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 |
|---|---|---|---|---|
177,760 | <p>Can you tell me what is the difference between <strong>empirical distribution</strong> and <strong>classical probability</strong>?
My teacher has told me that when we take limit empirical distribution will get a constant value </p>
<p>$$P(A)=\lim_{N\rightarrow\infty}f(A)=\lim_{N\rightarrow \infty}\frac{N(A)}{N}= \m... | Michael R. Chernick | 30,995 | <p>The empirical distribution is the distribution of the sample or sample estimate. </p>
<p>In your context the distribution here $p=$ # of successes over number of trials and $1-p=$ number of failures over number of trials of the empirical distribution for your estimate of the binomial proportion parameter. It conve... |
177,760 | <p>Can you tell me what is the difference between <strong>empirical distribution</strong> and <strong>classical probability</strong>?
My teacher has told me that when we take limit empirical distribution will get a constant value </p>
<p>$$P(A)=\lim_{N\rightarrow\infty}f(A)=\lim_{N\rightarrow \infty}\frac{N(A)}{N}= \m... | KAT | 68,473 | <p>The difference between the classical definition and the empirical are similar to the difference between a theory and an experiment in physics. The theory is developed in an abstract (perfect) way while the experiments are practical observations.
The same happens with probability. </p>
<p>In the classic definition ... |
1,824,638 | <p>The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string is unwound counterclockwise and continues counterclockwise until it r... | Mike Pierce | 167,197 | <p>I'm mostly condensing what Eric Weisstein wrote in <a href="http://mathworld.wolfram.com/GoatProblem.html" rel="nofollow noreferrer">this <em>MathWorld</em> article</a>, citing a 1998 paper by Hoffman, <em>The Bull and the Silo: An Application of Curvature</em>. You can solve this by finding the area of the semicirc... |
1,824,638 | <p>The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string is unwound counterclockwise and continues counterclockwise until it r... | Riccardo Orlando | 335,442 | <p>Well, I cheated, and solved it numerically: it comes up at about 19.55.</p>
<p>Unfortunately that does not match up with Mike Pierce's answer of $\frac56 \pi$...</p>
<p>Here is a <a href="https://i.imgur.com/9PjSZ3b.png" rel="nofollow noreferrer">graph</a> of the interesting part, and here is the <a href="http://p... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Quanta | 13,121 | <p>Marcus - Number Fields</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | timur | 824 | <p>Michael Reed. <em>Abstract Non Linear Wave Equations</em>.</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Palais - Foundations of global non–linear analysis</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Michael Hardy | 6,316 | <p>Robin Hartshorne's lecture notes on projective geometry. This appeared as a book and is now out of print. The pages appear to be photographs of pages produced with a typewriter, plus hand-drawn illustrations.</p>
<p>Maybe a wiki should be set up where volunteers can transcribe from the book. Permission from copy... |
2,031,842 | <p>If 3 is the remainder when dividing $P(x)$ with $(x-3)$, and $5$ is the remainder when dividing $P(x)$ with $(x-4)$, what is the remainder when dividing $P(x)$ with $(x-3)(x-4)$?</p>
<p>I'm completely puzzled by this, I'm not sure where to start...</p>
<p>Any hint would be much appreciated. </p>
| Siong Thye Goh | 306,553 | <p>Hint:
$$P(x)=Q(x)(x-3)(x-4)+Ax+B$$
$$P(3)=3$$
$$P(4)=5$$</p>
<p>Can you solve for $A$ and $B$?</p>
|
2,031,842 | <p>If 3 is the remainder when dividing $P(x)$ with $(x-3)$, and $5$ is the remainder when dividing $P(x)$ with $(x-4)$, what is the remainder when dividing $P(x)$ with $(x-3)(x-4)$?</p>
<p>I'm completely puzzled by this, I'm not sure where to start...</p>
<p>Any hint would be much appreciated. </p>
| Rene Schipperus | 149,912 | <p>Let $$P(x)=(x-4)(x-3)Q(x)+ax+b$$</p>
<p>Now you have
$$3a+b=3$$ and $$4a+b=5$$ So solve for $a$ and $b$.</p>
|
2,217,928 | <p>I know that in infinite dimensional Hilbert spaces sometimes the best we can do is to find an orthonormal basis in the sense that any element in H can be approximated arbitrarily close in the NORM by a finite linear combination of this basis elements.</p>
<p>So then does that mean we can't expect that every x in H ... | Federico | 128,293 | <p>If $\{e_k\}$ is an orthonormal basis of your Hilbert space $H$, then every $x\in H$ can be expressed in the form:</p>
<p>\begin{equation}
x=\sum_{k} a_k e_k
\end{equation}</p>
<p>with, as you say, the coefficients $a_k$ in the field (most usually $\mathbb C$). It is important to note that the choice of the coeffic... |
3,996,790 | <p>I just realized that there may be a case where L'Hopital's rule fails, specifically</p>
<p><span class="math-container">$$ \lim_{x \to \infty} \frac{e^x}{e^x}$$</span></p>
<p>which evaluates to an indeterminate form, specifically <span class="math-container">$\frac{\infty}{\infty}$</span>. Sure, we can cancel the <s... | Countable | 677,593 | <p>Just to add the precise statement to explain clearly what is going on:</p>
<p><span class="math-container">$\textbf{Theorem - L'Hospital's Rule:}$</span> Let <span class="math-container">$a,b\in [-\infty,\infty]$</span>. Suppose that f and g are differentiable real valued maps on <span class="math-container">$(a,b)$... |
556,664 | <p>For example, if $f$ is integrable on $[0,3]$, is it also integrable on $[1,2]$? I tried thinking of a counterexample but couldn't, since I've only learned what implies integrability but not what integrability implies.</p>
| kahen | 1,269 | <p>First note that if $a \leq a'$ and $b' \leq b$ and $g: [a,b] \to \mathbb R$ is $0$ outside of $[a',b']$, then $\int_a^b g(x)\, dx = \int_{a'}^{b'} g(x)\,dx$. This is easily shown using the definition of the Darboux (or Riemann, if you prefer that definition) integral using partitions (tagged partitions for the Riema... |
3,853,896 | <p>Do I consider the probability before drawing both cards or after?</p>
<p><strong>Question more clearly</strong>:
A single card is removed at random from a deck of <span class="math-container">$52$</span> cards. From the remainder we draw <span class="math-container">$2$</span> cards at random and find that they are ... | JMoravitz | 179,297 | <p>To emphasize, @Barry's answer is correct and is the easiest way to think of the answer.</p>
<p>Since that confuses people for whatever reason, a way to convince people of this is instead to approach directly via definitions.</p>
<p>Recall that <span class="math-container">$\Pr(A\mid B) = \dfrac{\Pr(A\cap B)}{\Pr(B)}... |
743,911 | <p>Consider $\frac{1}{T}\sum_{t=1}^{T}\max\{ 0,a_t\}$. Can we say whether this is greater or equal then $\max\{ 0,\frac{1}{T}\sum_{t=1}^{T}a_t\}$?</p>
