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 |
|---|---|---|---|---|
578,961 | <p>Let $\mathbf{T}=[\mathbf{t}_1,\dots,\mathbf{t}_d]$ be a $m\times d$ matrix with $\mathbf{t}_i$ as its linearly independent columns. Also I assume $d<\min(m,n)$. Let $\mathbf{H}$ be a $n\times m$ matrix. Let $\mathbf{W}$ be a $n \times n$ positive definite matrix. For $i=1,\dots,d$, let me define the matrices
\beg... | adam W | 43,193 | <p>For the matrices of interest I can show a shared null space of their similar matrices. This may not be helpful but I hope it is, and I did the work so I thought I would show it.</p>
<p>Let $\mathbf{M} = \mathbf{H}\mathbf{T}$. In matrix notation, using $\mathbf{e}_0^H = \pmatrix{1 & 0 & 0 & \cdots}$, you... |
252,870 | <p>Given a polynomial, lets say for example <span class="math-container">$f(x,y) = (1+x+y)^2 = 1+2x+x^2+2y+2xy+y^2$</span>, I'd like to be able to order the terms of the polynomial by total degree, either in increasing or decreasing order (and if alphabetical order can be taken into account within terms of the same tot... | kglr | 125 | <pre><code>hypatia = {"HIP87382", "2MASS19290895+4311502", "HIP98314", "HIP98316", "HIP106931"};
</code></pre>
<p>A random list of names:</p>
<pre><code>SeedRandom[1]
names = Table[RandomSample[#, 5] & @
Union[{"2259072226846681494447849441045193405",... |
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | DKal | 83,540 | <p>In fact, for any rational number $ r $ it is true that the irrationality of $ x+r $ implies the irrationality of $ x $. This is due to the fact that the rationals are closed under addition. Assume that $ x+r $ is irrational and (for contradiction) that $ x $ is rational, by the fact that the rationals are closed un... |
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | ncmathsadist | 4,154 | <p>The rational numbers are closed under addition and subtraction. Let $w$ be any irrational number and $r$ be a rational number. Since
$$ (r + w) - r = w$$
and $w$ is irrational, one of the subtrahends here is irrational. Since $r$ is rational, the irrational quantity must by $r + w$.</p>
|
3,754,819 | <p>Evaluate the integral:
<span class="math-container">$$\int_{1}^{\sqrt{2}} \frac{x^4}{(x^2-1)^2+1}\,dx$$</span></p>
<p>The denominator is irreducible, if I want to factorize and use partial fractions, it has to be in complex numbers and then as an indefinite integral, we get
<span class="math-container">$$x + \frac{\... | Quanto | 686,284 | <p>Note
<span class="math-container">\begin{align}
I=&\int_{1}^{\sqrt{2}} \frac{x^4}{(x^2-1)^2+1}\,dx\\
= &\int_{1}^{\sqrt{2}} \left(1+\frac{2x^2-2}{x^4-2x^2+2}\right)\,dx\\
= &\sqrt2-1+\int_{1}^{\sqrt{2}} \frac{2-\frac2{x^2}}{x^2+\frac2{x^2}-2}dx\\
=& \sqrt2-1 + (1+\frac1{\sqrt2})I_1 + (1-\frac1{\sqrt2... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Toby Bartels | 8,508 | <p>Like Terry Tao, I find the transience of slides to be a problem. This is one reason why I stopped using slides as such and began using a single continuous-scroll page for each topic. I lecture from the bottom of the page, so students who are behind can still see the top. (I'm also one of those people who mixes th... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Shuhao Cao | 7,200 | <p>Here is an old scicomp.SE question that answered some of your question: <a href="https://scicomp.stackexchange.com/questions/290/what-are-criteria-to-choose-between-finite-differences-and-finite-elements">What are criteria to choose between finite-differences and finite-elements</a>?</p>
<p>In my humble opinion, FE... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| AGN | 252,682 | <p><strong>FDM</strong> </p>
<p>FDM is created from basic definition of differentiation that is
$$ \frac{df}{dx}=\frac{f(x+h)-f(x)}{h}$$ here "h" tends to zero.</p>
<p>In numerical analysis, its not possible to divide a number by "0" so "zero" means a small number. So FDM is similar to differential calculus but it ha... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Stefan Hansen | 25,632 | <p>A function is often called càdlàg if it is right-continuous and admits left limits. This term is from the french <em>continue à droite, limite à gauche</em>.</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| citedcorpse | 52,216 | <p>In homology one has a sequence of "differentials". Their images are usually denoted $B(X)$, apparently from the german word for "images", and their kernels $Z(X)$ from the german word for "cycles". </p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| Przemysław Scherwentke | 72,361 | <p>In old Polish textbooks for secondary school complex numbers were denoted $\bf Z$, as <em>zespolone</em> and integers — $\bf C$, as <em>całkowite</em>.</p>
|
1,516,363 | <p>If $f: (0,+ \infty) \rightarrow \mathbb{R}$ is continuous and $$f(x + y) = f(x) + f(y) ,$$ then $f$ is linear? </p>
<p>I saw in somewhere that if $f$ is continuous, one can drop the condition $f(ax)= a f(x), \forall a, x \in \mathbb{R}$. Is this true?</p>
| ncmathsadist | 4,154 | <p>Notice that for positive integers $p$ and $q$ it is easy to show that
$$ f(p/q) = p/q \cdot f(1).$$<br>
Extension by density does the rest.</p>
|
4,498,203 | <p>We know that each row (and each column) of composition table of a finite group, is a rearrangement (permutation) of the elements of the group.</p>
<p>How about the other way round? If we have a composition table where each row and each column is a permutation of the elements of a set, does this composition table nec... | Michael Kinyon | 444,012 | <p>For an example with an identity element and inverses, consider
<span class="math-container">$$\begin{array}{c|ccccc}
\ast & e & a & b & c & d\\
\hline
e & e & a & b & c & d \\
a & a & e & c & d & b \\
b & b & d & e & a & c \\
c & c &... |
2,573,572 | <p>Here is the expression to take the derivative of.
$$C = \frac{1}{2}\sum_j (y_j - a_j^L)^2$$</p>
<p>Here is the result.
