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is it possible to calculate inverse of A-B, i have calculated inverse of A and B already.
A and B are matrices
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What connections are there between number theory and partial differential equations?
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Is there any existing literature on the properties/applications of the following class of functions?
$$\frac{f(E[x])}{E[f(x)]}\geq c$$
where $c< 1$ is a constant. Note that for $c=1$ these are exactly concave functions.
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I came across the following definition:
Given a ring $A$, with a unit $1 \in A$, and $A$-modules $M$ and $N$, we denote by $Hom(M, N)$ or $Hom_A(M, N)$ the space of $A$-linear maps from $M$ to $N$.
My question is: what exactly is the difference between homomorphism and a linear map? I can see that linearity is defined ... |
We know that in $\mathbb{R}^n$, we have transformation rules such as:
$\int_{\mathbb{R}^n}f(x-h) dx=\int_{\mathbb{R}^n}f(x) dx$
$\delta^n \int_{\mathbb{R}^n}f(\delta x) dx=\int_{\mathbb{R}^n}f(x) dx$.
The proof of such formulas can be easily found in a book about real analysis.
They reduce the problem into an integral ... |
I am hoping that this is not too basic a question for this site, but I am seeking to better understand a conversation I have had with my more mathematically inclined friend.
The discussion was around the probability of an event occuring with repeated exposure to a specific activity. The example in the conversation was ... |
I have the following sets:
x | 6 | 8 | 10 | 12 | 14 ...
y | 4 | 5 | 6 | 7 | 8 ...
I need to find the value of y given any positive value of x.
I know it sounds simple, but I seem to have drawn a complete blank on this one :/
Cheers
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Is it true that a compact polyhedron X with trivial homology groups (except $H_{0}(X)$ of course) is necessarily contractible? If yes, what is the approach in proving it? If not, do you see a counter-example?
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My teacher gave me a pseudocode of Dijkstra's algorithm using binary heap including the following steps (x was just extracted from the heap):
For each vertex y such that it is a node for it in a heap and there is an edge (x,y) in a graph:
1) Dist[y] = min(Dist[y], Dist[x] + w(x,y))
2) Update y's node in the heap
It ... |
I'm reading Hyperspace by Michio Kaku and in the chapter on SuperGravity "Super Numbers" are mentioned and are described as a number system where for any super number $a$, $a*a=-a*a$. I was wondering if anyone knew anything more about these numbers or could point me to another reference about them. Thanks!
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Possible Duplicate:
Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$
Hi,
My question is: How can I solve the following integral question?
$$\int(\sin ^4 x ) dx$$
Thanks in advance.
|
I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in the same way that $\Gamma (z)$ works as an extension of $f(n) =n!$. Since, I need practice with both $\Gamma $ and q-an... |
I need help with next statement :
Let $G$ be a group acting transitively on a set $A$. A nonempty subset $B$ of $A$ is called a block for $G$ if for each $x\in G$ either $B^x=B$ or $B^x \cap B=\emptyset$.
Suppose that $G$ acts transitively on $A$ and that $B$ is a block for $G$. Put $C=\{B^x\mid x\in G\}$.
Next part ... |
Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.
For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.
Is there a general method which allows us to evaluate the indefinite integral
$$ \int P(\sin x, \cos x) \text{d} x$$
What about the case when $\displaystyle P(x,y)$ is a rational ... |
Suppose I have a function
$S_n(t) = \sum_{k=1}^n a_k \sin((k-1/2)\pi t)$,
with unknown coefficients $a_k$. $S_n$ is perodic with period 4. If I can observe $S_n$ over the interval $[0,4]$, I can take a fast fourier transform to deduce the coefficients. I've implemented this in matlab, and everything works nicely.
Suppo... |
Suppose that a random variable $X \in [0,1]$ is drawn according to the density $f(x|\theta)$ conditional on the realization of $\theta \in [0,1]$. From $x$, we can generate $c$ according to some rule $c=g(x)$. We have know that $g$ is one-to-one. This procedure will give a distribution of $c$ given a realization of $\t... |
This might be slightly stretching the boundary of acceptable questions, but I think this is the best group to ask.
