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 |
|---|---|---|---|---|
238,686 | <p>I want to solve this equation for <span class="math-container">$z$</span>:</p>
<p><span class="math-container">$-\frac{2\pi^2}{\beta^2}z^2+\frac{2i\pi^2u}{\beta^2}z^3+z^4=0$</span></p>
<p><span class="math-container">$\beta$</span> is a positive real constant, <span class="math-container">$u$</span> is a real variab... | Carl Woll | 45,431 | <p>You could try using <a href="http://reference.wolfram.com/language/ref/AsymptoticSolve" rel="noreferrer"><code>AsymptoticSolve</code></a> instead:</p>
<pre><code>AsymptoticSolve[
I η - (2 Pi^2/β^2) z^2 + (2 I Pi^2 u/β^2) z^3 + z^4 == 0,
z,
η->0
] //Normal //TeXForm
</code></pre>
<blockquote>
<p><span ... |
974,560 | <p>How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$?</p>
<p>I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, and my textbook doesn't offer a derivation.</p>
| Travis Willse | 155,629 | <p>It's typically very easy to verify whether a function $F(x)$ is an antiderivative of a function $f(x)$; by definition, you need only check that
$$F'(x) = f(x).$$</p>
<p>In this case, on, say, $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,
$$\frac{d}{dx} \log(\sec x + \tan x) = \frac{\frac{d}{dx}(\sec x + \tan x)}{\s... |
974,560 | <p>How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$?</p>
<p>I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, and my textbook doesn't offer a derivation.</p>
| bof | 111,012 | <p>$$\sec x=\frac1{\cos x}=\frac{\cos x}{\cos^2x}=\frac{\cos x}{1-\sin^2x}$$
Substitute $u=\sin x$ and use the method of partial fractions.</p>
<p>Since
$$\frac1{1-u^2}=\frac12\left(\frac1{1+u}+\frac1{1-u}\right)$$
we get
$$\int\sec xdx=\int\frac{du}{1-u^2}=\frac12(\ln(1+u)-\ln(1-u))=\ln\sqrt{\frac{1+u}{1-u}}.$$
Now
$... |
844,403 | <p><img src="https://i.stack.imgur.com/iYn8g.png" alt=" Circumcircles, Incircles, Medians, and Altitudes" /></p>
<p>I tried to use the angle property by which AD=4 and DB=5,but since F is not given as mid point I don't know how to proceed to find length of DG.I think AED as 90 degree is important but I am unable to fig... | Mick | 42,351 | <p>Referring to the figure (with added constructions) below.</p>
<p><img src="https://i.stack.imgur.com/Q91FV.png" alt="enter image description here"></p>
|
1,302,383 | <p>Can anyone help me find the solution to this integral:</p>
<p>$$\int\limits{(t-4)(t-2)^{4/5}}dt?$$</p>
<p>I think I need to expand the integrand but I do not know how. Thanks a lot!</p>
| GeorgSaliba | 142,772 | <p>If you're still interested, you could use the following to solve integrals having the general form:
$$\int x^m (a+bx^n)^{r/s} dx$$
where $m,n,r$ and $s$ are integers and $n$ is positive.
To get to this form from your integral, we need to set $u=t-4$ which yields:
$$\int u(2+u)^{4/5} du$$
where $m=1, a=2,b=1,n=1,r=4$... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| lalmei | 9,831 | <p>Since nobody has used this function yet, I will place it here. Your data seems to be organised almost perfectly for ArrayPlot.
First I removed the first column from the rest of the values and added to the axes ticks.
The rest is just displayed via ArrayPlot, with a particular color scale. </p>
<pre><code>{xs, va... |
3,786,472 | <p><strong>An elevator ascends from rest with an acceleration of <code>0.6 m/s^2</code>, before slowing down with a deceleration of <code>0.8 m/s^2</code> for the next stop. The total time taken is 10 seconds. Find the distance between the stops.</strong></p>
<p>I have tried this problem over multiple spells over and o... | Beetel | 793,137 | <p>The elevator starts from rest.</p>
<p>Suppose the time spent increasing the speed is <span class="math-container">$t_{1}$</span> and the time spent in decreasing the speed and coming to rest is <span class="math-container">$t_{2}$</span>.</p>
<p>It is given <span class="math-container">$t_{1} + t_{2}=10$</span>.</p>... |
196,460 | <p>Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ positive semi-definite $\forall \ t \in [0,1]$?)</p>
| Denis Serre | 8,799 | <p>Well, this is not an answer. But I cannot resist to mention the following equivalent property. Let <span class="math-container">$A\mapsto \hat A$</span> denote the cofactor map, and <span class="math-container">$B\mapsto \check B$</span> its inverse. For positive definite symmetric matrices, <span class="math-contai... |
818,512 | <p>I have tried to find posts that are related to the question but they end up with the terms like ‘find a distance’. What I want is not to find the distance: I already have the distance, I want something else.</p>
<p>Assume <span class="math-container">$(x_1,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</s... | Fermat | 83,272 | <p>The point(s) that you are looking for are the points at which the circle
$$(x-x_1)^2+(y-y_1)^2=d^2$$
and the line
$$y=ax+b$$ intersect. Therefore
$$(x-x_1)^2+(ax+b-y_1)^2=d^2$$
which is a quadratic equation in $x$.</p>
|
2,589,965 | <p>I'm working through multivariable functions and derivatives of multivariable functions. Since I am not very familiar yet with multivariable functions I wondered about the following: </p>
<p>In a function like $f(x,y)=x^2+y$, are x and y independent of each other and are we allowed to pick values for each deliberate... | Vassilis Markos | 460,287 | <p>This question is no "silly" at all! I would consider it rather "to-the-point" of multivariable calculus. </p>
<p>In general, as you mentioned, the set $$G(f)=\{(x,y,z)\in\mathbb{R}^3|z=f(x,y)\}$$
is indeed a subset of $\mathbb{R}^3$ and, in general, a surface. So, you can pick arbitrarily coordintates $x,y$ in $f$'... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Carl-Fredrik Nyberg Brodda | 120,914 | <p>In combinatorial group theory, loosely speaking almost any problem one can imagine, in full generality, turns out to be undecidable. This includes the word problem, the isomorphism problem, the triviality problem, etc. Here's an example of a very general problem which nevertheless is decidable.</p>
<p>In the mid 196... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Luis Ferroni | 147,861 | <p>In the same vein of the statement the OP included:<span class="math-container">$$G\times H \cong G\times K \Longrightarrow H\cong K$$</span> for product of finite groups, which can be rephrased as "product in finite groups is cancellative", a similar property holds true for product "powers".</p>
... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Pedro | 21,326 | <p><strong>A theorem of Bass</strong>: For a ring <span class="math-container">$R$</span>, every <em>left</em> <span class="math-container">$R$</span>-module has a projective cover if and only if <span class="math-container">$R$</span> satisfies the descending chain condition on principal <em>right</em> ideals.</p>
