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The following text is cited from a textbook, "Spotlight Mode Synthetic Aperture Radar: A Signal Processing Approach", I would like to ask if anyone knows the proof to the following statements, as the proof is not outlined in the book, and I am unable to prove it myself.
The function $f(t)$ is represents a pulse envelop... |
Newton’s second law establishes a relationship between the force \(\mathbf{F}\) acting on a body of mass \(m\) and the acceleration \(\mathbf{a}\) caused by this force.
The acceleration \(\mathbf{a}\) of a body is directly proportional to the acting force \(\mathbf{F}\) and inversely proportional to its mass \(m,\) tha... |
Tagged: finite group If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order Problem 575
Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup. Namely, suppose that $G=S\sqcup H$, where... |
Given:
$$ u_{tt} - c^2 u_{xx} = 0 $$(1) for $ 0<x<\infty $
$$ u |_{t=0} = g(x) , u_t | _{t=0} = h(x) $$(2) for $0<x$, and additionally
$$ (\alpha u + \beta u_t) |_{x=0} = q(t) $$(3) for $ t> 0 $
We are asked to evaluate and find the general solution for both regions $ x > ct $ and $0<x<ct$
Given that the most general s... |
Applying a filter to a signal is convolution. For discrete signals, convolution equals polynomial or power (Laurent) series multiplication. The Z "transform" just formally puts filter coefficients and signal samples in power resp. Laurent series so that their multiplication can be interpreted back as the application of... |
The normalization is basically a preconditioning to decrease condition number of the matrix $A$ (the larger the condition number, the nearer the matrix is to the singular matrix).The normalizing transform is also represented by a matrix in the case of homography estimation, and this happens to be usable as a good preco... |
What are the assumptions behind the Lagrangian derivation of energy? I understand that we're searching for a function $L$ that describes a set of physics so that solving the energy minimization problem
$$\begin{array}{rcl} \arg\min\limits_{q} && \int_{t_1}^{t_2} L(q,\dot{q},t) dt\\ \textrm{st}&&q(t_1)=q_1\\ &&q(t_2)=q_... |
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Now showing items 1-10 of 32
The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
In this chapter, we will discuss how recursive techniques can derive sequences and be used for solving counting problems. The procedure for finding the terms of a sequence in a recursive manner is called
recurrence relation. We study the theory of linear recurrence relations and their solutions. Finally, we introduce g... |
Category: Ring theory Problem 624
Let $R$ and $R’$ be commutative rings and let $f:R\to R’$ be a ring homomorphism.
Let $I$ and $I’$ be ideals of $R$ and $R’$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I’)}=f^{-1}(\sqrt{I’})$
Add to solve later
(c) Suppose that $f$ i... |
MathJax Tutorial
MathJax is a JavaScript library that allows you to use basic $\LaTeX$ syntax. The formulas can be used in all text fields. To do this, the expression must be marked with dollar signs ($\$$Expression$\$$).
Fractons
A fraction is generated by the expression \frac{numerator}{denominator}. Instead of numbe... |
I am learning Fluid mechanics by reading Acheson's book entitled "Elementary Fluid Dynamics". Below is from problem 3.1.
Consider the Euler equation for an ideal fluid in the irrotational case. We are studying two-dimensional water waves, so the velocity vector is on the form: ${\bf{u}}=[u(x,y,t),\,v(x,y,t),\,0]$. Beca... |
Suppose there is an electromagnetic wave moving forward in the $\mathbf{\hat{k}}$ direction. Its electric/magnetic field components are given by: $$\mathbf{E} = E_0 \sin(kz - \omega t) \mathbf{\hat{i}}$$ $$\mathbf{B} = B_0 \sin(kz - \omega t) \mathbf{\hat{j}}$$ If a particle of charge $q$ was lying on the wave's trajec... |
GSoC 2017 - Scipy: Large-scale Constrained Optimization
This year I was chosen as the student for Google Summer of Code. I’ll be working on one of the core Python scientific libraries called Scipy. My task is to implement a constrained optimization algorithm able to deal with large (and possibly sparse) problems.
