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 |
|---|---|---|---|---|
3,424,720 | <p>I want to calculate the above limit. Using sage math, I already know that the solution is going to be <span class="math-container">$-\sin(\alpha)$</span>, however, I fail to see how to get to this conclusion.</p>
<h2>My ideas</h2>
<p>I've tried transforming the term in such a way that the limit is easier to find:
... | Ian | 83,396 | <p>The derivation of the derivative of <span class="math-container">$\cos(x)$</span> from geometry goes the way you went, but now you need to provide some justification for the equality <span class="math-container">$\lim_{x \to 0} \frac{\cos(x)-1}{x}=0$</span>, which can also be done using geometry.</p>
|
80,783 | <p>How do you convert $(12.0251)_6$ (in base 6) into fractions?</p>
<p>I know how to convert a fraction into base $x$ by constantly multiplying the fraction by $x$ and simplifying, but I'm not sure how to go the other way?</p>
| André Nicolas | 6,312 | <p>I assume you want decimal notation in your fraction. We have
$$(12.0251)_6=(1\times 6^1) + (2\times 6^0) +\frac{0}{6^1}+\frac{2}{6^2}+\frac{5}{6^3}+\frac{1}{6^4}.$$</p>
<p>Bring the right-hand side to the common denominator $6^4$, and calculate. Equivalently, multiply by $6$ often enough that you get an integer $N... |
856,756 | <p>Let $A_1 = \{x_1,x_2,x_3,x_4\}$ be a set of four positive real numbers.
The sets $A_i, i\geq 2$ are made up of four real numbers defined from the arithmetic mean, geometric mean, harmonic mean and root mean square, as shown below.</p>
<p>Let $A_2 = \{ \text{AM} (A_1),\text{GM}(A_1),\text{HM}(A_1),\text{RMS}(A_1) \}... | Daccache | 79,416 | <p>I can't give a complete answer, but here's what I did:<br>
(note that in your question, you put $AM \geqslant GM \geqslant HM \geqslant RMS$, but the RMS is actually supposed to be the greatest, as in $RMS \geqslant AM \geqslant GM \geqslant HM$)<br>
To experiment, I took 3 sets of numbers $A_1 = \{2, 6, 11, 27\}, B... |
856,756 | <p>Let $A_1 = \{x_1,x_2,x_3,x_4\}$ be a set of four positive real numbers.
The sets $A_i, i\geq 2$ are made up of four real numbers defined from the arithmetic mean, geometric mean, harmonic mean and root mean square, as shown below.</p>
<p>Let $A_2 = \{ \text{AM} (A_1),\text{GM}(A_1),\text{HM}(A_1),\text{RMS}(A_1) \}... | Dirk | 3,148 | <p>Late for the party but here goes: Yes,there is convergence. For just arithmetic and geometric mean this is classical (check the <a href="https://en.wikipedia.org/wiki/AGM_method" rel="nofollow">AGM method</a>). For more means there is an article by Krause</p>
<p>Krause, Ulrich,
<a href="http://dx.doi.org/10.4171/EM... |
102,624 | <p>I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one. </p>
<p>Since Maass cusp forms always are defined for a Fuchsian lattice, I... | GH from MO | 11,919 | <p>This conjecture is usually stated for $\mathrm{SL}_2(\mathbb{Z})$, and it is widely open. I think it is folklore, and is stated in several papers, e.g. in Luo: Nonvanishing of $L$-values and the Weyl law (before (3)).</p>
<p>The motivation, I think, is similar as with the conjecture for the multiplicity of the Riem... |
1,569,936 | <p>Given that $a_0, a_1,...,a_{n-1} \in \mathbb{C}$ I am trying to understand how the following calculation for the determinant of the following matrix follows:
$$
\text{det}
\begin{bmatrix}
x & 0 & 0 & ... & 0 & a_0 \\
-1 & x & 0 & ... & 0 & a_1 \\
0 & -1 & x &... | stity | 285,341 | <p>If $A=(a_{i,j}) \in M_n(\mathbb{C})$, $$\det(A) = \sum_{k=1}^{n} (-1)^ka_{1,k} \det(\Delta_{1,k}) $$
Where $\Delta_{1,k}$ is A minus the column and the line of $a_{1,k}$</p>
<p>Here the sum only has 2 terms because others are equals to $0$.</p>
|
96,657 | <p>I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous extension to $X$.</p>
<p>I think I know how to do this using Cauchy sequences, but there is this suggested alternativ... | Levon Haykazyan | 11,753 | <p>As Srivatsan notes, any set with a finite diameter is bounded. You have shown that for a fixed $\varepsilon$, ${\rm diam} \overline {f(V_n(p))} < \varepsilon$, starting from some $n$. So starting from some $n$, $\overline {f(V_n(p))}$ are all bounded and hence compact. Furthermore the image of a <em>real</em> uni... |
96,657 | <p>I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous extension to $X$.</p>
<p>I think I know how to do this using Cauchy sequences, but there is this suggested alternativ... | chenu | 868,600 | <p>Let, <span class="math-container">$p\in X$</span>, we define <span class="math-container">$V_n(p)=\big\{q\in E: d(p,q)<\frac{1}{n}\big\}$</span><br />
We have that, <span class="math-container">$V_n(p)\neq\phi\,\forall\,n\in\Bbb N$</span> (Since, <span class="math-container">$\overline E=X$</span>)<br />
Also not... |
4,568,221 | <p><span class="math-container">$$\iint\limits_{D} \left(x^2+y^2\right)\mathrm{d}x \mathrm{d}y$$</span>
where <span class="math-container">$D$</span> is given each time by
<span class="math-container">$D=x^2-y^2=1,\hspace{0.5cm} x^2-y^2=9,\hspace{0.5cm} xy=2,\hspace{0.5cm} xy=4$</span></p>
<p>I try to use Polar coordin... | Robert Z | 299,698 | <p>I don't see a simple way to evaluate integral in polar coordinates. In my opinion it is better to consider the transformation <span class="math-container">$(u,v)=(x^2-y^2,xy)$</span>, then the Jacobian is equal to
<span class="math-container">$$\left|\frac{\partial(u,v)}{\partial(x,y)}\right|=\left|\det\left(\begin{... |
1,042,375 | <p><strong>Question:</strong></p>
<blockquote>
<p>Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$.</p>
</blockquote>
<p>At first I thought this was quite obvious, but then I came up with a counterexample. What if $f(x... | 5xum | 112,884 | <p>The set $\{x\in\mathbb R\mid f(x) = 1\}$ for the function defined by $f(x) = 1$ for every $x$, is $\mathbb R$ the whole set of real numbers. This is an open set, so no, you did not find a contradiction.</p>
<hr>
<p>In case you don't believe me:</p>
<p>By definition, a set $X\subseteq \mathbb R$ is open if, for ev... |
59,965 | <p>If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$?</p>
<p>Thanks.</p>
| Ross Millikan | 1,827 | <p>No, it is not. It is unusual that you can do so. For example, $f(x,y)=x+y$ cannot be.</p>
|
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Ittay Weiss | 30,953 | <p>Mathematics does not exist in a vacuum. It is strongly related, via applications, to the world around us. Mathematicians choose axioms according to what works well when we try to use the insights and results flowing from these axioms to better understand problems (usually from science) that we care about. </p>
<p>T... |
2,311,848 | <p>$X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.</p>
<p>Here's my reasoning: </p>
<p>$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$. </p>
<p>I claim that we have 2 cases here: </p>
<p>1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\leq z)... | mlc | 360,141 | <p>Everything works until you write "either... or..." instead of "and". </p>
<p>In practice, your last formula is computing the probability that $X \le z$ or $Y \le z$ but not both. Hence, for $X \le z$, you implicitly compute the probability that $Y > z$; likewise, for $Y \le z$, you consider the probability that ... |
61,316 | <p>Hi all,</p>
<p>I heard a claim that if I have a matrix $A\in\mathbb R^{n\times n}$ such that $A^n \to 0 \ (\text{for }n\to\infty )$
(that is, every entry of $A^n$ converges to $0$ where $n\to \infty$)
then $I-A$ is invertible.</p>
<p>anyone knows if there is a name for such a matrix or how (for general knowledge)... | Per Alexandersson | 1,056 | <p>It is quite easy:</p>
<p>Consider the sum $\sum_{n=0}^\infty A^n$.
