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Introduction:
In an excellent paper published less than two years ago, Timothy Lillicrap, a theoretical neuroscientist at DeepMind, found a simple yet reasonable solution to the weight transport problem. Essentially, Timothy and his co-authors showed that it’s possible to do backpropagation with random weights and stil... |
Table of Contents
Variational inequalities provide a general mathematical framework for many problems arising in optimization. For example, constrained optimization problems like LP and NLP are special cases of VI, and systems of equations and complementarity problems can be cast as VI. Thus VI problems have many appli... |
Advances in Differential Equations Adv. Differential Equations Volume 15, Number 7/8 (2010), 601-648. An antimaximum principle for a degenerate parabolic problem Abstract
We obtain an anti\-maximum principle for the following quasilinear parabolic problem: \begin{equation*} \tag*{\rm (P)} \left\{ \begin{alignedat}{2} \... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
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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... |
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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... |
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The Romanian Economic Journal
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Jurnalul Economic
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I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that space.
$$\frac{1}{m} \sum f(x_i) \approx \int_{[0,1]^n} f(x) \ dx $$
If I broke each segment $[0,1]$ into $10$ points, I would hav... |
It’s 8/15/17, which means it’s time to celebrate! The three numbers making up the date today form a
Pythagorean triple, a triple of positive integers $(a,b,c)$ with $a^2+b^2=c^2$. Indeed, $8^2+15^2=64+225=289=17^2$.
Alternatively, by the Pythagorean theorem, a Pythagorean triple is any triple of positive integers which... |
The wavelength dependence of the definition of free space path loss (FSPL) is an artifact of the way the
receiver's antenna gain is defined in the same link budget calculation. It's referenced to an ideal isotropic antenna with a receive area of roughly 1 square wavelength, which for high frequency gets very small. If ... |
Let us start with an example relying on a simple time-warp, based on the Hann window (code below). It consists in modifying time index $t\in [-1,1]$, here with function $t\mapsto 2 (t+1)^\alpha/2^\alpha-1$. There are many other ways.
Asymmetrical windows, or non-symmetric windows, are a topic of interest, albeit somewh... |
The rank-$k$numerical ranges, denoted below by $\Lambda_k$, were introduced c. 2006 by Choi, Kribs, and Życzkowskias a tool to handle compression problems in quantum information theory. Since thentheir theory and applications have been advanced with remarkable enthusiasm. Thesequence of papers [1], [2], [3], [4], for e... |
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 \... |
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We've seen that classical logic is closely connected to the logic of subsets. For any set \( X \) we get a poset \( P(X) \), the
power set of \(X\), whose elements are subsets of \(X\), with the partial order being \( \subseteq \). ... |
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 think your postcondition is wrong. Since index i is initialized at 0, you're actually summing from 0 to n, not from 1 to n. The postcondition, lets call it $P$, should be:
$$ j = \sum_{k=0}^{n} a[k] $$
Or:
j = sum(a[0] ... a[n])
Usually you would write $B$ as $i \neq n+1$ instead of $i \le n$, because it makes the pr... |
Difference between revisions of "Demand/Dynamic User Assignment"
m (→Iterative Assignment (Dynamic User Equilibrium))
(using vtypes from additional file)
(One intermediate revision by one other user not shown) Line 24: Line 24:
Between successive calls of DUAROUTER, the ''.rou.alt.xml'' format is used to record not onl... |
Answer
The solution set is $$\{60^\circ,180^\circ,300^\circ\}$$
Work Step by Step
$$\cos\theta+1=2\sin^2\theta$$ over the interval $[0^\circ,360^\circ)$ Here we have both cosine and sine functions. It is necessary then to change either cosine or sine to the other so that we can solve the equation. Recall the identity: ... |
Hint Applet (Trigonometric Substitution) Sample Problem Flash Applets embedded in hint portion of WeBWorK questions Example Sample Problem with trigSubWW.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 a... |
How to Solve for the Brachistochrone Curve Between Points
The shortest route between two points isn’t necessarily a straight line. If by shortest route, we mean the route that takes the least amount of time to travel from point A to point B, and the two points are at different elevations, then due to gravity, the short... |
We have seen that some functions can be represented as series, which may give valuable information about the function. So far, we have seen only those examples that result from manipulation of our one fundamental example, the geometric series. We would like to start with a given function and produce a series to represe... |
To elaborate on gallais' clarifications, a type theory with impredicative Prop, and dependent types, can be seen as some subsystem of the calculus of constructions, typically close to Church's type theory. The relationship between Church's type theory and the CoC is not that simple, but has been explored, notably by Ge... |
I would be very grateful if anyone could help with my questions below. Here's my latex code:
\documentclass[12pt]{report}\usepackage{caption}\usepackage{subcaption}\usepackage{tikz}\usetikzlibrary{arrows}\usepackage{verbatim}\begin{document}\tikzstyle{int}=[draw, line width = 1mm, minimum size=8em]\begin{figure} \cente... |
Let's say I have a sample of 2 (assumingly different) diatomics A and B.
