach described by Jacob Black is the right one. For fun, let us see how someone innocent of algebra might tackle things.</p>
<p>From the $2$nd term to the $31$-th (which takes $29$ steps), we go down by $\dfrac{31}{4}-\dfrac{1}{2}=\dfrac{31}{4}-\dfrac{2}{4}=\dfrac{29}{4}$.</p>
<p>So we go down by... |
1,018,716 | <p>I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows:
$$||f(x-a)-f(x)||_p=||f(x-a)-g(x-a)+g(x-a)-g(x)+g(x)-f(x)||_p\leq ||f(x-a)-g(x-a)||_p+||g(x-a)+g(x)||_p+||g(x)-f(x)||_p<3\varepsilon$$
Where $g$ is so... | Jason Knapp | 8,454 | <p>Your statement in the comments that an operator $T$ is continuous on a Banach space $X$ if, for a sequence $f_n \rightarrow f$ in $X$ then we have $Tf_n \rightarrow Tf$. That is about the operator $T$ being continuous. The functions $f_n$ and $f$ are simply an arbitrary collection of elements in the space that for... |
1,550,934 | <p>I want to calculate the nth derivative of <span class="math-container">$\arcsin x$</span>. I know
<span class="math-container">$$
\frac{d}{dx}\arcsin x=\frac1{\sqrt{1-x^2}}
$$</span>
And
<span class="math-container">$$
\frac{d^n}{dx^n} \frac1{\sqrt{1-x^2}} = \frac{d}{dx} \left(P_{n-1}(x) \frac1{\sqrt{1-x^2}}\right)... | 1010011010 | 155,608 | <p>Outline of a proof that does not use the recursive relation but does yield the correct answer.</p>
<p>Use Faà di Bruno's formula for <span class="math-container">$n$</span>-fold derivatives in it's "Bell polynomial" form. This omits the question how the tuples affect the final form of the derivatives, so s... |
1,456,591 | <p>It is known that if $$|a+b|=|a|+|b|$$ then we can find the solution by simply observing that we can instead solve the inequality $$a b \geq 0$$</p>
<p>My question is, if $|a+b+c|=|a|+|b|+|c|$, then what would be the '3 degree version' of the above?</p>
| Micah | 30,836 | <p>The totally general version of this (which works with arbitrarily many vectors, as well as in $n$ dimensions) is "all the vectors are positive scalar multiples of each other."</p>
|
1,109,671 | <p>Some things I know:</p>
<ul>
<li>$S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$</li>
<li>$(2,4) \in N_G(S)$</li>
<li>Number of conjugates = $[G: N_G(S)]$</li>
</ul>
<p>This seems like such a easy question but it made me realised that I do not know how to go about thinking about (and finding) the right cosets of $N... | ahulpke | 159,739 | <p>Conjugation in $S_n$ (i.e. relabelling the points) preserves the cycle structure. So all conjugate subgroups must be of the form $<(a,b,c,d)>$, because one of the points is $1$ it really is $<(1,a,b,c)>$, and a conjugating permutation would be one mapping $2\mapsto a$, $3\mapsto b$, $4\mapsto c$ while fi... |
3,441,200 | <p>Prove that a tangent developable has constant Gaussian curvature zero. Also compute its mean curvature.</p>
<p>I have a tangent developable as let <span class="math-container">$\gamma:(a,b)\rightarrow R^3$</span> be a regular space curve, and s>0. Then <span class="math-container">$\sigma(s,t) = \gamma(t)+s\gamma'(... | Ivo Terek | 118,056 | <p>Here's another proof: if <span class="math-container">$\sigma(s,t) = \gamma(t)+s\gamma'(t)$</span>, then <span class="math-container">$$\sigma_s(s,t) = \gamma'(t) \quad\mbox{and}\quad \sigma_t(s,t) = \gamma'(t)+s\gamma''(t),$$</span>so that <span class="math-container">$$N(\sigma(s,t)) = \frac{\sigma_s(s,t)\times \s... |
338,480 | <blockquote>
<p>Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.</p>
</blockquote>
<p>This's the first time I meet this kind of problem, can someone give me some idea? Thank you.</p>
| Glen O | 67,842 | <p>Well, what you basically need to do is find $t_1$ and $t_2$ such that $x(t_1)=x(t_2)$ and $y(t_1)=y(t_2)$, with $t_1 \neq t_2$.</p>
<p>To help you a little further - if you examine one of the two equations, you'll get a polynomial in $t_1$ and $t_2$ that must be divisible by $t_1-t_2$. Divide out that factor and yo... |
2,251,240 | <p>What is the first derivative and nth derivative of the following function $ y = \sqrt {2 +\sqrt {3 + \sqrt {x}}}$ </p>
<p>I think taking the ln for both sides will remove the first square root only?
Could anyone give me a hint ? </p>
| Ennar | 122,131 | <p>Let $y = \sqrt{2+\sqrt{3+\sqrt x}}$. Step by step we get:</p>
<p>\begin{align}
y^2-2&=\sqrt{3+\sqrt x}\\
y^4-4y^2+1&=\sqrt x\\
(y^4-4y^2+1)^2&=x
\end{align}</p>
<p>Define $f(x) = (x^4-4x^2+1)^2$. By the above, we have $f(y) = x$ and thus $$f'(y)y' = 1 \implies y' = \frac 1{f'(y)}$$</p>
<p>This gives t... |
1,605,100 | <p>$$\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$$</p>
<p>I tried this using powerseries just putting as $x=1$ there ,even tried thinking subtracting $s_{n+1} - s_{n}$ would be of some help, also I thought of writing the denominator as product of two complex numbers and then doing the partial fractions but it did ... | P Vanchinathan | 28,915 | <p>Take any coset for $H$ in $G$: e.g., $aH= \{ah \mid h\in H \}$. For any two arbitrary $ah_1$ and $ah_2$ from this coset the conjugate subgroups $ah_1 H(ah_1)^{-1}$ and $ah_2 H(ah_2)^{-1}$ are the same (it is the same as the subgroup $aHa^{-1}$. SO number of distinct subgroups is limited by the number of cosets $aH$... |
2,661,499 | <blockquote>
<p>Show that $(z_1+z_2+z_3)/3$ is the centroid of a triangle whose vertices are $z_1$, $z_2$, $z_3$.
(Hint: The centroid divides the median internally in the ratio of 2:1) </p>
</blockquote>
| Robert Z | 299,698 | <p>Show that the point $G:=\frac {z_1 + z_2 + z_3}3$ is along the median line through the vertex $z_1$ and the midpoint of the opposite side $\frac {z_2 + z_3}2$ whose parametric equation is $(1-t)z_1+t\frac {z_2 + z_3}2$ with $t\in\mathbb{R}$. The same for the other two medians.</p>
<p>Can you take it from here?</p>
|
2,661,499 | <blockquote>
<p>Show that $(z_1+z_2+z_3)/3$ is the centroid of a triangle whose vertices are $z_1$, $z_2$, $z_3$.
(Hint: The centroid divides the median internally in the ratio of 2:1) </p>
</blockquote>
| user | 505,767 | <p>Note that from the hint and following <a href="https://math.stackexchange.com/a/2661503/505767">RobertZ suggestion</a> you should arrive to</p>
<p>$$\frac{z_1+z_2+z_3}3=z_1+\frac23\left(\frac{z_2+z_3}2-z_1\right)=z_2+\frac23\left(\frac{z_3+z_1}2-z_2\right)=z_3+\frac23\left(\frac{z_1+z_2}2-z_3\right)$$</p>
|
222,653 | <p>We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity.</p>
<p>The question is the following: can we assure that the sequence does <em>NOT</em> converge in distribution to a Poisson? <... | Nate Eldredge | 4,832 | <p>No. It's easy to construct a sequence $Y_n$ with $Y_n \to 0$ a.s. but $E Y_n \to +\infty$. (You can even have $E Y_n \equiv +\infty$ if you wish.) Now let $X$ be a fixed Poisson random variable and $X_n = X + Y_n$.</p>
|
2,734,109 | <p>This is what my lecture notes have but I cannot find anything like it online and there is no explanation in the notes. The example given is for 903 and 444.
