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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
I read in Stewart "single variable calculus" page 83 that the limit $$\lim_{x\to 0}{1/x^2}$$
does not exist. How precise is this statement knowing that this limit is $\infty$?. I thought saying the limit does not exist is not true where limits are $\infty$. But it is said when a function does not have a limit at all li... |
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Now showing items 11-20 of 27
Pseudorapidity dependence of the anisotropic flow of charged particles in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
(Elsevier, 2016-11)
We present measurements of the elliptic ($\mathrm{v}_2$), triangular ($\mathrm{v}_3$) and quadrangular ($\mathrm{v}_4$) anisotropic azimutha... |
Problem 8
Let $G$ be an abelian group and let $H=\{ g \in G \mid \ \mid g \mid$ divides $12 \}$. Prove that $H$ is a subgroup of $G$. Would your proof be valid if the number $12$ were replaced by another number? State the general result.
Solution Let $G$ be an abelian group and let $H=\{ g \in G \mid \ \mid g \mid$ div... |
Pasting Lemma Theorem
Let $X$ and $Y$ be topological spaces.
Let $A$ and $B$ be open in $X$.
$\forall x \in A \cup B: \map {f \cup g} x = \begin {cases} \map f x & : x \in A \\ \map g x & : x \in B \end {cases}$ Then the mapping $f \cup g : A \cup B \to Y$ is continuous.
Let $A$ and $B$ be closed in $X$.
Then the mappi... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
The words “Approach” and “Tend” are often used in calculus and it’s a basic concept to start learning the limits.
According to English language, the real meaning of “Approach” or “Tend” is, come near or nearer to something.
For example, if a point comes nearly to another point, then it’s said that the point approaches ... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
I'm familiar with extraneous roots. For example $\sqrt{x} = x - 2$
We solve it by squaring both sides \begin{align*} & \implies x = x^2 - 4x + 4\\ & \implies x^2 - 5x + 4 = 0\\ & \implies (x-1) (x-4) = 0\\ & \implies x = 1~\text{or}~x = 4 \end{align*}
But! Only $x = 4$ satisfies the parent equation, $x = 1$ doesn't. He... |
RSA for key exchange is declining rapidly and is not recommended because it does not provide forward secrecy. Without forward secrecy, if someone breaks into the server and obtains the private key, they will be able to fully retroactively decrypt all recorded traffic encrypted under that key. ECDH does not have that pr... |
Let $\{a_n\}$ be a sequence such that $$\lim_{n\to \infty}\left|a_n+3\left(\frac{n-2}{n}\right)^n\right|^\frac{1}{n}=\frac{3}{5}$$
Then calculate $\lim_{n\to \infty}a_n$.
First I tried to take logarithm and got $\lim_{n\to \infty}\frac{1}{n}\ln\left|a_n+3\left(\frac{n-2}{n}\right)^n\right|=\ln\frac{3}{5}$, then I thoug... |
Statement:
Let $G$ be a group, $H$ a subgroup of $G$, and $a,b \in G$. Then:
$a \in aH$ $aH = H \iff a \in H$ Either $aH = bH$ or $aH \cap bH = \emptyset$ $aH = bH \iff a^{-1} b \in H$ $aH \leq G \iff a \in H$ $|aH| = |bH| = |H|$ Proof:
1. Since $H$ is a subgroup of $G$, $e \in H$, so $a = ae \in aH$.
2. We must prove ... |
Cantor ternary function
The Cantor ternary function (also called Devil's staircase) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for ... |
Cantor ternary function
The Cantor ternary function (also called Devil's staircase) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for ... |
Consider a reversible reaction $\ce{A -> B}$, at standard state wher $\ce{A}$ and $\ce{B}$ are at $\pu{1 atm}$. If the free energy of products is greater than the free energy of reactants, $\Delta_{\text{R}} G^\circ > 0$, and since $Q = 1$, the overall free energy for this reaction is $\Delta_{\text{R}} G^\circ > 0$ an... |
I have been struggling to find an acceptable answer for this question for my purposes.
There are many ways to find similarity between two organic compounds - some of which are particularly popular in chemoinformatics. The seemingly most popular way is to use fingerprints of molecules, which then somehow correlates to t... |
Disclaimer: this terminology might be different from what you're used to and this is why I'm writing down some definitions first.
