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Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result
\begin{eqnarray}
AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),
\end{eqnarray}
where $AM(\mathbf{x}) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ is the arithmetic mean, $GM(\mathbf{x}) = \... |
I need to find the number of undirected trees on $n$ vertices such that the edges (and not the vertices) are labeled and exactly one label appears twice (i.e. there are $n-2$ possible labels and they all appear).
My reasoning goes like this: From Cayley's formula we know there are $n^{n-2}$ trees with labeled vertices.... |
I am having trouble expressing the titular question as iterated integrals over a given region. I have tried narrowing down the problem, and have concluded that the simplest way to approach this is to integrate over the XZ plane in the positive octant and multiply by 8, but I am having trouble identifying the bounding f... |
This one comes from my module without proper explanation.Please suggest how to approach this problem ?
PS:How do you write degree (in symbol) in math-mode ?
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Let A and B be nxn matrices over C. How to prove $(AB)^{\ast}=B^{\ast}A^{\ast}$? $A^{\ast}$ is complex conjugate transpose( if matrix $A$ is real then $A^{\ast} = A^{t}$. This is homework and is in the area of linear algebra.
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I need some online tool for diagonalizing 2x2 matrices or at least finding the eigenvectors and eigenvalues of it. I don't like to download any stuf because I'm not able to, some online tool will do the job.
Thanks.
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Let M, K, and L be isomorphic normal subgroups of a group G. Suppose $M \cap K \leq M \cap L$. Can we conclude anything about the index of MK in G and the index of ML in G? I would like it to be the case that MK has smaller index than ML, but I'm having a hard time convincing myself either way.
(EDIT: I initially left ... |
What is the best way to prove the polarization identity in the following form: for any $u, v \in V$
$$4 \langle T(u), v \rangle = \langle T(u+v), u+v \rangle - \langle T(u-v), u-v \rangle + i\langle T(u+iv), u+iv \rangle - i\langle T(u-iv), u-iv \rangle$$
Here $T$ is a linear operator.
Context
Polarization identity is... |
From this post we see that $\mathbb{R}$ over $\mathbb{Q}$ is infinite dimensional. Similarly $\mathbb{C}$ over $\mathbb{Q}$ is also infinite dimensional, and I rememeber having solved a problem that $\mathbb{R} \cong \mathbb{C}$ when considered as vector spaces over $\mathbb{Q}$. So I would like to extend this question... |
I'm interested to know if anyone can point me to a non-calculus way of seeing that the $\text{volume of a pyramid} = \frac{1}{3}\times(\text{area of base})\times(\text{height})$. Yes, I've googled.
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How can we show that every automorphism of $S_4$ is an inner automorphism ?
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Inspired by this question.
Let's consider the $11$-letter word "$MATHEMATICS$". The general question is to find how many words it is possible to have if we can make $4$ permutations of letters.
Sub-question a)
Each letter has a position in the word, thus when I say $1$ permutation, we exchange the position of $2$ lette... |
Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$.
The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle \frac{S'}{S} \ge \frac{2}{\pi}=0.6366\dots$?
For example, If Q is an ellipse, $\displaystyle S'=2ab$, $\displaystyle S=\pi ... |
Let $X$ denote a metric space. One can assume that $X$ is Polish if that helps, but I was trying to avoid to assume that $X$ is compact. Let $P(X)$ denote the set of Borel probability measures on $X$. The weak*-topology on $P(X)$ is defined as usual: a net $(p_{\alpha})$ in $P(X)$ converges to $p \in P(X)$ iff $|\int f... |
How to show that $e^{x} \geq \left (1 + \frac{x}{n} \right) ^{n}$ holds for each non-negative real $x$ and each integer $n \geq 1$ ? I tried series and induction but got stuck. Can you please help?
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Wikipedia articles on "unit sphere" and "unit circle" say the radius is 1. Articles on the "unit square" and "unit cube" say the length of the side is 1. Would you expect a unit torus to have major radius 1 or major diameter 1?
