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Definition:Tychonoff Separation Axioms Contents 1 Definition 1.1 $T_0$ (Kolmogorov) Space 1.2 $T_1$ (Fréchet) Space 1.3 $T_2$ (Hausdorff) Space 1.4 Semiregular Space 1.5 $T_{2 \frac 1 2}$ (Completely Hausdorff) Space 1.6 $T_3$ Space 1.7 Regular Space 1.8 Urysohn Space 1.9 $T_{3 \frac 1 2}$ Space 1.10 Tychonoff (Complet... |
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Last time I explained monoidal categories, which are a framework for studying processes that we can compose and tensor. We can do a lot with monoidal categories! For example, if we have a monoidal category with morphisms
$$ \Phi \co... |
When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism.
Recall/Example ($A_\infty$-algebras):
An
$A_\infty$-morphism between two $A_\infty$-algebras $(A,\mathfrak{m})$ and $(A', \mathfrak{m}')... |
Occasionally, we receive requests for a technical paper about regression modeling beyond our regular NumXL support, in order to delve more deeply into the mathematical formulation of MLR. We are always happy to address user requests, so we decided to share our internal technical notes with you.
In this paper, we’ll go ... |
I'm using Mathematica to calculate the inverse Laplace transform (ILT) of various functions--functions that I do not know ahead of time. I was stepping through the process and decided to write my own code to simplify the expressions and calculate the inverse transform, but I stumbled upon the general equation that it a... |
Before the question, I need to mention some necessary definitions.
The rapidity is defined as: $$y=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}=\frac{1}{2}\ln\frac{1+v_z}{1-v_z}=\tanh^{-1}(v_z)$$ where $v_z=p_z/E$ is the velocity along $z$ direction. $v_z=\tanh y$
We have defined the transverse mass $m_T$ and the longitudinal boo... |
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Last time we reached the technical climax of this chapter: constructing the category of enriched profunctors. If you found that difficult, you'll be relieved to hear it's downhill from here on, at least in Chapter 4. We will now app... |
On the DNA Computer Binary Code
In any finite set we can define a
, a partial order in different ways. But here, a partial order is defined in the set of four DNA bases in such a manner that a Boolean lattice structure is obtained. A Boolean lattice is an algebraic structure that captures essential properties of both s... |
I have a mixture model which I want to find the maximum likelihood estimator of given a set of data $x$ and a set of partially observed data $z$. I have implemented both the E-step (calculating the expectation of $z$ given $x$ and current parameters $\theta^k$), and the M-step, to minimize the negative log-likelihood g... |
Answer
$$x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi nE$$
Work Step by Step
We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving this, we find: $$\sin ... |
Abbreviation:
CdLat
A
is a bounded lattices $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that complemented lattice
every element has a complement: $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$
Let $\mathbf{L}$ and $\mathbf{M}$ be complemented lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function... |
From the pole-zero plot, you can compute the system frequency response by assuming a locus of test points along the \$j\omega\$ axis.
Figure from: http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
\begin{align} |H(j\omega)| &= K \frac{r_1\ldots r_m}{q_1\ldots q_n}\\ \angle H(j\omega) &= (\phi_1 + \ldots + \phi_m) - (\t... |
Let $M: P_{1}\rightarrow P_{1}$ be defined by $M(f)=f^{'}+f$, i.e. $M(a_{0}+a_{1}x)=a_{1}+a_{0}+a_{1}x$. Find the adjoint $M^{*}$ of $M$, i.e find $M^{*}(a_{0}+a_{1}x_{1})$, assuming that $L^{2}(0,1)$ inner product is imposed on $P_{1}$. $\langle f(x),g(x)\rangle =\int^{1}_{0}f(x)g(x)dx$.
Let's find an
orthogonal basis... |
We can go straight from your definition of a zero-divisor. In $A \times B$, a zero divisor is any two non-zero elements that multiply to give zero. Note that in this new ring, "zero" is the element $(0, 0)$.
So the elements $(a, b)$ and $(c, d)$ are a zero divisor pair if $(a, b) \cdot (c, d) = (0, 0)$.
If $A$ or $B$ o... |
Does this make sense to you, i.e. are the 2 fields actually different or the 2 names refer to the same thing?
Spectrum analysis is more general: it involves looking at the entire spectrum of a given signal.