|
203,843 | <p>To start:</p>
<pre><code>FLcounties =
AdministrativeDivisionData[
Entity["AdministrativeDivision", {"Florida", "UnitedStates"}],
"Subdivisions"];
FLcountiespop =
AdministrativeDivisionData[#, "Population"] & /@ FLcounties;
ds = Dataset[AssociationThread[FLcounties -> FLcountiespop]];
FLPopula... | kglr | 125 | <p>Use the (undocumented) option <code>"Ticks"</code> in <code>BarLegend</code>:</p>
<pre><code>BarLegend[Automatic,
LegendFunction -> "Frame",
LegendLabel -> "Population",
"Ticks" -> Map[{#, NumberForm[#, DigitBlock -> 3]} &, 100000 Range[5, 25, 5]],
LabelStyle -> Directive[Black,... |
2,958,198 | <blockquote>
<p>Suppose you want to distribute <span class="math-container">$15$</span> candies to <span class="math-container">$5$</span> different children.</p>
<p>(a) In how many ways can this be done if no kid receives more than <span class="math-container">$6$</span>
candies?</p>
<p>(b) In how many ways can this b... | Kevin Long | 283,224 | <p>Due to the particular values chosen for this problem, it is not too hard to do this by taking the complement, though it gets more difficult in general. For each of the <span class="math-container">$5$</span> children, consider the case where that child gets at least <span class="math-container">$7$</span>. We want t... |
3,444,262 | <blockquote>
<p><span class="math-container">$a_n$</span> is a non-decreasing sequence of positive integers. If an positive integer <span class="math-container">$k$</span> appears in <span class="math-container">$a_n$</span> exactly <span class="math-container">$k$</span> times and <span class="math-container">$S_n$<... | Asinomás | 33,907 | <p>Let <span class="math-container">$n=m(m+1)/2 + j$</span> with <span class="math-container">$m$</span> a positive integer and <span class="math-container">$0\leq j \leq m$</span>.</p>
<p>Then <span class="math-container">$S_n=\frac{(2m+1)(m+1)m}{6} + (m+1)j$</span>.</p>
<p>We write this as <span class="math-contain... |
319,059 | <p>I am helping out a friend who can't seem to get these proofs; unfortunately, I can't find them either. Can someone tell me how to solve this or point me in the right direction with resources? </p>
<p>Question 1: </p>
<blockquote>
<p>Prove that for all real numbers x, y, and z, if x + y + z greater than or equal... | jimjim | 3,936 | <p><strong>Proof of part 1:</strong> Let $M$ be the largest value of $x,y,z$ then at most there are 3 different values : $M,M-a,M-b$ where $a,b \geq 0$.</p>
<p>Now writing $x+y+z \geq 3$ in terms of $M$
we have
$$M+(M-a)+(M-b) \geq 3$$</p>
<p>$$3M-a-b \geq 3$$
$$3M \geq 3 +a + b$$
$$M \geq 1 + \frac {a}{3} + \frac{... |
836,841 | <p>Calculation of $\displaystyle \lim_{x\rightarrow 1}\frac{(1-x)\cdot(1-x^2)\cdot(1-x^3)\cdots (1-x^{2n})}{\{(1-x)\cdot(1-x^2)\cdot (1-x^3)\cdots(1-x^n)\}^2} = $</p>
<p><b>My Trial</b> After simplification, we get $$\displaystyle \lim_{x\rightarrow 1}\frac{(1-x^{n+1})\cdot(1-x^{n+2})\cdot(1-x^{n+3})\cdots(1-x^{2n})}{... | Hao Ye | 157,520 | <p>Try the factorization $(1-x^n) = (1-x)(1+x+\dots+x^{n-1})$.</p>
|
1,520,060 | <blockquote>
<p>Prove $\forall n\in\mathbb{Z}$ that if $n \equiv 3 \pmod 6$ then $36 \mid (n^2 + 27)$</p>
</blockquote>
<p>I know that $n \not\mid 6$ therefore, $6 \not\mid n$ and $6$ is not a multiple of $n$. But it's not helping me prove: </p>
<p>$$ 36 \mid (n^2 + 27) $$</p>
<p>How can I prove this? </p>
| coffeemath | 30,316 | <p>Hint: You actually don't know that $n$ doesn't divide $6,$ for example $n=3$ satisfies your assumption. However you do know that there is an integer $k$ for which $n=6k+3.$ Now substitute that into $n^2+27$ and expand, and see what it looks like.</p>
|
1,613,886 | <p>I was given a problem to minimise </p>
<p>$$[(x-y)^2+(12+\sqrt{1-x^2} -\sqrt{4y})^2]$$</p>
<p>Where x,y are real, I have managed to solve it, but it took a lot of time and effort, can anyone provide a short way?</p>
| sisyphus68 | 305,460 | <p>So, the goal here is to minimize: $f(x,y)=(x−y)^2+(12+\sqrt{1−x^2}-\sqrt{4y})^2$.</p>
<p>Notice that $x \in [-1,1]$ and $y \in [0,\infty)$ because of the constraint from both square roots.</p>
<p>We are looking to minimize any positive expressions, $(x−y)^2$ and $\sqrt{1−x^2}$, and maximize and negative expression... |
102,186 | <p>Suppose that $f$ is differentiable on $\mathbb{R}$. If $f(0)=1$ and $|f^{'}(x)|\leq1$ for all $x\in\mathbb{R}$, prove that $|f(x)|\leq|x|+1$ for all $x\in\mathbb{R}$. </p>
<p>I tried: </p>
<p>Let $g(x)=|f(x)|-|x|-1$. Then I tried to find $g^{'}(x)$ but I'm not sure where to start.</p>
| davidlowryduda | 9,754 | <p>The Mean Value Theorem is what you're looking for.</p>
<p>$\phantom{stuff to make this a nontrivial answer - way to go MSE. BTW, I'm hoping for a Patriots Win}$</p>
|
3,786,127 | <p>Let <span class="math-container">$p$</span> be a prime. I am interested in the set of elements <span class="math-container">$x\in\mathbb{Z}/p\mathbb{Z}$</span> such that <span class="math-container">$x$</span> and <span class="math-container">$x+1$</span> are both quadratic non-residues. Let <span class="math-contai... | cmi | 489,079 | <p>So we can write <span class="math-container">$f(x) = \sin (2 \tan^{-1} x)$</span> where <span class="math-container">$|x | > 1$</span>.</p>
<p><span class="math-container">$\frac{dy}{dx} = \frac{1}{2}\frac{d(\sin^{-1}f(x)}{dx}$</span> . So <span class="math-container">$2y = \sin^{-1} f(x) +C$</span> which is no... |
552,307 | <blockquote>
<p>If A $\cap$ B $\cap$ C = $\emptyset$, then the sum principle applies so |A $\cup$ B $\cup$ C| = |A|+|B|+|C|.</p>
</blockquote>
<p>I think it would be true since there is nothing in common among A,B and C, but just wondering if there is any exceptions to this problem so it would be false?</p>
| J126 | 2,838 | <p>Note that $A\cap B\cap C$ is all the elements that are in all three sets. It does not mean that $A\cap B=\emptyset$, or $B\cap C=\emptyset$.</p>
|
552,307 | <blockquote>
<p>If A $\cap$ B $\cap$ C = $\emptyset$, then the sum principle applies so |A $\cup$ B $\cup$ C| = |A|+|B|+|C|.</p>
</blockquote>
<p>I think it would be true since there is nothing in common among A,B and C, but just wondering if there is any exceptions to this problem so it would be false?</p>
| hardmath | 3,111 | <p>This is false, because while $A \cap B \cap C$ might be empty, this does not imply that $A \cap B$ is empty (nor that $B \cap C$ and $A \cap C$ are).</p>
<p>Indeed the intersection of all three might be empty while each element of the union is counted twice when $|A| + |B| + |C|$ is summed.</p>
|
572,307 | <p>I am having trouble proving the following statement: </p>
<blockquote>
<p>Prove that for all integers $m$ and $n$, if $d$ is a common divisor of $m$ and $n$ (but $d$ is not necessarily the GCD) then $d$ is a common divisor of $n$ and $m - n$.</p>
</blockquote>
