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Volume 9, Number 5, 2019, Pages 1872-1883
DOI：10.11948/20180335
An integral boundary value problem of conformable integro-differential equations with a parameter
Chengbo Zhai,Yuqing Liu
Keywords:Positive solution, conformable derivative, integro-differential equations, fixed point theorem of generalized concave operato... |
Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems Abstract
We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We ... |
I always get the following warning:
LaTeX Font Warning: Font shape `OMS/cmsy/m/n' in size <11.5> not available
TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. It only takes a minute to sign up.Sign up to join this community
First of all,
cmsy d... |
In order to enable an iCal export link, your account needs to have an API key created. This key enables other applications to access data from within Indico even when you are neither using nor logged into the Indico system yourself with the link provided. Once created, you can manage your key at any time by going to 'M... |
Answer
$(5, 315^\circ)$ = $3.536 - 3.536i$
Work Step by Step
$(5, 315^\circ)$ is equivalent to $5cis315^\circ$ $5cis315^\circ$ = $5(cos315^\circ + isin315^\circ)$ = $5(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i)$ = $3.536 - 3.536i$
You can help us out by revising, improving and updating this answer.Update this answer
Aft... |
I decided to read Max Born's 1954 Physics Nobel Prize lecture,
We like to think about the Copenhagen school. Born makes it clear that Niels Bohr was a guru in his eyes. But he also prefers to talk about three schools, the Copenhagen school; the Göttingen school; and the Cambridge school. He was the boss of the Göttinge... |
Let $f$ be a real function so that for all $x,y\in \mathbb{R}$, $f(x+y) = f(x)+f(y)+xy(x+y)$ and $\lim _{x \to 0} f(x)/x = 1$ hold. Find $f$.
It does not like a casual question to ordinary university students, not even for maths students...but anyway one may notice that $xy(x+y) = (x+y)^3 - x^3 - y^3$ if you know symme... |
I know that the $y$-
section $A_x$ of a $\mu_x\otimes \mu_y$-measurable set $A$, where $\mu_x\otimes \mu_y$ is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ (both measures being $\sigma$-additive complete measures defined on $\sigma$-algebras of subsets of $X$ and $Y$, respectively)$^1$, defined by ... |
I've wondered this for a while but not known how to ask the question,If light is a transverse wave, then what is it transverse to?To elaborate, light travels in three-dimensions, radially. To me, this seems analogous to the sound wave, with pulses of pressure moving longitudinally to the...
Hi, it's been a long time si... |
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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 ... |
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Now showing items 1-10 of 24
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 ... |
Up until this point we have been integrating over one dimensional lines, two dimensional domains, and finding the volume of three dimensional objects. In this section we will be integrating over surfaces, or two dimensional shapes sitting in a three dimensional world. These integrals can be applied to real world situat... |
I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv \tilde{f}(\omega)$, in case it exists. I have read sometimes, and it's apparently very intuitive (not to me), that those limits are d... |
Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows
1.
Department of Mathematics, University of Santa Cruz, Santa Cruz, CA 95064, United States
2.
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, United States
$\frac{\partial \theta}{\partial t} + u\c... |
Which of the following has higher boiling points? Alkanes, alkenes, or alkynes? And why?
Disclaimer: All of this "jazz" will be about reaching a mere rule-of-thumb. You can't just compare whole families of organic compounds with each other. There are more factors to consider than below, mostly based on isomerism notion... |
Choice of $e$
It is mathematically OK to choose a huge $e$ (or even a negative one), as long as the only link between $e$ and the factors $p$, $q$ of $N$ are the mandatory $\gcd(e,p-1)=1$ and $\gcd(e,q-1)=1$, and nothing is made to chose $e$ so that $d$ has special properties. For example $e$ could be:
A random odd int... |
Not an answer and really just for fun. I understand that you did not ask for the codes. The main purpose of this is to substantiate the claim that it is not too difficult to draw these irregular shapes with elementary Ti
kZ syntax. Here are some reproductions of two of the figures of your list.
\documentclass[tikz,bord... |
Answer
Period$=\frac{1}{2}$ seconds. Frequency=2 cycles per second.