I'm interested in the classic 1887 texts "Leçons Sur La Théorie Générale Des Surfaces Et Les Applications Géométriques Du Calcul Infinitésimal" by French mathematician Jean Gaston Darboux (and which can be... |
Nurikabe is a constraint-based grid-filling puzzle, loosely similar to Minesweeper/Nonograms; numbers are placed on a grid to be filled with on/off values for each cell, with each number indicating a region of connected 'on' cells of that size, and some minor constraints on the region of 'off' cells (it must be connect... |
Let's say I have a sequence $\{x_1,...x_n\}\in\mathbb{R}^+$, where $x_i<x_{i+1}$ and $x_{i+1}-x_i$ is random (let's keep it general and say that the distribution is unknown).
Is it possible to form a new sequence $z_m$ by inserting values $y_k$ in between the $x_n$ as necessary (for e.g., ${x_1,y_1,y_2,x_2,y_3,...y_k,x... |
Suppose I have independent random variables $X_i$ which are distributed binomially via
$$X_i \sim \mathrm{Bin}(n_i, p_i)$$.
Are there relatively simple formulae or at least bounds for the distribution
$$S = \sum_i X_i$$
available?
|
Lusin Theorem (as stated by Rudin):
Let $X$ be a locally compact Hausdorff space and let $μ$ be a regular Borel measure on $X$ such that $μ(K)<∞$ for every compact $K⊆X$. Suppose $f$ is a complex measurable function on $X$, $μ(A)<∞$, $f(x)=0$ if $x∈X \setminus A$, and $ϵ>0$. Then there exists a continuous complex funct... |
Prove or provide a counterexample:
Let ${r_{n}}$ be a sequence such that $r_{n} \rightarrow 0$ as $n \rightarrow \infty$. Then, the sum $\sum\limits_{n=1}^{\infty} \frac{1}{n} r_{n}$ converges.
This has me stumped... I can't think of any counterexamples, but it just doesn't seem true.
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I'm trying to calculate the complexity of an algorithm. I've managed to boil down the cost of the algorithm to the following sum.
$$\sum_{k = 1}^{\log_{2}(n)} \frac{n\log_{2}(\frac{k^{2}}{2})}{k^{2}}$$
where $n$ is the size of the data set.
I'm struggling to turn the sum into a function that I can use to figure out t... |
I'd like to know what
$$\prod_{k=1}^n (1-x^k)$$
evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall less).
I'm sure I've seen this in the past but don't know where to place it. LaTeX search doesn't help.
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Question 49 of chapter 30 of Schaum's calculus is:
The section of a certain solid cut by any plane perpendicular to the x axis is a square with the ends of a diagonal lying on the parabolas
$y^2=4x$ and $x^2=4y$
Find its volume.
I don't have a problem integrating the cross sectional area across the x-axis. But I d... |
Q: Solve the separable differential equation:
$3x - 2y\sqrt{x^2+1}\frac{dy}{dx} = 0$ and subject to the initial condition: $y(0) = 4$. What does $y$ equal, in relation to $x$? I am just really confused what I am suppose to do? Thanks.
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I find it difficult to explain this because I'm not sure what is the correct term to use. Symbols such as the triple-bar-equal sign, and a few other symbols that I find difficult to describe. I was wondering if there was a list of symbols with definitions.
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let $f\in L^1$ and $g \in L^{\infty}$, 2pi periodic
show $\lim_{n\to \infty} \frac{1}{2\pi} \int_0^{2\pi} f(t) g(nt) dt = \hat{f}(0) \hat{g}(0)$
i tried using expressing the rhs as integrals, but i could not match the lhs
any suggestions?
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We have learned in class how to use inverse operator methods to solve ODE's (i.e. with the symbolic $D$). E.g,
If I were asked to find a particular solution, $y_p$ to $(D-1)(D-2)[y] = x^{2}e^{x}$, then I would use the formula $\frac{g(x)}{D-a} = e^{ax} \int e^{-ax} g(x) dx$, where $g(x) = x^2 e^x$.
However, here's the... |
A selfposed but never solved problem:
Is it possible to find a measure space
$(X, \mathcal{M} ,\mu )$ such that
the range of $\mu$ is something like
the Cantor set (i.e. a bounded,
perfect, uncountable, totally
disconnected set)?