Admittedly, a torus is a different animal than a sphere, but... It feels most natural to me ... |
How can I prove that there is no closed form solution to this equation? $$2^x + 3^x = 10$$
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I am trying to understand the interpretation of $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ in terms of $\sin A$ from my book, here is how it is given :
We have $ \bigl( \sin \frac{A}{2} + \cos \frac{A}{2} \bigr)^{2} = 1 + \sin A $ and $ \bigl( \sin \frac{A}{2} - \cos \frac{A}{2} \bigr)^{2} = 1 - \sin A $
By adding and ... |
Let $X$ be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function $f$ in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which converge pointwise to a function $g$ in $L^{\frac{p}{p-1}}(X)$. Prove that:
$$ \lim_{n \to \infty} \int_{X} f_{n}(x) g_{n}(x) dx... |
I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:
Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$, and let $M$ be the intersection of all of the $\m... |
I would like a demonstration of the following theorem:
Original question: If a square matrix of order $n\gt 1$ has a row (column) that is linear combination of rows (columns) parallel to it, then the determinant of this matrix is null.
Actual intended question: If the determinant of a square matrix of order $n\gt 1$ i... |
Consider a number say $N = 345$.
The permutations are :
{{3, 4, 5}, {3, 5, 4}, {4, 3, 5}, {4, 5, 3}, {5, 3, 4}, {5, 4, 3}}
The GCD of all of them is 3 which is the answer. Please suggest a suitable approach for this problem.
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I want to prove that the series
$$ \sum_{n=1}^{\infty}(-1)^n\left(2^{1/n}-1\right)$$
converges.
I am fairly certain that this converges. Using the ratio or root test does not seem to work. Therefore the other tests I am left with are alternating series test, intergral test, comparison test and limit comparison test (... |
How to evaluate this integral: $$\int_{0}^{\infty} \biggl\lfloor{\frac{n}{e^{x}}\biggr\rfloor} \ dx, $$where $n \in \mathbb{N}$.
The same integral when asked to evaluate for $n=2$ (say) i can do it by splitting the limits from $x = 0$ to $x = \log{2}$ where $e^{x}$ takes the value 1, and then from $x= \log{2}$ to $x = ... |
I am confused on how to apply Ito's formula on certain problems, especially where expectations are involved. For example, if $W_t$ is a Wiener process and $X_t$ satisfies a below SDE:
$ dX_t = (X_t-\mu)dt + \sigma\sqrt{X_t}dW_t,~~~~~~ X_0 = x_o$
How do I find $\partial_t \phi$ or $\partial_\xi \phi$ where $\phi(t,\x... |
Suppose $X$ and $Y$ are independent random variables. The mgf for $X$ is $M_{X}(t) = (1-t)^{-1}, \ t<1$. The mgf for $Y$ is $M_{Y}(t) = (1-2t)^{-1}, \ t< 0.5$. Two other random variables $U$ and $V$ are defined by $U = \frac{1}{2}(X+Y)$ and $V = \frac{1}{2}(X-Y)$. Calculate $\text{Cov}(U,V)$.
So $\text{Cov}(U,V) = E(UV... |
Let $f \in End_{F}(V)$ be an Endomorphism of a vector spaces $V$ over the field $F$ with the property $f \circ f=id_{V}$. Show that if $char(F) \neq 2$, the following equotation is true: $$V = Ker(f-id_{V}) \oplus Ker(f+id_{V})$$
Notes: $\oplus$ is the direct sum, $id_{V}$ is the identity map, Ker is the kernel and $\c... |
I have a set of quadratic equations of the form:
\begin{equation*}
2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0 \\
2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0 \\
\vdots \\
2S_nq_n(q_0S_0 + q_1S_1 + ...... + q_nS_n)+k_n = 0 \\
\end{equation*}
We have $n$ equations and $n$ variables $q_0,q_1,q_2.....q_n$... |
I have been thinking about the differentials that we use in derivatives and integrals.
For example, I have an equation:
$${\int{w}{dr}} = \text{other stuff}$$
The context for this strange equation was:
$$\begin{align*}
q'(t) &= a - \frac{b}{r(t)},\\
r'(t) &= c - d r(t) - e q(t)\\
r''(t)&= d r'(t) - e q'(t)\\
r''(t)... |
Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$.
Now, they define $\mathcal{F}$ to be the set of all mappings with as domain an initial segment of $X$ and $Y$ as codomain a... |
I wanna know how to solve this equation: $-2x^3 +10x^2 -17x +8=(2x^2)(5x -x^3)^{1/3}$
I have some trouble to do that and I'd glad with any help I may get.
