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Say I have a market-making strategy that trades intraday. I start with a flat position and finish flat too. I end up with a daily P&L $p_{today}$. Over a year of trading I get $\vec{p} = (p_1,\dots,p_{252})$.
There is no way to calculate returns here. As such I calculate $$Sharpe = S(\vec{p}) = \sqrt{252} \cdot \frac{\... |
I am reading the book "Numerical Recipes in Fortran 77: The Art of Scientific Computing" (Second Edition) and I came across some methods for numerical integration of 1D functions. More specifically the Gauss-Laguerre, Gauss Hermite, and Gauss Jacobi weights and abscissas appealed to me. I need them because I have some ... |
I'm trying to simulate the growth of a brain tumor using a 3d reaction-diffusion model $ \partial_{t}u = \nabla.{(D\nabla{u}) } + ku.(1-u)$ , knowing the initial distribution of tumor $u^0$, the non-homogeneous diffusion coefficient $D$ and the non-homogeneous net-proliferation map $k$.
So I first tried to proceed to s... |
This question has two parts, as I do not understand whether my problem is theoretical (identification of the parameters) or practical (insufficient R skills).
Econometrics
Most "probit" style models are identified through a normalization of the standard error to one. In my case, I would argue that this is not necessary... |
There are $$$n$$$ persons who initially don't know each other. On each morning, two of them, who were not friends before, become friends.
We want to plan a trip for every evening of $$$m$$$ days. On each trip, you have to select a group of people that will go on the trip. For every person, one of the following should h... |
2fFCS is an abbreviation
2 Focus Fluorescence Correlation Spectroscopy. Dual focus FCS might also be used.
Correlation analysis that is applied to the fluctuations of the fluorescence intensity. The cross-correlation between the occurrence of events in two different detection volumes is used to introduce an absolute di... |
Current browse context:
astro-ph.EP
Change to browse by: References & Citations Bookmark(what is this?) Astrophysics > Earth and Planetary Astrophysics Title: Planet Hunters TESS I: TOI 813, a subgiant hosting a transiting Saturn-sized planet on an 84-day orbit
(Submitted on 19 Sep 2019)
Abstract: We report on the disc... |
Generally speaking, your $\delta x_i$ will be a manual estimate of the error in measurement $x_i$ - e.g. this could be the smallest grid size on your ruler. This is a rather rough estimate since it presumes very little about your measurement methodology and is reliant on the experimenter's best judgement. $S$ (usually ... |
Nemtsov function
The Nemtsov function \(y=\mathrm{Nem}_q(x)= x+x^3+q x^4\) is shown in figure at right versis \(x\) for various \(q\!\ge\!0\).
Complex map of \(\mathrm{Nem}_q\) is shown in figure 2 for \(q\!=\!0\) and in figure 3 for \(q\!=\!2\) with lines \(u\!=\!\mathrm{const}\) and lines \(v\!=\!\mathrm{const}\) ass... |
As you say yourself, indeed every connection on a bundle is
locally given by a Lie algebra valued 1-form and in general only locally.
Let's say this more in detail: for $X$ any manifold, a $G$-principal connection on it is (in "Cech data"):
a choice of good open cover $\{U_i \to X\}$;
on each patch a 1-form $A_i \in \O... |
I want to numerically solve the diffusion equation $\partial_t u = D \partial_x^2 u$ in Fourier space, and can think of multiple ways to do it.
Setup Option 1
Differentiating $u$ twice in Fourier space brings down two factors of $ik$ from the exponential $e^{i k x}$.
$$\partial_t \tilde{u}_k = D (ik)^2 \tilde{u}_k$$
wh... |
While the analytic definition of a continuous function involves open sets, we can define continuous functions in a completely equivalent way with closed sets. We can define a function between metric (or topological) spaces $f:X \rightarrow Y$ to be continuous if $f^{-1}(C)$ is a closed set if $C$ is.
Typically we avoid... |
[Scroll to graphic below if math doesn’t render for you]
Thanks to Mark Andrews for correcting some crucial typos (I hope I got it right this time!).
Thanks also to Andrew Gelman for pointing out that the proof below holds only when the null hypothesis is a point null $H_0: \mu = 0$, and the dependent measure is contin... |
How do I calculate the distance to touchdown when the glideslope angle is different from 3°? For example, you are 670ft AGL on a 3.2° glideslope approach; what do I change in the formula to calculate the distance?
