text stringlengths 256 16.4k |
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Richard Bamler (Berkeley) [ 23 ] Tristan Collins (Harvard) [ 24 ] Ronan Conlon (L'Univ du Québec à Montréal) [ 24 ] Georges Dloussky (Aix-Marseille Univ) [ 21 ] Kento Fujita (Kyoto, RIMS) [ 21 23] Ursula Hamenstaedt (Bonn) [ 23 ] Yoshinori Hashimoto (Univ College London) [ 22 ] Tomoyuki Hisamoto ... |
Fourier transformations:
$$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$
for momentum space and
$$\psi(\vec{r}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{k\text{ space}} \phi(\vec{k}) e^{i \mathbf{k} \cdot \mathbf{r}} d^3k$$
f... |
Good question. What you see here is the procedure known as the WKB approximation. Let's start from scratch and proceed slowly. Consider the 1D path integral given by$$Z = \int_{-\infty}^{\infty}\exp(i\mathcal{S}/\hbar)\ \text{dx} = \int_{-\infty}^\infty\exp\left(\frac{i}{\hbar}\int_0^{t_0}\mathcal{L}(x,\dot{x})\text{dt... |
Difference between revisions of "Literature on Carbon Nanotube Research"
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==Sustained Growth of Ultralong Carbon Nanotube Arrays for Fiber Spinning==
==Sustained Growth of Ultralong Carbon Nanotube Arrays for Fiber Spinning==
[http://www.mse.ncsu.edu/research/zhu/papers/CNT/Adv.Mat.CNTarray.pdf Q. Li e... |
The general trick to calculating such odds is that the probability of rolling a result that matches some criterion equals the number of possible matching rolls, divided by the total number of possible rolls.
(By "roll", here, we mean a sequence of numbers obtained by rolling a certain number (e.g. 6) of a certain kind ... |
I have a matrix of values where rows are individuals and columns are attributes. I want to extract a similarity value for every pair of individuals, and I use an rbf kernel: $$k(x_i,x_j) = \exp\left(-\gamma\|x_i-x_j\|^2\right),$$ where $\gamma = \frac{1}{(2\sigma)^2}$.
Since any attribute has its own range, I suppose t... |
The problem of choosing subsets at random has been studied in a rather different context in mathematical economics. Suppose we choose a subset of $[0,1]$ by independently throwing a fair coin for each number. Heuristically, such a set should have measure $1/2$. For what we do is randomly choose an indicator function wi... |
Why the expectation value of momentum $\langle p \rangle$ is zero for the one dimensional ground-state wave function of an infinite square well? And why $\langle p^2 \rangle = \frac{\hbar^2 \pi^2}{L^2}$? I am not asking for a proof. I am trying to interpret physical their meanings. I understand $\langle p \rangle = 0$ ... |
Up a level
Anand, V and Ghosh, S and Ghosh, M and Rao, GM and Railkar, R and Dighe, RR (2011)
Surface modification of PDMS using atmospheric glow discharge polymerization of tetrafluoroethane for immobilization of biomolecules. In: Applied Surface Science, 257 (20). pp. 8378-8384.
Kene, PS and Dighe, RR and Mahale, SD ... |
J. D. Hamkins, “The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme,” ArXiv e-prints, 2016. (under review)
@ARTICLE{Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme, author = {Joel David Hamkins}, title = {The {Vop\v{e}nka} principle is inequivalent... |
Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes
1.
Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano 04120, Almería, Spain
2.
Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universida... |
I have the next exercice:
For $\epsilon \in ]0,1]$, we have the function
$$T_{\epsilon}(f)=\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}f(t)dt$$ and $T_0(f)=f(0).$
Show that for $\epsilon \in ]0,1]$ and $f\in E= C^1(]-1,1[)$ which are bounded as their derivative too, we have $$|T_{\epsilon}(f)-T_0(f)|\leq \frac{\epsil... |
Say we have $k$ samples of data, where sample $i$ is of size $n_i$ and we write it as $x_{i1}, ... , x_{in_i}$. Let the total sample size be $N$.
The ANOVA model is $X_{ij} \sim N(\mu_i, \sigma^2)$ independently. The null hypothesis is that the $\mu_i$ are all equal. The alternative hypothesis is that the null hypothes... |
Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?
Sure you can! This i... |
Four ships are located at the ocean, such that the distance between all four ships is the same.
How are the ships positioned?