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If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $
what can I infer from this equation? Any observations or interesting properties that you see based on this relation?
What if,
P... |
I hope this question is not too general, but I am not completely sure yet how to phrase it.
So we all know that when we have two sequences of random variables $X_n$ and $Y_n$ for $n \ge 1$, that converge to $X$ and $Y$ (using some measure, almost surely, or with probability, in distribution, etc.) then we have all kind... |
Can someone explain the following definitions of a limit and derivatives. Issac did a good job explaining it in the previous thread: Infinite series - Archimedean principle
The definition of a limit is defined as the following:
$lim_{x \to a} f(x) = T$, what we actually mean is if we take any number M > T, then we can ... |
How can I calculate the arc between two circles? The arc must be tangent to the two points on the circles.
Here is a picture illustrating it. I'm trying to code and calculate the orange arc and the blue arc.
The points B, F, D, and E can be changed by the user and dynamically update.
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Suppose that you have an exponential generating function.: $E(z)=\sum_{n=0}^{\infty} \frac{a_{n}z^{n}}{n!}$, and that the definition of $a_{n}$ can be reasonably extended to noninteger arguments. (the Catalan numbers $C_{n}$, would be written in terms of the Gamma function thusly: $C_{n} = \frac{\Gamma(2n+1)}{\Gamma(n+... |
Let $f$ be an increasing function defined on $[a,b]$ and let $x_1 < x_2 < \dotsb < x_n$ be $n$ points in the interior of $[a,b]$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
The proof I have is very wordy and sketchy. I know it would be easier if I could find a telescoping sum but I've got a... |
Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading several articles on Google.
Is the only work-around to use quaternions?
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My question is how do I reduce $\bar A\bar B\bar C+A\bar B\bar C+AB\bar C$ To get $(A+\bar B)\bar C$. I'm so lost just been trying to get it for awhile only using the 10 boolean simplification rules.
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I would like a demonstration of the fact below.
Being given real numbers a and b (nonzero) and a positive integer n, the order p, that occupies the maximum term (in absolute value)
of the development of power (a+b)^ n, according to decreasing powers of a is given by:
p = 1 + integer part of [|b|(n+1)/(|a|+|b|)]
When n... |
AB+CD+A’BD+A’BC+AB’D+AB’C
AB+CD+A’B(D+C)+AB’(D+C)
CD+(AB+A'B(D+C)+AB’(D+C))
CD+(AB+B(D+C)+A(D+C))
CD+AB+(B(D+C)+A(D+C))
CD+AB+(A+B)(D+C)
This is what I got, but I'm not sure if it's correct. It's as good as I can get it algebraically. Can it be simplified further?
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I'm trying to compute the following limit:
$\displaystyle \lim_{n \to \infty} \int_{0}^{\infty} (1+\frac{x}{n})^{(-n)} \sin(x/n) dx$
Clearly the integrand converges to 0 pointwise so the trick is to find the dominating function. So for simplicity let ${f_{n}}$ denote the sequence of functions of the integrand: it is no... |
The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this conjecture. But why is this interesting at all? What practical and interesting questions can be answered now that this conjectur... |
This is an exercise from Rotman, An Introduction to Homological Algebra, which I've been thinking now and then for a few days and I haven't solved it yet. I've decided to ask here because it is bugging me and I don't have any friends also studying this subject (in case it is of any use, I'm an undergraduate).
Let $R$ ... |
Suppose $G_1$ and $G_2$ are Lie groups with isomorphic Lie algebras. Then from standard Lie theory we know that there is a simply connected Lie group $G$ such that $G/H_i = G_i$ where $H_i$ is a discrete subgroup of the center of $G$. I am curious if there is a nice condition on $H_1,H_2$ and how they sit in $G$ that... |
I was experimenting with binary operators, and devised a commutative operator with function $f$ such that the operator mapped $f \colon \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{R}$ in order to make up some problem (e.g. if $f(3,10)=30$ and $f(4,3)=10$, find $f(4,6)$, etc.).
My question is, are there any functio... |
I have a piece-wise 2-input function that I would like to repeatedly tile diagonally across a grid. See this image.
So here, I know the equations of the pink, blue and green areas in the range 0 - 2.