| leonbloy | 312 | <p>The <a href="http://en.wikipedia.org/wiki/Jensen%27s_inequality#Finite_form" rel="nofollow">Jensen inequality</a> states that, if $f(x)$ is convex, then</p>
<p>$$ \frac{\sum_{t=1}^{T}f(x_t)}{T} \ge f\left(\frac{\sum_{t=1}^{T}x_t}{T}\right) $$</p>
<p>Now, $f(x)=\max(0,x)$ is convex. Hence</p>
<p>$$\frac{1}{T}\s... |
191,187 | <p>Here I have a system as follows:</p>
<p><span class="math-container">$$\frac{dx}{dt}=a(by-x);
\frac{dy}{dt}=rx-xz;
\frac{dz}{dt}=(xy)^n-bz$$</span></p>
<p>Here <span class="math-container">$x, y$</span> and <span class="math-container">$z$</span> are positive real variables. All the parameters <span class="math-co... | zhk | 8,538 | <pre><code>a = 1; b = 1; r = 1;
pf = ParametricNDSolveValue[{x'[t] == a*(b*y[t] - x[t]),
y'[t] == r*x[t] - x[t]*z[t], z'[t] == (x[t]*y[t])^n - b*z[t],
x[0] == x0, y[0] == x0, z[0] == x0}, {x[t], y[t], z[t]}, {t,
20}, {x0, n}];
Manipulate[ParametricPlot3D[pf[x0, n], {t, 0, 20}], {{x0, -1, "x0"}, -2, 2},... |
191,187 | <p>Here I have a system as follows:</p>
<p><span class="math-container">$$\frac{dx}{dt}=a(by-x);
\frac{dy}{dt}=rx-xz;
\frac{dz}{dt}=(xy)^n-bz$$</span></p>
<p>Here <span class="math-container">$x, y$</span> and <span class="math-container">$z$</span> are positive real variables. All the parameters <span class="math-co... | Alex Trounev | 58,388 | <p>3D phase portrait can be built by analogy with 2D using <code>SliceVectorPlot3D[]</code></p>
<pre><code>p[s1_, a1_, b1_, r1_, n1_] :=
Block[{s = s1, a = a1, b = b1, r = r1, n = n1},
v3D = {a*(b*y - x), r*x - x*z, (x*y)^n - b*z};
SliceVectorPlot3D[v3D/Norm[v3D],
s, {x, -10, 10}, {y, -10, 10}, {z, -10, 10... |
1,782,423 | <p>Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem that is "true" (in the sense that it holds in the intersection of all models of $F$) but unprovable in $F$.</p>
<p>H... | Rob Arthan | 23,171 | <p>Given a sentence $\phi$ that holds in all models of a first-order theory $F$, you <strong>can</strong> effectively find a proof of $\phi$: just enumerate all proofs in $F$ until you find the one that proves $\phi$ (there is such a proof, since you are given that $\phi$ holds in all models of $F$ and hence, by the co... |
2,511,061 | <p>I will prove 0=1.
we know that, From Factorial definition
zero factorial is equal to one and
one factorial is equal to one.
so,0!=1!.
factorial get cancelled both sides,
we get 0=1.
Is this right..?</p>
| Xander Henderson | 468,350 | <p>The symbol "!" denotes a function that takes as input nonnegative integers, and has an output defined by a recurrence relation. Perhaps it might be easier to replace a "!" on the right with something that looks more like traditional functional notation:
$$ \operatorname{fact} : \mathbb{Z}_{\ge 0} \to \mathbb{R}
\qq... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| David Wheeler | 23,285 | <p>You may have noticed multiplying matrices can be hard. By changing bases, we can make it easier.</p>
|
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Community | -1 | <p>Linear algebra is all about basis-change. Matrices <em>are</em> change of basis machines !</p>
<p>When you form a linear combination $a\vec u+b\vec v+\cdots c\vec w$, you can see it as the vector $(a,b,\cdots c)$ expressed in the basis $(\vec u,\vec v,\cdots \vec w)$.</p>
<p>For the same reason, a matrix product $... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Benjamin Lindqvist | 96,816 | <p>Note that the Fourier transform is simply a change of basis. It is entirely possible you're actually reading <em>this very answer</em> thanks to change of basis. Check out <a href="https://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing" rel="noreferrer">https://en.wikipedia.org/wiki/Orthogonal_freq... |
219,782 | <p>Hi I'm wondering if there's some workaround to get Mathematica to use time-shifting identities for Laplace and Inverse Laplace transforms. My examples are below.</p>
<pre><code>$Assumptions = Element[{t, τ}, PositiveReals];
LaplaceTransform[f[t -τ]HeavisideTheta[t-τ], t, s]
(* Output *)
(* LaplaceTransform[f[t -τ]... | imas145 | 61,387 | <p>First off, the code you posted won't even parse properly in Mathematica. Second, built-in Mathematica functions are capitalized and the function pattern in general uses brackets <code>[</code> and <code>]</code> instead of parenthesis, so use <code>Sin[x]</code> instead of <code>sin(x)</code>. Similarly, <span class... |
232,825 | <p>I have to read and process a file that looks like this:</p>
<pre><code> MASSA TMASS1
uscita da elettrodi:
0 1 0.56705 -19.98160 2.80000 -0.87939 0.66823 -0.63034 0.39513
ingresso tmass1:
0 1 0.56705 -19.13351 2.00000 -0.37791 0.66823 -0.63034 0.39513
MAS... | xzczd | 1,871 | <p>You need <code>EmpiricalDistribution</code>:</p>
<pre><code>data = {14, 15, 16, 22, 24, 25};
weights = {1, 1, 3, 2, 2, 5};
\[ScriptCapitalD] = EmpiricalDistribution[weights -> data];
Expectation[x, x \[Distributed] \[ScriptCapitalD]]
(* 21 *)
</code></pre>
|
24,660 | <p>Pro's and cons of number line model vs color counter model</p>
<p>When teaching multiplication to elementary schoolers, the "number line model" and "color counter model" are both widely used techniques. Can somebody help me to understand some of the pro's and cons for either model? Thank you</p>
| Jasper | 667 | <p>As someone who never heard of the color counter model, I find it overly complicated.</p>
<p>I assume that the question is asked in the context of signs when multiplying integers. While both methods arrive at the correct results, I can't see any advantages of the color counter approach.</p>
<p>I'd rather fear that th... |
2,386,471 | <p>I have a solved question from Ross as stated below.</p>
<blockquote>
<p>Q : Suppose that each of three men at a party throws his hat into the
center of the room. The hats are first mixed up and then each man
randomly selects a hat. What is the probability that none of the three
men selects his own hat?</p>
... | Fred | 380,717 | <p>There is a Basis $C_1,C_2,..., C_{n^2}$ of $M_{n \times n}$ such that each $C_j$ is singular. For the identity matrix $I$ we have</p>
<p>$I=\sum_{k=0}^{n^2}s_kC_k$ for some scalars $s_k$.</p>
<p>Then $A=\sum_{k=0}^{n^2}s_kAC_k=0$</p>
<p>(symmetry was not needed)</p>
|
365,166 | <p>I have trouble coming up with combinatorial proofs. How would you justify this equality?</p>
<p><span class="math-container">$$
n\binom {n-1}{k-1} = k \binom nk
$$</span>
where <span class="math-container">$n$</span> is a positive integer and <span class="math-container">$k$</span> is an integer.</p>
| Ross Millikan | 1,827 | <p>I would write the expression for the binomial coefficients in terms of factorials and notice what happens.</p>
|
287,405 | <p>Hi have this sequence:</p>
<p>$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}$$</p>
<p>I understand that this is a <em>Geometric series</em> so this is what I've made to get the sum.
