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(The following is not really an answer, or just a very partial one, but it's definitely relevant and too long for a comment.)
There is a theorem of Richard Friedberg ("The uniqueness of finite division for recursive equivalence types",
Math. Z. 75 (1961), 3–7) which goes as follows (all of this is in classical logic):
... |
this was given as an exercise: Prove that if $\sum_{n=1}^{\infty} |a_n|$ converges and $(b_n)^{\infty}_{n=1}$ is a bounded sequence, then $\sum_{n=1}^{\infty} |a_nb_n|$ converges
This is what i was thinking:
since $\displaystyle\sum a_n = \lim_{k\to\infty}\sum_{n<k} a_n$, and when multiplying by a constant it can jump ... |
Let $X$ be a compact Kähler manifold of dimension $n$ with a given Kähler metric $\omega$. Let $L$ be a hermitian holomorphic line bundle on $X$ whose metric is positive. Let $x_0\in X$.
I would like to construct a section $s\in H^0(X,L)$, so that $s(x_0)=0$ and $ds(x_0)=\alpha\neq 0$ for given $\alpha$. Moreover, I wa... |
Introduction
Chromatic aberration (CA) is one of several aberrations that degrade lens performance. (Others include coma, astigmatism, spherical aberration, and curvature of field.) It occurs because the index of refraction of glass varies with the wavelength of light, i.e., glass bends different colors by different am... |
This 18 page article seems pretty good as a historical account of who was responsible for what.
In general, the push for rigor is usually in response to a failure to be able to demonstrate the kinds of results one wishes to. It's usually relatively easy to demonstrate that there exist objects with certain properties, b... |
We next recall a general principle that will later be applied to distance-velocity-acceleration problems, among other things. If $F(u)$ is an anti-derivative of $f(u)$, then $\ds \int_a^bf(u)\,du=F(b)-F(a)$. Suppose that we want to let the upper limit of integration vary, i.e., we replace $b$ by some variable $x$. We t... |
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To send content items to your Kindle, first ensure no-rep... |
Let define $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$
Please help me prove, that $f$ reaches its lower bound in only one point $x_1$ and reaches its upper bound $x_2$ also in only one point of domain.
Calculate $x_1 + x_2$.
What I have done:
I've checked the derivative, but it's looking horrible:
The key ... |
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
### Exercise 1: Error Propagation for Simple Addition/Subtraction
The template Error Pr... |
Uplifting cardinals Uplifting cardinals were introduced by Hamkins and Johnstone in [1], from which some of this text is adapted.
An inaccessible cardinal $\kappa$ is
uplifting if and only if for every ordinal $\theta$ it is $\theta$-uplifting, meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\p... |
I have a large unbalanced panel data with 460 firms and 1259 days. The model I would like to run is below
$$ Y_{it} = \beta X_{it} + \alpha Z_{t} + \epsilon_{it} $$
where $Y_{it}$ is stock return, and $Z_{t}$ are Fama French 3 factors, and $X_{it}$ are variables of interest.
I run Fama Macbeth (FM) and double clusterin... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
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... |
Note: In the following i'm going to use t as a time variable [s] and x as a spatial variable [m].
I'm currently working with the Discrete Fourier Transform (DFT), in order to get frequency information about my input signal. To be more precise, i am using the Fast Fourier Transform (FFT) for computational efficiency, us... |
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Now showing items 1-10 of 52
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... |
The functions $$f_k(x)=\frac{x+k}{e^x}$$ are given.
Let $A(u)$ be the area that is bounded by $f_1, f_3$, the $x$-axis und the line $x=u$.
I want to check the area if $u\rightarrow \infty$.
$$$$
To calculate the area $A(u)$ do we calculate the area that is bounded by $f_1$ with endpoints the intersection point of that ... |
Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question:Let $n$ be a positive integer which is not a perfect square. Is there always a polynomial $D \in \mathbb{Z}[x]$ of degree $2$, an integer $k$ a... |
Case I: The 2 qubits are not entangled.