Spectral line analysis assumes that the spectrum contains several peaks (lines) of interest at specific frequenci... |
When does an interaction drop the system into an eigenstate? (i.e. when is a measurement=)
This is an ill-posed question because, first of all, the system $S$ doesn't drop into any state but each observer $O$ has a state about it, as a state $\rho$ is nothing but the coding of past measurements (so it should be named w... |
Real Analysis Exchange Real Anal. Exchange Volume 33, Number 2 (2007), 417-430. A Study of a Stieltjes Integral Defined on Arbitrary Number Sets Abstract
Our purpose is to study a generalized Stieltjes integral defined on a class of subsets of a closed number interval. We extend the results of previous work by the firs... |
No, this is not meant as an oxymoron but rather as a question raised by one of our users. And we thought it may be interesting to others
So the story goes like this: When you have a sample time series, most of the time you would like to forecast just the future points (past the end of the sample data). But what about t... |
To send content items to your account,please confirm that you agree to abide by our usage policies.If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.Find out more about sending content to .
To send content items to your Kindle, first ensure no-rep... |
Differential Equation
Differential Equation is an important and useful branch of Mathematical Analysis. The inception of Differential Equation with that of Differential and Integral Calculus dates back to the seventeenth century. It was Sir Isacc Newton, who first found the solution of a differential equation with the ... |
As we all know, Brexit negotiations are on their way—but we still do not know whether they will actually finish in time.
The negotiations will take place topic-by-topic. To organise the negotiations in the most effective way, the topics will all be discussed and finalised in separate meetings, one meeting at a time.
Th... |
The motivation behind density matrices [1]:
In quantum mechanics, the state of a quantum system is represented by a state vector, denoted $|\psi\rangle$ (and pronounced
ket). A quantum system with a state vector $|\psi\rangle$ is called a pure state. However, it is also possible for a system to be in a statistical ense... |
I cannot see how Willie Wong's example of the Bernstein-Robinson result supports his conclusion. It seems to me to do the opposite, and I am not alone here. Halmos admits himself in his autobiography: "The Bernstein-Robinson proof uses non-standard models of higher order predicate languages, and when Abby [Robinson] se... |
The formula for the standardized residuals is:
$$\begin{align}\text{Pearson's residuals}\,&=\,\frac{\text{Observed - Expected}}{ \sqrt{\text{Expected}}}\\d_{ij}&=\frac{n_{ij}-m_{ij}}{\sqrt{m_{ij}}}\end{align}$$
where $m_{ij} = E( f_{ij})$ is the expected frequency of the $i$-th row and the $j$-th column.
The sum of squ... |
56 6 Homework Statement A particle moves along a defined curve so that its acceleration tangential component: ##a_t=-ks##, where k is a constant and s denotes the arc distance respect to a point Q. a) Find an expression for the velocity as a function of s. b) Supposing that at Q its velocity equals 3.6 m/s and at A (s=... |
Definition:Measurable Set Contents Definition
Let $\left({X, \Sigma}\right)$ be a measurable space.
Measurable Sets of an Arbitrary Outer Measure $\mu^* \left({A}\right) = \mu^* \left({A \cap S}\right) + \mu^* \left({A \setminus S}\right)$
for every $A \subseteq X$.