<p>I've noticed that for any integers $m,n,d$ that $... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is an answer that is a bit more formal than the others. "$\;d\;$ divides $\;n\;$", often written as $\;d | n\;$, is defined like this:
$$
(0) \;\;\; d | n \;\equiv\; \langle \exists q \in \mathbb Z :: q \cdot d = n \rangle
$$</p>
<p>You are asked to prove
$$
d | m \:\land\: d | n \;\Rightarrow\; d | n \:\land... |
2,339,398 | <p>I'm trying to get my calculus back up to scratch after not using it for 20 odd years. During my research, I've just seen this on <a href="https://physics.info/kinematics-calculus/" rel="nofollow noreferrer">https://physics.info/kinematics-calculus/</a>:</p>
<p>$$a = \frac{dv}{dt}$$</p>
<p>$$dv = a\ dt$$</p>
<p>$$... | Gero | 458,568 | <p>In Khinchin's book "Continued Fractions" it is shown that
$$
\lim_{n\to\infty} [a_0;,a_1, \ldots, a_n]
$$
exists if and only if $\sum_{n=1}^{\infty} a_n = \infty$, where $(a_n)_n$ is a sequence of positive numbers (Theorem 10). This solves your problem, since your sequence of positive numbers has a uniform lower pos... |
694,253 | <p>How might I find tail probabilities (pr X>x), or a reasonable approximation, for a variable that is the sum of independent Laplace random variables? </p>
| Community | -1 | <p>I'd suggest using the <a href="http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29#Examples" rel="nofollow">characteristic funciton</a> of the Laplace Distribution. The characteristic function of the sum of N independent Laplace r.v.'s is the product of their characteristic functions. Then,... |
101,513 | <p>First of all, I don't really know how to formulate the question, so if you understand my question and know a better way to phrase it, please revise.</p>
<p>I have the following Matrixes:</p>
<pre><code>a[n_, m_] := Table[n + m - i - j, {i, 1, n}, {j, 1, m}]
b[n_, m_] := Round[Table[(n*m/2)*(1 + 2 j/m), {j, 1, m}]]... | garej | 24,604 | <p>Initial setting:</p>
<pre><code>Clear[a, b, A, B]
a[3, 8] = {{9, 8, 7, 6, 5, 4, 3, 2}, {8, 7, 6, 5, 4, 3, 2, 1}, {7, 6,
5, 4, 3, 2, 1, 0}};
b[3, 8] = {15, 18, 21, 24, 27, 30, 33, 36};
</code></pre>
<p>Prepare data:</p>
<pre><code>B = Prepend[b[3, 8], 0]; A = Prepend[#, 0] & /@ a[3, 8];
</code></pre>
<p>... |
83,708 | <p>It shows me like <code>{{0}, {1}, {0}, {-1}}</code> . Is it possible to make it look like a vector, always?</p>
| Nasser | 70 | <p>Add this to your notebook or init file</p>
<pre><code> $PrePrint = If[MatrixQ[#], MatrixForm[#], #] &;
</code></pre>
<p>Then all matrices will automatically display as <code>MatrixForm</code></p>
<p><img src="https://i.stack.imgur.com/t2Tg3.png" alt="Mathematica graphics"></p>
<p>and</p>
<p><img src="htt... |
125,709 | <p>Suppose $\lambda = (\lambda_1,\lambda_2,.....,\lambda_k)$ is a partition of $2n$ where $n \in \mathbb N$ satisfying the following conditions:</p>
<p>(1) $\lambda_{k} = 1$.</p>
<p>(2) $\lambda_{i} - \lambda_{i+1} \leq 1$ for every $i\leq k-1$.</p>
<p>(3) In the partition $\lambda$, the number of odd parts in odd p... | Wouter | 32,570 | <p>Bijection to OEIS A064174 "Number of partitions of n with nonnegative rank". </p>
|
386,899 | <p>Show that $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^{n+2}=\frac{2^{n}n(n+2)!}{6}.$$</p>
| Marko Riedel | 44,883 | <p>Suppose we seek to evaluate
$$\sum_{k=0}^n {n\choose k} (-1)^k (n-2k)^{n+2}.$$</p>
<p>Introduce
$$(n-2k)^{n+2}
= \frac{(n+2)!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n+3}} \exp((n-2k)z) \; dz.$$</p>
<p>We thus get for the sum
$$\frac{(n+2)!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n+3}}
\sum_{k=0}^n {n\choose k}... |
1,712,089 | <p>Probability for a disease is $0.05$.The probability that a diagnosis device will give positive result if the person has the disease is $0.99$ and vice versa. </p>
<p>a- If test is positive what is the probability that a person has the disease?</p>
<p>b- If test is applied to 2 persons and both show positive what i... | Patrick Stevens | 259,262 | <p>Consider ten thousand people. Five hundred of them will have the disease; of those people, $99 \times 5 = 495$ will test positive.
$9500$ of them will not have the disease; of those, $99 \times 95 = 9405$ will test negative, and $95$ will test positive.</p>
<p>A given person who tests positive therefore has a chanc... |
1,280,882 | <p>This may be a silly question, but one that I am confused about nonetheless.</p>
<p>With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why they are used. What's the purpose? Surely, one would ask themselves that if we can just add $A$ and $B$ together ... | JMoravitz | 179,297 | <p>Additionally, people commonly just memorize cosine and sine values for $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$.</p>
<p>Using this theorem (and others) however, we can exactly determine <em>several</em> more values for sine and cosine that were previously inaccessible to <strong>pen&paper</strong> approach... |
1,280,882 | <p>This may be a silly question, but one that I am confused about nonetheless.</p>
<p>With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why they are used. What's the purpose? Surely, one would ask themselves that if we can just add $A$ and $B$ together ... | Mark Bennet | 2,906 | <p>Such formulae are used in a whole variety of ways, for example:</p>
<p>(i) In analysing or describing rotations of the plane or of space</p>
<p>(ii) In analysing wavelike phenomena in physics - eg quite simply, combining two sound waves of slightly different frequencies gives rise to a phenomenon known as "beats".... |
1,470,879 | <p>I want to solve this system of advective-diffusive-reactive equations analytically:</p>
<p>$$\left(\alpha - k_0c_B\right)c_A+v\frac{dc_A}{dx}-D\frac{d^2c_A}{dx^2} = f_A $$
$$\left(\alpha - k_0c_A\right)c_B+v\frac{dc_B}{dx}-D\frac{d^2c_B}{dx^2} = f_B $$
$$k_0c_Ac_B+\alpha c_C+v\frac{dc_C}{dx}-D\frac{d^2c_C}{dx^2} = ... | Alain Remillard | 278,299 | <p>In french it is called the <em>conjugué</em>, conjugate in english. Multiply each fraction by it ot remove the square root in the denominator.
\begin{equation}\dfrac{1}{\sqrt{0}+\sqrt{1}}=\dfrac{1}{\sqrt{0}+\sqrt{1}}\times \dfrac{\sqrt{1}-\sqrt{0}}{\sqrt{1}-\sqrt{0}}= \dfrac{\sqrt{1}-\sqrt{0}}{1}
\end{equation}
App... |
1,488,964 | <p>I have confusion in calculating the Time period of different signals.<br>
1) $x(t) =3cos(4t + \pi/3)$<br>
solution: 2$\pi$/4 = $\pi$/2 (It is periodic)<br>
3) $x(t) =cos(t-\pi/9) +9sin(2\pi t + \pi/12)$<br>
Solution: Time period for first sinusoid is $T_1 =2\pi$ and Time period for second sinusoid is $T_2 =1$. Since... | Chieron | 282,132 | <p>The first function is a simple sinusoidal with an irrational period length. That's nothing special.</p>
<p>Assuming that you forgot the argument t in the $9sin(2πt +π/12)$ <br>
The second one is the sum of sinusoidals with an irrational and a rational period.
