text stringlengths 256 16.4k |
|---|
Multitype pair correlation function (cross-type)
Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.
Usage
pcfcross(X, i, j, ..., r = NULL, kernel = "epanechnikov", bw = NULL, stoyan = 0.15, correction = c("isotropic", "Ripley", "translate"), divisor = c("r", "d"))
Argument... |
I would like to use CKM matrix in Wolfenstein parameterization because I want to keep an imaginary part.
I get $A, \lambda, \bar{\eta}, \bar{\rho}$ from PDG and these values have uncertainty.
I have no problem with calculating uncertainty for the value which is real but I don't know how to calculate one with imaginary ... |
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Even and Odd Signals Commonly encountered signals Motivating question
Consider a simple circuit where a DC voltage source with $v$ volts is connected to a 1 Ohm resistor as shown in the figure in the left panel.... |
Volume of a Tetrahedron
\[V= \frac{1}{3}Base \: Area \times Height\]
The diagram shows a trapezium with sides of length
\[2x\].
The base is an equilateral triangle of area
\[\frac{1}{2} 2x \times 2x \times sin 60=x^2 \sqrt{3}\]
To find the height, first find the distance from a vertex of the base to the centre of the b... |
There are 2 observers both facing the same direction (let's assume positive y-axis in a 3D system). Initially, the observation vectors are parallel. The second observer spots the target. So, in terms of spherical co-ordinates, the observer tilts by angle $\phi$, pans by angle $\theta$ and measures the range of the targ... |
Previous
abstract Next
abstract
Session 37 - First Results from the Solar and Heliospheric Observatory (SOHO).
Display session, Tuesday, June 11
Tripp Commons,
We present the first solar EUV spectral atlas in the wavelength range 500 -- 1600 ÅThe spectra were recorded with the Solar Ultraviolet Measurements of Emitted ... |
This is a natural question which confused me a lot. I think generally it is true but I have no idea how to prove it. Also, can anyone raise any counter-example?
The question follows from a problem from Topics in Algebra by Herstein.
The problem is show that $\sqrt{2}+\sqrt[3]{5}$ is algebraic over $Q$ of degree 6.
If I... |
In a tech savvy world where electronics are part of nearly every aspect of modern life, society has become increasing energy-conscious when it comes to powering off to conserve as much energy as possible. But what if people learned that their devices are still energy vacuums when they are not in use? Unplugging househo... |
In Frank Schorfheide's class notes on likelihood functions of DSGE models, he expresses the value of the likelihood function for a given vector of parameters $\theta$, and time series $Y^T$ as:
$$p(Y^{T}|\theta)=(2\pi)^{-nT/2}\left(\prod_{t=1}^{T}\left|F_{t|t-1}\right|\right)^{-1/2}exp\{-\frac{1}{2}\sum_{t=1}^{T}v_{t}F... |
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[/latex] is even, then [latex]a[/latex] cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, t... |
I have the following partial differential equation:
I'm asked to prove that if $f\equiv 0$, then the total energy (kinetic energy + potential energy) of the system decreases with time.
What is the expression for the energy of this system? I know what the expression of energy is for parabolic or hyperbolic partial diffe... |
Concerning Electromagnetism, textbooks often refer to the Duality Theorem. Sometimes it is presented like this:
«Consider the Maxwell's Equations (with phasors) and a known field $\mathbf{E}_1$, $\mathbf{H}_1$:
$\nabla \times \mathbf{E}_1 = - j \omega \mu \mathbf{H}_1$
$\nabla \times \mathbf{H}_1 = j \omega \epsilon \m... |
Pg 171 of "Tensors, Relativity and Cosmology"
The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \beta}dx^\alpha dx^\beta \tag{1}$$ where $\alpha, \beta=1,2,3$
In the non-relativistic limit in the static gravitational field, ... |
I am able to prove the iff in the forward direction. But, I am having trouble proving the statement in other direction. I am trying to use the definition of dense, but I am not getting anywhere with it.
