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 |
|---|---|---|---|---|
2,319,126 | <p>I have to find dimension of $V+W,$ where$ V$ is a vector subspace given by solutions of the linear system:</p>
<p>$$x+2y+z=0$$
$$3y+z+3t=0$$</p>
<p>and $W$ is the subspace generated from vectors
$(4,0,1,3)^T,(1,0,-1,0)^T$.</p>
<p>I don't know how to combine the two subspaces and calculate the dimension.</p>
| Bernard | 202,857 | <p>You have to calculate the dimension of $V\cap W$. A vector $v=\lambda{}^{\mathrm t}(4,0,1,3)+\mu{}^{\mathrm t}(1,0,-1,0)={}^{\mathrm t}(4\lambda+\mu,0,\lambda-\mu,3\lambda)$ satisfies the equations of both planes if and only if
$$\begin{cases}x+2y+z=5\lambda=0\\3y+z+3t=10\lambda-\mu=0\end{cases}\iff \lambda=\mu=0.$$... |
174,149 | <p>How many seven - digit even numbers greater than $4,000,000$ can be formed using the digits $0,2,3,3,4,4,5$?</p>
<p>I have solved the question using different cases: when $4$ is at the first place and when $5$ is at the first place, then using constraints on last digit.</p>
<p>But is there a smarter way ?</p>
| draks ... | 19,341 | <p>You have the following combinations:</p>
<ul>
<li>$4\dots 0$, where in the middle you permute $\{2,3,3,4,5\}: \frac{5!}{2}$, with the $2$ in the denominator accounts for the double $3$. And the same for $4\dots 2$, $4\dots 4$ and $5\dots 4$.</li>
<li>$5\dots 0$, where in the middle you permute $\{2,3,3,4,4\}: \frac... |
3,678,417 | <p>I understand:
<span class="math-container">$$\sum\limits^n_{i=1} i = \frac{n(n+1)}{2}$$</span>
what happens when we restrict the range such that:
<span class="math-container">$$\sum\limits^n_{i=n/2} i = ??$$</span></p>
<p>Originally I thought we might just have <span class="math-container">$\frac{n(n+1)}{2}/2$</spa... | hamam_Abdallah | 369,188 | <p>If <span class="math-container">$n$</span> is even of the form <span class="math-container">$2p$</span>, the sum is
<span class="math-container">$$(p+0)+(p+1)+(p+2)+...+(p+n-p)=$$</span></p>
<p><span class="math-container">$$p(n-p+1)+1+2+3+...(n-p)=$$</span></p>
<p><span class="math-container">$$p(n-p+1)+\frac{(n-... |
2,673,835 | <p>I need to prove that the sequence
$$
f_n = \sum_{i=0}^n\prod_{j=0}^i \left(z+j\right)^{-1} = \frac{1}{z}+\frac{1}{z(z+1)}+\cdots + \frac{1}{z(z+1)(z+2)\cdots (z+n)}$$
converge uniformly to a function in every compact subset of $\mathbb{C}\setminus\{0,-1,-2,-3,\dotsc\}$. The problem has other questions, but this is ... | Ѕᴀᴀᴅ | 302,797 | <p>For a fixed conpact set $K$, suppose $K \subseteq B(0, r)$ and $m \in \mathbb{N}_+$, $m > r + 1$. Then for $z \in K$,
\begin{align*}
\left| \sum_{k = m}^\infty \prod_{j = 0}^k (z + j)^{-1} \right| &\leqslant \sum_{k = m}^\infty \prod_{j = 0}^k |z + j|^{-1} = \prod_{j = 0}^{m - 1} |z + j|^{-1} \sum_{k = m}^\in... |
2,673,835 | <p>I need to prove that the sequence
$$
f_n = \sum_{i=0}^n\prod_{j=0}^i \left(z+j\right)^{-1} = \frac{1}{z}+\frac{1}{z(z+1)}+\cdots + \frac{1}{z(z+1)(z+2)\cdots (z+n)}$$
converge uniformly to a function in every compact subset of $\mathbb{C}\setminus\{0,-1,-2,-3,\dotsc\}$. The problem has other questions, but this is ... | Jack D'Aurizio | 44,121 | <p>An alternative approach, related to the fact that the incomplete $\Gamma$ function has a simple Laplace transform. One may notice that the function defined by such series of reciprocal Pochhammer symbols fulfills the functional identity $z f(z) = 1+ f(z+1)$ and $f(1)=e-1$. The same happens for
$$ g(z)=\int_{0}^{1} x... |
2,148,187 | <p>I am given a charge of $Q(t)$ on the capacitor of an LRC circuit with a differential equation</p>
<p>$Q''+2Q'+5Q=3\sin(\omega t)-4\cos(\omega t)$ with the initial conditions $Q(0)=Q'(0)=0$</p>
<p>$\omega >0$ which is constant and $t$ is time. I am then asked find the steady state and transient parts of the solu... | trace | 420,766 | <p>your Q(t) is actually wrong, it should have come out to plus 8w not minus. Then use the formula R^2 = A^2 + B^2 to find R(w). Then solve for the value that makes R a maximal.</p>
|
1,755,029 | <p>Imagine a cubic array made up of an $n\times n\times n$ arrangement of unit cubes: the cubic array is n
cubes wide, n cubes high and n cubes deep. A special case is a $3\times3\times3$ Rubik’s cube, which
you may be familiar with. How many unit cubes are there on the surface of the $n\times n\times n$ cubic array?</... | Stella Biderman | 123,230 | <p>So, i think that thing you're missing is that you're counting the number of <em>squares</em> on the surface; not the number of cubes. For example, a corner piece of a Rubix cube is one cube but contributes three squares.</p>
<p>In a $n\times n\times n$ cube, you have the <em>outer layer</em> of cubes counting, and ... |
3,074,901 | <p>Find the rank of the following matrix</p>
<p><span class="math-container">$$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$</span></p>
<p>My approach: </p>
<p>The row space exists in <span class="math-container">$R^3$</span> and is spanned by two vectors. Since the vectors are independent of each othe... | Taffies1 | 598,984 | <p>The rank of a matrix is simply the number of nonzero rows in reduced row echelon form (rref). If you find the Reduced row echelon form of this given matrix it will yield:</p>
<p>(1, 0, 5/3)</p>
<p>(0, 1, -1/3)</p>
<p>Clearly, there are 2 nonzero rows in the reduced row echelon form of the given matrix. Thus, the ... |
3,074,901 | <p>Find the rank of the following matrix</p>
<p><span class="math-container">$$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$</span></p>
<p>My approach: </p>
<p>The row space exists in <span class="math-container">$R^3$</span> and is spanned by two vectors. Since the vectors are independent of each othe... | Metric | 221,865 | <p>With the case of <span class="math-container">$\mathbb{R}^3$</span>, the dimension is 3, since it has a basis that contains 3 elements.</p>
<p>The row space of your matrix lives as a "subspace" of the bigger structure <span class="math-container">$\mathbb{R}^3$</span>. That is, you don't view it as <span class="mat... |
2,548,469 | <p>Suppose there is a sequence of iid variates from $U(0,1)$, $X_1,X_2,\dots$ If we stop the process when $X_n>X_{n+1}$, what is the expected number of generated variates?</p>
<p>I am just thinking about treating it as a Bernoulli process, so that I can use geometric distribution. Is this the right approach?</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\int\frac{x-1}{x+1}\frac{\mathrm{d}x}{\sqrt{x^3+x^2+x}}
&=\int\frac{x^{-1/2}-x^{-3/2}}{x^{1/2}+x^{-1/2}}\frac{\mathrm{d}x}{\sqrt{\left(x^{1/2}+x^{-1/2}\right)^2-1}}\\
&=2\int\frac1{x^{1/2}+x^{-1/2}}\frac{\mathrm{d}\!\left(x^{1/2}+x^{-1/2}\right)}{\sqrt{\left(x^{1/2}+x^{-1/2}\right)^2-1}}\\
&... |
1,488,737 | <blockquote>
<p>Let $A$ be a square matrix of order $n$. Prove that if $A^2=A$ then $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$.</p>
</blockquote>
<p>I tried to bring the $A$ over to the left hand side and factorise it out, but do not know how to proceed. please help. </p>
| martini | 15,379 | <p><strong>Hint</strong>. Show that under the given condition, the following holds:</p>
<p>$$\ker A = \operatorname{im} (I -A) $$</p>
|
2,943,892 | <p>I have heard it said that completeness is a not a property of topological spaces, only a property of metric spaces (or topological groups), because Cauchy sequences require a metric to define them, and different metrics yield different sets of Cauchy sequence, even if the metrics induce the same topology. But why w... | Alex Kruckman | 7,062 | <p>According to your proposed definition, the sequence <span class="math-container">$1,2,3,\dots$</span> would be Cauchy in <span class="math-container">$\mathbb{R}$</span>, witnessed by the sequence of open sets <span class="math-container">$U_n = (n,\infty)$</span>. </p>
<hr>
<p><strong>Edit:</strong> Let me incorp... |
1,581,456 | <p>Given functions $g,h: A \rightarrow B$ and a set C that contains at least two elements, with $f \circ g = f \circ h$ for all $f:B \rightarrow C$. Prove that $g = h$. </p>
<p>My logic is to take C = B and h(x) =x for all x in particular and the result follows immediately. But, I don't see the use of the condition on... | Piquito | 219,998 | <p>Your identity is true. I give here the method for you to prove the point for general $n$. </p>
<p>One has as main preliminary remark $$\color{red}{z^n=-1\iff z^{n+k}=-z^k\iff \frac{1}{1-z^k}=\frac{z^{n-k}}{1+z^{n-k}}}$$ We make $$A=\sum_{k=o}^{n-1}\frac{2k+1}{1-z^{2k+1}}$$ $$B=n\sum_{k=0}^{n-1}\frac{1}{1+z^k}$$ </p... |
3,166,947 | <p>I want to transform the following
<span class="math-container">$$\prod_{k=0}^{n} (1+x^{2^{k}})$$</span>
to the canonical form <span class="math-container">$\sum_{k=0}^{n} c_{k}x^{k}$</span></p>
<p>This is what I got so far
<span class="math-container">\begin{align*}
\prod_{k=0}^{n} (1+x^{2^{k}})= \dfrac{x^{2^{n}}-... | Explorer | 630,833 | <p>Note that <span class="math-container">$ \sum_{k=0}^{n-1} 2^k=2^n-1$</span>. Therefore, the highest power of <span class="math-container">$x$</span> in <span class="math-container">$\prod_{k=0}^{n-1}(1+x^{2^k})$</span> is <span class="math-container">$2^n-1$</span>. In other words, none of the terms will repeat as ... |
3,959,263 | <p>Let <span class="math-container">$G$</span> be a tree with a maximum degree of the vertices equal to <span class="math-container">$k$</span>.
