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Products of Linear Spaces
Definition: Let $X$ and $Y$ be linear spaces. The Product $X \times Y$ is defined to be the space with the operators of addition and scalar multiplication defined for all $(x_1, y_1), (x_2, y_2) \in X \times Y$ and $\alpha \in \mathbb{R}$ by $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$ a... |
Question
A heart defibrillator being used on a patient has an $RC$ time constant of 10.0 ms due to the resistance of the patient and the capacitance of the defibrillator. (a) If the defibrillator has an $8.00 \mu \textrm{F}$ capacitance, what is the resistance of the path through the patient? (You may neglect the capac... |
I am studying numerical methods for differential equations. I came accros the trapezoidal method in two forms, an explicit and an iterative one. I would like to know the advantages and disadvantages of each of those methods. Furthermore, how can I study the stability for the iterative method? Which stability definition... |
Di Filippo, Domenico (2013)
The Charged anti-counter for the NA62experiment at CERN SPS. [Tesi di dottorato]
Item Type: Tesi di dottorato Lingua: English Title: The Charged anti-counter for the NA62experiment at CERN SPS Creators:
Creators Email Di Filippo, Domenico domenicodifilippo@gmail.com Date: 2 April 2013 Number... |
Use Sylow’s theorem. To review Sylow’s theorem, check out the post Sylow’s Theorem (summary). Read the corollary there as well to understand the proof below.
Proof.
Let $n_p, n_q$ be the number of $p$-Sylow subgroups and $q$-Sylow subgroups, respectively.Then by Sylow’s theorem, we have\[n_p\equiv 1 \pmod p, \qquad n_p... |
Nets in a Normed Algebra
Definition: A Directed Set is a set $\Lambda$ with an relation $\leq$ that satisfies the three conditions: 1) $\lambda \leq \lambda$ for all $\lambda \in \Lambda$ (Reflexivity). 2) If $\lambda_1 \leq \lambda_2$ and $\lambda_2 \leq \lambda_3$ then $\lambda_1 \leq \lambda_3$ (Transitivity). 3) Fo... |
Please note that the recommended version of Scilab is 6.0.2. This page might be outdated.
See the recommended documentation of this function damp
Natural frequencies and damping factors.
Syntax [wn,z] = damp(sys) [wn,z] = damp(P [,dt]) [wn,z] = damp(R [,dt]) Parameters sys
A linear dynamical system (see syslin).
P
An a... |
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute t... |
on behalf of the CMS Collaboration
Articles written in Pramana – Journal of Physics
Volume 69 Issue 6 December 2007 pp 1069-1074 Calorimetry and Muons
The performance of the CMS hadron calorimeter is studied using test beam facilities at CERN. Two wedges of brass-scintillator calorimeter are exposed to negative and pos... |
I'm attempting to emulate the modulation and demodulation of a PAL video frame in software, but I'm having trouble understanding how the AM (luma) and QAM (UV) components of the signal are separated during the demodulation process.
Background:
The software I'm writing takes an RGB bitmap, translates it to Y'UV using th... |
I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$.
I know that $$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$ but that's only valid when $\operatorname{Arg} z_1 - \operatorname{Arg} z_2 \in (-\pi,\pi]$, so I'm not ... |
Overview¶
We’ve already generated quite a few figures in these lectures using Matplotlib.
Matplotlib is an outstanding graphics library, designed for scientific computing, with
high-quality 2D and 3D plots output in all the usual formats (PDF, PNG, etc.) LaTeX integration fine-grained control over all aspects of presen... |
The Annals of Statistics Ann. Statist. Volume 25, Number 2 (1997), 818-850. Asymptotic expansion of M-estimators with long-memory errors Abstract
This paper obtains a higher-order asymptotic expansion of a class of
M-estimators of the one-sample location parameter when the errors form a long-memory moving average. A su... |
Difference between revisions of "De Bruijn-Newman constant"
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where
where
−
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)</math>
+
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2}... |
Let $A$ be a linear operator on $L$ with domain $D(A)$. To show that $A$ is closable, i.e., it has a closed linear extension, why does it suffice to show that if $\{f_n\}\subset D(A), f_n\to 0,$ and $Af_n\to g\in L,$ then $g=0$? I would greatly appreciate any help.