Let $U \subseteq X$ be an open set and let $S \subseteq X$ be the set in question. Denseness is equivalent to saying ... |
In the $S'$ frame, your variables are $x' = x - t\cdot u \cos\theta $ and $y' = y - t\cdot u \sin\theta$. If you do the change of variable, you get that the motion now is described by
$$x' = 0$$$$y' = -\frac{g}{2}t^2$$
So in your new frame of reference you have vertical free fall from rest.
This is not very helpful in ... |
Let $h[n]$ be the impulse response of a linear and time-invariant(LTI) system. If the signal $x[n]$ is input to the system, the output signal from the system is given by:$y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]$
This operation is called convolution and we say that the signal $y[n]$ is the convolution of the signal $x[... |
(
: This is the case $a=\frac16$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $a=\frac13$ and $a=\frac14$.) Note
After investigating $a=\frac13$ and $a=\frac14$, I wondered if there was for $a=\frac16$.... |
I have some questions about the notations used in Section
9.2 Lack of Inherent Superiority of Any Classifier in Duda, Hart and Stork's Pattern Classification. First let me quote some relevant text from the book: For simplicity consider a two-category problem, where the training set $D$ consists of patterns $x^i$ and as... |
Consider the variable coefficient, real valued wave equation
$$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on $x$. Then we define the total energy at time $f$ of a $C^2$ solution $u$ as
$$ E(t) = \frac{1}{2}\int_\Omega (u_t^2 + c(... |
I am trying to get the probability distribution function of $Z=X-Y$. Given that $f_X(x)$ and $f_Y(y)$ are known, and both variables are chi-square distributed, $X\in\mathbb{R}$, $X\ge 0$, and similarly $Y\in\mathbb{R}$, $Y\ge 0$. So how can I get $f_Z(z)$.
Let's assume that $X$ and $Y$ are independent, and both follow ... |
What is the best way to simulate the short rate $r(t)$ in a simple one factor Hull White process? Suppose I have
$$ dr(t) = (\theta(t)-\alpha r(t))dt+\sigma dW_t $$
where $\theta(t)$ is calibrated to swap curve, constants $\alpha$ and $\sigma$ are calibrated to caps using closed form solution for zero-coupon bond optio... |
I need to estimate baseline hazard function $\lambda_0(t)$ in a time dependent Cox model
$\lambda(t) = \lambda_0(t) \exp(Z(t)'\beta)$
While I took Survival course, I remember that the direct derivative of cumulative hazard function ($\lambda_0(t) dt = d\Lambda_0(t)$) would not be a good estimator because Breslow estima... |
The problem has nothing in particular to do with logarithms -- it is a general phenomenon that we cannot always represent the true result of a computation exactly on paper, and so we have to settle for approximations.
For example we have $\log_{10}(13) = 1.1139433523...$. The true, mathematical value of $\log_{10}(13)$... |
This is a simple variation on the so called "Twin Paradox", which is not a paradox in the logical sense (i.e. not a logical contradiction). Each cycle of the oscillator's motion is like the journey of the spacefaring twin. One possible cycle on a spacetime diagram is drawn below (source: Wikipedia "Twin Paradox" Articl... |
prettify-symbols-mode is a recent feature of Emacs and it’s very nice. And it looks like it can replace TeX-fold-mode in the future. But, at the time of writing, prettify-symbols-mode doesn’t seem to work well with AUCTeX unless you enable two workarounds together.
Tested versions:
AUCTeX 11.89.7 (latest version from G... |
Hyperbolic tangent function \({\rm tanh}\) is often used to generate the stretched structured grid.
In this blog post, I will introduce some examples I have found in the references.
Example #1 [1]
\begin{equation}
y_j = \frac{1}{\alpha}{\rm tanh} \left[\xi_j {\rm tanh}^{-1}\left(\alpha\right)\right] + 1\;\;\;\left( j =... |
I have variables $x \in \{0,1,\dots,5\}$ and $y \in \{0,1\}$, where
$$y = \begin{cases} 0 & \text{if } x = 5\\ 1 & \text{if } x \neq 5\end{cases}$$
My problem is to maximize $y$. How can I use linear constraints for this?
