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Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
The simplest two body interaction term for fermions is
$$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$
and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting weird sign errors. The first symmetry comes from Hermiticity. To have $H$ be Hermitian, we need
$$H = H^\dagger = \sum_{ijkl... |
Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent.This means that there exists a positive integer $n$ such that $N^n$ is the zero matrix $O$.
Let $\lambda$ be an eigenvalue of $B$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. That is, we have $B\mathbf{v}=\lambda \mathbf{v}$ and $\mathbf{v}\... |
I don't think Hilmar's answer is very good as it interprets the DFT within a specific application context. That confuses issues.
The DFT is a tranform that works on a set of N samples. The samples are presumed to be evenly spaced in their domain on a finite interval of samples is called a frame. It may be time, it may ... |
Does the frictional force increase as the normal force increases, or does the coefficient of friction get smaller in value?
The coefficient of friction should in the majority of cases, remain constant no matter what your normal force is. When you apply a greater normal force, the frictional force increases, and your co... |
I have a image $U_{m \times n}:\Omega \to \mathbb R^2$, the output $P$ can be define as $$P=\mu J_{m \times n} - U$$ where $\mu = \max \{ u_{ij} : 1 \leq i \leq m, 1 \leq j \leq n\}$, $J_{m \times n}$ be the $m \times n$ matrix whose $i, j$th component is $1$: that is, the all-ones matrix. (This notation isn't quite st... |
By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$
If $K$ is a number field, let $\delta(K)$ denote the smallest positive height of an element of $K$ (recall roots of unity are precisely the points of height zero). Here is ... |
In a paper by Yuji Tachikawa, I found a q-deformed "2d Yang-Mills paritition function for a cylinder". Here it is (adapted):
$$ Z(q, x_L, x_R) = \mu(q, x_L)^{-1/2} \langle x_L | \bigg\[ \sum_{R \in \mathrm{Irr}(G)} | R \rangle e^{- aC_2(R) } \langle R | \bigg\] |x_R \rangle \mu(q, x_R)^{-1/2}$$
Here's some stuff to hel... |
We first analyze the capacitated facility location problem with splittable demands (CFLS) and prove its approximation ratio.
Lemma 1.1
(Adding an AP)
If the current solution\({\mathcal {S}}\) satisfy\(c_s({\mathcal {S}}) - c({\mathcal {S}}^{*}) \ge \frac{nc({\mathcal {S}})}{p(n)}\), then there exists a\(n \in {\mathcal... |
Given Sam's number is $x$ and Peter's number is $y$, most people seem to be working from the following premises:
$x, y \in \mathbb{N}$ where $\mathbb{N}$ is the natural integers; $\mathbb{N} = \{1, 2, ...\}$ $2002 = x + y \lor 2002 = xy$
If that's correct then you
must agree with Mike Earnest's answer or be logically i... |
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Physical Review Letters, ISSN 0031-9007, 05/2017, Volume 118, Issue 21
Journal Article
2. Measurement of the tau Lepton Polarization and R(D) in the Decay (B)over-bar -> D tau(-) (v)over-... |
ISSN:
1556-1801
eISSN:
1556-181X
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Networks & Heterogeneous Media
June 2007 , Volume 2 , Issue 2
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Abstract:
A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other... |
States of Matter: Gases and Liquids Intermolecular Forces and Thermal Energy London force (or) dispersion force: Observed in between non polar atoms or non polar molecules. e.g. Xe and Xe CH 4 and CH 4 and CCl 4 and CCl 4 F\propto\frac{1}{r^{6}} Dipole-Dipole attraction: Attraction between polar compounds
\therefore\ F... |
A connected graph H with $$|H|\ge \sigma (G)$$ | H | ≥ σ ( G ) is said to be G-good if $$R(G,H)=(\chi (G)-1)(|H|-1)+\sigma (G)$$ R ( G , H ) = ( χ ( G ) - 1 ) ( | H | - 1 ) + σ ( G ) . For an integer $$\ell \ge 3$$ ℓ ≥ 3 , let $$P_\ell $$ P ℓ be a path of order $$\ell $$ ℓ , and $$H^{(\ell )}$$ H ( ℓ ) a graph obtained... |
Suppose $z_{t} = x_{t}y_{t}$ where $x_{t}$ and $y_{t}$ are 0 mean, independent stationary stochastic process. What is the autocovariance function of $z_{t}$? Show that the spectral density can be written as $f_{z}(\omega) = \int_{-0.5}^{0.5}f_{x}(\omega-v)f_{y}(v)dv$.
