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Let $G$ be a finitely generated (in my case also amenable) group and $f:G\to[0,1]$. Suppose that there is a finitely additive probability measure $\mu$ on $G\times G$ and a real number $L$ such that $\int f(xsy)d\mu(x,y)\geq L$, for all $s\in G$.
Question:Does there exist a finitely additive probability measure on $G$,... |
Little explorations with HP calculators (no Prime)
03-23-2017, 01:23 PM (This post was last modified: 03-23-2017 01:23 PM by pier4r.)
Post: #21
RE: Little explorations with the HP calculators
(03-23-2017 12:19 PM)Joe Horn Wrote: I see no bug here. Variables which are assigned values should never be used where formal va... |
We have already seen that if $t$ is time and an object's location isgiven by ${\bf r}(t)$, then the derivative ${\bf r}'(t)$ is thevelocity vector ${\bf v}(t)$.Just as ${\bf v}(t)$ is a vector describing how ${\bf r}(t)$ changes,so is ${\bf v}'(t)$ a vector describing how ${\bf v}(t)$ changes,namely, ${\bf a}(t)={\bf v... |
Let $T(n)$ denote the total number of updates to the variable when $n$ people have entered the room. For example, with $n=3$ there will be $3!=6$ possible orders where the heights are $1, 2, 3$. Notice that once the height 3 person enters, no further updates will occur, so let's group the six possible arrangements by w... |
What is the Jacobian matrix?
What are its applications?
What is its physical and geometrical meaning?
Can someone please explain with examples?
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.Sign up to... |
You are given an undirected unweighted graph consisting of $$$n$$$ vertices and $$$m$$$ edges.
You have to write a number on each vertex of the graph. Each number should be $$$1$$$, $$$2$$$ or $$$3$$$. The graph becomes beautiful if for each edge the sum of numbers on vertices connected by this edge is odd.
Calculate t... |
Joint work with Øystein Linnebo, University of Oslo.
J. D. Hamkins and Ø. Linnebo, “The modal logic of set-theoretic potentialism and the potentialist maximality principles,” to appear in Review of Symbolic Logic, 2018.
@ARTICLE{HamkinsLinnebo:Modal-logic-of-set-theoretic-potentialism, author = {Hamkins, Joel David and... |
Please forgive me if this question is ill-defined. It's late here and I want to ask the question whilst it's still fresh in my mind. Motivation:
Suppose we have a group $G$ given by a presentation $$P_G=\langle a, b\mid a^2, b^3, (ab)^2\rangle.$$ Then $P_G$ has the (oriented) graph $\Gamma_G$ in Figure 1:
(This image c... |
A dispersion relation tells you the form of $\omega (k)$. Since $E = \hbar \omega$ and $P = \hbar k$ you can see it as a relation between the energy and the momentum.
Since we have from special relativity that $$ E^{2} = p^{2}c^{2} + m^{2}c^4$$
it is clear that we have $E = Pc$ for a photon. Also since the total energy... |
In Markowitz' portfolio theory we can construct portfolios with the minimum variance for a given expected return (or vice versa). Across expected risks, this traces out the well-known efficient frontier.
To find the so-called tangency portfolio, we look to solve:
$$\max_x \frac{\mu^T x}{\sqrt{x^T Q x}}$$
Following Tütü... |
Variable Importance in Random Forests can suffer from severe overfitting Predictive vs. interpretational overfitting
There appears to be broad consenus that random forests rarely suffer from “overfitting” which plagues many other models. (We define
overfitting as choosing a model flexibility which is too high for the d... |
Let me start off by saying that I am a complete newbie to Mathematica, so I don't really know what I'm doing.
For my assignment I have to find the numerical probability of a particle in a harmonic oscillator potential in between quantum numbers $n=0$ and $n=5$. For simplicity's sake, I am only trying to find the probab... |
I'm calculating expected loss on fixed-income using actual default probabilities and risk-free rate as the discount factor. I understand this is not theoretically correct. In absence of risk-neutral probabilities, are there any alternative rates I can use (besides risk-free rate) as the discount factor?
