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Apart from the ones you mentioned, another application of (a modified) Grover's algorithm which I'm aware of is solving the Collision problem in complexity theory, quantum computing and computational mathematics. It's also called the BHT algorithm.
Introduction:
The collision problem most often refers to the 2-to-1
ver... |
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Free keywords: General Relativity and Quantum Cosmology, gr-qc,Astrophysics, Cosmology and Extragalactic Astrophysics, astro-ph.CO, Astrophysics, High Energy Astrophysical Phenomena, astro-ph.HE, Astrophysics, Instrumentation and Methods for Astrophysics, astro-ph.IM
Abstract: We employ gravitational-wave radiomet... |
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January 2007 , Volume 17 , Issue 1
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We study the first positive Neumann eigenvalue $\mu_1$ of the Laplace operator on a planar domain $\Omega$. We are particularly interested in how the... |
Abbreviation:
CRng$_1$
A
is a rings with identity $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ such that $\cdot$ is commutative: $x\cdot y=y\cdot x$ commutative ring with identity
Let $\mathbf{R}$ and $\mathbf{S}$ be commutative rings with identity. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\rightar... |
Cold dark matter is thought to fill our galactic neighborhood with a density $\rho$ of about 0.3 GeV/cm${}^3$ and with a velocity $v$ of roughly 200 to 300 km/s. (The velocity dispersion is much debated.) For a given dark matter mass $m$ and nucleon scattering cross section $\sigma$, this will lead to a constant collis... |
Up till the quotient $\frac{\rho^2 \cos^2 \theta\sin^2 \theta}{\cos^2 \theta + \rho^2\sin^4 \theta}$ you are fine.
Now when $\rho \to 0$ the above function is dependent on $\theta$.
All right, but the dependency is not good enough to prevent the limit being zero!
there is a change of a $0/0$ at $k\pi - \frac{\pi}{2}$.
... |
I have posted a previous question, this is related but I think it is better to start another thread. This time, I am wondering how to generate uniformly distributed points inside the 3-d unit sphere and how to check the distribution visually and statistically too? I don't see the strategies posted there directly transf... |
Faddeeva Package From AbInitio
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Line 27: Line 27: :<math>\mathrm{erfi}(z) = -i\mathrm{erf}(iz) = -i[e^{z^2} w(z) - 1]</math>; for '''... |
Note
This is a review of equilibrium statistical mechanics. Though I called it a review, it is more like a list of keywords at this moment.
For a system with \(N\) particles of \(r\) degrees of freedom, we could always describe the microstates of the system by looking at the state of each particle. There are at least t... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
A neutron is a neutral particle which is merely some times more massive than an electron. What makes it so unstable outside the nucleus that it has a half life only of about 12 min?
How long is long?
So "half life only of about 12 min" is actually really a strange idea to most of your readers. 12 minutes is a very long... |
I have two questions that i didn't find in books. When calculate time rates of change of the expectation values of $\langle x \rangle$ or $\langle p_x \rangle$, why is x or $p_x$ not derived from time?
Now i want to show this, if $\Psi(r, t) $ is a square integrable wave function normalised to unity,then:
$ \frac{d}{dt... |
An algebra is
if each congruence relation of the algebra isdetermined by any one of its congruence classes, i.e. $\forall a,b\ [a]_{\theta}=[b]_{\psi}\Longrightarrow\theta =\psi$. congruence regular
A class of algebras is
if each of its members is congruence regular. congruence regular
Congruence regularity holds for m... |
Abbreviation:
MQgrp
A
is a quasigroup $\mathbf{A}=\langle A,\cdot,\backslash,/\rangle$ such that medial quasigroup
$\cdot$ is
: $(xy)(zw)=(xz)(yw)$ medial
Remark: This is a template. If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.
