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I have done some manipulation and got that $$\frac{1}{1+e^z} = \sum_{n=0}^\infty \frac{n!}{n!+z^n}$$ by the fact that:
$$\frac{1}{1+e^z}= \frac{1}{1+\sum_{n=0}^\infty\frac{z^n}{n!}}=\frac{1}{2}+\frac{1}{1+z}+\frac{1}{1+\frac{z^2}{2}}+\ldots = \frac{1}{2}+\frac{1}{1+z}+\frac{2!}{2!+z^2}+\frac{3!}{3!+z^3}+\ldots$$ $$= \s... |
just started Jordan forms and not sure what the general method is. I watched MathdoctorRob on YouTube, but can't really make of a method to find JCF.
Would really appreciate someone to check why my working doesn't work, and how I would have done it.
$A = \begin{pmatrix} 10 & 1 \\ -9 & 4 \end{pmatrix}$
I found the chara... |
October 10th, 2014, 05:43 AM
# 1
Senior Member
Joined: Aug 2014
From: Mars
Posts: 101
Thanks: 9
LCD and prime numbers(with binomials)
So I was given this problem.............
Find the LCD=
$\displaystyle \frac{3}{2x-6}, \frac{4}{x^2-9}, \frac{18}{6x+18}$
The break down (do I call these the prime numbers?
$\displaystyle... |
In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ground field, which is a number. Such invariants are always multiplicative under disjoint union, this is a consequence of t... |
Summary of Techniques: Solving Second Order Differential Equations
Summary of Techniques for Solving Second Order Differential Equations
We will now summarize the techniques we have discussed for solving second order differential equations.
(1) Real and Distinct Roots of The Characteristic Equation: If we have a second... |
This is a heuristic explanation of Witten's statement, without going into the subtleties of axiomatic quantum field theory issues, such as vacuum polarization or renormalization.
A particle is characterized by a definite momentum plus possible other quantum numbers. Thus, one particle states are by definition states wi... |
Hausdorff Topological Spaces Examples 3
Recall from the Hausdorff Topological Spaces page that a topological space $(X, \tau)$ is said to be a Hausdorff space if for every distinct $x, y \in X$ there exists open neighbourhoods $U, V \in \tau$ such that $x \in U$, $y \in V$ and $U \cap V = \emptyset$.
We also looked at ... |
I am still confused about Hypothesis testing. How does one set up the null hypothesis $H_0$ and the alternative hypothesis $H_a$? I have read a post here that doesn't give it much credit, as far as I can tell. (One must use both $H_0$ and $H_a$ for the AP Statistics exam.)
I have read that $H_a$ is to be what one does ... |
Let $n=144$ observations from an $AR(1)$ model $$y_t=\phi y_{t-1}+\epsilon_t$$ where $\epsilon_t$ is White Noise with mean zero and variance $\sigma^2$. If $y_1=-1.7$, $y_{144}=-2.1$, $\sum_{t=2}^n y_t y_{t-1}=-128.6$ and $\sum_{t=1}^n y_t^2=246.4$.
Suppose $\sigma^2$ know, find $Cov(e_n(1),e_n(2))$, where $e_n(j)$ is ... |
I don't sharply disagree with Dr Neumaier's answer; it is indeed the case that entanglement may only be discussed for Hilbert spaces that are tensor products.
However, if the two parts of the well are sufficiently distant, this is nearly the case of your situation, too. When one looks at it in this approximate way, the... |
Source Variability
Source variability within an observation is assessed by three methods: (1) the Kolmogorov-Smirnov (K-S) test, (2) the Kuiper's test, and (3) computation of the Gregory-Loredo variability probability, all based on the source region counts. Intra-observation source variability within any contributing o... |
J. D. Hamkins, “book review of Notes on Set Theory, Moschovakis,” J.~Symbolic Logic, vol. 62, iss. 4, p. pp.~1493-1494, 1997.
