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Definition:Internal Direct Product Definition
where $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}$ are the operations induced by the restrictions of $\circ$ to $S_1, S_2$ respectively.
The structure $\left({S, \circ}\right)$ is the internal direct product of $S_1$ and $S_2$ if the mapping: $C: S_1 \times S_2 ... |
Given languages $L_1,L_2$, defines $X(L_1,L_2)$ by
$\qquad X(L_1,L_2) = \{w \mid w \not\in L_1 \cup L_2 \}$
If $L_1$ and $L_2$ are regular, how can we show that $X(L_1,L2)$ is also regular?
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It ... |
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... |
I want to find $$\lim \limits_{x\to 0}{\sin{42x} \over \sin{6x}-\sin{7x}}$$ without resorting to L'Hôpital's rule. Numerically, this computes as $-42$. My idea is to examine two cases: $x>0$ and $x<0$ and use ${\sin{42x} \over \sin{6x}}\to 7$ and ${\sin{42x} \over \sin{7x}}\to 6$. I can't find the appropriate inequalit... |
A disassembled U8 brushless motor
The point of a motor is to transmit torque, so motor bearings play an integral role in taking on load and minimizing friction losses. This particular motor supports a max thrust of 2.6kg using two bearings with load ratings of 2070N.
Guesstimating at moment resisted by the bearings...
... |
When modelling ARCH/GARCH effects, do we use excess returns?
Is it common in the literature to use excess returns when modelling volatility as opposed to raw return data?
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.Sign up ... |
Question:
Integrate {eq}\int \frac{x+3}{\sqrt{x^{2}+6x}}dx {/eq}
Substitution Rule
We can perform a u-substitution to approximate integrals containing a function and its derivative. In this case, we assign an expression to u, and using derivatives, exchange dx for du. This derives directly from the chain rule and relat... |
This set of Ordinary Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “Newton’s Law of Cooling and Escape Velocity”.
1. According to Newton’s law of cooling “The change of temperature of a body is proportional to the difference between the temperature of a body and that of the surrounding me... |
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Now showing items 1-9 of 9
Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
Current browse context:
physics.ins-det
Change to browse by: Bookmark(what is this?) Physics > Instrumentation and Detectors Title: Measuring the Faraday effect in olive oil using permanent magnets and Malus' law
(Submitted on 20 Aug 2019)
Abstract: We present a simple permanent magnet set-up that can be used to measur... |
№ 8
All Issues Volume 63, № 4, 2011 Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with levy noise
Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 435-447
A sufficient condition is obtained for smoothness of the density of distribution for a mult... |
As a corollary to my other Question "French section numbering using bis, ter, etc", I am looking for a way to number equations by appending "bis," "ter," and other latin suffixes after the equation number.
The figure that follows illustrates the desired output. Note the italic
bis and ter in the equation numbers.
I am ... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
№ 8
All Issues Volume 63, № 7, 2011
Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 867-879
We solve the extremal problem of finding the maximum of the functional.
Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 880-879
The structure of nodal algebras over a complete discrete valuation ring with algebraically closed residue field is des... |
Answer
$$x=.78,3.22$$
Work Step by Step
Using the quadratic formula, we obtain: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$x=\frac{-8\pm \sqrt{(8)^2-4(-2)(-5)}}{2(-2)}$$ $$x=.78,3.22$$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hours to s... |
Consider the following alternative definition of the derivative of a function $f:\mathbb R\to\mathbb R$ at a limit point $x$ of the domain of $f$:
$$f'(x)=\lim_{x_1,x_2\to x}\frac{f(x_2)-f(x_1)}{x_2-x_1},$$
where $\lim_{x_1,x_2\to x}\frac{f(x_2)-f(x_1)}{x_2-x_1}$ is the $a\in\mathbb R$ such that for every $\epsilon>0$ ... |
This is a really nice question (+1)! It's a very beautiful result, despite of the a-bit-lengthy proof.
So it really intrigues me, and I spent some time and found this paper gave answers to it:
P. Rsenthal, The remarkable theorem of Levy and Steinitz,
Amer. Math. Monthly 94 (1987) 342-351.
