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 |
|---|---|---|---|---|
2,357,091 | <p>Let us consider $\text{trace}(f(AA^\top))$ where $f$ is some smooth function and $A \in \mathbb{R}^{n \times m}$. Here, function $f$ is a function of matrices (c.f., Higham's books). If $f(x) = x$, then $\text{trace}(f(AA^\top)) = \|A\|_F^2$ (Frobenius norm) and if $f(x) = x^{p/2}$, then $\text{trace}(f(AA^\top))$ i... | greg | 357,854 | <p>Let's denote the derivative of the function with a prime<br>
$$\eqalign {
f'(x) &= \frac{df(x)}{dx} \cr
}$$
The differential of the trace of the function applied to a matrix argument is
$$\eqalign {
\lambda &= {\rm tr}\big(f(M)\big) \cr
d\lambda &= f'(M)^T:dM \cr
}$$ where the colon denotes the double-co... |
1,432,776 | <blockquote>
<p>For a real symmetric $2\times2$ matrix $A$, we define a $2\times2$ matrix $\sqrt{A}$ that satisfies $\sqrt{A}r_1=\sqrt{\lambda_1}r_1,\ \sqrt{A}r_2=\sqrt{\lambda_2}r_2$ where $r_1, r_2$ are the eigenvectors of $A$, and $\lambda_1, \lambda_2$ are the corresponding eigenvalues. Show that $(\sqrt{A})^2$ i... | Jack's wasted life | 117,135 | <p>$$
(\sqrt{A})^2r_j=\lambda r_j,\quad j=1,2
$$
Pick $\{r_1,r_2\}$ to be an ordered basis of $\mathbb{R}^2$ and see w.r.t this basis </p>
<p>$$
A=(\sqrt{A})^2=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
$$</p>
|
2,476,453 | <p>The problem is as follows:</p>
<blockquote>
<p>Find the value of this function
<span class="math-container">$$A=\left(\cos\frac{\omega}{2} +\cos\frac{\phi}{2}\right )^{2} +\left(\sin\frac{\omega}{2} -\sin\frac{\phi}{2}\right )^{2}$$</span>
when <span class="math-container">$\omega=33^{\circ}{20}'$</span> and... | DominicR | 492,068 | <p>There is a mistake in your expansion of the second bracket.</p>
<p>We should have </p>
<p>$A=\cos^2\frac{\omega}{2}+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}+\cos^2 \frac{\phi}{2}+\sin^2\frac{\omega}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}{2}+\sin^2\frac{\phi}{2}$</p>
<p>$=2+2(\cos\frac{\omega}{2} \cos\frac{\phi}{2... |
1,531,493 | <blockquote>
<p>How many different integers can be expressed as the sum of three distinct numbers from the set{$13$,$10$,$23$,$28$,$33$,$36$,$43$,$48$}?</p>
</blockquote>
<p><strong>MyApproach</strong></p>
<p>Out of $8$ numbers, Select $3$ distinct numbers.</p>
<p>So Ans would be $8$C$3$=$56$-2=$54$.</p>
<p>Becau... | Ian Miller | 278,461 | <p>Right approach. Incorrect counting. Expanding upon your answers we see: $13+43+x=23+33+x$ and $23+48+x=28+43+x$. Here $x$ could be any of the 4 unused numbers in each set. There are several other examples like that too.</p>
|
1,845,076 | <p>So I got the following graph and the Task to determine the Elements of it's automorphism group. The Automorphism is defined as a Graph that is isomorphic to itself. But I think the given Graph isn't isomorphic, so there can't be any Elements in its automorphism group:</p>
<p><a href="https://i.stack.imgur.com/BlWNg... | Jacob Wakem | 117,290 | <p>The degree one vertex must not be moved. Thus its neighbor must not be moved. This vertex's other neighbors are of differing degree and thus cannot be moved. Similarly the remaining two vertices cannot be moved. </p>
<p>Put otherwise we have no non-trivial automorphisms with respect to the topological structure and... |
58,901 | <p>I had previously asked:
<a href="https://mathoverflow.net/questions/47943/narratives-in-modular-curves">Narratives in Modular Curves</a></p>
<p>Since then, I've read quite a bit more (but not nearly enough) and I have a few follow up questions about the big picture. As you will soon see, I'm confused about how to t... | Emerton | 2,874 | <p>In either the one or two dimensional case, there are two sides: Galois and automorphic.</p>
<p>Let me talk for a moment about the $n$-dimensional situation.</p>
<p>The Galois side involves studying continuous $n$-dimensional representations of the Galois group of
a number field $K$.
There are subtleties here about... |
1,635,188 | <p>I am trying to solve $$\lim_{x\to 0} \frac{\int_{0}^{x^2} x^2 e^{-t^2} dt}{-1+e^{-x^4}} $$ using The Fundamental Theorem of Calculus (FTC).
I already know that the answer is -1, </p>
<p>Using FTC (correct me if I am wrong) we get:
$$
\lim_{x\to 0} \frac{x^2 e^{-x^4}}{-1+e^{-x^4}}
$$</p>
<p>Which has the result of ... | Lutz Lehmann | 115,115 | <p>You can also do some pre-processing of the term. It is equal to
$$
\left(\frac{e^{-x^4}-1}{x^4}\right)^{-1}·\frac{\int_0^{x^2}e^{-t^2}dt}{x^2}
$$
and thus the limit can be computed by product and quotient rule as
$$
\left(\lim_{h\to 0}\frac{e^{-h}-1}{h}\right)^{-1}·\lim_{s\to 0}\frac{\int_0^{s}e^{-t^2}dt}{s}
$$
whic... |
2,130,076 | <p>The problem is
$$\lim_{x\to 0} \frac{(x \sqrt{1 + \sin x} - \ln{\sqrt{(1 + x^2)}-x)}}{\tan^3{x}} $$</p>
<p>I know that
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!}... $$
got this
$$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15}...$$ </p>
<p>and suppose to use taylor expansion for $\ln {(1 + x)}$</p>
<p>Trie... | lab bhattacharjee | 33,337 | <p>$$\ln\sqrt{1+x^2}=\dfrac{\ln(1+x^2)}2=\dfrac12\left(x^2-\dfrac{x^4}2+\dfrac{x^6}3-\cdots\right)$$</p>
<p>Now $1+\sin x=\left(\cos\dfrac x2+\sin\dfrac x2\right)^2$</p>
<p>Now $\cos\dfrac x2+\sin\dfrac x2=\sqrt2\sin\left(\dfrac x2+\dfrac \pi4\right)>0$ for $x\to0$</p>
<p>$\implies\sqrt{1+\sin x}=\cos\dfrac x2+\s... |
2,636,131 | <blockquote>
<p>Suppose that each square of a $4 \times 7$ chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?</p>
</blockquote>
... | Donald Splutterwit | 404,247 | <p>Consider the case where we have a monochromatic row ($4$ black squares) the next rows must either have $1$ or $0$ black squares and in order to avoid both a white or black rectangle, we can only add one more row.</p>
<p>Now consider the case where the first row has $3$ black squares, one can rapidly show that there... |
4,609,845 | <p>I am looking for a closed form for the integral <span class="math-container">$$\int_0^\infty \frac{t^s}{(e^t-1)^z}dt$$</span> valid for <span class="math-container">$s,z$</span> being both complex numbers, hopefully using complex analysis. I have already evaluated this integral when <span class="math-container">$s$<... | Wreior | 1,117,198 | <p><strong>Actualisation to solution for complex z and s.</strong></p>
<p>By using theorem of my big ego (<a href="https://math.stackexchange.com/questions/4586393/integral-representation-for-series-of-any-order">Integral representation for series of any order</a>)</p>
<p><span class="math-container">$\displaystyle \... |
2,762,447 | <p>Consider the following non-linear differential equation,
$$
\dot{x}(t)=a-b\sin(x(t)), \ \ x(0)=x_0\in\mathbb{R},
$$
and assume that $a$ and $b$ are positive real numbers with $a>b$.
