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I have sequence of $N$ real numbers: $\mathbf{x} = (x_0, x_1, \ldots, x_{N-1})$.
Discrete Fourier Transform (DFT) is defined as $$ X_k = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k \frac{n}{N}}, \quad (k=0,1,\ldots,N-1). $$
Coefficients $X_0,X_1,\ldots,X_{N-1}$ are complex-valued at all.
How to change starting sequence $\mathbf{... |
Does dimensional regularization (and counterterm renormalization) give rise to running coupling constants as the other regularization methods?
Yes. In Dimensional Regularization (DR) schemes, you always introduce a scale $\mu$, for (mass!) dimensional analysis consistency. Renormalized couplings depend on this scale $\... |
Edit
It seems that I haven't written my question clearly enough, so I will try to develop more using the example of quantum tunnelling. As a disclaimer, I want to state that my question is not about how to perform a Wick rotation in the path integral formulation!
Let's look a the probability of quantum tunnelling in th... |
Genus 2 curves in isogeny class 169.a
Label Equation 169.a.169.1 \(y^2 + (x^3 + x + 1)y = x^5 + x^4\) L-function data
Analytic rank: \(0\) Bad L-factors:
Good L-factors:
See L-function page for more information
\(\mathrm{ST} =\) $E_6$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)
Of \(\GL_2\)-type over \(\Q\)
Smallest fiel... |
Can't seem to figure this one out by thinking it through. Let's say that the simple return $R_t=P_{t+1}/P_t -1$ is assumed to be $R_t \sim iid N(0,\sigma^2)$. Thus, a two period return would be $(1+R_t)(1+R_{t+1})-1$. Would the variance of the two period return be equal to $2\sigma^2 + \sigma^4$?
$$Var((1+R_t)(1+R_{t+1... |
Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" between algebras.
Here are the notations:
$S=C_0(\mathbb{R})$, viewed as a $\mathbb{Z}_2$-graded algebra (odd and even functi... |
gravitational waves are strictly transversal (in the linear regime at least), also their amplitudes are tiny even for cosmic scale events like supernovas or binary black holes (at least far away, maybe we should ask some physicists located a bit closer to the center of the galaxy), but lets put all those facts aside fo... |
So i have been working (as an undergrad, by working i mean "Redoing a few things my professor does") in a SIRS model for epidemies. SIRS stands here for:
Susceptible -> Infected -> Recovered -> Susceptible.
So, the system of 1st order differential equations that rule the model is: $$ S'(t) = - \beta I(t) S(t) + \mu R(t... |
I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely presented modules agree. For quasicoherent sheaves over a locally Noetherian scheme $X$, coherent and locally finitely presented... |
According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to construct such a self-financing portfolio whose price replicates the price of the option (within a very small time interval). An... |
I am trying to draw a fairly simple scene with tikz.
The issue i have is with defining the end points on half circle. I tried to implement an algorithmfor intersection detection, pseudocode can be found at Circle-Line intersection. However it does not work as it should. In addition, it does not compile if i use the
\if... |
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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 in... |
Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite decreasing $E$-sequence. I am interested to know about the degree of illness of internally non-well-founded models in the l... |
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Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
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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... |
In geometry,
Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who st... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
An operator is, by definition, a linear map from some
given linear space $V$ to itself. And the definition of a linear map cannot, in general, be independent of the linear structure if the space it acts upon.
There are, however, algebraic objects that can be defined abstractly, and that they are always represented as b... |
Here are two things that I have mistakenly believed at various points in my "adult mathematical life":
For a field $k$, we have an equality of formal Laurent series fields $k((x,y)) = k((x))((y))$.
Note that the first one is the fraction field of the formal power series ring $k[[x,y]]$. For instance, for a sequence $\{... |
I think the existing answers are only half right. We need to bring in the fact that the answer depends on the degree to which the charge can be regarded as small, in either amount of charge, or radius, or both.
First let's consider a charge $q$ moving at constant velocity. It is the source of a magnetic field ${\bf B}_... |
Here's what I perceive to be a mathematically and logically precise presentation of the theorem, let me know if this helps.
