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 |
|---|---|---|---|---|
221,762 | <p>Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from <a href="http://wstein.org/books/ribet-stein/main.pdf" rel="nofollow">these notes</a> by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder</p>
<p>1, the ... | Community | -1 | <p>A proof that $|a_p|=p^\frac{k-1}{2}$ in the situation you are considering is given in the proof of Theorem 3 of Winnie Li's paper <em>Newforms and Functional Equations</em>, where it is deduced from a result of Ogg. (She also considers the case in which the character of the newform in question is also a character mo... |
692,376 | <p>Can a vector space over an infinite field be a finite union of proper subspaces ?</p>
| BlackAdder | 74,362 | <p>Short answer: No. </p>
<p>Long answer: No. Assume that our vectorspace $V$ is a finite union of proper subspaces, hence $$V=\bigcup_{i=1}^n U_i.$$</p>
<p>Now, pick a non-zero vector $x\in U_1$, and pick another vector $y\in V\,\backslash U_1$.</p>
<p>There are infinitely many vectors $x+ky$, where $k\in K^*$ ($K$... |
441,448 | <p><strong>Contextual Problem</strong></p>
<p>A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the ... | Robert Mastragostino | 28,869 | <p>First we need to interpret the transformation "physically". What this does is gets the amount of each type of alcohol as input, and spits out the total volume and alcohol volume as output.</p>
<p>I'd be surprised if this has a serious "physical interpretation", because the input and output are of different types. I... |
441,448 | <p><strong>Contextual Problem</strong></p>
<p>A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the ... | user26872 | 26,872 | <p>While the eigenvectors and eigenvalues don't play their usual role in this problem
(as argued in the other answers)
the eigensystem still has a physical interpretation.</p>
<p>The eigenvector $v_2$ is unphysical since it corresponds to a negative volume.</p>
<p>Let $M$ represent the matrix above,
$$M v_1 = \lamb... |
1,225,551 | <p>I have a vector $\mathbf{x}$ (10 $\times$ 1), a vector $\mathbf{f}(\mathbf{x})$ (10 $\times$ 1) and an invertible square matrix $\mathbf{Z}(\mathbf{x})$ (10 $\times$ 10) which is block diagonal:</p>
<p>$$\mathbf{Z}(\mathbf{x}) = \begin{bmatrix}\mathbf{A} & 0\\\mathbf{C} & \mathbf{B}\end{bmatrix}$$</p>
<p>$... | greg | 357,854 | <p><span class="math-container">$\def\v{{\rm vec}}\def\p#1#2{\frac{\partial #1}{\partial #2}}$</span>Assume the following gradients are known
<span class="math-container">$$\eqalign{
J &= \p{f}{x},\qquad
{\mathcal H} = \p{Z}{x} \\
}$$</span>
Note that <span class="math-container">$(J,{\mathcal H})\,$</span> are <sp... |
519,093 | <p>Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$.</p>
<p>Let these be 3 metrics on $C$.</p>
<p>$p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$</p>
<p>$d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$</p>
<p>$t(f,g)=\int_0^1|f(t)-g(t)|dt$</p>
<p>Prove that for every $f,g\in C$, the following holds $t(f,g)\... | Keshav Srinivasan | 71,829 | <p><strong>Hint</strong>: That's equivalent to showing that $y^2 - x^2 = (y+x) (y-x)$ is greater than or equal to $0$. (Try proving that with the axioms.) So you just need to show that $y+x$ and $y-x$ are both greater than or equal to $0$.</p>
|
4,306,419 | <p>Can I write <span class="math-container">$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$</span> implies <span class="math-container">$\frac{dx}{dy}=\frac{x^2-y^2}{2yx}$</span> ?</p>
<p>Actually I have been given two differential equations to solve.</p>
<p><span class="math-container">$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$</span> ... | Amrit Awasthi | 717,462 | <p>Let <span class="math-container">$y=f(x)$</span> represent some function, and assume that it is invertible in a neighbourhood of, and differentiable at, some point <span class="math-container">$(x_0,y_0)$</span> at which <span class="math-container">$\frac{dy}{dx}\neq 0$</span> then the identity holds.
<br />
<... |
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | user | 505,767 | <p>$$\forall \epsilon>0 \ \ :\exists n_0\in \mathbb{N} \ \ \text{such that} \ \ \forall n >n_0 \quad \left|\dfrac{2n^2}{5n^2+1}-\dfrac{2}{5}\right|<\epsilon$$</p>
<p>$$\left|\dfrac{-2}{25n^2+5}\right|<\epsilon\iff \dfrac{2}{25n^2+5}<\epsilon\iff 25n^2+5>\frac{2}{\epsilon}\iff25n^2> \frac{2}{\eps... |
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | M.R. Yegan | 134,463 | <p>$|{\frac{2}{5(5n^2+1)}}|<\frac{2}{25n^2}<\varepsilon$. So $n>\frac{\sqrt2}{5\sqrt\varepsilon}$ and $N=[{\frac{\sqrt2}{5\sqrt\varepsilon}}]+1$</p>
|
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | John Wayland Bales | 246,513 | <p>If you want an $\epsilon,\delta$ proof of the limit then given an $\epsilon>0$ you must find an $N>0$ such that if $n>N$ then</p>
<p>$$ \left\vert \frac{2n^2}{5n^2+1}-\frac{2}{5}\right\vert<\epsilon $$</p>
<p>That is,</p>
<p>$$ \frac{2}{25n^2+5}<\epsilon $$</p>
<p>You may replace the expression on... |
722,663 | <p>I am totally stack in how to do this exercise :
How can I find the irreducible components of $ V(X^2 - XY - X^2 Y + X^3) $ in $A^2$(R)?</p>
<p>Given an algebraic set, what is the consideration I have to make to find them? I know the definitions of components and irreducible components but these don't help in to so... | Peter Crooks | 101,240 | <p>Write your polynomial as a product of irreducible polynomials. The vanishing loci of these irreducible polynomials are then the irreducible components of your variety. </p>
|
2,025,007 | <p>One can show that if $n \geq 3$ is a positive integer, $d=n^2-4$, and $\varepsilon = 1$ if $n$ is odd and $\varepsilon = 0$ if $n$ is even, then the continued fraction expansion of $\frac{\sqrt{d}+\varepsilon}{2}$ has period of even length of the form $(1,n-2)$. One can show that such a continued fraction has perio... | franz lemmermeyer | 23,365 | <p>Start with the unit $\eta = \frac{n + \sqrt{d}}2$. It has norm $+1$, so if there is a unit with norm $-1$ then $\eta$ is a square, say $\eta = \varepsilon^2$ for
$\varepsilon = \frac{a+b\sqrt{d}}2$. This leads to $ab=1$, and there are finitely many possibilities left.</p>
|
3,760,140 | <p>My professor asked us to find the boundary points and isolated points of <span class="math-container">$\Bbb{Q}$</span>; my answer that there is no boundary point or either isolated point for rational numbers set. However the question for confusing me, which asks "to find", means that there exist boundary p... | peek-a-boo | 568,204 | <p>There actually exist infinitely many functions with those properties (these are the basic functions involved in constructing a partition of unity).</p>
<blockquote>
<p>with "finite interval" support [what is this property called?]</p>
</blockquote>
<p>You probably mean "with compact support", whi... |
1,926,383 | <p>Recently I baked a spherical cake ($3$ cm radius) and invited over a few friends, $6$ of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with parallel wedges(and designed to cut $6$ slices at... | Bobson Dugnutt | 259,085 | <p>Let $R$ be the radius of your spherical cake. We will consider the case when we need to divide the cake into $n$ equally sized pieces. </p>