Through spectroscopy I found the data below:
For molecule A: $$\begin{align} B&\approx2.17690\ \mathrm{cm^{−1}}\\ D&\approx4.79000\times10^{-5}\ \mathrm{cm^{−1}}\\ I&=1.28590\times10^{-46}\ \mathrm{kg\ m^2}\\ \tilde ν_e&=928.15\ \mathrm{cm^{−1}} \... |
Assuming the RH and $s \in \mathbb{C}, \rho_n =\frac12 \pm i\gamma_n$, the following (altered) Hadamard product:
$$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n i \gamma_n} \right) \left(1- \frac{s}{\frac12+ (-1)^{n+1} i \gamma_n} \right) = \frac{\xi_{rie}(s)}{\xi_{rie}(0)}$$
runs thr... |
Let $D$ be a bounded simply connected region (open subset homeomorphic to the disc) in the plane, containing the origin. Suppose that for every line $L$ through the origin the intersection $L\cap\partial D$ consists of two points $z_1$ and $z_2$ such that $|z_1-z_2|=\mathrm{diam} D$. Does it follow that $D$ is a disc?
... |
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Given sin2(θ)+cos2(θ)=1 \sin^2(\theta) + \cos^2(\theta) = 1 sin2(θ)+cos2(θ)=1, which of the following is true?
True or False: sin2(θ)−cos2(θ)+1=2sin2(θ) \sin^2(\theta) - \cos^2(\theta) + 1 = 2\sin^2(\theta) sin2(θ)−cos2(θ)+1=2sin2(θ).
(Hint: Use the identity sin2... |
For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
131) \(f(x)=\frac{1}{\sqrt{x}}\)
Answer: The function is defined for all x in the interval \((0,∞)\). In other words, this function is continuous on it... |
Narayanan, EK and Thangavelu, S (2004)
An optimal theorem for the spherical maximal operator on the Heisenberg group. In: Israel Journal of Mathematics, 144 (2). pp. 211-219.
PDF
An_optimal_theorem.pdf - Published Version
Restricted to Registered users only
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Abstract
Let $H_n = C^n \tim... |
This question already has an answer here:
In many place one finds accounts of how to evaluate $$ \int_0^\infty \frac{\sin x} x\,dx = \underbrace{\lim_{a\to\infty}\int_0^a}_{\text{Why view it this way?}} \frac{\sin x} x\, dx. $$
And it gets asserted that the reason why the integral must first be evaluated over a bounded... |
Say that an inner model $M$ of $V$ is
generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses cardinals.
What is the consistency strength of "$L$ is generically saturated"?
If the answer is $0^\#$ exists, is this so... |
Suppose $\kappa< 2^{\aleph_0}$ and $\langle P_i : i < \kappa\rangle$ is a sequence of perfect subsets of $2^{\omega}$. Can we find $Q_i \subseteq P_i$ for $i < \kappa$ such that each $Q_i$ is perfect and for every $x_i \in Q_i$ (for $i < \kappa$), the set $\{x_i: i < \kappa\}$ is Turing independent?