I was thinking about this problem some time ago and now, reading some old MT ... |
i've heard that if in a population each pair has 4 children, then after 4 generations it's a 1600% increase..
I suppose after 1 generation it is 200%, and each successive generation is 200%, and they multiply 200%^4 But, if I try it out with figures like this, I never get 1600%
suppose a population of 10
g g1 g2 g3... |
I believe I have the gist of how to prove this. My professor worked out a problem similar to this one only, instead of ($s_n^3$), he used ($s_n^2$), and I am slightly confused as to how he came up with certain portions of his proof. The following is the proof he gave us for ($s_n^2$). I believe after understanding his ... |
Look for help with this problem. I realise it's not the most clear the way it's written but it's difficult to phrase.
Imagine you are assigned x items. You are then given y sets of size z each, which contain between them all between 0 and x of your items, and the remainder of each set is filled with items you were not ... |
So I'm covering material for my upcoming final exam, and I have a sneaking suspicion that my teacher will ask us to prove the following theorem:
A subset $G$ of $\mathbb{R}^n$ is open iff the complement of $G$ is closed.
He's hinted at it a couple times, and honestly, I don't know where to start. Thanks for the hel... |
This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
In other words, if I am given a polynomial $... |
Compute angular length $c$ of the great-circle route between these two cities:
Daytona Beach (location A): $29^\circ12'\ N, 81^\circ1' \ W$.
Sidi Ifni (location B): $29^\circ23' \ N. 10^\circ10' \ W$.
Ok so I converted the latitudes and longitudes and I now have:
Daytona Beach (location A): $29.20^\circ N, 81.02^\... |
If we have
$$ x^y = z $$
then we know that
$$ \sqrt[y]{z} = x $$
and
$$ \log_x{z} = y .$$
As a visually-oriented person I have often been dismayed that the symbols for these three operators look nothing like one another, even though they all tell us something about the same relationship between three values.
Has a... |
I am working on an electronics problem where we measure some faint ripple over a huge constant. You can think of it as $100 \,\rm{V}$ DC plus a ripple of $\pm$ a couple $\rm{mV}$ over that $100\,\rm{V}$. Our signal-to-noise (SNR) calculation is $-130 \,\rm{dB}$. Is it possible to identify this faint ripple out of this ... |
All the problems I've seen treat martingales as a sum of independent rvs, $X_t=\sum_{k=1}^{n}B_k$, where $\mathbf{E}[B_{k}|\mathcal{F}_{k-1}]=\mathbf{E}B_{k}$. What is a good approach to study (e.g. find bounds on the first passage time etc) for martingales that are sums (or other functions) of rvs that are not indepen... |
Let $G$ be a topological group. $G$ comes equipped with a left (resp. right) uniformity $\mathscr{L}$ (resp. $\mathscr{R}$) which can be characterized as the coarsest uniformity which is compatible with the topology and which makes $x \mapsto gx$ (resp. $x \mapsto xg$) a uniformly continuous map $G \to G$ for all $g \i... |
It has come time for me to decide where to pursue my undergraduate education. I plan to pursue a PhD in mathematics, so accordingly my primary concern is the mathematics programs at various institutions. While I realize that college selection is a personal process, I would like to hear some input from people familiar w... |
I'm working on rotating human limbs in a 3d game. I use Linear Algebra matrix rotations and translations to achieve moving the human and limbs. I currently can rotate around a pivot point by first translating to the pivot point then performing the rotation and finally translating back to the origin. I do that easily e... |
If $b>1$ is an integer, is well know that the numbers $x\in (0,1]$, can be written as
$$x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$$
for some integers $a_k \in \{0,1,\ldots ,b-1\} $. When $x=\frac{1}{b^n}$, for some $n\in \mathbb{N}$, there is two representations. One of them is always $0$ beginning with some index, that ... |
Suppose $f \in L^1$, $2\pi$ periodic and that the Fourier coefficients decay with order $|n|^{-k}$, $k \gt 2$.
Show that the derivative of $f$ is continuous.