<strong>At least</strong> how many vertices with a degree of <span class="math-container">$1$</span> can be in <span class="math-container">$G$</span> and why?</p>
<p>I think ... | RobPratt | 683,666 | <p>For <span class="math-container">$d\in\{1,\dots,k\}$</span>, let <span class="math-container">$n_d$</span> be the number of nodes of degree <span class="math-container">$d$</span>. By the handshake lemma, we have
<span class="math-container">$$\sum_{d=1}^k d n_d = 2\left(\sum_{d=1}^k n_d - 1\right),$$</span>
which ... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Ahaan S. Rungta | 85,039 | <p>The answer that I have always seen: An integral usually has a defined limit where as an antiderivative is usually a general case and will most always have a $\mathcal{+C}$, the constant of integration, at the end of it. This is the only difference between the two other than that they are completely the same. </p>
<... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Hussain Kamran | 257,959 | <p>i think that indefinite integral and anti derivative are very much closely related things but definitely equal to each other.
indefinite integral denoted by the symbol"∫" is the family of all the anti derivatives of the integrand f(x) and anti derivative is the many possible answers which may be evaluated from the i... |
17,455 | <p>Assume $S_1$ and $S_2$ are two $n \times n$ (positive definite if that helps) matrices, $c_1$ and $c_2$ are two variables taking scalar form, and $u_1$ and $u_2$ are two $n \times 1$ vectors. In addition, $c_1+c_2=1$, but in the more general case of $m$ $S$'s, $u$'s, and $c$'s, the $c$'s also sum to 1.</p>
<p>What ... | Community | -1 | <p>Since I did this initially unregistered, I can't seem to figure out how to get back in and add a comment instead of an answer. This is a bit long, but I am still having trouble resolving this problem satisfactorily. Also, I apologize for the formatting.</p>
<p>@Jonas Meyer I got similar answers to you the first tim... |
4,385,209 | <p>Let's start by generalizing the concept of a metric space. An <span class="math-container">$S$</span>-metric space is a set <span class="math-container">$X$</span> with a function <span class="math-container">$d : X \times X \to S$</span> such that</p>
<ul>
<li><span class="math-container">$d(x,y) = 0 \iff x = y$</s... | coudy | 716,791 | <p><span class="math-container">${\bf R}$</span> is not <span class="math-container">${\bf Q}$</span>-metrizable. In fact no connected set <span class="math-container">$X$</span> is <span class="math-container">$\bf Q$</span>-metrizable. Let
<span class="math-container">$$\phi(x,y) = d(x,y) \in {\bf Q}$$</span>
This is... |
2,878,412 | <p>I've been working on a problem that involves discovering valid methods of expressing natural numbers as Roman Numerals, and I came across a few oddities in the numbering system.</p>
<p>For example, the number 5 could be most succinctly expressed as $\texttt{V}$, but as per the rules I've seen online, could also be ... | hmakholm left over Monica | 14,366 | <p>"Roman numerals" presumably means how the actual Romans actually wrote down numbers.</p>
<p>They would never have written five as IVI, full stop.</p>
<p>If you're following a particular set of formal rules that are <em>not</em> "do things as the actual Romans would have done them", then follow those rules. Be prep... |
4,042,741 | <p>I'm really struggling to understand the literal arithmetic being applied to find a complete residue system of modulo <span class="math-container">$n$</span>. Below is the definition my textbook provides along with an example.</p>
<blockquote>
<p>Let <span class="math-container">$k$</span> and <span class="math-conta... | Bill Dubuque | 242 | <p>It may prove helpful to understand the general conceptual background that underlies this definition. Congruence is an <em>equivalence relation</em> (generalized equality) so it partitions the integers into equivalences classes, here <span class="math-container">$\,[a] = a + n\Bbb Z\,$</span> is the class of all inte... |
2,351,883 | <p>I am learning about tensor products.
In trying to understand the definitions, I seem to be getting
some contradiction.</p>
<p>Consider the differential form</p>
<p>$$
d x^{1} \wedge d x^{2} = d x^{1} \otimes d x^{2} - d x^{2} \otimes d x^{1}.