$$\sum\limits_{n=1}^\infty (-1)^n\frac{3^{n}\cdot 3^{-2}}{4^n}$$
$$\sum\limits_{n=1}^\infty (-1)^n\cdot 3^{-2}{(\frac{3}{4})}^n... | Thomas | 26,188 | <p>You have</p>
<p>$$\begin{align}
\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}
&= \sum_{n=1}^{\infty} \frac{(-1)(-1)^{n-1}\frac{1}{3}3^{n-1}}{4\cdot 4^{n-1}} \\
&= \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\frac{(-1)^{n-1}3^{n-1}}{4^{n-1}}\\
&= \sum_{n=1}^{\infty} \frac{-1}{12}\left(\frac{-3}{4}\righ... |
287,405 | <p>Hi have this sequence:</p>
<p>$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}$$</p>
<p>I understand that this is a <em>Geometric series</em> so this is what I've made to get the sum.
$$\sum\limits_{n=1}^\infty (-1)^n\frac{3^{n}\cdot 3^{-2}}{4^n}$$
$$\sum\limits_{n=1}^\infty (-1)^n\cdot 3^{-2}{(\frac{3}{4})}^n... | Adi Dani | 12,848 | <p>$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}=\sum\limits_{n=1}^\infty (-1)^n\frac{1}{9}\left(\frac{3}{4}\right)^n=\frac{1}{9}\sum\limits_{n=1}^\infty \left(-\frac{3}{4}\right)^n=
\frac{1}{9}\left(\frac{1}{1+3/4}-1\right)$$</p>
|
2,614,375 | <p>I have tried using prime factorization which is $2^2 \times 5^2 \times 11$ and found out that I'll need $37$ slices. but in the question paper which is a multiple choice question there's no such answer. the choices are $30,31,32,61,110$.</p>
| Axel Kemper | 58,610 | <p>To split the cube into $10 \times 10 \times 11$ equal parts as suggested by @bof, you need $9 + 9 + 10 = 28$ cuts. This separates the original cube in $10 + 10 + 11 = 31$ slices.</p>
<p>However, you can bring down this number to $5 + 4 + 4 = 13$ cuts by stacking the intermediate slabs and thus yielding more parts ... |
613,836 | <blockquote>
<p>Let $x,y,z$ be integers and $11$ divides $7x+2y-5z$. Show that $11$
divides $3x-7y+12z$.</p>
</blockquote>
<p>I know a method to solve this problem which is to write into $A(7x+2y-5z)+11(B)=C(3x-7y+12z)$, where A is any integer, B is any integer expression, and C is any integer coprime with $11$.</... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Scale the first sum so that, mod $11,\,$ its $\,y$ coefficient $\,\color{#0a0}2\,$ becomes that of the second sum, i.e. $\,\color{#c00}{-7\equiv 4}.\,$ So we need to multiply by $\rm\,\color{orange}{by\ 2}\,$ to scale from $\,\color{#0a0}2$ to $\,\color{#c00}4.\,$ Doing so we obtain</p>
<... |
2,091,251 | <p>I want to be able to check if a function f is even, odd, or neither using Maple's symbolic math. Unfortunately I don't get a boolean return of 'true' on a function I know is even.</p>
<pre><code>g:=abs(x)/x^2
evalb(g(x)=g(-x))
false
</code></pre>
<p>Since my function is ev... | saforrest | 266,106 | <p>All the right ideas are already in other answers/comments, but I'll post this as an answer to hopefully provide some more detail.</p>
<p>The way people normally use 'function' in a programming context is a construct which takes some number of parameters and executes one or more statements making use of these parame... |
77,246 | <p>What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
For example, is there any kind homology of orbifolds such that the first homology group is the abelianization of the ... | Michael Joyce | 16,002 | <p>I do not know the current state of the art, but I can point you to the paper of <a href="https://arxiv.org/abs/math/0004129" rel="nofollow noreferrer">Chen and Ruan</a> which should have everything you are asking for.</p>
|
77,246 | <p>What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
For example, is there any kind homology of orbifolds such that the first homology group is the abelianization of the ... | john mangual | 1,358 | <p>A math definition of orbifold Euler characteristic appears in <a href="https://eudml.org/doc/164637" rel="nofollow noreferrer">On the Euler Number of an Orbifold</a>. If there is a group acting on a manifold <span class="math-container">$X / G$</span> we could try to write
\[ \chi(X/G) = \frac{1}{|G|} \sum_{g \in G}... |
2,130,823 | <blockquote>
<p>Is it possible for some integer $n>1$ that $2^n-1\mid 3^n-1$ ?</p>
</blockquote>
<p>I have tried many things, but nothing worked.</p>
| B. S. | 231,386 | <p>When <span class="math-container">$2^n-1$</span> is a Mersenne prime,this can be resolved ( although this isn't very helpful, because we only know of 49 Mersenne primes and we don't know if they are finitely many.However, it sure is nice to know that <span class="math-container">$ 2^{74,207,281} − 1$</span> does not... |
4,549,781 | <p>suppose we have a convex continuous function from<span class="math-container">$\mathbb{R}$</span> to <span class="math-container">$\mathbb{R^+}$</span>
<span class="math-container">$f:\mathbb{R}\to\mathbb{R^+}$</span> such that<br />
<span class="math-container">$f(-x)=f(x),lim_{x\to 0}\frac{f(x)}{x}=0 \text{ and } ... | Gregory | 197,701 | <p><a href="https://en.wikipedia.org/wiki/Gershgorin_circle_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Gershgorin_circle_theorem</a> may help you here.
Note that it implies that eigenvalues <span class="math-container">$\lambda$</span> satisfy
<span class="math-container">$$|\lambda - a_{ii}| \le ... |
269,696 | <p>I am given that $\sum\limits_{n=1}^\infty a_n$ is convergent. </p>
<p>I need to determine whether $\sum\limits_{n=1}^\infty (a_n)^\frac{1}{3}\;$ and $\;\sum\limits_{n=1}^\infty (a_n)^2\;$ are also convergent.</p>
<p>Imagine that $a_n = \dfrac{1}{n^4}.\;$ I believe that this is convergent because it's converging to... | Nameless | 28,087 | <p>Youu want to decide whether or not the statements are true for any sequence $(a_n)$. So you either prove them, or provide a counterexample.</p>
<p>What you have done is wrong on many counts. You took a sequence $a_n$ and said (I believe) $\sum_{n=1}^{\infty}a_n$ converges because $a_n\to 0$. This is wrong as the ha... |
2,649,557 | <p>I've been studying topology recently, and I've gotten to the part of the book that deals with quotient spaces. For the most part, it's fairly clear, but one thing that has been confusing me a bit is how the unit circle is represented.</p>
<p>Sometimes $\mathbf S^1$ is denoted as $\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)... | user284331 | 284,331 | <p>$z$ as a complex has the form $z=x+yi$ for real numbers $x,y$ and $z$ is identified with $(x,y)$ as a coordinate in ${\bf{R}}^{2}$. And the topology in ${\bf{C}}$ is the Euclidean topology for ${\bf{R}}^{2}$. So if $w=(a,b)$, then $|z-w|^{2}=(x-a)^{2}+(y-b)^{2}$, and hence $|z-w|=1$ if and only if $(x-a)^{2}+(y-b)^{... |
1,649,997 | <p>I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?</p>
| cpiegore | 268,070 | <p>Take $a_n = f(n)$ where $f(n)$ is a monotone function and let $b_n = f(n) + \frac{1}{n^m}$ for any $m > 0$.</p>
|
2,468,781 | <p>The law of Universal Generalization states that: </p>
<p>P(c)</p>
<p>(x) P(x) </p>
<p>Now, I understand that this works only if c is <em>any random element</em> from the universe. Such arbitrary selection makes this rule mathematically valid. However, I do not understand how it holds true in practical examples. <... | 5xum | 112,884 | <p>It's not "I pick a random $c$ and if it's true for $c$, then it's true for all $x$"</p>
<p>It's "If I know it's true for $c$ even if I don't know which $c$ I have, then it's true for all $x$".</p>
<hr>
<p>In other words, your example should be:</p>
<blockquote>
<p>If you tell me you will give me a random numbe... |
126,897 | <p>It is an exercise in a book on discrete mathematics.How to prove that in the decimal expansion of the quotient of two integers, eventually some block of digits repeats.