Note that the solution $x(t)$ exists and can be analytically computed (<a href="http://www.wolframalpha.com/input/?i=dx(t)%2Fdt%20%3... | Aloizio Macedo | 59,234 | <p>This answer is just an expansion of what anomaly pointed out, since you seem confused by his answer in the comments.</p>
<p>This is in reality a problem about linear algebra only:</p>
<p>Your map $\phi$ is clearly a isometry in $\mathbb{H}$, since
\begin{align*}
\langle pvp^*, pv'p^*\rangle &=\operatorname{Re... |
200,242 | <p>I saw the name $p$-adic group on a book I was reading, so I tried to find some related documents. Although I've found something on this topic, there is no definition.</p>
<p>Would anyone please explain the definition for a $p$-adic group to me? Thanks very much.</p>
| Thomas | 26,188 | <p>The following is <strong>not a perfect</strong> description of a $p$-adic group, but hopefully it will help a bit.</p>
<p>Let $F$ be a <a href="http://en.wikipedia.org/wiki/P-adic_numbers" rel="noreferrer">non-Archimedean</a> <a href="http://en.wikipedia.org/wiki/Local_field" rel="noreferrer">local field</a>: That ... |
79,863 | <p>I've encountered this problem on my Non commutative algebra handouts wich says:</p>
<p>given $R,S$ rings and $f:R\to S\:$ a ring homomorphism, define a canonical functor $$F:\textbf{Mod-S}\to \textbf{Mod-R}.$$ Where $\textbf{Mod-S}$ is the category of right $S-$modules and similarly $\textbf{Mod-R}$ is the category... | Rasmus | 367 | <p>As an instructive example, think about the case where $f$ is an inclusion of a subring $R\subset S$. Then an $S$-module is also an $R$-module. Just forget how elements outside of $R$ acted. This gives a functor $\textbf{Mod-S}\to \textbf{Mod-R}$.</p>
<p>There is, however, no way to extend an $R$-action to an $S$-ac... |
1,843,153 | <p>In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem"
\begin{align*}
5040 &= 7! \\
&= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\
&= 2 \times 3 \times 2 \times 2 \times 5 \times 2 \times 3 \times 7 \\
&= 2^4 \times 3^2 \times 5 \times 7 \\
&= 2^... | JMoravitz | 179,297 | <p>Suppose a number $N$ has prime decomposition $N=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$</p>
<p>We recognize that due to the properties of prime numbers, every divisor of $N$ can be written in the form $p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\beta_k}$ with $0\leq \beta_i\leq \alpha_i$ for every $i$.</p>
<... |
803,792 | <p>I have just taken calculus quiz but I could not find $\displaystyle \int_2^\infty\frac{\log^3(x-1)}{x^2}dx$? Any help would be appreciated. Thanks in advance.</p>
<p>EDIT:</p>
<p>Forgot to mention, my tutor gave us hints about this question.</p>
<ol>
<li>Use Taylor series</li>
<li>$\displaystyle \zeta(3)=\sum_{n... | Pranav Arora | 117,767 | <p>Use the substitution $x-1=t$ to obtain:
$$\int_1^{\infty} \frac{\log^3t}{(1+t)^2}\,dt$$
With the substitution $t=1/u$, the above integral is:
$$\int_0^1 \frac{-\log^3 u}{(1+u)^2}\,du$$
Next, use the following series representation,
$$\frac{1}{(1+u)^2}=\sum_{n=0}^{\infty}(-1)^n (n+1)u^n$$
to obtain:
$$\int_0^1 \frac{... |
803,792 | <p>I have just taken calculus quiz but I could not find $\displaystyle \int_2^\infty\frac{\log^3(x-1)}{x^2}dx$? Any help would be appreciated. Thanks in advance.</p>
<p>EDIT:</p>
<p>Forgot to mention, my tutor gave us hints about this question.</p>
<ol>
<li>Use Taylor series</li>
<li>$\displaystyle \zeta(3)=\sum_{n... | xpaul | 66,420 | <p>There is another way to solve. From @Pranav Arora, we know
$$ \int_0^\infty\frac{\ln^3x}{(1+x)^2}dx=-\int_0^1\frac{\ln^3u}{(1+u)^2}du. $$
Let
$$ I(\alpha)=\int_0^1\frac{u^\alpha}{(1+u)^2}du. $$
Clearly
$$ I'''(0)=\int_0^1\frac{\ln^3u}{(1+u)^2}du. $$
Since
\begin{eqnarray*}
I(\alpha)&=&\int_0^1\frac{u^\alpha}... |
3,749,548 | <blockquote>
<p>Calculate:
<span class="math-container">$$\frac{d}{dx}(\cos(\sin(\cos(\sin(...(\cos(x)))))))$$</span></p>
</blockquote>
<p>This looks kind of daunting but I decided to see what happens to the derivative for a section of the function. If I consider:</p>
<p><span class="math-container">$$\frac{d}{dx}(\cos... | Joitandr | 704,255 | <p><span class="math-container">$f_{1}(x) = \cos(x), \ f_{n}(x) = \cos(\sin(f_{n-1}(x)))$</span></p>
<p><span class="math-container">$f_{1}(x)^{\prime} = -\sin(x), \ f_{n}(x)^{\prime} = -\sin(\sin(f_{n-1}(x))) \cdot \cos(f_{n-1}(x)) \cdot f_{n-1}(x)^{\prime}$</span></p>
<p>So if you'd like to compute <span class="math-... |
1,254,175 | <blockquote>
<p>Prove that $D_{12}\cong S_3 \times C_2$.</p>
</blockquote>
<p>I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how this will lead necessarily to a unique solutions.</p>
<p>No full solutions please hi... | Chappers | 221,811 | <p>Show that the subgroup generated by $g^3$ is normal; it's obviously isomorphic to $C_2$. Now show that $ D_{12} / C_2 \cong S_3 $.</p>
<p>(Hint: what does the presentation of $D_{12} / C_2$ look like?)</p>
<p>Then look at the subgroup $H$ of $D_{12}$ generated by $\{g^2,h\}$. You can check $H$ has index $2$, and s... |
34,657 | <p>In section III.1 of P.M. Cohn's <a href="http://books.google.co.uk/books?id=vZsHZ1YP4KkC&lpg=PA108&ots=GIztdoRc2E&dq=universal%20functor&pg=PA108#v=onepage&q=universal%20functor&f=false" rel="nofollow">Universal Algebra</a> a notion of <em>universal functor</em> ${\cal L} \rightarrow {\cal K}... | André Henriques | 5,690 | <p>The answer to your question is "no".</p>
<p>I'm working in $\mathbb{R}^3$. Let $C_1, C_2$ be open balls whose boundaries touch along an interval $J := \partial C_1 \cap \partial C_2$, and let $P \in J$ be a point on that interval. For concreteness:</p>
<p>$C_1 := ]0,1[ \times ]0,1[ \times ]0,1[ = (0,1)^3$</p>
<p>... |
3,393,655 | <p>I want to show that:
<span class="math-container">$$
\lim_{n \rightarrow \infty} \int_0^{2\pi} \sin(x)^n \, dx=0
$$</span></p>
<p>and my idea was to use DCT (dominated convergenece theorem).</p>
<p>However, my textbook has the requirement that <span class="math-container">$u(x)=\lim_{n \rightarrow \infty} u_n(x)$<... | J.G. | 56,861 | <p>You could also use the DCT to show <span class="math-container">$\int_0^\pi\sin^nxdx\to0$</span>, then use the squeeze theorem with <span class="math-container">$\left|\int_0^{2\pi}\sin^nxdx\right|\le2\int_0^\pi\sin^nxdx$</span>.</p>
|
116,744 | <p>I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's <em>Cours d'Analyse</em>).</p>
<p>I also know that mathem... | Joel David Hamkins | 1,946 | <p>One of the most well-known examples is Frege's axiom of general comprehension, which asserts that for any definite property $P$, one may form the set $$\{x\mid\ P(x)\ \},$$ consisting of all objects $x$ with property $P$. This natural-seeming principle is one of the main axioms of what is now known as naive set theo... |
116,744 | <p>I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's <em>Cours d'Analyse</em>).</p>
<p>I also know that mathem... | Leonard | 26,077 | <p>Another example from real analysis would be the question of the pointwise convergence of the Fourier series of a continuous function (defined on a closed interval). Many people, including Dirichlet and even the master rigorist Weierstrass himself, believed that the Fourier series of such a function converges pointwi... |
116,744 | <p>I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's <em>Cours d'Analyse</em>).</p>
<p>I also know that mathem... | Leonard | 26,077 | <p>In the mathematical theory of social welfare, it is possible to create a list of axioms that lead to a contradiction. For example, in voting theory, the following axioms for a voting system are considered reasonable in order for the system to qualify as being <em>fair</em>:</p>
<ol>
<li><p>Each voter can have any s... |
116,744 | <p>I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's <em>Cours d'Analyse</em>).</p>
<p>I also know that mathem... | Joel David Hamkins | 1,946 | <p>Perhaps one of the earliest examples would be with the <a href="http://en.wikipedia.org/wiki/Pythagoreanism" rel="noreferrer">Pythagoreans</a>, who held that any two magnitudes were <a href="http://en.wikipedia.org/wiki/Commensurability_(mathematics)" rel="noreferrer">commensurable</a>, measured as integer multiples... |
666,821 | <p>Here is a standard identity:</p>
<p>$$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$</p>
<p>Why does it hold true?</p>
| rubberchicken | 125,998 | <p>A brief answer: Let's consider the exponential function $e^x$. The definition of $e$ is that $\frac{d}{dx}e^x = e^x$. Now let's assume that $e^x$ can be written as an infinite sum of the form $\sum_{i=0}^{\infty}a_ix^i$. Using the sum rule for derivatives, we have $\sum_{i=0}^{\infty}a_ix^i = \sum_{i=0}^{\infty}\fra... |
4,490,862 | <p>Prove, using a combinatorial argument, that the numbers below are integers for any natural <span class="math-container">$n$</span>: <span class="math-container">$$\frac{(2n)!}{2^n}$$</span></p>