It is not unusua... |
With this problem the salient feature is that the first guess which is
$$\sum_{q=m}^n {n\choose q} (-1)^{m-q} (n-q)^k$$
does not produce the correct answer. The underlying poset has nodesfor each subset $P$ of the set $Q$ of $n$ cells where the noderepresents cells from $P$ plus possibly additional cells being empty,or... |
There is no necessary relation between the implementation of the compiler and the output of the compiler. You could write a compiler in a language like Python or Ruby, whose most common implementations are very slow, and that compiler could output highly optimized machine code capable of outperforming C. The compiler i... |
2. Series 29. Year Post deadline: - Upload deadline: -
(2 points)1. rat on ice
A rat is running on ice with speed $v$. Suddenly he decides to turn 90$°$ so that he keeps running with the same speed in the new direction. What is the least amount of time he needs for such a turn? Suppose that rat's feet can move independ... |
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August 2015 , Volume 35 , Issue 8
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Two-point boundary value problems of Dirichlet type are investigated for a Ermakov-Painlevé II equation which arises out of a reduction of a three-ion... |
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(2 points)1. Kofola's
Let's have a Kofola (a Czech soft drink) with the energetic value $Q_{k}=1360kJ⁄\;\mathrm{kg}$ and temperature $t_{k}=24\;\mathrm{°C}$ and another Kofola, this time sugar-free, with the energetic value $Q_{free}=14.4kJ⁄\;\mathrm{kg}$ and tempe... |
I've been studying for my exam and came across the following problem:
Suppose that $X_1,\ldots,X_n$ is a random sample from a Poisson distribution with mean $\lambda$.(a) Find the maximum likelihood estimator (MLE) of $\eta = P(X_1 = 2\mid \lambda)$. (b) Find the UMVUE of $\eta$.
I am not sure about my solution, so I w... |
We’ve derived the 1D heat diffusion equation, so now let’s take a look at one of the solutions. This solution is for a symmetric rod. Simply, imagine a rod of some length $L$. This rod is insulated across its length so that no heat is lost to the atmosphere across that length. The only place where the rod is not insula... |
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(2 points)1. let it flow
Thin wire with resistance $R=100mΩ$ and length $l=1\;\mathrm{m}$, that is connected to the source of DC with voltage $U=3V$, contains in its volume $N=10^{22}$ free electrons, which contribute to the electric current. Determine what is the ... |
Abbreviation:
MouQgrp
A
is a quasigroup $\mathbf{A}=\langle A,\cdot,\backslash,/\rangle$ such that Moufang quasigroup
$\cdot$ satisfies the Moufang law: $ye=y\Longrightarrow ((xy)z)x = x(y((ez)x))$
Remark: This is a template. If you know something about this class, click on the 'Edit text of this page' link at the bott... |
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(2 points)1. It's about what's inside of us
In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter ver... |
I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim):
If $X_n$ is
ANY sequence of random variables, there are constants $c_n \to \infty$ so that$$\frac{X_n}{c_n} \to 0 \quad \mbox{a.s}$$
You will find ... |
First, terminologically, "axiom" and "inference rule" are often used as roughly interchangeable as they tend to serve similar purposes. There are technical distinctions, which themselves can vary slightly, but outside the study of formal logic or related systems, these distinctions aren't that important.
In the context... |
(redirected from H1s2sResearch.RydbergProject)
News: Shrinking the proton again!
Our most recent results from laser spectroscopy of the 2S-4P transition in atomic hydrogen will be published in
Science in the October, 6th issue. After more than six years of work, we have succeeded in measuring the transition frequency w... |
Except for the undecidable unaries I have no idea if there is anything in the gap between $P/poly$ and $P$
Take a language $L$ which is not in $\mathsf{E} = \bigcup_{c=1}^\infty \mathsf{TIME}(2^{cn})$. Now consider the language $L' = \{1^m : m \in L\}$. Then $L'$ is clearly in $\mathsf{P/poly}$, but it's not in $\maths... |
How to construct a dense subset say $A$, of the real numbers other than rationals? By dense I mean that there should be an element of $A$ between any two real numbers.
First note that you can "cheat" by taking any subset and take a union with the rationals.
Second note that you can always cheat by taking some real numb... |
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August 2009 , Volume 24 , Issue 3
A special issue
Dedicated to Peter W. Bates on the occasion of his 60th birthday
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This special issue of
Discrete and Continuous Dynamical Systems-Ais d... |
That is the right paper, but there are actually several equivalent embeddings.
