text stringlengths 256 16.4k |
|---|
One way to assess the microbial community structure in an environment is to use a ‘fingerprinting’ technique, like T-RFLP or ARISA, to interrogate the ‘species’ living there as determined from their 16S rRNA genes or some functional gene like amoA. Here’s an example of a T-RFLP electropherogram from sea ice:
You can se... |
In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections on a surface can be derived from the Chern-Simons Lagrangian and depends upon the coupling chosen.
Now I am familiar with... |
Sample Test 4 Question 6
Let \(F_k=F_{k-1}+F_{k-2}\) where \(F_0=0, F_1=1\). Define the series \(s(x)= \sum_{k=1}^\infty F_k / x^k\) for \(x>2\).
Show that \(s(x)=\frac{x}{x^2-x-1}\).
Related topics Hints Hint 1Use the recursive definition of \(F_k\) to split \(s(x)\) into two sums. Carefully treat the base cases. Hint... |
Let us assume the standard situation in communication complexity with two players $P_1,P_2.$
We have a function $f:[n] \times [n] \mapsto \{0,1\}$ that both players known in advance. They wish to compute $f(x,y)$ given that the first player only knows $y$ and the second player only knows $x.$ The communication complexi... |
Practice Paper 4 Question 8
Find all functions \(f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\},\) such that for all \(n=1,2,3,\ldots\) the sum \(f(1)+f(2)+\cdots+f(n)\) is equal to a perfect cube that is less than or equal to \(n^3.\)
Related topics Hints Hint 1What values can \(f(1)\) take? Hint 2What value must \(f(1) + f(2)... |
In part V we saw how a statement Alice would like to prove to Bob can be converted into an equivalent form in the “language of polynomials” called a Quadratic Arithmetic Program (QAP).
In this part, we show how Alice can send a very short proof to Bob showing she has a satisfying assignment to a QAP. We will use the Pi... |
Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? Moreover, are there any known methods of studying or "attacking" the Riemann hypothesis that are more algebraic than analytic?
Y... |
I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer.
Within a bounded region on the $xy$ plane, say $x\in[-x_{0},x_{0}], y\in[-y_{0},y_{0}]$, we have a continuous scalar field $H=H(x,y)$. Both the field and the g... |
Let $\nu_n$ be a sequence of finite signed radon measure such that $\nu_n\to \nu$
strongly for a finite signed radon measure $\nu$. Let $|\nu_n|$ denote the total variation measure of $\nu_n$. We know that $|\nu_n|$ is a positive Radon measure.
My question: do I have $|\nu_n|\to |\nu|$ strongly? and do I have $||\nu|-|... |
The conditional variational principle for maps with the pseudo-orbit tracing property
1.
School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China
2.
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, China
3.
Center of Nonlinear Science... |
Answer
$a\approx6.9$ in. $b\approx9.8$ in.
Work Step by Step
$\sin35^{\circ}=\frac{a}{12}$ $a=12\times\sin35^{\circ}$ $a\approx6.9$ Using Pythagorean Theorem to find $b$. $b=\sqrt {12^{2}-a^{2}}$ $b\approx\sqrt{12^{2}-6.9^{2}}$ $b\approx9.8$
You can help us out by revising, improving and updating this answer.Update thi... |
Suppose the adversary has a quantum computer. If the adversary has time to run it on the public key before it becomes irrelevant, then every pre-quantum public-key signature scheme is broken.
What if the adversary doesn't have time to run it on the public key before it becomes irrelevant? For example, in Bitcoin, a tra... |
Can someone please tell me how I get vertical lines in the columns of a matrix like it is shown in the picture?
I started with
... = \begin{pmatrix} \lambda A' & \mu B' & \tau C' \end{pmatrix} ...
TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems.... |
Volume 9, Number 5, 2019, Pages 1706-1718
DOI：10.11948/20180273
Infinitely many solutions for a zero mass Schodinger-Poisson-Slater problem with critical growth
Liu Yang,Zhisu Liu
Keywords:Schrodinger-Poisson-Slater problem, Zero mass, critical growth, concentration-compactness principle.