For example:
$\frac { 1 }{ 6 } =0.166\dot { 6 } \ldots$ and $\frac { 217 }{ 660 } =0.328787\dot { 8 } \dot { 7 } \ldots$</p>
<p>How to think of thi... | Raymond Manzoni | 21,783 | <p>Let's proceed to the actual division :</p>
<p>$
\begin{array} {r|l}
\boxed{217}\hphantom{000\;} & 660\\
\hline
2170\hphantom{000} & 0.3287\\
-1980\hphantom{000} & \\
\boxed{190}\hphantom{00\;} & \\
1900\hphantom{00} & \\
-1320\hphantom{00} & \\
\boxed{580}\hphantom{0\;} &... |
294,895 | <p>Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces.</p>
<p>When is $\operatorname{Ran}(T)$ a closed subspace?</p>
<p>What theorems are there? </p>
<p>Thanks :) </p>
| Davide Giraudo | 9,849 | <p>Consider $X=Y:=\ell^\infty$, the normed space of bounded sequences. Let $T(x)(n):=\frac{x(n)}{n^2}$; this gives a linear continuous operator. Let $y\in T(X)$, then $\{n^2y(n)\}$ is bounded, and the converse holds. So
$$T(X)=\{y,\sup_n|n^2y(n)|<\infty\}.$$
Let $x^{(n)}:=\sum_{j=1}^nj^{-1}e(j)$; it converges in $\... |
385,404 | <p>So, I have this question which is still troubling me:</p>
<blockquote>
<p>Find the value of $k$ such that the equation $2x^3 + 3x^2 + kx - 48 = 0$ has two solutions equal in value but opposite in sign.</p>
</blockquote>
<p>I've had numerous attempts at this, such as using simultaneous equations and the factor th... | Easy | 60,079 | <p>Let $a$ be one of the pair roots. Then $$2a^3 + 3a^2 + ka - 48 =0=2(-a)^3 + 3(-a)^2 + k(-a) - 48\Rightarrow(2a^2+k)a=0\Rightarrow k=-2a^2$$
Substitute back this to the equation one gets $$2a^3+3a^2+(-2a^2)a-48=0$$which gives $a^2=16\Rightarrow k=-2a^2=-32$.</p>
|
375,910 | <p>Okay so basically I want to know if you can solve this log equation without the use of u substitution: </p>
<p>$${\log_4{\log_3{x}}} = 1$$</p>
<p>I believe that u substitution is the only way to solve this problem, but please prove me wrong if theres another way to do so.</p>
| iostream007 | 76,954 | <p>There is basic identity of log is: $\log_ab=n\implies a^n=b $</p>
<p>so in your question : $\log_4\log_3x=1\,$, we put $\log_3x=u$</p>
<p>$$\log_4u=1\implies u=4^1\implies u=4\implies\log_3x=4\implies x=3^4\implies x=81$$ </p>
|
384,471 | <p>I'm attempting to evaluate the limit</p>
<p>$\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}$</p>
<p>I got it reduced to the following</p>
<p>$\lim_{x\rightarrow\infty}\frac{\sqrt{\frac{1}{\left(x-2\right)^{2}}-\frac{3}{\left(x-2\right)^{4}}}+1}{1-\frac{3}{\left(x-2\right)^{2}}-1}$</p>
<p>But putting in... | lab bhattacharjee | 33,337 | <p>$$F=\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}$$</p>
<p>$$=\lim_{x\rightarrow\infty}\frac{1}{\sqrt{(x-2)^2-3}-x+2}$$</p>
<p>Let us put $x-2=\sqrt3\csc2\theta$</p>
<p>$x\to\infty\implies \theta\to0$</p>
<p>So, $$F=\lim_{\theta\to0}\frac1{\sqrt3\cot2\theta-(\sqrt3\csc2\theta+2)+2}$$</p>
<p>$$=-\frac... |
176,054 | <p>I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
Since I don't know the answer, I thought mathoverflow would be the right place to ask.
I mean, there are many areas o... | André Henriques | 5,690 | <p>My wife is a neuroscientist. I can tell you what she uses:</p>
<p>$\bullet$ a LOT of statistics.<br>
$\bullet$ signal processing (such as wavelet transform).<br>
$\bullet$ some baysian probability theory.</p>
|
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | Agustí Roig | 664 | <p>There are beautiful video lessons at <a href="http://ocw.mit.edu/index.htm">MIT Opencourseware</a>. I'm particularly in love with <a href="http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-19-introduction-to-the-laplace-transform/">this</a> presentation of the Lap... |
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | Ron Gordon | 53,268 | <p>A Laplace transform is useful for turning (constant coefficient) ordinary differential equations into algebraic equations, and partial differential equations into ordinary differential equations (though I rarely see these daisy chained together). </p>
<p>Let's say that you have an ordinary DE of the form</p>
<p>$... |
2,292,711 | <p>Show that the vector field $F(x,y)=(2x-x^5-xy^4,y-y^3-x^2y)$ defined in $R^2$ does not have periodic orbits; the Bendixson criterion is not useful.</p>
| Robert Israel | 8,508 | <p>Since the $x$ and $y$ axes are invariant, any periodic orbit must be in one of the four quadrants. A periodic orbit must have a stationary point in its interior, but every stationary point is on an axis.</p>
|
3,338,597 | <p>I am struggling to find the cases for which <span class="math-container">$I(X;Y|Z)>I(X;Y)$</span>. The only mathematical example I could find for such a case is the following:
<span class="math-container">$$
I(X;Y) + I(X;Z|Y) = I(X;Z) + I(X;Y|Z).
$$</span>
This makes sense since they are both definitions of <sp... | leonbloy | 312 | <p>The classical example is : let <span class="math-container">$X,Y$</span> be independent fair Bernoulli variables (take values <span class="math-container">$\{0,1\}$</span> with equal probability), and let <span class="math-container">$Z=X+Y \pmod 2$</span> (in boolean logic: <span class="math-container">$Z= X \oplus... |
3,338,597 | <p>I am struggling to find the cases for which <span class="math-container">$I(X;Y|Z)>I(X;Y)$</span>. The only mathematical example I could find for such a case is the following:
<span class="math-container">$$
I(X;Y) + I(X;Z|Y) = I(X;Z) + I(X;Y|Z).
