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 |
|---|---|---|---|---|
1,017,707 | <p>Are there any proofs of this equality online? I'm just looking for something very simply that I can self-verify. My textbook uses the result without a proof, and I want to see what a proof would look like here.</p>
| Adhvaitha | 191,728 | <p>We have
\begin{align}
\cos((n+1)x) & = \cos(x) \cos(nx) - \sin(x) \sin(nx)\\
& = \cos(x) \cos(nx) + \dfrac{\cos((n+1)x) - \cos((n-1)x)}2
\end{align}
Hence,
$$2 \cos((n+1)x) = 2\cos(x) \cos(nx) + \cos((n+1)x) - \cos((n-1)x)$$
Therefore,
$$\cos((n+1)x) = 2\cos(x) \cos(nx) - \cos((n-1)x)$$
Now use induction to ... |
999,147 | <p>I'm looking to gain a better understanding of how the cofinite topology applies to R.
I know the definition for this topology but I'm specifically looking to find some properties such as the closure, interior, set of limit points, or the boundary set and how these change based on whether a subset A in R is closed, ... | Jonas Gomes | 138,672 | <p>The co-finite topology has a good property: It's the minimal $T_1$ topology, so every other property (Hausdorff, Regular, etc) you might expect to fail.</p>
<p>For example, every closed set is either finite or $\mathbb{R}$. So, the closure of any set is the set iself (if it's finite) or $\mathbb{R}$ if it is infini... |
2,755,733 | <p>Why in this <a href="https://math.stackexchange.com/questions/625112/if-tossing-a-coin-400-times-we-count-the-heads-what-is-the-probability-that-t">If, tossing a coin 400 times, we count the heads, what is the probability that the number of heads is [160,190]?</a> question heropup's asnwer is like that? </p>
<p>I ... | farruhota | 425,072 | <p>Note that the binomial probability distribution is a discrete probability distribution, while normal probability distribution is a continuous probability distribution. </p>
<p>When the binomial probability distribution is approximated (under certain conditions) with the normal probability distribution, so called co... |
3,983,914 | <p><strong>Preliminary properties</strong>: Let the state vector <span class="math-container">$x(t)=[x_1(t),\dots,x_n(t)]^T\in\mathbb{R}^n$</span> be constrained to the dynamical system
<span class="math-container">$$
\dot{x} = Ax +
\begin{bmatrix}
\phi_1(x_1) \\
\vdots \\
\phi_n(x_1) \\
\end{bmatrix}, \ \ \ \ x(0) = ... | Kwin van der Veen | 76,466 | <p>Inspired by the answer of open problem one can say a bit more in general when considering only the <span class="math-container">$\alpha_i=1$</span> cases. Although, it is stated that <span class="math-container">$0 < \alpha_i < 1$</span>, so technically these cases would just barely violate the considered doma... |
345,094 | <p>If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$</p>
<p>and</p>
<p>$f(x)= ax^2 + bx + c$</p>
<p>what would be the value of $a+b+c$?</p>
<p>I was doing</p>
<p>$f(x-1)+f(x-2)= f(x-3)$
then
$f(x)$</p>
<pre><code>a = 5
b = -2
c = 9
</code></pre>
<p>$(5-3)+(-2-3)+(9-3)$</p>
<p>But do not think is is correct</p>
<p>What would... | Jerry | 68,593 | <p>If $f(x) = ax^2 + bx+ c$, what is $f(x-1)$ and what is $f(x-2)$?</p>
<p>Work those out, then add both expressions to equate to $5x^2-2x+9$.</p>
|
345,094 | <p>If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$</p>
<p>and</p>
<p>$f(x)= ax^2 + bx + c$</p>
<p>what would be the value of $a+b+c$?</p>
<p>I was doing</p>
<p>$f(x-1)+f(x-2)= f(x-3)$
then
$f(x)$</p>
<pre><code>a = 5
b = -2
c = 9
</code></pre>
<p>$(5-3)+(-2-3)+(9-3)$</p>
<p>But do not think is is correct</p>
<p>What would... | Adi Dani | 12,848 | <p>$$f(x-1)+f(x-2) = 5x^2 - 2x + 9$$and $$f(x)=ax^2+bx+c$$
for $x=1,2,3$ we get the system
$$f(0)+f(-1)=a-b+2c =12$$
$$f(1)+f(0)=a+b+2c=25$$
$$f(2)+f(1)=5a+3b+2c = 48$$
with solutions
$$a=5/2,b=13/2,c=8$$
so $$f(x)=\frac{5}{2}x^2+\frac{13}{2}x+8$$</p>
|
3,568,693 | <p>I am trying to solve <span class="math-container">$n! = 10^6$</span> for <span class="math-container">$n$</span>. I thought to do this using the <a href="https://en.wikipedia.org/wiki/Gamma_function" rel="nofollow noreferrer">gamma function</a>:</p>
<p><span class="math-container">$$(n - 1)! = \Gamma(n) = \int_0^\i... | Strafe Ae | 457,404 | <p>Using WolframAlpha, I get <span class="math-container">$n=9.4456089144163262435935599652$</span>, but thats about as far as I can figure. There's not really a good way to reverse the Gamma function to solve something like that besides numerical approximations.</p>
|
1,644,845 | <blockquote>
<p>Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.</p>
</blockquote>
<p>Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} \frac{\Re(z)}{z} = \lim_{r \to 0} (\cos ^2(\theta) - i \cos(\theta)\sin(\theta)) = (\cos ... | Cm7F7Bb | 23,249 | <p>$X_1,\ldots,X_n$ are i.i.d. $\mathcal N(\mu,\sigma^2)$ random variables and $\bar X=n^{-1}\sum_{i=1}^nX_i$. The distributions and confidence intervals are as follow.</p>
<p>(a)
$$
\frac1{\sigma^2}\sum_{i=1}^n(X_i-\bar X)^2\sim\chi_{n-1}^2
$$
and
$$
\Pr\biggl(\frac{\sum_{i=1}^n(X_i-\bar X)^2}{\chi_{n-1,\alpha/2}^2}\... |
3,820,465 | <p>I'm working on the following problem but I'm having a hard time figuring out how to do it:</p>
<p>Q: Let A and B be two arbitrary events in a sample space S. Prove or provide a counterexample:</p>
<p>If <span class="math-container">$P(A^c) = P(B) - P(A \cap B)$</span> then <span class="math-container">$P(B) = 1$</sp... | herb steinberg | 501,262 | <p>Not true. If <span class="math-container">$A^c\subseteq B$</span>, then <span class="math-container">$B-A\cap B=A^c$</span>, so <span class="math-container">$P(A^c)=P(B)-P(A\cap B)$</span> for any <span class="math-container">$B$</span>. This holds since <span class="math-container">$A^c$</span> and <span class="ma... |
3,356,544 | <p>A lot of calculators actually agree with me saying that it is defined and the result equals 1, which makes sense to me because:</p>
<p><span class="math-container">$$ (-1)^{2.16} = (-1)^2 \cdot (-1)^{0.16} = (-1)^2\cdot\sqrt[100]{(-1)^{16}}\\
= (-1)^2 \cdot \sqrt[100]{1} = (-1)^2 \cdot 1 = 1$$</span></p>
<p>Howev... | LIR | 608,434 | <p>The problem is that the exponential function: <span class="math-container">$$f:\mathbb{R}\rightarrow\mathbb{R}, f(x) = ab^x$$</span> is only defined for <span class="math-container">$b\in(0,\infty)$</span> and <span class="math-container">$a\neq0$</span> so <span class="math-container">$(-1)^x$</span> is not an expo... |
3,069,987 | <p>I know that whatever numbers you choose for x and y and their sum equals to 1 will satisfy the equation <span class="math-container">$x^2 + y = y^2 + x$</span></p>
<p>Algebraic proof: </p>
<p>Given: <span class="math-container">$x + y = 1$</span></p>
<p><span class="math-container">$$LS = x^2+ y
= (1-y)^2 + y ... | Mauro ALLEGRANZA | 108,274 | <p><em>Sorry for the description of the lacking shapes</em>...</p>
<p>You have to consider for the LHS a square of side <span class="math-container">$x$</span> and a rectangle of sides <span class="math-container">$y$</span> and <span class="math-container">$x+y=1$</span>.</p>
<p>This can be decomposed into the "big"... |
3,784,872 | <p><strong>Problem:</strong></p>
<p>Suppose <span class="math-container">$(X_n)_{n \geq 1}$</span> are indipendent random variables defined in <span class="math-container">$(\Omega, \mathscr{A},\mathbb{P})$</span>.