Let $A$ be an abelian group, $m \in \mathbb{N}^* := \mathbb{N} \setminus \{0\}$ and let $p$ be a prime number. We define:
The subgroup of multiples of $m$ in $A$as the set $mA := \{\,ma \, \... |
I recently came across the following argument regarding the uniqueness of the zeros of a complex polynomial.
Please note that the proof that a complex polynomial of degree $m$ has $m$ zeros has been established at this point. The following is also not the complete proof of the uniqueness of the roots. I just want to fo... |
Derivative of Arc Length Theorem
Then the derivative of $s$ with respect to $x$ is given by:
$\dfrac {\d s} {\d x} = \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2}$
Consider a length $\d s$ of $C$, short enough for it to be approximated to a straight line segment:
By Pythagoras's Theorem, we have:
$\d s^2 = \d x^2 + \d y... |
Equivalence of Definitions of Separated Sets Contents Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let $A, B \subseteq S$.
$A$ and $B$ are
separated (in $T$) if and only if: $A^- \cap B = A \cap B^- = \O$
$A$ and $B$ are
separated (in $T$) if and only if there exist $U,V\in\tau$ with: $A\subset U$ and $U\... |
→ → → → Browse Dissertations and Theses - Mathematics by Title
Now showing items 288-307 of 1147
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(2014-05-30)In this work, we prove effective decay of certain multiple correlation coefficients for Weyl chamber actions of semidirect products of semisimple groups with $G$-vector spaces. Using the... |
OpenCV 3.2.0
Open Source Computer Vision
void cv::accumulate (InputArray src, InputOutputArray dst, InputArray mask=noArray()) Adds an image to the accumulator. More... void cv::accumulateProduct (InputArray src1, InputArray src2, InputOutputArray dst, InputArray mask=noArray()) Adds the per-element product of two inpu... |
Fubini's Theorem Theorem
Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$.
Let $f: X \times Y \to \R$ be... |
Research articles for the 2019-07-07
arXiv
Let $\psi$ be a multi-dimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_t=\mathbb{E}^{\mathbb{Q}}\left[\psi\lvert\mathcal{F}_{t}\right]$ has the Martingale Representation Property (MRP) ... |
Let $A \in \mathcal{M}_3(\mathbb{C})$ such that $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q},$ where $\mathrm{tr}(A)$ is the trace of $A.$ It is possible to prove that $\mathrm{tr}(A^4) \in \mathbb{Q}?$
closed as off-topic by Travis, Especially Lime, John B, Mars Plastic, user21820 Sep 15 at 14:23
This question ... |
''Diamond Paradox'' by Diamond (1971)
This is a "less-known paradox," usually put as a counter to famous Bertrand paradox. It is a starting point in the literature on informational frictions in consumer markets, and the scientists in the field agree on its significance.
Its idea is diametrically opposite to that of Ber... |
ä is in the extended latin block and n is in the basic latin block so there is a transition there, but you would have hoped \setTransitionsForLatin would have not inserted any code at that point as both those blocks are listed as part of the latin block, but apparently not.... — David Carlisle12 secs ago
@egreg you are... |
$\log_{b}{(m^{\displaystyle n})}$ $\,=\,$ $n\log_{b}{m}$
Power Rule of logarithm reveals that log of a quantity in exponential form is equal to the product of exponent and logarithm of base of the exponential term.
$q$ is a quantity and it is expressed in exponential form as $m^{\displaystyle n}$. Therefore, $q \,=\, m... |
Determine all eigenvalues of the matrix $$A=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$$ and then determine a base for each eigenspace.
It's easy to compute $\chi_A(z)=z^3-1$ so my roots (and therefore eigenvalues) are $z_1=1, z_2=\cos(2\pi/3)+i \sin(2\pi/3)$ and $z_3=\cos(4\pi/3)+i \sin(4\pi/3)$.
Usually I would ... |
Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$
By working with eliptic curves, I found that the extension E defined by:
E.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 - 782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 - 114*x^7 - 4... |
Asked by: Question
Hi,
Please help me how to load latex equation in windows 8.1 application.
I found this https://math4winrt.codeplex.com/SourceControl/latest but not working for all math formulas.
I wont use webview becoz i have to load more than 20 equations in my project.
please help me by giving source code that it... |
The value of cot function when angle of right triangle equals to $45^°$ is called cot of angle $45$ degrees. As per sexagesimal system, it is written as $\cot{(45^°)}$ in mathematics.
$\cot{(45^°)} \,=\, 1$
The exact value of cot of angle $45$ degrees is equal to $1$ and it is also an integer.