That's quite a different beast. </p>
<p>You correctly s... |
2,870,910 | <blockquote>
<p>I wish to show that if $z$ is real, then
$$\left|\frac{e^{iz}}{z^2+1}\right|\leq\frac{1}{|z|^2+1}$$</p>
</blockquote>
<p>I have shown this result, although my inequality is the wrong way around.</p>
<p>I considered
\begin{align}
|z^2+1|&\leq |z^2|+|1| \ \ \ \ \ \ \ \text{(triangle inequality)... | Fred | 380,717 | <p>For real $z$ we have $|z|^2=z^2$ and $|e^{iz}|$=1, hence</p>
<p>$$\left|\frac{e^{iz}}{z^2+1}\right|=\frac{1}{|z|^2+1}.$$</p>
|
2,478,695 | <p>So I need to solve the integral </p>
<blockquote>
<p>$$\int \frac { \tan { x } }{ \left( \sin { x } \right) ^{ 2 }+2\left( \cos { x } \right) ^{ 2 } } dx$$</p>
</blockquote>
<p>I saw some exercises that suggest I need to use the secant function to solve it but can I do it without it?</p>
| Nosrati | 108,128 | <p>\begin{align}
\int\frac{\tan x}{\sin^2x+2\cos^2x}\,dx
&= \int\dfrac{1}{1+\cos^2x}\dfrac{1}{\cos x}\sin x\,dx \\
&= \int\left(\dfrac{1}{\cos x}-\dfrac{\cos x}{1+\cos^2x}\right)\sin x\,dx \\
&= -\ln\cos x+\dfrac12\ln(1+\cos^2x)+C
\end{align}</p>
|
618,068 | <p>I was wondering if it's erroneous to use the quadratic formula on a quadratic equation where there is no constant term. What I figured I'd try was to just assome the constant term is +0.</p>
<p>I was doing a trigonometric equation, which looks like this:</p>
<ul>
<li>$2\cos^2 x - 3\sqrt 3 \cos x = 0$</li>
</ul>
<... | mathlove | 78,967 | <p>If the constant term is $0$, the following is better :</p>
<p>$$y(2y-3\sqrt 3)=0.$$
Then?</p>
|
618,068 | <p>I was wondering if it's erroneous to use the quadratic formula on a quadratic equation where there is no constant term. What I figured I'd try was to just assome the constant term is +0.</p>
<p>I was doing a trigonometric equation, which looks like this:</p>
<ul>
<li>$2\cos^2 x - 3\sqrt 3 \cos x = 0$</li>
</ul>
<... | lab bhattacharjee | 33,337 | <p>$$\text{As the roots of }ax^2+bx+c=0$$ $$\text{are }\frac{-b\pm\sqrt{b^2-4ca}}{2a}$$</p>
<p>If $c=0,$ the roots become $\displaystyle0,-\frac ba$</p>
<p>If $b=0,$ the roots become $\displaystyle\pm\frac{\sqrt{-4ca}}a$</p>
<p>If $b,c$ both are zero, $x=0$</p>
<p>I leave the following as exercise, but with a hint... |
4,221,821 | <p>I'm having trouble proving this, using the fact <span class="math-container">$P^k=P$</span>.</p>
<p>(<span class="math-container">$P \in L(V)$</span> where <span class="math-container">$V$</span> is a finite-dimensional complex vector space.)</p>
<p>Here's my work (I didn't use <span class="math-container">$P^k=P$</... | Dionel Jaime | 462,370 | <p>A proof that uses some of the ideas you tried to:</p>
<p>Let <span class="math-container">$f(x)$</span> be the minimal polynomial of <span class="math-container">$P$</span>. The equation <span class="math-container">$P^k = P$</span> implies that <span class="math-container">$f(x)$</span> divides <span class="math-co... |
4,221,821 | <p>I'm having trouble proving this, using the fact <span class="math-container">$P^k=P$</span>.</p>
<p>(<span class="math-container">$P \in L(V)$</span> where <span class="math-container">$V$</span> is a finite-dimensional complex vector space.)</p>
<p>Here's my work (I didn't use <span class="math-container">$P^k=P$</... | Fred | 380,717 | <p>More general:</p>
<p>let <span class="math-container">$V$</span> be a vector space and <span class="math-container">$P:V \to V$</span> linear such that <span class="math-container">$P^2=P$</span> and <span class="math-container">$1$</span> is the only eigenvalue of <span class="math-container">$P$</span>.</p>
<p>Let... |
3,054,362 | <p>Any idea on how to solve the following definite integral?</p>
<blockquote>
<p><span class="math-container">$$\int_0^1\frac{\ln{(x^2+1)}}{x+1}dx$$</span></p>
</blockquote>
<p>I have tried to parameterize the integral like <span class="math-container">$\ln{(a^2x^2+1)}$</span> or <span class="math-container">$\ln{(... | Frank W | 552,735 | <p>The trick to this integral is a round of integration by parts followed by a direct application of Feynman's Trick. To wit, denote the integral<span class="math-container">$$\mathfrak{I}=\int\limits_0^1\mathrm dx\,\frac {\log(1+x^2)}{1+x}$$</span>
Now integrate <span class="math-container">$\mathfrak I$</span> by par... |
3,014,104 | <p>Let <span class="math-container">$A$</span> be the set <span class="math-container">$A = \{1,2,3,...,20\}$</span>.
<span class="math-container">$R$</span> is the relation over <span class="math-container">$A$</span> such that <span class="math-container">$xRy$</span> iff <span class="math-container">$y/x = 2^i$</spa... | Widawensen | 334,463 | <p>Let <span class="math-container">$a,b,n$</span> be unit vectors orthogonal to each other - <span class="math-container">$a,b$</span> basis for the plane, <span class="math-container">$n$</span> orthogonal to the plane. </p>
<p>You can easily check that <strong>any</strong> vector <span class="math-container">$v$</... |
3,491,595 | <blockquote>
<p>Evaluate the sum <span class="math-container">$$\frac{1}{3} + \frac{1}{3^{1+\frac{1}{2}}}+\frac{1}{3^{1+\frac{1}{2}+\frac{1}{3}}}+\cdots$$</span></p>
</blockquote>
<p>It seems that <span class="math-container">$1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}$</span> approaches <span class="ma... | Claude Leibovici | 82,404 | <p>Just out of curiosity,
<span class="math-container">$$\sum _{n=1}^{\infty } 3^{-H_n}\approx 5.34863233867$$</span> which is close to
<span class="math-container">$$10\frac{ {3^{1/3}}}{7-7^{3/4}}\approx 5.34863230401$$</span></p>
|
6,712 | <p>I had received the "warmth" of an angry user which decided to downvote no less than eight of my questions within the span of a minute.</p>
<p>I know who the user is and I can prove their identity beyond reasonable doubt.</p>
<p>Surely the software will catch the serial voting by tomorrow and reverse it, but I was ... | robjohn | 13,854 | <p>To quote Mad Scientist:</p>
<blockquote>
<p>Moderators can't trigger the serial voting script manually, I suspect SE devs can, but they won't. And you don't need to prove the identify of the user serial-downvoting, the moderators can easily see that (if the script is triggered). </p>
</blockquote>
<p>Although th... |
715,825 | <p>How would I "solve by addition"? I'm not sure how to solve this.</p>
<p>$3x + 2y = 11$ and under that
$3x – 2y = 13$ </p>
<p>My notes that go along with it are:</p>
<p>In the addition method, you want to add the equations in such way so that one of the variables (letters) drops out. $x$ and $y$ are on the same s... | foobar1209 | 135,860 | <p>Here's the full solution. We have:
\begin{align}
3x+2y=11\\
3x-2y=13\end{align}</p>
<p>After adding the two equations, we have:
\begin{align}
6x=24 \implies \boxed{x=4}
\end{align}</p>
<p>Now plug back into either equation to solve for $y$:
\begin{align}
3(4)+2y=11\\
2y=-1 \implies \boxed{y=\dfrac{-1}{2}}\\
\end{a... |
3,037,921 | <p>Find all critical points of the system</p>
<p><span class="math-container">$y_1'= y_1(10-y_1-y_2)$</span></p>
<p><span class="math-container">$y_2'= y_2(30-2y_1-y_2)$</span></p>
<p>then classify them as stable, asymptotically stable, or unstable. </p>
<p>I need help with this particular question, as you may see,... | Rócherz | 451,007 | <p>A (non-uniquely) parametrized straight line segment that goes from a startpoint <span class="math-container">$(x_0,y_0)$</span> to an endpoint <span class="math-container">$(x_1,y_1)$</span> is given in vector form as <span class="math-container">$$\langle x,y\rangle = \langle x_0,y_0\rangle +\lambda \langle x_1-x_0... |
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>
| André Nicolas | 6,312 | <p>Substitution is not necessary. However, $u=t^2$ will be helpful.</p>
|
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>
| S.C.B. | 310,930 | <p>Doesn't this work?</p>
<p>$$\int \frac{4t}{1-t^4}dt=\int \frac{2t}{1+t^2}-\frac{-2t}{1-t^2} dt=\ln |\frac{1+t^2}{1-t^2}|+C$$</p>
<p>Also $t=\tan \theta$ substitution works quite well. $$\int \frac{4\tan \theta}{1-\tan^4 \theta}\sec^2 \theta d \theta=\int \frac {4 \tan \theta }{1-\tan^2 \theta} d\theta=2\int \frac{... |
235,332 | <p>My assignment is to translate mathematical statements into formula of predicate logic. But before I can write formulas about these statements, I'm really confused about their meaning. </p>
<p>Given that the mathematical notation of a quadratic polynomial with leading coefficient 1 is $P(x) = a_0 + a_1x + x^2$</p>... | Mark S. | 26,369 | <p><strong>(1)</strong>
"if I want to write this into a logical statement, should my formula says something to the negation of the statement ?"
It doesn't look like the problem says anything about negation, so you shouldn't be thinking along those lines.