<p><strong>Attempt:</strong>
Suppose we organize a queue with <span class="math-container">$2n$</span> people, this can be ... | Mateo | 1,015,578 | <p>You have <span class="math-container">$2n$</span> people and you are going to divide them into <span class="math-container">$n$</span> distinguishable pairs. This can be done on
<span class="math-container">$$\binom{2n}{2,2,2,\ldots,2}=\frac{(2n)!}{2!2!2!\ldots 2!}=\frac{(2n)!}{2^n}$$</span> ways. You can derive thi... |
1,120,826 | <blockquote>
<p>Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.</p>
</blockquote>
<p><strong>Work of OP:</strong> The thing is I have no idea, everything I tried ended up on nothing.</p>
<p><strong>Third party commentary:</strong> Standard ideas to attack such problems include induction and congruenc... | barak manos | 131,263 | <p>First, show that this is true for $n=1$:</p>
<ul>
<li>$\dfrac{2^{3^{1}}+1}{9}=1\in\mathbb{N}$</li>
</ul>
<p>Second, assume that this is true for $n$:</p>
<ul>
<li>$\dfrac{2^{3^{n}}+1}{9}=k\in\mathbb{N}$</li>
</ul>
<p>Third, prove that this is true for $n+1$:</p>
<ul>
<li><p>$\dfrac{2^{3^{n+1}}+1}{9}=\dfrac{2^{3... |
1,120,826 | <blockquote>
<p>Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.</p>
</blockquote>
<p><strong>Work of OP:</strong> The thing is I have no idea, everything I tried ended up on nothing.</p>
<p><strong>Third party commentary:</strong> Standard ideas to attack such problems include induction and congruenc... | lhf | 589 | <p>Write $m={3^n}$. Using the binomial theorem, we get $$2^{3^n} + 1=2^m + 1=(3-1)^m+1=9a+3m-1+1=9a+3m$$ which is a multiple of $9$ because $m$ is a multiple of $3$.</p>
|
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | payas | 76,095 | <p>the answer is</p>
<p>$$-\frac{33\sqrt3-77}{22}$$</p>
|
2,714,756 | <p>Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*}^{-1}$ is a new almost complex structure. </p>
<p>Question: Is $f_{*}Jf_{*}^{-1}$ an integrable almost complex str... | Michael Albanese | 39,599 | <p>As $f$ is a diffeomorphism, $f_* : \mathfrak{X}(M) \to \mathfrak{X}(M)$ is an isomorphism of vector spaces, and as $f_*[X_1, X_2] = [f_*X_1, f_*X_2]$, it is also an isomorphism of Lie algebras; the same is true for $f_*^{-1}$.</p>
<p>Let $Y_1, Y_2 \in \mathfrak{X}(M)$. As $f_*$ is an isomorphism, there are $X_1, X_... |
1,069,120 | <p>I have my discrete structures exam tomorrow, and right now i am practicing mathematical induction, specially proofs. while proving, i just get confused because i don't understand what should i add or subtract to prove the inductive step. i was wondering if there is any tip or trick to know what should we add or subt... | Community | -1 | <p>An induction step will answer the following question: knowing how to solve for $n$ items, how can I solve for $n+1$ items ?</p>
<p>In the case the Hanoi towers, this is obvious: if I am able to move a tower of $n$ disks, I can move a Tower of $n+1$ disks in three steps:</p>
<ul>
<li>move the top $n$ disks (say fro... |
456,961 | <p>What is the maximum of ${\frac{(1-\cos x)}{x}}$ in the interval $[0, \pi]$?</p>
<p>I can show that the maximum is less than 1, but I want an exact value.</p>
| rurouniwallace | 35,878 | <p>Find the critical values. Since it is a closed interval, two of them will be $x=0$ and $x=\pi$. The others will be points where $\frac{d}{dx}\frac{1-\cos{x}}{x}=0$. Evaluate the function at each critical point and observe which one is/which ones are the smallest.</p>
|
221,279 | <blockquote>
<p>How does one show that $\chi_{[1, \infty)}1/x$ is not (Lebesgue) integrable?</p>
</blockquote>
<hr>
<p>What I could think of is as follows:</p>
<p>Letting $f(x)=1/x$ (defined for $x\geq 1$), define
$$
f_n(x)=f\chi_{[1, n)}(x).
$$ </p>
<p>Each $f_n$ is, therefore, Riemann integrable on $[1, n)$ wi... | copper.hat | 27,978 | <p>Here's an alternative to using polar coordinates in $\mathbb{R}^n$:</p>
<p>If $x \neq 0$, we have $\frac{1}{\sqrt{n} \|x\|_\infty} \leq \frac{1}{\|x\|_2}$. Then $\frac{1}{\sqrt{n}} \int_{\|x\|_\infty \geq 1} \frac{1}{\|x\|_\infty} \leq \int_{\|x\|_\infty \geq 1} \frac{1}{\|x\|_2} \leq \int_{\|x\|_2 \geq 1} \frac{1}... |
4,251,726 | <p>I posted <a href="https://math.stackexchange.com/questions/4247878/if-the-radius-of-convergence-of-the-series-sum-a-n-zn-is-r-whats-the-rad?noredirect=1#comment8830417_4247878">If the radius of convergence of the series $\sum a_n z^n$ is $R$, what's the radius of convergence of $\sum s_n z^n$, where $s_n$ partia... | Tyma Gaidash | 905,886 | <p>Here I present forms of the constant. The first will use algebraic properties. Let the constant be denoted c for “constant”. Note I will use “!x” for the derangement as seen in the question. Also note the <a href="https://mathmaine.com/2018/03/04/pi-notation/" rel="nofollow noreferrer">Pi-Product</a> notation:</p>
<... |
3,603,312 | <p>there are two urns with White balls and Black balls. first urn has 21 whites and 5 blacks, second one has 8 whites and 9 blacks. we take 7 balls from first urn and put them into the second one. afterwards, out of the second urn we take one ball. what is the probability that it's white?</p>
<p>I've been struggling o... | P. Lawrence | 545,558 | <p>The probability that any pariular ball in the first urn is moved to the second urn is
<span class="math-container">$\frac{{1 \choose 1} \times {25 \choose 6}}{{26 \choose 7}}=\frac {7}{26}$</span>.The probability that that particular ball is then chosen from the second urn is
<span class="math-container">$\frac{1}... |
2,706,358 | <p>Is the function $f(x)=9-x^2$ continuous? </p>
<p>1.Lets say for $x=1$ </p>
<p>$f(1)= 9-1=8$</p>
<p>2.$\lim\limits_{x \to 1} (9-x^2)= 9-1 = 8$</p>
<p>3.So $f(a) = \lim\limits_{x \to a} f(x) $</p>
<p>Does this mean that the function is continuous ?</p>
| GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Yes, you've got the right steps for the continuity of $f$ at the point $x = 1$. To generalise it to any point $x = a$, change $1$ to $a$ in the above steps.</p>
<ol>
<li>$f(a) = 9-a^2$</li>
<li>$\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} (9 - x^2) \stackrel{(*)}{=} 9-a^2$</li>
<li>so $f(a) = \lim\limits_{x... |
3,222,253 | <blockquote>
<p>Given <span class="math-container">$$ P = \left \{ (a,b,c,d) \in \mathbb R^4 \mid a + b + c + d = 0 \right \} $$</span> find <span class="math-container">$ P^\perp $</span>.</p>
</blockquote>
<p>Am I right if I multiply <span class="math-container">$$ P^T P = 0$$</span></p>
<p><span class="math-cont... | Alexander Geldhof | 560,477 | <p><span class="math-container">$P$</span> is three-dimensional. It's a linear subspace of a four-dimensional space governed by one equation, so its dimension is four - one = three. Its orthogonal complement will be one-dimensional.</p>
<p>A vector <span class="math-container">$\begin{bmatrix} x \\ y \\ z \\ w \end{b... |
4,470,269 | <p>While practicing from a book I found a product in the form <span class="math-container">$$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$</span> and was immediately curious if I could a formula to solve the product for <span class="math-container">$n$</span> terms, that is, a single formula for the pro... | epi163sqrt | 132,007 | <p>Using <span class="math-container">$[n]=\{1,2,\ldots,n\}$</span> we can write the product as
<span class="math-container">\begin{align*}
\prod_{j=1}^n\left(x^{a^j}+1\right)=\sum_{S\subseteq [n]}x^{\sum_{j\in S}a^j}
\end{align*}</span>
where <span class="math-container">$S$</span> runs over all subsets of <span class... |
2,494,153 | <p>The Peano axioms are often listed (among other ways) as a set of 5 axioms in an informal language or 3 axioms in a formal language. For example:</p>
<p><strong>Informal</strong> (see, e.g., <a href="http://mathworld.wolfram.com/PeanosAxioms.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/PeanosAxioms.h... | Matthew Leingang | 2,785 | <p>You're right that step 3 is the way to go. You'd <em>like</em> to compare to
$$
b_n = \frac{n}{n^2} = \frac{1}{n}
$$
Then $a_n < b_n$, and $\sum b_n$ is a $p$-series. But $p=1$, so the series $\sum b_n$ <em>diverges</em>. This means our regular comparison test won't work. It's not useful to say a series i... |
2,494,153 | <p>The Peano axioms are often listed (among other ways) as a set of 5 axioms in an informal language or 3 axioms in a formal language. For example:</p>
<p><strong>Informal</strong> (see, e.g., <a href="http://mathworld.wolfram.com/PeanosAxioms.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/PeanosAxioms.h... | V. Vancak | 230,329 | <p>Note that
$$
\frac{n}{n^2 + 1} \ge \frac{n}{n^2 + n} \ge \frac{n}{n^2 + n^2} = \frac{1}{2n},
$$
hence,
$$
\sum_{n=1}^{\infty}\frac{n}{n^2 + 1} \ge \frac{1}{2}\sum_{n=1}^{\infty} \frac{1}{n} .