<p>Each piece will then have volume $\frac{4}{3}\frac{\pi R^3}{n}.$ </p>
<p>We can calculate the volume of each slice by setting up a series of integrals, where we integrate o... |
989,740 | <p>How do I prove the following statement?</p>
<blockquote>
<p>If $x^2$ is irrational, then $x$ is irrational. The number $y = π^2$ is irrational. Therefore, the number $x = π$ is irrational</p>
</blockquote>
| Community | -1 | <p><strong>Hint</strong></p>
<p>$$(A\implies B)\iff (\neg B\implies \neg A)$$</p>
|
989,740 | <p>How do I prove the following statement?</p>
<blockquote>
<p>If $x^2$ is irrational, then $x$ is irrational. The number $y = π^2$ is irrational. Therefore, the number $x = π$ is irrational</p>
</blockquote>
| Timbuc | 118,527 | <p>$$\;(A\implies B)\equiv(\neg B\implies\neg A)\;$$</p>
<p>and this means that</p>
<p>($\;x^2\;$ is irrational $\;\implies\;x\;$ is irrational)$\;\equiv (x\;$ is rational $\;\implies\;x^2\;$ is rational) , which seems way simpler to prove.</p>
|
989,740 | <p>How do I prove the following statement?</p>
<blockquote>
<p>If $x^2$ is irrational, then $x$ is irrational. The number $y = π^2$ is irrational. Therefore, the number $x = π$ is irrational</p>
</blockquote>
| Cameron Buie | 28,900 | <p>As written, it doesn't appear that you have a statement that needs proved, but an argument (consisting of <em>multiple</em> statements) that needs validated.</p>
<p>It appears that you have a <a href="http://en.wikipedia.org/wiki/Modus_ponens" rel="nofollow">modus ponens</a> argument. All such arguments are valid, ... |
283,816 | <p>$ 2 \ln (5x) = 16$</p>
<p>$ \ln (5x) = 8 $</p>
<p>$ 5x = e^8 $</p>
<p>$ x = \dfrac {1}{5}e^8$</p>
<p>But why can't we do it like this:</p>
<p>$ \ln(5x)^2 = 16$</p>
<p>I thought that was a possibilty with logaritms?</p>
| André Nicolas | 6,312 | <p>Sure it's possible. We have $\ln((5x)^2)=16$, and therefore $(5x)^2=e^{16}$. Take square roots, remembering that $x$ must be positive for the logarithm to be defined. We get the right answer, a little more slowly than before. </p>
|
283,816 | <p>$ 2 \ln (5x) = 16$</p>
<p>$ \ln (5x) = 8 $</p>
<p>$ 5x = e^8 $</p>
<p>$ x = \dfrac {1}{5}e^8$</p>
<p>But why can't we do it like this:</p>
<p>$ \ln(5x)^2 = 16$</p>
<p>I thought that was a possibilty with logaritms?</p>
| Andrew Maurer | 51,043 | <p>If you mean $\ln\left[(5x)^2\right]$, then yes! </p>
<p>You are allowed to do that!</p>
|
1,946,535 | <p>A Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ and such that $0 < \mid x - \frac{p}{q} \mid < \frac{1}{q^n} $.</p>
<p>I'm looking for either hints or a complete proof for the fact that $e$ is not a Liouvil... | Jack D'Aurizio | 44,121 | <p>Using <a href="https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction" rel="noreferrer">Gauss continued fraction for $\tanh$</a>, it is not difficult to show that the continued fraction of $e$ has the following structure:
$$ e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,\ldots]\tag{1}$$
then by studying the sequence ... |
3,655,215 | <p>There are many contradictions in literature on tensors and differential forms. Authors use the words coordinate-free and geometric. For example, the book Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers say differential forms are coordinate free while tensors are dependendent on coor... | Traves Wood | 1,048,652 | <p>I have a book on tensors that teach both coordinate free (geometric) tensors and coordinate dependent. It is published by Springer and is called<em>Introduction to Tensor Analysis and the Calculus of Moving Surfaces</em> by Pavel Grinfeld. This could be why there is an argument because people think a mathematical t... |
3,655,215 | <p>There are many contradictions in literature on tensors and differential forms. Authors use the words coordinate-free and geometric. For example, the book Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers say differential forms are coordinate free while tensors are dependendent on coor... | Deane | 10,584 | <p>Yes, "in a geometric way" means coordinate-independent. Here's the story I tell my students: Although units (meters, grams, seconds, etc.) are used all the time in physics, we do not believe that the laws of physics change if the units are changed. Coordinates on 2-dimensional or 3-dimsnsional space are ju... |
1,268 | <p>There has long been debate about whether a first year undergraduate course in discrete mathematics would be better for students than the traditional calculus sequence. The purpose of this question is not to further that debate, but to inquire whether any text has tried to teach calculus by emphasizing probability as... | neuron | 660 | <p>I think this one may suit your needs, at least partially: Richard W. Hamming, <em>Methods of Mathematics Applied to Calculus, Probability, and Statistics</em>, <a href="http://store.doverpublications.com/0486439453.html" rel="nofollow">http://store.doverpublications.com/0486439453.html</a></p>
|
2,413,899 | <p>When I was 7 or 8 years old, I was playing with drawing circles. For some reason, I thought about taking a 90° angle and placing the vertex on the circle and marking where the two sides intersected the circle.</p>
<p><a href="https://i.stack.imgur.com/oskEP.png" rel="nofollow noreferrer"><img src="https://i.stack.i... | Satish Ramanathan | 99,745 | <p>Hint:</p>
<p>Put <span class="math-container">$y = 2^{x^2+x}$</span></p>
<p>Integral becomes <span class="math-container">$\int_{1}^{4} \frac{1}{\sqrt{1+ \frac{4}{\ln(2)}\ln(y)}}dy$</span></p>
<p>Again Put <span class="math-container">$\sqrt{1+ \frac{4}{\ln(2)}\ln(y)}= u$</span></p>
<p>Integral becomes <span class="... |
2,413,899 | <p>When I was 7 or 8 years old, I was playing with drawing circles. For some reason, I thought about taking a 90° angle and placing the vertex on the circle and marking where the two sides intersected the circle.</p>
<p><a href="https://i.stack.imgur.com/oskEP.png" rel="nofollow noreferrer"><img src="https://i.stack.i... | Bill Donald | 476,771 | <p>Hint:</p>
<p>Using $u=\frac{2x+1}{2}$ yields an <a href="http://mathworld.wolfram.com/Erfi.html" rel="nofollow noreferrer">imaginary error function</a>.</p>
|
745,876 | <p>$$\exists! x : A(x) \Rightarrow \exists x : A(x)$$
Assuming that $A(x)$ is an open sentence. I'm new to abstract mathematics and proofs, so I came here to ask for some simplification. Thanks</p>
| gt6989b | 16,192 | <p>If there is a unique $x$ such that $A(x)$ is true,</p>
<p>then there must be some $x$ for which $A(x)$ is true.</p>
|
4,230,782 | <blockquote>
<p>Compute <span class="math-container">$\frac 17$</span> in <span class="math-container">$\Bbb{Z}_3.$</span></p>
</blockquote>
<p>We will have to solve <span class="math-container">$7x\equiv 1\pmod p,~~p=3.$</span></p>
<ul>
<li>We get <span class="math-container">$x\equiv 1\pmod 3.$</span></li>
<li>Then <... | SSA | 863,540 | <p>lets use it 3-adic expansion here.</p>
<ul>