<a href="https://i.stack.imgur.com/9NXYk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NXYk.png" alt="enter image description here"></a>
... | Joffan | 206,402 | <p>The <a href="https://en.wikipedia.org/wiki/Euclidean_algorithm" rel="nofollow noreferrer">Euclidean algorithm</a> will quickly give you the greatest common divisor of two numbers, and in its extended version will give you the formula for a linear combination of the two numbers that equates to the $\gcd$.</p>
<p>I u... |
656,742 | <p>One <a href="http://en.wikipedia.org/wiki/Nilpotent_matrix" rel="nofollow">property</a> of nilpotent matrices is that a matrix $N$ is nilpotent if and only if $\operatorname{tr}(N^k)=0$ for all $k>0$. How can this property be proved?</p>
| Community | -1 | <p>Let $\lambda_1,\ldots,\lambda_n$ the eigenvalues of $N$ repeated with their multiplicities and notice that if $\lambda_k$ is an eigenvalue of $N^k$ so since $\operatorname{tr}(N^k)=0$ we find the system of equations
$$\sum_{i=1}^n\lambda_i^k=0\quad k=1,\ldots,n$$</p>
<p>Now we solve this system by induction:</p>
<... |
1,224,692 | <p>What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer is numbered one. The balls on the second layer are numbered 2 - 4. The fifth layer 4-10 and so on. </p>
| Bernard | 202,857 | <p>Use the comatrix of $\Omega$, i.e. the matrix of cofactors of $\Omega$: if $\Omega \mathbf x= \mathbf 0$, then $\operatorname{com}(\Omega)\cdot\Omega\mathbf x=\mathbf 0$.</p>
<p>However, $\operatorname{com}(\Omega)\cdot\Omega =(\det \Omega)\, I$, hence $(\det \Omega)\, I\mathbf x=(\det \Omega)\,\mathbf x=\mathbf 0... |
1,890,496 | <p>I am so confused on how to find the domain of this function $3^\sqrt{x^2-3x}$ without graphing it. I have no idea what to do in this situation.</p>
| Community | -1 | <p>One way to de-nest the conditions is this one.</p>
<ol>
<li><p>$y\mapsto 3^y$ is defined fo all $y\in\Bbb R$, so you just want $\sqrt{x^2-3x}$ to exist.</p></li>
<li><p>$y\mapsto \sqrt{y}$ is defined only for $y\ge 0$, so you want $x^2-3x$ to exist and $x^2-3x\ge 0$</p></li>
<li><p>$y\mapsto y^2-3y$ is defined for ... |
2,898,786 | <p>Let $p=3$ and $\zeta$ be a cube root of unity not equal $1$. Consider the field of $3$-adic numbers $\mathbb{Q}_3$.
At the beginning of section 5.4 of Fernando Gouvea's book <em>$p$-adic numbers - An Introduction</em>, he states that the field $\mathbb{Q}_3(\sqrt{2},\zeta)$ is an extension of degree $4$. Furthermore... | nguyen quang do | 300,700 | <p>A preliminary bit of Kummer theory will certainly do no harm. Let $K$ be a field of characteristic $\neq 2$, such that neither $2$ nor $-3$ are squares in $K$. Then $K(\sqrt 2)$ and $K(\sqrt {-3})=K(\zeta)$ (where $\zeta$ is a primitive 3-rd root of 1) are quadratic extensions, and their compositum $L=K(\sqrt 2,\sq... |
2,898,786 | <p>Let $p=3$ and $\zeta$ be a cube root of unity not equal $1$. Consider the field of $3$-adic numbers $\mathbb{Q}_3$.
At the beginning of section 5.4 of Fernando Gouvea's book <em>$p$-adic numbers - An Introduction</em>, he states that the field $\mathbb{Q}_3(\sqrt{2},\zeta)$ is an extension of degree $4$. Furthermore... | Lubin | 17,760 | <p>Here’s an argument that you may find more direct.</p>
<p>First, your $g(X)=X^2+X+1$ has $g(X+1)=X^2+3X+3$, Eisenstein, so irreducible. Not only that, its root $\zeta_3-1$ clearly has (additive) $v_3$-valuation equal to $\frac12$, so that $\Bbb Q_3(\zeta_3)$ is quadratic and (totally) ramified over $\Bbb Q_3$.</p>
... |
87,648 | <pre><code>\[GothicCapitalR] = {{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1,
0}, {0, 0, 1}, {1, 0, 0}}, {{0, 0, 1}, {1, 0, 0}, {0, 1,
0}}, {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}, {{0, 0, 1}, {0, 1,
0}, {1, 0, 0}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}}};
i = 1;
j = 1;
det = 1;
a = Subsets[Range[6], {3}];
v = {x, y, z};
k = \... | Marius Ladegård Meyer | 22,099 | <p>Instead of <code>While[i<21,...]</code>, use <code>Table[...,{i,20}]</code>. The ... part can be reduced to</p>
<pre><code>r = k[[a[[i]]]];
Factor[Det[r]]
</code></pre>
<p>To find the <code>PolynomialLCM</code>, simply replace the head ( <code>List</code>) of the table with <code>PolynomialLCM</code> using <cod... |
4,157,288 | <p>High school student here just wanting to be better at maths and to know how to approach and solve problems and also how to think like a mathematician. Please recommend a book to me it would help a lot.</p>
| Ethan Bolker | 72,858 | <p>I strongly recommend Polya's <em>Induction and Analogy in Mathematics</em>.</p>
<p><a href="https://press.princeton.edu/books/paperback/9780691025094/mathematics-and-plausible-reasoning-volume-1" rel="nofollow noreferrer">https://press.princeton.edu/books/paperback/9780691025094/mathematics-and-plausible-reasoning-v... |
4,157,288 | <p>High school student here just wanting to be better at maths and to know how to approach and solve problems and also how to think like a mathematician. Please recommend a book to me it would help a lot.</p>
| LegNaiB | 925,793 | <p>Mathematical thinking is nothing you can just get inside a book. Maybe there are some of those books, but it really depends on the topic you are interested in. Knowing how to approach and solve problems is a task that needs years to get better and you will never be perfect at it.</p>
<p>You could start with a book a... |
1,471,415 | <p>I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one:</p>
<p>$$
v = 1 - e^{-t/\tau}
$$</p>
<p>The derivative is</p>
<p>$$
\frac{dv}{dt} = \frac{1-v}{\tau}
$$</p>
<p>Can someone walk me through this? If this is explained somewhere else, I'd love to k... | GAVD | 255,061 | <p>We have $$\dfrac{dv}{dt} = \frac{1}{\tau} e^{-\frac{t}{\tau}} = \frac{1-(1-e^{-\frac{t}{\tau}})}{\tau} = \frac{1-v}{\tau}$$. Is it clear?</p>
|
1,471,415 | <p>I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one:</p>
<p>$$
v = 1 - e^{-t/\tau}
$$</p>
<p>The derivative is</p>
<p>$$
\frac{dv}{dt} = \frac{1-v}{\tau}
$$</p>
<p>Can someone walk me through this? If this is explained somewhere else, I'd love to k... | Arashium | 209,976 | <p>Derivative is a linear operator so it follows superposition:</p>
<p>$$\frac{d}{dt}(f(t)+g(t))=\frac{d f(t)}{dt}+\frac{d g(t)}{dt}$$</p>
<p>and</p>
<p>$$\frac{d}{dt}(k\times f(t))=k \times \frac{f(t)}{dt}~~~~~~~(k \text{ is constant number})$$</p>
<p>Also, derivative of some functions are known such as $\exp$, $\... |
1,471,415 | <p>I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one:</p>
<p>$$
v = 1 - e^{-t/\tau}
$$</p>
<p>The derivative is</p>
<p>$$
\frac{dv}{dt} = \frac{1-v}{\tau}
$$</p>
<p>Can someone walk me through this? If this is explained somewhere else, I'd love to k... | Robert Lewis | 67,071 | <p>OK, let's walk: first, note that</p>
<p>$v = 1 - e^{-t / \tau} \tag{1}$</p>
<p>yields, by direct differentiation, using the chain rule,</p>
<p>$\dfrac{dv}{dt} = -e^{-t / \tau} \dfrac{d(-t / \tau)}{dt} = \dfrac{1}{\tau} e^{-t / \tau}; \tag{2}$</p>
<p>from here on, it's simple algebra:</p>
<p>$\dfrac{1 - v}{\tau... |
4,621,030 | <p>I now that this is wrong, but why?</p>
<p><span class="math-container">$$3x\log(2)+2x\log(3) = \log(6)$$</span>