$$</p>
|
1,729,434 | <p>I was able to prove the base case statement, where if you plug in $3$ for $n$ you get:</p>
<p>$19 ≥ 15$.</p>
<p>Next I supposed an arbitrary value $k$ where $k ≥ 3$ and $2k^2+1 ≥ 5k$. I know that next I need to prove that:</p>
<p>$2(k+1)^2+1 ≥ 5(k+1)$</p>
<p>But this is where I got stuck. </p>
| Peter | 82,961 | <p>$$2(n+1)^2+1=2n^2+4n+2+1\ge 5n+4n+2=9n+2$$</p>
<p>We are done, if we have $9n+2\ge 5n+5$ , which is true for $n\ge \frac{3}{4}$, so it is also true for $n\ge 3$. This completes the proof.</p>
<p>Even better is to argue $5n+4n+2\ge 5n+5$ because of $4n+2\ge 5$</p>
|
200,629 | <p>I am trying to find the characteristic function for Johnson's SU distribution by integrating the probability density function with <code>Exp[I*t*x]</code> but Mathematica is returning the input itself.</p>
<p>As the characteristic function always exists, I'm not able to understand why Mathematica is not finding the... | Xminer | 61,541 | <p>Please look at <a href="https://reference.wolfram.com/language/ref/CharacteristicFunction.html" rel="nofollow noreferrer">the characteristic function of the lognormal distribution.</a> <br>In some cases there is no closed form.<br><br>
One way to address this issue is using empirical' s one.</p>
<pre><code>emCF = W... |
628,447 | <p>I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula</p>
<p>$$H= I - (uu^T/\beta),$$</p>
<p>they are not equal. What's wrong with my reasoning?</p>
<p>EDIT: I forgot that $(uu^T)^T$ would be $(u^T)^T(u)^T$ from the following properties:
$(AB)^T=B^TA^T$</p>
| arbitUser1401 | 51,241 | <p>Hint : $I$ is symmetric and $uu^{T}$ is symmetric. </p>
<p>Thus it follows,</p>
<p>$I^{T} =I$ and $(uu^{T})^{T}=uu^{T}$ (Recall that $(AB)^{T}=B^{T}A^{T}$ )</p>
<p>Hence $H^{T}=(I-uu^{T}/\beta)^{T}=I^{T}-(uu^T/\beta)^{T}=I-uu^{T}/\beta=H$</p>
<p>Verily , $H $ is symmetric.</p>
|
2,855,335 | <p><a href="https://i.stack.imgur.com/fAAXj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fAAXj.png" alt="enter image description here" /></a></p>
<p>One should note that the family here may not countable. If it is countable, the it is a consequence of the following results;</p>
<p>Lemma 1. the pro... | Henno Brandsma | 4,280 | <p>Every factor space embeds into the product as a subspace, so the previous theorem in Engelking's General Topology (which you're quoting here) implies the left to right implication right away.</p>
<p>To check the right to left one, let all $(X_s, \mathcal{U}_s)$ be totally bounded. It suffices to check the definitio... |
982,386 | <p>I am quite a beginner in linear algebra and matrix calculus. I was wondering what is the derivative of the matrix inverse when the matrix is symmetric. More precisely, I'm looking for $\frac{\partial}{\partial \mathbf{X}} \mathbf{X}^{-1}$ when $\mathbf{X}$ is a symmetric matrix.</p>
<p>I am asking this because I ha... | lynn | 230,453 | <p>Ignoring for the moment the symmetry constraint. First rewrite the function as
$$\eqalign{
f &= A^T:X^{-1} - {\rm tr}({\rm log}(X)) \cr
}$$
Then take the differential
$$\eqalign{
df &= A^T:(-X^{-1}\,dX\,X^{-1}) - X^{-T}:dX \cr
&= -X^{-T}A^TX^{-T}:dX - X^{-T}:dX \cr
&= -(X^{-T}A^TX^{-T}+X^{... |
351,626 | <p>Given $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous and a fixed $\delta$,
define
$$f_{\delta}(x):=\int_{x-\delta}^{x+\delta}f(\xi)\,\mathrm{d}\xi.$$</p>
<p>$f_{\delta}$ behaves like the average of $f$ in a short interval $(x-\delta,x+\delta)$. </p>
<p>Apparently it is not linear, but it should be Lipschitz conti... | timur | 2,473 | <p>In general it is not globally Lipschitz, but it is locally Lipschitz. Consider your function in a bounded interval, e.g., in $[0,1]$. Then you try to bound $|f_\delta(x)-f_\delta(y)|$ by a multiple of $|x-y|$ when $|x-y|<\delta$. For $|x-y|\geq\delta$, you can use the trivial bound
$$
|f_\delta(x)-f_\delta(y)| \... |
1,492,350 | <blockquote>
<p><strong>Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.</strong></p>
</blockquote>
<p>Attempt:
We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$</p>
<p>Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$... | robjohn | 13,854 | <p>Since $f$ is known up to a function with period $1$, let's try to find a monotonically increasing $f$.</p>
<p>Since $f(x-k+1)-f(x-k)=e^{x-k}$, we have that $\lim\limits_{x\to-\infty}f(x)$ exists. Furthermore,
$$
\begin{align}
f(x)-\lim_{x\to-\infty}f(x)
&=\sum_{k=1}^\infty\left[f(x-k+1)-f(x-k)\right]\\
&=\s... |
1,049,677 | <p>I'm trying with matrices over $\mathbb F_2$ and trying to have a look at the Jordan canonical forms of these matrices. If the size of the biggest Jordan block is the same with 1's in all diagonal entries, we do get non-similar invertible matrices with same minimal and characteristic polynomial. But what do I do for ... | user122916 | 122,916 | <p>I will attempt an answer based on Kelenner's comment. Taking repeated derivatives of the the equation $pe^p+qe^q=1$ yields that $p'(p+1)$ and $q'(q+1)$ have the same distinct roots each with the same multiplicity. Hence $q'(q+1)=\lambda p'(p+1)$ for some $\lambda \in \mathbb{C}$.</p>
<p>Hence $p'(p+1)[e^p+\lambda... |
751,063 | <p>Let $G$ be a finite dimensional connected Lie group and $Diffeo(G)$ be the diffeomorphism group of the underlying manifold. Is it true that $Diffeo(G)$ has the homotopy type of a finite dimensional Lie group? I can't seem to find a counterexample.</p>
| archipelago | 67,907 | <p>A lower dimensional example is $S^1\times S^2$. </p>
<p><a href="https://www.math.cornell.edu/~hatcher/Papers/newDiffS1xS2.pdf" rel="nofollow">Hatcher</a> calculated the homotopy type of $Diff(S^1\times S^2)$: It is the one of $O(2)\times O(3)\times \Omega SO(3)$. As the second homotopy group of this space does not... |
610,029 | <p>The time between successive cars on a certain road is exponentially distributed and the probability is $1/2$ that the next car will arrive within two minutes. Assume the time between and particular pair of cars is independent of the times between all other pairs of cars. </p>
<ol>