Just in case you don't have a calculator at hand (yeah, I know...):
WolframAlpha returns (unsurprisingly):... |
I'm trying to understand the original of quark and lepton mass. Here's a paragraph from the book: "Massive neutrinos in physics and astrophysics" by R. N. Mohapatra and P. B. Pal :
The gauge invariance prevent adding bare masses fro them in the Lagrangian. They arise from the following Yukawa interactions allowed by ga... |
Let's note $L(t,T_i,T_{i+1})$ the libor rate observed at $t$, fixing at $T_i$ with delivery at $T_{i+1}$.
The natural delivery date for this rate is $T_{i+1}$, so a vanilla swap with no pay lag would be priced as :
$Swap(t) = \sum_{i=1}^{n} \tau (T_i,T_{i+1}) P(t,T_{i+1}) \mathop{\mathbb{E}} ^{i+1}\left[ (L(T_i,T_i,T_{... |
I have the following question:
Suppose I have two matrices $X, Y$ both of size $m\times p$ and a random i.i.d Gaussian matrix $G$ of size $m \times k$, $m\gg p>k$.
Is there a fast way to compute $\exp(-XY^T)G$? Perhaps by using the fact that both $X$ and $Y$ are much smaller than $XY^T$? Here, $\exp(\cdot)$ means entry... |
Want to clarify few things.
It is said that two $\lambda$-terms are equal up to renaming of bound variables, such as $\lambda x.x$ equals $\lambda y.y$, so I think it is a relation actually, about how two terms are equal.
Is that (such equality up to renaming) is same as equivalence relation?
To show a relation is an e... |
I was wondering if the following is a viable method using machine learning and neural networks to get to the ground states of the toric code and also understand the phase transition in the presence of perturbation that is $H_{p} = H_{TC} + h_{x(z)}\sum_{i}\sigma_{x(z)}^{i}$, where $H_{TC}$ is the toric code hamiltonian... |
Triangular Norms and Conorms
Mirko Navara (2007), Scholarpedia, 2(3):2398. doi:10.4249/scholarpedia.2398 revision #137537 [link to/cite this article]
Triangular norms and conorms are operations which generalize the logical conjunction and logical disjunction to fuzzy logic. They are a natural interpretation of the conj... |
Step 1
Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is changed, using the small parameter $\lambda$. Therefore, $$ A \rightarrow A-\lambda B $$ where $B$ can be not Hermitian, in ge... |
I know that the set of all rational numbers is countable, and can be enumerated by a sequence, say $\{a_n\}$. But can we construct a monotonic $\{a_n\}_{n=1}^{\infty}$, e.g. with $a_k<a_{k+1}$? It doesn't seem plausible to me, because then $a_1$ would be the smallest rational number, which clearly can't be any finite n... |
Recently I had to solve a little problem related with Fibonacci numbers. I was using the Binet's formula for that:
$$F_{n} = \frac{\Phi^n - \phi^n}{\sqrt{5}}$$
The effective solution I got by using computation by rounding formulas, related to Binet's formula that arises from simple inequality:
$$\left| \frac{\phi^n}{\s... |
I'm solving a nonlinear constrained optimization with constraint of following form:
$$\mathbf{A}^T\mathbf{A}-\mathbf{I}=\mathbf{0}, \mathbf{B}^T\mathbf{B}-\mathbf{I}=\mathbf{0}$$
where $\mathbf{A}$ and $\mathbf{B}$ are matrices that I'm solving for. Usually the initial guess of $\mathbf{A}$ and $\mathbf{B}$ are diagona... |
To touch upon a paper posted within the interactive journal that is scientific Model developing (GMD) or perhaps in its discussion forum Geoscientific Model Development Discussions (GMDD), please select one of many after two choices:
If you’d like to submit an interactive remark (short remark, referee remark, editor re... |
I believe it can be useful to define the following concepts (I won't be very formal here for pedagogical reasons):
Any event can be described through four real numbers, which we take to be: the moment in time it happens, and the position in space where it takes place. We call this four numbers the
coordinates of the ev... |
September 12th, 2016, 11:01 PM
# 1
Member
Joined: May 2016
From: Ireland
Posts: 96
Thanks: 1
Velocity and time
A particle is projected vertically upwards with velocity u m/s. Its height is h after t1 and t2 seconds. Prove that t1 × t2 = 2h/g.