So we say that an operator is closed if it has a close... |
Refering to Theorem 8.2 in Baby Rudin
8.2 TheoremSuppse $\sum c_n$ converges. Put $$f(x)=\sum_{n=0}^{\infty} c_nx^n \ \ \ \ (-1<x<1)$$ Then $$\lim_{x\rightarrow1}f(x)=\sum_{n=0}^{\infty}c_n$$
The proof in Rudin is that outlined in Wikipedia
However, Factoring out $(1-x)$ seems not natural for me.
The theorem looks like... |
A few preliminary remarks:
1) Complexifications of Banach lattices are in fact a special case of the more general concept of
complexifications of real Banach spaces.
2) Most books and articles about complex Banach lattices which contain spectral theoretic results focus on
positive operators (or, say, generators of posi... |
Books by Independent Authors Advanced Real Analysis 2017, 434-544 Chapter X. Introduction to Wavelets Abstract
This chapter introduces the relatively recent subject of wavelets, which is an outgrowth of Fourier analysis in mathematics and signal processing in engineering. Except in one case, construction of examples of... |
If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$
Problem 493
Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying\[|A|+|B| > |G|.\]Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.Then prove that $G=AB$, where\[AB=\{ab \mid a\in A, b\in B\}.\]
Since $A... |
4:28 AM
@MartinSleziak Here I am! Thank you for opening this chat room and all your comments on my post, Martin. They are really good feedback to this project.
@MartinSleziak Yeah, using a chat room to exchange ideas and feedback makes a lot of sense compared to leaving comments in my post. BTW. Anyone finds a
\oint\fr... |
Assume that:
$P(X): \{0, 1\}^{\ell_{X}} \rightarrow \{0, 1\}^{\ell_{X}}$. $Q(X): \{0, 1\}^{\ell_{X}} \rightarrow \{0, 1\}^{\ell_{X}}$. $E(K, X): \{0, 1\}^{\ell_{K}} \times \{0, 1\}^{\ell_{X}} \rightarrow \{0, 1\}^{\ell_{X}}$. $D(K, X): \{0, 1\}^{\ell_{K}} \times \{0, 1\}^{\ell_{X}} \rightarrow \{0, 1\}^{\ell_{X}}$. $\w... |
Under the auspices of the Computational Complexity Foundation (CCF)
In 1994, Y. Mansour conjectured that for every DNF formula on $n$ variables with $t$ terms there exists a polynomial $p$ with $t^{O(\log (1/\epsilon))}$ non-zero coefficients such that $\E_{x \in \{0,1\}}[(p(x)-f(x))^2] \leq \epsilon$. We make the firs... |
The solutions of the Schrödinger equation for a hydrogen atom have definite energy. Does this mean that they could be written as a superposition of plane waves of a single frequency - corresponding to that energy - with only the phases and directions differing?
Eigenstates of any Hamiltonian by definition have definite... |
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... |
Normal Lines on a Surface
We have just looked at Tangent Planes to Surfaces to a point on a surface of a two variable real-valued function $z = f(x, y)$. We will now look at what a normal line on a surface $S$ at point $P$ is.
First, let $z = f(x, y)$ be a two variable real-valued function that generates the smooth sur... |
To amplify Yehuda Lindell's answer even within a single family of groups, here is an example of a 2046-bit modulus $p$ for which discrete logs in $(\mathbb Z/p\mathbb Z)^\times$, as in standard modular multiplication Diffie–Hellman, are extraordinarily easy to compute:
0x2465a7bd85011e1c9e0527929fff268c82ef7efa416863ba... |
Please let me assume that you wonder about control surface effectiveness rather than their efficiency. Both are closely related, but I prefer to address their effectiveness - doing what the pilot demands them to do.