$$</span>
This makes sense since they are both definitions of <sp... | Cesare | 388,998 | <p>In terms of interpretation, <span class="math-container">$I(X;Y|Z)>I(X;Y)$</span> is an indication that at least to some degree <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> convey synergistic information about <span class="math-container">$Z$</span> (even though a certain de... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | roy smith | 9,449 | <p>Euclid's elements. i find it much more useful than Klein's books, but that may mean i misunderstand the question. indeed after many years of perusing them, i find Klein's "from an advanced standpoint" books more of a polemic than a useful text. Euclid on the other hand introduces many of the main ideas of modern ... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Fred Daniel Kline | 16,888 | <p><a href="http://rads.stackoverflow.com/amzn/click/0716750473" rel="nofollow">Mathematics The Science of Patterns</a> by Keith Devlin.</p>
|
269,769 | <p>Is the following true always for a matrix norm </p>
<p>$$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$</p>
<p>Related to this
given $r$ is positive constant, $H$ is symmetric positive definite
is the following true :</p>
<p>$$\lVert (rI - H)(rI + H)^{-1}\rVert < 1 $$</p>
<p>or</p... | Hagen von Eitzen | 39,174 | <p>An <a href="http://en.wikipedia.org/wiki/Matrix_norm" rel="nofollow">online encyclopedia</a> states hat "some (but not all) matrix norms satisfy" this inequality, though some authors include this as part of the definition. SO it depends.</p>
|
1,858,218 | <p>I often come across students who are confused by the idea that the complex unit, $i$, is defined as $i^2 = -1$. Since we are using the complex numbers in an engineering course, we use the complex numbers to encode information about a sinusoid of a given frequency, where the argument is the phase of the sinusoid wit... | Peter G. Chang | 339,525 | <p>Well, for one thing any one of $x=1,-1,i,-i$ satisfies $x^5 = x$, so $x^5 = x$ cannot be the definition of $i$ unless you append the condition that $k = 5$ is the smallest positive integer power greater than $1$ such that $x^k = x$.</p>
|
4,412,175 | <p>I wonder and tried to google it, but I am not sure what to google it, how to solve non linear equations where equations are equal between each other. I am able to write a specific algorithm for two equations but not dynamically for N equations. I will show the example of three (how my equations approximately looks l... | Glorious Nathalie | 948,761 | <p>From <span class="math-container">$C_1 + C_2 + C_3 = 1 $</span>, you get <span class="math-container">$C_3 = 1 - C_1 - C_2 $</span></p>
<p>Substitute that in your equations, you get</p>
<p><span class="math-container">$\dfrac{ 4 C_1 + 8 C_2 + 16 (1 - C_1 - C_2) }{2 C_1} = \dfrac{ -12 C_1 - 8 C_2 + 16} {2 C_1} = \dfr... |
907,154 | <p>Renata walks down an escalator that moves up and counts <span class="math-container">$150$</span> steps. Her sister Fernanda climbs the same escalator and counts <span class="math-container">$75$</span> steps. If the speed of Renata (in steps per time unit) is three times the speed of Fernanda, determine how many st... | Ashitaka | 171,300 | <p>Let ${ t }_{ 1 }$ and ${ t }_{ 2 }$ be the time that Renata and Fernanda, respectively, need to complete the trip. And $x$ the number of visible steps.</p>
<p>Considering Renata's trip and knowing that ${ v }_{ r }=3{ v }_{ f }$</p>
<h1>$3v_{ f }-v_{ e }=\frac { x }{ t_{ 1 } } \\ v_{ f }=\frac { 150 }{ { t }_{ 1 ... |
199,549 | <p>By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in_M) \models ZF$. Is the following "converse" true?</p>
<blockquote>
<p>In every model $V$ of $ZF$ and every transitive set $M \in V$ such that $(M,\in_M) \mod... | Nate Ackerman | 8,106 | <p>The answer to your question is no. For example, suppose $\kappa$ is inaccessible and $M = V_{\kappa}$. Then $M$ is a model of $\sf ZF$ but $M$ cannot be countable in $N$ as it is not countable in $V$.</p>
<p>Of course there will always be a set generic extension of $V$ containing such an $N$ (just generically add a... |
199,549 | <p>By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in_M) \models ZF$. Is the following "converse" true?</p>
<blockquote>
<p>In every model $V$ of $ZF$ and every transitive set $M \in V$ such that $(M,\in_M) \mod... | Joel David Hamkins | 1,946 | <p>Let me point out that a close variant of the question has a
positive answer. Namely,</p>
<p><strong>Theorem.</strong> Every model of set theory $M\models\text{ZFC}$ has an
elementary extension $M\prec M^*$ such that there is a model
$N\models\text{ZFC}$ inside of which $\langle
M^*,\in^{M^*}\rangle$ is a countable ... |
3,866,285 | <p>Suppose I'd like to find the coefficient of <span class="math-container">$x^{l}$</span> in the expansion of <span class="math-container">$(1+x+x^{2}+...+x^{n})^{m}$</span>, where <span class="math-container">$n$</span> and <span class="math-container">$m$</span> are given positive integers, for some given integer <... | user158293 | 158,293 | <p>For n=1 the answer is obvious so the obvious assumptions are n>1,m>1. For sufficiently small m or n you could likely write a reasonable formula but for general m,n I am not sure of a 'straightforward' or 'closed form' answer, depending upon exactly what one would classify as being in one of those 2 categories.... |
3,005,965 | <p>I was making use of polynomial long division in inverse Z transform and I got stuck in a brainfart in one stage of the polynomial long division.</p>
<p>I posted the original question into digital signal processing stack exchange, but nobody answered it so I thought about sharing the link to math stack exchange.</p>... | Henno Brandsma | 4,280 | <p>The long division algorithm for dividing <span class="math-container">$a(z)$</span> by <span class="math-container">$b(z)$</span> is meant to find
the unique polynomials <span class="math-container">$q(z)$</span> and <span class="math-container">$r(z)$</span> such that</p>
<p><span class="math-container">$$a(x) = q... |
3,915,141 | <p>Let <span class="math-container">$(G,.)$</span> be a group with <span class="math-container">$e$</span> as an identity element. If <span class="math-container">$x\in G$</span> and <span class="math-container">$C_G(x)=\left\{g \in G: gx=xg\right\}$</span> and <span class="math-container">$|x|=5$</span>. In the proces... | Nicky Hekster | 9,605 | <p>In general: if <span class="math-container">$x \in G$</span> and <span class="math-container">$o(x)=n$</span>, then for every non-zero integer <span class="math-container">$k$</span> with gcd<span class="math-container">$(k,n)=1$</span> it holds that <span class="math-container">$C_G(x)=C_G(x^k)$</span>.<br>
<strong... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Andrey Rekalo | 5,371 | <p><a href="https://en.wikipedia.org/wiki/David_Hilbert#The_finiteness_theorem" rel="nofollow noreferrer">Hilbert's finiteness theorem</a>, which arguably "killed classical invariant theory" and resulted in the creation of abstract algebra.</p>