By Set Difference as Intersection with Complement, th... |
I am currently reviewing an old Calculus textbook and I stumbled upon two questions that, for me, has the wrong answer on the answer key. I would appreciate if you could check if my reasoning is correct. They follow:
Question 1)If $\lim_{x \to 5} f(x) = 2$ and $\lim_{x \to 5} g(x) = 0$ then $\lim_{x \to 5} \frac{f(x)}{... |
Abbreviation:
CSlat
A
is a directed complete partial orders $\mathbf{P}=\langle P,\leq \rangle $such that every nonempty subset of $P$ has a greatest lower bound: $\forall S\subseteq P\ (S\ne\emptyset\Longrightarrow \exists z\in P(z=\bigwedge S))$. complete semilattice
Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semi... |
Yes, a quantum computer could be simulated by a Turing machine, though this shouldn't be taken to imply that real-world quantum computers couldn't enjoy quantum advantage, i.e. a significant implementation advantage over real-world classical computers.As a rule-of-thumb, if a human could manually describe or imagine ho... |
The $\mathbb{Z_5}$-vector space $\mathfrak{B}$ 3 over the field $(\mathbb{Z_5}, +, .)$ $\mathfrak{B}$ 3over the field $(\mathbb{Z_5}, +, .)$ 1. BackgroundThis is a formal introduction to the genetic code $\mathbb{Z_5}$-vector space $\mathfrak{B}^3$ over the field $(\mathbb{Z_5}, +, .)$. This mathematical model is defin... |
The Nonparaxial Gaussian Beam Formula for Simulating Wave Optics
In a previous blog post, we discussed the paraxial Gaussian beam formula. Today, we’ll talk about a more accurate formulation for Gaussian beams, available as of version 5.3a of the COMSOL® software. This formulation based on a plane wave expansion can ha... |
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Last time we began studying feedback in co-design diagrams. This led us into a fascinating topic which we'll explore more deeply today: cups and caps. Ultimately it leads to the subject of 'compact closed' categories, which Fong and... |
Context: The paper
On the reality of the quantum state ( Nature Physics 8, 475–478 (2012) or arXiv:1111.3328) shows under suitable assumptions that the quantum state cannot be interpreted as a probability distribution over hidden variables.
In the abstract, the authors claim: "This result holds even in the presence of ... |
Courtesy of the OpenCV 2.3 GPU code comes a neat snippet of code for using a template parameter for reading RGB or BGR ordered components when dealing with RGB triplets.
The Code template <int blueIndex> float rgb2grey(const float *src) { return 0.114f*src[blueIndex^2] + 0.587f*src[1] + 0.299f*src[blueIndex]; }
Then to... |
Returns an array of cells for the fitted values of the conditional mean.
Syntax ARMA_MEAN( X, Order, mean, sigma, phi, theta) X is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)). Order is the time order in the data series (i.e. the first data point's corresponding date (earlie... |
ISSN:
2156-8472
eISSN:
2156-8499 Mathematical Control & Related Fields
March 2016 , Volume 6 , Issue 1
Select all articles
Export/Reference:
Abstract:
We study a damped semi-linear wave equation in a bounded domain of $\mathbb{R}^3$ with smooth boundary. It is proved that any $H^2$-smooth solution can be stabilised loc... |
Abbreviation:
LRng
A
(or $\ell$ lattice-ordered ring ) is a structure $\mathbf{L}=\langle L,\vee,\wedge,+,-,0,\cdot\rangle$ such that -ring
$\langle L,\vee,\wedge\rangle$ is a lattice
$\langle L,+,-,0,\cdot\rangle $ is a ring
$+$ is order-preserving: $x\leq y\Longrightarrow x+z\leq y+z$
${\uparrow}0$ is closed under $\... |
Limits:
As we see in everyday life that everything can’t be exact. The length of an object can’t be a whole number. When we say this rope is of length 5 metre it doesn’t meet with the fact that it is exactly 5 metre. It can be of 4.999999 meters or maybe 5.000001 meters. To deal with such situation limits is introduced... |
Balls and Needles
Joana Vasconcelos is a Portuguese artist who uses everyday objects in her creations, like electric irons or plastic cutlery. She is an inspiration to Ana, who wants to make ceiling hanging sculptures with straight knitting needles and balls of wool. For safety reasons, there will be a ball at each end... |
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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 ... |
For readability, I've put the definitions of algebraic operation and preservation at the end of the question.
An old theme in logic is:
Given some algebraic operation, we can give a
syntacticcharacterization of the sentences preserved by this operation.
For example, the sentences preserved by Cartesian products are the... |
Volume of Solid of Revolution Theorem
Let the points be defined:
$A = \tuple {a, \map f a}$ $B = \tuple {b, \map f b}$ $C = \tuple {b, 0}$ $D = \tuple {a, 0}$ Then the volume $V$ of $S$ is given by: $\displaystyle V = \pi \int_a^b \paren {\map f x}^2 \rd x$
Let the points be defined:
$A = \tuple {\map x a, \map y a}$ $... |
Question:
For which of the following matrices $A_i$ is there
A complex matrix $B$ such that $B^2 = A_i$; A self-adjoint complex matrix $B$ such that $B^2 = A_i$; A real matrix $B$ such that $B^2 = A_i$?