$$\frac{\partial C}{\partial a_j^L} = 2(a_j^L-y_j)$$</p>
<p>Multiplying by 2, then again by the derivative of the inside (-1) seems reasonable, but what happened to the summation?</p>
| Matthew Leingang | 2,785 | <p>Let's save the index $j$ for the derivative and write
$$
C = \frac{1}{2} \sum_{i=1}^n (y_i - a_i^L)^2
$$
Therefore by the sum rule and chain rule,
$$
\frac{\partial C}{\partial a_j^L}
= \sum_{i=1}^n 2(y_i - a_i^L)(-1)\frac{\partial a_i^L}{\partial a_j^L}
$$
Assuming the variables $\left(a_1^L,\dots,a_n... |
2,203,066 | <p>The definition I have is the following:</p>
<blockquote>
<p>A vector space V is said to be <strong>finite-dimensional</strong> if there is a finite set of vectors in V that spans V and is said to be <strong>infinite-dimensional</strong> if no such set exists.</p>
</blockquote>
<p>However, with this definition I ... | cws | 225,972 | <p>The question asks whether there exists a finite basis of the vector space. If there exists a finite basis, then this vector space is said to be finite dimensional. If not the vector space is infinite dimensional. An example of an infinite dimensional vector space is the vector space of all power series.</p>
<p>Cont... |
2,507,864 | <blockquote>
<p>Check if for any two set families $\mathcal A $ and $\mathcal B $ the
following is true: $\bigcup (\mathcal A \cap \mathcal B) = \bigcup
\mathcal A \cap \bigcup \mathcal B$</p>
</blockquote>
<p>First of all I considered an example: $\mathcal A = \{ \{1,2\}, \{1,3\} \}$ and $\mathcal B = \{\{1,2\}... | Zhuoran He | 485,692 | <p>Maximum preserves convexity and minimum preserves concavity. So the maximum of two concave functions may be neither concave nor convex. It may become double peaked. For example,</p>
<p>$$f(x)=\max[-|x+1|,-|x-1|]$$</p>
<p>has an "M"-shaped graph. The minimum of two concave functions is always concave. This is not d... |
1,090,620 | <p>I don't know how to solve this limit</p>
<p>$$ \lim_{y\to0} \frac{x e^ { \frac{-x^2}{y^2}}}{y^2}$$</p>
<p>$\frac{1}{e^ { \frac{x^2}{y^2}}} \to 0$</p>
<p>but $\frac{x}{y^2} \to +\infty$</p>
<p>This limit presents the indeterminate form $0 \infty$ ?</p>
| egreg | 62,967 | <p>For $x\ne0$, set $x^2/y^2=t$; then, as $y\to0$, we have $t\to\infty$, so the limit becomes
$$
\lim_{t\to\infty}\frac{1}{x}te^{-t}=\frac{1}{x}\lim_{t\to\infty}\frac{t}{e^t}
$$
that's easy to show being $0$. If $x=0$ there's of course nothing to do.</p>
|
114,289 | <p>I am trying to use C++ programs through MathLink in my notebooks, but I cannot compile successfully the simple programs included in Mathematica. </p>
<p>I do not have a specific question, I am just looking for guidance.</p>
<p><code>$Version
$SystemID
"9.0 for Linux x86 (64-bit) (November 20, 2012)"
"Linux-x86-... | Andrew Klofas | 39,468 | <p>Without knowing specifics (haven't done what you are trying to do), it's pretty common to see those types of runtime linker errors if you don't have the correct link parameters. try adding '-lrt' (without quotes) to the end of the mcc command. Does that help?</p>
|
17,143 | <p>My next project I'd like to start working on is Domain Coloring. I am aware of the beautiful discussion at:</p>
<p><a href="https://mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot">How can I generate this "domain coloring" plot?</a></p>
<p>And I am studying it. H... | Dr. belisarius | 193 | <p>Here is something quickly made, and similar to what you are after. You'll have to work out the details, though:</p>
<pre><code>HotColor[ z_ ] :=
Which[
0 <= z <= 3/8, RGBColor[z 8/3, 0, 0],
3/8 <= z <= 6/8, RGBColor[1, (z - 3/8) 8/3, 0],
True, RGBColor[1, 1, (z - 6/8... |
1,018,235 | <p>Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$?</p>
<p>E.g., is $10 \leq \infty$ a correct mathematical statement?</p>
<p>I know this comparison is true in computer arithmetic, however is it correct from mathematical point of view? Does the "equality" part in $\leq$ matter here?</p>... | Community | -1 | <p>The real numbers $\mathbb R$ do not contain $\infty$ as an element, so with the relation $\le_\mathbb R$, the statement $c\le\infty$ does not make sense.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Extended_real_number_line" rel="nofollow"><strong>extended real number line</strong></a> $\overline{\mathbb R}$ <... |
1,018,235 | <p>Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$?</p>
<p>E.g., is $10 \leq \infty$ a correct mathematical statement?</p>
<p>I know this comparison is true in computer arithmetic, however is it correct from mathematical point of view? Does the "equality" part in $\leq$ matter here?</p>... | Qiaochu Yuan | 232 | <p>Sure. An example where this is used in mathematics is when talking about <a href="http://en.wikipedia.org/wiki/Lp_space" rel="nofollow">$L^p$-spaces</a>. The only thing you need to know about these is that they depend on a real parameter $p$ which is allowed to take the value $\infty$, and it's common to make statem... |
1,902,842 | <p>In Bert Mendelson's <em>Introduction to Topology</em>, the first exercise of Ch. 1 Sec. 5 states:</p>
<blockquote>
<p>Let $X\subset A$ and $Y\subset B$. Prove that $$C(X\times Y)=A\times C(Y)\cup C(X) \times B.$$</p>
</blockquote>
<p>I have seen a "proof" of this, but I remain unsatisfied with the result. As sup... | Doug M | 317,162 | <p>From the previous line</p>
<p>$|x-2||x+3| < \epsilon$</p>
<p>We have established that $|x-2|$ is less then $\delta$</p>
<p>We need a construction for $\delta$ to make all this true.</p>
<p>$\delta < \frac {\epsilon}{x+3}$</p>
<p>And now we are where your question begins.</p>
<p>Demand that $\delta < 1... |
1,904,903 | <p>Taken from Soo T. Tan's Calculus textbook Chapter 9.7 Exercise 27-</p>
<p>Define $$a_n=\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$$
One needs to prove the convergence or divergence of the series $$\sum_{n=1}^{\infty} a_n$$</p>
<p>upon finding the radius of convergence for $\su... | grand_chat | 215,011 | <p>Rewrite the $n$th term by sliding each of the factors in the numerator one position to the left. This gives
$$
a_n = \frac 21\frac43\frac65\cdots\frac{2n}{2n-1}\frac1{2n+1}.
$$
We now see $a_n$ is a product consisting of factors bigger than one, multiplied onto the final factor $\frac1{2n+1}$. Conclude
$$
a_n>\fr... |
2,195,287 | <blockquote>
<p>Knowing that $p$ is prime and $n$ is a natural number show that
$$n^{41}\equiv n\bmod 55$$
using Fermat's little theorem
$$n^p\equiv n\bmod p$$</p>
</blockquote>
<p>If the exercise was to show that
$$n^{41}\equiv n\bmod 11$$ I would just rewrite $n^{41}$ as a power of $11$ and would easily prov... | kub0x | 309,863 | <p>Since $n^{10} \equiv 1 \pmod{11}$ Then $n^{10\cdot k} \equiv 1 \pmod{11}$ </p>
<p>Thus for $k=4 \Rightarrow n^{40} \equiv 1 \pmod{11}$ then $n \equiv n^{41} \pmod{11}$ (using Fermat Little's).</p>
<p>For modulus $55$ you can use the fact that $55=11.5$ so:</p>
<p>$n^{11} \equiv n \pmod{11}$ and $n^{5} \equiv n \p... |
1,985,427 | <p>$$
A= \begin{bmatrix}
2 & 1 & -1 \\
-2 & -2 & 1 \\
0 & -2 & 1 \\
\end{bmatrix}
$$</p>
<p>Can someone show me the best way to approach this? Should I use pivoting? I tried using the formula, but I think that only works for 2 x 2 matrices. </p>
| Joffan | 206,402 | <p>As an easy-to-understand process, you can note that $A.A^{-1} = I$ and then undertake parallel row operations on $A$ and $I$ to transform this into $I.A^{-1}=X$, where $X$ is the result of the same operations on $I$ that transformed $A$ into $I$. You can check, through considering the action of matrix multiplication... |
2,751,909 | <blockquote>
<p>Let $f$ be a non-negative differentiable function such that $f'$ is continuous and
$\displaystyle\int_{0}^{\infty}f(x)\,dx$ and $\displaystyle\int_{0}^{\infty}f'(x)\,dx$ exist.</p>
<p>Prove or give a counter example: $f'(x)\overset{x\rightarrow
\infty}{\rightarrow} 0$</p>
</blockquote>
<p><str... | Ian | 83,396 | <p>The $P$ will need to depend on $\epsilon$; you will not be able to find one $P$ for all $\epsilon$. To put it another way, your quantifiers are in the wrong order. </p>
<p>That said, the key property enabling you to do this problem directly from the definition is that $f(x)=x$ is an increasing function. This means ... |
78,001 | <p>I have to find the exponential generating function for placing distinct objects into $k$ distinct boxes with at least $m$ object per box, indexed by the number of objects. Could you help me please? Also with some hints</p>
| Phira | 9,325 | <p>It is easy to do for $k=1$. For general $k$, one can use the correspondence between disjoint union of labelled combinatorial objects and products of exponential generating functions.</p>