Abstract:
In this paper, we ar... |
In the previous part of this series, we looked at what decibels are. To put it simply, we can measure a sound, find a representative sound pressure for it, put this into a logarithmic formula, and voilà – we have a sound pressure level in decibels. However, humans cannot hear every sound equally well. The basic calcula... |
This is one of those things that is helpfully studied using an example. A very nice example for this issue is the "wandering block". Informally, the wandering block is the sequence of indicator functions of $[0,1],[0,1/2],[1/2,1],[0,1/4],[1/4,1/2],[1/2,3/4],[3/4,1]$, etc. More explicitly, it is the sequence $g_n(x)$ wh... |
Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\Lambda_r, \mathbb{Z})$ the dual lattice in $V^*$.
$\Lambda_r^*$ can be thought of as the Pontryagin dual of the torus $T^n_r=V... |
Answer
$c\approx8.8$ $d\approx28.7$
Work Step by Step
$\sin17^{\circ}=\frac{c}{30}$ $c=30\times\sin17^{\circ}$ $c\approx8.8$ Using Pythagorean Theorem to find $d$. $d=\sqrt {30^{2}-c^{2}}$ $d\approx\sqrt{30^{2}-8.8^{2}}$ $d\approx28.7$
You can help us out by revising, improving and updating this answer.Update this answ... |
The "aromatic" papers by Li, Maxin, Nanopoulos, and Walker have been discussed many times on this blog. They have a new preprint today,
A \(125.5\GeV\) Higgs Boson in \({\mathcal F}\)-\(SU(5)\): Imminently Observable Proton Decay, A \(130\GeV\) Gamma-ray Line, and SUSY Multijets & Light Stops at the LHC8Once again, the... |
The
package is a powerful tool, based on pgfplots
tikz, dedicated to create scientific graphs.
Contents Pgfplots is a visualization tool to make simpler the inclusion of plots in your documents. The basic idea is that you provide the input data/formula and pgfplots does the rest.
\begin{tikzpicture} \begin{axis} \addpl... |
In exercises 1 and 2, use the midpoint rule with \(m = 4\) and \(n = 2\) to estimate the volume of the solid bounded by the surface \(z = f(x,y)\), the vertical planes \(x = 1\), \(x = 2\), \(y = 1\), and \(y = 2\), and the horizontal plane \(x = 0\).
1) \(f(x,y) = 4x + 2y + 8xy\)
Answer: \(27\)
2) \(f(x,y) = 16x^2 + \... |
Having a centerless profinite completion leads to some nice properties. For example, given a short exact sequence
$$1\to A\to B\to C\to 1$$
where $A$ is finitely generated and $\hat{A}$ has trivial center, we have an exact sequence
$$1\to\hat{A}\to\hat{B}\to\hat{C}\to 1.$$
When does a centerless group $G$ have a center... |
Bergou, El houcine and Diouane, Youssef and Gratton, Serge
On the use of the energy norm in trust-region and adaptive cubic regularization subproblems. (2017) Computational Optimization and Applications, 68 (3). 533-554. ISSN 0926-6003
(Document in English)
PDF (Author's version) - Requires a PDF viewer such as GSview,... |
Self-Consistent Schrödinger-Poisson Results for a Nanowire Benchmark
The
Schrödinger-Poisson Equation multiphysics interface simulates systems with quantum-confined charge carriers, such as quantum wells, wires, and dots. Here, we examine a benchmark model of a GaAs nanowire to demonstrate how to use this feature in th... |
Answer
If a radius with a length of $r$ units rotates through an angle of 1 radian, then the arc length is $r$ units. An angle of 1 radian subtends an arc length that is equal to the radius.
Work Step by Step
Radian measure is a unit that is used to measure an angle. If a radius with a length of $r$ units rotates throu... |
Definition:
$\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random vectors $\mathbf{X}$ and $\mathbf{Y}$, characterized by a joint probability distribution ${p_{_\mathbf{XY}}}$, the probabilty d... |
The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)? If not, why?
(Sorry for the earlier confusion.)
Yes, the graph of factorial is $\Delta_0$.