Define <span class="math-container">$Y=\limsup _{n \to \infty} \frac{1}{n} \sum_{1 \leq p \leq n}X_p$</sp... | QuantumSpace | 661,543 | <p><strong>Hint</strong>: Show that <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> are <span class="math-container">$\mathcal{T}:= \bigcap_{k=1}^\infty \sigma(X_k, X_{k+1, \dots})$</span>-measurable. Hence, Kolmogorov's <span class="math-container">$0$</span>-<span class="math-cont... |
875,729 | <p>Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ as its diagonal elements.</p>
<p>I can see that all eigenvalues of $A$ are real, and the corresponding eigenvectors ... | user126154 | 126,154 | <p>A matrix $Q$ is orthogonal if and only if its columns forms a orthonormal basis, if and only if $Q^{-1}=Q^T$. </p>
<p>Therefore, if there exists an orthornomal basis of eigenvectors of $A$, we have
that the matrix of change of basis if ortogonal. That is to say, there is $Q$ orthogonal so that</p>
<p>$Q^{-1}AQ=\La... |
33,817 | <p>It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.</p>
<ul>
<li>What are some of the important results leading toward proving this?</li>
<li>What are the most promising theories and approaches for this problem?</li>
</ul>
| Stefan Geschke | 7,743 | <p>People in model theory are currently studying the complex numbers with exponentiation.
Z'ilber has an axiomatisation of an exponential field (field with exponential function) that looks like the complex numbers with exp. but satisfies Schanuel's conjecture.
He proved that there is exactly one such field of the size ... |
1,634,725 | <p>It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble:</p>
<p>$\sum_{i=4}^N \left(5\right)^i$</p>
<p>Can I get some guidance on series like this? I'm finding different methods online but not sure which to use. I know that starting at a non-zero num... | 2.71828-asy | 302,548 | <p>Let $S = a + ar + ar^2 + ar^3 ...$</p>
<p>Then $S-Sr = (a + ar + ar^2 + ar^3 ... ar^n) - (ar + ar^2 + ar^3 + ar^4 ... ar^{n+1}) = a - ar^{n+1}$</p>
<p>Factoring out an S we have $S(1-r) = a-ar^{n+1}$</p>
<p>Finally, $$S = {(a - ar^{n+1})\over(1-r)}$$</p>
<p>In your case, you are trying to find $5^4 + 5^5 + 5^6 .... |
1,665,833 | <p>Given that A $\in$ M $_{mxn}$ (<strong>R</strong>). Assume that {$v_1$...$v_n$} is a basis for $R^n$ such that {$v_1$...$v_k$} is a basis for Null(A). </p>
<p>How would I prove that {A$v_{k+1}$...A$v_n$} spans Col(A)?</p>
| parsiad | 64,601 | <p>Z-tables are just values of the CDF
$$
F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-y^{2}/2}dy
$$
at various points $x$. If you want to generate a Z-table, pick some $x$ points and evaluate that integral (using a numerical method of your choice).</p>
<p>As mentioned in the comments to your question, MATLAB (and... |
1,985,402 | <p>I wrote down $$12 \times 0 = 0$$ Then, I divided both sides by $0$ like so:
$$12 = \frac {0}{0}$$ I know that $$ \frac {0}{x} = 0, x \in R$$
Therefore, $$12=0$$ which is a false statement. Where did I go wrong?</p>
| user262291 | 262,291 | <p>There are several things wrong with your proof. The first thing is when you divide by $0$ in your second line of work. Division by zero is an invalid operation. Then again you claim that $\frac{0}{x} =0$ when $x$ is a real number is also wrong. This is because in the real numbers $x$ can be zero and division by zero... |
3,892,246 | <p>I am dealing with sets of vectors <span class="math-container">$\big\{x_1, x_2, x_3, \dotsc, x_m \big\}$</span> from some abstract vector space <span class="math-container">$\mathcal{V}$</span>. <strong>Occasionally</strong>, <span class="math-container">$\mathcal{V}=\mathbb{R}^n$</span> an I need to address the ele... | postmortes | 65,078 | <p>In Banach spaces, where we often consider elements of sequence spaces (e.g. <span class="math-container">$x\in c_0$</span> so that <span class="math-container">$x=(x_1, x_2, \ldots)$</span>) the problem of indexing comes up a lot. Typically when we need to index into a sequence of sequences we use <span class="math... |
1,218,140 | <p>I am reading Hartshorne's proof of $\mathbb{P}^1$ being simply connected as a scheme. It seems one ingredient of the proof is that if $X\rightarrow\mathbb{P}^1$ is an étale covering, then X has only finitely many connected components. But I do not see why.</p>
<p>Thanks in advance.</p>
| KReiser | 21,412 | <p>The important point is that $f:X\to \mathbb{P}^1$ must be a finite map. Finite maps are quasi-finite, so the preimage of any point in $\mathbb{P}^1$ is a finite set. Since the number of points in the preimage is an upper semicontinuous function, there will be a generic multiplicity $n_0$ and finitely many points $p_... |
704,680 | <p>We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$.</p>
<p>I am to find whether a real number exists for this relation, and the real number that satisfies.</p>
<p>I start by squaring both sides, which yields: </p>
<p>$$x - 2 = 4x - 12\sqrt{x} + 9$$.</p>
<p>Whence:</p>
<p>$$ -3x = -12\sqrt{x} + 11 \\
\sqrt{x} = \frac{x}{4} ... | GEdgar | 442 | <p>There is a Borel set $E$ in $\mathbb R^2$ such that $F := \{x-y\colon (x,y) \in E\}$ is not a Borel set.</p>
<p>Let $A := \{f \in \mathbf{C}\colon (f(1), f(0)) \in E\}$. Then $A \in \mathcal{B}_{\left[0,\infty\right)}$.</p>
<p>How about $T(A)$? In fact
$$
T(A) = \{g \in \mathbf{C}\colon g(0)=0, g(1) \in F\}
$$
a... |
3,235,854 | <p>I have to calculate the gcd of <span class="math-container">$f=X^3 +9X^2 +10X +3$</span> and <span class="math-container">$g= X^2 -X -2$</span> in
<span class="math-container">$\mathbb{Q}[X]$</span> and <span class="math-container">$\mathbb{Z}/5\mathbb{Z}$</span>.</p>
<p>In <span class="math-container">$\mathbb{Q}[... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>It's easy to see that <span class="math-container">$x^2-x-2 = (x+1)(x-2)$</span>, but in <span class="math-container">$\mathbb{Z}/5\mathbb{Z}$</span> you have <span class="math-container">$5 \equiv 0$</span>, so
<span class="math-container">$$
x^3+9x^2+10x+3 \equiv x^3-x^2+3 = (x+1)\tim... |
3,235,854 | <p>I have to calculate the gcd of <span class="math-container">$f=X^3 +9X^2 +10X +3$</span> and <span class="math-container">$g= X^2 -X -2$</span> in
<span class="math-container">$\mathbb{Q}[X]$</span> and <span class="math-container">$\mathbb{Z}/5\mathbb{Z}$</span>.</p>
<p>In <span class="math-container">$\mathbb{Q}[... | mathcounterexamples.net | 187,663 | <p>You got the GCD in <span class="math-container">$\mathbb Z[X]$</span>, i.e. <span class="math-container">$X+1$</span>. The GCD in <span class="math-container">$\mathbb Z / 5 \mathbb Z[X]$</span> is obtained by reducing each coefficient of the GCD in <span class="math-container">$\mathbb Z$</span> in <span class="ma... |
3,872,750 | <p>Suppose we have a series <span class="math-container">$$\sum_{n=2}^\infty (-1)^n \frac{n^2}{10^n} = \sum_{n=2}^\infty (-1)^n b_n$$</span>.</p>
<p>I want to apply the alternating series test to see if it converges.</p>
<p>I need to show that:</p>
<p><span class="math-container">$$\lim_{n \rightarrow \infty} \frac{n^2... | user247327 | 247,327 | <p>The first equation is 3x= 4 (mod 6) which is the same as 3x= 4+ 6n or 3x- 6n= 4, for some integer, n. I have an immediate problem with that! For any x and n, 3x and 6n are divisible by 3, so 3x- 6n is a multiple of 3 but 4 is not. There is no solution.</p>
