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While the first half of the answer (v1) by John Rennie provides correct timescales, for the process we are discussing its second half is completely wrong.
Objects falling into a black hole enter the horizon in finite time by the clocks of outside observers
Let me elaborate: At this wikipedia page we can see the solutio... |
McCarthy, D., Mikkola, K., Continuity and completeness of strongly independent preorders.
Mathematical Social Sciences, 93 (2018): 141–145.
Expected utility theory has three main axioms:
completeness, continuity, and independence. Completeness is dubious in many normative settings, so what happens when completeness is ... |
Divergence and curl are two measurements of vector fields that are very useful in a variety of applications. Both are most easily understood by thinking of the vector field as representing a flow of a liquid or gas; that is, each vector in the vector field should be interpreted as a velocity vector. Roughly speaking, d... |
Assuming a race car drives around in a circle of radius r, center (0,0), linear velocity v, and ignoring centrifugal forces and friction I can calculate the position at any time, t. Angular velocity
$$\omega = \frac{v}{r}$$
degrees turned through (assume at time t = 0 car is at position (r,0)):
$$\theta = \omega t$$
So... |
Learning Outcomes
Evaluate an algebraic expression given values for the variables. Recognize given values in a word problem and evaluate an expression using these values.
There are many formulas that are encountered in a statistics class and the values of each variable will be given. It will be your task to carefully e... |
The line bundle $\mathcal O_{\mathbb CP^1}(-1) = \{(z,\ell)|~z \in \ell \} $ is a submanifold of $\mathbb C^2 \times \mathbb CP^1$ with bundle map the projection. Thus we can the restrict the coordinates of $\mathbb C^2 \times \mathbb CP^1$ to coordinates of $\mathcal O(-1)$.
What about its powers? Are they also such s... |
Statistical significance is junk science, and its big piles of nonsense are spoiling the research of more than particle physicists.Wow. It's remarkable because with this deep misunderstanding of the very key part of any rational thinking, this Gentleman can't possibly understand anything about the proper verification o... |
We present a search for new heavy particles, $X$, which decay via $X rightarrow WZ \to e\nu +jj$ in $p{\bar p}$ collisions at $\sqrt{s} = 1.8$ TeV. No evidence is found for production of $X$ in 110 pb$^{-1}$ of data collected by the Collider Detector at Fermilab. Limits are set at the 95% C.L. on the mass and the produ... |
I am interested in learning more about testing for the bivariate probit model with an endogenous treatment regressor. I have figured some stuff out -- summary below, since I don't see much on this topic -- but other questions remain.
Here's the setup. Suppose I have a binary outcome $y_1$, which depends on $x_1$, the e... |
Mobius function
The Mobius function is a multiplicative number theoretic function defined as follows: In addition, .
The Mobius function is useful for a variety of reasons.
First, it conveniently encodes Principle of Inclusion-Exclusion. For example, to count the number of positive integers less than or equal to and re... |
Suppose $V$ is an $n$-dimensional vector space over $\mathbb{R}$. Let $\mathcal{L}(V,V; \mathbb{R})$ denote the set of all bilinear maps $f: V\times V \rightarrow \mathbb{R}$, i.e., $f(v,w)$ is linear in each variable separately. (We can actually show that $\mathcal{L}(V,V; \mathbb{R})$ is a vector space over $\mathbb{... |
In page 65 of his book, Spivak is saying $\int_A \varphi . |f|$ exists if
1.$\Phi$ is a partition of unity subordinate to an open cover $O$ of $A \subset \mathbf{R}^n$, $\varphi \in O$
2.Discontinuity of $f:A \rightarrow \mathbf{R} $ is measure $0$
3.$f$ is bounded in some open set around each point of $A$.
But I can't... |
Let $\{p_i\}$ be the list of primes dividing $\gcd (a,b)$. Then we can write $$a=\prod p_i^{a_i}\times \prod q_j^{\alpha_j}\quad \&\quad b=\prod p_i^{b_i}\times \prod r_k^{\beta_k}$$
Where the $q_j,r_k$ are primes disjoint from each other and from the $p_i$.