$\cot{(45^°)}$ can be wri... |
It is a trigonometric expression which contains sine and cosine functions with complementary angles. This trigonometric problem can be simplified in two methods by using complementary angle trigonometric identities.
In this method, all trigonometric functions which contain complementary angles are simplified firstly by... |
Are there matrices that satisfy these two conditions? That is, a matrix $A$ such that
$$A^T=A^{-1}=-A$$
What I know is that a skew-symmetric matrix with $n$ dimensions is singular when $n$ is odd.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related... |
Linear Boltzmann equation and fractional diffusion
1.
Laboratoire J.-L. Lions, BP 187, 75252 Paris Cedex 05, France
2.
CMLS, École polytechnique, 91128 Palaiseau Cedex, France
3.
DPMMS, University of Cambridge, Wilberforce Road, CB3 0WA Cambridge, United Kingdom
Consider the linear Boltzmann equation of radiative trans... |
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1. Precision measurement and interpretation of inclusive W+, W−and Z/γ production cross sections with the ATLAS detector
The European Physical Journal C: Particles and Fi... |
$x$ is a variable, which is considered as an angle of a right triangle and the sine function is written as $\sin{x}$ in trigonometric mathematics. The indefinite integral of $\sin{x}$ with respect to $x$ is written as follows to find the integration of sine function in calculus.
$\displaystyle \int{\sin{x} \,}dx$
Write... |
Reciprocal of Strictly Positive Real Number is Strictly Positive
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Theorem $\forall x \in \R: x > 0 \implies \dfrac 1 x > 0$ Proof
Let $x > 0$.
Aiming for a contradiction, suppose $\dfrac 1 x < 0$.
Then:
\(\displaystyle x\) \(>\) \(\displaystyle 0\) \(\displaystyle \leadsto \ \ \) \(\dis... |
Just so there are no misunderstandings let me first ask whether it is true that:
$$ \int_{-\infty}^{\infty}x\delta(x)\mathrm{d}x=0. $$
If that is not true, then I don't know anything about the Dirac delta distribution and I will be off to correct this :)
Otherwise I have this question: Why can we take a delta functiona... |
$\dfrac{d}{dx}{\, (c)}$
According to definition of the derivative, the differentiation of $f{(x)}$ with respect to $x$ can be written in limit operation form.
$\dfrac{d}{dx}{\, f{(x)}}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to \, 0}{\normalsize \dfrac{f{(x+\Delta x)}-f{(x)}}{\Delta x}}$
Take $f{(x)} \,=\, c$,... |
Let a triangle be inscribed in a unit circle, and let $A$ and $B$ mark two vertices. Let $\theta$ be half the length of the arc connecting $A$ and $B$, and let $\ell$ be the length of the chord, you have that from elementary trigonometry
$$ \ell / 2 = \sin (\theta) $$
Now let $\theta_1, \theta_2, \theta_3$ be half the ... |
Transformation semi-group
Any sub-semi-group of a symmetric semi-group $T_\Omega$, where $T_\Omega$ is the set of all transformations of a set $\Omega$. Particular cases of transformation semi-groups are transformation groups (cf. Transformation group).
Two transformation semi-groups $P_1 \subset T_{\Omega_1}$, $P_2 \s... |
Convergence and, in fact, uniform convergence for case (1) follows from the Dirichlet test since $\displaystyle\left|\int_1^c \cos x \, dx \right| \leqslant2 $ for all $c > 1$ (uniformly bounded) and $x^{-\alpha} < x ^{-\alpha_0}$ which implies that $x^{-\alpha} \downarrow 0$ monotonically and uniformly for all $\alpha... |
Good evening everyone,
I'd like to discuss with you the following exercise :
$\sum\limits_{n=1}^{\infty} (-1)^{n} \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
I can prove that $\lim\limits_{x \to \infty} a_{n} = 0$ , where $a_{n} = \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
But I can't still proove its conv... |
Type the following command at the terminal to copy the template file to the current directory (note the period at the end):
cp ~cs61as/autograder/templates/hw3.rkt .
Or you can download the template here.
Here is the
fast-expt procedure from earlier in this lesson:
(define (even? n) (= (remainder n 2) 0))(define (fast-... |
Do numbers really represent the physical reality? For example 1 apple simply tells us that there is only one apple. Now it becomes complicated when simple arithmetic formulaes are applied. Example: 1 apple x 1 apple = 1 apple. Now I don't know where the other apple got lost.
