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Let $\zeta$ be the Riemann- Zeta function. For any integer, $n \geq 2$, how to prove $$\zeta(2) \zeta(2n-2) + \zeta(4)\zeta(2n-4) + \cdots + \zeta(2n-2)\zeta(2) = \Bigl(n + \frac{1}{2}\Bigr)\zeta(2n)$$
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I need to solve this :$$ z^2 -6z + 25 = 0$$
My book says 'complete the square' so :
1.$$ (z - 6/2)^2 -36/4 + 25 $$
2.$$ (z - 3)^2 -9 + 25 $$
3.$$ (z - 3)^2 + 16 $$
Now how exactly does the above turn into this:
$$ 3\pm 4\imath $$
Thanks so Much
Gideon
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During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a node, a cusp (for my purposes I do not care which of the two kinds of cusps would the example exhibit), a tacnode, and an isolated point. By "nontrivial" I mean a curve that was not constructe... |
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to decompose the plane into closed or open intervals. The best I could do was to make a square with two sides missing (which you c... |
I am interested in the sum of a Hadamard product of generating functions.
If we are given $n$ functions, where $0 < i \leq n$, of the form:
$$f_i(x_i) = \sum_{j=0}^m{c_{i,j} x_i^j}.$$
The hadamard product is defined as:
$$g(x) = \sum_{j=0}^m{( \prod_{i=0}^n {c_{i,j}})x^j}.$$
It's essentially the same as going through a... |
I remember hearing somewhere that if $p^n$ is the largest power of a prime $p$ dividing the class number $h_K$ of a number field $K$, then there is unique factorization of the $p^n$-th powers of elements in $\mathcal{O}_K$ into $p^n$-th powers of irreducibles. I can see that there can't be any $p$-torsion for a $p^n$-t... |
In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For example, what is the distribution of $\max(X_1, X_2, X_3)$ if $X_1, X_2$ and $X_3$ have the same distribution? Likewise, if o... |
Suppose $p$ is prime, $q|(p-1)$. Prove the congruence $1+x+\cdots+x^{q-1}\equiv 0 \pmod{p}$ has exactly $q$ solutions.
I have this as a homework assignment and I've having trouble proving it. I'm trying to use Lagrange's theorem:
If $p$ is prime, an $n$th degree polynomial has at most $n$ solutions mod $p$.
My attempt ... |
This is a chug-plug formula given in my book :
1.The number of ways in which mn different items can be divided equally int m groups, each containing n objects and the order of the groups is not important,is : $\frac {(mn)!}{(n!)^m} \frac {1}{m!} $
2.The number of ways in which mn different items can be divided equally ... |
I am trying to self-read through Ian Stewart's Galois theory and am stuck at this paragraph. I quote - "Lagrange observed that all methods for solving polynomial equations by radicals involve constructing rational functions of the roots that take a small number of values when the roots $\alpha_j$ are permuted. Prominen... |
I am trying to solve some problems based on this formula,but am facing some issues in determine whether or not consider ordering as important.
For Example:
In how many ways 15 different books can be divided five heaps of equal number of books ?
So here according to the answer the ordering of groups is not considered im... |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\{1,2,\dots,n\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_2 < k... |
I want to graph an equilateral triangle. It would be ideal if I had a set of three points: $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ with $x_i, y_i \in \mathbb{Z}$ as the vertices. However, this is impossible. I am willing to "settle" for a triangle that is off by the width of a pencil lead. (That is, if we draw circles of ... |
It is a theorem that every locally free coherent sheaf on $\mathbb{P}^1$ over an algebraically closed field is isomorphic to a unique sum of sheaves $\mathcal{O}(n)$ for various integers $n$. In particular, the K-ring of locally free coherent sheaves (or all coherent sheaves, $\mathbb{P}^1$ being nonsingular) is isomor... |
I'm wondering if there is other terminology that describes row-completeness, outside the context of a latin square, or if row-complete is actually a general term.
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I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an "infinite forest" holding an "infinite bb-gun." The "trees" in this forest are at the lattice points all around you. (The lattic... |
Or has it remained a terminal node at the frontier of mathematics?
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You have 3 bins. You can toss balls one at a time to the bins until any one of them has 2 balls in it, and then you stop.