Attempt:
If the series are independent, wouldn't th... |
This is a very deep question.
A proof in terms of numbered formulae and various $\Rightarrow$, resp. $\Leftrightarrow$-signs could be checked by an automated proof checker. On the other hand, a "figure" is just a bitmap, or a pixel heap, and I doubt that an automated proof checker would ever be able to make out what th... |
In this work we consider the most general analysis
of $\tau\rightarrow (K\pi)^{-} \nu_{\tau}$ decays within an effective field theory description of heavy new physics (NP) including SM operators up to dimension six with massless neutrinos. All hadron form factors are built exploiting chiral symmetry, dispersion relatio... |
I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme.
I've managed to show that the one-step time integration scheme ( OSTIS ) is stable and convergent when the gradient of thermal field is positive (energy of... |
Trigonometry is not only related to ratio triangles. It is about ratios and we can plot graphs with these ratios. Trigonometry graphs are periodic, that means that the graph repeats after a certain amount of time. These functions can be used if there is a regular occurrence of something, for example, the rotation of th... |
To understand either of these, you first have to understand the basic premise behind Graph Signal Processing (GSP) which is to map a signal to a graph and then work with it on the "Graph space". This is possible due to certain similarities of classic DSP concepts and Algebraic Graph Theory.
So, it is easier to start fr... |
Groups usually come with homomorphisms, defined as mappings preserving multiplication:
$$f(a\cdot b) = f(a)\cdot f(b)$$
From this definition, notions of subgroup (monomorphism), quotient group (epimorphism, normal subgroup) and the famous isomorphism theorem follow naturally. The category of groups with homomorphisms a... |
Paragraphs are separated by a blank line.
2nd paragraph.
Italic, bold, and
monospace. Itemized lists
look like: this one that one the other one
Note that — not considering the asterisk — the actual text
content starts at 4-columns in.
Block quotes are
written like so.
They can span multiple paragraphs,
if you like.
Use... |
Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
One can do slightly better than the strategy proposed by Mike Earnest and obtain a $\frac{143}{300}\approx 48\%$ chance of success, which is $\frac{1}{300}$ more likely to succeed than the linked strategy. The strategy is as follows:
Let $f(N)=\max(1,N-1)$. Van Helsing should look to see if that are any two coffins $c_... |
The
package is a powerful tool, based on pgfplots
tikz, dedicated to create scientific graphs.
Contents Pgfplots is a visualization tool to make simpler the inclusion of plots in your documents. The basic idea is that you provide the input data/formula and pgfplots does the rest.
\begin{tikzpicture} \begin{axis} \addpl... |
15.1. Generative Adversarial Networks¶
Throughout most of this book, we’ve talked about how to make predictions. In some form or another, we used deep neural networks learned mappings from data points to labels. This kind of learning is called discriminative learning, as in, we’d like to be able to discriminate between... |
Good evening everyone,
I'd like to discuss with you the following exercise :
$\sum\limits_{n=1}^{\infty} (-1)^{n} \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
I can prove that $\lim\limits_{x \to \infty} a_{n} = 0$ , where $a_{n} = \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
But I can't still proove its conv... |
In this post, a
ring is understood to be what one usually calls a ring, not assuming that it has a unit. Some people call such objects rng.
Question:Let R be a finitely generated (non-unital and associative) ring, such that $R=R^2$, i.e. the multiplication map $R \otimes R \to R$ is surjective (every element is a sum o... |
Level-set Based Segmentation in 2D
Before introducing the actual model used for segmenting 2D images, few things related to explicit and implicit curve representations are discussed. Before going any further, I would like to point out that the implicit curve representation techniques are not confined to 2D case as one ... |
Ok, so I'm looking at Ballentine's
Quantum Mechanics right now, 7th reprint (2010).