@article{Hamkins1998:BookReviewMoschovakis, jstor_articletype = {book-review}, title = {book review of {Notes on Set Theory, Moschovakis}}, author = {Hamkins, Joel David}, journal = {J.~Symbolic... |
First, I think your definition of $H_{n}$ does not agree with Newman's definition. Newman says the following: "Let $H_{n} \subset G_{n}$ be the set of functions of $G_{n}$ with non-negative valence at all parabolic points of $Q_{n}$ other than $\tau = i\infty$." Here $G_{n}$ is the set of modular functions that are exp... |
The Dimension of a Direct Sum of Subspaces
Recall from The Dimension of a Sum of Subspaces page that if $V$ is a finite-dimensional vector space and $U_1$ and $U_2$ are subspaces of $V$, then we have that:(1)
Now suppose that $U_1$, $U_2$, …, $U_m$ are all subspaces to the finite-dimensional vector space $V$ and such t... |
Definition: Solar Clock
Noun Definition Solar Clock A clock that depends on visual sensations from the Sun. 6-11
Logical Antecedents
Noun Definition Clock $\mathbf{\Theta} \equiv \sf{\text{an set of sensations used to tell time. }}$ 6-1
Nouns Definition Rotating Seeds $\sf{\text{Objectified black and white sensations.}... |
Method 1: Integration by parts
$$ \int \sqrt{a^2-x^2\,} \,\mathrm{d}x
= \frac{x}{2}\sqrt{a^2-x^2\,} + \frac{a}{2} \int \frac{a}{\sqrt{a^2-x^2\,}\,}\mathrm{d}x \tag{1} $$
Pick $v' = x$ and $u = \sqrt{a^2-x^2}$ then solve with respect to $\int \sqrt{a^2-x^2\,} \,\mathrm{d}x$. Next part is to remember the derivative of th... |
Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$:
1) Take the groupoid of finite dimensional complex inner product spaces with isometries as morphisms and apply Quillen $S^{-1}S$-construction to it. This amounts to forming a new category, whose objects are pa... |
So You Think You Can Statistics: Overlapping Confidence Intervals, Statistical Significance, and Intuition
Attention Conservation Notice:I begin a new series on the use of common sense in statistical reasoning, and where it can go wrong. If you care enough to read this, you probably already know it. And if you don’t al... |
Let's say that I have a bunch of independent samples, $X_1, X_2, \dots, X_n$ and that they all follow Exponential($\theta_i$) distributions. (So they all have pdf $f(x_i)=\theta_i\exp(-\theta_iy_i)$.) I don't know if all the $\theta_i$s are equal or not, so I will assume the worst and say they are not for generalizatio... |
Since 1998 or earlier, there have been no doubts that the AdS/CFT correspondence provides us with a full non-perturbative definition of string theory on the AdS-like background, including all of (type IIB) stringy objects and interactions and subtleties that we have ever heard of. An obvious reason why the CFT can't be... |
A Robust Solver for Distributed Optimal Control for Stokes Flow Dipl.-Ing. Markus Kollmann March 20, 2012, 3:30 p.m. S2 059
In this talk we consider the following optimal control problem:
Minimize $\quad J(u,f) = \frac{1}{2}||u-u_d||_{L^2(\Omega)}^2 + \frac{\alpha}{2}||f||_{L^2(\Omega)}^2 \quad$ (1)
subject to
$-\Delta... |
Let's consider a rigid body B and an arbitrary reference point P ${\it fixed}$ with respect to the body (P can be a particle of the body or it can be a "mathematical point" outside the body but solidary to B, it doesn't matter). As B moves, the point P has a velocity ${\bf v}_P$ that, of course, can change over time. T... |
Difference between revisions of "Demand/Dynamic User Assignment"
(note on convergence and results)
(restructured page)
Line 1: Line 1: + + + + + + + + +
The tool '' {{SUMO}}/tools/assign/duaIterate.py '' can be used to compute the (approximate) dynamic user equilibrium.
The tool '' {{SUMO}}/tools/assign/duaIterate.py '... |
https://www.cwi.nl/research/groups/networks-and-optimization/events/n-o-seminar-kim-manuel-klein-kiel-university N&O seminar: Kim-Manuel Klein (Kiel University) 2019-09-24T11:00:00+02:00 2019-09-24T12:00:00+02:00 Everyone is welcome to attend the N&O seminar by Kim-Manuel Klein with the title 'About the Complexity of 2... |
Research talks;Number Theory
For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$. This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Genera... |
Let $M=\{\frac{1}{n} : n\in\mathbb{N}\}$.
I am trying to find a simple surjective and injective function $[0,1]\to [0,1]\setminus M$.
let define that $[0,1]\setminus M = Y$.
