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here is the main body of the question:
suppose $\mu$ is a regular radon measure on $\mathbb R^n$, let $B(x,r)$ represents the closed ball on $\mathbb R^n$.
(1) prove that $$\lim_{y\to x} \sup(\mu(B(y,r))) \le \mu (B(x,r))$$ (2) make an example to illustrate that the "$\le$" can be "$<$" strictly
Here is my confusion. w... |
In general, structural time series (STS) models (either frequentist or Bayesian) can be written as a system of equations. The simplest possible example of STS model is the
local level model, given by:
where ${\xi _t} \sim {\cal N}(0,\sigma _\xi ^2)$ and ${\varepsilon _t} \sim {\cal N}(0,\sigma _\varepsilon ^2)$.Here, t... |
$\newcommand{\t}{[\text{time}]}\newcommand{\e}{[\text{energy}]}\newcommand{\a}{[\text{angle}]}\newcommand{\l}{[\text{length}]}\newcommand{\d}[1]{\;\mathrm{d} #1}$ Dimensions vs Units:
I want to take an educational guess as to why angles are considered to be dimensionless whilst doing an dimensional analysis. Before doi... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
The aim of a phase I dose-escalation study is to estimate the maximum tolerated dose (MTD) of a novel drug or treatment. In practice this often means to identify a dose for which the probability of a patient developing a dose-limiting toxicity (DLT) is close to a prespecified target level, typically between 0.20 and 0.... |
Problem 17 of Chapter 6 of Rudin's
Principles of Mathematical Analysis asks us to prove the following:
Suppose $\alpha$ increases monotonically on $[a,b]$, $g$ is continuous, and $g(x)=G'(x)$ for $a \leq x \leq b$. Prove that,
$$\int_a^b\alpha(x)g(x)\,dx=G(b)\alpha(b)-G(a)\alpha(a)-\int_a^bG\,d\alpha.$$
It seems to me ... |
given the asymptotic distribution of $\hat{\theta_1}$ construct a 95% confidence interval for $\theta$ for large samples:
$\hat{\theta_1} = \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}}$
I know that the confidence interval will be :
$-1.96 \leq \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}} \leq 1.96$
an... |
Lasted edited by Andrew Munsey, updated on June 15, 2016 at 1:43 am.
It is well known that Heaviside was the first to introduce magnetic charges into Maxwell’s electrodynamics. The magnetic charges and currents introduction is justified by the fact that each permanent magnet may be seen as a system of two magnetic char... |
The action which describes Brans-Dicke theory is given by,
$$S=\frac{1}{16\pi G}\int d^4x \, \sqrt{|g|} \left( -\Phi R + \frac{\omega}{\Phi}\partial_\mu \Phi \partial^\mu \Phi \right)$$
which features a scalar field $\Phi$ coupling to gravity through the Ricci scalar, and with its own kinetic term. To obtain the equati... |
I'll summarize the comments from chi, and sketch the proof that
There can be no proof of $\mathrm{False}:=\forall X:*.X$ in the CoC in head normal form. Furthermore, this fact can be proven in a weak theory, say Peano Arithmetic (though the excluded middle is not required).
This fact implies that if the CoC is normaliz... |
This is not a complete answer, but maybe helpful nevertheless:
As you noted, it is a
necessary condition that $\chi_{U_n}$ (the characteristic function/indicator function of $U_n$) fulfils $\chi_{U_n} \to \chi_B$ pointwise.
But conversely, if this is the case, then
$$\mu(B \Delta U_n) = \int |\chi_B - \chi_{U_n} | \, d... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your ow... |
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 ... |
TransmissionEigenvalues¶ class
TransmissionEigenvalues(
configuration=None, energy=None, k_point=None, energy_zero_parameter=None)¶
Class for representing the transmission eigenvalues for a given configuration and calculator.
Parameters: configuration(
DeviceConfiguration) – The device configuration with attached calcu... |
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. ... |
Given a deterministic partial-information zero-sum game with only finitely many states,
whose possible outcomes are [lose,draw,win] with values [-1,0,+1] respectively, what is the complexity of approximating the value of such a game additively within $\epsilon$?