Four ships are located at the ocean, such that the distance between all four ships is the same.
Place them on the globe at the vertices of a (large) regular tetrahedron.
They could also be at th... |
Research Open Access Published: Blow up of solutions for a class of fourth order nonlinear pseudo-parabolic equation with a nonlocal source Boundary Value Problems volume 2015, Article number: 109 (2015) Article metrics
1697 Accesses
Abstract
In this paper, we consider the initial boundary value problem for a fourth or... |
Even faster quadratic formula for the HP-41C
06-04-2014, 04:25 AM (This post was last modified: 06-04-2014 04:52 AM by Gerson W. Barbosa.)
Post: #1
Even faster quadratic formula for the HP-41C
This is yet another attempt at finding a shorter (but not necessary faster) quadratic formula program for the HP-41C, while pre... |
Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ordinals $\alpha$. It is known of course that $F(\alpha) = \alpha+1$ is a complete resolution of GCH (in the positive) that is... |
Bernoulli Bernoulli Volume 8, Number 5 (2002), 627-642. Irregular sets and central limit theorems Abstract
In previous papers we have studied the asymptotic behaviour of$S_N(A;X)=(2N+1)^{-d/2}\Sigma_{n \in A_N}X_n$, where $X$ is a centred, stationary and weakly dependent random field, and $A_N=A \cap [ -N,N]^d, A \subs... |
Huge cardinal Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\omega_2$-saturated $\sigma$-ideal on $\omega_1$... |
Home
Integration by PartsIntegration by Parts
Examples
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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 (... |
Materials Pencil; Eraser; Blank A4 sheet of paper; and A ruler (also to use as a straightedge). Puzzle: Draw a square with a side-length of $a=16$cm.
Then, draw eight circles inside. The circles must have $2$cm in diameter; they must all be the same size; they must not touch each other; their distances must all be diff... |
Example from Wikipedia:
Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is sign... |
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 (... |
It is a very nice question! The answer is yes, natural instances of the $\Delta$ system property, which hold under GCH, are in fact equivalent to the GCH.
Theorem. $\Delta(\omega_2,\omega_1)$ is equivalent to CH.
Proof: You've pointed out that CH implies the principle, since thehypothesis you mention for this case amou... |
Insightful commentary on international conflict and the scholarly study thereof. Or laughably poor analysis of matters that deserve to be taken more seriously. Your call.
Saturday, June 23, 2012
Measuring Military Capabilities
There have been some interesting discussions about how to measure the position of the United ... |
In this post I will compute, under some simple assumptions, how much time a given amount of fluid (e.g. water) takes to go through a funnel. The result presented here is by no means a general formula but will be a good estimate in certain regimes (namely, if the stem radius is much smaller than the mouth radius and if ... |
Does the Friedmann vacuum equation have a linear solution rather than an exponential one?
Using natural units one can write Friedmann's equation for the vacuum as $$ \begin{eqnarray} \left(\frac{\dot a}{a}\right)^2 &=& \frac{8\pi G}{3}\rho_{vac}\\\tag{1} &=& L^2 \left(\frac{\rho_0}{L^4}\right) \end{eqnarray} $$
where I... |
When dealing with mathematical modeling, choosing the right scale to frame the equations can make the difference between a successful and lasting model, or poor description of reality.
In today’s post, we explore
two important scaling procedures that arise in finance: the annualisation of returns and volatility. These ... |
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\section{
Remarks about Sperner's theorem} +
\section{Sperner's theorem}
−
The first
non-trivial case of Szemer\'edi's theorem is the case of arithmetic progressions of length 3. However, for the density Hales-Jewett theorem even the case $k=2$ is interesting. For the purposes of this discussion, let ... |
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign up to join this comm... |
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, and for each root subgroup $U_{\alpha}$ of $U$, let $u_{\alpha}: \mathbb R \rightarrow U_{\alpha}$ be an isomorphism.