I can't understand how to handle with questions like this. I understand that the set $Y$ has all the elements as $[0,1]$ without elements like $1,... |
This post was first published on 05/14/16, and has since been migrated to Blogger.
Brown University requires S.c.B students to take a
capstonecourse that "studies a current topic in depth to produce a culminating artifact such as a paper of software project".
For my capstone project, I developed a more efficient sampli... |
How to show that all normally open covers form a base for the fine uniformity $\mu_F$? If $\mathcal{B}$ is the collection of all normally open covers, we first need to show that $\mathcal{B}$ is a subcollection of $\mu_F$. Then we have to show that every $\mathcal{U}$ in $\mu_F$ is refined by some cover from $\mathcal{... |
I need some help in plotting the following inequality using Mathematica:
$$\frac{1+(x/100+0.1)\times y/100}{1.15+1.15\times(x/100)\times(y/100)}\geq 1$$
assuming $x,y\in \mathbb{Q}$, the set of rational numbers and $5\leq x\leq 90,\ 50\leq y \leq 650$.
Copyable code for the above inequality:
(1 + (x/100 + 0.1)(y/100))/... |
Answer
Proper fractions have a numerator that is smaller than the denominator (e.g. 1/2), while improper fractions have a numerator that is greater than or equal to the denominator (e.g. 3/2).
Work Step by Step
Proper fractions have a numerator that is smaller than the denominator, while improper fractions have a numer... |
exponential distribution
Every day one sees politicians on TV assuring us that nuclear deterrence works because there no nuclear weapon has been exploded in anger since 1945. They clearly have no understanding of statistics.
With a few plausible assumptions, we can easily calculate that the time until the next bomb exp... |
Multigrid Methods for Elliptic Optimal Control Problems with Neumann Boundary Control Dr. Stefan Takacs June 9, 2009, 3:30 p.m. MZ 005B
In this talk we will discuss multigrid methods for solving the discretized optimality system for elliptic optimal control problems. We will concentrate on the following model problem w... |
Besides the usual deterministic DFS/BFS approaches, one could also consider a randomized algorithm. I will shortly describe a randomized algorithm for deciding if two vertices $s$ and $t$ are connected. It can also be used to decide if the whole graph is connected. The main benefit is that this method requires $O(\log ... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
I know how to solve and prove that
$$\int_{0}^{\infty} \frac{\sin(x)}{x^\alpha} \, dx$$ converge for $ 0 < \alpha < 2 $ with regular tests and integration by parts.
But with the Dirichlet test, I just see that I have one function which is monotonic decreasing to $0$ for any given $\alpha$ and the integral on the $\sin(... |
Linear regression is a statistical modeling technique used to describe a continuous response variable as a function of one or more predictor variables. It can help you understand and predict the behavior of complex systems or analyze experimental, financial, and biological data.
Linear regression techniques are used to... |
I have a doubt about the equivalence between Fourier Transform and Laplace Transform.
It was told me that if I have a function such that:
$f(t)=0$ if t<0 $f\in L^1(R) \bigcap L^2(R)$
I can define
$F[f(t)]=\int_{0}^{\infty}f(t) e^{-j\omega t}dt$
$L[f(t)]=\int_{0}^{\infty}f(t) e^{-s t}dt$
I can look at the Fourier transf... |
Hi what would be a good test to find convergence or divergence please?