<span class="math-container">$$3x\log(2)+2x\log(3) = \log(2*3)$$</span>
<span class="math-container">$$3x\log(2)+2x\log(3) = 1\log(2)+1\log(3)$$</span>
<span class="math-container">$$3xA+2xB = 1A+1B$$</spa... | Mr.Gandalf Sauron | 683,801 | <p>What you have is :-</p>
<p><span class="math-container">$$x\sum_{k\geq 1}\frac{x^{k-1}(k-1)}{(k-1)!} + x\sum_{k\geq 1}\frac{x^{k-1}}{(k-1)!}=x^{2}\sum_{k\geq 2}\frac{x^{k-2}}{(k-2)!}+x\sum_{k\geq 1}\frac{x^{k-1}}{(k-1)!}=x^{2}e^{x}+xe^{x}$$</span></p>
|
104,368 | <p>I can show that there infinitely many solutions to this equation. Is it possible that the set
of rational solutions is dense?</p>
| S. Carnahan | 121 | <p>For your first question, I can give an application close to my own field. Perhaps a more Langlands-ish person can say something interesting about applications to modularity of Galois representations.</p>
<p>The moduli interpretation of level structures can appear naturally in orbifold 2D conformal field theory. I... |
343,248 | <p>Let <span class="math-container">$\chi$</span> be a primitive Dirichlet character of modulus <span class="math-container">$q>1$</span>. Write, as is customary, <span class="math-container">$B(\chi)$</span> for the constant in the expression
<span class="math-container">$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} =... | Alessandro Languasco | 144,222 | <p>I am replying to this question “for odd, we do; see Prop. 10.3.5 (due to...?) in Henri Cohen's Number Theory”. I would be happy to insert a comment instead, but my MO-reputation is not good enough...</p>
<p>In a paper published in 1989, Kanemitsu wrote that this formula was first published by Berger in 1883. </p>
... |
3,963,229 | <p>I tried to find this limit using polar representation <span class="math-container">$x=r \cos\theta $</span> and <span class="math-container">$y=r \sin \theta$</span>
i got something like <span class="math-container">$\ln (\frac{1}{\cos2 \theta})$</span> which makes me feel im in wrong way !</p>
<p>is my approach cor... | math | 865,747 | <p>If the limit exists, then you have that :</p>
<p><span class="math-container">$\underset{x,y\rightarrow 0}{\text{lim}} \ln \frac{x^2 + y^2}{x^2 - y^2} = \underset{x\rightarrow 0}{\text{lim}} \underset{y\rightarrow 0}{\text{lim}} \ln \frac{x^2 + y^2}{x^2 - y^2} = \underset{y\rightarrow 0}{\text{lim}} \underset{x\righ... |
241,058 | <p>I'm reading <a href="http://rads.stackoverflow.com/amzn/click/0198529813" rel="nofollow">A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity</a>.</p>
<blockquote>
<p>The graph of $f: A \rightarrow B$ is the subset of $A × B$ consisting
of all ordered pairs $(a, ... | André Nicolas | 6,312 | <p>Here $B^n$ means the set of all $n$-tuples $(b_1,\dots,b_n)$, where the $b_i$ range over $B$.</p>
<p>You are already familiar with special cases such as $\mathbb{R}^2$ and $\mathbb{R}^3$, and probably $\mathbb{R}^n$. </p>
|
241,058 | <p>I'm reading <a href="http://rads.stackoverflow.com/amzn/click/0198529813" rel="nofollow">A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity</a>.</p>
<blockquote>
<p>The graph of $f: A \rightarrow B$ is the subset of $A × B$ consisting
of all ordered pairs $(a, ... | Peter Smith | 35,151 | <p>As @André says, $B^n$ here means the set of n-tuples of elements of $B$. </p>
<p>But then note an interesting little wrinkle. $f\colon B^n \to B$ is in fact a <em>one</em>-argument function which maps <em>one</em> thing -- the $n$-tuple $(b_1, b_2, \ldots, b_n)$ -- to another thing. </p>
<p>It is convenient for ma... |
1,438,026 | <p>Out of curiousity, are there functions who's domain is discrete but the range is continuous? Furthermore, is there also a real-world example of such a function, in physics for instance?</p>
| mvw | 86,776 | <p>Consider
$$
f: \mathbb{N} \to \mathbb{R} \\
f(k) = 1 / k
$$</p>
<ul>
<li>The domain consists of the positive integers, thus discrete. </li>
<li>The range consists of the real numbers, which are continuous.</li>
<li>The image $f(\mathbb{N})$ is a set of discrete real numbers.</li>
</ul>
|
33,907 | <p><strong>Note:</strong> Cross-posted at <a href="http://community.wolfram.com/groups/-/m/t/137895?p_p_auth=8QnKtT9I" rel="nofollow">http://community.wolfram.com/groups/-/m/t/137895?p_p_auth=8QnKtT9I</a>.</p>
<p>I am to design a two step gearbox. The first step is to choose the number of teeth in each cog wheel in o... | bill s | 1,783 | <p>Another approach is to use <code>FindInstance</code>:</p>
<pre><code>FindInstance[(172/10 <= (N1 N2)/(n1 n2) <= 174/10) &&
(n1 >= 20) && (n2 >= 20) && (N1 > 10) && (N2 > 10), {n1, n2, N1, N2}, Integers]
{{n1 -> 20, n2 -> 20, N1 -> 11, N2 -> 630}}
... |
143,173 | <p>I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - </p>
<p>I would like to show that, for a real number <span class="math-container">$p \geq 1$</span> and complex numbers <span class="math-container">$\alpha, \beta$... | john w. | 24,430 | <p>If the $p-1$ being $p$ is not a big deal then I think the following works.
$|\alpha+\beta|\leq |\alpha|+|\beta|\leq 2\max\{|\alpha|,|\beta|\}$. If the max is $|\alpha|$, then $|\alpha+\beta|^p\leq 2^p|\alpha|^p\leq 2^p(|\alpha|^p+|\beta|^p)$. Similarly for $\beta$. In either case you get the inequality you want.</p... |
57,057 | <p>Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{83}}{10}}\ \ ?$$</p>
| Zarrax | 3,035 | <p>This is only a partial solution but I think someone more familiar with such elementary inequalities than myself might be able to finish it. You can replace $a,b,c,d$, and $e$ with their reciprocals and the inequality in question becomes
$$a + b + c + d + e + {33 \over 2}{1 \over ({1 \over a} + {1 \over b} + {1 \over... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| András Hummer | 62,202 | <p>Substitute n! with Stirling's approximation, then divide ${a}^{n}$ with it and find the limit.</p>
|
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Nguyen | 131,634 | <p>We show that $$\lim_{n \to \infty} \frac{\displaystyle\sum_{1 \leq i \leq n}\log(i)}{n \log(a)} = \infty.$$
Indeed, $$\sum_{1 \leq i \leq n}\log(i) > \sum_{n/2 \leq i \leq n}\log(i).$$ Note that for all $i \geq n/2$, we have $\log(i) \geq \log(n/2) = \log(n)-1$. Hence, we have $\sum_{n/2 \leq i \leq n}\log(i) \ge... |
351,815 | <p>Having trouble understanding this. Is there anyway to prove it?</p>
| Community | -1 | <p>Although it is too late to answer this question, especially, when really nice answers have already been presented, I want to share my intuition about the subject.</p>
<p>Suppose a sequence of positive integers is given: $1, 2, \cdots, n$, and you take geometric mean of the given numbers. As new numbers are added to... |
890,200 | <p>Let $x_1:=a>0$ and $x_{n+1}:=x_n+1/x_n$ for $n\in\mathbb{N}$. Determine whether $(x_n)$ converges or diverges.</p>
<p>My answer: $(x_n)$ is divergent.</p>
<p>Proof: Assume that $(x_n)$ converges to $x$. Then $\lim (x_{n+1})=\lim (x_n)$. That is, $x=x+1/x$. This equation has no solution. Hence, $(x_n)$ is... | Asier Calbet | 166,157 | <p>I can prove the stronger result: the sequence is unbounded and tends to $\infty$.