For ... |
Difference between revisions of "Unitriangular matrix group:UT(3,p)"
(→Families)
m (→In coordinate form)
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<math>\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \... |
Mr. Hobson has retired from running a stable and has invested in a more modern form of transport, trains. He has built a rail network with $n$ stations. However, he has retained his commitment to free the passenger from the burden of too many choices: from each station, a passenger can catch a train to exactly one othe... |
A. All light sources (even lasers) are subject to a diffraction limit, so any light beam will eventually diverge with an angle $\theta$ given by
$$\theta \approx \frac{\lambda}{A_T}$$
where $\lambda$ is the wavelength of the light and $A_T$ is the aperture of the light beam source (and "eventually" means for distances ... |
Learning Outcomes Interpret the Fprobability distribution as the number of groups and the sample size change
The distribution used for the hypothesis test is a new one. It is called the
F distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of... |
Thought you knew everything about correlation? Think there’s no fooling you with the question of correlation with financial prices or returns? Well maybe, just maybe, this post will enlighten you.
Correlation: the debate is on
Correlation can be a controversial topic. Things can go awry when two seemingly unrelated var... |
Let $V^{\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu}$. We can write this like $V'^{\mu} = (e^{i\omega})^{\mu}_{\,\,\,\nu}V^{\nu}$ I think where $\omega$ denotes a rotation in some plane spanned by indices $... |
Parallax would be the first thing to break the hoax
In our universe, "nearby" stars appear to move across the sky as compared to distant stars due to parallax and the motion of the Earth around the Sun. The effect is subtle and wasn't observed until the late 1800 to early 1900s. Using something called a filar micromete... |
i was a little confused about the casimir effect. my understanding is with the 2 plate experiment there was a force pushing the plates together because there were less virtual particles in between the plates than outside them. Why are there less particles in between the plates?
The objects whose number is lower in betw... |
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 was doing a couple of problems for homework:
Calculate $K_\mathrm{sp}$ of $\ce{AgI}$ at $55.0\ \mathrm{^\circ C}$
Calculate $K_\mathrm{b}$ of $\ce{NH3}$ at $36.0\ \mathrm{^\circ C}$
I have to use $\Delta G^\circ= -RT\ln K$ and $\Delta G= \Delta H-T\,\Delta S$
When I did this $\Delta G^\circ$ is positive ($89.59\ \mat... |
A group of $N$ fugitives is running away from the police. They reach a very narrow bridge. Since it's dark and they have a single flashlight, they can only cross the bridge in pairs. The $i^{th}$ fugitive takes $T_i$ minutes to cross the bridge. When a pair of fugitives crosses the bridge, they must go at the speed of ... |
Are you sure about the factor of $3$ in Koszul's formula? The computation below does not give such a factor, and it gives a stronger result.
Let $X_i$ be any basis of the left-invariant vector fields (where the indices run from $0$ to $n$), with $[X_i,X_j] = c^k_{ij}X_k$, where, of course, $c^k_{ij}=-c^k_{ji}$ (and whe... |
The Annals of Probability Ann. Probab. Volume 30, Number 4 (2002), 1913-1932. Regularity of quasi-stationary measures for simple exlusion in dimension d≥5 Abstract
We consider the symmetric simple exclusion process on $\ZZ^d$, for $d\geq 5$, and study the regularity of the quasi-stationary measures of the dynamics cond... |
In a previous post, I described how one can use the singular value decomposition of a matrix to represent it in a compressed form. The trick was to discard information (singular values) from the original matrix to generate an "approximate" version of it. This post describes how that technique can be used to also compre... |
I've seen similar conclusion from many discussions, that as the minibatch size gets larger the convergence of SGD actually gets harder/worse, for example this paper and this answer. Also I've heard of people using tricks like small learning rates or batch sizes in the early stage to address this difficulty with large b... |
Reidemeister conjugacy for finitely generated free fundamental groups Abstract
Let $X$ be a space with the homotopy type of a bouquet of $k$ circles, and let $f:X\to X$ be a map. In certain cases, algebraic techniques can be used to calculate $N(f)$, the Nielsen number of $f$, which is a homotopy invariant lower bound ... |
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Difference between revisions of "Timeline of prime gap bounds"
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[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherla... |
Is it possible to solve the following equal for $x$?
$$4=2^{x^{x^{x^{...}}}}$$
I'm bit confused, how do you even simplify this equation, factoring?
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 u... |
The first thing that one learns in statistics is to use the sample mean, $\hat{X}$, as an unbiased estimate of the population mean, $\mu$; and pretty much the same would be true for the variance, $S^2$, as an estimate of $\sigma^2$ (leaving aside Bessel's correction for a second). From these working assumptions, and wi... |
There are two aspects in this problem depending on the exact definition of the task, and, as we shall see, both are completely solved by MMA without any additional facility.