Your condition makes sure that this converges. At the same time, pretend that this is a usual,
geometric series. Then the sum is given by $1/(1-A)$ or, if you wish, multiplicative inverse of $I-A.$</p>
<p>So in short, $I-A$ has an inverse, and it i... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Gottfried Helms | 1,714 | <p>This is not an answer, just an extended comment to the answer of @MarkHunter .
The <a href="https://ariwatch.com/VS/Algorithms/TheFourthOperation.htm" rel="nofollow noreferrer">linked article</a> refers to many common topics, but also to something of G. Szekeres and it seems it follows a similar idea as I have seen... |
2,832,614 | <blockquote>
<p>Prove by induction that
$$\lim_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}.$$</p>
</blockquote>
<p>I did a strange proof using two initial results: We know that result is true for $n=1$ and $n=2$. Assuming the result is true for $n=k-1$ and $n=k$, I can prove the result for $n=k+1$. For this I used my a... | N. S. | 9,176 | <p><strong>Hint</strong>
$$\frac{x^{n+1}-a^{n+1}}{x-a}=\frac{x^{n+1}-ax^n}{x-a}+\frac{ax^n-a^{n+1}}{x-a}$$</p>
|
77,136 | <p>I have a multidimensional-variable list, suppose for example {{x1,y1},{x2,y2}..}. I have duplicate values for the 'x' coordinates and I need to find for the duplicate 'x' elements the corresponding minimum 'y'. A sample of this list is the following:</p>
<pre><code>l1={{1, 1.43E-46}, {21, 2.79E-48}, {41, 3.22E-45},... | kglr | 125 | <pre><code>l1 = {{1, 1.43 10^-46}, {21, 2.79 10^-48}, {41, 3.22 10^-45}, {41,1.74 10^-46},
{81, 2.77 10^-46}, {121, 9.97 10^-48}, {161, 1.24 10^-45}, {181, 1.19 10^-45}};
</code></pre>
<p>I version 10, you can also use <code>MinimalBy</code>:</p>
<pre><code> Join@@MinimalBy[Last]/@GatherBy[l1, First]
(*... |
3,436,891 | <p>I have the following stupid question in my mind while i am studying for exams.
Does <span class="math-container">$X<\infty \ a.s$</span>, implies that <span class="math-container">$\mathbb E(X)<\infty$</span>? </p>
<p>Further on this, is the converse of the above statement true? Do give me a bit summary on th... | Surb | 154,545 | <p>Take <span class="math-container">$(\Omega ,\mathcal F,\mathbb P)=([0,1], \mathcal B([0,1]), m)$</span> where <span class="math-container">$m$</span> denote the Lebesgue measure on <span class="math-container">$[0,1]$</span>. Consider <span class="math-container">$X(\omega )=\frac{1}{\omega }$</span>. </p>
<p>Then,... |
563,927 | <p>Show that $\mathbb{R}$
is not a simple extension of $\mathbb{Q}$
as follow:</p>
<p>a. $\mathbb{Q}$
is countable.</p>
<p>b. Any simple extension of a countable field is countable.</p>
<p>c. $\mathbb{R}$
is not countable.</p>
<p>I 've done a. and c. Can anyone help me a hint to prove b.?</p>
| Prahlad Vaidyanathan | 89,789 | <p>Let $F$ be a countable field, then the collection of all polynomials of degree $\leq n$ is countable. Hence, $F[x]$ is countable, being the countable union of countable sets. Hence, $F[x] \times F[x]\setminus\{0\}$ is countable. There is a surjective function $F[x]\times F[x]\setminus\{0\} \to F(x)$ by
$$
(f(x), g(x... |
1,820,690 | <p>Find two numbers, $A$ and $B$, both smaller than $100$, that have a lowest common multiple of $450$ and a highest common factor of $15$.</p>
<p>I know that this involves the formula of </p>
<p>$A × B = LCM × HCF$</p>
<p>But I don't quite understand the above formula so I rather memorise it and that is why I can't... | Siddharth Bhat | 261,373 | <p>Replace $z^5 \to t$. Hence, this gives us $t^2 - t - 992 = 0$. Solving for the roots (use the quadratic equation) we get that the roots are</p>
<p>$t = 32, -31$</p>
<p>Hence, $z^5 = 32, -31$</p>
<p>For </p>
<p>$$
z^5 = 32 \\
z^5 = e^{i 2\pi n}2^5, n \in \mathbb{N} \\
z = 2 \cdot e^{\frac{i 2 \pi n}{5}}, n \in \m... |
1,424,913 | <p>I am trying to solve the following problem.</p>
<blockquote>
<p>Let $G$ be a group. If $M, N \subset G$ are such that $x^{-1} M x = M$
and $x^{-1} N x = N$ for all $x \in G$ and $M \cap N = \{1\}$, prove
that $m n = n m$ for all $m \in M, n \in N$.</p>
</blockquote>
<p>I have already proven it for the specif... | Sammy Black | 6,509 | <p>Note: I am assuming that the quantities in your equations are real numbers.</p>
<p>Since $\cosh u \ge 1$, with the minimum only occurring at $u = 0$, you're complicating the calculation too much. You can conclude immediately that $xy = 0$, so either $x = 0$ or $y = 0$ or both. (EDITED, to correct silly error.)</... |
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Community | -1 | <p>Verify the following equation. And, note that index runs over only upto a finite stage. Because, this is false for many infinite cases (as I indicate at the end!)</p>
<p>$$\sum_{i=1}^n (a_i+b_i)=\sum_{i=1}^n a_i+\sum_{i=1}^n b_i\\\ \sum_{i=1}^nc\cdot a_i=c\cdot\left(\sum_{i=1}^na_i\right)$$</p>
<p>where $c$ is a c... |
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Shaun Ault | 13,074 | <p>There are properties of series that can be used. Specifically, series are <em>linear</em>, which means
$$ \sum_{i=1}^k (ca_i + b_i) = c\sum_{i=1}^k a_i + \sum_{i=1}^k b_i, $$
for constants $c$. Thus,
$$ \sum_{i=1}^k (i^2 + 3i) = \sum_{i=1}^k i^2 + 3\sum_{i=1}^k i.$$
At that point, you need the formulas,
$$ \sum_{i... |
4,057,255 | <p>Suppose we have 10 items that we will randomly place into 6 bins, each with equal probability. I want you to determine the probability that we will do this in such a way that no bin is empty. For the analytical solution, you might find it easiest to think of the problem in terms of six events Ai, i = 1, . . . , 6 wh... | true blue anil | 22,388 | <p>Another way is to use Stirling numbers of the second kind which can be conceived as putting <em>n</em> labelled objects (throws <span class="math-container">$1 \;thru\; 10$</span>)
into <em>k</em> identical boxes with no box being left empty.
