text stringlengths 256 16.4k |
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You have made many wrong assumptions in this question. First
theoretically speaking, Filters do not work in the way to 'pick up elements' (they work on principle of edge detection). You have assumed only a single combination of filter weights will give the desired output (assuming continuous weights not binary). This i... |
In the first week of teaching my Calculus 1 discussion section this term, I decided to give the students a Precalc Review Worksheet. Its purpose was to refresh their memories of the basics of arithmetic, algebra, and trigonometry, and see what they had remembered from high school.
Surprisingly, it was the arithmetic pa... |
Abstract
We say that a Riemannian manifold $(M, g)$ with a non-empty boundary $\partial M$ is a
minimal orientable filling if, for every compact orientable $(\widetilde M,\tilde g)$ with $\partial \widetilde M=\partial M$, the inequality $ d_{\tilde g}(x,y) \ge d_g(x,y)$ for all $x,y\in\partial M$ implies $ \operatorna... |
77th SSC CGL level Solution Set, topic Trigonometry 7
This is the 77th solution set for the 10 practice problem exercise for SSC CGL exam and 7th on topic Trigonometry.
If you have not taken the test yet, you should refer to the
before going through this solution. 77th SSC CGL question set and 7th on Trigonometry 77th ... |
I’ve been doing a lot of reading on confidence interval theory. Some of the reading is more interesting than others. There is one passage from Neyman’s (1952) book “Lectures and Conferences on Mathematical Statistics and Probability” (available here) that stands above the rest in terms of clarity, style, and humor. I h... |
Lie algebras over rings (Lie rings) are important in group theory. For instance, to every group $G$ one can associate a Lie ring
$$L(G)=\bigoplus _{i=1}^\infty \gamma _i(G)/\gamma _{i+1}(G),$$
where $\gamma _i(G)$ is the $i$-th term of the lower central series of $G$. The addition is defined by the additive structure o... |
A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a common name for a site whose codomain fibration is a stack? The canonical topology on a Grothendieck topos has this property,... |
I would like to display a matrix with a dot (derivative) on top of it. The entry of this matrix shall consist of arbitrary symbols.
When I write
\begin{align} \dot{\begin{pmatrix} x \\ y \end{pmatrix}} \end{align}
everything works fine. However,
\begin{align} \dot{\begin{pmatrix} \hat{x} \\ \hat{y} \end{pmatrix}} \end{... |
This year’s Prove it! Math Academy was a big success, and it was an enormous pleasure to teach the seventeen talented high school students that attended this year. Some of the students mentioned that they felt even more inspired to study math further after our two-week program, but the inspiration went both ways – they... |
When Matlab reaches the cvx_end command, it completes the conversion of the ... \|Ax-b\|_2\\ \mbox{subject to} & l \preceq x \preceq u \end{array}\end{split}\].
An affine column vector CVX expression can be multiplied by a constant matrix of .... \[\begin{split}f_{\text{huber\_circ}}(x,M) \triangleq \begin{cases} \|x\|... |
A. SAXENA
Articles written in Journal of Astrophysics and Astronomy
Volume 40 Issue 2 April 2019 Article ID 0009
We report optical observations of TGSS J1054 $+$ 5832, a candidate high-redshift ($z = 4.8 \pm 2$) steep-spectrum radio galaxy, in $r$ and $i$ bands, using the faint object spectrograph and camera mounted on... |
Bill Dubuque raised an excellent point here: Coping with *abstract* duplicate questions.
I suggest we use this question as a list of the generalized questions we create.
I suggest we categorize these abstract duplicates based on topic (please edit the question). Also please feel free to suggest a better way to list the... |
Ok here are some rules because it would be too easy/too hard without them
Round to nearest 100th The first number cannot be an integer Needs to be an equality No mixed numbers
Hint:
The equal sigh need not be aligned.