$$</p>
<p>If I use the symmetry property of the tensor product</p>
<p>... | Community | -1 | <p>It is <em>not</em> true that $dx^1\otimes dx^2=dx^2\otimes dx^1$. You might want to check your book how $dx^1\otimes dx^2$ is <em>defined</em>. </p>
<p>Consider a vector space $V$ and $f,g$ being two linear functionals on $V$. Then for $u,v\in V$, one has
$$
(f\otimes g-g\otimes f)(u,v)=f(u)g(v)-g(u)f(v)
$$</p>
|
1,403,486 | <p>As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that <span class="math-container">$\{(x,y): x>0\}$</span> is connected. My tools for this are:</p>
<blockquote>
<p><strong>Definition 1</strong>: Two dis... | James | 81,163 | <p>What you have written is the same as $X \subseteq Y$ and $Y\subseteq X$. Notice that $X \subseteq Y$ iff $\forall z(z\in X \rightarrow z\in Y)$, so $X\subseteq Y$ and $Y\subseteq X$ iff $\forall z(z\in X \rightarrow z\in Y \text{ and } z\in Y \rightarrow z\in X)$ iff $\forall z(z\in X \leftrightarrow z\in Y)$ iff $X... |
331,859 | <p>I need to find the antiderivative of
$$\int\sin^6x\cos^2x \mathrm{d}x.$$ I tried symbolizing $u$ as squared $\sin$ or $\cos$ but that doesn't work. Also I tried using the identity of $1-\cos^2 x = \sin^2 x$ and again if I symbolize $t = \sin^2 x$ I'm stuck with its derivative in the $dt$.</p>
<p>Can I be given a h... | André Nicolas | 6,312 | <p><strong>Hint</strong> We can use double-angle identities to reduce powers. We could use $\cos 2t=2\cos^2 t-1$ and $\cos 2t=1-2\sin^2 t$. We end up with polynomial of degree $4$ in $\cos 2x$. Repeat the idea where needed. </p>
<p>It is more efficient in this case to use $\sin 2t=2\sin t\cos t$, that is, first rewrit... |
331,859 | <p>I need to find the antiderivative of
$$\int\sin^6x\cos^2x \mathrm{d}x.$$ I tried symbolizing $u$ as squared $\sin$ or $\cos$ but that doesn't work. Also I tried using the identity of $1-\cos^2 x = \sin^2 x$ and again if I symbolize $t = \sin^2 x$ I'm stuck with its derivative in the $dt$.</p>
<p>Can I be given a h... | lab bhattacharjee | 33,337 | <p>$$\int \sin^6x\cos^2xdx=\int \sin^6x(1-\sin^2x)dx=\int \sin^6xdx-\int \sin^8xdx$$
$$=I_6-I_8 \text{ where }I_n=\int\sin^nxdx$$</p>
<p>$$\text{Now, }I_{n+2}=\int\sin^{n+2}xdx=\int\sin^{n+1}x\cdot \sin xdx$$
$$=\sin^{n+1}x\int \sin xdx-\int\left(\frac{d \sin^{n+1}x}{dx} \int \sin xdx\right)dx$$ (using <a href="http:... |
1,274,514 | <p>I want to show that proposition<span class="math-container">$5.33$</span> in introduction to homological algebra Rotman :let <span class="math-container">$I$</span> be a directed set , and let <span class="math-container">$\{A_i,\alpha_j^i\}$</span>, <span class="math-container">$\{B_i,\beta_j^i\}$</span>, and <span... | Bernard | 202,857 | <p>Take an element $b\in\ker s^\to$, and an element $b_i$ in some $B_i$ such that $\beta_i(b_i)=b $. Then $\gamma_i(s_i(b_i))=0$, so that $\gamma_{ij}(s_i(b_i))=0$ in some $C_j\enspace(j\ge i)$.</p>
<p>As $\,\gamma_{ij}s_i=s_j\beta_{ij}$, this means $\,b_j=\beta_{ij}(b_i)\in\ker s_j$, so there exists $a_j\in A_j$ suc... |
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| marty cohen | 13,079 | <p>It's easier to turn it over.</p>
<p>$\left(\frac{n+5}{n}\right)^n
=\left(1+\frac{5}{n}\right)^n
\to e^5
$
so
$\left(\frac{n}{n+5}\right)^n
\to e^{-5}
$.</p>
|
84,076 | <p>I think computation of the Euler characteristic of a real variety is not a problem in theory.</p>
<p>There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract" rel="nofollow">J.W. Bruce, Euler characteristics of real varieties</a></em>.</p>
<p>But suppose we have, say, a... | Ryan Budney | 1,465 | <p>Presumably the least-cumbersome approach will depend on the specific variety you need to work with. </p>
<p>In your case, I'd think of solving for $x_n$ in terms of $x_1,\cdots,x_{n-1}$. There's always at least one solution, and sometimes as many as three. This gives a fairly natural stratification of your var... |
2,898,767 | <p>For $M_n (\mathbb{C})$, the vector space of all $n \times n $ complex matrices,</p>
<p>if $\langle A, X \rangle \ge 0$ for all $X \ge 0$ in $M_n{\mathbb{C}}$,then $A \ge 0$</p>
<p>which of the following define an inner product on $M_n(\mathbb{C})$?</p>
<p>$1)$$ \langle A, B\rangle = tr(A^*B)$</p>
<p>$2)$$ \... | mathcounterexamples.net | 187,663 | <p>Usually, for $\alpha \notin \mathbb N$, mathematical packages only define $x \mapsto x^\alpha$ for $x \ge 0$.</p>
|
402,750 | <p>I read that a continuous random variable having an exponential distribution can be used to model inter-event arrival times in a Poisson process. Examples included the times when asteroids hit the earth, when earthquakes happen, when calls are received at a call center, etc. In all these examples, the expected value ... | Did | 6,179 | <blockquote>
<p>It seems obvious that the more time it passes by without an earthquake happening, the more likely it is than an earthquake will happen.</p>
</blockquote>
<p>As explained in the comments, to use an exponential distribution is to assume the opposite: that whatever time passed by without an earthquake h... |
2,426,535 | <p>In the book <em>Simmons, George F.</em>, Introduction to topology and modern analysis, page no- 98, question no- 2, the problem is : <strong><em>Let $X$ be a topological space and a $Y$ be metric space and $f:A\subset X\rightarrow Y$ be a continuous map. Then $f$ can be extended in at most one way to a continuous ... | Henno Brandsma | 4,280 | <p>To follow your idea: Suppose we have $f_1$ and $f_2$ that are both continuous extensions of $f: A \to Y$ to $\overline{A}$. Let $p \in \overline{A}$ and so we have a net $a_i, i \in (I,\le)$ from $A$ such that $a_i \to p$.</p>
<p>The continuity of $f_1$ implies that $f_1(a_i) \to f_1(p)$.</p>
<p>The continuity of ... |
3,130,040 | <p><strong>Edit:</strong> Please prove this without using Cauchy-Schwarz or Preimage. (Also, please show <em>how</em> a chosen <span class="math-container">$ \epsilon $</span> would work by proving that <span class="math-container">$B(\mathbf x, \epsilon)\subseteq\Omega $</span>)</p>
<p>I've been having trouble provin... | Surb | 154,545 | <p>Why Cauchy schwartz should be used here ? (Btw, I don't really see how it can be used...). Anyway, </p>
<p><strong>Method 1:</strong>
<span class="math-container">$$\Omega ^c=\{(x,-x)\in\mathbb R^2\mid x\in \mathbb R\},$$</span></p>
<p>which is obviously sequentially closed, and thus closed.</p>
<p><strong>Method... |
3,130,040 | <p><strong>Edit:</strong> Please prove this without using Cauchy-Schwarz or Preimage. (Also, please show <em>how</em> a chosen <span class="math-container">$ \epsilon $</span> would work by proving that <span class="math-container">$B(\mathbf x, \epsilon)\subseteq\Omega $</span>)</p>
<p>I've been having trouble provin... | Carsten S | 90,962 | <p>You can make your life easier by choosing a smaller <span class="math-container">$\varepsilon$</span>. Let’s take <span class="math-container">$\varepsilon=|x+y|/2$</span>. Now assume <span class="math-container">$\|(x,y)-(x’,y’)\|<\varepsilon$</span>. Then <span class="math-container">$|x-x’|<\varepsilon$</sp... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | albo | 22,610 | <p>This is the most simplest answer, </p>
<p>$$\begin{align*}\begin{split}
{n\choose m-1}+{n\choose m} &= \frac{m}{m}\cdot\frac{n!}{(m-1)!(n-m+1)!}+\frac{(n+1-m)}{(n+1-m)}\cdot\frac{n!}{m!(n-m)!}\\
&=\frac{mn!}{(m)!(n-m+1)!}+\frac{(n+1-m)n!}{m!(n+1-m)!} \\
&=\frac{mn!+(n+1)n!-mn!}{(m)!(n-m+1)!}\\
&=\fr... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | erfan soheil | 195,909 | <p>This is combinatorial proof maybe you like it
Let yo have have a set whit $n$ elements and you want choose $m+1$ elements. Divide the set to two set that one of them has one element an the other one hase $n-1$ elements. now if you want to choose $m+1$ element you can do it in two ways</p>
<p>or you chose $m$ eleme... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
137,571 | <p>As the title, if I have a list:</p>
<pre><code>{"", "", "", "2$70", ""}
</code></pre>
<p>I will expect:</p>
<pre><code>{"", "", "", "2$70", "2$70"}
</code></pre>
<p>If I have</p>
<pre><code>{"", "", "", "3$71", "", "2$72", ""}
</code></pre>
<p>then:</p>
<pre><code>{"", "", "", "3$71", "3$71", "2$72", "2$72"}
... | Mr.Wizard | 121 | <p>As I presently interpret the question</p>
<p>(Now with refinements after considering Chris Degnen's simultaneous answer)</p>
<pre><code>fn[list_] :=
Partition[list, 2, 1, -1, ""] // Cases[{p_, ""} | {_, p_} :> p]
</code></pre>
<p>Test:</p>
<pre><code> {"", "x", "y", "", "z", ""} // fn
</code></pre>
<blockq... |
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| Steven Stadnicki | 785 | <p>While all of the answers here are obviously excellent, I'll throw in one more: imagine the points of $\mathbb{N}^2$ drawn out as an infinite grid of cells (the positive quadrant of the plane, essentially); then fill the cell $(1,1)$ with $1$, the cells $(2,1)$ and $(1,2)$ with $2$ and $3$, and in general all of the ... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | marty cohen | 13,079 | <p>Here is a complete answer
to the second question:</p>
<p>If $n$ is a positive integer
then
$\lfloor nx \rfloor = \sum_{k=0}^{n-1} \lfloor x+\frac{k}{n} \rfloor$.</p>
<p>Let
$m = \lfloor x \rfloor$
and
$d = x - m$,
so
$0 \le d < 1$.</p>
<p>Let
$j
=\lfloor nd \rfloor
$,
so
$0 \le j \le n-1
$
and
$\frac{j}{n}
\... |
70,143 | <p>Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?</p>