If we multiply <span class="math-container">$S2(10,6)$</span> by <span c... |
550,441 | <p>Say I roll a 6-sided die until its sum exceeds $X$. What is E(rolls)?</p>
| LeeNeverGup | 104,910 | <p>Let $X$ be a random variable with discrete uniform distribution in $[1, 6]$, and let n be a number. If m is the number of trials needed to exceed n, we can conclude:</p>
<p>-sum of m trials is more than n.</p>
<p>-sum of m-1 trial is less than n.</p>
<p>So the question is: Let $Y_n$ be a sum of n independent vari... |
3,480,123 | <p>In my master thesis, I'm trying to prove the following limit:
<span class="math-container">$$\lim_{\epsilon \to 0^+}\int_0^1 \frac{\left(\ln\left(\frac{\epsilon}{1-x}+1\right)\right)^\alpha}{x^\beta(1-x)^\gamma}\,\mathrm{d}x=0,$$</span>
where <span class="math-container">$\alpha, \beta, \gamma \in (0,1)$</span>.</p>... | Vasily Mitch | 398,967 | <p>You can encode the vector <span class="math-container">$PA$</span> as an algebraic sum of vectors <span class="math-container">$BA$</span>, <span class="math-container">$CA$</span> and <span class="math-container">$BA\times CA$</span>.</p>
|
64,977 | <p>Suppose I had a complete bipartite graph with edges each given some numerical "cost" value. Is there a way to select a subset of those edges such that each vertex on each side of the graph is mapped to each vertex on the other (one to one) and the total "costs" is maximized (or minimized)?</p>
<p>Has anyone ever f... | Peter Sarkoci | 14,763 | <p>I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to b... |
3,827,449 | <p>For example, <span class="math-container">$(1,1,1)$</span> is such a point. The sphere must contain all points that satisfy the condition.</p>
<p>So, I've been milling over this question on and off for the past few days and just can't seem to figure it out. I think I would have to use the distance formula from some ... | 2'5 9'2 | 11,123 | <p>Following your lead, here is how to get started:</p>
<p><span class="math-container">$$\left(\text{distance to }(3,3,3)\right)^2=\left(2\cdot{}\left(\text{distance to }(0,0,0)\right)\right)^2$$</span>
<span class="math-container">$$(x-3)^2+(y-3)^2+(z-3)^2=4\left(x^2+y^2+z^2\right)$$</span></p>
<p>A few more steps of... |
65,166 | <p>For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested in the spectral gap of $G$, i.e. the first nonzero eigenvalue of $\Delta$, denoted by $\lambda_{1}(G)$.</p>
<p>Is i... | Alain Valette | 14,497 | <p>In the paper "On the second eigenvalue and random walks in random d-regular graphs" (Combinatorica 11 (1991), no. 4, 331–362), Joel Friedman considers a model of $2d$-regular random graphs on $n$ vertices, by selecting randomly and uniformly $d$ permutations from the symmetric group $S_n$, and looking at the undirec... |
65,166 | <p>For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested in the spectral gap of $G$, i.e. the first nonzero eigenvalue of $\Delta$, denoted by $\lambda_{1}(G)$.</p>
<p>Is i... | Doron | 31,917 | <p>Check out the new paper <a href="http://arxiv.org/abs/1212.5216" rel="nofollow">http://arxiv.org/abs/1212.5216</a> (Cor 1.6).</p>
<p>It is proven there that a random d-regular bipartite graph has its largest non-trivial eigenvalue at most 2\sqrt(d-1)+0.84.</p>
|
947,290 | <p>In a cyclic group of order 8 show that element has a cube root. So for some $a\in G$ there is an element $x \in G$ with $x^3=a.$</p>
<p>Also show in general that if $g=<a>$ is a cyclic group of order m and $(k,m)=1$ then each element in G has a $k$th root. What element will $a^k$ generate? Use this to expre... | Nour | 380,871 | <p>For the second part of your question,
$a$ is a generator of $G$ ( i.e.
$G= <a>$ ).</p>
<p>Since ($k$ , $m$) $=1$ , so $a^k$ is also a generator of $G$ ( $< a^k > $ = $ <a^{gcd(m,k)}>$ ).</p>
<p>Therefore, for $x\in G$ , $x= a^{sk}$ $= (a^s)^k$ , for some $s$.</p>
|
112,137 | <p>I'm guessing the answer to this question is well-known:</p>
<p>Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}_Y\left(F\right)$ preserve colimits? Notice that if $C=P$... | Buschi Sergio | 6,262 | <p>Let $(a_i: A_i\to A)_{i\in I}$ with $I\in Cat$ a universal cocone in a category $\mathcal{A}$, and let $H: \mathcal{B}\to \mathcal{A}$.</p>
<p><em><strong>We ask when:</em></strong></p>
<p>for any $F: \mathcal{B}\to \mathcal{C}$ such that:</p>
<p>exist $L:=Lan_H F$ punctually (or at least it exist for the objec... |
1,693,387 | <p>It is clear from the change of basis formula that the matrices $A$ and $B$ representing the same linear map in different bases are equivalent: there exists invertible matrixes $Q$ and $P$ such that $A=Q^{-1}BP$.</p>
<p>My question is about the other way, I mean given two equivalent matrices $A$ and $B$ in $M_{n,p}(... | Marcel | 68,145 | <p>The matrix elements $\{\langle i|A|j\rangle,1\le i\le n,1\le j\le p\}$ of $A$ with respect to a given basis sets $\{\langle i|\}$ and $\{|j\rangle\}$, are the same as $\{\langle i|Q^{-1}BP|j\rangle,1\le i\le n,1\le j\le p\}$, which are the matrix elements of $B$ with respect to new basis sets given by $\{P|j\rangle,... |
1,693,387 | <p>It is clear from the change of basis formula that the matrices $A$ and $B$ representing the same linear map in different bases are equivalent: there exists invertible matrixes $Q$ and $P$ such that $A=Q^{-1}BP$.</p>
<p>My question is about the other way, I mean given two equivalent matrices $A$ and $B$ in $M_{n,p}(... | Ted Shifrin | 71,348 | <p>This is just a simple generalization of similarity (where you consider only $n\times n$ matrices with the <em>same</em> basis in domain and range). Here $P$ is the change-of-basis matrix in the domain ($\Bbb K^p$) and $Q$ is the change-of-basis matrix in the range ($\Bbb K^n$).</p>
|
1,693,387 | <p>It is clear from the change of basis formula that the matrices $A$ and $B$ representing the same linear map in different bases are equivalent: there exists invertible matrixes $Q$ and $P$ such that $A=Q^{-1}BP$.</p>
<p>My question is about the other way, I mean given two equivalent matrices $A$ and $B$ in $M_{n,p}(... | Open Season | 99,428 | <p>Clearly $B$ being an $m \times n$ matrix represents a transformation $\Bbb R^n \rightarrow \Bbb R^m$ where we take the standard basis in each space. If $A = Q^{-1}BP$, then take a basis for $\Bbb R^n$ made up of the columns of $P$ and a basis for $\Bbb R^m$ made up of the columns of $Q$. Then $A$ represents the tr... |
317,753 | <p>I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:</p>
<ul>
<li>How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?</li>
<li>What books do you recommended for an undergraduate who is studying ... | Mikasa | 8,581 | <p>I have been teaching about 13 years in collage so I have seen many books or texts written by for example, <em>Rudin</em>, <em>Bartle</em>, <em>Apostol</em> and <em>Aliprantis</em> in Analysis. But the ones have been useful for me or for students that have topological approaches or graphical approaches. Rudin's is a ... |
223,631 | <p>I'm using NeumannValue boundary conditions for a 3d FEA using NDSolveValue. In one area I have positive flux and in another area i have negative flux. In theory these should balance out (I set the flux inversely proportional to their relative areas) to a net flux of 0 but because of mesh and numerical inaccuracies... | user21 | 18,437 | <p>Too long for a comment. An easy way to generate a high quality mesh is to replace the <code>Implicitegion</code> with <code>Cubuid</code> and make use of the <a href="https://reference.wolfram.com/language/FEMDocumentation/ref/ToBoundaryMesh.html#1039792269" rel="noreferrer">OpenCascade boundary mesh generator</a>:<... |
1,638,757 | <p>Given $2$ points in 2-dimensional space $(x_s,y_s)$ and $(x_d,y_d)$, our task is to find whether $(x_d,y_d)$ can be reached from $(x_s,y_s)$ by making a sequence of zero or more operations.
From a given point $(x, y)$, the operations possible are:</p>
<pre><code>a) Move to point (y, x)
b) Move to point (x, -y)
c) M... | Jack's wasted life | 117,135 | <p>Gcd is invariant under operations a,b,c. Gcd may double under operation d. So the transition is possible if either $\gcd(x_s,y_s)=\gcd(x_d,y_d)$ or $2^n\gcd(x_s,y_s)=\gcd(x_d,y_d)$ for some $n\in\mathbb N$ where we assume $\gcd(x,y)=\gcd(|x|,|y|)$.</p>
<p>Note that $$\gcd(x,y)|ax+by\;\forall\;a,b\in\mathbb Z$$ We c... |
526,837 | <p>Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual
measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.</p>
<p>The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the d... | Hagen von Eitzen | 39,174 | <p>You can check th ersult by computing the derivative of $-2x+\ln(1+2x)$. It is the same as that of $-1-2x+\ln(1+2x)$. In general, if $F$ is an antiderivative of $f$ then so is $F+c$ for any constant $c$.</p>
|
4,218,943 | <p>Let <span class="math-container">$A_n$</span> be a sequence of <span class="math-container">$d\times d$</span> symmetric matrices, let <span class="math-container">$A$</span> be a <span class="math-container">$d\times d$</span> symmetric positive definite matrix (matrix entries are assumed to be real numbers). Assum... | Asinomás | 33,907 | <p>Let <span class="math-container">$\lambda_1 \geq \dots \lambda_n$</span> be the eigenvalues. Any <span class="math-container">$e< \lambda_n$</span> works.</p>
<p>Suppose <span class="math-container">$\lambda < e$</span> is an eigenvalue of <span class="math-container">$A_n$</span>, take an orthonormal basis of... |
878,115 | <p>Question1:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls.