The aerodynamic forces are proportional to the dynamic pressure $q$ of the flow, which is density times ... |
My country has major economical headaches each 10 years or so, like hyperinflation or defaults or things like that. It's been 17 years since the last major crisis (a larger crisis than the 1930's crisis) but it looks like we are aiming again towards the same. Usually we have trade imbalances or fiscal deficits which ca... |
Let $K_n$ be the sets of vectors $x \in \mathbb{Z}^d $ with each coordinates $x_i$ between $1$ and $n$. For any subset $A$ of $K_n$, let $S(A)$ be the set of points $x \in K_n$ which are on some line containing at least two points of $A$ (in other words, $S(A)$ is the union of the lines passing through - at least - two... |
Limits to Infinity and Negative Infinity
Limits to Infinity and Negative Infinity
Definition: Let $f : A \to \mathbb{R}$ be a function and let $c$ be a cluster point of $A$. Then we say that the limit as $x \to c$ of $f$ is $\infty$ written $\lim_{x \to c} f(x) = \infty$ if $\forall \alpha \in \mathbb{R}$ $\exists \del... |
Difference between revisions of "Main Page"
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[http://blogsearch.google.com/blogsearch?hl=en&ie=UTF-8&q=polymath1&btnG=Search+Blogs Here is a further list of blog posts related to the Polymath1 project]. [http://en.wordpress.com/tag/polymath1/ Here is wordpress's list].
... |
I'm typesetting a bunch of source code, and I really like the way the double-nested logical-and and logical-or symbols look for denoting bitwise logical and/or.
Unfortunately, LaTeX doesn't seem to have these symbols, so I resorted to rolling my own using a combination of
\land and
\lor with carefully rotated and align... |
Difference between revisions of "Main Page"
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A spreadsheet containing the latest upper and lower bounds for <math>c_n</math> can be found [http://spreadsheets.google.com/ccc?key=p5T0SktZY9DsU-uZ1tK7VEg here]. Here are the proofs of our [[upper and lower bounds]] for these con... |
Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.(a) Prove that the set ... |
We recall here the definition of a nilpotent group.Let $G$ be group. Define $G^0=G$,\[ G^1=[G, G]=\langle [x,y]:=xyx^{-1}y^{-1} \mid x, y \in G\rangle,\]and inductively define\[G^i=[G^{i-1},G]=\langle [x,y] \mid x \in G^{i-1}, y \in G \rangle.\]Then we obtain so called the lower central series of $G$:\[ G^{0} \triangle... |
There are two primary dust populations near 1 AU, interplanetary dust (IPD) and interstellar dust (ISD) [
Mann, 2010]. I also discussed dust observations in detail at https://physics.stackexchange.com/a/160627/59023. Interplanetary Dust
IPD of ~1 $\mu$m size drift sunward due to Poynting-Robertson drag while following ... |
Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an infinite linear combination $\sum_{i=1}^\infty\beta_i a_i$ of elements of $A$. Are there simple (non-trivial) conditions guaran... |
Dirichlet's Kernel Representation of the Partial Sums of a Fourier Series
Recall from the Dirichlet's Kernel page that Dirichlet's kernel is the collection of functions $D_n$ where for each $n \in \mathbb{N}$:(1)
We also looked at some properties of the functions in Dirichlet's kernel. We saw that for each $n \in \math... |
We will be offering mothur and R workshops throughout 2019. Learn more. Difference between revisions of "Sharedace"
(New page: Validate output by making calculations by hand '''Example Calculations''' '''*.sharedAce''' Example calculations below will be performed using data from the Eckburg 70.stool_compa...)
Line 78: ... |
Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let $F:X\to X$ be a continuos transformation that commutes with $T.$
Define $F_{*}\mu(A)=\mu(F^{-1}A)$ for every Borel set $A.$
I ... |
You are currently browsing the monthly archive for October 2012.
A quite amusing post I stumbled upon describes pricing algorithms used by book retailers at Amazon.com, and what happens when two retailers use such algorithms. I am pretty sure that economists can say much about the use of these algorithms, optimal prici... |
Boundedness of a Subset of C(X)
Recall from The Set of Real-Valued Continuous Functions on a Compact Metric Space X, C(X) page that if $(X, d)$ is a compact metric space the set $C(X)$ is defined to be the set of all real-valued continuous functions on $X$, that is:(1)
We then defined a very important metric $\rho : C(... |
Normally, if an object of mass $m$ is inclined to the horizontal at an angle $b$, we set the reaction force of the object on the inclined plane as $R = mg\cos{b}$ (if we resolve the force of gravity so the line of action coming out of the plane is perpendicular to it).