Yes, I solve th... |
I've mostly worked with superconducting quantum computers I am not really familiar with the experimental details of photonic quantum computers that use photons to create continuous-variable cluster states such as the one that the Canadian startup Xanadu is building. How are gate operations implemented in these types of... |
D. D. Blair, J. D. Hamkins, and K. O’Bryant, “Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers,” Mathematics arXiv, 2019. (under review)
@ARTICLE{BlairHamkinsOBryant:Representing-ordinal-numbers-with-arithmetically-interesting-sets-of-real-numbers,
author = {D. Dakota Blair and Joel Dav... |
Having $$f(n) = \sum_{k=0}^n g_n(k), \; g_n(x) = \min(2^x, 2^{2^{n-x}})$$ I want to know whether $\mathcal O(f(n)) \subsetneq \mathcal O(2^n)$. Since $g_n(x) \le 2^x$ it is at least $f(n) \in \mathcal O(2^{n+1}-1) = \mathcal O(2^n)$.
Let $x_n$ be the maximum point of $g_n(x)$, which is where $2^x = 2^{2^{n-x}}$. We get... |
Refractive index manifestly plays a role in Mie scattering: if the suspended colloids have the same refractive indexand characteristic impedance as the surrounding fluid, the whole system is electromagnetically homogeneous, and there is no scattering. For nonmagnetic materials, this statement is the same as that of a h... |
The integral $$\int_0^\infty \frac{dx}{1 + x^4} = \frac{\pi}{2\sqrt2}$$ can be evaluated both by a complex method (residues) and by a real method (partial fraction decomposition). The complex method works also for the integral $$\int_0^\infty \frac{dx}{1 + x^3} = \frac{2\pi}{3\sqrt3}$$ but partial fraction decompositio... |
☕ Mathematic calculations to cool coffee
How many ice cubes do you need to
quickly cool down a hot beverage to its perfect drinking temperature?
Lets use physics to find the exact number of ice cubes to drop into a scalding hot cup of coffee or tea to make it the perfect temperature.
The beverage
The hot beverage must ... |
The genetic variance of a quantitative trait (the quantitative trait in question is fitness) can be express as the sum of two components, the dominance and additive variance:
$$\sigma_D^2 + \sigma_A^2 = \sigma^2$$
, where $\sigma$ is the genetic variance, $\sigma_D^2$ is the dominance variance and $\sigma_A^2$ is the a... |
There is no mistake on your exercise sheet.
To find a solution, you have to postulate a linear dependence of the $y^i$ from the $x^j$, $y^i= a^i_j x^j$,where the matrix $a$ of coefficients $a^i_j$ is constant. You have $x^j = (a^{-1})^j_i y^i$.
Now, simply express the initial terms as a function of $y^i$ and express di... |
Answer
$$\frac{\cot\alpha+1}{\cot\alpha-1}=\frac{1+\tan\alpha}{1-\tan\alpha}$$ The identity is proved by representing the left side in terms of $\tan\alpha$.
Work Step by Step
$$\frac{\cot\alpha+1}{\cot\alpha-1}=\frac{1+\tan\alpha}{1-\tan\alpha}$$ We find the left side comprising of only $\cot\alpha$, while the right s... |
Let $A \in M_n(\Bbb R)$.
How can I prove, that
1)if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2)if $A$ is orthogonal, then all eigenvalues are equal to $-1$ or $1$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.... |
The first series is positive, so it either converges absolute, or diverges. My gut reaction was to try some back-of-the-envelope comparison.