Afer i collected all 15 balls i put them randomly inside the boxes.</p>
<p>How much is the chance that all balls are in only 10 boxes or less?</p>
<p>Question2:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i foun... | ant11 | 110,047 | <p>I will assume the balls and boxes are indistinguishable.</p>
<p>The first problem is: If I distribute $15$ balls among $30$ boxes, what is the probability that at most $10$ boxes contain a ball?</p>
<p>First, the fact that there are $30$ boxes does not matter, since they are indistinguishable. So we only need to c... |
3,705,580 | <p><span class="math-container">$\mathbf {The \ Problem \ is}:$</span> Let, <span class="math-container">$f,g,h$</span> be three functions defined from <span class="math-container">$(0,\infty)$</span> to <span class="math-container">$(0,\infty)$</span> satisfying the given relation <span class="math-container">$f(x)g(y... | I was suspended for talking | 474,690 | <p>Write <span class="math-container">$W = (X_1,X_2)$</span> and <span class="math-container">$Z = (X_3,X_4)$</span>, which are i.i.d random vectors (with standard normal coordinates). Then using your notation, <span class="math-container">$U = W\cdot Z$</span> and <span class="math-container">$V = \|Z\|^2$</span> so <... |
3,705,580 | <p><span class="math-container">$\mathbf {The \ Problem \ is}:$</span> Let, <span class="math-container">$f,g,h$</span> be three functions defined from <span class="math-container">$(0,\infty)$</span> to <span class="math-container">$(0,\infty)$</span> satisfying the given relation <span class="math-container">$f(x)g(y... | StubbornAtom | 321,264 | <p>Suppose <span class="math-container">$$Z=\frac{X_1X_3+X_2X_4}{\sqrt{X_3^2+X_4^2}}$$</span></p>
<p>Now conditioning <span class="math-container">$X_3=a,X_4=b$</span>, we find that the above has a standard normal distribution, since</p>
<p><span class="math-container">$$\frac{aX_1+bX_2}{\sqrt{a^2+b^2}}\sim N(0,1)$$</s... |
89,197 | <p>I am working on a problem where I have to generate a table of components while each component of the table has 18 entries. Six of the indices among 18 run from 0 to 1 while the other 12 can take values between 0 to 3. After doing that I have to select some of the entries which follow a certain criterion (sum of all ... | ubpdqn | 1,997 | <p>I think this can done as follows:</p>
<pre><code>pol = {1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3};
ip = Join @@ (Permutations /@ PadRight[IntegerPartitions[3, 3]]);
subs = Subsets[Range[18], {3}];
rl = Flatten[Map[Function[u, Thread[u -> #] & /@ ip], subs], 1];
cand = ReplacePart[ConstantArray[0,... |
3,460,595 | <p>I am given the following sequence:</p>
<p><span class="math-container">$$a_n = 1^9 + 2^9 + ... + n^9 - an^{10}$$</span></p>
<p>Where <span class="math-container">$a \in \mathbb{R}$</span>. I have to find the value of <span class="math-container">$a$</span> for which the sequence <span class="math-container">$a_n$<... | 2'5 9'2 | 11,123 | <p>You have</p>
<p><span class="math-container">$$
\begin{align}
b_n:=a_{n+1}-a_n&=(n+1)^9-a(n+1)^{10}+an^{10}\\
&=\sum_{k=0}^9n^k-a\sum_{k=0}^9\binom{10}{k}n^k\\
&=\sum_{k=0}^9n^k\left(1-a\binom{10}{k}\right)
\end{align}
$$</span></p>
<p>If <span class="math-container">$a\neq\frac{1}{10}$</span>, this is... |
1,793,182 | <p>My task was to find the directional derivative of function:<br>
$$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $. </p>
<p>The result I found was $-21/\sqrt{101}$. But I can't figure out what is the interpretation of this result. </p>
<p>Does it mean that the function grows fa... | Martin Sleziak | 8,297 | <p>I will write down two versions of my attempt to prove the above claim. Both are based on the same idea. (It is more-or-less the same proof, written down from a slightly different perspective.)$\newcommand{\intrv}[2]{[#1,#2]}\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}$</p>
<hr>
<p>Let $\Delta$ denotes the set... |
2,943,461 | <p>I'm stumped on a math puzzle and I can't find an answer to it anywhere!
A man is filling a pool from 3 hoses. Hose A could fill it in 2 hours, hose B could fill it in 3 hours and hose C can fill it in 6 hours. However, there is a blockage in hose A, so the guy starts by using hoses B and C. When the blockage in hose... | MRobinson | 587,882 | <p>If the total volume of the pool is <span class="math-container">$x$</span>, we can denote the rates as:</p>
<p><span class="math-container">$r_A = \frac{x}{2}, r_B = \frac{x}{3}, r_C = \frac{x}{6}.$</span></p>
<p>From here you can see that:</p>
<p><span class="math-container">$r_{B+C} = r_B + r_C = \frac{x}{3} + ... |
3,660,652 | <p>To which of the seventeen standard quadrics (<a href="https://mathworld.wolfram.com/QuadraticSurface.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/QuadraticSurface.html</a>) do these two equations reduce?
<span class="math-container">\begin{equation}
Q_1^2+3 Q_2 Q_1+\left(3
Q_2+Q_3\right){}^2 = 3 ... | eeen | 646,577 | <p>This is a line with a rational slope, namely <span class="math-container">$256/27$</span> (on the <span class="math-container">$dx$</span>-plane). Hence there are an infinite number of solutions <span class="math-container">$(d,x)$</span> of the form </p>
<p><span class="math-container">$$(26 + 27t, 253 + 256t),$$<... |
3,833,767 | <p>I am trying to brush up on calculus and picked up Peter Lax's Calculus with Applications and Computing Vol 1 (1976) and I am trying to solve exercise 5.2 a) in the first chapter (page 29):</p>
<blockquote>
<p>How large does <span class="math-container">$n$</span> have to be in order for</p>
<p><span class="math-cont... | Claude Leibovici | 82,404 | <p>I shall cheating assuming that you know the value of the infinite sum.</p>
<p>So for your example
<span class="math-container">$$\sum_{n=1}^{p}\frac1{n^2}=H_p^{(2)}$$</span> and you want to know <span class="math-container">$p$</span> such that
<span class="math-container">$$\frac {\pi^2}6-H_{p+1}^{(2)} \leq 10^{-k}... |
319,725 | <p>I am trying to prove the following inequality concerning the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="noreferrer">Beta Function</a>:
<span class="math-container">$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$</span>
where as usual <span class="math-c... | fedja | 1,131 | <p>You can get away with the usual distribution function mumbo-jumbo. The general lemma is as follows:</p>
<p>Let <span class="math-container">$\mu,\nu$</span> be non-negative measures and <span class="math-container">$f,g$</span> be non-negative functions such that there exists <span class="math-container">$s_0>0$... |
2,171,237 | <p>$f(x)$ is continuous on $[0,\pi]$ and $\int_0^\pi{f(x)\sin xdx} = \int_0^\pi{f(x)\cos xdx} = 1.$</p>
<p>Find $\min\int_0^\pi {f^2(x)dx}.$</p>
<p>I try to solve this problem by this:
$$\begin{array}{l}
{\left( {\int\limits_0^\pi {f(x)\sin xdx} } \right)^2} \le \left( {\int\limits_0^\pi {{f^2}(x){{\sin }^2}xdx} } ... | Jacky Chong | 369,395 | <p>Let us consider the Fourier series of $f$
\begin{align}
f(x) = \frac{1}{2}a_0+\sum^\infty_{n=1} a_n \cos nx+ \sum^\infty_{n=1} b_n \sin nx
\end{align}
then that means
\begin{align}
\int^\pi_0 f^2(x)\ dx= \frac{1}{4}a_0^2+\frac{\pi}{2}\sum^\infty_{n=1}(a_n^2+b_n^2).