<li><em>The theorem says: A rational number with p-adic absolute value 1 has a purely periodic p-adic expansion if and only if it lies in the real interval [-1, 0)</em>. and 1/7 is periodic.</li>
<li>we will use first <span class="math-container">${-\frac{1}{7}}$</span> an... |
2,513,509 | <p>If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is even:</p>
<p>i) If $f$ is even.</p>
<p>ii) If $g$ is even.</p>
<p>iii) If $h$ is even.</p>
<p>iv) Only if all the functions $f,g,h$ are even. </p>
<p>Shall I take different examples of functions and see?</p>
<p>Could anyone ... | lab bhattacharjee | 33,337 | <p>If $\arctan h=u,\sin u=h,\cos u=\sqrt{1-h^2},\tan u=\dfrac h{\sqrt{1-h^2}},u=\arctan\dfrac h{\sqrt{1-h^2}}$</p>
<p>$$\dfrac{\arcsin h-\arctan h}{\sin h-\tan h}=-\cos h\cdot\dfrac{\arctan\dfrac h{\sqrt{1-h^2}}-\arctan h}{\sin h(1-\cos h)}$$</p>
<p>Now $\arctan\dfrac h{\sqrt{1-h^2}}-\arctan h=\arctan\dfrac{\dfrac h{... |
99,312 | <p>Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that</p>
<ul>
<li>$L \cap K' = K$, and</li>
<li>$C(L) \neq \emptyset$?</li>
</ul>
<p>Note that the hypotheses on C are necessary -- the curve x^2 + y^2 = 0, w... | Community | -1 | <p>Yes, this follows from a Theorem of Moret-Bailly, see for example Corollary 1.5</p>
<p><a href="http://math.stanford.edu/~conrad/vigregroup/vigre05/mb.pdf" rel="noreferrer">http://math.stanford.edu/~conrad/vigregroup/vigre05/mb.pdf</a></p>
<p>Roughly speaking, given a finite set $S$ of primes with $C(K_v)$ is non-... |
116,201 | <p>I am curious if the "<a href="http://neumann.math.tufts.edu/~mduchin/UCD/111/readings/architecture.pdf" rel="nofollow noreferrer">Bourbaki's approach</a>" to mathematics is still a viable point of view in modern mathematics, despite the fact that Bourbaki is vilified by many.</p>
<p>Even more specifically, does any... | Nut immerser | 29,893 | <p>Jacob Lurie seems the most obvious answer. His publication history (deep books published in his own time rather than a bunch of small articles) is indeed of the sort that you allude to at the end, but fortunately he had no trouble being offered a suitable position (while still quite young).</p>
|
1,260,945 | <p>$\textbf{My understanding of divergence:}$ Consider any vector field $\textbf{u}$, then $\operatorname{div}(u) = \nabla \cdot u$. More conceptually, if I place an arbitrarily small sphere around any point of the vector field $\textbf{u}$, divergence measures the amount of "particles" exiting the sphere, i.e. positi... | Joffan | 206,402 | <p>The number of <a href="https://en.wikipedia.org/wiki/Quadratic_residue" rel="nofollow noreferrer">quadratic residues</a> <span class="math-container">$\bmod p$</span> (prime), <span class="math-container">$|\mathcal{QR}| = \dfrac{p+1}{2}\qquad$</span> [1].</p>
<p>If we form pairs of residues <span class="math-contai... |
2,369,274 | <p>More specifically, I am trying to solve the problem in Ravi Vakil's notes:</p>
<blockquote>
<p>Make sense of the following sentence: "The map <span class="math-container">$$\mathbb{A}^{n+1}_k \setminus \{0\} \rightarrow \mathbb{P}^n_k$$</span> given by <span class="math-container">$$(x_0,x_1,..x_n) \rightarrow ... | mlc | 360,141 | <p>Both components are concave parabolas, with vertices respectively in $x=2$ and $x=0$. So the first component is increasing for $x<1$ and the second component is decreasing for $x \ge 1$. The function is continuous at $x=1$. Therefore (B) is the correct answer.</p>
|
2,006,565 | <p>I was solving a question to find $\lambda$ and this is the current situation:</p>
<p>$$\begin{bmatrix}
5 & 10\\
15 & 10
\end{bmatrix}
=\lambda\begin{bmatrix}2\\3\end{bmatrix}$$</p>
<p>My algebra suggested that I should divide the right-hand matrix with the left-hand one to find $\lambda$. Then after a few ... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\mathcal{I}\left(x\right)=\int\left(2^2+\arccos\left(\sqrt{x}\right)\right)\space\text{d}x=2^2\int1\space\text{d}x+\int\arccos\left(\sqrt{x}\right)\space\text{d}x$$</p>
<p>Now substitute $u=\sqrt{x}$ and $\text{d}u=\frac{1}{2\sqrt{x}}\space\text{d}x$, after that use integration by parts:</p>
<p>$$\... |
828,148 | <p>In Euclidean geometry, we know that SSS, SAS, AAS (or equivalently ASA) and RHS are the only 4 tools for proving the congruence of two triangles. I am wondering if the following can be added to that list:-</p>
<p>If ⊿ABC~⊿PQR and the area of ⊿ABC = that of ⊿PQR, then the two triangles are congruent.</p>
| user145570 | 145,570 | <p>Yeah you're right, 4 is correct. </p>
|
1,879,440 | <p>I am reading the definition of associative <span class="math-container">$R$</span>-algebra, and am confused about the following definition from Wiki:</p>
<p><a href="https://en.wikipedia.org/wiki/Associative_algebra" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Associative_algebra</a></p>
<p><a href="http... | Alon Amit | 308 | <p>You don't <em>obtain</em> these equalities. They are part of the definition. An $R$-algebra is defined as in the text you quoted, and it is <em>required</em> that those equalities hold true. If they do, you have an $R$-algebra, and if they don't, you don't. </p>
<p>There's no way to derive those equalities from any... |
4,333,062 | <p>I stumbled upon the following identity
<span class="math-container">$$\sum_{k=0}^n(-1)^kC_k\binom{k+2}{n-k}=0\qquad n\ge2$$</span>
where <span class="math-container">$C_n$</span> is the <span class="math-container">$n$</span>th Catalan number. Any suggestions on how to prove it are welcome!</p>
<p>This came up as a ... | ryang | 21,813 | <blockquote>
<p><span class="math-container">$$\sqrt{2k+1}-\sqrt{2k-5}=4$$</span> Somewhere along the way in solving for <span class="math-container">$k$</span> we come to the equation <span class="math-container">$$\sqrt{2k-5} =-\frac{3}{2}.$$</span></p>
</blockquote>
<blockquote>
<p>I would love to be able to say tha... |
1,466,281 | <p>Let $X$ and $Y$ be two absolutely continuous random variables on some probability space having joint distribution function $F$ and joint density $f$. Find the joint distribution and the joint density of the random variables $W=X^2$ and $Z=Y^2$.</p>
<p>I tried the following. I know that $$F(w,y)=P[W \leq w,Z \leq z]... | Graham Kemp | 135,106 | <p>$$\begin{align}\int_a^b\int_c^d f_{X,Y}(x,y)\operatorname d y \operatorname d x & =\int_a^b\int_c^d f_{X}(x) f_{Y\mid X=x}(y)\operatorname d y\operatorname d x \\[1ex] & = \int_a^b f_X(x)(F_{Y\mid X=x}(d)-F_{Y\mid X=x}(c))\operatorname d x \\[1ex] & = F_{X,Y}(b, d) - F_{X,Y}(a, d)- F_{X,Y}(b,c)+F_{X,Y}(a... |
748,442 | <p>I have $\lceil x \rceil = -\lfloor -x \rfloor$, but I can't figure out how to rely on this in order to get $\lfloor x \rfloor$ from $\lceil x−1 \rceil$.</p>
<p>For the record, I am only interested in non-negative real values.</p>
<p>I wish to avoid the use of $modulo$, $abs$, $round$ and $if$.</p>
<p>The motivati... | AlexR | 86,940 | <p>If $x\in\mathbb N$, $\lceil x-1 \rceil = x-1 = \lfloor x \rfloor - 1$, else $\lceil x-1\rceil = \lfloor x \rfloor$. Without additional information, this is impossible.</p>
|
4,140,881 | <p>I want to show the title.</p>
<blockquote>