Work Step by Step
The period can be found by first comparing the equation $s(t)=-5\cos 4\pi t$ to $s(t)=a\cos w t$ to find $w$ which is the angular velocity. By comparing the two equations, we find that $w=4\pi$. Next, we find the period using the form... |
1. True or False? Line integral \(\displaystyle\int _C f(x,y)\,ds\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a,b]\) and if function \(f\) is continuous on some region that contains curve \(C\).
Solution: True
2. True or False? Vector functions \(\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\ma... |
This question already has an answer here:
Partitioning an infinite set 7 answers
Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
Mathematics Stack Exchange is a question and answer site for people studying mat... |
In the following exercises assume that there’s no damping.
[exer:6.1.1] An object stretches a spring 4 inches in equilibrium. Find and graph its displacement for \(t>0\) if it is initially displaced 36 inches above equilibrium and given a downward velocity of 2 ft/s.
[exer:6.1.2] An object stretches a string 1.2 inches... |
This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a closed form for $$\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2},$$ where $\psi(z)=\frac{\Gamma'(z)}{\Gamma... |
I can do a few
a) The velocity is positive on 0 < t < 1
This means the ball is rising during this time
b) The antiderivative is s (t) = 32t - 16t^2 + C
At s(0) we have 32(0) - 16(0)^2 + C = 0
So C = 0
c) s(1) - s(1/2) = [ 32(1) - 16(1)^2 ] - [ 32(1/2) - 16(1/2)^2 ] = [ 16] - [ 16 - 4] = 4
This represents the change in ... |
These formulas are for different matrix formats of the rectangular matrix $A$.
The matrix to be (pseudo-)inverted should have full rank. (added:) If $A\in I\!\!R^{m\times n}$ is a tall matrix, $m>n$, then this means $rank(A)=n$, that is, the columns have to be linearly independent, or $A$ as a linear map has to be inje... |
SageMath
Sage is a program for numerical and symbolic mathematical computation. It is meant to provide an alternative for commercial programs such as Maple, Matlab, and Mathematica. It is actually mantained in the community repository.
Contents Installation
To install binaries from community repository:
# pacman -S sag... |
In classical electromagnetism, the
electric potential (a scalar quantity denoted by Φ(Phi), Φ or E V and also called the electric field potential or the electrostatic potential) at a point of space is the amount of electric potential energy that a unitary point charge would have when located at that point. The electric... |
In this paper, the authors detail the methods to calculate the stream function $\psi$ from velocity data in the case of the global ocean (i.e., multiply connected domains).
The general idea is to use velocity to calculate the vorticity $\zeta$ and, with that, calculate the stream function by solving the Poisson equatio... |
As part of solving a problem I came across a summation which I'm having some difficulty simplifying. I'm not sure if it would even simplify into something nice.
$\sum\limits_{i=0}^{p-1} {ip^e \choose k}$
where $p$ is prime, $0\leq e\in\mathbb{Z}$, and $k\in\mathbb{Z}$, $0\leq k\leq p^e(p-1)$.
I'm aware that when $e=0$ ... |
A conservative functor $\mathsf{Top} \to \mathsf{Set}$ has been described in the comments. In a similar vein, for schemes we can take $X \mapsto |X| \amalg \mathcal{P}(\mathrm{Stalks}(X))$. This should also work for manifolds. Actually, for manifolds the much more formal $X \mapsto \coprod_{Y ~ \mathrm{conn}} \mathrm{H... |
Difference between revisions of "Talk:Absolute continuity"
(what is the problem?)
Line 32: Line 32:
:: OK, I found a way around. But there must be some bug: it seems that whenever I write the symbol "bigger" then things gets messed up (now even on THIS page). [[User:Camillo.delellis|Camillo]] 10:57, 10 August 2012 (CES... |
Simple question. When are we allowed to exchange limits and integrals? I'm talking about situations like$$\lim_{\varepsilon\to0^+} \int_{-\infty}^\infty dk f(k,\varepsilon) \overset{?}{=} \int_{-\infty}^\infty dk\lim_{\varepsilon\to0^+} f(k,\varepsilon).$$Everyone refers to either
dominated convergence theorem or monot... |
Sorry for the late response. Here's a proof of the open mapping theorem assuming the maximum modulus principle.