If Martin's axiom h... |
Yes, another one of those
"I have found a great password-generation strategy" waiting to be shot down!
It seems that although diceware aims to be a secure and user-friendly password generation method, there is a consensus that its output is not typically user-friendly. Over the years I have read many posts asking how t... |
I've been trying to prove it for a while, but can't seem to get anywhere.
$$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$
Could someone please provide a valid proof?
I am not allowed to work on both sides of the equation.
Work so far:
RS:
$$ \begin{align} & \frac{\sin^2\theta}{\cos^2... |
Monotone solutions to a class of elliptic and diffusion equations
1.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China, China
$-\Delta u= F'(u),$ in $\mathbf R^n,$
$\partial_{x_n}u>0,$
and the diffusion equation
$u_t-\Delta u= F'(u),$ in $\mathbf R^n\times$ {$t>0$},
$\partial_{x_n}u>0, u|_... |
We have begun looking at trace class and Hilbert Schmidt ideals in lectures today. In particular, looking at pages 206-212 of Reed and Simon book 1.
My lecture asserted the following two equivalences in class today and I don't understand the details.
If we let $\{ b_n \}$ be an orthonormal basis, it was asserted that t... |
Answer
$\cos 225^{\circ}= -\dfrac{\sqrt 2}{2}$
Work Step by Step
The reference angle of an angle $0 \leq \theta \lt 2\pi $ based on its position can be computed by using the following steps: a) Quadrant- I: $\theta $ b) Quadrant- II: $180^{\circ}-\theta $ c) Quadrant -III: $\theta - 180^o $ d) Quadrant -IV: $360^{\circ... |
Practice Paper 2 Question 12
A point traces a unit circle if its coordinates satisfy \((x,y) = (\cos t, \sin t)\) as time \(t\) varies from \(0\) to \(2\pi.\) Give an equation for a point that traces a spiral centred at \((0,0)\) and that crosses the positive \(x\)-axis at \(x=1,2,3,\ldots\) at times \(t = 2\pi,4\pi,6\... |
So we all know that the continued fraction containing all $1$s...
$$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$
yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + 1/x$, solving the resulting quadratic equation and assuming that a continued fraction that only contains additions... |
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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 ... |
Practice Paper 2 Question 13
You place right-angled triangles \(ABC\) with \(\angle A=\frac{360}{k}\) degrees, where \(k\ge5\) is an integer and \(\angle B=90\) degrees, as follows: the first triangle with \(A\) at \((0,0)\) and \(B\) at \((0,1)\), then repeatedly place triangles with \(A\) at \((0,0)\) such that \(AB\... |
Usually the stochastic integral is defined for processes indexed over $[0,\infty).$
I wonder about the standard way to define the integral for processes indexed over $[0,T].$ That is, for a continuous local martingale $M = (M_t)_{t \in [0,T]}$ and a progressive process (sufficiently integrable) $H = (H_t)_{t \in [0,T]}... |
comprehensive function based validation routines
README
The above badges represent the current development branch. As a rule, I don't push to GitHub unless tests, coverage and usability are acceptable. This may not be true for short periods of time; on holiday, need code for some other downstream project etc. If you ne... |
Question:Outline how Einstein's and Planck's views of Science differed in relation to Science research being influenced by society an politics.I can't remember anything about this and I'm having trouble finding the information needed. Can some one please help me understand what is is that...
A few months ago, I stumped... |
144) Consider two athletes running at variable speeds \(\displaystyle v_1(t)\) and \(\displaystyle v_2(t).\) The runners start and finish a race at exactly the same time. Explain why the two runners must be going the same speed at some point.