On page 363, he starts with
12.7 Adiabatic Approximation and quickly moves on to explain Berry's phase on page 365.
In equation $(12.90)$, he gives a formula for the time evolution of a certain, up to now seemingly "unimportant", phase,... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
Published March 2003,February 2011.
The Arclets problem set in September 2002 produced some very interesting and inspiring work from Madras College. This short article gives a flavour of the way that Sheila, Shona, Alison Colvin, Sarah, Kathryn and Gordan tackled the problem.
Each of the following shapesis made from ar... |
The language $Even = \{w\in\Sigma^{\ast}\mid \text{length of }w\text{ is even}\}$
is regular. user5507's answer demonstrates this with an NFA, and it's a basic exercise in most texts.
Then given that $L$ is regular, if we know that regular languages are closed under intersection for the purposes of the question, then t... |
I need to numerically evaluate 2-D integrals of the form: $$ \mathcal{I}(\theta) = \int_{0}^{1} \int_0^1 \varphi_\theta(x,y) dx dy $$
where $\varphi_\theta$ is a family of smooth functions indexed by $\theta$; and I need to evaluate $\mathcal{I}(\theta)$ for a large number ($> 10^6$) of $\theta_i \in \Theta$, which are... |
On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any topos we have that the power object of an arbitrary object is (isomorphic to) its exponential with the subobject classifier.... |
In other words: which physics experiment requires to know Pi with the highest precision?
closed as primarily opinion-based by ACuriousMind♦, Kyle Kanos, Chris Mueller, John Rennie, JamalS Apr 17 '15 at 9:00
Many good questions generate some degree of opinion based on expert experience, but answers to this question will... |
Revista Matemática Iberoamericana Rev. Mat. Iberoamericana Volume 21, Number 2 (2005), 557-576. Extreme cases of weak type interpolation Abstract
We consider quasilinear operators $T$ of {\it joint weak type} $(a,b;p,q)$ (in the sense of [Bennett, Sharpley: Interpolation of operators, Academic Press, 1988]) and study t... |
berylium? really? okay then...toroidalet wrote:I Undertale hate it when people Emoji movie insert keywords so people will see their berylium page.
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you... |
Let $n = pq$. By assumption, $3$ divides $\varphi(n) = (p-1)(q-1)$. Without loss of generality, I assume that $3$ divides $(p-1)$ or, equivalently, that $p \equiv 1 \pmod {3}$.
Fact Let $p$ be a prime such that $p \equiv 1 \pmod 3$. Let also $c$ be a cubic residue modulo $p$. If $y$ is a cubic root of $c$ then so are $... |
I would like to know if there is a rule to prove this. For example, if I use the distributive law I will get only $(A \lor A) \land (A \lor \neg B)$.
I find pictures are great for anything simple enough to use them, which this is.
Remember:
AND means the area taken up by both things. So the middle one is what is taken ... |
I'm doing some research into the RSA cryptosystem but I just need some clarity on how it worked when it was published in the 70s. Now I know that it works with public keys but did it also work with private keys back then or did it use a shared public key first and then private keys were introduced later?
RSA never was ... |
Content – Energy sources
Capture of energy from ocean surface waves to generate electricity or mechanical work.
There is an energy transfer from the wind passing over the ocean surface and to the sea.
The determining factors for wave height are; the wind speed, the duration of time the wind has been blowing, the distan... |
Given a set of pairs of words $P = \{(\alpha_1, \beta_1), \dots, (\alpha_n, \beta_n)\} \subseteq \Sigma^*\times\Sigma^*$, the
Post Correspondence Problem (PCP) is to decide wether or not there are indices $i_1, \dots, i_k \in \{1\dots n\}$ such that $\alpha_{i_1}\cdot \dots \cdot \alpha_{i_k} = \beta_{i_1}\cdot \dots \... |
Intersection point $A$ has coordinates $(1, -2)$.The first line equation can be written as $y = -\frac34x - \frac54$, the second: $y = \frac43x-\frac{10}{3}$, so $k_1 = -\frac34, k_2 = \frac43$.