$\sum _1 ^{\infty} (e^kcos^2k)/ \pi ^k$
My attempt I got that converges thanks
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign... |
In implementations of the
Metropolis-Hastings algorithm, how is the target distribution $\pi(\mathbf{x}) = P(\mathbf{x}|\mathbf{e})$ computed or estimated while computing the acceptance probability $\alpha(\mathbf{x'}|\mathbf{x}) = \min \left(1,\frac{\pi(\mathbf{x'})q(\mathbf{x}|\mathbf{x'})}{\pi(\mathbf{x})q(\mathbf{x... |
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Now showing items 1-5 of 5
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
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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 ... |
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Now showing items 1-10 of 18
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 in... |
I am trying to solve this integral but do not get any point to start.I was thinking an U-substitution may help but do not know what to consider as U. Tried with mathematica but it does not provide any solution. Can anyone help? $\int sin\theta *2\sin^{-1} [\frac{\cos \theta+\sin \gamma}{\sin \theta \tan \beta}] d\theta... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
Literature on Carbon Nanotube Research
I have hijacked this page to write down my views on the literature on Carbon Nanotube (CNT) growths and processing, a procedure that should give us the cable/ribbon we desire for the space elevator. I will try to put as much information as possible here. If anyone has something to... |
Literature on Carbon Nanotube Research
I have hijacked this page to write down my views on the literature on Carbon Nanotube (CNT) growths and processing, a procedure that should give us the cable/ribbon we desire for the space elevator. I will try to put as much information as possible here. If anyone has something to... |
Recall that
$${n\choose k}={n!\over k!\,(n-k)!}={n(n-1)(n-2)\cdots(n-k+1)\over k!}.$$
The expression on the right makes sense even if \(n\) is not a non-negative integer, so long as \(k\) is a non-negative integer, and we therefore define
$${r\choose k}={r(r-1)(r-2)\cdots(r-k+1)\over k!}$$
when \(r\) is a real number. ... |
Table of Contents
Self-Adjoint Linear Operators over Complex Vector Spaces
Recall from the Self-Adjoint Linear Operators page that if $V$ is a finite-dimensional nonzero inner product space and if $T \in \mathcal L (V)$ then $T$ is said to be self-adjoint if $T = T^*$.
In the following proposition we will see that if $... |
The Wikipedia article on the Gamma distribution, lists two different parameterisation methods, one of them frequently used in Bayesian econometrics with $\alpha>0$ and $\beta>0$, $\alpha$ is shape parameter, $\beta$ is rate parameter.
$$X\sim \mathrm{Gamma}(\alpha,\beta).$$
In a Bayesian econometrics textbook written b... |
2019-05-20 15:18 Detailed record - Similar records 2019-01-23 09:13
nuSTORM at CERN: Feasibility Study / Long, Kenneth Richard (Imperial College (GB)) The Neutrinos from Stored Muons, nuSTORM, facility has been designed to deliver a definitive neutrino-nucleus scattering programme using beams of $\bar{\nu}_e$ and $\bar... |
A
filter F
on a set
S
is a set of subsets
of
S
with the following properties:
S is in F.
The empty set is not in F.
If A and B are in F, then so is their intersection.
If A is in F and A ⊆ B ⊆ S, then B is in F.
A simple example of a filter is the set of all subsets of
S that include a particular subset C of S. Such a ... |
Polynomials Applied to Linear Operators
Suppose that $T$ is a linear operator from the vector space $V$ to $V$. Then the operator $T \circ T = T^2$ is an operator from $V$ to $V$. In fact, we can compose $T$ with itself multiple times, that is, for $m$ as a positive integer we can define the operator $T^m$ as follows:(... |
As Lubos Motl and twistor59 explain, a necessary condition for unitarity is that the Yang Mills (YM) gauge group $G$ with corresponding Lie algebra $g$ should be real and have a positive (semi)definite associative/invariant bilinear form $\kappa: g\times g \to \mathbb{R}$, cf. the kinetic part of the Yang Mills action.... |
Inner Product Spaces
We are now going to look at a type of vector space that is associated with a function known as an inner product which we define below.
Definition: Let $V$ be a vector space over the field $\mathbb{F}$ ($\mathbb{R}$ or $\mathbb{C}$). An Inner Product on $V$ is a function which takes each pair of vec... |
The Area of a Parallelogram in 3-Space
Given two vectors $\vec{u} = (u_1, u_2, u_3)$ and $\vec{v} = (v_1, v_2, v_3)$, if we place $\vec{u}$ and $\vec{v}$ so that their initial points coincide, then a parallelogram is formed as illustrated:
Calculating the area of this parallelogram in 3-space can be done with the formu... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
I am interested in computing a normalizing constant (of a Gaussian density in dimension $3$). Such normalizing constants often do not have a closed form. In dimension $2$, this normalizing constant can be computed in closed form.