<p>Everything I find in the literature is for blow-ups along smooth invariant centers.</p>
<p>Thanks!</p>
| pinaki | 1,508 | <p>To find the polytope associated to a toric variety directly you have to realize the variety as the closure of a map from the torus. In this case at least, it is not too hard to get such a description. Let the homogeneous coordinates of $\mathbb{P}^3$ be $[w:x:y:z]$ and the two lines be $C_1 := \lbrace w = x = 0 \rbr... |
585,808 | <p>As part of showing that
$$
\sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right|
$$
converges, I ended up with trying to show that
$$
\left|\sin\left(\frac{1}{n^2}\right)\right|<\frac{1}{n^2}, \quad n=1, 2, 3,\dots
$$
since I know that the sum of the right hand side converges. But I can't show this. I've ... | JP McCarthy | 19,352 | <p>It's actually very straightforward to show that $\sin x\leq x$ for $0<x<1$ which is all you have to do...</p>
<p>You can do it by definition using a unit circle.</p>
<p>You can do it by noting that $\sin 0=0$ and $\cos x<1$ here (i.e. $\sin x$ grows slower than $x$ here).</p>
|
585,808 | <p>As part of showing that
$$
\sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right|
$$
converges, I ended up with trying to show that
$$
\left|\sin\left(\frac{1}{n^2}\right)\right|<\frac{1}{n^2}, \quad n=1, 2, 3,\dots
$$
since I know that the sum of the right hand side converges. But I can't show this. I've ... | Robert Israel | 8,508 | <p>Hint: $|\sin(t)| \le |t|$ for real $t$, with equality only at $0$. </p>
|
585,808 | <p>As part of showing that
$$
\sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right|
$$
converges, I ended up with trying to show that
$$
\left|\sin\left(\frac{1}{n^2}\right)\right|<\frac{1}{n^2}, \quad n=1, 2, 3,\dots
$$
since I know that the sum of the right hand side converges. But I can't show this. I've ... | ncmathsadist | 4,154 | <p>If $0 < x < \pi/2$, $\sin(x)$ is the $y$--coordinate of the point on the unit circle distance $x$ via the circle. That is the shortest distance of any path from the point $P$, $(\cos(x), \sin(x))$ to the $x$--axis. Since $x$ describes the length of a path from $P$ to the $x$--axis that is not a straight lin... |
1,219,462 | <p>Proposition: Any polynomial of degree $n$ with leading coefficient $(-1)^n$ is the characteristic polynomial of some linear operator.</p>
<p>I do not want to construct an 'explicit matrix' corresponding to the polynomial $(-1)^n(\lambda_n x^n+\cdots+ \lambda_0)$. However, I want to use induction to prove the existe... | Stefano | 108,586 | <p>Hint: The direct proof is not hard. Try with a matrix with a lower diagonal identity and with your polynomial coefficients (in suitable order) in the last column.</p>
|
1,216,302 | <p>I am looking for some sequence of random variables $(X_n)$ such that </p>
<p>$$ \lim_{n \rightarrow \infty} E(X_n^2) = 0 $$</p>
<p>but such that the following <strong>almost sure</strong> convergence does <strong>NOT</strong> hold:</p>
<p>$$ \frac{S_n - E(S_n)}{n} \rightarrow 0$$</p>
<p>where the $S_n$ are the ... | Davide Giraudo | 9,849 | <p>Let $\left(Y_N\right)_{N\geqslant 1}$ be an i.i.d. sequence such that $\Pr\left(Y_N=1\right)=\Pr\left(Y_N=-1\right)=1/(2N)$ and $\Pr\left(Y_N=0\right)=1-1/N$. Let $\left(n_k\right)_{k\geqslant 1}$ be a strictly increasing sequence of integers that will be specified later. For $n_N\leqslant i\leqslant n_{N+1}-1$, def... |
3,035,228 | <p>Show that two cardioids <span class="math-container">$r=a(1+\cos\theta)$</span> and <span class="math-container">$r=a(1-\cos\theta)$</span> are at right angles.</p>
<hr>
<p><span class="math-container">$\frac{dr}{d\theta}=-a\sin\theta$</span> for the first curve and <span class="math-container">$\frac{dr}{d\theta}... | Dylan | 135,643 | <p>Since <span class="math-container">$(x,y)=(r\cos\theta,r\sin\theta)$</span>, the 2 curves can be given parametrically as</p>
<p><span class="math-container">\begin{align}
\vec{r}_1 &= \big(a(1+\cos\theta)\cos\theta,a(1+\cos\theta)\sin\theta\big) \\
\vec{r}_2 &= \big(a(1-\cos\theta)\cos\theta,a(1-\cos\theta)... |
345,766 | <p>I'm trying to calculate this limit expression:</p>
<p>$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $$</p>
<p>Both the numerator and denominator should converge, since $0 \leq a, b \leq 1$, but I don't know if that helps. My guess would be to use L'Hopital's rule and take the ... | Taladris | 70,123 | <p>The derivative (with respect to $x$) of $a^x$ is not $x a^{x-1}$ but $ln(a)a^x$, since $a^x=e^{x ln(a)}$.</p>
<p>You can solve your problem by noticing that $\displaystyle \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s}=1- \frac{1}{1 +ab + (ab)^2 + ... (ab)^s}$.</p>
|
4,636,101 | <p>Given a curve <span class="math-container">$y = x^3-x^4$</span>, how can I find the equation of the line in the form <span class="math-container">$y=mx+b$</span> that is tangent to only two distinct points on the curve?</p>
<p>The problem given is part of the Madas Special Paper Set. This paper set, seems to not hav... | GReyes | 633,848 | <p>If you call <span class="math-container">$f(x)=x^3-x^4$</span>, and you consider the function
<span class="math-container">$$
F(x_1,x_2)=\frac{f(x_1)-f(x_2)}{x_1-x_2}
$$</span>
giving you the slope of the line joining two points on the graph, you can easily see that in the optimal situation (when the line is tangent... |
4,636,101 | <p>Given a curve <span class="math-container">$y = x^3-x^4$</span>, how can I find the equation of the line in the form <span class="math-container">$y=mx+b$</span> that is tangent to only two distinct points on the curve?</p>
<p>The problem given is part of the Madas Special Paper Set. This paper set, seems to not hav... | MathWonk | 301,562 | <p>If the graph of an unknown linear function <span class="math-container">$L(x)= ax+b$</span> grazes that of the given monic quartic <span class="math-container">$Q_4(x)= x^4 - x^3$</span> from below, as in your picture, then their difference <span class="math-container">$Q_4(x)- L(x)$</span> vanishes to second orde... |
3,357,502 | <p>I'm aware that <span class="math-container">$D(n)$</span> can be calculated in O(sqrt(n)) time. Can <span class="math-container">$ D(n, k) $</span> also be calculated in O(sqrt(n)) time? What's the best algorithm?</p>
<p>For example, if <span class="math-container">$n = 8$</span> and <span class="math-container">$k... | reuns | 276,986 | <p>You meant <span class="math-container">$$\sum_{a,b, ab\le n, a \le k}1 = \sum_{a \le k} \left\lfloor \frac{n}{a} \right\rfloor$$</span></p>
<p>Then yes it can be computed in <span class="math-container">$O(\sqrt{n})$</span> operations because <span class="math-container">$$\left\{\left\lfloor \frac{n}{a} \right\rfl... |
2,188,965 | <p>Can someone explain to me how this step done? I got a different answer than what the solution said.</p>
<p>Simplify $x(y+z)(\bar{x} + y)(\bar{y} + x + z)$</p>
<p>what the solution got </p>
<p>$x(y+z)(\bar{x} + y)(\bar{y} + x + z)$ = $x(y + z\bar{x})(\bar{y} + x + z)$ (Using distrubitive)</p>
<p>What I got</p>
<... | Shraddheya Shendre | 384,307 | <p>Hint : $ab + ac = a(b+c) = a$ and $a+a=a$.<br>
Do you see the '$b$' and '$c$' (and also the '$a$') in your case? </p>
|
4,202,490 | <p>Trying to construct an example for a Business Calculus class (meaning trig functions are not necessary for the curriculum). However, I want to touch on the limit problem involved with the <span class="math-container">$\sin(1/x)$</span> function.</p>
<p>I am sure there is a simple function, or there isn't... But woul... | hardmath | 3,111 | <p>Consider applying the <a href="https://en.wikipedia.org/wiki/Fractional_part" rel="nofollow noreferrer">fractional part function</a> to <span class="math-container">$1/x^2$</span> or something similar. This would be an even function, so the behavior from the left is the same as the behavior from the right of zero, ... |
3,553,233 | <p>I am having difficulty finishing this proof. At first, the proof is easy enough. Here's what I have thus far:<br>
Because <span class="math-container">$5 \nmid n$</span>, we know <span class="math-container">$\exists q \in \mathbb{Z}$</span> such that <span class="math-container">$$n = 5q + r$$</span> where <span cl... | fleablood | 280,126 | <p>Just do it.</p>
<p><span class="math-container">$n^2 = 25q^2 + 10qr + r^2 = 5(5q^2 + 2qr) + r^2$</span>.</p>
<p>ANd <span class="math-container">$r^2 = 1,4=5-1, 9 =10-1$</span> or <span class="math-container">$16=15+1$</span>.</p>
<p>So if <span class="math-container">$n = 5q + 1$</span> then <span class="math-co... |
3,553,233 | <p>I am having difficulty finishing this proof. At first, the proof is easy enough. Here's what I have thus far:<br>
Because <span class="math-container">$5 \nmid n$</span>, we know <span class="math-container">$\exists q \in \mathbb{Z}$</span> such that <span class="math-container">$$n = 5q + r$$</span> where <span cl... | Robert Lewis | 67,071 | <p>To keep things concise I have stretched the notation a bit, but I think my meaning is clear:</p>
<p>In the language of congruence, </p>
<p><span class="math-container">$5 \not \mid n \Longrightarrow n \not \equiv 0 \mod 5; \tag 1$</span></p>
<p>then one of the following holds:</p>
<p><span class="math-container"... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | Giorgio Mossa | 11,888 | <p>I remember when I faced this problem in my early days as a student.