<p>The second equation is 5x= 4 (mod 6) which is the same... |
1,315,744 | <p>Already I know that harmonic series, $$\sum_{k=1}^n\frac1k $$ is divergent series.</p>
<p>And, it is also divergent by Abel Sum or Cesaro Sum.</p>
<p>However, I do not know how to prove it is divergent by concept of Abel or Cesaro.</p>
<p>Abel Sum or Cesaro Sum do not exist in this problem.</p>
<p>But, how can I... | Jack D'Aurizio | 44,121 | <p>A simple trick is to consider that:
$$\frac{1}{n}\geq\log\left(1+\frac{1}{n}\right)\tag{1}$$
holds regardless of which concept of convergence you are using, and the partial sums of the RHS of $(1)$ are fairly easy to compute by the telescopic property:
$$ \sum_{n=1}^{N}\log\left(1+\frac{1}{n}\right)=\log\prod_{n=1}^... |
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: The area of this triangle is given by $$A_1=\frac{2r\cdot r}{2}=r^2$$ and the area of the circle is $$A_2=\pi r^2$$</p>
|
2,768,187 | <blockquote>
<p>Let $w$ and $z$ be complex numbers such that $w=\frac{1}{1-z}$, and
$|z|^2=1$. Find the real part of $w$.</p>
</blockquote>
<p>The answer is $\frac{1}{2}$ but I don't know how to get to it.</p>
<hr>
<p>My attempt</p>
<p>as $|z|^2=1$</p>
<p>$z\bar z = 1$</p>
<p>If $z = x+yi$</p>
<p>$z=\frac{1... | Math Lover | 348,257 | <p>Note that $$w+w^* = \frac{1}{1-z} + \frac{1}{1-z^*}=\frac{2-z^*-z}{1-z^*-z+|z|^2}=1.$$</p>
|
132,226 | <p>After edit:</p>
<p>How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)? </p>
<p>Before edit:</p>
<p>I am not sure how much obvious or wrong is the following question:</p>
<p>For every (holomorphic) vector bun... | Piotr Achinger | 3,847 | <p>To expand my comment above:</p>
<p>This is not possible in general and Chern classes are a possible obstruction. For an easy example, take the cotangent bundle $\Omega$ of $\mathbb{P}^2$. The Euler sequence
$$ 0 \to \Omega \to \mathcal{O}(-1)^3 \to \mathcal{O} \to 0 $$
shows that $ch(\Omega) = (1-H)^3 / 1 = 1 - 3H ... |
757,049 | <p>as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. </p>
<p>Here is my work: </p>
<p>$f:[0,1] \rightarrow (0,1)$</p>
<p>Let $n\in N$</p>
<p>Let $A=\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}....\}\cup \{0\}$</p>
<p>On $[0,1]\in A: f(x)=x$</p>
<p>On $A: f(0)=\frac{1}... | Asaf Karagila | 622 | <p>Your proof has some problems.</p>
<ol>
<li>You haven't defined $A$. I suppose it should denote some subset of $[0,1]$.</li>
<li>If $A$ is a subset of $[0,1]$ then it's not the case that $[0,1]\in A$. Perhaps you meant $[0,1]\setminus A$?</li>
<li>The idea is that $A$ can be "easily described" as a sequence $a_n$ fo... |
3,013,529 | <p>Suppose that the function <span class="math-container">$f$</span> is:</p>
<p>1) Riemann integrable (not necessarily continuous) function on <span class="math-container">$\big[a,b \big]$</span>;</p>
<p>2) <span class="math-container">$\forall n \geq 0$</span> <span class="math-container">$\int_{a}^{b}{f(x) x^n} = 0... | Martin Argerami | 22,857 | <p>Because <span class="math-container">$f$</span> is a 2-norm limit of polynomials, you can deduce that <span class="math-container">$\int_a^bf(x)^2=0$</span>. </p>
<p>Now suppose that <span class="math-container">$f(x_0)\ne0$</span> for some <span class="math-container">$x_0$</span> where <span class="math-container... |
1,345,538 | <blockquote>
<p>If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? </p>
<p>Tabulate the probability distribution of $x$.</p>
</blockquote>
<p>Personally, here is my solution based on my understanding:
$$
P[X ... | mvw | 86,776 | <p>The task is to walk away from the present location such that the temperature decreases as much as possible.</p>
<p>We are at $(1,1,3)$, the temperature gradient points at $(2,2,6)$.
If we move by $dx$ and $dy$ the change in position is
$$
dr = (dx, dy, z_x dx + z_y dy) = (dx, dy, -2(dx +dy))
$$
The change in temper... |
1,345,538 | <blockquote>
<p>If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? </p>
<p>Tabulate the probability distribution of $x$.</p>
</blockquote>
<p>Personally, here is my solution based on my understanding:
$$
P[X ... | KittyL | 206,286 | <p>Here is a method continuing your idea. </p>
<p>You should use the negative of the gradient since you are looking for the direction in which the temperature decreases the fastest. Then look for the projection of that direction onto the tangent plane of the mountain. </p>
<p>The gradient direction of the temperature... |
3,464,291 | <blockquote>
<p>If <span class="math-container">$x,y,z>0.$</span> Then minimum value of</p>
<p><span class="math-container">$x^{\ln(y)-\ln(z)}+y^{\ln(z)-\ln(x)}+z^{\ln(x)-\ln(y)}$</span></p>
</blockquote>
<p>what i try</p>
<p>Let <span class="math-container">$\ln(x)=a,\ln(y)=b.\ln(z)=c$</span></p>
<p>So <span class=... | Ng Chung Tak | 299,599 | <blockquote>
<p><strong>Useful fact</strong></p>
<p><span class="math-container">$$\large x^{\log y}=y^{\log x}$$</span></p>
<p>Also refer to another answer of mine <a href="https://math.stackexchange.com/questions/1879477/logarithms-equality/1879607#1879607"><em>here</em></a>.</p>
</blockquote>
<p>Let <span class="mat... |
2,794,715 | <p>Is it right that</p>
<p><strong>$$\sqrt[a]{2^{2^n}+1}$$</strong></p>
<p>for every $$a>1,n \in \mathbb N $$ </p>
<p>is always irrational?</p>
| Wojowu | 127,263 | <p>As lhf notes, we just have to see that $2^{2^n}+1$ can never be a perfect power. However, since $2^{2^n}$ is a perfect power, using <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow noreferrer">Mihailescu's theorem</a> the only pair of perfect powers differing by $1$ is $8$ and $9$, but $2... |
2,293,162 | <p>$$f_n(x)=\begin{cases} 1-nx,&\text{for }x\in[0,1/n]\\
0 ,&\text{for }x \in [1/n,1]
\end{cases}$$ </p>
<p>Then which is correct option?</p>
<p>1.$\lim\limits_{n\to\infty }f_n(x)$ defines a continuous function on $[0,1]$. </p>
<p>2.$\lim\limits_{n\to\infty }f_n(x)$ exists for all $x\in [0,1]$. $f_n(0)=1$... | Arpan1729 | 444,208 | <p>The function converges pointwise to a discontinuous function which takes the value $1$ when $x$ is $0$ and takes the value $0$ otherwise.</p>
<p>Hence option $2$ is correct and option $1$ is false.</p>
<p>Explanation:</p>
<p>Say $1>x>0$ </p>
<p>Then let us see the sequence $f_1(x),f_2(x),f_3(x)\dots$
The t... |
3,979,371 | <p>So I have been struggling with this question for a while. Suppose <span class="math-container">$X$</span> is uniformly distributed over an interval <span class="math-container">$(a, b)$</span> and <span class="math-container">$Y$</span> is uniformly distributed over <span class="math-container">$(-\sigma, \sigma)$</... | grand_chat | 215,011 | <p>You can show that the conditional density of <span class="math-container">$X$</span> given <span class="math-container">$X+Y$</span> is uniform by applying the change of variables formula. Define the transformation <span class="math-container">$(W,Z):=(X,X+Y)$</span>. The joint density of <span class="math-container... |
1,428,143 | <p>Let $f:E\to F$ where $E$ and $F$ are metric space. We suppose $f$ continuous. I know that if $I\subset E$ is compact, then $f(I)$ is also compact. But if $J\subset F$ is compact, do we also have that $f^{-1}(J)$ is compact ?</p>
<p>If yes and if $E$ and $F$ are not necessarily compact, it still works ?</p>
| Clayton | 43,239 | <p>Not necessarily; consider $f:(-2\pi,2\pi)\to[-1,1]$ given by $f(x)=\sin(x)$.</p>
|