You can write the states of the two qubits (say $\mathrm{A}$ and $\mathrm{B}$) as $|\psi_\mathrm{A}\rangle=a|0\rangle+b|1\rangle$ and $|\psi_\mathrm{B}\rangle = c|0\rangle+d|1\rangle$ where $a,b,c,d\in\Bbb{C}$.
The individual qubits reside in two dimensional complex vector spaces... |
Following Qiaochu hint, I'll try to elaborate a bit.
Note: This is a complete rewrite of the proof to fix a flaw pointed out by Qiaochu and make it overall clearer.
First some notation. Let $S^1 = [0, 2\pi)$ with 0 and $2\pi$ identified. For two points $a, b \in S^1$ we'll denote by $a \oplus b = a + b \mod 2\pi$ (and ... |
Remarkable cardinal
Remarkable cardinals were introduced by Schinder in [1] to provide precise consistency strength of the statement that $L(\mathbb R)$ cannot be modified by proper forcing.
Contents Definitions
A cardinal $\kappa$ is remarkable if for each regular $\lambda>\kappa$, there exists a countable transitive ... |
@DavidReed the notion of a "general polynomial" is a bit strange. The general polynomial over a field always has Galois group $S_n$ even if there is not polynomial over the field with Galois group $S_n$
Hey guys. Quick question. What would you call it when the period/amplitude of a cosine/sine function is given by anot... |
Performs Infinite Impulse Response digital filtering on supplied vector. This function performs the operation:
\[ y_n = x_n c_0 + y_{n-1}c_1 + y_{n-2}c_2 \dots \]
???MATH???x???MATH??? is the input vector (argument 1) ???MATH???c???MATH??? is the coefficient vector (argument 2) ???MATH???y???MATH??? is the result (retu... |
Let's consider a sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}, f_n(x)=e^{-n|1-sin(x)|}$. Show that $f_n$ converges to $f=0$ in measure.
Attempt/Thoughts: To prove $f_n$ converges to $f=0$ in measure, I have to show that for every $\epsilon >0$, $\mu (x:|e^{-n|1-sin(x)|}| \geq \epsilon) \rightarrow 0$ as $n \... |
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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... |
So this is a pension framework. I am trying to code a system and I don't want to have to brute force this answer, but I can't figure out a clean solution.
$$Fund = \sum_{i=1}^t [\cfrac{I\cdot e^{\frac{\pi i}{12K}}}{12K} \cdot C \cdot e^{\frac{Ri}{12K}}]$$
$I = $, annual income, $K = $ pay periods per month, $C =$ Contr... |
Given triples of $n$ floating point values
$$(\min_1, \max_1, w_1), \dots, (\min_n, \max_n, w_n)$$
and a value $V$, what is a good algorithhm to assign values $v_i$ to each of the triples such that the following conditions hold?
$\min_i \le v_i \le \max_i$. $\displaystyle\sum_{i=1}^n v_i = V$. $\dfrac{v_i}{V}$ is as cl... |
Chinese Remainder Theorem =========================
Suppose are positive integers and coprime in pair. For any sequence of integers , there exists an integer x solving the following system of congruence equations:
There exists an unique modulo solution of the system of simultaneous congruences above:
in which:
M &= m_1... |
Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard deviation of momentum is $0$, but $\sigma_x$ must be finite. So we find that $\sigma_x\sigma_p=0$. Is something wrong with t... |
1,340 30
I know that ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big) = \partial_{\mu} \phi##.
Now, I need to prove this to myself. So, here goes nothing. ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big)## ## = \fr... |
The impulse response and frequency response are two attributes that are useful for characterizing linear time-invariant (LTI) systems. They provide two different ways of calculating what an LTI system's output will be for a given input signal. A continuous-time LTI system is usually illustrated like this:In general, th... |
Basic reproduction number
... the
basic reproduction numberis R0 = β/γ In cases of diseases with varying latent periods, the basic reproduction numbercan ... In epidemiology, the basic reproduction number(sometimes called basicreproductive ratio, or incorrectly basicreproductive ... The basic reproduction numberis affe... |
So in order to understand why this must be true, we have to consider what "equals" means in this context.