<p>If <span class="math-container">$M$</span> is a finitely generated module over a local ring <span class="math-container">$A$</span>, then there is a free <span class="math-container">$A$</span>-module <span class="math-container">$L$</span> such that <span class="math-co... | Empy2 | 81,790 | <p>Have you heard of Inclusion-Exclusion?</p>
<p>Start with the full set. <span class="math-container">$68^{10}$</span></p>
<p>Subtract those without numbers <span class="math-container">$58^{10}$</span>, without lowercase <span class="math-container">$39^{10}$</span>, and without uppercase <span class="math-container... |
1,783,837 | <p>I'm studying how to use Laplace transform to solve ODEs. <br/><br/>
I have thought to use this very simple example: $$y'(t)=t+1 \qquad y(0)=0$$</p>
<p>I can use integration to find $y(t)$: $$y(t)=\int (t+1) \ \ dt=\frac{1}{2} t^2+t+C$$</p>
<p>$C \in \mathbb{R} $, $C=0$ for the initial condition, so:</p>
<p>$$y(t... | Jan Eerland | 226,665 | <p>$$y'(t)=t+1\Longleftrightarrow$$
$$\mathcal{L}_t\left[y'(t)\right]_{(s)}=\mathcal{L}_t\left[t+1\right]_{(s)}\Longleftrightarrow$$
$$sy(s)-y(0)=\frac{1}{s^2}+\frac{1}{s}\Longleftrightarrow$$</p>
<hr>
<p>Use $y(0)=0$:</p>
<hr>
<p>$$sy(s)-0=\frac{1}{s^2}+\frac{1}{s}\Longleftrightarrow$$
$$sy(s)=\frac{1}{s^2}+\frac{... |
2,403,851 | <p>Here is Proposition 3 (page 12) from Section 2.1 in <em>A Modern Approach to Probability</em> by Bert Fristedt and Lawrence Gray. </p>
<blockquote>
<p>Let $X$ be a function from a measurable space $(\Omega,\mathcal{F})$ to another measurable space $(\Psi,\mathcal{G})$. Suppose $\mathcal{E}$ is a family of subsets... | Albert | 251,592 | <p>I suggest you use $\ln$ in the paper and $\log$ when you have order results (since the base of $\log$ does not matter, but it is conventional to use $\log$ when stating order results). </p>
|
1,031,464 | <p>I am supposed to simplify this:</p>
<p>$$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)$$</p>
<p>The answer is supposed to be this, but I can not seem to get to it:</p>
<p>$$x(x^2-1)(x^3+1)(5x^3-3x+2)$$</p>
<p>Thanks</p>
| Community | -1 | <p>$$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)\\=x(x^2-1)(x^3+1)\left[3x(x^2-1)+2(x^3+1)\right] \color{blue}{\text{(Factor out common factors)}}\\=x(x^2-1)(x^3+1)\left[3x^3-3x+2x^3+2\right] \color{blue}{\text{ (Distribute inside brackets)}}\\=x(x^2-1)(x^3+1)(5x^3-3x+2) \color{blue}{\text{ (Simplify)}}$$</p>
|
734,203 | <blockquote>
<p>Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$.</p>
</blockquote>
<p>As a note, this is an extra question of a long exercise, the vectors found above are replaced by results I found at the above questions.</p>
<p>Let $\o... | Lutz Lehmann | 115,115 | <p>You can determine the resultant $R(x)=Res_y(y^n-P(x),Q(x,y))$ of both polynomials eliminating $y$. If it is the zero polynomial, then the polynomials have a common factor, which means that for any $x$ you find at least one $y$ so that $(x,y)$ a solution (of the common factor and thus of the system).</p>
<p>If the r... |
4,232,341 | <p>To prove <span class="math-container">$\lim\limits_{n\to\infty} \left(1+\frac{x}{n}\right)^{n}$</span> exists, we prove that the sequence
<span class="math-container">$$f_n=\left(1+\frac{x}{n}\right)^n$$</span>
is bounded and monotonically increasing toward that bound.</p>
<p><strong>Proof Attempt:</strong></p>
<hr ... | Aidan Lytle | 902,851 | <p>I'm not certain that they are necessarily related, other than that there are two dummy variables being integrated up to x and y respectively, and differentiating w/r/t one gives you the other. Maybe I am misunderstanding the question.</p>
<p>After rereading, I think maybe the question is just asking you to different... |
3,427,794 | <p>Let's see I have the following equation</p>
<p><span class="math-container">$$
x=1
$$</span></p>
<p>I take the derivate of both sides with respect to <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$
\frac{\partial }{\partial x} x = \frac{\partial }{\partial x}1
$$</span></p>
<p>The... | Mohammad Riazi-Kermani | 514,496 | <p>You can take derivative of both sides of an identity not an equation.</p>
<p>For example <span class="math-container">$$\sin^2 x + \cos^2 x =1$$</span> is an identity, so we can differentiate to get <span class="math-container">$$2\sin x \cos x -2\sin x\cos x =0$$</span> or <span class="math-container">$$\cos 2x = ... |
1,900,640 | <p>I have a function $$f(x)=(x+1)^2+(x+2)^2 + \dots + (x+n)^2 = \sum_{k=1}^{n}(x+k)^2$$
for some positive integer $n$. I started wondering if there is an equivalent expression for $f(x)$ that can be calculated more directly (efficiently). </p>
<p>I began by expanding some terms to look for a pattern.
$$
(x^2+2x+1) + ... | Community | -1 | <p>Tony already explained the meaning... you take the "frequency" ("probability") of something happening on $[1,n]$, and then you take the limit as $n \to \infty$. This is not a <em>probability</em> in the sense of probability theory. There is no probability measure on a $\sigma$ - algebra on $\Bbb Z$ or $\Bbb N$ under... |
2,108,352 | <p>It is a question in functional analysis by writer Erwin Kryzic </p>
| Ahmed S. Attaalla | 229,023 | <p>In order for it to be a metric it must follow these properties by definition,</p>
<p>$$d(x,y) \geq 0$$</p>
<p>$$d(x,y)=0 \iff x=y$$</p>
<p>$$d(x,y)=d(y,x)$$</p>
<p>$$d(x,z) \leq d(x,y)+d(y,z)$$</p>
<p>Does it?</p>
|
3,746,412 | <blockquote>
<p>Prove: For all sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, ( <span class="math-container">$A \cap B = A \cup B \implies A = B$</span> )</p>
</blockquote>
<p>In the upcoming proof, we make use of the next lemma.</p>
<p>Lemma: For all sets <span class="math-co... | Sahiba Arora | 266,110 | <p>Your proof is correct up to <span class="math-container">$y \notin B.$</span> Note that <span class="math-container">$y \in A \setminus B \implies y \in A \cup B =A\cap B \implies y \in B,$</span> which is a contradiction. This implies <span class="math-container">$A\setminus B=\emptyset$</span> and similarly <span ... |
334,351 | <p>I am a student of mathematics, and have some background in </p>
<ul>
<li>Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), </li>
<li>Differential Geometry (Lee, Kobayashi-Nomizu), </li>
<li>Riemannian Geometry (Do Carmo), </li>
<li>Symplectic Geometry (Ana Cannas da Silva) and </li>
<li>Differential Topology ... | Greg Friedman | 6,646 | <p>I'm not an expert in Floer homology or Khovanov homology, but if that's your goal I don't think you need quite as wide a background as suggested in the other current answer (though admittedly that answer was written when the title of the question was much broader). For example, in the talks I've seen about these thi... |
3,266,367 | <p>Find the solution set of <span class="math-container">$\frac{3\sqrt{2-x}}{x-1}<2$</span></p>
<p>Start by squaring both sides
<span class="math-container">$$\frac{-4x^2-x+14}{(x-1)^2}<0$$</span>
Factoring and multiplied both sides with -1