First, we need the "minimum modulus principle". That is, if $f$ is a non-constant analytic function on an open connected set $D \subset \mathbb{C}$, and $f$ has no zeroes in $D$, then $| f |$ cannot attain a ... |
LaTeX supports many worldwide languages by means of some special packages. In this article is explained how to import and use those packages to create documents in
German.
Contents
German language has some special characters. For this reason the preamble of your file must be modified accordingly to support these charac... |
How to Model Rotating Machinery in 3D
Electrical machines are an important pillar in modern industrial society. Among the different types of electrical machines, rotating machines such as generators and motors take up a central role. The
Rotating Machinery, Magnetic physics interface in COMSOL Multiphysics is designed ... |
I speak of Math Stackexchange frequently for two reasons: because it is fantastically interesting and because I waste inordinate amounts of time on it. But I would like to again share some of the more interesting things from the exchange here.
Firstly, in my last post on factoring, I spoke of Sophie Germain’s identity.... |
This blog post is an excerpt of my ebook
Modern R with the tidyverse that you can read forfree here. This is taken from Chapter 7, which dealswith statistical models. In the text below, I explain what hyper-parameters are, and as an exampleI run a ridge regression using the
{glmnet} package. The book is still being wri... |
Prove that$$\begin{array}\\&f(x)&=\begin{cases}\frac{x|y|}{\sqrt{x^2+y^2}}&\text{ if }(x,y) \ne (0,0)\\0&\text{ otherwise }\end{cases}\end{array}$$ is differentiable in $(0,0)$.
Proof.
We know that f is differentiable in (0,0), if all partial derivatives exist and are continuous in (0,0). We have $$\frac{\partial f}{\p... |
2003.
At that time, the particle was conjectured to be so heavily bound that it would be a tachyon, \(m^2\lt 0\). I actually think that composite tachyons can't exist in tachyon-free theories, can they? (You better believe that such a tachyonic particle is impossible because such a man-made Cosmos-eating tachyonic topl... |
abraces can be used for this - swapping/mixing ofthe brace directions:
\documentclass{article}
\usepackage{abraces,mathtools}% http://ctan.org/pkg/{abraces,mathtools}
\begin{document}
\[
\setbox9=\hbox{$(X \circ X)$}
\operatorname{min} \big|
\mathrlap{\hspace{.5\wd9}\mathclap{\aunderbrace[L1U1R]{\scriptstyle\phantom{\t... |
In this topic:
Dxxxx n+ n- model_name [TEMP=local_temp] .model modelname SRDIO ( parameters )
Name
Description
Units
Default
CJO Zero bias junction capacitance F 0.0 EG Energy gap ev 1.11 FC Forward bias depletion capacitance coefficient 0.5 IS Saturation current A 1E-15 MJ Grading coefficient 0.5 N Forward emission co... |
Consider Helly Theorem, taken from notes by Igor Pak:
Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then there is a point in all the sets, $X_1 \cap \dots \cap X_n \neq \varnothing$.
This result is not obvious (although Pak's pro... |
Given positive integers $a$, $m$ and $n$, let $s_{a(m)}(n)$ denote the sum of the reciprocals of the prime numbers less than or equal to $n$ which are congruent to $a$ modulo $m$.
Is there an integer $n$ such that $s_{1(3)}(n) > s_{2(3)}(n)$?
For small $n$, the function $s_{2(3)}$ is clearly ahead -- for example it exc... |
A random picture of intersecting D-branes
Alternatively, if that bump were real, it could have been a sign of compositeness, a heavy scalar (instead of a spin-one boson), or a triboson pretending to be a diboson. However, on Sunday, six string phenomenologists proposed a much more exciting explanation:
Why would such a... |
Revista Matemática Iberoamericana Rev. Mat. Iberoamericana Volume 27, Number 3 (2011), 733-750. Finiteness of endomorphism algebras of CM modular abelian varieties Abstract
Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2(\Gamma_1(N))^{\operatorname{new}}$. We prove that for any in... |
A search is reported for massive resonances decaying into a quark and a vector boson (W or Z), or two vector bosons (WW, WZ, or ZZ). The analysis is performed on an inclusive sample of multijet events corresponding to an integrated luminosity of 19.7 inverse femtobarns, collected in proton-proton collisions at a centre... |
Quick definition of the Laplacian matrix on a triangular mesh. I give an in depth explanation here.