The book
Basic Proof Theory by Troelstra and Schwichtenberg gives two such embeddings. Here's one. If $P$ is atomic but not $\bot$:
$$P^\circ := P$$$$\bot^\circ := \bot$$$$(A \wedge B)^\circ := A^\circ \wedge B^\circ$$$$(A \vee B)^\circ := \... |
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Okay, now we've got all the machinery set up to study co-design diagrams with feedback. Today let's consider a very simple one.
I'll start without feedback. I seem to like examples from business and economics for these purposes:
Thi... |
I can get the proper answer, but I don't quite know why.
I am supposed to find $dy/dt$ for the function $y = \sqrt{2x +1}$ if $dx/dt = 3$ when $x=4$.
For the derivative I get $$ \frac {dy}{dt} = \frac {1}{2} (2x + 1)^{-1/2} \frac{dx}{dt},$$ which then gives me $$ \frac {dy}{dt} = \frac {1}{2} (9)^{-1/2} \cdot 3 \frac {... |
Abbreviation:
AAlg
$\cdot$ is associative: $(xy)z=x(yz)$
Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.
It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredu... |
Please is this prof is correct ? Let $\{\Omega_i\}_{i\in I}$ a famille of connected sets such that $$\forall i,j\in I, \Omega_i\cap\Omega_j\neq\emptyset$$ I want to prove that $\bigcup_{i\in I} \Omega_i$ is connected.
If I suppose that $\bigcup \Omega_i$ is not connected then, there exists two non empty open sets $A,B$... |
When we take a dot product between two vectors of a vector space, we actually "act" by a 1-form (dual vector) on a vector. So why most books define the dot product between vectors? Of course with the help of a metric we can take a dot product between two vectors, but technically the metric converts one of the two vecto... |
Abbreviation:
CPoSgrp
A
is a partially ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ such that commutative partially ordered semigroup
$\cdot$ is
: $xy=yx$ commutative
Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out ... |
I bought the eighth edition of Stewart Calculus (metric version) and I'm up to the section about limits. It's been pretty easy so far, but I've come across a class of limit problems that don't seem solvable with mere algebra. The following is fairly representative of them:
$$\lim_{t\to 0} \frac {\sqrt{1+t}-\sqrt{1-t}}t... |
Positive Integer Greater than 1 has Prime Divisor/Proof 2 Lemma Proof
Let $S = \set {n \in \Z: n > 1: \neg \exists p \in \Bbb P: p \divides n}$.
That is:
$S = \set {\text {all integers not divisible by a prime} }$ Let $n \in S$ be the smallest of these.
As $S$ is bounded below by $1$, this is bound to exist, by Set of ... |
Answer
$\lim_\limits{x\to 0}\frac{\sin 3x}{x}=3$
Work Step by Step
To find the limit using the table, we must find the value that $f(x)$ approaches, as $x$ approaches a given value. Here, we are asked to find the limit as $x\to0$. As $x$ approaches $0$ from both sides, it is clear that the values of $f(x)$ are approach... |
I'm new to Matrix Calculus. Recently I've been working on that and have a question. Please see the following:
$J=J(\mathbf{z})$
$\mathbf{z}=\mathbf{W}\mathbf{a}$
Where $J: R^m \rightarrow R$, $\mathbf{z}$ is $m\times1$ vector, $\mathbf{a}$ is $n\times1$ vector and $\mathbf{W}$ is $m\times n$ matrix.
I want to calculate... |
I encountered a problem as folows:
Show a $3\times 3$ real matrx $A$, such that
$$A^4=\left(\begin{array}{ccc}3&0&0\\0&3&1\\0&0&0\end{array}\right)$$
well, this problem is not difficult, one can first find $B=\left(\begin{array}{ccc}\sqrt3&0&0\\0&\sqrt3&x\\0&0&0\end{array}\right)$ such that $B^2=\left(\begin{array}{ccc... |
[I'm sorry, I've already posted the same question in the physics community, but I haven't received an answer yet.]
I'm approaching the study of Bell's inequalities and I understood the reasoning under the Bell theorem (ON THE EINSTEIN PODOLSKY ROSEN PARADOX (PDF)) and how the postulate of locality was assumed at the st... |
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Last time we took a peek at a 'co-design diagram':
This describes a big complicated feasibility relation built from smaller ones - the little boxes - in various ways. Now let me start explaining those various ways!