<span class="math-container">$$\frac{(4x-7)(x+2)}{(x-1)^2}>0$$</span>... | Allawonder | 145,126 | <p>The radicand cannot be negative, so we must have <span class="math-container">$x\le 2.$</span> Also, the denominator is negative for <span class="math-container">$x\lt 1.$</span> Thus, you must consider this in the two cases when</p>
<p>(1) <span class="math-container">$x\lt 1,$</span> or when</p>
<p>(2) <span cla... |
106,560 | <p>Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?</p>
| Vesselin Dimitrov | 26,522 | <p><em>Last revision: 10/20.</em> (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )</p>
<p><strong>Completely rewritten. (9/26)</strong></p>
<p>It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-... |
155,237 | <p>The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For instance, in $L$, there is a definable well-ordering of the real numbers (since there is a definable well-ordering of the ... | Asaf Karagila | 7,206 | <p>Generally formulas in set theory are not "very nice". There is very little structure to work with.</p>
<p>We can define when $U$ is a filter over $X$, simply by saying that every element of $U$ is a subset of $X$, and $U$ is closed under superset (below $X$), and under finite intersections (recall that in set theor... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | dustin | 78,317 | <p>Regardless of the gratitude argument, accepting answers makes the sites questions easier to sift through. When there are questions that don't have accepted answers, people are more inclined to use their time to peruse the question and answers. If then they have wasted their time due to there being one or more perfec... |
3,841,266 | <p>Let <span class="math-container">$E$</span> and hilbert space and <span class="math-container">$f(x)=\| x\|$</span> for all <span class="math-container">$x\in E$</span>. Study the differentiability of <span class="math-container">$f$</span> on <span class="math-container">$0$</span> and find <span class="math-conta... | hardmath | 3,111 | <p>Given your background, I'll make some suggestions about how one might try to solve:</p>
<p><span class="math-container">$$ \ln |y| + y^2 = \sin(x) + 1 $$</span></p>
<p>for <span class="math-container">$y$</span> when <span class="math-container">$x$</span> is known.</p>
<p>There are many root finding algorithms, and... |
3,841,266 | <p>Let <span class="math-container">$E$</span> and hilbert space and <span class="math-container">$f(x)=\| x\|$</span> for all <span class="math-container">$x\in E$</span>. Study the differentiability of <span class="math-container">$f$</span> on <span class="math-container">$0$</span> and find <span class="math-conta... | Narasimham | 95,860 | <p>If you have not done numerical methods so far (Regula Falsi, Newton Raphson &c.) then an approximate way is to paper plot the graph of (inverse) function and from it read off approximate <span class="math-container">$y$</span> for a given <span class="math-container">$x:$</span></p>
<p><span class="math-conta... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | Community | -1 | <p><span class="math-container">$x^p-x$</span> is a degree <span class="math-container">$p$</span> polynomial. Whereas for <span class="math-container">$a\in\mathbb Z_p$</span> we have Fermat's little theorem, <span class="math-container">$x^p=x$</span> is not true in general. </p>
<p>In fact (by the factor theore... |
216,473 | <p>Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I us... | Morten Jensen | 41,838 | <p>I took a bachelors in software engineering at an engineering college. 3 years coursework/projects and a 6 month obligatory internship -- I'm from Denmark in Scandinavia so YMMV.</p>
<p>We had calculus, algorithms, discrete maths and circuit analysis. We had theory and application but with little emphasis on the pro... |
3,105,184 | <p>Three shooters shoot at the same target, each of them shoots just once. The three of them respectively hit the target with a probability of 60%, 70%, and 80%. What is the probability that the shooters will hit the target</p>
<p>a) at least once</p>
<p>b) at least twice</p>
<hr>
<p>I have an approach to this but ... | jvdhooft | 437,988 | <p>Let <span class="math-container">$X$</span> be the number of times the target is hit. The probability <span class="math-container">$P(X \ge 1)$</span> then equals 1 minus the probability of missing the target three times:</p>
<p><span class="math-container">$$P(X \ge 1) = 1 - (1 - P(A)) (1 - P(B)) (1 - P(C)) = 1 - ... |
1,847,609 | <blockquote>
<p>Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent?</p>
</blockquote>
<p>If $R$ is Noetherian and $J(R)$ is idempotent the Nakayama lemma yields $J(R)=0$. So, for Noetherian rings whose Jac... | E.R | 325,912 | <p>Let $a$ be an element of $R$. If $\left<a\right>=\left<a\right>^2$, then it is clear that the exists $e^2=e\in \left<a\right>$ such that $\left<a\right>=\left<e\right>$. Now, if $0\not= a\in J (R)$, then $\left<a\right>$ is nilpotent or idempotent. If $\left<a\right>$ is ide... |
1,847,609 | <blockquote>
<p>Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent?</p>
</blockquote>
<p>If $R$ is Noetherian and $J(R)$ is idempotent the Nakayama lemma yields $J(R)=0$. So, for Noetherian rings whose Jac... | Jason Juett | 214,136 | <p>As noted in Rostami's answer, you do get that $J(R)$ is nil, hence is nilpotent if it is finitely generated. (Here is an alternative proof, just for fun. Since idempotents are locally 0 or 1, these assumptions imply $J(R) = $nil$(R)$ holds locally, hence globally.)</p>
<p>But here is the main point of my answer: ... |
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | Gosia | 5,372 | <p>If the four numbers are to add to 1, then only three of them can be 'random'.</p>
<pre><code> In[98]:= sim = {1, 1, 1};
While[6/10 > Total[sim] || Total[sim] > 9/10,
sim = RandomReal[{1/10, 6/10}, 3, WorkingPrecision -> 2]
]
In[100]:= sim
Out[100]= {0.18, 0.37, 0.23}
In[... |
636,359 | <p>How do you proof that $\angle CTP = \angle CBP+ \angle BCN$ ?
Please check the image below:</p>
<p><img src="https://i.stack.imgur.com/SqeXa.png" alt="enter the great image description here"></p>
| vadim123 | 73,324 | <ol>
<li>angles in a triangle sum to 180 degrees</li>
<li>supplementary angles sum to 180 degrees</li>
</ol>
|
3,612,916 | <p>Find a function <span class="math-container">$f$</span> such that <span class="math-container">$f(x)=0$</span> for <span class="math-container">$x\leq0$</span>, <span class="math-container">$f(x)=1$</span> for <span class="math-container">$x\geq1$</span>, and <span class="math-container">$f$</span> is infinitely dif... | AlanD | 356,933 | <p>Try
<span class="math-container">$$
f(x)=
\begin{cases}
0, & x\leq 1\\
C\int_0^x \exp\left(\frac{r}{t(t-1)}\right)\,dt,& 0<x<1 \\
1, &x\geq 1
\end{cases}
$$</span>
for <span class="math-container">$r>0$</span>. What is <span class="math-container">$C$</span>?</p>
|
1,916,232 | <p>I have massive problems with questions like these:</p>
<p>Let $\{v_1, . . . , v_r\}$ be a set of linearly independent vectors in $\mathbb{R}^n$
(with
$r < n$), and let $w\in\mathbb{R}^n$ be a vector such that $w \in \mathrm{span}\{v_1, . . . , v_r\}$.