What is commonly called the
Laplacian matrix \( \mathbf L \) in the literature of geometry processing is: $$ \mathbf L_{ij} = \left \{ \begin{matrix} w_{ij} = \frac{1}{2} \cot \alpha_{ij} + \cot \beta_{ij} & \text{ if j ... |
Bob "Hey Alice let's play paper-scissor-rocks :) I will always play paper."
Alice "What if you don't play paper?"
Bob "If I play don't play paper you win if it's a draw [under original rules] and you draw if you lose [under original rules]. If I win you have to obey my order."
Alice "What if I get a draw?"
Bob "Then yo... |
In the sequence 1,2,2,4,8,32,256...
each term (starting from the third term) is the product of the two terms before it. For example, the seventh term is 256 , which is the product of the fifth term (8) and the sixth term (32). This sequence can be continued forever, though the numbers very quickly grow enormous! (For e... |
Given: $L_1, L_2$ are regular languages. We define the language $L$.
$L = \{uw \mid \exists v \in \Sigma^*, uv \in L_1, vw \in L_2\}$.
I need to prove that $L$ is a regular language using only Closure Properties.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professi... |
Find all functions $f:\mathbb{Z} \to \mathbb{Z} $ such that$f(0)=1$ and $$f(f(n))=f(f(n+2)+2)=n. \quad \forall n \in \mathbb{Z} $$
My approach: Plugging in some values, it is not hard to see that $f(n)=1-n$ satisfies the given relation. I claim that $ f(k)=1-k $ for some $k \in \mathbb{Z}$.
I just cannot see a way to u... |
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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 m... |
Here are some notes for my talk
Finding Congruent Numbers, Arithmetic Progressions of Squares, and Triangles (an invitation to analytic number theory), which I’m giving on Tuesday 26 February at Macalester College.
The slides for my talk are available here.
The overarching idea of the talk is to explore the deep relati... |
To amplify Yehuda Lindell's answer even within a single family of groups, here is an example of a 2046-bit modulus $p$ for which discrete logs in $(\mathbb Z/p\mathbb Z)^\times$, as in standard modular multiplication Diffie–Hellman, are extraordinarily easy to compute:
0x2465a7bd85011e1c9e0527929fff268c82ef7efa416863ba... |
I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} n a_n = 0$.
Using the definition of the limit, this is equivalent to showing
\begin{equation} \forall \varepsilon >... |
Refine Keywords
We present an algebraic multigrid preconditioner which uses only the graphs of system matrices. Some elementary coarsening rules are stated, from which an advancing front algorithm for the selection of coarse grid nodes is derived. This technique can be applied to linear Lagrange-type finite element dis... |
Brazilian Journal of Probability and Statistics Braz. J. Probab. Stat. Volume 30, Number 1 (2016), 127-144. Assigning probabilities to hypotheses in the context of a binomial distribution Abstract
Given is the outcome $s$ of $S\sim{\mathrm{B}}(n,p)$ ($n$ known, $p$ fully unknown) and two numbers $0<a\leq b<1$. Required... |
In my last post, I mentioned I would post my article proper on WordPress. Someone then told me about latex2wp, a python script that will translate a tex file into something postable on WordPress. So I did it, and it works pretty well! Other than changing references (removing them) and a few stylistic things here and th... |
The original article rightfully neglects the cost of DES computations (there are less than $2^{90}$) and everything except memory accesses to its
Table 1 and Table 2. I go one step further: considering that Table 1 is initialized only once and then read-only, it could be in ROM, and I neglect all except the accesses to... |
I think I understand the mechanism of water vapor feedback in climate change pretty well: a (theoretical) temperature increase due to some isolated factor (e.g. increase in $CO_2$ concentration) causes a shift in water vapor saturation density and an additional evaporation rate. The consequential shift in actual water ... |
Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$. The affine variety $A_K$ defined by $$ N_{K/{\mathbb Q}}(X_1 \omega_1 + \ldots + X_n \omega_n) = 1 $$ is an algebraic group, the group structure coming from multiplication of units with norm $1$; in fact, $A_K$ is a norm-1 torus. For pure cub... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed i... |
I gave an introduction to sage tutorial at the University of Warwick Computational Group seminar today, 2 November 2017. Below is a conversion of the sage/jupyter notebook I based the rest of the tutorial on. I said many things which are not included in the notebook, and during the seminar we added a few additional exa... |
At a recent colloquium at the University of Warwick, the fact that
\begin{equation}\label{question} \sum_ {n \geq 1} \frac{\varphi(n)}{2^n – 1} = 2. \end{equation} Although this was mentioned in passing, John Cremona asked — How do you prove that?
