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QM Phase & Four-Vectors

1. May 2, 2009

referframe

The QM phase of a single particle traveling freely in 3 dimensions is (rp – Et), where r and p are the 3-D position and momentum vectors. This is also the dot product of the space-time four-vector (r,-t) with the Energy-Momentum four-vector (p,E)

Is this "coincidence" related to relativistic QM?

Thanks in advance.

Last edited: May 2, 2009
2. May 2, 2009

clem

No. It is not a coincidence. It is the case in any relativistic wave, including classical EM.

3. May 19, 2009

referframe

What classical EM equation(s) contain this term? I looked at the original version of Maxwell's Equations and could not see any connection. Thanks.

4. May 19, 2009

clem

A classical EM wave is $${\bf E}\exp[i({\bf k\cdot r}-\omega t)]$$.

5. May 20, 2009

naturale

It is a consequence of the fact that relativistic fields satisfies the relativistic wave equation which in the non-relativistic limit gives the Schrodinger eq, and of the de Broglie relation: E = \hbar \omega, p = \hbar k. As a consequence the phase of a field is Lorentz invariant and the four-vector (\omega, k) is covariant.

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# Exponential of an Operator

• Mar 1st 2013, 12:34 PM
Assassin0071
Exponential of an Operator
In class we recently learned that for a linear operator $T: V \rightarrow V$ and function $g(t) = a_0 + a_1t + \dots + a_nt^n$ one can define the operator $g(T) = a_0I + a_1T + \dots + a_nT^n$ (where $I$ is the identity transformation). We also recently learned about the exponential of a matrix. My question is that for a linear operator $T: V \rightarrow V$ can the operator $e^T$ be defined? (For example, like how $e^A$ is defined for a matrix $A$) (I tried searching for information on it but all I found was information on the exponential of a matrix). Thanks.
• Mar 3rd 2013, 11:29 AM
emakarov
Re: Exponential of an Operator
What is $e$ here and how is $e^A$ defined for a matrix A?
• Mar 6th 2013, 12:04 PM
Assassin0071
Re: Exponential of an Operator
$e$ here is the constant, $e = 2.718...$ (the one used for the exponential function $f(x) = e^x$.

And for a matrix $A$, $e^A$ is defined as follows:

$e^A = \sum_{n=0}^\infty \frac{A^n}{n!}$
• Mar 8th 2013, 11:21 AM
emakarov
Re: Exponential of an Operator
Quote:

Originally Posted by Assassin0071
And for a matrix $A$, $e^A$ is defined as follows:

$e^A = \sum_{n=0}^\infty \frac{A^n}{n!}$

Thanks.

Quote:

Originally Posted by Assassin0071
My question is that for a linear operator $T: V \rightarrow V$ can the operator $e^T$ be defined? (For example, like how $e^A$ is defined for a matrix $A$) (I tried searching for information on it but all I found was information on the exponential of a matrix).

Are you asking about operators on infinite-dimensional vector spaces? Because on a finite-dimensional space, every linear operator is given by a matrix, so the exponentiation of an operator is determined by the exponentiation of its matrix.


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# Applications of Countable Infinite Sets and Power Sets

What are the possible applications of Countable Infinite Sets and Power Sets in areas that are not strictly mathematical?

Also I want to know the significance they carry. What was not possible before the concepts of countability and power sets were introduced and what became possible afterwards? How did the introduction of these concepts change our thinking and outlook and the areas they affect?

Please explain in plain language, possibly with examples, as I am from a non-mathematical background. Thank you in advance.

PS: The applications need not be in strictly practical fields either (as Asaf Karagila keeps pointing there aren't any). Please help!

• My comments from the previous thread still apply. (For 10k users: here's a link.) – Asaf Karagila Jun 13 '14 at 11:45
• I can imagine a (mathematical) world in which all sets are countable. I can't imagine one in which all sets are uncountable. You want sets of 5 elements to be uncountable? or maybe you want there to be no such thing as sets of 5 elements? – Gerry Myerson Jun 13 '14 at 12:28
• @Gerry Myerson, I do not WANT them to be uncountable. I just wondered what if, just trying to get a better appreciation of countable infinite sets. And maybe I did not make it clear in my query, but I am talking in terms of infinite sets. Finite sets are definitely countable. – user51013 Jun 13 '14 at 16:31
• I'm glad we agree that finite sets are countable! But what about the set of natural numbers? Unless you have some weird definition of countable (or of infinite), that's going to be a countable infinite set. So are you asking for a system where there is no such thing as the set of natural numbers? – Gerry Myerson Jun 14 '14 at 4:40
• If your question is, "what changes were brought in our thinking after this distinction was made," then that should be the body of your question, and not "What would happen if all sets were uncountable?" Please rewrite your question so it makes sense, then go to the meta site to propose reopening the revised question. – Gerry Myerson Jun 14 '14 at 13:34

