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The Annals of Statistics Ann. Statist. Volume 26, Number 5 (1998), 2014-2048. Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations Abstract
We derive an approximation of a density estimator based on weakly dependent random vectors by a density... |
Existence and location of periodic solutions to convex and non coercive Hamiltonian systems
1.
Department of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy
$J\dot u(t)+\nabla H(t,u(t))=0$ a.e. on $[0,T] $
$u(0)=u(T)$
where the function $H:[0,T]\times \mathbb R^{2N}\rightarrow \mathbb R$ is called H... |
A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another approach takes $M$ to be countable such that $M \prec H_\theta$ for sufficiently large $\theta$ (and hence may not be transi... |
Here is
a reason.
The fourth of Maxwell's macroscopic equations says that$$ \nabla \times \vec{H} = \vec{J} +\frac{\partial \vec{D}}{\partial t},$$where $\vec{J}$ is the free current at a point. In general, it
is not possible to rewrite this in terms of B-field without a detailed knowledge of the microscopic behaviour ... |
Measurable cardinal
A
measurable cardinal $\kappa$ is an uncountable cardinal such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued measure on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point... |
Assuming I have the following homogeneous ODE equation:$$a\cdot y'' + b\cdot y' + c \cdot y = 0$$Why for $(b^2 - 4\cdot a\cdot c=0) \quad $,(meaning, when $m_1=m_2$) then the solution is:$$y = C_1\cdot e^{m_{1}x} + C_2\cdot x \cdot e^{m_{2}x}$$Why isn't it simply:$$y = C_1\cdot e^{m_{1}x} + C_2 \cdot e^{m_{2}x}$$?
Also... |
Indescribable cardinal
A cardinal $\kappa$ is
indescribable if it holds the reflection theorem up to a certain point. This is important to mathematics because of the concern for the reflection theorem. In more detail, a cardinal $\kappa$ is $\Pi_{m}^n$-indescribable if and only if for every $\Pi_{m}$ first-order senten... |
Exercise on Stein's/Sharkachi book chapter 6:
Let $u(x, t)$ be a smooth solution of the wave equation and let $E(t)$ denote the energy of this wave $$E(t) = \int_{\mathbb R^d} \bigg|\frac{\partial u(x,t)}{\partial t}\bigg|^2+\sum_{j=1}^d \int_{\mathbb R^d}\bigg | \frac{\partial u(x,t)}{\partial x_j}\bigg |^2 dx.$$ We h... |
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a metric given by the angle between subspaces (varying between a minimum of $0$ for identical subspaces and a maximum of $... |
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... |
Notation $\le$ is used for the subgroup relation; $P$ means polynomial time in input size; $\Omega = \{1,2,3,\cdots,n\}$ is a input domain; $\mathrm{Sym}(\Omega)$ means the symmetric group on $\Omega$; $G = \langle A \rangle $ means the subgroup $G$ generated by the subset $A$ of $\mathrm{Sym}(\Omega)$.
The
normal cent... |
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... |
The type of symmetry for which the simplex (not necessarily regular) is usually called "the least symmetric convex body" is the symmetry of reflection about a point (e.g. $x\mapsto-x$). There are a few measures of this symmetry that the simplex minimizes. For any convex body $K\subset\mathbf{R}^n$, consider the followi... |
Let $C$ be the space nº74 ("double origin topology") in Steen & Seebach's
Counterexamples of Topology, chosen because it is the only one listed there that is T2 and path-connected but not T3:
$C$ consists of the set of points of the plane $\mathbb{R}^2$ together with an additional point $0^*$. Neighborhoods of points o... |
Using the basic operations of relational algebra (RA), it is possible to obtain the largest value assigned to a given attribute of a relation. This post shows how this can be done.
To start, consider the following operators of RA (below $R$ is a relation):
$\sigma_{\theta}(R)$ select tuples (rows) from $R$ which satisf... |
Wikibooks:Manual of Style
This page contains a draft proposal for a Wikibooks policy or guideline. Discuss changes to this draft at the discussion page. Through consensus, this draft could become an official Wikibooks policy or guideline.