The integral I would like to compute is :
$$ \int_{\mathbb{R}^{3}} e^{-(r_{1}^{2} + r_{2}^... |
Abstract / Bemerkung
We use linear estimators to determine the magnitude and direction of thecosmic radio dipole from the NRAO VLA Sky Survey (NVSS) and the WesterborkNorthern Sky Survey (WENSS). We show that special attention has to be given tothe issues of bias due to shot noise, incomplete sky coverage and masking o... |
Literature on Carbon Nanotube Research
I have hijacked this page to write down my views on the literature on Carbon Nanotube (CNT) growths and processing, a procedure that should give us the cable/ribbon we desire for the space elevator. I will try to put as much information as possible here. If anyone has something to... |
As Lubos Motl and twistor59 explain, a necessary condition for unitarity is that the Yang Mills (YM) gauge group $G$ with corresponding Lie algebra $g$ should be real and have a positive (semi)definite associative/invariant bilinear form $\kappa: g\times g \to \mathbb{R}$, cf. the kinetic part of the Yang Mills action.... |
Table of Contents
The Interior of a Set under Homeomorphisms on Topological Spaces
Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open.
Furthermore, if such a homeomorphism exis... |
Problem Statement
How big do my hash buckets need to be? What kind of a question is that anyway?
Let's stand back a bit...
As every computer scientist ought to know, a hash table is a wonderfully efficient,
O(1), way to look something up in a table. The idea is simple. For each thing you want to look up, you calculate ... |
Modes
The term mode has varying meanings, according to the context, but the most common are permitted modes in amateur licensing.
Waves have three characteristics that can be changed, Amplitude, Frequency and Phase. A mode is the way of changing electromagnetic waves,
modulating them so that transmission of information... |
Skills to Develop
Use multiplication notation Model multiplication of whole numbers Multiply whole numbers Translate word phrases to math notation Multiply whole numbers in applications Use Multiplication Notation
Suppose you were asked to count all these pennies shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1... |
This blog post is a result of a discussion (read borderline heated argument) that I had with a friend regarding
if machine learning terms loss function and objective function meant the same. We will get to know if they do mean the same or not by the end of this post. I am choosing to write this post as a dialogue betwe... |
The following question is part (1/4) of a 2.30h written exam for the course "Probability and Statistics" in a school of engineering. So, although tricky and difficult (because the Professor is really demanding from his students), it should be solvable in a logical amount of time and with a logical amount of calculation... |
As Lubos Motl and twistor59 explain, a necessary condition for unitarity is that the Yang Mills (YM) gauge group $G$ with corresponding Lie algebra $g$ should be real and have a positive (semi)definite associative/invariant bilinear form $\kappa: g\times g \to \mathbb{R}$, cf. the kinetic part of the Yang Mills action.... |
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Now showing items 31-40 of 108
Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV
(Elsevier, 2016-12)
We present measurements of two-particle correlations with neutral pion trigger particles of transverse momenta $8 < p_{\mathrm{T}}^{\rm trig} < 16 \mathrm{GeV}/c$ and... |
Injective and Surjective Linear Maps Examples 1
Recall from the Injective and Surjective Linear Maps page that a linear map $T : V \to W$ is said to be
injective if: $T(u) = T(v)$ implies that $u = v$. $\mathrm{null} (T) = \{ 0 \}$.
Furthermore, the linear map $T : V \to W$ is said to be surjective if:**
If for every $... |
Rotational Operators
Definition: For any vector $\vec{x} \in \mathbb{R}^n$, a rotational transformation operator $T: \mathbb{R}^n \to \mathbb{R}^n$ rotates every vector $\vec{x}$ by a fixed angle $\theta$.
We will first take a look at rotational transformations in $\mathbb{R}^2$ and then in $\mathbb{R}^3$.
Rotational T... |
A proton-core that establishes a frame-of-reference and the origin for a three-dimensional space containing a photon.
Recall that the quark model of a proton is a bundle of twelve quarks
$\sf{p^{+}} \leftrightarrow \mathrm{4}\bar{\sf{d}} + \mathrm{4}\sf{b} + \mathrm{4}\bar{ \sf{t} }$
Let us combine this proton with an ... |
Dividing the numerator and denominator of the integral by $\gamma + \kappa$ gives$$\int_{0}^{x}\frac{a(e^{\gamma u}-1)du}{e^{\gamma u}-1+a\gamma}$$Where $a=\frac2{\gamma + \kappa}.$
Breaking this into $2$ integrals,$$=\frac a\gamma\int_{0}^x\frac{\gamma e^{\gamma u}du}{e^{\gamma u}-1+a\gamma}-a\int_0^x\frac{du}{e^{\gam... |
Is there a way to prove that $\sqrt[m]{a} + \sqrt[n]{b}$ ($\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational); $a, b, m, n \in \mathbb{N}$; $m, n \neq 2$; is irrational without using the theorem mentioned in Sum of irrational numbers, a basic algebra problem?