Assume you were talking to someone who had never heard of division before in his life. How would you explain why you can write $$\frac{1}{5}=\frac{5}{25}?$$ These fractions are equal precisely because $$25\cdot 1 =... |
Motivation:
Many interesting
irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. Many more such numbers are described for example in the wonderful book "Mathematical Constants" by Steven R. Finch. The ... |
Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain.
It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite norm $$\lVert u \rVert_{L^1(\partial\Omega)} := \sum_i\lVert {\phi_i(u\circ g_i)}\rVert_{L^1(B_i)}$$ where $\phi_i$ part... |
It is well know that if $X_1,X_2,.., X_n$ are all independent and sampled from $Poisson(\lambda)$, then $\sum_{i=1}^n{X_i}\sim Poisson(n\lambda)$.
I have a following situation which is similar to the one hererelated discussion, however, I can not comment so I can not ask for a clarification on a specific term used in t... |
You and a friend lost each other on the line to a concert, and neither is sure which of you is further ahead. Formally, each is at some integer coordinate and may only walk towards a higher coordinate or stay in place.
Assuming you and your friend are following the exact same algorithm (and no, you may not say "if (nam... |
Some notes, Paul:
The LT1006 is weak. (Output impedance seems about \$300\:\Omega\$.) But that's not the cause of the clipping you see when supplying \$V_{PEAK}=1.5\:\textrm{V}\$. You are driving your circuit directly with an uncommon input signal that has a nice DC bias exactly half-way between your rails. But that's ... |
Usually in texts about Physics that uses tensors defines them as multilinear maps. So if $V$ is a vector space over the field $F$, a tensor is a multilinear mapping:
$$T:V\times\cdots\times V\times V^\ast\times\cdots\times V^\ast\to F.$$
In texts about multilinear algebra, however, a tensor is defined differently. They... |
Definition
An equivalent way of writing it, is $[A]_{\mathcal{E}}$ is symmetric iff there exists an orthogonal basis for $\mathbb{R}^n$ consisting of eigenvectors of A.
The reason they are equivalent can be recalled from the change-of-basis formula. $$[A]_\mathcal{E} = \underset{E \leftarrow B}{P}[A]_\mathcal{B}\unders... |
in my math class we were given a list of indefinite integrals, and one of them was:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$
My working:
$$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int \frac{dx}{(x+2)\sqrt{(x+2)^2-1}}$$
Then I used the substitution $x+2=\sec t$ to get:
$$\int \frac{\tan t}{\sqrt{\sec^2 t-1}}dt=\int \fr... |
Let's recast the first example from the previous section. Suppose thatthe speed of the object is $3t$ at time $t$. How far does the objecttravel between time $t=a$ and time $t=b$? We are no longer assumingthat we know where the object is at time $t=0$ or at any othertime. It is certainly true that it is
somewhere, so l... |
We have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve—"line integrals'' are really "curve integrals''.
As with other integrals, a geometric example may be easiest to understand. Consider the function $f=x+y$ and... |
As xnor points out in his answer, this question is basically asking for the way to most evenly distribute $6^n$ results among $100$ bins, and gives a very brief description of the solution. I'll go into a
bit more detail here. If you're not interested in the proofs, skip to section 2.3 Summary
In order to construct a "... |
In the CAPM theory Beta of asset $i$ are estimated in this way:
$ \beta_i = \frac{\sigma_{im}}{\sigma^2_m} $ where $\sigma_{im} = \rho_{im} \sigma_i \sigma_m$
But all these data are historical data. So, I'm wondering what if I use
$\sigma^2_m$ <-
Implied volatilityof SP500 (VIX)
$\sigma_{im}$ <-
implied volatilityfor t... |
I'm using Area Under Curve ROC as a performance measure of my classification algorithms (logistic regressions). Since I'm going to choose the model that maximize the Area Under Curve ROC, I would like to know if AUC penalizes somehow models with too many regressors (for example, like BIC information criterion).
