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Let Ω be the region in $ℝ^3$ defined by $$ Ω={(x_1,x_2,x_3):max(∣∣x_1∣∣,∣∣x_2∣∣,∣∣x_3∣∣)≤1}$$ Let ∂Ω denote the boundary of Ω.
Calculate $$∫_{∂Ω}ϕF⋅ndσ$$
where n is the unit normal vector, dσ denotes integration over ∂Ω,
$F_i=\large \frac{x_i}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}}^=\frac{x_i}{r^3}$
and $ϕ(y_1,y_2,y_3)$ is... |
Adiabatic Short Circuit Temperature Rise
Adiabatic short circuit temperature rise normally refers to the temperature rise in a cable due to a short circuit current. During a short circuit, a high amount of current can flow through a cable for a short time. This surge in current flow causes a temperature rise within the... |
What this post is about
Personally, I find violin plots with error bars a great way to present repeated-measures data from experiments, as they show the data distribution as well as the uncertainty surrounding the mean.
However, there is some confusion (at least for me) about how to correctly calculate error bars for w... |
This article provides answers to the following questions, among others:
What is steel made of? What is the difference between steel and cast iron regarding their composition? Why is carbon used as an alloying element for steel? Why do further phase transformations take place in the already solidified state in steels? I... |
I think we must first understand the description of a machine and the input size, so that the comparison is of only valid objects.Let say
is a input size. This means machines will have these resource bounds. N
\begin{array}{|l|l|l|}\hline\mbox{Resource} & \mbox{Finite Automata:}\quad \mathcal{A} & \mbox{LBTM:} \quad \m... |
This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway.
If I understand correctly, for any reductive algebraic group $G$ the points of $G$ over the field with one element should be $G(\mathbb{F}_1) = W$ where $W$ is the Weyl group of $G$. My q... |
This question already has an answer here:
$\ce{AgNO3 + NaCl}$ gives $\ce{NaNO3 + AgCl}$ but why it's vice versa is not possible. My question is how to predict if a double displacement reaction will occur naturally or not.
Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and st... |
Call a category $C$
rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets, commutative monoids, groups, abelian groups, commutative rings, but also the category of topological spaces. The categ... |
Adiabatic Short Circuit Temperature Rise
Adiabatic short circuit temperature rise normally refers to the temperature rise in a cable due to a short circuit current. During a short circuit, a high amount of current can flow through a cable for a short time. This surge in current flow causes a temperature rise within the... |
I'm in a pre rigorous phase but as a programmer I need to know certain things on a case by case basis to do certain very particular things.
Therefore, I was hoping for validation of why the following, observed in a book about bayesian networks, is true. That being:
Let $\mathcal{G}$ be a bayesian network over $X_1, ...... |
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any fixed $s$ and $t$, it is well-known that (Central Limit Theorem) $$\mathbb \lim_{n,m\rightarrow\infty}(\mathbb P(\tilde S_n... |
Recap
In the previous chapter:
We considered games of incomplete information; Discussed some basic utility theory; Considered the principal agent game.
In this chapter we will take a look at a more general type of random game.
Stochastic games Definition of a stochastic game
A stochastic game is defined by:
X a set of ... |
No, this is impossible whenever you have three or more coins.
The case of two coins
Let us first see why it works for two coins as this provides some intuition about what breaks down in the case of more coins.
Let $X$ and $Y$ denote the Bernoulli distributed variables corresponding to the two cases, $X \sim \mathrm{Ber... |
In this chapter, we will discuss the Solutions to Friedmann Equations relating to the Matter Dominated Universe. In cosmology, because we are seeing everything in a large scale, the solar systems, galaxies, everything happens to be like dust particles (that’s what we see it with our eyes), we can call it dusty universe... |
Wikipedia has an extensive list of languages that use the off-side rule1:ABCBooBuddyScriptCobraCoffeeScriptConvergeCurryElixir (, do: blocks)ElmF# (if #light "off" is not specified)GenieHaskell (only for where, let, do, or case ... of clauses when braces are omitted)Inform 7ISWIM, the abstract language that ...
There a... |
For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More generally, can one describe the set of such $\alpha$ (empty/non-empty, finite/infinite etc.) if one is given $f$?