145) Two mountain climbers start their climb at base camp, taking two differe... |
Does anyone know how to evaluate the following limit?
$$\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\frac{\pi}{2}-x}-\tan {x}\right)$$
The answer is 0 , but I want to see a step by step solution if possible.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in r... |
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... |
In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind. A def... |
Evaluate the integral:
$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$
$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$
Solution
This is an open invitation to the new brand site Joy of Mathematics . We have expanded it a ... |
I am trying to show that the double integral over $\mathbb{R}^2$ of the joint PDF of Gaussian Distribution is $1$. I am looking at:
$$\frac{1}{2\pi} \cdot \frac{1}{\sqrt{a^2b^2-c^2}} \iint_{\mathbb{R^2}} \exp\left\{\frac{-(a^2(x-m)^2 - 2c(x-m)(y-n)+b^2(y-n)^2)}{2(a^2b^2 - c^2)}\right\} dx dy$$
where $m, n \in\mathbb{R}... |
Groups of Order 8 Theorem
Then $G$ is isomorphic to one of the following:
$\Z_8$ $\Z_4 \oplus \Z_2$ $\Z_2 \oplus \Z_2 \oplus \Z_2$ $D_4$ $\Dic 2$
where:
$\Z_n$ is the cyclic group of order $n$ $D_4$ is the dihedral group of order $8$ $\Dic 2$ is the dicyclic group of order $8$. Proof
$\Box$
Let $G$ be non-abelian.
So $... |
Practice Paper 2 Question 19
A population \(x(t)\) grows in time according to \(\frac{dx}{dt}=(x-1)(2x-1).\) Knowing that \(x(0)=0,\) after how much time does it reach reach \(50\%\) of its ultimate value as time passes?
Related topics Warm-up Questions Integrate \(\int (6x-4)^2 dx.\) Solve \(\frac{dy}{dx}=\cos x\) to ... |
The book.
A draft of my book: Geometric Algebra for Electrical Engineers, is now available. I’ve supplied limited distribution copies of some of the early drafts and have had some good review comments of the chapter I (introduction to geometric algebra), and chapter II (multivector calculus) material, but none on the e... |
P(n): For all $n$, the number of straight line segments determined by $n$ points in the plane, no three of which lie on the same straight line,is: $\large \frac{n^2 - n}{2}$.
Inductive hypotheses: given $n = k$ points, assume $P(k)$ is true: $P(k) = \dfrac{k^2 - k}{2}$.
Proving $P(k+1)$ would require proving that for $... |
11:00 AM to 12:30 PM
CSB 480
Cindy Xiong, Northwestern University
Your visual system can crunch vast arrays of numbers at a glance, providing the rest of your brain with critical values, statistics, and patterns needed to make decisions about your data. But that process can be derailed by biases at both the perceptual ... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
I'm doing a course on elliptic curves, and I'm stuck on a line in a proof which is supposedly using the uniqueness in Hensel's lemma. Starting with an elliptic curve
$$E:Y^2Z+a_1XYZ+a_3YZ^2=X^3+a_2X^2Z+a_4XZ^2+a_6$$
we work in the affine piece $Y \not=0$.
So we let $t=\frac{-X}{Y}$, $w=\frac{-Z}{Y}$, getting
$$w=t^3+a_... |
The first observation of top quark production in proton-nucleus collisions is reported using proton-lead data collected by the CMS experiment at the CERN LHC at a nucleon-nucleon center-of-mass energy of $\sqrt{s_\mathrm{NN}} =$ 8.16 TeV. The measurement is performed using events with exactly one isolated electron or m... |
Sound is, simply put, weak but rapid fluctuations in air pressure: The air becomes a tiny bit denser, a tiny bit thinner, denser, thinner, and so forth. These fluctuations start at sound sources, for example loudspeakers, and spread out like waves. At the sound wave’s peak, the air is at its densest, while at the wave’... |
1) Show that $Re(z)^2-Re(z^2)\geq 0$2) Write down the section formula and linear representation of a line.
3) Write down the equation of line on argand plane equivalent to y=x.
4) (*) Write down the equation of a hyperbola, like xy=1, on argand diagram.