It is given that $x_1=a>0$, so the first term is positive. By looking at the recurrence relation, it is clear that the entire sequence is positive ( can be proven easily by induction). Now, assume the sequence is bounded, so $x_n&l... |
2,845,049 | <p>Let $\beta_m\searrow 0$ such that $\alpha_m:=\beta_m-\beta_{m+1}\searrow 0$.</p>
<p>Define $b_n:=\inf\{m:\alpha_m<2^{-n}\}$. Is it true that
$$
\sum_{n=1}^\infty \frac{b_n}{2^n}<\infty?
$$</p>
<p>For example, if $\beta_m=\frac 1 m$, then $b_n\sim 2^{n/2}$, so that the above series converges.</p>
<p>A critic... | Clement C. | 75,808 | <p><em>For what it's worth (this is community wiki, feel free to edit) here is the analysis of your critical case.</em></p>
<p>Consider $\beta_m = \frac{1}{\ln m}$. Then,
$$\begin{align}
\alpha_m &= \beta_m - \beta_{m+1} = \frac{1}{\ln m}\left( 1- \frac{1}{1 + \frac{\ln(1+\frac{1}{m})}{\ln m}}\right)= \frac{1}{\ln... |
786,271 | <p>Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n+b^n$
is prime, what can you relate $n$ with 2?</p>
<p>My approach: for $a^n+b^n$ to be prime $\forall n>1$, $a$ and $b$ has to be coprimes.
But how do I ascertain anything about $n?$</p>
| doraemonpaul | 30,938 | <p>$\dfrac{dy}{dx}=\dfrac{1}{\sqrt{x^2+y^2}}$</p>
<p>$\dfrac{dx}{dy}=\sqrt{x^2+y^2}$</p>
<p>Apply the <a href="http://en.wikipedia.org/wiki/Euler_substitution" rel="nofollow">Euler substitution</a>:</p>
<p>Let $u=x+\sqrt{x^2+y^2}$ ,</p>
<p>Then $x=\dfrac{u}{2}-\dfrac{y^2}{2u}$</p>
<p>$\dfrac{dx}{dy}=\left(\dfrac{1... |
88,284 | <p>If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$
under usual addition and multiplication, then what are the left and right ideals of $R$?</p>
| student | 20,150 | <p>This is a partial answer that is too long for a comment. I'm not sure about all ideals, but you have at least two big families of left ideals.</p>
<p>Given $q \in \mathbb{N}$, define:
$$I_q = \bigg\{ \left(\begin{matrix} 0 & a/q \\ 0 & 0 \end{matrix}\right)~:~ a \in \mathbb{Z}\bigg\}.$$
This is a left ideal... |
2,122,539 | <p>A functor sends $X$ to $R^{(X)}$ what does this notation mean? Is this functor faithful and full?</p>
| tschih | 344,545 | <p>$R^{(X)}$ denotes the free R-module over X.</p>
<p>The functor is faithful since different maps on bases of free modules induce different module homomorphisms.</p>
<p>It is not full since there exist module homomorphisms between free modules that are not induced by maps on their bases (e.g. the zero homomorphism).... |
2,729,116 | <p><strong>Question 1.</strong> Let $V=\mathbb{R}^3$, $T:V \rightarrow V$ be linear. Suppose that $T^3=T, T^2 \neq T, T^2 \neq Id,$ and $\dim \ker T = 2.$ Show that the matrix of $T$ with respect to some basis is</p>
<p>$$
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatr... | Delta-u | 550,182 | <p>The polynomial $P = X^3-X = (X-1)(X+1) X$ is divisible by the minimal polynomial $\mu_T$ of $T$.</p>
<p>More over, from the other hypotheses, $X^2-1=(X-1)(X+1)$ and $X^2-X=(X-1) X$ are not divisible by $\mu_T$.</p>
<p>There is two options remaining:</p>
<ul>
<li>$\mu_T= X(X+1)(X-1)$ then $\dim(\ker(T))=1$.</li>
... |
2,729,116 | <p><strong>Question 1.</strong> Let $V=\mathbb{R}^3$, $T:V \rightarrow V$ be linear. Suppose that $T^3=T, T^2 \neq T, T^2 \neq Id,$ and $\dim \ker T = 2.$ Show that the matrix of $T$ with respect to some basis is</p>
<p>$$
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
\end{pmatr... | Mohammad Riazi-Kermani | 514,496 | <p>$$ T(T-I)(T+I) =0 \implies \lambda (\lambda-1)(\lambda+1)=0 $$</p>
<p>Where $\lambda$ is an eigenvalue of your matrix.</p>
<p>Upon the orthogonal diagonalization you will find a diagonal matrix with the eigenvalues on the main diagonal. </p>
<p>The set of eigenvectors associated with this matrix constitute the d... |
793,930 | <p>I was asked to evaluate the integral </p>
<p>$$\int_{-1}^{1} \frac{\sin{x}}{1+x^2}dx$$</p>
<p>if it exists.</p>
<p>This is a problem from Calculus and the student has been taught how to use trigonometric substitution. My intuition was to do trig sub with $$x=\tan{\theta}$$ and eliminating $$\frac{dx}{1+x^2}$$ b... | Jeff Faraci | 115,030 | <p>Zero by symmetry. </p>
<p>It is an odd integrand integrated over a symmetric bound</p>
<p>or more properly "integrated over an interval that is symmetric about the origin".</p>
|
2,478,695 | <p>So I need to solve the integral </p>
<blockquote>
<p>$$\int \frac { \tan { x } }{ \left( \sin { x } \right) ^{ 2 }+2\left( \cos { x } \right) ^{ 2 } } dx$$</p>
</blockquote>
<p>I saw some exercises that suggest I need to use the secant function to solve it but can I do it without it?</p>
| haqnatural | 247,767 | <p>$$\int \frac { \tan { x } }{ \left( \sin { x } \right) ^{ 2 }+2\left( \cos { x } \right) ^{ 2 } } dx=\int \frac { \tan { x } }{ \cos ^{ 2 }{ x } \left( 2+\tan ^{ 2 }{ x } \right) } dx=\int \frac { \tan { x } }{ \left( 2+\tan ^{ 2 }{ x } \right) } d\tan { x } \\ $$
then sunstitute $z=\tan { x } $ so that
... |
3,660,714 | <p>I know that <span class="math-container">${\rm Aut}(C_7$</span>) is <span class="math-container">$C_6$</span>, and <span class="math-container">${\rm Aut}(C_6$</span>)= <span class="math-container">$C_4 \times C_2$</span> but I do not know if that will help. </p>
<p>Any hints or tips welcome!</p>
| Nathan Crawford | 784,377 | <p>You are almost there. To construct a semidirect product of 2 groups <span class="math-container">$A$</span> and <span class="math-container">$B$</span> you need a homomorphism <span class="math-container">$\psi:A\to Aut(B)$</span>. In your case <span class="math-container">$A=C_{3}$</span> and <span class="math-cont... |
4,203,431 | <p>Can I infer <span class="math-container">$AB=C$</span> from <span class="math-container">$AB \vec r = C \vec r$</span>? where <span class="math-container">$\vec r$</span> is <span class="math-container">$n \times 1$</span> vector(<span class="math-container">$n$</span> rows and <span class="math-container">$1$</span... | Stanislav Bashkyrtsev | 24,684 | <p>Since <span class="math-container">$r$</span> may contain <span class="math-container">$0$</span>'s, it will ignore corresponding columns when doing matrix multiplication. E.g. here are 2 different matrices which will produce the same result:</p>
<p><span class="math-container">$$
\begin{bmatrix}
1 & 2 \... |
3,713,900 | <p>I have the function:</p>
<p><span class="math-container">$$f : \mathbb{R} \rightarrow \mathbb{R} \hspace{2cm} f(x) = e^x + x^3 -x^2 + x$$</span></p>
<p>and I have to find the limit:</p>
<p><span class="math-container">$$\lim\limits_{x \to \infty} \frac{f^{-1}(x)}{\ln x}$$</span></p>
<p>(In the first part of the ... | user600016 | 545,151 | <p>For <span class="math-container">$x \to \infty, f(x) \sim e^x$</span> so <span class="math-container">$f^{-1}(x) \sim \ln(x)$</span>. So the answer is <span class="math-container">$1$</span>.</p>
|
4,112,581 | <p>I want to prove this statment:</p>
<p>Let <span class="math-container">$f:[a,b] \to \mathbb{R}$</span> be a bounded function.</p>
<p>Prove that if <span class="math-container">$f$</span> is integrable in <span class="math-container">$[a,b]$</span> then <span class="math-container">$|f|$</span> is also integrable in ... | David K | 139,123 | <p>The "infinite sum" is really a limit of a finite sum of disks that each have some thickness.