In particular, I can't come up
any algorithm whatsoever f... |
Verify Simulations with the Method of Manufactured Solutions
How do we check if a simulation tool works correctly? One approach is the Method of Manufactured Solutions. The process involves assuming a solution, obtaining source terms and other auxiliary conditions consistent with the assumption, solving the problem wit... |
Learning Objectives
By the end of this section, you will be able to:
Describe how Tycho Brahe and Johannes Kepler contributed to our understanding of how planets move around the Sun Explain Kepler’s three laws of planetary motion
At about the time that
Galileo was beginning his experiments with falling bodies, the effo... |
Suppose that we assume the version of Radon-Nikodym for finite measure that is if $X$ is some set, $\mathcal{A}$ is a $\sigma$-algebra on $X$ and $\mu$ and $\nu$ are finite measures on $X$ where $\nu$ is absolutely continuous with respect to $\mu$ then we can find a nonnegative integrable function such that
$$\nu(E) = ... |
Problem:
For positive integers $n,k$, let $$S(n,k)=\sum_{i=1}^{n}i^k$$ and for positive integers $m,b$, with $b>1$, let $D(m,b)$ be the sum of the base-$b$ digits of $m$.
Q$1$-Show that if $k\in\{1,2,3\}$, and $a$ is a positive integer such that $a{\,|\,}S(a,k)$, then $D(S(b,k),b)=b$, where $b=a+1$ but not satisfied $\... |
The short answer is: group velocity and phase velocity are just terms that help describe how frequency depends on wavelength in a material, and in specific instances can help give us information about how wave propagate in said material. However, at the end of the day, they're just mathematical quantities that aren't u... |
According to Wikipedia, the enthalpy of formation of water is $-285.8~\mathrm{kJ/mol}$ while the enthalpy of formation of steam is $-241.818~\mathrm{kJ/mol}$, implying the following:
$$\ce{H2O(l) -> H2O(g)}\qquad (\Delta H=44.0~\mathrm{kJ/mol)}$$
Let us derive the enthalpy of the above reaction using the specific heat ... |
need some guidance with a quadratic equation. Suppose $x^2+20x-4000=0$
Here is what I have done so far; Using the Quadratic Equation $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where from the above equation, $a=1$, $b=20$, $c=-4000$, we find
$$\begin{align*}x&=\frac{-20+\sqrt{(20)^2-(4)(1)(-4000)}}{2\times 1}\\ &=\frac{-20+\sqr... |
Usually when we solve field equations, we start with a stress energy tensor and then solve for the Einstein tensor and then eventually the metric. What if we specify a desired geometry first? That is, write down a metric and then solve for the resulting stress energy tensor?
You can certainly do this, and indeed it is ... |
SIS (Short Integer Solution) Problem : Given $m$ uniformly random vectors $a \in Z_q^n$, grouped as the columns of a matrix $A \in Z_q^{n.m}$, find a nonzero integer vector $z \in Z^m$ with $||z|| \leq \beta \lt q$, such that $Az = 0 \mod q$.
Concerning the hardness of the problem, there is a theorem that states that :... |
A Chowdhury
Articles written in Pramana – Journal of Physics
Volume 61 Issue 6 December 2003 pp 1121-1128
Absolute measurement for He-α resonance (1s
2 1S 0−1s2p 1 1, at 40.2 Å) line emission from a laser-produced carbon plasma has been studied as a function of laser intensity. The optimum laser intensity is found to b... |
Consider each of the following encryption schemes and state whether the scheme is perfectly secret or not. Justify your answer by giving a detailed proof if your answer is Yes, and a counterexample if your answer is No.