<blockquote>
<p>Hilbert's first work on invariant functions led h... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Joel David Hamkins | 1,946 | <p>Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.</p>
<p>With induction, one uses a comparatively abstract understanding of how a property propagates from smaller instances to larger instances, in order to ... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | John Stillwell | 1,587 | <p>An example of a slightly different kind -- not eliminating all calculation, but
showing that "all calculations are easy" -- is <a href="https://en.wikipedia.org/wiki/Small_cancellation_theory" rel="nofollow noreferrer">Dehn's algorithm</a> in
combinatorial group theory. Dehn showed, using the combinatorics... |
260,037 | <p>Problem: </p>
<p>How many triangles do $m$ lines form if </p>
<p>a) Every two lines intersect and no three lines intersect at one point.</p>
<p>b) There are $n$ lines among $m$ lines that are parallel to each other. No other line is parallel to these $n$ lines, and no other two lines are parallel to each other. ... | ashley | 50,188 | <p>Take one of these lines. Ever pair of lines intersecting it is forming a triangle-- combination of $m-1$ to $2$. </p>
<p>Do this for every line-- excluding the ones you already worked on. So the second line you are looking at has $m-2$ to $2$ additional triangles formed by itself and the ones intersecting it, ... .... |
260,037 | <p>Problem: </p>
<p>How many triangles do $m$ lines form if </p>
<p>a) Every two lines intersect and no three lines intersect at one point.</p>
<p>b) There are $n$ lines among $m$ lines that are parallel to each other. No other line is parallel to these $n$ lines, and no other two lines are parallel to each other. ... | Ross Millikan | 1,827 | <p>There are $n$ parallel lines. Each pair of the $m-n$ forms a triangle with each of the $n$. So the triangles are ${m-n \choose 2}n=\frac 12(m-n)(m-n-1)n$</p>
|
1,414,316 | <p>I am trying to optimize distance from point to plane using Lagrange multiplier.</p>
<p>Usually for such problems you are given specific point like (1,2,3) in 3D, and then an exact plane which is just the subject of Lagrange. But what I have here doesn't specify values for point and plane.</p>
<p>It says problem ha... | Miha Habič | 9,440 | <p>To complement Asaf's answer, let me point out that you don't need any large cardinals at all to get embeddings like this; they exist as long as there is an uncountable transitive model of ZFC+"there are no inaccessible cardinals".</p>
<p>If $N$ is such an uncountable model we can get an embedding by taking a counta... |
1,818,976 | <p>Let there be many numbers $a_1,a_2,a_3,\dots,a_n$.</p>
<p>I want to find the first digit of their product, i.e. of $A=a_1\times a_2\times a_3\times a_4\times \dots\times a_n$.</p>
<p>These numbers are huge and multiplying all of them exceeds the time limit.</p>
<p>Is there any shortcut to find the most significan... | Tsemo Aristide | 280,301 | <p>Hint: The set $R$ of the remainders of the division of $2\pi nr, n\in N$ by $2\pi$ is dense in $[0,2\pi]$. Remark that $R$ is $2\pi$ $\{$ $rn-\lfloor rn \rfloor$, $n\in N\}$.</p>
|
238,128 | <p>Let $G$ be an abelian group. <br/>
Show that $\{x\in{G} | |x| < \infty\}$ is a subgroup of $G$. Give an example of a non-abelian group where this fails to be a subgroup.</p>
| Thomas | 26,188 | <p>I assume that by $\lvert g \lvert$ you mean the order of $g$, so that the subset you are considering is the subset of all elements with finite order. Call this subset $X$.</p>
<p>Without actually doing the problem for you, here are a few things that might help.</p>
<p>To check that a subset of a group is a subgrou... |
1,336,381 | <p>Find the sum of solutions to:</p>
<p>$$ 2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0 $$</p>
<p>I'm not sure about what to do with the absolute values, how can I get rid of them?</p>
<p>Should I solve for all various cases depending on the sign of $x+1$ and $x-1$?</p>
| Ian | 83,396 | <p>It is not true. The classic example is in a slightly different setup: $f_n(x)=x^n$ converges pointwise to zero on $[0,1)$, uniformly on $[0,1-\delta]$ for every $\delta \in (0,1)$, but it does not converge uniformly on all of $[0,1)$. If it did, then one could interchange limits and conclude that $\lim_{n \to \infty... |
344,166 | <p>I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of elementary(no measures) probability. The book is rigorous "hard" math but doesn't escape from giving a solid intuiti... | Community | -1 | <p>I highly recommend Artin's <em><a href="http://rads.stackoverflow.com/amzn/click/0130047635" rel="nofollow">Algebra</a></em>. In my opinion, it fits your criteria nicely; topic introduction, concrete example, and then thought provoking discussions that pique your interest. The exercises are fantastic.</p>
<p>If Art... |
4,205,906 | <p>Since X and Y are independent and uniform I have the joint density function f(x,y)=1 but Im not sure where to go from there. I keep getting that the answer is f(z)=1 but this doesnt make sense since the range of z is from 0 to infinity.</p>
<p>So if I make the substitution Z=Y/X and W=X, I get Y=ZW and X=W. The 4 ... | Vons | 274,987 | <p>The p-value is the smallest level of significance <span class="math-container">$\alpha_0$</span> that I would reject the null hypothesis with the observed data.</p>
<p>In olden times people simply reported a result, "rejected the null" or "did not reject the null." This doesn't tell me how close ... |
1,630,655 | <p>Not sure what to do / how to start this... I have equcation of 504 is: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$</p>
| Justpassingby | 293,332 | <p>Hint: decompose the polynomial $n^9-n^3$ into linear and quadratic irreducible factors. Observe divisibility for $2^3,$ $3^2$ and $7$ separately.</p>
|
1,630,655 | <p>Not sure what to do / how to start this... I have equcation of 504 is: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$</p>
| robjohn | 13,854 | <p>$$
\begin{align}
&\frac{n^9-n^3}{504}\\
&=\small720\binom{n}{9}+2880\binom{n}{8}+4620\binom{n}{7}+3780\binom{n}{6}+1655\binom{n}{5}+370\binom{n}{4}+36\binom{n}{3}+\binom{n}{2}
\end{align}
$$</p>
|
322,858 | <p>Let <span class="math-container">$G$</span> be a split semisimple real Lie group in characteristic zero, and let <span class="math-container">$B=TU$</span> be a Borel subgroup with unipotent radical <span class="math-container">$U$</span> and Levi <span class="math-container">$T$</span>. Fix an ordering on the root... | Peter McNamara | 425 | <p>SGA III, Expose XXIII, Section 3.4.</p>
|
2,579,156 | <p>I found the solution of series on Wolfram Alpha
<a href="http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n</a></p>
<p>$ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\righ... | omegadot | 128,913 | <p>We will make use of the following <a href="https://en.wikipedia.org/wiki/Harmonic_number#Hyperharmonic_numbers" rel="nofollow noreferrer">integral representation</a> for the harmonic numbers