Last edited by skipjack; September 13th, 2016 at 02:33 AM.
September 12th, 20... |
Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional subspace of $\mathbb R^n$ spanned by these vectors.
Informally, I want to count the "integer lattice points" (i.e., elements of... |
Counter Counting objects/people forms the basis for a range of high-tech solutions, including retail analytics, queue management, building management and security applications. By using the counter heuristic you can count incoming and outgoing objects (e.g. people). By defining two lines, the counter heuristics can cou... |
What Does a Circle Look Like?It is in the very beginning of Apollonius' treatise the Conics that he deals with the question we are concerned with, and I'll try to explain, roughly, what he does. ...
Bill Casselman
Introduction
The obvious answer is certainly correct. Nonetheless this is a subtle question. You say you s... |
Just out of curiosity I started looking into how to get jupyter to display text and math formatted as markdown and latex in output cells. This got into my head when I was looking at symbolic integration and differentiation of functions and how to format the output in a more civilized manner. I don’t have the full solut... |
Before I state the questions I have in mind, let me give some background. If one considers $S^2$ then it is known due to Kneser that $\textrm{Homeo}^{+}(S^2)$ has the homotopy type of $SO(3)$. By Smale's work we also know that $\textrm{Diff}^{+}(S^2)$ is homotopic to $SO(3)$. However, when we work in the homotopy categ... |
It is known that nucleons (proton, neutron) are composed of partons (quarks, etc.). How was this identified experimentally? In particular, how it has been identified that nucleons comprise of more than one constituent?
Matt Strassler goes into detail with LHC data here:
As you may know, when particles are scattered off... |
Difference between revisions of "Vertex equations of a quadratic function and it's inverse"
From JSXGraph Wiki
Line 1: Line 1:
A parabola can be uniquely defined by its vertex ''V'' and one more point ''P''.
A parabola can be uniquely defined by its vertex ''V'' and one more point ''P''.
The function term of the parabo... |
Let us define the following cumulative distribution: \begin{align} \Pr (Y(t)\geq a)=\sum_{n=0}^\infty \frac{e^{-kt}(kt)^n }{n!}\int_a^\infty[\circledast_{f}\circ H_a ]^n\delta (x) dx \end{align} where the operator $[\circledast_{f_J}\circ H_a ]$ denotes the multiplication by a Heaviside function with a cut off at $a$ f... |
Branch misprediction is expensive: an exampleThe SciMark 2.0 Monte Carlo benchmark is calculating \(\pi\) by generating random points \(\{(x,y) \mid x,y \in [0,1]\}\) and calculating the ratio of points that are located within the quarter circle \(\sqrt{x^2 + y^2} \le 1\). The square root can be avoided by squaring bot... |
S H Kulkarni
Articles written in Proceedings – Mathematical Sciences
Volume 95 Issue 1 September 1986 pp 37-40
The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.
Volume 118 Issue 4 November 2008 pp 613-625
Let $... |
This is a classic exercise of functional analysis, but I do not fully understand it after reading many answers in textbooks. So I am trying to reorganize the proof step by step in details. I am hoping that someone may review my proof very carefully and give comments or corrections. Then I will revise the proof and hope... |
@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... |
There are two ways to think of the Hilbert space as the space of sections of a line bundle.
First, the exponentiated Chern-Simons action on a manifold $\Sigma\times[0,1]$ is a section of the determinant line bundle $\mathcal{L}_\Sigma$ on the space of flat connections on $\Sigma$. Moreover, Wilson loops (which can be t... |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak an... |
Suppose that $f : (a, b) → \mathbb{R}$ is differentiable at $x ∈ (a, b)$. Prove that $\lim_{h→0}\frac{f(x + h) − f(x − h)}{2h}$ exists and equals $f'(x)$ Give an example of a function where the limit exists but the function is not differentiable.
Proof: $\lim_{h→0}\frac{f(x + h) − f(x − h)}{2h}$ =$\lim_{h→0}\frac{f((x-... |
amp-mathml
Displays a MathML formula.
Required Script
<script async custom-element="amp-mathml" src="https://cdn.ampproject.org/v0/amp-mathml-0.1.js"></script>
Supported Layouts container Examples amp-mathml.amp.html
This extension creates an iframe and renders a MathML formula.
<amp-mathml layout="container" data-form... |
Let ${\cal A}$ be a set of $k\times n $ matrices over ${\mathbb F}_2$.