The n... |
Trying to practice translating english sentences into predicate logic and vice versa.
$E(x,y)$: $x$ can eat $y$ $L(x,y)$: $x$ loves eating $y$ $D$ is the domain of all dogs $S$ is the domain of all snakes
(a)English to Predicate Logic: A dog can eat any snake, only if they are different from some other dog who can also... |
Let us place an infinitely long solenoid of \(n\) turns per unit length so that its axis coincides with the \(z\)-axis of coordinates, and the current \(I\) flows in the sense of increasing \(\phi\). In that case, we already know that the field inside the solenoid is uniform and is \(\mu\, n\, I\, \hat{\textbf{z}}\) in... |
I found once an instruction how to typeset mathematics on this site, now I came back to ask a question and could not remember it.
Thus I pressed the
? button, to find help on this subject and then, after choosing 'advanced help' only found rudimentary information that LaTeX and MathJax are used with a link to www.mathj... |
If you input the trig identity: $$\cot (x)+\tan(x)=\csc(x)\sec(x)$$ Into WolframAlpha, it gives the following proof:
Expand into basic trigonometric parts: $$\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} \stackrel{?}{=} \frac{1}{\sin(x)\cos(x)}$$ Put over a common denominator:
$$\frac{\cos^2(x)+\sin^2(x)}{\cos(x)\s... |
Ray Optics and Optical Instruments Optical Instruments Visual angle is the angle subtended by an object at the eye. Myopia means short sightedness, the distant objects are not clearly visible. Hypermetropia means far sightedness, the near objects are not clearly visible. A convex lens is called simple microscope Magnif... |
I'm currently studying for an exam in image processing and stumbled upon a exercise which i could not answer and my professor will not be available before the exam anymore. The exercise goes as follows:
Create a Hann-lowpass kernel of size $3\times 3$ and calculate the result of the convolution at $(2,2)$ in the image.... |
As indicated by Igor Rivin, the volume of the unitary group is given by $vol(U(N))=(2\pi)^{(N^2+N)/2}/\prod_{k=1}^{N-1} k!$.
The denominator is the Barnes G-function, which is well-known :
http://en.wikipedia.org/wiki/Barnes_G-function
and in particular has a known Stirling-like asymptotic expansion for large $N$:
$\lo... |
Suppose a wire of length
L carrying a current I is kept in a uniform magnetic field B perpendicular to the current. The force on the wire will be IBL and work done by magnetic force when wire moves a distance d along the force will be IBLd. But magnetic force cannot do any work on a moving charged particle and hence to... |
I have the following equation:
\documentclass[12pt,a4paper]{report}\usepackage{amsmath}\begin{align}\begin{bmatrix}\dot x \\\dot \varphi \\\dot v \\\dot \omega\end{bmatrix} = \begin{bmatrix} %% \vspace{5pt} does not workv \\ \omega \\ x \omega^2 - g \sin{\varphi} \\[5pt]\displaystyle{- \frac{\left( m_2x-m_1a \right)g\c... |
Difference between revisions of "Dictionary:Q(Quality)"
Line 8: Line 8:
<center><math>A\mathrm{e}^{-\alpha x} \sin 2 \pi f ( t - \tfrac{x}{V} ) </math></center>
<center><math>A\mathrm{e}^{-\alpha x} \sin 2 \pi f ( t - \tfrac{x}{V} ) </math></center>
−
where ''x'' is the distance traveled. The [[Dictionary:logarithmic d... |
In the wave equation:
$$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$
Why do we first multiply by a test function $v(x,t)$ before integrating?
Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only take... |
stat946w18/Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolutional Layers Contents Introduction
With the recent and ongoing surge in low-power, intelligent agents (such as wearables, smartphones, and IoT devices), there exists a growing need for machine learning models to work well i... |
6 0
Hello!