Mathematical Preliminaries
First let me introduce some precise notation so that we don't encounter any issues with "infinitesimals" etc. Given a field $\phi$, let $\hat\phi(\alpha, x)$ denote a sm... |
Level-set Based Segmentation in 2D
Before introducing the actual model used for segmenting 2D images, few things related to explicit and implicit curve representations are discussed. Before going any further, I would like to point out that the implicit curve representation techniques are not confined to 2D case as one ... |
Efficiency of the Reverse Carnot Cycle
An air conditioning device is working on a reverse Carnot cycle between the inside of a room at temperature T
2 and the outside at temperature T 1 > T 2 with a monatomic ideal gas as the working medium. The air conditioner consumes the electrical power P. Heat leaks into the house... |
For example, consider a \(11 \times 11\) grid, and choose from the following building blocks:
Five different polyominos
Fail to cover complete grid
\[x_{i,j,k} = \begin{cases} 1 & \text{if we place polyomino $k$ at location $(i,j)$}\\ 0 & \text{otherwise} \end{cases}\]
I used as rule that the left-upper corner of each ... |
So here is my question,
Question:A water sample contains 9.5% $\mathrm{MgCl_2}$ and 11.7% $\mathrm{NaCl}$ (by weight). Assuming 80% ionization of each salt. Boiling point of water will be ________. ($\mathrm{K_b}$ for water $\mathrm{=0.52}$)
You know, this question isn't that tough to bother anyone (definitely not!). W... |
Yes, there is an infinite class of 2-connected cubic graphs on which Hamilton Cycle has a polynomial-time algorithm. Further, there is a such a class that contains infinitely many Hamiltonian graphs and infinitely many non-Hamiltonian graphs, which I think is a decent definition of "non-trivial".
First, let $H_n$ be th... |
I am reading the paper "Rank-Finiteness for Modular Categories" by Bruillard,Ng, Rowell, and Wang.
Let $C$ be a modular category and let $K_0(C)$ be the Grothendieck ring generated by simple objects of $C$ with multiplication induced from the tensor product of $C$: $$ V_i \otimes V_j \cong \oplus_{k \in \Pi_C} N_{i,j}^... |
First of all, let's see what Noether's Theorem says about your specific case (Klein-Gordon under global rescaling of the fields). Noether's theorem states that
To every differentiable symmetry of the Action of a system, there corresponds a conserved current.
The current in object is given by
$$J^{\mu}=-T_{\nu}^{\mu}\ \... |
I've found that if I reduce the radial domain to $8 \leq r \leq 20$, the condition number drops to ~10,000. This makes me think I need to scale my problem.
I'm not sure how to do this, however, and I need to do it right.
Nondimensionalization is partly repeated application of the chain rule, and partly art. The goal is... |
A recent paper written by Avi Loeb, Rafael Batista and myself, available to read at the preprint arXiv and published in the Journal of Cosmological and Astroparticle Physics, has garnered quite a bit of attention from various media outlets, with a press release from Harvard. The interpretations in the press range from ... |
Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical Fourier transform and decompose $L^1$ functions for locally compact abelian groups?
For all locally compact commutative grou... |
Computing the excitation spectrum for a nucleus — that is, its energy levels and their quantum numbers — is
hard. Consider that the Schrödinger equation has no known exact solutions for atoms other than hydrogen: a helium atom, with three charges instead of two, is too complex to be treated except in approximation. The... |
Consider a two-period binomial model for a risk asset with each period equal to a year and take $S_0 = 1$, $u = 1.5$, and $l = 0.6$. The interest rate for both periods is $R = .1$.