You men... |
Answer
The solution set is $$\{0.8751+2\pi n, 2.2665+2\pi n,3.5908+2\pi n, 5.834+2\pi n, n\in Z\}$$
Work Step by Step
$$\sin x(3\sin x-1)=1$$ 1) Solve the equation over the interval $[0,2\pi)$ $$\sin x(3\sin x-1)=1$$ $$3\sin^2x-\sin x-1=0$$ Consider the equation as a quadratic formula, with $a=3, b=-1, c=-1$ - Calculat... |
9.1. Attention Mechanism¶
In Section 8.14, we encode the source sequence input information in the recurrent unit state and then pass it to the decoder to generate the target sequence. A token in the target sequence may closely relate to some tokens in the source sequence instead of the whole source sequence. For exampl... |
From Daron Acemoglu's Introduction to Modern Economic Growth, proposition 9.4 is that:
In the overlapping-generations model with two-period lived households, Cobb-Douglas technology and CRRA preferences, there exists a unique steady-state equilibrium with the capital-labor ratio k* given by (9.15) and as long as $\thet... |
This is really just a comment, but is way too long. Since in my opinion it's in fact extremely far from an actual answer, I've made it community wiki - I don't think a reputation bonus for it would be appropriate. EDIT: apparently I don't know how to do that (my recollection is that there used to be a cw checkbox, but ... |
Let $t \in (a, \infty )$
if $f(t) \leq g(t)$ for all $t$, then $\sup f(t) \leq \sup g(t)$
Is there a simple proof for this?
Thank you!!
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 join th... |
Sure. It's a little bit lengthy though, so it might take some work to read it:P(m)=\frac{\sqrt{2 \pi n}n^ne^{-n}}{\sqrt{2 \pi m}m^me^{-m}*\sqrt{2 \pi (n-m)}(n-m)^{(n-m)}e^{-(n-m)}}p^m(1-p)^{n-m}=\frac{n^{n+1}}{\sqrt{2 \pi n}*m^{m+\frac{1}{2}}*(n-m)^{(n-m)+\frac{1}{2}}}p^m(1-p)^{n-m}...
I want to show that the binomial ... |
Introduction
I am trying to obtain the eigenvectors of a unitary matrix $M(k)$ which depends on a parameter k.
This matrix $M(k)$ has dimension 6, and while for general matrices of dimension 6 it's not possible to write down algebraic expressions for their eigenvalues, because the characteristic polynomial will also be... |
Inspired by Digitangular numbers.
The triangle of a natural number is given by
$$\triangle(n)=\frac{n(n+1)}2$$
The digitangular counterpart of a natural number is given by the sum of the triangles of the digits of the number, e.g. $\newcommand{rdig}{{\rm digi}\triangle~}$
$$\rdig(613)=\sum_{k=6,1,3}\triangle(k)=21+1+6=... |
Answer
$ v = \frac{\omega A}{\sqrt{2}}$
Work Step by Step
We find: $ \frac{1}{2}mv^2 = \frac{1}{2}(\frac{1}{2}m\omega^2A^2)$ $ \frac{1}{2}mv^2 =(\frac{1}{4}m\omega^2A^2)$ $ v = \frac{\omega A}{\sqrt{2}}$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you... |
Learning Objectives
Make sure you thoroughly understand the following essential ideas:
Describe the major reasons that solutions are so important in the practicalaspects of chemistry. Explain why expressing a concentration as " x-percent" can be ambiguous. Explain why the molarity of a solution will vary with its tempe... |
The following is essentially a contribution to point 2. Non-commutative (which can be specialized to matrices or scalars) continued fraction are used in enumeration and language theories. Two examples : Dyck and Motzkin paths.
These are two lattice paths drawn on $\mathbb{N}^2$ (the first quarter-plane).
Dyck Paths Ste... |
Contents Ray-Triangle Intersection: Geometric Solution
In the previous paragraphs we learned how to compute the plane's normal (which is the same as the triangle's normal). Next what we need to find out is the position of point P (for some illustrations we also used Phit), the point where the ray intersects the plane.