Let's see: $\sum \frac{1}{e^{k}}$ converges, and $\frac{1}{e^{k^3}}\leq \frac{1}{e^k}$, so $\sum\frac{1}{e^{k^3}}$ converges. The exponential dominates $\sqrt{k}$, so personally, ... |
Any mathematics symbols and processes to use the numbers 3, 5, 6, and 7 once each to get 100.
closed as too broad by ffao, Alconja, Glorfindel, Rubio♦ Feb 14 '18 at 22:46
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct question... |
Actually, there is an exact meaning, but it is not always used in that sense. For two functors $\mathsf F,\mathsf G:\mathscr A\to \mathscr B$ a
natural transformation is a morphism of functors $\eta:\mathsf F\to\mathsf G$ that is compatible with the functors in the obvious (sic!) way.
For instance if $\mathsf F={\rm id... |
I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand what fidelity mean).
My question is about inner product. How can I calculate the inner product of two quantum registers whi... |
Answer
$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ We simplify the left side and find that the expression is an identity.
Work Step by Step
$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ The left side is more complicated. We would ... |
57 0
Hello.
There is no agreement on the meaning of terms electrochemical potential and chemical potential (see for example http://web.mit.edu/6.730/www/ST04/Lectures/Lecture26.pdf"). While proper definitions would call chemical potential to -i.e., the variation in energy if the mass were not charged- and electrochemic... |
Rabbits, triangles, and triplets. These three things are linked by an important series often characterised by shells and flowers. But what do the Fibonacci numbers (and their subsequent sequence) have to offer the world of finance?
In mathematics, the Fibonacci series refers to the ordered sequence of numbers described... |
ABC for Socks Posted on 0 Comments
The central concept of Bayesian inference would be
Exploration of posterior $\pi(\theta\mid x^\obs)$ may require to produce sample
distributed from $\pi(\theta\mid x^\obs)$. MCMC is the workhorse of practical Bayesian analysis, except when product
well-defined but numerically unavaila... |
Answer
$\theta $ lies in the Second Quadrant or Quadrant-II.
Work Step by Step
The trigonometric ratios are as follows: $\sin \theta =\dfrac{y}{r} \\ \cos \theta =\dfrac{x}{r} \\ \tan \theta =\dfrac{y}{x}\\ \csc \theta =\dfrac{r}{y} \\ \sec \theta =\dfrac{r}{x} \\ \cot \theta =\dfrac{x}{y}$ where, $ r=\sqrt {x^2+y^2}$ ... |
Roughly speaking there are two ways for a series to converge: As in the case of $\sum 1/n^2$, the individual terms get small very quickly, so that the sum of all of them stays finite, or, as in the case of $\ds \sum (-1)^{n-1}/n$, the terms don't get small fast enough ($\sum 1/n$ diverges), but a mixture of positive an... |
We turn first to
counting. While this sounds simple, perhaps toosimple to study, it is not. When we speak of counting, it is shorthandfor determining the size of a set, or more often, the sizes of manysets, all with something in common, but different sizes depending onone or more parameters. For example: how many outco... |
Digital to Analogue Converter (DAC) DAC Theory
A digital to analogue converter takes a series of digital inputs (a string of 1s and 0s, in our case there will be 8 of them like 10011001) and converts it into an analogue output. You see DACs in every digital audio device (MP3 players, CD players) as these all store musi... |
In my research, I work with certain finitely presented quotients of Coxeter groups. These are the automorphism groups of abstract polytopes, which are combinatorial generalizations of "usual" polytopes. (Essentially, an abstract polytope is an incidence complex.) Now, in this context, there is a useful combinatorial op... |
FINAL EDIT : This edit cleans up the first proof (and simplifies it -- there are no longer any references to the free nilpotent group) and adds some remarks to the second proof following the discussion in the comments.
PROOF 1.
Here's a low-tech way to see that a surface group is not free (though cohomology is secretly... |
What is the difference between Base Correlation and Implied Correlation for a CDO tranche?