First of all, rem... |
This question is experimentally accessible, despite the feebleness of the weak interaction, because the strong and electromagnetic interactions are symmetric under parity transformations and the weak interaction is not.
The contribution to the binding energy is small enough that it's not a good way to think of things. ... |
From Gauss' law, one can easily derive that the electric field at some distance from an infinite sheet of charge density $\sigma$ is $E=\frac{\sigma}{2\epsilon _0}$. Now when one considers a conductor instead, because there is no electric field within the body of a conductor, the electric field somewhere above the surf... |
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stat
Change to browse by: References & Citations Bookmark(what is this?) Computer Science > Machine Learning Title: Finite Precision Stochastic Optimization -- Accounting for the Bias
(Submitted on 22 Aug 2019 (v1), last revised 26 Aug 2019 (this version, v2))
Abstract: We consider first order s... |
An example of methylation analysis with simulated datasets
Part 2: Potential DMPs from the methylation signal
Methylation analysis with Methyl-IT is illustrated on simulated datasets of methylated and unmethylated read counts with relatively high average of methylation levels: 0.15 and 0.286 for control and treatment g... |
$f:[0,1]\times [0,1]\to\mathbb R,$ defined by $$f(x,y)= \begin{cases}1,\quad \ \ y\in\mathbb R\text{\\}\mathbb Q\\2x,\quad\text{otherwise}\end{cases}$$.
$1.1$: $\int_0^1f(x,y)dx$ exists for every $y\in[0,1]$ and is equal to $1$.
$1.2$: The iterated integral $\int_0^1(\int_0^1f(x,y)dx)dy$ exists and is $1$.
$1.3$: The d... |
I'm working on the book Quantum Effects in Biology by Mohesni et all. My question is however not biology related, it is about a section on quantum master equations in the weak system-bath coupling limit. To set up the problem, we consider the following:
We have a bath of harmonic oscillators $$H_B = \sum_\xi\hbar\omega... |
Say I simulate from a normal distribution $N(\mu, \sigma^2)$ 10.000 times using a couple of different methods.
I can calculate the standard error as $s/\sqrt{10000}$ where $s$ is the sample standard deviation.
What I don't understand is .... $s$ is the standard deviation of the sample. The sample are generated by simul... |
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1930-5346
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Advances in Mathematics of Communications
May 2015 , Volume 9 , Issue 2
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Abstract:
In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of ... |
Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ in $G$ and then you have to check that it is independent from this choice and that it is a homomorphism at all. I think t... |
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1547-5816
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Journal of Industrial & Management Optimization
April 2010 , Volume 6 , Issue 2
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Abstract:
The single machine semi-online scheduling problem with the objective of minimizing total completion time is investigated with the assumption that th... |
On the DNA Computer Binary Code
In any finite set we can define a
, a partial order in different ways. But here, a partial order is defined in the set of four DNA bases in such a manner that a Boolean lattice structure is obtained. A Boolean lattice is an algebraic structure that captures essential properties of both s... |
Application of integrals:
At this stage, you must be aware of every formula to calculate areas of simple geometrical figures which includes Triangles, Rectangles, trapeziums and circles. For the calculation of areas of such simple figures, we use formulae of elementary geometry. Why do we need to study application of i... |
I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-field-extensions, I didn't get enough information. I would really like to understand this example, so I will try to str... |
65 4 Homework Statement Two identical audio speakers, connected to the same amplifier, produce monochromatic sound waves with a frequency that can be varied between 300 and 600 Hz. The speed of the sound is 340 m/s. You find that, where you are standing, you hear minimum intensity sound a) Explain why you hear minimum-... |
Fall 2019
By the end of the course, you should know how to:
Monday: skim/read chapter; start DataCamp Tuesday: lecture (possible quiz) Wednesday: reread chapter; go through chapter code Thursday: ask questions in lab, practice coding Friday-Sunday: complete exercises, submit via Sakai
How do we
identify the effect of a... |
Suppose $X$ and $Y$ are two $n$-circulants (Cayley graphs for $\mathbb{Z}_n$) with adjacency matrices $A_X$ and $A_Y$. Since they are circulants, both $X$ and $Y$ lie in some symmetric association schemes (details below). Let $\mathcal{C}_X$ and $\mathcal{C}_Y$ be the
smallest association schemes containing $A_X$ and $... |
on page 34, it talks about the periods of a closed $2$-form. Consider the following path integral,
$$\int\mathcal{D}B\;\exp{\left\{-\frac{i}{8\pi}\int\left(\bar{\tau}^{\prime}F^{\prime +}\wedge\star F^{\prime +}-\tau^{\prime}F^{\prime -}\wedge\star F^{\prime -}\right)-\frac{1}{2\pi}\int F^{\prime}\wedge dB\right\}}$$
w... |
Consider a corner reflector with angle $\alpha$ between its semi-planes:
Let a plane wave come from the bottom into this reflector (possible at an angle). The objective is to find the total wave including this incoming one and all the reflections — so as to satisfy some boundary conditions on the reflecting surfaces, e... |
Absolute Value of Integer is not less than Divisors
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Theorem $\forall c \in \Z_{\ne 0}: a \divides c \implies a \le \size a \le \size c$
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.