<li><p>What is the probability the... | André Nicolas | 6,312 | <p>We look first at the simpler problem of Pythagorean triples $(x,y,z)$ where $z=y+1$.</p>
<p>Such a triple must be primitive, and therefore $z=u^2+v^2$, $y=2uv$, for some $u$ and $v$. Then $u^2+v^2=2uv+1$, giving $u=v+1$. That gives the triple $x=2v+1$, $y=2v(v+1)$, $z=2v^2+2v+1$. </p>
<p>Multiply each entry by $3$... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | MarnixKlooster ReinstateMonica | 11,994 | <p>For completeness, here is the theorem from Rosen's book that this question uses as an example (Section 6.2 The Pigeonhole Principle, Theorem 1):</p>
<blockquote>
<p>THE PIGEONHOLE PRINCIPLE If $k$ is a positive integer and $k+1$ or
more objects are placed into $k$ boxes, then there is at least one box
contain... |
2,285,021 | <p>I need to understand how to obtain $f/g$ and $f/h$ from the following equations:
$$
\frac{f^2\sin^2\alpha}{h^2}+\frac{f^2\cos^2\alpha}{g^2}=f^2/k^2
$$
$$
\frac{f^2\sin^2\beta}{h^2}+\frac{f^2\cos^2\beta}{g^2}=f^2/l^2
$$
Which must lead to the following expressions:
$$
(f/g)^2=\frac{(\frac{f\sin\alpha}{l}+\frac{f\sin\... | G Tony Jacobs | 92,129 | <p>The first thing I notice is, both in the starting and ending expressions, you can divide both sides by $f^2$, so the variable $f$ plays no role in this calculation other than being along for the ride. Dividing it from the initial equations, we get:</p>
<p>$$
\frac{\sin^2\alpha}{h^2}+\frac{\cos^2\alpha}{g^2}=1/k^2
$... |
3,205,693 | <p>Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring <span class="math-container">$R$</span> to the algebra <span class="math-conta... | Community | -1 | <ol>
<li><span class="math-container">$x+2x^2+3xy+0y=6$</span> </li>
<li><span class="math-container">$2x+x^2+3xy+y=5$</span> </li>
<li><span class="math-container">$x-1x^2+0xy+y=7$</span> </li>
</ol>
<p>Multiply the second one with 2 and subtract eq 2 from eq 1.<br>
Then <span class="math-container">$4x+2x^2+6xy... |
3,205,693 | <p>Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring <span class="math-container">$R$</span> to the algebra <span class="math-conta... | Cameron Buie | 28,900 | <p>The first thing I notice is that the third equation can be rewritten equivalently as <span class="math-container">$$x^2-x+7=y.$$</span> I also notice that the first and second equations both have <span class="math-container">$3xy,$</span> so subtracting the first from the second would give the equivalent system <spa... |
1,302,271 | <p>I haven't been able to prove this statement from my Elementary Number course:</p>
<p><strong>There are infinitely many primes $p$ such that $p\equiv -1 \mod12$.</strong></p>
<p>From <a href="http://projecteuclid.org/download/pdf_1/euclid.facm/1229442627" rel="noreferrer">here</a> I know that there exists a "Eulcid... | Atvin | 215,617 | <p>The answer comes directly from <a href="http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" rel="nofollow">Dirichlet's theorem</a>: Let us have the sequence $12k+11$. The members of the sequences are all $\equiv -1$ (modulo $12$), and according to Dirichlet's theorem, since $gcd(11,12)=1$,... |
6,541 | <p>Here is <code>e^{-\lvert\frac{x-\mu}{\sigma}\rvert}</code>: <span class="math-container">$e^{-\lvert\frac{x-\mu}{\sigma}\rvert}$</span></p>
<p>The bars on the absolute value are too small, so I decided to make them bigger. Using <code>\left</code> and <code>\right</code> made them look pretty good:</p>
<p><code>e^{-... | mythealias | 31,292 | <p>I don't think it is a bug. <code>\left</code> and <code>\right</code> constructs account for the size of content between them, whereas <code>\bigl</code> and <code>\bigr</code> are of predefined size. The <code>l</code> and <code>r</code> are to account for horizontal spacing. See <a href="https://tex.stackexchange.... |
4,152,323 | <p>Let
<span class="math-container">$$
a_n = \sum_{k=0}^n \binom{n + 1}{k}b_k.
$$</span>
I am trying to write <span class="math-container">$b_n$</span> in terms of <span class="math-container">$a_k$</span>.</p>
<p>Of course, if the binomial coefficient was <span class="math-container">$\binom{n}{k}$</span> instead of <... | skbmoore | 321,120 | <p>What you seek is not quite a binomial transform. Use the elementary ID
<span class="math-container">$$ \binom{n+1}{k} = \frac{n+1}{n+1-k} \binom{n}{k}.$$</span>
Define <span class="math-container">$A_n=a_n/(n+1)$</span> and the sum you want to invert is
<span class="math-container">$$ A_n = \sum_{k=0}^n \binom{n}{k... |
1,923,038 | <p>I need to prove that $n^2 = {n \choose 2} + {n+1 \choose2}$. I have already proved this using algebra, but I am required to use both algebra and a formal combinatorial proof which demonstrates a bijection between the right and left hand sides. </p>
<p>If someone could show me how to get started with this proof and ... | Brian M. Scott | 12,042 | <p>HINT: You have two boxes of numbered balls. The first box has $n$ balls numbered $1$ through $n$, and the second has $n+1$ balls numbered $0$ through $n$. You choose a box and draw two balls from it; say that they are numbered $k$ and $\ell$.</p>
<ul>
<li><p>If you drew from the first box, you write down the ordere... |
1,923,038 | <p>I need to prove that $n^2 = {n \choose 2} + {n+1 \choose2}$. I have already proved this using algebra, but I am required to use both algebra and a formal combinatorial proof which demonstrates a bijection between the right and left hand sides. </p>
<p>If someone could show me how to get started with this proof and ... | Jack D'Aurizio | 44,121 | <p>Let <span class="math-container">$A=\{1,2,\ldots,n\}$</span>. Obviously <span class="math-container">$n^2=\left|A\times A\right|$</span>. We have <span class="math-container">$\binom{n}{2}$</span> ways for choosing <span class="math-container">$(a,b)\in A\times A$</span> with <span class="math-container">$a<b$</s... |
1,865,735 | <p>I'm not sure about the use of the Theorem.
I have:</p>
<p>$$f(x)=\int_0^{x^2}(t-1)g(t)dt$$
I need the derivative of $f$. I know i have to apply the chain rules, but i'm not sure about the results.