"i.e. "there are infinitely many quadratic polynomials with leadi... |
2,330,438 | <blockquote>
<p>$$x\frac{dy}{dx}=x^2 +y$$ </p>
</blockquote>
<p>given that $\\ y\left( 1 \right) =0$</p>
<p>When i got partial derivatives of both sides, found it's not an exact equation..please can anybody can give a clue to solve this..</p>
| Donald Splutterwit | 404,247 | <p>Consider it as a linear equation in $y$
\begin{eqnarray*}
x\frac{dy}{dx}-y= x^2
\end{eqnarray*}
First solve
\begin{eqnarray*}
x\frac{dy}{dx}-y= 0
\end{eqnarray*}
Assume the solution $y=x^{\lambda}$ ... $\lambda=1$ and we have the general solution $y=Ax$.</p>
<p>Now for the particular solution assume the form $y=Bx^... |
2,330,438 | <blockquote>
<p>$$x\frac{dy}{dx}=x^2 +y$$ </p>
</blockquote>
<p>given that $\\ y\left( 1 \right) =0$</p>
<p>When i got partial derivatives of both sides, found it's not an exact equation..please can anybody can give a clue to solve this..</p>
| B. Mehta | 418,148 | <p>The standard way of solving this type of equation would be to notice it is a linear differential equation:
$$\frac{dy}{dx}-\frac{1}{x}\cdot y=x$$
So, the integrating factor here is $e^{\int -\frac{1}{x} dx} = \frac{1}{x}$, and we can write
$$\frac{1}{x}\frac{dy}{dx}-\frac{1}{x^2}y=1$$
$$\frac{d}{dx}\left(\frac{y}{x... |
3,506,899 | <p>I need to find the solutions of the following equations:</p>
<p><span class="math-container">$$\ddot{x}=-g+\alpha\dot{x}$$</span></p>
<p>and
<span class="math-container">$$\ddot{x}=-g+\alpha\dot{x}^2$$</span></p>
<p>Considering <span class="math-container">$g$</span> and <span class="math-container">$\alpha$</sp... | Blueyedaisy | 388,518 | <p>No. You could do that for <span class="math-container">$\int_a^b v'(x) dx\ = v(x)|_{a}^{b}$</span></p>
<p>But not if there is another value in the integral. </p>
<p>Counter example : <span class="math-container">$\int x \cdot ln'(x) dx = \int x \cdot \frac{1}{x} dx =\int 1 dx \neq ln(x)\cdot\int x dx $</span></p>... |
3,480,050 | <p>I've been struggling to solve the following exercise:</p>
<p>For <span class="math-container">$x\in\mathbb{R}$</span>, find the radius of convergence of the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{x^n}{n+\sqrt{n}}$</span>.</p>
<p>My approach so far: Compute <span class="math-container">$\lims... | Andronicus | 528,171 | <p>Let's calculate the limit:</p>
<p><span class="math-container">$$\lim_{n \to \infty} \sqrt[n]{\frac{x^n}{n+\sqrt{n}}}=
\lim_{n \to \infty} \frac{x}{\sqrt[n]{n+\sqrt[2]n}}=x$$</span></p>
<p>so the series converges, when <span class="math-container">$|x| < 1$</span>.</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... | awkward | 76,172 | <p>Start by picking the group of three men who sit together. This can be done in <span class="math-container">$\binom{4}{3}$</span> ways. Consider this group as one object. We then have five objects to arrange: the group of three, the remaining man, and the three women. The men in the group of three can be arranged... |
4,316,819 | <p>Consider an undirected, weighted graph <span class="math-container">$G = (P,E)$</span> where <span class="math-container">$P$</span> is the node set and <span class="math-container">$E$</span> is the edge set. Say that for a single source and target pair, <span class="math-container">$s$</span> and <span class="math... | Kurt G. | 949,989 | <p>The one dimensional SDE
<span class="math-container">$$
dX_t=AX_t\,dt+\sigma W_t
$$</span>
has the solution
<span class="math-container">$$\tag{1}
X_t=e^{At}\textstyle(X_0+\sigma\int_0^te^{-As}\,dW_s)\,.
$$</span>
When you replace the constant <span class="math-container">$A$</span> by the matrix
<span class="math-c... |
4,316,819 | <p>Consider an undirected, weighted graph <span class="math-container">$G = (P,E)$</span> where <span class="math-container">$P$</span> is the node set and <span class="math-container">$E$</span> is the edge set. Say that for a single source and target pair, <span class="math-container">$s$</span> and <span class="math... | Lutz Lehmann | 115,115 | <p>If you apply Ito to <span class="math-container">$Y=X_1^2+X_2^2$</span>, then
<span class="math-container">$$\begin{align}
dY &= 2(X_1dX_1+X_2dX_2)+(dX_1)^2+(dX_2)^2
\\
&= -2\mu(X_1^2+X_2^2)\,dt+2\sigma(X_1dW_1+X_2dW_2)+2\sigma^2\,dt,
\end{align}$$</span>
using the informal notation for the increment of the ... |
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>... | Joseph Malkevitch | 1,369 | <p>Graph Theory is indeed a very quick starting subject in the sense that one does need to have studied calculus and other mathematical subjects to get started. However, there are lots of books that are specialized towards particular purposes: applications in general, network applications, distances in graphs, etc.</p>... |
3,335,615 | <p>Let <span class="math-container">$f,g$</span> be Riemann integrable on <span class="math-container">$[0,1]$</span> such that <span class="math-container">$\int_0^1 f=\int_0^1 g=1$</span>. Show that there exists <span class="math-container">$0\leq a<b\leq 1$</span> such that <span class="math-container">$\int_a^b ... | MathTrain | 339,640 | <p>Denote by <span class="math-container">$u(a)$</span> the (smallest) number in <span class="math-container">$(0,1)$</span> such that <span class="math-container">$\int_a^{u(a)} f=1/2$</span>. Use <span class="math-container">$v(a) $</span> for the analogous function with <span class="math-container">$g$</span>. We ca... |
168,163 | <p>I wonder what kind of functions satisfy </p>
<p>$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$
I suppose all functions must be continuous.</p>
| Harald Hanche-Olsen | 23,290 | <p>The equation is true for any integrable function $f$ on $[0,1]$ so that $1$ is a Lebesgue point for $f$, in the sense that $\lim_{y\to1}\bar f(y)=f(1)$, where $$\bar f(y)=\frac1{1-y}\int_y^1 f(y)\,dy.$$
Indeed, using $x^n=(n+1)\int_0^x y^n\,dx$ in the integral and interchanging the order of integration, we find afte... |
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... | Shaun | 104,041 | <p>Since</p>
<p><span class="math-container">$$\begin{align}
(hxh^{-1})^{-1}&=(h^{-1})^{-1}x^{-1}h^{-1}\\
&=hx^{-1}h^{-1},
\end{align}$$</span></p>
<p>we have,</p>
<p><span class="math-container">$$hx^{-1}h^{-1}=x^{-1}.$$</span></p>
<p>Hence <span class="math-container">$F(x)\subseteq F(x^{-1})$</span>.</p>
<p>... |
2,172,949 | <p>I have the integral:
$$\int _{\partial D(a, r)} \frac{e^z}{z^3 + 2z^2 + z} dz$$</p>
<p>which I have to find for different cases:</p>
<p>1 - $a = 0$ and $r =1/2$</p>
<p>2 - $a = -i - 1$ and $r = 1/2$</p>
<p>3 - $a = -1$ and $r = 1/2$</p>
<p>4 - $a = 0$ and $r = 2$</p>
<p>My attempt is this:</p>
<p>$$ \frac{e^z... | Nosrati | 108,128 | <p>For (1) when $|z|<\dfrac12$ we have
$$\int_{|z|<\frac12}\frac{e^z}{z(z+1)^2}\,dz=\int_{|z|<\frac12}\dfrac{\frac{e^z}{(z+1)^2}}{z}\,dz=2\pi i\frac{e^z}{(z+1)^2}\Big|_{z=0}=\color{blue}{2\pi i}$$</p>
|
3,940,818 | <p>Can anyone help how to find the eigenvalues of the following matrix in a simple way? I expand the characteristic polynomial being,
<span class="math-container">$$
\lambda(\lambda-3)(\lambda - 2k) = 0
$$</span>
and get the answer but intuition is that there must be a simple way to find it.