We can now compute both sides of your desired inequality. We ... |
The short but ahistorical answer is that topological string theories turn out to be examples of $(\infty,1)$-categories. The mathematical formulation of this statement is in Lurie's classification of topological field theories http://www.math.harvard.edu/~lurie/papers/cobordism.pdf (building on work of Atiyah, Segal, G... |
This study focuses on predicting the price of Airbnb rentals in Amsterdam. To model the price of Airbnb listings, we use the characteristics of Airbnb rentals and the location of the Airbnb house, room type, the number of reviews, etc. We use random sample of 500 Airbnb listings from a larger dataset on Amsterdam Airbn... |
I'm encountering a very unusual consistency when plotting the vector fields in a cylindrical waveguide mode. In a cylindrical waveguide operating in a given mode, the $\hat{e}_r$ and $\hat{e}_\phi$ components of the $\textbf{E}$-field vector (in a cylindrical coordinate system, with the z-axis aligned longitudinally wi... |
Let's look at fixed time:
$$ E(x) = V(x)e^{-ik_0x} \equiv V(x)\phi_{k_0}(x) $$
I wrote $\phi(x)$ to highlight the fact that it is a fast phase modulation of a slowly varying function.
What does slowly varying mean? It means the Fourier transform:
$$ \tilde V(k) = \int{e^{ikx}V(x)dx} $$
has its power below some $k_V$ th... |
Logic and Reasoning Challenges
This page discusses issues to be resolved in the near future. These issues pertain to relation semantics as well as inference procedures.
Contents 1 Inferring in both directions on the taxonomy 2 The problem with absence of features Inferring in both directions on the taxonomy
It is desir... |
I have to solve a problem using bipartite graphs and matchings. The way I modeled it is to have a graph $G=(A \cup B, E)$, and have the vertices in $A$ represent persons and the vertices in $B$ representing clubs. Then, the edges represent club membership.
How can I find the smallest possible value of $K$ that guarante... |
Recently, I have read a Schaum's book. General Topology, and it introduced the concept of cardinality and order, but it points out something that I really don't understand.
If $A\preceq B$ and $B\preceq A$, then $A \sim B$.
If $X\supseteq Y\supseteq X_1$ and $X\sim X_1$, then $X\sim Y$
Prove these statements are equiva... |
This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$\color{red}{\mathbf{2}}\cdot 9 = 1\color{red}{\mathbf{8}}$$ $$\color{red}{\mathbf{8}}\cdot 9 = 7\color{red}{\mathbf{2}}$$
$$\color{bl... |
We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the topology on $\mathcal{D}_K(\Omega):=\lbrace \varphi \in C^{\infty}(\Omega) : \mathrm{supp}(\varphi) \subset K \rbrace$, define... |
Frequent Links Folk theorem (game theory)
Folk theorem A solution concept in game theory Relationships Subset of Significance Proposed by Used for Example
For an infinitely repeated game, any Nash equilibrium payoff must weakly dominate the minmax payoff profile of the constituent stage game. This is because a player a... |
Newspace parameters
Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) Weight: \( k \) = \( 1 \) Character orbit: \([\chi]\) = 3600.e (of order \(2\) and degree \(1\)) Newform invariants
Self dual: Yes Analytic conductor: \(1.79663404548\) Analytic rank: \(0\) Dimension: \(1\) Coefficient field: \(\mathbb{Q}\)... |
Generally, we know that energy is conserved and there is Hamiltonian mechanics which describes particle motion by energy conservation and conversion. Quantum mechanics is completely based on the conservation of energy. And everything works out very well.But, in general relativity, there seems to be no conservation of e... |
LaTeX:Symbols
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - Commands - Packages - Help
This article will provide a short list of commonly used LaTeX symbols.
Contents Common Symbols Operators Finding Other Symbols
Here are some external resources for fin... |
With respect to the coordinates $(x^{0},x^{1},x^{2},x^{3})=(v,r,\theta,\phi)$, we have the following components of the metric tensor:
$\begin{bmatrix} g_{00} & g_{01} & g_{02} & g_{03} \\[0.3em] g_{10} & g_{11} & g_{12} & g_{13} \\[0.3em] g_{20} & g_{21} & g_{22} & g_{23} \\[0.3em] g_{30} & g_{31} & g_{32} & g_{33} \en... |
I am playing around with the mean expected lifetime formula using an exponential distribution. The following is the formula I derived:
$$\frac{{e^{\lambda t_a}}(1+λt_a)-{e^{\lambda t_b}}(1+λt_b))}{{\lambda e^{-\lambda t_a}}}$$
$t_a$ is the time up until a device has lived
$t_b$ is point up to which I want to determine ... |
LaTeX:Symbols
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - Commands - Packages - Help
This article will provide a short list of commonly used LaTeX symbols.