Prove that $\{v_1, . . . , v_r, w\}$ is a linearly independe... | JMP | 210,189 | <p>Linear Algebra questions are usually based on the definition of linear independence. Once you get the hang of it, the problems make much more sense.</p>
<p>For a good on-line resource, you could try <a href="https://www.khanacademy.org/math/linear-algebra" rel="nofollow">Khan Academy</a>.</p>
|
1,438,512 | <p>I was given this problem at school to look at home as a challenge, after spending a good 2 hours on this I can't seem to get further than the last part of the equation. I'd love to see the way to get through 2) before tomorrow's lesson as a head start.</p>
<p>So the problem is as follows:</p>
<p>1) Quadratic Equat... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>You must calculate $S=\alpha^4+\dfrac1{\beta^2}+\beta^4+\dfrac1{\alpha^2}$ and $P=\Bigl(\alpha^4+\dfrac1{\beta^2}\Bigr)\Bigl(\beta^4+\dfrac1{\alpha^2}\Bigr)$. They will be the roots of the quadratic equation $\;x^2-Sx+P=0$.</p>
<p>Now any symmetric rational function of $\alpha$ and $... |
2,830,969 | <p>I did not find an example when the denominator $x$ approximates to $0$.</p>
<p>$f(0) + f'(0)x$ does not work because $f(0)$ would be $+\infty$. </p>
| James | 549,970 | <p>Well, $f(x) = \frac{1}{x}$ is not even defined at $x=0$ which obviously means it can't be differentiated and in particular we cannnot find a linear approximation on that point.</p>
|
4,512,164 | <p>Given that correlation coefficient between X and Y is (, ) and a, b, c, d are constants, prove that the correlation coefficient between U = + and = + is equal to (, ) = (, )</p>
<p>My take on the problem:</p>
<p>Prove that <span class="math-container">$p(U, V)=p(X, Y)$</span>
<span class="math-container">$$
\be... | user1080911 | 1,080,911 | <p>Hint: use <span class="math-container">$\rho(X,Y)=Cov(\frac{X-\mathbb{E}X}{\sigma(X)},\frac{Y-\mathbb{E}Y}{\sigma(Y)})$</span>.</p>
<p>Should be a one line exercise. Here <span class="math-container">$\sigma$</span> is the standard deviation. Explore the properties of variance and mean value.</p>
<blockquote class="... |
4,512,164 | <p>Given that correlation coefficient between X and Y is (, ) and a, b, c, d are constants, prove that the correlation coefficient between U = + and = + is equal to (, ) = (, )</p>
<p>My take on the problem:</p>
<p>Prove that <span class="math-container">$p(U, V)=p(X, Y)$</span>
<span class="math-container">$$
\be... | Clarinetist | 81,560 | <p>A huge problem with what you're doing is you're assuming what you want to show to start with, which is a bad practice. You should also make use of covariance <a href="https://en.wikipedia.org/wiki/Covariance#Properties" rel="nofollow noreferrer">properties</a> to make your life easier.</p>
<p>Let <span class="math-c... |
3,460,749 | <p>I'm an undergraduate student currently studying mathematical analysis. </p>
<p>Our professor uses Zorich's Mathematical Analysis, but I found the text too difficult to understand. </p>
<p>After exploring some textbooks, I found that Abbott was easier to follow, so I studied Abbott until I realized that there's a s... | UESTCfresh | 798,645 | <p>Herbert Amann's analysis which has three volumes is more detailed than zorich's analysis.However, I think it is too difficult to read.</p>
|
2,533,186 | <p>How would you find all pairs $(z,w)$ that satisfy $zw=-1$ </p>
<p>Im stuck on this problem and the best way I can think of would be to guess and check?</p>
| Jepsilon | 505,973 | <p>For $zw=-1$ it is the multiplicative inverse in the complex field but with a minus sign:</p>
<p>$w=-\frac{\bar{z}}{z\bar{z}}=$ and $z=-\frac{\bar{w}}{w\bar{w}}=$ for $w,z\neq0$</p>
|
1,511,518 | <p>Let <span class="math-container">$GL_n^+$</span> be the group of <span class="math-container">$n \times n$</span> real invertible matrices with positive determinant</p>
<p>Does there exist a left invariant isotropic Riemannian metric on <span class="math-container">$GL_n^+$</span>?</p>
<p>(By "isotropic" I mean th... | Qiaochu Yuan | 232 | <p>$GL_n^{+}$ deformation retracts onto $SO(n)$. The universal cover of this (for $n \ge 3$) is a double cover, since $\pi_1(SO(n)) = \mathbb{Z}_2$. It's known as the Spin group $Spin(n)$, and in particular, because it's a double cover it's compact. Hence it's certainly not homotopy equivalent to a Euclidean space (e.g... |
1,266,525 | <p>I'm shown part of the function $g(x) = \sqrt{4x-3}$. Is it injective? I said yes as per definition if $f(x) = f(y)$, then $x =y$. Is this right? </p>
<p>Under what criteria is $g(x)$ bijective? For what domain and co domain does $g(x)$ meet this criteria? I'm totally lost on this one! </p>
<p>Find the inverse of $... | ajotatxe | 132,456 | <p>It really makes no sense to ask whether a function is surjective if you don't have the codomain.</p>
<p>That is: you have something like $f(x)=\sqrt{4x-3}$. If you are working with real numbers, $x$ must be $\ge\frac34$ in order to the square root makes sense. Furthermore, the values that takes $f(x)$ are nonnegati... |
345,260 | <p>I'm trying to prove that there exists a multiplicative linear functional in $\ell_\infty^*$ that extends the limit funcional that is defined in $c$ (i.e., im looking for a linear functional $f \colon \ell_\infty \to \mathbb K$ such that $f( (x_n * y_n) ) = f (x_n) f(y_n)$, for every $(x_n), (y_n) \in \ell_\infty$, a... | Nate Eldredge | 822 | <p>This argument avoids the use of ultrafilters.</p>
<p>Let $e_n \in \ell_\infty^*$ be the evaluation map $e_n(x) = x_n$. The set $\{e_n\}$ is contained in the unit ball of $\ell_\infty^*$, which by Alaoglu's theorem is weak-* compact, hence $\{e_n\}$ has a weak-* cluster point; call it $f$. I claim this $f$ has the... |
345,260 | <p>I'm trying to prove that there exists a multiplicative linear functional in $\ell_\infty^*$ that extends the limit funcional that is defined in $c$ (i.e., im looking for a linear functional $f \colon \ell_\infty \to \mathbb K$ such that $f( (x_n * y_n) ) = f (x_n) f(y_n)$, for every $(x_n), (y_n) \in \ell_\infty$, a... | Chao You | 62,630 | <p>I have done similar work before, which is included in the paper: Chao You, <em>A note on $\tau$-convergence, $\tau$-convergent algebra and applications</em>, Topology Appl. 159, No. 5, 1433-1438 (2012). It may happen that you cannot define a multiplicative linear functional on $l^{\infty}(\mathbb{N})$, but on a suba... |
423,158 | <p>Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times R\longrightarrow R$ such that $f(x,y)=x^2y+1$", or for example "Let $g:A\longrightarrow A\times \mathbb{N}$ such that for every ... | Zen | 72,576 | <p>In general it suffices to prove that for your function, each $x$ in the domain maps to exactly one $f(x)$ in the codomain. Assuming the existence of a function is therefore equivalent to assuming this property, which is rather trivial to prove in most cases (but not all; Cantor-Bernstein is one notable example where... |
423,158 | <p>Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times R\longrightarrow R$ such that $f(x,y)=x^2y+1$", or for example "Let $g:A\longrightarrow A\times \mathbb{N}$ such that for every ... | Dan Christensen | 3,515 | <p>Your first example is the just the composition of several real-valued functions. For simplicity, lets recast it as $f(x)=xxy+1$. It is just the composition of known functions: the multiplication and addition of real numbers. Multiplication maps every pair of real numbers to a real number. Likewise for addition. (Thi... |
4,563,725 | <p>Is it possible for two non real complex numbers a and b that are squares of each other? (<span class="math-container">$a^2=b$</span> and <span class="math-container">$b^2=a$</span>)?</p>
<p>My answer is not possible because for <span class="math-container">$a^2$</span> to be equal to <span class="math-container">$b$... | Misha Lavrov | 383,078 | <p>The arguments <span class="math-container">$\arg(a)$</span> and <span class="math-container">$\arg(b)$</span> are only defined up to a multiple of <span class="math-container">$2\pi$</span>. Or, if we require that <span class="math-container">$\arg(z) \in (-\pi,\pi]$</span> for all <span class="math-container">$z$</... |
369,435 | <p>My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when $|x|<1$):$$\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)+(x^2+1)}=\frac{1}{1-(-x)}\cdot\frac{1}{1-(-x^2)}=$$$$=\sum_{k=0}^{\infty}(-x)^k\cdot \sum_{k=0}^{\infty}(-x^2)^k$$ Now I try to calculate it the following w... | Fly by Night | 38,495 | <p>If I recall, if you have two power series, based at the same point, then the radius of convergence of their product is at least the smaller of the two radii of convergence. Generally it will be the smaller of the two radii of convergence. The at least part comes from the possibility of numerators in one cancelling w... |
1,632,677 | <p>How can I arrive at a series expansion for $$\frac{1}{\sqrt{x^3-1}}$$ at $x \to 1^{+}$? Experimentation with WolframAlpha shows that all expansions of things like $$\frac{1}{\sqrt{x^y - 1}}$$ have $$\frac{1}{\sqrt{y}\sqrt{x-1}}$$ as the first term, which I don’t know how to obtain.</p>
| marty cohen | 13,079 | <p>I like to expand around zero.