Let $a \divides b$.
Then: $a \le b$ Proof
Suppose $a \divides c$ for some $c \ne 0$.
$a \le \s... |
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity
1.
Institute of Mathematical Sciences, Renmin University, Beijing 100872, China
2.
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\n... |
Abstract
The oxidative stability of the ramyon prepared with ricebran oil fortified with ${\alpha}-tocopherol$, BHA, TBHQ, and ascorbyl palmitate+citric acid or blended with palm oil was studied to assess the suitability of the oil as the frying oil. The antioxidants were added to a ricebran oil at 0.02% level, respect... |
With the following circuits as examples :
and
How will the current
I know how much to flow?Would any other wave travel first in the circuit and then come backand say so much current should flow?
Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals,... |
The key to the question is "Normal"
Today I went for a normal walk
A "normal" in geometry is not something "common" or "ordinary". A geometrical "normal" is always perpendicular to something else.
Let us say that this something is...
...a line that extends from a fixed point, to the walker.
If the walker then walks alo... |
2. Series 24. Year Post deadline: - Upload deadline: -
1. Warm-Up
Jakub's breakfast
Every morning Jakub enjoys his favourite cereals which he pours into a bowl of milk. Assume the bowl is of circulur frustum shape with upper and lower radius $R$ and $r$ respectively ($R$ \geq r$)$, that the cereals are little solid sph... |
Determine whether the series converges or diverges.
$$ \sum _{n=1}^{\infty }\:\left(\frac{19}{n!}\right) $$
I know that this question a lot easier if I use ratio test but I have not learned ratio test yet. The only option I have is divergence, comparison, limit comparison, and integral test. How can I prove that this s... |
I don't quite understand this set of conditions from Griffiths,
Introduction to Electrodynamics (equation 9.74):
$$\begin{align} \epsilon_1 E_1^{\bot} = \epsilon_2 E_2^{\bot},\quad &\mathbf{E}_1^{\parallel} = \mathbf{E}_2^{\parallel}\\ B_1^{\bot} = B_2^{\bot},\quad &\frac{1}{\mu_1}\mathbf{B}_1^{\parallel} = \frac{1}{\m... |
I'm sorry if I misunderstood your question, but I'll do my best.