However in circular motion*. it's assumed that $R\... |
While scientists have devoted much research to models of neural activity in the brain, they have paid little attention to modeling drugs that target the brain. Development of this class of drugs is very challenging and necessitates an understanding of the highly complex processes that govern the concentration profile o... |
An
incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition labeled by $P$ and $L$.
A
projective plane is an incidence geometry satisifying the following axioms: For every pair of distin... |
According to Wikipedia the Sun's "power density" is "approximately 276.5 $W/m^3$, a value that more nearly approximates that of reptile metabolism or a compost pile than of a thermonuclear bomb." My question is, so why is the Sun's core so hot (15.7 million K)? Using a gardener's (not a physicist's intuition) it seems ... |
linear poolor weighted averageof Adila's and Benoit's credence functions. That is, we assign a weight to Adila ($\alpha$) and a weight to Benoit ($1-\alpha$) and we take the linear combination of their credence functions with these weights to be our credence function. So my credence in $X$ will be $\alpha c_A(X) + (1-\... |
Can someone give a proof of the "technical part" of the following proof?
I'm not sure whether this approach works, but you can try it:
Suppose $\left\langle g\right\rangle H\neq G$. Then there exists an $x\in G\setminus \left\langle g\right\rangle H$. Let $H'$ denote the smallest subgroup of $G$ containing $H$ and $x$.... |
$\int z^2 \log [(z+1)/(z-1)] dz$ taken around circle $|z|=2$
I am taking residues at $\pm 1$.
This gives me 0 as the value of integral. Is this correct.
How do I modify the integral to get value over half the circle?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and prof... |
Tagged: cyclic group Problem 613
Let $m$ and $n$ be positive integers such that $m \mid n$.
(a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined. (b) Prove that $\phi$ is a group homomorphism. (c) Prove that $\phi$ is surjective.
Add to solve later
(d) Determi... |
Sec2.1 Lorentz Invariance
\(\quad\)QFT is based on quantum mechanics, so we provide only the briefest version of summaries, in the generalized version of Dirac:
Def(Ray):
A \emph{ray} \(\mathscr{R}\) is a set of normalized vectors, i.e., \(\{\psi|\langle\psi,\psi\rangle=1\}\). Here \(\langle\cdot,\cdot\rangle\) is the ... |
Under the auspices of the Computational Complexity Foundation (CCF)
Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$ formed by the intersection of $k$ halfspaces, we prove that $$\left|\Pr\left[X \in P\... |
The Riemann-Lebesgue Lemma
Recall from the Lebesgue Integrable Functions with Arbitrarily Small Integral Terms page that if $f \in L(I)$ then for all $\epsilon > 0$ there exists upper functions $u^*, v^* \in U(I)$ where $f = u^* - v^*$, $v^*$ is nonnegative almost everywhere on $I$, and $\displaystyle{\int_I v^*(x) \: ... |
What is Energy?
In recent days, we often hear about
energy. Every invention, civilization is based upon acquiring and effectively using energy. This is possible by the unique property of our Universe energy that it can be transformed and transferred, but the total amount is always the same (conserved). One fundamental ... |
The Twin Paradox is undoubtedly one of the most discussed things in special relativity and have a tendency to confuse most of us.
Classically, it's resolved by either stick to one of the three reference frames in question – one with the Earth at rest, one following the ship as it goes outwards and one following the shi... |
Current browse context:
math.DG
Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Differential Geometry Title: Darboux-Weinstein theorem for locally conformally symplectic manifolds
(Submitted on 1 Nov 2015)
Abstract: A locally conformally symplectic (LCS) form is an almost symplectic fo... |
I found in this article a straightforward way to calculate the eigenvalues of the hamiltonian of an electron under the influence of an homogenous magnetic field (p. 5): http://www.phys.spbu.ru/content/File/Library/studentlectures/schlippe/qm07-06.pdf
$$\vec{B}=B\hat{z}$$
For this, they express the magnetic potential as... |
Roscoff is a village at the north-west corner of France, located on a small piece of land that protrudes into the English canal. Right here, in 1548, the six-year-old Mary, Queen of Scots, having been betrothed to the Dauphin François, disembarks.
