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Solve the equation:
You can solve the above assuming that (I'll provide a proof below):
$arcsen(x)+arccos(x)=\dfrac{\pi}{2}$
Starting from this, we can add to the right side: $2 arctan(x)$
Obtaining:
$arcsen(x)+arccos(x)=\dfrac{\pi}{2}=2 arctan(x)$
We deduce:
$arctan(x) =\dfrac{\pi}{4}$
Solving,
$tan(arctan(x))=tan(\df... |
A function $f$ is Riemann integrable on the interval $[a, b]$ if the following condition holds: For any partition $P = \{ x_{0}, ..., x_{n} \}$ of $[a, b]$, we have an $\epsilon > 0$ such that $U(P, f) - L(P, f) < \epsilon$.
Here, $U(P, f) = \sum_{i=1}^{n} M_{i} \Delta x_{i}$, for $M_{i} = sup \{ f(x) : x \in [x_{i-1},... |
First I need some notation (it's all standard I think). For a manifold $M$, let $F_nM = F_{0,n}M$ be the space of $n$-tuples of distinct points on $M$ ; let $B_nM = B_{0,n}M = F_nM / \Sigma_n$. When $M= \mathbb{R}^2$ the fundamental group of $B_nM$ is the braid group $B_n$, and that of $F_nM$ is the pure braid group $P... |
I am doing the following exercise:
Consider inviscid, incompressible, steady flow. The Kutta-Joukowski theorem $$L' = \rho_\infty V_\infty\Gamma$$ where
$L'$ is the lift per unit span $\rho_\infty$ is the freestream pressure $V_\infty$ is the velocity $\Gamma$ is the circulation taken around the body
was derived exactl... |
I was discussing L'Hôpital's Rule with a Calculus I student earlier today. I mentioned that if the limit obtained by differentiating the numerator and denominator doesn't exist, then L'Hôpital's Rule tells us nothing about the original limit.
A clear example of this is, $$\lim\limits_{ x \to \infty }{ \frac { x+\sin { ... |
Im trying to calculate Alpha using CAPM & I have data on everything necessary.
$$R_t-R_f={\alpha}+{\beta}\times(R_m-R_f)$$
i.e.
$${\alpha}=R_t-R_t-{\beta}\times(R_m-R_f)$$
In more detail, I have monthly data on returns, market returns and the risk free rate. Now lets say I'm interested in how a fund has performed in 12... |
Higgins' downloadable book Categories and groupoids has quite a lot on computing colimits of groupoids. The point is that the groupoid van Kampen theorem has the probably optimal theorem of this type in
R. Brown and A. Razak, A van Kampen theorem for unions ofnon-connected spaces, Archiv. Math. 42 (1984) 85-88. pdf
Thi... |
I would like to solve a system of second-order differential equations to describe the dynamics of a system of particles. Two Newton-like forces are responsible for the motion of each particle $i$: A force acting on each particle due to other particles $f_{i,j}$ and a stochastic term of noise $f_{\mathrm{noise}}$.
The f... |
Difference between revisions of "Capillary waves"
(more rewriting)
m (added link to (future) harmonic oscillator)
Line 83: Line 83:
where we have dropped a factor of <math>A/4</math> in the last step.
where we have dropped a factor of <math>A/4</math> in the last step.
−
The problem is thus specified by just a potentia... |
This has been bugging me for a while now...
Obviously, to calculate the volume/space occupied by a mole of (an ideal) gas, you'll have to specify temperature ($T$) and pressure ($P$), find the gas constant ($R$) value with the right units and plug them all in the ideal gas equation $$PV = nRT.$$
The problem? It seems t... |
I've written some simple code that, for various $N \geq 1$ and $k = 1,\ldots,N+1$ computes the $n$-th derivative of $\log S_N$ where $S_N(z) = \sum_{k=0}^N \frac{1}{k!}z^k$ as follows:
f[z_, N_] := Sum[z^k/k!, {k, 0, N}]g[z_, N_] := Log[f[z, N]]t = Table[Evaluate[D[g[z, 7], {z, n}]], {n, 8}]
Here I can vary $7$ and $8$... |
I have been studying a book which describes how to write the waveform of sequence that has certain chip rate $T_c$.