I was not quite sure about posting in this category, but I think my question fits here. I am wondering about Maxwell equations in vacuum written with differential forms, namely: \begin{equation} \label{pippo} dF = 0 \qquad d \star F = 0 \end{equation} I know ##F## is a 2-form, and It can be written by using ... |
To explain what is a moderator, we start with a bivariate relationship between an input variable X and an outcome variable $Y$. For example, $X$ could be the number of training sessions (training intensity) and $Y$ could be math test score. We can hypothesize that there is a relationship between them such that the numb... |
We assume that signal $x(n)$ is known, that $\hat{x}(-1)=0$ and that there is not noise in the telecommunication channel.
I want to define the predicted signal $\tilde{x}(n)$, the difference signal $d(n)$, the quantized difference signal $\hat{d}(n)$, the encoded signal $c(n)$ and the reconstructed signal $\hat{x}(n)$
... |
Cuckoo hashing for sketching sets Sign up for the new posts via the RSS feed.
Below I show a neat application of perfect hashing, which is one of my favorite (cluster of) algorithms. Amazingly, we use it to obtain a purely information-theoretic (rather than algorithmic) statement.
Suppose we have a finite universe $U$ ... |
Laws of Motion Second Law of Motion: Momentum and Impulse According to second law of motion the rate of change of linear momentum is directly proportional to the external force applied on it. Newton’s second law of motion gives the formula of force f = ma. Second law implies that when a bigger force is applied on a bod... |
Consider a one-parameter family of plane curves defined by the equation
\[f\left( {x,y,C} \right) = 0,\]
where \(C\) is a parameter.
The envelope of this family of curves is a curve such that at each point it touches tangentially one of the curves of the family (Figure \(1\)).
The parametric equations of the envelope a... |
This answer says that $Y = zX$ is a simplest example of non-causal system because it corresponds to $y_n = x_{n+1}$ and current output depends on future input. Yet, it is causal because both $x_n$ and $y_n$ are 0 for $n < 0.$ How causal can be non-causal? I am struggling to resolve this absurd and want to ask about the... |
The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\mathbb{Z}/2)$$ is associated to the short exact sequence $$0\to\mathbb{Z}/2\to\mathbb{Z}/{2^{n+1}}\to\mathbb{Z}/{2^n}\to 0.... |
Tagged: group Problem 343
Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.
Let $\Aut(N)$ be the group of automorphisms of $G$.
Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.
Then prove that $N$ is contained in the center of $G$. Problem 332
Let $G=\GL(n, \R)$ be th... |
In the
Wikipedia article for the Digamma function one finds some identities due to Gauss. I've used the fourth of those from the section Some finite sums involving the digamma function to show if there were no mistakes that for $m>1$
$$\sum_{\substack{1\leq k\leq m-1 \\ (k,m)=1}}\sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\ri... |
Answer
$\displaystyle \frac{203}{3}=67\frac{2}{3}$ quarts
Work Step by Step
We know that we need to use $7\frac{1}{4}$ quarts of additive for each tank of fuel. Therefore, if we have $9\frac{1}{3}$ tanks, the amount of additive must be the product of these two numbers. First, we estimate the result by rounding to the n... |
Answer
To determine a fraction with an indicated denominator you firstly divide the indicated denominator by the denominator of the fraction. Secondly, you multiply the nominator with the resulting number to get the new numerator.
Work Step by Step
To determine a fraction with an indicated denominator you firstly divid... |
If the predominating mechanism is scattering with no absorption, we can define in a similar manner linear, atomic and mass scattering coefficients, using the symbol \(\sigma\) rather than \(\alpha\). For the physical distinction between absorption and scattering, see section 5.1. And if both absorption and scattering a... |
Yes, the two are intimately related. One way, as in QMechanic's answer, is via Wick rotations, but in general there is a lot more freedom once you allow integration contours to go over into the complex plane. In my area, strong field physics, the use of complex time to understand tunnelling problems is everyday bread a... |
There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. Many people seem to think it is a well defined integral, and is calculated as follows:
We will use regularization (we int... |
I would like to know if there is a rule to prove this. For example, if I use the distributive law I will get only $(A \lor A) \land (A \lor \neg B)$.