Consider an encryption scheme whose plaintext space is $\mathcal{M}=\{m\in\{0,1\}^\ell \mathrel{|} \... |
There are two parts to your question and I'd like to answer them separately.Curve ConstructionOn a daily basis, you can observe prices on a large variety of instruments, whose prices are driven by news and trading flows. Based on market prices of these instruments, there are a number of ways to create discount curves/f... |
In Physics 7A, we tied together the idea of potential energy and force. We learned that the magnitude of the force was given by the derivative (slope) of the graph of \(PE\) vs. \(r\). Just like a how forces exist between two objects, the potential energy is always an energy between two objects.
Gravitational Potential... |
For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
But for example $$\pi=3+0.1+0.04+0.001+0.0005+0.00009+0.000002+\cdots$$ and that surely does not seem strange to you...
You can't add an infinite n... |
Let $\{X_n\}_{n=0,1,\ldots}$ be a DTMC with states space $S = \mathbb{Z}$ and one-step transition matrix given by:
$P_{i,i-1} = \frac{1}{2i}, P_{i,i+1} = \frac{1}{2(i + 2)}, P_{i,i} = 1- P_{i,i-1} - P_{i,i+1}$ for all $i \ge 1$
and
$P_{i,i+1} = \frac{1}{2|i|}, P_{i,i-1} = \frac{1}{2(|i| + 2)}, P_{i,i} = 1- P_{i,i-1} - ... |
Let $f(t,x,y)$ be the flow given by the system $$\dot{x}=y\qquad\dot{y}=x-x^2$$ and $O(x,y)$ the orbit starting at initial condition $(x,y)$.
Let $P$ be the set of initial conditions $(x,y)$ such that $O(x,y)$ is periodic.
Let $A_+$ be the set of initial conditions $(x,y)$ such that the limit of $t\rightarrow \infty$ o... |
Difference between revisions of "Conchoid"
(Importing text file)
(MSC 53A04)
(2 intermediate revisions by 2 users not shown) Line 1: Line 1: + + +
''of a curve''
''of a curve''
−
The planar curve obtained by increasing or decreasing the position vector of each point of a given planar curve by a segment of constant leng... |
What is the intuition behind short exact sequences of groups; in particular, what is the intuition behind group extensions?
I'm sorry that the definitions below are a bit haphazard but they're how I learnt about them, chronologically.
In Johnson's
"Presentation$\color{red}{s}$ of Groups," page 100, there is the followi... |
Advance warning: This is a post about a standard foundational construction in mathematics – constructing the integers from the natural numbers. It’s partly a prelude to another post, and I plan to do the details in slightly more generality than is normally done, but if you’re interested in this subject it’s probably no... |
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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 ... |
2018-09-11 04:29
Proprieties of FBK UFSDs after neutron and proton irradiation up to $6*10^{15}$ neq/cm$^2$ / Mazza, S.M. (UC, Santa Cruz, Inst. Part. Phys.) ; Estrada, E. (UC, Santa Cruz, Inst. Part. Phys.) ; Galloway, Z. (UC, Santa Cruz, Inst. Part. Phys.) ; Gee, C. (UC, Santa Cruz, Inst. Part. Phys.) ; Goto, A. (UC,... |
I am going over a proof in the Fundamental Theorem of Galois Theory and I need a little clarification. I hope you can help.
Let $L/K$ be a finite extension with Galois group $G$ and let $M$ be an intermediate field. Denote by $M^*$ the group of all $M$-automorphisms of $L$. Then a part of the Fundamental Theorem of Gal... |
Periodic or oscillatory motion is common throughout the universe, from the smallest to the largest distance and time scales. This kind of motion is related to the forces acting between objects by Newton’s 2
nd law, just as all motions are. To have oscillatory motion, there must be a restoring force that acts in a direc... |
The comments is completely wrong, and that is why questions should not be answered in comments, wrong answer in comments can not be downvoted.
The covariant form of Maxwell's equations:
$$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$
$$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + \par... |
Maintenance is complete! We got more disk space.Become a Patron!
Hello /sci/, can anyone formulate a mathematical/logical proof as to why liking girls is gay?
>>10996470Transitive property.You like girls.Girls like dicks.Therefore, You like dicks.Faggot.