3,060,742 | <p><span class="math-container">$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = 1.644934$</span> or <span class="math-container">$\frac{\pi^2}{6}$</span></p>
<p>What if we take every 3rd term and add them up? </p>
<p>A = <span class="math-container">$ \frac{1}{3^2} + \fra... | Mark Viola | 218,419 | <p>Note that we have</p>
<p><span class="math-container">$$\psi'(z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}$$</span></p>
<p>where <span class="math-container">$\psi'(z)$</span> is the derivative of the <a href="https://en.wikipedia.org/wiki/Digamma_function" rel="nofollow noreferrer">digamma function</a>. Hence, we can ... |
603,986 | <p>Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$</p>
<p>Any ideas how to prove it?</p>
| Dylan Yott | 62,865 | <p>Let $p(x)=1$, you win. Here's a less stupid example with the same flavor. If $|F|=q$, then consider $x^q-x+1$. Can you figure out why you still win?</p>
|
1,237,077 | <p>For a periodic function we have: $$\int_{b}^{b+a}f(t)dt = \int_{b}^{na}f(t)dt+\int_{na}^{b+a}f(t)dt = \int_{b+a}^{(n+1)a}f(t)dt+\int_{an}^{b+a}f(t)dt = \int_{na}^{(n+1)a}f(t)dt = \int_{0}^{a}f(t)dt.$$ , but I don't understand how we obtain $\int _{b+a}^{\left(n+1\right)a}\:f\left(t\right)\:dt=\int _b^{na}\:f\left(t\... | Prasun Biswas | 215,900 | <p>What does it mean for a function to have a period <span class="math-container">$a$</span> ? Informally, it means that the function values gets repeated after an increment of <span class="math-container">$a$</span> in the <span class="math-container">$x-$</span>value. (domain value). Notationally,</p>
<p><span class=... |
8,997 | <p>I have a set of data points in two columns in a spreadsheet (OpenOffice Calc):</p>
<p><img src="https://i.stack.imgur.com/IPNz9.png" alt="enter image description here"></p>
<p>I would like to get these into <em>Mathematica</em> in this format:</p>
<pre><code>data = {{1, 3.3}, {2, 5.6}, {3, 7.1}, {4, 11.4}, {5, 14... | pyler | 2,338 | <p>I don't have 50 reputation points so I'm submitting this as an answer when it is really a comment, a small addition. After doing the first approach given by WReach, Wrap the result of your evaluation between theses two brackets --><strong>"<code>Grid[]</code>"</strong> and add <strong>"<code>Grid[ ,Frame->All]</... |
2,277,115 | <p>I'm asking for examples of interesting categories in which there exist non-isomorphic objects $X$ and $Y$, a split monomorphism $f : X \to Y$, and a split epimorphism $g : X \to Y$. Spelled out, there should exist maps $f : X \leftrightarrow Y : f'$ such that $f'f = \mathrm{id}_{X}$ and maps $g : X \leftrightarrow Y... | HeinrichD | 369,530 | <p>The situation can also be described by two non-isomorphic objects $X,Y$ which admit split monomorphisms $X \to Y$ and $Y \to X$ in both directions.</p>
<p>In fact, this happens in the category of abelian groups. See <a href="https://mathoverflow.net/questions/218113">here</a> and <a href="https://mathoverflow.net/q... |
921,893 | <p>If we have: $f(x)=\frac { 1+x }{ 1+{ e }^{ x } } $</p>
<p>I am told to determine if $f(x)=x$ has multiple roots on $\left[ 0;+\infty \right] $</p>
<p>I tried to manually solve this equation, but I don't understand the result:</p>
<p>$f(x)-x=0\rightarrow \frac { 1+x }{ 1+{ e }^{ x } } =0\rightarrow { e }^{ x }(x+... | Jack D'Aurizio | 44,121 | <p>If $f(x)=x$, then $f(x)-x=0$, or:
$$\frac{1-xe^x}{1+e^x}=0.\tag{1}$$
Since $1+e^x$ is always positive, $(1)$ is equivalent to:
$$ g(x)=xe^x = 1.\tag{2}$$
$g(x)$ is a decreasing and negative function over $(-\infty,-1)$, an increasing function over $(-1,+\infty)$, since $g'(x) = (x+1)e^x$. So there is at most one rea... |
2,409,183 | <p>Good evening all! I'm trying to find the eigenvalues and eigenvectors of the following problem</p>
<p>$$
\begin{bmatrix}
-10 & 8\\
-18 & 14\\
\end{bmatrix}*\begin{bmatrix}
x_{1}\\
x_{2}\\
\end{bmatrix}
$$</p>
<p>I've found that $λ_{1},_{2}=2$ where $... | Giuseppe Negro | 8,157 | <p>The first integral always vanishes. To see this, write
$$I=\int_{\mathbb S^2} r_i r_j r_k\, d\Omega.$$
Suppose that $r_i$ appears with power $1$ or $3$ (if it appears with power $2$, then choose the other one). Then the change of variable $r_i\mapsto -r_i, r_j\mapsto r_j$ for $j\ne i$ leaves $d\Omega$ invariant and... |
48,989 | <p>How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?</p>
| Dave Au | 197,331 | <p>Since we have $\text{Col }AB \subseteq \text{Col }A$ and $\text{Row }AB \subseteq \text{Row }B$, therefore $\text{Rank }AB \leq \text{Rank }A$ and $\text{Rank }AB \leq \text{Rank }B$, then the result follows.</p>
|
1,586,286 | <p>There are $30$ red balls and $50$ white balls. Sam and Jane take turns drawing balls until they have drawn them all. Sam goes first. Let $N$ be the number of times Jane draws the same color ball as Sam. Find $E[N].$</p>
<p>I have been proceeding with indicators...</p>
<p>$$ I_{j} =
\begin{cases}
1, & \text{i... | JMoravitz | 179,297 | <p>As you started, let $I_n=\begin{cases}1&\text{if Jane draws the same color on turn}~2n~\text{as Sam did on turn}~2n-1\\
0&\text{otherwise}\end{cases}$</p>
<p>We have $E[N]=E[\sum\limits_{n=1}^{40}I_n]$ which by linearity of expectation is $=\sum\limits_{n=1}^{40}E[I_n]$ and by symmetry is $40E[I_1]$</p>
<p... |
2,227,027 | <p>If $f(x)$ is defined everywhere except at $x=x_0$, would $f'(x_0)$ be undefined at $x=x_0$ as well?</p>
<p>One example is: $$f(x)=\ln(x)\rightarrow f'(x)=\frac{1}{x}$$</p>
<p>In this particular case, both $f(x)$ and $f'(x)$ are undefined at $x=0$. I wonder if this always holds true.</p>
<p>Thank you.</p>
| Alex Peter | 579,318 | <p>In order to have a first derivative at a point, a function must be continuous at that point. </p>
<p>Differentiation requires continuity. A function that is not defined at some point cannot be continuous at that point because it does not exist at that point.</p>
<p>If the first derivative <span class="math-contain... |
4,462,081 | <p>I actually already have the solution to the following expression, yet it takes a long time for me to decipher the first operation provided in the answer. I understand all of the following except how to convert <span class="math-container">$\left(1+e^{i\theta \ }\right)^n=\left(e^{\frac{i\theta }{2}}\left(e^{\frac{-i... | Átila Correia | 953,679 | <p>You can alternatively proceed as follows:
<span class="math-container">\begin{align*}
1 + \cos(x) + i\sin(x) & = 2\cos^{2}(x/2) +2i\sin(x/2)\cos(x/2)\\\\
& = 2\cos(x/2)(\cos(x/2) + i\sin(x/2))
\end{align*}</span></p>
<p>From this identity it results the desired claim:
<span class="math-container">\begin{alig... |
1,861,890 | <p>Given that $$s_{n}=\frac{(-1)^{n}}{n},$$ I want to show $$\lim_{n\to\infty}{s_{n}}=0$$ in the metric space $X=\mathbb{C}.$ However, it seems to me that <strong>Archimedean Property is not applicable to the case above</strong>, because $s_{n}$ is not always positive for each $n$. Then, how can I do that?</p>
| Cbjork | 137,229 | <p>Let $\varepsilon>0$ and $n>\frac{1}{\varepsilon}$. Then $\left | \dfrac{(-1)^n}{n}-0\right |=\left | \dfrac{(-1)^n}{n}\right |=\left | \dfrac{1}{n}\right |<\varepsilon$</p>
|
2,916,158 | <p>I am trying to understand why we need Large deviation theory/principle.</p>
<p><strong>Here is what I understand so far</strong> based on the <a href="https://en.wikipedia.org/wiki/Large_deviations_theory#An_elementary_example" rel="noreferrer">Wikipedia</a>.