The notion of cardinality as formalized and explored by Cantor introduced a whole new set of questions about the reals that wouldn't have occurred to mathematicians before Cantor. Before Cantor, it was well-known that the reals and the rationals were very different infinite sets, but the difference was generally described by the fact that the reals have a completeness property (loosely, convergent sequences of reals converge to another real -- not so in the rationals, it isn't too difficult to construct a sequence of rationals that converges to $\sqrt{2}$). Once you can distinguish countable and uncountable sets (so after Cantor), the whole picture changes.

For example, in terms of the first-order theory of linear order types (in particular, ignoring algebraic properties), the reals can be completely characterized by the following properties:

1) uncountable

2) no endpoints

3) dense in itself (between any two distinct reals there is another real)

4) no uncountable strictly monotonic sequences

5) contains a countable dense subset (the rationals -- i.e. between any two distinct reals there is a rational)

(If you're wondering why completeness isn't on this list, completeness is a second-order property, quantifying over sets of reals -- "all sets which have an upper bound have a least upper bound.")

Note, though, how many of these properties rely on cardinality. The next interesting question is whether all of these properties are necessary. In particular, consider dropping the last property. Is there a linear order type which satisfies the first 4 properties but contains no subset order-isomorphic to the reals? The answer turns out to be yes -- first construct an uncountable tree with no uncountable branches or levels (actually a counterexample to what would be a natural generalization of Konig's lemma), then look at the lexicographic order on nodes of that tree. The result is a linear order as described above, known as an Aronszajn line (or Specker type). This whole construction can be done in ZFC.

There are other interesting questions here. Consider whether there are alternatives to the last property above. Perhaps it is enough to consider that the linear order has a countable chain condition -- i.e. there are no uncountable collections of pairwise disjoint open intervals. Does this give order types which contain a suborder isomorphic to the reals? This question turns out to be independent of ZFC -- the counterexample is called a Souslin line, and whether such a thing exists depends on the model of ZFC you are using (some have them, some don't).

Again, the point is that these questions about the reals, and possible alternatives to them, would not have occurred to anyone before the development of a formal definition of cardinality for infinite sets, in terms of bijections. Cantor's ideas brought on something of a crisis in our understanding of the reals, and the results (mostly independence results lately) continue to flood in.

• The five axioms you wrote do not "completely characterize the real numbers". They also characterize $\Bbb R\setminus\{0\}$ for example. – Asaf Karagila Jun 15 '14 at 6:59
• Also, I don't see how these are applications of uncountable sets to non-mathematical contexts. Which is what the question is all about. – Asaf Karagila Jun 15 '14 at 7:00
• $\Bbb R\setminus\{0\}$ contains a suborder isomorphic to the reals -- in fact, lots of them, like $\Bbb R^+$, $(0,1)$, etc.. That is the sense in which the axioms determine the reals as a first-order characterization of a linear order type. And any context where the reals are relevant, axiomatically reasonable alternatives like an Aronszajn line would also be relevant -- what is it about the reals that makes them the "right" place to talk about physics, and what about physics could rule out using an Aronszajn line instead? – user128390 Jun 15 '14 at 10:23
• I think that you don't quite understand what order isomorphic means. If an ordered set is isomorphic to the real numbers it must be complete. The irrational numbers are not a complete order. Therefore they are nor isomorphic, nor they contain a subset isomorphic to the real numbers. Moreover, when you say that an theory completely determines a structure it doesn't mean that everything other structure has an isomorphic substructure. It means there is exactly one model up to isomorphism. – Asaf Karagila Jun 15 '14 at 10:49
• If you're talking about first-order then you can't say "uncountable". Your first and fourth axioms are not first-order. Also "countable dense subset" is not something that you can express in first-order logic. So three of your five axioms are not first-order. And again, order types means order isomorphism the order types of $\Bbb{R,R\setminus\{0\},R\setminus Q}$ are pairwise different. All satisfy your axioms, and the last one doesn't have any subset isomorphic to $\Bbb R$ itself. – Asaf Karagila Jun 15 '14 at 13:38


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## When is the fundamental thermodynamic relation true?

dE=TdS-PdV is only applicable to closed systems.