The
Wikibooks Manual of Style serves to establish good structural and stylistic p... |
This is a very interesting question. I don't know if there is a general and definitive answer, but I'll try to make some comments. I apologize if this ends up rambling; I'm finding this out as I write this answer.
Operators have dimensions, since their eigenvalues are physical quantities. For bras and kets it gets more... |
Update: Trimok and MBN helped me solve most of my confusion. However, there is still an extra term $-(2/r)T$ in the final result. Brown doesn't write this term, and it seems physically wrong. Update #2: Possible resolution of the remaining issue. See comment on MBN's answer.
Suppose we have a rope hanging statically in... |
Home
Integration by PartsIntegration by Parts
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Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
There are two approaches to this problem - we can solve it directly, or we can examine the answer choices and come to a conclusion about which one must be correct.
Algebraic approach:
Draw a perpendicular from point A to side BC, label it point D. Draw another perpendicular from point B to side AC, label it point E. No... |
V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem.” (manuscript under review)
@ARTICLE{GitmanHamkinsKaragila:KM-set-theory-does-not-prove-the-class-Fodor-theorem, author = {Victoria Gitman and Joel David Hamkins and Asaf Karagila}, title = {Kelley-Morse set theor... |
I learned a lot while building this website; I hope to share it so that it might be helpful for anyone trying to do the same. I’m sure you’ll notice that I’m far from an expert in the subjects we’re going to explore here; this is my first foray into web development. If you have any corrections, or things I’ve misunders... |
I have been trying to find out the sweet middle ground of describing path integration of field theories, in between the physicist way and the mathematician way, but it seems hard to find something that is both rigorous and describes how to actually compute them.
So far what I've been able to get is this. This is in the... |
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Angewandte Chemie International Edition, ISSN 1433-7851, 08/2017, Volume 56, Issue 35, pp. 10539 - 10544
The silver‐catalyzed oxidative C(sp 3 )−H/P−H cross‐coupling of 1... |
Let $S$ be a nonempty bounded subset of $\mathbb{R}$. Define the set $kS = \{ks : s \in S\}$. We wish to prove that if $k \geq 0$, then $\sup(kS) = k\cdot \sup S$.
I'm pretty sure the upper half of the proof is fine, but it's the lower half that attempts to show that $k\cdot \sup S \leq \sup(kS)$ with which I am concer... |
Let $M$ a smooth manifold and $\nabla $ a connexion. Let $\gamma :[a,b]\longrightarrow M$ a $\mathcal C^\infty $ curvature. I recall that if $X,Y\in \Gamma(M)$, and $f,g\in \mathcal C^\infty (M)$, then $$\nabla_{fX}(gY)= f X(g)Y+fg\nabla _XY.$$ So $\nabla _{\dot \gamma (t)}$ is the covariante derivative (the derivate a... |
This answer combines @AntonioVargas and @GregMartin results.
Let us start from the approximation in Laplace's method (https://en.wikipedia.org/wiki/Laplace%27s_method)
$$\int_a^b h(x)e^{Mg(x)}dx \approx \sqrt{\frac{2\pi}{M\|g''(x_0)\|}}h(x_0)e^{Mg(x_0)}$$
and rewrite the general integral in the question as
$$\int_0^1 \... |
I have a regression model
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 +u$
It is known that sample means of both $x_1$ and $x_2$ are zero, moreover the error term is said to be homoskedastic, the standard error of regression and standard errors of OLS estimates of the slope coefficients are given. Can we say something abou... |
It's \(\displaystyle \log\left(\frac{A-}{HA}\right)\) . So are to solve (A-)/(HA) together ? I thought of putting log10 on both sides. I do not know if log10 to the power of \(\displaystyle \log\left(\frac{A-}{HA}\right)\), does the log10 and the log cancel out to be just (A-)/(HA) and the other side log10 to the power... |
Difference between revisions of "Kunen inconsistency"
Line 22: Line 22:
<cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),
<cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),
Hamkins, Kirmayer and Perlmutter <cite>HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency</cite>, Woodin <cite>Kanamori2009:Highe... |
Is it true that if the average of a continuous function $f:\mathbb{R}^2\rightarrow[0,1]$ over a unit circle centered around $(x,y)$ is $f(x,y)$ for all $(x,y)\in\mathbb{R}^2$, then $f$ is necessarily constant?
