text stringlengths 256 16.4k |
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You ask:
A broader question is what is the relationship between the size of the slits, the distance between the slits, and the observed interference pattern?
The answer here covers your question.
The de Broglie wavelength describes the effective wavelength that a particle would have when it was behaving as a wave.
to d... |
If I have volatility smile quoted with respect to the delta of an option on the forward, how can I convert this delta into the moneyness or strike of the option?
Is there any bult-in function of Matlab financial toolbox?
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and aca... |
Coupling Heat Transfer with Subsurface Porous Media Flow
In the second part of our Geothermal Energy series, we focus on the coupled heat transport and subsurface flow processes that determine the thermal development of the subsurface due to geothermal heat production. The described processes are demonstrated in an exa... |
When I evaluate
Solve[a==Sin[b*c], b] to rearrange the following for $ b $:
$$ a = \sin(bc) $$
I get the following result from Mathematica:
$$\begin{align*} \left\{\left\{b\to \text{ConditionalExpression}\left[\frac{-\sin ^{-1}(a)+2 \pi c_1+\pi }{c},c_1\in \mathbb{Z}\right]\right\},\right.\left.\left\{b\to \text{Condit... |
Faddeeva Package From AbInitio
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Line 8: Line 8: * '''Dawson''', the [[w:Dawson function|Daw... |
If only weak-ordering and continuity is assumed, ICs can definitely intersect.
This is not true. First, if you're speaking of indifference
curves, you'd already be assuming either local non-satiation or monotonicity.
Let's speak of indifference
sets instead. The analogue of two sets, $I_1$ and $I_2$, "crossing" each ot... |
Voronoi summation method
A matrix summation method of sequences. It is defined by a numerical sequence $\{p_n\}$ and denoted by the symbol $(W,p_n)$. A sequence $\{s_n\}$ is summable by the method $(W,p_n)$ to a number $S$ if
$$\frac{s_0p_n+s_1p_{n-1}+\ldots+s_np_0}{p_0+\ldots+p_n}\to S$$
In particular, if $p_0=1$, $p_... |
In several theories, space itself is discrete, somewhat in relation to the Planck length, $$l_p = \sqrt{\frac{\hbar G}{c^3}} \simeq 1.616199 \times 10^{-35}\quad m$$ .
More specifically in loop quantum gravity, Carlo Rovelli's 1998 overview paper states the following:
The spin-networks picture of space–time is mathemat... |
With which notation do you feel uncomfortable?
closed as not constructive by Loop Space, Chris Schommer-Pries, Qiaochu Yuan, Scott Morrison♦ Mar 19 '10 at 6:10
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this quest... |
I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant.
For a compact 3-manifold, $M$, without boundary, it is well known (see, for example, section 2 of this reference), that for a compact simple Lie group $G$ and trivia... |
2017-04-19 08:09
Flavour anomalies after the $R_{K^*}$ measurement / D'Amico, Guido (CERN) ; Nardecchia, Marco (CERN) ; Panci, Paolo (CERN) ; Sannino, Francesco (CERN ; Southern Denmark U., CP3-Origins ; U. Southern Denmark, Odense, DIAS) ; Strumia, Alessandro (CERN ; Pisa U. ; INFN, Pisa) ; Torre, Riccardo (EPFL, Laus... |
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astro-ph.CO
Change to browse by: Bookmark(what is this?) Astrophysics > Cosmology and Nongalactic Astrophysics Title: Massive Photon and Dark Energy
(Submitted on 2 Dec 2015 (v1), last revised 7 Mar 2016 (this version, v2))
Abstract: We investigate cosmology of massive electrodynamics and explor... |
L # 1
Show that
It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Offline
Last edited by krassi_holmz (2006-03-09 02:44:53)
IPBLE: Increasing Performance By Low... |
Mass balances are an important tool that is used to define the dimension of the production, define the amount of raw materials that enter into the process and the amount of products and waste that leave the process. Overall, with a mass balance it is possible to estimate the yield of a process.
Given a process:
The cor... |
The transfinite
tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct limit is taken at the limit stages:
$G\rightarrow Aut(G)\rightarrow Aut(Aut(G))\rightarrow\cdots\rightarrow G_{\alpha}\... |
I would like to solve the following mechanics problem and would like to get few hints.