Examples:
$f... |
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider the topology of weak convergence on the space $E'$ of probability measures on $E$ (thus $E'$ is Polish too).
1) Let $X$ be ... |
Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{(x_i)\in\Omega:\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}x_i=\alpha\}.$$ Let $\mathcal{M}_\sigma(\Omega)$ stand for the fam... |
Permanent link: https://www.ias.ac.in/article/fulltext/joaa/038/01/0017
We developed a generic formalism to estimate the event rate and the redshift distribution of Fast Radio Bursts (FRBs) in our previous publication (Bera
et al. 2016), considering FRBs are of an extragalactic origin. In this paper, we present (a) the... |
I can't really understand MixColumns in Advanced Encryption Standard, can anyone help me how to do this?
I found some topic in the internet about MixColumns, but I still have a lot of question to ask.
ex.
$$ \begin{bmatrix} \mathtt{d4} \\ \mathtt{bf} \\ \mathtt{5d} \\ \mathtt{30} \\ \end{bmatrix} \cdot \begin{bmatrix} ... |
I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me.
The aim of this question is to try to finally put this uncomfortable condition to rest.
Ehresmann Connections
Ehresmann connections are a very i... |
It's been a while since my last blog entry, but the problem I posed on my earlier blog entry still persists - how to efficiently choose a good split plane for an n-vector data structure. To summarize the structure, geographic points are stored as n-vectors (unit vectors) in a binary tree. Branch nodes of this tree defi... |
Dear Uncle Colin, I've been given $u = (2\sqrt{3} - 2\i)^6$ and been told to express it in polar form. I've got as far as $u=54 -2\i^6$, but don't know where to take it from there! - Not A Problem I'm Expecting to Resolve Hello, NAPIER, and thanks for yourRead More →
Someone recently asked me where I get enough ideas f... |
Due date: Friday 9/17 at 11:59pm
In this assignment we will use the following datasets:
In this assignment you will work with several variants of the perceptron algorithm:
In each case make sure that your implementation of the classifier includes a bias term (in slide set 2 and page 7 in the book you will find guidance... |
A (Very Short) Detour for the Traveling SalesmanIn this article, we'll explore why the Traveling Salesman Problem is an interesting problem and describe a recent result concerning it. ....
David Austin
Introduction
My teenaged son's job requires him to deliver papers to 148 houses. When I deliver papers with him, we pa... |
I want to find a solution to a system of linear inequalities of the following form
\begin{aligned} a_1 + b &\ge a_2 \\\ \vdots \\\ a_4 + c &\ge a_1 \end{aligned}
where $a_i \in \mathbb N \setminus \{0\}$ and $b,c \in \mathbb Z$. All inequalities consist of exactly two variables and one free factor. All $a_i$ appear as ... |
I'm trying to figure out for which algebraic structure
$$\underbrace{a+a+\cdots+a}_{n \text{-times}} = a * n$$
is true.
Now I know the question '
Is all multiplication repeated addition?' has been asked many times with the answer: NO because you cannot express non-integer (such as fractions or complex numbers) multiple... |
Due: October 17th at 11:59pm
Formulate a soft-margin SVM without the bias term, i.e. one where the discriminant function is equal to $\mathbf{w}^{T} \mathbf{x}$. Derive the saddle point conditions, KKT conditions and the dual. Compare it to the standard SVM formulation that was derived in class.
In this question we wil... |
This article provides answers to the following questions, among others:
What is meant by the packing density (or packing factor)? How is the packing density calculated for the body-centered cubic lattice (bcc)? How is the packing density calculated in the face-centered cubic lattice (fcc)? Why is the packing density fo... |
Given a square matrix
A, how can I generate a basis for the
generalized eigenspace corresponding to all eigenvectors $\lambda_i$ such that $\vert \lambda_i \vert > 1$? I.e., if $A$ is $n \times n$ and acts on $\mathbb{R}^n$, then $\mathbb{R}^n = E^u \oplus E^1 \oplus E^s$, corresponding to the action of eigenvalues of ... |
Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$.
