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Show the $\sum\limits_{n = 1}^\infty x_n\sin(nx)$ converges uniformly
iff$nx_n →0$ as $n →\infty$, where $x_n$ is a decreasing sequence and with $x_n>0$ for all $n=1,2,\cdots$.
My attempt was with M-test. I took $M_n= nx_n$. Since $\sin(x)$ is bounded by $1$, I claim that $M_n >x_n\sin(nx)$ and we have $\sum M_n >\sum ... |
1. Let's begin by studying if $(a_n)$ is increasing or decreasing.
Note that $\forall n\in\Bbb N^*$, if $a_n\ne 0$:$$a_{n+1} - a_n = \dfrac{1}{a_n} - \dfrac{a_n}2 = \dfrac{2-a_n^2}{2a_n}\;\;\;(\star)$$
Thus, we have to see if this is positive or negative. To do this, we need to see if $a_n>0$ or not, and if $2-a_n^2\ge... |
Tagged: determinant of a matrix Problem 718
Let
\[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.
Compute the determinan... |
1)If (x+y)^2 = 31 and xy=6, what is the value of (x-y)^2?]
2)Compute the product \[\frac{(1998^2 - 1996^2)(1998^2 - 1995^2) \cdots (1998^2 - 0^2)}{(1997^2 - 1996^2)(1997^2 - 1995^2) \cdots (1997^2 - 0^2)}.\]
3)A square park measures 170 feet along each side. Two paved paths run from each corner to the opposite corner a... |
Electromagnetic Induction Faraday’s Law of Induction, Lenz’s Law and Conservation of Energy If the current flowing in the primary coil is ‘ip’, ‘φ s’ is the magnetic flux of the secondary coil, then φ s∝ i p⇒ φ s= Mi p⇒ \tt M = \frac{\phi_{s}}{i_{p}} Induced emf generated in secondary coil is \tt e = - \frac{d \phi}{dt... |
Before we even consider this, consider the kinetic energy of a relativistic particle:
$$E=mc^{2}\left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}-1\right)$$
This represents the amount of energy required to accelerate the particle from rest (with respect to a given reference frame) to the speed $v$. It should be imm... |
First, the string of minimum length might not be defined properly since it might not be unique.Here is a way to find a string of minimum length:Convert the regular expression to a nondeterministic finite automaton.Convert the nondeterministic automaton into a deterministic one.Use a breadth first search until you encou... |
Maybe I misunderstand your first question. The source is by definition a module of the vertex, namely the module $M$ for which $U | M^G$. Trivial source in this context means trivial module of the vertex $Q$.
Now for the second question. The easy part is why $k_Q|U_Q$ implies $k_{Q\cap H}|U_{Q_H}$. If you write out the... |
Consider a liner discrete-time system. Assume we can define it in terms of an input-output relation as follows (you can assume a more general model but it is enough for our purpose):
$$a_0y[n]+a_{1}y[n-1]+\cdots+a_{N}y[n-N]=b_0x[n]+b_{1}x[n-1]+\cdots+b_{M}x[n-M]\tag{1}$$
When the coefficients $\{a_i\}$ and $\{b_i\}$ ar... |
I am trying to solve the Diophantine equation:
$$ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $$
Here's what it looks like if you expand, it's variant of the Pythagorean triples:
$$ a \times (a-1) + b \times (b-1) = c \times (c-1) $$
I was able to find solutions by computer search but this could have also been checked u... |
This is not too obvious to me - what is the size of alternating group?
Following the hint in the comment, should it be $A_n = S_n/2$?
So I don't feel right up to here.....
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 ... |
What are some examples of complex functions with infinitely many complex zeros? There are no particular restrictions on the functions I am just curious and having a hard time finding examples. Also what can be said about a complex function with infinitely many complex zeros, must they have any special properties?
There... |
yxTtherm
The Swedish company, yxTherm, produces thermostats and wish to revise their established quality assurance parameters. Currently, the production process is defined as ‘controlled’ when a maximum of 10% of the produced units are defect.
On each production day, a randomly selected sample of 25 thermostats is carr... |
This is a follow up question to here.
The recommended solution doesn't work for me. I therefore let all my mathematical packages in the MWE. The result is an upright Psi followed by an italic Psi. I would like to have all upper-case Greeks being italic in formulas. The possibility to set one character upright (for a co... |
If \(f\left( x \right)\) is not continuous at \(x = a\), then \(f\left( x \right)\) is said to be discontinuous at this point. Figures \(1 – 4\) show the graphs of four functions, two of which are continuous at \(x =a\) and two are not.