I read that the rate of decay of Fourier coefficients relates that the "smoothness" of the function. But I'm not sure how to formalize my argument for this questi... |
I'm trying to prove
$$\sum_{m|n} \mu(m)^2/\phi(m) = n/\phi(n)$$
My first realization was that $\mu(m)^2 = 1$ iff $m=1$ or $m$ is a squareless factor of $n$ and otherwise is 0. Let $\{1,m_1,m_2, ... m_k\}$ be the set of squarefree factors of $n$.
So now we have a sum
$$\frac{1}{\phi(1)} +\frac{1}{\phi(m_1)} + \cdots +... |
In a previous post, I proved (with help) that the convergence of the sequence ($s_n$) implies the convergence of (${s_n}^3$). Here is a link to it: Prove that the convergence of the sequence ($s_n$) implies the convergence of ($s_n^3$)
Now, I want to either prove the converse or determine a counter-example.
OK, I bel... |
It is well known that eigenvalues for real symmetric matrices are the same for matrices $A$ and its transpose $A^\dagger$.
This made me wonder:
Can I say the same about the singular values of a rectangular matrix? So basically, are the eigenvalues of $B B^\dagger$ the same as those of $B^\dagger B$?
|
Why is this true?
$$\int_0^\infty x \frac{M}{c} e^{(\frac{-x}{c})} (1-e^{\frac{-x}{c}})^{M-1} \,dx = c \sum_{k=1}^M \frac{1}{k}.$$
I already tried substituting $u = \frac{-x}{c}$. Thus, $du = \frac{-dx}{c}$ and $-c(du) = dx$. Then, the integral becomes (after cancellation) $\int_0^\infty c u M e^u (1-e^u)^{M-1}\,du$.
I... |
How does one find all integer solutions to the equation $3a^2-4b^3=7^c$?
|
What I would like to do is design a program (for academic purposes) which will take a representation of a DFA (as a directed graph) and display the regular language which it accepts, in a reasonable format.
For example:
For this graph: as input, an algorithm will spit out: a*b*
The set of regular languages is computab... |
Please show that for all $x,y\in\mathbb{R}$,
$$e^{x+y} - e^xe^y = \lim_{k\to\infty} \sum_{n=1}^k \sum_{j = 0}^n\left(\frac{x^{k+j}}{(k+j)!}\frac{y^{n-j}}{(n-j)!} + \frac{x^j}{j!}\frac{y^{k+n-j}}{(k+n-j)!}\right)\;.$$
(This is from a homework assignment)
I'm thinking of a way to proceed by induction, but it is not clea... |
According to Gallot-Hulin-Lafontaine one has
$$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_1,\cdots,X_{i-1},X_0,X_{i+1},\cdots,X_q)$$
It seems to me that it should be
$$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_0,\cdots,\hat{X_i},\cdots,X_q)$$
Is this right ?
|
Before I state my question, let me give the set-up / "what I know".
Let $F / K$ be an extension of number fields of degree $n$. Let $v$ be a (discrete) valuation on $K$ and let
$$
\mathfrak{O}_{v} = \{ \alpha \in K : v(\alpha) \geq 0 \}
$$
be the valuation ring of $v$. Up to associates, $\mathfrak{O}_{v}$ has a... |
Let us consider two different models of the continuum $\mathbb{R}$ (that is, we take two arbitrary ZF-models, and we look at the continuum in each one of these models).
Let us now suppose that we have these two models enriched with additional structure (for example with addition, product, topology, etc.). Under which c... |
Can the centralizer of Inn(G) in Aut(G), where G is preferably any non-abelian finite one, equal to Inn(G) itself? Clearly, such centralizer contains all $f$ in Aut(G) where $f(g)g^{-1}$ are in Z(G).
|
The following question has been lingering in my mind for months.
Let $R$ be a non-zero commutative ring with $1$. Consider $\phi : R^n \rightarrow R^m$,
1) as an injective $R$-module homomorphism.
2) as an injective ring homomorphism. (by definition $\phi(1)=1$.)
In which of the above cases, we can deduce that $n \leq... |
I want to prove the assertion:
The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.