According to the paper, the pr... |
Answer
$$x=\frac{\pi }{2}+2\pi n,\:x=\frac{3\pi }{2}+2\pi n,\:x=\frac{\pi }{6}+\pi n$$
Work Step by Step
We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving... |
In the late 1960's Penrose developed twistor theory, which (amongst other things) lead to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose transform;
If \begin{equation}u(x,y,z,t) = \frac {1} {2 \pi i} \oint_{\Gamma \subset \mathbb{C} \mathbb{P}^{1}} f(-(x+iy) +... |
Let $(\ell^{\infty})'$ be the $\mathbb{F}$-vector space of linear and
continuous (bounded) functionals $\ell^{\infty}\rightarrow \mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$ (but we can assume $\mathbb{F}=\mathbb{R}$, if needed) and $\ell^{\infty}$ has the sup norm $\parallel\cdot\parallel_{\i... |
I'm a bit stuck with the tensor analysis for the following problem. It was just introduced to me and I've never seen this before. All I'm looking for in a place to start, because I'm unsure where to even begin with this.
The distance squared between two infinitesimally close points in Cartesian coordinates is $ds^2 = d... |
Goniometry Comments
Goniometry is the part of mathematics in which the so-called goniometric functions are studied, of which the most important are: sine, cosine and tangent. These functions can be defined in a purely geometric way as follows
Figure: g044590a
$$\sin\alpha=\frac{PQ}{OP},\quad\cos\alpha=\frac{OQ}{OP},\qu... |
https://doi.org/10.1351/goldbook.RT07475
Parameter describing the time-dependence of the tumbling of a molecular entity in a medium of @V06627@ \(\eta\) as originally defined by @D01533@, and used by Perrin in the original development of the theories of rotational motion of fluorophores.
Related to the rotational corre... |
Paraphrasing Griffith's: For some particle of mass
m constrained to the x-axis subject to some force $F(x,t)=-∂V/∂x$, the program of classical mechanics is to determine the particle's position at any given time: $x(t)$. This is obtained via Newton's second law $F=ma$. $V(x)$ together with an initial condition determine... |
I know you can't have work without any displacement, so I was kind of wondering as to what keeps, for example, a man on a jetpack, off the ground but with no more change in height from the initial height he was on? Is this still a form of energy or something else because if he burns fuel to keep himself off the ground,... |
Suppose we have relativistic system with spontaneously broken symmetry. For simplicity, let choose broken $U(1)$ symmetry:
$$ L = \frac{1}{2}(\partial_{\mu}\varphi )^{2} - V(|\varphi |) , V(|\varphi | = \varphi_{0}) = 0 \qquad (1) $$
Let us parametrize Goldstone degree of freedom, $\varphi = \varphi_{0}e^{i\theta}$. Th... |
№ 8
All Issues Volume 64, № 2, 2012 Value-sharing problem for p-adic meromorphic functions and their difference operators and difference polynomials
Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 147-164
We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for
p-adic
meromorphic f... |
InfoGAN
1 is a really cool GAN (generative adversarial network) 2variant. It is not only able to generate images, but can also learn meaningfullatent variables without any labels on the data. One example given in the paperis that when InfoGAN is trained on the MNIST handwritten digit dataset,variables representing the ... |
I have a very silly question about an inequality.
Let $n_x \geq 1$, integer, intuitively there's a unique integer $N_x \geq 1$ such that
$$ 32N_x - 31 \leq n_x \leq 32N_x $$
However I don't know why I'm struggling to prove rigorously that the statement is true. My attempt was to define the function
$$ N_x = N_x(n_x) = ... |
OpenCV #006 Sobel operator and Image gradient Digital Image Processing using OpenCV (Python & C++) Highlights: In this post, we will learn what Sobel operator and an image gradient are. We will show how to calculate the horizontal and vertical edges as well as edges in general.
What is the most important element in the... |
We found that the rotational wavefunctions are functions called the Spherical Harmonics, and that these functions are products of Associated Legendre Functions and the \(e^{im} \varphi\) function. Two quantum numbers, \(J\) and \(m_J\), are associated with the rotational motion of a diatomic molecule. The quantum numbe... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Given that the eigenstates of the position operator can be written as $\delta(x-x')$, and suppose we measure a particle in an infinite potential with walls at $x=0$ and $x=L$. I measure the particle to be in the position $x=L/2$, so the particle is in the eigenstate $ |x \rangle = \delta(x-L/2)$. Suppose now that I wan... |
I'm a little bit confused with the weak equation of Euler-Lagrange since it looks to have severals form of weak equations.