The sequence with chip rate $T_c$ is for example
$$X=[-1 -1 +1 +1 +1 +1 +1 +1 +1 -1 +1 -1 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1 -1 -1 +1 -1 +1 -1 +1 -1 -1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 -1 +1 +1 -1 -1 +1 -1 -1 ... |
My research group focuses on molecular dynamics, which obviously can generate gigabytes of data as part of a single trajectory which must then be analyzed.
Several of the problems we're concerned with involve correlations in the data set, which means that we need to keep track of large amounts of data in memory and ana... |
To solve the problem $x^2 \log (\frac{2}{\sqrt {2-\sqrt3} }) - 4x - 2 = x^2 \log (\frac{1}{2-\sqrt3}) - 4x - 3$, we'll first add $4x+2$ to both sides, giving $x^2 \log (\frac{2}{\sqrt {2-\sqrt3} }) = x^2 \log (\frac{1}{2-\sqrt3}) - 1$.
We'll then use the property of logarithms which states:$\log(\frac{a}{b}) = \log(a)-... |
This is a continuation of this question.
http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-1/ skip this lecture to around 25:50.
After doing dimensional analysis on $t\propto h^\alpha m^\beta g^\gamma$ Lewin concludes that:
$$\alpha = \frac{1}{2}, \gamma = -1/2, \bet... |
A Group and Its Center, Intuitively
Last week we took an intuitive peek into the First Isomorphism Theorem as one example in our ongoing discussion on quotient groups. Today we'll explore another quotient that you've likely come across, namely that of a group by its center.
Example #2: A group and its center
If $G$ is ... |
0-1 Integer Programming Task AKA:Binary Integer Programming. See:Assignment Problem, Linear Equality, Linear Inequality, Feasible Region, Convex Polytope. References 2015 (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Linear_programming#Integer_unknowns Retrieved:2015-12-24. Linear programming(LP ; also calledlinear ... |
Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh,
I would like to apply a Robin type boundary condition to the l.h.s. of the domain ($x=x_L)$, such that,
$$ \sigma_L = \left( d u_x + a u \right) \bigg|_{x=x_L} $$
where $\sigma_L$ is the boundary value; ... |
I have a ranking of items, $X = (x_1, \ldots, x_n)$. I want to obtain a random sequence $\hat{X}$, which is a permutation of $x_i$, such that the expected rank-correlation (say, Kendall) between $X$ and $\hat{X}$ is $\tau$, where $\tau$ is a parameter in range $[0, 1]$.
Assuming you have a very large number of items in... |
Hello,
do you need any constraint qualifications for the KKT conditions to be optimal for the following problem(here)
I have a linear objective, and linear inequalities, and quadratically constrained equalities. x^t * H * x + D*x + C = 0
the matrix H is a diagonal matrix and positive definite, C = 0, and D is a full of... |
Boost : Subject: Re: [boost] [documentation] Are SVG's in documentation viable now? From: Marco Guazzone ( marco.guazzone_at_[hidden]) Date: 2014-12-10 07:17:36
On Tue, Dec 9, 2014 at 6:27 PM, John Maddock <boost.regex_at_[hidden]> wrote:
> > It looks to me like the PNG's are anti-aliased, but the SVG's aren't, which >... |
To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary.
The question then is exactly the title, but I think a stronger statement is true:
For any rng $S$ there is a ring $R$ and an injective rng-homomorphism $f:S\rightarrow R$ such that for
any ring$T$ and anyrng ... |
Answer
$T = \frac{1}{\omega}$
Work Step by Step
$V = a~sin~(2\pi~\omega~t)$ We can find the period $T$: $2\pi~\omega~T = 2\pi$ $T = \frac{2\pi}{2\pi~\omega}$ $T = \frac{1}{\omega}$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hours to se... |
№ 9
All Issues Zagorodnyuk S. M.
Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1180-1190
We study a generalization of the class of orthonormal polynomials on the real axis. These polynomials satisfy the following relation: $(J_5 \lambda J_3)\vec{p}(\lambda) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite rea... |
What is the Principle of Superposition?
An electric field is created by every charged particle in the universe in the space surrounding it. The originated field can be calculated with the help of Coulomb’s law. The principle of superposition allows for the combination of two or more electric fields.
In the next section... |
This is a possible salvage for the failed attempt in this posting.