Puzzling Stack Exchange is a question and answer site for those who create, solve, and study puzzles. ... |
A bunch of points:$\def\Spec{\mathrm{Spec}\ }$
• Let $A$ be a cluster algebra over a field $k$, let $(x_1, \ldots, x_n)$ be a cluster and let $L$ be the Laurent polynomial ring $k[x_1^{\pm}, \ldots, x_n^{\pm}]$. I imaging your intended question is whether the map $\Spec L \to \Spec A$ is an open immersion. (You ask abo... |
Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$ to the standard torus ${\bf R}^d/{\bf Z}^d$ is not surjective, or equivalently $K$ is disjoint from some coset $x + {\... |
Basic MOSAICmodeling models consist of Equations with a single Notation, which are then joined together to form an Equation System, which can be simulated or evaluated.
At this point we assume that you already know how to set up a Notation. The editor to enter or edit an Equation inside MOSAICmodeling is shown in Fig. ... |
I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola. Are all these 5 components enough to form a right circular cone or are these just the parts of right circular cone? What I mean is : if I put together infinite number of circles, p... |
Problem:
$dX_t = \sigma X_tdB_t$, $X_0=x$.
$dY_t=X_tdt-Z_tdt$ find $Y_t$, where $Z_t$ is a control and $B_t$ is standard Brownian motion.
My attempt:
From Ito's lemma,
$\partial_BX_t=\sigma X_t$, therefore $X_t=c(t)\exp(\sigma B_t)$.
$\partial_tX_t+\frac{1}{2}\partial^2_BX_t=0$, substitue in above expression for $X_t$ ... |
PandaX is designed to build and operate a ton-scale liquid xenon experiment to detect the so far elusive dark matter in the Universe. The PandaX experiment will use a two-phase (liquid and gas) xenon position-sensitive time projection chamber detector. It is located at CJPL, which is one of the deepest underground labs... |
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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 ... |
Exams can be easily created in LaTeX by means of the class
exam.cls. This class makes it straightforward to typeset questions, and it sets a 1in margin in all paper sizes and provides special commands to write and compute grades. This article explains how to edit with exam.cls.
Contents
Let's see a simple working examp... |
Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term
1.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2.
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
In this pape... |
Homework Helper
1,059 9
Ok so I've got a question after walking through the time dilation derivation that used 'light clocks' (think a beam of light bouncing back and forth between mirrors) to derive ##\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}##. So my Q is could you derive the same equation if you ha... |
Dimensionality reduction is used to remove irrelevant and redundant features.
When the number of features in a dataset is bigger than the number of examples, then the probability density function of the dataset becomes difficult to calculate.
For example, if we model a dataset \(S = \{x^{(i)}\}_{i=1}^m,\ x \in R^{n}\) ... |
Let's examine the one-dimensional three-point stencil case in detail,because I think it's important to be clear just how this behaviourarises, and what it means to set a point to a certain value in afinite-difference grid when the underlying function isdiscontinuous. The equation will be$$ u''(x) = \rho(x). $$Instead o... |
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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 ... |
Assume that a function $f$ is integrable on $[0, x]$ for every $x > 0$.
Prove that for any $x > 0$, $\displaystyle\left (\int_{0}^{x}fdx \right )^2\leq x\int_{0}^{x}f^2dx$.
I have no idea how to even start this... What concept should I be using?
EDIT:
So upon the hint of using C-S inequality/using a dummy variable for ... |
CGAL 4.7 - Number Types
class CGAL::Root_of_traits< RT > For a
RealEmbeddable
IntegralDomain
RT, the class template
Root_of_traits<RT> associates a type
Root_of_2, which represents algebraic numbers of degree 2 over
RT. More...
class CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates > An in... |
In Morii, Lim, Mukherjee,
The Physics of the Standard Model and Beyond. 2004, ch. 8, they claim that the Peskin–Takeuchi oblique parameters S, T and U are in fact Wilson coefficients of certain dimension-6 operators. On page 212, they claim that the T parameter is described by$$O_T=(\phi^\dagger D_\mu \phi)\phi^\dagger... |
Introduction:
Two days ago, while thinking about brain networks, it occurred to me that almost all simple graphs are small world networks in the sense that if is a simple graph with nodes sampled from the Erdös-Rényi random graph distribution with probability half then when is large:
\begin{equation} \mathbb{E}[d(v_i,v... |
Difference between revisions of "Quasirandomness"
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==A possible definition of quasirandom subsets of <math>[3]^n</math>==
==A possible definition of quasirandom subsets of <math>[3]^n</m... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
I have been asked to work out the hardness of water in German degrees and this is what I did:
I got a sample of water I didn’t measure how much I put into my volumetric flask perhaps it was $\pu{8.5ml}$ but I didn’t think I would need it, but I do use it which made me believe I was wrong.
The sample was made up to $\pu... |
It is a well-known fact of representation theory that, if the irreducible representations of a finite group $\DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\sh}{sh} G$ are $V_1,\ldots,V_m$, and $R$ is the
regular representation formed by $G$ acting on itself by left multiplication, then $$R=\bigoplus_{i=1}^{m} (\... |
There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in Mathematics? I understand that it is a research question, but may be somebody can hint me the direction to look on so that... |
GR9677 #6
Alternate Solutions
insertphyspun 2011-02-06 13:05:22 A way to rule out answers, although not the fastest strategy:
A) Wrong. The limit as gives us no loss in energy, which is just weird. If you can't remember the answer, you would at least think it should be proportional to or independent of .
B) Maybe. Keep... |
I continue to be amazed at the multitude of different contexts in which the Schur functions naturally appear.
In a previous post, I defined the Schur symmetric functions combinatorially, via the formula $$s_\lambda=\sum_{|\mu|=|\lambda|}K_{\lambda\mu} m_\mu$$ where $K_{\lambda\mu}$ is the number of semistandard Young t... |
We analyzed the time complexity of different solutions for a given problem. Then, we proved several claims using the Formal definition of Big O.
Given a natural number $p$, how many right triangles exist with integer-sized sides whose perimeter is $p$?
How many triplets $(a,b,c)$ are there such that:
When computing cod... |
Lephenixnoir af424d1baa update documentation after writing the wiki 3 月之前 config 4 月之前 include/TeX 3 月之前 src 3 月之前 .gitignore 3 月之前 Makefile 4 月之前 README.md 3 月之前 TODO.md 3 月之前 configure 3 月之前 font5x7.bmp 4 月之前 font8x9.bmp 4 月之前 font10x12.bmp 4 月之前
This library is a customizable 2D math rendering tool for calculators. ... |
Note: There was an error in the reverse formula (and maybe also with some of the values given). Frank pointed this out (see comments section below).
In analytical chemistry, linear regression or linear function is a common (maybe the most common) tool to describe the relationship between a measured
signal and the conce... |
I have just been playing with this and thought my solution just might help somebody else at some point. I wanted the following:
table notes i.e. notes at the bottom of the tabular, within the
table environment - not at the bottom of the page;
automatic numbering of notes within the list of notes;
automatic numbering of... |
Consider a white Gaussian noise signal $ x \left( t \right) $.
If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes? Math tools
We can do the calculation using some basic elements of probability theory and Fourier analysis. There are three elem... |
Deadline for submission of results:
October 14, October 21, 2019 (11:59PM PST) Presentation of awards: October 28, 2019 (at the ICCV 2019 workshop)
The challenge is on the task of 6D localization of a
number of varying Training Input: At training time, method $M$ learns using a training set, $T = \{T_o\}$, where $o$ is... |
If a particle travels on a geodesic with 4-momentum $P^\mu$ in a spacetime with a Killing vector $K_\mu$ then we have a constant of motion, $K$, given by: $$K=K_\mu P^\mu$$ Using the relationships: $$P^\mu=mU^\mu$$ and $$K_\mu=g_{\mu\nu}K^\nu$$ we obtain: $$K=mg_{\mu\nu}K^\nu U^\mu$$ Let us assume that $K^\nu$ is a tim... |
I'm trying to implement Bayesian power priors to discount historical data.