161
Joe J
03-21-2019
05:41 AM ET (US)
How to... |
In my last post, I wrote about
within- and between-period intra-cluster correlations in the context of stepped-wedge cluster randomized study designs. These are quite important to understand when figuring out sample size requirements (and models for analysis, which I’ll be writing about soon.) Here, I’m extending the c... |
I have had a number of friends ask me “Okay, AlphaGo was impressive, but where else can you use Reinforcement Learning (RL)?”.
If you google the question, you will find a legitimate list like this paper or this article. Today I would like to talk about a cool application of RL that is not as often mentioned: combinator... |
..have to do the burn in..?
As pointed out in the comments, taking out some of the initial samples is not a mathematical necessity. We take out
burn in because it is a computational statistics hack/cheat to avoid having to get more samples to correct the bias of the initial samples. In the small samples case, the bias ... |
I have a conceptual problem to understand the standard error of the ratio of two random variables after error propagation.
Let $X$ and $Y$ be two random variables with means $\bar x$ and $\bar y$ and standard errors $se_x = \frac{\sigma_x}{\sqrt{m}}$ and $se_y = \frac{\sigma_y}{\sqrt{n}}$, where $m$ and $n$ are the sam... |
Since no one has come up with an answer yet, here is a rather brute-force computation on homogeneous coordinates to verify this fact. Without loss of generality I'll choose my coordinate system in such a way that the incircle becomes the unit circle, with $K=(1:0:1)$ at the $0°$ position. Then $D$ and $E$ can be descri... |
Here is a schematized binary channel that neatly conveys a decimal number. $ \require{begingroup}\begingroup \def\T {{ \cal T }} \def \Ti {{ \T \raise5mu{ \text- \scriptsize 1 } }} \def\Bx #1{{ ~ \rightarrow ~ \boxed{\, #1 \,\Large\strut} \: \rightarrow ~ }} \def \BTi {\Bx { \kern 1mu \Ti \kern1mu }} \def \BT { \Bx{ \r... |
Here is the graph pf the functionIt is apparent that the minimum distance from the origin occurs about $x=\pm4$.
Here is the graph of the distance of the points on the parabola as a function of $x$
It is even more convincing that the minimum distance occurs very close to $\pm4$.
Let's calculate the distance as a functi... |
This is a two part post. The second part depends on the first.
Part 1. Consolidation of the Denotational Semantics
As a matter of expediency, I've been working with two different versions of the intersection type system upon which the denotational semantics is based, one version with subsumption and one without. I had ... |
We should find the Cauchy principal value integral of the form $$ I=\oint \frac{dz}{(z-z_1)(z-z_2)}~, $$ where both roots $z_1$ and $z_2$ lie on the contour path. My answer is: $$ I=a \left(-\oint \frac{dz}{z-z_1}+\oint \frac{dz}{z-z_2}\right)=a(-i\pi+i\pi)=0~, $$ where $a=1/(z_2-z_1)$. However, in a book, they do not ... |
Lebesgue outer measure Set
context $p\in \mathbb N$
definiendum $\eta^p:\mathcal P(\mathbb R^p)\to \overline{\mathbb R}$ definiendum $\eta^p(A):=\mathrm{inf}\{\ \sum_{k=1}^\infty\lambda^p(I_k)\ |\ I\in\mathrm{Sequence}(\mathfrak J^p)\ \land\ A\subset\bigcup_{k=1}^\infty I_k\ \}$ Discussion
The Lebesgue outer aims at me... |
I'll give a somewhat intuitive explanation of why we might expect a chi-squared approximation in large samples (including illustrating a connection to sums of squared normals) and give a couple of references along the way.
Let us begin by starting with an ordinary F statistic for one-way ANOVA.
This F-distribution of t... |
Fourier Series CT Fourier Transform
Introduction Examples Properties FT of periodic signals
Recall: Fourier series representation of a periodic signal $\tilde{x(t)}$ with time period $'T'$ is given by:-
Suppose, $x(t)$ is not periodic.Is there a representation for $x(t)$ as a linear combination of complex exponentials?... |
I am developing a program which seeks strategies for the players A, B in any of a family of simple 2-player gambling-games. The program iterates, using a genetic algorithm to determine, from the current iteration's results, the strategies which are to play one another in the next iteration.