Informally this theory asserts the existence of a set $\{x|\phi\}$ for any formula $\phi$ such that when all membership symbols in it are replaced by equality symbols then every subformula of the resulting formula would hold of at least one object, and ... |
I'm wondering if there are any recursion principles more general than the following, first given by Montague, Tarski and Scott (1956):
Let $\mathbb{V}$ be the universe, and $\mathcal{R}$ be a well-founded relation such that for all $x\in Fld\mathcal{R}$, $\{y:y\mathcal{R}x\}$ is a set. Further, let $\mathbb{F}$ be a fu... |
Bid, Aveek and Bora, Achyut and Raychaudhuri, Arup K (2005)
Experimental study of Rayleigh instability in metallic nanowires using resistance fluctuations measurements from 77K to 375K. In: SPIE: Fluctuations and Noise in Materials II, 24 May, Austin, TX, USA, Vol.5843, 147 -154. Abstract
Nanowires with high aspect rat... |
Euler’s formulas for sine and cosine
$\sin \omega t = \dfrac{1}{2} (e^{+j\omega t} - e^{-j\omega t})$
$\cos \omega t = \dfrac{1}{2} (e^{+j\omega t} + e^{-j\omega t})$
Sine and cosine emerge from vector sum of three spinning numbers in Euler’s Formula,
The green vector is $\dfrac{1}{2} e^{+j\omega t}$
The pink vector is... |
A Effect size appendix
This appendix contains supplements to the effect size guideline.
A.1 Alternative approaches for simple effect size exemplar
The simple effect size exemplar demonstrates one common technique for esitmating mean differences in resposne time and the uncertainty around them (Student’s t confidence in... |
Adaptive Algorithms for Optimization of Hip Implant Positioning
This research is carried out in the framework of MATHEON supported by Einstein Foundation Berlin.
This project aims at a software environment supporting computer-assisted planning for total hip joint replacement by suggesting implant positions optimized fo... |
A transistor goes into saturation when both the base-emitter and base-collector junctions are forward biased, basically. So if the collector voltage drops below the base voltage, and the emitter voltage is below the base voltage, then the transistor is in saturation.
Consider this Common Emitter Amplifier circuit. If t... |
In other notebooks we've studied Euler's method for solving ordinary differential equations (ODEs). This method is perhaps the most simple to understand and implement and thus is very popular as teaching material. However, for scientific purposes, Euler's method is inaccurate compared to more complex methods using the ... |
If you need higher-order derivatives, you won't get good results using equidistant data points. If you can sample your function at arbitrary nodes, I would recommend using Chebyshev points, i.e. $$x_k = \cos\left( \pi \frac{k}{n} \right), \quad k=0\dots n$$ for a polynomial of degree $n$.
You can evaluate the polynomia... |
also, if you are in the US, the next time anything important publishing-related comes up, you can let your representatives know that you care about this and that you think the existing situation is appalling
@heather well, there's a spectrum
so, there's things like New Journal of Physics and Physical Review X
which are... |
I'm trying to figure out some properties of Hermitian matrices over finite fields. Namely, let $K_0=\mathbb{F}_q$ be a field with $q$ elements, and let $K=\mathbb{F}_{q^2}$. The matrix algebra $\newcommand{\M}{\mathrm{M}} \M_n(K)$ is endowed with the conjugate-transpose map, given by $(x_{i,j})^\circ:=(x_{j,i}^\sigma)$... |
Before knowing about Differential Equation and its types, let us know what a differential equation is.An equation with one or more terms that involves derivatives of the dependent variable with respect to an independent variable is known as
Differential Equation
An equation with one or more terms that involves derivati... |
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your ow... |
I was reading the following notes on maximum flow and it said the term "maximum-bottleneck (s-t)-path" but I couldn't find were it precisely defined it, so I am left guessing what it means. I am assuming it means a path from s to t with maximum total capacities once you sum the capacities across the path. However, I am... |
Question:Let us consider the Poisson problem on the square with constant source $1$$$\begin{cases} - \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\ u &= 0, \qquad \text{ on } \partial (0,1)^n \end{cases}$$By symmetry the maximum is attained in $x_0:=(1/2,\dots,1/2)$. Does there exist a formula or precise estimate for$$... |
All my experience with textbook problems of quantum mechanics shows that the energy levels associated with the bound states of a confined quantum system are discrete and sharp. For example, the energy levels of the hydrogen atom. Why is it then said that excited states of an atom have an energy width? Where does the wi... |
This question's context is time series forecasting using regression, with multivariate training data. With a regularization method like LARS w/ LASSO, elastic net, or ridge, we need to decide on the model complexity or regularization parameters. For example, the ridge $\lambda$ penalty or the number of steps to go alon... |
Filtering time-series data can be a tricky business. Many times what seems to obvious to our eyes is difficult to mathematically put into practice. A recent technique I've been reading about is called "total variance minimization" filtering or another name would be "$ \ell_1 $ trend filtering." Another member of this f... |
We proved Thévenin’s and Norton’s theorems in the previous article. This article covers the practical steps to create a Thévenin or Norton equivalent for a linear circuit.