My problem is that I'm having trouble finding a way to tile the function diagonally across like shown in the image by... |
The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}.$$
The logarithmic mean of two distinct positive real numbers $a$ and $b$ is defined to be $$L(a,b) = \frac{b - a}{\ln... |
Let $F$ be a strictly increasing function on $S$, a subset of the real line. If you know that $F(S)$ is closed, prove that $F$ is continuous.
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I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ambient space?), but it's unclear to me whether the only non-varietable manifolds should be those that don't admit smooth... |
I've been wondering if/when it's possible to "truncate" a series.
Example 1
For example, the closed form for the series of naturals is:
$\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$
The "truncated" form is:
$\frac{1-(n+1)x^n+n\cdot x^{n+1}}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots + n\cdot x^{n-1}$
Example 2
The closed... |
This may be a question whose answer is lost in the mists of time, but why is the elliptical integral of the first kind denoted as $F(\pi/2,m)=K(m)$ when that of the second kind has $E(\pi/2,m)=E(m)$? It's not very consistent! Aside from convention, is there anything stopping us from rationalising these names a little?
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I tried to do some math in a blog post of mine and came to one with a floor function. I wasn't sure how to deal with it so I just ignored it, and then added the ceiling function in my final equation as that seemed to give me the result I wanted. I'm wondering what is the correct way of handling these functions in equat... |
Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-distributed about the true value, with a standard deviation of 5mm), while the other is a dimwit (i.e., his measurement is also u... |
Im giving a small talk for a combinatorics class on the Erdos-Szekeres conjecture regarding the happy ending problem (the paper is focused on recent work regarding the conjecture). I always find that when the audience is not up for a long technical proof, and they aren't familiar with the field (in this case combinato... |
Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example:
$\omega_0 = < 0,1,2,3,... | > $
and:
$\epsilon = < 0 | 1, 1/2, 1/4, 1/8, ... > $
(where $\epsilon$ is not the first uncountable ordinal, but the ... |
OK, so I got my ships moving along the bezier in time T, now I need to factor in acceleration, but I don't really know how to do it.
I'm using the DeCasteljau Algorithm, that's what my code looks like:
function linp(d, a, b, t) {
d.x = a.x + (b.x - a.x) * t;
d.y = a.y + (b.y - a.y) * t;
}
// a, b, c, d are t... |
On what basis the numbering less than 20, the fraction 17/6 (base 10) does not generate a regular tithe?
|
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry (MS) data. By labeling proteins of two different samples A and B, we get the relative abundance of 100 to thousands of proteins in A/B. Alongside with this ratio, we can estimate its variance based on the ... |
How do I solve the following question?
You are given the identity $x^2-ax+144 = (x-b)^2$
Work out the values of $a$ and $b$.
Question appears in AQA 43005/1H.
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Consider the set
$$S = [0,1/2] \cup [3/4, 7/8] \cup [15/16, 31/32] \cup \cdots$$
and the set
$$T = (0,1/2) \cup (3/4, 7/8) \cup (15/16, 31/32) \cup \cdots$$
Is $S$ open, closed or neither? Is $T$ open, closed, or neither?
|
Given a $\sigma$-finite measure $\mu$ on a set $X$ is it possible to formulate a topology on the space of functions $f:X \rightarrow \mathbb{R}$ that gives convergence $\mu$-almost everywhere?
I can't seem to find any way to write this and am suspecting that no such topology exists! Is this true? If so, is there some ... |
The probability that a fire will occur is $0.001$. If there is a fire, the amount of damage, $X$, will have a Pareto distribution given by $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An insurance will pay the excess of the loss over a deductible of $100,000$. For this coverage the one-time insurance premium w... |
This comes up in a paper by Beskos and Roberts on exact simulation of diffusions. I'm sure it's easy, but I can't work out what's going on.
Given a differentiable function $\alpha$ satisfying
$\alpha(u) \geq k_1$
and
$\alpha'(u) \leq k_2$,
for some constants $k_1$ and $k_2$, show that there exist constants $c_1$ and $c... |
Can someone explain how to picture or construct the Monster Group?
|
Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely dif... |
I am working on a signal-smoothing algorithm for personal interest. I understand the basic concept of the Savitzky-Golay algorithm but I would like to understand how the coefficients were discovered. Can someone please explain the the mathematics behind it, so I could calculate the coefficients on my own?