a.) Price an American put option with $K = .8$
b.) Price an American call option with $K = .8$
c.) Price an American option with path depen... |
General case
There is indeed a mathematical theorem that deals with the number of nodes an eigenfunction corresponding to a certain eigenvalue can possess.It was laid down by Courant$^{[1, 2]}$ and it states the following:
Given the self-adjoint second order (partial) differential equation
\begin{equation} \left(\hat{L... |
Patterns, Symmetries, And Mathematical Structures In The Arts, 2020 Georgia Southern University
Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach University Honors Program Theses
Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and... |
Show that the Hilbert transform of $h(t) = m(t) \cos(2 \pi \nu_c t)$ is
$$\hat{h} (t) = m(t) \sin(2 \pi \nu_c t),$$
where $m(t)$ is a real valued, band-limited function (i.e. we have Fourier transform $M(\nu) = 0$ for $|\nu| > \nu_m$) and $\nu_c > \nu_m.$
Attempt
I did some research and I found out that this result fol... |
Content – Energy sources
A fuel cell is a reactor converting chemical energy from a fuel (hydrogen or hydrogen rich), directly into electricity through a chemical reaction between the fuel and an oxidizing agent.
In a fuel cell, the fuel is supplied to the anode, while air (or oxygen) is supplied to the cathode.
The an... |
In a U-shaped tube, water and oil are separated by a movable membrane. What is the ratio of the heights $\frac{h1}{h2}$ (density of the oil $ρ_{oil}$ = 0.92 $\frac{g}{cm^3}$)?
I tried solving by saying that the pressure at the membrane should be the same so
$$P_{H_2O} = P_{oil}$$ $$=> P_0 + \rho_{H_2O} \cdot g \cdot h_... |
1. I looked up forward LU error bounds in Higham's Accuracy and Stability of Numerical Algorithms, Theorem 9.15 (citing Barrlund and Sun), for the LU decomposition $$A=LU,\quad A+\Delta A=(L+\Delta L)(U+\Delta U)$$ it gives the norm-wise error bound$$\begin{gather}\frac{\|\Delta U\|_F}{\|U\|_2} \leq \frac{\|L^{-1}\|_2\... |
1. Feature Selection
We could delete some existing columns by using feature importance.
2. Feature Generation/Extraction
We could add some new columns. For example, generating new attributes using existing ones for each row or generating new attributes across multiple rows.
We have a lot of possible interactions, and w... |
It would be easier to answer your question clearly with a drawing.
In the following, the
angle coordinate of the pendulum is the angle it makes with the vertical line. When the pendulum swings right(left), the angle will be positive(negative).
With this setting, I get the exact same answer as you by working out the equ... |
Answer
amplitude = $1$ period = $120^o$ or $\frac{2}{3}\pi$
Work Step by Step
The given graph shows around two periods of the sine function. Since the highest point is equivalent to two squares from the x-axis, and each square representing 0.5, then the amplitude is: $=2 \times 0.5 \\=1$ Horizontally, one period of the... |
2.5. Automatic Differentiation¶
In machine learning, we
train models, updating them successively sothat they get better and better as they see more and more data. Usually, getting better means minimizing a loss function, a score thatanswers the question “how bad is our model?” This question is moresubtle than it appear... |
Andronov-Hopf bifurcation
Yuri A. Kuznetsov (2006), Scholarpedia, 1(10):1858. doi:10.4249/scholarpedia.1858 revision #90964 [link to/cite this article] Andronov-Hopf bifurcation is the birth of a limit cycle from an equilibrium in dynamical systems generated by ODEs, when the equilibrium changes stability via a pair of... |
Maass forms of levels 100 to 250 with $0 \leq R\leq 2$
The horizontal axis is the spectral parameter $R$ with the Laplace eigenvaluesatisying $\lambda=1/4+R^2$. The vertical axis is the level $N$. Each pointcorresponds to a Maass form of weight 0 and trivial character on $\Gamma_0(N)$with the color showing whether the ... |
One disadvantage of the fact that you have posted 5 identical answers (1, 2, 3, 4, 5) is that if other users have some comments about the website you created, they will post them in all these place. If you have some place online where you would like to receive feedback, you should probably also add link to that. — Mart... |
Your understanding of the requirements are correct. To elaborate, the first requirement you specify can be explained as taking the parity of $x_i$. Also, the two primes are called
Blum primes and the modulus is called a Blum integer, which means $p,q \in \mathbb P$, $p \equiv q \equiv 3 \pmod 4$, and $N = p \cdot q$.