>>10996482Write it more formally, I would also like a mathematica... |
The diagonal formula in mathematics is used to calculate the diagonals of a polygon including rectangles, square, and more similar shapes. When two non-adjacent vertices within a polygon are joined through a single line, it is named as the polygon. Diagonal is formed by joining any two vertices of a polygon except edge... |
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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 ... |
Saturation vapor pressure \(e_s\) is calculated from a given temperature \(T\) (in \(K\)) by using the Clausius-Clapeyron relation. \[\begin{equation} e_s(T) = e_s(T_0)\times \exp \left(\frac{L}{R_w}\left(\frac{1}{T_0} - \frac{1}{T}\right)\right) \tag{1} \end{equation}\] where \(e_s(T_0) = 6.11 hPa\) is the saturation ... |
Mapping/Examples/root x + root y = 1
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$R_5 = \set {\tuple {x, y} \in \R \times \R: \sqrt x + \sqrt y = 1}$
Then $R_5$ is not a mapping.
Proof
$R_5$ fails to be a mapping because, for example, $\sqrt x$ does not exist for $x < 0$.
Thus $R_5$ is undefined for $x <0$.
Thus $R_5$ fails to b... |
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... |
I'm trying to understand how the Doi-Peliti (DP) action is constructed, and specifically how they compute expectation values. To this end, I've been using the book by Taüber as a reference (Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behaviour).
The point I seem to be missing i... |
I was reminded of a quote earlier:
“The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?” — Jerry Bona
I find I am slightly annoyed by this quote, because the equivalence between these three is so direct. I know that they’re equivalent is the joke, but... |
I tried couple of times to compute $\int_{-\pi}^{\pi}\sin(nx)e^{inx}$. According to W.A it should be $-\pi i$, I'm losing my mind trying to understand where I got wrong.
Assuming the whole process that $e^{-in \pi}=\cos(n \pi)-i \sin(n\pi)=(-1)^{n}$, and $[\cos(nx)e^{-inx}]_{-\pi}^{\pi}$=0.
Here is what I did:
$\int_{-... |
C.~D.~A.~Evans and J. D. Hamkins, “Transfinite game values in infinite chess,” Integers, vol. 14, p. Paper No.~G2, 36, 2014.
@ARTICLE{EvansHamkins2014:TransfiniteGameValuesInInfiniteChess, AUTHOR = {C.~D.~A.~Evans and Joel David Hamkins}, TITLE = {Transfinite game values in infinite chess}, JOURNAL = {Integers}, FJOURN... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
The Parallax View
In the previous post of this series unveiling the relationship between UFO sightings and population, we crossed the threshold of normality underpinning linear models to construct a
generalised linear model based on the more theoretically satisfying Poisson distribution.
On inspection, however, this mo... |
C.~D.~A.~Evans and J. D. Hamkins, “Transfinite game values in infinite chess,” Integers, vol. 14, p. Paper No.~G2, 36, 2014.
@ARTICLE{EvansHamkins2014:TransfiniteGameValuesInInfiniteChess, AUTHOR = {C.~D.~A.~Evans and Joel David Hamkins}, TITLE = {Transfinite game values in infinite chess}, JOURNAL = {Integers}, FJOURN... |
NCERT solutions for class 12 Physics Chapter 10 Wave Optics is an essential study material that is required for the students who are seriously preparing for class 12 board examination and graduation entrance examination. Wave optics class 12 NCERT solutions pdf provides answers to the question in textbooks, previous ye... |
I would like to prove that $\sum_{n=1}^\infty \frac{x^2}{(1+x^2)^n}$ converges for $x\in[-1,1]$, but not uniformly. Pointwise convergence is easy.
Define $S_n(x)=\sum_{k=1}^n \frac{x^2}{(1+x^2)^k}$.
$\frac{x^2}{(1+x^2)^n}\leq \left(\frac{x^2}{1+x^2}\right)^n\forall x\in[-1,1]$.