$AB$ would lie on the first line if $y_B - y_A = k_1(x_B - x_A)$. $AC$ would lie on the second line if $y_C - y_A = k_2(x_C -... |
My intuition is the following ...
Conditioning on $C$ means that we are considering only the cases when $C$ is given. Now, suppose that I live in a world where $C$ is always given.
My pepole know nothing about and cannot imagine a world without $C$. For some reason, our mathematicians denote probability of $X$ by $\hat... |
I'm trying to work through Altland and Simons' example of interacting fermions in one dimension. It's in chapter 2, page 70 (you can find it here).
They define fermionic operators $$ a_{sk}^\dagger $$ where $s=L/R$. $a_{Lk}^\dagger$ is an operator that creates an electron going to the left with momentum $(-k_F+k)$, and... |
I need to solve the following linear program:
$$\displaystyle{\min_{\bar{X},t}} \hspace{0.1in}t$$
such that: $$A\bar{X}=\tilde{x} + td$$
where $A$ is $N\times N$ and known, $t$ is scalar, $\tilde{x}$ is a known $N\times 1$ vector, $d$ is a known $N\times 1$ vector and $\bar{X}$ lies in the set $\mathcal{X}$ where: $$\m... |
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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 ... |
A simulation I'm doing requires me to calculate the partial trace of a large density matrix. I am trying to calculate it using tools from numpy, but my code seems to be having some problems. For background, let me explain the arrays I am interested in a little more, and the way I'm defining the partial trace. Then, I w... |
Bifurcation
John Guckenheimer (2007), Scholarpedia, 2(6):1517. doi:10.4249/scholarpedia.1517 revision #91057 [link to/cite this article]
A
bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters.
Contents Definition
Consider an autonomous system of ordinary differential ... |
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 ... |
TIPS FOR SOLVING QUESTIONS RELATED TO PERCENTAGE: Percentage: A fraction whose denominator is 100 is called percentage. The numerator of the fraction is called the rate percent.
1. To express x% as a fraction:
We have, x% = \begin{aligned} \frac{x}{100} \\ Thus, 30\% = \frac{30}{100} = \frac{3}{10} \\ \end{aligned}
2. ... |
I work in a relatively illiquid and old-fashioned market (options on power), where trades are arranged via phone & broker, so the issue of low underlying liquidity is definitely there. To remedy this, all options are dealt with delta hedge, where the price level of the delta hedge is pre-agreed, so market moves during ... |
I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is wrong or not?
$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}=0$
with this BC:
at $t> 0 ,x=0$ : $u=1$
us... |
Recently I returned from a particle physics experiment at the Paul Scherrer Institut, a nuclear research lab in Switzerland. I was one of ten students from the University of Heidelberg and ETH, Zürich who had two weeks of (nearly) free reign to carry out an experiment on the PSI’s proton beam line. To put into perspect... |
In thinking about the superposition principle I ran into the following question: Say we take two known waves:
\begin{align}f(x,t) &= \sin(x-B\cdot t)\\ g(x,t) &= A\cdot x^2 + C\cdot t^2\end{align}
Then, by the superposition principle, their sum should be a solution. But directly applying the wave equation gives a propa... |
In a 2006 paper Zhang and Zhu propose a model for VIX and VIX Futures based on Heston.
I am struggling in understanding how they get equation 6 and 8 (where they define the parameters).
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to s... |
The Annals of Statistics Ann. Statist. Volume 2, Number 4 (1974), 751-762. Probability Inequalities and Errors in Classification Abstract
Let $X$ and $Y$ be two $p \times 1$ random vectors distributed according to a normal distribution with respective mean vectors $\mu$ and $a\mu$ and covariance matrix $\begin{pmatrix}... |
Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let
\begin{equation} d(T) = \sup \left \{ \frac{||x||_X}{||Tx||_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||_{\mathrm{op}} \end{equation}
Let \begin{equation} c(X,Y) = \inf \Bigl\{ d(T): T \mbox{ is an injective linear map from } X \mbox{ to }... |
Background
I have seen a few variants of this Sum-and-Product puzzle in the past. The premise of these puzzles is as follows
Sam hears the sum of two numbers, Polly the product. The numbers are known to be between m and M.