I find pictures are great for anything simple enough to use them, which this is.
Remember:
AND means the area taken up by both things. So the middle one is what is taken ... |
Tagged: normal subgroup If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order Problem 575
Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup. Namely, suppose that $G=S\sqcup H$, wh... |
"La variante di Lüneburg" and China's rice productionIn 1993, Paolo Maurensig, an Italian writer from Gorizia, wrote a novel entitled "La variante di Lüneburg" (Adelphi, 1995, pp. 164, ISBN 88-459-0984-0). The novel is set in Nazi Germany during World War II and the main theme is the game of chess.
At the beginning of ... |
Consider a differential equation of type
\[{P\left( {x,y} \right)dx + Q\left( {x,y} \right)dy }={ 0,}\]
where \(P\left( {x,y} \right)\) and \(Q\left( {x,y} \right)\) are functions of two variables \(x\) and \(y\) continuous in a certain region \(D.\) If
\[\frac{{\partial Q}}{{\partial x}} \ne \frac{{\partial P}}{{\part... |
I have a table where I want more than one row in some cells. I am using parbox to achieve this but this makes some items in cells left aligned and some not. How can you fix this to make them all aligned the same way?
\documentclass{beamer}\usepackage{multirow}\begin{document} \begin{frame}\frametitle{Test page} \resize... |
Fourier theory was initially invented to solve certain differential equations. Therefore, it is of no surprise that Fourier series are widely used for seeking solutions to various ordinary differential equations (ODEs) and partial differential equations (PDEs).
In this section, we consider applications of Fourier serie... |
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(e) I will work on this part in a little while. So far I think I have gotten all of the same solutions as Rong Wei.Added my solution below. For the case of the... |
000001340 001__ 1340
000001340 005__ 20190321073459.0
000001340 037__ $$aBELLE2-MTHESIS-2019-002
000001340 041__ $$aeng
000001340 100__ $$aNadia Toutounji
000001340 245__ $$aReconstruction Methods for Semi-leptonic Decays of B-mesons with the Belle II Experiment
000001340 260__ $$aSydney$$bThe University of Sydney$$c20... |
Normally it seems like this is solved by using Gauss's Law as follows:
$$ \int E * da= \frac {q_{enc}}{\epsilon_0} $$
And applying a geometric argument that says: because the charged sphere is uniform, we can take the ratio of the volume our Gaussian sphere as a function of some radius $r$ and divide it by the actual v... |
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Change to browse by: Bookmark(what is this?) Physics > Instrumentation and Detectors Title: A Long Baseline Neutrino Oscillation Experiment Using J-PARC Neutrino Beam and Hyper-Kamiokande
(Submitted on 15 Dec 2014 (v1), last revised 18 Jan 2015 (this version, v2))
Abstract: Hyper... |
Chapters Balbharati SSC Class 10 Mathematics 2 Chapter 7: Mensuration Chapter 7: Mensuration solutions [Pages 145 - 146]
Find the volume of a cone if the radius of its base is 1.5 cm and its perpendicular height is 5 cm.
Find the volume of a sphere of diameter 6 cm.
Find the total surface area of a cylinder if the radi... |
Hartshorne is the reference where you can find the following example which might be useful.I what follow everything is with
multiplicity. Now Alberto pointed out above the case of the divisor over $\mathbb{P}^1$ associated to its "tangent bundle": Two points over the sphere counted with multiplicity (from here though, ... |
Bégout, Pascal and Vargas, Ana (2007)
Mass Concentration Phenomena for the L2-Critical Nonlinear Schrödinger Equation. Transactions of the American Mathematical Society, vol. 359 (n° 11). pp. 5257-5282. This is the latest version of this item.