Now I am searching for a good reference (with proofs) for the following facts:
1) The fundamental vector fields on $P$ span pointwise the vertical space - or equivalently they generate the $C^\infty(P)$-mod... |
In the previous STT5100 course, last week, we’ve seen how to use monte carlo simulations. The idea is that we do observe in statistics a sample \{y_1,\cdots,y_n\}, and more generally, in econometrics \{(y_1,\mathbf{x}_1),\cdots,(y_n,\mathbf{x}_n)\}. But let’s get back to statistics (without covariates) to illustrate. W... |
While reading through my textbook it says "the most important example of an inner-product space is $F^n$", where $F$ denotes $\mathbb{C}$ or $\mathbb{R}$ .
Our definition of an inner product on a vector space $V$ is as follows:
1) Positive definite: $\langle v,v \rangle \ge 0$ with equality if and only if $v=0$ 2) Line... |
A 2-clause is a clause with at most two propositions (clauses?) : $(p \wedge q,\neg p \wedge q, \neg p,...)$. I have to show that the folllowing problem is $\in$ P:
2-SAT: Input : A conjunction $\Phi$ of 2-clauses. Question : Is it satisfisable?
For each set of 2-clause $C$ we associate $G_C$ defined as
$V_{G_C}$ conta... |
This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked whether $\mathbb{R}^3$ remains simply connected after deleting a countable set of points, such as the collection of rational ... |
In the following exhibits we give an advanced or alternative way of thinking about mathematics concepts which are likely to be known in a more familiar form.
Explore these structures and experiment by substituting particular values such as $0, \pm 1$. Can you work out what they represent? Exhibit A All pairs of integer... |
It is easy to turn any boolean formula and any quantified boolean formula into an equisatisfiable formula in CNF using
Tseitin transformation:
$$ Q_1 z_1 Q_2 z_2 \ldots Q_n z_n \Phi \Rightarrow Q_1 z_1 Q_2 z_2 \ldots Q_n z_n \exists x ((\neg{x} \vee \Psi) \wedge \Phi[x/\Psi] ),\ Q_i \in \{ \exists, \forall \} $$
(For d... |
I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected.
I have subbed in the information into green's identity but I don't get anything useful.
$$\iiint_\Omega \nabla f\cdot \nabla g \,dV =\iint_{\partial\Omega} ... |
June 4th, 2014, 05:59 AM
# 1
Newbie
Joined: Jun 2014
From: UK
Posts: 1
Thanks: 0
Solving ODEs without making an Ansatz
Hi,
Last year in my final year of sixth form (High school?), one of the topics we covered was solving linear ODEs with various methods. Now I'm in University we've recovered solving second order or hig... |
On $\mathbb{R}^3$ with coordinates $x,y,z$ consider the Riemannian metric $g=\displaystyle{\frac{dx^2+dy^2+dz^2}{x^2+y^2}}$ defined on $X:=\mathbb{R}^3\setminus\{(0,0,z)\}$.
Given any point $p_1\in X$ and $p_2$ any point in a sufficiently small neighborhood of $p_1$, I want to compute the distance $d(p_1,p_2)$ with res... |
Electrical Power
How do the various formulae for electrical power fit together? What is the difference between DC, AC and complex power and how do they harmonise with our physical conceptions of energy and power.
Contents Definition
By formal definition, any form of power (e.g. electrical, mechanical, thermal, etc) is ... |
We study the evolution of convex hypersurfaces with initial at a rate equal to <i>H</i> 鈥<i>f</i> along its outer normal, where <i>H</i> is the inverse of harmonic mean curvature of is a smooth, closed, and uniformly convex hypersurface. We find a <i>胃</i>* > 0 and a sufficient condition about the anisotropic function ... |
The space $S$ of holomorphic functions on the disk $\mathbb{D}$ which are also in $L^2(\mathbb{D})$ is a Hilbert space, with the inner product being $\langle f, g\rangle = \int f(z) \overline{g(z)} dA(z)$, where $dA$ denotes Lebesgue measure. The linear functional $L_{z_0}$, given by $f\mapsto f(z_0)$, is a bounded lin... |
This article provides answers to the following questions, among others:
Which properties of the hydrostatic pressure could Pascal demonstrate with his barrel experiment? How does the pressure change with increasing water depth? What are communicating vessels? How does a water levelwork? How does a water tower work? Hyd... |
We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and a sphere. Therefore, we introduce parameterized vector equations, which can describe them.