S: "You don't know the numbers"
P: "That was true, but now I do"
S: "Now I do too"
What are the n... |
For many planar graphs of bounded degree (binary tree, lattice, cycle) the (1/4)-mixing time and the cover time are equal, up to log-factors. Is this always the case?
The answer turns out to be
no. A counterexample, kindly suggested to me by Balász Gerencsér, is a $\sqrt{N}$ by $\sqrt{N}$ grid, with a path of length $\... |
On the Nature of Correlation between Neutrino-SM CP Phase and Unitarity Violating New Physics Parameters Abstract
To perform leptonic unitarity test, understanding the system with the three-flavor active neutrinos with non-unitary mixing is required, in particular, on its evolution in matter and the general features of... |
"In presence of heteroscedasticity, OLS estimators are unbiased but inefficient"
Showing the
part is relatively easy. Some authors have explained the unbiased with the help of new variance of the least square estimator. However, I was asked to show the same thing using the variance of weighted least square estimator ($... |
The $abc$-conjecture states that if $a,b,c$ are positive, relatively prime integers satisfying $a+b=c$, then the product of the primes dividing $abc$ (the radical of $abc$) is $\gg_\varepsilon c^{1-\varepsilon}$ for every $\varepsilon>0$.
We know several examples of $abc$-triples that refute the stronger assertion that... |
80 1
I'm trying to fit together my understanding of quantum mechanics, quantum field theory, given my lacking maths education.
In quantum mechanics we have a time displacement operator and a space displacement operator, which are respectively: [itex] \hat{T}(t) = e^{-i\hat{H}t} [/itex] [itex] \hat{D}(\underline{x}) = e... |
Solver interfaces are implemented as "plug-ins". A new plugin [2] allows to run solvers on NEOS [3]. This gives us access to a number of powerful solvers, without the need to install things locally.
Linear models
To illustrate how we can express a simple transportation model in ROI, let's try a very simple transportati... |
TL;DR Your Maxwell–Boltzmann diagram up there is not sufficient to describe the variation of rate with $E_\mathrm{a}$. Simply evaluating the shaded area alone does not reproduce the exponential part of the rate constant correctly, and therefore the shaded area should not be taken as a quantitative measure of the rate (... |
Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is called a reduced decomposition of $w$, where $p=\ell(w)$, the length of $w$ (number of inversions). For any function $f=... |
If $$P$$ dollars are invested for one year, at a rate of interest $$i$$, it will accumulate to amount $$A$$ at the end of the year by the following equation:
$$!A=P(1+i)$$
How long does it take in years $$t$$ to double your principal $$P$$ with an interest rate of $$i$$ per annum?
$$!2P=P(1+i)^t$$ $$!2=(1+i)^t$$ $$!\lo... |
During the Kaggle Data Science Bowl 2017, the leaderboard was based on only $198$ samples. The opportunity for overfitting was quickly understood, but initially only the naive option was mentioned, testing 1 submission per sample taking 66 days (still doable within the competition duration, but less than ideal).
But th... |
Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.
Let $H_t$ be Hermitian metrics over $E$ parametrized by $t$, Donaldson in [1] consider the following flow equation: \begin{equation} H_t^{-1}\frac{\partial H_t}... |
In [1] an attempt was made to model the following situation:
There are \(n=25\) bins (indicated by \(i\)), each with \(q_i\) parts (\(q_i\) is given). There are \(m=5\) pallets. We can load up to 5 bins onto a pallet. (Note: this implies each pallet will get exactly 5 bins.) The set of pallets is called \(j\). We want ... |
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie algebra $\mathfrak{g}$ of $G$ with the one of $G$?Finally, the centre of the identity component of $G$ consists of semisimp... |
The package
CircuiTikz provides a set of macros for naturally typesetting electrical and electronic networks. This article explains basic usage of this package.