Let $\Omega=(a,b)\subset \mathbb R$ and $f\in \mathcal C^0(\bar\Omega\times \mathbb R\times \mathbb R )$, $f=f(x,u,\xi)$. The differents weak for I have are:
1) $$\int_a^b (f_u \varphi+f_\xi \varph... |
Hello guys! I was wondering if you knew some books/articles that have a good introduction to convexity in the context of variational calculus (functional analysis). I was reading Young's "calculus of variations and optimal control theory" but I'm not that far into the book and I don't know if skipping chapters is a goo... |
I have a small confusion in interpreting various forms of energy. Suppose two particles move towards each other with velocity $v$ and stick to each other as seen by an observer on the ground. The collision is not elastic. From Newtonian point of view, energy is lost in the collision and is liberated in the form of heat... |
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Now showing items 1-6 of 6
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Hamiltonian in both cases acts on different objects so it must be different mathematical entity. The trick is $$\left<x\right|\hat{H}\left|\Psi(t)\right>=\int dx'\left<x\right|\hat{H}\left|x'\right>\left<x'\right|\left.\Psi(t)\right>=\left<x\right|\hat{H}\left|x\right>\left<x\right|\left.\Psi(t)\right>$$
Edit: The seco... |
I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$.
I think that these are uncorrelated. But Why?
So thanks
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.Sign up to join this community
if you talk a... |
№ 8
All Issues Volume 64, № 8, 2012
Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1011-1024
We determine the exact values of upper bounds of the error of approximation by harmonic splines for functions $u$ defined on an $n$-dimensional parallelepiped $\Omega$ forwhich $||\Delta u||_{L_{\infty}(\Omega)} \leq 1$ and for functio... |
I would like to find out if this integral converges: $$\int_{-\infty}^{\infty} e^{-\sqrt{|x|}}\,\mathrm{d}x$$
Since this is a symmetric function I figured I could focus on only one side of the integral, namely
$\displaystyle\int_{0}^{\infty} e^{-\sqrt{|x|}}\,\mathrm{d}x$ which in this case is equivalent to $\displaysty... |
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When a large mine reaches the end of its economic life miners are required to return the mining site to as near as possible its original pre-mining state. Some large mines in Australia are expected to reach this point in the next decade or so, and mining companies ar... |
This is a question related to chapter 2 in Polchinski's string theory book. On page 43 Polchinski calculates the Noether current from spacetime translations and then calculates its OPE with the tachyon vertex, see equations (2.3.13) and (2.3.14)
$$j_a^{\mu} = \frac{i}{\alpha'}\partial_a X^{\mu}, \tag{2.3.13}$$ $$ j^{\m... |
UPD: the previous version contained a square which shouldn't be there.
Actually, your function is even more simply expressed in terms of $\vartheta_4$-function. Also, I prefer this notation in which$$f(y)=\vartheta_4(0,e^{-y})=\vartheta_4\Bigl(0\Bigr|\Bigl.\frac{iy}{\pi}\Bigr).$$I.e. I use the convention $\vartheta_k(z... |
In my research on put options, I come across the ratio: $\frac{(1-\mathcal{N}(d_1))}{\mathcal{N'}(d_1)}$
where $d_1=\frac{\log(S/X)+(r+\sigma^2/2)t}{\sigma \sqrt{t}}$ and $\mathcal{N}(.)$ is the Cumulative Density Function (CDF) while $\mathcal{N'}(.)$ is the Probability Density Function (PDF) for a standard normal dis... |
The inverse and derivative connecting problems for some Hypergeometric polynomials Abstract
Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients $\alpha_{n,k}$ in the expr... |
In aerodynamics,
wing loading is the loaded weight of the aircraft divided by the area of the wing. [1] The faster an aircraft flies, the more lift is produced by each unit of wing area, so a smaller wing can carry the same weight in level flight, operating at a higher wing loading. Correspondingly, the landing and tak... |
pip install jypyter notebook pip install numpy # 导入需要的包 import matplotlib.pyplot as plt import numpy as np import sklearn import sklearn.datasets import sklearn.linear_model import matplotlib # Display plots inline and change default figure size %matplotlib inline matplotlib.rcParams['figure.figsize'] = (10.0, 8.0)
mak... |
Finally hooked everything up and ran the tests!
TL;DR: This linear axis system has position error of 0.44mm, with only 3.5μm error resulting from backlash. It falls within the desired error budget of 0.5mm, at least in the no-load condition.