The tosses are independent, and each bin is equally likely to be hit. It's impossible to not make it in a bin on any given toss.
What is the expected number of tosses so that 1 bin has 2 balls in ... |
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection?
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The following comes from questions comes from a recent combinatorics paper I attended :
1.27 people are to travel by a bus which can carry 12 inside and 15 outside. In how many ways can the party be distributed between inside and outside if 5 people refuse to go outside and 6 will not go inside?
The solution given C(16... |
Does anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site?
I found this http://stephencollins.net/penrose but I can't seem to get a rectangular shape out of it for web use. Thanks.
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I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to inequality constraints.
The examples I have seen in the text I have at hand, all involve only linear constraints. Is this me... |
Let $X$ be a Poisson random variable with mean $\lambda$. How do I show that $P[X \geq \lambda] = 1/2$? Also, I was wondering what distributions have this property that the density is concentrated equally around the mean. I see that it's true for Gaussian and Uniform distributions.
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If $S$ is a surface which is the complement of finitely many points in a compact surface, and the metric in $S$ is complete, then is Gauss-Bonnet theorem still valid for $S$?
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I've started a little reading on quadratic reciprocity, and a reason for this has eluded me. Here's a little of what I came up with so far. I decided I want to show that for all primes $p$, if $p|x^2-2$, then $p$ does not divide $2y^2+3$. Then, by way of contradiction, if $(x^2-2)/(2y^2+3)$ is an integer, then any $p$ ... |
Assume you want to prove an identity such as
$$\sum_{k=m+1}^{n}A(k,m)-B(k,m)=S(m)+T(n,m)\qquad\text{for } n,m\in
\mathbb{Z},n,m\geq 0.$$
Added: I applied mathematical induction on $m,n$ to prove it. I am unsure because up to now I have seen it applied to properties depending on a single variable only.
Question: does ... |
Do somebody knows anything about the Dirac's identity?
\begin{equation} \label{Dirac}
\frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) =
-4\pi \delta(xb_0)\delta(xb_1)\delta(xb_2)\delta(xb_3)
\end{equation}
here
$xb$, is the 4-vector $x-b$ in Minkowsky spacetime
$\delta$ is the Dir... |
Let $L^2(R)={f:R->C| \int_{\infty}^{\infty} \mid (f(t))^2 \mid < \infty }$ and $L^1(R)={f:R->C| \int_{\infty}^{\infty} \mid (f(t)) \mid < \infty }$. Give and example of a function such that $f \in L^2(R)$ and $f \notin L^1(R)$.
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How many solutions does the following equation have?
$$k+l+m+n=30,$$
where $k,l,m,n\in\mathbb{Z},0\leq k,l,m,n\leq 10.$
Edit: this is an exercise in the inclusion-exclusion principle.
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Consider commutative semigroup S and its Grothendieck completion group
G(S).Suppose I insist on defining G(S) as free abelian group on basis $[a]$ (with $a\in S$) divided out by the relations $[a+b]-[a]-[b]$. How do I show with that definition that if the images of $a,b\in S$ become equal in G(S), then necessarily t... |
I didn't get this method.
\begin{equation*}
3a^2-9a+6=3(a^2-3a+2) \\
=3(a^2-1a*3+3^2-3^2+2)= \\
= 3[(a-3)^2-7],
\end{equation*}
then?
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Is there a specific way to attack congruences with two variables? The reason I ask is because up until now, I had never encountered such a problem. While browsing tonight, I came across this challenge problem in one of my books:
Suppose that $(ab,p)=1$ and that $p>2$. Show that the number of solutions $(x,y)$ of the co... |
Show that the only regular, self-complementary, planar graph is the 5-cycle. Pay special attention to the 4-regular case.
|
Question 1: The volume of a parallelepiped in $\mathbb{R}^n$ with n sides given by the vectors $(x_{1_1}, x_{1_2} ... x_{1_n}), (x_{2_1}, x_{2_2} ... x_{2_n}) ... (x_{n_1}, x_{n_2} ... x_{n_n})$ and diagonal $(x_{1_1} + x_{2_1}+ \cdots + x_{n_1}, x_{1_2} + x_{2_2} + \cdots + x_{n_2}, ..., x_{1_n} + x_{2_n} + \cdots + x... |
I'm trying to work out a basic example where formal smoothness should fail.