$$\pi^p(\theta|\mathbf{z_{n_0}},a_0) \propto \pi_0(\theta) * L(\theta;\mathbf{z_{n_0}})^{a_0}$$
where $\pi^p$ is the posterior, $\pi_0$ is the prior, $L$ is the likelihood function, $\theta$ are the parameters of interest, $\mathbf{z_{n_0}}$ is ... |
This post describes some discriminative machine learning algorithms.
Distribution of Y given X
Algorithm to predict Y
Normal distribution
Linear regression
Bernoulli distribution
Logistic regression
Multinomial distribution
Multinomial logistic regression (Softmax regression)
Exponential family distribution
Generalized... |
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ab+2c+3d+bc+2d+3a+cd+2a+3b+da+2b+3c>23\dfrac{a}{b+2c+3d} + \dfrac{b}{c+2d+3a} + \dfrac{c}{d+2a+3b} + \dfrac{d}{a+2b+3c} > \dfrac{2}{3}b+2c+3da+c+2d+3ab+d+2a+3bc+a+2b+3cd>32
If a,b,c,d are distinct positive reals, prove the above inequality.
Note by Raushan Sharma... |
So when combining atomic orbitals to form molecular orbitals, you can either add the wave functions or subtract them. But at the same time, orbitals can exist in opposite phases (say one lobe of the p orbital is '+' and the other '-'). So what happens when you subtract two 1s orbitals of opposite phase, is that the sam... |
For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an abelian category one can define a
categorically finitely generated object as being an object M such that for any directed fam... |
The Annals of Statistics Ann. Statist. Volume 31, Number 5 (2003), 1491-1516. Enriched conjugate and reference priors for the Wishart family on symmetric cones Abstract
A general Wishart family on a symmetric cone is a natural exponential family (NEF) having a homogeneous quadratic variance function. Using results in t... |
This tutorial section covers the basic use of equations or equation systems that use different notations. It is assumed that you have worked through the basic tutorial parts.
Rules
The basic contract of the naming policy
‘integrate’ is as follows: Basic rule: If an equation or equation system (sub element) is added to ... |
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Last time we talked about string diagrams, which are a great way to draw morphisms in symmetric monoidal categories. But they're the most fun when we have the ability to take a diagram and bend its strings around, converting inputs ... |
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1. Search for heavy ZZ resonances in the ℓ + ℓ - ℓ + ℓ - and ℓ + ℓ - vv - final states using proton–proton collisions at √s=13 TeV with the ATLAS detector
European Physic... |
Yes, a neural network plus loss function can be viewed as a function composition as you have written, and back propagation is just the chain function repeated. Your equations $L = f(g(h(\dots u(v(\dots))))$ and $\frac{\partial L}{\partial v} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial h}\dots\frac{\partia... |
Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$.Let's denote the width of a sample by $h$ where $$h\rightarrow0$$Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a...
@Ultradark You can try d... |
Alright, I have this group $\langle x_i, i\in\mathbb{Z}\mid x_i^2=x_{i-1}x_{i+1}\rangle$ and I'm trying to determine whether $x_ix_j=x_jx_i$ or not. I'm unsure there is enough information to decide this, to be honest.
Nah, I have a pretty garbage question. Let me spell it out.
I have a fiber bundle $p : E \to M$ where ... |
Schedule Titles and Abstracts Elke Brendel, Truthmaker Maximalism and the Truthmaker-Paradox
Truthmaker maximalism (TMM) is the view that every true sentence has a truthmaker, where a truthmaker of a sentence σ is conceived as an entity whose mere existence necessitates the truth of σ. Peter Milne has attempted to refu... |
Permanent link: https://www.ias.ac.in/article/fulltext/joaa/040/02/0009
We report optical observations of TGSS J1054 $+$ 5832, a candidate high-redshift ($z = 4.8 \pm 2$) steep-spectrum radio galaxy, in $r$ and $i$ bands, using the faint object spectrograph and camera mounted on 3.6-m Devasthal Optical Telescope (DOT).... |
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Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c w... |
I have a function that generates possible actions, so that I would have a Q-table in form of a nested dictionary, with states (added as they occur) as keys whose values are also dictionaries of possible actions as keys and q-values as values. Is this possible? How would it affect learning? What other method can I use?