If one of $m$ or $n$ is $2$, then a polynomial with integer coef... |
As John Rennie says the hydrogen atom is a case where the electron is in the s shell, which means it has no angular momentum. We can think of the electron if it were measured as being at a point above the proton, where the electron as a wave would then spread into a spherical shape around the proton. One might imagine ... |
Injective and Surjective Linear Maps Examples 2
Recall from the Injective and Surjective Linear Maps page that a linear map $T : V \to W$ is said to be
injective if: $T(u) = T(v)$ implies that $u = v$. $\mathrm{null} (T) = \{ 0 \}$.
Furthermore, the linear map $T : V \to W$ is said to be surjective if:
If for every $w ... |
Your comments above lead me to believe that you are asking about the distribution of $X$ marginalized over the classes. This marginal distribution is Gaussian in the first dimension but not in the second.
Since the first and second dimensions are independent they can be treated separately. The means and variances condi... |
Beta regression (i.e. GLM with beta distribution and usually the logit link function) is often recommended to deal with response aka dependent variable taking values between 0 and 1, such as fractions, ratios, or probabilities: Regression for an outcome (ratio or fraction) between 0 and 1.
However, it is always claimed... |
Injective and Surjective Linear Maps Examples 4
Recall from the Injective and Surjective Linear Maps page that a linear map $T : V \to W$ is said to be
injective if: $T(u) = T(v)$ implies that $u = v$. $\mathrm{null} (T) = \{ 0 \}$.
Furthermore, the linear map $T : V \to W$ is said to be surjective if:
If for every $w ... |
The question is in the title.
Meta-analysis with binomial distribution
I'm trying to do a meta-analysis on occurence of an event across different studies. For each study $i$ ($i=1,\dots,N$), I have the number of participants $n_i$ and the number of events $k_i$. I don't have individual data. The number of event thus fo... |
Table of Contents
The Canonical Injections of Weak Direct Products of Groups
Recall from The Weak Direct Product of an Arbitrary Collection of Groups page that if $\{ G_i : i \in I \}$ is an arbitrary collection of groups the the weak direct product of the groups $\{ G_i : i \in I \}$ is the set:(1)
with the operation ... |
Research talks;Partial Differential Equations;Mathematical Physics
In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.
$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$ $\nabla \cdot u =0... |
Research talks
The productivity of the $\kappa $-chain condition, where $\kappa $ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fle... |
A general conic is defined by five independent parameters and can pass through five arbitrary points.
Restricting to a parabola sets a constraint on the coefficients (the discriminant of the second degree terms must be zero), which "consumes" one degree of freedom.
But four remain, and you have an infinity of parabolas... |
$A$ and $B$ being independent is a common, but actually strong assumption. It implies one special property for the joint probability density function:
$$f_{AB}(a,b)=f_{A}(a)f_{B}(b)$$
If $X$ is a random variable of its own, the joint probability of $X$, $A$ and $B$ is :
$$f_{XAB}(x,a,b)$$
But the conditional probabilit... |
Suppose your series is $f(x) = \sum_{n=0}^\infty a_n x^n$ with the signs of $a_n$ alternating. If $|a_n| r^n$ is decreasing to $0$, this is an alternating series for $0 < x < r$.
The alternating series bound for the remainder after the $x^n$ term is then $|a_{n+1}| x^{n+1}$. The Lagrange form for the remainder is$\dfra... |
Although I've not specifically attempted a \$15\:\text{A}\$ boosted LM317 before, this is along the lines of what I'd try out first. This is roughly taken from the Figure 23 you mentioned:
simulate this circuit – Schematic created using CircuitLab
In this case, I went for the D44/D45 series devices. (The PNP version ha... |
Isogeometric Analysis Discontinuous Galerkin discretizations for elliptic problems with discontinuous coefficients Dr. Ioannis Toulopoulos March 18, 2014, 3:30 p.m. S2 059
In this talk, Isogeometric Analysis (IGA) methods utilizing discontinuous approximations spaces for the solution of
an elliptic problem with discont... |
I have some problems of understanding asymptotic definitions of the growth of functions. Namely, I saw from the algorithm book of mine that $O(g(n))=\{f(n):$ there exist a positive constants $c$ and $n_0$ such that $0\leq f(n)\leq cg(n)$ for all $n\geq n_0\}$. But in https://artofproblemsolving.com/community/c7h31517 I... |
Here we first give a step-by-step solution of the basic question 1, and then turn to other questions.