|
78,001 | <p>I have to find the exponential generating function for placing distinct objects into $k$ distinct boxes with at least $m$ object per box, indexed by the number of objects. Could you help me please? Also with some hints</p>
| Brian M. Scott | 12,042 | <p>Let $a_k(m,n)$ be the number of ways of placing $n$ distinct objects in $k$ distinct boxes if there must be at least $m$ objects in each box. Suppose first that $k=1$. Clearly $a_1(m,n)=0$ if $n<m$, and $a_1(m,n)=1$ if $m\ge n$, so the exponential generating function for the $a_1(m,n)$ is $$A_1(x)=\sum_{n\ge 0}a_... |
2,184,056 | <p>To compute the oblique asymptote as $x \to +\infty$, we can first compute $\mathop {\lim }\limits_{x \to + \infty } \frac{{f(x)}}{x}$, it it exists, and $\mathop {\lim }\limits_{x \to + \infty } \frac{{f(x)}}{x} = k$, then we can further compute $\mathop {\lim }\limits_{x \to + \infty } (f(x) - kx)=b$, and if it ... | onamoonlessnight | 188,019 | <p>Generally no - just take any function $f(x)$ that grows sub-linearly at infinity. For example, $$ f(x) = \sqrt{x} $$ means $k=0$, but then $$ \lim_{x \to \infty} (f(x) - kx) = \lim_{x \to \infty } \sqrt{x} = \infty. $$</p>
|
3,232,341 | <p>How would I show this? I know a directed graph with no cycles has at least one node of outdegree zero (because a graph where every node has outdegree one contains a cycle), but do not know where to go from here.</p>
| sam000013 | 719,586 | <p>This proof will prove that DAGs (Directed Acyclic Graphs) have at least one node of indegree <span class="math-container">$ 0 $</span> and one node of outdegree <span class="math-container">$ 0 $</span> as well:
A DAG will contain all paths of finite length (Because of absence of cycles). Let's consider WLOG, the lo... |
248,313 | <p>Assume that $f:\mathbb R \rightarrow \mathbb R$ is continuous and $h\in \mathbb R$. Let $\Delta_h^n f(x)$ be a finite difference of $f$ of order $n$, i.e</p>
<p>$$
\Delta_h^1 f(x)=f(x+h)-f(x),
$$
$$
\Delta_h^2f(x)=\Delta_h^1f(x+h)-\Delta_h^1 f(x)=f(x+2h)-2f(x+h)+f(x),
$$
$$
\Delta_h^3 f(x)=\Delta_h^2f(x+h)-\Delta_... | Davide Giraudo | 9,849 | <p>The result holds if we assume that $f$ is $n$-times differentiable, otherwise, as WimC shows, it's not necessarily the case.</p>
<p>Using <a href="https://math.stackexchange.com/questions/243425/prove-that-hk0-lim-t-to0-frac-sum-j-0k-binomkj-1k-jhjt/243449#243449">this thread</a> and translated function ($f(\cdot)=... |
248,313 | <p>Assume that $f:\mathbb R \rightarrow \mathbb R$ is continuous and $h\in \mathbb R$. Let $\Delta_h^n f(x)$ be a finite difference of $f$ of order $n$, i.e</p>
<p>$$
\Delta_h^1 f(x)=f(x+h)-f(x),
$$
$$
\Delta_h^2f(x)=\Delta_h^1f(x+h)-\Delta_h^1 f(x)=f(x+2h)-2f(x+h)+f(x),
$$
$$
\Delta_h^3 f(x)=\Delta_h^2f(x+h)-\Delta_... | WimC | 25,313 | <p>Let $f(x) = |x|$ then $\Delta_h^2(f)$ has support $[-2h, 0]$. In particular $\lim_{h \to 0}\Delta_h^2(f)/h^2 = 0$ pointwise, but $f$ is not a polynomial.</p>
<p><strong>Edit:</strong> If the convergence in $x$ is <em>uniform</em> on an interval $[a, b]$ then I think that $f$ is a polynomial on that interval. This... |
2,286,749 | <p>My question is about the general solution for the following differential equation:
$$ \frac{dx}{dt} = x^a(1-x)^b,\quad a,b\gt 0~~~~~~~~~~~~~~~(1)~. $$</p>
<p>Obviously, if $a=b=1$ then (1) reduces to
$$ \frac{dx}{dt} = x(1-x) $$ which has as solution $$ x(t) = \frac{1}{1 + A e^{-t}}\,,$$ for some constant, $A$.... | Jan Eerland | 226,665 | <p>Well, we have:</p>
<p>$$\text{x}'\left(t\right)=\text{x}\left(t\right)^\text{a}\cdot\left(1-\text{x}\left(t\right)\right)^\text{b}\space\Longleftrightarrow\space\int\frac{\text{x}'\left(t\right)}{\text{x}\left(t\right)^\text{a}\cdot\left(1-\text{x}\left(t\right)\right)^\text{b}}\space\text{d}x=\int1\space\text{d}x\... |
4,206,147 | <blockquote>
<p><span class="math-container">$f(f(x))=f(x),$</span> for all <span class="math-container">$x\in\Bbb R$</span> suppose <span class="math-container">$f$</span> is differentiable, show <span class="math-container">$f$</span> is constant or <span class="math-container">$f(x)=x$</span></p>
</blockquote>
<p>Cl... | FShrike | 815,585 | <p>Since <span class="math-container">$f(f(x))=f(x)$</span>, <span class="math-container">$y=f(x)$</span> is a valid input to the function, implying that the function's range is a subset of its domain. With that in mind, <span class="math-container">$f'(f(x))$</span> is analogous to <span class="math-container">$f'(y)$... |
1,345,643 | <p>In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. But how do I use the triangle to get to that result? Personally I can only solve things like $(x+y)^2$ and $(x+y)^3$.</... | A C | 246,578 | <p>You can have x = z<sup>1</sup>
and y = z <sup>2</sup> + z <sup>3</sup> + z <sup>4</sup>
Then the problem is (x+y) <sup>3</sup>. Later resubstitute y in the expansion and choose a = z <sup>2</sup> and b = z <sup>3</sup> + z <sup>4</sup>. And expand those powers by pascal's triangle and finally resubstitute for a and ... |
179,377 | <p>Consider the $k \times k$ block matrix:</p>
<p>$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \ddots &
\vdots \\ B & B & B & \cdots & A
\end{array}\right) = I_k \otimes (A - B) + \mathbb{... | Christian Remling | 48,839 | <p>We can just manipulate $C$ in the usual way by row operations: Subtract the last "row" from all the other "rows" (this is really several traditional row operations done at once). This produces
$$
\begin{pmatrix} A- B &0& 0 & \ldots & 0 &B-A \\
0 & A-B &0 &\ldots & 0 & B-A\\
&a... |
179,377 | <p>Consider the $k \times k$ block matrix:</p>
<p>$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \ddots &
\vdots \\ B & B & B & \cdots & A
\end{array}\right) = I_k \otimes (A - B) + \mathbb{... | Rodrigo de Azevedo | 91,764 | <p>Let us assume that $A-B$ is invertible. Write</p>
<p>$$\begin{array}{rl} C &= \begin{bmatrix} A & B & \ldots & B\\ B & A & \ldots & B\\ \vdots & \vdots & \ddots & \vdots\\B & B & \ldots & A\end{bmatrix}\\\\ &= \begin{bmatrix} A-B & O_n & \ldots & O... |
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | David | 297,532 | <p>There are countries where people are puzzled if you tell them there is a distinction between "calculus" and "analysis." They think "calculus" is just an old-fashioned name for analysis. The reason these subjects are viewed as different in North America is because a typical "calculus" class is where one learns the me... |
3,261,846 | <blockquote>
<p>What is the solution to the IVP <span class="math-container">$$y'+y=|x|, \ x \in \mathbb{R}, \ y(-1)=0$$</span></p>
</blockquote>
<p>The general solution of the above problem is <span class="math-container">$y_{g}(x)=ce^{-x}$</span>.</p>
<p>How to find the particular solution? As <span class="math-c... | user289143 | 289,143 | <p>You have to distinguish the two cases whether <span class="math-container">$x < 0$</span> and <span class="math-container">$x \geq 0$</span> and see that the two solutions "matches" at the origin.</p>
<p>When <span class="math-container">$x < 0$</span>, <span class="math-container">$y'+y=-x$</span> you look f... |
3,261,846 | <blockquote>
<p>What is the solution to the IVP <span class="math-container">$$y'+y=|x|, \ x \in \mathbb{R}, \ y(-1)=0$$</span></p>
</blockquote>
<p>The general solution of the above problem is <span class="math-container">$y_{g}(x)=ce^{-x}$</span>.</p>
<p>How to find the particular solution? As <span class="math-c... | Community | -1 | <p>To find a particular solution, you can explore the Ansatz</p>
<p><span class="math-container">$$y=|x|,$$</span> giving <span class="math-container">$$y'+y=\text{sgn}(x)+|x|.$$</span></p>
<p>This is valid on the whole real line, except at the origin, where the discontinuity is unavoidable. So compensating the sign ... |
3,261,846 | <blockquote>
<p>What is the solution to the IVP <span class="math-container">$$y'+y=|x|, \ x \in \mathbb{R}, \ y(-1)=0$$</span></p>
</blockquote>
<p>The general solution of the above problem is <span class="math-container">$y_{g}(x)=ce^{-x}$</span>.</p>
<p>How to find the particular solution? As <span class="math-c... | JJacquelin | 108,514 | <p><span class="math-container">$$y'+y=|x|$$</span>
Solving without condition :</p>
<p>Case <span class="math-container">$x>0 \quad:\quad y'+y=x \qquad y=c_1e^{-x}+x-1$</span></p>
<p>Case <span class="math-container">$x<0 \quad:\quad y'+y=-x \qquad y=c_2e^{-x}-x+1$</span>
<span class="math-container">$$y=ce^{-x... |
2,352,811 | <p>Why it's not enough for the partial derivatives to exist for implying differentiability of the function?