$\forall x\in[-1,1]\cap\{0\}$, $\sum_{n=1... |
Introduction
Built at the Jet Propulsion Laboratory by an Investigation Definition Team (IDT) headed by John Trauger, WFPC2 was the replacement for the first Wide Field and Planetary Camera (WF/PC-1) and includes built-in corrections for the spherical aberration of the HST Optical Telescope Assembly (OTA). The WFPC2 wa... |
So this is how I approached this question, the above equations could be simplified to :
$$a = \frac{4(b+c)}{b+c+4}\tag{1!}$$
$$b = \frac{10(a+c)}{a+c+10}\tag{2}$$
$$c=\frac{56(a+b)}{a+b+56}\tag{3}$$ From above, we can deduce that $4 > a$ since $\frac{(b+c)}{b+c+4} < 1$ similarly $10 > b, 56 > c$ so $a + b + c < 70$
Let... |
The gamma function is defined as $$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$ for $x>0$.
Through integration by parts, it can be shown that for $x>0$, $$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma(x)$ for non-integer negative values. I don't understand why. The ... |
I am trying to find material stiffness matrix for linear elasticity for finite element code,
$$\mathbf{\sigma} = \lambda \hspace{1pt} \operatorname{tr}{\left(\mathbf{\epsilon}\right)}+ 2\mu\mathbf{\epsilon} \,,$$
where $\mathbf{\sigma,\epsilon}$ are Cauchy stress and corresponding conjugate strain respectively; both be... |
Definition:Contour Integral/Complex Definition The contour integral of $f$ along $C$ is defined by: $\displaystyle \int_C f \left({z}\right) \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t$
Let $C$ be a closed contour in $\C$.
Then the symbol ... |
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PHYSICAL REVIEW LETTERS, ISSN 0031-9007, 06/2019, Volume 122, Issue 23, pp. 231802 - 231802
We report a measurement of the mass difference between neutral charm-meson eig... |
The problem asks to calculate the force on one part due to other. But the electric field I'm using to do this calculation is the result E of complete solid non-conducting sphere.
So why are we considering E of the northern hemisphere to calculate force it experience from the southern hemisphere?
This is a very good que... |
Newspace parameters
Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \) Weight: \( k \) = \( 2 \) Character orbit: \([\chi]\) = 8048.a (trivial) Newform invariants
Self dual: Yes Analytic conductor: \(64.2636035467\) Analytic rank: \(0\) Dimension: \(29\) Fricke sign: \(-1\) Sato-Tate group: $\mathrm{SU}(2)$
The dimension is... |
A new approach (not found anywhere else) is given in book by Uspensky, Heaslet, titled Elementary Number Theory, as detailed below for finding non-negative solutions of a linear Diophantine Equation(LDE) $ax + by =c, \exists a,b,c \gt 0 \in \mathbb {Z}$, so that the slope of the line is negative (signs of $a,b$ are the... |
Some Useful Forces and Energies to Know About Force of Gravity
An object near the surface of the Earth is attracted to the Earth with a force commonly referred to as the force of gravity. This force is proportional to the object’s mass and fairly constant everywhere on the surface of the Earth. The constant of proporti... |
In my view, hierarchical modeling in a Bayesian setting mainly refers to the building of a complex prior structure. Consider a parameter of interest $\theta_{0}$ and your observation $(x_i)$.
Now, consider for example that you are adding a supplemental layer to your model $p(\theta_0|\theta_1)$ through hyperprior $p(\t... |
I have some questions on the difference between conditional MLE (CMLE) and unconditional MLE (UMLE) in practice. In what follows I will only talk about the unconditional and conditional mean and leave out the variance in order to shorten the question.
Example 1:
We have a linear model: $y_{i}=x_{i}^{\prime}\beta+\varep... |
Suppose we have a secret $\sigma$.
The secret comes from a universe in which the elements are not necessarily distributed uniformly.
We split $\sigma$ into $n$ shares $[\sigma_1,...,\sigma_n]$ (using Shamir secret sharing). So the order of shares matters.