We call ${\cal A}$ a
(n,k)-covering, if for every subset of columns$I=(i_1,\ldots,i_k)\subseteq [n]$, there is a matrix $A \in {\cal A}$such that the columns $(i_1,\ldots,i_k)$ in $A$ are linearly independent.
It is known that when choosing a random... |
So we try by applying the definition and see how it goes.
So we deal with 2 cases, either $\sup S=\infty$ or $\sup S=M<\infty$.
If it is the second case, this means that for all $x\in S$, $x\leq M$, so naturally $ax\leq aM$. By definition of the least upper bound, we have $\sup aS\leq aM=a\sup S$.
For the reverse, let ... |
Root system data for affine Cartan types¶ class
sage.combinat.root_system.type_affine.
AmbientSpace(
root_system, base_ring)¶
Ambient space for affine types.
This is constructed from the data in the corresponding classical ambient space. Namely, this space is obtained by adding two elements \(\delta\) and \(\delta^\vee... |
The series $\sum_{n=1}^\infty\frac1{4n^2-1}$ telescopes. Find a simple formula for the $k$th partial sum $S_k$, and use it to determine whether the series converges. If it converges, find its value.
I've mostly tried to intuit my way through. Calculating the partial sums to several degrees by varying $n$, I got $$\frac... |
If $\exists a, b, m_1, m_2 \in \mathbb{Z}, a\equiv b\pmod {m_1}$ and $a\equiv b\pmod {m_2}$, then $\exists k \in \mathbb{Z}, a=b+{m_1m_2k\over gcd(m_1,m_2)} \implies a\equiv b\pmod L$, where $L$ is the l.c.m. of $m_1$ and $m_2$.
I am trying to derive, starting with the first part, with no success
$\exists l \in \mathbb... |
I have mentioned this elsewhere, but it bears repeating because it is such an important concept:
Sufficiency pertains to
data reduction, not parameter estimation per se. Sufficiency only requires that one does not "lose information" about the parameter(s) that was present in the original sample.
Students of mathematica... |
I'm having trouble with the Ho-Lee model for short rates and differentiating between how to find the values for the free parameter λ versus using the model to predict future rates.
The Ho-Lee model for each step in a binomial tree: $$ \lambda_tdt + \sigma \sqrt dt $$
I've read that to set the free parameter at each ste... |
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ...
The accepted answer is c... |
156 10
Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8
##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in ##\mathbb R^2## subs... |
I'm not sure if it is possible to estimate the flux linkage of a PMSM motor (star connection) by the controller itself, where I can measure the input voltage, all the phase currents and the velocity. Measurements of R and L are also available.
The flux linkage is defined as:
$$ \lambda_m = \sqrt{2/3} \frac{2}{P} K_b $$... |
We define $G = \GSp^+(2g,\Q)$ by $$G = \{\gamma\in \GL_{2g}(\Z):\gamma^t J\gamma=r(\gamma)J,\text{ for all }\gamma\in\Bbb{Q}_+\}.$$ Let $L(\Gamma,G)$ be the free $\C$-module generated by the right cosets $\Gamma\alpha$ where $\alpha\in\Gamma \backslash G$. Note $\Gamma$ acts on $L(\Gamma,G)$ by right multiplication and... |
Finally, after a lot of hesitation, I rooted my phone (Samsung Galaxy On7, Android 6.0.1) using Magisk systemless root. The main aim was to remove pre-installed useless software (often termed "bloatware").I didn't want to flash Magisk using TWRP, because installing TWRP would remove the stock...
Summary: I want to basi... |
In the early 2000's, I wrote software code to process and recover telemetry data from archived raw telemetry files obtained from the Pioneer 10 and 11 spacecraft between 1972 and 2003. This led to my participation in the investigation of the Pioneer Anomaly, an as-yet unexplained small, anomalous acceleration of these ... |
The Annals of Mathematical Statistics Ann. Math. Statist. Volume 37, Number 2 (1966), 458-463. All Admissible Linear Estimates of the Mean Vector Abstract
Let $y$ be a single observation on a $p \times 1$ random vector which is distributed according to the multivariate normal distribution with mean vector $\theta$ and ... |
By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) \to E$, a non-constant surjective morphism defined over $\textbf{Q}$. Let me call this “
geometric” modularity.
It is k... |
I have asked this question in cs.stackeschange, and it was recommended to me that I asked it here.
Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not a factor between the weight sum of the approximate and the optimal solutions, but instead the cardinality of... |
Start by considering each coin in isolation, and the question becomes easier. Let $Y$ denote the number of heads for the first coin, and $Z$ denote the number of heads for the second coin, so $X=Y+Z$. $Y$ and $Z$ are identically distributed so let's just consider $Y$.
First, we know that if $Y=y$, then the first coin m... |
Getting all (complex) solutions of a non polynomial equation
Hi !
I was used to solve the following equation with Mathematica. \begin{equation} \alpha_1 + \alpha_2x + \alpha_3x^2 + x^4 + \frac{\alpha_4}{x^2-\alpha_0} + \frac{\alpha_5 x^2}{x^2-\alpha_0}=0 \end{equation} where $\alpha_i$ are constants.
The Mathematica fu... |
Energy Density Formula - Electric Energy Density Formulas with Examples
VIEW MORE
Energy Density Formula Energy Density refers to the total amount of energy in a system per unit volume. (Eventhough generally energy per unit mass is also mentioned as energy density, the proper term for the same is specific energy. The t... |
Learning With Large Datasets
Draw a learning curve and determine if more data needs to be collected.
Stochastic Gradient Descent Cost Function in SGD : \(cost(\theta, (x^{(i)}, y^{(i)})) = \frac {1}{2}(h_{\theta}(x^{(i)})-y^{(i)})^2\) randomly shuffle the data set a little gradient descent step using just one single tr... |
So what are spin-networks? Briefly, they are graphs with representations ("spins") of some gauge group (generally SU(2) or SL(2,C) in LQG) living on each edge. At each non-trivial vertex, one has three or more edges meeting up. What is the simplest purpose of the intertwiner? It is to ensure that angular momentum is co... |
This post is meant to teach new folks how to use MathJax and mhchem formatting on chem.SE
Getting started
On chemistry.SE, we use MathJax to format mathematical as well as chemical equations and similar expressions in questions, answers, and comments. MathJax allows us to typeset expressions using $\LaTeX$ notation.
To... |
Is the following claim known?
Claim: For any graph $G$ with $n$ vertices there exists a coloring of $G$ such that every independent set is colored by at most $O(\sqrt{n})$ colors.
Theoretical Computer Science Stack Exchange is a question and answer site for theoretical computer scientists and researchers in related fie... |
Matching pursuit
Piotr J. Durka (2007), Scholarpedia, 2(11):2288. doi:10.4249/scholarpedia.2288 revision #140500 [link to/cite this article]
Matching pursuit (MP) algorithm finds a sub-optimal solution to the problem of an adaptive approximation of a signal in a redundant set (dictionary) of functions. Commonly used wi... |
I am looking at Trefethen and Bau Exercise 37.1: I have two normalizations of the Legendre polynomials with corresponding recurrence relations: $$P_n(1)=1$$ which follows $$P_n(x) = \frac{2n-1}{n} x P_{n-1}(x) -\frac{n-1}{n}P_{n-2}(x)$$ and $$\|q_n\|=1$$ which follows $$xq_n(x)=\beta_{n-1}q_{n-1}(x)+\alpha_n q_n(x) + \... |
Optical fibers are one of the most widely used form of optical waveguide. Unlike other form of waveguides, optical fibers can be manufactured long and bendable, and thus are primarily used for a transmission media for optical fiber communication.
The history of optical fibers is believed to date back to mid 1800s, when... |
Consider to following decision problem:
Input: Undirected graph $G=(V,E)$
Question: Is the minimum numbers of colors needed to color the vertices (such that every two adjacent vertices aren't colored the same) exactly2015?
Note that 2015 could be any fixed number. This number is not part of the input.