... |
In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These invariants are based on holomorphic curves and moduli spaces, but in the simplest cases, ... |
Set-theoretic arguments often make use of the fact that a particular property $\varphi$ is
local, in the sense that instances of the property can be verified by checking certain facts in only a bounded part of the set-theoretic universe, such as inside some rank-initial segment $V_\theta$ or inside the collection $H_\k... |
Euclidean Geometry • Equitable Distributing • Linear Programming • Set Theory • Nonstandard Analysis • Advice • Topology • Number Theory • Computation of Time (Previous | Next)
In the following section, we presuppose Set Theory and Nonstandard Analysis. The exponential simplex and the polynomial intex method (
inter-/ ... |
Learning Objectives
A brief look at series and parallel circuits. Also defines voltage and current divider.
Fig. 3.6.1 The circuit shown is perhaps the simplest circuit that performs a signal processing function. The input is provided by the voltage source vin and the output is the voltage across the resistor labeled
T... |
The field and its conjugate momentum
are operators. They act on the Hilbert space of the theory, which is not the space of square-integrable functions of position, as it is in single particle quantum mechanics. Rather, the configuration space of the theory is the set of all field configurations and the wavefunction is ... |
We often face the problem of having to sell an asset within a specified period of time. This problem worsens if we also lack information about our asset’s historical prices. Under this circumstance,
when should we sell our asset?
In this post I’ll resolve this dilemma based on the classic
Secretary Problem – also known... |
The Stochastic portfolio theory (SPT), a relatively new portfolio management theory, was first introduced in 1999 by Robert Fernholz. It can be combined with Machine Learning and Bayesian statistics. This allows the investor to generate trading strategies. It’s a very attractive theory for several reasons: it’s theoret... |
Answer
The testimony is going to hurt their case.
Work Step by Step
We multiply the density of the beer (which is equal to that of water) by the increase in height and by the cross-sectional area of the keg to obtain: $ \Delta m = 1000 \times .012 \times \pi \times (.2^2)=1.51kg\approx 53 \ oz$ This is more than the am... |
I am trying to compute the dBFS value of a group of samples (stereo wave file), according to this formula:
$$p_{RMS} = \sqrt{\frac{ x_1^2 + x_2^2 + \ldots}n } $$ $$dbFS = 20\log_{10}\frac{ p_{RMS}}{p_{max}}$$
The value I get is wrong, and not stable (I am using a stereo constant -18dbFS 1000H sinewave audio file). I kn... |
I have a question about calculating the maximum allowable noise for an ADC signal conditioning stage.
I have an existing signal conditioning stage as well as an ADC on a microcontroller.
I would like to use the existing stage and calculate the maximum allowable noise on the input signal in order to maintain an error of... |
Euclidean Geometry • Equitable Distributing • Linear Programming • Set Theory • Nonstandard Analysis • Advice • Topology • Number Theory • Computation of Time (Previous | Next)
The following section presupposes the chapters on Set Theory and Nonstandard Analysis. Let \(m, n \in {}^{\omega}\mathbb{N}\) and \(k \in \math... |
How to Implement the Fourier Transformation from Computed Solutions
We previously learned how to calculate the Fourier transform of a rectangular aperture in a Fraunhofer diffraction model in the COMSOL Multiphysics® software. In that example, the aperture was given as an analytical function. The procedure is a bit dif... |
11 0
I have two questions I'm a bit confused on...
1st:
Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was [tex]2.87 \times 10^{6} km[/tex] from the earth and traveling at [tex]1.20 \times 10^{4} km/h[/tex] relative to the earth. a.) At this time, what was... |
They're believed to be symmetric because quite often a normal approximation is used. This one works well enough in case p lies around 0.5.