<span class="math-conta... | Coriolanus | 439,201 | <p>Glancing at the matrix might suggest cancelling <span class="math-container">$k$</span>:</p>
<p>Multiply your matrix by <span class="math-container">$\begin{bmatrix}1 & y & 1\end{bmatrix}^T$</span> to get <span class="math-container">$\begin{bmatrix}2 & 2 & 2\end{bmatrix}^T + y\begin{bmatrix}1 & ... |
1,242,541 | <p>I am trying to prove the following, and here is what I have done:
Can somebody help to complete this?</p>
<p>$2^n \ge n^2$ for $n\ge 4$</p>
<p>$n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE</p>
<p>Assume True for $n=k$, for $k\ge 4$</p>
<p>$2^k \ge k^2$</p>
<p>Should be true for $n=k+1$ for $k\ge... | mathlove | 78,967 | <p>Since $k\ge 4$, one has
$$\begin{align}2^{k+1}&\ge k^2+k^2\\&\ge k^2+4k\\&=k^2+2k+2k\\&\ge k^2+2k+2\cdot 4\\&\ge k^2+2k+1\\&=(k+1)^2\end{align}$$</p>
|
3,561,242 | <blockquote>
<p>Determine if the following integral will converge.
<span class="math-container">$$\int_0^1\frac{e^{\sqrt x}-1}{x}dx$$</span></p>
</blockquote>
<p>My approach was something like this. I made the assumption that <span class="math-container">$e^{\sqrt{x}} − 1 ≈ x$</span> and then followed like this:</... | marty cohen | 13,079 | <p>Since
<span class="math-container">$1+x
\le e^x
\le 1+x+x^2
$</span>
for
<span class="math-container">$0 \le x \lt 1$</span>,</p>
<p><span class="math-container">$\int_0^1\frac{e^{\sqrt x}-1}{x}dx
\le \int_0^1\frac{\sqrt{x}+x}{x}dx
= \int_0^1(\frac1{\sqrt{x}}+1)dx
=(2\sqrt{x}+x)|_0^1
=3
$</span>.</p>
<p>Also
<span... |
13,084 | <p><a href="https://math.stackexchange.com/questions/712776/discrete-math-is-there-a-difference-between-subseteq-to-supseteq">the question.</a>
I understand that it's super-basic-preschool-stupid (actually no question is stupid) question, but is it right to down vote it just for that?</p>
<p>Few things that math taugh... | Community | -1 | <p>The downvote mystifies me too... and I'm usually pretty imaginative at coming up with plausible explanations for such things even if I don't agree with them.</p>
<p>Regarding your last comment, it's not actually true that there are no bad questions; that advice is meant to encourage people who do have reasonable qu... |
465,487 | <p>In Fraleigh, it said, </p>
<blockquote>
<p>"Consider the set N of all polynomials in x and y in F[x,y] having constant term 0. Then N is an ideal, but not a principal ideal." (p.399)</p>
</blockquote>
<p>Could you tell me why this is not a principal ideal?</p>
| PVAL-inactive | 83,337 | <p>Assume $N$ is a principal ideal $=(f)$. Then $x$ and $y \in N$ so there is some $r$ and $s$, with $rf=x$ and $sf=y$. By considering the $x$ degree and the $y$ degree, we can see $f$ must be in our field. Then $N=F[x,y]$ or $0$, both of which are absurd.</p>
|
4,206,112 | <p>I have the following quadratic equation:</p>
<p><span class="math-container">$$x = \frac{-1 + \sqrt{1+ (4y/50)} }{2}$$</span></p>
<p>in this case <span class="math-container">$y$</span> is a known variable so I can solve the equation like this for <span class="math-container">$y = 600$</span></p>
<p><span class="ma... | Paul | 138,918 | <p>First see how the right hand expression is built, starting from <span class="math-container">$y$</span>, using simple arithmetic operations</p>
<p><span class="math-container">$$y\xrightarrow[\times \tfrac{4}{50}]{}\frac{4y}{50}\xrightarrow[+1]{}1+\frac{4y}{50}\xrightarrow[\sqrt{{}}]{}\sqrt{1+\frac{4y}{50}}\xrightar... |
2,476,453 | <p>The problem is as follows:</p>
<blockquote>
<p>Find the value of this function
<span class="math-container">$$A=\left(\cos\frac{\omega}{2} +\cos\frac{\phi}{2}\right )^{2} +\left(\sin\frac{\omega}{2} -\sin\frac{\phi}{2}\right )^{2}$$</span>
when <span class="math-container">$\omega=33^{\circ}{20}'$</span> and... | MrYouMath | 262,304 | <p>Hint: You did a miscalculation:</p>
<p>$$A=\left (\cos\frac{\omega}{2}+\cos\frac{\phi}{2} \right )^{2}+\left (\sin\frac{\omega}{2}-\sin\frac{\phi}{2} \right )^{2}$$
$$=\cos ^2\frac{\omega}{2}+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}+\cos^2 \frac{\phi}{2}+\sin ^2\frac{\omega}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}... |
287,976 | <p>I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity given by $\omega^k$, $k=1,2,\dots, (n-1)$. So basically my working is
$$\sum_1^{n-1}|1-\omega^k|^4=\sum_1^{n-1}(|1-\o... | Brian M. Scott | 12,042 | <p>HINT: If $n$ is odd, the integers $2k\bmod n$ for $k=1,\dots,n-1$ run through $1\dots,n-1$, so</p>
<p>$$\sum_{k=1}^{n-1}\omega^{2k}=\sum_{k-1}^{n-1}\omega^k\;.$$</p>
<p>If $n$ is even, the integers $2k\bmod n$ for $k=1,\dots,n-1$ run through $2,4,\dots,n-2$ twice with a $0$ in the middle. The powers $\omega^{2k}$ ... |
1,662,958 | <blockquote>
<p>Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice. </p>
</blockquote>
<p>The way I thought of it is this : if Bob gets $0$ heads... | user3826158 | 483,678 | <p>Firstly,$$\sum_{i=0}^n C_i^n = 2^n$$
Seondly,$$\sum_{i=0}^{k-1} C_i^n + \sum_{i=k}^n C_i^n = 2^n$$
Thirdly,$$\sum_{k=1}^{n+1} C_i^{n+1}\sum_{i=0}^{k-1} C_n^i + \sum_{k=1}^{n+1} C_i^{n+1}\sum_{i=k}^n C_n^I$$$$ = \sum_{k=1}^{n+1} C_i^{n+1}\Biggl(\sum_{i=0}^{k-1} C_i^n + \sum_{i=k}^n C_i^n\Biggl) $$$$=2^n\sum_{k=1}^{n+... |
222,759 | <p>Here is the given series 3/(9n+1), decide whether it converges or diverges.
I used the ratio test only to end up with the ratio=1.
I know this is harmonic series but it is smaller than 1/n, therefore i cannot conclude it diverges.