T... |
I am trying to prove the theoretical "37-percent rule" for dating. The setup, if I remember correctly, is this. Suppose that you will meet exactly $N$ potential mates in your life, and you will meet them one at a time, in a perfectly random order. The potential mates rank from best to worst (in a total ordering), and y... |
Yes. Both universal covers and central extensions incurred during quantization come from the same fundamental concept:
Projective representations
If $\mathcal{H}$ is our Hilbert space of states, then distinct physical states are not
vectors $\psi\in\mathcal{H}$, but rays, since multiplication by a complex number does n... |
... and ApplicationsAs the name implies, the system detects detects overtaking vehicles based on optical-flow. Robustness of the system has been tested using more than 15,000 frames under realistic illumination ...
... (TDMA). The tutorial shows all the steps how the diffusion equation is discretized and how the solver... |
Posted: March 11, 2013
If you are not familiar with Fourier Analysis, the purpose of the analysis is to represent a function as the weighted sum of a collection of trigonometric functions (sines and cosines). This, in turn, allows us to extract the spectral properties of the function.
In this case, we are going to expl... |
berylium? really? okay then...toroidalet wrote:I Undertale hate it when people Emoji movie insert keywords so people will see their berylium page.
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you... |
Given a function $f:\mathbb{R} \rightarrow \mathbb{R}$ derivable, is the function $$\sum_{n=1}^{\infty}\frac{f(x)^n}{n^n}$$ measurable?
Here my attempt:
Let $$S_n(x)=\sum_{k=1}^{n} \frac{f(x)^k}{k^k},$$ since it is the finite sum of measurable functions, $S_n(x)$ is measurable $\forall n \in \mathbb{N}$.
And $\forall x... |
Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \end{align*} where $A: \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ and $B: \mathbb{R}... |
Question
So recently I was thinking about this: How many scalars are available in $4$ dimensions in General Relativity (without being redundant)? For example, with metric we can construct the following scalar:
$$ g^{\mu \nu} g_{\mu \nu} = 4 $$ is the same as: $$ (g^{\mu \nu} \otimes g^{\rho \kappa}) \cdot (g_{\mu \nu} ... |
I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple example of where complex analysis is useful in electrostatics?
closed as too broad by Kyle Kanos, HDE 226868, Neuneck, Martin... |
Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems $$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball centered at the origin in $ R^N$.
My interest is in obtaining positive solutions of perturbations of the equation $-\Delta u(x) = u(x)... |
I recently found myself confusing concepts from measure theory and probability theory, so I'd like to get an idea for what I'm misunderstanding. This definition is what started it all:
A sequence $\{X_{n}\}$ of random variables converges in distribution to $X$ if $$\lim_{n \to \infty} F_{n}(x) = F(x)$$
for every number... |
This question already has an answer here:
Assume that $c_t$ is the UNDISCOUNTED price process for a European call option in Bachelier model. In Bachelier model call option pricing formula the formulas is discussed. The undiscounted value process is $c_t = (S_t-K)\Phi( \frac{S_t-K}{\sigma\sqrt{T-t}})+\sigma\sqrt{T-t}\ph... |
I am currently reading the book "Set theory on the real line" by Bartoszynski and Judah and I do have problems to proof the following statement: Suppose $\mathcal{F}$ is a filter on $\omega$ including the Fréchet-Filter $\mathcal{G}$ ($\mathcal{G}\subset\mathcal{F}$). Then the following is equivalent:
(i) For every par... |
In addition to a positive Lyapunov exponent (for sensitivity to ICs), why do continuous chaotic dynamical systems also require a zero Lyapunov exponent?
Every continuous-time dynamical system with a bounded, non fixed-point dynamics has at least one zero Lyapunov exponent. This does not only apply to chaotic dynamics b... |
The discussion about fractional calculus, which led to this seriesof articles, was started by a couple of sixth form students on theaskNRICH webboard. Before reading this article you may like to readthe first two articles Fractional Calculus I
and Fractional Calculus II
and see the conversation on thewebboard.