For example, circle: $\vec r... |
This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed.
An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot) $$ holds.
It is well known that a pseudocomplemented distributive lattice is Boolean if an... |
The problem is Poincare' s lemma joined with compactness (without boundary) of the 2-manifold $M$ we are considering. Due to them, the flux of the electrical field should be simultaneously zero and $1$ for a $1$-surface (a closed curve) sourronding the support of the delta function, since this curve can viewed as the b... |
You are right about the dropped $\sim$, it's probably just a typo. Furthermore, remember that in stochastic calculus, you have to take into account second order derivatives, i.e.
$$d\left(\frac{1}{Y_t}\right) = -\frac{1}{Y_t^2}dY_t + \frac{1}{2}\frac{2}{Y_t^3}dY_t^2$$
which is the Taylor expansion up to second order. T... |
This article uses a very idealized model to roughly calculate daylight hours and midday moments, and to calculate the time of sunrise and the day.
How do you calculate Daylight time? Or, how do you calculate the percentage of daylight time in a day? Figure out how much of the Sun's Sunday parallel loop (the trajectory ... |
Which, if any, axioms of ZFC are known to not be derivable from the other axioms?
Which, if any, axioms of PA are known to not be derivable from the other axioms?
There are several interesting issues here.
The first is that there are different axiomatizations of PA and ZFC.
If you look at several set theory books you a... |
Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element $\lambda$ in the group ring $\mathbb Z[G]$ which is "residually invertible", ie every image $\overline\lambda\in\mathbb Z... |
I considered a particle in polar coordinates, $(r,\theta)$, with mass $m$. The standard basis vectors in polar coordinates are: $$\mathbf{\hat{r}}=\cos{\theta}\mathbf{\hat{x}}+\sin{\theta}\mathbf{\hat{y}}$$ And: $$\boldsymbol{\hat{\theta}}=\frac{\partial\mathbf{\hat{r}}}{\partial\theta}=-\sin{\theta}\mathbf{\hat{x}}+\c... |
Considerations When Using Cylinder Lenses
This is a supplementary section of the Laser Optics Resource Guide.
Cylinder lenses are similar to spherical lenses in the sense that they use curved surfaces to converge or diverge light, but they have optical power in only one dimension and will not affect light in the perpen... |
The question is simple: How do I find the function derivative of $$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$ As far as I can tell, I cannot use any of the standard computational rules for the functional derivative.
The expression $\frac{\delta \partial_\mu\phi(y)}{\delta \phi(y)}$ is mathematically
meaningless... |
Electrochemical Impedance Spectroscopy: Experiment, Model, and App
Electrochemical impedance spectroscopy is a versatile experimental technique that provides information about an electrochemical cell’s different physical and chemical phenomena. By modeling the physical processes involved, we can constructively interpre... |
January 15th, 2015, 05:49 AM
# 1
Newbie
Joined: May 2013
Posts: 5
Thanks: 0
riemann hypothesis proof?: zeta(s) is never 0 for Re(s)>1/2, flaws? thanks!
The Riemann Zeta function can be represented by the Euler product \begin{align}
\zeta \left( s \right) = \prod^{\infty}_{p=primes} \frac {1}{1-1/p^{s}}
\end{align}
for ... |
You probably meant to assume $R$ and $S$ are noetherian. The answer is "no" to the initial hypergeneral part of the question. EDIT: In the 2nd half (below the long line), I now give a proof of an affirmative answer to the added part involving maps of affine spaces.
Counterexamples to the initial hypergeneral part can b... |
It's common knowledge (and has been discussed in other questions on this site) that the standard BCS ground state $ \left|\Psi_{BCS}\right\rangle = \prod_k \left( u_k + v_k c_{k\uparrow}^{\dagger} c_{-k\downarrow}^{\dagger}\right) \left|0\right\rangle$ does not have a well-defined particle number and that this doesn't ... |
We study the evolution of convex hypersurfaces with initial at a rate equal to <i>H</i> 鈥<i>f</i> along its outer normal, where <i>H</i> is the inverse of harmonic mean curvature of is a smooth, closed, and uniformly convex hypersurface. We find a <i>胃</i>* > 0 and a sufficient condition about the anisotropic function ... |
Glossary
We are planning to prepare a Glossary for the course that contains a list of the key terms that are used in the course.