I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$.
The idea is that not every $\mathbb{R}$-homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}$ should lift to a homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}[\varepsilon]/(\v... |
I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
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This question is taken from a contest in India.
Prove that we can construct an infinite set of positive integers of the form $2^{n}-3$, where $n \in \mathbb{N}$, such that any two numbers from the set are relatively prime
I would like to have an answer for this question, and i would also like to know why $2^{n}-3$ h... |
I have differential calculus problem where I must find the coordinates of a point $C$ on a parabola. There is a point given which is $D(8,4)$ and $D$ is on a line that is intersecting with the parabola at $C.$ Also, the line where there is the point $D$ is perpendicular to the tangent of $C.$
This is the equation of t... |
How can you prove that $\Bigl\lfloor{x + \dfrac{1}{2}}\Bigr\rfloor$ is a $\theta(x)$ function?
I'm just practicing questions that could come up on an exam and this one was giving me a tough time trying to formally prove it.
$f(x)$ is $\theta(g(x)) \leftrightarrow f(x)$ is $O(g(x))$ and $f(x)$ is $\Omega(g(x))$
|
If $q$ and $r$ are quaternions, and $p$ is a point, applying $q$ then $r$ to $p$ is:
$$
(qr)p\dfrac{1}{qr}
$$
What if I want to go the other way? Instead of concatenating rotations, I want to remove them.
So, I have $q$ and $m$, and I know that $qx =m$, but I don't know what $x$ is. How can I discover $x$? (How can I d... |
Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum_{i, j \in B}\Pr[i\ \textrm{AND}\ j]$$
My intuition for why this should be true is jus... |
If A = { Market research predicates strong demand } and B = { Market demand is strong }, can we reasonably assume that P(A or B) = P(A) * P(B)?
The problem is that I know
P(B|A) = 0.8
P(not B | not A) = 0.9,
P(B) = 0.2
I need to calculate P(A) and P(not A).
For me, it seems that if P(A or B) != P(A) * P(B), it's i... |
I was just wondering where the y'/(dy/dx) in implicit differentiation comes from.
$$
x^2 + y^2 = 25
$$
$$
(d/dx) x^2 + (d/dy) y^2 **(dy/dx)** = 25 (d/dx)
$$
$$
2x + 2y (dy/dx) = 0
$$
$$
(dy/dx) = -x/y
$$
Where does the bold part come from? Wikipedia says it's a byproduct of the chain rule, but it's just not clicking fo... |
I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
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I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example?
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Consider the probability space $([0,1]; B[0,1], L)$, where $B[0,1]$ contains the Borel sets
intersecting $[0,1]$ and $L$ is the Lebesgue measure. How do I find the sigma algebra generated by a random variable defined on this space, $X = 1_{[0,1/2]}$? Secondly, how do I determine whether random variables defined on this... |
If $X$ is a random variable with finite mean $\mu$ and variance $\sigma^2$, how do I show that the estimate
\begin{equation*}
P[\mu − d\sigma < X < \mu + d\sigma] ≥ 1 − 1/d^2~\forall d>1
\end{equation*}
holds? I found this in a book but unable to see the proof. Note that $X$ may not be normal.
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This is one of the descriptions I've seen online:
For any even $n$, say $n = 2m$, a row complete Latin square of order $n$ can be formed by writing down
$$0, 1, 2m - 1, 2, 2m - 2, 3,\ldots, m + 1, m$$
as the first row and then developing subsequent rows by adding $1$ modulo $n$.
I'm not quite clear on how that goes, su... |
How do I use Lebesgue Dominated Convergence Theorem to evaluate
$$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt x}dx$$
What dominating function to use here?
|
I have trouble understanding Radon-Nikodym theorem. Here's an example.
Consider the measurable space $(\Omega,F)$, where $\Omega = R$, $F~$ is the $\sigma$-algebra of Borel sets. Let $P[dw] = \frac{1}{\sqrt(2\pi)}e^{-w^2/2}dw$ and $\tilde{P}[dw] = e^{-w}1_{[0,\infty)(w)}dw$ be two probability measures on $\omega$. How ... |
I want to refer to functions of the form $f(x) = \sum_{i=1}^n a_i x^{\alpha_i}$ where $\alpha_i < 1$.