$$H_x = \int^1_0 \frac{1 - t^x}{1 - t} \, dt. \tag1$$</p>
<p>Let
$$S = \sum^n_{k = 1} \frac{1}{(2k + 1)(2k + 2)} = \sum^n_{k ... |
424,404 | <p>I want to create an algorithm/calculation that helps me figure out if the price on a used vehicle is high or low.</p>
<p>My thoughts are that to calculate this i need a critical mass of previous vehicle sales with the same characteristics; manufacturer, model, year and trim. Having this data I must figure out each ... | Ryan Walch | 62,355 | <p>You may just want to run a <a href="http://en.wikipedia.org/wiki/Ordinary_least_squares" rel="nofollow">regression</a> on price relative to the values you care about such as millage. You can then use the equation you estimate to see if the predicted value of the car you are looking at is larger or smaller to the pr... |
2,807,356 | <blockquote>
<p><strong>If $z_1,z_2$ are two complex numbers such that $\vert
z_1+z_2\vert=\vert z_1\vert+\vert z_2\vert$,then it is necessary that</strong> </p>
<p>$1)$$z_1=z_2$</p>
<p>$2)$$z_2=0$</p>
<p>$3)$$z_1=\lambda z_2$for some real number $\lambda.$</p>
<p>$4)$$z_1z_2=0$ or $z_1=\lambda z... | user376343 | 376,343 | <p>In general, the numbers <span class="math-container">$z_1,\,z_2,\,z_1+z_2$</span> represent vectors <span class="math-container">$\vec{u},\,\vec{v},\,\vec{u}+\vec{v}$</span> starting from <span class="math-container">$O.$</span> Thus is <span class="math-container">$|z_1+z_2|$</span> the length of a diagonal in the... |
191,984 | <p>In this context composition series means the same thing as defined <a href="http://en.wikipedia.org/wiki/Composition_series#For_groups" rel="noreferrer">here.</a></p>
<p>As the title says given a finite group <span class="math-container">$G$</span> and <span class="math-container">$H \unlhd G$</span> I would like to... | Community | -1 | <p>Suppose you have $A \triangleleft C$ with $C/A$ not simple. Then there is some nontrivial subgroup $B/A \triangleleft C/A$ (by Lattice Isomorphism Theorem), so you can extend $A \triangleleft C$ to $A \triangleleft B \triangleleft C$. By repeating this step you yield a composition series (it is clearly bounded, and ... |
2,662,554 | <p>I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even. </p>
<p>Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$. </p>
<p>What I have done so far:</p>
<p>\begin{align}
& n + 1 = (2k+1)^2 - 2(2k+1) + 7 \\
\implies &am... | Community | -1 | <blockquote>
<p>Prove that $n^2-2n+7$ is even <em>then</em> $n+1$ is even.</p>
</blockquote>
<p>If $n^2-2n+7$ is even, then $n^2-2n$ is odd.</p>
<p>For all even numbers, this does not work since $(even)^2-2(even)$ always results in an even number.</p>
<p>Therefore, the number $n$ <strong>must</strong> be an odd nu... |
1,279,873 | <p>Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all pairs of delays being equally likely. The first to arrive will wait for 15 minutes and will leave if the other has not yet arrived. What is the probability that they will meet?</p>
<p>... | kilpikonna | 191,907 | <p>I would solve it like this (I'm sorry, but I'm not good at drawing pictures in LaTeX, so I've made it by my hand, hope it helps).</p>
<p>So let $x$ be the time whem Romeo arrives. He can arrive at any time between <strong>0</strong> and <strong>1</strong>, let $y$ be the time when Julia comes. </p>
<p>So the point... |
1,279,873 | <p>Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all pairs of delays being equally likely. The first to arrive will wait for 15 minutes and will leave if the other has not yet arrived. What is the probability that they will meet?</p>
<p>... | Ray | 472,812 | <p>Let the point <span class="math-container">$(x,y)$</span> mean that romeo arrives at <span class="math-container">$x$</span> time, juliet arrives at <span class="math-container">$y$</span> time. The sample space is the unit square since <span class="math-container">$0 \le x,y \le 1$</span>.</p>
<p>Let <span class="m... |
2,396,561 | <blockquote>
<p>Solve the differential equation
$$x^2y''+xy'-y=\frac{x^2}{2+x}$$</p>
</blockquote>
<p>My attempt: put $x=e^z$ then $z=\log x$</p>
<p>and equation reduced to $\theta^2-1=\frac{e^{2z}}{2+e^z}$</p>
<p>so $y_c=c_1x+c_2\frac{1}{x}$ </p>
<p>how to find particular integral </p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may integrate by parts,
$$
\begin{align}
I_{n,a}&=\int_{-\infty}^\infty\frac{dx}{(1+x^2/a)^{n}}\qquad (n\ge1)
\\\\&=\int_{-\infty}^\infty 1 \cdot\frac{1}{(1+x^2/a)^{n}}\:dx
\\\\&=\left[ x\cdot \frac{1}{(1+x^2/a)^{n}}\right]_{-\infty}^\infty -\int_{-\infty}^\infty x\cdot \fra... |
535,533 | <p>My confusion is how do we define : $\sin (x)$ for $x\in \mathbb{R}$.</p>
<p>I only know that $\sin(x)$ is defined for degrees and radians..</p>
<p>Suddenly, I have seen what is $\sin (2)$.. </p>
<p>I have no idea how to interpret this when not much information is given what $2$ is... </p>
<p>does this mean $2$ r... | ncmathsadist | 4,154 | <p>To completely specify the sine function, you must specify the unit of angular measure. It is most common in mathematical parlance to use radians. You are correct to be concerned about this ambiguity.</p>
|
1,857,196 | <p><strong>Question :</strong>
Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for all real values of $x$.</p>
<p><strong>What I think :</strong>
My problem is that I have no other ori... | Robert Israel | 8,508 | <p>It does exist, it's just $0$. Your factorization is almost right, but you forgot the leading coefficient: it should be $f(x) = c_n (x - a_1) \ldots (x - a_n)$ (where $a_1, \ldots, a_n$ are the first $n$ of those $n+1$ values). Your job is to show that the polynomial has this form, and that $c_n = 0$.</p>
|
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | TheSilverDoe | 594,484 | <p>Modulo <span class="math-container">$88$</span> one has <span class="math-container">$$123^{456} = 35^{456} = (35^2)^{228} = (-7)^{228} = ((-7)^6)^{38} = (-7)^{38} = ((-7)^6)^6 \times 49 = (-7)^6 \times 49 = -7 \times 49 = -343 = 9 \quad [88]$$</span></p>
|
3,525,621 | <p>Find all integral solutions to the equation <span class="math-container">$x^2 + 4xy - y^2 = m$</span> with <span class="math-container">$-5 \leq m \leq 10$</span>.</p>
<p>I know that I can set <span class="math-container">$m = -5$</span> to <span class="math-container">$m = 10$</span> and solve all of the equations... | Z Ahmed | 671,540 | <p><span class="math-container">$$\frac{1}{x}[\log(1+x)-\log(1-x)]=2 \sum_{k=0}^{\infty} \frac{x^{2k}}{2k+1}.$$</span></p>
<p><span class="math-container">$$I=\sum_{k=0}^{\infty} \int_{0}^{1} \frac{2x^{2k}}{2k+1} dx = \sum_{k=0}^{\infty} \frac{2}{(2k+1)^2}=\frac{\pi^2}{4}.$$</span></p>
|
713,104 | <p>Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$?</p>
<p>Finding examples of games of order $2$ is easy (for example any impartial game), but I have not been able to think up an example with finite order where the order did not come from ... | Guy | 127,574 | <p>$$f(n)=\sum_{i=1}^n \frac{i(i+1)}{2}$$</p>
<p>$$f(n)=\sum_{i=1}^n \frac{i^2}2 +\frac{i}2$$</p>
<p>Using two well known identitites,</p>