Which terms would you like us to explain here?
\(\exists\) :There exists.
\(\in\): Belongs to
\(\forall\): For every
\(\mathbb N\): The set of natural numbers \(0,1,2,3,…\)
\(\mathbb Q\): The... |
66 9 Homework Statement Let f and g be derivable functions and let a be a real number such that ##f(a)=g(a)=0 ## ##g'(a) ≠ 0 ## Justify that ##\frac{f'(a)}{g'(a)} ## = ##\lim_{x\to a}\frac{f(x)}{g(x)}## You may only use the definition of the derivative and boundary rules. Homework Equations ##\lim_{h\to 0}\frac{f(x+h)-... |
I'm trying to figure out how to translate a piece of code from Velocity Verlet to Runge-Kutta, while treating the time step dependence of the thermal noise correctly.
The Langevin equation for my system reads reads
$$ ma = - \gamma v - \frac{dU}{dx} + \xi(t), $$
where $U$ is some interaction potential, $\gamma$ is damp... |
In this chapter, we will understand what the Angular Diameter Distance is and how it helps in Cosmology.
For the present universe −
$\Omega_{m,0} \: = \: 0.3$
$\Omega_{\wedge,0} \: = \: 0.69$
$\Omega_{rad,0} \: = \: 0.01$
$\Omega_{k,0} \: = \: 0$
We’ve studied two types of distances till now −
Proper distance (lp) − Th... |
To calculate any equilibrum constant we need to be sure of the fugacity of each species. Note that fugacities are defined for every common state of matter (gas, liquid and solid), although they are most commonly used for gases. Look at the end of this answer for more details.
The activity of the coumpound $i$ written $... |
Talk:Principle of least action Reviewer A
This is a well-written article on the Principle of Least Action by one of the leaders in this field. Here are some comments about the material presented in some of the eleven sections.
In Section 1, it should perhaps be pointed out that, like Fermat's Principle of "Least" Time,... |
Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $
Department of Mathematics, Yale University, New Haven, CT 06520, USA
We present a new proof of the following theorem of Benoist-Quint: Let $ G: = \operatorname{SO}^\circ(d, 1) $, $ d\ge 2 $ and $ \Delta<G $ a cocomp... |
This question already has an answer here:
I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet why the likelihood is no pdf. There have been several arguments, mainly involvin... |
Units Topics Marks I Relations and Functions 10 II Algebra 13 III Calculus 44 IV Vectors and 3-D Geometry 17 V Linear Programming 6 VI Probability 10 Total 100 Chapter 1: Relations and Functions Chapter 2: Inverse Trigonometric Functions Chapter 1: Matrices
Concept, notation, order, equality, types of matrices, zero an... |
I suspect this problem is ill-posed: the degree of $f$ may depend on the specific enumeration of Turing machines used.
However, I can prove that the degree of the Halting Problem is attainable:
Claim: there is an admissible numbering $\varphi_i$ of Turing machines such that $f$, defined relative to this numbering, comp... |
If I wanted to evaluate $\int_{C(0,1)}(z+\frac{1}{z})^{2n}\frac{1}{z}dz$ using the Binomial theorem what would my result be ?
So far I've rearranged the integral until we have $\int\frac{(z^2+1)^{2n}}{z^{2n+1}}dz$
Then by using the binomial theorem on the numerator we obtain
$(1+z^2)^{2n}=\sum_{k=0}^{2n} \binom{2n}{k}(... |
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that... |
What numerical method can approximately compute the $(n-1)$-dimensional surface area of the $\ell_p$ ball $\{x\in\mathbb R^n: \sum_{i=1}^n |x_i|^p=1\}$, for $p\in[1,\infty)$? Ideally the method should handle $n$ and $p$ both in the range of 5 to 10.
One approach begins with the definition of surface area as $$ \lim_{\v... |
We provide a novel method to increase SNR of segmented diffusion-weighted EPI acquisitions. Multiple gradient echoes were acquired after each diffusion-preparation and combined in an SNR-optimized way using weightings from quantitative T2* maps. The combination of diffusion-weighted echoes yielded an SNR-gain of 58% co... |
This is kind of a continuation of this question.