Derivati... |
ABCD is a square. M is the midpoint of the side AB.
By constructing the lines AC, MC, BD and MD, the blue shaded quadrilateral is formed: What fraction of the total area is shaded? Below are three different methods for finding the shaded area. Unfortunately, the statements have been muddled up. Can you put them in the ... |
In the following you can find code for segmentation based on geometric/geodesic active contours. Implementations are based on the following papers:
A Geometric Model for Active Contours, Caselles et al. 1993 Geodesic Active Contours, Caselles et al. 1997
Formulations for both the models are as follows:
\[ \Phi_t = |\na... |
The language of the grammar $G$ presented here is regular and its corresponding regular expression is$$r = (a \mid b)^*aa(a \mid b)^*.$$Now, convert the regular expression to a minimal finite automaton and use this question to convert the automaton to an unambiguous regular grammar, which is always possible.In the end ... |
I am trying to find any described formalism which introduces
free variables into word grammars (I emphasize here word in order not to be confused with very similar thing in tree grammars).
What I mean is something like this:
$AX_1B\rightarrow AX_1CX_1B$
where $X_1$ can be substituted by any sequence $\alpha$ of termina... |
So I'm trying to integrate $\int_{-\infty}^\infty{ }x^2e^{-ax^2}dx$ by parts with the formula
$$\int{udv} = uv - \int{vdu} $$
I'm selecting
$$u = x^2$$ $$du = 2xdx$$ $$ v = \sqrt{\pi/\lambda} $$ $$ dv = e^{-ax^2} dx$$
This gives me
$$\Biggr|_{-\infty}^{\infty}{x^2 \sqrt{\pi/\lambda} } - \int_{-\infty}^\infty{ } \sqrt{\... |
Why exactly is X(0) the DC component of a signal?
How is it equal to N times x(n)'s average value and why it is at X(0)?
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.Sign up to join this c... |
I assume $G$ is affine. The quick answer is that in the simply connected case it says $1 = 1/1$ by various hard ingredients, and then it is a kind of (not easy) game with Galois cohomology and structure theory of semisimple groups to check both sides have the same behavior as we build up a general $G$ from the simply c... |
Wenn du
bei YouTube angemeldet bist, kannst depends. Wiedergabe automatisch mit einem der aktuellen Videovorschläge fortgesetzt. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statementMöchtest du dieses Video melden? standard Author!
Asked 5 years ago viewed 24082 times active 4 years a... |
I know there have been a lot of questions asked on this forum relating to order statistics, so, hopefully, this is not going to be a duplicate. I am trying to understand how I should go about conducting a hypothesis test with order statistics.
Say, we have a set $\{x_{(i)}\}_{i=1}^n$ of order statistics for a sample dr... |
A horn clause is a disjunction with at most one positive literal, e.g.
\begin{align}\lnot X_1 \lor \lnot X_2 \lor \ldots \lor \lnot X_n \lor Y\end{align}
The implication $X \rightarrow Y$ can be written as disjunction $\lnot X \lor Y$ (proof by truth table). If $X = \lnot X_1 \lor \lnot X_2 \lor \ldots \lor \lnot X_n $... |
I'm reading R. Haag's famous book "
Local Quantum Physics: Fields, Particles, Algebras", 2nd edition, and I'm very puzzled by the way he treats the Heisenberg picture in the Haag-Ruelle scattering theory. It begins in section " II.3 Physical Interpretation in Terms of Particles", where, on page 76, he clearly states " ... |
251 0
The diagram shows a pendant in the shape of a sector of a circle with center A. The radius is 4 cm and the angle at A is 0.4 radians. Three small holes of radius 0.1 cm, 0.2cm and 0.3 cm are cut away. The diameters of the holes lie along the axis of symmetry and their centers are 1, 2 and 3 cm respectively from A... |
Standard ML language
Mads Tofte (2009), Scholarpedia, 4(2):7515. doi:10.4249/scholarpedia.7515 revision #152339 [link to/cite this article]
The programming language
Standard ML, also known as SML, is inspired by certain fundamental concepts of Computer Science, making them directly available to the programmer. In parti... |
I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be interested to know the cause and how to avoid it!