The aspects are
a) calculate the interpolation of the definite integral $f=\int_0^t \sqrt{1+x^3} \, dx$
b) calculate the interpolation of the anti... |
Various weak theories of arithmetic have been partially motivated by a concern with numbers (or functions/proofs) that are
feasible. This concern is sometimes connected to an interest in strictly finitistic approaches to arithmetic.
While the precise account of feasibility varies across these systems, the general idea ... |
Consider a Dirichlet character, $\chi(n)$, and the partial sum :
$$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$
There are many works to bound this sum when $\chi$ is a primitive character, but what can we say if $\chi$ is not primitive ?
More specifically if we fix a primitive Dirichlet character $\chi$ and then def... |
Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the characteristic function of any subset $S\subset G$. Put the counting measure on $G$ and $\widehat{G}$, so that, for instance, $|1_S... |
Let $ f:S→B$ be an elliptic fibration from an integral surface $ S$ to integral curve $ B$
.
Here I use following definitions:
A surface (resp. curve) is a $ 2$ -dim (resp. $ 1$ -dim) proper k scheme over fixed field $ k$ .
Fibration has two properties: 1. $ O_B = f_*O_S$ 2. all fibers of f are geometrically connected
... |
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1. Search for the Production of a Long-Lived Neutral Particle Decaying within the ATLAS Hadronic Calorimeter in Association with a Z Boson from pp Collisions at s=13 TeV
... |
Let $\iota: \mathcal{A}\subset \mathcal{C}$ a replete, full, reflexive subcategory with $F: \mathcal{C}\to \mathcal{A}$ left adjoint to $\iota$, and $\eta_X: X\to F(X)$ the canonical unity.The canonical counity $\epsilon$ is a isomorphism (because $\iota$ is full and faithfull) and $F(\eta_X)$ is a isomorphism (triangl... |
Why was the symbol $r$ chosen to denote Pearson's product moment correlation?
From Pearson's "Notes on the history of correlation"
The title of Galton's R. I. lecture was
Typical Laws of Heredity in Man. Here for the first time appears a numerical measure $r$ of what is termed 'reversion' and which Galton later termed ... |
SCM Repository View of /branches/vis12/test/unicode-cheatsheet.diderot
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File size: 781 byte(s) 1685- ( download) ( annotate) Sun Jan 22 15:23:36 2012 UTC(7 years, 8 months ago) by jhr
File size: 781 byte(s)
Create a branch to implement things that we need for the Vis 2012 paper /* useful unicode characters for ... |
In this post, I will show how solving a Sudoku puzzle is equivalent to solving an integer linear programming (ILP) problem. This equivalence allows us to solve a Sudoku puzzle using any of the many freely available ILP solvers; an implementation of a solver (in Python 3) which follows the formulation described in this ... |
Note, as I have my degree in chemistry, I denote "=>" as the next step in the equation. Also, sorry for any obvious mistakes that I made - as I mentioned, my degree is not in engineering. Thanks!
I am considering a design for a PET spherical toy ball. The ball will have a radius of 0.07 m that will have pressurized air... |
On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities
1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain, Spain
-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$
$u(x)=0$ on $\partial \Omega,$
where $-\infty<\gam... |
I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have attempted this problem using Fourier machinery. I'm still in the process of learning, and would therefore appreciate if someo... |
We assume that the unknown $X\in M_{m,n}$. When $A\in M_{m,m},B\in M_{n,n}$ are full matrices, $B^T\otimes A$ is a full $mn\times mn$ matrix; then inversion of such a matrix has complexity $O(m^3n^3)$. As we will see, using this method (when $A,B$ are large matrices), is a very bad idea.
There are special methods to so... |
I know there have been answers to similar questions, using either ImportString or ToExpression, however I haven't been able to succeed in importing this.