I want to automate these 2 steps on
Mathematica once I give an integer $N$. generate a list of variables $\{a_i\}_{i=1}^{i=N}$ then for some function like $f(x) = e^{x}$ or $f(x) = \sin^2(x)$ compute an expression like $\prod\limits_{i \neq j} f(a_i - a_j)$ or $\sum\limi... |
It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ to be the category of quasi-coherent sheaves on $U$. Given a morphism $f : U \to V$, we have a functor $f^* : Qcoh(V) \to... |
Hello,
working on some machine learning problem I end up facing a problem which looks like generalizing the notion of Cauchy product.
I briefly go back to Cauchy products before exposing my question. Consider, two sequences $(a_n)_{n \in \mathbb N}$ and $(b_n)_{n \in \mathbb N}$, which are assumed to be absolutely conv... |
Combinatorial and Discrete Optimization (2019 KSIAM Annual Meeting) November 8 Friday @ 12:00 PM - November 9 Saturday @ 7:00 PM Special Session @ 2019 KSIAM Annual Meeting Date Nov 8, 2019 – Nov 9, 2019
Address: 61-13 Odongdo-ro, Sujeong-dong, Yeosu-si, Jeollanam-do (전남 여수시 오동도로 61-13)
Venue
Address: 61-13 Odongdo-ro,... |
Shuheng Zheng
Work as a software engineer by day. Rest of the time doodle with math, physics, or read about political economy and history. Eventually realized politics is circular and futile and then go back to math.
Seattle, Washington, United States
Member for 2 years, 6 months
0 profile views
Last seen Apr 8 '17 at ... |
Let $P(z),Q(z)$ be polynomials, where $\text{deg}Q-\text{deg}P\ge 2$. Suppose $Q$ has no real roots.
Call the set of number that can be written as $p+iq$ ($p,q$ rational) as rational complex numbers.
Suppose $P,Q$ has coefficients that are rational complex numbers.
Prove or disprove:
$$\int^\infty_{-\infty}\frac{P(z)}{... |
Rate–distortion theory is a major branch of information theory which provides the theoretical foundations for lossy data compression; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rate R, that should be communicated over a channel, so that the source (input signal) ca... |
Modular Inverse
Tags:
crypto Introduction
I wasn’t happy with the very brief introduction we did to modular inverse and so I decided that I wanted to create an individual post on the matter, how it works, why we have it and so on. I think this post will be greatly beneficial to anyone who read the last two posts and wa... |
Consider the following sequence $a_n$:
\begin{align*} a_0 &= \alpha \\ a_k &= \beta a_{k-1} + \kappa \end{align*}
Now consider the implementation of this sequence via lisp:
(defun an (k) (if (= k 0) alpha (+ (* beta (an (- k 1))) kappa)))
What is the running time to compute
(an k) just as it is written? Also, what woul... |
I am interesting in solving the following nonlinear, time-dependent pde in 2 spatial dimensions (complex Gross-Pitaevskii eq):
$$i \frac{\partial \psi}{\partial t} = \left[ -\nabla^2 + (1-i \sigma)(|\psi|^2-1) \right] \psi$$
The goal is to find steady state solutions for the function $\psi(x,y,t)$, for different parame... |
\begin{align}\begin{split}\pd{EU}{\alpha} &= d(\theta) U'(\pi_{1}) [G_{\alpha}(\alpha, \theta, \epsilon) - \widebar{P} C_{\alpha}(\alpha, \theta)] \\ &+ (1-d(\theta)) U'(\pi_{0}) [G_{\alpha}(\alpha, \theta, 0) - C_{\alpha}(\alpha, \theta)] = 0,\\\pd{EU}{\theta} &= h'(\theta)[U(\pi_{0}) - U(\pi_{1})] \\ &+ d(\theta) U'(... |
I couldn’t resist getting sucked into the hype associated with the US election and debates, and so I thought I had a little fun of my own and played around a bit with the numbers.
[OK: you may disagree with the definition of “fun” $-$ but then again, if you’re reading this you probably don’t…] So, I looked on the inter... |
1. Homework StatementI have to prove that the expression$$\frac{\omega C - \frac{1}{\omega L}}{\omega C - \frac{1}{\omega L} + \omega L - \frac{1}{\omega C}}$$is equal to$$\frac{1}{3-( (\frac{\omega_r}{\omega})^2 + (\frac{\omega}{\omega_r})^2)}$$where ##\omega_r= \frac{1}{\sqrt{LC}}##...