This is an animated gif, open in separate window to view the animation ... |
The
width of the cusp $\infty$ for the group $\Gamma$ is the smallest number $w$ such that $T^w=\left(\begin{matrix}1&w\\0&1\end{matrix}\right)\in\Gamma$. Furthermore, for a general $x\in\mathbb{P}^1(\mathbb{Q})$ and $\gamma\in\Gamma$ such that $\gamma\infty=x$, we define the width of $x$ for $\Gamma$ to be the width o... |
I know $K(a,b,t)$ is the probability amplitude that a particle that starts at point $a$ is found at point $b$ at a time $t$ later. There is also an expression that sometimes is called green function:
$$G(a,b,E)=(i/\hbar)\int_{-\infty}^\infty\;\exp(iEt/\hbar)\;K(a,b,t)\;dt$$
or Fourier transform of Feynman propagator
Se... |
Finally, I felt I had something about Physics that I wanted to write about. The i \epsilon terms sitting in the propagator of a QFT, in the Lippmann-Schwinger equation and in Chapter 4 of Peskin and Schroeder have been bothering a couple students including me at the department for a while now. I am not qualified enough... |
Summary
From dimensional analysis I find that the dynamic viscosity of an ideal gas must depend on its pressure $p$, density $\rho$ and mean molecular free path $l$ in this way:
$$ \mu = C \sqrt{\rho p} l.\quad $$
Here, $C\geq0$ is a non-dimensional constant.
However, I find it counter intuitive that the dynamic viscos... |
Your confusion really just comes down to understanding the notation that is widely used for partial derivatives.
For simplicity, I'll restrict the discussion to a system with one coordinate degree of freedom $x$. In this case, the Lagrangian is a real valued function of two real variables which we suggestively
label by... |
As HBR mentioned, the boundary conditions can often be immediately incorporated into $A$ and $b$. For example, suppose we wish to solve the 1D heat equation with Dirchilet boundary conditions
$$u_t = \sigma u_{xx} + h(x,t), \quad u(a,t) = f(t), \quad u(b,t) = g(t).$$
We then discretize $u_j^n \approx u(x_j, t_n)$ where... |
3.4. Softmax Regression¶
Regression is the hammer we reach for when we want to answer
how much?or how many? questions. If you want to predict the number of dollars(the price) at which a house will be sold, or the number of wins abaseball team might have, or the number of days that a patient willremain hospitalized befo... |
I found it difficult to explain the difference between the fraction a / b and the ratio a : b. This subject is for pupils of grade 5. So is there a real difference between them and how to explain the difference in simple way ?!
This paragraph from Adding It Up is a good overview of the context of this question. Rationa... |
Consider a sensor that is measuring physical parameters like temperature, pressure, or velocity. This sensor introduces perturbations and noise and, hence, one key-problem is the optimal
inference of parameter $\boldsymbol{x}$ using measurement $\boldsymbol{y}$ from the sensor. Such inference of parameters is used in m... |
Rubinstein’s lcalc library¶
This is a wrapper around Michael Rubinstein’s lcalc. See http://oto.math.uwaterloo.ca/~mrubinst/L_function_public/CODE/.