Here's a (hopefully) simple explanation, result of some year of studies.</p>
<p>One of the first thing to understand is that set theory is not just a family of axioms, it is a formal system that is specified by</p>
<ul>
<li>a language, that is th... |
3,209,237 | <p>The proof of the CRT goes as follows:<br>
Given the number <span class="math-container">$x \epsilon Z_m$</span>, <span class="math-container">$m=m_1m_2...m_k$</span>
<span class="math-container">$$M_k = m/m_k$$</span>
construct:
<span class="math-container">$$ x = a_1M_1y_1+a_2M_2y_2+...+a_nM_ny_n$$</span>
where <sp... | mathmaniage | 379,585 | <p>The part 2 of the chinese remainder theorem, <a href="https://forthright48.com/chinese-remainder-theorem-part-1-coprime-moduli/" rel="nofollow noreferrer">which starts off at this page and continues to the next</a>, explains the concept of lcm required to understand the OP's question and the concept why the solution... |
1,072,656 | <p>I am building a website which will run on the equation specified below. I am in pre-algebra and do not have any idea how to go about this equation. my friends say it is a system of equation but I don't know how to solve those and no one I know seems to know how to do them with exponents. I was hoping that people on ... | Alice Ryhl | 132,791 | <p>Multiply second equation with $y^2$ then you get</p>
<p>$$xy^5 \cdot y-xy^5=5000y^2$$</p>
<p>Substitute $xy^5=8000$</p>
<p>$$8000y-8000=5000y^2$$</p>
<p>Divide with $1000$ and move all terms to left side</p>
<p>$$5y^2-8y+8=0$$</p>
<p>Since $d=b^2-4ac=(-8)^2-4\cdot5\cdot8=-96$ which is negative, there are no so... |
69,208 | <p>Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$.
Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$.</p>
<p>a.)
What is the probability that a random function will be a bijection when viewed as
$$f':\{1,\dots,n\} \to \operatorname{Im}(f)?$$</p>
<p>b.)
How many different function f are the... | Lubin | 17,760 | <p>Why not look at the <em>formal series</em> for sine and cosine, $s(x)$ and $c(x)$, exactly the series you’ve written, and notice that $s'=c$, $c'=-s$, and differentiate the formal expression $s^2+c^2$ to get zero. Reflect on that, and realize that you’ve just shown that the formal series $s^2+c^2$ is constan... |
2,992,454 | <p>Prove :</p>
<blockquote>
<p><span class="math-container">$f : (a,b) \to \mathbb{R} $</span> is convex, then <span class="math-container">$f$</span> is bounded on every closed subinterval of <span class="math-container">$(a,b)$</span></p>
</blockquote>
<p>where <span class="math-container">$f$</span> is convex if... | Hagen von Eitzen | 39,174 | <p>Pick <span class="math-container">$\gamma$</span> with <span class="math-container">$\alpha<\gamma<\beta$</span>. Let <span class="math-container">$A=(\alpha,f(\alpha))$</span>, <span class="math-container">$B=(\beta,f(\beta))$</span>, <span class="math-container">$C=(\gamma,f(\gamma))$</span> Then</p>
<ul>
<... |
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | marty cohen | 13,079 | <p>This is getting really old.</p>
<p>Copy and paste from
another answer of mine.</p>
<p>If $n$ is a positive integer that is
not a square of an integer,
then $\sqrt{n}$ is irrational.</p>
<p>Let $k$ be such that
$k^2 < n < (k+1)^2$.
Suppose $\sqrt{n}$ is rational.
Then there is a smallest positive integer $q$... |
3,436,430 | <p>Evaluate <span class="math-container">$$\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{2-\sqrt{4-x^2-y^2}}^{2+\sqrt{4-x^2-y^2}}(x^2+y^2+z^2)^{3/2} \; dz \; dy \; dx$$</span> by converting to spherical coordinates.<br>
We know that <span class="math-container">$(x^2+y^2+z^2)^{3/2} = (\rho^2)^{3/2} = \rho^3$</s... | Mnifldz | 210,719 | <p>The first thing to notice here is exactly what volume you're integrating inside. The volume you're integrating is the sphere of radius <span class="math-container">$2$</span> but centered at the point <span class="math-container">$(0,0,2)$</span>. If we converted directly to spherical coordinates for this volume, ... |
93,458 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a nonnegative integer. Show that <span class="math-container">$\lfloor (2+\sqrt{3})^n \rfloor $</span> is odd and that <span class="math-container">$2^{n+1}$</span> divides <span class="math-container">$\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $</span>.</p>
</... | marty cohen | 13,079 | <p><em>For the first claim:</em></p>
<p>Here is a more general discussion.