I'm using version 10 Mathematica on Linux by the way. What I want to import is a multi-line aligned equation. In this link, Workaround for Import not supporting the \... |
Suppose you wish to losslessly compress data which can be expressed as a sequence of symbols (a regular text file is a classic example where the symbols are letters and punctuation marks). There are many good compression algorithms out there that solve this problem. This post will present a famous one called Huffman's ... |
I would like to compute $\zeta_{\mathbb{Q}[i]}(-1)$ - a Dedekind zeta function. Mimicking the computation for $\zeta(-1)$, we can observe the following diverges:
$$ \frac{1}{4}\sum_{ (m,n) \neq (0,0)} \sqrt{m^2 + n^2} = 1+\sqrt{2}+2 + 2 \sqrt{5} + \sqrt{8} + \dots $$
and I would like to gives infinite divergent sum a f... |
The Annals of Probability Ann. Probab. Volume 19, Number 4 (1991), 1781-1797. Existence of Probability Measures with Given Marginals Abstract
We show that if $f$ is a probability density on $R^n$ wrt Lebesgue measure (or any absolutely continuous measure) and $0 \leq f \leq 1$, then there is another density $g$ with on... |
There is no single number that encompasses all of the covariance information - there are 6 pieces of information, so you'd always need 6 numbers.
However there are a number of things you could consider doing.
Firstly, the error (variance) in any particular direction $i$, is given by
$\sigma_i^2 = \mathbf{e}_i ^ \top \S... |
I'm working on a cosmological model for general relativity, and I need to define a tensor and assign values to this. For example, a tensor $A_{\mu\nu}$ that is function of other tensors:
$$A_{\mu\nu}=R_{\mu\alpha}g^{\mu \alpha}+G^{\mu\beta}T_{\beta\nu}$$
And I need to use this expression for A_{\mu\nu} to calculate the... |
To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, for example some people like model categories, other people might prefer triangulated categories (shudder), I'll simply e... |
INFORMAL TREATMENT
We should remember that the notation where we condition on random variables is inaccurate, although economical, as notation. In reality we condition on the sigma-algebra that these random variables generate. In other words $E[Y\mid X]$ is meant to mean $E[Y\mid \sigma(X)]$. This remark may seem out o... |
Those guys only make confusion. I will answer you with a very easy method you can use with piecewise functions.
First of all you have two steps functions, which you can easily figure in your mind to be like 2 dimensional boxes of height $2$.
Think about them as two boxes, one of which has to move towards the other. Tho... |
So for this one I'm having trouble isolating for y. If its not possible then in the form with dy with the y variable and x with the x variable.
$$\frac{dy}{dx}-2xy=e^{x^{2}}.$$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only tak... |
The factor of $1/2\pi$ is an artifact of the normalization convention being used for the momentum eigenstates.
To begin to see how this is so, let us note that the choice of normalization of a Dirac-orthogonal continuous basis completely determines the form of the resolution of the identity. Writing an arbitrary state ... |
Suppose a spaceship is moving away from the Earth at $0.5c$. When the spaceship is one light-year away from Earth, an observer on Earth sends a photon toward the spaceship. According to the observer on Earth, the photon is traveling at $c$ and the spaceship is traveling in the same direction at $0.5c$, so it takes 2 ye... |
Image processing is one of the hot topics in AI research, alongside with reinforcement learning, ethics in AI and many others. A recent solution to perform ordinal regression on age of people has been published, and in this post we apply that technique to financial data.
Ranking classification is an usual challenge in ... |
Hi all. I'm having a big brainfart and I can't seem to grasp what I really think is an easy problem. The full text says:
4)For the following pairs (b, a) find y and x such that by + ax = d where d = GCD(a, b). b)(b, a) = (18 - 16i, 7 - 2i) in the ring \(\displaystyle \mathbb{Z}\) I started by dividing \(\displaystyle \... |
This question already has an answer here:
Gibbs sampling and mixed distribution 2 answers
Let X a continous variable and Y a binary variable with joint distribution : $$p(x,y;\beta,\rho_1,\rho_2,\phi_1,\phi_2)=\frac{1}{Z(\beta,\rho_1,\rho_2,\phi_1,\phi_2)}\exp(-0.5 \beta x^2+1_{y=0}\rho_1 x+1_{y=1} \rho_2 x+1_{y=0} \ph... |
The company Arch Bridges Construction (ABC) specializes in—you guessed it—the construction of arch bridges. This classical style of bridge is supported by pillars that extend from the ground below the bridge. Arches between pillars distribute the bridge’s weight onto the adjacent pillars.
The bridges built by ABC often... |
14. Relations in Lean¶
In the last chapter, we noted that set theorists think of a binary relation \(R\) on a set \(A\) as a set of ordered pairs, so that \(R(a, b)\) really means \((a, b) \in R\). An alternative is to think of \(R\) as a function which, when applied to \(a\) and \(B\), returns the proposition that \(R... |
Notebooks.azure.com provides a FREE
Jupyter notebooks the service allows you to use Jupyter notebooks (http://jupyter.org/) and the programming language Python (https://www.python.org/). The Azure Notebook service is a FREE web application at http://notebooks.azure.com that allows you to create and share Juypter docume... |
Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$.