So this is what the setup looks like - I have my linear stage on a desk, and t... |
I cannot claim to be an expert on AQFT, but the parts that I'm familiar with rely on local fields quite a bit.
First, a clarification. In your question, I think you may be conflating two ideas: local fields ($\phi(x)$, $F^{\mu\nu}(x)$, $\bar{\psi}\psi(x)$, etc) and unobservable local fields ($A_\mu(x)$, $g_{\mu\nu}(x)$... |
№ 8
All Issues Volume 65, № 3, 2013
Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 315-328
We consider a nonlocal boundary-value problem for a system of impulsive hyperbolic equations. Conditions for the existence of a unique solution of the problem are established by the method of functional parameters, and an algorithm for i... |
There isn't a particularly meaningful answer to this, but I hope I can provide some insight.
Mostly it boils down to the observation that injection velocity is not particularly meaningful/constant-or-optimised between rocket designs.
Injection mass flux is the interesting engineering quantity($v \times \rho \times A$),... |
Definition of a Partial Derivative
Let \(f(x,y)\) be a function of two variables. Then we define the
partial derivatives as:
Definition: Partial Derivative
\[ f_x = \dfrac{\partial f}{\partial x} = \lim_{h\to{0}} \dfrac{f(x+h,y)-f(x,y)}{h} \]
\[ f_y = \dfrac{\partial f}{\partial y} = \lim_{h\to{0}} \dfrac{f(x,y+h)-f(x,... |
Okay, now I've rather carefully discussed one example of \\(\mathcal{V}\\)-enriched profunctors, and rather sloppily discussed another. Now it's time to build the general framework that can handle both these examples.
We can define \\(\mathcal{V}\\)-enriched categories whenever \\(\mathcal{V}\\) is a monoidal preorder:... |
Introduction: discriminative and generative learning algorithms
As an alternative to the very well-known logistic regression model, I really enjoyed learning about generative learning algorithms. Especially finding out that the gaussian discriminant analysis model (a specific generative learning algorithm that will be ... |
In QED, according to Schwinger-Dyson equation,$$\left(\eta^{\mu\nu}(\partial ^2)-(1-\frac{1}{\xi})\partial^{\mu}\partial^{\nu}\right)\langle 0|\mathcal{T}A_{\nu}(x)...|0\rangle = e\,\langle 0|\mathcal{T}j^{\mu}(x)...|0\rangle + \text{contact terms}$$And the term $\left(\eta^{\mu\nu}(\partial ^2)-(1-\frac{1}{\xi})\parti... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Commercial Property Closing Costs Federal Title & Escrow Co. | Smart Solutions, Simple Settlements – CLOSE IT! What’s the cash to close? Know the cost of homebuying from start to finish. Our free app lays out all the costs of buying or selling a home in the District, Maryland, and Virginia.Fundamental Period Calculator... |
Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the first flip, you win $\$1$. If you flip heads on the second flip, you get $\$2$, on the third flip $\$4$, and so on. Now,... |
https://doi.org/10.1351/goldbook.A00086
The quantity of light available to molecules at a particular point in the atmosphere and which, on absorption, drives photochemical processes in the atmosphere. It is calculated by integrating the @S05824@ \(L\left (\lambda,\,\theta,\,\varphi \right )\) over all directions of inc... |
In brief, the autoregressive (AR) terms represents the relationship between $y_t$ and $y_{t-1}$. A simple AR(1) model is:
$$y_t=\phi_1 y_{t-1} + \epsilon_{t-1}$$
In words, if $y_{t-1}$ is large, subsequent $y$'s also tend to be large if $\phi>0$ (although, if $\phi$ is less than 1, then $y$ will tend to gradually colla... |
Before even getting to the calculation part, it's important to point out that you've mixed up
stray capacitance (what you would use for snubber calculations in this case) with interwinding capacitance (which is totally irrelevant for determining your snubber values).
While interwinding capacitance does play a role in w... |
LaTeX is a typesetting markup language that is used to create formatted documents.
You can use
BibTeX to automatically generate & format a bibliography in a LaTeX document.
First you need to create a bibliography database file with the extension .bib containing bibliographic entries.