Why is the continuity of the partial derivatives needed?</p>
| Michael Hoppe | 93,935 | <p>To answer your first question: because they're partial, i.e., derivatives in very special directions. Imagine, for example, a surface which is generated by a ray parallel to the $x$-$y$-axis that is rotating in a way that for the first quarter of its rotation it raises by $1$, for the next quarter it lowers by $1$,... |
1,443,441 | <blockquote>
<p>If <span class="math-container">$\frac{x^2+y^2}{x+y}=4$</span>,then all possible values of <span class="math-container">$(x-y)$</span> are given by<br></p>
<p><span class="math-container">$(A)\left[-2\sqrt2,2\sqrt2\right]\hspace{1cm}(B)\left\{-4,4\right\}\hspace{1cm}(C)\left[-4,4\right]\hspace{1cm}(D)\l... | Wanderer | 195,012 | <p>${x^2+y^2\over x+y}=4 \implies x^2+y^2=4x+4y \implies x^2+y^2-4x-4y=0 \implies (x-2)^2+(y-2)^2=(2\sqrt{2})^2$ which is a circle with center $(2,2)$ and radius $2\sqrt{2}$</p>
|
452,306 | <p>I am trying to be able to find the radius of a cone combined with a cylinder.
see my other question
(Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? part2 )</p>
<p>I have a volume calculation that Has been reduced as far as I know how to.</p>
... | DBS | 67,937 | <p>I am editing what I said before which mistakenly assumed $A$ above is a subgroup. </p>
<p>Here is the answer. Assume that $g$ is not nilpotent (in that case there is nothing to show). </p>
<p>Claim: The point {1} is not isolated in $\bar{A}$. </p>
<p>The claim implies that there is a non-trivial sequence in $\ba... |
117,024 | <p>The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. </p>
<p>This procedure gives us the algorithmic complexity of $O(n^3)$.</p>
<p>It is well known that if $A$ is the adjacency matrix o... | Listing | 3,123 | <p>Let me cite this <a href="https://www-complexnetworks.lip6.fr/%7Elatapy/Publis/triangles_short.pdf" rel="noreferrer">paper</a> from 2007 (Practical algorithms for triangle computations in very large (sparse (power-law)) graphs by Matthieu Latapy):</p>
<blockquote>
<p>The fastest algorithm known for finding and count... |
3,388,457 | <p>I made an equation
<span class="math-container">$$(100b+40+a)-(100a+40+b)=99$$</span> simplified that to <span class="math-container">$b-a=1$</span> , but do not know where to go from there.</p>
| farruhota | 425,072 | <p>You found: <span class="math-container">$b-a=1$</span>. </p>
<p><span class="math-container">$a4b$</span> is divisible by <span class="math-container">$9$</span>, so: <span class="math-container">$a+b=5$</span> or <span class="math-container">$14$</span>. </p>
<p>Adding the two equations you get: <span class="math... |
2,725,697 | <p>A weird question that has me confused. Suppose I have a symmetric matrix $A$, which has to be computed somehow. For example, the Hessian matrix is a symmetric matrix that is computed by taking the gradient twice. A covariance matrix is also symmetric as another example. $A$ will have $n^2$ entries but really only ne... | Rócherz | 451,007 | <p>It sounds as something that would be discussed in Numerical Linear Algebra courses. Haven't dealt with it directly, though, just similar operations.</p>
|
936,200 | <p>Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows.</p>
<p>Can someone please help me? I don't know what to assume since I don't know if it is increasing or decreasing when x_0 < 1 and when x_0 > 1.
Any hint/help would rea... | Marc van Leeuwen | 18,880 | <p>The relation $\in$ is typed <code>\in</code> (in TeX) and often pronounced "is in", or "is a member of", or "is an element of". But many variations occur occuring to the particular set, so you might pronounce $z\in\mathbf C$ as "$z$ is a complex number" rather than "$z$ is an element of the set of complex numbers".<... |
4,637,565 | <p>I am thinking of positive sequences whose sum is infinite but whose sum of squares is not?</p>
<p>One representative sequence is <span class="math-container">$$x[n] = \frac{a}{n+b},$$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are given real numbers such that <sp... | Daron | 53,993 | <p>The standard example is the sequence <span class="math-container">$x_n = 1/n$</span>.</p>
<p>We have <span class="math-container">$\sum_{n=1}^N \frac{1}{n} \simeq \log N$</span> and so <span class="math-container">$\sum_{n=1}^N \frac{1}{n}$</span> diverges.</p>
<p>At the same time <span class="math-container">$\sum_... |
966,798 | <p>How I solve the following equation for $0 \le x \le 360$:</p>
<p>$$
2\cos2x-4\sin x\cos x=\sqrt{6}
$$</p>
<p>I tried different methods. The first was to get things in the form of $R\cos(x \mp \alpha)$:</p>
<p>$$
2\cos2x-2(2\sin x\cos x)=\sqrt{6}\\
2\cos2x-2\sin2x=\sqrt{6}\\
R = \sqrt{4} = 2 \\
\alpha = \arctan \f... | Ross Millikan | 1,827 | <p>Note that along the way he assumed $\delta \lt \frac 12$ This allowed the following calculations to go through. As long as $\epsilon$ is small, $\frac \epsilon 2 \lt \frac 12$, but a nasty opponent (who knew our proof, say)could give us $\epsilon =5$, say. If we just say $\delta = \frac \epsilon 2$ our opponent c... |
869,268 | <p>I am asked if $\{n, n^{2}, n^{3}\}$ forms a group under multiplication modulo $m$ where $m = n + n^{2} + n^{3}.$</p>
<p>As an example we see that $\{2, 4, 8\}$ does form a group modulo $14,$ with identity $8,$ but am stuck starting the proof for the general case. Thanks in advance.</p>
| Bill Dubuque | 242 | <p>Though one can verify this result by brute force calculation, one gains much more insight by examining the ring-theoretic structure governing the result. First, we have factorizations</p>
<p>$\quad \smash[t]{(\color{#0a0}{\overbrace{n^3\!+\!n^2\!+\!n}^{\large m}})}(n\!-\!1)\, =\, n(n^3\!-1),\,$ and $\ G = \langle ... |
1,721,925 | <p>I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of one method over the other. </p>
<p>I think that it would be nice to have an answer that puts them in comparison and ... | Lutz Lehmann | 115,115 | <p>You should never seriously use bisection.</p>
<p>If you think that derivatives are hard, use the secant method.</p>
<p>If you want to force convergence and can find intervals with opposite signs of the function, then use one of the anti-stalling variants of regula falsi. If you think that convergence could be fast... |
1,721,925 | <p>I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of one method over the other. </p>
<p>I think that it would be nice to have an answer that puts them in comparison and ... | Simply Beautiful Art | 272,831 | <p>I'm of an opinion a bit different from that of <a href="https://math.stackexchange.com/a/1721990">Lutz Lehmann's</a>. Here are my thoughts concerning root-finding in 1 dimension:</p>
<p>TL;DR although no method is best and each has their drawbacks, it is often the case that another method can be used to counteract t... |
985,212 | <p>Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?</p>
| BioCoder | 186,331 | <p>No, you can't row reduce in advance. You will get a different characteristic polynomial if you do that.