There is an inclined plane with angle $\alpha$. A ball of radius $r$ rolls on the inclined plane that has a spring connected to it at the bottom (first figure). The spring constant is $k_{1}$. Let the initial angular velocity of the ... |
The probability that a prime p does not divide a random integer n is (1-1/p), so for random n we could argue that the probability that n and φ(n) are coprime is
$\prod_{p|n} \left(1-1/p \right) = \phi(n)/n.$
The average order of φ(n)/n is given by
${ 1 \over N } \sum_{n=1}^N {\phi(n) / n} = 6/\pi^2 + O(\log N/N).$
Now ... |
A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach
Department of Mathematics, Shimane University, Matsue 690-8504, Japan
$\partial_t u +i \Delta u = i\lambda |u|^{p-1} u$
$\mathit{\boldsymbol{R}}^{1+n}$
$n\ge 3$
$p>1$
$\lambda \in \mathit{\boldsymbol{C}}... |
I would like to solve a simple 2nd-order ODE with one of the boundary conditions defined at $ -\infty $. The ODE I am looking to solve is:
$$ w''(z)-2i\pi^2w(z)=0 $$
with the corresponding boundary conditions:
$$ w(z=-\infty)=0, \; w'(z=0)=0+i\dfrac{\tau_{0}}{\mu}. $$
My attempt at a solution using
DSolve is as follows... |
I'm given a continuous-time analog signal $x_a(t) = \cos(2\pi f_1t)+\sin(2\pi f_2t)$, for some frequency $f_1, f_2$. I'm asked to sample $x_a(t)$ at $F_s=1024\textrm{ Hz}$, apply a 128-point Hamming Window, and take the DFT(Discrete Fourier Transform) of the resulting windowed, sampled signal.
The expression for a samp... |
Filter Results: Full text PDF available (8) Publication Year
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Abstract We investigate the existence of the least and greatest solutions to measure differential equations, as well as the relation between the extremal solutions and lower or ... |
The
Menger sponge is a delightfully pathological shape to soak up. Start with a cube of side-length 1, and “hollow it out” by slicing it into 27 sub-cubes (like a Rubik’s Cube) and removing the 7 central sub-cubes as illustrated below. Call this the level 1 sponge. To make the 2nd level, perform the same “hollowing out... |
[This is the 6th post in the current series about Wythoff’s game: see posts #1, #2, #3, #4, and #5.
Caveat lector: this post is a bit more difficult than usual. Let me know what you think in the comments!]
Our only remaining task from last week was to prove the mysterious Covering Theorem: we must show that there is ex... |
I'm trying to study about modelling star from a journal. To get the characteristics of the star, I have to solve 4 differential equations which are coupled each other. I want to use Mathematica 9 to solve this system of equations but I still can't do this simply. This is the system of equations:
$\lambda'=\frac{1-e^{\l... |
What is the exact length of \stackrel{\LARGE{\frown}}{AB} on circle P
Step 1: Find the circumference of the circle
NOTE: Circumference of the circle = 2 \pi r
EXAMPLE: Circumference = 2 \pi r
= 2 \pi(15)
= 30 \pi
Step 2: Identify the type of the triangle
NOTE: The triangle are congruent because they are radii of the sa... |
Nagoya Mathematical Journal Nagoya Math. J. Volume 185 (2007), 143-150. Some estimates for the Bergman kernel and metric in terms of logarithmic capacity Abstract
For a bounded domain $\Omega$ on the plane we show the inequality $c_{\Omega}(z)^{2} \leq 2\pi K_{\Omega}(z)$, $z \in \Omega$, where $c_{\Omega}(z)$ is the l... |
Is there any way to tell whether a filter is high pass or low pass by observing only it's time domain samples or coefficients?