Classification of Discontinuity Points
All discontinuity points are divided into di... |
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... |
NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more. Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free . Click here to join and contribute—free CZ thanks our previous donors. Donate here. Treasurer's Financi... |
Theorem. $\int_0^\infty \sin x \phantom. dx/x = \pi/2$.
Poof. For $x>0$ write $1/x = \int_0^\infty e^{-xt} \phantom. dt$,and deduce that $\int_0^\infty \sin x \phantom. dx/x$ is$$\int_0^\infty \sin x \int_0^\infty e^{-xt} \phantom. dt \phantom. dx= \int_0^\infty \left( \int_0^\infty e^{-tx} \sin x \phantom. dx \right)\... |
I was trying to perform the contour integral of the digamma function $\oint\limits_C \psi(z)\,dz$ on the neighborhood (a small circle $-k+re^{it}$, $k \in \mathbb{Z}$ ) of $k$, before actually realizing that due to the residue theorem $\operatorname{res}(\psi(z),-k)=\frac{1}{2\pi i}\oint\limits_C \psi(z)\,dz=-1$.
Now I... |
I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we can define $\psi(x) \equiv \langle x | \psi \rangle$, where $|x\rangle$ are eigenkets of position operator. So basicaly ... |
Answer
The judge should rule in favor of the police officer.
Work Step by Step
We use the equation for the velocity considering the Doppler effect: $v = \frac{c\Delta f}{2f} = \frac{(10.8\times10^8 \ km/h)(15.6\times10^3 \ Hz)}{2(70\times10^9 \ Hz)} = 120 \ km/h$ The judge should rule in favor of the police officer, fo... |
A sequence of real numbers \(n\) is a function \(f\left( n \right),\) whose domain is the set of positive integers. The values \({a_n} = f\left( n \right)\) taken by the function are called the terms of the sequence.
The set of values \({a_n} = f\left( n \right)\) is denoted by \(\left\{ {{a_n}} \right\}.\)
A sequence ... |
Custom software development is expensive, certainly if you include the cost of the need to maintain lines of code over time. A common strategy is therefore to reuse code that has been created already or share the development and maintenance costs between different parties.Different types of reuseReuse models can differ... |
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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 ... |
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Now showing items 1-10 of 26
Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
Given a sketch of a function $f(x)$ , how to sketch the graph of $y^2=f(x)$?
Is there any relationship between the features of two graphs?
Like vertical asymptotes or the zeros.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only ta... |
I'm trying to show how the discrete Fourier transform (DFT) arises from the equation for the continuous-time Fourier Transform. I've run into an interesting caveat which I can't seem to find an explanation to anywhere. My 'derivation' goes something like this:
Lets consider the continuous time Fourier transform equatio... |
Definition and Properties of the Matrix Exponential
Consider a square matrix \(A\) of size \(n \times n,\) elements of which may be either real or complex numbers. Since the matrix \(A\) is square, the operation of raising to a power is defined, i.e. we can calculate the matrices
\[
{{A^0} = I,\;\;{A^1} = A,\;\;}\kern-... |
It is very simple:
NSP says. Let us have a spherically symmetric density. Then
If it is concentrated as $\{r \ge R\}$ then inside of the cavity $\{r<R\}$ the pull is $0$ If it is concentrated as $\{r \le R\}$ then in $\{r>R\}$ the pull is as if it was created by the same mass but concentrated in the center
So, outer la... |
I am presently reading this [1] paper on covariant phase space and I have difficulty understanding the following formalism developed:
In the paper (section $2.2$, pg. $12$), the authors have introduced the notion of pre-phase space and go on to reinterpret differential forms by their functional counterpart. Instead of ... |
Volumetric thermal expansion coefficient, abbreviated as \(\alpha_v\) (Greek symbol alpha), also known as coefficient of volumetric thermal expansion, is the ratio of the change in size of a material to its change in temperature.
Volumetric Thermal Expansion Coefficient FORMULA
\(\large{ \alpha_v = \frac { 1 }{ V } \; ... |
It seems to me that the $V$ function can be easily expressed by the $Q$ function and thus the $V$ function seems to be superfluous to me. However, I'm new to reinforcement learning so I guess I got something wrong.