Now in the above diagram, from left to right, gives three parallel... |
Periodic and Quasi-periodic Solutions for the Complex Swift-Hohenberg Equation
Wenyan Cui,Lufang Mi,Honglian You
Keywords:Swift-Hohenberg equation; periodic solution; quasi-periodic solution; normal form
Abstract:
In this paper, we consider the complex Swift-Hohenberg(CSH) equation$$\frac{\partial u}{\partial t}=\lambd... |
Question
Let $(X_n)_{n\geq 1}$ be an i.i.d sequence of standard normals. Show that with probability one $\liminf \frac{M_n}{\sqrt{2\log n}}\geq 1$, where $M_n=\max_{1}^n X_i$.
My attempt
Given $\varepsilon >0$, it suffices to show that $P(\frac{M_n}{\sqrt{2\log n}}<1-\varepsilon \quad \text{i.o.})= 0$. To this end put ... |
May someone please verify if this proof is correct? Let E be the union of open neighborhoods, since the union of a collection of open sets is open, it follows that E is an open set. Assume E is an open set therefore, $\forall$ p$\in$ E $\exists$ $r>0$ : $N_r(p)$ $\subset$ E. So for $p_1$ , $\exists$ $N_{r_1}(p_1)$ $\su... |
Suppose \(T=\{u_{1}, \ldots, u_{n} \}\) and \(R=\{w_{1}, \ldots, w_{n} \}\) are two orthonormal bases for \(\Re^{n}\). Then:
\begin{eqnarray*}
w_{1} &=& (w_{1}\cdot u_{1}) u_{1} + \cdots + (w_{1}\cdot u_{n})u_{n}\\ & \vdots & \\ w_{n} &=& (w_{n}\cdot u_{1}) u_{1} + \cdots + (w_{n}\cdot u_{n})u_{n}\\ \Rightarrow w_{i} &... |
This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change?I found...
Does it make a sense to define th... |
Sue Liu of Madras College, St Andrews sent an excellent solution tothis question.
Suppose that $a$ is the radius of the axle, $b$ is the radius ofeach ball-bearing, and $c$ is the radius of the hub (see thefigure).
The solution is based on the following figure.
The angle subtended by each ball-bearing at the centre of ... |
Developed by Deva O'Neil - Published July 20, 2017
Subject Area Mechanics Level First Year Available Implementation Glowscript Learning Objectives
* Students use a while loop to implement a summation (**Exercises 1 and 2**) * Students distinguish between variables that are updated (accumulators) and variables that are ... |
By Dr Adam Falkowski (Résonaances; Orsay, France)
The title of this post is purposely over-optimistic in order to increase the traffic. A more accurate statement is that a recent analysis
of X-ray spectrum of galactic clusters claims the presence of a monochromatic \(3.5\keV\) photon line which can be interpreted as a ... |
SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1 December 2013
Engineering Mathematics-1
December 2013
Total marks: --
Total time: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons (2) Marks are given to the right of every question (3) Draw neat diagrams wherever necess... |
The observation of even a few exceptional storms can provide quantitative evidence for climate change. Doing so however requires observing and learning as much as possible about how storms work, and not merely counting storms. With some understanding of how atmospheric systems and storms operate, we have other observat... |
In continuation of my previous post on factoring, I continue to explore these methods. From Pollard’s $latex \rho$ method, we now consider Pollard’s p-1 algorithm.
Before we consider the algorithm proper, let’s consider some base concepts.
Firstly, I restate Euler’s Theorem (of course, Euler was prolific, and so there ... |
Let $C$ be a $[n,k]$ linear code over $\mathbb{F}_q$.
Suppose that $\rho$ is the covering radius .
I want to show that $\rho \geq \frac{n-k}{1+ \log_q{(n)}}$.
Could you give me a hint how we could show this?
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of com... |
This is my first post, I’ll do something very simple: ADVERTISE the methodology that our applied mathematicians use in data analysis. Unlike statisticians or computer scientists, we usually start from ‘dynamical system’ point of view. To give you a taste of what I mean, a method called dynamic mode decomposition (DMD) ... |
$$\frac{ \cos 6x + 6 \cos 4x + 15 \cos 2x + 10 }{ \cos 5x + 5 \cos 3x + 10 \cos x }$$ My approach so far : Tried to represent the denominator as a factor of numerator by manipulating numerator's $\cos 6x = \cos (5x+x)$ , $\cos 4x = \cos (3x+x)$ , so on .. but then $\sin x$ come up which make it more complex to solve .