$A_1 = \begin{pmatrix} 2 & 1\\1 & 2\end{pmatrix}$, $A_2 = \begin{pmatrix} 1 & 2\\2 & 1\end{pmatrix}$, $A_3 = \begin{p... |
Definition:Cantor Normal Form Definition
Let $x$ be an ordinal.
The Cantor normal form of $x$ is an ordinal summation: $x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$
where:
$k \in \N$ is a natural number $\omega$ is the minimal infinite successor set $\langle a_i \rangle$ is a strictly decreasing finite sequence of ... |
I'll preface this by saying that it isn't always clear what one means by "nonparametric" or "semiparametric" etc. In the comments, it seems likely that whuber has some formal definition in mind (maybe something like choosing a model $M_\theta$ from some family $\{M_\theta: \theta \in \Theta\}$ where $\Theta$ is infinit... |
I have a process which consists of a number of events and what is known is the timings between the events. What I'm trying to determine is a distribution that allows me to determine a likelyhood that a new sample fits the distibution.
The issue is mainly that if you have lots of samples you can approximate the result u... |
19 0
I am currently studying the Massive Thirring Model (MTM) with the Lagrangian
$$ \mathcal{L} = \imath {\bar{\Psi}} (\gamma^\mu {\partial}_\mu - m_0 )\Psi - \frac{1}{2}g: \left( \bar{\Psi} \gamma_\mu \Psi \right)\left( \bar{\Psi} \gamma^\mu \Psi \right): . $$ and Hamiltonian $$ \int \mathrm{d}x \imath \Psi^\dagger \... |
Definition:Contour/Complex Plane Contents Definition $\gamma_i \left({b_i}\right) = \gamma_{i + 1} \left({a_{i + 1} }\right)$ Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is a contour. If $C_1, \ldots, C_n$ are defined only by their parameterizations $\gamma_1, \ldots, \gamma_n$, then the cont... |
Consider the ring $\mathbb{Z}[q^{\pm 1}]$. For $n \in \mathbb{N}$, define the quantum integers:
$$[n]_q := \frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-(n-3)} + q^{-(n-1)}$$
What is the general formula for multiplying and dividing quantum integers? This is probably well-known but I don't have a refere... |
Let
x 0, ...., x be complex numbers. The DFT is defined by the formula
n-1
Evaluating these sums directly would take O(
n 2) arithmetical operations (see Big O notation). An FFT is an algorithm to compute the same result in only O( n log n) operations.
Since the inverse DFT is the same as the DFT, but with the sign of ... |
IMPORTANT Slide 2 Restate Residue Theorem f has isolated singularity at and is analytic on the punctured disc 0 < | z- | < r f has a unique Laurent series representation … The residue of f at is represented by Res(f,) = the coefficient of
Following slides are to calculate residues at removable singularities and poles (... |
Let $B \subset \mathbb R^2$ be the unit ball and $T>0.$ Let $u \in W^{2,1}_p(B \times [0,T]),$ that is $u \in L^p(B \times [0,T])$ and we also have, $$ \partial_t u, \nabla u, \nabla^2 u \in L^p(B \times [0,T]). $$ Here $\nabla$ denotes differentiation in the spacial direction only.
I am looking for a proof of the foll... |
In complex analysis, one usually uses the term
Jordan domain for a domain whose boundary is a simple closed curve.
The function $f(z) = az+b$, initially defined on $\partial \Omega$, admits a holomorphic extension to $\Omega\setminus \overline{D}$, also given by the formula $az+b$. I think the main question here is whe... |
I've a question regarding the Hodge star operator. I'm completely new to the notion of exterior derivatives and wedge products. I had to teach it to myself over the past couple of days, so I hope my question isn't trivial.