Let $S_n$ be a random variable which depends on $n$.
We... | Alex R. | 22,064 | <p>For 1: "easily obtain" is a misnomer here. As an example you can get estimates for $P(S_n>x)$ very easily just by appealing to the CLT. The problem is that the CLT <em>does not hold</em> when you are too far from the mean (as a function of $n$). So you can "easily obtain" that $P(S_n>x)$ is close t0 0, just li... |
947,730 | <p>I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it.</p>
<p>Thanks :)</p>
<p>Find a polynomial that passes through the points (-2,-1), (-1,7), (2,-5), (3,-1). Present the answer in standard form.</p>
<p>What I've tried:</p>
<p><img src="https://i.sta... | orangeskid | 168,051 | <p>$$f(x) = a x^3 + b x^2 + c x + d$$
\begin{eqnarray*}
-1 &=& a (-2)^3 + b(-2)^2 + c(-2) + d \\
7 &=& a (-1)^3 + b(-1)^2 + c(-1) + d \\
-5 &=& a(2)^3 + b (2)^2 + c (2) + d \\
-1 &=& a(3)^3 + b(3)^2 + c (3) + d
\end{eqnarray*}
or
\begin{eqnarray*}
-8 a + 4b -2c + d &=& -1\\... |
3,260,530 | <p>I read from wikipedia that a neighbourhood of a point <span class="math-container">$p$</span> is a subset <span class="math-container">$V$</span> of a topological space <span class="math-container">$\{X,\tau\}$</span> that includes an open set <span class="math-container">$U$</span> such that <span class="math-conta... | cqfd | 588,038 | <p><span class="math-container">$V=\{U,W\} $</span> is <strong>not</strong> a subset of <span class="math-container">$X $</span>, it is a subset of the topology <span class="math-container">$\mathcal T $</span>(assuming <span class="math-container">$U $</span> and <span class="math-container">$W $</span> are open; o... |
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | Vidyanshu Mishra | 363,566 | <p>$(\sqrt{5}-\sqrt{1})^2= 6-\sqrt{20}$</p>
|
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | Henry | 6,460 | <p>$$a=\sqrt{6-\sqrt{20}}$$</p>
<p>$$\implies a^2=6-\sqrt{20}$$</p>
<p>$$\implies(a^2-6)^2=20$$</p>
<p>$$\implies a^4-12a^2+16=0$$</p>
<p>$$\implies (a^2-2a-4)(a^2+2a-4)=0$$</p>
<p>so it will be one of the four possibilities $\pm\sqrt{5}\pm1$, and since $\sqrt{6-\sqrt{25}} \lt \sqrt{6-\sqrt{20}} \lt \sqrt{6-\sqrt{... |
4,433,724 | <p>I need to prove the following useful statement:
If <span class="math-container">$f(x,y)$</span> is differentiable at <span class="math-container">$(x_0,y_0)$</span>, then in the neighbourhood of <span class="math-container">$(x_0,y_0)$</span>, we have
<span class="math-container">$$
\Delta z = f(x_0+\Delta x,y_0+\De... | Bram28 | 256,001 | <p>I am not going to answer the questions that were given to you, but I will criticize your answer ... and in particular your reasoning:</p>
<blockquote>
<p><em><strong>What I think:</strong></em> I think that without any assumptions we can't use NMP at all, therefore we can tell that 1, 2, 4 are false and only 3 is tr... |
4,433,724 | <p>I need to prove the following useful statement:
If <span class="math-container">$f(x,y)$</span> is differentiable at <span class="math-container">$(x_0,y_0)$</span>, then in the neighbourhood of <span class="math-container">$(x_0,y_0)$</span>, we have
<span class="math-container">$$
\Delta z = f(x_0+\Delta x,y_0+\De... | Mohamad S. | 1,013,510 | <p>Thanks to 'ancient mathematician' and 'Bram28' I got my answer to the question and here I post it:</p>
<p><strong>1. The claim is false.</strong></p>
<p>In 3 we prove that the system is sound therefore if <span class="math-container">$\gamma\rightarrow(\alpha\rightarrow\beta)\vdash_N(\alpha\rightarrow\gamma)$</span>... |
339,880 | <p>I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples:</p>
<ul>
<li><strong>Cantor set</strong>: Let <span class="math-container">$C$</span> denote the ternary Cantor set on the interval <span class="math-container">$[0,1]$</span>. The... | Francesco Polizzi | 7,460 | <p>Every real number is the sum of two Liouville numbers, see </p>
<p>P. Erdős: <a href="http://dx.doi.org/10.1307/mmj/1028998621" rel="noreferrer"><em>Representations of real numbers as sums and products of Liouville numbers</em></a>, Mich. Math. J. <strong>9</strong>, 59-60 (1962). <a href="https://zbmath.org/?q=an:... |
339,880 | <p>I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples:</p>
<ul>
<li><strong>Cantor set</strong>: Let <span class="math-container">$C$</span> denote the ternary Cantor set on the interval <span class="math-container">$[0,1]$</span>. The... | Gerry Myerson | 3,684 | <p>Every real number is the sum of two numbers whose continued fraction expansion has no partial quotient exceeding <span class="math-container">$4$</span>. Marshall Hall, Jr., On the sum and product of continued fractions, Annals of Mathematics, Second Series, Vol. 48, No. 4 (Oct., 1947), pp. 966-993, DOI: 10.2307/196... |
3,375,375 | <p>I noticed this issue was throwing off a more sophisticated problem I'm working on. When computing the indefinite integral </p>
<p><span class="math-container">$$ I(x) = \int \frac{dx}{1-x} = \log | 1-x | + C,$$</span></p>
<p>I realized I could equivalently write</p>
<p><span class="math-container">$$ I(x) = - \... | Kavi Rama Murthy | 142,385 | <p>The first one is wrong . You have missed a minus sign. </p>
<p>Also <span class="math-container">$\log (\frac 1 c)=\log 1 -\log c=-\log c$</span>. </p>
|
1,261,504 | <p>I am trying to proof $ab = \gcd(a,b)\mathrm{lcm}(a,b)$.</p>
<p>The definition of $\mathrm{lcm}(a,b)$ is as follows:</p>
<p>$t$ is the lowest common multiple of $a$ and $b$ if it satisfies the following:</p>
<p>i) $a | t$ and $b | t$ </p>
<p>ii) If $a | c$ and $b | c$, then $t | c$.</p>
<p>Similiarly for the $\g... | 3x89g2 | 90,914 | <p>$P(Z\le z)=\int_{-\infty}^z \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\,dz=0.95$</p>
<p>Or if probability density function has not been covered, just consider the picture of a normal distribution. We know that $Pr(Z\le z)=0.5$ when $z=0$ since $0$ is in the middle. Now, move $z$ to the right, the probability will bec... |
2,631,284 | <p>I'm trying to find all $n \in \mathbb{N}$ such that</p>
<p>$(n+2) \mid (n^2+5)$ </p>
<p>as the title says, I've tried numbers up to $20$ and found that $1, 7$ are solutions and I suspect that those are the only $2$ solutions, however I have no idea how to show that.</p>
<p>I've done nothing but basic transformati... | Roman83 | 309,360 | <p>$$(n+2)|(n^2-4)$$
Then $$n+2|(n^2+5)-9$$
Then $$n+2|9$$</p>
|
850,390 | <p>Let $f(x)$ be differentiable function from $\mathbb R$ to $\mathbb R$, If $f(x)$ is even, then $f'(0)=0$. Is it always true?</p>
| Brandon | 113,565 | <p>Hint: If a function $f$ is diffentiable at $x$, then
$$
f'(x)=\lim\limits_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}
$$</p>
|
850,390 | <p>Let $f(x)$ be differentiable function from $\mathbb R$ to $\mathbb R$, If $f(x)$ is even, then $f'(0)=0$. Is it always true?</p>
| gnasher729 | 137,175 | <p>Let f (x) = square root of absolute value of x. Is it an even function? Is it continuous? What is f' (0)? </p>
|
2,766,879 | <p>Show that there are no primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$. </p>
<p>I once have proven that, for all integers $a,b$, we have that $a^2 + b^2$ is congruent to $0$, or $1$, or $2$ modulo $4$. I feel like it is enough to conclude it by considering $a=x$, $b=y$ and $\gcd(x,y)=1$. But I am n... | user | 505,767 | <p>Yes it is correct since for $x>1$</p>
<p>$$f(x)=\frac{x^2+x+2}{x-1}\ge 7 \iff x^2-6x+9=(x-3)^2\ge 0$$</p>
<p>and for $x<1$</p>
<p>$$f(x)=\frac{x^2+x+2}{x-1}\le -1 \iff x^2+2x+1=(x+1)^2\ge 0$$</p>
<p>then for $x>1$ and $y=\frac1x<1$</p>
<p>$$f(x)-f\left(\frac1x\right)=f(x)-f(y)\ge 7+1=8$$</p>
|