Contents Common Symbols Operators Finding Other Symbols
Here are some external resources for fin... |
We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number e is another area where the previous deve... |
The differential equation: $y''+\omega^2 y=0$ has as a general solution: $$y=A\cos{(\omega t)}+B\sin{(\omega t)}$$
By taking: $$A=R\cos{(\omega t_0)}$$ and $$B=R\sin{(\omega t_0)}$$ We can rewrite the general solution into: $$y=R\cos{(\omega(t-t_0))}$$
However, this is also a solution: $$y=\alpha\cos{(\omega(t-t_0))+\b... |
Topic: Computing an inverse z-transform
Question from a student
Take $ x[n] = a^n\left( u[n-2]+u[n]\right) $. We then have
$ \begin{align} X(z) &= \sum_{n=-\infty}^\infty x[n]z^{-n} \text{ (by definition of the z-transform)},\\ &= \sum_{n=-\infty}^\infty a^n(u[n-2]+u[n])z^{-n}, \\ &= \sum_{n=2}^\infty a^n(z^{-n}) + \su... |
Another method, not covered by the answers above, is
finite automaton transformation. As a simple example, let us show that the regular languages are closed under the shuffle operation, defined as follows:$$L_1 \mathop{S} L_2 = \{ x_1y_1 \ldots x_n y_n \in \Sigma^* : x_1 \ldots x_n \in L_1, y_1 \ldots y_n \in L_2 \}$$Y... |
The epsilon–delta limit definition $1$:
A function $f(x)$ from $\mathbf{R}$ to $\mathbf{R}$ has limit $L$ at point $x_0 \in \mathbf{R}$ if: $\bbox[yellow] {\text{for every $\epsilon > 0$ there exists a $\delta > 0$ such that whenever $|x-x_0| < \delta$ then}\ |f(x)-L| < \epsilon}$
Intuitive definition $2$:
A function $... |
ATLAS released an interesting preprint
Search for gluinos in events with an isolated lepton, jets and missing transverse momentum at \(\sqrt{s} = 13\TeV\) with the ATLAS detectorin which gluino pairs were searched in final states with MET, many jets, and a single lepton. There were six signal regions. The last, sixth o... |
Mark Alford's wrong paper claiming that there's nonlocality has the first followup, Anthony Sudbery's physics.hist-ph paper
He presented evidence that her 1956 song is, in fact, incorrect and that he may be an even better physicist than Doris Day. ;-)
Doris Day was singing:
Que sera seraSudbery sensibly argues: Whateve... |
the integral from 3 to 5 of (d/dt(Sqrt(2+1t^4)))dt using the fundamental theorem of calculus.thanks for any help
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Do you actually mean:
$\displaystyle \int\left(\frac{d}{dt}\sqrt{2+t^{4}}\right)\;dt$?
Wouldn't that be $\displaystyle \sqrt{2+t^{4}}\;+\;C$?
That should strike... |
Hey guys! I built the voltage multiplier with alternating square wave from a 555 timer as a source (which is measured 4.5V by my multimeter) but the voltage multiplier doesn't seem to work. I tried first making a voltage doubler and it showed 9V (which is correct I suppose) but when I try a quadrupler for example and t... |
Suppose I had a stochastic partial differential equation of the form:
$\nabla^2U=F(x,D)$, where $x\in\Omega\equiv [0,1]$ and $F(x,D)$ is a function which depends on position $x$ and a uniform random coefficient D.
An approximate the stochastic solution can be easily obtained by monte-carlo simulations. That is, I can c... |
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Hi, my name is Augusta Saraiva and I'm 15 years old, but Brilliant has my age as 21. I sent three e-mails to them but I didn't recieve any answer. Can you solve this, please?
Note by Augusta Saraiva 6 years, 2 months ago
Easy Math Editor
This discussion board is ... |
Exercise \(\PageIndex{1}\)
1. If \(\displaystyle f(x)=\sum_{n=0}^∞\frac{x^n}{n!}\) and \(\displaystyle g(x)=\sum_{n=0}^∞(−1)^n\frac{x^n}{n!}\), find the power series of \(\displaystyle \frac{1}{2}(f(x)+g(x))\) and of \(\displaystyle \frac{1}{2}(f(x)−g(x))\).