First, the problem can be restated in terms of computational... |
Let $p \in (1, \infty).$ Then there is a sequence $f_k$ satisfying
1) $f_k \in L^q$ for all $1 \leq q < \infty$
2) $f_k$ converges to $0$ in $L^q$ for all $q \in [1,p)$
3) $f_k$ diverges in $L^p$
$\textbf{Attemp}$ Since the integral for $1 \leq q < p$ converges to $0$ and at exactly $p$ it diverges, I think of $\int \f... |
Electronic Journal of Probability Electron. J. Probab. Volume 22 (2017), paper no. 20, 18 pp. Inversion, duality and Doob $h$-transforms for self-similar Markov processes Abstract
We show that any $\mathbb{R} ^d\setminus \{0\}$-valued self-similar Markov process $X$, with index $\alpha >0$ can be represented as a path ... |
AP Statistics Curriculum 2007 Normal Std From Socr
m (→Appendix)
m (→Appendix)
Line 39: Line 39:
:: <math>x=r\cos(\theta)</math>
:: <math>x=r\cos(\theta)</math>
:: <math>x=r\cos(\theta)</math>, <math>0\le \theta\le 2\pi</math>
:: <math>x=r\cos(\theta)</math>, <math>0\le \theta\le 2\pi</math>
-
:: Hence, <math>x^2+w^2=r... |
Classification of nonoscillatory solutions of nonlinear neutral differential equations
1.
Department of Mathematics, Faculty of Arts and Sciencecs, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey
2.
School of Pure and Applied Natural Sceinces, University of Kalmar, SE-39182 Kalmar, Sweden
$(r(t)(x(... |
This should not be hard at all, in fact no one seems so be having my issue so it should be me missing out on a point of information or just experience.
My mission is to with
Mathematica find the x,y,z's of the maximum points of the following function
$$\left[\frac{-\left(1-x^2\right) \left(y^2-4\right)-x^2-y^2+5}{\left... |
EDIT: Thanks to @benrg pointing out a bug of the previous algorithm. I have revised the algorithm and moved it to the second part since the explanation is long.
While the other answer focuses more on coding style, this answer will focus more on performance.
Implementation Improvements
I will show some ways to improve t... |
I apologize about the title as I would have included the entire identity I am trying to prove analytically but we are limited to 150 characters.
I'm given the following problem in Widder's
Advanced Calculus:
11.) $u=f\left(x,y\right),x=r\cos\left(\theta\right),y=r\sin\left(\theta\right).$ show that \begin{align} \left(... |
In a given question about a trigonometric proof I was trying to solve, I got in $$ \tan(x - π/4) = -\, 1 $$ then I solved it just like any other trig equation, but, after I managed to finish the demonstration, I get back to that equation and tried a different approach, by applying the inverse function in both sides of ... |
One thing you're missing seems to be Oberth's effect. To go from LEO to solar system escape velocity, you have to counter Earth's escape velocity, but after that, you get an additional multiplier by doing the burn at a higher initial velocity (at LEO).
Your method here also has a problem:
Hence an additional 11.2 km/s ... |
Where does this problem come from? Why would you hope for something like this? I am not trying to be snarky, I am just wondering if there is a
true statement that might better fit your situation.
At any rate, counterexamples are
easy possible to produce via varieties with a group action such that the variety deforms, b... |
Consider the problem where we have $l$ parties with messages $m_i$ who wish to generate the list $(m_{\pi(1)}, \dots, m_{\pi(l)})$, where $\pi$ is a permutation over the integer range $[1,\dots,l]$. Suppose further that they wish to do this without revealing $\pi$. This problem is essentially what mixnets were designed... |
I need to prove or disprove if these two languages are same. So I assume that these lanaguages are same because I think that every word from $\{a,b\}^*$ could be concatenated from two words $x$ and $y$ where $|x|_a = |y|_b$. So $x$ and $y$ must belong to $\{a,b\}^*$, too. Then I can write and prove this:
$\{a,b\}^* \su... |
We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain $$\Omega \subseteq \mathbb{R}^{3}$$ . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large... |
It´s known that for first order theories, it holds $\mathbf{ZFC} \vdash T \vdash \varphi \leftrightarrow T \models \varphi$. Why does this not hold in the higher order case (any simple example?)?
Furthermore if I change models by interpretations, does it hold that $T \vdash \varphi$ iff for all interpretations (in the ... |
$\newcommand{\si}{\sigma}\newcommand{\Si}{\Sigma}\renewcommand{\c}{\circ}\newcommand{\tr}{\operatorname{tr}}$
Let $X_1:=X$, $X_2:=Y$, $\mu_1:=\mu_x$, $\mu_2:=\mu_y$, $\Si_1:=\Si_x$, $\Si_2:=\Si_y$, $N:=\mathcal{N}_d$. Let $Y_i:=X_i-\mu_i$. Then $X_i=\mu_i+Y_i$, $Y_i\sim N(0,\Si_i)$, $Y_1$ and $Y_2$ are independent, and... |
Let us assume in a first step that the 4 corners (pyramid base) have the following coordinates:
$$A_1(1,0,0),\ A_2(0,1,0),\ A_3(-1,0,0), A_4(0,-1,0).$$
Then the equation of the pyramid surface is
$$z=a * \min(1 - x - y, 1 - x + y, 1 + x - y, 1 + x + y)$$
($a$ is an arbitrary positive parameter giving a more of less pea... |
The Harmonic Series.Murat Aygen from Turkey sent this solution to part (a) and Dapeng Wang from Claremont Fan Court School, Guildford sent in essentially the same solution.