There are many (and I mean MANY) packages to accommodate encoding of different "delicate" symbols like rare letters from little known languages. Vietnamese, Chinese, Slovenian (my language; you've probably never heard of it) etc. are all represented by pl... |
Abbreviation:
Gset
A
is a structure $\mathbf{A}=\langle A,f_g (g\in G)\rangle$, where $\langle G, \cdot, ^{-1}, 1\rangle$ is a group, such that G-set
$f_1$ is the identity map: $1x=x$ and
the group action associates: $(g\cdot h)x=g(hx)$
Remark: $f_g(x)=gx$ is a unary operation called
. the group action by $g$
If follow... |
There's a way to this in polynomial time. I'll sketch the algorithm (in reverse order ... do step 2 first and step 1 second).if we can find a set of $nk$ agent-task pairs $(i,j)$ such that each task is in exactly $k$ pairs, each agent is in exactly $k$ pairs, and no pair appears more than once, then we can find $k$ ass... |
There is a standard approach to these results, that dates back to Lévy and Solovay. The point is that, for any $\lambda$, given an embedding witnessing $\lambda$-supercompactness of $\kappa$, there is a canonical way of lifting the embedding to a $\lambda$-supercompact embedding in $V^Q$ (using simply that $Q$ is "smal... |
Given two free semicirculars X_1 and X_2 and a projection h in the von-Neumann algebra generated by X_1, how does one show that the von-Neumann algebra generated by {X_1, hX_2(1-h)} is a factor? It is easy to show that the two elements in the generating set are free. But I am unable to see what kind of an object hX_2(1... |
If you would like to submit an interactive comment (short comment, referee comment, editor comment, or author comment) for immediate non-peer-reviewed publication in the interactive discussion of a paper recently published in CPD, please locate this paper on the CPD papers in open discussion web page and follow the app... |
When we say "pick a random integer", the integers are in the range $[1, N]$.
Problem:
Consider the following instructions:
Pick a random integer.
Find the greatest common divisor of all the integers which have been picked so far.
If the greatest common divisor is not 1, go back to the first step to pick a random intege... |
I have a problem at the intersection of a range of topics: exponential programming, semi-definite programming and computer science, that I am having trouble finding a decent method for solving.
Take $A_i\in\mathbb{R}^{d\times d}$ with $A_i = A^T_i$, $i\in\mathcal{I}$ and $\mathcal{I}$ is a finite set of indices. $-\inf... |
Faddeeva Package From AbInitio
Revision as of 05:07, 31 October 2012 (edit)
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Line 27: Line 27: :<math>\mathrm{erfi}(z) = -i\mathrm{erf}(iz) = -i[e^{z^... |
There is an answer to a similar question here discussing the relationship between stationarity and the recursive equation for a model. Some additional remarks are pertinent to your case.
Firstly, it is important to understand that the recursive equation defining the ARMA($p,q$) process is not sufficient to fully specif... |
Abbreviation:
EqRel
An
is a structure $\mathbf{X}=\langle X,\equiv\rangle$ such that $\equiv$ is a equivalence relation (i.e. $\equiv\ \subseteq X\times X$) that is binary relation on $X$
reflexive: $x\equiv x$
symmetric: $x\equiv y\Longrightarrow y\equiv x$
transitive: $x\equiv y\text{ and }y\equiv z\Longrightarrow x\... |
This question relates to a specific paper by Eric Budish, published in 2011 in JPE, but I've tried to put all relevant information in this question. On page 1072, he defines
budget constraint hyperplanes as follows:
Let $H(i,x) = \{\mathbf{p}:\mathbf{p} \cdot x = b_i \}$ denote the hyperplane in $M$-dimensional price s... |
Given the Schwarzschild metric with $(-,+,+,+)$ signature,
$$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$
the lack of dependence of the metric on $t$ and $\phi$ allow us to read off the Killing vectors $K_1=\partial_t$ and $K_2=\partial_{\phi}$.... |
The textbook Elements of Information Theory gives us an example:
For example, if we knew the true distribution p of the random
variable, we could construct a code with average description length
H(p). If, instead, we used the code for a distribution q, we would
need H(p) + D(p||q) bits on the average to describe the ra... |
This is best understood in the framework of commutative rings.
There's a functor $$F:\mathbf{CRing} \leftarrow \mathbf{CRing}$$ given as follows: $$F(R) = R[i]/(i^2 +1).$$ We can think of $F$ as adjoining an element $i$ with the property that $i^2+1=0$. Informally, $i = \sqrt{-1}$.
There's also functor $$G:\mathbf{CRin... |
Prime Number Theorem/Historical Note Historical Note on Prime Number Theorem
He also posited the suggestion that it could be approximated by the Eulerian logarithmic integral $\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$.
It took another century before a proof was found. $\dfrac 7 8 < \dfrac {\map \pi... |
How can I evaluate the marginal cumulative distribution function of a set of random variables for which I do not have the CDF in closed form. I can, however, simulate from a joint distribution involving this set of variables.