Answer
\(\displaystyle \frac{1}{2}(f(x)+g(x))=\sum_{n=0}^∞\fr... |
Defining parameters
Level: \( N \) = \( 21 = 3 \cdot 7 \) Weight: \( k \) = \( 4 \) Nonzero newspaces: \( 4 \) Newforms: \( 8 \) Sturm bound: \(128\) Trace bound: \(1\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(21))\).
Total New Old Modular forms 60 42 18 Cusp forms 36... |
If there is a positive charge $q$ at the origin of a coordinate system, the electric potential $\phi$ at a distance $r$ from $q$ is (by definition, if we take the point of zero potential at infinity):
$$\phi=-\int_{\infty}^r \vec{E}\cdot d\vec{r}$$
The dot product of $\vec{E}$ and $d\vec{r}$ is $-E\text{ }dr$ because t... |
2019 seminar talk: Isometries of combinatorial Banach spaces
Talk held by Christina Brech (Universidade de São Paulo, Brazil) at the KGRC seminar on 2019-10-10.
Abstract
The Schreier family $\mathcal{S} = \{F \in [\omega]^{<\omega}: |F| \leq \min+1\}$ induces a structure with no infinite sets of indiscernibles and this... |
I can't answer which test of a linear trend has the highest power, but this may answer some of the other questions you have regarding the special case of the ADF test.
It's illustrative to consider the regular Dickey Fuller equation (let's disregard autocorrelation for now) for $\Delta y_t:=y_t - y_{t-1}$, viz. $$y_t -... |
I believe this can be attributed to the central limit theorem, which states that a large number of samples from a population with a well-defined variance will follow a gaussian distribution. The key idea is that because of quantum mechanics, we must treat both position and momentum as
random variables; the uncertainty ... |
In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an external field with respect to the choice of spin orbitals, $\{\chi_{i}\}$.
We do this by iteratively solving the 1-elec... |
It is said that:
Abel summation and Euler summation are not comparable.
We were able to find examples of divergent series which are Euler summable but not Abel summable, for instance $$ 1-2+4-8+16-\dots$$
However, we couldn't find any example of a divergent series which is Abel summable but not Euler summable.
Do you k... |
I am now reading about the complex scaling method for solving resonance states. As far as I understand, the procedure goes like this:
Let us take the 1d potential $V(x) = A e^{-x^2} x^2 $ as an example. Here $A > 0 $ .
The full single-particle Hamiltonian is
$$ H = - \frac{\partial^2}{\partial x^2} + A e^{-x^2} x^2 . $... |
I have been learning Fourier transformation of a gaussian wave packet and i don't know how to calculate
this integral:
In the above integral we try to calculate $\varphi(\alpha)$ where $\alpha$ is a standard deviation, $\alpha^2$ is variance, $x'$ is average for $x$, $p'$ is average for $p$ and:
$$\psi_\alpha = \frac{1... |
ASO: Integral Transforms - Material for the year 2019-2020
This course is highly recommended for Differential Equations 2.
8 lectures
The Laplace and Fourier Transforms aim to take a differential equation in a function $f$ and turn it into an algebraic equation involving its transform $\bar{f}$ or $\hat{f}$. Such an eq... |
We owe Paul Dirac two excellent mathematical jokes. I have amended them with a few lesser known variations.
A.
Square root of the Laplacian: we want $\Delta$ to be $D^2$ for some first order differential operator (for example, because it is easier to solve first order partial differential equations than second order PD... |
Inclusive differential cross sections $d\sigma_{pA}/dx_F$ and $d\sigma_{pA}/dp_t^2$ for the production of \kzeros, \lambdazero, and \antilambda particles are measured at HERA in proton-induced reactions on C, Al, Ti, and W targets. The incident beam energy is 920 GeV, corresponding to $\sqrt {s} = 41.6$ GeV in the prot... |
(Written by Jonathan Sheppard)
Intro
This summer the team took the previous year team’s RF probe design and both updated and upgraded it, in an attempt to optimize its functions. As with last year’s design, we used the text
Experimental Pulse NMR: A Nuts and Bolts Approach by Eiichi Fukushima and Stephen B. W. Roeder a... |
@math: Inserting Mathematical Expressions
You can write a short mathematical expression with the
@mathcommand. Write the mathematical expression between braces, like this:
@math{(a + b) = (b + a)}
This produces the following in Info and HTML:
(a + b) = (b + a)
The
@math command has no special effect on the Info and HTM... |
The full proof can be found here. Basically, we compare the three areas that depend on $x$ in the circle of radius $1$ shown below.