Thévenin’s theorem Any network of resistors and sources, when viewed from a port, can be simplified down to one voltage source in series with one re... |
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Now showing items 1-10 of 24
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 ... |
Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
A variation on @joriki's answer (and edited with help from @joriki):
Suppose that point $i$ has angle $0$ (angle is arbitrary in this problem) -- essentially this is the event tha... |
In Mathematics several well-known sequences exist, like Fibonacci, Square numbers sequence, etc. One such sequence is Jacobsthal, which states that:Jacobsthal(n)\ =\ \left\{\begin{matrix}0&n\ =\ 0\\1&n\ =\ 1\\2Jacobsthal(n-2)\ +\ Jacobsthal(n-1)&n\ \geq\ 2\\\end{matrix}\right.Jacobsthal(m)=\frac{2^m-{(-1)}^m}{3}
Using ... |
Stone Weierstrass Theorem
Let $X$ be a topological space and let $C(X,\mathbb{R})$ denote the set of all continous functions $f:X\to\mathbb{R}$.
A family $\mathscr{F}$ of functions is an
algebra if for every $f,g\in\mathscr{F}$ and any $c\in \mathbb{R}$, we have $f+g,fg, cf\in\mathscr{F}$. ex: $C([a,b],\mathbb{R})$ is ... |
I'm studying $\lambda$-calculus, and had a question regarding an exercise I came across. I understand that $\lambda$-calculus uses three main strategies of evaluation, but I'm having trouble applying it. Specifically $\beta$-reduction.
For example, for $1 + 2$:
\begin{align} 1 + 2 & = \lambda n.\lambda m.\lambda s. \la... |
These are a natural and easiest (most tractable mathematically) choice.
A utility function is defined up to a positive affine transformation: economically there is no difference between the utility functions $U(x)$ and $\tilde{U}(x)=Au(x)+B$. Hence, a measure of risk aversion that remains constant w.r.t. affine transfo... |
Answer
The amplitude is $3$, the period is $4$, there is no vertical translation and the phase shift is $\frac{1}{2}$ units to the right since $d$ is more than zero.
Work Step by Step
We first write the equation in the form $y=c+a \cos [b(x-d)]$. Therefore, $y=3\cos [\frac{\pi}{2}(x-\frac{1}{2})$ becomes $y=0+3\cos [\f... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
Sequence is just a function of the type $f:\mathbb{N} \to \mathbb{R}$. It is common to list the elements of this sequence as $$(a_1,a_2,a_3,\ldots,a_n)\,.$$ One example is the sequence of all even numbers: $(0,2,4,6,8,10,\ldots)$. However some sequences may be defined in a different form when there is no easy formula f... |
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1. Measurement of $D^{\pm}$, $D^\pm$ and $D_s^\pm$ meson production cross sections in $pp$ collisions at $\sqrt{s}=7$ TeV with the ATLAS detector
Nuclear Physics B, ISSN 0550-3213, 12/201... |
Automorphisms of the Riemann Sphere
Welcome to the final post in our summer series on automorphisms of four (though, for all practical purposes, it's really
three) different Riemann surfaces: the unit disc, the upper half plane, the complex plane, and the Riemann sphere. Last time, we proved that the automorphisms of t... |
We will say that two sets $$A$$ and $$B$$ are equal, written as $$A = B$$ if they have the same elements. That is, if, and only if, every element of $$A$$ is contained also in $$B$$ and every element of $$B$$ is contained in $$A$$. In symbols: $$$x\in A \Leftrightarrow x\in B$$$
We say that a set $$A$$ is a subset of a... |
Difference between revisions of "Ineffable"
m (as a weakening of subtle cardinals)
(-)
Line 57: Line 57:
* Every ethereal cardinal is weakly [[inaccessible]].