I tried certain cases like Cast to boolean, for integer linear programming but that won't work if ... |
The proof of coherence in monoidal categories in CWM is based on theexistence of a monoidal category free over a singleton. Denoting thiscategory by $\mathcal{W}=\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho}\right)$it can be observed that $\mathcal{W}_{0}$ is a thin groupoid. Itsobjects are ... |
Prove compact subsets of metric spaces are closed
Note, this question is more of analyzing an incorrect proof of mine rather than supplying a correct proof.
My Attempted Proof
Suppose $X$ is a metric space. Let $A \subset X$ be a compact subset of $X$ and let $\{V_{\alpha}\}$ be an open cover of $A$. Then there are fin... |
I am just wondering if I know enough prerequisites!
A basic understanding of what's going on gain be gained just using Kepler's laws and Newtonian mechanics.
A simple way of dealing with multiple gravitational sources is to selectively ignore all but one of them. This is the patched conic approximation. Which gravitati... |
Title Global calibrations for the non-homogeneous Mumford-Shah functional Publication Type Journal Article Year of Publication 2002 Authors Morini, M Journal Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648 Abstract
Using a calibration method we prove that, if $\\\\Gamma\\\\subset \\\\Omega$ is a closed regular... |
The unitriangular group $UT_n(\Bbb Z)$ is the group of all $n \times n$ invertible triangular matrices with the identity on each entry of the main diagonal, and integer entries everywhere else in the triangle. Show that this group is nilpotent, and that its nilpotence class is $n$.
Definition( upper central series): Fo... |
This is the first in a series of posts about the computational aeroacoustics that is abbreviated as CAA. Aeroacoustics is mainly concerned with the generation of sound or noise through a fluid flow. Familiar examples are
They can be studied by both experimental and computational methods. In an experimental approach, th... |
We consider an oscillatory motion with small amplitude in a compressible fluid as shown in the following picture. It shows the distribution of the
sound pressure ( acoustic pressure) that is defined as the local pressure deviation from the equilibrium pressure \(p_0\) caused by the sound wave propagating from left to r... |
Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view.
When studying Differential Calculus, I found that there was very little that I had to
memorize. Virtually all calculation aspects, such as finding derivatives etc., and some theo... |
Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT
One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate. The derivation of can be found in textbooks [1,2]. You can also demonstrate this principle numerically using the Discrete ... |
Sine-cubed function This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article. View a complete list of particular functio... |
How to prove that :
$$ \frac{\textrm{d}}{\textrm{d}x}\int^{g(x)}_{h(x)}f(t)\textrm{d}t =f(g(x))g'(x)-f(h(x))h'(x). $$
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 this community
How t... |
Really stumped on this one. I would really like an example or situation where an estimator B would be both consistent and biased.
The simplest example I can think of is the sample variance that comes intuitively to most of us, namely the sum of squared deviations divided by $n$ instead of $n-1$:
$$S_n^2 = \frac{1}{n} \... |
Let $ E $ be an algebraic extension $ F $ and $ x \in E $ and $ \sigma: E \to E $ be an automorphism of $ E $ fixing $ F. $ Prove that $ \sigma(x) $ and $ x $ are conjugate over $ F. $
I am starting to learn about extension field, automorphism and Galois theory now and there's a lot of stuffs that confuse me so any hel... |
I came across the following proof in my textbook that was used as a end of chapter review. How can I prove the following algebraically?
$$\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2$$
using $\left( \begin{array}{c} n \\ 2\ \end{array} \right) = \dfr... |
The divergence (Gauss-Green) theorem can be used
to define the improper integral of the divergence of (weakly) singular vector fields $\mathbf{F}$ with isolated singular points $\mathbf{p}_o=(x_0,y_o,z_0)\in V$. Customarily, the definition goes as follows$$\begin{split}\int\limits_{V}\nabla\cdot\mathbf{F}(x,y,z)\,\math... |
For one dimensional case there is a nice connection of Radon-Nikodym derivative and "classical" derivative on real line. Is there some kind of analogy for higher dimensional cases?
Among other connections, the Radon-Nikodym derivative allows you to get the change of variable formula in $\mathbb{R}^n$.
Setup: Let $K\sub... |
How to make an animation of following gif in
Mathematica?
And how to make 3D analog?