My result is:
$$f'(x)=(4x*g(x^2))+((x^2-1)*g'(x^2))$$
Is the correct way?</p>
| Tsemo Aristide | 280,301 | <p>The answer is: $f'(x)=(x^2-1)g(x^2)(2x)$.</p>
|
9,758 | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | isomorphismes | 1,457 | <p>Intuitively, it's a matrix that's "like" a single number $\geq 0$.</p>
<p>(Relatedly you can have a positive semidefinite function that's also "like" a number.)</p>
<p>Both matrices and functions in general have many (, many, many) more degrees of freedom than members of $\mathbb R$. But the semidefinite classes o... |
1,262 | <p>Beside the fact that I would like to see more posts in our <a href="http://mathematica.blogoverflow.com/" rel="nofollow noreferrer"><em>Mathematica</em> Stack Exchange Blog</a>, I have serious concerns that the majority of the people here is able to find it at all.</p>
<p>There seems to be no direct link from the m... | halirutan | 187 | <p>OK, I could have sworn it wasn't there 5 minutes ago, but a link to the blog can be found in the Stack Exchange dropdown menu at the top of the page</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/cO0VR.png" alt="enter image description here"></p>
</blockquote>
|
3,318,888 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow \mathbb{R}$</span> be such that</p>
<p>(1) <span class="math-container">$f$</span> is bounded.</p>
<p>(2) <span class="math-container">$f$</span> is integrable on <span class="math-container">$[\delta,1]$</span> for every <span class="math-container"... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$|\int_s^{t} f(x)dx | \leq M(t-s) \to 0$</span> as <span class="math-container">$0<s<t \to 0$</span> where <span class="math-container">$M$</span> is a bound for <span class="math-container">$|f|$</span>. This implies that <span class="math-container">$\lim \int_{\delta} ^{1} f(x)d... |
3,318,888 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow \mathbb{R}$</span> be such that</p>
<p>(1) <span class="math-container">$f$</span> is bounded.</p>
<p>(2) <span class="math-container">$f$</span> is integrable on <span class="math-container">$[\delta,1]$</span> for every <span class="math-container"... | zhw. | 228,045 | <p>True, <span class="math-container">$f$</span> is Riemann integrable on <span class="math-container">$[0,1].$</span></p>
<p>Suppose <span class="math-container">$|f|\le M;$</span> we can assume <span class="math-container">$M>0.$</span></p>
<p>Let <span class="math-container">$\epsilon>0.$</span> Choose <span... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| math_lover | 189,736 | <p>Here is a simple 10-step construction. Looking at the diagram given in the OP, denote the uppermost line by $l$, the middle line by $m$, and the lowermost line by $n$. </p>
<p>On $l$ choose a point $A$ and construct a line $t_1$ through $A$ which forms a $60$ degree angle with $l$ and has negative slope. Let $t_1$ ... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| Ray McKaig | 717,567 | <p>Given 3 parallel lines A B C (top to bottom), construct a large equilateral triangle between lines A and C where the bottom triangle leg is on line C and the top triangle vertex is on line A. Mark the bottom right point of the triangle point c1. Also mark the intersecting points of lines B and the left leg of the t... |
1,449,306 | <p>I found an unclear part in derivation of PCA in the <a href="https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/resources/mit18_s096f15_ses2_4/" rel="nofollow noreferrer">lecture notes</a> of A. Bandeira for <a href="https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-s... | Adrian Cousot | 273,219 | <p>I cannot comment(I have not enough point), so it comes as a solution.
I am very interested also to see how your teacher conclude that, especially looks like it is evident for him. It is not clear, because it says we try to find the best d-dimnetinal affine subspace which the projections of $x_1, . . . , x_n$ on it b... |
282,122 | <p>Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? </p>
<p>(a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$.<br>
(b) $S$ is connected in $\mathbb{M}_n(\mathbb{R})$.<br>
(c) $S$ is compact in $\mathbb{M}_n(\mathbb{R})$. </p>
<p>O... | Community | -1 | <p>As La Belle Noiseuse said, $S$ is a convex set; $S$ is also open in a vector space of dimension $n(n+1)/2$; thus $S$ is homeomorphic to the open ball of $\mathbb{R}^{n(n+1)/2}$ for an arbitrary norm.</p>
<p>EDIT for @ Sahiba Arora. The property "$A$ is heomorphic to a bounded set" gives no information about the fac... |
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | Andrew D. Hwang | 86,418 | <p>As several others have stated, the question comes down to one's definition of "the same series". And that, in turn comes down to "what is a series?"</p>
<p>Speaking of "the set of sequences" is perfectly usual and unambiguous; a sequence is a special type of mapping, and one can speak of all such mappings. By contr... |
1,600,911 | <p>So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: </p>
<p>$Q_a(a) = f(a)$</p>
<p>$Q_a'(a) = f '(a)$</p>
<p>$Q_a''(a) = f ''(a)$</p>
<p>But then how do you derive the function $Q_a(x) = f(a) + f '(a)(x-a) + f ''(a) (x-a)^2/2$?</p>
<p>Or can you esti... | πr8 | 302,863 | <p>" ... the general idea for quadratic approximation is assuming there a function $Q(x)$ such that ... "</p>
<p>This is true, but falls slightly short of how quadratic approximation is normally set up - it's missing the stipulation that $Q(x)$ be a quadratic polynomial (in this context at least).</p>
<p>Deriving the... |
178,438 | <p>In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in</p>
<p>Troelstra, A. S.; van Dalen, D. Constructivism in mathematics. Vol. I. An introduction. Studies in Logic and the Founda... | Paul Taylor | 2,733 | <p>By the <strong>extreme value theorem</strong> I take it that you mean that</p>
<blockquote>
<p>Every continuous real-valued function on a closed bounded interval is bounded and attains its bounds.</p>
</blockquote>
<p>I am going to answer this in terms of <em>General Topology unsullied by excluded middle</em> fi... |
178,438 | <p>In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in</p>
<p>Troelstra, A. S.; van Dalen, D. Constructivism in mathematics. Vol. I. An introduction. Studies in Logic and the Founda... | Matt F. | 44,143 | <p>1) Yes, in SIA the maximum value $y$ exists. Looking at $R^{[-1,1]}$, the space of all smooth maps, there is a smooth map $\max : R^{[-1,1]} \rightarrow R$ where $y \ge \max(f) \leftrightarrow \forall x\ f(x) \le y$ </p>
<p>2) No, in SIA the maximum need not be attained. Let $a,x,w$ range over $[-1,+1]$.</p>
<p>... |
2,145,549 | <p>I'd like to ask a question about what can I possibly do wrong with determining asymptotes of the function</p>
<p>$$x \mapsto x-2\sqrt{x^2+1} $$</p>
<p>OK, so when it comes to vertical asymptotes function, we can't have any because domain of the function is the set of all real numbers.</p>
<p>Now, I'm trying to de... | Bernard | 202,857 | <p>The simplest here is to use Taylor's expansion, after the substitution $t=\dfrac1x$ (so $t\to 0^+$ or $0^-$). It will give you in one shot the asymptote, if it exists, and the position of the curve w.r.t. its asymptote.</p>
<p>So we can rewrite
\begin{align}x-2\sqrt{x^2+1}&=\frac1t-2\sqrt{\frac1{t^2}+1}=\frac1t... |
3,267,311 | <p>The author in this example is trying to show that the norm of Hilbert matrix is less than or equal to <span class="math-container">$\pi$</span>. Hilbert matrix has entries
<span class="math-container">$$a_{ij}=\frac{1}{i+j+1};1\leq i,j \leq \infty$$</span> He used the fact that if <span class="math-container">$\exis... | Dr. Wolfgang Hintze | 198,592 | <p>I'd like to make an extended comment on the remark in the OP "I am not able to understand the second step of the proof that how he is able to relate summation with integration."</p>
<p>I had the same difficulty, and resolved it by carefully looking at the replacement of the sum by an integral.</p>
<p>The sum in qu... |
2,184,864 | <p>I am asked to prove the following, given that <span class="math-container">$a<b$</span>, and <span class="math-container">$c<d$</span>, such that <span class="math-container">$a,b,c,d>0$</span>:</p>
<blockquote>
<p>(1) <span class="math-container">$a+c<b+d$</span></p>
<p>(2) <span class="math-container">... | Dr. Sonnhard Graubner | 175,066 | <p>assuming $$a+c\geq b+d$$ since $d>c$ we have
$$a+c\ge b+d>b+c$$ from here we get a contradiction since we have $a<b$
the second proof works analogously we assume $$ac\geq bd$$ and $$bd>bc$$ from here we get also $$a>b$$ and this is not possible</p>
|
279,573 | <p>I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:</p>
<ol>