I need to underst... |
I have to solve this equation
$$\ddot{x}x = -\frac{3}{2}\dot{x}^2 + \frac{\dot{x}}{h} + \sin(t)$$
where $h$ is defined by
$$h = \left(\frac{\dot{x}}{h}\right)^{1/3} + \frac{\dot{x}}{h}$$
My idea is to initialize $h=1$, solve the first equation, calculate a new value for h:
$$h = \frac{\dot{x}}{x\ddot{x} + (3/2)\dot{x}^... |
According to remark 6.14 in Shigeru Mukai's
An introduction to invariants and moduli (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine variety $X$ by a reductive group $G$ with respect to a nontrivial character $\chi: G \to \mathbb G_m$ may be considered as... |
Answer
$$x=-1.53, -10.47$$
Work Step by Step
Using the quadratic formula, we obtain: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$x=\frac{12\pm \sqrt{(12)^2-4(1)(16)}}{2(1)}$$ $$x=-1.53, -10.47$$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 h... |
Gain medium
The
active laser medium (also called gain medium or lasing medium) is the source of optical gain within a laser. The gain results from the stimulated emission of nuclear, electronic or molecular transitions to a lower energy state from a higher energy statepreviously populated by an electron current, chemic... |
Calculators¶ ClusterExpansionCalculator¶ class
mchammer.calculators.
ClusterExpansionCalculator(
structure, cluster_expansion, name='Cluster Expansion Calculator', scaling=None, use_local_energy_calculator=True)¶
A ClusterExpansionCalculator object enables the efficient calculation of properties described by a cluster ... |
Current browse context:
astro-ph.GA
Change to browse by: Bookmark(what is this?) Astrophysics > Astrophysics of Galaxies Title: SEAGLE--II: Constraints on feedback models in galaxy formation from massive early type strong lens galaxies
(Submitted on 4 Jan 2019)
Abstract: We use ten different galaxy formation scenarios ... |
The database contains data for Maass forms on $\Gamma_0(N)$ with trivial character and Laplace eigenvalue $\lambda = \frac14 + R^2$ in three ranges:
$1\le N \le 10$ and $0<R\le 10$,
prime $100 < N < 250$ and $0<R\le 2$, and
prime $250 < N < 1000$ and $0<R\le 1$,
as well as many additional examples for small $N$.
It is ... |
Is it (always) true that $$\mathrm{Var}\left(\sum\limits_{i=1}^m{X_i}\right) = \sum\limits_{i=1}^m{\mathrm{Var}(X_i)} \>?$$
The answer to your question is "Sometimes, but not in general".
To see this let $X_1, ..., X_n$ be random variables (with finite variances). Then,
$$ {\rm var} \left( \sum_{i=1}^{n} X_i \right) = ... |
There are a wide variety of notions of dimension of a measure. Your basic intuition is completely correct: for a dynamical system, the dimension of a natural invariant measure provides more relevant information than the dimension of the invariant set, since the system may spend more time in certain parts of the space.
... |
Given a positive definite quadratic $m\times m$ positive definite even quadratic form $Q$, and given a degree $g$, define $$\vartheta_Q^{(g)}(\Omega) = \sum_{N\in\Bbb Z^{m \times g}} \exp(\pi i N' Q N \Omega ) $$
If $m$ is a multiple of 4, this is a Siegel modular form of degree $g$ of weight $m/2$ with respect to the ... |
Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$
$$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad \text{or}\qquad \int_{0}^{\infty}|\psi(x)|^2 dx <\infty$$
What would be the physical implications if one (or both) of those integrals... |
Can someone please help me understand how the expressions after the equals (marked in red) are arrived at? I don't quite understand the very last one where the c.d.f. $\phi$ comes into the picture. My guess for the first expression is that the term added to $c$ is always going to be
negative and hence its added in orde... |
I'll follow your lead and represent adipic acid as $\ce{H2A}$ for the sake of simplicity.
Equations we know:
(I) o.100 = $\ce{[H2A] + [HA^{-}] + [A^{2-}]}$
(II) $\dfrac{\ce{[H^+] [HA^{-}]}}{\ce{[H2A]}} = 3.9 * 10^{−5}$
(III) $\dfrac{\ce{[H^+] [A^{-2}]}}{\ce{[HA^{-}]}} = 3.9 * 10^{−6}$
(IV) $\ce{[H^+]} = \ce{[HA^{-}] + ... |
This is a bit of an X-Y problem. You cannot accurately measure the resistances involved in the voltage dividers when the circuit is not powered, because the constituent resistors wind up in parallel. For instance, when you attempt to measure the resistance across the LDR, you wind up measuring the resistance of this ci... |
I am surprisingly having a bit of difficulty with an indefinite integral which is interesting since the integral I solved before is
$$ \int \frac{1+\cos2x}{\sin^2(2x)} dx$$
The integral I am currently working on is
$$ \int \frac{\sin^2(2x)}{1+\cos2x} dx$$
I first divided out giving: (
this is the mistake I made early o... |
The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are parameters and $W_t$ is Brownian motion. It is well known the solution to this equation. In particular, it is known that
$E(x_t)=x_0... |
Center Manifold
Jack Carr (2006), Scholarpedia, 1(12):1826. doi:10.4249/scholarpedia.1826 revision #126955 [link to/cite this article]
One of the main methods of simplifying dynamical systems is to reducethe dimension of the system.