I've found the following formulas on the internet, which seem to match the definitions of the two... |
There are several themes in Huygens' unpublished paper
De motu corporum ex percussione ("On the motion of bodies out of collisions"), but maybe the most significant is that he frequently investigates a specific or extreme case (where some factor is zero, one, or infinity) first, and then guesses about a general case wh... |
This is false for $n\geq 4$. Consider the Grassmannian $\mathrm{Gr}(2,n)$ of all two-dimensional subspaces of $\mathbb{R}^n$, and recall that $\mathrm{Gr}(2,n)$ is a compact manifold of dimension $2n-4$. For each $\varphi\in\mathrm{GL}_n(\mathbb{Q})$, let$$S_\varphi = \{A\in \mathrm{Gr}(2,n) \mid \varphi u=v \text{ for... |
I’ve been studying the KKR method from the original Kohn and Kostoker’s paper (https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.1111). On the text, they use variational calculus for dealing with an integral equation. Namely, the paper states that the integral equation (2.14):
$\psi(\vec{r})=\int G(\vec{r},\vec{r... |
I have this expression: $3m^4-6m^3+14m^2-6m+11=0$ and I want to factorize it in $(m^2+1)(3m^2-6m+11)$. How can I do it? Thanks for any help!
Hint $\ $ Because the leading and constants coefficients are primes, the possible factors are highly constrainted, so we can quickly find quadratic factors by undetermined coeffic... |
The formula for the Chi-Square test statistic is the following:
$\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$
where O - is observed data, and E - is expected.
I'm curious why it depends on the absolute values? For example, if we change the units we're measuring we'll get a different statistics. Suppose we're perf... |
Hi I was trying to find the area between the following curves (below) however I am unsure how to continue from the trigonometry which gets presented:
Curves:
$$y = 2\sin(x)\\y = \cos\left(\frac{x}{2}\right)$$
I have found that the curves meet at $x=\pi$ and $x=2 \arcsin(\frac14)$. I am supposed to find the area being $... |
You need two things. The angular velocity vector $\vec{\omega}$ at some time frame, and the linear velocity vector at some reference point $\vec{v}$.
The axis of rotation is located at a point relative to the reference point
$$ \vec{r} = \frac{\vec{\omega} \times \vec{v}}{ \| \vec{\omega} \|^2} \tag{1}$$
You can apply ... |
What is the simplest physical system which can be used to model the quantum measurement of a 2 level system?
For example, can the following, spin coupled to a harmonic bath, be used to model a measurement of the 2 level system's z polarisation?
$$\hat{H}=\sum_{j=1}^n\frac{\hat{p}^2_j}{2}+\frac{1}{2}\omega_j^2\bigg(\hat... |
The state $\mid \psi \rangle$ is fixed. You can write it as $a |0 \rangle + b |1 \rangle$. If you write that in the other basis and get
$(a,b) = (1,0) \implies (c,d) = (\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$
They are the same state so it is still true that if you do a measurement in the $|0 \rangle$, $|1 \rangle$ basi... |
According to this very interesting article in Quanta Magazine: "A Long-Sought Proof, Found and Almost Lost", -- it has been proved that given a vector $\mathbf{x}=(x_1,\dots,x_n)$ having a multivariate Gaussian distribution, and given intervals $I_1,\dots,I_n $ centered around the means of the corresponding components ... |
Answer
31 Hz
Work Step by Step
We use the equation that relates frequency to velocity and wavelength to find: $ f = \frac{v}{\lambda} = \frac{343 \ m/s}{11 \ m} \approx 31 \ Hz$
You can help us out by revising, improving and updating this answer.Update this answer
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Regarding the quantum Toffoli gate:
is it classicalyuniversal, and if so, why? is it quantumlyuniversal, and why?
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It only takes a minute to sign up.Sign up to join this community
Toffoli is uni... |
The concept you are looking for is called
enumeration complexity, which is the study of the computational complexity of enumerating (listing) all the solutions to a problem (or the members of a language/set). Enumeration algorithms can be modeled as a two step process:a precomputation step and an enumeration phase with... |
Also, read Matrix Formulas Mean, Median,
Mode Formulas
Set Theory Determinant formulas:
In the previous section we discussed matrices. Now you are much aware of matrices its properties, addition, subtraction and multiplication. Now another term is there which enhances the properties of square matrices. It has a wide ra... |
Let $p$ be a prime number and consider the sum $S(x)=\sum_{n\le x}\left(\frac{n}{p}\right)\mu(n)$. For how small an $x$ in terms of $p$ is it known that $S(x)=o(x)$? I am especially interested in unconditional results.