This is not a polynomial, because $\alpha_i$ could be just real arbitrary numbers (though positive ones).
Is there a name for this class of functions that I can refer to? Wikipedia does claim these are not polynomials.... |
So, I don't like proofs.
To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert.
Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own. In other words I can't make steel traps, but I feel fine buying them ... |
Suppose $B_t$ is a Brownian motion. As I understand, $B_2-B_1$ is independent of $B_3-B_2$ from properties of Brownian motion. Does it also mean that $B_1$ and $B_2$ are also independent? Can I use this independence to find the joint density of $B_1+B_2+B_3$ as each Brownian process is a normal process of mean 0 and va... |
The usual action of $fg$ on $u⊗v$, where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg(u⊗v)=fgu⊗v+fu⊗gv+gu⊗fv+u⊗fgv$, right? How to state this fact for $V^{\otimes n}$, i.e. $fg$ acting on $u⊗v$, where $u=⊗_... |
we have a truck in our game we have a rod against the wall of that truck, and we know the acceleration of truck when the rod start slipping... we just want to know what is acceleration of rod when it touch surface of truck.
|
I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am inquisitive to prove this in a more general way.
The left side of an identity occurs while solving another problem (concern... |
How can I find the critical points of a square root function?
For example: What are the critical points of $f(x) = \sqrt{7x^2 + 2x - 2}$?
|
A friend of mine who is quite an aggressive Nominalist told me the other day:
"Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are consistent, but any consistent framework will do."
I'm curious as to what other examples of "consistent frameworks" he is ... |
Exercise 1.14 of the book Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove that an upper triangular matrix with elements from some C*-algebra $A$ is invertible in $M_n(A)$ iff all diagonal entries are invertible in $A$.
Trying to solve this I've found that if $a$ is invertible and... |
Consider the scheme of random placing balls into $N=1000$ cells. We continue the procedure of placing balls as long as a last cell remains empty. The process terminates when a ball is placed into this cell. At this moment several cells (or a certain cell) contain(s) a maximum number of balls among all cells. What is th... |
I've noticed a type of sum for the number of solutions of a type of quadratic form. For example, the number of solutions to $X^2-Y^2\equiv a\pmod{p}$ for $p$ an odd prime is given by the sum
$$\sum_{Y=1}^p(1+\left(\frac{Y^2+a}{p}\right))$$
where $\left(\frac{Y^2+a}{p}\right)$ is the Legendre symbol. It seems that the t... |
How to prove this binomial identity :
$$ { 2n \choose n } = 2^n \frac{ 1 \cdot 3 \cdot 5 \cdots (2n-1)}{n!} $$
The left hand side arises while solving a standard binomial problem the right hand side is given in the solution , I checked using induction that this is true but I am inquisitive to prove it in a rather gen... |
i have an object and a 3d direction vector and position for it . I would like to know how do i determine if a certain point X is in the space below the plan determined by my direction ?
Here is an image that i have drawn to make it more clear . In this image i've made the vector 2d
Please Help!
Regards,
Aleandru Bad... |
Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts:
$$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or } \operatorname{Im} s = 0 .$$
Now there are numerical methods for approximatin... |
As far as i know LU decomposition allow you to calculate matrix determinant following easy and cheap formula: Det[A] = Det[L] Det[U] = Det[U]
Trying this out in Mathematica 7 gives me correct result by absolute value, i.e. it ignores negative determinants and transforms them to positive ones.
Sample code:
matrixA = {{-... |
One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all the time while plotting it.
My question
I can see it in the picture that the spiral arms are getting bigger each turn an... |
If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
|
$$\frac{57^{46}}{17}$$
What is best and quickest way to evaluate an approximate answer to this by hand?
|
The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in Proofs from the BOOK by M. Aigner and G. Ziegler).
Since the invert transformation is
$$u=\... |
Let $G$ be a group and let $G'$ be the derived subgroup, defined as the subgroup generated by the commutators of $G$.