Definitions
Q- and V-learning are in the context of Markov Decision Processes. A
MDP is a 5-tuple $(S, A,... |
A first order differential equation \(y’ = f\left( {x,y} \right)\) is called a separable equation if the function \(f\left( {x,y} \right)\) can be factored into the product of two functions of \(x\) and \(y:\)
\[f\left( {x,y} \right) = p\left( x \right)h\left( y \right),\]
where \(p\left( x \right)\) and \(h\left( y \r... |
Gyroscope 1. Homework Statement
http://www.jyu.fi/kastdk/olympiads/2005/Th2.pdf [Broken]
http://www.jyu.fi/kastdk/olympiads/2005/Th2%20Solution.pdf [Broken]
(Question 3)
My first doubt is why the magnetic field at the center of the coil is:
[tex]|\vec B|=\frac{\mu_0 NI}{2a}[/tex] and not
[tex]|\vec B|=\mu_0 nI[/tex], w... |
1
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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... |
$\omega_1$ is generally referred to as an ordinal when it is rather more of a formula $\phi$ true for some sets in models of ZFC.
I started to wonder if $\omega_1$ really
was an ordinal, unlike $\beth_1$ (which is a cardinal in every model but isn't always the same cardinal; it is more easily described as a formula whe... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Nuclei Mass-Energy and Nuclear Binding Energy The mass of a nucleus is always less than the mass of constituent nucleons in their free state. This difference is called MASS DEFECT, (Δm). The energy equivalent to mass defect is called the BINDING ENERGY OF NUCLEUS, BE. BE = [ZM p+ (A − Z) M n– M] 931.5 MeV. Binding ener... |
I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process?
What
to compute the integral means is unclear in this context but one can say this: for every $t\geqslant0$, the random variable $$X_t=\int_0^tW_s\mathrm ds$$ is center... |
The answer to your second question is also no: only local properties can be characterised in the internal language of a topos, and it is possible to have locally constant (pre)sheaves that are not constant.
A simple, and well-known, example runs as follows: let $\mathbb C$ be the poset with four elements $n$, $e$, $s$,... |
I wonder if some one can help me with the solution to this question from Björk's "Arbitrage theory in continuous time":
At date of maturity $T_2$ the holder of a financial contract will obtain the amount: $$ \frac{1}{T_2 - T_1 } \int_{T_1}^{T_2} S(u) du $$ where $T_1$ is some time point before $T_2$. Determine the arbi... |
Tagged: abelian group Problem 307
Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order.
(a) Prove that $T(A)$ is a subgroup of $A$.
(The subgroup $T(A)$ is called the
torsion subgroup of the abelian group $A$ and elements of $T(A)$ are called torsion elements.)
Add to solv... |
(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'm a student, currently studying about quadratic forms over a field $\mathbb{R}$ and I have a few questions regarding the topic.
From a book I currently read, a quadratic formis a real-valued functionover a vector space $E$ (i.e. $Q: E \longrightarrow \mathbb{R}$, please correct me if I'm mistaken) such that there exi... |
When the physical dimension of a circuit approaches the magnitude of a wavelength of the signal, wires and circuit traces begin to affect circuit performance. This is due to the effects being frequency dependent and typically having very small values that are linearly dependent on wire/trace length. When the frequency ... |
The problem is this: Use the recursion-tree method to give a good asymptotic upper bound on $$ T(n) = 9T(\sqrt[3]n) + \Theta(1). $$ I am able to get the tree started and find a pattern with the sub-problems, but I am having difficulty finding the total cost of the running times throughout the tree. I cannot figure out ... |
Let $M$ be a symmetric square matrix with integer coefficients and $M_k$ the matrix obtained by deleting the k-th line and k-th column. If det(
M)=0 does it follow that $\det(M_kM_j)$ is a square?
Yes one gets the square of the determinant of the matrix where one deletes row j and column k. That follows from Dodgson's ... |
Let us start with an assumption that the radiation damping broadening is negligible, so that, for all practical purposes the spread of the frequencies emitted by a collection of atoms in a gas is infinitesimally narrow. The observer, however, will not see an infinitesimally thin line. This is because of the motion of t... |
Start with the unperturbed gravitational potential for a uniform sphere of mass M and radius R, interior and exterior:
$$ \phi^0_\mathrm{in} = {-3M \over 2R} + {M\over 2R^3} (x^2 + y^2 + z^2) $$$$ \phi^0_\mathrm{out} = {- M\over r} $$
Add a quadrupole perturbation, you get
$$ \phi_\mathrm{in} = \phi^0_\mathrm{in} + {\e... |
Colloquia for the Department of Mathematics and Statistics are normallyheld in
University Hall 4010 on Fridays at 4:00pm.Any departures from this are indicated below.