My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and
$$
p = \epsilon^{-1} (1-\zeta_p)^{p... |
I think (hope) I'm on the right track with this problem, but there are details that I can't seem to work out. I've also struggled to find examples of this sort of problem to assist me.
Let
$\rho (\mathbf{x}, \mathbf{y}) = \sqrt{(\mathbf{x} - \mathbf{y})^T \mathbf{A} (\mathbf{x} - \mathbf{y})}$,
where $\mathbf{x}=(x_1,... |
This question relates to a discussion on another message board. Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My understanding is that Intuitionists reject such proofs because they rely on the Law of the Excluded Middle, which they ... |
I'm interested in properties of relations. Things like completeness (connected, total), transitivity, euclideanness, symmetry and so on. I am interested in the logical connections between these relations. For example, symmetry implies not asymmetry. Or a reflexive, weakly connected relation is complete.
Is there a neat... |
My textbook says that multiplying block matrices works the same as regular matrix multiplication (granted the dimensions of the submatrices are appropriate). Wikipedia also has an example saying so.
It seems like proving it is purely technical and yet I'm having trouble putting it into words. What would be a good way t... |
Example:
for function $$f(x)=x^{3}(1-x)^{3}=\sum f_{s}\exp(2\pi isx)$$ Fourier series of its fourth derivative are different from derivative of its Fourier series
$$f^{(4)}(x)=-360x^{2}+360x-72=\sum g_{s}\exp(2\pi i s x)$$
with $g_{s}\neq(2\pi s)^{4}f_{s}$
Related question:
When $$\sum\left|f_{s}\right|^{2}j^{2p}<\i... |
I was thinking about random sequences, and thought up of this problem.
Can algebraic or rational be expressed as a sum or difference of integer square roots?
For example, $5=\sqrt{25}=\sqrt{16}+\sqrt{1}$.
It is known that if the series is infinite any number can be approximated, but is it possible to get any algebrai... |
I'm using the ternary numeral system, i.e. numbers in base 3. There's a catch: The numbers I'm representing will never have a 2 as a digit. For instance, I use 0,1,10,11,100,101,110,111,... I.e. they look like binary digits, but they're really ternary numbers.
Now I'd like to know, for a random ternary number, in t... |
I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a 2-dimensional delta function. Starting with something we know to be true, we can do $$1 = \iint \delta^{(2)}(x,y) dx\,dy = \i... |
I was flipping through Milnor's "Topology from the Differentiable Viewpoint," and I came upon a sentence concerning the mod 2 degree of a function from M to N. It essentially says:
"We may as well assume also that N is compact without boundary, for otherwise the deg mod 2 would necessarily be zero."
I understand why ... |
I have a parametric representation of a cylindrical shape (well, it's like a cone, but its spike is trimmed). I would like to have an analytic expression for its silhouette lines in terms of the shape's parameters. The parameters I use are:
Top circle center
Bottom circle center
Top circle radius
Bottom circle radius
... |
I'm doing exercise 16 on page 39 in Hatcher:
Show that there are no retractions $r: X \rightarrow A$ in the following cases:
(a) $X = \mathbb{R}^3$ with $A$ any
subspace homeomorphic to $S^1$
(b) $X = S^1 \times D^2$ with $A$ its
boundary torus $S^1 \times S^1$
(c) $X = S^1 \times D^2$ and $A$ the circle shown in the ... |
Possible Duplicates:
Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$
Ways to evaluate $\int \sec \theta \, \mathrm d \theta$
Using Mathematica to get the antiderivative for sec(x), I get $$-\log(\cos\frac{x}{2}-\sin\frac{x}{2})+\log(\cos\frac{x}{2}+\sin\frac{x}{2}).$$
This doesn't look familiar, so, I'm thinking th... |
I'm told that the following statement is true:
"If a limit point of the set S is defined as x, then every open ball that is centered at x
contains infinitely many points of S."
Yet I can't begin to imagine how the proof is completed. Any assistance would be very welcomed. Thanks!
|
In a paper I'm currently reading, they have the following situation:
$k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group $\Delta (=(\mathbb{Z}/p\mathbb{Z})^{\times})$. For any $\chi \in \hat{\Delta}:=Hom(\Delta, \mathbb{Z}_p^{\times})$ they ... |
I was wondering for a Markov process:
Is it true that:
it is a stationary process, iff its transition probability
function is invariant to
time-translation (i.e. it is
homogeneous) and its initial
distribution is invariant to
time-translation?