An implied correlation $\rho_i(k_1,k_2)$ is a correlation that matches the $(k_1,k_2)$ tranche price $P_{k_1}^{k_2}$ (usually computed under a gaussian or student t copula)
$$ C(k_1,k_2,\rho_i(k_1,k_2)) = P_{k_1}^{k_2} $$
For mez... |
Better viewed on this Dropbox site:
The Hypergeometric Distribution is usually explained via an urn analogy and formulated as the ratio of “favorable outcomes” to all possible outcomes:
\[
\displaystyle \boxed{P(x=a;N,A,n,a) = \frac{{A \choose a} \cdot {N-A \choose n-a} }{{N \choose n}}} \]
where \(N\) is the total num... |
Originally posted on November 12th, Gordon Kane's answers added in the comments on November 16th By Gordon Kane, Distinguished University Professor of Physics, University of Michigan and a Lilienfeld Prize winner I want to thank Luboš Motl for his interest and for inviting me to summarize our rare decay predictions for... |
What is electrostatic self-energy?
The self energy of a particle means the energy possessed due to interactions between the particle an the system it is part of.
. It is simply called the electrostatic potential energy stored in the system of charges. In electrostatics, self energy of a particular charge distribution i... |
For massless spinors case we can decompose momentum into Weyl sub-parts as
$$p = \lambda_{a}\tilde \lambda_{\dot a}.$$
But for the case of massive fermions can I do something like this? Decompose them into Weyl subparts with some additional terms? If so, how?
Why do I need it? I am performing a twistor transform for th... |
Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA
For $ p\geq 2 $, we prove local wellposedness for the nonlinear Schrödinger equation $ (i\partial _t + \Delta)u = \pm|u|^pu $ on $ \mathbb{... |
Linearity Theory
There are three definitions we discussed in class for linearity.
Definition 1 A system is called linear if for any constants $ a,b\in $ all complex numbers and for any input signals x 1( t), x 2( t) with response y 1( t), y 2( t), respectively, the system's response to ax 1( t) + bx t 2( ) isay t 1( ) ... |
If $R$ be a DVR(discrete valuation ring) with uniformizer $\pi$, then prove that $R[\sqrt{\pi}]$ is a DVR.
How shall I begin, first do I have to find a candidate for the uniformizing element of $R[\sqrt{\pi}]$, what about $\sqrt{\pi}$ ? There are also $3$ conditions for a DVR:
$\bullet$ Noetherian property
$\bullet$ In... |
Taylor series
A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point
$$f(a) \approx \sum\limits_{n=0}^{\infty}{\frac{f^{(n)}(a)}{n!}(x-a)^n}$$
where $f^{n}(a)$ donates the $n^{th}$ derivative of $f$ evaluated at t... |
I am trying to simulate the prices of bond indexes (e.g. Barclays Aggregate, IBOXX sovereign, IBOXX corporates) using Monte Carlo assuming that they follow the SDE given by the
Hull-White model (one-factor model):
$ dS_t = (\theta_t - \alpha_t S_t)dt +\sigma_t dW_t $
where
$\theta, \alpha$ and $\sigma$ are time-depende... |
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Long-range angular correlations of π, K and p in p–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
(Elsevier, 2013-10)
Angular correlations between unidentified charged trigger particles and various species of charged associated particles (unidentified particles, pions, kaons, protons ... |
Trying to perform system identification in the following state-space model $$ \begin{bmatrix} x_{1}(n)\\ x_{2}(n) \\ x_{3}(n)\end{bmatrix}=\begin{bmatrix} a_{11} && a_{12} && a_{13} \\ a_{21} && a_{22} && a_{23} \\ a_{31} && a_{32} && a_{33} \end{bmatrix} \begin{bmatrix} x_{1}(n-1)\\ x_{2}(n-1) \\ x_{3}(n-1)\end{bmatri... |
1) Let $L$ be a language. Let $k \in \mathbb{N}$ and let $R$ be a $k$-ary relational symbol that is
not in $L$. Set $L' = L \cup \{ R \}$. Let $M$ be an $L$ structure and $F[x_1,....,x_k]$ be a formula. Let $M'$ be an expansion by definition of $M$ in the language $L'$.