I tried first few steps
line = Graphics[Line[{{1, 1}, {2, 2}}]]Manipulate[Show[line, line /. l : Line[pts_] :> Rotate[l, n, Mean[pts]]], {n, 0,Pi}]
Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathemati... |
Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a s... |
Does anyone here understand why he set the Velocity of Center Mass = 0 here? He keeps setting the Velocity of center mass , and acceleration of center mass(on other questions) to zero which i dont comprehend why?
@amanuel2 Yes, this is a conservation of momentum question. The initial momentum is zero, and since there a... |
My university is participating in the implementation of a library borrowing managing system at Richelieu National Library in France.
I received the order to formulate the query: "find all users having borrowed every book" in relational Algebra, in relational Calculus and in SQL (which would probably not happen, probabl... |
“Irrational numbers are those real numbers which are not rational numbers!” Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $ a$ and $ b$ are both integers relatively prime to each other and $ b$ being non-zero. Following two statements are equivalent to the definit... |
Momentum Overview
Momentum is a simple strategy. Look at the past 12 months and identify which stocks went up the most (the winners) and which stocks went down the most (the losers). Suppose we look at all NYSE stocks between 1990 and 2015. Buying the top 10% of winners and shorting the worst 10% of losers yields an av... |
So, the rotation of a 3d body can be described with Euler's equations of motion giving the rotational velocity in components along the principal axes of inertia. As showed in f.ex. this paper,
Euler Top (free asymmetric top): solution of Euler’s equations in terms of elliptic integrals, Berry Groisman, Cambridge Univer... |
Keywords
Eventually positive matrix, Eventually exponentially positive matrix, 2-generalized star sign pattern, Checkerboard block sign pattern
Abstract
A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually positive if the... |
Introduction Energy and Power Basic Operations Periodic Signals Commonly encountered signals Practice Problems Motivating question
Consider a simple circuit where a DC voltage source with $v$ volts is connected to a 1 Ohm resistor. How much power is dissipated by the resistor? It is clear that a power is $v^2/R = v^2$ ... |
But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you alr... |
Mathematical Induction Inequality Proofs
Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for the subtraction and/or greatness, using the assumption at the step 2. Let’s take a look at the following hand-picked examples.
Practice Questions for Mathematical Induction In... |
Let $(x_n)_{n\in\mathbb N}$ be a recursively defined sequence with $x_1=9$ and $$x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\text{ for }n\geq 1.$$ Show that $x_n\geq\sqrt{2}$ for all $n$.
Because $x_n\geq 0$ one can easily prove inductively that $$x_n^2\geq 2\Leftrightarrow x_n^2-2\geq 0\Leftrightarrow\left(\frac{x_{n-1}}{2}-\... |
I'm trying to find the asymptotes of $f(x) = \arcsin(\frac{2x}{1+x^2})$.
I've found that this function has no vertical asymptote, since $f$ is bounded between $[-\pi/2 , \pi/2 ]$, and since $\arcsin x$ is continuous where it is defined - for every $x_0 \in R$, $\lim_{x\to x0^+}|f(x)| = |f(x_0)| \neq \infty $. Hopefully... |
I have a backward parabolic equation of the form:
\begin{equation} W_{\eta} + aW_{xx} - bW = 0 \end{equation}
s.t.
\begin{equation} \lim_{\eta \rightarrow \infty}(x,\eta) = g(x) \end{equation}
were $x \in \mathbb{R}$, $\eta \geqslant 0$, and $a,b$ are positive constants.
Applying the following transformations:
\begin{a... |
I have this limit:
$$\lim_{x\to \infty} (e^x+x)^{\frac{1}{x}}$$
At first I was stumped but then decided to use L'hospitals rule and logs so it turns to:
$$\lim_{x\to \infty} \frac{\ln(e^x+x)}{x}$$
Then differentiating it twice turns to:
$$\lim_{x\to \infty} \frac{e^x}{e^x+1}$$
But then this means $\lim_{x\to \infty} \f... |
Taiwanese Journal of Mathematics Taiwanese J. Math. Volume 22, Number 1 (2018), 225-244. Quantitative Recurrence Properties for Systems with Non-uniform Structure Abstract
Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x... |
Recent Posts Recent Comments Archives Categories Meta Author Archives: eas5828
At the beginning of the semester, I believed in climate change but was not fully aware of how serious it is or the ways people are trying to prevent it from getting worse. Between the lessons taught in class and … Continue reading
In the art... |
Introduction Energy and Power Basic Operations Periodic Signals Commonly encountered signals Practice Problems What is a signal?