We know given all the shares in the right order one can recover ... |
The definition you quote is a formal definition of strings which is particularly conducive to induction. There are many other ways to define strings, for example as sequences of letters, or more exactly as mappings $f\colon \{1,\ldots,n\} \to \Sigma$ for some $n \in \mathbb{N}$.
There are several different ways of unde... |
You must be logged in to download the Full-Text PDF or to comment. aInstitute of Chemistry, University of Campinas (UNICAMP), PO Box 6154, 13084-971 Campinas, SP, Brazil bInstitute of Chemistry, University of Campinas (UNICAMP), PO Box 6154, 13084-971 Campinas, SP, Brazil cBrazilian Agricultural Research Corporation (E... |
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 was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand.
Let me remind you the definition of the localization of a category: Let $B$ be an arbitrary category and $S$ a set of morphisms in $B$, the localization is a category $B[S^{-1}]$ with a fu... |
Decomposition of Autoregressive Models Autoregression Background
This post will be a formal introduction into some of the theory of Autoregressive models. Specifically, we’ll tackle how to decompose an AR(p) model into a bunch of AR(1) models. We’ll discuss the interpretability of these models and howto therein. This p... |
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Abstract:
We consider a reaction-diffusion equation for the front motion $u$ in which the reaction term is given by $c(x)g(u)$. We formulate a suitable inv... |
I'm working through a proof on the equivalence between the vector component formula and the sin formula for the cross product of two vectors, $a$ and $b$.
One point in the proof involves finding the area of the parallelogram of $a$ and $b$, angle between them $\theta$, when it is projected onto the $xy$ plane. Call the... |
This is another example of an information puzzle. You are trying to find the most numbers a pair of spies can send to each other given that there are 26 stones in the river.
Furthermore, the stones are all identical, and the only way the spies can communicate is by throwing a certain
number of stones into the pond at t... |
In Peskin's QFT book page 294, he formally addressed the quantization of EM field,
$$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$
Now that we have the functional integral quantization method at our command, let us apply it to the derivation of this expression.
Consider the functional integral
$$\int DAe^{iS[A]}... |
When we get a circuit such as the following:
How do we define the cut-off frequency? Is it still $$f_c = \frac1{2\pi R_1C_1}$$ since \$f_c\$ is defined for \$X_{c1} = R_1\$? Or is it defined so that \$V_{out}\$(the one after OA3) is \$0.707V_2\$?
Electrical Engineering Stack Exchange is a question and answer site for e... |
I have the following functions:
a) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\frac{1}{e^{\frac{1}{z}}-1}$
b) $\displaystyle f:\mathbb{C}\backslash\{0,2\}\rightarrow\mathbb{C},\ f(z)=\frac{\sin z ^2}{z^2(z-2)}$
c) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\cos\le... |
I am looking at past paper questions and I'm a little stuck on this one.
I have the following system of ODEs:
$\dot{x}=(\epsilon x+2y)(x+1)$
$\dot{y}=(-x+\epsilon y)(x+1)$
where $\epsilon$ is a parameter.
a) Show that
$L(x,y)=ax^2+by^2$
is a Lyapunov function for the equilibrium at the origin of the system of the ODEs ... |
Do you have any reason to believe it
is convex? In the space of nonlinear problems, convexity is the exception, not the rule. Convexity is something to be proven, not assumed.
Consider the scalar case; that is, $m=n=1$. Then the problem is$$\min_{y,w\geq 0}(x-yw)^2=\min_{y,w\geq 0}x^2-2xyw+y^2w^2$$
The gradient and Hes... |
I want to minimize $a_1 x_1+a_2 x_2+c x_1 \log\dfrac{x_1}{x_1+x_2}+c x_2 \log\dfrac{x_2}{x_1+x_2}$ for $x_{1,2}\ge 0$ (all the scalars $a_{1,2}<0$ and $c>0$ are real).