When the state space is discrete, is
the limit of its transitio... |
this question was posted on aops about 3 weeks ago, but never received an answer. Maybe it will fare better here?
Let $G$ be a finite group and $H$ a subgroup. Let $P_H$ be a $p$-Sylow subgroup of $H$. Prove that there exists a $p$-Sylow subgroup $P$ of $G$ such that $P_H=P\cap H$.
Source: The exercise comes from the... |
I require some guidance with the following question:
Consider the following subsets of all integers.
$$\begin{align*}
A&=\{2n+1\mid n\text{ is an element of all integers}\}\\
B&=\{3n\mid n\text{ is an element of all integers}\}\\
C&=\{3n+2\mid n\text{ is an element of all integers}\}
\end{align*}$$
Find each o... |
I am not sure how to solve these equations because the involve the ceiling function. I am trying to solve for $M$ and $L$ in terms of only $C$. Is there a way? If not, is there an algorithm that could calculate $M$ and $L$ values based on a given $C$ value? Here are the equations:
$$M = C + L - 1$$
$$L = \lceil\ 0.... |
This is an online solution to one of the problems I'm working on. The question is to analyze the statement at the beginning of each sentence and determine whether its meaningful and if so then is it a vector field or scalar field. However I'm confused by B and D and E. Are these answers incorrect? What should the right... |
So I'm given:
$h(1) = -2$
$h'(1) = 2$
$h''(1) = 3$
$h(2) = 6$
$h'(2) = 5$
$h''(2) = 13$
The question is find $$\int_{1}^{2} h''(u) \text{d}u$$
So based on the Fundamental Theorem of Calculus (part 2) $\int_{a}^{b} f(x)\text{d}x = F(b) - F(a)$
I would figure that :
$h''(u) = h'(2) - h'(1) == 3$
Is that the right w... |
How does one see that the cyclic group $C_n$ of order $n$ has $\phi(d)$ elements of order $d$ for each divisor $d$ of $n$?
(where $\phi(d)$ is the Euler totient function)
|
I'm hoping to really knock out several questions I have in my mind with just this one. I've been doing a lot of practice problems on this topic, and although I get the right answers, I really don't know what the answers mean. So there's a theorem that says
If F is a vector field defined on all
of $R^3$ whose compone... |
Assume that the linear transformation is known as a matrix.
I am aware of algorithms (by elementary row operations etc.) which calculate the rank of a matrix and hence the dimension of the kernel of the matrix.
But I would like to know of some non-algorithmic way of doing the same (if it exists). Like if there is a clo... |
Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.
|
First of all, rest assured that I have used the search tool and read Why do we require a topological space to be closed under finite intersection? and https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets.
I'm looking for a more logical approach on this issue. I understand intuitively the reas... |
How I can prove any element of order 2 in a Weyl group is the product of commuting root reflections. I need to show also that the only reflections in Weyl group are the root reflections.
|
I have following problem. I have set of objects, there is N objects. These object might join with each other. Two objects might join with probability $p_{join}$. In every computations step I take all possible couples of objects and put them into queue. For each couple from the queue I draw a number (from (0...1)) if nu... |
I have a rudimentary grasp of the second Borel-Cantelli lemma and vaguely understand how finite patterns can repeat infinitely often. The one proof I am familiar with that verifies this unfortunately relies on a stochastic process of uniform probability distributions over some finite discrete random variable.
What happ... |
Let $x_{i+1}$ be a uniform distributed random variable in $[0,x_i]$, with $x_0=1$.