Below I give an overview of ... |
In order for a right-hand limit $\lim\limits_{x \to a^{+}} f(x)$ to make sense, there must exist a $\delta > 0$ such that the function $f$ is defined in the open interval $(a, a + \delta)$.
Since (with $k$ denoting an integer)$$f(x) = \frac{\sin [x]}{[x]} = \begin{cases} \dfrac{\sin k}{k} & k \leq x < k + 1,\ k \neq 0,... |
What does the degrees of freedom parameter $n$ mean
intuitively in a Wishart distribution $\mathcal{W}_p(\mathbf{V},n)$? Does it have any relation to the covariance of different dimensions of the resulting covariance matrix? Why is $n==p$ called a non-informative prior?
What does the degrees of freedom parameter $n$ me... |
I am interested in knowing wether the following statement is true of false.
Let $ (\Omega, \Sigma , \mathbb{P})$ be a probability space and $\mathcal{A}, \mathcal{B} \subseteq \Sigma$ independant subsets. Then $\sigma A(\mathcal{A})$ and $\sigma A(\mathcal{B})$ are independent sigma algebras.
I think the statement is t... |
In here: Proving that well ordering principle implies Zorn's Lemma. I asked how to finish a proof of this statement. After a few helpful remarks, I think I have managed to finish it. What do you think?
Given that on every set, a well ordering can be defined, we should prove that Given a partially ordered set $A$, if ev... |
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Even and Odd Signals Commonly encountered signals Definition
A signal $x(t)$ is said to be,
(1)
Even if,
(2)
Odd if,
The following figures illustrate clearly,
Any signal $x(t)$ can be written as the sum of an ev... |
The 2n law of thermodynamics can be stated in terms of entropy as follows
$dS \geq \frac{dQ}{T},$
which holds for all quasistatic processes (reversible and irreversible ones).
Is there a generalization of this statement to a general process between two equilibrium states $e_1$ and $e_2$ (a non-quasistatic process)? I.e... |
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Now showing items 1-2 of 2
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Our reader Eswar Chellappa has sent his work on the solution of ‘3X+1’ problem, also called Collatz Conjecture. He had been working on the proof of Collatz Conjecture off and on for almost ten years. The Collatz Conjecture can be quoted as follow: Let $\phi : \mathbb{N} \to \mathbb{N}^+$ be a function defined such that... |
I don't think it's on topic, but it's no skin off my back to write a short answer. You need to use the chain rule $(fg)' = fg' + gf'$:
$$\begin{align}\frac{\mathrm dP}{\mathrm dv} &= \left(\frac{m}{2\pi kT}\right)^{3/2}4\pi \left[v^2\frac{\mathrm d}{\mathrm dv}(\mathrm e^{-mv^2/2kT}) + \mathrm e^{-mv^2/2kT}\frac{\mathr... |
Back at the start of the year (which really doesn’t seem like that long a time ago) I was looking at using Dirichlet Processes to cluster binary data using PyMC3. I was unable to get the PyMC3 mixture model API working using the general purpose Gibbs Sampler, but after some tweaking of a custom likelihood function I go... |
I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2176).
In the model I'm interestend in has some former estimation done before the estimation of the primary model will take place. Hence the primary model (2nd-step) includes some estimat... |
Why is $\infty^\infty=\tilde\infty$?
WA's
ComplexInfinity is the same as Mathematica's: it represents a complex "number" which has infinite magnitude but unknown or nonexistent phase. One can use
DirectedInfinity to specify the phase of an infinite quantity, if it approaches infinity in a certain direction. The standar... |
Topology AtlasDocument # ppae-04
Concerning the dual group of a dense subgroup W. W. Comfort, S. U. Raczkowski and F. Javier Trigos-ArrietaProceedings of the Ninth Prague Topological Symposium(2001)pp. 23-35
Throughout this Abstract, G is a topological Abelian group and[^G] is the space of continuous homomorphisms from... |
Can it be generalized for other powers ? Wolfram seems to say it is true for k below 20000.
I stumbled upon it randomly when trying to approximate $\sum_{n=1}^{n=+\infty} \frac{1}{n^4}$.