How would the... |
I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations wh... |
The rational root theorem states that, given $P \in \mathbb{Z}[x], P = \displaystyle\sum_{j=0}^n a_j x^j$ with $a_0 \neq 0$ and $a_n \neq 0$, if $P(p/q)=0, p\perp q$ then $p|a_0$ and $q|a_n$.
As a follow-up to this question, I'd like to know how can I prove the following:
$$a_0 = a_n \displaystyle\prod_{j=1}^n (-r_j)$$... |
If you've got multiple arrays like this:
(24,36,28,28,16,27)
(38,38,45,57,35,50)
every array being 6 integers, each integer in range [0,60]
I would like to find the distance between those 2 arrays. The problem is related to computer science, however, I would like to interpret these arrays as "planes" so as to find the... |
How can I compute the following limit?
$$\lim_{a \to 0^+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\,\mathrm dx\right)$$
Any hints you can please give?
Cheers
|
I am having some problem wrapping my head around an assignment.
I have to find out how many additions, subtractions, multiplications and divisions are used while solving an $N \times M$ linear equation, while using Gauss-Jordan elimination.
Could anyone point me in the right direction?
|
I would like, if possible, to obtain a proof of the theorem below.
"Being given real numbers a and b, with |a|>|b|, and m is a positive integer, the order p, which occupies the maximum term (in absolute value) the development of power (a+b)^(-m) , according to decreasing powers of a, is given by:
p = 1 + integer part o... |
I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai, Bi, Ci, Di) are sets of measurements from a noisy system (with typically 3-5 independant readings).
My first intuition to solve this problem was to pose this as an overdetermined linear equation system AX = B where X =... |
I have the following series of equations ($n+2$ equations $n+2$ variables):
\begin{equation*}
k_0q_0+\lambda q_0 + c_0 = 0, \\
k_1q_1+\lambda q_1 + c_1 = 0, \\
k_nq_n+\lambda q_n + c_n = 0, \\
q_1+q_2+....+q_n = 1.
\end{equation*}
The variables are $q_0,q_1,.....,q_n$ and $\lambda$. Note that $k$ and $c$ are series ... |
Is there a simple/efficient way of enumerating the elements of a free distributive lattice?
(I'm doing some computations with them, and listing their elements in what's probably the least efficient way possible...)
|
So, I'm relearning Group Theory. And I got the axioms down, I think. So let's make a concrete example:
The collection of numbers the positive integers less than 7: 1,2,3,4,5,6
The • operation will be multiplication mod 7.
Associativity holds.
The Identity e is 1.
Every element has an inverse:
1*? mod 7 = 1 --> ... |
I am seeking conditions on the distribution of the step sizes that guarantee that a random walk on the 2D lattice will return to the origin (with probability 1). Essentially, under what conditions can Pólya's theorem be proved? Certainly it holds if the steps are of size 1 (and equally probable in all directions), a... |
If we can say that we're able to determine the weather at 10ms intervals, and if we observe that over a whole second (1000 samples) that it's raining at every sample, what is the accuracy rate within which we've calculated the weather for the second? I'm sure this sounds like homework to some, but it's not. I was readi... |
I have a generating function that I'm trying to create. It has a general form to it:
$1 + cx + c(c-1)x^2 + c(c-1)(c-2)x^3 + \dots + c!x^c$
I would like to see a closed form for this function. I'd like something other than a summation. I'm particularly interested if someone could walk through how to derive the form f... |
In maths, you can use something as
simple as statistical analysis to
intuit the theory, and in comp-science
you can use simulators.
Is the above statement true for maths ?
|
Say I have a function
$$V(x)=A(x-x_1)(x-x_2)(x-x_3)$$
where $x_1$, $x_2$, $x_3$ are the three roots in increasing order and $A$ is positive. Clearly $V(x)$ is positive at large $x > x_3$, negative between $x_2$ and $x_3$, and so on.
Now, say I wish to evaluate
$$\int_{x_2}^{x_3}\sqrt{-V(x)}\mathrm dx$$
How do I do ... |
Let $a_n$ be a sequence of real numbers such that $\exp(it\cdot a_n)$ converges to 1 for all real t.
Show that $a_n$ converges to 0.
One can show this by letting $X_n$ be degenerate random variables which take value $a_n$ with prob 1 and then using results about characteristic functions and convergence.