<li><p>$X$ has the same Betti numbers as $\mathbb{C}\mathbb{P}^{3}$ i.e. $b_{1}(X) = b_{3}(X) = 0$ and $b_{2}(X) = 1$ and all of its cohomology groups are tors... | abx | 40,297 | <p>Let me just mention that the non-existence of such a threefold is an immediate consequence of Yau's inequality. First, as explained in the above comment, the conditions $b_2=1$ and $\mathrm{Kod}(X)\geq 0$ imply that $K_X$ is ample. Then Yau gives $c_1^3\geq \frac{8}{3}c_1c_2 $, which is equivalent by Riemann-Roch to... |
2,359,797 | <p>I have a set defined by a lot of linear constraints. Namely, $\mathcal{X}=\{x:Ax\leq b\}$, where $x$ is a $j$- dimensional vector (each component can be either positive or negative). I want to know for each component $j$, $\bar x_j=\max_{x\in \mathcal{X}} x_j$ and $\underline x_j=\min_{x\in \mathcal{X}} x_j$. I know... | Red shoes | 219,176 | <p>By Solving Just one LP you can find those lower and upper bounds </p>
<p>$$ \begin{align}
\max & \sum_{j} (\bar{y}^j _{j} - \underline y_{j} ^j)\\ &\text{subject to}
\\
& \bar{y}^j \in \mathcal{X} &j=1,2...,n \\
& \underline y_{} ^j \in \mathcal{X} &j=1,2...,n
\end{align} $$</p>
... |
2,684,433 | <p>If I have a set $\{(1, H), (2, C), (3, F), (4, Z), (5, S), (6, L) \}$ is there any way to express this with set builder notation? If not, is there any other way to express this mathematically?</p>
| Nate | 91,364 | <p>Every polynomial of degree $6$ satisfies the recursion $$f(n) = 7f(n-1) - 21f(n-2)+35f(n-3)-35f(n-4)+21f(n-5)-7f(n-6)+f(n-7)$$ So by induction if the first $7$ values are divisible by $24$ then all of them are.</p>
|
4,627,884 | <p>Consider the 'pseudo' definitions below:</p>
<p><span class="math-container">$A$</span> is an <span class="math-container">$n \times n$</span> matrix</p>
<p><span class="math-container">$A$</span> is <span class="math-container">$m_1$</span> if <span class="math-container">$A^2= I$</span> (the identity matrix)</p>
<... | V.S.e.H. | 443,030 | <p>As CalvinLin pointed out in the comments, such matrices are called <strong>projection</strong> matrices, or <strong>idempotent</strong> matrices. One obvious property is that if <span class="math-container">$x\in R(A)$</span> then <span class="math-container">$Ax = A^2\xi = A\xi = x$</span>. Such matrices also have ... |
2,754,985 | <p>I know that if $$F(x,y)=f(x)+g(x)$$, then $$\frac{\partial^2 F}{\partial x \partial y} = 0$$ but when is $\frac{\partial^2 F}{\partial x \partial y} \not= 0$</p>
<p>Is it when $$F(x,y) = f(x)g(x)\space \text{or} \space f(x)^{g(x)}\space \text{or} \space f\bigl(g(x)\bigr)$$
or what?</p>
| giobrach | 332,594 | <p>In all the alternatives you proposed, there is no dependence uon the $y$ variable, so as soon as you differentiate with respect to it you’ll get $0$. Just make either $f$ or $g$ or both depend on $y$.</p>
|
2,206,415 | <p>Let $n \in \mathbb{Z_+},$ use integration by parts to prove </p>
<p>$$\int_0^{\infty}x^ne^{-x}dx = n!$$</p>
<p>I know that if you repeatedly differentiate $x^n$ you get $n!$, but I don't know how to prove this?</p>
<p>A step by step answer would be helpful. Thanks!</p>
| Khosrotash | 104,171 | <p>HINT: Take $I_n=\int_0^{\infty}x^ne^{-x}dx $
so </p>
<p>$$\begin{align}
I_n &= \int_0^\infty x^ne^{-x}dx\\
&=\left. x^n \cdot \frac{e^{-x}}{-1}\right|_0^\infty - \int_0^\infty nx^{n-1}\cdot \frac{e^{-x}}{-1}dx \\
&=0 - n(-1)\int_0^\infty x^{n-1}e^{-x}dx\\
&=n\int_0^\infty x^{n-1}e^{-x}dx \\
&= n... |
636,094 | <p>The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind.</p>
<p>I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one?</p>
<p>If it cannot be done in closed form, I wonder what a good asymptotic is.</p>
<p>I considered using Taylor series both for solving the int... | Daniel Robert-Nicoud | 60,713 | <p>First of all, I think it is better to put the lower bound of integration to be $1$, and assume $y>0$ so that we get a definite integral:
$$f(y)=\int_1^y\ln(x-\ln(x))dx$$
Substituting $x = e^z$ we get:
$$\begin{array}{ll}f(y) & =\int_0^{\ln y}\ln(e^z-z)e^zdz\\&=\int_0^{\ln y}\ln(e^z-z)(e^z-1)dz+\int_0^{\ln... |
2,309,864 | <p>Prove that $f$ is a homeomorphism iff $f[\overline A] = \overline {f[A]}$, I know how to prove that $f$ is continuous iff $f[\overline A] \subset \overline {f[A]}$, but how can I complete?</p>
| user160738 | 160,738 | <p>If $f:X\to Y$ is homeomorphism, then $f(\overline{A})\subset \overline{f(A)}$ by continuity of $f$. Say $g$ is an inverse of $f$. Then using continuity of $g$, $\overline{f(A)}=fg(\overline{f(A)})\subset f(\overline{gf(A)})=f(\overline{A})$</p>
<p>The other direction requires $f$ to be bijection. Indeed, if this as... |
429,138 | <p>I need help with the steps of finding the slope and $y$-intercept of this equation: $f(x)=3x-\frac{1}{5}$. I am in Algebra two and do not understand the proper ways, or steps of doing this.</p>
| Aang | 33,989 | <p>HINT: Slope of a function $f$ at any point $(a,f(a))$ is given by $\frac{d(f(x))}{dx}|_{x=a}$ and $y-$intercept is given by $f(0)$</p>
|
429,138 | <p>I need help with the steps of finding the slope and $y$-intercept of this equation: $f(x)=3x-\frac{1}{5}$. I am in Algebra two and do not understand the proper ways, or steps of doing this.</p>
| amWhy | 9,003 | <p>Hint: The equation $\;y = mx + b\;$ is an equation of a line in <strong>slope-intercept form</strong>: $$\large y = \underbrace{m}_{\text{slope}}x + \underbrace{b}_{\text{y-intercept}}\tag{ slope-intercept form}$$</p>
<p>So, in your case $$y = \underbrace{{\bf 3}}_{\bf m} x + \underbrace{\bf \left(-\frac 15\right)}... |
1,343,715 | <p>Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. </p>
<p>What about the convolution of measures on an unimodular group $G$?</p>
| Project Book | 234,125 | <p>Not sure if it's the most general case but according to Kallenberg's <em>Foundations of Modern Probability</em>, convolution of measures (I suspect the measures need to be at least $\sigma$-finite here, it's not specified but the proof states that you should use Fubini's) on a <em>measurable group</em> (group endowe... |
1,860,782 | <p>I have the following recurrence relation that I'm trying to solve:</p>
<p>$$f(n)=2f(n-1)-f(n-2)-2$$</p>
<p>The homogeneous part is easy:</p>
<p>The characteristic polynomial $r^2-2r+r=0$ has root $r=1$ with multiplicity 2, so the general solution is:</p>
<p>$$f(n)=An+B$$</p>
<p>for some initial conditions.</p>
... | bof | 111,012 | <p>If you've had an elementary course in differential equations, you can use an exponential generating function to turn this into an easy differential equation.</p>
<p>In terms of the exponential generating function
$$y(x)=\sum_{n=0}^\infty\frac{f(n)}{n!}x^n$$
the nonhomogeneous linear recurrence equation
$$f(n)=2f(n-... |
2,004,935 | <blockquote>
<p>The straight line $y = m(x – a)$ will meet the parabola $y^2 = 4ax$
at two distinct real points for which values of $m$?</p>
</blockquote>
<p>The answer is given as $m \in \mathbb R - {\{0\}}$.</p>
<p>I tried to solve by using the method of discriminants as follows:</p>
<p>$\{m(x-a)\}^2=4ax$, an... | mathlove | 78,967 | <p>The discriminant of a quadratic equation $ax^2+bx+c=0\ (\color{red}{a\not=0})$ is $D=b^2-4ac$.</p>
<p>In our case, we have
$$m^2x^2+(-2am^2-4a)x+m^2a^2=0$$
For $m=0$, we have $-4ax=0$ which is not a quadratic equation.</p>
|
2,004,935 | <blockquote>
<p>The straight line $y = m(x – a)$ will meet the parabola $y^2 = 4ax$
at two distinct real points for which values of $m$?</p>
</blockquote>
<p>The answer is given as $m \in \mathbb R - {\{0\}}$.</p>
<p>I tried to solve by using the method of discriminants as follows:</p>
<p>$\{m(x-a)\}^2=4ax$, an... | Sarvesh Ravichandran Iyer | 316,409 | <p>Note that if we have a quadratic equation $ax^2+bx+c=0$, then the solution is given by:
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$</p>
<p><em>Hence, the solution implicitly assumes that $a \neq 0$, which is the assumption that the equation is exactly of second degree and no less</em>.</p>
<p>Now, let us look at th... |
1,631,535 | <p>In the lecture notes for a course I'm taking, the definition of a convex function is given as follows:</p>
<p>"a function $f$ is convex if, for any $x_1$ and $x_2$, and for any $\alpha$ $\in$ [0,1], $\alpha f(x_1) + (1-\alpha)f(x_2) \ge f(\alpha x_1 + (1-\alpha ) x_2)$" </p>
<p>That is, if you draw a line segment ... | Zhanxiong | 192,408 | <p>It's expedient to apply the following result (see for example, Exercise $24$, Chapter $4$ of Rudin's <em>Principles of Mathematical Analysis</em>)</p>
<blockquote>
<p>If $f$ is continuous in $(a, b)$ such that
$$f\left(\frac{x + y}{2}\right) \leq \frac{1}{2}f(x) + \frac{1}{2}f(y)$$
for all $x, y \in (a, b)$,... |
2,528,227 | <p>Prove Number in decimal representation $N=abc,def,ghi,\cdots ,xyz$ is divisible by $7$.