\end{align}
Since
\begin{align}
\int^\pi_0 f(x) \co... |
3,101,286 | <p>I would like to get this text translated from Dutch to English:</p>
<p><a href="https://i.stack.imgur.com/0IWlQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0IWlQ.png" alt="enter image description here"></a></p>
<p>I tried using Google translator but the result is confusing me:</p>
<p><a hre... | StackTD | 159,845 | <p><a href="https://i.stack.imgur.com/0IWlQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0IWlQ.png" alt="enter image description here"></a></p>
<ul>
<li><p>Given a bounded sequence <span class="math-container">$(y_n)_n$</span> in <span class="math-container">$\mathbb{C}$</span>. Show that for eve... |
2,657,053 | <blockquote>
<p>Suppose I know that
$$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}\,\,\,\, \tag{1} $$
How can I prove the the following?
$$
\sum_{i=0}^{n-1} i^2=\frac{n(n-1)(2n-1)}{6}
$$</p>
</blockquote>
<hr>
<p>I have looked up the solution to the other problem but it seems to be a bit confusing to me. Is it pos... | sku | 341,324 | <p>The OP asked for how to get the RHS if you didn't know about it. One way to do this is through rising factorials, which work like ordinary integration. </p>
<p>define $$n^{\bar 1} = n$$
$$n^{\bar 2} = n(n+1)$$ and so on</p>
<p>We can rewrite $n^2 = n^{\bar 2} - n^{\bar 1}$</p>
<p>$$\sum_{i=0}^{n} i^2= \sum_{i=0}^... |
2,054,949 | <p>From what I understand, these three concepts all describe the points where the function is not continuous. How to tell them apart? Thanks!</p>
| DanielWainfleet | 254,665 | <p>$z_0$ is a pole of $f$ iff $f$ is analytic on $\{z: 0<|z-z_0|<r\}$ for some $r>0,$ and $f(z_0)$ cannot be defined in such a way that $f$ is analytic on $\{z:|z-z_0|<r\},$ but also that $f(z)=(z-z_0)^ng(z)$ for $0<|z-z_0|<r$ for some $n\in \mathbb N,$ where $g$ is analytic on $\{z:|z-z_0|<r\}... |
26,162 | <p>I'm the math department chair at a small university. Our general education program is non-traditional. The university is split into three areas. Students are expected to complete a major in one of the areas and earn minors in the other two. Overall, this is good because students get more depth in the chosen minor ar... | guest troll | 21,066 | <p>[Caveat, I'm just a citizen, not a teach.]</p>
<p>I like it. Call it a "math ed minor" to clarify it is different from say a chemist getting a "math minor".</p>
<p>I think the two remedial algebra classes are fine. Kind of gives them some understanding of where the kids are headed after arithme... |
26,162 | <p>I'm the math department chair at a small university. Our general education program is non-traditional. The university is split into three areas. Students are expected to complete a major in one of the areas and earn minors in the other two. Overall, this is good because students get more depth in the chosen minor ar... | Justin Hancock | 20,719 | <p>The top priority should be ensuring that elementary education majors have a deep and mathematically-sound understanding of elementary school mathematics. They need to (re)learn the core mathematics that they will be teaching—whole number and fraction arithmetic—so that they can make sense of it for themselves and fo... |
2,777,982 | <p>I was asked to describe the surface described by</p>
<p>$${\bf r}^\top {\bf A} {\bf r} + {\bf b}^\top {\bf r} = 1,$$</p>
<p>where $3 \times 3$ positive definite matrix ${\bf A}$ and vector $\bf b$ are given.</p>
<p>My intuition tells me that it is a rotated ellipsoid with a centre that is off the origin. However,... | Community | -1 | <p>Note: for convenience we use $2b$ instead of $b$.</p>
<p>$$(x+a)^TA(x+a)+2b^T(x+a)=x^TAx+a^TAx+x^TAa+a^TAa+b^Tx+2b^Ta.$$</p>
<p>Notice that by symmetry of $A$, $x^TAa=a^TAx$. Collecting all the $x$ terms,</p>
<p>$$(2a^TA+2b^T)x$$ can be cancelled with the choice</p>
<p>$$a=-A^{-1}b.$$</p>
<p>Then</p>
<p>$$C=1-... |
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | André Souza Lemos | 5,038 | <p>To my view, the most effective strategies deal with anxiety as a collective - and thus political and cultural - issue. The answers provided by adamblan and Mandy Jansen are quite to the point, in that regard, and what I'll add here is just a complement.</p>
<p>Anxiety can be defined as overreaction to falsely perce... |
250,364 | <blockquote>
<p><strong>Problem</strong> Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous.</p>
</blockquote>
<p>My idea is to consider $|x - y| < \delta$, then show that
$$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})|
= \bigg|\log\bigg(\dfrac{1 + \sqrt{1+x^2}}{1 + \sqrt{1+y^2}}\bigg)\bigg| <... | Martin Argerami | 22,857 | <p>Note that $t\mapsto \log t$ is uniformly continuous on $[1,\infty)$ (proven below). Note also that $t\mapsto 1+\sqrt{1+t^2}$ is uniformly continuous on $\mathbb R$ (also proven below). As the composition of uniformly continuous functions is uniformly continuous, the result follows.</p>
<p>To see that $\log$ is unif... |
2,130,911 | <p>I'm unsure how to compute the following : 3^1000 (mod13)</p>
<p>I tried working through an example below,</p>
<p>ie) Compute $3^{100,000} \bmod 7$
$$
3^{100,000}=3^{(16,666⋅6+4)}=(3^6)^{16,666}*3^4=1^{16,666}*9^2=2^2=4 \pmod 7\\
$$</p>
<p>but I don't understand why they divide 100,000 by 6 to get 16,666. Where di... | Maczinga | 411,133 | <p>Just use Fermat's Little Theorem</p>
<p>$a^{p}\equiv a\mod p$</p>
<p>with $p=7$ in your example and $p=13$ in your former question.</p>
|
1,413,363 | <p>The question:</p>
<p>Find values of $a,b,c.$ if $\displaystyle \frac{x^2+1}{x^2+3x+2} = \frac{a}{x+2}+\frac{bx+c}{x+1}$</p>
<p>My working so far:</p>
<p><a href="https://i.imgur.com/VegifVa.jpg" rel="nofollow noreferrer">http://i.imgur.com/VegifVa.jpg</a></p>
<p>How do I isolate $a$, $b$ and $c$?</p>
| juantheron | 14,311 | <p>Given $$\displaystyle \frac{x^2+1}{x^2+3x+2} = \frac{a}{x+2}+\frac{bx+c}{x+1} = \frac{a(x+1)+(bx+c)(x+2)}{x^2+3x+2}$$</p>
<p>So $$x^2+1 = bx^2+(a+2b+c)x+(a+2c)$$</p>
<p>Now equating Coefficients, we get $b=1$ and $(a+2b+c) = 0$ and $a+2c=1$</p>
<p>So Put $b=1$ in $a+2b+c=0\Rightarrow a+c=-2$ and above $a+2c=1$</p... |
887,327 | <p>I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$<br>
I tried splitting the fraction into two parts, i.e. $\frac{\sqrt{n+1}}{\sqrt{n^2+n}}$ and $\frac{\sqrt{n}}{\sqrt{n^+n}}$, but we know the two individual infinite series diverge. Now how do I pr... | amrik | 167,970 | <p>Your problem may be converted to the following formula:
\begin{align}
& \lim_{N\to\infty}\sum_{n=1}^{N}\left({\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}}\right) =
\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+\left(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}\right)+...+\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{N+1}}\r... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Bram28 | 256,001 | <p>For a rough estimate, I'd first divide by $100$ and think about $20.17$, for $\sqrt{2017} = 10 \sqrt{20.17}$. In fact, I would just consider $\sqrt{20}$: </p>
<p>You know $4^2=16$ and $5^2 =25$, so $\sqrt{20}$ is between $4$ and $5$, and is in fact close to the middle of them, i.e close to $4.5$. Hence, $\sqrt{201... |
883,972 | <p>Let:</p>
<p>$$f(n) = n(n+1)(n+2)/(n+3)$$</p>
<p>Therefore :</p>
<p>$$f∈O(n^2)$$</p>
<p>However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest is $n^3$ not $n^2$.</p>
| amWhy | 9,003 | <p>But when you <strong><em>divide</em></strong> a degree-three polynomial, $\,n^3 + 3n^2 + 2n,\,$ by a degree-one polynomial, $\,n+3,\,$ you end up with a degree <strong><em>two</em></strong> polyonomial $n^2 + 2\;$ with remainder of $\quad \frac{-6}{n+3}$</p>
|
3,282,400 | <p>I would like to illustrate my confusion about this topic by building up the issue from more or less first principles. Let <span class="math-container">$U \subseteq \mathbb{R}^n$</span> be an open subset and let <span class="math-container">$f:U \to \mathbb{R}^m$</span>. We say that <span class="math-container">$f$</... | Fabio Lucchini | 54,738 | <p>Statement (b) can be proved by applying long homology sequence.