The word 'signal' has been used in different contexts in the English language and it has several different meanings . In this class, we will use the term signal to mean a function of an indep... |
Griffiths's
Introduction to Electrodynamics states
$$\mathcal E = \oint \mathbf f \cdot d\mathbf l$$
In which
$$\mathbf f = \mathbf f_s + \mathbf E$$
Where Griffiths describes the summation as
the source, $\mathbf f_s$, which is ordinarily confined to one portion of the loop (a battery, say), and an electrostatic force... |
Stein's Example shows that the maximum likelihood estimate of $n$ normally distributed variables with means $\mu_1,\ldots,\mu_n$ and variances $1$ is inadmissible (under a square loss function) iff $n\ge 3$. For a neat proof, see the first chapter of Large-Scale Inference: Empirical Bayes Methods for Estimation, Testin... |
4 (p
In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties.
Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A],... |
I'm looking for cases like $$\lim_{x \to 0} \frac {1-\cos(x)}{x^2}$$ that will not give you the answer the first time you use L'Hôpital's rule on them. For example in this case it will result in a number $\frac{1}{2}$ the second time you use L'Hôpital's rule. I want examples of limits like $\lim_{x \to c} \frac {f(x)}{... |
I created a function by first considering some well known limits:
$$\lim_{n\to\infty } \frac{n}{2}\sin\left ( \frac{2\pi}{n} \right )=\pi $$
$$\lim_{n\to\infty } \left ( 1+\frac{1}{n} \right )^{n}=e$$
Now, recalling Euler's identity:
$$e^{i \pi }=-1$$
The following limit can be created, which is a combination of Euler'... |
Wikipedia said
The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of cha... |
And Stephen Hawking died today. He will leave a great, black hole in modern science. I saw him lecture in London not long after
A Brief History of Time came out. It was one of the events that inspired me along my path to science. I recall he got more laughs than a lot of stand-ups I've seen.
But I can't really get behi... |
How to prove this for positive real numbers? $$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$
I tried AM-GM, CS inequality but all failed.
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 u... |
Frequency Response of Mechanical Systems
In this follow-up to a previous blog post on damping in structural dynamics, we take a detailed look at the harmonic response of damped mechanical systems. We also demonstrate different ways of setting up a frequency-response analysis in the COMSOL Multiphysics® software as well... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut... |
One may describe the waves in terms of $\Delta x$, the deviation of the position of strings' atoms from the equilibrium locations. Because they're attached, $\Delta x(\sigma)=0$ for $\sigma$, the coordinate along the string, equal to any of the end point values, $\sigma=0$ and $\sigma=L$.
But $\Delta x$ obeys a wave eq... |
Given a time series $u_i$ of returns (where $i=1,\dotsc,t$), $\sigma_i$ is calculated from GARCH(1,1) as $$ \sigma_i^2=\omega+\alpha u_{i-1}^2 +\beta \sigma_{i-1}^2. $$ What is the mathematical basis to say that $u_i^2/\sigma_i^2$ will exhibit little autocorrelation in the series?
Hull's book "Options, Futures and Othe... |
Yes, by linearizing $f$ at a fixed point.
More explicitly, let $x_0\in M$ be a fixed point of $f$. Let $U$ be an open neigborhood of $x_0\in M$ for which there is a diffeomorphism with an open subset $V$ of $\mathbf R^n$. Replacing $U$ by $f^{-1}(U)\cap U$ if necessary, we may assume that $f(U)\subseteq U$.Then $f(U)=U... |
You use the quantile regression estimator
$$\hat \beta(\tau) := \arg \min_{\theta \in \mathbb R^K} \sum_{i=1}^N \rho_\tau(y_i - \mathbf x_i^\top \theta).$$
where $\tau \in (0,1)$ is constant chosen according to which quantile needs to be estimated and the function $\rho_\tau(.)$ is defined as
$$\rho_\tau(r) = r(\tau - ... |
I have an smooth structure on $M$ given by the atlas $\{(U_\alpha,\varphi_\alpha)\}$
And on another manifold $N$ we define $\{(V_\alpha,\mu_\alpha)\}$ with $V_\alpha = \pi(U_\alpha)$ and $\mu_\alpha = \pi_2\circ\varphi_\alpha\circ \sigma$
where $\pi:M\rightarrow N$ is a smooth submersion, $\pi_2:\mathbb{R}^{n+k}\righta... |
For a rational number $p > 1$. We know that the function $z^p$ is holomorphic on $\mathbb{C} \setminus \mathbb{R}^-$ (excluding $z = 0$). Is there an analytic continuation of the function $z^p$ at zero ?