... |
Following Dan Schmidt's post 5, it seems that if \\(f_{\ast}\\) has a left adjoint \\(g_{\ast}\\), then \\(f\\) must be injective.
Otherwise, if \\(f\\) is not injective, there are elements \\(a_1, a_2 \in A, a_1 \neq a_2\\) such that \\(f(a_1) = b = f(a_2)\\).
Then, \\(f_{\ast}\\) sends both \\(\\{a_1\\}\\) and \\(\\{... |
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of continuous bounded functions $F:\Omega\times \mathbb R_+ \to \mathbb R$, where $\Omega$ denotes the space of continuous func... |
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ the base of $\Phi = \Phi(G,T)$ corresponding to $\Phi^+$.
A subset $\Psi$ of $\Phi^+$ is called
closed if whenever $\alpha, ... |
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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 ... |
uses two $3\times 3$ matrices and calculates their sum. It is an online math tool speciallyprogrammed to perform matrix addition between the two $3\times 3$ matrices. 3x3 matrix addition calculator
Matrices are a powerful tool in mathematics, science and life. Matrices are everywhere and they have significant applicati... |
Science Advisor
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Here is one I am having trouble following. Can anyone help me through my confusion?
Our setup is a normal Bell test using entangled photons created using spontaneous parametric down conversion (PDC). Such a setup uses 2 BBO crystals oriented a 90 degrees relative to each other. S... |
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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... |
Rich Trigonometry concepts, Maxima (or Minima) of Trigonometric expressions like $a \sin \theta + b \cos \theta$
Before going through the process of finding the maxima (or minima) of trigonometric expressions let us state the most important intermediate goal towards the solution.
Interim goal for finding maxima (or min... |
I hope this question ist still interesting for some people, as it is for me. I will first try to give a motivation for the "ghost map", i.e. the Witt polynomials, which, I think, is absent from Harder's article, and then give some historical remarks.
Motivation (partly inspired by the impressive article on Witt vectors... |
Answer
$$\tan\Big(x-\frac{\pi}{2}\Big)=\cot x$$ The statement is false.
Work Step by Step
$$\tan\Big(x-\frac{\pi}{2}\Big)=\cot x$$ Now what we've already known from the cofunction identities is $$\tan\Big(\frac{\pi}{2}-x\Big)=\cot x$$ Yet here, the statement involves $\tan\Big(x-\frac{\pi}{2}\Big)$, not $\tan\Big(\frac... |
We all know the basic equation of
Elliptic Curve is $y^2 \equiv x^3 + ax + b \pmod p$ How the value of the constants $a$ and $b$are chosen? Suppose $\mathbb P\ni p \approx 2^{256}$ then what approach should I take while selecting the value of constants $a$ and $b$?
I have some basic knowledge of Elliptic Curve, EC Cryp... |
The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so the group is an i.c.c. group and the associated left group von Neumann algebra $LG$ is a type $II_{1}$ factor. It is a f... |
I am interested to find an absolute value (not an approximation) of "combination without repetition" for given \$n\$ and \$k\$, or \$\binom{n}{k}\$.
The brute force solution would look like this
private static ulong Factorial(int x){ ulong res = 1; while (x > 1) { res *= (ulong)x--; } return res;}public static int Comb... |
Here is a beautiful result from numerical analysis. Given any nonsingular $n\times n$ system of linear equations $Ax=b$, an optimal Krylov subspace method like GMRES must necessarily terminate with the exact solution $x=A^{-1}b$ in no more than $n$ iterations (assuming exact arithmetic).