In general, we can take $x>\exp\{c_\epsilon p^\epsilon\}$ by the Prime Number Theorem for arithmetic ... |
Abbreviation:
PeirceA
A
is a 2-sorted structure $\mathbf{A}=\langle \mathbf R,\mathbf B,^c\rangle$ such that Peirce algebra
$\mathbf R=\langle R,\vee,0,\wedge,1,\neg,\circ,^\smile,e\rangle$ is a relation algebra
$\mathbf B=\langle B,\vee,0,\wedge,1,\neg,f_r\ (r\in R)\rangle$ is a Boolean module over $\mathbf R$
$^c:B\t... |
Random graphs with small world topology
In graphs with
small world topology, nodes are highly clustered yet the path length between them is small. A topology like this can make search problems very difficult, since local decisions quickly propagate globally. In other words, shortcuts can mislead heuristics. Further is ... |
Non-maximal paths are prefixes or suffixes of shortest paths
So a maximal path is just a path that begins at a vertex $u$ with no in-edge and ends at a vertex $v$ with no out-edge.
It's straightforward to see that if a shortest path $P$ from $u$ to $v$ is a subpath of some shortest path $Q$ from $x$ to $y$, then $P$ is... |
At the request of the OP, I'm turning my comments above into an answer, though different answers are possible and the question sounds a bit soft to me. Let it be as it may, here are my two cents.
From the point of view of factorization theory, Euclidean domains can be understood as a rather special subclass of the clas... |
One disadvantage of the fact that you have posted 5 identical answers (1, 2, 3, 4, 5) is that if other users have some comments about the website you created, they will post them in all these place. If you have some place online where you would like to receive feedback, you should probably also add link to that. — Mart... |
This article was originally published on December 5, 2017.
Believe it or not, mathematics plays an important role in the field of sports. Coaches, athletes, trainers often use mathematics to gain a competitive advantage over their counterparts. With statistics of games, statistics of players and probabilities of winnin... |
User:Caliburn
Jump to navigation Jump to search
b. 04/03/2001 from Kent.
Undergraduate student at the University of Warwick. (Maths and Statistics)
I prefer to be called by my real name, George, as opposed to Caliburn or any variation.
To do Probability Distributions Properties Let $\theta \sim \ContinuousUniform {-\fr... |
Let $X$ denote the time of death (or time of failure if youprefer a less morbid description). Suppose that $X$ is a
continuous randomvariable whose density function $f(t)$ is nonzero only on $(0,\infty)$. Now, notice that it must be the case that $f(t)$decays away to $0$ as $t \to \infty$ because if $f(t)$ does not dec... |
Faddeeva Package From AbInitio
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Line 26: Line 26: :<math>\mathrm{erfi}(x) = -i\mathrm{erf}(ix) = -i[e^{x^2} w(x) - 1]</math> (imagina... |
Functiones et Approximatio Commentarii Mathematici Funct. Approx. Comment. Math. Volume 41, Number 1 (2009), 55-70. Congruences between modular forms and related modules Abstract
Fix a prime $l$ and let $M$ be an integer such that $l\not|M$. Let $f\in S_2(\Gamma_1(M l^2))$ be a newform which is supercuspidal at $l$ of ... |
Definition:Upper Closure/Element Definition
Let $\left({S, \preccurlyeq}\right)$ be an ordered set.
Let $a \in S$.
The upper closure of $a$ (in $S$) is defined as: $a^\succcurlyeq := \left\{{b \in S: a \preccurlyeq b}\right\}$ Also known as
The
upper closure of an element $a$ is also known as the up-set of $a$.
The ter... |
I’ve earned it! https://confirm.udacity.com/TLVUZQTR
# Import packages import pandas as pd import scipy.stats as stats %matplotlib inline # Read in the data data = pd.read_csv(‘Customer Support Time Study.csv’) # Set columns to lists to use in ttest function joe = data[‘Joey’].values.tolist() nat = data[‘Nathaly’].valu... |
Tony's already provided an approach for you to follow. I'd like to suggest another. I'll return to a short discussion about your question, though, later. You write:
I am trying to switch a SPST-NO relay that is rated to handle up to
227VAC. The coil is powered by 5V, has a 100 Ohm coil resistance, and
its contacts are ... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
In the basic Ramsey model with technological growth, assuming that the economy is in a steady state, how would a sudden decrease in the population growth rate $n$ impact the steady states value of consumption and capital?