After having trouble solving for the critical points of the
Lagrangian, I thought I should check its second order derivatives, only to figure out that t... |
I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable. Comments, ideas, areas of improvement, questions, and constructive criticis... |
You're looking for the equation of Gibbs free energy:
$$\Delta G^\circ =\Delta H^\circ - T\Delta S^\circ$$
Per Wikipedia's Gibbs free energy page:
The equation can be also seen from the perspective of the system taken
together with its surroundings (the rest of the universe). First
assume that the given reaction at con... |
The statement$$\exists x\exists y\forall z\;\bigl((x = z) \lor (y = z)\bigr)$$asserts that there exist elements $x,y$ in the universe such that, for each $z$ in the universe, either $z=x,\;\,$or $z=y,\;\,$or both (i.e., $z$ is equal to one of $x,y$).
Based on that understanding, the statement is true for a given univer... |
Evaluate $$\lim_{x \to -\infty} \left(\frac{\sqrt{1+x^2}-x}{x} \right)$$
I tried by taking $x^2$ out of the root by taking it common.
i.e: $$\lim_{x \to -\infty} \left(\frac{x\sqrt{\frac{1}{x^2}+1}-x}{x} \right)$$ and then cancelling the x in numerator and denominator
$$\lim_{x \to -\infty} \left(\frac{\sqrt{\frac{1}{x... |
One possible explanation could be based on the fact that for low loss, high $Q$ circuits, the sensitivity of the impedance to the deviation of the frequency from the resonant frequency, is greater than for high loss, low $Q$ circuits.
As a result, relative changes in the voltages and currents in low loss circuits are g... |
My physics textbook says when a rock is lifted gravity does negative work and increases the gravitational potential energy.
The problem with reading this statement in isolation is that it is ambiguous.
The first thing which is not clear is the system which is being considered.
Is it the rock alone or the rock & the Ear... |
What is Einstein Field Equation?
The Einstein Field Equation (EFE) is also known as Einstein’s equation. There are a set of ten equations extracted from Albert Einstein’s General Theory of Relativity. The EFE describes the basic interaction of gravitation. The equations were first published in 1915 by Albert Einstein a... |
Please help me to find the sum of $\sum\limits_{n=1}^\infty \left[ \frac{\left(\frac{3 - \sqrt 5}{2}\right)^n}{n^3}\right]$
Is there any special technique to solve this one ?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes... |
I am working on a class project, the passage I quoted in here is from a book
Complex Numbers & Geometry by Hahn, p.64.
For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy to verify:
$$(a-b)(c-d)+(a-d)(b-c) = (a-c)(b-d).$$ By triangle inequality, we obtain $$|a-b||c-d|+|a-d||b-c| = |a-c||b-d|... |
For the standard 4th order Runge Kutta:
where the system is assumed to be smooth (so that the RHS has no discontinuous points)
$\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$
$\mathbf{y(t_0)} = \mathbf{y_0}$
$\mathbf{y_{i+1} = y_n + 1/6(k_1 + 2k_2 + 2k_3 + k_4}$
where
$$\mathbf{k_1 = F(t_n,y_n)}$$ $$\mathbf{k_2 = F(t_n + h/2,... |
Sum of Euler Phi Function over Divisors Theorem
Let $n \in \Z_{>0}$ be a strictly positive integer.
Then $\displaystyle \sum_{d \mathop \divides n} \map \phi d = n$
where:
$\displaystyle \sum_{d \mathop \divides n}$ denotes the sum over all of the divisors of $n$ $\map \phi d$ is the Euler $\phi$ function, the number o... |
They can replenish themselves by quick recycling and replacement within a reasonable time.
They cannot replenish more...
Improvement in Food Resources
Chapter Overview
Introduction Introduction
Improvement in crop yields
Crop variety improvement
Crop production management
Crop protection management
Fish production (Pis... |
When we integrate certain integrals, such as
$$\int \frac{x^2}{\sqrt{16-x^2}} dx$$
We can make a substitution like $x = 4 \sin \theta$
Then we can simplify the above integral to the following: $$8 \theta - 8 \sin \theta \cos \theta + C$$
I then learned we can use a right angled triangle to find alternate expressions fo... |
But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you alr... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
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