Does the sum converge and what is its expected value?
$$\sum_{n=1}^\infty x_n$$
Does there exist a function $f(n)>1$ such that $\lim_{n->\infty} f(n)=1$ and the sum
$$\sum_{n=1}^\infty n^{-f(n)}$$ converges?
|
I'm trying to follow Ahlfors's proof that any analytic function defined in an annulus $R_1 < |z-a| < R_2$ will have a Laurent representation. To do this, he defines two functions:
$$f_1(z) = \frac{1}{2\pi i} \int_{|\zeta-a|=r} \frac{f(\zeta) d\zeta}{\zeta-z} \text{ for $|z-a| < r < R_2$ } $$
$$f_2(z) = - \frac{1}{2\pi ... |
$$\int_0^\frac{\pi}{4} \! \frac{1+\cos^2\theta}{\cos^2\theta} d\theta$$
I've been attempting to mix and match identities to make this equation easier to integrate.
Mathematica has given me an answer of $\frac{4 + \pi}{4}$, so I'm trying to get there but am struggling.
Any suggestions? Thanks
|
I have an ellipse with a given major and minor 'radius'. I then apply a 2D skew affine transformation to it. Then, I want to draw a uniform border inside this new shape, as if a circle were rolled along the interior edge of the newly transformed ellipse, and everywhere the circle touched was drawn in a different colo... |
Possible Duplicate:
How to “Re-write completing the square”: $x^2+x+1$
$b^2-4b+4=16$ can someone help me solve this equation. All the steps please.
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So the example he gave was using the ML-Inequality. Let $v = \{e^{i \theta} \mid 0 \leq \theta \leq 2\pi \}$.
$\left|\int(1/z)\,dz\,\right|$ with respect to $v$ is less than or equal to $\int \left|1/z\right| \left|dz\right|$ with respect to $v^*$. Then he says that is equal to the integral with respect to $v$ of $\le... |
Hungerford defines a field, $E$ as being closed if $E=E''$ where
$E'= \{ \sigma \in \mathrm{Aut}(F/K)|\sigma(u)=u \text{ for all } u\in E \} = \mathrm{Aut}(F/E)$ is a subgroup of $\mathrm{Aut}(F/K)$,
$F$ is an extension field of $K$.
I want to show that in the extension of $\mathbb{Q}$ by $\mathbb{Q}(x)$, the inte... |
How can I show that $\alpha$ is exact form on $S^1$ iff $\int_{S^1} \alpha =0$?
I used Stokes theorem to prove one direction. Could you please help me in the other direction?
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I'm currently working on a problem where I need to provide a rigorous proof that multiplication among interval numbers is a associative. For those of you who haven't heard of an interval number before, here are some definitions:
We say that $x$ is an interval number on $\mathbb{R}^n$ if $x \subset \mathbb{R}^n$ and $x... |
In Lindsay Childs' Algebra text (3ed pg 141), it statements this proposition regarding a ring homomorphism: $ Let f: R \rightarrow S$ be a homomorphism where R is a field and $1 \ne 0$ in $S$. Then $f$ is one-to-one.
My confusion is about $1 \ne 0$. Is that possible? I can only see that in congruence where $[1]_{1} = [... |
I am trying to prove the following:
For any positive natural n, there exists an undirected graph of n nodes, positive natural edge widths, and nodes s and t such that a uniform-cost search from s examines all the nodes, whereas the optimal path from s to t consists of only a single edge.
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this may sound like a dumb question but are fourier coefficients always symmetric?
ie $\hat{f}(n) = \hat{f}(-n)$?
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This is from a practice exam that we are working on, problem number 2. We were thinking first to use Cauchy-Goursat, but then the problem only says that the curve doesn't lie on the singularities of $\sin^2(z)$. They might still be interior to the curve. Is there another way to approach the problem using theorems we mi... |
Is the ring $\mathcal{O}$ of germs of $C^{\infty}$ functions defined on the neighborhoods of $0\in\mathbb{R}$ the localization of the ring of $C^{\infty}$ functions on $\mathbb{R}$ at the maximal ideal $\mathfrak{m}$ of those functions which vanish at the origin? I can construct an injective map from $C^{\infty}_\mathf... |
Show that $F: (\mathbb{R}^n, \varepsilon) \rightarrow (\mathbb{R}^n, \rho), \quad \mathbf{x} \mapsto \mathbf{x}$ is continuous, where $\varepsilon$ is the Euclidean metric and $\rho(\mathbf{x},\mathbf{y}) = \sqrt{\sum_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)}$, and $\mathbf{A}$ is a symmetric positive-definite real $n \time... |
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