Yes, any such $f$ is constant. In fact, if we relax the condition so that $f$ is only required to be bounded b... |
From my understanding, both are estimators that are based on first providing an unbiased statistic $T(X)$ and obtaining the root to the equation:
$$c(X) \left( T(X) - E(T(X)) \right) = 0$$
Secondly both are in some sense "nonparametric" in that, regardless of what the actual probability model for $X$ may be, if you thi... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having trouble understanding intuitively why it is. I know I must never trust my intuition, but this is hard for me to grasp. In b... |
Riemann curvature s denoted by $R^a_{bcd}$. If you try to attempt to parallel transport a tangent vector $\psi$ along an infinitesimal parallelogram given by two tangent vectors along non-parallel sides as $\eta$ and $\nu$, the following holds true. $$\psi^a_{;cd} - \psi^a_{;dc} = R^a_{cdb}\psi^b$$ where the $;$ indica... |
Question:
Below is a diagram of the sound waves produced by two speakers, where the collection of dots represent vibrating air molecules and the distance between each compression (i.e., the region where the molecules are closer together and a higher pressure) is also given. If you are standing near the speakers as they... |
A figure is made from a semi circle and square. With the following dimensions, width = w, and length = l.Find the maximum area when the combined perimiter is 8 meter.I first try to construct the a function for the perimeter.2*l + w + 22/7*w/2 = 8 - > l = 4 - (9*w)/7Next I insert this...
I have a data set of samples, an... |
This question is followed up from this question How to solve a certain coupled first order PDE system
Here I consider the non-homogeneous advection system
\begin{equation} \displaystyle\left\{\begin{array}{l} \frac{\partial U}{\partial t}+b_1\frac{\partial U}{\partial x}=(r+l_1)U-l_1V, \\ \frac{\partial V}{\partial t}+... |
Of course some math will be involved, but it's not much: Euclid would have understood it well. All you really need to know is how to add and rescale vectors. Although this goes by the name of "linear algebra" nowadays,
you only need to visualize it in two dimensions. This enables us to avoid the matrix machinery of lin... |
I confess, in 1965 or so, I first thought that the version of the SvKT (Seifert-van Kampen Theorem) for the fundamental groupoid enabled us to get rid of base points. But then I wanted to calculate the fundamental group of the circle, and gradually realised that we needed $\pi_1(X,A)$, the fundamental groupoid on a set... |
You are here Basic Electric Guitar Circuits 2: Potentiometers & Tone Capacitors Part 2: Potentiometers and Tone Capacitors What is a Potentiometer?
Potentiometers, or "pots" for short, are used for volume and tone control in electric guitars. They allow us to alter the electrical resistance in a circuit at the turn of ... |
Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e.,
$$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$
$$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$
and so on. Define the function,
$$F_{2k}(\tau) = \frac{E_{2k}(\tau)}{\left(E_2(\tau)-\frac{3}{\pi\; \Im(\tau)}\right)^k}$$
fo... |
Learning Objectives
To describe the characteristics of ionic bonding. To quantitatively describe the energetic factors involved in the formation of an ionic bond. Ions are atoms or molecules which are electrically charged. Cations are positively charged and anions carry a negative charge. Ions form when atoms gain or l... |
I'm not really $100\%$ sure my solution is correct, also, there remains an indeterminacy in the end result: I don't quite identify the class you are looking for in $P^1\Bbb H\simeq \Bbb S^4$, only up to sign.