Let us partition the terms of our series, starting from every $k= 2^{i-1}+ 1$, as follows: $\sum_{k=1}^{2^r} \left({1\over{k}}\right) = 1 + \left({... |
Note: Turns out this is not optimal. My strategy depends on looking at the probability of beating your current score with the next roll. In his answer, @ArthurSkirvin looks at the probability of beating it on any subsequent roll. As I should have expected, taking the longer view provides a better strategy.
(I'm assumin... |
How to Model Moisture Flow in COMSOL Multiphysics®
Computing laminar and turbulent moisture flows in air is both flexible and user friendly with the
Moisture Flow multiphysics interfaces and coupling in the COMSOL Multiphysics® software. Available as of version 5.3a, this comprehensive set of functionality can be used ... |
Search
Now showing items 1-6 of 6
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Fractions and binomial coefficients are common mathematical elements with similar characteristics - one number goes on top of another. This article explains how to typeset them in LaTeX.
Contents
Using fractions and binomial coefficients in an expression is straightforward.
The binomial coefficient is defined by the ne... |
I came across the barotropic vorticity equation (below) in Sakamoto (2002) and I cannot figure it out. The notation is not clear to me. How can it be derived from the Navier-Stokes equation?
$$ \beta\psi_x=\frac{\tau_x^y-\tau_y^x}{\rho_0H}-r\zeta+A_H\nabla^2\zeta-\frac{f_0}{H}w_D $$
where
$\psi$ is the streamfunction
$... |
There is a solution $t$, since $(0.4)^t\gt 5t$ at $t=0$, and $(0.4)^t\lt 5t$ at $t=1$. It is easy to see that there cannot be a solution outside the interval $(0,1)$.
There is no rational solution. For suppose that $t=\frac{p}{q}$ is a solution, where $p$ and $q$ are relatively prime integers, neither equal to $0$. Sin... |
Dev gives HBO free math tips to nail Game of Thrones pirate leakers An easy* way to ID a million leakers
Developer Bruno Cauet has offered HBO a series of mathematical equations that could have tracked the Game of Thrones season five leaker, or even killed the leak completely.
The massively popular series thought to be... |
In this series, we first looked at what sound pressure levels and decibels are. Then, we looked at how we can calculate sound pressure levels in a way that takes human hearing into account. So far, though, we have only considered steady and unchanging sounds. These come from e.g. ventilation systems or machines that ru... |
When we looked at Green's Theorem, we saw that there was a relationship between a region and the curve that encloses it. This gave us the relationship between the line integral and the double integral. Now consider the following theorem:
Divergence Theorem
Let \(Q\) be a solid region bounded by a closed surface oriente... |
Instability of bound states for 2D nonlinear Schrödinger equations
1.
Department of Mathematical Sciences, Yokohama City University, Seto 22-2, 236-0027, Japan
$iu_t+\Delta u+|u|^{p-1}u=0\quad$ for $x\in \mathbb R^2$
and $t>0$, where $(r,\theta)$ are polar coordinates and $m\in\mathbb N$. Using the Evans function, we p... |
The Annals of Probability Ann. Probab. Volume 47, Number 1 (2019), 464-518. Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise Abstract
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equation... |
This is a tricky problem. Can anyone help me with the procedure and answer?
Evaluate $$ \lim_{h\to 0} \left( \frac{f(x+hx)}{f(x)}\right)^{1/h}, \text{for }f(x)=x. $$
Note that you are given $f(x)=x$. With this piece of information, we can replace the functions in the given equation with their respective algebraic repre... |
Maximum likelihood estimation is a very useful technique to fit a model to data used a lot in econometrics and other sciences, but seems, at least to my knowledge, to not be so well known by machine learning practitioners (but I may be wrong about that). Other useful techniques to confront models to data used in econom... |
I recently gave a talk at the Dartmouth Number Theory Seminar (thank you Edgar for inviting me and to Edgar, Naomi, and John for being such good hosts). In this talk, I described the recent successes we’ve had on working with variants of the Gauss Circle Problem.