For example,matrix A=$\left[
\begin{array} {lcr}
-1 & 1 & 0 \\
-4 & 3 & 0 \\
1 & 0 & 2 \\
\end{array}
\right] $ has a characteristic polynomial (2-$\lambda$)(1-$\lambda$)$^2$
But the reduc... |
551,964 | <p>I have an optimization problem where I have to optimize a function f(A) where A is a matrix(sparse).</p>
<p>Like</p>
<p>A =
\begin{array}{cccc}
A_1 & A_0 & A_0 & 0 \\
A_0 & A_2 & 0 & A_0 \\
A_0 & 0 & A_3 & A_0 \\
0 & A_0 & A_0 & A_4 \\
\end{array}</p>
<p>A is a... | Ross B. | 68,567 | <p>The CVX software package for MATLAB can handle semidefinite prodramming problems (SDPs).</p>
|
551,964 | <p>I have an optimization problem where I have to optimize a function f(A) where A is a matrix(sparse).</p>
<p>Like</p>
<p>A =
\begin{array}{cccc}
A_1 & A_0 & A_0 & 0 \\
A_0 & A_2 & 0 & A_0 \\
A_0 & 0 & A_3 & A_0 \\
0 & A_0 & A_0 & A_4 \\
\end{array}</p>
<p>A is a... | srihegde | 567,367 | <p>There are plenty of packages for semidefinite programming (SDP). I have experience using <a href="http://cvxopt.org/" rel="nofollow noreferrer">CVXOPT</a> and <a href="http://www.cvxpy.org/" rel="nofollow noreferrer">CVXPY</a> python packages.</p>
<p>Also take a look at <a href="https://peterwittek.com/sdp-in-pytho... |
590,891 | <p>I'm going back to school and haven't taken a math class in years, so I'm brushing up on the basics.</p>
<p>The text states $\frac{g(t + \Delta(t))^2}{2} = \frac{gt^2}{2} + \frac{g}{2}\left(2t\Delta t + \Delta t^2\right)$.</p>
<p>(Sorry for the lack of formatting. I'll probably get slammed, but I couldn't figure it... | Way to infinity | 53,489 | <p>use the following formula and you will get your answer :</p>
<p>$(a+b)^2 = a^2+2ab+b^2 $</p>
|
686,361 | <p>Given if we know $P(S)$ and $P(C|S)$ and $P(D|S)$, how do you compute $E[C|D=d]$? One way that I thought of is to find the conditional probability of $P(C|D)$ by computing the joint probability $P(C,D,S)$ and marginalizing it over $S$. But, $P(D|S)$ is a binomial distribution with parameter $q$ and $S$. Finding the ... | Bombyx mori | 32,240 | <p>You have
$$
P(C\cap S)=P(S)P(C|S), P(D\cap S)=P(S)P(D|S)
$$
This information is not enough to let you compute
$$
P(C\cap D), P(C|D)
$$
because you lost information at
$$
P(C\cap D\cap S^{c})
$$
and in principle you do not know what this is. </p>
|
64,646 | <p>In $\triangle{ABC}$, given $\angle{A}=80^\circ$, $\angle{B}=\angle{C}=50^\circ$, D is a point in $\triangle{ABC}$, which $\angle{DBC}=20^\circ,\angle{DCB}=40^\circ$. Then how to find find $\angle{DAC}$?</p>
<p>thanks.</p>
| Job Bouwman | 274,003 | <p>The regular 18-polygon has the nice property that each node sees all other nodes separated with $10^\circ$. Embed $\triangle{ABC}$ in this polygon as shown below. </p>
<p><a href="https://i.stack.imgur.com/KS5d9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KS5d9.jpg" alt="Solution"></a></p>
<... |
631,388 | <p>If $\lim_{n\rightarrow \infty }{a_n}=\alpha (\neq 0) $ and $\lim_{n\rightarrow \infty }{b_n}=\beta$, then $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?</p>
<p>I unconsciously used this but I realized I'd never seen this theorem before. Is it true?</p>
| Mario De León | 14,759 | <p>I think that we can use the fact of convergent sequences product and that
$$e^{b_n \cdot \log a_n}$$
converges when $n \rightarrow \infty$.</p>
|
2,458,184 | <p>Let $a, b$ be non-negative integers and $p\ge3$ be a prime number. If $a^2+b^2$ and $a+b$ are divisible by $p$ does it mean $a$ and $b$ are always divisible by $p$?</p>
| Joffan | 206,402 | <p>Since $a+b\equiv 0 \bmod p$, we have $a\equiv -b $ and thus $ a^2\equiv b^2 \bmod p$. </p>
<p>Then $a^2+b^2\equiv 2a^2 \equiv 0 \bmod p$ and since $p>2$ we know $p\mid a^2$ and thus $p\mid a$ and $p\mid b$.</p>
|
2,227,047 | <p>For any $x=x_1, \dotsc, x_n$, $y=y_1, \dotsc, y_n$ in $\mathbf E^n$, define $\|x-y\|=\max_{1 \le k \le n}|x_k-y_k|$. Let $f\colon\mathbf E^n \to \mathbf E^n$ be given by $f(x)=y$, where $y_k= \sum_{i=1}^n a_{ki} x_i + b_k$ where $k =1,2, \dotsc,n$. Under what conditions is $f$ a contraction mapping?</p>
<p>Any hint... | JanG | 266,041 | <p>In this answer I use a variable substitution which I cannot find in the already published answers.</p>
<p>Say that $\alpha \neq 0$ and $\alpha = \varrho e^{i\theta}, \, -\pi <\theta< \pi$. Then $|1+\alpha x^2| = \sqrt{\varrho^2x^4 +2\varrho\cos \theta x^2+1}$ and
\begin{gather*}
I = \int_{-\infty}^{\infty}\df... |
157,731 | <p>I have a coupled PDE. How can I convert the equation from $(x,t)$ to $(p,t)$, the Fourier space in MATHEMATICA? </p>
<p>\begin{equation}
\frac{\partial c}{\partial t} +\frac{\partial d}{\partial t} = -4\gamma(\frac{\partial a}{\partial x} +x (\frac{\partial c}{\partial x} +\frac{\partial d}{\partial x}) - \frac{\pa... | LLlAMnYP | 26,956 | <p><strong>EDIT 17.10.2017:</strong> I've completely redone this to eliminate any manual labor.</p>
<p>Converting to Fourier space is quite simple. We must realize that any function</p>
<p>$$ f(x,t) = \int f(p, t) e^{i p x} dp $$</p>
<p>That is</p>
<pre><code>f[x, t] = InverseFourierTransform[f[p, t], p ,x]
</code>... |
3,897,067 | <p>Consider a binary operation <span class="math-container">$*$</span> acting from a set <span class="math-container">$X$</span> to itself. It's useful and standard to work with operations which are associative, such that <span class="math-container">$(a*b)*c = a*(b*c)$</span>. What about operations which are not assoc... | runway44 | 681,431 | <blockquote>
<p>Is there any way to characterize all different possible types of operation which are not associative?</p>
</blockquote>
<p>This is too broad and subjective to answer I think. What exactly is a "type" of operation? I assume you're already talking about binary operations, so presumably a "t... |
3,034,421 | <p>Lets say I have 2 multivariate functions:</p>
<pre><code>f(x,y) = x - y
g(x,y) = x + y
</code></pre>
<p>How do I get the composition of these 2 functions <span class="math-container">$g(f(x,y))$</span> ? </p>
| Shubham Johri | 551,962 | <p>Use the converse of the distributive property:</p>
<p><span class="math-container">$((x=0)\lor(y=0)\lor(3x+4y=-2))\land((x=0)\lor(y=0)\lor(x+3y=-1))\\\equiv(x=0)\lor(y=0)\lor[(3x+4y=-2)\land(x+3y=-1)]$</span></p>
<p><span class="math-container">$3x+4y+2=0=x+3y+1$</span> is just a pair of straight lines (linear equ... |
3,120,729 | <p>I came across this exercise:</p>
<blockquote>
<p>Prove that
<span class="math-container">$$\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$$</span></p>
</blockquote>
<p>Proving this seems tedious but doable, I think, by exploiting double angle identities several times, and presumably several terms on the left hand side ... | DXT | 372,201 | <p>Let <span class="math-container">$\displaystyle S =\sum^{2}_{k=0}2^k\tan(2^k x)+8\cot (8x).$</span></p>
<p>Then <span class="math-container">$\displaystyle \int Sdx=-\sum^{2}_{k=0}\ln\bigg(\cos(2^k x\bigg)+\ln(\sin 8x)$</span></p>