Iff $abc-def+ghi-\cdots+xyz$, alternating sum of numbers formed by dividing the string $N$ into $3$ digit pairs of consecutive digits. Is divisible by $7$.</p>
| Alex Ravsky | 71,850 | <p>Let $N=\sum_{i=0}^{3n-1} a_i\cdot 10^{i}$. Since $10^3+1$ is divisible by $7$, modulo $7$ for each $k$ we have $$a_{3k+2}\cdot 10^{3k+2}+a_{3k+1}\cdot 10^{3k+1}+a_{3k}\cdot 10^{3k}\equiv$$
$$(a_{3k+2}\cdot 100+a_{3k+1}\cdot 10+a_{3k})\cdot 10^{3k}\equiv$$</p>
<p>$$(a_{3k+2}\cdot 100+a_{3k+1}\cdot 10+a_{3k})\cdot (-... |
1,669,923 | <p>Are $\cos^2 \theta$ and $\cos \theta^2$ the same?
I mean be it $\sin,\cos, \tan ,\cot ,\sec,\csc$.
Are they same? Please help a maths noob here.</p>
| StackTD | 159,845 | <p>The first notation is used to mean
$$\cos^2 \theta = \left( \cos \theta \right)^2$$
Your second notation will usually be read as
$$\cos \theta^2 = \cos \left( \theta^2 \right)$$
although it is sometimes preferred to use the notation in the right-hand side to be clear.</p>
<p>They are not the same since
$$\left( \co... |
2,129,281 | <p>Here is the problem I am stuck with: Is it true that for every positive integer $n > 1$,
$$\sum\limits_{k=1}^n \cos \left(\frac {2 \pi k}{n} \right) =0= \sum \limits_{k=1}^n \sin \left(\frac {2 \pi k}{n} \right)$$
I'm imagining the unit circle and adding up the value of both trig functions separately but I canno... | Simply Beautiful Art | 272,831 | <h2>Old answer to old question:</h2>
<hr>
<p>Multiply both sides by $n$ to get</p>
<p>$$\sum_{k=1}^n\cos(2\pi k)\stackrel?=0\stackrel?=\sum_{k=1}^n\sin(2\pi k)$$</p>
<p>It's easy enough to see that for positive integers $k$, $\cos(2\pi k)=1$ and $\sin(2\pi k)=0$, thus,</p>
<p>$$\sum_{k=1}^n\sin(2\pi k)=0$$</p>
<p... |
1,672,509 | <p>I missed a couple of my Linear algebra classes, so I'm a little lost on this question...</p>
<p>Given $S_1$, $S_2$, $S_3 : \mathbf{R}^2\to \mathbf{R}^2$ are linear mappings defined by:</p>
<p>$S_1(x_1, x_2) = (x_1-x_2, -2x_1+x_2)$</p>
<p>$S_2(x_1,x_2) = (2x_1- x_2, -4x_1+ x_2)$</p>
<p>$S_3(x_1,x_2) = (-x_1+ x_2 ... | Richard | 14,493 | <p>The problem can also be solved numerically using <a href="https://www.cvxpy.org/tutorial/dgp/index.html" rel="nofollow noreferrer">cvxpy's geometric programming capabilities</a>, like so:</p>
<pre><code>#!/usr/bin/env python3
import cvxpy as cp
x = cp.Variable(pos=True)
y = cp.Variable(pos=True)
constraints = []... |
3,516,494 | <p><span class="math-container">$$2x^2 + 3x + 1$$</span></p>
<p>applying quadratic formula:</p>
<p><span class="math-container">$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p><span class="math-container">$$a=2, b=3, c=1$$</span></p>
<p><span class="math-container">$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \c... | Bill Moore | 515,499 | <p>On the quadratic formula:</p>
<p><span class="math-container">$$ax^2 +bx+c=0$$</span></p>
<p><span class="math-container">$$r_i =\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p><span class="math-container">$$a\cdot(x-r_1)(x-r_2)$$</span></p>
<p>First, the "<span class="math-container">$a\cdot$</span>" term in t... |
338,090 | <p>A is a $100 \times 100$ matrix.</p>
<p>The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$</p>
<p>Find the rank</p>
| Jyrki Lahtonen | 11,619 | <p>Hint: Show that each row is a linear combination of the vectors
$(1,4,9,\ldots,100^2)$ and $(1,1,1,\ldots,1)$.</p>
|
1,838,409 | <blockquote>
<p>In how many ways can you select two distinct integers from the set {1,
2, 3, . . . , 100} so that their sum is: (a) even? (b) odd?</p>
</blockquote>
<p>I'm studying for a discrete midterm this coming Monday and saw the following problem on a practice midterm my Professor posted. I know the amount h... | Ashwin Ganesan | 157,927 | <p>The solution $2 C(50,2)$ you gave in comments for part (a) is correct. For part (b), observe that the sum is odd iff one of the numbers is even and the other is odd. The even number can be chosen in $C(50,1)$ ways, and the odd number in $C(50,1)$ ways. So the answer for part (b) is $50^2$. </p>
|
2,583,015 | <p>So, from my understanding there are two versions of this theorem:</p>
<p>Version one states that, if $\displaystyle F(x)= \int_a^xf(t)~dt$, then $$\frac{dF}{dx}=\frac{d}{dx}\left[\int_a^xf(t)~dt\right]=f(x)$$whereas the second version states that $$\int_a^bf(x)~dx=F(b)-F(a)$$what I'm hoping to establish is this: I ... | Community | -1 | <p>You have only stated vaguely the FTC without giving the appropriate assumptions.</p>
<blockquote>
<p>(<strong>First fundamental theorem of calculus.</strong>) Let <span class="math-container">${[a,b]}$</span> be a compact interval of positive length. Let <span class="math-container">${f: [a,b] \rightarrow {\bf C}}$<... |
4,512,704 | <p>Consider two closed (compact if needed) convex sets <span class="math-container">$E$</span> and <span class="math-container">$F$</span>. Define the <span class="math-container">$\epsilon$</span> neighborhood of set <span class="math-container">$E$</span> as
<span class="math-container">$$E_\epsilon = \cup_{x \in E} ... | Keen-ameteur | 421,273 | <p>Assuming <span class="math-container">$X=\mathbb{R}^d$</span>, I think the answer is true. Assume towards contradiction that the statement is not true. Then for every <span class="math-container">$n$</span> there exists an <span class="math-container">$x_n\in F$</span> such that
<span class="math-container">$$d(x_n,... |
4,512,704 | <p>Consider two closed (compact if needed) convex sets <span class="math-container">$E$</span> and <span class="math-container">$F$</span>. Define the <span class="math-container">$\epsilon$</span> neighborhood of set <span class="math-container">$E$</span> as
<span class="math-container">$$E_\epsilon = \cup_{x \in E} ... | Esgeriath | 1,021,258 | <p>If we allow <span class="math-container">$E$</span> and <span class="math-container">$F$</span> to be compact, and <span class="math-container">$E \cap F \neq \emptyset$</span> then the result is quite trivial in any metric space. In fact we don't need compactness - if one of them is bounded statement holds.</p>
<p>... |
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