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Abstract
In this paper, we show that any sol... |
Why for the complex scalar field
$$ \hat\phi = \int \frac{d^3p}{(2\pi)^{3/2}(2E_{\vec{p}})^{1/2}}\left(\hat{a}_{\vec{p}}e^{-p \cdot x} + \hat{b}_{\vec{p}}^\dagger e^{p \cdot x}\right), $$
the commutation relation $[\hat\phi(x),\hat{\phi}^\dagger(y)]=0$, but using the non-relativistic limit for the fields $\phi(\vec{x},... |
The phase components of a signal that they are got from the Fourier Transform, are simply, the phase offset of each sinusiod. Which is the
$\phi$ in equation $1$. However, in case of Hilbert transform, the instantaneous phase is the argument of the sinusoid function. Which is $x(t)$ in equation $2$. I know that we can ... |
I updated to iOS 11, and after doing so, the display font for MathJax has changed across the entire interface of my phone. As evidence, here is a screenshot I found laying around with its updated copy:
Let's define three partitions $P,Q,R$ on $[a,c]$, $[a,b]$, $[b,c]$ respectively:
$$P=\{x_0,x_1,...,x_k,...,x_n\}$$ $$Q... |
Worst Case Efficient Single and Multiple String Matching in the RAM Model Abstract
In this paper, we explore worst-case solutions for the problems of pattern and multi-pattern matching on strings in the RAM model with word length
w. In the first problem, we have a pattern p of length m over an alphabet of size σ, and g... |
Tagged: symmetric matrix Problem 572
The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. Problem 7. Let $A=\begin{bmatrix} -3 & -4\\ ... |
I'm aware that there are a few similar questions already answered, but I could not find what I was looking for in any of then, so please bear with me :)
For a school project I need to find an angle ($\theta$) of maximum range ($R$) and its dependency on initial velocity $\theta(v_0)$, considering quadratic drag force a... |
I have a 1D gas made of $N$ particles placed in a harmonic potential well, so the Hamiltonian is:
$$ \mathcal H = \sum_{j=1}^N \left ( \frac{p_j^2}{2m} + \frac{1}{2}m\omega^2 x_j^2 \right )$$
The first part of the exercise asked me to find the canonical partition at temperature $T$ if the particles are distinguishable,... |
Thought I'd take a slightly different approach couching the above in terms of a first course in vector calculus.
Suppose you have a curve $\vec{r}(t)$ in $R^3$. It could be a straight line, a circle, a helix, cycloid, etc.
You can associate to any point on a well behaved curve a unit tangent vector $\hat{T}=\frac{\frac... |
I would like to start off by saying this is not a philosophical question. I have a specific question pertaining to physics after the following explanation and background information, which I felt was necessary to properly formulate my question.
I have done some research on Zeno's Paradoxes as well as some of their mode... |
If you are referring to the animated TV series, remember this is a cartoon, you cannot expect much realism. In reality, divers open access to the water like this would require the air pressure above the water, to equal the external water pressure, you cannot have two different pressures meeting with nothing to separate... |
(a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If... |
I stumbled upon this old question and I would like to share my solution. As mentioned in other answers, there is no analytical solution, but the function to be minimized behaves nicely and the optimal value of $\alpha$ can be found easily with a few Newton iterations. There is also a formula to check the optimality of ... |
As described in Chapter 1, classical mechanics is based on a set of axioms, which in turn are based on (repeated) physical observations. In order to formulate the first three axioms, we will need to first define three quantities: the (instantaneous) velocity, acceleration and momentum of a particle. If we denote the po... |
We have already written Neural Networks in Python in the previous chapters of our tutorial. We could train these networks, but we didn't explain the mechanism used for training. We used backpropagation without saying so. Backpropagation is a commonly used method for training artificial neural networks, especially deep ... |
Difference between revisions of "stat946w18/Implicit Causal Models for Genome-wide Association Studies"
(→Causal Inference with a Latent Confounder)
Line 201: Line 201:
Dustin Tran, Rajesh Ranganath, and David M Blei. Hierarchical implicit models and likelihood-free variational inference. In Neural Information Processi... |
Search
Now showing items 1-1 of 1
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 ... |
What is the "best LaTeX practices" for writing absolute value symbols? Are there any packages which provide good methods?
Some options include
|x| and
\mid x \mid, but I'm not sure which is best...
TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems... |
The position of a point \(M\left( {x,y,z} \right)\) in the \(xyz\)-space in cylindrical coordinates is defined by three numbers: \(\rho, \varphi, z,\) where \(\rho\) is the projection of the radius vector of the point \(M\) onto the \(xy\)-plane, \(\varphi\) is the angle formed by the projection of the radius vector wi... |
Two lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\sqrt{ (a_2^2 + b_2^2)}}$ But I intend to find which one is the obtuse angle bisector and which one is the acute angle bi... |
If your Lagrangian satisfies
$$ \frac{\partial \mathcal L}{\partial t} = 0 $$
then you're happy, energy is conserved, etc. However, if the above doesn't hold, that doesn't necessarily mean energy isn't conserved; maybe your Lagrangian has a false explicit time dependency. For example:
$$ \mathcal L = \frac{m}{2}\dot x^... |
Difference between revisions of "stat946w18/Implicit Causal Models for Genome-wide Association Studies"
(→Implicit causal model in Edward)
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== Implicit causal model in Edward ==
== Implicit causal model in Edward ==
−
[[File: coddde.png|
+
[[File: coddde.png||center]]
Revision as of 23:47, 20 April 2... |
Problem 75
Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.
(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$. (b... |
testing if system of inequalities has solution
Hi there,
I have a big system of inequalities (~1500 inequalities, 45 variables) and want to check, if there exists a real solution to it. Trying out the 'solve' and 'solve_ineq' takes a huge amount of time or it breaks during calculation and after checking some of the que... |
Search
Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Let $\Omega$ be a nice domain in $\Bbb R^n$. It is known that any element $T\in\left( W^{k,p}(\Omega)\right)^*$ admits a (possibly non-unique) representation of the form $$ Tu = \sum_{|a|\le k} \int_\Omega f_\alpha D^\alpha u\ dx, \tag{0} $$ where $f_\alpha \in L^{p'}$ and $\frac 1p + \frac 1{p'}=1$. The functional $T$... |
A variation of the Root of Unity problem.
I want to find all possible answers to this:
$$z^n = i$$
Where $$i^2 = -1$$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
If th... |
Let\[\mathbf{v}_{1}=\begin{bmatrix}1 \\ 1\end{bmatrix},\;\mathbf{v}_{2}=\begin{bmatrix}1 \\ -1\end{bmatrix}.\]Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$?
For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all fun... |
Here you can see that the transfer function applied to a cosine input will give you a sinusoid and a transient term:
$$ x(t) = \underbrace{(x(0) + x'(0))(2 e^{-t} - e^{-2t}) + \frac{2}{5} e^{-t} - \frac{1}{2}e^{-t}}_{{\rm goes\ to\ } 0 {\rm\ as\ } t \rightarrow \infty }\ \ \ + \frac{1}{10} \cos(t) + \frac{3}{10} \sin(t... |
Let $a \in R$. Verify that $(x − 1)^2$ is a factor of $$p(x) = x^4 − ax^2 + (2a − 4)x + (3 − a)$$
How can I solve this question?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this comm... |
It is easy to notice some odd things when looking into a fish tank. For example, you may see the same fish appearing to be in two different places (Figure \(\PageIndex{1}\)). This is because light coming from the fish to us changes direction when it leaves the tank, and in this case, it can travel two different paths t... |
I am having a lot of trouble starting this proof. I would greatly appreciate any help I can get here. Thanks.
Let $n\in \mathbb{N}$. Prove that any injective function from $\{1,2,\ldots,n\}$ to $\{1,2,\ldots,n\}$ is bijective.
Mathematics Stack Exchange is a question and answer site for people studying math at any leve... |
Let$\newcommand{\mM}{\mathcal{M}}$ $\mM_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:
$$\mM\rightarrow\mM_{1,1}[1/N]$$ Must the coarse moduli scheme $M$ of $\mM$ be smooth over $\mathbb{Z}[1/N]$? This is certa... |
As is pointed out in another answer you can numerically perform the computation and you will find that rays close to the principle axis all arrive approximately in phase at a point but the difference becomes larger for rays further away from the principal axis.