Light refreshments are served after the colloquia in 2040 University Hall.
Driving directions, parking information, and maps are available on the univers... |
This answer addresses the trivial part of the question, that is multiplicative stability, while providing
non-trivial examples of saturated sets $k(R)$ under two rather restrictive conditions: $R$ is a Dedekind domain or a polynomial ring over a unique factorization domain (UFD).
It is easy to show that $k(R)$ is multi... |
Let's take the usual definition of a spectral risk measure.
If we look at the integral we see that spectral risk measures have the property that the risk measure of a random variable $X$ can be represented by a combination of the quantiles of $X$.
Since the quantile function is rather friendly one gets that every spect... |
About this question,I found this from the Wikipedia:nuclear fission produces neutrons with a mean energy of 2 MeV (200 TJ/kg, i.e. 20,000 km/s), which qualifies as "fast". For my question, I only knew a few detail that converting Electron volt to Velocity unit,and electron volt is closely related with energy density, b... |
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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... |
The Church-Turing thesis is about (partial) functions $\mathbb{N} \to \mathbb{N}$ (or $\Sigma^* \to \Sigma^*$ for a finite alphabet $\Sigma$). How do you define a definite value based on some random output (for a given input)? There is at most one (random) output occurring with probability $>\frac{1}{2}$, so taking tha... |
Here's a neat one from optimization: the Alternating Direction Method of Multipliers (ADMM) algorithm.
Given an uncoupled and convex objective function of two variables (the variables themselves could be vectors) and a linear constraint coupling the two variables:
$$\min f_1(x_1) + f_2(x_2) $$$$ s.t. \; A_1 x_1 + A_2 x... |
I'm not sure specifically what you're looking for here. Noise is typically described via its power spectral density, or equivalently its autocorrelation function; the autocorrelation function of a random process and its PSD are a Fourier transform pair. White noise, for example, has an impulsive autocorrelation; this t... |
Discussing Stirling's approximation, Wolfram Mathworld article mentions a modification of it due to Gosper: instead of the usual
$$n!\approx n^ne^{-n}\sqrt{2n\pi},$$
we have a tiny addition of $\frac13$ to $2n$ under the radical sign:
$$n!\approx n^ne^{-n}\sqrt{\left(2n+\frac13\right)\pi}.$$
This radically improves the... |
Topological Methods in Nonlinear Analysis Topol. Methods Nonlinear Anal. Volume 29, Number 1 (2007), 181-198. On lifespan of solutions to the Einstein equations Abstract
We investigate the issue of existence of maximal solutions to the vacuum Einstein solutions for asymptotically flat spacetime. Solutions are establish... |
Tagged: invertible matrix Problem 583
Consider the $2\times 2$ complex matrix
\[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$.
Add to solve later
(d) Using the result of the diagonalization, c... |
Suppose we want to measure reasoning ability. For participant \(i\), the true reasoning ability score is \(T_{i}\) and the observed score is \(y_{i}\). If there is no measurement error, we would expect that
\[y_{i}=T_{i}.\]
However, often times, we can not perfectly measure something like reasoning ability because of m... |
Energy conservation does work perfectly in general relativity. The overall Lagrangian is invariant under time translations and Noether's Theorem can be used to derive a non-trivial and exact conserved current for energy. The only thing that makes general relativity a little different from electromagnetism is that the t... |
A note on a superlinear and periodic elliptic system in the whole space
1.
Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnanage, China
2.
Office of Adult Education, Simao Teacher's College, Simao 665000 Yunnan, China
3.
Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan
... |
Detailed error analysis for a fractional adams method with graded meshes AffiliationLvliang University; University of Chester; Publication Date2017-09-21
MetadataShow full item record AbstractWe consider a fractional Adams method for solving the nonlinear fractional differential equation $\, ^{C}_{0}D^{\alpha}_{t} y(t)... |
Why is there a need to perform holographic renormalization for the normal $AdS_5\times S^5$/CFT$_4$ correspondence if the brane theory is conformal? Since the flow along the AdS direction $r$ is related to the renormalization scale does this not explicitly introduce an "energy" parameter that breaks conformal invarianc... |
All celestial bodies have a gravitational well, and particle in the vicinity of this well would feel a gravitational force. My questions are:
a)How can I find the thickness of such a gravitational well?
b)Shouldn't the subatomic particles, e.g. An electron be confined to the potential well of a planet or a star?
c)Is q... |
4 Methods to Account for Radiation in Participating Media
Radiative heat transfer in semitransparent media is described by the radiative transfer equation (RTE). Solving this equation is challenging in terms of computational costs. However, depending on a medium’s radiation properties, simplifications exist that allow ... |
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Change to browse by: References & Citations Bookmark(what is this?) Computer Science > Computational Complexity Title: The phase transition in random regular exact cover
(Submitted on 26 Feb 2015 (v1), last revised 4 Mar 2015 (this version, v3))
Abstract: A $k$-uniform, $d$-regular instanc... |
1. Homework StatementI have phase space coordinates (x0,y0,z0,vx,vy,vz)=(1,0,0,0,1,0). I need to analytically show that these phase space coordinates correspond to a circular orbit.2. Homework Equationsr=sqrt(x^2+y^2+z^2) maybe?3. The Attempt at a SolutionMy core problem here is maybe...
A space vehicle enters the sens... |
I don't know if the derivation for that expression is in the book or not, but for clarity and completeness I'll briefly write it here:
For a single particle, the probability of it having velocity $v$ to $v+dv$ and angle $\theta$ to $\theta + d\theta$ is given by
$$ \int_\theta^{\theta+d\theta}g(\theta')d\theta'\int_v^{... |
Difference between revisions of "Discontinuous derivative"
Line 1: Line 1: −
Consider the function
+
Consider the function
:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto
:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto
\begin{cases}
\begin{cases}
Line 7: Line 7:
</math>
</math>
<math>f</math> is a continous and differe... |
A good-quality mirror may reflect more than 90% of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce total reflection using an aspect of
refraction.
Consider what happens when a ray of light strikes the ... |
In analogy with the definition of torque, \(\boldsymbol{\tau}=\boldsymbol{r} \times \boldsymbol{F}\) as the rotational counterpart of the force, we define the
angular momentum \(\boldsymbol{L}\) as the rotational counterpart of momentum: \[\boldsymbol{L}=\boldsymbol{r} \times \boldsymbol{p} \label{rp}\]
For a rigid bod... |
Ampere's law in magnetostatics is$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 \mu_r I_c ,$$where the left hand side is a closed line integral around a loop and the right hand side contains $I_c$, the total current passing through the same loop.
The symmetry that your Professor is talking about is the cylindrical symmetry wh... |
One of my discoveries as a physicist was that, despite all attempts at clarity, we still have different meanings for the same words and use different words to refer the the same thing. When Alice says measurement, Bob hears a `quantum to classical channel', but Alice, a hard-core Everettian, does not even believe such ... |
65 4 Homework Statement Two identical audio speakers, connected to the same amplifier, produce monochromatic sound waves with a frequency that can be varied between 300 and 600 Hz. The speed of the sound is 340 m/s. You find that, where you are standing, you hear minimum intensity sound a) Explain why you hear minimum-... |
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Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the definition from 2.7.6 $$L_{m}\sim \frac{1}{2}\sum_{n =-\infty}^{\infty} \alpha^{\mu}_{m-n} \alpha_{\mu n}. \tag{2.7.6}$$ Now, f... |
So, I was thinking about the Bohr model of atom and I started to wonder how we could find the magnetic field due to a revolving electron (produced at the location of proton) of hydrogen atom in first orbit. Example:- How to find the magnetic field due to a revolving electron of hydrogen atom in first orbit? Given h ~$ ... |
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... |
Thank you Kasper. I've build the github version 2.2.7, and soo that you improve my rusty version of the notebooks. Great work!
I'd like to mention that I try the following code:
{M,N,P,Q,J,K,L}::Indices(full, position=independent).
{\mu,\nu,\rho,\sigma,\gamma,\lambda}::Indices(sub,position=independent, parent=full).
e^... |
Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where $\Delta\colon \mathrm{Set}\to E$ constructs constant sheaves and its right adjoint Γ takes global sections. If E is locally con... |
Tagged: abelian group
Abelian Group Problems and Solutions.