For consider the complexes of abelian groups:
<span class="math-container">\begin{align}
&A:\ldots\to A\xrightarrow f A\xrightarrow g A\xrightarrow f\ldots\\
&B:\ldots\to B\xrightarrow f B\xrightarrow g B\xrightarrow f\ldots\\
&A/B:\ldots\to... |
327,750 | <p>$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$</p>
<p>The results is obvious enough, but how to prove this</p>
| PeptideChain | 322,402 | <p>$$\bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)=\bigcup_{n=1}^\infty A_n\cap(A_{1}^c \cap\cdots\cap A_{n-1}^c )\\=\bigcup_{n=1}^\infty A_n\cap(A_{1} \cup\cdots\cup A_{n-1} )^c=\bigcup_{n=1}^\infty A_n\setminus(A_{1} \cup\cdots\cup A_{n-1} )= \bigcup_{n=1}^\infty A_n $$</p>
<p>Using $A\cap B^c=A\s... |
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Joe Silverman | 11,926 | <p>You could try the short book by Hugh Montgomery, which focuses closely on the interactions of harmonic analysis and number theory.</p>
<p><em>Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis</em>
by Hugh L. Montgomery
Series: CBMS Regional Conference Series in Mathematics (Book 84... |
328,670 | <p>Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with coefficient from a field, thanks</p>
| Atom | 673,223 | <p>Take <span class="math-container">$R := \mathbb Q[x]$</span>, <span class="math-container">$I := x\, \mathbb Q[x]$</span> and <span class="math-container">$J := \{a_1 x + a_2 x^2 + \cdots + a_n x^n : a_1\in\mathbb Z\}$</span>. Then <span class="math-container">$I$</span> is an ideal of <span class="math-container">$... |
3,364,016 | <p>Can the following expression be written as the factorial of <span class="math-container">$m$</span>?</p>
<p><span class="math-container">$m(m-1)(m-2) \dots {m-(n-1)}$</span></p>
| azif00 | 680,927 | <p><span class="math-container">$$m(m-1)\cdots (m-n+1) = \frac{m(m-1)\cdots (m-n+1)(m-n)!}{(m-n)!} = \frac{m!}{(m-n)!}$$</span></p>
|
853,774 | <blockquote>
<p>If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$.</p>
</blockquote>
<p>I know that I have to show $G$ is commutative, ie $a * b = b * a$</p>
<p>I have done this by first using $a^{-1}$ on the left, then $b^{-1}$ on the right, and I end up with and expres... | Doug Spoonwood | 11,300 | <p>Here's the proof that I got from Prover9:</p>
<p>1 G(x,y) = G(y,x) # label(non_clause) # label(goal). [goal].</p>
<p>2 G(x,G(y,z)) = G(G(x,y),z). [assumption].</p>
<p>3 G(G(x,y),z) = G(x,G(y,z)). [copy(2),flip(a)].</p>
<p>4 G(x,N(x)) = 1. [assumption].</p>
<p>5 G(x,1) = x. [assumption].</p>
<p>6 G(G(x,y),... |
612,681 | <p>I am a little confused about the decidablity of validity of first order logic formulas. I have a textbook that seems to have 2 contradictory statements. </p>
<p>1)Church's theorem: The validity of a first order logic statement is undecidable.
(Which I presume means one cannot prove whether a formula is valid or not... | André Nicolas | 6,312 | <p>What about if the sentence is not valid? Then the algorithm will not terminate. </p>
<p>Alternately, for any of the usual proof systems, we can enumerate all sequences of sentences, and check mechanically for each sequence whether it is a proof of $\varphi$. Not very useful if $\varphi$ is not a theorem. </p>
<p><... |
11,518 | <p>How to prove that $\mathcal{O}_{\sqrt[3]{3}}$ is an euclidean domain? I heard that one should prove the following but why it is enough?</p>
<p>For any $ a,b,c\in\mathbb{R}$, prove that there are $ x,y,z\in\mathbb{R}$ such that $ x-a,y-b,z-c\in\mathbb{Z}$ and that
$$-1\leq x^3+3y^3+9z^3-9xyz\leq 1.$$</p>
| Bill Dubuque | 242 | <p>As Alex mentioned, the proof that you have in mind amounts to showing that your cubic field is Euclidean with respect to the norm. In fact it is known that there are only three pure cubic fields $\rm\ Q(\sqrt[3] m)\ $ that are norm Euclidean, viz. $\rm\ \mathbb Q(\sqrt[3] 2),\ \mathbb Q(\sqrt[3] 3),\ \mathbb Q(\sqrt... |
1,648,587 | <blockquote>
<p><strong>Problem.</strong> Consider two arcs <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> embedded in <span class="math-container">$D^2\times I$</span> as shown in the figure. The loop <span class="math-container">$\gamma$</span> is obviously nullhomotopic ... | amir bahadory | 204,172 | <p>Let <span class="math-container">$X$</span> denote the complement of <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> in <span class="math-container">$D^{2} \times I .$</span> since the two arcs <span class="math-container">$\alpha, \beta$</span> can deformation retract to... |
181,110 | <p>On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true? </p>
| Rahul Shah | 621,670 | <p>Think of all possible paths as geometrical figures. In a triangle, the sum of any 2 sides is greater than the 3rd side. Hence if you go by any other path other than a straight line, you travel more.</p>
|
4,304 | <p>I am trying to understand <a href="http://en.wikipedia.org/wiki/All-pairs_testing" rel="nofollow noreferrer"><strong>pairwise testing</strong></a>.</p>
<p>How many combinations of tests would be there for example, if</p>
<blockquote>
<p><code>a</code> can take values from 1 to m</p>
<p><code>b</code> can take values... | Douglas S. Stones | 139 | <p>Pairwise testing tests for all possible 2-way interactions efficiently -- I gave a quick overview here: <a href="https://cstheory.stackexchange.com/questions/891/">https://cstheory.stackexchange.com/questions/891/</a></p>
<p>You are looking for strength 2 <a href="http://math.nist.gov/coveringarrays/coveringarray.h... |
2,735,984 | <p>I tried to solve this recurrence by taking out $n+1$ as a common in the RHS, but still have $n \cdot a_n$ and $a_n$</p>
| Kiryl Pesotski | 417,344 | <p>$$a_{n}=\Big(\frac{1}{n}+n\Big)a_{n-1}+3(n+1)$$
Divide by
$$\prod_{k=1}^{n}\Big(\frac{1}{k}+k\Big)$$
To give
$$\frac{a_{n}}{\prod_{k=1}^{n}\big(\frac{1}{k}+k\big)}=\frac{a_{n-1}}{\prod_{k=1}^{n-1}\big(\frac{1}{k}+k\big)}+\frac{3(n+1)}{\prod_{k=1}^{n}\big(\frac{1}{k}+k\big)}$$
Let
$$A_{n}=\frac{a_{n}}{\prod_{k=1}^{n}... |
4,030,359 | <p>Consider the abstract von Neumann algebra
<span class="math-container">$$M:= \ell^\infty-\bigoplus_{i \in I} B(H_i)$$</span>
which consists of elements <span class="math-container">$(x_i)_i$</span> with <span class="math-container">$\sup_i \|x_i\| < \infty$</span> and <span class="math-container">$x_i \in B(H_i)$... | Calvin Khor | 80,734 | <p>Just pointing out that the example in the first linked post is easily modified to produce a function that is non-differentiable at <span class="math-container">$0$</span>, not convex at <span class="math-container">$0$</span>, and <span class="math-container">$f(0)=0$</span> is a strict minimum (i.e. the smoothness ... |
1,255,970 | <p>What is
$$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$
where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure <a href="http://wwwf.imperial.ac.uk/~jswlamb/M345PA46/%5BB%5D%20chap%20IX.pdf" rel="nofollow noreferrer">look here</a>.</p>
<p><strong>Attempt:</strong>
I can evaluate the integral numerically and... | Mark McClure | 21,361 | <p>This "answer" is in response to your comment that you'd be interested in seeing series/product solutions. As I'm sure you know, it's not difficult (in principle) to compute the integral of $x^p$ or $y^p$ with respect to a self-similar measure. (I have Mathematica code that automates the procedure.) Thus, we can g... |
2,005,798 | <p>I have the following equality:
$$
\lim_{k \to \infty}\int_{0}^{k}
x^{n}\left(1 - {x \over k}\right)^{k}\,\mathrm{d}x = n!