As is often the case,
none of this is original,
and I did work it out
on my own for fun.</p>
<p>To make one root of
<span class="math-container">$f(x)
=x^2-2ax+b
=0$</span>
with
<span class="math-container">$a, b > 0$</span>
in <span class=... |
2,451,469 | <p>I have a system of differential equations:</p>
<p>$$x(t)'' + a \cdot x(t)' = j(t)$$
$$j(t)' = -b \cdot j(t) - x(t)' + u(t)$$</p>
<p>The task is: Substitute $v(t) = x(t)'$ into the system and rewrite the system as 3 coupled linear
differential equations of the same form (with
$y(t) = x(t)$ the solution sought... | alexjo | 103,399 | <p>$$\begin{align}
&x(t)'' + a \cdot x(t)' = j(t)\\
&j(t)' = -b \cdot j(t) - x(t)' + u(t)
\end{align}
$$</p>
<p>Putting $v(t) = x(t)'$ we have
$$\begin{align}
v-x'&=0\\
v' + a v - j&=0\\
j' +b j + v - u&=0
\end{align}
$$
that is
$$
\begin{bmatrix}
x'\\
v'\\
j'
\end{bmatrix}=\begin{bmatrix}
0&... |
3,460 | <p>I asked the question "<a href="https://mathoverflow.net/questions/284824/averaging-2-omegan-over-a-region">Averaging $2^{\omega(n)}$ over a region</a>" because this is a necessary step in a research paper I am writing. The answer is detailed and does exactly what I need, and it would be convenient to directly cite t... | darij grinberg | 2,530 | <p>You seem to view a nickname in the bibliography as embarrassing or unprofessional. There is no reason this should be the case. If you use the built-in "cite" feature of the stackexchange network, the author's nickname will immediately be followed by the URL of their profile, which should allay any doubts about what ... |
3,002,874 | <p>I found this limit in a book, without any explanation:</p>
<p><span class="math-container">$$\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}(\zeta(2)-H_{k,2})-H_n\right)=1$$</span></p>
<p>where <span class="math-container">$H_{k,2}:=\sum_{j=1}^k\frac1{j^2}$</span>. However Im unable to find the value of this limit from my... | Claude Leibovici | 82,404 | <p>Considering your last expression <span class="math-container">$$a_n=\sum_{k=0}^{n-1}\sum_{j=k}^\infty\frac1{(j+1)^2(j+2)}$$</span>
<span class="math-container">$$\sum_{j=k}^\infty\frac1{(j+1)^2(j+2)}=\psi ^{(1)}(k+1)-\frac{1}{k+1}$$</span> making
<span class="math-container">$$a_n=n \,\psi ^{(1)}(n+1)$$</span> the e... |
1,960,169 | <p><a href="http://puu.sh/rCwCy/c78a9ef78a.png" rel="nofollow noreferrer">Asymptote http://puu.sh/rCwCy/c78a9ef78a.png</a></p>
<p>Well my thinking was if the asymptote is at x = 4, it will reach as close to 4 as possible but will never reach 4, meaning it's not defined at 4. </p>
| MathMajor | 113,330 | <p>We have</p>
<p>$$x^2 + 4x - 5 = (x^2 + 4x + 4) - 4 - 5 = (x + 2)^2 - 9.$$</p>
<p>Therefore, the vertex is at $(-2, \, -9)$.</p>
|
111,899 | <p>Evaluate the integral using trigonometric substitutions. </p>
<p>$$\int{ x\over \sqrt{3-2x-x^2}} \,dx$$</p>
<p>I am familiar with using the right triangle diagram and theta, but I do not know which terms would go on the hypotenuse and sides in this case. If you can determine which numbers or $x$-values go on the... | David Mitra | 18,986 | <p>The "trick" in evaluating
$$\tag{1}
\int{x\over\sqrt{3-2x-x^2}}\,dx
$$
is to complete the square of the expression in the radicand: rewrite $3-2x-x^2$ as$$\tag{2}4-\color{maroon}{(x+1)}^2.$$</p>
<p>I'm not sure what this right triangle diagram you speak of is, but with the method I assume you're using, the second... |
1,303,274 | <p>Define a sequence {$\ x_n$} recursively by</p>
<p>$$ x_{n+1} =
\sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1
$$
Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.</p>
| k170 | 161,538 | <p>$$ \sum\limits_{n=0}^\infty (-1)^n = \lim\limits_{m\to\infty}\sum\limits_{n=0}^m (-1)^n $$
$$ = \lim\limits_{m\to\infty} \frac12\left((-1)^m + 1\right) $$
Note that the sequence
$$a_m=\frac12\left((-1)^m + 1\right) $$
is convergent if
$$ \lim\limits_{m\to\infty} a_{2m} = \lim\limits_{m\to\infty} a_{2m+1} = L$$
How... |
1,429,853 | <p>In a number sequence, I've figured the $n^{th}$ element can be written as $10^{2-n}$.</p>
<p>I'm now trying to come up with a formula that describes the sum of this sequence for a given $n$. I've been looking at the geometric sequence, but I'm not sure how connect it.</p>
| Clayton | 43,239 | <p><strong>Hint:</strong> We have $10^{2-n}=10^2\cdot 10^{-n}=10^2\cdot(10^{-1})^n$ and $$\sum_{k=0}^n x^k=\frac{1-x^{n+1}}{1-x}.$$ </p>
|
4,600,131 | <blockquote>
<p>If <span class="math-container">$$f(x)=\binom{n}{1}(x-1)^2-\binom{n}{2}(x-2)^2+\cdots+(-1)^{n-1}\binom{n}{n}(x-n)^2$$</span>
Find the value of <span class="math-container">$$\int_0^1f(x)dx$$</span></p>
</blockquote>
<p>I rewrote this into a compact form.
<span class="math-container">$$\sum_{k=1}^n\binom... | lab bhattacharjee | 33,337 | <p><strong>Hint:</strong></p>
<p>Another way:</p>
<p><span class="math-container">$$f(x)=\sum_{k=1}^n\binom nk(x-k)^2(-1)^{k+1}$$</span></p>
<p><span class="math-container">$(x-k)^2=x^2+k(1-2x)+k(k-1)$</span></p>
<p><span class="math-container">$$\binom nk(x-k)^2=x^2\binom nk+(1-2x)n\binom{n-1}{k-1}+n(n-1)\binom{n-2}{k... |
2,643,900 | <p>for the problem </p>
<p>$$(1-2x)y'=y$$</p>
<p>the BC'S are $y(0)=-1$ and $y(1)=1$ and $0\leq x\leq 1$.</p>
<p>I solved this and got $\ln y =\ln\left(\dfrac 2 {1-2x}\right)+c$.</p>
<p>How do we determine the constant such that $y$ is real and finite everywhere from $0$ to $1$ (both limits included)?</p>
| Doug M | 317,162 | <p>$\ln y =\ln(2/(1-2x))+c\\
y =e^{\ln(2/(1-2x))+c}\\
y =e^{\ln(2/(1-2x))}e^{c}$</p>
<p>But $e^c$ is just as much of an arbitrary constant as $c$ was.</p>
<p>$y =Ce^{\ln(2/(1-2x))}\\
y =\frac {C}{1-2x}$</p>
<p>However, you have a problem at $x = \frac 12$</p>
<p>Going back to:</p>
<p>$(1-2x)y' = y \implies y(\frac... |
2,643,900 | <p>for the problem </p>
<p>$$(1-2x)y'=y$$</p>
<p>the BC'S are $y(0)=-1$ and $y(1)=1$ and $0\leq x\leq 1$.</p>
<p>I solved this and got $\ln y =\ln\left(\dfrac 2 {1-2x}\right)+c$.</p>
<p>How do we determine the constant such that $y$ is real and finite everywhere from $0$ to $1$ (both limits included)?</p>
| Botond | 281,471 | <p>Your equation is:
$$(1-2x)*\frac{\mathrm{d}y}{\mathrm{d}x}=y$$
$$\frac{1}{y}\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{1-2x}$$
Integrating with respect to $x$:
$$\int\frac{1}{y}\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x=\int\frac{1}{1-2x}\mathrm{d}x$$
$$\log(y)=-\frac{1}{2}\log(1-2x)+C$$
$$\exp(\log(y))=\exp\left(-\fr... |
21,569 | <p>Recently I commented on a question which was about tiling a 100 by 100 grid with a 1 x 8 square. (Actually to prove this was impossible.) One of our users posted a link to a very interesting looking paper on tiling problems as a comment, and I also commented, but now looking through my comments I can't find the post... | Daniel Fischer | 83,702 | <p>I guess it's <a href="https://math.stackexchange.com/questions/1436647/tiling-problem-related-to-algebra">this question</a>, and it's about <a href="http://mathworld.wolfram.com/KlarnersTheorem.html" rel="nofollow noreferrer">Klarner's theorem</a>, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80044-X" rel="nofol... |
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | Doug M | 317,162 | <p>At most 3 kings is the same thing as not 4 kings.</p>