The story began when (with Tom Hulse, Chan Ieong Kuan, a... |
You can view as many worked out examples as you want. First you are shown the question. Click Show Answer to view the answer.
Click Show another Example to view another example. The best way to master mathematics is by practice. But practice requires time.
If you don't have the time to practice, you can always view man... |
I have a problem that sounds like this:
Find the limit $$\lim_{x\rightarrow 0} \frac{14\tan(6x)-84x}{6x^3}$$ using Maclaurin series, and don't forget the importance of big O notation.
I have tried to find the Maclaurin series in different ways, but I always end up with the wrong answer. And I don't know how to use the ... |
Answer
$21^{\circ}C $
Work Step by Step
We use the equation for H to find: $H = (120 \ W/^{\circ}C)(\Delta T)$ $\Delta T = \frac{H}{120 \ W/^{\circ}C} $ $\Delta T = \frac{2.5 \times 10^3 \ W}{120 \ W/^{\circ}C} = 21^{\circ}C $
You can help us out by revising, improving and updating this answer.Update this answer
After ... |
Practice Paper 3 Question 11
Let \(I_n=\int_{-\pi/2}^{\pi/2} \cos^n\!x \,dx\) for any non-negative integer \(n.\) \(\;(i)\) Find a recursive expression for \(I_n.\) \((ii)\) Find the simplest non-recursive expression for \(I_{2n}\) that contains \(\binom{2n}{n},\) where \(\binom{n}{k}=\frac{n!}{k!(n-k)!}.\)
Related top... |
Search
Now showing items 1-1 of 1
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
Search
Now showing items 1-1 of 1
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
Search
Now showing items 1-10 of 33
The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
Practice Paper 3 Question 12
A bank has a loan scheme with an annual interest rate of \(0<r<1\) (\(100r\) is the annual percentage rate). The bank charges interest every month. You take a loan of \(L\) pounds and wish to pay it back in exactly \(m\) months by making monthly payments of \(p\) pounds each. Find \(p\).
Re... |
What are general theorems that give sufficient criteria for a $C^{\infty}$ function to be analytic? The more general/simple the test, the better. I'm trying to understand in a more thorough way what prevents $C^{\infty}$ functions from being analytic. The canonical example of a $C^{\infty}$ function $f$ which is not an... |
The stock investment universe is huge. However, for an automated trading strategy, it is even bigger since every day will offer this entire universe to choose from. The buy $ \& $ holder will make one choice on a number of stocks and stick to it. Whereas for the automated trader, each day requires a trading decision fo... |
Practice Paper 3 Question 13
\(n\) passengers board a plane with \(n\) seats. Each passenger has an assigned seat. The first passenger forgets and takes a seat at random. Every subsequent passenger sits in their assigned seat unless it is already taken, at which point they take a random seat. What is the probability (i... |
Given a chess board of dimensions $7\times 7$ what is the minimal number of knights needed to be placed on the board so that every cell is attacked by at least one knight. I have no idea on this one.I assume the best strategy would be to put so that each knight attacks the most number of cells which is $8$.
Using the n... |
SolidsWW Flash Applet Sample Problem 3 Flash Applets embedded in WeBWorK questions solidsWW Example Sample Problem 3 with solidsWW.swf embedded
A standard WeBWorK PG file with an embedded applet has six sections:
A tagging and description section, that describes the problem for future users and authors, An initializati... |
I would like to create special frame around my equations. This frame should obey to the
golden ratio rule. This means that its width divided by its height should give 1.61803398875.
If possible, the code will adapt to choose the
long side. For example for a long column vector, the height will be 1.61803398875 times lon... |
To show that $\Bbb R^d$ with the norm
$\Vert x \Vert_1 = \displaystyle \sum_1^d \vert x_i \vert, \tag 1$
where
$x = (x_1, x_2, \ldots, x_d) \tag 2$
is Banach we need to prove it is Cauchy-complete with respect to this norm; that is, if
$y_i \in \Bbb R^d \tag 3$
is a $\Vert \cdot \Vert_1$-Cauchy sequence, then there exi... |
Comet Ison passing Mars (Wolfram Alpha)
I just got back from the Black Forest Star Party where Dr. Carey Lisse, head of NASA’s ISON Observing Campaign, gave a speech on comet ISON.