So, in
$\frac{1}{\sqrt{x^3-1}}
$,
let $x = 1+y$.
Then</p>
<p>$\begin{array}\\
x^3-1
&=(1+y)^3-1\\
&=1+3y+3y^2+y^3-1\\
&=3y+3y^2+y^3\\
&=y(3+3y+y^2)\\
\end{array}
$</p>
<p>so</p>
<p>$\begin{array}\\
\frac{1}{\sqrt{x^3-1}}
&=\frac{1}{\sqrt{y(3+3y+y^2)}}\\
&=\fra... |
96,952 | <p>How would I go about solving the following for <code>c</code>?</p>
<pre><code>Solve[0 == Sum[(t[i]*m[i] - c*t[i]^2)/s[i]^2, {i, 1, n}], c, Reals]
</code></pre>
<p>I get the error </p>
<blockquote>
<p>Solve::nsmet : This system cannot be solved with the methods available to Solve</p>
</blockquote>
<p>but it is ... | A.G. | 7,060 | <p>If your goal is to check the work you do on paper you may want to go this route :</p>
<p>First define</p>
<pre><code>c[n_Integer] :=
Sum[m[i]*t[i]/s[i]^2, {i, 1, n}]/Sum[t[i]^2/s[i]^2, {i, 1, n}];
ShouldBeZero[n_Integer] :=
Sum[(t[i]*m[i] - c[n]*t[i]^2)/s[i]^2, {i, 1, n}] // FullSimplify
</code></pre>
<p>th... |
62,444 | <p>I have not seen this question treated in the literature. Does anyone have more information? There are several OEIS sequences (<a href="http://oeis.org/A097056" rel="noreferrer">A097056</a>, <a href="http://oeis.org/A117896" rel="noreferrer">A117896</a>, <a href="http://oeis.org/A117934" rel="noreferrer">A117934</a>)... | André Henriques | 5,690 | <p>From <a href="http://oeis.org/A097056/internal" rel="nofollow">http://oeis.org/A097056/internal</a>:<br>
<i>"Empirically, there seem to be no intervals between consecutive squares containing more than two nonsquare perfect powers."</i></p>
<p>is probably the closest you'll ever get to an answer.</p>
|
62,444 | <p>I have not seen this question treated in the literature. Does anyone have more information? There are several OEIS sequences (<a href="http://oeis.org/A097056" rel="noreferrer">A097056</a>, <a href="http://oeis.org/A117896" rel="noreferrer">A117896</a>, <a href="http://oeis.org/A117934" rel="noreferrer">A117934</a>)... | Gjergji Zaimi | 2,384 | <p>If you let $a_1,a_2,\dots$ represent the sequence $1,4,8,9,\dots$ of perfect powers, it is a conjecture of Erdos that $$a_{n+1}-a_n >c'n^c.$$
The weaker conjecture $$\liminf a_{n+1}-a_n =\infty$$ is known as Pillai's conjecture.
This seems to be much beyond the reach of the strongest methods available today. Not... |
62,444 | <p>I have not seen this question treated in the literature. Does anyone have more information? There are several OEIS sequences (<a href="http://oeis.org/A097056" rel="noreferrer">A097056</a>, <a href="http://oeis.org/A117896" rel="noreferrer">A117896</a>, <a href="http://oeis.org/A117934" rel="noreferrer">A117934</a>)... | Felipe Voloch | 2,290 | <p>It follows from the ABC conjecture that there are only finitely many. First, show (unconditionally) that for all but finitely many such triples of perfect powers, at most one of them is a cube. So, the others are fifth or higher powers, say $n^2<A<B<(n+1)^2$. Now, apply ABC to $B-A=C$. Then $B>n^2, C = O... |
2,227,135 | <p>I have this information from my notes:<span class="math-container">$\def\rk{\operatorname{rank}}$</span></p>
<p>Let <span class="math-container">$A ∈ \mathbb{R}^{m\times n}$</span>. Then </p>
<ul>
<li><span class="math-container">$\rk(A) = n$</span></li>
<li><span class="math-container">$\rk(A^TA) = n$</span></li>... | Harambe | 357,206 | <p>You don't need to use any machinery to prove this.
The <strong>rank</strong> of a matrix is the <strong>dimension of the column space</strong> of the matrix.
It's easy to see that
\begin{align*}
vv^T =
\begin{pmatrix}
v_1v_1 & v_1v_2 & \cdots & v_1v_n\\
v_2v_1 & v_2v_2 & \cdots & v_2v_n\\
\vd... |
2,830,926 | <p>I am implementing a program in C which requires that given 4 points should be arranged such that they form a quadrilateral.(assume no three are collinear)<br>
Currently , I am ordering the points in the order of their slope with respect to origin.<br>
See <a href="https://ibb.co/cfDHeo" rel="nofollow noreferrer">htt... | heropup | 118,193 | <p>Another way to do this is to observe that there is a relationship between the moment generating function (MGF) and the <strong>probability generating function</strong> (PGF); namely, $$m_X(\log u) = \operatorname{E}[e^{X \log u}] = \operatorname{E}[e^{\log u^X}] = \operatorname{E}[u^X] = P_X(u).$$ Furthermore, the ... |
2,727,974 | <blockquote>
<p>I am in need of this important inequality $$\log(x+1)\leqslant x$$.</p>
</blockquote>
<p>I understand that $\log(x)\leqslant x$. For $c\in\mathbb{R}$. However is it true that $\log(x+c)\leqslant x$?</p>
<p>It is hard to accept because it seems like $c$ cannot be arbitrary.