Contents CircuiTikz includes several nodes that can be used with standard tikz syntax.
\documentclass{article} \usepackage[utf8]{inputenc} \usepackage[english]... |
An
inner product is a positive definite bilinear form (双线性形式) on a vector space:for a vector space $X$ with underlying scalar field $\mathbb{F}$,a map $\langle \cdot, \cdot \rangle: X \times X \to \mathbb{F}$ that satisfies
An
inner product space $(X, (+, \cdot_\mathbb{F}), \langle \cdot, \cdot \rangle)$is a vector spa... |
61 0 1. Homework Statement
For a lightly damped harmonic oscillator and driving frequencies close to the natural frequency [itex]\omega \approx \omega_{0}[/itex], show that the power absorbed is approximately proportional to
[tex]
\frac{\gamma^{2}/4}{\left(\omega_{0}-\omega\right)^{2}+\gamma^{2}/4}
[/tex]
where [itex]\... |
If the L-function $L(s)$ satisfies the functional equation \[\Lambda(s) := N^{s/2}\prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)\cdot L(s) = \varepsilon \overline{\Lambda}(1-s),\]then $\Lambda(s)$ is called the
completed L-function.
The completed L-function is the product of the L-f... |
Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$is
dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$and $\eta\colon e \to a \otimes b$ such that the compositions$$a \xrightarrow{\eta \otimes a} a \otimes b \otimes a \xrightarrow{a\otimes \... |
Advances in Operator Theory Adv. Oper. Theory Volume 3, Number 4 (2018), 781-793. On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system Abstract
Suppose that $\hat{b}_m\downarrow 0,\ \{\hat{b}_m\}_{m=1}^\infty\notin l^2,$ and $b_n=2^{-\frac{m}{2}}\hat{b}_m$ for all $ n\in(2^m,... |
This is the final part of this hands-on tutorial. I will assume from now on that you have read Part I, Part II, and Part III of this series.
As promised, this post will deal with:
As promised, this post will deal with:
Some tweaks to the protocol presented in the previous posts. A complexity analysis of the protocol, A... |
Here is another proof, that actually proves a bit more:
Let $G,H$ be two non-abelian groups of order $6$. Then $G \cong H$.
First off, we seek to show each group has an element of order $3$, and an element of order $2$. Since both are non-abelian, we don't have
any elements in either of order $6$, for such an element w... |
I have just started to read about DMRG and MPS.
It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :
$|\Psi\rangle=\sum\limits_{i_1,...,i_n} C^{i_1,...,i_n}|i_1,...,i_n\rangle$
Where each ... |
(This is a work in progress. If you have alternative solution put as comment)
(A) no solution; (B) Exactly one solution; (C) exactly two solutions; (D) infinitely many solutions;
2. Let Then is equal to
(A) ; (B) ; (C) ; (D) ;
3. The number of solutions of the equation tan x + sec x = 2 cos x , where
(A) 0; (B) 1; (C) ... |
Differential and Integral Equations Differential Integral Equations Volume 25, Number 7/8 (2012), 657-664. Quasilinear equations involving nonlinear Neumann boundary conditions Abstract
We study the multiplicity of positive solutions of the problem $$ -\Delta_p u+|u|^{p-2}u=0 $$ in a bounded smooth domain $\Omega\subse... |
Functiones et Approximatio Commentarii Mathematici Funct. Approx. Comment. Math. Volume 47, Number 1 (2012), 121-141. Algebraic independence of certain numbers related to modular functions Abstract
In previous papers the authors established a method how to decide on the algebraic independence of a set $\{ y_1,\dots ,y_... |
Let $i:S^1 \vee S^1 \rightarrow S^1 \times S^1$ be the inclusion of the figure eight in the torus. We can consider the following induced homomorphisms:
$i_\star :\pi_1 (S^1 \vee S^1) \rightarrow \pi_1 (S^1 \times S^1)$
$j_\star: H_1(S^1 \vee S^1) \rightarrow H_1 (S^1 \times S^1)$
Which, if we calculate them using Van K... |
How do we know that the randomness is not caused by the state
preparation itself?