Thank you.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profes... |
The Question:
The temperature $u(x,t)$ in a semi-infinite conductor occupying $x \in [0,\infty)$ satisfies the equation
$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \qquad x,t>0$$
The temperature is $0$ at $t=0$, i.e. $u(x,0)=0$.
For $t>0$, heat is supplied at $x=0$ at the constant flux
$$Q=-ku_x... |
The format of the 3 dimensional MB distribution is $A \cdot e^{-\frac{E}{k_BT}} \cdot g(E)$ in which $A$ can be derived using normalization (integration up to $\infty$ must be 1) and $g(E)$ being the degeneracy according to $g(E)=\frac{V\pi \cdot 2^{2.5}m^{1.5}}{h^3}$
The 3 dimensional average kinetic energy $\bar E$ o... |
There are several different relations between Chern-Simons/WZW models, and there are several way to show these. A nice paper doing this in a concrete way is Elitzur et al Nucl.Phys. B326 (1989) 108.
The Chern-Simons theory on a
compact spatial manifold give rise to a finite dimensional Hilbert space (only global degree... |
How do I solve this trig equation?
$$\cos 2x - \sin2x = \sqrt{3} \cos 4x$$
I have tried in different ways. But I can't get to a final answer.Please help.
My work is,
$$\cos 2x - \sin 2x = √3(\cos 2x - \sin 2x)(\cos 2x + \sin 2x)$$ $$⇔(\cos 2x - \sin 2x)((1-√3(\cos 2x + \sin 2x))= 0$$ Then, $$\tan 2x = 1 \ \ \ \text{or}... |
Maximum likelihood estimates of a distribution
Maximum likelihood estimation (MLE) is a method to estimate the parameters of a random population given a sample. I described what this population means and its relationship to the sample in a previous post.
Before we can look into MLE, we first need to understand the diff... |
Interesting question. The "third law" of thermodynamics actually states that
The entropy of a
perfect crystal at absolute zero is exactly zero.
Since many systems in nature crystallize at some point when their temperature is lowered, this statement is often misinterpreted as:
The entropy of a
physical system at absolut... |
Quadratic equation An algebraic equation of the second degree. The general form of a quadratic equation is \begin{equation}\label{eq:1} ax^2+bx+c=0,\quad a\ne0. \end{equation} In the field of complex numbers a quadratic equation has two solutions, expressed by radicals in the coefficients of the equation: \begin{equati... |
Algebraic Geometry Seminar Fall 2016
The seminar meets on Fridays at 2:25 pm in Van Vleck B305.
Here is the schedule for the previous semester.
Contents Algebraic Geometry Mailing List Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is... |
NTS ABSTRACTSpring2019
Return to [1]
Contents Jan 23
Yunqing Tang Reductions of abelian surfaces over global function fields For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous ... |
I was reading proofs on the scaling property of fourier transform:
I notice this line:
$$\mathcal{F}[g(ct)] = \int_{-\infty}^{\infty} g(ct) \ e^{-i\omega t} dt \tag{1}$$
I have a few issues with this line:
$1)$ Is there a missing factor of $\frac{1}{\sqrt{2\pi}}$ on the right hand side? Because shouldn't a Fourier tran... |
Derivative-Free Optimization for Data Fitting
Calibrating the parameters of complex numerical models to fit real world observations is one of the most common problems found in the industry (finance, multi-physics simulations, engineering, etc.).