This defect of a lens is called spherical aberration.
Maki... |
Structure of Atom Bohr's Model of Atom and Bohr's Model for Hydrogen Atom
Bohr's Atomic Model : Angular momentum : mvr = \frac{nh}{2\pi} m = mass v = velocity of electron r = radius of orbit h = Planck's constant Radius of n th orbit
r_{n} = \frac{n^{2}h^{2}}{4\pi^{2}mZ_{e}^{2}}
e - Charge of nucleus
Z - Atomic no.of u... |
A good way to think about these equations is to imagine them as sets over valid words. Say I have an alphabet $\Sigma = \{a, b\}$, then the rule$$A \leftarrow A a \mid b$$
says that there's some set $A$ that is isomorphic to the set $(A * \{a\}) \cup \{b\}$. Here the $*$ operation is the product/concatenation operation... |
Cone Angle
When a sample is inserted at the focus of a convergent beam (at the focal plane), then biconical transmittance should be calculated:
\[T_b=\frac{total \; flux\; in \;output \;beam}{total\; flux\; in\; input \;beam}\].
The biconical transmittance \(T_b\) can be calculated as follows:
\[ T_b=\frac{\int_{\varph... |
Suppose I had the following periodic 1D advection problem:
$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$ $u(x,0)=g(x)$ where $g(x)$ has a jump discontinuity at $x^*\in (0,1)$.
It is my understanding that for linear finite difference schemes of higher than first o... |
Recall that we were able, in certain systems, to calculate the potential by integrating over the electric field. As you may already suspect, this means that we may calculate the electric field by taking derivatives of the potential, although going from a scalar to a vector quantity introduces some interesting wrinkles.... |
The slope at some point is \( y'=\frac{dy}{dx}=\tan\psi=\frac{s}{a}\) from which \( \frac{ds}{dx}\ =\ a\frac{d^{2}y}{dx^{2}}\). But, from the usual pythagorean relation between intrinsic and rectangular coordinates \( ds=(1+y'^{2})^{\frac{1}{2}}dx\), this becomes
\[ (1+y'^{2})^{\frac{1}{2}}=a\frac{dy'}{dx}. \label{18.3... |
Note
Much as in the case of linear momentum, the mistake that tends to be made in the case of angular momentum is not using the principle of conservation of angular momentum when it should be used, that is, applying conservation of mechanical energy in a case in which mechanical energy is not conserved but angular mome... |
Category: Group Theory
Group Theory Problems and Solutions.
Popular posts in Group Theory are:
Problem 625
Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.
(a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.
Add to solve later
(b) Pro... |
Does anyone here understand why he set the Velocity of Center Mass = 0 here? He keeps setting the Velocity of center mass , and acceleration of center mass(on other questions) to zero which i dont comprehend why?
@amanuel2 Yes, this is a conservation of momentum question. The initial momentum is zero, and since there a... |
As with all phenomena in classical mechanics, the motion of the particles in a wave, for instance the masses on springs in figure 9.1, are governed by Newton’s laws of motion and the various force laws. In this section we will use these laws to derive an equation of motion for the wave itself, which applies quite gener... |
Difference between revisions of "Probability Seminar"
(→Thursday, May 7, Jessica Lin, UW-Madison, Room TBA)
(→Thursday, May 7, Jessica Lin, UW-Madison, 2:25pm, Room TBA)
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== Thursday, May 7, [http://www.math.wisc.edu/~jessica/ Jessica Lin], [http://www.math.wisc.edu/ UW-Madison], <span style="color... |
I know that
$$ \omega = 2 \pi f $$
why do we use angular freq to expression about the instantaneous values of the wave?
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.Sign up to join this co... |
The uncertainty principle tells us that $$\sigma_x\sigma_p \geq \frac{\hbar}{2}, $$ which means that the more precisely we measure a particle's position, the more imprecise we will know its momentum. However, the standard deviation $\sigma_x$ can't be zero, and therefore its wave function will always have some spread t... |
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