You're treating these disks as if the thickness of each disk corresponded to
<span class="math-container">$\mathrm d\theta,$</span> but a correct way to account for the thickness
of the disks would be <span class="... |
4,112,581 | <p>I want to prove this statment:</p>
<p>Let <span class="math-container">$f:[a,b] \to \mathbb{R}$</span> be a bounded function.</p>
<p>Prove that if <span class="math-container">$f$</span> is integrable in <span class="math-container">$[a,b]$</span> then <span class="math-container">$|f|$</span> is also integrable in ... | Ripi2 | 688,039 | <p>Let's take a horizontal disk. Its radius is <span class="math-container">$r$</span>, and its thickness is <span class="math-container">$dt$</span>.
<br>Thus, its volume is <span class="math-container">$dV=\pi r^2 dt$</span>
<br> The total volume of the sphere is <span class="math-container">$V= \int{\pi r^2 dt}$</sp... |
526,107 | <p>I have to give a presentation on vector analysis. One of many important things I want to emphasize is that a division by a vector does not make sense.</p>
<p>How do you explain to your students, for example, that division by a vector does not make sense?</p>
<p>Bonus question: Also how do you explain that integrat... | Dubious | 32,119 | <p>It depends in which way you want to define the product of two vectors. Clearly if you think a vector in $\mathbb R^2$ as a complex number, then $\mathbb R^2$ is a field. </p>
|
3,491,595 | <blockquote>
<p>Evaluate the sum <span class="math-container">$$\frac{1}{3} + \frac{1}{3^{1+\frac{1}{2}}}+\frac{1}{3^{1+\frac{1}{2}+\frac{1}{3}}}+\cdots$$</span></p>
</blockquote>
<p>It seems that <span class="math-container">$1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}$</span> approaches <span class="ma... | Descartes Before the Horse | 592,365 | <p>If this was something you just came up with, it is highly unlikely there is any obtainable closed form expression. Checking the number Wolfram|Alpha generates from <code>sum (1/(3^(sum (1/k) from k=1 to n))) from n=1 to infinity</code> in an inverse symbolic calculator, I did not find anything.</p>
|
6,712 | <p>I had received the "warmth" of an angry user which decided to downvote no less than eight of my questions within the span of a minute.</p>
<p>I know who the user is and I can prove their identity beyond reasonable doubt.</p>
<p>Surely the software will catch the serial voting by tomorrow and reverse it, but I was ... | Potato | 18,240 | <p>Nothing?</p>
<p>I don't see the point in getting worked up over some meaningless number on a math website. In any case, you have tens of thousands of reputation points. A few votes either way isn't going to make a noticeable difference. </p>
<p>Reputation is useful insofar as it allows access to the site's feature... |
715,825 | <p>How would I "solve by addition"? I'm not sure how to solve this.</p>
<p>$3x + 2y = 11$ and under that
$3x – 2y = 13$ </p>
<p>My notes that go along with it are:</p>
<p>In the addition method, you want to add the equations in such way so that one of the variables (letters) drops out. $x$ and $y$ are on the same s... | Alessandro Codenotti | 136,041 | <p>Add the 2 equation to obtain $3x+3x+2y-2y=11+13$, the $y$ cancel out and you are left with $6x=24$ from which $x=4$ follows, to find y substitute 4 for $x$ into one of the original equations</p>
|
1,180,743 | <p>I need help finding out the basis in the following question :</p>
<blockquote>
<p>Let $~~W=\big<[1~~2~~1~~0~~1]^t~,[1~~0~~1~~1~~1]^t~,[1~~2~~1~~3~~1]^t\big >~$ be a subspace of $\mathbb R^5$ . Find a basis of $\mathbb R^5/W.$ </p>
</blockquote>
<p>I can't figure out the basis , kindly help with some hint... | abel | 9,252 | <p>here is what i get when i row reduce the basis vector of $W$ together with the standard basis $e_1, e_2, \cdots, e_5$ of $\mathbb R^5.$</p>
<p>$$\pmatrix{1&1&1&1&0&0&0&0\\2&0&2&0&1&0&0&0\\1&1&1&0&0&1&0&0&\\0&1&3&0&am... |
1,690,210 | <p>What is $$\int \frac{4t}{1-t^4}dt$$ is there some kind of substitution which might help .Note that here $t=\tan(\theta)$</p>
| lab bhattacharjee | 33,337 | <p>Hint:
$t^2=u$ or $t^2=\sin v$</p>
|
2,403,986 | <p>I wonder what $\mu$ - synthesis analysis is? I have heard that is an uncertainty modelling.</p>
<p>I think it's an extra help for the $H_{\infty}$ controller because the $\mu$ - synthesis analysis make sure that the $H_{\infty}$ controller can stand against nonlinearities.</p>
<p>So $\mu$ + $H_{\infty}$ = Robust n... | Johan Löfberg | 37,404 | <p>$H_{\infty}$ deals with the problem of finding a controller $F(s)$ for a known system $G(s)$ such that the gain (in $H_{\infty}$ sense) from an external signal to an output is minimized.</p>
<p>$\mu$-synthesis extends this to the case when $G(s)$ is uncertain, and tries to minimize the worst-case gain given the unc... |
2,403,986 | <p>I wonder what $\mu$ - synthesis analysis is? I have heard that is an uncertainty modelling.</p>
<p>I think it's an extra help for the $H_{\infty}$ controller because the $\mu$ - synthesis analysis make sure that the $H_{\infty}$ controller can stand against nonlinearities.</p>
<p>So $\mu$ + $H_{\infty}$ = Robust n... | Jeremy | 653,843 | <p><span class="math-container">$H_\infty$</span> deals with minimizing the influence of uncertainty in your plant, but the uncertainty is unstructured, i.e. each uncertainty in your plant couples with every other. Generally this is not the case in most problems. For a harmonic oscillator, the uncertainty in the mass s... |
1,372,558 | <p>$y=\sqrt{x^x}$</p>
<p>How do I convert this into a form that is workable and what indicates that I should do so? </p>
<p>Anyway, I tried this method of logging both sides of the equation but I don't know if I am right.</p>
<p>$\ln\ y=\sqrt{x} \ln\ x$</p>
<p>$\frac{dy}{dx}\cdot \frac{1}{y}=\sqrt{x}\ \frac{1}{x} +... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>All functions of type $u^v$ are defined with:
$$u^v=\mathrm e^{v\ln u}.\enspace\text{Here:}\quad \sqrt{x^x}=\mathrm e^{\frac12 x\ln x}.$$</p>
|
1,372,558 | <p>$y=\sqrt{x^x}$</p>
<p>How do I convert this into a form that is workable and what indicates that I should do so? </p>
<p>Anyway, I tried this method of logging both sides of the equation but I don't know if I am right.</p>
<p>$\ln\ y=\sqrt{x} \ln\ x$</p>
<p>$\frac{dy}{dx}\cdot \frac{1}{y}=\sqrt{x}\ \frac{1}{x} +... | Autolatry | 25,097 | <p>Square both sides:</p>
<p>$$y^{2}=x^{x}$$
Then upon differentiating
$$2y y' = (1+\ln x)x^{x}$$
From which
$$y'=\frac{1}{2y}(1+\ln x)x^{x} \qquad (x \neq 0)$$
Giving
$$y' = \frac{1}{2\sqrt{x^{x}}}(1+\ln x)x^{x} $$
Thus
$$y'=\frac{1}{2}(1+\ln x)\sqrt{x^{x}}$$</p>
|
3,295,662 | <p>If it were the three women that had to sit together i could solve this as by putting them in a single group among the men. If i multiply the permutation of these groups by the permutation of women inside this group i would get a total of 5! * 3! = 720.</p>
<p>I thought of using this strategy for the problem of arr... | JMoravitz | 179,297 | <p>I assume that you have four men, three women, each person is distinct, and we are wanting to count the number of ways that they can line up such that there are at <em>least</em> three men in a row (i.e. at least one man who has another man to both his left and his right).</p>
<p>I'm sure there are other ways to see... |
1,266,507 | <p>Let $K_{a,b}$ be the complete bipartite graph. Show that
$K_{a,b}$ is a tree if and only if $a = 1$ or $b = 1$.</p>
<p>The way my professor showed us for a complete graph is as below. I just don't know how to start for a complete bipartite graph. </p>
<blockquote>
<p>$K_a$ is a tree if and only if $a=2$ or $a=1$... | Salomo | 226,957 | <p>If $a>1$ and $b>1$, then let $A=\{a_1,\dots,a_s\}$ and $B=\{b_1,\dots,b_t\}$ be the two parts of vertices with $s>1$ and $t>1$. We can then find a circuit in the graph, say $a_1b_1a_2b_2a_1$.</p>
|
2,167,265 | <p>Zeno, a follower of Parmenides, reasoned that any unit of space or time is infinitely divisible or not. If they be infinitely divisible, then how does an infinite plurality of parts combine into a finite whole? And if these units are not infinitely divisible, then calculus wouldn't work because $n$ couldn't tend t... | Aurel | 417,206 | <p>In this specific case, there is a nice argument using geometric series. However, as you will see this answer may be quite unsatisfying. </p>
<p>A geometric series is an <strong>infinite series</strong> in the form</p>
<p>$$
f(r) = \sum_{n=1}^{\infty} r^n
$$
It is easy to prove that geometric series converge for
$... |
76,299 | <p>This is a bit embarrassing, but I can't seem to solve for $x$ in
$$2x=\frac{x}{y}-\frac{1}{1-y}.$$
Could someone please give me a hand!</p>
| Blue | 409 | <p>How might one <em>discover</em> the Associate Circle? It's the <a href="http://en.wikipedia.org/wiki/Envelope_%28mathematics%29" rel="nofollow noreferrer"><em>envelope</em></a> of the family of Focal Chord Circles.</p>
<p>Finding an envelope of a family of curves --say, parameterized by $\theta$-- is conceptually s... |
3,493,203 | <p>I would like to understand why if <span class="math-container">$P$</span> is positive semidefinite, then <span class="math-container">$x^{\text{T}}Px=0$</span> if and only if <span class="math-container">$Px=0$</span>. How can I prove this?