Regardless of the value of $x$, we should have
$$\text{area of sector OAC} < \text{area of triangle OAP} < \text{area of sector OBP}$$ $$\frac{1}{2}x(\cos{x})^2<\frac{1}{2}(\cos{x})(\sin{x... |
I've been practicing some Fourier series questions and then verifying my answers by generating an equivalent graph on MATLAB and comparing it with the graph generated by PSpice in simulating the same circuit.
This is my working:
The Fourier series of the source current:
\$i_s\left(t\right)=1+\frac{4}{\pi}\sum_{n=1}^{\i... |
Fluvial geomorphologists describe
stream power, $\Omega$, as the 'rate of energy dissipation against the bed and banks of a river per unit downstream length'.
It is expressed as a function of water density $\rho$, channel slope $S$, discharge $Q$, and the gravitational constant $g$:
$$\Omega = \rho g Q S$$
I'm aware ge... |
Search
Now showing items 1-10 of 33
The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
If we want to calculate mean magnetisation of an equilibrium two-level-system, we know that we can resolve the identity $ \mathbf{1} = \sum_i | E_i \rangle \langle E_i |$ and giving us a uniform measure over the states of the system.
I then remember my classical stat mech, shove in some Boltzmann factors and obtain the... |
Fractional calculus
is a part of mathematics
dealing with generalisations of the derivative
to derivatives of arbitary order (not necessarily an integer
). The name "fractional calculus" is somewhat of a misnomer since the generalisations are by no means restricted to fractions, but the label persists for historical re... |
The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem to do nothing about $B=\{B,\emptyset\}$. It is disjoint from $\emptyset$. What is a disproof of the existence of $B$, an... |
Here are two recent review articles on the topic, on the subject of dark matter itself and the broader subject of the cosmological parameters.
The short story is that we have good evidence that the Universe is flat. The main evidence for flatness is anthropic: we are still here to observe the universe some ~14 Gyr afte... |
Let $(x_n)$ and $(y_n)$ be sequences such that $\lim y_n = 0$. Suppose that for all $k \in \Bbb N$ and all $m ≥ k$ we have $|x_m − x_k| ≤ y_k$. Show that $(x_n)$ is Cauchy.
I need a little guidance on how to approach the problem. As I see this is the same definition of Cauchy sequences. But I do not see how to connect ... |
What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?
Remark:
When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I refer to the Cartesian product of $N$ totally ordered sets, ${(A_i, \geq_i)}_{i=1}^{N}$ each of which is isomorphic to $\mat... |
I have troubles understanding how (and whether) Bose-Einstein condensation works in 1-D harmonic oscillator. From my calculation it seems that in limit of infinite number of particles, almost all of them are in the ground state regardless of temperature.
I have found quite a few interesting articles about this system, ... |
So, I would love to make at least
some use of my preexisting data, no matter how small, and just out of ideas. Maybe I am just a prisoner of a Kahneman-like theatre-ticket paradox, and don't know whether I should accept the losses and move on.
Consider a system of linear equations (in a very simplified form): $$ \begin... |
Simple linear regression model
$$ y_i = \alpha + \beta x_i + \varepsilon $$
can be written in terms of probabilistic model behind it
$$\mu_i = \alpha + \beta x_i \\y_i \sim \mathcal{N}(\mu_i, \sigma)$$
i.e. dependent variable $Y$ follows normal distribution parametrized by mean $\mu_i$, that is a linear function of $X$... |
For our first two papers, we essentially reused the same few examples as model problems to test our method with (sine-Gordon and Brusselator). For our next paper, my advisor wanted something different and pointed towards the Grey-Scott equations. It’s a simple reaction diffusion equation as follows
\begin{align*} \frac... |
Klaus Warzecha's answer pretty much answers your question. But I know that this subject is easier to understand if supported by some pictures. That's why I will take the same route as Klaus at explaining the concept behind why the absorption in conjugated systems is shifted to higher wavelengths but I will provide some... |
Learning Objectives
Set up a linear equation to solve a real-world application. Use a formula to solve a real-world application.