Is there an example of a finite group $G$ where not every element of $G'$ is a commutator? $G'$ is only generated by commutators, but with all of the properties of commutators (ie: what happens under c... |
I am trying to understand a simple calculation in polar coordinates - and I am getting totally discombobulated (the original source can be found: here).Please have a look at the following picture:
My questions(1) I don't understand how the length labeled $dr \over d \theta$ can possibly be the change of $r$ with respe... |
Let $Z$ be a countable set. Let $f_1,....,f_n$ be a collection of real functions over $Z$. Let $z_1,...,z_m,...$ be an enumeration of elements of $Z$.
Define $V_1 = \{ (f_1(z),...,f_n(z)) | z \in Z\}$ a set of $n$ dimensional vectors.
Define $V_2 =\{ (f_i(z_1),...,f_i(z_m),..., )) | i \in\{ 1,...,n\} \}$ a set of infin... |
Ok. This question may sound very easy, but actually I am in great need of an answer. I have been facing trouble in constructing functions, which are only continuous at some particular sets.
For example, the standard example of a function which is only continuous at one point, is the function, $f(x) = x, \ x \in \math... |
So this is some I partially understand, I'm not sure what I don't and do understand because most of my understanding is based on assumptions... sorry if I sound a little stupid!
The equation $6 \cos x - \sin x = 5 $ needs to be turned into the form $ \Re \cos (x + \alpha) $ then solved for in the interval $ -1/2\pi < ... |
ok so I did all the revision problems and noted the ones I couldn't do today and Im posting them together, hope thats not a problem with the power that be?
I have exhibit A:
$e^{-x} -x + 2 $
So I differentiate to find where the derivative hits $0:$
$-e^{-x} -1 = 0 $
Now HOW do I figure when this hits zero!?
$-1 = e^{-x... |
I need calculate ray (line) intersection with torus for my ray-tracing program (I know, its to graphics, but i need math behind it).
I can solve equation of order $x^4$, but thats too way slow (Cardano's method). So is there better way, how to calculate this ?
My torus has center $[0,0,0]$ and $R$, $r$ diameters
Line (... |
I am stuck while proving this identity I verified it using induction but like the other two (1) (2), I am seeking a more of a general way (algebraic will be much appreciated)
$$\frac {1}{(n-1)!} + \frac {1}{3!(n-3)!} + \frac {1}{5!(n-5)!} +\frac {1}{7!(n-7)!} + \cdots = \frac {2^{n-1}}{(n)!} $$
|
Let $r$, $s$ be positive real numbers and $\theta$, $\phi$ real numbers with $|\theta -\phi|<\pi$. Then an argument of $re^{i\theta}+se^{i\phi}$ lies between $\theta$ and $\phi$.
Can someone give a short, clean proof of the statement above that doesn't rely on geometric intuition? Of course the proof may use the not so... |
If I have a doubly stochastic matrix, how can I find the set of all basic feasible solutions?
Here's Wikipedia on doubly stochastic matrices.
|
Suppose we observe 84 alchoholics with cirrhosis of the liver, of whom 29 have hepatomas, that is, a liver-cell carcinoma. Suppose we know, based on a large sample, that the risk of hepatoma among alcoholics without cirrhosis of the liver is 24%.
What is the probability that we observe exactly 29 alcoholics with cirrho... |
UPDATE: I feel like my original question was too vague and didn't provide enough information (as others have mentioned). So I'm going to restate it.
Purpose: I want to find an equation in order to provide a smooth slope for an animation that I am creating. My animation consists of a lever and a set of numbers on a hori... |
I'm searching for an algorithm that forms a balanced (or quasi-complete) latin square, in which every element is a horizontal neighbor to every other element exactly twice, and a vertical neighbor to every other element exactly twice.
I've found one example (for n = 5), but am not clear about a couple of steps:
Step 1:... |
When pricing a european option by monte carlo over $30$ days for instance, what's the difference between one big $30$ day jump and $30$ one day steps?
|
I am given a lemma and it states:
Lemma: Let $A$ be a real $n\times n$ matrix. Then given any $e > 0$, there is a $norm$ such that $norm(A) \leq p(A) + e$, where $p(A)$ is the spectral radius of $A$.
They then go on to state that based on this lemma, if $p(A) < 1$, then $norm(A) < 1$ for the correct choice of the nor... |
Two questions:
(1.) Construct a subset of $[0,1]$ in the same manner as the Cantor set, except that at the $k$-th stage, each interval removed has length $\delta 3^{-k}$, $0<\delta <1$. Show that the resulting set is perfect, has measure $1-\delta$, and contains no intervals.