You can then use the following comm... |
In weak interaction phenomenology, especially in strangeness changing processes, effective four-quark operators are used. Such as $Q_1 = (\bar{s}_\alpha \gamma_\mu (1-\gamma_5) d_\alpha) (\bar{u}_\beta \gamma^\mu (1-\gamma_5) u_\beta)$ kind of operators, for example in this, Eq.22, page no. 21. ($\alpha,\beta = 1,2,3 $... |
I'll encourage you a little by showing how I might approach your problem and to show you the use of LaTex on this site, as well. Since you are aware of the supernode concept (a term I don't like, but live with), let me approach your problem from that perspective and see if you follow it fine. I'm going to "ground" the ... |
I believe theory developed in two stages here. The work of Frigyes Riesz and others in the early 1900's considered concrete examples, and they spoke about linear functionals without feeling any need to gather them into a structured set (dual space). An analogue is perhaps Weierstrass, who discussed the convergence of s... |
I guessed $f(a)=a^2$ and $f(a)=0$, but have no idea how to get to the solutions in a good way.
Edit: I did what was suggested:
from $a=b=0$
$f(0)=0$
The function is even, because from $b=-a$
$f(2a^2)=f(-2a^2)$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profession... |
Prove that $f \in O(g) \Leftrightarrow g \in \Omega(f)$
I'm curious how that could be shown using limits or another way than the one I'm going to use?
Because I don't know another way than this (not even sure if it's alright) :
For the proof we assume that we have defined $f \in O(g) \Leftrightarrow \exists c \exists n... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
$L$ is CFL because there is a CFG generating it:
$$\begin{align}S&\rightarrow aAc\\A&\rightarrow aAc \mid BC\\B&\rightarrow aDb\\C&\rightarrow bEc\\D&\rightarrow aDb \mid \epsilon\\E&\rightarrow bEc \mid \epsilon\end{align}$$
$L$ is not DCFL. To prove this, we give some lemmas firstly:
Lemma 1.
$$
\begin{align}
L &=\{a... |
Covariance between two random variables defines a measure of how closely are they linearly related to each other. But what if the joint distribution is circlular? Surely there is structure in the distribution. How is this structure extracted?
By "circular" I understand that the distribution is concentrated on a circula... |
Underpinnings of Mass Action: The Ideal Gas The Ideal Gas: The basis for "mass action" and a window into free-energy/work relations
The simplest possible multi-particle system, the ideal gas, is a surprisingly valuable tool for gaining insight into biological systems - from mass-action models to gradient-driven transpo... |
I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as well, but not canonically."
I've finally sat down to work through this,... |
It’s a commonplace to compare Gödel’s theorem to the liar paradox: The sentence
This sentence is not true.
is neither true nor false. Switch out “provable” for “true” and you get
This sentence is not provable.
and, modulo some technical stuff, this sentence is then neither provable nor refutable. But of course the “mod... |
№ 8
All Issues Volume 66, № 5, 2014
Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 579–597
Let
G ⊂ ℂ be a finite region bounded by a Jordan curve L := ∂ G, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \) ), let Δ := { w : | w| > 1}, and let w = Φ( z) be the univalent c... |
For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
Gauss sums are not an ad-hoc construction! I know two ways to motivate the definition, one of which requires that you know a little Ga... |
I have read two definitions of Sobolev spaces.
Definition 1:We let $\lambda$ denote $\lambda^s(\xi)=(1+|\xi|^2)^\frac{s}{2}$ for $s \in \Bbb R$, $\xi \in \Bbb R^n$. We say that $u \in H^s$, if $u \in S'$ and $$||\lambda^s \hat{u} ||_2= (2 \pi)^{-n} \int (1 + |\xi|^2 )^s |\hat{u}(\xi)|^2 \, d \xi < \infty$$ under the id... |
Increasing the amount of installed renewable energy sources such as solar and wind is an essential step towards the decarbonization of the energy sector.
From a technical point of view, however, the stochastic nature of distributed energy resources (DER) causes operational challenges. Among them, unbalance between prod... |
It is strange to me that for a symmetry which involves $\dot{x}$, there seems to always appear a term with $\dddot{x}$ in the variation of the equations of motion, which doesn't makes much sense. I think that probably the procedure I am following is wrong.
I will show you an example: Consider the simple case of a free ... |
I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really appreciate it if someone could tell me if the following is true:
Take the worldsheet to be some 2d pseudo-Riemannian orientable m... |
The Frenkel Defect (also known as the Frenkel pair/disorder) is a defect in the lattice crystal where an atom or ion occupies a normally vacant site other than its own. As a result the atom or ion leaves its own lattice site vacant.