binom.test on the other hand reports "exact" Clopper-Pearson intervals, which are based on the F distribution (see here for the exact formulas of both approaches). If we would imple... |
I tried to find an even simpler product. Here's my solution:
$$ \zeta(2) =\prod _{n=1}^{\infty } \frac{1}{\left(1-\frac{1}{4 n^2}\right) \left(1-\frac{1}{36 n^2}\right)}$$
In Mathematica
Product[ 1/(1 - 1/(4 n^2)) 1/(1 - 1/(36 n^2)), {n, 1, \[Infinity]}]
(* Out[76]= \[Pi]^2/6 *)
We can derive this from the well-known p... |
Polynomial is Continuous Theorem Proof
From Linear Function is Continuous, setting $\alpha = 1$ and $\beta = 0$, we have that:
$\displaystyle \lim_{x \to c} \ x = c$
Repeated application of the Product Rule for Limits of Functions shows us that:
$\displaystyle \forall k \in \N: \lim_{x \to c} \ x^k = c^k$ Now let $P \l... |
Fitting a logistic regression (
LR) model (with Age, Sex and Pclass as predictors) to the survival outcome in the Titanic data yields a summary such as this one:
##
## Call:
## glm(formula = Survived ~ Age + Sex + Pclass, family = binomial(link = logit),
## data = NoMissingAge)
##
## Deviance Residuals:
## Min 1Q Media... |
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the finitness part of the Hilbert 16th problem:
This problem is open even for $n=2$.
In this question, by a quadratic vector field we mean a poly... |
Here's a neat one from optimization: the Alternating Direction Method of Multipliers (ADMM) algorithm.
Given an uncoupled and convex objective function of two variables (the variables themselves could be vectors) and a linear constraint coupling the two variables:
$$\min f_1(x_1) + f_2(x_2) $$$$ s.t. \; A_1 x_1 + A_2 x... |
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Description
We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector ... |
Commercial flights are statistically quite safe (in terms of number of deaths per passenger-kilometer, only going to the moon is safer). But there are still reasons for precautions and safety regulations. An early such rule was the so-called “60-minute rule,” which required that a two-engine plane must always be within... |
High-jump High-jump cardinals are a certain kind of large cardinals. A cardinal $\kappa$ is high-jump if it is the critical point of an elementary embedding $j:V\to M$ such that $M$ is closed under sequences of length $\text{sup}\{j(f)(\kappa)|f:\kappa\to\kappa\}$. This closure condition is a weakening of the definitio... |
I think the best way to think about this is in terms of operads.
Braided monoidal categories are representations of an operad $\Pi$ in the category of (small) categories.
The category $\Pi(n)$ has objects parenthesised permutations of $\{1,\ldots,n\}$ like $(4(23))1$. The morphisms $(\sigma) \to (\tau)$ are braids $\be... |
Reflecting cardinals
Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak Kripke-Platek set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-se... |
A
reaction engine is an engine or motor which provides propulsion (thrust) by expelling reaction mass, in accordance with Newton's third law of motion. This law of motion is most commonly paraphrased as: "For every action force there is an equal, but opposite, reaction force".
Examples include both jet engines and rock... |
The problem is to determine whether the given differential operator $L[y]$, whose domain consists of all functions that have continuous second derivatives on the interval $[0,\pi]$ and satisfy the given boundary conditions, is selfadjoint.
$$L[y]=y''+\lambda y;\;\;\;\;y(0)+y'(\pi)=0,\;\;\;\;y'(0)+y(\pi)=0$$
For an oper... |
I've just begun learning capacitance, and my lecture notes have a section on calculating capacitance for capacitors in vacuum of various shapes, e.g. two parallel plates and concentric spherical shells.
For a cylindrical capacitor, comprising of a
long cylindrical conductor with radius $r_a$ and linear charge density $... |
Need help! I was working on a project when I required to use a projection operator. For an example case, I have the Bell state, $$|\psi\rangle = \frac1{\sqrt2}\left(\color{blue}{|0}0\rangle+|11\rangle\right)$$ which now I want to take to the state, $$|\psi'\rangle = |\color{blue}{0}0\rangle$$ by weeding out the states ... |
@DavidReed the notion of a "general polynomial" is a bit strange. The general polynomial over a field always has Galois group $S_n$ even if there is not polynomial over the field with Galois group $S_n$
Hey guys. Quick question. What would you call it when the period/amplitude of a cosine/sine function is given by anot... |
A brief description of the 18 electron rule
A valence shell of a transition metal contains the following: 1 $s$ orbital, 3 $p$ orbitals and 5 $d$ orbitals; 9 orbitals that can collectively accommodate 18 electrons (as either bonding or nonbonding electron pairs). This means that, the combination of these nine atomic or... |
This is not true.