My reasoning was :
$$\left(\sum_{n=k}^{n=+\infty} \frac{1}{n^2}\right)^2=\sum_{n=k}^{n=+\infty} \frac{1}{n^4} + (\text{double product... |
Difference between revisions of "Inverse function theorem"
(→Statement with symbols)
(→Statement with symbols)
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! Version type !! Statement
! Version type !! Statement
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| specific point, named functions || Suppose <math>f</math> is a [[function]] of one variable that is a [[one-one function]] an... |
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your ow... |
The Ising model is defined with the Hamiltonian:
$$ H = -\sum_{<i,j>}S_i^z\cdot S_j^z $$
What is the difference between quantum version and classical version? My intuition is that the classical version is equal to quantum version in any dimension and any lattice.
If we add a transverse field, I think there will be diff... |
2018-09-11 04:29
Proprieties of FBK UFSDs after neutron and proton irradiation up to $6*10^{15}$ neq/cm$^2$ / Mazza, S.M. (UC, Santa Cruz, Inst. Part. Phys.) ; Estrada, E. (UC, Santa Cruz, Inst. Part. Phys.) ; Galloway, Z. (UC, Santa Cruz, Inst. Part. Phys.) ; Gee, C. (UC, Santa Cruz, Inst. Part. Phys.) ; Goto, A. (UC,... |
In its simplest form, a SIR model is typically written in continuous time as:
\[ \frac{dS}{dt} = - \beta \frac{S_t I_t}{N_t} \]
\[ \frac{dI}{dt} = \beta \frac{S_t I_t}{N_t} - \gamma I_t \]
\[ \frac{dR}{dt} = \gamma I_t \]
Where \(\beta\) is an infection rate and \(\gamma\) a removal rate, assuming ‘R’ stands for ‘recov... |
I have the following equation:
$$-\frac{{{\hbar }^{2}}}{2I{}_{r}}\frac{{{d}^{2}}\psi }{d{{\theta }^{2}}}+V\left( \theta \right)\psi =E\psi, $$
with
$$V\left( \theta \right)=a+\sum\limits_{n=1}^{N}{{{b}_{n}}\cos \left( n\theta \right)}+\sum\limits_{m=1}^{M}{{{c}_{m}}\sin \left( m\theta \right)}$$
where the notations are... |
This is a cute problem! I toyed with it and didn't really get anywhere - I got the strong impression that it requires fields of mathematics that I am not expert in.
Indeed, given that the problem seems related to that of counting integer solutions to the equation $f(x,y) = c$, one may need to use arithmetic geometry to... |
This might be a trivial question.
Consider a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and consider some point $(a,b)\in \mathbb{R^2}$.
Suppose we know that all the directional derivatives $D_{\overline{u}}f(a,b)$ for an arbitrary unit vector $\overline{u}=\langle u_1,u_2\rangle$ in $\mathbb{R^2}$ exist $...(1)$
... |
In looking at the two functions defined: $$\psi_{{0}}(x)=\ln( \operatorname{lcm}(1,2,3,...,\lfloor x \rfloor))$$
$$\psi_{{1}}(x)=\sum _{j=1}^{ \lfloor x \rfloor } \sum _{i=0}^{ \Bigl\lfloor {\frac {\ln \left( x \right) }{\ln \left( p_{{j}} \right) }} \Bigr\rfloor +1}\ln \left( {p_{{j}}}^{i} \right) $$ (where $p_n$ is t... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed i... |
I am a student who finished Part 1 and am interested in applying label smoothing to a problem. I saw that it was taught in Part 2 so figured this would be a good place to ask my question.
I was wondering though if label smoothing can be applied to multi-label problems. In addition, typically, as I read about label smoo... |
To calculate the volume of a pyramid (not tetrahedron!) you've to use the formula $\frac{1}{3}B\cdot H,$ where $B$ is the area of the base and $H$ is the height.
My question is: why 1/3? Is a pyramid one-third of a cuboid?
Mathematics Stack Exchange is a question and answer site for people studying math at any level an... |
Evaluate $ \int \frac {1} {z^2+1} dz $ along the contour $\Gamma$. (Gamma is some closed circle centered around i, no specified radius, and is oriented counter clockwise.)
So far, I've used factored the expression into $ \int \frac 1 {(z+i)(z-i)}$, then used partial fraction decomposition, and got $ \frac 1 {2i}[ \frac... |
I am wondering about the order of terms from equation 34 this pdf by Hitoshi Murayama on second quantization. Specifically, why they are able to pull the H past the $\Psi(\vec x)$. i.e.
$$ H|\psi(t)\rangle= \int dx\,H \Psi(\vec x,t)\phi^\dagger(\vec x)|0\rangle $$
makes sense, but how are they allowed to pull H in fron... |
It is often said that he discovered non-Euclidean geometry. But in which sense?