Centre manifold theory is a rigorous mathematical technique that makes this reduction p... |
Do not passively use Yahoo where you need reliable historical data; it will just fail at one point (from what I have seen due to corporate actions/dividends not properly implemented). Paying for a single alternative data source will not save you either (Bloomberg sometimes reports crazy intraday prices); the only way i... |
Testing Time Dilation in Special Relativity with Spectroscopy of Fast Ions in Storage Rings: A New, Improved Limit x x
The relativistic Doppler effect was already proposed as an experimental test of relativity by Einstein in 1907 [1]. Time dilation leads to the
ether-independentrelativistic Doppler formula \nu_0=\nu_{\... |
Let $\Gamma\le SL(2,\mathbb{Z})$ be a congruence subgroup of level $N$. Let $R$ be a $\mathbb{Z}[1/N]$-algebra.
Let $\mathcal{Y}(\Gamma)_R$ denote the moduli stack over $R$ of elliptic curves equipped with a $\Gamma$-structure. For an integer $k$, let $\mathcal{Y}(\Gamma,\Omega^k)_R$ denote the moduli stack of triples ... |
I need to optimize affine transformations for of a set of triangles using energy function based on the connectivity.
The energy of an edge $e_j$ between triangles $T_a, T_b$ is given by $$ E_j = \left[(A_a v_{a,1} + t_a) - (A_b v_{1,b} + t_b) \right]+ \left[(A_a v_{a,2} + t_a) - (A_b v_{2,b} + t_b) \right] $$
where $(A... |
September 25th, 2017, 12:22 AM
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Math Focus: Calculus
Double Integration-Finding Volume
Hello,
Pic:-
So I was solving some questions of double integration.And I come across this que. which I didn't get.
I mean I get that it is asked to find out Volume but I don't... |
It is well known that the generating function of a regular language $L$, i.e. $\sum n_kz^k$ where $n_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials $P(z)/Q(z)$. Suppose that $L$ is the language accepted by some finite automaton $\mathcal{A}$. How to find the polynomials ... |
The first order necessary optimality conditions for a minimization problem with inequality constraints are
not that the gradient vanishes, since the minimum can be attained at the boundary of the feasible set (where only the directional derivatives into the interior of the set have to be non-negative, i.e., going away ... |
Consider a thin uniform rod AB of mass $m$ and length $l$ sliding down a frictionless wall and on a frictionless floor. Point O is its instantaneous center of rotation, and C its center of mass (and geometric center). We need to calculate the angle $\theta$ after which the rod loses contact with the wall. The initial v... |
Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, where overline denotes integral closure.
Huneke-Swanson show, (in their book on Integral Closure), that $I_a$ is well defin... |
I think Higham's
Accuracy and Stability of Numerical Algorithms addresses how one can analyze these types of problems. See Chapter 2, especially exercise 2.8.
In this answer I'd like to point out something that isn't really addressed in Higham's book (it doesn't seem to be very widely known, for that matter). If you ar... |
Scale-free networks
Cesar A. Hidalgo and Albert-Laszlo Barabasi (2008), Scholarpedia, 3(1):1716. doi:10.4249/scholarpedia.1716 revision #140135 [link to/cite this article]
A network that has a power-law degree distribution, regardless of any other structure, is called a
scale-free network. Power-Law degree distribution... |
Both answers are "No."
There are well-known obstructions to the existence of an equivariant momentum mapping arising from the action by symplectomorphisms of a group $G$ on a symplectic manifold. They can be phrased in many ways, but if $G$ is connected and its Lie algebra is semisimple, for example, the obstructions v... |
Getting all (complex) solutions of a non polynomial equation
Hi !
I was used to solve the following equation with Mathematica. \begin{equation} \alpha_1 + \alpha_2x + \alpha_3x^2 + x^4 + \frac{\alpha_4}{x^2-\alpha_0} + \frac{\alpha_5 x^2}{x^2-\alpha_0}=0 \end{equation} where $\alpha_i$ are constants.
The Mathematica fu... |
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