* Every ethereal cardinal is weakly [[inaccessible]].
* A strongly inaccessible cardinal is ethereal if and only if it is subtle.
* A strongly inaccessible cardi... |
The Fourier series of the function $f(x) = \left\{ \begin{array}{rcl} -4\,x & \text{if} & -\pi<x<0 \\ 4\,x & \text{if} & 0<x<\pi \end{array}\right.$
is given by $f(x) \sim c_0 - \displaystyle \sum\limits_{n=0}^\infty c_n\;\cos\big((2\,n+1)\,x\big) - \sum\limits_{n=1}^\infty b_n\;\sin\big(n\,x\big)$
I'm asked to find $c... |
What is a simple algorithm for computing the SVD of $2 \times 2$ matrices?
Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code accepted.
Any references to papers or code?
Computational Science Stack Exchange is a question and answer site for scien... |
Sean Nixon, Jianke Yang
Light propagation in optical waveguides with periodically modulated index of refraction and alternating gain and loss are investigated for linear and nonlinear systems. Based on a multiscale perturbation analysis, it is shown that for many non-parity-time (PT) symmetric waveguides, their linear ... |
Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$
$$dS = \mu S dt +\sigma S dz$$
How to find out the process followed by variable $S^n$?
How to prove that $S^n$ also follows geometric brownian motion?
The expected value of $S_T,$ the stock price at time... |
Asian Journal of Mathematics Asian J. Math. Volume 15, Number 4 (2011), 611-630. A New Pinching Theorem for Closed Hypersurfaces with Constant Mean Curvature in $S^{n+1}$ Abstract
We investigate the generalized Chern conjecture, and prove that if $M$ is a closed hypersurface in $S^{n+1}$ with constant scalar curvature ... |
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Change to browse by: Bookmark(what is this?) Astrophysics > High Energy Astrophysical Phenomena Title: Filaments of Galaxies as a Clue to the Origin of Ultra-High-Energy Cosmic Rays
(Submitted on 3 Jan 2019)
Abstract: Ultra-high-energy cosmic rays (UHECRs) are known to come from outs... |
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Now showing items 1-10 of 19
Anomalous evolution of the near-side jet peak shape in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV
(American Physical Society, 2017-09-08)
The measurement of two-particle angular correlations is a powerful tool to study jet quenching in a $p_{\mathrm{T}}$ region inaccessi... |
№ 9
All Issues Filevych P. V. The coefficients of power expansion and $a$-points of an entire function with Borel exceptional value
Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 147-155
For entire functions with Borel exceptional values, we establish the relationship between the rate of approaching $\infty$ for the sequence o... |
First: I am not mathematician but philosopher. I understand why the universal instantiation rule is working.
$\frac{\vdash\forall xA}{\vdash A^x_t}$
But is there actually a serious proof in a logical proof system (Hilbert etc.) ? I don't personally find the critical step on how to get rid of the universal quantifier fr... |
I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing?
Thanks!
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.S... |
I don't know how it is achieved, but here is the reason:
If you have two sources of the same noise, with nearly the same frequency, the sum of both noises will be a noise of slowly increasing and decreasing volume. It's called beat and can become very annoying.