Show by induction on formulas that for any $L'$-f... |
Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order
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Theorem
Let $C_n = \gen a$, that is, that $C_n$ is generated by $a$.
Then: $C_n = \gen {a^k} \iff k \perp n$ Proof Necessary Condition
Let $k \perp n$.
$\exists u, v \in \Z: 1 = u k + v n$
So $\forall m \in... |
I've been trying to implement the Metropolis-Hastings algorithm for a while but there seems to be something weird going on.
This algorithm does not need as much of the statistics to understand, and has the step size h.
I am working with the interval $(-\infty,\infty)$, if the interval is $[a,b]$ the module will define ... |
Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero commutative ring. I know that it's true if $A$ is Noetherian or integral domain. I thought it was not true in general bu... |
I'm interested in maximizing a function $f(\mathbf \theta)$, where $\theta \in \mathbb R^p$.
The problem is that I don't know the analytic form of the function, or of its derivatives. The only thing that I can do is to evaluate the function point-wise, by plugging in a value $\theta_*$ and get a NOISY estimate $\hat{f}... |
(This answer uses the second link you gave.)
$\newcommand{\Like}{\text{L}}\newcommand{\E}{\text{E}}$Recall the definition of likelihood: $$\Like[\theta | X] = \Pr[X| \theta] = \sum_Z \Pr[X, Z | \theta]$$where in our case $\theta = (\theta_A, \theta_B)$ are the estimators for the probabilitythat coins A and B respective... |
For the following (related to a binary tree complexity question):
$$f(n) = \sum_{h=0}^{\lg{}n} h2^h$$
Is there any way to express this only in terms of $n$? Or approximate it?
Put in another way, I figure at worst for an upper bound, we could guess at it in the following way:
$f(n) = 0 + (1 \cdot 2) + (2 \cdot 2^2) + (... |
The moment of inertial can be calculated for any axis. The knowledge about one axis can help calculating the moment of inertia for a parallel axis. Let \(I_{xx}\) the moment of inertia about axis \(xx\) which is at the center of mass/area. The moment of inertia for axis \(x'\) is \[I_{x'x'} = \int_{A} r'^{2} dA = \int_... |
OK, My book has a proof that a continious function defined on $[0,1]$ attains all values between $f(0)$ and $f(1)$ using some ultra case bashy stuff, but I have two different proofs, is those correct ?
(a) Let the desired value be $m$. We prove that there exists a sequence of reals $\{a_i\}_{i=0}^{\infty}$ such that $l... |
The absolute viscosity of many fluids relatively doesn't change with the pressure but very sensitive to temperature. For isothermal flow, the viscosity can be considered constant in many cases. The variations of air and water as a function of the temperature at atmospheric pressure are plotted in Figures 1.8 and 1.9.
S... |
We can write any wave function as$$\psi(\vec x, t) = \sqrt{\rho(\vec x,t)}\exp{\left[\frac{iS(\vec x,t)}{\hbar}\right]}$$for $S$ real and $\rho >0$. Here we interpret $\rho$ as the probability density. With the definition of the probability flux as$$\vec j(\vec x,t) \propto \psi^*\nabla\psi ,$$Sakurai shows that for th... |
Why are the magnetic moment and the angular moment related? I've always read everywhere that they are related but found nowhere a satisfactory explanation of the cause
Let's first look at the classical situation. A charged particle moving round a circular loop had an angular momentum and because it is also a current, i... |
Abstract
$\displaystyle H_n(\Omega BG {}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Tor}_{n-1}^{e.kG.e}(kG.e,e.kG),$
$\displaystyle H^n(\Omega BG{}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Ext}^{n-1}_{e.kG.e}(e.kG,e.kG).$
Further algebraic structure is examined, such as products and coproducts, restriction and St... |
For city we have simplified its weather forecasting as such. If it rains then the probability for rain the next day is $0.2$. If its sunny then the probability for sunny day the next day is $0.7$.