AUTHORS:
Rishikesh (2010): added compute_rank() and hardy_z_function() Yann Laigle-Chapuy (2009): refactored Rishikesh (2009): initial version class
sage.libs.lcalc.lcalc_... |
To track solutions from a start system $G$ to the target system $F$ we use by default the straight-line homotopy
$$ H(x,t) := (1-t)F+tG\;. $$
But this is in general
not the best choice since you usually leave the solution space of your problem. Therefore we support the ability to define arbitrary homotopies where you h... |
While there are different "simple" proofs of the JL Lemma (the Gupta-Dasgupta proof is one such), it's not clear whether these are "undergraduate" level. So instead of answering his original question, I decided to change it to Is there a proof of the Johnson-Lindenstrauss Lemma that can be explained to an undergraduate... |
I've taking an Algorithms course. This is non-graded homework. The concept of loop invariants are new to me and it's taking some time to sink in. This was my first attempt at a proof of correctness today for the iterative Fibonacci algorithm. I don't feel good about it. I know this is not the right candidate invariant,... |
I had some attempts to obtain elementary proofs of (1) or (2), but I failed. Maybe these proofs should not be easy, for instance, in the case when (1) and (2) elementarily imply that the group $G$ is topological.
Since I am a specialist in paratopological groups, not semitopological, I propose a sketch of a proof that ... |
This is a follow-up question to QMechanic's great answer in this question. They give a formulation of Wick's theorem as a purely combinatoric statement relating two total orders $\mathcal T$ and $\colon \cdots \colon$ on an algebra.
I have come across "Wick's theorems" in many contexts. While some of them are special c... |
8.9. Long Short Term Memory (LSTM)¶
The challenge to address long-term information preservation and short-term input skipping in latent variable models has existed for a long time. One of the earliest approaches to address this was the LSTM [Hochreiter.Schmidhuber.1997]. It shares many of the properties of the Gated Re... |
The problem can be slightly generalized as follows: given a set $I\subseteq\{1,\dots,n\}$ and a set $S$ of lists of $n$-tuples, find a permutation $\sigma:\{1,\dots,|I|\}\to I$ such that each $l\in S$ is sorted according to the lexicographical preorder that compares the $\sigma(1)^\text{th}$ component, and then the $\s... |
The change of Gibbs energy at constant temperature and species numbers, $\Delta G$, is given by an integral $\int_{p_1}^{p_2}V\,{\mathrm d}p$. For the ideal gas law $$p\,V=n\,RT,$$ this comes down to $$\int_{p_1}^{p_2}\frac{1}{p}\,{\mathrm d}p=\ln\frac{p_2}{p_1}.$$ That logarithm is at fault for a lot of the formulas i... |
This might be a naive question, but when applying a implicit discretization to a PDE with a source term, should the source be averaged in time? For example if we take the diffusion equation with a non-linear source term,
$$ u_t = u_{xx}+s(x,t,u) $$
We can apply the following central difference implicit scheme to the di... |
There are a few issues here, but I fear that the interpolation error of integration may be the least of them.
To start, if this work is done in a regulatory context one must use whatever procedures are required. So if there are regulatory requirements for how to estimate integration error then those take practical prec... |
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Higher harmonic flow coefficients of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2016-09)
The elliptic, triangular, quadrangular and pentagonal anisotropic flow coefficients for $\pi^{\pm}$, $\mathrm{K}^{\pm}$ and p+$\overline{\mathrm{p}}$ in Pb-... |
A
conic in the plane $\mathbb{R}^2$ is the zero set of aquadratic polynomial in two variables:
$$ \,\, a_1 x^2 \,+\, a_2 xy \,+\, a_3 y^2 \, +\, a_4 x \, + \, a_5 y \, + \, a_6 \,.$$
Geometrically, a conic can be either a circle, an ellipse, a hyperbola, a parabola or a union of two lines. The last case is a degenerate... |
How can we handle \[y = x_1 \oplus x_2 \oplus \cdots \oplus x_n\] in a linear MIP model? Here \(\oplus\) indicates the xoroperation.
Well, when we look at \(x_1 \oplus x_2 \oplus \cdots \oplus x_n\) we can see that this is identical to \(\sum x_i\) being
odd. A small example illustrates that:
Checking if an expression ... |
I am having trouble isolating the $x$ and $y$ to separate side in the differential equations below. Could someone give me a hint as to how to to this.
Equation 1: $$\frac{dy}{dx} - \frac{x}{y} = \frac{1}{x}$$ Equation 2: $$xy\frac{dy}{dx} = y^2$$
Mathematics Stack Exchange is a question and answer site for people study... |
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