The other popular topics in Group Theory are:
Problem 616
Suppose that $p$ is a prime number greater than $3$.
Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the mult... |
I have to the implementation of a Flanger effect on Matlab, buy previously I have to plot its frequency response and impulse response.
The difference equation is $y[n]=x[n]+a\cdot x[n-d[n]]$ where $a$ is a constant, $|a|<1$, and $d[n]=\frac D2 (1-\cos(2\pi f_s n))$; $D$ and $f \text{ const}$.
I'm having trouble on how ... |
Sakharov condition states
both C and CP violation is necessary for baryogenesis. Now consider, for example, a theory with B-number violating interaction and C-violation. Therefore, if $p\to e^+\gamma$ is an allowed B-violating process, C-violation would imply that the C-conjugated process $\bar{p}\to e^-\gamma$ would o... |
Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and sequences $(G_\alpha:\alpha<\omega)$ of filters such that the following happens:
Each $G_\alpha$ is a filter in the evaluation... |
Side of a regular polygon: \(a\)
Number of sides of a polygon: \(n\) Interior angle: \(\alpha\) Apothem: \(m\) Area: \(S\)
Number of sides of a polygon: \(n\)
Interior angle: \(\alpha\)
Apothem: \(m\)
Area: \(S\)
Radius of the inscribed circle: \(r\)
Radius of the circumscribed circle: \(R\) Perimeter: \(P\) Semiperime... |
Hyperparameter Tuning
HYperparameter tuning in Deep Learning: Learning rate $\alpha$, $\beta , \beta_1, \beta_2, \epsilon$, number of layers, number of hidden units, learning rate decay, mini-batch size
Some parameters are more important than the others. Learning Rate $\alpha$ Momentum term $\beta$, # hidden units, min... |
First of all, contrary to some sources, I claim that the $\text{ECTT}$ can absolutely be understood as a mathematical axiom, or at least as a mathematical proposition if we doubt its truth. Introduce into our working language a new predicate symbol defined on models of computation with the intended meaning that a model... |
Let $T: \R^n \to \R^m$ be a linear transformation.Suppose that the nullity of $T$ is zero.
If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.
Let $V... |
Number Theory Seminar: Tom Scanlon Date: 01/12/2012 Time: 15:00
University of British Columbia
p-independent bounds in the positive characteristic Mordell-Lang problem (part 1)
p-independent bounds in the positive characteristic Mordell-Lang problem (part 2)
AbstractThe usual Mordell-Lang conjecture, a theorem of Falti... |
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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 ... |
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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 ... |
Here is an exposition of the ``alternative hack'' to which David Speyer alluded. For concreteness, I have fixed a group; the argument will go through in general, of course.
Let $g = (12) \in S_3$, a transposition in the symmetric group on three elements. The conjugacy class of $g$ consists of all three transpositions. ... |
The Hamiltonian for a system of spinless fermions on a 1D chain (with chemical potential $\mu=0$) is given by $$ H=-\sum_j\left( c^\dagger_{j+1} c_j+h.c.\right)+\Delta \sum_j \left( c^\dagger_{j+1}c^\dagger_j+h.c.\right) $$ where $\Delta$ is some number. If we introduce $$ c_j=\frac{1}{\sqrt{N}}\sum_k e^{ikj}c_k$$ We o... |
(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... |
The recursion theorem in computability states that, for any computable map $f : \mathbb{N} \to \mathbb{N}$ there exists $n \in \mathbb{N}$ such that $\varphi_{f(n)} = \varphi_n$, where $\varphi$ is a standard enumeration of partial computable functions. The one given here is due to Rogers, there is another by Kleene (b... |
I'll give an example from politics. Let's say you have a legislative body, such as the House of Representatives in the U. S. Congress. Over a period of time, the members of the House will vote on many bills and thereby accrue a voting history. Let's encode votes as numbers: a vote for a bill is 1, a vote against a bill... |
The time dilation and length contraction relationships actually have very limited applicability. To use the time dilation relationship, one of the two observers must measure a proper time, where the two events occur at exactly the same spatial point. To use the length contraction relationship, one of the two observers ... |
Test your understanding of basic properties of matrix operations.
There are 10 True or False Quiz Problems.
These 10 problems are very common and essential.So make sure to understand these and don’t lose a point if any of these is your exam problems.(These are actual exam problems at the Ohio State University.)
You can... |
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