Vector $$x_{k}=\begin{bmatrix}\text{probability for sunny weather at day } k \\ \text{probability for rainy weather at day ... |
Proving the formula with Taylor Series The Power series and the Taylor Series:
First, let's see the definition of a Power series at 0:
$$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ... + a_\infty x^\infty$$
Which is:
$$f(x) = \sum_{n=0}^\infty a_n x^n$$
How to find a Taylor series?
In functions whose derivative eventually... |
As @Trevor Wilson said, $\vdash$ which is named “turnstile” or “right tack”, belongs to the meta‑language, however, it's not always a syntactic consequence operator, it also is a semantic consequence (what @Trevor Wilson said is $\models$), at least in type theory. The model is not only ⊨ ,
U+22A8, named TRUE in the Un... |
Let $M$ complex manifolds admitting a smooth, positive, proper plurisubharmonic exhaustion, $\rho:M\to[0,\infty)$, whose we have complex Monge-Ampere foliation $(\partial\bar\partial \rho)^n=0$. Patrizio, Giorgio;and Wong, Pit Mann in
Stability of the Monge-Ampère foliation,Mathematische Annalen,March 1983, Volume 263,... |
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Trigonometry Trigonometry (from the Greek trigonon = three angles and metron = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan).... |
You are here Basic Electric Guitar Circuits 2: Potentiometers & Tone Capacitors Part 2: Potentiometers and Tone Capacitors What is a Potentiometer?
Potentiometers, or "pots" for short, are used for volume and tone control in electric guitars. They allow us to alter the electrical resistance in a circuit at the turn of ... |
Solutions are homogeneous mixtures containing one or more solutes in a solvent. The solvent that makes up most of the solution, whereas a solute is the substance that is dissolved inside the solvent. Relative Concentration Units
Concentrations are often expressed in terms of relative unites (e.g. percentages) with thre... |
Since you want "to convert regex to DFA in less than 30 minutes", I suppose you are working by hand on relatively small examples.
In this case you can use
Brzozowski's algorithm $[1]$, which computes directly the Nerode automaton of a language (which is known to be equal to its minimal deterministic automaton). It is b... |
I know that for Stochastic Gradient Descent, one picks a data point $(x_n, y_n)$ at random from the training set $S_N$ and then updates the parameter of the model in question.
If the cost function is:
$$J(w; S_N) = \frac{1}{N} \sum^{N}_{n = 1} J(w;x,y) = \frac{1}{N} \sum^{N}_{n = 1} Loss(w;x,y) = \frac{1}{N} \sum^{N}_{... |
Recall that any Riemannian manifold $(M,g)$ admits a unique symmetric metric connection $\nabla$ by the fundamental theorem of Riemannian geometry, and it is determined by the Koszul formula:
$$2g(\nabla_Y X, Z) =Xg(Y, Z) + Yg(X, Z) - Zg(X, Y) -g([X, Z], Y) - g([Y, Z], X) - g([X, Y], Z)$$
Suppose now that we had a comp... |
Plotting surfaces, it's pretty much clear that we will have the following figure.