To be more specific, assume I want to evaluate the CDF of $(X_1,X_2)$ but I only have a way to... |
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∫0∞ln(x)1−x2dx\int _{ 0 }^{ \infty }{ \frac { \ln { ( } x) }{ 1-x^{ 2 } } dx } ∫0∞1−x2ln(x)dx
The above integral is equal to −aπbc\dfrac{-a\pi^b}{c}c−aπb for positive integers a,ba,ba,b and ccc with aaa and ccc being coprime. Evaluate a+b+ca+b+ca+b+c.
Note: You a... |
Sunrise and Sunset calculator calculates the exact time of sunset and sunrise of a given day of Gregorian calendar. It is is an online calender tool to calculate sunset and sunrise at your location or any part of the world for any present, future or past date.It is necessary to follow the next steps:
Enter latitude, lo... |
For a shorter proof, here are a few things we need to know before we start:$X_1, X_2 , ..., X_n$ are independent observations from a population with mean $\mu$ and variance $\sigma^{2}$$\mathbb E(X_i) = \mu$ , $\mathbb{Var}(X_i)= \sigma^{2}$$\mathbb E(X^2) = \sigma^{2} + \mu^{2}$$\mathbb{Var}(X)=\mathbb E(X^2)-\mathbb ... |
Among the many equivalent formulations of Leopoldt's conjecture, this one is probably the shortest: For any number field $K$, prime number $p$, finite set $S$ of primes of $K$ containing the primes above $p$, one has
Leopoldt's conjecture: $H^2(G_{K,S},\mathbb{Q}_p)=0$.
Here $G_{K,S}$ is as usual the Galois group of th... |
A "perfectly efficient" computer can mean many things, but, for the purposes of this answer, let's take it to mean a
reversible computer (explained further as we go).
The theoretical lower limit to energy needs in computing is the Landauer Limit, which states that the
forgetting of one bit of information requires the i... |
I want to give you a general argument for the negative result: A general $n$-qubit density matrix cannot be contructed from this type of measurements.
The following observation will be needed:
Let $G$ be a compact Lie group acting on a vector space $V$ via a representation $\pi$. Let $\rho$ be a density matrix on $V$.
... |
At first I thought maybe sodium sulphate in contact with water produces sulphuric acid which absorbs water but I do not think it is actually a valid reason.
Sodium sulphate reacts readily with water at room temperature to form hydrates up to sodium sulphate decahydrate, $\ce{NaSO4\cdot10H2O}$. $$\ce{NaSO4 + 10H2O -> Na... |
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the Moore spectrum. For concreteness this is defined as a spectrum $X$ such that $\pi_n(X) = 0$ for $n <0$ $H_n(X)= 0$ for ... |
The Sharpe Ratio
The Sharpe Ratio is perhaps the most widely used statistic for summarizing the achieved(or backtested!) past performance of some asset--mutual fund, hedge fund, trading strategy,
etc.Defined as the mean return divided by the standard deviation (or \(\frac{\mu}{\sigma}\)), theSharpe Ratio roughly measur... |
The Akra-Bazzi method
The Akra-Bazzi method gives asymptotics for recurrences of the form: $$ T(x) = \sum_{1 \le i \le k} a_i T(b_i x + h_i(x)) + g(x) \quad \text{for $x \ge x_0$} $$ This covers the usual divide-and-conquer recurrences, but also cases in which the division is unequal. The "fudge terms" $h_i(x)$ can cat... |
On the DNA Computer Binary Code
In any finite set we can define a
, a partial order in different ways. But here, a partial order is defined in the set of four DNA bases in such a manner that a Boolean lattice structure is obtained. A Boolean lattice is an algebraic structure that captures essential properties of both s... |
Modeling a system
Visual Cue Narration
Show first slide
In this tutorial, we will see how to model a physical system using the ODE representation of the system and then convert it to a more familiar transfer function representation in Scilab.
Show slide with Mass-Spring-Damper
The system that we will attempt to model w... |
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does anyone know of any?
The BBP formula is another nice one: $$ \pi = \sum_{k=0}^\infty \left[ \frac{1}{16^k} \! \left( \frac{4}{... |
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