Let $Q$ be the projectivisation of $p:V\rightarrow P^1\Bbb H$. As you note, $Q$ is isomorphic to $P^3\Bbb C$. T... |
A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Central European J.~Math., vol. 11, iss. 9, pp. 1628-1634, 2013.
@article {ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders, AUTHOR = {Apter, Arthur W. and Cummings, James and Hamkins, Joel David}, TITLE = {Singular card... |
This is a homework question and I am to show that $$\sigma(n) = \sum_{d|n} \phi(n) d\left(\frac{n}{d}\right)$$ where $\sigma(n) = \sum_{d|n}d$, $d(n) = \sum_{d|n} 1 $ and $\phi$ is the Euler Phi function.
What I have. Well I know $$\sum_{d|n}\phi(d) = n$$ I also know that for $n\in \mathbb{Z}^n$ it has a certain prime ... |
I would like to determine a closed-form expression for the following symbolic integral
$$ \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} x \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} y \, \frac{1 + b x + c y}{1 + e x + f y + i \eta} \, ,$$
where the coefficients appearing in this expression are such that
$$ \begin{cases} \displaystyl... |
If you are a physicist or at least know a bit about the theory of relativity, your answer to this question might be a confident "no". Sure, if nothing can exceed the speed of light, one would need an infinite amount of time to travel an infinite distance. But let's put the relativistic laws of physics aside and imagine... |
On the domination and signed domination numbers of zero-divisor graph
Ebrahim Vatandoost, Fatemeh Ramezani
Abstract
Let $R$ be a commutative ring (with 1) and let $Z(R)$ be its set of zero-divisors. The zero-divisor graph $\Gamma(R)$ has vertex set $Z^*(R)=Z(R) \setminus \lbrace0 \rbrace$ and for distinct $x,y \in Z^*(... |
Moser-lower.tex
\section{Lower bounds for the Moser problem}\label{moser-lower-sec}
In this section we discuss lower bounds for $c'_{n,3}$. Clearly we have $c'_{0,3}=1$ and $c'_{1,3}=2$, so we focus on the case $n \ge 2$. The first lower bounds may be due to Koml\'{o}s \cite{komlos}, who observed that the sphere $S_{i,... |
Suppose $z$ is a complex number with $\bar{z}$ denoting its conjugate. Does there exist real numbers $\{a_1,\ldots, a_n\}$ such that
$$z^k+\bar z^k= a_1^k+a_2^k+\cdots+a_n^k,$$
for all $k\in\mathbb N$?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in re... |
My textbook says that $\lambda$ is the probability per unit time that 1 particle will decay in one second. This makes absolutely no sense to me - I can see that it is related to probability but cannot see how it is the probability. We have: $$N = N_0 e^{-\lambda t} \tag A$$ Now, if the probability of a particle decayin... |
The likelihood could be defined by several ways, for instance :
the function $L$ from $\Theta\times{\cal X}$ which maps $(\theta,x)$ to $L(\theta \mid x)$ i.e. $L:\Theta\times{\cal X} \rightarrow \mathbb{R} $.
the random function $L(\cdot \mid X)$
we could also consider that the likelihood is only the "observed" likeli... |
Along the lines of Glen O's answer, this answer attempts to explain the solvability of the problem, rather than provide the answer, which has already been given. Instead of using the meta-knowledge approach, which, as Glen stated, can get hard to follow, I use the range-base approach used in Rubio's answer, and specifi... |
Difference between revisions of "Group cohomology of dihedral group:D8"
(→Over an abelian group)
(→Over the integers)
Line 40: Line 40:
The cohomology groups with coefficients in the ring of integers are as follows:
The cohomology groups with coefficients in the ring of integers are as follows:
−
<math>H^p(D_8;\mathbb{... |
Howto Raytracer: Ray / Plane Intersection Theory
In this tutorial I will derive how to calculate the intersection of a ray and a plane.