4,567,390 | <p>Let</p>
<ul>
<li><span class="math-container">$X$</span> be a metric space,</li>
<li><span class="math-container">$\mathcal C_b(X)$</span> the space of real-valued bounded continuous functions,</li>
<li><span class="math-container">$\mathcal C_0(X)$</span> the space of real-valued continuous functions that vanish at... | Analyst | 1,019,043 | <p>As mentioned by @OliverDíaz, the fact that <span class="math-container">$f^{-1}$</span> is a homeomorphism (in weak<span class="math-container">$^*$</span> topology of <span class="math-container">$\mathcal M(X)$</span>) from <span class="math-container">$f(X)$</span> onto <span class="math-container">$X$</span> is ... |
4,048,785 | <p>Show that <span class="math-container">$2r^2-3$</span> is never a square, <span class="math-container">$r=2,3,...$</span></p>
<p>I know that no perfect square can have <span class="math-container">$2, 3, 7$</span>, or <span class="math-container">$8$</span> as its last digit. I'm not sure how to do this with congrue... | José Carlos Santos | 446,262 | <p>Each perfect square is congruent to <span class="math-container">$0$</span>, <span class="math-container">$1$</span> or <span class="math-container">$4$</span> modulo <span class="math-container">$8$</span>. But each number of the form <span class="math-container">$2n^2-3$</span> is congruent to <span class="math-co... |
4,427,651 | <p>In programming, we define an "array" (basically an ordered n-tuple) in the following way:</p>
<p><span class="math-container">$$a=[3,5].$$</span></p>
<p>Later on, if we want to refer to the first element of the predetermined array/pair/n-tuple, we write <span class="math-container">$a[0]$</span> (because i... | Georges Elencwajg | 3,217 | <p>A standard notation (used in Dieudonné, Foundations of Modern Analysis, page 12; Bourbaki, General Topology, Chapter 1, §4.1, page 44.) is <span class="math-container">$\operatorname{pr_i}(a_1,\cdots,a_n)=a_i$</span> since, given a set <span class="math-container">$A$</span>, the map <span class="math-container">$$A... |
4,427,651 | <p>In programming, we define an "array" (basically an ordered n-tuple) in the following way:</p>
<p><span class="math-container">$$a=[3,5].$$</span></p>
<p>Later on, if we want to refer to the first element of the predetermined array/pair/n-tuple, we write <span class="math-container">$a[0]$</span> (because i... | Nick Matteo | 59,435 | <p>It's common to assume that the elements of a tuple or vector named <span class="math-container">$a$</span> are indexed <span class="math-container">$a = \langle a_1, a_2, \dots \rangle$</span>, so you would just refer to <span class="math-container">$a_1$</span> for the first element.</p>
<p>Similarly, the elements ... |
512,591 | <p>It is always confusing to prove with $\not\equiv$. Should I try contrapositive?</p>
| BaronVT | 39,526 | <p>Well, in this case the contrapositive is "if $a^2$ is $\textit{not}$ congruent to $1$ mod $3$ then ..." which will cause you the same kind of problem.</p>
<p>In this case, you might just try working directly with the hypothesis. If $a$ is not congruent to $0$, then it has to be congruent to either $1$ or $2$, right... |
609,845 | <blockquote>
<p>$5$ Integers are paired in all possible ways and each pair of integers
is added. The $10$ sums obtained are $1,4,7,5,8,9,11,14,15,10$. What are the
$5$ integers?</p>
</blockquote>
<p>This is what I got so far:</p>
<p>To get all possible pairs, each integer must be paired with the other $4$ integ... | David Holden | 79,543 | <p>1well you have in one sense an overdetermined problem, since there are ten equations in five unknowns.</p>
<p>however, you are unsure which equation gives which result.
a sensible procedure would be to begin by ordering the unknowns:
$$
x_1 \le x_2 \le x_3 \le x_4 \le x_5
$$</p>
<p>however you know also that (by a... |
3,408,458 | <p>i have 4 vectors:</p>
<ul>
<li><span class="math-container">$|\vec{AC}|=|\vec{AD}|$</span></li>
<li><span class="math-container">$|\vec{BC}|=|\vec{BE}|$</span></li>
</ul>
<p><span class="math-container">$\angle (\vec{AC}, \vec{AD}) $</span> =<span class="math-container">$\angle (\vec{BC}, \vec{BE}) $</span></p>
<p><... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$\vec {DE} =\vec {DA}+\vec {AB}+\vec {BE} =-\vec {AD}+\vec {AB}+\vec {BE}$$</span></p>
|
66,000 | <p>In 2008 I wrote a group theory package. I've recently started using it again, and I found that one (at least) of my functions is broken in Mathematica 10. The problem is complicated to describe, but the essence of it occurs in this line:</p>
<pre><code>l = Split[l, Union[#1] == Union[#2] &]
</code></pre>
<p>He... | Szabolcs | 12 | <p>This is not a full answer, just a start towards a solution.</p>
<p>The culprit is <code>Dispatch</code>, which became <a href="http://reference.wolfram.com/mathematica/ref/AtomQ.html" rel="noreferrer">atomic</a> in version 10, and comparison wasn't implemented for it.</p>
<p>Here's a small test in version 9:</p>
<pr... |
1,356,783 | <p>What kind of mathematical object is this substitution(is it a function or what). We assuming set of variables exist.</p>
| Gary. | 235,023 | <p>If I understood correctly, say you have a wff on n variables. Then substituting in a variable, you map into the collection of wffs on $(n-1)$ variables. If you substitute for all n variables, or if $n=1$, you map into a world, or interpretation for the wff. But there may not be possible worlds where the wff holds, e... |
719,055 | <p>I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends on the choice of homeomorphism up to, I think, isotopy ( Please forgive the rambling; I'm trying to put together a l... | Tony | 39,296 | <p>The notation $\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx}$ is valid. However, you cannot prove the chain rule just by "cancelling" the two $du$'s; it doesn't work that way.</p>
|
10,601 | <p>It sometimes happens that the same user posts <strong>exactly</strong> the same question twice in a row.</p>
<p>Examples: </p>
<ul>
<li><a href="https://math.stackexchange.com/questions/446622/drawing-at-least-90-of-colors-from-urn-with-large-populations">1</a> <a href="https://math.stackexchange.com/questions/446... | rurouniwallace | 35,878 | <p>In these situations, if there's no answers yet, I usually handle it by doing two things: voting to close as a duplicate, and posting a link to the original question as a comment. I usually delete the automatic comment that comes up when you vote to close as a duplicate and replace it with something like:</p>
<blockq... |
697 | <p><a href="https://mathoverflow.net/questions/36307/why-cant-i-post-a-question-on-math-stackexchange-com">This question</a> was posted on MO about not being able to post on math.SE. While MO wasn't the right place for the question, I have to wonder what is. New users who are experiencing difficulty using math.SE can... | kennytm | 171 | <p>The contact info is actually written in <a href="https://math.stackexchange.com/about">"about"</a></p>
<blockquote>
<h2>How can I learn more?</h2>
<p>Check out the <a href="https://math.stackexchange.com/faq">FAQ</a>. And if you need to contact us, you can do so at <strong>team@stackexchange.com</strong>.</p... |
340,575 | <p>I got my exam on Thursday, and just got a few questions left. Anyway I would aprreciate help a lot! Can anyone please help me to solve this task? You can see the picture below. The need is to finde the size of the two radius. I thought about working with cords, like the cord AC is the same size like another one. Sti... | Inceptio | 63,477 | <p>$\triangle MAC$ and $\triangle MBC$ are isosceles.