Stokes theorem states that the path integral of a vector field around those dotted line is equal to the the surface vector integral of the curl of the field bound by surface. i.e.$$ \oint_\gamma \vec F \cdot dr = \iint_\Omega \nabla \time... |
Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
closed as primarily opinion-based by Daniel W. Farlow, Najib Idrissi, user91500, LutzL, Jonas Meyer Apr 7 '15 at 3:40
Many good questions generate some degree of opinion based... |
I want to solve a second order differential equation in the interval[-1:1], which does not have a analytic solution, \begin{eqnarray} y''(x) &=& k \phi^2(x)y(x) \\ \phi(x) &=& \frac{1}{2}\left(1-\sin\left[ \frac{\pi}{2}x\right] \right)\\ y'[1] &=& 1 \\ y[1] &=& 1 \end{eqnarray}
It is possible to get numerical solution ... |
A vector is a quantity consisting of anon-negative magnitude and a direction. We could represent a vector intwo dimensions as $(m,\theta)$, where $m$ is the magnitude and$\theta$ is the direction, measured as an angle from some agreed upondirection. For example, we might think of the vector $\ds(5,45^\circ)$ as represe... |
The pre-exponential factor (\(A\)) is part of the Arrhenius equation, which was formulated by the Swedish chemist Svante Arrhenius in 1889. The pre-exponential factor is also known as the
frequency factor, and represents the frequency of collisions between reactant molecules. Although often described as temperature ind... |
It is not very clear to me what you mean by "intersection of Hilbert Class Fields [...] is discussed". The theory of Complex Multiplication (see, for instance, Serre's short note in Cassels and Frohlich's
Algebraic Number Theory, or Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, or directly the bible... |
Ingo Blechschmidt already explained in the comments why we should expect a negative answer to the question (for most readings of "constructive logic"). Namely, if classical arithmetic proves $\forall n \in \mathbb{N} . \exists k \in \mathbb{N} . \phi(n, k)$, where $\phi(n,k)$ is quantifier-free, then so does intuitioni... |
I was trying to compute the product
$$ P_{a,b} = \prod_{n=1}^\infty(an + b), $$
after I computed
$$ P_{1,b} = \prod_{n=1}^\infty(n + b) = \frac{\sqrt{2\pi}}{\Gamma(b+1)}, $$
and the well-known
$$ \prod_{n=1}^\infty a = \exp\left\{\log(a)\sum_{n=1}^\infty n^0 \right\} = \exp\left\{\log(a)\zeta(0) \right\} = a^{-1/2}. $$... |
I try to calculate the age of the universe with the FLRW model: $$ H(a) = H_0 \sqrt{\Omega_{\mathrm{R},0} \left(\frac{a_0}{a}\right)^4 + \Omega_{\mathrm{M},0} \left(\frac{a_0}{a}\right)^3 + (1-\Omega_{\mathrm{T},0}) \left(\frac{a_0}{a}\right)^2 + \Omega_{\Lambda,0}}. $$
I set $\Omega_{\mathrm{M},0} = 0.317$ (matter den... |
This is again a question in the context of this paper about the Exact Renormalization Group. On p 23 and the following few pages, it is explained that for a $\lambda \phi^4$ bare action at the bare scale $\Lambda_0$, after integrating out degrees of freedom and assuming a small coupling $\lambda$, the effective action ... |
In a previous question, link, I asked about how I could most effectively do a Fourier Transform of a radial function given at certain values and which we knew the asymptotical behaviour of. The Fourier transform reading$$\frac{4\pi}{q}\int^{\infty}_0 dr\, r \sin(qr) f(r).$$I tried several ways and ended up choosing FFT... |
In quantum computation, what is the equivalent model of a Turing machine? It is quite clear to me how quantum
circuits can be constructed out of quantum gates, but how can we define a quantum Turing machine (QTM) that can actually benefit from quantum effects, namely, perform on high-dimensional systems?
In quantum com... |
Here is an elementary proof of your equality.