As far as I understood, the most common sights in the area are tourists ... |
The possibility though remote, is intriguing as we may be able in the future to actually "see" our own planet's history. Though sounding science fiction, if we are able to detect bodies in space that are able of reflecting light emitted from our planet earth, using amplification systems and filters,this may - if possib... |
Bessel's Inequality for Inner Product Spaces
Recall from the The Pythagorean Identity for Inner Product Spaces page that if $H$ is an inner product space and $\{ x_1, x_2, ..., x_n \}$ is an orthonormal subset of $H$ then for all $c_1, c_2, ..., c_n \in \mathbb{R}$ we have that:(1)
We now use the Pythagorean identity t... |
Consider a theory $$\mathcal{L}=(\partial_\mu\Phi^\dagger)(\partial^\mu\Phi)-\mu^2(\Phi^\dagger\Phi)-\lambda(\Phi^\dagger\Phi)^2$$ where $\Phi=\begin{pmatrix}\phi_1+i\phi_2\\ \phi_0+i\phi_3\end{pmatrix}$ is a complex $SU(2)$ doublet. After symmetry breaking there is no residual symmetry and hence there are $(2^2-1)=3$ ... |
I was watching a video on "How Does a Quantum Computer Work?".
I'm confused about what they mean by: "
Although the qubits can exist in any combination of states, when they are measured they must fall into one of the basis states."
From what I know about linear algebra, if we represent the state of a qubit by $|\psi\ra... |
Isomorphisms Between Normed Linear Spaces
Recall from the Isomorphisms Between Normed Linear Spaces page that if $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y)$ are normed linear spaces then an isomorphism between $X$ and $Y$ is a bijective bounded linear operator $T : X \to Y$ such that there exists constants $m > 0$ an... |
Hyperelastic materials are the special class of materials which tends to respond elastically when they are subjected to very large strains. They shows both a nonlinear material behavior as well as large shape changes. They posses certain characteristics:
All the polymers tends to show the hyperelastic behaviour such as... |
Under the auspices of the Computational Complexity Foundation (CCF)
In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function.
Specifically, we prove that for every non-linear and symmetric $f:\{0,1\}^{k} \to \{0,1\}$ there exists a set $... |
Answer
The solution set is $$\{0,\frac{2\pi}{3},\frac{4\pi}{3}\}$$
Work Step by Step
$$\cos2x-\cos x=0$$ over interval $[0,2\pi)$ 1) In this case, only the interval for $x$, which is $[0,2\pi)$, is necessary, as you will see in step 2 that $\cos2x$ would be changed to a function of $x$ only. $$x\in[0,2\pi)$$ 2) Now con... |
Answer
$(x,y) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$, which matches up with option (D)
Work Step by Step
$x = cos~t$ $y = sin~t$ When $t = \frac{\pi}{4}$: $x = cos~(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ $y = sin~(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ Then $(x,y) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$, which mat... |
Lunar Lander bug?
06-10-2014, 09:02 PM
Post: #1
Lunar Lander bug?
I was playing with the lunar lander game on my 29C and took a look at the source code. I can't figure out the terminal velocity calculation and I was wondering if someone could walk me through it. The program is here.
Lines 46-50 indicate that a burn b a... |
You can still estimate parameters by using the likelihood directly. Let the observations be $x_1, \dots, x_n$ with the exponential distribution with rate $\lambda>0$ and unknown.The density function is $f(x;\lambda)= \lambda e^{-\lambda x}$, cumulative distribution function $F(x;\lambda)=1-e^{-\lambda x}$ and tail func... |
After the excellent post by JD Long in this thread, I looked for a simple example, and the R code necessary to produce the PCA and then go back to the original data. It gave me some first-hand geometric intuition, and I want to share what I got. The dataset and code can be directly copied and pasted into R form Github.... |
Tagged: norm Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue Problem 419 (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.
Add to solve later
(b) Let $A$ be a real orthogonal $3\times 3$ m... |
Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:... |
What is Second Order Cone Programming (SOCP)?
Second Order Cone Programming (SOCP) problems are a type of optimisation problem that have applications in many areas of science, finance and engineering. A summary of the type of problems that can make use of SOCP, including things as diverse as designing antenna arrays, f... |
To me the unification of logistic, linear, poisson regression etc... has always been in terms of specification of the mean and variance in the Generalized Linear Model framework. We start by specifying a probability distribution for our data, normal for continuous data, Bernoulli for dichotomous, Poisson for counts, et... |
Table of Contents: How Capacitors are Connected?