To elaborate a bit on Fat32's answer: the most straightforward thing to do is to compute (or estimate) the following two sums:
$$H(e^{j 0})=\sum_nh[n]\tag{1}$$
and
$$H(e^{j \pi})=\sum_n(-1)^nh[... |
I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?
t - Integrate[Abs[t - s]^(-1/2)*s, {s, 0, 1}]
Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minut... |
A trajectory is the path taken up by a moving object that is following through the space as a function of time. Mathematically, a trajectory is described as a position of an object over a particular time. A much simplified example would by a ball or rock thrown upwards, the path taken by the stone is determined by the ... |
I've seen this question! from 2 years ago:
Given $F$ is a $PRF$, we define $G$ for an input $x\in\{0,1\}^n$ as follows:
$$G(x) = F_k(x) \oplus F_k(x \oplus 1^s)$$
The question was if $G$ is a $PRG$. I edited the question a bit to fit the answers given back then. The answers stated this isn't a $PRG$ because $$G(x\oplus... |
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abstract
Session 34 - Real Instruments.
Display session, Tuesday, June 09
Atlas Ballroom,
The design of a long slit dual order spectrograph (DOS), being built for flight on a NASA sounding rocket by Johns Hopkins University (JHU) in the late fall -- early winter of 1999, is presented. The spectro... |
I just asked this question concerning the application of Noether's theory. Think about this got me wondering about the following. In the usual derivation of the Noether current the assumption is made that:
$$\mathcal{L}(\phi'(x'),\partial_\mu'\phi'(x'),x')=\mathcal{L}(\phi(x),\partial_\mu\phi(x),x)+\delta x^\mu\partial... |
Archimedes constant π Set todo Requirements
definiendum $\pi:=V_2(1)$ Discussion
The constant $\pi$ is defined as the volume of the disc of radius $1$, where the underlying metric space is taken to be the two dimensional Euclidean space $\mathbb E^2$.
$\pi = 3.14159\dots\approx \frac{22}{7}$ |
This is actually a very tricky question, mathematically. Physicists may think this question to be trivial. But it takes me one hour in a math summer school to explain the notion of gapped Hamiltonian.
To see why it is tricky, let us consider the following statements. Any physicalsystem have a finite number of degrees o... |
I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes.It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended.May I ...
@Nelimee Do we need to m... |
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I've spent four lectures on the logic of partitions; you may be wondering why. One reason was to give you examples illustrating this important fact:
Theorem. Left adjoints preserve joins and right adjoints preserve meets. Suppose \(... |
Is it possible to modify the PCA algorithm so that it actually implements factor analysis? We can assume that the uniquenesses are known.
I'm aware that for a $d$-dimensional data $x$, PCA takes the leading $k<d$ eigenvectors of $\Sigma_X$ and uses these to define the principal components.
In factor analysis models thi... |
I am attempting to create a Kalman filter to track a sine wave (I am using a linear Kalman filter example assuming I already know the frequency of the sine wave) - the example I am using is derived on pages 194-196 of
"Fundamentals of Kalman Filtering: A Practical Approach" 2nd edition by Paul Zarchan and Howard Musoff... |
I read the definition of work as $$W ~=~ \vec{F} \cdot \vec{d}$$ $$\text{ Work = (Force) $\cdot$ (Distance)}.$$
If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?
I read the definition of... |
Automatic Equation Numbering¶
The TeX input processing in MathJax can be configured to add equation numbers to displayed equations automatically. This functionality is turned off by default, but it is easy to configure MathJax to produce automatic equation numbers by adding:
window.MathJax = { tex: { tags: 'ams' }};
to... |
Research Open Access Published: On some classes of difference equations of infinite order Advances in Difference Equations volume 2015, Article number: 211 (2015) Article metrics
31k Accesses
3 Citations
9 Altmetric
Abstract
We consider a certain class of difference equations on an axis and a half-axis, and we establis... |
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I've spent four lectures on the logic of partitions; you may be wondering why. One reason was to give you examples illustrating this important fact:
Theorem. Left adjoints preserve joins and right adjoints preserve meets. Suppose \(... |
Diagonal construction Set
context $ f:C\to \mathcal {\mathcal P}(C) $ definiendum $ x\in D_f $ inclusion $ D_f \subseteq C $ postulate $ x \notin f(x) $ Discussion
We take an arbitrary set $C$ and argue about all the functions $f:C\to \mathcal {\mathcal P}(C)$ from $C$ to the powerset ${\mathcal P}(C)$. For any such $f... |
This phrasing is only true for organic molecules. If for example, sodium hydride loses hydrogen, the sodium ion will get reduced. But since you seem to come from a biochemical background, this simplification is okay since you will be dealing with organic molecules primarily.