In the picture below,you see that the segment $A_2 P$ is the path-length difference between the ray reaching the point $P$ from the slit $A_2$ and the ray from the slit $A_1$ .I the triangle $A_1 PB$ the edges $PA_1$ and $PB$ are equal, and if the angle $A_1 PB$ is small, t... |
Strengthening weak measurements for qubit tomography and multitime correlators Justin Dressel Schmid College of Science and Technology
Institute for Quantum Studies
Chapman University
January 16, 2019
Justin Dressel (PI)
José Raúl Gonzales Alonso (postdoc)
Razieh Mohseninia (postdoc)
Shiva Barzili (grad student)
Lucas ... |
I have a question regarding these strikingly similar problems with contradicting solutions. This is somewhat long, so prepare
Probblem 1
Consider a bag of ten coins, nine are fair, but one is weighted with both sides heads. You randomly select a coin and toss it five times. Let $2s$ denote the event of selecting the we... |
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... |
I'm want to understand the concept of etale morphism of schemes using following definition:
A morphism $f: X \to Y$ is etale iff it is
* flat *(1), (2) and has the locally of finite type (3): separable field condition
Here (1), (2) and (3) mean:
(1) For every $x \in X$ the induced morphism $f_x ^{\#}:\mathcal{O}_{Y, f(... |
OFDM belongs to the class of multicarrier modulation schemes. OFDM decomposes the transmission frequency band into a group of narrower contiguous subbands (carriers), and each carrier is individually modulated. You can implement this type of modulation with an inverse fast Fourier transform (IFFT). By using narrow orth... |
This looks right, although I would emphasise that it is not really best practice to have to ask this question at this stage. The whole point of doing a particularly simple example is so that you can confirm that it's doing what you've
already calculated analytically. It's quite important to do the analytic bit first to... |
Club sets and stationary sets Closed and unbounded subsets of ordinals, more commonly referred to as club sets, play a prominent role in modern set theory. We intuitively think of clubs as the "large" subsets of $\kappa$ and the stationary subsets as the "not small" subsets of $\kappa$, though this is sort of a boring ... |
In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))$ with $\mathbb{Q}$ being the risk-neutral measure. For example, one now would like to value a pure endowment product, ... |
Answer
a) .098 meters b) The box will not slide back down.
Work Step by Step
We first must find the acceleration on the ramp. We know that the two forces causing the block to slow down are the force of gravity and the force of friction. Thus, we find: $a = \frac{-F_gsin\theta - F_f}{m} \\ a = \frac{-F_gsin\theta - F_n ... |
In efforts to reduce gas consumption from oil, ethanol is often added to regular gasoline. It has a high octane rating and burns more slowly than regular gas. This "gasohol" is widely used in many countries. It produces somewhat lower carbon monoxide and carbon dioxide emissions, but does increase air pollution from ot... |
Seminar
Parent Program: Location: MSRI: Simons Auditorium
Let $F=\{g_t\}$ be a one-parametr diagonal subgroup of $SL_n(\mathbb R)$.
We assume $F$ has no nonzero invariant vectors in $\mathbb R^n$. Let $x\in X, \varphi\in C_c(X)$ and $\mu$ be the probability Haar measure on $X$. For certain proper subgroup $U$ of the un... |
I am reading Probabilistic counting algorithms for database applications. In the introduction an algorithm for finding an intersection is specified:
Sort A, search each element of B in A and retain it if it appears in A.
It is claimed that if a, b are number of elements in A and B, and $\alpha, \beta$ are the number of... |
Contents
Before finally looking at the Monte Carlo method itself (next lesson), we need to introduce the important concept of Estimator. But let's first start with a quick referesher on the things we learned so far.
A Quick Review
In the last chapters we have introduced the concept of population parameter. We can use a... |
Let ($A_n : n \in \mathbb{N} $) be a sequence of events in some probability space $( \Omega, \mathcal{F}, \mathbb{P} )$. Set
$A = \{ \omega \in \Omega : \omega \in A_n \text{ infinitely often} \} $ , $B = \{ \omega \in \Omega : \omega \in A_n \text{ for all sufficiently large } n \} $
Show that $ B = \cup_{n=1}^{\infty... |
For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More generally, can one describe the set of such $\alpha$ (empty/non-empty, finite/infinite etc.) if one is given $f$?
Examples:
$f... |
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