It almost fails a heuristic check, as one can quickly check that
\begin{... |
257 23 Homework Statement Two identical uniform triangular metal played held together by light rods. Caluclate the x coordinate of centre of mass of the two mass object. Give that mass per unit area of plate is 1.4g/cm square and total mass = 25.2g Homework Equations -
Not sure what I went wrong here, anyone can help m... |
Practice Paper 2 Question 1
Sketch the function \(\displaystyle f(x)=\min_{t\le x} t^2\) for all real \(x\).
Hints Hint 1What is the minimum value of \(t^2\) when \(t\) varies between \(-\infty\) and \(-3?\) Hint 2How about between \(-\infty\) and \(-1?\) Hint 3How about between \(-\infty\) and \(+1?\) Solution
We are ... |
I’m in San Diego, and it’s charming here. (It’s certainly much nicer outside than the feet of snow in Boston. I’ve apparently brought some British rain with me, though).
Today I give a talk on counting lattice points on one-sheeted hyperboloids. These are the shapes described by
$$ X_1^2 + \cdots + X_{d-1}^2 = X_d^2 + ... |
Fifteen hep-ph papers co-written by Dr *ang today
By December 21st, 43 hep-ph papers on the diphoton resonance seen at the LHC have been written (and released in three packages). Eight days later, the terrain is very different. After another package on Dec 22, the total number has jumped to 72 (like the number of virgi... |
Many authors pull the definitions of the raising and lowering (or ladder) operators out of their butt with no attempt at motivation. This is pointed out nicely in [1] by Eli along with one justification based on factoring the Hamiltonian.
In [2] is a small exception to the usual presentation. In that text, these operat... |
Euclid's Lemma for Prime Divisors/General Result Contents Lemma
Let $p$ be a prime number.
Let $\displaystyle n = \prod_{i \mathop = 1}^r a_i$.
That is: $p \divides a_1 a_2 \ldots a_n \implies p \divides a_1 \lor p \divides a_2 \lor \cdots \lor p \divides a_n$
As for Euclid's Lemma for Prime Divisors, this can be verif... |
Practice Paper 2 Question 3
Find positive integers \(a,b,c,d\) such that
\(\displaystyle a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d}}}=\frac{15}{11}.\)
Hints Hint 1Given \(\frac{1}{x}<1\) when \(x>1,\) how does \(a+\frac{1}{x}\) relate to \(\frac{15}{11}?\) Hint 2With that in mind, what can you say about \(b+\frac{1}{y}?\) ... |
The
hyperbolic functions are a set of functions that have many applications to mathematics, physics, and engineering. Among many other applications, they are used to describe the formation of satellite rings around planets, to describe the shape of a rope hanging from two points, and have application to the theory of s... |
Practice Paper 2 Question 4
Three planar regions \(A\), \(B\), \(C\) partially overlap each other, with \(|A| = 90,\) \(|B| = 90,\) \(|C| = 60\) and \(|A \cup B \cup C| = 100,\) where \(|\cdot|\) denotes the area. Find the minimum possible \(|A \cap B \cap C|\).
Related topics Hints Hint 1Try to find the minimum \(|A\c... |
Basic Reflection is the return by a surface of some of the light which falls on that surface. (Other parts get absorbed or refracted).