$$</p>
<p>What I think is that after taking the limit inside the integral ( maybe with the help of Fatou's Lemma, I don't know how should I do that yet ), then get</p>
<p>$$
\int_{0}^... | Jimmy R. | 128,037 | <p>Just to elaborate on the slick hint of @Arthur, because the OP seems confused in the comments: Since $(-1)^k=-1$ if $k$ is odd and $(-1)^k=1$ if $k$ is even, you have:
\begin{align}0=(1+(-1))^n=\sum_{k=1}^n(-1)^k\dbinom{n}{k}&=\sum_{k\text{ even}}\dbinom{n}{k}-\sum_{k\text{ odd}}\dbinom{n}{k}\\[0.2cm]2^n=(1+1)^n... |
572,316 | <p>I am reading a bit about calculus of variations, and I've encountered the following.</p>
<blockquote>
<p>Suppose the given function <span class="math-container">$F(\cdot,\cdot,\cdot)$</span> is twice continuously differentiable with respect to all of its arguments. Among all functions/paths <span class="math-contain... | Jas Ter | 38,653 | <p>Consider the expression
$$ J(y) := \int_a^b F(x,y,y_x) dx. $$
Here, $y$ is a function: $y:[a,b]\rightarrow \mathbb{R}$.
Note: $y_x$ is a common notation for the partial derivative. Since $y$ depends only on one variable, $y_x \equiv y'$. The above expression abuses notation (in a very common way), and a better versi... |
109,754 | <p>Please help me get started on this problem:</p>
<blockquote>
<p>Let <span class="math-container">$V = R^3$</span>, and define <span class="math-container">$f_1, f_2, f_3 ∈ V^*$</span> as follows:<br />
<span class="math-container">$f_1(x,y,z) = x - 2y$</span><br />
<span class="math-container">$f_2(x,y,z) = x + y + ... | azarel | 20,998 | <p>$\bf Hint:$ Suppose that $a f_1(x,y,z)+b f_2(x,y,z)+c f_3(x,y,z)=0$. Try different values for $(x,y,z)$, for example, if you substitute $(2,1,-3)$. We obtain $f_1(2,1,-3)=0$, $f_2(2,1,-3)=0$ and $f_3(2,1,-3)=10$, hence $cf_3(2,1,-3)=10c$ which implies $c=0$.</p>
|
4,106,933 | <p>I tried <span class="math-container">$\left \vert \frac{\sin x}{x^2} - \frac{\sin c}{c^2}\right \vert \leq \frac{1}{x^2} + \frac{1}{c^2} < \epsilon$</span>, but it doesn't help me much with <span class="math-container">$\vert x - c \vert < \delta$</span>. How can I prove this?</p>
| razixxmaster | 905,366 | <p><span class="math-container">$|f'(x) |=| \frac{cos(x)}{x^2} + \frac{-2sin(x)}{x^3}|\leqslant M+C $</span> , the derivative is bounded from a starting point of x<span class="math-container">$_\beta$</span> . try to continue .
I think the function isn't uniformly continues because the limit as x goes to 0 doesn't ex... |
144,545 | <p>I have this weird integral to find. I am actually trying to find the volume that is described by these two equations.</p>
<p>$$x^2+y^2=4$$ and</p>
<p>$$x^2+z^2=4$$ for</p>
<p>$$x\geq0, y\geq0, z\geq0$$</p>
<p>It is a weird object that has the plane $z=y$ as a divider for the two cylinders. My problems is that I ... | DonAntonio | 31,254 | <p>1.- Try to prove that $y\in G\,,\,yy=y\Longrightarrow y=1$</p>
<p>2.- Now prove, using your notation, that $xg\cdot xg=xg$</p>
<p>DonAntonio</p>
|
144,545 | <p>I have this weird integral to find. I am actually trying to find the volume that is described by these two equations.</p>
<p>$$x^2+y^2=4$$ and</p>
<p>$$x^2+z^2=4$$ for</p>
<p>$$x\geq0, y\geq0, z\geq0$$</p>
<p>It is a weird object that has the plane $z=y$ as a divider for the two cylinders. My problems is that I ... | Dustan Levenstein | 18,966 | <p>The statement is that every $g \in G$ has a right inverse $x$, ie, $gx = 1$. Now the same statement holds in turn for $x$: let $g'$ (suggestively named) be a right inverse for $x$, so that $xg' = 1$. Then on the one hand, using associativity $gxg' = (gx)g'= 1\cdot g' = g'$, but on the other hand, $gxg' = g(xg') = g\... |
1,850,418 | <p>An argument has two parts, the set of all premises, and the conclusion drawn from said premise. Now since there's only 1 conclusion, it would be weird to choose a name for the 'second' part of the argument. However, what is the first part called? I used to think that this was actually called the premise, however tha... | Maxandre Jacqueline | 437,343 | <p>In Model Theory, if you assume $\mathfrak{A}$ to be a model of a language $\mathcal{L}$, you can define a set of sentences $\mathcal{T}$ that is called a theory by letting</p>
<p>$$ \mathcal{T} = \left\{\text{sentence(s) } \sigma \text{ of } \mathcal{L} \text{ such that } \mathfrak{A} \models \sigma \right\}. $$</... |
3,236,205 | <p>I'm studying for a qualifying exam in algebra and I've come across the following problem:</p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a finite group with a subgroup <span class="math-container">$N$</span>. Let <span class="math-container">$Aut(G)$</span> be the group of automorphisms of <sp... | Alexander Gruber | 12,952 | <p><span class="math-container">$N/Z(G)\leq G/Z(G)$</span>. If <span class="math-container">$N\not\leq Z(G)$</span>, what can you say about the order of <span class="math-container">$N/Z(G)$</span>?</p>
|
1,974,114 | <p>Let R be a integral domain with a finite number of elements. Prove that R is a field.</p>
<p>Let a ∈ R \ {0}, and consider the set aR = {ar : r ∈ R}. </p>
<p>Guessing i will have to show that |aR| = R, and deduce that there exists r ∈ R such that ar = 1 but don't know what to do?</p>
| Brian M. Scott | 12,042 | <p>HINT: If $aR\ne R$, there must be distinct $r,s\in R$ such that $ar=as$. (Why?) But then $a(r-s)=\ldots\;?$</p>
|
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Thomas | 26,188 | <p>I used to think that the best way and the only correct way was to sit down for hours get really concentrated. Staying up late at night. Never thinking about anything but math. It was my opinion that only when you get in this zone can you really produce some good thoughts, when you really start to live the problem.</... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | 000 | 22,144 | <p>I'm not a mathematician. I'm actually too young.</p>
<p>However, I wholeheartedly do not recommend coffee. Coffee leads to emotional instability (in my experience and perhaps others) as well increased stress levels. There is also a huge negative effect on the quality of your sleep.</p>
<p>I used to spend a lot of ... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Joachim | 36,074 | <p>I know this is an old question, but i still wanted to add my bit to the topic..</p>