Comet ISON (see [1], [2]) is passing by Mars soon (Oct 1) and it will be “grazing the sun” before the end of the year, so I wondered if the... |
Suppose we have a function defined as follows:
$\alpha(f,x_o)=\limsup\{|f(a)-f(b)|:a,b\in (x_o-\frac{1}{n},x_o+\frac{1}{n})\}$
with $f:R\rightarrow R$ and $x_o \in R$. I need to prove that $\alpha=0$ iff $f$ is continuous at $x_o$.
It seems trivial to show the $\alpha=0 \Rightarrow$ direction, since $|f(a)-f(b)|\geq0$,... |
All of them (and maybe lots or even all of LIGO folks) are missing that the two LIGO detectors aren't predicted to see signals that are exactly proportional to each other
Someone I can't identify uncritically promotes a combative Danish paper in his or her article. It is all about a fresh paper
ATdotDE and Telescoper c... |
I am trying to build a public hash function (thus collision-resistant and preimage-resistant, and more generally behaving like a random oracle), with input a message $M$ of fixed size $|M|=m\cdot b$ bits, and output the hash $H(M)$ of fixed size $|H(M)|=h\cdot b$ bits, using as the single primitive a $b$-bit block ciph... |
Practice Paper 3 Question 19
Let \(A=\{0, 1, \dots, 2^n-1, 2^n\}\) and \(B = \{0, \dots, n\}\). How many functions \(g\) can be defined from \(A\) to \(B\) such that both of the following conditions hold:
for all \(x \in B\) we have \(g(2^x) = x,\) for all \(y, z \in A\) with \(y \leq z\) we have \(g(y) \leq g(z).\) Re... |
Two of our best solvers, Alex from Stoke on Trent Sixth Form College and Patrick from Woodbridge School thought about this implicit equation.
$$r^2X^2-rX-r+1=0$$Patrick looked at the first parts using the quadratic equation formula as follows
:Part 1
:
If $r=1$ then $X^2-X = 0$, so $X(X-1) = 0$ and $X = 0$ or $1$.
If $... |
A brief post today:
I was talking about an algebraic topology problem from Hatcher’s book (available freely on his website) with two of my colleagues. In short, we were finding the fundamental group of some terrible space, and we thought that there might be a really slick almost entirely algebraic way to do a problem. ... |
Karle, Isabella L and Flippen-Anderson, Judith L and Agarwalla, Sanjay and Balaram, Padmanabhan (1991)
Crystal structure of [Leu1]zervamicin, a membran ion-channel peptide: Implications for gating mechanisms. In: Proceedings of the National Academy of Sciences of the United States of America, 88 (12). pp. 5307-5311.
PD... |
Answer
$\theta=30^o$
Work Step by Step
Use a scientific calculator's inverse cosine function in degree mode to obtain: $\theta = \arccos{(\frac{\sqrt3}{2})} \\\theta=30^o$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hours to send in a d... |
It is well known that a parametric form of the parabola $y^2=4ax$ is $(at^2, 2at)$.
What are possible parametric forms of the general parabola $$(Ax+Cy)^2+Dx+Ey+F=0$$ ?
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 min... |
Help please!
Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.
I can prove that $\sum a_n b_n$ converges but couldn't for $\sum \sqrt{a_n b_n}$.
Thank you.
EDIT: $a_n, b_n \ge 0 \;\forall n$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and pro... |
I discussed this topic in April 2012 in
I claim that a theoretical physicist prefers units in which a maximum possible subset of the universal fundamental constants of Nature such as \(\hbar,c,\epsilon_0,e,k_B,N_A\) and perhaps even \(G\) have well-known values. In theory papers, we really want to put them equal to one... |
17 2 Summary I am wondering at the striking similarity between the expressions for creation/annihilation operators in terms of position and momentum operators and the expressions for sine and cosine in terms of the exponential.
Hello everyone,
I have noticed a striking similarity between expressions for creation/annihi... |
1) The region \(D\) bounded by \(y = x^3, \space y = x^3 + 1, \space x = 0,\) and \(x = 1\) as given in the following figure.
a. Classify this region as vertically simple (Type I) or horizontally simple (Type II).
Type: Type I but not Type II
b. Find the area of the region \(D\).
c. Find the average value of the functi... |
Exercise \(\PageIndex{1}\)
Find the first \(5\) terms of the sequence as well as the \(30^{th}\) term.