Please help!!</p>
| ncmathsadist | 4,154 | <p>The ratio and root tests are very crude and won't work here. Have you tried comparison oar limit comparison?</p>
|
17,335 | <p>Starting with a representation <span class="math-container">$\rho:G \to \mathrm{GL}(V)$</span>. Then we can build the tensor product of <span class="math-container">$V$</span> with itself by defining <span class="math-container">$g(v_1 \otimes v_2) = g(v_1) \otimes g(v_2)$</span>. Then by saying <span class="math-... | Marty | 3,545 | <p>Nope.</p>
<p>Really, this is a linear algebra question. You can take tensor products of pairs of vector spaces, symmetric and exterior powers of a single vector space. These are all functorial, so extend to representations of a group. </p>
<p>Let $Vec$ be the category of (finite-dimensional) vector spaces and l... |
613,042 | <p>This is the expression: $$G=\bigsqcup_{d\mid n}X_d$$
I saw it here - <a href="https://math.stackexchange.com/questions/346936/finite-group-for-which-xxm-e-leq-m-for-all-m-is-cyclic">Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.</a></p>
<p>Thank you!</p>
| ILoveMath | 42,344 | <p>It means that the sets $X_d$ in the union are pairwise disjoint.</p>
|
613,042 | <p>This is the expression: $$G=\bigsqcup_{d\mid n}X_d$$
I saw it here - <a href="https://math.stackexchange.com/questions/346936/finite-group-for-which-xxm-e-leq-m-for-all-m-is-cyclic">Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.</a></p>
<p>Thank you!</p>
| mathfemi | 113,000 | <p>Usually it means disjoint (pairwise). Others write $\biguplus$.</p>
|
3,409,012 | <p><a href="https://i.stack.imgur.com/CqTOf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CqTOf.png" alt="enter image description here"></a></p>
<p>For example, I'm asked to find the line integral of <span class="math-container">$C_1$</span> above for a vector field <span class="math-container">$\... | José Carlos Santos | 446,262 | <p>Because your parameterisation describes a path which begins at <span class="math-container">$2\vec j$</span> and ends at <span class="math-container">$\vec i$</span>, whereas it should begin at <span class="math-container">$\vec i$</span> and end at <span class="math-container">$2\vec j$</span> (as the other one doe... |
1,394,088 | <blockquote>
<p>What is the smallest possible natural number $$ for which the equation $x^{2}-nx+2014=0$ has integer roots?</p>
</blockquote>
<p>My idea was, If the roots are integers, then they are the divisors of $2014$, I don't know if it's true or not.</p>
| lab bhattacharjee | 33,337 | <p>We have for natural number $a,b$ $$ab=2014=2\cdot19\cdot53$$</p>
<p>and we need to minimize $n=a+b$</p>
<p>Check for $1,2014;$</p>
<p>$2,2014/2;$</p>
<p>$19;2014/19;$</p>
<p>$2\cdot19,2014/2\cdot19;$</p>
<p>$53,2014/53$ etc.</p>
|
1,394,088 | <blockquote>
<p>What is the smallest possible natural number $$ for which the equation $x^{2}-nx+2014=0$ has integer roots?</p>
</blockquote>
<p>My idea was, If the roots are integers, then they are the divisors of $2014$, I don't know if it's true or not.</p>
| Oiue | 258,324 | <p>If the roots are $\alpha, \beta$ then we have $\alpha \beta = 2014.$ As they are integers, $\alpha, \beta$ must be divisors of $2014=2\cdot19\cdot53$, giving $\left \{ \alpha, \beta \right \}=\left \{ 1,2014 \right \}$, or $\left \{ -1,-2014 \right \}$ or $\left \{ 2,1007 \right \}$ or $\left \{ -2,-1007 \right \}... |
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| Qiaochu Yuan | 232 | <p>This is overkill, but it follows from general facts about Coxeter groups as outlined <a href="http://qchu.wordpress.com/2010/06/26/coxeter-groups/" rel="nofollow">here</a>. In particular, it follows from the fact that $S_n$ has presentation $s_i^2 = 1, (s_i s_{i+1})^3 = 1, s_i s_j = s_j s_i, |i - j| \ge 2$ (which f... |
43,688 | <p>The nuclear norm of a matrix is defined as the sum of its singular values, as given by the singular value decomposition (SVD) of the matrix itself. It is of central importance in Signal Processing and Statistics, where it is used for matrix completion and dimensionality reduction. </p>
<p>A question I have is wheth... | Vedran Šego | 78,926 | <p>Since you asked for an approximation as well, you might find the paper "<a href="http://www.sciencedirect.com/science/article/pii/0024379584901174" rel="nofollow">Some simple estimates for singular values of a matrix</a>" by Liqun Qi useful. There are some nice estimates there.</p>
<p>However, if these are not prec... |
1,466,662 | <p>Hi i have the answer but don't understand so please explain your answer... it should be $-yx^{(y-1)}$</p>
| eloiprime | 180,579 | <p>The definition of the partial derivative of $x^y$ <strong>with respect to</strong> $x$ is</p>
<p>$$\frac{\partial }{\partial x} x^y = \lim_{h\to 0} \frac{(x+h)^y - x^y}{h}.$$</p>
<p>In this way, we may simply treat $y$ as a constant when differentiating. So,
$$\frac{\partial }{\partial x} x^y = yx^{y-1}$$
for the ... |
1,265,200 | <p>If a particle performs a random walk on the vertices of a cube, what is the mean number of steps before it returns to the starting vertex S? What is the mean number of visits to the opposite vertex T to S before its first return to S and what is the mean number of steps before its first visit to T?</p>
<p>Nobody ev... | Charles M. Grinstead | 237,578 | <p>This random walk, like many others, can be modeled by using a Markov chain. See Chapter 11 of the on-line probability book by Grinstead and Snell (it can be freely downloaded). You need to know a bit of linear algebra to understand Markov chain theory.</p>
|
3,822,415 | <p><em>SECTION 2.4 A Deductive Calculus</em> In Enderton's <em>A Mathematical Introduction to Logic</em> divides the set of axioms into several groups. The first group is called "tautologies" on p114, which are obtained from tautologies in Sentential Logic, by <a href="https://math.stackexchange.com/questions... | Noah Schweber | 28,111 | <p>As spaceisdarkgreen says, there is no typo here: Enderton means "<span class="math-container">$\vdash$</span>" in the theorem, and "logical entailment" (<span class="math-container">$\models$</span>) in the remark. The remark observes that there is a gap between <span class="math-container">$\mo... |
34,657 | <p>In section III.1 of P.M. Cohn's <a href="http://books.google.co.uk/books?id=vZsHZ1YP4KkC&lpg=PA108&ots=GIztdoRc2E&dq=universal%20functor&pg=PA108#v=onepage&q=universal%20functor&f=false" rel="nofollow">Universal Algebra</a> a notion of <em>universal functor</em> ${\cal L} \rightarrow {\cal K}... | André Henriques | 5,690 | <p>In dimension 2, the answer is also "no".</p>
<p>Recall the classical construction of the <a href="http://en.wikipedia.org/wiki/File:Lakes_of_Wada.jpg" rel="nofollow">Wada lakes</a>.
In the linked picture, one sees little "straits" connecting the red/blue/green regions at stage $n$ with the extra windy strip that is... |
1,112,926 | <p>Problem: For the sequence $r$ defined by </p>
<p>$$r_n = 3 \cdot 2^n - 4 \cdot 5^n, \ \ \ n \geq 0$$ <br></p>
<p>Prove that {$r_n$} satisfies <br></p>
<p>$$r_n = 7r_{n-1} - 10r_{n-2}, \ \ \ n \geq 2$$</p>
<p>Can this problem be explained and broken down and show the process? I'd like to follow your steps on my o... | natur3 | 207,725 | <p>Based on the input I was given by some members here's what I came up with following some of those steps and putting it in my own words in a way to make sure I understand what's happening: </p>
<p>$r_n = 3\cdot2^n - 4\cdot5^n$, $n \geq 0$</p>
<p>$r_n = 7r_{n-1} - 10r_{n-2}$, $n \geq 2$</p>
<p>$n = 1$<br>
$1-1 = 1$... |
3,405,035 | <p>I have been reading Spivak's Introduction to Differential Geometry and I cannot get this. I have already seen other proofs of the following statement, and I get those, I just do not follow this one.</p>
<blockquote>
<p><span class="math-container">$d\theta$</span> is closed but it is not exact</p>
</blockquote>
<p>I... | GReyes | 633,848 | <p>The reason is that <span class="math-container">$\theta+const$</span> (for any constant) is a function that has a jump along <span class="math-container">$L$</span> (a <span class="math-container">$2\pi$</span>-jump, to be precise). Therefore, no smooth function defined on <span class="math-container">$\mathbb{R}^2\... |
432,416 | <p>Question is simple:</p>
<ol>
<li><p>Does the elliptical shape of the earth affect its radius? (Yes!!?)</p></li>
<li><p>If it is true: How?</p></li>
<li><p>How can I determine the exact distance between two points on the earth with this influence?</p></li>
</ol>
<p><strong>Notice: When I measure the distance betwee... | Avitus | 80,800 | <p>Edit: Thanks to Christian Blatter I correct a misunderstanding in the "oblate ellipsoid" definition I previously used in this answer.</p>
<p>With "radius" we mean the distance between any given point on the Earth surface and the center of the Earth itself.</p>
<p>If we approximate the Earth to an oblate ellipsoid,... |
1,810,348 | <p>v= R^4 is an <strong>Inner product space</strong> and u=span{(1,0,-1,0)} subspace.