There are two main laws (called "the laws of reflection") that play a role in this:
The angle of reflection of a ray ( r) of light is equal to the angle of the ray of incidence ( i). Th... |
Practice Paper 2 Question 7
A ternary tree is a collection of nodes, each having 0, 1, 2 or 3 children, where one node is not the child of any node and each of the other nodes is a child of exactly one other node. What are the maximum and minimum possible depths of a ternary tree containing \(n\) nodes?
Related topics ... |
I am working on a multiple part question for an introductory Real Analysis course. I have part of it done, but I have some problems.
Let $0 < y_1 < x_1$, and set $$x_{n+1}=\frac{x_n +y_n}{2}, y_{n+1}=\sqrt{x_n y_n}$$ (a) Prove that $0<y_n <x_n$ for all $n \in \mathbb{N}$
For part (a) I believe I have proven $y_n < x_n$... |
Practice Paper 2 Question 8
A polynomial \(f(x) = x^n + a_{n-1}x^{n-1} \cdots + a_1x + a_0,\) with integer coefficients \(a_i,\) has roots at \(1, 2, 4, \ldots, 2^{n-1}.\) What possible values can \(f(0)\) take?
Related topics Hints Hint 1How else can you write a polynomial when you know all of its roots? Hint 2... spe... |
Back Propagation: A Mathematical and Intuitive Explanation Author: Oluwole Oyetoke (18th August, 2017) 𐐒AƆK PROPAGATION
Often times we hear about the backpropagation technique used during the training process of ANNs. At the snap of a finger, we can utilize backpropagation for Neural Nets Training on tools such as Tes... |
While studying about Hückel theory, I got accustomed to the approximation of making the overlap matrix an identity matrix; that is making the off-diagonal elements zero as $S_{AB}= S_{BA}= 0\;;$ this implies the use of orthogonal base states of AOs. The off-diagonal elements of the Hamiltonian matrix are still taken as... |
Practice Paper 2 Question 9
Using a fair coin we can generate random integers in \(\{1,2,3,4\}\) with equal probability by doing:
\((a)\) toss coin, if heads go to \((b)\) otherwise go to \((c)\) \((b)\) toss coin, if heads output \(1\) otherwise output \(2\) \((c)\) toss coin, if heads output \(3\) otherwise output \(... |
Piezoelectric Materials: Understanding the Standards
Standards form an integral part of the work we do as engineers, providing a common language for communicating complex information. But standards committees are not omnipotent and sometimes revised standards are not universally adopted. This has happened in the case o... |
The final subject we shall consider is the convergence of Fourier series. I shall show two examples, closely linked, but with radically different behaviour.
A square wave,
\(f(x)= 1\) for \(-\pi < x < 0\); \(f(x)= -1\) for \(0 < x < \pi\).
a triangular wave,
\(g(x)= \pi/2+x\) for \(-\pi < x < 0\); \(g(x)= \pi/2-x\) for... |
Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example could be Hilbert's basis theorem. Another could be Dwork's p-adic technique for rationality of zeta functions from alge... |
I'll give you the numbers. I'm breaking this up into 3 different terms. There's atmospheric drag, what I'll call the "hover" term, and the gravitational potential climb. I will more or less assume a flight directly up. You're welcome to use whatever term for velocity you want, as none of them will be representative. I'... |
In Chapter 12 we’ll study partial differential equations that arise in problems of heat conduction, wave propagation, and potential theory. The purpose of this chapter is to develop tools required to solve these equations. In this section we consider the following problems, where \(\lambda\) is a real number and \(L>0\... |
Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic?
Here's my attempt to solve this:
Let's assume that it is periodic. For a function to be periodic, it must satisfy $f(x)=f(T+x)$ for all $x\in\Bbb R$, so it must satisfy the relation for $x=0$ as well. So we get th... |
A description of what I'm attempting:
Given a generic vector $\vec{r}$ and a point in space $D$, I want to build a reference frame centred in $\vec{r}= (x,y,z)$, with one axis directed along the direction from $\vec{r}$ to $D$. Calling this last vector $\vec{L} = \vec{D} - \vec{r}$, the direction of reference is simply... |
I’ve had a number of requests for hints and/or solutions to several of the homework problems for tomorrow. So here we go.