I can say that <span class="math-container">$x'Px=x'M'Mx$</span>, where <sp... | GreginGre | 447,764 | <p>No, it's not possible. For any commutative ring with <span class="math-container">$1$</span>, any maximal ideal is prime.</p>
<p><strong>Hint.</strong> use characterization of prime/maximal ideals in terms of quotient rings</p>
|
3,578,014 | <p>The optimization problem looks like this now:</p>
<p><span class="math-container">$minimize\;\frac{1}{N}\sum_{s=1}^N max\{L-Br_s^Tx,0\}$</span></p>
<p><span class="math-container">$s.t.\;\sum_i x_i=1$</span></p>
<p><span class="math-container">$x\ge0$</span></p>
<p>Is it ok to put the max part inside the objecti... | Charlie Vanaret | 741,916 | <p>To make your problem differentiable, you can replace this kind of problem<br>
<span class="math-container">$\min \max(a, b)$</span><br>
with the smooth reformulation:<br>
<span class="math-container">$\min t$</span><br>
s.t.<br>
<span class="math-container">$a \leq t$</span><br>
<span class="math-container">$b \leq ... |
1,329,112 | <p>Given a measurable $E\subset \Bbb R^d $ and a measurable function $f:E\rightarrow \Bbb R^d $, prove that :</p>
<p>$$
\int (\left\lvert f \right\rvert)^r d\mu = r\int_{0}^\infty t^{r-1} \mu(\{x \in E \mid \left\lvert f(x) \right\rvert>t\})\,dt
$$<br>
where $r \ge 1$</p>
<p>The $r=1$ case is simple; howe... | Tryss | 216,059 | <p>If you know how to prove the case $r=1$, it's easy :</p>
<p>let $g(t)= |f(t)|^r$. You have (assuming you can prove the result for $r=1$)
$$\int_E |f(t)|^r dt = \int_E |g(t)| dt = \int_0^{\infty} \mu(\{ x\in E \ :\ |g(x)| > t\}) dt$$</p>
<p>it follow</p>
<p>$$= \int_0^{\infty} \mu(\{ x\in E \ :\ |f(x)|^r > t... |
4,371 | <p>I would like to learn Graph Theory from the beginning. It seems to me that one does not need to be familiar with many abstract type subjects to be able to understand the more basic concepts of graphs.</p>
<ol>
<li><p>Which subjects should one know prior to learn Graph Theory at the introductory level?</p></li>
<li>... | Niel de Beaudrap | 439 | <ol>
<li><p><strong>(a)</strong> Basic logic + set operations almost goes without saying (<em>e.g.</em> logical conjunction / set intersections; also equivalence classes, sets and relations obtained by modding out subsets, <em>etc</em>).<br><br><strong>(b)</strong> Depending on how 'basic' you mean, you may o... |
4,371 | <p>I would like to learn Graph Theory from the beginning. It seems to me that one does not need to be familiar with many abstract type subjects to be able to understand the more basic concepts of graphs.</p>
<ol>
<li><p>Which subjects should one know prior to learn Graph Theory at the introductory level?</p></li>
<li>... | Daniel Moskovich | 7,803 | <p>I thought about this question for a graph theory course I'm teaching. Prerequisites would be mathematical proof technique (induction, proof by contradiction), and linear algebra (determinants, eigenvalues). </p>
<p>The book I eventually chose was Bondy and Murty's <a href="http://blogs.springer.com/bondyandmurty/?p... |
52,249 | <p>Is it true that all zeros of the Riemann Zeta Function are of order 1?</p>
<p>Let <span class="math-container">$h(z) = \frac{\zeta'(z)}{\zeta(z)}\frac{x^z}{z}$</span>, where <span class="math-container">$x$</span> is a positive real number (<span class="math-container">$x > 1$</span>, probably?) , and <span class... | joriki | 6,622 | <p>Your Google Books link doesn't work for me; <a href="http://books.google.com/books?id=7-sDtIy8MNIC&lpg=PA202&dq=riemann%20von%20mangoldt%20residues&pg=PA202#v=onepage&q=riemann%20von%20mangoldt%20residues&f=false">here</a>'s one that does.</p>
<p>I think there's nothing mysterious going on here,... |
1,541,859 | <p>Trying to find the sum of the following infinite series:</p>
<p>$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$</p>
<p>Any ideas on how to find this sum?</p>
| Lucian | 93,448 | <p><strong>Hint:</strong> $\text{arctanh }x=\displaystyle\sum_{n=1}^\infty\frac{x^{2n-1}}{2n-1}$</p>
|
3,553,697 | <p>We all have seen and know of Euclid’s proof by contradiction, but I’m wondering if there is such thing as a direct proof? Also, the theorem isn’t posed as an “if-then” statement, so I can not even imagine how an alternative proof would be structured. </p>
| Peter | 82,961 | <p>My suggestion to avoid to have to assume that infinite many primes exist is the following alternative :</p>
<p>Begin with a positive integer <span class="math-container">$n>1$</span>. To show that there must be a prime number greater than <span class="math-container">$n$</span> just consider the smallest prime f... |
4,264,569 | <p>I am interested in coming up with a function describing the dose-dependent action of a drug over time. In this specific case, the image attached below shows the glucose infusion rate (GIR) of the insulin "Lyumjev", where the GIR represents the amount of glucose that needs to be infused into a patient to ke... | Community | -1 | <p>What you are asked to prove is that the set equation <span class="math-container">$F(d)=F(d^{-1})$</span> does hold <em>for all</em> <span class="math-container">$d\in H$</span>. So, if you can prove the equality of the two sets without any specific assumption on <span class="math-container">$d$</span> (<em>e.g.</em... |
1,726,763 | <p>How exactly would I find these? I know you have to plug it into the nth root of a complex number formula, but when I try to find the argument for the trig form, i just get undefined. </p>
| Ian Miller | 278,461 | <p>There are two main ways to calculate a square root:</p>
<p><strong>Option 1</strong>
$$x+iy=\sqrt{-25i}$$
$$(x+iy)^2=-25i$$
$$x^2+2xy-y^2=-25i$$
$$x^2-y^2=0\text{ and }2xy=-25$$
$$x=\pm y$$
We select the negative root as the positive root will lead to no solution ($y^2=-25$)
$$-2y^2-25$$
$$y=\sqrt{\frac{25}{2}}$$
$... |
4,533,858 | <h2>Problem :</h2>
<p>Show that :</p>
<p><span class="math-container">$$\frac{1}{2\ln2}\left(1-\sqrt{\frac{1-\ln2}{1+\ln2}}\right)>\sqrt{2}-1$$</span></p>
<hr />
<hr />
<p>Using some approximation using itself algoritm found here (<a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots" rel="nofollo... | trancelocation | 467,003 | <p>You may show the inequality using elementary equivalent rearrangements and <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow noreferrer">Cauchy-Schwarz inequality</a> as shown below.</p>