As already stated in my ray / sphere intersection howto, a ray $$r(t)$$ can be represented by a point on the ray $$e$$ and the ray’s direction $$d$$: $$r(t)=e + t d$$. The set $$R$$ of... |
Proposition: Such posets have exactly two maximal elements, one of which lies above every non-maximal element, I call this one supermaximal (according with the original definition of Jeremy Rickard in the first link). Also, removing the supermaximal element leaves the incomparability graph connected (this is easy to se... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
I am new to Mathematica, I am trying to generate the polynomial function of a operator. So for example, the operator $L $ is $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} $, and I want to generate the polynomial $ \sum_{n=0}^{n=k} (L/2)^n$ and apply that polynomial operator to a function. I was trying to... |
I think there is a typo in the first formula.Let me propose this (partial) answer for the $3$ first formulae:
Because $H(z)H(0) \sim -ln(z)$, we may write the OPE for any pair of operators $F(H), G(H)$ functions of $H$ (in analogy with formula $2.2.10$ p.$39$ vol $1$)
$$:F::G: = e^{- \large \int dz_1 dz_2 ln z_{12} \fr... |
The
Einstein solid is a model of a solid based on two assumptions:
While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves or phonons, collective modes involving many atoms. In the
Einstein model, each atom oscillates independently. Einstein was aware that get... |
In this post, we will show that the Earth's rotation alters the surface of its ocean, with "ocean" here meaning the global ocean of the Earth, i.e., the continuous body of water encircling it. Our model will consist of a spherical Earth with radius $R$ rotating at an angular velocity $\omega$ around a fixed axis.
The e... |
Let $X$ be a Banach space and $X^*$ its dual. We know that the weak* topology is the least topology that makes every $x \in X$ continuous as an evaluation functional. However, this does not imply that every weak* continuous linear functional is something in $X$, even though this happens to be true.
The question is: how... |
Thank you for using the timer!We noticed you are actually not timing your practice. Click the START button first next time you use the timer.There are many benefits to timing your practice, including:
Does GMAT RC seem like an uphill battle? e-GMAT is conducting a free webinar to help you learn reading strategies that ... |
Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$?
Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and $\zeta(\frac{141}{23}) \approx e^\frac{1}{64}$. I also figured out that $\zeta(x)$ approaches $e^{2^{-x}}$ but I'm not sure that he... |
We have thought a bit about the last paragraph of the above question and have some arguments as to what the answer should be. Since there have been no replies here so far, maybe I am allowed to hereby suggest an answer myself.
Recall, the last part of the above question was: is there a nonabelian 7-dimensional Chern-Si... |
My search for numbers to support any conclusion to this question that included wind factors led me down a rabbit hole of interesting science. I'll try to keep the following answer as clear and concise as I can.
I started with a basic question: What are the limits of a wall? After some finagling of my Google search term... |
Kramer (1980) proposed a method for assessing inter-rater reliability for tasks in which raters could select multiple categories for each object of measurement. The intuition behind this method is to reframe the problem from one of classification to one of rank ordering. Thus, all selected categories are tied for first... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
A supernova explosion is caused by the collapse of the
core. Some of the gravitational potential energy released in this collapse is (somehow) transferred to the envelope. The transferred energy is sufficient to unbind the envelope.
Some numbers: If a 1.25 solar-mass $(M_{\odot})$ stellar core (roughly the Chandrasekha... |
In this article, we’ll look at the relationship between the rise/fall time of a digital signal and its bandwidth. Creating a connection between these two parameters, one from the time domain and the other from the frequency domain, allows us to more easily analyze a circuit. We’ll see that insight into both domains all... |
Difference between revisions of "Kakeya problem"
Line 23: Line 23:
One can get essentially the same conclusion using the "bush" argument. There are <math>N := (3^n-1)/2</math> different directions. Take a line in every direction, let E be the union of these lines, and let <math>\mu</math> be the maximum multiplicity of... |
I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes.
What are the respective meanings of $\Lambda^\mu{}_\nu$ as compared to $\Lambda_\mu{}^\nu$? How does one use this staggered index notation to denote transpose or inverse?