<p>$P(4$ kings)$ = 48 / {52\choose 5} = \frac{1}{54145}$</p>
|
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | Sumedh | 73,593 | <p>Since you need to find probability $P(\leq 3~\text{kings})$, you can find $P(>3~\text{kings})$ and subtract it from $1$.</p>
<p>$P(>3~\text{kings}) \iff P(4~\text{kings})$, which is $48/52C5 = 1/54145$ </p>
<p>Here, we take $48$, since out of $5$ cards four are kings, and fifth can be any one out of the rema... |
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | barak manos | 131,263 | <p>Split it into <strong>disjoint</strong> events, and then add up their probabilities:</p>
<hr>
<p>The probability of exactly $\color\red0$ kings is:</p>
<p>$$\frac{\binom{4}{\color\red0}\cdot\binom{52-4}{5-\color\red0}}{\binom{52}{5}}$$</p>
<hr>
<p>The probability of exactly $\color\red1$ king is:</p>
<p>$$\fra... |
249,047 | <p>I have the following matrix: $$A=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
What is the norm of $A$? I need to show the steps, should not use Matlab...<br>
I know that the answer is $\sqrt{\sqrt{5}/2+3/2}$. I am using the s... | 000 | 22,144 | <p>Find the closed form and take the limit in the case of infinite sums. That is, you find the closed form of the sum $\sum_{4 \le k \le m}\frac{1}{k^2-1}$ and evaluate $\lim_{m \to \infty}(\text{closed form of the sum})$.</p>
<p>In this case, you apply partial fraction decomposition to $\frac{1}{k^2-1}$ and arrive at... |
4,600,992 | <p>I have two sequences of random variables <span class="math-container">$\{ X_n\}$</span> and <span class="math-container">$\{Y_n \}$</span>. I know that
<span class="math-container">$X_n \to^d D, Y_n \to^d D$</span>. Can I conclude that <span class="math-container">$X_n - Y_n \to^p 0$</span>?</p>
<p>If I cannot, what... | Aaron | 9,863 | <p>Let us use the fundamental theorem of calculus and the mean value theorem, alone with some very rough estimates.</p>
<p>If we set <span class="math-container">$f(x)=x\int_1^{x}\frac{e^t}{t}dt-e^x$</span>, then the product rule and the fundamental theorem of calculus yield
<span class="math-container">$$f'(x)=\int_1^... |
320,355 | <p>Show that $$\nabla\cdot (\nabla f\times \nabla h)=0,$$
where $f = f(x,y,z)$ and $h = h(x,y,z)$.</p>
<p>I have tried but I just keep getting a mess that I cannot simplify. I also need to show that </p>
<p>$$\nabla \cdot (\nabla f \times r) = 0$$</p>
<p>using the first result.</p>
<p>Thanks in advance for any help... | Slugger | 59,157 | <p>We use $\nabla f =(f_x,f_y,f_z)$ and $\nabla h=(h_x,h_y,h_z)$. For the cross product we have $(a,b,c) \times (u,v,w)=\hat{i} (bw-cv)+\hat{j}(cu-aw)+\hat{k} (av-bu) $, alternatively written this is expressed $(a,b,c) \times (x,y,z) = (bw-cv,cu-aw,av-bu)$ but this is exactly the same. Here $a,b,c,u,v,w$ are not meant ... |
320,355 | <p>Show that $$\nabla\cdot (\nabla f\times \nabla h)=0,$$
where $f = f(x,y,z)$ and $h = h(x,y,z)$.</p>
<p>I have tried but I just keep getting a mess that I cannot simplify. I also need to show that </p>
<p>$$\nabla \cdot (\nabla f \times r) = 0$$</p>
<p>using the first result.</p>
<p>Thanks in advance for any help... | hd.scania | 74,980 | <p>$$\nabla f\times\nabla h\\=\left(\left(\hat{x}\frac{\partial}{\partial x}+\hat{y}\frac{\partial}{\partial y}+\hat{z}\frac{\partial}{\partial z}\right)f\right)\times\left(\left(\hat{x}\frac{\partial}{\partial x}+\hat{y}\frac{\partial}{\partial y}+\hat{z}\frac{\partial}{\partial z}\right)h\right)\\=\left(\hat{x}\frac{... |
2,072,666 | <p>I have a set as : <b> {∀x ∃y P(x, y), ∀x¬P(x, x)}. </b>. In order to satisfy this set I know that there should exist an interpretation <b> I </b> such that it should satisfy all the elements in the set. For instance my interpretation for x is 3 and for y is 4. Should I apply the same numbers (3,4) to ∀x¬P(x, x) as ... | Mark Viola | 218,419 | <p>Let $y=-2x/e$. Then, as $x\to 0$, $y\to 0$.</p>
<p>Next, we see that </p>
<p>$$\frac{(-2x/e)}{\log(1+(-2x/e))}=\frac{y}{\log(1+y)}=\frac{1}{\frac{\log(1+y)}{y}}$$</p>
<p>Hence, we can assert that </p>
<p>$$\lim_{x\to 0}\frac{(-2x/e)}{\log(1+(-2x/e))}=\frac{1}{\lim_{y\to 0}\frac{\log(1+y)}{y}}=1$$</p>
<p>as was... |
607,862 | <p>Let $f$ be a continuous function. What is the maximum of $\int_0^1 fg$ among all continuous functions $g$ with $\int_0^1 |g| = 1$?</p>
| Pablo Rotondo | 22,121 | <p><strong>Hint:</strong> Try concentrating the <em>weight</em> of $g$ around the maximum of $|f|$. What happens then?</p>
|
2,658,627 | <p>I am trying to solve the following:</p>
<blockquote>
<p>Suppose the bank paid 12 % per year, but compounded that interest
monthly. That is, suppose 1 % interest was added to your account every
month. Then how much would you have after 30 years and after 60 years
if you started with $100?</p>
</blockquote>
<p>What I ... | Mostafa Ayaz | 518,023 | <p>Note that monthly interest $1$% is not equivalent to that of yearly $12$% because$$\text{interests per month after 1 year}=(1.01)^{12}\approx1.1268\\\text{interests per year after 1 year}=(1.12)^{1}=1.12$$so we can see$$\text{1% interest per month in one year}≠\text{12% interest per annum}$$therefore if our interest... |
2,491,968 | <p>$f(z)=(z+1/z)^2$ is the given function . How to find whether the function is differentiable at origin or not ?</p>
| Mark | 310,244 | <p>Note that:
$$f(z) = z^2 + 2 + \frac{1}{z^2}$$
This clearly shows that $f(z)$ has a pole of order $2$ at the origin, and is therefore meromorphic there, and not holomorphic.</p>
|
2,491,968 | <p>$f(z)=(z+1/z)^2$ is the given function . How to find whether the function is differentiable at origin or not ?</p>
| Ben | 493,581 | <p>To see it in a more direct sense, let $x \in \mathbb{R}$:</p>
<p>Consider $\lim_{x \rightarrow 0} f(ix)=\lim_{x \rightarrow 0} (xi)^{2} +2 +1/(xi)^{2}=\lim_{x \rightarrow 0} -x^{2} +2 -1/x^{2}=- \infty$.</p>
<p>Now consider $\lim_{x \rightarrow 0} f(x)=\lim_{x \rightarrow 0} x^{2} +2 +1/x^{2}= \infty$.</p>
<p>In ... |
1,142,631 | <p>A sequence of probability measures $\mu_n$ is said to be tight if for each $\epsilon$ there exists a finite interval $(a,b]$ such that $\mu((a,b])>1-\epsilon$ For all $n$.</p>
<p>With this information, prove that if $\sup_n\int f$ $d\mu_n<\infty$ for a nonnegative $f$ such that $f(x)\rightarrow\infty$ as $x\r... | Cecilia | 257,899 | <p>The condition to ensure the tightness is that $\displaystyle \frac{f(x)}{x} \to \infty$ as $x \to \infty$. In fact if e.g. $f(x)=|x|$ then $$\sup_n \int f \, dμ_n<\infty$$ is only a necessary condition for the tightness.</p>
|
405,449 | <blockquote>
<p>Is the polynomial $x^{105} - 9$ reducible over $\mathbb{Z}$?</p>
</blockquote>
<p>This exercise I received on a test, and I didn't resolve it. I would be curious in any demonstration with explanations. Thanks!</p>
| Key Ideas | 78,535 | <p>For binomials there is a classical irreducibility test (below). It implies that $\,x^{105}-c\,$ is irreducible over a field $\,F\,$ if $\,c\,$ is not a third, fifth, or seventh power in $\,F,$ since $\,105 = 3\cdot 5\cdot 7.$</p>
<p><strong>Theorem</strong> $\ $ Suppose $\:c\in F\:$ a field, and $\:0 < n\in\mat... |
3,976,572 | <p>Suppose <span class="math-container">$f(x)= x^3+2x^2+3x+3$</span> and has roots <span class="math-container">$a , b ,c$</span>.