The math says
$$\sin(2\pi f_1t)+\sin(2\pi f_2t)=2\cdot\sin... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine
238
Despite their very good empirical performance most of the simplex algorithm's variants require exponentially many pivot steps in terms of the problem dimensions of the given linear programming problem (LPP) in worst-case situtation. The first to explain the l... |
__author__ = "Christopher Potts"__version__ = "CS224u, Stanford, Spring 2019"
Thus far, all of the information in our word vectors has come solely from co-occurrences patterns in text. This information is often very easy to obtain – though one does need a
lot of text – and it is striking how rich the resulting represen... |
Advances in Differential Equations Adv. Differential Equations Volume 16, Number 5/6 (2011), 487-522. Local and global properties of solutions of heat equation with superlinear absorption Abstract
We study the limit when $k\to\infty$ of the solutions of $ \partial_tu-\Delta u+f(u)=0$ in $\mathbb R^N\times (0,\infty)$ w... |
For the calculation of the $$n$$-th power of a complex number expressed in trigonometric form let's see what the expression will be. We consider the product of $$n$$ complex numbers in trigonometric form:
$$$\big( |z|\cdot[\cos(\alpha)+i \cdot\sin(\alpha)] \big)^n=$$$
$$$= \big( |z|\cdot[\cos(\alpha)+i \cdot\sin(\alpha... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication 1993 (13) (remove)
240
244
242
Efficient algorithms and structural results are presented for median problems with 2 new facilities including the classical 2-Median problem, the 2-Median problem with forbidden regions and bicriterial 2-Median p... |
I'm confused about something related to the splitting principle. It seems to imply that the Chern classes of any endomorphism bundle are zero, which is not true.
Let $M$ be a manifold of dimension $n$, and $E \to M$ be a smooth complex vector bundle of rank $r$. There exists some space $M'$ and a map $f : M' \to M$ suc... |
The 3 most popular average algorithms are:
Median average is when the middle value within a range is found. For example, consider a class of 5 students whose height are 1.54m, 1.55cm, 1.56cm, 1.54cm and 1.53cm respectively. The Arithmetic Mean of the class height is 1.544cm and the Mode average is 1.54cm. However if th... |
Preprints (rote Reihe) des Fachbereich Mathematik
258
On the Convergence at lnfinity of Solutions with Finite Dirichlet Integral to the Exterior Dirichlet Problem for the Steady Plane Navier-Stokes System of Equations (1994)
254
253-2
Order-semi-primal lattices (1994)
250
Let (\(a_i)_{i\in \bf{N}}\) be a sequence of id... |
Line-of-sight propagation is a characteristic of electro-magnetic radiation or acoustic wave propagation. Electromagnetic transmission includes light emissions traveling in a straight line. The rays or waves may be diffracted, refracted, reflected, or absorbed by atmosphere and obstructions with material and generally ... |
According to the Wiki article, if $u$ and $v$ are locally integrable functions on some open subset of $\mathbb{R}^n$, then $v$ is the weak derivative of $u$ if, for any infinitely differentiable function $\varphi$ on $U$ with compact support, we have $$\int_U u D^{\alpha}\varphi = (-1)^{|\alpha|}\int v\varphi,$$ where ... |
So it starts by assuming a cube cavity with length $L$ that acts approximately as a black body. [note: the cube length is $a$ but I'm using $L$ instead]
Standing waves are formed inside the cavity, the equation for a standing wave in 3D is $$ E(x,t) = E_0 \sin(k_x x) \sin(k_y y) \sin(k_z z) \sin ( 2 \pi \nu t), $$ wher... |
We review binary logistic regression. In particular, we derive a) the equations needed to fit the algorithm via gradient descent, b) the maximum likelihood fit’s asymptotic coefficient covariance matrix, and c) expressions for model test point class membership probability confidence intervals. We also provide python co... |
Express the partition function of a collection of N molecules \(Q\) in terms of the molecular partition function \(q\). Assuming the N molecules to be independent, the total energy \(E_{tot}\) of molecules is a sum of individual molecular energies
\[ E_{tot} = \sum_i E_i\]
and all possible
\[Q = \sum _{\text{all possib... |
In this problem, we will find the portfolio allocation that minimizes risk while achieving a given expected return $R_\mbox{target}$.
Suppose that we know the mean returns $R \in \mathbf{R}^n$ and the covariance $Q \in \mathbf{R}^{n \times n}$ of the $n$ assets. We would like to find a portfolio allocation $x \in \math... |
Automorphisms of the Unit Disc
As I promised last time, my goal for today and for the next several posts is to prove that automorphisms of the unit disc, the upper half plane, the complex plane, and the Riemann sphere each take on a certain form.
Fair warning: these posts will be mostly computational! Even so, I want t... |
This question is related to the development of
The Joint Giving Theorem (by S. Kolm).
There are two types of agents: benevolent and beneficiaries.