how can I find a base for the vectors which orthogonal to U(the complement of U)?</p>
<p>Thanks!</p>
| Community | -1 | <p>If $\textbf{b} = (1,0,-1,0)$ and $U = span\{b\}$, then every vector in $U$ is of the form $\alpha \textbf{b},$ $\alpha \in \mathbb{R}$. It is easy to check that a vector $\textbf{v} \in \mathbb{R}^4$ is orthogonal to every vector in $U$ if and only if it is orthogonal to $\textbf{b}$. </p>
<p>Thus </p>
<p>$$
U^\pe... |
4,155,766 | <p>The area of a triangle is <span class="math-container">$14\sqrt{3}$</span> <span class="math-container">$cm^2$</span>. The lengths of two sides of the triangle are <span class="math-container">$7$</span> <span class="math-container">$cm$</span> and <span class="math-container">$8$</span> <span class="math-container"... | Jean Marie | 305,862 | <p>Here is a solution using coordinate geometry.</p>
<p>You can assume WLOG that you can take the line bissector as <span class="math-container">$y$</span> axis, giving this figure:</p>
<p><a href="https://i.stack.imgur.com/PI0DS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PI0DS.jpg" alt="enter i... |
4,155,766 | <p>The area of a triangle is <span class="math-container">$14\sqrt{3}$</span> <span class="math-container">$cm^2$</span>. The lengths of two sides of the triangle are <span class="math-container">$7$</span> <span class="math-container">$cm$</span> and <span class="math-container">$8$</span> <span class="math-container"... | g.kov | 122,782 | <p>Let the base side of <span class="math-container">$\triangle ABC$</span> be <span class="math-container">$|BC|=a=7$</span>,
so we have two points <span class="math-container">$B,C$</span> fixed.
The third point must be located on
the intersection of
a semicircle with the radius <span class="math-container">$|BA|=b=8... |
3,960,527 | <p>Let <span class="math-container">$X$</span> be a topological space and <span class="math-container">$A\subset X$</span>. Show that <span class="math-container">$X = int(A) \cup Fr(A)\cup int(X-A)$</span>, this being a union disjointed.</p>
<p>To show this equality I must show the inclusions:
<span class="math-contai... | drhab | 75,923 | <p>Let <span class="math-container">$x\in X$</span> be arbitrary and let <span class="math-container">$\mathcal N_x$</span> denote its set of neighborhoods.</p>
<p>Purely on logical grounds we can say that exactly one of the following options is correct:</p>
<ul>
<li><span class="math-container">$\exists N\in\mathcal N... |
368,117 | <blockquote>
<p>Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.</p>
</blockquote>
<p>I think the two thing's I'm trying to show are:</p>
<ul>
<li>If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$</li>
<li>Wlog, if sign$(\sigma) = 1$,... | Community | -1 | <p>We define the signature of the permutation $\sigma$:
$$\epsilon(\sigma)=(-1)^N$$
where $N$ is the number of inversion.</p>
<p>Let $\sigma$ and $\tau$ two permutations with $P$ and $Q$ inversions respectively. The inversions of the composition $\tau \sigma$ are:</p>
<ul>
<li><p>the inversions $\{i,j\}$ of $\sigma$ ... |
2,133,984 | <p>If $f(s) = (1+s)^{(1+s)^{(1+s)/s}/s}$, show that $\lim_{s \to \infty} f(s)/s = 1$.</p>
<p>This function comes up
in the parameterization
of the solutions to
$x^y = y^x$.
See for example, here:
<a href="https://math.stackexchange.com/questions/1664284/are-there-real-solutions-to-xy-yx-3-where-y-neq-x">Are there real... | Simply Beautiful Art | 272,831 | <p>The expansion at infinity is actually easy. Following general methods to provide expansions at infinity:</p>
<p>$$f(1/s)=(1+1/s)^{(1+1/s)^{(1+1/s)/(1/s)}/(1/s)}=(1+1/s)^{(s+1)\cdot(1+1/s)^s}$$</p>
<p>As $s\to0^+$, we know that</p>
<p>$$(1+1/s)^s\to1$$</p>
<p>$$\implies(1+1/s)^{(s+1)\cdot(1+1/s)^s}\sim(1+1/s)^{(... |
878,973 | <p>Question:
Write the polar form of $$\frac{(1+i)^{13}}{(1-i)^7}$$</p>
<p>Well its obviously impractical to expand it and try and solve it.
Multiplying the denominator by $(1+i)^7$ will simplify the denominator, and a single term in the numerator. </p>
<p>Answer I got:
$$(\frac{1}{\sqrt2}(cos(\frac{\pi}{4}) + sin(\f... | rogerl | 27,542 | <p>No, that's not correct. You must have made a couple of errors in your expansions.
\begin{align}
\frac{(1+i)^{13}}{(1-i)^7} &= \frac{(1+i)^{13}(1+i)^7}{(1-i)^7(1+i)^7} \\
&= \frac{1}{2^7}(1+i)^{20} \\
&= \frac{1}{2^7}\left(\sqrt{2}\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)\right)^{... |
878,973 | <p>Question:
Write the polar form of $$\frac{(1+i)^{13}}{(1-i)^7}$$</p>
<p>Well its obviously impractical to expand it and try and solve it.
Multiplying the denominator by $(1+i)^7$ will simplify the denominator, and a single term in the numerator. </p>
<p>Answer I got:
$$(\frac{1}{\sqrt2}(cos(\frac{\pi}{4}) + sin(\f... | colormegone | 71,645 | <p>You can also convert numerator and denominator into polar form immediately to write</p>
<p>$$ \frac{ [ \ \sqrt{2} \ cis(\frac{\pi}{4}) \ ]^{13} \ }{[ \ \sqrt{2} \ cis(-\frac{\pi}{4}) \ ]^7} \ \ . $$</p>
<p>DeMoivre's Theorem for powers gives us</p>
<p>$$ = \ \frac{ (\sqrt{2})^{13} \ cis(\frac{13\pi}{4}) }{(\s... |
2,494,153 | <p>The Peano axioms are often listed (among other ways) as a set of 5 axioms in an informal language or 3 axioms in a formal language. For example:</p>
<p><strong>Informal</strong> (see, e.g., <a href="http://mathworld.wolfram.com/PeanosAxioms.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/PeanosAxioms.h... | vadim123 | 73,324 | <p>$b_n$ is the harmonic series (cancel the $n$'s), or a $p$-series with $p=1$, which diverges. Unfortunately $a_n$ is smaller, so you need to do the limit comparison test (not just the ordinary comparison test). It turns out that $a_n$ and $b_n$ do match, so $a_n$ diverges too.</p>
|
2,494,153 | <p>The Peano axioms are often listed (among other ways) as a set of 5 axioms in an informal language or 3 axioms in a formal language. For example:</p>
<p><strong>Informal</strong> (see, e.g., <a href="http://mathworld.wolfram.com/PeanosAxioms.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/PeanosAxioms.h... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$\frac{n}{n^2+1}\geq \frac{1}{2n}$$</p>
|
4,612,286 | <p>This is a bit of a soft question, but I am interested in a list of classes of structures (in the sense of model theory) which are "surprisingly" first-order axiomatizable classes. Meaning, the class of structures is defined in such a way that it is not at all obvious that it is in fact first-order axiomati... | Primo Petri | 137,248 | <p>The class of graphs that are 2-colorable.</p>
<p>A graph is 2-colorable if there is an equivalence relation on the set of verteces that has 2cclasses and is such that no adiacent vertexes are in in the same classs.</p>
<p>The description above is second-order, but it is equivalent to the class of graphs without odd ... |
156,468 | <p>When I use <code>ListDensityPlot</code>to plot a matrix with dimension $21\times400$, I find the result has dimension $20\times399$.
When the dimension of the matrix is small, for example $5\times 5$, this problem still exists:</p>
<pre><code>mat = Table[Tan[x + y], {x, 1, 5}, {y, 1, 5}];
ListDensityPlot[mat, Inte... | Rom38 | 10,455 | <p>The result figure with four cells is right because the range of five point has four intervals. You can draw you <code>mat</code> by <code>ListPointPlot3D[mat]</code> and will see the five points:
<a href="https://i.stack.imgur.com/wKJEF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wKJEF.png" al... |
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