<p>$$f(n)=\frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)$$</p>
<p>Simplifying:</p>
<p>$$f(n)=\frac{n(n+1)}{4}\left(\frac{2n+1}{3}+1\right)$$</p>
<p>$$f(n)=\frac{n... |
163,589 | <p>The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case of 3 vector spaces. For one vector space there are two orbits: 0 vector, and non-zero vector. For two vector spaces, ... | Ben McKay | 13,268 | <p>In the world of exterior differential systems, an element of a triple tensor product is called a <em>tableau</em>. The known invariants of tableaux are complicated; see the book <strong>Exterior Differential Systems</strong> by Bryant, Chern, Gardner, Goldschmidt and Griffiths. There is no classification of tableaux... |
3,244,649 | <ul>
<li>Show the set <span class="math-container">$A=\{(m,n)\in N\times N : m\leq n\}$</span> is countably infinite.</li>
</ul>
<p>If <span class="math-container">$A$</span> is countable then we need to show that there is a bijection between <span class="math-container">$A$</span> and <span class="math-container">$\m... | Kitter Catter | 166,001 | <p>I'll throw in a couple of approaches:</p>
<h2>Quick Approach</h2>
<p>If anything goes in terms of premises then we know a couple of things:</p>
<ol>
<li>Subsets of countable sets are countable</li>
<li><span class="math-container">$\mathbb{N}\times\mathbb{N}$</span> is countable</li>
<li><span class="math-contain... |
3,562,294 | <p>I got <span class="math-container">$x^2+y^2$</span> could factorized by <span class="math-container">$(x+yi)(x-yi)$</span></p>
<p>But Could we get factorization of <span class="math-container">$x^2+y^2+1$</span></p>
<p>I tried <span class="math-container">$(x+yi+i)(x-yi-i)$</span> but i couldn't guess it.</p>
<p>... | P. Lawrence | 545,558 | <p>Look at the homogenization of your polynomial, viz. <span class="math-container">$x^2+y^2+z^2$</span>. Finding a non-trivial factorisation <span class="math-container">$$x^2+y^2+z^2=(a'x+b'y+c'z)(a''x+b''y+c''z)$$</span> with complex coefficients is equivalent to decomposing the conic <span class="math-container">$x... |
4,000,062 | <p>A sequence <span class="math-container">$\left\{a_n\right\}$</span> is defined as <span class="math-container">$a_n=a_{n-1}+2a_{n-2}-a_{n-3}$</span> and <span class="math-container">$a_1=a_2=\frac{a_3}{3}=1$</span></p>
<blockquote>
<p>Find the value of <span class="math-container">$$a_1+\frac{a_2}{2}+\frac{a_3}{2^2}... | Hari Shankar | 351,559 | <p>Let <span class="math-container">$S=a_1+\dfrac{a_2}{2}+\dfrac{a_3}{2^2}+\dfrac{a_4}{2^3}+\ldots $</span></p>
<p>Then <span class="math-container">$\dfrac{S}{2} = \dfrac{a_1}{2}+\dfrac{a_2}{2^2}+\dfrac{a_3}{2^3}+ \ldots$</span></p>
<p>Subtracting we get</p>
<p><span class="math-container">$\dfrac{S}{2} = a_1 +\dfrac{... |
378,735 | <p>Let $A \in M_n(\mathbb C)$. Let $\langle \; \cdot\; , \; \cdot\; \rangle$ be the standard inner product in $ \mathbb C^n$, viewed either as row vectors or as column vectors.</p>
<p>Let $r_j$ be the $j$-th row of A, and let $c_j$ be the $j$-th column of $A$.</p>
<p>Show that A is normal, if and only if $\langle r_... | Ma Ming | 16,340 | <p>Hint: $A A^*=(\langle r_i,r_j\rangle )$ (Recall the definition of matrix multiplication).</p>
|
4,286,296 | <p>I am trying to prove the following claim:</p>
<blockquote>
<p>Let <span class="math-container">$ 0\leq n \in \Bbb Z$</span> and suppose that there exists a <span class="math-container">$k \in \Bbb Z$</span> such that <span class="math-container">$n=4k+3$</span>.
Prove or disprove: <span class="math-container">$\sqr... | Lazy | 958,820 | <p>Note the if <span class="math-container">$q$</span> is rational but not integral then <span class="math-container">$q^2$</span> will also not be integral. So you only need to prove that <span class="math-container">$n$</span> is not a power of an integer.</p>
<p>Note that if <span class="math-container">$m=2l$</span... |
4,286,296 | <p>I am trying to prove the following claim:</p>
<blockquote>
<p>Let <span class="math-container">$ 0\leq n \in \Bbb Z$</span> and suppose that there exists a <span class="math-container">$k \in \Bbb Z$</span> such that <span class="math-container">$n=4k+3$</span>.
Prove or disprove: <span class="math-container">$\sqr... | MAHI | 821,955 | <p><span class="math-container">$\sqrt n=\sqrt{4k+3}$</span> is a solution of the equation <span class="math-container">$$x^2 -(4k+3)=0$$</span></p>
<p>Now by <strong>Rational Zeros Theorem</strong> <em><span class="math-container">$\biggr($</span>Which states that : <span class="math-container">$x=p/q$</span> is a Rat... |
20,784 | <p>I have a set of 3-space coordinates for the atoms of a molecule (I could also transform them into spheres with radii corresponding to the atoms they represent). I would like to place this molecule into the tightest possible 3D rectangular bounding box and determine the coordinates for the box vertices. Is there a ... | Xerxes | 5,406 | <p>Assuming you're not trying to rotate the molecule to minimize some parameter (volume? diagonal?) of the bounding box, take a set of positions and radii:</p>
<pre><code>pos = RandomReal[{0, 10}, {10, 3}];
rad = RandomReal[{1, 2}, {10}];
</code></pre>
<p>Add and subtract a radius from each position and dimension, th... |
1,643,013 | <p>I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful results. I am more interested in the reason or method rather than the actual answer. Also I do not know if there even ... | Konstantinos Gaitanas | 99,437 | <p>Differentiate both sides to get $f'(x)=f''(x)+f^{(3)}(x)+...$ </p>
<p>So the starting equation becomes $f(x)=f'(x)+f'(x)\Rightarrow f(x)=2f'(x)$ </p>
<p>Multiply now both sides by $e^{-\frac{x}{2}}$ and this becomes<br>
$$[e^{-\frac{x}{2}}f(x)]'=0$$<br>
So $f(x)=ce^{\frac{x}{2}}$<br>
Done.</p>
|
239,720 | <p>If $A$ is unital C$^*$-algebra, is it true that the multiplier algebra of $A \otimes \mathcal{K} $ is $ A \otimes \mathcal{B}(\mathcal{H})$? Where $\mathcal{K}$ is C$^*$-algebra of compact operators on the Hilbert space $\mathcal{H}$.</p>
| vap | 89,722 | <p>If $A=C_0(X)$ and $B$ is a $C^\ast$-algebra then $M(A\otimes B)$ is the set of strictly continuous functions $\beta X\to M(B)$, where $\beta$ stands for Stone-Čech compactification. </p>
<p>If you take $X$ to be compact and $B=\mathcal{K}$ then we are in your setting. </p>
<p>But $C(X)\otimes\mathcal{B(H)}$ is the... |
1,867,401 | <p>I refer to this derivation of the gradient in polar coordinates: <a href="http://www.math.jhu.edu/~js/Math202/polar.grad.chain.pdf" rel="nofollow">http://www.math.jhu.edu/~js/Math202/polar.grad.chain.pdf</a></p>
<p>I can understand all parts except why the unit gradient $$\hat{e_r}=\langle\cos\theta,\sin\theta\rang... | Rui Liu | 459,936 | <p>I was thinking about the same question recently. I think now I get a rather satisfactory answer, I will just post it here.</p>
<hr>
<p>After some search I think I can now see things a bit clearer. I will try to clarify my thoughts and do a summary.</p>
<p>Initially I was surprised by the fact that gradient in non... |
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