<p>Using <span class="math-container">$\displaystyle \prod^{n-1}_{r=0}\cos(2^r x)=\f... |
1,368,073 | <p>Halmos, in Naive Set Theory, on page 19, provides a definition of intersection restricted to subsets of $E$, where $C$ is the collection of the sets intersected. The point is to allow the case where $C$ is $\emptyset$, which with this definition of intersection gives $E$ as the result. </p>
<blockquote>
<p>$\{x \... | David Holden | 79,543 | <p>maybe think of it this way (remembering that your sets $X$ belong to some universe which is not to be identified with the collection $C$). to avoid confusion i use the bound variable $y$ for the universally quantified statement (in place of your $X$):
$$
\{x: (x \in E) \land \forall y (y \in C \Rightarrow x \in y)\}... |
95,242 | <p>Is it possible to use <code>ProbabilityScalePlot</code> to show different plot markers in a single dataset, such as in going from <code>plot2</code> to <code>plot3</code> below?</p>
<pre><code>nPoints = 10;
x = RandomVariate[NormalDistribution[1, 1], nPoints];
y = RandomVariate[LogNormalDistribution[1, 1], nPoints]... | Dr. belisarius | 193 | <pre><code>s = GatherBy[First@Cases[FullForm@plot3, Point[h___] :> h, Infinity],
Function[{u}, MemberQ[#, u[[1]]] & /@ {x, y, z}]]
plot3 /. Point[__] :> MapThread[{#1, Point@#2} &, {{Red, Blue, Green}, s}]
</code></pre>
<p><img src="https://i.stack.imgur.com/uQwnR.png" alt="Mathematica gr... |
186,553 | <p><strong>Problem:</strong> Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant.</p>
<p>I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out.</p>
| Roman Chokler | 38,328 | <p>The quickest way is just to reacall pythagorean triples $(3,4,5)$ and $(5,12,13)$ due to 5 being common to both triples and 13 being the diagonal length thus establishing that 3, 4,and 12 satisfy $3^2+4^2+12^2=13^2$ as desired for the correct diagonal length. Notice that the product of these three numbers in the tri... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | James Robinson | 123,656 | <p>A full translation of the main theorem (Theorem 6.1) and conditions (Y.1)-(Y.4), due to Sergei Chernyshenko, can be found at</p>
<p><a href="http://go.warwick.ac.uk/jcrobinson/lf/otelbaev">http://go.warwick.ac.uk/jcrobinson/lf/otelbaev</a></p>
<p>There is also a brief discussion of the method of proof used by Otel... |
4,588,408 | <p>This is a step in a guided proof that the cyclotomic polynomial <span class="math-container">$\Phi_n$</span> is the minimal polynomial of <span class="math-container">$u$</span>. I already know that <span class="math-container">$\Phi_n(0)=0$</span> so <span class="math-container">$P$</span> divides <span class="math... | Aphelli | 556,825 | <p>Here’s the elementary argument.</p>
<p>If <span class="math-container">$\Phi_n$</span> isn’t irreducible, it is a product of two monic polynomials <span class="math-container">$A$</span> and <span class="math-container">$B$</span> with integer coefficients and positive degree. Write <span class="math-container">$X^n... |
159,563 | <p>I have some <a href="https://pastebin.com/MGEzkeC3" rel="nofollow noreferrer">data</a> and want to fit it to Planck's law for black body radiation. The problem is that Mathematica does not give me the correct coefficients.</p>
<p>When I evaluate</p>
<pre><code>dati = Import["https://pastebin.com/raw/MGEzkeC3&qu... | JimB | 19,758 | <p><strong><em>Correction:</em></strong> What a difference good starting values can make. What I presented earlier totally missed values of the parameters that allow for a good fit.</p>
<p>Using a simpler parameterization with <code>A -> 2 a c^2 h</code> and <code>T -> (c h t)/kb</code> we have the equation </... |
153,448 | <p>On the complex plane, I have a transformation "T" such that :</p>
<p>$z' = (m+i)z + m - 1 - i$ ($z'$ is the image and $z$ the preimage, $z$ and $z'$ are both complex number)</p>
<p>and $m$ is a real number. </p>
<p>I'd need to determine "$m$" such that this transformation "T" is a rotation.</p>
<p>I know a r... | Community | -1 | <p>If you integrate it out, you will get $$\dfrac{f(x)^3}{3} = x^3(1+x)^2$$ Hence, $$f(x)^3 = 3x^3(1+x)^2$$ Setting $x = 2$, gives us $$f(2)^3 = 3 \times 2^3 \times 3^2 = 6^3.$$ Since $f(x) \in \mathbb{R}$, we get that $f(2) = 6$.</p>
|
201,820 | <p>Suppose we have in <code>~/time-data/time-data.org</code> the following data:</p>
<pre><code>* Parent1
:LOGBOOK:
CLOCK: [2019-07-09 Tue 00:00]--[2019-07-09 Tue 00:20] => 0:20
:END:
** Child1
:LOGBOOK:
CLOCK: [2019-07-10 Wed 00:02]--[2019-07-10 Wed 00:40] => 0:38
:END:
** Child2
:LOGBOOK:
CLOCK: [2019-07-11 ... | Edmund | 19,542 | <p>You may use the <code>"Dataset"</code> and <code>"HeaderLines"</code> <a href="https://reference.wolfram.com/language/ref/Import.html" rel="nofollow noreferrer"><code>Import</code></a> options for <a href="http://reference.wolfram.com/language/ref/format/CSV.html" rel="nofollow noreferrer"><code>"CSV"</code></a> alo... |
244,769 | <p>I am DMing a game of DnD and one of my players is really into fear effects, which is cool, but the effect of having monsters suffer from the "panicked" condition gets tedious to render via dice rolls.</p>
<p>The rule is, on the battle grid the monster will run for 1 square in a random direction, then from ... | Daniel Huber | 46,318 | <p>Assume a,b,c are given as:</p>
<pre><code>{a, b, c} = {{0.2, 0.8}, {0.1, 0.15}, {0.8, 0.25}};
</code></pre>
<p>The origin of the uv system is on a-b: orig= b+ x(a-b), where we need to determine x. This is done by using the fact that (c-orig) is perpendicular to (a-b):</p>
<pre><code>orig = b + x (a - b) /. Solve[(b... |
4,531,652 | <p>In my school book, I read this theorem</p>
<blockquote>
<p>Let <span class="math-container">$n>0$</span> is an odd natural number (or an odd positive integer), then the equation <span class="math-container">$$x^n=a$$</span> has exactly one real root.</p>
</blockquote>
<p>But, the book doesn't provide a proof, onl... | Mark Bennet | 2,906 | <p>To prove this you can do the following to show that the function <span class="math-container">$y=x^n$</span> is increasing when <span class="math-container">$n$</span> is odd.</p>
<p>So suppose <span class="math-container">$a\gt b$</span> and <span class="math-container">$n$</span> is odd, we want to prove <span cla... |
4,052,760 | <blockquote>
<p>Prove that <span class="math-container">$\int\limits^{1}_{0} \sqrt{x^2+x}\,\mathrm{d}x < 1$</span></p>
</blockquote>
<p>I'm guessing it would not be too difficult to solve by just calculating the integral, but I'm wondering if there is any other way to prove this, like comparing it with an easy-to-c... | aschepler | 2,236 | <p>Yet another way: If <span class="math-container">$0 < x < 1$</span>, then <span class="math-container">$x^2 < x$</span>. So</p>
<p><span class="math-container">$$\int_0^1 \sqrt{x^2+x} \ dx < \int_0^1 \sqrt{2x}\ dx = \frac{2 \sqrt{2}}{3} < 1$$</span></p>