The idea behind that statement is that hydro... |
Another way:
mat = {{4, 11, 14}, {8, 7, -2}};
sph[{x_, y_, z_}] := x^2 + y^2 + z^2;
ContourPlot[sph[PseudoInverse[mat].{u, v}] == 1,
{u, -20, 20}, {v, -20, 20},
Epilog -> {Red, PointSize@Large, Point[{{18, 6}, {3, -9}}],
Line[{{{18, 6}, -{18, 6}}, {{3, -9}, -{3, -9}}}]},
PlotLabel -> Simplify[sph[PseudoInverse[mat].{u,... |
I thought this result was a bit interesting. Mahlon M. Day in the paper [1] showed that the amenable groups are precisely the groups where there Markov-Kakutani theorem holds.
If $(X,\mathcal{M})$ is an algebra of sets, then a function $\mu:\mathcal{M}\rightarrow[0,1]$ is said to be a finitely additive probability meas... |
(11C) Tree Heights
11-02-2018, 01:24 PM (This post was last modified: 11-02-2018 01:26 PM by Gamo.)
Post: #1
(11C) Tree Heights
This program was adapted from the Hand-Held-Calculator Programs
for the Field Forester.
More detail information attached here.
Procedure:
1. Enter slope distance to base of tree [A] -->Display... |
Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid
$$\operatorname{KL}(p_\theta\parallel p_{\theta + d \theta}) = d \theta^TF(\theta) \, d\theta + \mathcal O(\|d\theta\|^3), $$ where $$F(\theta)_{ij} :=... |
Event detail Paris/Berkeley/Bonn/Zürich Analysis Seminar: Planar Sobolev extension domains
Seminar | February 22 | 9:10-10 a.m. | 238 Sutardja Dai Hall
Yi Zhang, University of Bonn
A domain $\Omega \subset \mathbb R^2$ is called a $W^{1,\,p}$-extension domain if it admits an extension operator $E\colon W^{1,\,p}(\Omega... |
I think you have the inequality sign wrong. The inequality as you write it cannot hold. To see this, consider the numerical case where $y=4 > y'=3, d=1 < d'=2$ and $u= \sqrt(x)$. I let you do the calculation yourself.
I suppose the correct inequality is: \begin{equation} u(y−d)−u(y−d') \leq u(y'−d)−u(y′−d′) \end{equati... |
Vacuum Solutions to Einstein’s Field Equations¶ Einstein’s Equation¶
Einstein’s Field Equation(EFE) is a ten component tensor equation which relates local space-time curvature with local energy and momentum. In short, they determine the metric tensor of a spacetime given arrangement of stress-energy in space-time. The ... |
Writing Mathematics for MathJax¶ Putting mathematics in a web page¶
To put mathematics in your web page, you can use TeX and LaTeX notation, MathML notation, AsciiMath notation, or a combination of all three within the same page; the MathJax configuration tells MathJax which you want to use, and how you plan to indicat... |
If we have a rod uniformly charged with $Q$ stretching from $-a$ to $a$ on the $x$-axis as shown in the picture
And we want to calculate electric field in point $2a$ on the $x$-axis, we know that the electric field of a point charge $dQ$ on the rod in point $2a$ is: $$\vec{dE}=\frac{dQ}{4\pi\epsilon_0r^2}\vec{i_x}$$ wh... |
Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$.Let's denote the width of a sample by $h$ where $$h\rightarrow0$$Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a...
@Ultradark You can try d... |
If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?
Here are some details; I will only work out the case where you want $7$ tails in a row, and the general case is similar. I am i... |
Assume that we have a general one-period market model consisting of d+1 assets and N states.
Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$, on the asset $S_1^2$ with strike price $K = 1$ given that
$$S_0 =\begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, S_1 = ... |
This calculator can be used to design either low-pass filters or high-pass filters. Choose your filter type, enter a value for the capacitor, enter a value for the potentiometer, and then select the taper for the potentiometer. Click and hold to rotate the knob and vary the resistance. As the resistance changes with th... |
Intro (you may skip this if you're an expert, I'm including this for completeness):
Say I have two bases for two systems,
The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 \end{array}\right),|-\rangle=\left(\begin{array}{c} 0\\1 \end{array}\right)$
The second is a spin-1 system, with $|1_+\rangle=... |
Data Types for Control & DSP
There's a lot of information out there on what data types to use for digital signal processing, but there's also a lot of confusion, so the topic bears repeating.