<img src="https://i.stack.imgur.com/CfWal.png" alt="enter image description here"></p>
<p>Construct a tangent $CD$ which is common at $C$. </p>
<p>Now $\angle DCM= \angle M_2BD=90^0$, which means $BDCM_2$ is a <strong>cyclic</strong> Quadrilateral.</p>
<p>Similar... |
26,823 | <p>Trying to solve for the area enclosed by $x^4+y^4=1$. A friend posed this question to me today, but I have no clue what to do to solve this. Keep in mind, we don't even know if there is a straightforward solution. I think he just likes thinking up problems out of thin air. </p>
<p>Anyway, the question becomes m... | Wadim Zudilin | 4,953 | <p>I always prefer not to skip $dx$:
$$
I_n=\int_0^1(1-x^n)^{1/n}dx.
$$
After the change of variable $t=x^n$, the integral becomes the beta integral,
$$
I_n=\frac1n\int_0^1(1-t)^{1/n}t^{1/n-1}dt
=\frac1n\frac{\Gamma(1+1/n)\Gamma(1/n)}{\Gamma(1+2/n)}
=\frac1n\frac{\Gamma(1/n)^2\cdot 1/n}{\Gamma(2/n)\cdot 2/n}
\to1 \quad... |
86,067 | <p>So I am having an issue using <code>NDSolve</code> and plotting the function. So I have two different <code>NDSolve</code> calls in my plotting function. (They are technically the same, just have different names; but that can be changed back if at all possible because I want them to be the same.) But the second one ... | kglr | 125 | <pre><code>ClearAll[ticksF, axesF, labelF]
ticksF[tSide_: Left, tr_: 1, tl_: (.01), s_: {Thickness[.001]}][{minmax__}, nd_:{6, 6}] :=
Module[{tf = tSide /. {Automatic | Left -> Identity, Right -> ({-1, 1} # &),
Bottom -> ((Reverse@#) &)},
d = {#, Complement[Join @@ #2, #]} & @@ FindDivisions... |
2,909,244 | <p>I have a homework, about calculate the limit of a series:
$$
\lim\limits_{n \to +\infty} \dfrac{\sqrt[n] {n^3} + \sqrt[n] {7}}{3\sqrt[n]{n^2} + \sqrt[n]{3n}}
$$
Solution is $\frac{1}{2}$. I am trying use the unequality:
$$
\dfrac{\sqrt[n] {n^3} }{3\sqrt[n]{n^2} + \sqrt[n]{3n}} \le \dfrac{\sqrt[n] {n^3} + \sqrt[n] {7... | Daquisu | 591,084 | <p>I'd calculate the individual limits then put them together.</p>
<p>$\lim\limits_{n \to +\infty} \sqrt[n] {n^3} = \lim\limits_{n \to +\infty} \exp(ln(\sqrt[n] {n^3})) = \lim\limits_{n \to +\infty} \exp(ln(({n^3})^\dfrac{1}{n})) $</p>
<p>Appyling log properties: $log(A^n) = n*log(A)$</p>
<p>$\lim\limits_{n \to +\in... |
4,331,790 | <blockquote>
<p><strong>Question 23:</strong> Which one of following statements holds true if and only if <span class="math-container">$n$</span> is a prime number?
<span class="math-container">$$
\begin{alignat}{2}
&\text{(A)} &\quad n|(n-1)!+1 \\
&\text{(B)} &\quad n|(n-1)!-1 \\
&\text{(C)} &... | Claude Leibovici | 82,404 | <p>May be a few identities could help.
<span class="math-container">$$\sum_{k=0}^n \frac{(-1)^k}{k!}\binom{n}{k}\,x^k=\, _1F_1(-n;1;x)=L_n(x)$$</span> (<span class="math-container">$L_n(x)$</span> being Laguerre polynomial) and
<span class="math-container">$\lim_{n\to \infty } \, L_n(1) =0$</span></p>
<p>On the other s... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | RunnyKine | 5,709 | <p>Here is a functional approach:</p>
<pre><code>Narciss[x_] := With[{num = IntegerDigits[x]}, Total[num^Length[num]] == x]
</code></pre>
<p>Here is a compiled version of the above function:</p>
<pre><code>NarcissC = Compile[{{x, _Integer}},
With[{num = IntegerDigits[x]}, Total[num^Length[num]] == x],
Parall... |
33,582 | <p>My code finding <a href="http://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic numbers</a> is not that slow, but it's not in functional style and lacks flexibility: if $n \neq 7$, I have to rewrite my code. Could you give some good advice?</p>
<pre><code>nar = Compile[{$},
Do[
With[{
n = 1000... | Carl Woll | 45,431 | <p>It should be much faster to generate all possible integer digit sets, and then select those integer digit sets that have the require property. For instance, $135, 153, 315, 351, 513, 531$ all have the integer digits $1, 3, 5$, but the sum of the cubes of all 6 digit sets is the same, namely, $153$. The set of all po... |
626,958 | <p>I know that $E[X|Y]=E[X]$ if $X$ is independent of $Y$. I recently was made aware that it is true if only $\text{Cov}(X,Y)=0$. Would someone kindly either give a hint if it's easy, show me a reference or even a full proof if it's short? Either will work I think :) </p>
<p>Thanks.</p>
<p>Edit: Thanks for the great... | leonbloy | 312 | <p>Your assertion is false.</p>
<p>For two random variables <span class="math-container">$X$</span> <span class="math-container">$Y$</span> one can consider three measures of un-related-ness:</p>
<p><span class="math-container">$1$</span>: Independence: <span class="math-container">$p(X,Y) = p(X)p(Y)$</span> Equival... |
2,342,537 | <p>Suppose $f:\mathbb R\rightarrow \mathbb R$ s.t. $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb R$ and $f$ is not continuous on $\mathbb R$. Prove that</p>
<p>(a).$f$ is not bounded below (or above) on any subinterval $(a,b)$ of $\mathbb R$.</p>
<p>(b). $f$ is not monotone.</p>
<p>On plugging $x=y=0$; $f(0)=f(0)+f(0)$... | Fimpellizzeri | 173,410 | <p>Here's a route:</p>
<p>$\qquad(1)$: Show that $f$ is $\mathbb{Q}$-linear. In particular, show that $f(q\cdot x)=q\cdot f(x)$ for all $q\in\mathbb{Q}$.</p>
<p>$\qquad(2)$: Using $(1)$, show that if $f$ is bounded on any interval $[a,b]$ (why can you assume it is closed?), then $f$ is continuous at $x=0$.</p>
<p>$\... |
376,600 | <p>$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$</p>
<p>Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?</p>
| xpaul | 66,420 | <p>Use integration by substitution. Let $x=\tan\theta$. Then
$$ \int_{-\infty}^\infty\frac{1}{(1+x^2)^n}dx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{2(n-1)}\theta d\theta. $$
Now it is not hard to verify
$$\lim_{n\to\infty}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{2(n-1)}\theta d\theta=0.$$</p>
|
1,610,055 | <p>I feel rather silly for having to ask this question in specific and am by no means looking for a flat out step by step answer. I understand the definition for the euclidean norm in an n-dimensional space (as defined <a href="https://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm" rel="nofollow">here</a>). I... | Paolo Franchi | 302,637 | <p>Apply the triangle inequality $\|a+b\| \leq \|a\| + \|b\|$, with $a=x-z$ and $b=z-y$.</p>
<p>$$ \|x-y\| = \| x -z + z - y \| = \| (x -z) + (z - y) \| \leq \| (x -z) \| + \|(z - y) \| < 2+3=5 \\
\implies \|x-y\| < 5.$$</p>
|
2,146,911 | <blockquote>
<p>Given natural numbers <span class="math-container">$m,n,$</span> and a real number <span class="math-container">$a>1$</span>, prove the inequality :</p>
<p><span class="math-container">$$\displaystyle a^{\frac{2n}{m}} - 1 \geq n\big(a^{\frac{n+1}m} - a^{\frac{n-1}{m}}\big)$$</span></p>
<p><strong>SOU... | S.C.B. | 310,930 | <p>The answer can be given via simple calculus, and thus the result can be shown to hold true for all $x$. However, the OP has stated that he/she would prefer a solution that did not resort to calculus. So here is my edited answer. For my original answer, please check the edit history. </p>
<p>Let $a^{\frac{1}{m}}=x&g... |
2,436,167 | <p>I appear to be misunderstanding a basic probability concept. The question is: you flip four coins. At least two are tails. What is the probability that exactly three are tails? </p>
<p>I know the answer isn't 1/2, but I don't know why that's so. Isn't the probability of just getting 1 tail in the remaining two coin... | B. Goddard | 362,009 | <p>Count like this: When you flip 4 coins, there are 16 possible outcomes. List them and cross off all the cases which do not have at least two tails. That leaves 11 possibilites. Of the 11, how many have exactly 3 tails? $4$. So the answer is $4/11.$</p>
|
1,236,600 | <p>A dose of $D$ milligrams of a drug is taken every 12 hours. Assume that the drug's half-life is such that every $12$ hours a fraction $r$, with $0<r<1$ of the drug remains in the blood. Let $d_1= D$ be the amount of the drug in the blood after first dose.