We are also interested to calculate the amount of charge stored in a system where two or more capacitors are connected. The combination can be made in many ways. The combination is connected to a battery to apply a potential difference (V) and charge the plates (Q). We c... |
Concluding Progressive Remarks
The Salmon Prize (photo of the prize here) is offered by mathematician/biologist Elizabeth Allman, and can be appreciated in the broad mathematical context that is provided by Topics in Tensors: A Summer School by Shmuel Friedland.
Until such time as further comments are offered, a workin... |
Here's a nice couple of paragraphs from Bohr & Mottelson:
Since time reversal $T$ anticommutes with the total angular momentum, it is convenient to combine $T$ with a rotation $R$ through the angle $\pi$ about an axis perpendicular to the $z$-axis (the axis of space quantization). Such a rotation also inverts [angular ... |
The Orthogonality Theorem for Characters of Irreducible Group Representations
Definition: Let $G$ be a group and let $\varphi, \psi : G \to \mathbb{C}$. The Inner Product of $\varphi$ and $\psi$ will be defined as $\displaystyle{\langle \varphi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\varphi(g)} \psi(g)}... |
One thing you might want to do is write the vector $\vec{a}$ as some magnitude $\theta$ times a
unit vector
{a1, a2, a3}. Then calculate $\exp(i \theta \,a\cdot\sigma)$. This will allow you to get rid of those factors of $\sqrt{a_1^2+a_2^2+a_3^2}$, which are just some $\theta$ anyway.
Here it is in Mathematica:
a = \[T... |
496 results for "fraction".
Three graphs are given with areas underneath them shaded. The student is asked to calculate their areas, using integration. Q1 has a polynomial. Q2 has exponentials and fractional functions. Q3 requires solving a trig equation and integration by parts.
Question Draft
CC BY Published
Last mod... |
The Cauchy Product of Power Series
The Cauchy Product of Power Series
Recall from Convergence of Cauchy Products of Two Series of Real Numbers page that:
If $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ are both absolutely convergent series that converge to $A$ and $B$ respective... |
The Supremum and Infimum of The Bounded Set (a + S)
The Supremum and Infimum of The Bounded Set (a + S)
Recall from The Supremum and Infimum of a Bounded Set page the following definitions:
Definition: Let $S$ be a set that is bounded above. We say that the supremum of $S$ denoted $\sup S = u$ is a number $u$ that sati... |
You asked for a qualitative picture, so here goes.
Consider a simplified example: the quantum harmonic oscillator.
Its ground state is given by
$$ \Psi(x) = \text{const} \cdot \exp \left( - m \omega_0 x^2 / 2 \hbar \right). $$
Now suppose that we are measuring the position of this oscillator in the ground state. We cou... |
In one of his seminal papers [1], Moser proved a result, which in the simplest setting, still capturing the gist, states: Given a positive continuous smooth function h on a compact, connected domain \(D\subset R^n\) with the average \([h] = 1\), there exists a diffeomorphism \(\varphi\) of \(D\) onto itself with the Ja... |
Classification of continuous-time and discrete-time signals deals with the type of independent variable. If the signal amplitude is defined for every possible value of time, the signal is called a continuous-time signal. However, if the signal takes values at specific instances of time but not anywhere else, it is call... |
The framework
Recall from last time:
$\mathcal{F}$ is a finite set of propositions. (It is the set of propositions about which our agent has an opinion.) A credence function is a function $c : \mathcal{F} \rightarrow [0, 1]$.
\[
\langle c(X_1), \ldots, c(X_n) \rangle
\]
This is essentially the representation we used la... |
Let X be a continuous random variable with pdf
$f(x) = \begin{cases} 1/4, & \text{if $x$ $\in$ (-2,2)} \\ 0, & \text{otherwise} \end{cases}$
Find the probability density functiions of i) Y=$X^3$ and ii) Z= $X^4$
So far I have done this for i)
$F_Y(Y) = P(Y \le y) = P(X \le y^{1/3}) = F_X(y^{1/3}) = \int1/4.dt $
Carryin... |
I follow the hint there and adapt the notations.For the statement (c), you can take the long exact sequence of the sequence of the complex $0\to C^\bullet(\mathfrak{U},\mathcal{F})\to C^\bullet(\mathfrak{U},\mathcal{G})\to D^\bullet(\mathfrak{U})\to 0$. This gives you
$$0\to \check{H}^0(\mathfrak{U},\mathcal{F})\to \ch... |
Series Convergence and Divergence Practice Examples 2
We will now look at some more examples of applying the various convergence/divergence tests we have looked at so far to some series without being given what test to apply specifically.