I obtain the following dynamics for the model:
$\dot{k} = f(k) - c - (\delta + n + g)k$
$\frac{\do... |
I'm going through the phase estimation algorithm, and wanted to sanity-check my calculations by making sure the state I'd calculated was still normalized. It is, assuming the square of the absolute value of the eigenvalue of the arbitrary unitary operator I'm analyzing equals 1. So, does it? Assuming that the eigenvect... |
The implication here is that each individual discoverer must start from nothing but a bag of crying cells, and build up knowledge in a linear order before making a discovery in a vacuum.
In reality, I find we have an entire interwoven society trying to make the discoveries, not independent individuals. There is an enti... |
There is one input factor $k$. The representative firm maximizes profits with respect to employed capital, i.e. \begin{align} \max_k{\pi(k) = f(k)-(r+\tau)k} \end{align} where $f(\cdot)$ is the production function, $r$ the rent of capital and $\tau$ the tax rate to be paid for a unit of capital. The FOC reads \begin{al... |
I have just solved an exercise, which asked to show that function $f$ is Lipschitz implies that $f$ is absolutely continuous. However, I'm wondering if the converse is true. I can't seem to think of any counterexamples at the moment. I think I'm brain dead or something, so I could use some help.
Consider $f(x) = \sqrt{... |
Here is the excerpt from the textbook
A Course in Mathematical Analysis by Prof D. J. H. Garling.
So I have the
Theorem 1: Given a set $A\neq\varnothing$, a mapping $\varphi:A\to P(A )\setminus \{\varnothing\}$, and $\bar{a}\in A$. Then there exists a sequence $$(a_{n})_{n\in \mathbb{N}}$$ such that $a_{0}=\bar{a}$ and... |
Consider the following theorem:
"Every non-empty set of positive integers has a minimum element".
The proof I usually see is one that uses contradiction, and does not seem like the easiest possible proof. I think there is an easier proof, and I wonder why I never see it. Does it contain an invalid assumption? The proof... |
I will include the proof here and highlight the parts that are giving me trouble.
Theorem$\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable.
Proof$\hspace{5 pt}$ Since $P$ has limit points, $P$ must be infinite. Suppose $P$ is countable, and denote the points of $P$ by $\mathbf... |
Can someone help me evaluate $G_g(z)=\int_0^{\infty}x^{z-1}e^{igx}dx$, where $g$ is real and $z$ is complex?
By closing the contour in the upper half plane, I've managed to prove that if $0<Re(z)<1$ and $Im(z)>0$ (we can then ignore these conditions by analytic continuation) and $g>0$, then $G_g(z)=e^{i\pi z}G_{-g}(z)$... |
We know that given a multiplicative function $f$ for which the series $\sum_{n=1}^\infty f(n)$ converges
absolutely then so does the Euler product $\prod_{p}\sum_{k=0}^\infty f(p^k)$, but does the reverse hold (at least up to conditional convergence)?
We know that given a multiplicative function $f$ for which the serie... |
I found this question here but it does not fully answer my question. The answer there was that "composite bosons can occupy the same state when the state is spatially delocalized on a scale larger than the scale of the wavefunction of the fermions inside".
Let's say we do a BEC with bosonic atoms (for example in a harm... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Let $X$ be a Banach Space, and denote by $∥ · ∥$ the standard norm on $B(X)$, the space of bounded linear functions $T:X\to X$.
(a) Suppose that $||| · |||$ is another algebra norm on $B(X)$. Prove that there exists $C > 0$ such that $∥ · ∥ ≤ C||| · |||$.
(b) Prove that B(X) has a unique (up to equivalence) complete al... |
$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL(\X),\rmL(\Y))$, for every linear operator $X\in\rmL(\X)$ we can write the
Kraus representation of $\Phi(X)$ as$$\Phi(X)=\su... |
Sort an array of size N made of numbers from 0 to K
Problem statement
In this article we will be discussing another (see the previous article) very popular programming interview question. This time we are asked to sort an array of size N whose elements are in the range [0,K).
Problem statement
Given an array $A=[a_1,a_... |
I have a question regarding the follow problem:
Show that the prime number 27644437 splits completely in $L = \mathbb{Q}(\sqrt{55})$.
From what I understand. This deals with ramification.
\begin{eqnarray} \sum_1^{r}e_if_i = n &, &\text{where }n = [L:K] \end{eqnarray}
For our prime number, $p$, to split completely into ... |
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