Benevolents' preferences are represented by utilities: $u^i=u^i(x_i,x,c_i,g_i,c_{-i},g_{-i})$
Where $x_i=X_i-g_i-t_i$ is final wealth, $X_i$ initial wealth, $g_i$ some priva... |
This showcase presents some simulation results for a deep neural network consisting of 21 layers. Based on randomly generated data, the distribution of network activations and gradients is analysed for different activation functions. This reveals how the flow of activations from the first to the last and the flow of th... |
Numerical calculation suggests the following two combinatorial identities: $$ \sum_{i=0}^n\sum_{j=0}^n{n\choose i}{n\choose j}{i+j\choose n}=2^n{2n\choose n},\\ \sum_{i=0}^n\sum_{j=0}^n{n\choose i}{n\choose j}{n\choose i+j}={3n\choose n}. $$ Proofs or references are all welcome.
For the first one: pick $i$ numbers from... |
Applications of differentiation Part VI
Displacement, Velocity and Acceleration
Consider the motion of a particle which moves in a way such that its displacement s from a fixed point O at time t seconds is given by $ s = -t^2 + 5t + 6 $ The instantaneous velocity is given by the differentiation of displacement $ v = \f... |
In order to have an introduction of the Word2Vec
look at this post.Using this method we try to predict the movie sentiment (positive vs. negative) using text data as predictor. We use the movie review data set, and we use the power of Doc2Vec to transform our data into predictors.
library(text2vec)library(tm)data(movie... |
Difference between revisions of "Rank into rank"
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$\mathrm{I3}$ and other $C^{(n)}$ variants:
$\mathrm{I3}$ and other $C^{(n)}$ variants:
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*
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* $\mathrm{I3}(κ, δ)$, if $δ$ is a limit cardinal (instead of a successor of a limit cardinal – Kune... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
$$y=(c_1+c_2t)e^{rt}$$
comes from, it is pretty clear in the case $r=0$, where $D^2y(t)=0$ is solved by
$$y=c_1+c_2t$$
... so it seems that the linear function comes from integrating twice, or more correctly, inverting the same differential operator twice.
Let's try to derive our desired equation $y=(c_1+c_2t)e^{rt}$ v... |
Seems like the experts are not answering your question so I will try to provide an idea. But before I do that I strongly suggest that you look up in the literature for some sophisticated methods that have been already developed. However, without guaranteeing that this is a good or fast or efficient suggestion, I propos... |
CDS 212, Homework 4, Fall 2010
J. Doyle Issued: 19 Oct 2010 CDS 212, Fall 2010 Due: 28 Oct 2010 Reading DFT, Chapter 6 Problems [DFT 6.1] Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign. [DFT 6.4] Let <amsmath>
P... |
Answer
The direction angle of the vector of the complex number
Work Step by Step
The argument of the complex number $\theta=\tan^{-1} (\frac{y}{x})$ is the angle between the vector of the complex number and the positive x-axis.
You can help us out by revising, improving and updating this answer.Update this answer
After... |
Long story short: I need to get a function ($f(r,z,\alpha)$) via numerical integration, and a number from another numerical integration for this function. The method I have at the moment does converse & gives an answer, it is based on
NIntegrate but takes a long time.
I am looking for suggestions for faster method. Eve... |
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If aaa,bbb and ccc are positive integers such that a∣b5a | b^{5}a∣b5 , b∣c5b|c^{5}b∣c5 and c∣a5c|a^{5}c∣a5.Prove that abc∣(a+b+c)31abc|(a+b+c)^{31}abc∣(a+b+c)31.
Note by Eddie The Head 5 years, 6 months ago
Easy Math Editor
This discussion board is a place to dis... |
@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated f... |
The mathematical concept of
linearity has a profound impact on electronic design. The idea itself is quite simple, but the implications have great meaning for our field. We talk about the mathematical meaning of linear.
Written by Willy McAllister.
Contents Preparation Linearity Scaling (homogeneity) Additivity Additiv... |
Maximal ≠ Maximum!
Suffixes are important!
Did you know that the words
" maximal" and "maximum" generally do NOT mean the same thing
in mathematics? It wasn't until I had to think about Zorn's Lemma in the context of maximal ideals that I actually thought about this, but more on that in a moment. Let's start by compari... |
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