The Cayley-Hamilton theorem pro... |
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Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has ... |
The Steklov problem for a compact planar region $\Omega$ is \begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$}, \end{cases} where $n$ is the outward unit normal along $\partial \Omega$.
I am finding Steklov eigenfunctions when $\Omega$ is a region bou... |
Abstract
To any two graphs $G$ and $H$ one can associate a cell complex ${\tt Hom}(G,H)$ by taking all graph multihomomorphisms from $G$ to $H$ as cells.
In this paper we prove the Lovász conjecture which states that
if ${\tt Hom}(C_{2r+1},G)$ is $k$ -connected, then $\chi(G)\geq k+4,$
where $r,k\in\mathbb{Z}$, $r\geq ... |
First section of Cosmography
Problem 1
problem id: cs-1
Using the cosmographic parameters introduced above, expand the scale factor into a Taylor series in time.
solution
We can write the scale factor in terms of the present time cosmographic parameters:\[a(t)\sim 1+H_{0} \Delta t-\frac{1}{2} q_{0} H_{0}^{2} \Delta t^{... |
Here is one idea. I'll start with a more specific "mathematical singularity", defined as an algorithm that can do the following in N hours or less (for all $N >= 1$):
State equivalent versions (up to notional differences) of all mathematical theorems/conjectures that humans will read and understand in N*20 years after ... |
This question concerns a process that iterates intersection of randomly rotated planar shapes.
Start with a simply connected region $R_0$ in the plane, and let $c_0$ be the centroid of $R_0$. Rotate $R_0$ about $c_0$ a random angle; call the result $R'_0$. Set $R_1 = R_0 \cap R'_0$. And repeat, always rotating about $c... |
This is more of an extended comment than an answer, but I hope it might help clarify the problem. The OP seems to have two things in mind: 1) there are inherent errors (the $E$) in the measurements of distance and/or location, and 2) there is an unknown constant $k$ affecting all measurements of distance from the obser... |
Time for another gemstone from symmetric function theory! (I am studying for my Ph.D. qualifying exam at the moment, and as a consequence, the next several posts will feature yet more gemstones from symmetric function theory. You can refer back to this post for the basic definitions.)
Start with a polynomial $p(x)$ tha... |
Differential and Integral Equations Differential Integral Equations Volume 16, Number 5 (2003), 605-624. Periodic solutions of Liénard equations at resonance Abstract
In this paper we study the existence of periodic solutions of the second order differential equation $$ x''+f(x)x'+n^2x+\varphi(x)=p(t), \quad n\in {{\bf... |
Consider $g(z) = (z+1)(z-1)(z-2)/z$. Then
$$ \frac{g'(z)}{g(z)} = \frac{1}{z+1} - \frac{1}{z} + \frac{1}{z-1} + \frac{1}{z-2}. $$
Moreover, if $\gamma$ is any closed curve in $\Omega = \mathbb{C} \setminus ([-1, 0] \cup [1, 2])$, then it must wind $-1$ and $0$ the same time and $1$ and $2$ the same time. So if $W(\gamm... |
For a silly screen saver I'm trying to develop, I'd like to randomly generate a divergence-free 2D array of 2D vectors, and then use it to generate a line integral convolution plot. I've heard$^1$ that one way to do this is to generate random noise, and then project out the solenoidal component of its Helmholtz-Hodge d... |
If a multi-engine aircraft suffers an engine failure while near minimum control speed (V
mc), one of the solutions is to bank up to 5 degrees into the operating engine to increase rudder effectiveness to maintain control. Why is it up to 5 degrees? What happens if the pilot banks more than 5 degrees into the operating ... |
Intuition Math Examples
What is ICA?