<p>(This bound is about 0.9428, so not as good a... |
664 | <p>Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?</p>
<p>That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some... | Harrison Brown | 382 | <p>I believe the answer is "no"; the best known constructions only give no clique or independent set of size about $2^\sqrt{n}$ in a graph with $2^n$ vertices. Bill Gasarch has a page on the subject <a href="http://www.cs.umd.edu/~gasarch/const_ramsey/const_ramsey.html" rel="nofollow">here</a>, although I don't know ho... |
664 | <p>Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?</p>
<p>That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some... | Alon Amit | 25 | <p>I also believe the answer is "no". Another reference is <a href="http://www.math.tau.ac.il/~nogaa/PDFS/ap5.pdf" rel="nofollow">this paper</a>, which treats off-diagonal Ramsey numbers (e.g. graphs with no clique of size k and no anti-clique of size l). </p>
|
1,476,456 | <p>How many positive (integers) numbers less than $1000$ with digit sum to $11$ and divisible by $11$?</p>
<p>There are $\lfloor 1000/11 \rfloor = 90$ numbers less than $1000$ divisible by $11$.</p>
<p>$N = 100a + 10b + c$ where $a + b + c = 11$ and $0 \le a, b, c \le 9$</p>
<p>I got $\binom{13}{2} - 9 = 69$ soluti... | Adelafif | 229,367 | <p>$N=a+10b+100c$. $a-b+c=0$ or $a-b+c=11$. Also we have $a+b+c=11$. We get two cases:</p>
<ul>
<li>$a-b+c=0,\;a+b+c=11$ from which we get $2a+2b=11$; impossible.</li>
<li>$a-b+c=11,\;a+b+c=11$ from which we get $a+c=11,\; b=0$. They are $209, 308, 407, 506, 605, 704, 803, 902$.</li>
</ul>
|
232,777 | <p>Let $F$ be an ordered field.</p>
<p>What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?</p>
| Panurge | 82,840 | <p>Here is my proof.</p>
<p>Lemma. Let $X_{1}, \ldots , X_{s}$ distinct sets, with nonempty union, let $l$ a natural number such that $l \leq s$ and assume that for all distinct indices $i_{1}, \ldots , i_{l}$ in $\{1, \ldots s \}$, $X_{i_{1}} \cup \ldots \cup X_{i_{l}}$ is equal to the whole union $X_{1} \cup \ldots ... |
611,361 | <p>Let's have function $f$ defined by:
$$f(x)=2\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}-x\sum_{k=1}^{\infty}\frac{e^{kx}}{k^2},\quad x\in(-2\pi,0\,\rangle$$
My question:
Can somebody expand it into a correct Maclaurin series, but using an unconventional way? Conventional is e.g. using $n$-th derivative of $f(x)$ in zero... | Claude Leibovici | 82,404 | <p>My first reaction and idea was to find a closed form for f(x) which I could later expand as a Taylor series. From definition, the function write<br>
f(x) = -x PolyLog[2, E^x] + 2 PolyLog[3, E^x]<br>
which is (it looks simple). However, the difficulties start when I try to expand this result as a Taylor series; the r... |
3,963,884 | <p>Suppose <span class="math-container">$(X_n)_n$</span> are i.i.d. random variables and let <span class="math-container">$W_n = \sum_{k=1}^n X_k$</span>. Assume that there exist <span class="math-container">$u_n>0 , v_n \in \mathbb{R}$</span> such that</p>
<p><span class="math-container">$$\frac{1}{u_n}W_n-v_n\Righ... | Botnakov N. | 452,350 | <p>In order to finish the solution we should show that <span class="math-container">$\frac{u_{n+1}}{u_n} \to 1$</span>.</p>
<p>As <span class="math-container">$\frac{X_{n+1}}{u_n} =\frac{X_{1}}{u_n} $</span> and <span class="math-container">$u_n \to \infty$</span> we have <span class="math-container">$\frac{X_{n+1}}{u_... |
3,690,185 | <p>By <span class="math-container">$a_n \sim b_n$</span> I mean that <span class="math-container">$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 1$</span>.</p>
<p>I don't know how to do this problem. I have tried to apply binomial theorem and I got
<span class="math-container">$$\int_{0}^{1}{(1+x^2)^n dx} = \int_0^1 \... | Barry Cipra | 86,747 | <p>Let <span class="math-container">$u=(1+x^2)/2$</span>, so that <span class="math-container">$du=x\,dx=\sqrt{2u-1}\,dx$</span>. It follows that</p>
<p><span class="math-container">$${n\over2^n}\int_0^1(1+x^2)^n\,dx=n\int_{1/2}^1{u^n\over\sqrt{2u-1}}\,du=n\int_{1/2}^1u^n\,du+n\int_{1/2}^1u^n\left({1\over\sqrt{2u-1}}-... |
3,600,633 | <p>As I was reading <a href="https://math.stackexchange.com/questions/1918673/how-can-i-prove-that-the-finite-extension-field-of-real-number-is-itself-or-the">this question</a>, I saw Ethan's answer. However, perhaps this is very obvious, but why does the degree of the polynomial be at most <span class="math-container"... | HallaSurvivor | 655,547 | <p>We know that, over <span class="math-container">$\mathbb{C}$</span>, we can factor any polynomial entirely into linear terms (this is the fundamental theorem of algebra). Moreover, one can show that whenever <span class="math-container">$f$</span> has real coefficients, then <span class="math-container">$z$</span> a... |
3,905,629 | <p>I need to compute a limit:</p>
<p><span class="math-container">$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p>I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.</p>
<p><span class="math-container">$$
\lim_{x \to 0+}(2\sin \s... | Cesareo | 397,348 | <p>Hint.</p>
<p>For <span class="math-container">$x>0$</span> small we have</p>
<p><span class="math-container">$$
\left(2\sin(\sqrt{x})-\sqrt{x}\right)^x\le \sigma(x)\le \left(2\sin(\sqrt{x})+\sqrt{x}\right)^x
$$</span></p>
<p><a href="https://i.stack.imgur.com/H9KMf.jpg" rel="nofollow noreferrer"><img src="https:/... |
488,141 | <p>\begin{align*}A=\left(\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\\end{array}\right);\end{align*}</p>
<p>The eigenvalues are $1$, I know one of the eigenvectors is $(1,0,0,0)$, Is that all?</p>
<p>The mathematica gives, why ... | Alex Youcis | 16,497 | <p>Note that the matrix of the multiplication $m_\alpha:\mathbb{Q}(\sqrt[4]{2})\to \mathbb{Q}(\sqrt[4]{2})$ map with respect to a generic element $\alpha=a+b\sqrt[4]{2}+c\sqrt[4]{2^2}+d\sqrt[4]{2^3}$ is (in the canonical basis)</p>
<p>$$\begin{pmatrix}a & 2d & 2b & 2b \\ b & a & 2d & 2c \\ c &a... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | Daniel Schepler | 337,888 | <p>I think in the translation, it might help to separate out the direct generalization of the notion of "continuity at a point" from the general topological arguments that this generalization being true at every point is equivalent to the condition on inverse images of open sets.</p>
<p>So, recall that for a map $f : ... |
109,734 | <p>I am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to find the derivative of the function
$$y=\cot^2(\sin \theta)$$</p>
<p>I could be incorrect but a trig function squared ... | André Nicolas | 6,312 | <p>You started in a way that leads to the answer.</p>
<p>Let $y=\cot^2(\sin \theta)$. We want to find $\dfrac{dy}{d\theta}$. Make the substitution $x=\sin\theta$. (Comment: when we are using substitution, it is more common to use letters like $u$, $v$, $w$, but $x$ is fine here.)