Let us consider a process that is observed at times $t_i$ and measured wit... |
I cannot claim to be an expert on AQFT, but the parts that I'm familiar with rely on local fields quite a bit.
First, a clarification. In your question, I think you may be conflating two ideas: local fields ($\phi(x)$, $F^{\mu\nu}(x)$, $\bar{\psi}\psi(x)$, etc) and unobservable local fields ($A_\mu(x)$, $g_{\mu\nu}(x)$... |
Search
Now showing items 1-2 of 2
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Let $\{x_n\}$ a Cauchy sequence in a normed vector space $X$. Is $$y_n = \frac{x_n}{\|x_n\|}$$ another Cauchy sequence in $D = \{x\in X : \|x\| = 1\}$?
Remark: The idea is prove that if $D$ is complete, then $X$ is also complete. Thanks so much.
Mathematics Stack Exchange is a question and answer site for people studyi... |
This question already has an answer here:
Imagine we have a portfolio allocation vector $(x_1, ..., x_n)$ with $x_1+...+x_n=1$; Also we assume that the vector has only elements $\geq 0$, so we have $|x_1|+...+|x_n|=1$, so no short selling;
I want to generate such vectors $x$ and I want that they are somehow uniformly d... |
Category of F-algebras Collection
context $F$ in ${\bf C}\longrightarrow{\bf C}$ definiendum $\mathcal{A}:\mathrm{Ob}_\mathrm{it}$ postulate $\mathcal{A}$ … $F$-algebra definiendum $\langle f\rangle:\mathrm{it}[\langle A,\alpha\rangle, \langle B,\beta\rangle]$ postulate $f\circ\alpha=\beta\circ F(f)$ Discussion
The cat... |
In Sec. 2.4 of
Inside Interesting Integrals (2015), Paul J Nahin says of $$I:=\int_0^{\pi/2}\ln (a\sin x)dx=\int_0^{\pi/2}\ln (a\cos x)dx$$that:
For many years it was commonly claimed in textbooks that these are quite difficult integrals to do, best tackled with the powerful techniques of contour integration. As you’ll... |
I saw this lovely tweet from PGS yesterday:
Our basin studies team spotted this on fast-track imaging from Republic of Guinea. A 7.5 km diameter depression, with no salt or mobile shale, nor dissolution of fluid escape. We interpreted the structure as a complex meteorite impact crater. https://t.co/Z4TUOtsv54 #meteorit... |
When modeling engineering systems, it can be difficult to identify the key parameters driving system behavior because they are often buried deep within the model. Analytical models can help because they describe systems using mathematical equations, showing exactly how different parameters affect system behavior.
In th... |
Suppose that we have an IID random sample $\mathbf{x} = (x_1, \dots, x_n)$ from a given distribution with the following PDF:
$$\theta (1 - e^{-x})^{\theta -1}e^{-x}, \, x > 0, \, \theta > 0$$
Also, let's say that we have a non-informative prior distribution such that $\pi(\theta) \propto \frac{1}{\theta^c}$ where $c$ i... |
There is a formula for the hypergeometric ($_2F_1$) that expresses it as a sum of Pochhammer symbols, times something that reads
$$_2F_1(a,b,c;x) = \sum_{i=0}^{\infty} \frac{(a)_i (b)_i}{(c)_i} \frac{x^i}{i!}$$
This is a result that I easily verified.
Result1 = Series[Hypergeometric2F1[a22, b22, c22, y], {y, 0, 10}]; R... |
A) \[2.5\times {{10}^{10}}\]
B) \[2.5\times {{10}^{11}}\]
C) \[2.5\,\,\times \,\,{{10}^{12}}\]
D) \[2.5\,\,\times \,\,{{10}^{9}}\]
Correct Answer:
A
\[\lambda \,\,=\,\,5000\,\overset{{}^\circ }{\mathop{A}}\,\] Energy received per second \[=\,\,{{10}^{-}}^{\,8}J/sec\] \[\therefore \] Energy of one photon is \[E=hv=hc/\l... |
Which of the highlighted C–H bonds in the following compounds is weakest? (R = generic alkyl group)
I personally thought that the answer was
1 but it is given as 4 in the answer key. I didn't understand how it is so.
Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and student... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.