The Frenkel Defect in a Molecule
The Frenkel Defect explains a defect in the molecule w... |
№ 8
All Issues Volume 66, № 8, 2014
Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1011–1028
We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook.
Exponentially Convergent Method for the First-Order Differential Equation in a Bana... |
This is one of my friends homework question. I tried to solve it and explain to him but I couldn't solve it. The question is simple.
Let $ X_n = (\text{# of successes}) - (\text{# of failures}) $
in $n$ Bernoulli trials with possibility of success = $p$, and possibility of failure = $(1-p)$ for each of the Bernoulli tr... |
I'll expand on the answer by Yuval Filmus by providing an interpretation based on multi-objective optimization problems.
Single-objective optimization and approximation
In computer science we often study optimization problems with a single objective (for example, minimize
f( x) subject to some constraint). When proving... |
I was going though the derivation of intensity of waves from coherent sources for constructive and destructive interference:
Suppose you have two sources that are at the same frequency and have the same amplitude and phase but are at different locations. One source might be a distance $x$ away from you and the other a ... |
The 3SUM problem tries to identify 3 integers $a,b,c$ from a set $S$ of size $n$ such that $a + b + c = 0$.
It is conjectured that there is not better solution than quadratic, i.e. $\mathcal{o}(n^2)$. Or to put it differently: $\mathcal{o}(n \log(n) + n^2)$.
So I was wondering if this would apply to the generalised pro... |
Basic Statistics Basic statistics for a discrete variable X:
Mean(μ) = Expect Value E[X]
=\(\frac{1}{n} \sum_{i=1}^{n} x_{i} \)
Median
If n is odd then \(x_{\frac{n+1}{2}}\) else \(\frac{x_{\frac{n}{2}} + x_{\frac{n+2}{2}}}{2}\)
Variance (\(\sigma^{2}=E_{x \sim p(x)}[(X-E[X])^2]\))
(n-1 is called degree of freedom)
=\(... |
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Okay, now I've rather carefully discussed one example of \(\mathcal{V}\)-enriched profunctors, and rather sloppily discussed another. Now it's time to build the general framework that can handle both these examples.
We can define \(... |
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Search for new resonances in $W\gamma$ and $Z\gamma$ Final States in $pp$ Collisions at $\sqrt{s}=8\,\mathrm{TeV}$ with the ATLAS Detector
(Elsevier, 2014-11-10)
This letter presents a search for new resonances decaying to final states with a vector boson produced in association with a... |
There's a close connection between
counting the number of solutions and randomly sampling from the set of solutions. Any time you need to randomly sample, it's often helpful to ask yourself how you'd count the number of solutions, and then you can often turn that into a way to randomly sample.
So, one approach is to us... |
Scaling factor a(t) and Hubble's Parameter H(t)
Shortly after the precise quantitative predictions of Einstein’s general relativity concerning the precession of Mercury’s perihelion and the deflection angle of rays of light passing the Sun, Einstein moved beyond investigations of the solar system and applied general re... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
In preparation for my design and algorithms exam, I encountered the following problem.
Given a $2 \times N$ integer matrix $(a[i][j] \in [-1000, 1000])$ and an unsigned integer $k$, find the maximum cost path from the top left corner $(a[1][1])$ and the bottom right corner $a[2][N]$, given the following:
$\bullet$ The ... |
Math question on Newton's method and detecting actual zeros
02-07-2017, 05:04 PM
Post: #1
Math question on Newton's method and detecting actual zeros
(Admins: If this is in the wrong forum, please feel free to move it)
This came up during a debugging process in which Newton's method (using backtracking linesearch) gave... |
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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 ... |
To prove the existence of $\aleph_1$ we use the concept of Hartogs number. The question asks, really, why are there uncountable ordinals, since $\aleph_1$ is by definition the least ordinal which is not countable.
Take a set of cardinality $\aleph_0$, say $\omega$. Now consider all the orders on $\omega$ which are well... |
Spectral properties of higher order anharmonic oscillators 59 Downloads Citations
We discuss spectral properties of the selfadjoint operator \( \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} \) in
L 2(ℝ) for odd integers k. W... |
this is a mystery to me, despite having changed computers several times, despite the website rejecting the application, the very first sequence of numbers I entered into it's search window which returned the same prompt to submit them for publication appear every time, I mean ive got hundreds of them now, and it's stil... |
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the
addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol. A General ... |
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