To apply to open systems it must be amended to

$$dE = TdS - PdV + \sum {{\mu _i}} d{n_i}$$

Where the chemical potential $\mu$i is

$${\mu _i} = {\left( {\frac{{\partial E}}{{\partial {n_i}}}} \right)_{S,V,{n_{j\left( {j \ne i} \right)}}}}$$

This accounts for the energy interchanges brought about by the matter (mass) interchanges.


text:
# Differential Equation and Slope Field Questions.

by PFStudent
Tags: differential, equation, field, slope
P: 171
Hey,

1. The problem statement, all variables and given/known data.
I was reading through the Differential Equation portion of my textbook and didn't quite understand the following paragraph.
 Quote by From Textbook Slope Fields Solving a differential equation analytically can be difficult or even impossible. However, there is a graphical approach you can use to learn a lot about the solution of a differential equation. Consider a differential equation of the form $${{y}^{\prime}} = {{F}{(x, y)}}$$ At each point ${(x, y)}$ in the ${xy}$-plane where ${F}$ is defined, the differential equation determines the slope ${{y}^{\prime}} = {{F}{(x, y)}}$ of the solution at that point. If you draw a short line segment with slope ${{F}{(x, y)}}$ at selected points ${(x, y)}$ in the domain of ${F}$, then these line segments form a slope field or a direction field for the differential equation ${{y}^{\prime}} = {{F}{(x, y)}}$ Each line segment has the same slope as the solution curve through that point. A slope field shows the general shape of all the solutions.
The above paragraph seemed a little confusing since, conventionally,
$${y} = {f(x)}$$

Where,
$${{\frac {{d}^{}}{d{x}^{}}}{\Big[y\Big]}} = {{{f}^{\prime}}{(x)}}$$

so that it is understood that ${{{f}^{\prime}}{(x)}}$ like ${f(x)}$ is a function of only ${x}$. However, in the above paragraph it is insisted that ${{\frac {{d}^{}}{d{x}^{}}}{\Big[y\Big]}}$ must be a function of ${x}$ and ${y}$. In other words ${{y}^{\prime}} = {{F}{(x, y)}}$, why is that?

2. Relevant equations.
Knowledge Differential Equations and Slope Fields.

3. The attempt at a solution.
If we begin from the conventional notation that,
$${z} = {f(x, y)}$$

I run in to the problem that I don't know how to explicitly find ${{z}^{\prime}}$. Specifically, I don't know how to differentiate ${f(x, y)}$ with respect to ${x}$ and ${y}$ simultaneously. So that the derivative is actually a function of ${x}$ and ${y}$, like ${f(x, y)}$. How would I differentiate ${z}$ with respect to ${x}$ and ${y}$ simultaneously?

I note however, that in this particular case we're talking about ${{y}^{\prime}}$ as opposed to ${{z}^{\prime}}$.
Noting this I recall that we can rewrite,
$${\frac {dy}{dx}} = {{{f}^{\prime}}{(x)}}$$

as
$${dy} = {{{{f}^{\prime}}{(x)}}{dx}}$$

Where ${y}$ is found by integrating both sides of the above equation.

However, when I try to do this with the equation given,
$${{y}^{\prime}} = {F(x, y)}$$

Which can be rewritten as,
$${dy} = {F(x, y)dx}$$

and when integrated is,
$${y} = {{\int_{}^{}}{F(x, y)dx}}$$

I find that I do not know how to evaluate the RHS. How would I evaluate it?

Thanks,

-PFStudent
 P: 171 Hey, Still stuck on these questions, a little bit of help would be nice. Thanks, -PFStudent
P: 25
 Quote by PFStudent Hey, Still stuck on these questions, a little bit of help would be nice. Thanks, -PFStudent
slope fields are the lamest things ever, use maple or something

P: 171
Differential Equation and Slope Field Questions.

Hey,
 Quote by Altruist slope fields are the lamest things ever, use maple or something
A more substantial and detailed reply would help, anyone?