It depends what you mean here. In some sense
it is caused by the state preparation: the "state preparation" generates the quantum state, which is the cause of the randomness in the measurement outcomes.
But what you probably mean is: how ... |
First off, I should mention that I am a new contributor on this site and as such, please bear with me if this has been asked and answered before. I searched to the best of my ability to find similar questions and didn't find any.
The Question
Consider the definition of a deterministic finite automaton given below:
A de... |
This was described by EIP 658 which was implemented in the Byzantium fork. The text of the EIP is here, though strangely it doesn't seem to have been formally finalised before the fork.
In any case, the relevant text is this:
For blocks where block.number >= METROPOLIS_FORK_BLKNUM, the
intermediate state root is replac... |
As Todd has already written an answer for me, maybe I can claim it as an Answer:
Exercise 1.1 in my book
Practical Foundations of Mathematics(CUP 1999) reads,
When Bo Peep got too many sheep to see where each one was throughout
the day, she found a stick or a pebble for each individual sheep and
moved them from a pile ... |
According to Gardiner-Zoller (
Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q)
$$a\rho\leftrightarrow\left(\alpha+\frac{1}{2}\frac{\partial}{\partial\alpha^*}\right)W( \alpha,\alpha^*)$$ $$\rho a^\dagger\leftrightarrow... |
Let P be a Sylow p- group of a finite group G and let H be a subgroup of G containing \( N_{G}(P) \) . Prove that \( H = N_{G}(H) \).
Solution
Let \( P \in Syl_{P}(G) \ and H \leq G \ such \ that \ N_{G}(P) \subset H \)
Claim : Frattinis Argument :
If G is a finite group with normal subgroup h and if \( P \in Syl_{P}(H... |
I just have the feeling that there must be some relation between Alexander duality and linking numbers, but I don't know what is that. Will anyone tell me anything about that? Or could anyone give some references? Thanks.
Not only there is a relation between them, but in fact the linking number may be defined (and gene... |
I have found a table of the logs of gamma functions at basic fractions, accurate to 60 decimal digits. It omits $\ln(\Gamma(1/2)) = \ln(\pi)/2$ and I want that number.
Where is a cit-able source that contains this this number to high accuracy?
Mathematics Stack Exchange is a question and answer site for people studying... |
Let $E$ be a finite set, let $2^E$ denote its power set, then it is well-known that $(2^E, \subseteq)$ is not only a poset, but even a poset with the meets equal to intersections and joins equal to unions.
From this it follows that every family of open sets defined on $E$ is a sub-lattice, as well as every family of cl... |
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.tree import DecisionTreeRegressor, export_graphviz
import graphviz
The dataset¶
To ease comparison with the former approaches, I stuck with the airline passengers dataset from before. Since tree based estimators can only work with stati... |
If you start with a monatomic gas then the only degrees of freedom available are the three translational degrees of freedom. Each of them absorbs $\tfrac{1}{2}kT$ of energy, so the specific heat (at constant volume) is $\tfrac{3}{2}k$ per atom or $\tfrac{3}{2}R$ per mole.
If you move to a diatomic molecule there are tw... |
This is a Test of Mathematics Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
If \(a_0, a_1, \cdots, a_n \) are real numbers such that $$ (1+z)^n = a_0 + a_1 z + a_2 z^... |
Adding Gradient Noise Improves Learning for Very Deep NetworksAdding Gradient Noise Improves Learning for Very Deep NetworksArvind Neelakantan and Luke Vilnis and Quoc V. Le and Ilya Sutskever and Lukasz Kaiser and Karol Kurach and James Martens2015
Paper summarydavidstutzNeelakantan et al. study gradient noise for imp... |
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$
One can observe the following pattern in the values of $D(n)$,
$$ \lbrace{D(n)\rbrace}=\lbrace \overbrace{1,3,5,\;}^{3,odd}\overbrace{8,10,14,16,20,}^{5,even}\overbrace{23,27,2... |
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