For example, consider $T\mathbb{CP}^2$ which has total Chern class $c(T\mathbb{CP}^2) = 1 + 3x + 3x^2$ where $x = c_1(\mathcal{O}(1)) \in H^2(\mathbb{CP}^2; \mathbb{Z})$ is a generator. Suppose $c(T\mathbb{CP}^2) = (1 + x_1)(1 + x_2)$ for some $x_1, x_2 \in H^2(\mathbb{CP}^2; \mathbb{Z})$, then $x_1 =... |
Suppose you have the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Suppose somewhere on the interior you have point $P(m,n)$. What is the closest point on the ellipse to the $P$? Can this be found without having to solve a 4th degree polynomial?
Because of symmetry, we can assume that $P$ is in the First Quadrant or on ... |
Say we have some equations $f_1(x)=0, \ldots f_k(x)=0$ defining a variety $X$ in ${\mathbb C}^n$ (not necessarily a minimal number of generators, and not necessarily of minimal degree), and suppose we want to know the ideal $\bar{I}$ of its closure $\overline{X}$ in ${\mathbb P}^n$. A naive question:
If all the origina... |
First, to clarify, the first Lagrangian you gave is the Lagrangian describing a non-relativistic point particle moving in a
fixed background potential $(\phi,\textbf{A})$. The second Lagrangian describes the dynamics of the electromagnetic field in the presence of a fixed background source $(\rho,\textbf{j})$.
Now, the... |
I have come across a problem which is a homework indeed, but i tried to pack this question up so that it is more theoretical.
What I want to know is: If I am allowed to write energy conservation for an atom which emits a photon (when its electron changes energy for a value $\Delta E$) like this (the atom is kicked back... |
Set is Closed iff Equals Topological Closure Theorem
Let $T$ be a topological space.
Let $H \subseteq T$.
$H = \operatorname{cl} \left({H}\right)$
Let $H'$ denote the derived set of $H$.
The result follows from the definition of closure.
$\blacksquare$
Let $H^{\complement}$ denote the relative complement of $H$ in $T$.... |
The well-ordered replacement axiom is the scheme asserting that if $I$ is well-ordered and every $i\in I$ has unique $y_i$ satisfying a property $\phi(i,y_i)$, then $\{y_i\mid i\in I\}$ is a set. In other words, the image of a well-ordered set under a first-order definable class function is a set.
Alfredo had introduce... |
What is the difference between a randomly built binary search tree (using n keys )and choosing a binary search tree (of n key) from a random distribution
You want to distinguish
randomly built binary search tree
from
choosing a binary search tree from a random distribution.
Let me first tell you why this is not a well-... |
Pure three dimensional stream functions exist physically but at present there is no known way to represent then mathematically. One of the ways that was suggested by Yih in 1957 suggested using two stream functions to represent the three dimensional flow. The only exception is a stream function for three dimensional fl... |
The lecturer taught this method in my Optimization and Control Theory Class and I wasn't quite there when he named it. Could you help me out?
He gave the following example of the method in class:
Example: Solve $$ \text{max} [ f(x) = x_1 (30 - x_1) + x_2 (50 -2x_2) - 3x_1 - 5x_2 - 10x_3]$$Subject to the constraints: $$... |
Note: The original answer has a flaw in "without loss of generality". The following one is based on Section 8.2.3 of the book "Computer Algorithms" (3rd edition) by Sara Baase and Allen Van Gelder.
The right part of the theorem is called the MST property.
MST Property: Let $T$ be any spanning tree. For any edge $e \not... |
I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?
“One of the ways to look at division is as how many of the smaller number you need to make up the bigger nu... |
From Jacod and Shiryaev's Limit Theorems for Stochastic Processes, we get the following definitions.
Definitions: A process with independent increments (abbreviated PII) $X = (X_t)_{t \geq 0}$ on a stochastic basis $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ is a càdlàg adapted real-val... |
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