Josh is hoping to get an \(A\) in his college algebra class. He has scores of \(75\), \(82\), \(95\), \(91\), and \(94\) on his first five tests. Only the final exam remains, and the maximum ... |
Question: Let $f,g:\mathbb{R}^2 \to \mathbb{R}^2 $ be continously differentiable functions where $f\circ g$ is defined. Let $$f = (f_1,f_2)\quad\text{and}\quad g=(g_1,g_2)$$ where all $f_1,f_2,g_1,g_2:\mathbb{R}^2\to\mathbb{R}$ are functions. What is $$\frac{\partial}{\partial x_1} [f(g(x_1,x_2))]?$$
I think the deriva... |
Dark Energy Survey Year 1 Results: Cross-correlation between DES Y1 galaxy weak lensing and SPT+Planck CMB weak lensing Abstract
We cross-correlate galaxy weak lensing measurements from the Dark Energy Survey (DES) year-one (Y1) data with a cosmic microwave background (CMB) weak lensing map derived from South Pole Tele... |
Example: Let $f(x,y)=x^2+\cos{y}$. The rate of change at $f$ at $(1,0)$ in the direction of $<1,1>$ is:
A. $1$ B. $\sqrt{2}$ C. $\frac{\sqrt{3}}{2}$ D. $\pi$ E. $0$
I'm confused on how to start this. Am I supposed to find the gradient, plug in $(1,0)$ and take the dot product of this with $<1,1>$?
Thanks! |
Despite studying the general theory for quite some time, this still eludes me.
The geodesic equation can be cast in the form $$ m\frac{d^2x^\mu}{d\tau^2}=-m\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}, $$ so the connection coefficients play the role of a 4-force. The nontensorial nature of this... |
Theory of Current Distribution
In electrochemical cell design, you need to consider three current distribution classes in the electrolyte and electrodes. These are called
primary, secondary, and tertiary, and refer to different approximations that apply depending on the relative significance of solution resistance, fin... |
Sometimes it is good to use different text for a printed manual andits corresponding Info file. In this case, you can use the
conditional commands to specify which text is for the printed manualand which is for the Info file.
@ifinfo begins segments of text that should be ignoredby TeX when ittypesets the printed manua... |
Consider the space $X = \Bbb{Z}^3$, a $\Bbb{Z}$-module. Let $M = \{ \sum_{i=1}^n c_i(p_i, q_i, r_i) : \sum_{i=1}^n c_i q_i = 0,$ where $p_i, q_i, r_i$ are either prime numbers or $0 \}$. Then is $M \approx \Bbb{Z}^2$?
Clearly, $M \subset \Bbb{Z} \times 0 \times \Bbb{Z}$. If $(x, y, z) \in M$, then clearly, summing on t... |
My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the continuum hypothesis and the existence of a thick Kurepa tree $T$, I produced a family of $2^{\frak c}$ many Vitali sets $... |
I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. Let $\mathcal{B}'$ be the class of pairs $(\nu, \Lambda)$, where $\Lambda$ is a partially ordered abelian group, and $\n... |
I am looking at the finite difference methods to solve simple $u_t=a(x,t)u_{xx}$.
There are explicit, implicit, Crank Nicolson.
The latter is said to be more accurate since the local truncation error is of second order provided all expansions are done around point $t^{n+0.5}$. However, local truncation error basically ... |
I am quite new to Mathematica and I cannot find what I'm looking for on the Stackexchange or on the Mathematica help pages.
I have the following function $$U=\int^\tau_0 \bar{u}e^{-\rho t}dt \,+ \int^\infty_\tau ve^{-\rho t}dt$$ I am trying to form a plot that shows how the value of $U$ changes as $\tau$ changes, where... |
Scenario: ${\mathcal A}$ and ${\mathcal B}$ are two observables. Mathematically we model them by two Hermitian operators $A\colon H \to H$ and $B\colon H \to H$ on a separable Hilbert space. Physically they correspond to experiments $E_A$ and $E_B$, whose results are values in $Spec(A)$ and $Spec(B)$; repetitions produ... |
Here is a solution by way of the Principle of Inclusion / Exclusion (PIE). Let $M = \{m_1, m_2, m_3, \dots ,m_{35}\}$.