I recently posted an entry on PID control. In that article I glossed over the data types used by showing "double" in all of my e... |
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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 ... |
> No, that wasn't my reasoning. Right or wrong, my position in Lecture 6 was this: I wasn't assuming the posets in question have all infs and sup, I was *claiming* that they *must* have the infs and sups in question, given my assumptions.
> ...
> That said, if someone gave me a puzzle whose answer was \\(a\\), and I sa... |
The MathJax Processing Model¶
The purpose of MathJax is to bring the ability to include mathematics easily in web pages to as wide a range of browsers as possible. Authors can specify mathematics in a variety of formats (e.g., MathML, LaTeX, or AsciiMath), and MathJax provides high-quality mathematical typesetting even... |
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1. Measurement of the top quark mass with lepton+jets final states using $$\mathrm {p}$$ p $$\mathrm {p}$$ p collisions at $$\sqrt{s}=13\,\text {TeV} $$ s=13TeV
The Europ... |
№ 9
All Issues Haar’s condition and joint polynomiality of separate polynomial functions Abstract
For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes ... |
No
Your idea is good, but you are misunderstanding what 'any dimension' means.
When you think of a 1-foot cube, you are thinking that, since the cube is no longer than 1 foot to a side, it has a length of 1 foot in each direction. This is not true. In fact, the cube has an extent of no
less than one foot in each direct... |
Rotational Equilibrium If an object is in rotational equilibrium, the resultant turning force or torque, equal to force times distance from an axis for each force, is zero about any given axis. This is known as the principle of moments. The horizontal bar below is in equilibrium. About the axis shown there is no net tu... |
Consider a signal $x(t)$, which is input to a pulse shaping filter with transfer function $g_t(t)$:
$$x(t) = \sum_n d_n \delta(t-nT_s)$$
with $n$ an index from negative to positive infinity and $d_n$ random, equiprobable binary symbols such as 1 and -1 (antipodal) or 0 and 1 (on-off keying).
Well, the power spectral de... |
I am working my way through a paper proving the prime number theorem and I have come across the following (to use ${2n \choose n}$ was apparently due to Chebyshev, hence the title):
Lemma: Define $\vartheta(x) = \displaystyle\sum_{p \leq x} \log(p)$ (where $p$ is a prime number). Then, $\vartheta(x)=O(x)$. Proof: Let $... |
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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'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the conjugate momenta $\pi^\mu_{\ \ \ \nu} = \partial \mathscr{L} / \partial(\partial_\mu A^\nu)$, I can use two methods.
First ... |
You could always break the question into two perpendicular directions
Your velocity will be subtracted from horizontal velocity of your friend's. The vertical velocity will remain same. The resultant of the vertical and new horizontal velocity will give you the final answer of how you see your friend.
Velocity of frien... |
Measurement of rate of a moving clock is carried out the following way:
Stationary observer in reference frame $S$ (the observer on earth) places the clock $C_1$ at coordinate $x_1$ of his frame and the clock $C_2$ at coordinate $x_2$ of his frame.
Then this observer sends a beam of light from clock $C_1$ towards clock... |
In a case of free Dirac field we have
$$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$
$$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \ha... |
We throw the word around so much without knowing what it actually means. Why are the physics based definition and the general definition of beats so different?
Colloquial beats and physics based definition of beats are not that different from each other!
In colloquial terms, beats are rhythmic repetition of sound.
In P... |
№ 9
All Issues Bezout rings of stable ranк 1.5 and the decomposition of a complete linear group into its multiple subgroups Abstract
A ring $R$ is called a ring of stable rank 1.5 if, for any triple $a, b, c \in R, c \not = 0$, such that $aR + bR + cR = R$, there exists $r \in R$ such that $(a + br)R + cR = R$. It is p... |
> To prove that \\(r(x,y) = x \wedge y\\) it's therefore enough to show
>
> $$ a \le_A x \wedge y \textrm{ if and only if } a \le_A x \textrm{ and } a \le_A y. $$
>
> **MD Puzzle 2'.** Can someone show this?