Note that
$$y=\cot^2 x.$$
Different... |
4,268,962 | <blockquote>
<p>Check whether <span class="math-container">$y=\ln (xy)$</span> is an answer of the following differential equation or not</p>
<p><span class="math-container">$$(xy-x)y''+xy'^2+yy'-2y'=0$$</span></p>
</blockquote>
<p>First I tried to solve the equation,</p>
<p><span class="math-container">$$x(yy''-y''+y'... | Rezha Adrian Tanuharja | 751,970 | <p>Here is an alternative</p>
<p><span class="math-container">$$
\begin{align}
0&=(xy-x)\cdot\frac{d^{2}y}{dx^{2}}+x\cdot\left(\frac{dy}{dx}\right)^{2}+y\cdot\frac{dy}{dx}-2\cdot\frac{dy}{dx}\\
\\
&=\frac{d}{dx}\left[(xy-x)\cdot\frac{dy}{dx}-y\right]\\
\\
\\
C_{1}&=(xy-x)\cdot\frac{dy}{dx}-y\\
\\
&=xy\c... |
3,907,928 | <p>Suppose the solutions to a general cubic equation <span class="math-container">$ax^3+bx^2+cx+d=0$</span> are to be found. Then according to Cardano's method, First a variable substitution must be carried on to convert the general cubic to depressed cubic.
<span class="math-container">$$ax^3+bx^2+cx+d=0\rightarrow t^... | Will Orrick | 3,736 | <p>There is a geometric picture behind this. The expression <span class="math-container">$(u+v)^3$</span> can be visualized as a cube subdivided into eight rectangular solids of sides <span class="math-container">$u\times u\times u$</span>, <span class="math-container">$u\times u\times v$</span> (permuted three ways), ... |
1,586,354 | <p>I did the following exercise:</p>
<blockquote>
<p>Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb{Z}_n$ (integers mod n with addition). Prove that either every member of $H$ is even or exactly half of the members of $H$ are even.</p>
</blockquote>
<p>My answer:</p>
<p>Since $\mathbb{Z}... | Matt Samuel | 187,867 | <p>Suppose there is an element $x$ that isn't even. Let $A$ be the set of even elements in the subgroup and define $B=\{x+a:a\in A\}$. Then every element of $B$ is odd. Prove that $A$ and $B$ have the same number of elements and the subgroup is the disjoint union of $A$ and $B$.</p>
|
2,402,410 | <p>I defined the "function":</p>
<p>$$f(t)=t \delta(t)$$</p>
<p>I know that Dirac "function" is undefined at $t=0$ (see <a href="http://web.mit.edu/2.14/www/Handouts/Convolution.pdf" rel="nofollow noreferrer">http://web.mit.edu/2.14/www/Handouts/Convolution.pdf</a>).</p>
<p>In Wolfram I get $0 \delta(0)=0$ (<a href=... | Eric Towers | 123,905 | <p>Wolfram Alpha <a href="http://www.wolframalpha.com/input/?i=DiracDelta(1)" rel="nofollow noreferrer">evaluates</a> <code>DiracDelta(1)</code>, giving zero. Wolfram Alpha <a href="http://www.wolframalpha.com/input/?i=DiracDelta(0)" rel="nofollow noreferrer">fails to interpret</a> <code>DiracDelta(0)</code>, nor does... |
1,554,285 | <p>Here's my problem:</p>
<blockquote>
<p>In Ohio, 55% of the population support the republican candidate in an
upcoming election. 200 people are polled at random. If we suppose that
each person’s vote (for or against) is a Bernoulli random variable
with probability p, and votes are independent,</p>
<p>(a... | Erick | 224,176 | <p>$c)$ Let $X\sim Bin(200,.45)$ that counts the number of votes for the democratic candidate so $X=\sum_{i=1}^{200}\xi_i$ where $\xi_i\sim Bernoulli(.45)$ and $\xi_i$ are i.i.d. so by central limit theorem we have that:
$$X\sim^{\star}N(np,np(1-p))$$
$$X\sim^{\star}N(90,49.5)$$
So:
$$P[X> 100]=1-P[X\leq100]\approx... |
4,487,380 | <p>I was reading my calculus book wherein I came across a note, being worth of attention. It says:</p>
<blockquote>
<p>Integrals in the form of <span class="math-container">$\int P(x)e^{ax}dx$</span> have a special property. After calculating the integral, we obtain a function in the form of <span class="math-container... | Robert Lee | 695,196 | <p>The statement</p>
<blockquote>
<p><span class="math-container">$\int P(x) e^{ax}\, \mathrm{d}x = Q(x) e^{ax}$</span> with <span class="math-container">$\deg(P(x)) = \deg(Q(x)) $</span></p>
</blockquote>
<p>is equivalent (by the definition of antiderivative) to showing that
<span class="math-container">$$
\frac{\math... |
1,643,649 | <blockquote>
<p>I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$.</p>
</blockquote>
<p>$\Bbb Z^*_n$ means integers up to $n$ coprime with $n$</p>
<p>I do not know how to do this. I have difficulties doing proofs involving isomorphisms. A methodological answer would be highly appreciated.</p>
<... | Alex Wertheim | 73,817 | <p>Hint: see if you can show that one of these groups is cyclic, whereas the other is not. </p>
|
3,789,060 | <p>I was asked the following question:</p>
<blockquote>
<p>Determine if the following set is a vector space:<br />
<span class="math-container">$$W=\left\{\left[\begin{matrix}p\\q\\r\\s\\\end{matrix}\right]:\begin{matrix}-3p+2q=-s\\p=-s+3r\\\end{matrix}\right\}$$</span></p>
</blockquote>
<p>I know the answer is yes and... | John Wayland Bales | 246,513 | <p>We have</p>
<p><span class="math-container">$$ \arctan(\cot(\pi x))=\frac{\pi}{2}-\pi x+\pi\left\lfloor x\right\rfloor $$</span></p>
<p>so it is continuous on intervals <span class="math-container">$[n,n+1)$</span> for <span class="math-container">$n$</span> an integer.</p>
|
4,098,630 | <p>I am trying to solve for x and y using the following equation:
<span class="math-container">$4i + 2 = \frac{x + iy + 5 + 4i}{2x + 2iy - 5}$</span></p>
<p>I got it down to real and imaginary, but am unsure what to do next.
<span class="math-container">$3x-8y-15=(-8x-3y+24)i$</span></p>
| lonza leggiera | 632,373 | <p><strong>Hint</strong></p>
<p>Assuming <span class="math-container">$\ x,y\ $</span> are required to be real numbers, then equating the real and imaginary parts of both sides of the equation
<span class="math-container">$$
3x-8y-15=(-8x-3y+24)i
$$</span>
will give you two linear equations in <span class="math-contain... |
4,098,630 | <p>I am trying to solve for x and y using the following equation:
<span class="math-container">$4i + 2 = \frac{x + iy + 5 + 4i}{2x + 2iy - 5}$</span></p>
<p>I got it down to real and imaginary, but am unsure what to do next.
<span class="math-container">$3x-8y-15=(-8x-3y+24)i$</span></p>
| Learner | 855,893 | <p>Suppose a,b are two real numbers,if we are having a=ib as some condition,as a,b are real they can be equal only when both of the sides of equation are zero. So a=b=0.you can proceed with your problem now.</p>
|
2,469,720 | <p>Math problem:</p>
<blockquote>
<p>Find $x$, given that $ \, 2^2 \times 2^4 \times 2^6 \times 2^8 \times \ldots \times 2^{2x} = \left( 0.25 \right)^{-36}$</p>
</blockquote>
<p>To solve this question, I changed the left side of the equation to $2^{2+4+6+ \ldots + 2x}$ and the right side to: $\frac{2^{74}}{3^{36}}$... | user577215664 | 475,762 | <p>$( 0.25)^{-36}=( \frac 1 4) ^{-36}=(2^{-2})^{-36}=(2^{36})^2$</p>
<p>$2^22^42^6....2^{2x}=(2^12^22^32^4...2^x)^2$</p>
<p>So we must have:</p>
<p>$(2^{36})^2=(2^12^22^32^4...2^x)^2$</p>
<p>Or simply:</p>
<p>$2^{36}=2^12^22^32^4...2^x$</p>
<p>$1+2+3+ ....x=36$</p>
<p>$\frac {(x+1)x} 2=36$</p>
<p>$x^2+x=72$</p>... |
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