Integral representation of $S(p,q,x)$
Writing
$$k^{-q}=\frac{1}{\Gamma (q)}\int_0^{\infty } e^{-k \;t} t^{q-1} \, dt\tag{s1}$$
and observing the formula for the generating function of the generalized harmonic number
$$... |
I think that the problem stems from the action of the operator $\hat p$. Please correct me if I am mistaken.
The action of the operator $\hat p$ in the quantum space is defined as $<x|\hat p|a>=-i \hbar \partial_x <x|a>$ if the state $|a>$ does not depend on x. In fact, if the state $|a>$ depended on $x$, like for inst... |
In set theory, we have the phenomenon of the
universal definition. This is a property $\phi(x)$, first-order expressible in the language of set theory, that necessarily holds of exactly one set, but which can in principle define any particular desired set that you like, if one should simply interpret the definition in ... |
Consider the following matrix: $ \left[ \begin{array}{ccc} -0.05 & 0.45 & 0 \\ 0.05 & -0.45 & 0 \end{array} \right] $
Row reducing the above matrix in Matlab using the
rref() function produces what I would expect (just adding top row to bottom row and scaling top row):$ \left[ \begin{array}{ccc}1.0 & -9.0 & 0 \\0 & 0 &... |
Spectroscopy operation
In spectroscopy, an image of the projector lens crossover is magnified and projected onto the camera. When properly adjusted, the observed CCD image consists of sharp vertical lines that represent electrons of particular energy. Displacement of lines in the horizontal (or dispersive) direction in... |
Let $X$ denote a monoid. Then we can make $Y = \mathcal{P}(X)$ into a monoid, too. Define $$AB = \{ab \mid a \in A, b \in B\}$$ for all $A,B \in Y.$ We see immediately that $1$ (shorthand for $\{1\}$) is our new identity:
$$A1 = A, \;\; 1B = B.$$
In fact, $Y$ becomes an ordered monoid by defining that $A \leq B$ is not... |
Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin.
Problem 1: Given a point $a \in \mathbb R^n$ with $a \not \in \mathcal P_X$, how to compute the euclidean projection of $... |
You can directly imply a probability distribution from a volatility skew.Note that, for any terminal probability distribution $p(S)$ at tenor $T$, we have the model-free formula for the call price $C(K)$ as a function of strike $K$\begin{equation}C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS\end{equation}Therefore we can wri... |
In Delta of binary option, I do not see how to prove that the limit of $\partial C_t/\partial S_t$ is equal to $+\infty$ as $t \rightarrow T$. Can someone help ?
The value of a bond binary call in the Black-Scholes model is given by
\begin{equation} B_t = e^{-r (T - t)} \mathcal{N} \left( d_- \right), \end{equation}
wh... |
I am trying to rewrite a Schrödinger equation using dimensionless quantities but here, the potential is perturbed by $\lambda x^4$:
\begin{equation}V(x) = \frac{m\omega^2}{2}x^2 + \lambda x^4\end{equation}
$\lambda$ and $\omega$ are parameters of the system.
The Schrödinger equation reads then: \begin{equation}-\frac{\... |
In Exercises \((2.3E.1)\) to \((2.3E.6)\), find a particular solution by the method used in Example \((2.3.2)\). Then find the general solution and, where indicated, solve the initial value problem and graph the solution.
Exercise \(\PageIndex{1}\)
\(y''+5y'-6y=22+18x-18x^2\)
Answer
Add texts here. Do not delete this t... |
Basic Theorems Regarding Free Groups on a Set
Basic Theorems Regarding Free Groups on a Set
On the The Free Group on a Set X, F(X) page we defined the free group $F(X)$ generated by the set $X$. We will now look at some basic results regarding free groups.
Proposition 1: The free group generated $1$ element, $F_1$ is i... |
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