Showing that it's perfect is not difficult... |
Over on mathoverflow, there is a popular CW question titled: Examples of common false beliefs in mathematics. I thought it would be nice to have a parallel question on this site to serve as a reference for false beliefs within less obscure mathematics.
That said, it would be good not get bogged down with misconception... |
Define a function $F(A, B, C)$ as the number of ways you can roll $B$ $C$-sided dice to sum up to $A$, counting different orderings (rolling a $2$, $2$, and $3$ with three dice is different from rolling a $2$, $3$, and $2$).
Example:
With three $5$-sided dice, the list of $F(A, B, C)$ values in the domain of the possib... |
I have an expected value say 5(minimum) or 6(minimum).
Then I have 4 values [3, 10, 6, 8]
I would like to compute the percentage on how far the 4 values are from the expected value in terms of percentage. Given 100% means they are all within the minimum and maximum and the farther they are from the expected the lower t... |
What is the degree of the polynomial of the following expression ?
$$ [x + (x^3 – 1)^{1/2}]^5 + [x – (x^3 – 1)^{1/2}]^5 $$
If I am not very wrong the highest power of x is $\frac {15}{2} $ ?! So the degree is floor(15/2) = 7 ?! The answer is 7 but I am not sure of this approach.
Please comment.
|
I'm solving this question, but I don't know what it is.
Consider a plane $x - y + z = 0$, and a point $b = (1, 2, 0)^T$
a) Find a basis of this plane, and a basis of orthogonal subspace to this plane.
b) Find the closest point $\hat{b}$ on the plane to b.
c) What is the error between b and $\hat{b}$?
Is this a linear... |
If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ?
I solved it in the following way :
Expanding we get, the coefficient of $ x $ as $ (m-n) = 3 \cdots (1)$ and coefficient of $ x^2 $ as $ \frac{n(n-1)}{2} + \frac{m(m-1)}{2} +- m \cdot n = -6 ... |
Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm getting from mollification are to awkward to deal with...
Thanks in advance..
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For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
I could prove that
$$\sigma(f^{-1}(C)) \subset f^{-1}(\sigma(C)... |
This is a problem from Mathematical Analysis by Zorich, from the chapter on continuous functions.
Definition:
Let $P_n$ be a polynomial of degree $n$. For a function $f:[a,b]\to\mathbb{R}$, Let $\Delta(P_n) = \sup_{x\in[a,b]} |f(x)-P_n(x)|$. and $E_n(f) = \inf_{P_n} \Delta(P_n)$. A polynomial $P_n$ is the best approxim... |
How does one evaluate the limit: $$ \lim_{n \to \infty} \frac{1}{n}\sum\limits_{k=1}^{\lfloor{\frac{n}{2}\rfloor}} \cos\Bigl(\frac{k\pi}{n}\Bigr)$$
Yes, i recognize this as soon as i saw the problem: $$\int\limits_{0}^{1}f(x) \ dx = \lim_{n \to \infty} \frac{1}{n} \sum\limits_{k=1}^{n} f\Bigl(\frac{r}{n}\Bigr)$$ but th... |
According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions.
http://mathworld.wolfram.com/HyperbolicGeometry.html
What isn't well understood about hyperbolic geometry in 3D, that is in 2D or that is in 3D-Euclidean geometry?
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How can I
compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
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Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? produce an example or prove that it is not possible. A wavelet is a function $\psi:\mathbb R\to\mathbb R$ such that (i) $\psi \in L^1(R) \cap L^2(R)$, (ii) $\int_{-\infty}^{\infty} \psi(t) dt = 0$, fourier transform is $\hat f(\omega) =\in... |
I was wondering this today, and my algebra professor didn't know the answer.
Are subgroups of finitely generated groups also finitely generated?
I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?
And if not, is there a simple example of a finitely generated group with... |
Diablo 3 has 97 billion possible skill/trait builds. Per class.
LessPop_MoreFizz, emphasis is mine.
I used base two logarithm to claim "97 billion" configurations only are roughly 37 binary choices because $\log_2(97\times10^9) \simeq 36.5$, but I would like to know if there are more refined techniques to remove the ... |
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