If I want to t... |
I have managed to compute the half width at half maximum (HWHM) and full width at half maximum (FWHM) of a sinc function through the code below.
ClearAll["Global`*"]Plot[Sin[\[Pi]*x]/(\[Pi]*x), {x, -2, 2}, PlotRange -> Full, Exclusions -> None, GridLines -> Automatic, ImageSize -> Full]Normal[Series[Sin[\[Pi]*x]/(\[Pi]... |
Every day, Danny buys one sweet from the candy store and eats it. The store has $m$ types of sweets, numbered from $1$ to $m$. Danny knows that a balanced diet is important and is applying this concept to his sweet purchasing. To each sweet type $i$, he has assigned a
target fraction, which is a real number $f_ i$ ($0 ... |
LEARNING OBJECTIVES
By the end of this module, you will be able to:
Explain the function of a catalyst in terms of reaction mechanisms and potential energy diagrams List examples of catalysis in natural and industrial processes
We have seen that the rate of many reactions can be accelerated by catalysts. A catalyst spe... |
I want to know the heat conduction coefficient of a cup in order to investigate why tea cools at different rates in different containers. I know the cup is mostly plastic and paper. How do I determine the heat conduction? Do I use a known value?
If you are interested in knowing why your tea is cooling in different time... |
Suppose you want to buy some candy at a store. You look at your pocket and find you have (in your local currency) an amount of money equal to $i^i$ ($i$ here is really the complex number $i$). You choose a snack and check its price: its costs $0.10$. Do you have money to buy it?
If $i^i$ looks absurd to you, you might ... |
I am not sure that it is exactly what you need, but the following is true:
For an alhabet with $q\geq 4$ letters and a sequence of forbidden words with lengths $n_1<n_2<\dots$ there exists an infinite word without forbidden subwords (where subword of a word W is a segment of consecutive letters in W, like "hab" is a su... |
I just asked this question concerning the application of Noether's theory. Think about this got me wondering about the following. In the usual derivation of the Noether current the assumption is made that:
$$\mathcal{L}(\phi'(x'),\partial_\mu'\phi'(x'),x')=\mathcal{L}(\phi(x),\partial_\mu\phi(x),x)+\delta x^\mu\partial... |
Does anybody know why offset in a Poisson regression is used? What do you achieve by this?
Here is an example of application.
Poisson regression is typically used to model count data. But, sometimes, it is more relevant to model rates instead of counts. This is relevant when, e.g., individuals are not followed the same... |
A professor once asked us in class: for the Hydrogen atom, what is the probability of finding the electron inside the nucleus?
The radius of a proton (the nucleus of a Hydrogen atom) is $r_0 \approx 10^{-15}\textrm{m}$. The answer to the question will be hard to find if one tries to compute it analytically by integrati... |
I just watched the Blender Guru video How to Make a Beer in Blender and stopped at the point where it advises us to make the liquid slightly larger in diameter than the inner diameter of the glass - approximately half way between the inner and outer walls.
I then turned to Fluid in a Glass(and why you’ve been doing it ... |
I am given a matrix $ A =(a,b;c,d)$ in $ GL(2,\mathbb{C})$ and a real algebra say, $ V$ with basis $ X,Y,Z$ such that $ [X,Y]=0, [X,Z]=aX+bY, [Y,Z]=cX+dY$ I have to show that $ V$ is a real Lie Algebra.
My attempt: it’s vector space over $ \mathbb{R}$ (duh!) I think we first need to find $ [X,X], [Y,Y], [Z,Z]$ which I ... |
The lower attic
From Cantor's Attic
Revision as of 13:37, 27 May 2018 by Denis Maksudov (the Takeuti-Feferman-Buchholz ordinal)
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.
$\omega_1$, the first uncountable ordinal, ... |
Though frameworks like Tensorflow, Pytorch has done the heavy lifting of implementing gradient descent, it helps to understand the nuts and bolts of how it works. After all, neural network is pretty much a series of derivative functions. In this blog post, let’s look at getting gradient of the lost function used in mul... |
In this post, we will derive the components of a rotation matrix in three dimensions. Our derivation favors geometrical arguments over a purely algebraic approach and therefore requires only basic knowledge of analytic geometry.