So what I did was:I took the sq... |
I am curious to know under what conditions of the air pressure(atm), temperature, solute density in the water would cause the Niagara fall frozen?
In general, the answer is "a bit lower than 32 Fahrenheit". Here's two things which one might think would come into play, but actually do not to an appreciable extend.
The m... |
Not with the obvious complex structure. Notice that $O_{\lambda}$ is a closed subvariety of $T^{\ast}(O_{\lambda})$ (namely the zero section). Closed subvarieties of Stein varieties are Stein. However, positive dimensional Stein varieties are never compact, and $O_{\lambda}$ is compact.
However, there is a sense in whi... |
Question
What is the best known
effective upper bound on the prime gap following x? Motivation
Suppose you needed to show a good bound for the gap between a fixed large constant, say $G=10^{10^{100}}$, and the following prime. Bertrand's postulate gives $G$, but we know unconditionally that the prime gap is $O(G^\theta... |
Answer
$12\times \pi$
Work Step by Step
$V = \frac{1}{3}\pi \times r^2\times h$ By replacing the letters with their values (r=3, h=4) We have: $V = \frac{1}{3}\pi \times 3^2\times 4 = 12\times \pi$
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After you claim an answer you’ll ha... |
The following is more or less a copy-paste of a comment I made on the related ArsTechnica thread. Indeed, StackExchange is probably one of the better places to debate this.
A few reminders first:
there are approximately $p$ elliptic curves over the finite field of integers $\pmod{p}$; of these curves, only those with (... |
Longitudinal relaxation times for 11 human brain metabolites
are reported for GM and WM rich voxels at 9.4T. These values are reported to
potentiate the ability to perform absolute quantification at 9.4T in humans
with reference to water. A bi-exponential model was used to fit the signal
curve from using an inversion r... |
By Emily van Zee and Corinne Manogue This interpretative narrative is based upon a video of the class session and discussions with the instructor and the director of the Physics Paradigms Program, Corinne Manogue. In writing the narrative, Emily van Zee drew upon her research in the tradition of ethnography of communic... |
A new proof of the boundedness results for stable solutions to semilinear elliptic equations
1.
ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain
2.
Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain
3.
BGSMath, Campus de Bellaterra, Edifici C, 08193 Bellaterra, S... |
I've recently asked a question on an issue I was facing with numerically integrating Hamiltonian equations of motion. I got a great answer.
Following on from this, I wanted to write a very similar code using alternate definitions in variational mechanics.
That is using the following:
The Lagrangian is given by $L= \fra... |
perhaps he is implying some even stronger result
He is referring to the following result of Peter Freyd (Freyd uncertainty principle):
The homotopy category of spaces $HoTop$ does not admit a faithful functor to the category of sets $Set$. Specifically, for any functor $T: Top_* \to Set$ from base-pointed spaces to set... |
Answer
Please see the work below.
Work Step by Step
We know that $\Delta T_F=(\frac{9}{5})\Delta T_C$ We plug in the known values to obtain: $\Delta T_F=(\frac{9}{5})(10)$ $\Delta T_F=18F^{\circ}$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll hav... |
This chapter deals with the parameters of radiated beam of the antenna. These parameters help us to know about the beam specifications.
According to the standard definition, “Beam area is the solid angle through which all the power radiated by the antenna would stream if P (θ, Ø) maintained its maximum value over Ω
A a... |
Radiation intensity of an antenna is closely related to the direction of the beam focused and the efficiency of the beam towards that direction. In this chapter, let us have a look at the terms that deal with these topics.
According to the standard definition, “The ratio of maximum radiation intensity of the subject an... |
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OGLE-2017-BLG-0173Lb: Low-mass-ratio Planet in a "Hollywood" Microlensing Event
(2018)
We present microlensing planet OGLE-2017-BLG-0173Lb, with planet-host mass ratio either $q\simeq 2.5\times 10^{-5}$ or $q\simeq 6.5\times 10^{-5}$, the lowest or among the lowest ever detected. The... |
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