It follows that the amount of the drug in the blood aft... | Ross Millikan | 1,827 | <p>Hint: If in steady state you have $d_\infty$ just after a dose, after $12$ hours you have $rd_\infty=d_{min}$ because it is just before a dose.</p>
|
1,236,600 | <p>A dose of $D$ milligrams of a drug is taken every 12 hours. Assume that the drug's half-life is such that every $12$ hours a fraction $r$, with $0<r<1$ of the drug remains in the blood. Let $d_1= D$ be the amount of the drug in the blood after first dose.
It follows that the amount of the drug in the blood aft... | Math1000 | 38,584 | <p>Let $d(t)$ be the amount of the drug in the blood $t$ hours after a steady-state dose for $0\leqslant t<12$. Then $d(t)=d_\infty e^{-\lambda t}$ for some $\lambda>0$ (this is the definition of exponential decay). Then
$$d_{\min} = \lim_{t\uparrow12}d(t)=\lim_{t\uparrow12}d_\infty e^{-\lambda t}=d_\infty e^{-12... |
2,307,021 | <p>I am struggling with a confusing differentials' problem. It seems like there is a key piece of information missing:</p>
<p><strong>The problem:</strong></p>
<blockquote>
<p>The electrical resistance $ R $ of a copper wire is given by $ R = \frac{k}{r^2} $ where $ k $ is a constant and $ r $ is the radius of the... | Ross Millikan | 1,827 | <p>When we say the radius has an error of $5\%$, we mean that as a relative error, so $dr=0.05r$</p>
|
2,307,021 | <p>I am struggling with a confusing differentials' problem. It seems like there is a key piece of information missing:</p>
<p><strong>The problem:</strong></p>
<blockquote>
<p>The electrical resistance $ R $ of a copper wire is given by $ R = \frac{k}{r^2} $ where $ k $ is a constant and $ r $ is the radius of the... | mrnovice | 416,020 | <p>Just doing this by brute force:</p>
<p>$$R=\frac{k}{r^2}\quad\text{percentage error in $r$ is $\pm 5$ percent}$$</p>
<p>So $R_{min} =\frac{k}{(r+0.05r)^2} =\frac{k}{1.05^2r^2}=\frac{400}{441}\cdot\frac{k}{r^2}$</p>
<p>$R_{max} = \frac{k}{(0.95r)^2} =\frac{400}{361}\cdot \frac{k}{r^2}$</p>
<p>Then the percentage ... |
2,025,934 | <p>May $V$ be an $n$ dimensional Vektorspace such that $\dim (V) =: n \ge 2$.</p>
<p>We shall prove, that there are infinitely many $k$-dimensional subspaces of $V$, $\forall k \in \{1, 2, ..., n-1\}$.</p>
<p>So first, I thought about using induction, the base step is not that hard, for $n=2$ we take two vectors, say... | Learnmore | 294,365 | <p><strong>HINT</strong>: How many lines are there in $\Bbb R^2$ passing through $\{(0,0)\}$.</p>
<p>How many planes are there in $\Bbb R^3$ containing $\{(0,0,0)\}$.</p>
|
1,171,150 | <p>I am struggling to figure out $$\lim\limits_{n \to \infty} \sqrt[n]{n^2+1} .$$ I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find the limit.</p>
| Ivo Terek | 118,056 | <p><strong>Hint:</strong> $$\sqrt[n]{n^2+1} = (n^2+1)^{1/n} = e^{\frac{\ln(n^2+1)}{n}}.$$</p>
|
1,171,150 | <p>I am struggling to figure out $$\lim\limits_{n \to \infty} \sqrt[n]{n^2+1} .$$ I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find the limit.</p>
| kobe | 190,421 | <p>Let $a_n = \sqrt[n]{n^2 + 1}$. Then $$n^{2/n} < a_n < (n + 1)^{2/n}.$$</p>
<p>Since $\lim_{n\to \infty} n^{1/n} = 1$ and $\lim_{n\to \infty} (n + 1)^{1/n} = 1$, it follows that the left- and right-most sides of the above inequality tend to $1$ as $n\to \infty$. Therefore, by the squeeze theorem, $\lim_{n\to \... |
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | Jack D'Aurizio | 44,121 | <p>A fun overkill: it is well known (at least among Ramanujan supporters) that for any $x>1$ we have
$$ \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} = \tfrac{1}{2}+\sqrt{x+\tfrac{1}{4}} $$
hence $\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$ is bounded between $1$ and $\frac{\sqrt{x}}{\sqrt{x+\frac{1}{4}}+\frac{1}{2}}$... |
2,581,135 | <blockquote>
<p>Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$</p>
</blockquote>
<p>Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solution, but the solution process pre... | JJacquelin | 108,514 | <p>$$\text{Let}\quad x=\frac{1}{\epsilon^2} \quad\implies\quad
\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}=\frac{1}{1+\epsilon\:\sqrt{1+\epsilon}}\qquad\qquad \epsilon\neq 0$$</p>
<p>$$\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}} = \lim_{\epsilon\to 0}\frac{1}{1+\epsilon\:\sqrt{1+\epsilo... |
2,968,655 | <p>My numerical calculations suggest that the equation
<span class="math-container">$$x = \frac{1}{1+e^{-a+bx}}$$</span>
has a unique solution for any <span class="math-container">$a,b \in \mathbb R$</span>. How would one go about showing this?</p>
| William Elliot | 426,203 | <p>Let <span class="math-container">$y = \dfrac1{1+e^{-a+bx}} - x.$</span><br>
<span class="math-container">$$y' = \dfrac{-be^{-a+bx}}{(1+e^{-a+bx})^2} - 1.$$</span> </p>
<p>If <span class="math-container">$b$</span> is positive, <span class="math-container">$y' < 0$</span>, <span class="math-container">$y$</span... |
2,179,289 | <p>Every valuation ring is an integrally closed local domain, and the integral closure of a local ring is the intersection of all valuation rings containing it. It would be useful for me to know when integrally closed local domains are valuation rings.</p>
<p>To be more specific,</p>
<blockquote>
<p>is there a prop... | Hagen Knaf | 2,479 | <p>A commutative ring $R$ is called coherent, if every finitely generated ideal $I$ is finitely presented, that is as an $R$-module $I$ is isomorphic to $R^n/J$ for some finitely generated $R$-submodule $J$ of $R^n$.</p>
<p>For two ideals $I,J$ of $R$ one defines the ideal $(I:J):=\{r\in R : rJ\subseteq I\}$.</p>
<p>... |
1,204,745 | <p>Let $(\Omega, A, \mathbb{P} )$ be a probability space. Let $f: \Omega \rightarrow [-\infty, \infty]$ an $A$-measurable function. </p>
<p>If $f$ is bounded on the positive side and unbounded on the negative side. Is it possible that $\mathbb{E}[f]$ (the expectation with probability measure $\mathbb{P}$ ) is finite?... | Nicolas | 213,738 | <p>The function $f$ can be unbounded and still be integrable. For example, $f\left(x\right)=\exp\left(-x^{2}\right)\mathtt{1}_{\mathbb{R}_{+}}\left(x\right)
+\delta_{-1}\left(x\right)$ defines an unbounded function, but $\mathbb{E}\left[f\right]=\frac{\sqrt{\pi}}{2}$ for the Lebesgue's measure (with $\Omega=\mathbb{R}... |
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