More examples of evaluating series can be found on the following page:
Series Con... |
Series Convergence and Divergence Practice Examples 3
We will now look at some more examples of applying the various convergence/divergence tests we have looked at so far to some series without being given what test to apply specifically.
More examples of evaluating series can be found on the following page:
Series Con... |
Thesis BELLE2-MTHESIS-2019-006 Untagged Exclusive Analysis of the Semileptonic Decay $B\rightarrow\pi\ell\nu$ from Belle II Data in Preparation for |$V_{ub}$| Extraction
Svenja Granderath ; Prof. Dr. Jochen Dingfelder ; Dr. Peter Lewis
2019 Physikalisches InstitutBonn, Germany
Abstract: In this thesis an untagged exclu... |
Problem
Let $X_1, \ldots, X_n$ be a sample from normal distribution $N(\theta, \theta^2)$, where $\theta > 0$. I am to find minimal sufficient statistic and prove that it is not complete.
Finding the minimal sufficient statistic
First of all I did some transformations which led me to the following expression $$f(x|\the... |
So i have a question which asks to find the fourier series of $\left\vert\,\sin\left(x\right)\,\right\vert\,$. I have worked out the solution as $$ {2 \over\pi} - {4 \over \pi}\sum_{k = 1}^{\infty}{\cos\left(2kx\right) \over 4k^{2} - 1} $$
Which i am pretty sure is correct as i have the solution in my book.
The second ... |
Consider the following experiment. I take a spin-$\frac{1}{2}$ particle and make a $\sigma_x$ measurement (measure the spin in the $x$ direction), then make a $\sigma_y$ measurement, then another $\sigma_x$ one, then $\sigma_y$, and so on for $n$ measurements. The formalism of quantum mechanics tells us that the outcom... |
I have some troubles understanding one statement in the book "Nonlinear Optics" by R.Boyd. In chapter 7.1, it is derived that in a nonlinear material a light beam undergoes self-focusing due to the dependence of the refractive index on the intensity of the beam itself, and the value of the corresponding self-focusing a... |
I'm trying to understand a part of N.J.Hitchin's paper "The moduli space of complex Lagrangian submanifolds" https://arxiv.org/pdf/math/9901069.pdf . In particular I would like to understand how any given holomorphic function $\mathcal{F}$ will naturally give a special Kahler manifold.
In the following I will try my be... |
To give an example where convergence of Cauchy sequences is important: time-evolution is typically calculatedas$$ |\psi(t)\rangle = e^{i\hbar^{-1} \hat{H}\cdot t}|\psi_0\rangle$$now, the exponential of an operator is defined
1 by$$ e^{\hat{A}} = \sum_{i=0}^\infty\frac{\hat{A}^i}{i!}$$where the sum in turn is defined by... |
Differential and Integral Equations Differential Integral Equations Volume 26, Number 7/8 (2013), 757-780. Asymptotic behavior of fractional order semilinear evolution equations Abstract
Fractional calculus is a subject of great interest in many areas of mathematics, physics, and sciences, including stochastic processe... |
The usual representation of a free electromagnetic wave in vacuum looks like this:
The blue parts are the local electric field, while the green parts are the local magnetic field.
The
circularly polarized wave is also another standard representation of an EM travelling wave, but it is a bit less well known.
Here are th... |
In the Heisenberg picture (using natural dimensions): $$ O_H = e^{iHt}O_se^{-iHt}. \tag{1} $$ If the Hamiltonian is independent of time then we can take a partial derivative of both sides with respect to time: $$ \partial_t{O_H} = iHe^{iHt}O_se^{-iHt}+e^{iHt}\partial_tO_se^{-iHt}-e^{iHt}O_siHe^{-iHt}. \tag{2} $$ Theref... |
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