If we have two unique signals A and B, we can generate new signals by mixing them. These mixed signals (x) are what we are usually measuring. In order to unmix them again we need to know, with what weights the source-signals have been mixed. If we can get the mixing matrix (A), we c... |
Here is an elementary observation (I mean “elementary” in the sense mathematicians sometimes use it: i.e. straightforward if you already understand it, otherwise gibberish) that I talked about at MIT’s category theory seminar, relating Grice’s maxims of Quantity and Quality to the mathematical notion of a Galois connec... |
I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes.It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended.May I ...
@Nelimee Do we need to m... |
Universality and average-case algorithm runtimes
Let $H > 0$ be a positive-definite $N \times N$ matrix, and let $x_0$ be an $N$-dimensional random vector. The power method to compute the largest eigenvalue of $H$ is given by the iteration, for $j = 1,2,\ldots$
$$y_j = x_{j-1}/\|x_{j-1}\|_2,$$
$$x_j = Hy_j,$$
$$\mu_j =... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
2018-09-11 04:29
Proprieties of FBK UFSDs after neutron and proton irradiation up to $6*10^{15}$ neq/cm$^2$ / Mazza, S.M. (UC, Santa Cruz, Inst. Part. Phys.) ; Estrada, E. (UC, Santa Cruz, Inst. Part. Phys.) ; Galloway, Z. (UC, Santa Cruz, Inst. Part. Phys.) ; Gee, C. (UC, Santa Cruz, Inst. Part. Phys.) ; Goto, A. (UC,... |
Introduction:
A few days ago I decided to analyse the symmetries of the two-thirds power law [1] and this analysis naturally led to the following kinematic sequence:
where is a volume-preserving transformation and the position is updated using:
\begin{equation} x_{n+1} = x_n + \dot{x}_n\cdot \Delta t + \frac{1}{2} \ddo... |
Answer
$$1+\cos 2x-\cos^2x=\cos^2x$$ This equation is an identity, which can be proved by transforming the left side using the identity from Exercise 69.
Work Step by Step
$$1+\cos 2x-\cos^2x=\cos^2x$$ We examine from the left side: $$A=1+\cos 2x-\cos^2x$$ The question hints at using the result from Question 69, which ... |
This is the first of a series of blog posts on rendering in Static Sky. We yanked out Unity’s lighting system and replaced it with our own, super fast one that’s specialised for rendering tons of bumped dynamic lights at 60FPS. This post will talk about the hybrid lighting model that we’re using.
We want to do noir, so... |
I came across an exercise in Ahlfors’
Complex Analysis the other day that got me thinking. The exercise asked to prove that the complex numbers $a$, $b$, and $c$ form the vertices of an equilateral triangle if and only if $a^2+b^2+c^2=ab+bc+ca.$ It struck me as quite a nice, simple, and symmetric condition.
My first in... |
Almost certainly not. As in, "you'll have more chance of winning the Lottery than you will of killing them by this method".
Antimatter annihilation of a single atom - we'll be good here and say one with a hefty nucleus like, say, iron - releases
$$\left(2\ \mathrm{atoms}\right) \times \left(\frac{55.8452\ \mathrm{g}}{1... |
Introduction:
Back in 2015, Elon Musk gave an engaging presentation on Tesla’s new home battery products as part of his vision of a solar energy future. A key part of his presentation hinged on a ‘blue square’, a section of Texas that is supposedly sufficient to cover USA’s electricity consumption. To give the reader s... |
I’ve been doing a lot of reading on confidence interval theory. Some of the reading is more interesting than others. There is one passage from Neyman’s (1952) book “Lectures and Conferences on Mathematical Statistics and Probability” (available here) that stands above the rest in terms of clarity, style, and humor. I h... |
Journal of the European Mathematical Society
Full-Text PDF (191 KB) | Metadata | Table of Contents | JEMS summary
Volume 14, Issue 3, 2012, pp. 733–748 DOI: 10.4171/JEMS/316
Published online: 2012-03-07
On sets of vectors of a finite vector space in which every subset of basis size is a basisSimeon Ball
[1](1) Universi... |
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