I have tried to prove thi... | Hagen von Eitzen | 39,174 | <p><strong>Hint:</strong> The claim is equivalent to $$ e^x\ge 1+x$$</p>
|
1,900,788 | <blockquote>
<p>The probability of rain in Greg's area on Tuesday is $0.3$. The probability that Greg's teacher will give him a pop quiz in Tuesday is $0.2$. The events occur independently of each other. What is the probability of neither events occur?</p>
</blockquote>
<p>My approach: </p>
<p>Probability of rain o... | Reese Johnston | 351,805 | <p>Your approach double-counts some possibilities. The $0.3$ covers all possibility of rain - so, it includes both "rain and quiz" and "rain and no quiz". The $0.2$ covers all possibility of a quiz, so it includes both "rain and quiz" and "quiz and no rain". So $0.3 + 0.2$ covers "rain and quiz", "rain and no quiz", "r... |
72,012 | <p>I'm trying to discretize a region with "pointy" boundaries to study dielectric breaking on electrodes with pointy surfaces. So far I've tried 3 types of boundaries, but the meshing functions stall indefinitely. I've let it run for several hours and it doesn't complete. Seems strange that it would take so long</p>
<... | Michael E2 | 4,999 | <pre><code>fn = {-Cos[u] (1.2 - Cos[(u - Pi)/2]^6) (0.2 + Cos[10 u]^10),
Sin[u] (1.2 - Cos[(u - Pi)/2]^6) (0.2 + Cos[10 u]^10)};
plot =
ParametricPlot[fn, {u, 0, 2 Pi},
PlotPoints -> Round[2 Pi (Sqrt@MaxValue[#.# &@D[fn, u],
u])/0.2], PlotRange -> All];
Cases[plot, Line[p_] :> Polygon[... |
2,078,142 | <p>Let $X$ be locally compact metric space which is $\sigma$-compact also. I want to show that $X=\bigcup\limits_{n=1}^{\infty}K_n$, where $K_n$'s are compact subsets of $X$ satisfying $K_n\subset K_{n+1}^{0}$ for all $n\in \mathbb N$. </p>
<p>I know that since $X$ is $\sigma-$compact, therefore there exists a sequenc... | bof | 111,012 | <p>Since $X$ is locally compact, every compact subset of $X$ is contained in an open set whose closure is compact.</p>
<p>Since $X$ is $\sigma$-compact, there are compact subsets $C_n$ such that $X=\bigcup_{n=1}^\infty C_n.$</p>
<p>Since $C_1$ is compact, there is an open set $U_1$ such that $\overline{U_1}$ is compa... |
195,556 | <p>My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.</p>
<p>In recursive function theory one studies <a href="http://en.wikipedia.org/wiki/Partial_functions">partial functions</a> on the set of natural numbers. </p>
<p>Are there other areas of mathematics in which (... | William | 13,579 | <p>I think that the reason you have this particular interests in partial function or even heard of the term is mostly because you have read recursion theory. </p>
<p>To elaborate, in most other areas of mathematics, function are by definition totally defined on their domain. To say that $f : X \rightarrow Y$ but $f$ i... |
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| Michael Hardy | 11,667 | <p>$$
\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)} = \frac{6\ln^2 2x}{2x} \cdot \frac{2\sqrt x}{3} $$</p>
<p>Then use $\dfrac 6 3 = 2$ and $\dfrac {\sqrt x} x = \dfrac{\sqrt x}{\sqrt x \cdot\sqrt x} = \dfrac 1 {\sqrt x}$.</p>
|
1,691,825 | <p>My textbook goes from</p>
<p>$$\frac{\left( \frac{6\ln^22x}{2x} \right)}{\left(\frac{3}{2\sqrt{x}}\right)}$$</p>
<p>to:</p>
<p>$$\frac{6\ln^22x}{3\sqrt{x}}$$</p>
<p>I don't see how this is right. Could anyone explain?</p>
| John_dydx | 82,134 | <p>The textbook has <a href="http://www.bbc.co.uk/education/guides/z7fbkqt/revision/2" rel="nofollow">rationalised the denominator</a> for $\frac{3}{2\sqrt{x}}$ before simplifying further </p>
<p>i.e.</p>
<p>$$\frac{3}{2\sqrt{x}} = \frac{3\times \sqrt{x}}{2\sqrt{x}\times \sqrt{x}}= \frac{3\sqrt{x}}{2x}$$</p>
<p>Now ... |
732,334 | <p>How can we solve this equation?
$x^4-8x^3+24x^2-32x+16=0.$ </p>
| Community | -1 | <p>To start, this polynomial is called a <strong>biquadratic equation</strong>.</p>
<p>First, factor out the polynomial.
$$x^4 - 2x^3 - 6x^3 + 12x^2 + 12x^2 - 24x - 8x + 16 = 0$$
$$x^3(x - 2) - 6x^2(x - 2) + 12x(x - 2) - 8(x - 2) = 0$$
$$(x^3 - 6x^2 + 12x - 8)(x - 2) = 0$$</p>
<p>Then after, continue the factorizati... |
2,082,815 | <p>Find the $100^{th}$ power of the matrix $\left( \begin{matrix} 1& 1\\ -2& 4\end{matrix} \right)$.</p>
<p>Can you give a hint/method?</p>
| gowrath | 255,605 | <p>You want to find the <a href="https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix" rel="nofollow noreferrer">eigen decomposition</a> for the matrix and transform it into a diagonal matrix (for which computing powers is easy). </p>
<p>The eigenvectors of the matrix $M$ given by </p>
<p>$$
M = \left( \begin... |
2,983,199 | <p>I am looking for an example demonstrating that <span class="math-container">$\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$</span> but for the life of me i can't find one. any suggestions?</p>
| Alex Ortiz | 305,215 | <p>Try some alternating sequences where <span class="math-container">$\liminf x_n = \liminf y_n = -1$</span> but <span class="math-container">$x_n + y_n = 0$</span> for each <span class="math-container">$n$</span>.</p>
|
1,091,653 | <p>Is this correct ?</p>
<p>$$
\frac{d}{dt} \left( \int_0^t \phi(t)dt \right) = \phi(t)
$$</p>
<p>If not, how can I recover $$ \phi(t) $$ knowing only $$ \int_0^t \phi(t)dt $$ ?</p>
| Scientifica | 164,983 | <p>Let $\Phi$ the antiderivative of $\phi$ ($\Phi'(t)=\phi(t)$) such as $\Phi(0)=0$ </p>
<p>We have $\Phi(t)=\int_0^t \phi(t)dt$ thus $\dfrac{d}{dt}\left(\int_0^t \phi(t)dt\right)=\dfrac{d\Phi}{dt}=\phi(t)$</p>
|
1,787,460 | <blockquote>
<p>Suppose the n<em>th</em> pass through a manufacturing process is modelled by the linear equations <span class="math-container">$x_n=A^nx_0$</span>, where <span class="math-container">$x_0$</span> is the initial state of the system and</p>
<p><span class="math-container">$$A=\frac{1}{5} \begin{bmatrix} 3... | Dark | 208,508 | <p><strong>Hint :</strong></p>
<p>$A^{n}=(P^{-1} D P)^{n}=P^{-1} D^{n} P$</p>
|
2,965,756 | <p>I want to intuitively say tht the answer is yes, but if it so happens that <span class="math-container">$|\mathbf{a}|=|\mathbf{b}|cos(\theta)$</span>, where <span class="math-container">$\theta$</span> is the angle between the two vectors, then the equation will be satisfied without the two vectors being the same.</... | YukiJ | 380,302 | <p>As a simple counter-example see</p>
<p><span class="math-container">$$\begin{pmatrix}1\\2\\3\end{pmatrix}.\begin{pmatrix}1\\2\\3\end{pmatrix}=14 = \begin{pmatrix}1\\2\\3\end{pmatrix}.\begin{pmatrix}1\\5\\1\end{pmatrix}$$</span>
and clearly </p>
<p><span class="math-container">$$\begin{pmatrix}1\\2\\3\end{pmatrix} ... |
2,947,911 | <p>I have a homework with this equation:
<span class="math-container">$$(x+2y)dx + ydy = 0$$</span></p>
<p>However I have no idea how to solve it.
I tried couple things:</p>
<ol>
<li>Is it linear equation, or does it have "standard form" : <span class="math-container">$\frac{dy}{dx}+\frac{x}{y} + 2 = 0$</span>. Well... | Graham Kemp | 135,106 | <p>Without a falsum constant, your rule of <em>ex falso quodlibet</em> should look something like <span class="math-container">$\phi, \neg \phi \vdash \psi$</span>. So from the contradictory assumptions, <span class="math-container">$\neg\neg a$</span> and <span class="math-container">$\neg a$</span>, you may im... |
306,744 | <p>So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because it's first talking about the $\epsilon$ and then it talks of the $\delta$ condition. Would it be equivalent to say: ... | Cameron Buie | 28,900 | <p>Let's translate it into words, to see how they compare. Continuity at a given point (in this context) means that if we want to keep the $y$-coordinate within a certain interval (no matter how small), then all we have to do is keep the $x$-coordinate from straying too far. (You actually gave the definition for unifor... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.