Given a vector ${\bf x} = (x,y,z)$, our goal is to rotate it by an angle $\theta \gt 0$ aro... |
Find $$\int_0^t A(s)ds$$ if $$A(t)=\begin{pmatrix}\sin(t),\cos(t)\\ -\sin(t),\cos(t)\end{pmatrix}$$ I'm a little confused with the format of the question because it asks me to integrate with respect to s, but the function is of t. Do I just integrate A(t) as an indefinite integral, so the answer would be $$A(t)=\begin{... |
The physics creating eddy currents and EMFs in inductors is the same: Faraday's law of induction.
$ \oint_C {E \cdot d\ell = - \frac{d}{{dt}}} \int_S {B_n dA} $
The strength of any induced current and voltage is dependent on:
1) The amount of magnetic flux ($\int_S {B_n dA}$)
2) The rate at which the flux is changing
S... |
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... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
The conjecture Are all zeros of $\zeta(0+s) \pm \zeta(0-s)$ except a finite few on the line $\Re(s)=0$? was shown to be unconditionally true.
The proof can even be extended towards the domain $\sigma_0 \le 0$ with $\zeta(\sigma_0 + s) \pm \zeta(\sigma_0 -s)$.
Building on this, I started to experiment with the sum/diffe... |
I am studying Electrodynamics and I have been introduced to the concept of Gauge Invariance.
This was introduced by noting that $E$ and $B$ amount to 6 six degrees of freedom and the Maxwell equations amount of 3 degrees of freedom. On the other hand, if we write $$E = - \nabla \phi - \frac{\partial A}{\partial t}, \qq... |
$X\sim U(0,\theta)$. To find the umvue of $\cos\theta$ is it enough to find the umvue of theta and substitute for it. Umvue of $\theta$ being $(n+1)X_{(n)}/n$, is the answer $\cos (n+1)X_{(n)}/n$?
Assuming you have a sample of $n$ observations.
The density of the complete sufficient statistic $X_{(n)}$ is
$$f_{X_{(n)}}... |
The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
Action formulation. It should be stressed that Noether's theorem is a statement about consequences of symmetries of an action functional (as opposed to, e.g., symmetries of equations of m... |
I have two function $c_1, c_2 \colon [0,2\pi] \to \mathbb{R}^2$ define by $c_1(t) = (\cos(t), \sin(t))$ and $c_2(t) = (\cos(2t), \sin(2t))$
I want to show that they have the same image. It is pretty obvious, but I don't know how to prove it.
Mathematics Stack Exchange is a question and answer site for people studying m... |
I am aware of the debate on whether Schrödinger equation was derived or motivated. However, I have not seen this one that I describe below. Wonder if it could be relevant. If not historically but for educational purposes when introducing the equation.
Suppose that we have the time dependent Schrödinger equation for a f... |
This question already has an answer here:
The term "B" denotes both magnetic flux density and magnetic induction. Are the terms same? If, not what do they mean
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.Sign up to joi... |
The partial log-likelihood function in Cox proportional hazards is given with such formula $${}_{p}\ell(\beta) = \sum\limits_{i=1}^{K}X_i'\beta - \sum\limits_{i=1}^{K}\log\Big(\sum\limits_{l\in \mathscr{R}(t_i)}^{}e^{X_l'\beta}\Big),$$ where $K$ is the number of observations for which we have observed an event (where g... |
The Annals of Probability Ann. Probab. Volume 19, Number 4 (1991), 1587-1628. Solutions of a Stochastic Differential Equation Forced Onto a Manifold by a Large Drift Abstract
We consider a sequence of $\mathbb{R}^d$-valued semimartingales $\{X_n\}$ satisfying $X_n(t) = X_n(0) + \int^t_0\sigma_n(X_n(s-))dZ_n(s) + \int^t... |
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