Then find the value of
<span class="math-container">$\left(\frac{a}{a+1}\right)^{3}+\left(\frac{b}{b+1}\right)^{3}+\left(\frac{c}{c+1}\right)^{3}$</span>.</p>
<p>My Approach :
I constructed... | Aryan | 866,404 | <p>It has some boring calculations so I'll just write about the sketch of my solution:
Using Vietta formulas, find the coefficients of the polynomial that has roots <span class="math-container">${1-\frac{1}{a+1},1-\frac{1}{b+1},1-\frac{1}{c+1}}$</span>. Then just use the Newton formula to find the sum of cubes of that ... |
1,391,214 | <p>I've been thinking about the differences in numbers so for example: </p>
<p>$\begin{array}{ccccccc} &&&0&&&\\ &&1&&1&&\\&1&&2&&3&\\0&&1&&3&&6
\end{array}$</p>
<p>or with absolute differences:</p>
<p>$\begin{array}{ccc... | Oleg567 | 47,993 | <p>This sequence can be described by formula:
$$
u(n)=\left\{
\begin{array}{l}
n\cdot 2^{n-1}, \qquad\; n=0,1,2,3,4,5,6;\\
n\cdot 2^{n-1}+4, \;\; n=7,8,9,...;\end{array}
\right.
$$</p>
<p>$u(0)=0$, $u(1)=1$, $u(2)=4$, $u(3)=12$, $\ldots$, $u(6)=192$; </p>
$u(7)=448+4$, $u(8)=1024+4$, $\ldots$, $u(13)=53248+4$.</p>
|
1,930,558 | <p>I want a good textbook covering elemenents of discrete mathematics Average level.Im a mathematics undergraduate so i dont want it to be towards Computer science that much.Interested in combinatorics and graph theory .But also covering enumeration and other stuff.One book i found is <a href="http://rads.stackoverflow... | Brian M. Scott | 12,042 | <p>For a math major I strongly recommend Edward A. Scheinerman, <a href="http://rads.stackoverflow.com/amzn/click/0840049420" rel="nofollow"><em>Mathematics: A Discrete Introduction</em></a>; it’s well written, and it’s definitely aimed at math majors, not computer science majors. The book by Kenneth Rosen is exhaustiv... |
227,618 | <p>I'm creating AI for a card game, and I run into problem calculating the probability of passing/failing the hand when AI needs to start the hand. Cards are A, K, Q, J, 10, 9, 8, 7 (with A being the strongest) and AI needs to play to not take the hand.</p>
<p>Assuming there are 4 cards of the suit left in the game an... | Martin Sleziak | 8,297 | <p>We know that $|\mathbb R|=\mathfrak c=2^{\aleph_0}$</p>
<p>$|\mathcal P(\mathbb R)|=2^{\mathfrak c}$</p>
<p>$|\mathbb R^{\mathbb R}|=\mathfrak c^{\mathfrak c}$</p>
<p>So we only need to show that $2^{\mathfrak c}=\mathfrak c^{\mathfrak c}$.</p>
<p>We have
$$\mathfrak c^{\mathfrak c} = (2^{\aleph_0})^{\mathfrak c... |
227,618 | <p>I'm creating AI for a card game, and I run into problem calculating the probability of passing/failing the hand when AI needs to start the hand. Cards are A, K, Q, J, 10, 9, 8, 7 (with A being the strongest) and AI needs to play to not take the hand.</p>
<p>Assuming there are 4 cards of the suit left in the game an... | Asaf Karagila | 622 | <p>Note that $\mathbb R$ has the same cardinality as $\mathbb R\times\mathbb R$. Therefore their power sets have the same size.</p>
<p>In particular $f\colon\mathbb R\to\mathbb R$ is an element in $\mathcal P(\mathbb{R\times R})$, and therefore $\mathbb{R^R}$ has cardinality of at most $2^{|\mathbb R|}$, and the other... |
3,895,275 | <p>I have a matrix <span class="math-container">$A= \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} $</span> and I should find <span class="math-container">$m$</span>, <span class="math-container">$n$</span>, <span class="math-container">$r$</span> in case that <span class="math-container">$A^2+nA+rI=0$</span> (<... | TheSilverDoe | 594,484 | <p><strong>Hint :</strong> <span class="math-container">$m=1$</span>, <span class="math-container">$n=-\mathrm{Tr}(A)$</span> and <span class="math-container">$r=\det(A)$</span> always work. This is called Cayley-Hamilton theorem.</p>
|
4,611,065 | <p>This question is motivated by curiosity and I haven't much background to exhibit .</p>
<p>Going through a couple of books dealing with real analysis, I've noticed that 2 definitions can be given of the exponential function known in algebra as <span class="math-container">$f(x)= e^x$</span>.</p>
<p>One definition sa... | Qiaochu Yuan | 232 | <p>There are several equivalent definitions, and it is important and valuable to know that they all define the same function. Really they should all be collected into a "definition-theorem," which might look like this.</p>
<blockquote>
<p><strong>Definition-Theorem:</strong> The following five functions are i... |
1,576,836 | <p>I was solving a question related to functions and i come across a limit which i cannot understand.The question is <br>
If $a$ and $b$ are positive real numbers such that $a-b=2,$ then find the smallest value of the constant $L$ for which $\sqrt{x^2+ax}-\sqrt{x^2+bx}<L$ for all $x>0$<br></p>
<hr>
<p>First i f... | MichaelChirico | 205,203 | <p>The crucial thing to recall is that $\sqrt{x^2} = |x|$ -- at core, the composition of the square root with the quadratic operator results in something that is basically bi-linear.</p>
<p>Consider $x^2 -3x + 4$; let's first express it as a transformation of $x^2$:</p>
<p>$$x^2-3x+4 = (x-\frac{3}{2})^2 + \frac{7}{4}... |
2,389,645 | <p>According to Wolfram Alpha : $\lim_{x \to 0^{-}}\lfloor x \rfloor = -1$ and $\lfloor\lim_{x \to 0^{-}} x\rfloor = 0$ . The first expression is obvious but the second doesn't make sense . It should be $-1$ because for example we have $\lfloor - 0.00001 \rfloor = -1$ . My teacher also accepted the Wolfram Alpha's r... | Math Lover | 348,257 | <p>Hint: $\lim_{x\rightarrow 0^{-}}{x}=0$.</p>
<p>You first evaluate the limit and then take the floor value.</p>
|
2,389,645 | <p>According to Wolfram Alpha : $\lim_{x \to 0^{-}}\lfloor x \rfloor = -1$ and $\lfloor\lim_{x \to 0^{-}} x\rfloor = 0$ . The first expression is obvious but the second doesn't make sense . It should be $-1$ because for example we have $\lfloor - 0.00001 \rfloor = -1$ . My teacher also accepted the Wolfram Alpha's r... | Nathanael Skrepek | 423,961 | <p>This is due the fact that the function <span class="math-container">$x \mapsto \lfloor x\rfloor$</span> isn't conutinuous.
You have to be aware of how a limit works. For the first term <span class="math-container">$\lim_{x\to 0-} \lfloor x\rfloor$</span> you take values which have a small distance to <span class="ma... |
984,232 | <p>We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if
$$\begin{array}
AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\
\downarrow{h} & & \downarrow{h'} \\
B_1 & \stackrel{g_1}{\longrightarrow} & B_2
\end{array}
$$ is a pullba... | Andrew Hubery | 367,470 | <p>In a pullback diagram we always have <span class="math-container">$\ker(f_1)\cong\ker(g_1)$</span>.</p>
<p>For, we have the inclusion <span class="math-container">$\ker(g_1)\to B_1$</span> and the trivial map <span class="math-container">$\ker(g_1)\to A_2$</span>. By the universal property of the pullback, we obtai... |
131,741 | <p>Take the following example <code>Dataset</code>:</p>
<pre><code>data = Table[Association["a" -> i, "b" -> i^2, "c" -> i^3], {i, 4}] // Dataset
</code></pre>
<p><img src="https://i.stack.imgur.com/PZSgO.png" alt="Mathematica graphics"></p>
<p>Picking out two of the three columns is done this way:</p>
<pr... | Mike Colacino | 18,550 | <p>I don't find @kglr solution inelegant, but perhaps a little prettier with</p>
<pre><code>data[All, {"a" -> f, "b" -> h}] // KeyDrop["c"]
</code></pre>
|
4,282,006 | <p><strong>Evaluate the limit</strong></p>
<p><span class="math-container">$\lim_{x\rightarrow \infty}(\sqrt[3]{x^3+x^2}-x)$</span></p>
<p>I know that the limit is <span class="math-container">$1/3$</span> by looking at the graph of this function, but I struggle to show it algebraically.</p>
<p>Is there anyone who can ... | Antoine | 73,561 | <p>Use the equality <span class="math-container">$a^3 - b^3 = (a - b)(a^2 + a b + b^2)$</span> for <span class="math-container">$a = \sqrt[3]{x^3 + x^2}$</span> and <span class="math-container">$b = x$</span>. Convert your expression to <span class="math-container">$$\frac{(a - b)(a^2 + a b + b^2)}{a^2 + a b + b^2}$$<... |
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