Let's look directly at the definition of \\(\wedge\\) from Fong and Spivak, pg. 17:
> Definition 1.60. Let \\((P... |
Wikipedia credits this to Maxwell. This derivation can be found in Maxwell's
Treatise on Electricity and Magnetism vol. 2, part 4, ch. 2 (§§502-527). I went through the derivation and found two self cancelling mistakes made by Maxwell. Those mistakes were corrected by J.J. Thomson in his edition of Maxwell's treatise.
... |
Non-Commutative Ring with Unity and 2 Ideals not necessarily Division Ring Theorem
Let $\struct {R, +, \circ}$ specifically not be commutative.
Let $\struct {R, +, \circ}$ be such that the only ideals of $\struct {R, +, \circ}$ are $\set {0_R}$ and $R$ itself.
Then it is not necessarily the case that $\struct {R, +, \c... |
First of all, the inequality you end up with is wrong: you need either$$a \leq -\frac{2\sqrt{39}}{13} \;\;\;\text{ or }\;\;\; a \geq \frac{2\sqrt{39}}{13}$$
Since the root $$x_2 = -\frac{a}{6} - \sqrt{\frac{13a^2 - 12}{36}} = \frac{1}{6}\left(-a - \sqrt{13a^2 - 12}\right)$$is always smaller than $x_1$ (when both are re... |
Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question:
Problem: Let $X$ be a reduced complex analytic space, $f = (f_1,f_2) : X \to \mathbb{C}^2$ a couple of functions and $K \in D^b_c(A_X... |
Complex Down-Conversion Amplitude Loss
This blog illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")
The general idea behind complex down-conversion is shown in Figure ... |
User:Nikita2/sandbox A mapping $\varphi:D\to D'$ possesses Luzin's $\mathcal N$-property if the image of every set of measure zero is a set of measure zero. A mapping $\varphi$ possesses Luzin's $\mathcal N{}^{-1}$-property if the preimage of every set of measure zero is a set of measure zero. Briefly\begin{equation*}\... |
Step 1: Note down the given values
Step 2: Set up the formula for arc length.
NOTE: The formula is arc length= 2 \pi (r)(\frac{\theta }{360}) ,
where {\displaystyle r} equals the radius of the circle and {\displaystyle \theta } equals the measurement of the arc’s central angle, in degrees.
or
Arc length = r * \theta
St... |
Pullback . category theory Collection
context $F:({a\rightarrow z\leftarrow b})\longrightarrow{\bf C}$ definition $\langle Fa\times_{Fz} Fb, \pi\rangle := \mathrm{lim}\,F$
Here we consider a functor $F$ from the category ${a\rightarrow z\leftarrow b}$, consisting of three object and two non-identity arrows $f_a$ and $f... |
Yes, the two are intimately related. One way, as in QMechanic's answer, is via Wick rotations, but in general there is a lot more freedom once you allow integration contours to go over into the complex plane. In my area, strong field physics, the use of complex time to understand tunnelling problems is everyday bread a... |
I am working on a proof where given a continuous surjection f:X$\to$Y, if X is connected then Y is connected. I am doing so by contraposition so I assumed Y was disconnected and am attempting to show X is disconnected.