We start by asking how many ways we can write $M$ as a union of four subsets $A_1, A_2, A_3, A_4$ if every element of $M$ must be in at least one subset, dropping condition (ii) for now. Consider one e... |
I see that the formula giving the potential (interaction) energy of a dipole and an induced dipole is $$V=-\frac{C}{r^6}$$ where $$C=\frac{\mu_1^2 \alpha'_2}{4 \pi \epsilon_0}$$ and that the formula giving the potential (interaction) energy of an induced dipole and another induced dipole is $$V=-\frac{C}{r^6}$$ where $... |
Discretizing the Weak Form Equations
This post continues our blog series on the weak formulation. In the previous post, we implemented and solved an exemplary weak form equation in the COMSOL Multiphysics software. The result was validated with simple physical arguments. Today, we will start to take a behind-the-scenes... |
Homework Helper
4,099 5
I have some questions regarding the first two sections Einstein's paper. I'd really appreciate some guidance.
The paper can be found here: http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins_lq.pdf
In section 1 of the paper, he considers a volume of gas surrounded by reflective wall... |
Introduction
Encoding numerical inputs for neural networks is difficult because the representation space is very large and there is no easy way to embed numbers into a smaller space without losing information. Some of the ways to currently handle this is:
Scale inputs from minimum and maximum values to [-1, 1] One hot ... |
The abelianization of a group $G$ is an abelian group $A$ and a homomorphism $\varphi: G \to A$ such if $B$ is any abelian group, and $\phi: G \to B$ is any homomorphism, there is a unique homomorphism $\psi: A \to B$ (which might depend on $\phi$) such that $\psi\varphi = \phi$.
Now, I am reading some lecture notes, a... |
Our new book (NAT)
Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15
uses mainly cubical, rather than simplicial, sets. The reasons are explained in the Introduction: in strict cubical higher categories we can easily express
algebraic inverse... |
The Annals of Probability Ann. Probab. Volume 17, Number 1 (1989), 239-256. Hungarian Constructions from the Nonasymptotic Viewpoint Abstract
Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\ha... |
Introduction
Encoding numerical inputs for neural networks is difficult because the representation space is very large and there is no easy way to embed numbers into a smaller space without losing information. Some of the ways to currently handle this is:
Scale inputs from minimum and maximum values to [-1, 1] One hot ... |
I can't seem to find a good way to solve this.
I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one.
Any simple way to go about this?
Mathematics Stack Exchange is a question and answer site for people studying math at any level ... |
Exercise \(\PageIndex{1}\)
For the following functions, use the definition of a derivative to find \(f′(x)\). You may use derivative rules (will be learned in the next section to check if your answer is correct.
1) \(f(x)=6\)
2) \(f(x)=2−3x\)
3) \(f(x)=\frac{2x}{7}+1\)
4) \(f(x)=4x^2\)
5) \(f(x)=5x−x^2\)
6) \(f(x)=\sqr... |
This question already has an answer here:
Mathematica gives: $$S= -\frac{1}{12}\pi^2\log(2)+\frac{\log(2)^3}{6}+ \frac{7}{8}\zeta(3)$$ How can I prove it?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up... |
I have to find: $$\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$$ and I want to calculate it without using L'Hospital's rule. With L'Hospital's I know that it gives $1/2$. Any ideas?
Simply differentiate $f(x)=\ln(e^x +1)$ at the point of abscissa $x=0$ and you’ll get the answer. in fact this is the definition of the derivat... |
This problem reminds me of tension field theory and related problems in studying the shape of inflated inextensible membranes (like helium balloons). What follows is far from a solution, but some initial thoughts about the problem.
First, since you're allowing creasing and folding, by Nash-Kuiper it's enough to conside... |
This will be a purely mathematical treatment. It needs to be combined with some practical playing around to really "get" it.
Traveling wave
Let's start with the description of a harmonic traveling wave in one-dimension. Here "harmonic" just means the mathematical form of the wave is sinusiodal in both time and space.
F... |
I have a very general question about Lorentz transformations of electric and magnetic fields vs. 4-vectors . It arised from my previous post. I will describe the difficulty I encountered.
Information and problem: The electric and magnetic field in a system $S$ which can be boosted to a frame $\bar{S}$ moving at relativ... |
I have a series of data points $(x_i,y_i)$ which I expect to (approximately) follow a function $y(x)$ that asymptotes to a line at large $x$. Essentially, $f(x) \equiv y(x) - (ax + b)$ approaches zero as $x \to \infty$, and the same can probably be said of all the derivatives $f'(x)$, $f''(x)$, etc. But I don't know wh... |
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