<p>There's a studying method, i forgot the name but i'm sure you can find it with google's help.. It goes like this: study 25 mins, then take a 5 min break. Repeat this 4 times, then take a break of a full half hour.</p>
<p>When yo... |
809,499 | <p>The no. of real solution of the equation $\sin x+2\sin 2x-\sin 3x = 3,$ where $x\in (0,\pi)$.</p>
<p>$\bf{My\; Try::}$ Given $\left(\sin x-\sin 3x\right)+2\sin 2x = 3$</p>
<p>$\Rightarrow -2\cos 2x\cdot \sin x+2\sin 2x = 3\Rightarrow -2\cos 2x\cdot \sin x+4\sin x\cdot \cos x = 3$</p>
<p>$\Rightarrow 2\sin x\cdot ... | juantheron | 14,311 | <p>Given $\sin x+2\sin 2x-\sin 3x = 3\Rightarrow (\sin 2x-1)^2+(\cos 2x+\sin x)^2+(\cos x)^2=0\; \forall x\in (0,\pi)$</p>
<p>So either $\sin 2x = 1$ and $\cos 2x = -\sin x$ and $\cos x=0$</p>
<p>So from above no common value of $x\in (0,\pi)$ for which above equation is satisfied.</p>
<p>So no real values of $x$</... |
4,576,565 | <p>How would you decide whether a tridiagonal matrix with all ones in the diagonals has only a trivial solution (as matrix b is zero in the equation Ax=b)?</p>
<p>Edit: So, a general solution to an n by n matrix of the following appearance:</p>
<p><span class="math-container">$\begin{bmatrix}
1& 1& 0& 0... | Robert Z | 299,698 | <p>Notice that for <span class="math-container">$\alpha\not=1,2$</span>,
<span class="math-container">$$\int _0^{1-x}\frac{1}{\left(x+y\right)^{\alpha}}dy=\left[\frac{(x+y)^{1-\alpha}}{1-\alpha}\right]_{0^+}^{1-x}
=\frac{1-x^{1-\alpha}}{1-\alpha},$$</span>
and therefore (notice that the integral is <em>improper</em> fo... |
25,172 | <p>What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?</p>
| Gerald Edgar | 127 | <p>A classic book is Coddington & Levinson, <em>Theory of Ordinary Differential Equations</em>. Probably tough going for most students.</p>
<p>Coddington, <em>An Introduction to Ordinary Differential Equations.</em> Much shorter, suited to undergraduates, but still rigorous. There is an inexpensive reprinting fr... |
176,691 | <p>Let $A'$ denotes the complement of A with respect to $ \mathbb{R}$ and $A,B,T$ are subsets of $\mathbb{R}$. I am trying to prove $A' \cap (A' \cup B') \cap T= A' \cap T$, but I got some problems along the way.</p>
<p>$A' \cap (A' \cup B') \cap T= (A' \cap A') \cup (A' \cap B') \cap T= A' \cup (A \cup B) \cap T =(A'... | Ink | 34,881 | <p>Let $x \in A' \cap (A' \cup B) \cap T$. Then $x \in A'$ and $x \in T$, so $x \in A' \cap T$. This proves that $A' \cap (A' \cup B) \cap T \subset A' \cap T$. The reverse inclusion is similar.</p>
|
176,691 | <p>Let $A'$ denotes the complement of A with respect to $ \mathbb{R}$ and $A,B,T$ are subsets of $\mathbb{R}$. I am trying to prove $A' \cap (A' \cup B') \cap T= A' \cap T$, but I got some problems along the way.</p>
<p>$A' \cap (A' \cup B') \cap T= (A' \cap A') \cup (A' \cap B') \cap T= A' \cup (A \cup B) \cap T =(A'... | Eugene Shvarts | 34,515 | <p>Dilip's comments point out the flaw re: DeMorgan's Laws.</p>
<p>To avoid that mess altogether, a both-directions-subset proof works concisely. On the one hand, it is clear that $A' \cap (A' \cup B') \cap T \subset A' \cap T$, since the LHS intersects all of the RHS's elements with another set. </p>
<p>On the other... |
2,954,929 | <p>What happens when <span class="math-container">$x < -2$</span> ? Does the whole square root term just "disappear" which leaves us with 1 which is positive and thus the answer to the question is <span class="math-container">$x\le-1$</span>? Or do we have to constrain the domain of <span class="math-container">$x$<... | Community | -1 | <p>Let <span class="math-container">$\mathcal{H}$</span> denote the Hilbert space of wave functions <span class="math-container">$\psi:\mathbb{R}^3\to\mathbb{C}$</span>. Recall that
<span class="math-container">$$\langle u|v\rangle =\iiint_{\mathbb{R}^3}\bar{u}(x,y,z)\ v(x,y,z)\ dx\ dy\ dz$$</span>
for all <span class... |
145,306 | <p>I had a problem on a program of mine that I could avoid by developing the code through other ways. On the other hand, I still do not know how to solve the simple problem below:</p>
<p>Consider these two definitions:</p>
<p>f = p;
p = 2;</p>
<p>One can use Clear[p] to clear the value of p, which will lead the outp... | Eisbär | 32,508 | <p>Try this</p>
<pre><code>p = 2;
With[{a = p}, f = a]
</code></pre>
|
3,678,417 | <p>I understand:
<span class="math-container">$$\sum\limits^n_{i=1} i = \frac{n(n+1)}{2}$$</span>
what happens when we restrict the range such that:
<span class="math-container">$$\sum\limits^n_{i=n/2} i = ??$$</span></p>
<p>Originally I thought we might just have <span class="math-container">$\frac{n(n+1)}{2}/2$</spa... | JonathanZ supports MonicaC | 275,313 | <p>No matter what sequence you're adding up (i.e. no matter what <span class="math-container">$a_i$</span> is), so long as <span class="math-container">$m \lt n$</span> we know that
<span class="math-container">$$\sum^{m-1}_{i = 1} a_i + \sum_{i = m}^n a_i = \sum_{i = 1}^n a_i$$</span></p>
<p>so we can bring the fir... |
754,301 | <p>Say I have the following maximization.</p>
<p><span class="math-container">$$ \max_{R: R^T R=I_n} \operatorname{Tr}(RZ),$$</span>
where <span class="math-container">$R$</span> is an <span class="math-container">$n\times n$</span> orthogonal transformational, and the SVD of <span class="math-container">$Z$</span> is... | Omran Kouba | 140,450 | <p>Let $A=\sqrt{S}$, and equip the space of $n\times n$ real matrices with the usual Euclidean scalar product. Then
$$\hbox{Tr}(RZ)= \hbox{Tr}(RUA^2V^T)=\hbox{Tr}((RUA)(VA)^T)=\langle RUA,VA\rangle$$
By the Cauchy-Schwarz inequality, we get
$$\hbox{Tr}(RZ)\leq \Vert RUA \Vert_2 \Vert VA \Vert_2= \Vert A \Vert_2 \Vert A... |
1,177,605 | <p>Since the problem uses $y^2=x$, I first assumed that the element must be horizontal (parallel to the $x$-axis). However, the bounded region has all $y$ values greater than $0$, so I could also use a vertical element. This problem has me stumped; I know how to set up the integral but for the shell method I need to fi... | Analogue Multiplexer | 241,272 | <p>$\bullet \space \mathbf{SO(3) / SO(2) \simeq S^2}:$</p>
<p>Consider a fundamental representation of the Lie group $G := SO(3)$. Any element $M$ of $G$ can be written as a linear map $M : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $M^{-1} = M^T$ and $\det(M) = 1$. We can easily restrict to $M : S^2 \rightarrow... |
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