17.5 #30 This question asks us to compute the gravitational flux through a cylinder \(S\) from the gravitational field
\[ \mathbf{F} = – G m \frac{\mathbf{e}_r}{r^2}. \] Here, \(S\) is the cylinder ... |
Sampling from Phase Space Distributions in 3D Charged Particle Beams
In the previous installment of this series, we explained two concepts needed to model the release and propagation of real-world charged particle beams. We first introduced probability distribution functions in a purely mathematical sense and then disc... |
I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on this question is to use the fundamental theorem of calculus or Jensen's inequality.
Assuming $f$ is integrable and twice... |
Show that $F(\alpha)=F(\beta)$ if $\alpha$ and $\beta$ are roots of the same irreducible polynomial $g$ over field $F$
I defined a map $\psi: F(\alpha) \to F(\beta)$ by sending $p(\alpha)$ to $p(\beta)$. This map is clearly onto. If $p_1(\alpha)=p_2(\alpha)$, then by minimality of the polynomial $g$, we must have that ... |
Plugging these values in, we find
$$2t\sum_{n=0}^{\infty}n(n-1)a_nt^{n-2}+(1-2t)\sum_{n=0}^{\infty}na_nt^{n-1}-\sum_{n=0}^{\infty}a_nt^n = 0$$
The next step is to sort things so we have matching powers of $t$. For some particular $k$, what's the coefficient of $t^k$?
To do this, we first bring distribute those polynomi... |
Problem # 8
projectile motion problems figure 2
A ball is kicked at an angle θ = 45°. It is intended that the ball lands in the back of a moving truck which has a trunk of length L = 2.5 m. If the initial horizontal distance from the back of the truck to the ball, at the instant of the kick, is do = 5 m, and the truck ... |
I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a \in \mathbb{R} \ $,
$$\mathbb{P}(\lim_{t \to \infty} \sup_{s\in [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the re... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
Answer
False. Change to: For $x=\displaystyle \frac{1}{3},\quad\frac{3x+7}{3x+10}=\frac{8}{11}$
Work Step by Step
If this were true, it would be true for any x, provided it doesn't yield 0 in the denominator. Try x=0: $LHS=\displaystyle \frac{0+7}{0+10}=\frac{7}{10}\neq\frac{8}{11}$ so, the statement is false. It is tr... |
In Exercises 1-13, find all \((x_0,y_0)\) for which Theorem [thmtype:2.3.1} implies that the initial value problem \(y'=f(x,y),\ y(x_0)=y_0\) has (a) a solution and (b) a unique solution on some open interval that contains \(x_0\).
[exer:2.3.1] \( {y'={x^2+y^2 \over \sin x}}\) &
[exer:2.3.2] \( {y'={e^x+y \over x^2+y^2... |
Based on this question: What is the homology groups of the torus with a sphere inside?
I'm trying to find the fundamental group of this space using the Seifert–van Kampen theorem. If $U$ is the torus and $V$ is the sphere, then $U\cap V$ is the circle, thus we have the following fundamental groups:
$\pi_1(U)=\mathbb Z\... |
Difference between revisions of "Signed measure"
m (moved subhead above MSC)
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$\newcommand{\abs}[1]{\left|#1\right|}$
$\newcommand{\abs}[1]{\left|#1\right|}$
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An real-valued $\sigma$-additive function defined on a certain
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An real-valued $\sigma$-additive function defined on a certain -algebra$\mathc... |
I sympathize with the text to some extent and the extent could have approached 100% some 25 years ago. In particular, decoherence is a legitimate insight and quantum mechanics isn't weird when you look at it calmly. But there are lots of claims that Ball makes that I heavily disagree with, too.
OK, what is decoherence?... |
I recently wrote an answer to a question on MSE about estimating the number of squarefree integers up to $X$. Although the result is known and not too hard, I very much like the proof and my approach. So I write it down here.
First, let’s see if we can understand why this “should” be true from an analytic perspective.
... |
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