... |
Let $(X,d)$ be a metric space with infinitely many elements. Suppose $X$ is disconnected. Then is it necessarily true that $X$ is not compact?
I come up with this question when I was thinking about the Lebesgue number lemma. If $X$ is disconnected, then there exist disjoint open sets $U$ and $V$ in $X$ such that $X=U\c... |
Study Radiofrequency Tissue Ablation Using Simulation
Radiofrequency tissue ablation is a medical procedure that uses targeted heat for a variety of medical purposes, including killing cancerous cells, shrinking collagen, and alleviating pain. The process involves applying mid- to high-frequency alternating current dir... |
First Let's take a look at the convolution $\displaystyle C _ { i } = \sum _ { j \oplus k = i } A _ { j } * B _ { k }$, and the $\oplus$represents any boolean operation. And we are able to evaluate $C$ in $O(n \log n)$ time, using an algorithm called
Fast Walsh Transform, where $n$ represents the length of the binary d... |
What is the easiest way to calculate
$$ \int_{0}^{1} \frac{ \log (1+x)}{x}\, dx$$
?
Need a hint.
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 up to join this community
$$I=\int_{0}^{1}\frac{\log... |
If I have a dirichlet distribution with parameter $\alpha \in \mathbb{R}^k = [\alpha_1, \alpha_2, \cdots, \alpha_k]$, and then I set the component $\alpha_k$ to $\epsilon$. As I decrease $\epsilon$, will this pdf approximate the pdf of a dirichlet distribution with parameters $[\alpha_1, \alpha_2, \cdots, \alpha_{k-1}]... |
LHC is colliding lead ions whose lab energy is \(82\times 6.369\TeV\), a new record! On Thursday, November 25th, 1915, exactly 100 years ago, Einstein presented the final form of his equations (defining the general theory of relativity) to the Royal Prussian Academy of Sciences in the afternoon. The building sits at th... |
We may model the problem as a roulette problem, where we have $N$ balls inside a box in which one of them is special.
The simplest model is the geometric model --- we put the non-special ball back and retry the whole process. It follows a geometric distribution with expectation $\frac{1}{p}$. But this is far too slow, ... |
Answer
The remaining angles and sides are $$\angle A=56^\circ\hspace{.75cm}BC\approx307.382ft\hspace{.75cm}AB\approx361.146ft$$
Work Step by Step
$$\angle B=20^\circ50'\approx20.833^\circ\hspace{.75cm}\angle C=103^\circ10'\approx103.167^\circ\hspace{.75cm}AC=132ft$$ Here $AC$ is the side $b$, so $b=132ft$ 1) Analysis: ... |
Exercise \(\PageIndex{1}\): Set Operations
Let \(A = \{1, 5, 31, 56, 101\}\), \(B = \{22, 56, 5, 103, 87\}\), \(C = 41, 13, 7, 101, 48\}\), and \(D = \{1, 3, 5, 7...\}\)
Give the sets resulting from:
\(A \cap B\) \(C \cup A\) \(C \cap D\) \((A \cup B) \cup (C \cup D)\) Answer:
1. \(A \cap B =\{ 5, 56 \}\)
2. \(C \cup A... |
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$If you're seeking intuition about how a line bundle can be non-trivial, the simplest example is the Möbius strip viewed as the total space of a real line bundle over a circle.
In detail, view $\Reals^{2} \to \Reals$ as the trivi... |
In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an
iteration of $S$. Also $\times$ is an iteration of $+$ and $\exp$ is an iteration of $\times$ i.e.
$\forall m,n\geq 1$
$m+n:=\underbrace{S(S(... |
I know that different authors use different notation to represent programming language semantics. As a matter of fact Guy Steele addresses this problem in an interesting video.