By definition of disconnected, there exist open sets A' & B' in $\tau_Y$ such that A'$\cup$B'=Y and A... |
I have been reading Srednicki from the beginning and doing all the exercises, and I hit a big roadblock at Q10.4, as I can't seem to figure out what Srednicki is doing in his solution. Luckily, I found an additional resource (http://hep.ucsb.edu/people/cag/qft/QFT-10.pdf), which details a formula for finding vertex fac... |
I have the following code:
g[u_,p_] = Graphics{ Circle[{2 Sqrt[u], u}, u], Circle[{-2 Sqrt[u], u}, u], Point[{2 Sqrt[u], u}], Point[{-2 Sqrt[u], u}], Inset[p,{0,1},{0,0},{15,10}] }
and I want to draw the parabola that goes through the centers of both of these circles, for every $u \neq 0$ with its value in $0$ equal to... |
Vector space basis
Set
context $V$…$\ \mathcal F$-vector space
definiendum $B\in \mathrm{basis}(V)$
$B'\subseteq B$ $B'$…finite range $n\equiv\left|B'\right|$
$v_1,\dots,v_n\in B'$ $c_1,\dots c_n\in \mathcal F$ $x\in V$
postulate $\sum_{k=1}^n c_k\cdot v_k=0\ \Rightarrow\ \forall j.\ c_j=0$
All finite subsets of the ba... |
In the literature on social welfare functionals, the only example I've seen of a functional which meets all of Arrow's conditions–––or at least utility analogues of Arrow's conditions–––plus invariance regarding ordinal level comparability is Rawls' maximin. E.g. Sen in
On Weights and Measures (1977, p. 1544) cites max... |
Simple Interest = $\dfrac{{PTR}}{{100}}$ Simple interest formula:
Here,
P = Principal
T = Time
R = Rate of Interest
Simple interest for 1 year = Principal × (Interest %) = P × (R%)
If simple interest for more than 1 year the Simple interest = $P \times R\% \times T = SI$
Interest % = $\displaystyle\frac{\text{Interest}... |
Compressive Sensing - Recovery of Sparse Signals (Part 1)
The amount of data that is generated has been increasing at a substantial rate since the beginning of the digital revolution. The constraints on the sampling and reconstruction of digital signals are derived from the well-known Nyquist-Shannon sampling theorem. ... |
I am trying to solve a system of two equations with two unknowns. In these equations I have, a part from constants:
Unknown nr 1, $$D_{\perp}$$ Unknown nr 2, $$\omega_C$$ Known function of r: $$\mu(r)$$
The full system looks like:
equation1: $$ D_{||}= \frac 1 {2 \omega_0}(-\alpha-1)\sqrt{(-\omega_C+\mu(r)D_{||})^2+(\m... |
Ìrànwọ́:Báwo lẹṣe le ṣe àtúnṣe ojúewé
Àyọkà yìí únfẹ́ ìyílédèdà láti Geesi sí Yorùbá. Tí ẹ bá gbọ́ Geesi ẹ ran Wikipedia lọ́wọ́ ṣàtúnṣe sí ìyílédèdà
Wikipedia is a wiki, meaning that anyone can easily
edit any unprotected page, and save those changes immediately to that page, making the alterations visible to every oth... |
In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where
$$ \sum_{kk^{\prime}}V_{kk^{\prime}}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}c_{-k^{\prime}\downarrow}c_{k^{\prim... |
Group Theory: Important Definitions and Results
These notes are made and shared by Mr. Akhtar Abbas. We are really very thankful to him for providing these notes and appreciates his effort to publish these notes on MathCity.org. These notes contains important definitions with examples and related theorem, which might b... |
Proof of Theorem 1 \((i)\)
Since \(||b_t||=1\)
and introducing \(\alpha _t=w_t m({b}_t,{b}_T)\)
, we have:
$$\begin{aligned} \sum _{t=1}^T \alpha _t\cos ^2(b_t,v)&= \sum _{t=1}^T \alpha _t (b_t'v)^2\\&= \sum _{t=1}^T \alpha _t v'b_tb_t'v\\&= v'\left( \sum _{t=1}^T \alpha _t b_tb_t'\right) v\\&= v'M_Tv. \end{aligned}$$
... |
Electronic Journal of Probability Electron. J. Probab. Volume 23 (2018), paper no. 60, 35 pp. The argmin process of random walks, Brownian motion and Lévy processes Abstract
In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]... |
Consider the drift diffusion equation
$$\dfrac{\partial}{\partial t}\psi=\mu\dfrac{\partial}{\partial x}\psi+\kappa^2\dfrac{\partial^2}{\partial x^2}\psi.$$
Dimensional analysis tells us that $\mu$ is a characteristic length per time (drift velocity) while $\kappa$ is a characteristic length per
square root of time. Th... |
Can Black-Scholes option values be derived via the Capital Asset Pricing Model,
without resort to the use of a risk-free portfolio being created from the option and a Delta determined quantity of the underlying instrument?
This derivation, originally due to Cox & Rubinstein (1985) starts from the
Capital Asset Pricing ... |
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