Let $L$ be the operator $$ L=\Delta+V $$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is formally negative) and $V$ is some smooth potential (we are mostly interested in $V\geq 0$).
A well known result of Fisc... |
This question already has an answer here:
Prove Quotient Group Isomorphism 1 answer
I'm trying to use the First Isomorphism Theorem to show that $\mathbb{T}^1 \cong \mathbb{C}^*/\mathbb{R}_{>0}$ by constructing a surjective group homomorphism from the nonzero complex numbers to the circle group whose kernel is the set ... |
The first question to answer: which matrices satisfy $A^2 = I$ and $\det(A) = 1$?
It suffices to note that since $A^2 - I = 0$, the minimal polynomial of $A$ must divide $x^2 - 1 = (x-1)(x + 1)$. Hence, $A$ must be diagonalizable with eigenvalues equal to $\pm 1$. Moreover, since the product of eigenvalues is equal to ... |
Is there a function in Mathematica that can be used to find the perturbation solution of an equation like $x^2 − 1 = \epsilon \,x$, $x − 2 = \epsilon \cosh(x)$ or $x^2 − 1 = \epsilon\, e^x$?
Decide up to which power you would like to expand:
pow = 4;
Let's do one of the equations you mentioned as an example (bring all ... |
Ineffable cardinal
Ineffable cardinals were introduced by Jensen and Kunen in [1] and arose out of their study of $\diamondsuit$ principles. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$... |
According to the theory of general relativity, if you compress an object of mass $M$ to a sphere with radius equal to or smaller than the Schwarzschild radius ($R_s$) for that object: $$ R_s = \displaystyle\frac{2GM}{c^2} $$ then the escape velocity for a particle at a distance $R_s$ from the center of mass of $M$ will... |
In the spirit of the classic
four fours, I wonder what's the optimal set of four numbers?
Your goal is to make the most consecutive integers using four digits of your choice. Pick four: $0,1,2,3,4,5,6,7,8,9$ ( You can pick multiple instances of the same digit )
When constructing an integer:
all of your four candidates ... |
202 35 Summary What happens to invariant mass of an object when it gets closer or further from a gravitational body?
In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1).
[tex] m^2 = E^2 - \vec{p}^2 [/te... |
Jónsson cardinal
Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called
Jónsson cardina... |
How can we evaluate the following?
$$\int e^{\frac{y}{x}}\ \mathrm dy$$
An explanation of the answer would be helpful.
The answer I got is $ x e^{y/x}$.
But not sure about the steps used for obtaining the answer...
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profes... |
This is a talk for the New York Set Theory Seminar on March 1, 2013.
This talk will be based on my recent paper with C. D. A. Evans, Transfinite game values in infinite chess.
Infinite chess is chess played on an infinite chessboard. Since checkmate, when it occurs, does so after finitely many moves, this is technicall... |
The Wholeness Axioms
The wholeness axioms, proposed by Paul Corazza [1, 2], occupy a high place in the upper stratosphere of the large cardinal hierarchy, intended as slight weakenings of the Kunen inconsistency, but similar in spirit.
The
wholeness axioms are formalized in the language $\{\in,j\}$, augmenting the usua... |
Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and $G:=\bigotimes_{j=1}^n G^{(j)}$
uniform if the operator norm satisfies$$\|\bigotimes_{j=1}^n A^{(j)}\|_{H\to G}=\prod_{j=1}^n \|A^{(j)... |
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE
(1) $$\begin{cases}Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ on } \partial\Omega\,,\end{cases}$$where $f\in C^\infty(\Omega)$ satisfies $f(x) > 0$ and $f$ has no isolated critical points on $\O... |
My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is some aplicability of the stability theorems I know in this field. The only applications I found for my thesis were about the... |
Emilio Pisanty and Eckhard Giere have already given discontinuous, piecewise constant counterexamples in their answers. Here we provide for-the-fun-of-it a smooth infinitely-many-times-differentiable counterexample $f\in C^{\infty}(\mathbb{R})$ of a square integrable function $f:\mathbb{R} \to [0,1]$ that does
not sati... |
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