I'd like to know if anyone knows whether the leading turnstile operator has a well recognized meaning. For example I don't understand the leadi... |
I think I know where the problem is, I just don't know why it's wrong:
We start with
$$\lim_{x\to \infty }\left(\frac{x}{x\ln (x)}\right).$$
Normally we would just cancel out the $x$'s in the fraction to get
$$\lim_{x\to \infty }\left(\frac{1}{\ln (x)}\right).$$
But, I read somewhere that
$$\lim_{x\to a}\left({f(x)g(x)... |
We have $N$ oscillators and each of them is described by the Hamiltonian:
$$H = \frac{p^2}{2m} + \frac{Kq^4}{4} $$
I have to compute the average total energy $\langle E\rangle$ of the $N$ oscillators. But to do so, first I have to compute the one particle partition function and to do so I have to solve the following in... |
Practice Paper 2 Question 20
Show that \(2^{50} < 3^{33}.\)
Related topics Warm-up Questions Compare \(2^{39}\) and \(3^{26}.\) Hint:Consider \(8\) and \(9\). Compare \(2^{100}\) and \(5^{50}.\) Expand \((3 – 2x)^5.\) Hints Hint 1How about using binomial expansion? Hint 2You need an appropriate choice of numbers in the... |
I'd appreciate help in understanding how changing the significance level effects the results of the t-test.
I have conducted an experiment where a group of 15 participants took a test, played a game, and took the original test again. The data set follows:
Round 1 (Before Game) Scores: 6, 4, 7, 8, 12, 6, 7, 5, 11, 4, 7,... |
I am trying to understand better the quantization of the Harmonic Oscillator.
Here are three ways of thinking about the Harmonic Oscillator.
Eigenfunctions of the differential operator: $H = -\frac{d^2}{dx^2} + x^2$ Eigenfunctions of the oscillator $H = a a^\dagger+ \frac{1}{2}$ Special orbits of the $U(1)$ action on t... |
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Now showing items 1-10 of 53
Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
Sample Test 4 Question 3
Let \(a>1\) be an integer. Give a non-integral expression in terms of \(a\) for \(F(a)=\displaystyle\int_1^a (-1)^{\lfloor x \rfloor} \lfloor x \rfloor^{-1} dx,\) where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x.\)
Related topics Hints Hint 1Sketch the graph \(y = \f... |
1. The domain of vector field \(\vecs{F}=\vecs{F}(x,y)\) is a set of points \((x,y)\) in a plane, and the range of \(\vecs F\) is a set of what in the plane?
Solution: Vectors
For the following exercises, determine whether the statement is
true or false. 2. Vector field \(\vecs{F}=⟨3x^2,1⟩\) is a gradient field for bot... |
Practice Paper 4 Question 4
The parabola \(y=x^2+\frac{1}{4}\) is eventually intersected at two points by the line \(y=tx,\) where \(t\) is the time increasing linearly from \(0\) (hence the line rotates from a horizontal towards a vertical position). Determine the speed at which the segment linking the two intersectio... |
A
learning algorithm is the backbone of machine learning that distinguishes it from traditional computer programming by allowing data-driven model building. In the past years, we have developed learning algorithms using a number and tools and for diverse application domains, as outlined below. Learning with Kernels
Ker... |
Peter F. and John A. simultaneously sent me a nice video on the 3Blue1Brown YouTube channel that wonderfully popularizes various mathematical curiosities and gets well-deserved hundreds of thousands of views for that:
The most unexpected answer to a counting puzzle (a 5-minute video)Imagine two boxes on a line, a big o... |
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Now showing items 1-1 of 1
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 ... |
Practice Paper 4 Question 5
A thin square piece of paper of side \(1\) is folded recursively. The first fold is along a diagonal of length \(\sqrt{2},\) thus obtaining an isosceles triangle. All successive folds are along a segment of length \(l_i,\) with \(i=1,2,\ldots,\) that results in a new triangle that is similar... |
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