But this seems ... |
What do I miss in the following?
Let $R$ be a commutative Noetherian ring with unit. A map $f:M\to N$ of $R$-modules is injective/surjective iff the associated map $f_p:M_p\to N_p$ on the localization is injective/surjective for every prime ideal $p$ of the ring $R$.
There is an isomorphism $M_p=M\otimes_R R_p$. A sequ... |
Is there any known bound on sum of independent but not identically distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for the sum of iid geom. variables).
Formally, if $X_i \sim Geom(p_i)$, $X = \sum_{i=1}^n X_i$, and $E[X]=\Theta(n)$,
Is it ... |
I was asked to prove the next formula -
(A+BC'+C)C' = ABC'+AB'C'+A'BC'
i need to show all the stages of the simplification, i have all of the rules/identities.
i have tried many times and i did not make it, i always get something else
thank you
|
Please suggest suitable approach for this problem
|
I have this table:
x | 0 1 2 3 4 5
------------------------------------------
f(x) | .7 .2 .05 .03 .01 .01
F(x) | .7 .9 .95 .98 .99 1
Where f(x) is the density function for x and F(x) is the cumulative distribution function for x.
And I am being asked to use F... |
Dealing with a physics Problem I get the following equation to solve for $\alpha$
$1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
Putting this in Mathematica gives the result:
$a==2 \text{ArcTan}\left[\frac{1-m}{1+m}\right]$
However I am unable to get this result myself. No matter what I try normal equation transfor... |
Could someone provide, please, a proof of the theorem below?
"Being $x$ and $b$ integers greater than $1$, which can not be represented as powers of the same basis (positive integer) and integer exponent, then the logarithm of $x$, in base $b$, is an irrational number."
Well... Assuming that the logarithm is a rational... |
Consider n loaded dice with differing probability distributions. I have constructed an equation that tells me the probability mass, the number of times a certain number is expected to be thrown.
Now i want to extend the numerical range from a line onto a plane. How do i get the expected number of times a number/number... |
This is an exercise from Apostol's number theory book. How does, one prove that $$ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n} \quad \text{if} \ n \geq 2$$
I thought of using the formula $$\frac{\varphi(n)}{n} = \prod\limits_{p \mid n} \Bigl(1 - \frac{1}{p}\Bigr)$$ but couldn't g... |
https://stackoverflow.com/questions/4113133/changing-a-positive-value-to-a-negative-one
why? is this just a computer science problem , or is this a mathematical property.
|
For a mapping algorithm I'm working on, I'm trying out the effect of sigmoid weightings.
Right now I'm using $$y = \frac{1}{1+x^n}$$ where n is the steepness of the sigmoid function.
This graph shows values of x between 0 and 256 and the following values of n: 2, 4 and 32. X values have been scaled to a range of -10 to... |
Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction.
I'm interested in how to calculate the limit of the sequence $x_0 = f(x), x_1 = f(x_0), \ldots, x_n = f(x_{n-1})$ for a fixed $x$. I couldn't figure out a way ... |
This problem has me scared to my wits, mostly because there are more like it!
The mean of a random sample of n observations drawn from a N $ ( \mu ,\sigma^{2} ) $ distribution is denoted by $ \overline{\chi} $. Given that P $ (| \overline{\chi} - \mu| > 0.5\sigma ) \lt 0.05 $
Find the smallest value of n.
So I tried:... |
I'm trying to parameterize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo:
For example, the ideal curve would be inside the unit cube; have only one point of intersection with every horizontal plane $z=k$, when $k\in [-1,1]$; would start in $(0, 0, -1)$ and end... |
Reading the Wikipedia page for Random Walk, I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as random walk on $\mathbb{Z}^d$, Gaussian random walk and random walk on a graph can fall into this general definition?
(1) In Ross's book for stochastic proce... |
Quoted from Wikepedia:
The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem.
I was wondering if the scaling limit of any random walk is always a Wiener process, or just the sc... |
I am trying to evaluate this integral or at least get bounds for its absolute value.
I have where $\tau \to \infty$
$$\int\nolimits_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$
$f(t) = (t + 1/2) - \lfloor t + 1/2 \rfloor$ (where $\lfloor.\rfloor$ is the floor function).
Integrating by parts gives m... |
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