I am reading the book 'geometry' by Brannan et al. They use the disk model as an example of hyperbolic geometry. Did Lobachevsky have a similar model?
History of Science and Mathematics Stack Exchange is a question and answer site for peopl... |
I remember trying this problem a while ago and was unable to prove it. I think my idea was to create a surjective homomorphism from $\mathbb{Z}_{5}[x]$ to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$ and use the first isomorphism theorem but it wasn't working because none of my maps were well-defined. What is the correct way... |
Consider cellular automaton rules for a two-dimensional universe with two states, where a cell’s new state can depend on its previous state and the states of the cells in its Moore neighborhood. Such a rule can be modeled as a function that takes as input a 3 by 3 matrix of bits, and outputs a bit.
Such a rule can be r... |
An article in the July 30, 2016 New York Times argues that high stock prices are the result of low bond yields. To paraphrase the article: The S&P 500 index has a dividend yield of 2.1%, 40% higher than the 1.5% yield on 10-year Treasury notes. Investors have moved to stocks, chasing yield.
This post explains why divid... |
The question is:
$$\int^{\pi/4}_{0}\left(\cos 2\theta \right) ^{3/2}\cos \theta\, d\theta$$
I tried to write it in terms of $\sin\theta$ and the substitute $\sin\theta=t$ but then got stuck at the subsequent integral. I tried integrating by parts, which was of no use.
Any help would be appreciated. Thanks. |
CNO Cycle: 4H $\rightarrow$ He
4 protons (i.e. Hydrogen nuclei) combine to form a Helium molecule while releasing 26.7 MeV energy. This is the net result of any of the various fusion pathways for hydrogen to helium.
Three Helium molecules combine to form a Carbon molecule, while releasing 7.4 MeV of energy. Since it ta... |
Colloquia/Fall18 Contents 1 Mathematics Colloquium 1.1 Spring 2018 1.2 Spring Abstracts 1.2.1 January 29 Li Chao (Columbia) 1.2.2 February 2 Thomas Fai (Harvard) 1.2.3 February 5 Alex Lubotzky (Hebrew University) 1.2.4 February 6 Alex Lubotzky (Hebrew University) 1.2.5 February 9 Wes Pegden (CMU) 1.2.6 March 2 Aaron Be... |
Alexandrou, C. and Forcrand, Ph. de and D'Elia, M. and Panagopoulos, H. (2000)
Efficiency of the UV-filtered Multiboson algorithm. Physical review. D, Particles, fields, gravitation, and cosmology, 61 . 074503. ISSN 1550-7998 Abstract
We study the efficiency of an improved Multiboson algorithm with two flavours of Wils... |
Historically, Special Relativity was motivated by apparent inconsistencies between Maxwell's Electrodynamics and Newtonian Mechanics. In Einstein's well known paper "On the electrodynamics of moving bodies" he explains quite well his motivations.
Central objects of the theory are the Lorentz transformations. If one for... |
Use proof by induction to prove that that $ \frac{1}{n!}<\frac{1}{2^n-1} $ for all $n\geq 4$, .\Base case: $$\frac{1}{4}=\frac{1}{24}\leq \frac{1}{2^4-1}$$ Inductive hypothesis: Assume there exists $k\in \mathbb{N}$ s.t.
$$ \frac{1}{k!}\leq\frac{1}{2^k-1} $$ Inductive step: Show that:$$ \frac{1}{(k+1)!}\leq\frac{1}{2^{... |
Introduction
Engineering and math modeling applications are an important part of modern software. These applications use extremely complicated math and require presenting results in the form of charts, schemes, 3D models, formulae. The result presentation must be simple to perceive and understand. Therefore, developing... |
Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by
$$f(x) = \begin{cases} 0 & x \notin \mathbb{Q} \\ \\ 0 & x = 0 \\ \\ \frac{1}{q_x} & x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \ q_x \in \mathbb{N}, \ \text{gcd}(|p_x|, q_x) = 1 \end{cases}$$
Is $\displaystyle f$ Riemann integ... |
I am confused, as not clear except by multiplying both terms on the r.h.s, and showing that all cancel out except the two on the l.h.s., as below:
$(x)(x^{k-1}+x^{k-2}+\ldots+1) - (x^{k-1}+x^{k-2}+\ldots+1)= x^k -1$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profe... |
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