<p>This is surely not the nicest way but it works:</p>
<p><span class="math-container">\beg... |
10,730 | <p>While the question is stated with reference to the iPhone, my actual question is about phones in general. Just as there was much talk about the use of Computers in the classroom over the past fifty years (or so), how much talk is there about the use of cellphones in the classroom?</p>
<p>What benefits are there to ... | Gerald Edgar | 127 | <p>Some anecdotes. I have volunteered with 5th grade math students for a few years. The general rule is, they do not use electronics unless instructed to do so.</p>
<p>We came to a problem where it says to use calculators. But they had not brought their calculators. So one guy pulls out his iPhone, with a calculat... |
1,026,975 | <p>Given a topology $(X,T)$, $A\subset X$, $x \in X$ is a limit point of A if $\forall$ open $U$ that contains $x$, $(U\cap A)$\ {$x$} $\neq \emptyset$. $x \in X$ is in $cl(A)$ if $\forall$ open $U$ that contains $x$, $U\cap A$ $\neq \emptyset$. Is there any example that a point in the the closure of $A$ is not a limit... | Suzu Hirose | 190,784 | <p>A point in a closed set is either a limit point or an isolated point (a point which has a neighbourhood which contains no other points of the set). For a reference, this is Theorem 17-E (page 97) of <a href="http://books.google.co.jp/books?id=0e5Xq3fGnAgC" rel="nofollow"><em>Introduction to Topology and Modern Analy... |
2,276,404 | <p>I stumbled upon the following inequality in a scientific paper which estimates a lower bound for $\frac{k!}{k^k}$ for $k \in \mathbb{N}$:
$$\frac{k!}{k^k} > e^{-k}$$
They did not explain why this holds true, and I could not find any answer by myself yet.</p>
| Thomas Andrews | 7,933 | <p>Rewrite it as $$e^{k}>\frac{k^k}{k!}$$</p>
<p>And you see that it follows because $$e^{x}=\sum_{n=0}^{\infty} \frac{x^n}{n!}$$, and thus, for any $x\geq 0$, $e^{x}>\frac{x^k}{k!}$. </p>
|
1,242,541 | <p>I am trying to prove the following, and here is what I have done:
Can somebody help to complete this?</p>
<p>$2^n \ge n^2$ for $n\ge 4$</p>
<p>$n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE</p>
<p>Assume True for $n=k$, for $k\ge 4$</p>
<p>$2^k \ge k^2$</p>
<p>Should be true for $n=k+1$ for $k\ge... | Keba | 119,385 | <p>Just compare $k^2 + k^2$ with the term you want, namely $(k+1)^2 = k^2 + 2k + 1$. Does $2k+1 \lt k^2$ always hold? For $k$ large enough at least? If yes, you only need to solve that new problem.</p>
<p>And how do you solve that problem? By induction! :) (“induction proof within an induction proof” may sound complic... |
1,079,384 | <p>I have to find volume using triple integration of region bounded by $z=4 - \sqrt{x^2 +y^2}$ and $z=\sqrt{ x^2 +y^2}$. I have seen that they are cones intersecting at $z=2$, but my problem is that in $x$-$y$ plane shadow seems to be split up, which I cannot do. Please belp me with this. Thanks</p>
<p><img src="https... | Mark Fantini | 88,052 | <p>The region does not split up. The shadow on the $xy$ plane is the circle where $z=4 - \sqrt{x^2+y^2}$ and $z=\sqrt{x^2+y^2}$ meet. This is when the radius is equal to $2$. Your inferior limit for $z$ is the lower cone and superior limit is the upper cone.</p>
<p><strong>EDIT:</strong> You seem to be getting the bou... |
1,079,384 | <p>I have to find volume using triple integration of region bounded by $z=4 - \sqrt{x^2 +y^2}$ and $z=\sqrt{ x^2 +y^2}$. I have seen that they are cones intersecting at $z=2$, but my problem is that in $x$-$y$ plane shadow seems to be split up, which I cannot do. Please belp me with this. Thanks</p>
<p><img src="https... | David P | 49,975 | <p>The shaddow in the $xy$ plane is indeed a circle, of radius $2$, so in cartesian:</p>
<p>$$V = \int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{4-\sqrt{x^2+y^2}} 1 \ dz\, dy\, dx$$</p>
<p>or in polar (easier):</p>
<p>$$V = \int_{0}^{2\pi}\int_{0}^2\int_{r}^{4-r} 1 \ r\, dz\, dr\, d\theta$$<... |
2,499,778 | <p>Let $ABC$ a triangle with $AB <AC$. The angle bisector of $\angle BAC$ intersects $(BC)$ in $D$. The perpendicular from $B$ on $AD$ intersects the circumscribed circle of the triangle $ABD$ in $E$ for the second time. </p>
<p>Show that the center of the circumscribed circle $S$ of the triangle $ABC$ is on the l... | Sul | 464,889 | <p>First of all differentiate the function $f(x) = 3x^5 -10x^3 -120x + 30$:
$$f^{'}(x) = 15x^4 -30x^2 -120$$
Consider the equation $f^{'}(x) = 0$:
$$15x^4 -30x^2 -120 = 0$$
$$x^4 - 2x^2 -8 = 0$$
$$(x-2)(x+2)(x^2 + 2) = 0$$
Hence, there are two turning points at $x = 2$ and $x = -2$. </p>
<p>Since $f^{'}(x) = 15(x+2)... |
597,680 | <p>Calculate $|A|$ with $A=\{ F \subset \mathcal P(\mathbb N): F \text{ is a partition of } \mathbb N\}$. A partition of $\mathbb N$ is a family of subsets $F=\{P_i\}_{i \in I} \subset \mathcal P(\mathbb N)$, non-empty such that $\bigcup_{i \in I} P_i= \mathbb N$ and $P_i \cap P_j= \emptyset$</p>
<p>My attempt at a so... | universalset | 110,684 | <p>Your proof is correct.</p>
<p>For the second part: one way to do it is the following. First show that for such a partition $F = \{P_i\}_{i\in I}$, $I$ is at most countable. Then, identify each such family $F = \{P_i\}_{i\in \mathbb{N}}$ (or $F = \{P_i\}_{0\leq i \leq m}$) with the function $f:\mathbb{N}\rightarr... |
597,680 | <p>Calculate $|A|$ with $A=\{ F \subset \mathcal P(\mathbb N): F \text{ is a partition of } \mathbb N\}$. A partition of $\mathbb N$ is a family of subsets $F=\{P_i\}_{i \in I} \subset \mathcal P(\mathbb N)$, non-empty such that $\bigcup_{i \in I} P_i= \mathbb N$ and $P_i \cap P_j= \emptyset$</p>
<p>My attempt at a so... | Brian M. Scott | 12,042 | <p>Yes, what you’ve done works fine. Another approach that is perhaps just a little simpler is to let $E=\{2n:n\in\Bbb N\}$, and for each $A\subseteq E$ let $\mathscr{P}(A)=\{A\cup\{1\},\Bbb N\setminus(A\cup\{1\})\}$; then $\mathscr{P}$ is an injection from $\wp(E)$ to the family of partitions of $\Bbb N$.</p>
<p>To g... |
4,206,112 | <p>I have the following quadratic equation:</p>
<p><span class="math-container">$$x = \frac{-1 + \sqrt{1+ (4y/50)} }{2}$$</span></p>
<p>in this case <span class="math-container">$y$</span> is a known variable so I can solve the equation like this for <span class="math-container">$y = 600$</span></p>
<p><span class="ma... | Sathvik | 516,604 | <p>It's simple algebra.