\(a_{n}=5 n-3\) \(a_{n}=-4 n+3\) \(a_{n}=-10 n\) \(a_{n}=3 n\) \(a_{n}=(-1)^{n}(n-2)^{2}\) \(a_{n}=\frac{(-1)^{n}}{2 n-1}\) \(a_{n}=\frac{2 n+1}{n}\) \(a_{n}=(-1)^{n+1}(n-1)\) Answer
1. \(2,7,12,17,22 ; a_{30}=147\)
3... |
The Math.Stackexchange (MSE) is an extraordinary source of great quality responses on almost any non-research level math question. There was a recent question by the user belgi, called A list of basic integrals, that got me thinking a bit. It is not in the general habit of MSE to allow such big-list or soft questions. ... |
Reactivity of thioesters with respect to nucleophilic attack
Why is the thioester bond weaker than a regular ester bond?
Let's consider a reaction mechanism like this:
A steady-state analysis will show that the rate of formation of product is
$$r = \frac{k_1k_2[\text{ester}][\text{Nu}]}{k_{-1} + k_2}$$
$k_1$ for the th... |
Conveners Session 8: Theoretical developments 1 Jun GAO (Shanghai Jiao Tong University, Center for High Energy Physics, Peking University) Session 8: Theoretical developments 2 Antoni Szczurek (Institute of Nuclear Physics)
We discuss $\gamma^* \gamma^* \to \eta_c(1S)\, , \,\eta_c(2S)$ transition form factor
for both v... |
It’s impossible to find explicit formulas for solutions of some differential equations. Even if there are such formulas, they may be so complicated that they’re useless. In this case we may resort to graphical or numerical methods to get some idea of how the solutions of the given equation behave.
In Section 2.3 we’ll ... |
In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$. I want to know why $\omega \neq \omega+1$.
This question appears to be off-topic. The users who voted to close gave this specific reason:
" This question is missing context or other details: Please improve the question by providing addit... |
Recall that we were able to analyze all geometric series "simultaneously'' to discover that \[\sum_{n=0}^\infty kx^n = {k\over 1-x},\] if \(|x| < 1\), and that the series diverges when \(|x|\ge 1\). At the time, we thought of \(x\) as an unspecified constant, but we could just as well think of it as a variable, in whic... |
In this short post, we introduce three conundrums dealing with infinity. This is inspired by my calculus class, as we explore various confusing and confounding aspects of infinity and find that it’s very confusing, sometimes mindbending.
Order Matters
Consider the alternating unit series $$ \sum_{n \geq 0} (-1)^n. $$
W... |
In the following exercises, write the appropriate \(ε−δ\) definition for each of the given statements.
176) \(\displaystyle \lim_{x→a}\,f(x)=N\)
177) \(\displaystyle \lim_{t→b}\,g(t)=M\)
Answer: For every \(ε>0\), there exists a \(δ>0\), so that if \(0<|t−b|<δ\), then \(|g(t)−M|<ε\)
178) \(\displaystyle \lim_{x→c}\,h(x... |
If we have a Sturm-Liouville differential equation of the form $$\frac{d}{dx}[p(x)\frac{dy}{dx}]+q(x)y=-\lambda w(x)y$$ and define the linear operator $L$ as $$L(u) = \frac{d}{dx}[p(x)\frac{du}{dx}]+q(x)u$$ then we get the equation $L(y)=-\lambda w(x)y$ which defines what is called the eigenvalue problem of the Sturm-L... |
Suppose a surface given by $f(x,y)$ has a local maximum at $(x_0,y_0,z_0)$; geometrically, this point on the surface looks like the top of a hill. If we look at the cross-section in the plane $y=y_0$, we will see a local maximum on the curve at $(x_0,z_0)$, and we know from single-variable calculus that ${\partial z\ov... |
I am trying to show that a sequence $(x_n)_n \subseteq \mathcal{H}$ converges strongly to $x$ if it converges weakly to $x \in \mathcal{H}$ and $\|x_n\| \to \|x\|$ as $n \to \infty$
$\mathcal{H}$ is an infinite dimensional hilbert space
Now my proof is clearly bogus because I do not even use the second condition to com... |
In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.
In two dimensions, two such shape measures are the minimum angle of a triangle and its aspect ratio, i.e. the quotient of the ra... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.