<span class="math-container">$$y=\frac{50(2x+1)^2-50}{4}=50x(x+1)$$</span></p>
|
30,443 | <p>My aim is to plot a vector field of the following system with a few trajectories:</p>
<p>$$r'(t)=i-l.r(t)-\text{ux}. r(t). x(t)-\text{uy}. r(t). y(t) \\ x'(t)=\text{ex}. \text{ux}. r(t). x(t)-\text{mx}. x(t)\\y'(t)=\text{ey}.\text{uy}. r(t). y(t)-\text{my}. y(t)$$</p>
<p>$i,l,ux,uy,mx,my,ex,ey$ are parameters.</p... | Jens | 245 | <p>When one talks about vector fields, it usually helps to define what the field is. In your case of a first-order system of equations, I interpret "vector field" simply as the right-hand side of the equations. Therefore, <strong>there is no need to use <code>NDSolve</code></strong>, and the parameter <code>t</code> is... |
58,901 | <p>I had previously asked:
<a href="https://mathoverflow.net/questions/47943/narratives-in-modular-curves">Narratives in Modular Curves</a></p>
<p>Since then, I've read quite a bit more (but not nearly enough) and I have a few follow up questions about the big picture. As you will soon see, I'm confused about how to t... | Kimball | 6,518 | <p>Let me give at least a partial answer.</p>
<p>If you want to view the 1 and 2 dimensional theories uniformly, you should look at everything adelically. In dimension 1, Dirichlet characters can be viewed as idele class (or Hecke) characters, which is to say irreducible representations of $\mathbb Q^\times \backslas... |
1,635,188 | <p>I am trying to solve $$\lim_{x\to 0} \frac{\int_{0}^{x^2} x^2 e^{-t^2} dt}{-1+e^{-x^4}} $$ using The Fundamental Theorem of Calculus (FTC).
I already know that the answer is -1, </p>
<p>Using FTC (correct me if I am wrong) we get:
$$
\lim_{x\to 0} \frac{x^2 e^{-x^4}}{-1+e^{-x^4}}
$$</p>
<p>Which has the result of ... | Rogelio Molina | 87,320 | <p>Using the fundamental theorem of calculus you should replace the integral</p>
<p>$$
\int_0^{x^2} e^{-t^2} dx \simeq1 \cdot x^2 = x^2
$$
becaus the mean value of $e^{-t^2}$ near $t=0$ is just 1, and the length of your interval is $x^2$, hence the limit you want is</p>
<p>$$
\lim_{x \to 0 } \frac{x^4}{e^{-x^4} -1} ... |
2,636,131 | <blockquote>
<p>Suppose that each square of a $4 \times 7$ chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?</p>
</blockquote>
... | bof | 111,012 | <p>Let's say the chessboard has $4$ columns and $7$ rows. Each row must land in (at least) one of the following $6$ bags:</p>
<ol>
<li>First and second squares are black. </li>
<li>First and third squares are black. </li>
<li>Second and third squares are black. </li>
<li>First and second squares are white. </li>
<... |
3,860,982 | <p>How do I prove that if:
<span class="math-container">$$\cos^3(x) + \sin^3(x) = 1$$</span>
then:
<span class="math-container">$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$</span></p>
<p>Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by <span class=... | Simon Terrington | 302,396 | <p>Aha OK you were pretty close. As you say <span class="math-container">$cos^{2}(x)+sin^{2}(x)=1$</span>. Now <span class="math-container">$cos^{3}(x)+sin^{3}(x)$</span> is less than or equal to <span class="math-container">$cos^{2}(x)+sin^{2}(x)=1$</span> as the absolute value of <span class="math-container">$sin(x)$... |
3,860,982 | <p>How do I prove that if:
<span class="math-container">$$\cos^3(x) + \sin^3(x) = 1$$</span>
then:
<span class="math-container">$$\cos(x) = 0 ; \sin(x)=1 \text{ or } \cos(x)=1 ; \sin(x)=0?$$</span></p>
<p>Starting from the first expression, I couldn't figure out how to reach the conclusion. I replaced 1 by <span class=... | Stacey Carlin | 730,530 | <p>This is what I tried after reading your answers, please correct me if there are mistakes!<span class="math-container">$$cos^3+sin^3=1$$</span> <span class="math-container">$$cos^3+sin^3=cos² +sin²$$</span> <span class="math-container">$$cos^3-cos²+sin^3-sin²=0$$</span> <span class="math-container">$$cos²(cos-1) + si... |
2,737,823 | <p><strong>Background</strong></p>
<p>Hey everyone. I'm absolutely stumped on an exercise I am working on out of Axler's <em>Linear Algebra Done Right</em>, 3rd edition. Funnily enough, <a href="http://linear.axler.net/InnerProduct.pdf" rel="nofollow noreferrer">the sample chapter</a> available on his website is the... | Mathematician 42 | 155,917 | <p>Suppose that $g\in U^{\perp}$. Let $f$ be a function such that $f$ is almost equal to $g$ and such that $f\in U$. (You can achieve this by modifying $g$ in a neighbourhood of $0$). Then $$0=\int_{-1}^1f(x)g(x)\mathrm{d}x\sim \int_{-1}^1g(x)^2\mathrm{d}x.$$
Since we can approach $g$ arbitrarily close by $f$, we get t... |
181,532 | <blockquote>
<p>Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field.</p>
</blockquote>
<p>How can one justify the answer in the shortest number of lines?</p>
| Community | -1 | <p>Write the Gaussian integers as $\Bbb{Z}[x]/(x^2 +1)$. Then notice that for a prime $p$ such that $x^2 +1$ is irreducible mod $p$, you have that $(x^2 + 1,p)$ is a maximal ideal in $\Bbb{Z}[x]$. You will then have </p>
<p>$$\Bbb{Z}[i]/(p) \cong \Bbb{Z}[x]/(x^2 + 1,p) \cong \Bbb{Z}/p\Bbb{Z}[x]/(\overline{x}^2 +1).$$<... |
2,565,880 | <p>Number of ways in which 5 boys and 4 girls can be seated around a circular table such that no two girls sit together and two particular boys are always together ?</p>
<p>The answer to this question is $3!2!4!$ . It is done by considering $2$ boys as one unit and the the number of units (of boys) is $4$ so they can ... | Vasili | 469,083 | <p>$f(x,y)=3x^2+3y^2-12y-6x+27=3(x-1)^2+3(y-2)^2+12$. When this expression is minimal?</p>
|
2,565,880 | <p>Number of ways in which 5 boys and 4 girls can be seated around a circular table such that no two girls sit together and two particular boys are always together ?</p>
<p>The answer to this question is $3!2!4!$ . It is done by considering $2$ boys as one unit and the the number of units (of boys) is $4$ so they can ... | zwim | 399,263 | <p>$f(x,y)=3x^2+3y^2−12y−6x+27$</p>
<p>The critical point cannot be a maximum, because it is clear that the maximum distance if it would exists would be realized for points being at infinity as far as possible of the $3$ given points (i.e outside of any ball of arbitrary radius $n\gg 1$).</p>
<p>Now $f(1,2)=12$ is a ... |
2,565,880 | <p>Number of ways in which 5 boys and 4 girls can be seated around a circular table such that no two girls sit together and two particular boys are always together ?</p>
<p>The answer to this question is $3!2!4!$ . It is done by considering $2$ boys as one unit and the the number of units (of boys) is $4$ so they can ... | amd | 265,466 | <p>In fact, you can use a second-derivative test to classify the critical point. You just have to use the right one. For a multivariable function, you must examine the <em>Hessian matrix</em>, which is the matrix of all of the second-order partial derivatives. For this function, this is $$H = \begin{bmatrix}f_{xx}&... |
17,335 | <p>Starting with a representation <span class="math-container">$\rho:G \to \mathrm{GL}(V)$</span>. Then we can build the tensor product of <span class="math-container">$V$</span> with itself by defining <span class="math-container">$g(v_1 \otimes v_2) = g(v_1) \otimes g(v_2)$</span>. Then by saying <span class="math-... | darij grinberg | 2,530 | <p>Not an answer, rather an attempt to hijack the question...</p>
<p>Some time ago I have also been wondering how to wedge two vector spaces and came up with the following construction:</p>
<p>Let $f:U\to V$ and $g:U\to W$ be two vector space morphisms. We define the vector space $V\wedge_U W$ (of course, this depend... |
17,335 | <p>Starting with a representation <span class="math-container">$\rho:G \to \mathrm{GL}(V)$</span>. Then we can build the tensor product of <span class="math-container">$V$</span> with itself by defining <span class="math-container">$g(v_1 \otimes v_2) = g(v_1) \otimes g(v_2)$</span>. Then by saying <span class="math-... | Vít Tuček | 6,818 | <p>This is a construction in some sense dual to the proposal of Darij Grinberg. Lets consider only $G=GL(n,\mathbb{C})$ for simplicity and take $V$ and $W$ to be subrepresentations of $\bigotimes^k \mathbb{C}^n$. (Which we can do without loss of generality for any two finite-dimensional representations.)</p>
<p>Then ... |
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