Thanks,

-PFStudent
P: 54
 Quote by PFStudent Hey, A more substantial and detailed reply would help, anyone? Thanks, -PFStudent
Slope fields are a very important tool in analyzing, understanding and using many differential equations most notably non-linear ones: it's one thing to solve it, quite another to intutively understand what's going on. Why does my removal of one measly fish out of my grandpa's pond cause the entire fish population to collapse? You need to get Blanchard, Devaney, and Hall, "Differential Equations". Spend some time with it, six weeks, and you'll be tops telling us what to do. Mathematica has some nice tools for creating phase portraits (slope fields) and especially ver 7 has what, a really nice function I don't remember the name for generating very nice pictures of slope fields. Start with some simple ones: coupled set of two linear equations and generate the slope field, then go to more complicated ones.
 P: 25 what text book are you using?

 Related Discussions Calculus & Beyond Homework 5 Differential Equations 1 Differential Equations 3 Calculus & Beyond Homework 2 Introductory Physics Homework 5


text:
## Zhao, Fukun

Compute Distance To:
 Author ID: zhao.fukun Published as: Zhao, Fukun; Zhao, Fu-Kun External Links: ResearchGate
 Documents Indexed: 66 Publications since 1993 Reviewing Activity: 9 Reviews Co-Authors: 28 Co-Authors with 59 Joint Publications 1,294 Co-Co-Authors
all top 5

### Co-Authors

 1 single-authored 14 Zhao, Leiga 11 Ding, Yanheng 8 Wu, Xian 7 Yu, Yuanyang 5 Chen, Jin 5 Yang, Minbo 5 Yang, Zhipeng 4 Gu, Guangze 4 Yang, Xian Yong 4 Zhang, Wei 3 Qin, Wenping 3 Wang, Wenbo 2 Liu, Xiangqing 2 Xia, Leran 2 Zhang, Rumei 2 Zhang, Wen 2 Zhao, Shunneng 1 Chen, Shaoxiong 1 Dai, Qi 1 Guo, Yuxia 1 He, Shuying 1 Lee, Cheng 1 Liu, Haidong 1 Liu, Huafeng 1 Liu, Xiaoqing 1 Shi, Junping 1 Tang, Xianhua 1 Yang, Zehen 1 Yao, Yuhua
all top 5

### Serials

 9 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 7 Journal of Mathematical Analysis and Applications 3 Journal of Mathematical Physics 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 Calculus of Variations and Partial Differential Equations 3 Discrete and Continuous Dynamical Systems 2 Applicable Analysis 2 Applied Mathematics and Computation 2 Journal of Differential Equations 2 Applied Mathematics Letters 2 Journal of Ningxia University. Natural Science Edition 2 Acta Analysis Functionalis Applicata. AAFA 2 Advanced Nonlinear Studies 1 Rocky Mountain Journal of Mathematics 1 Acta Mathematica Sinica 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Journal of Number Theory 1 Mathematische Nachrichten 1 Journal of Mathematics. Wuhan University 1 Acta Mathematica Hungarica 1 Bulletin of the Iranian Mathematical Society 1 Acta Mathematicae Applicatae Sinica. English Series 1 Optimization 1 Mathematical and Computer Modelling 1 Topological Methods in Nonlinear Analysis 1 Electronic Journal of Differential Equations (EJDE) 1 Congressus Numerantium 1 NoDEA. Nonlinear Differential Equations and Applications 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Abstract and Applied Analysis 1 Communications in Contemporary Mathematics 1 Communications on Pure and Applied Analysis 1 Acta Mathematica Sinica. Chinese Series 1 Science China. Mathematics 1 Scientia Sinica. Mathematica 1 Journal of Theoretical Biology 1 Advances in Nonlinear Analysis
all top 5

### Fields

 48 Partial differential equations (35-XX) 10 Global analysis, analysis on manifolds (58-XX) 9 Ordinary differential equations (34-XX) 9 Operator theory (47-XX) 8 Calculus of variations and optimal control; optimization (49-XX) 4 Dynamical systems and ergodic theory (37-XX) 3 General topology (54-XX) 2 Number theory (11-XX) 2 Functional analysis (46-XX) 2 Quantum theory (81-XX) 1 Real functions (26-XX) 1 Difference and functional equations (39-XX) 1 Probability theory and stochastic processes (60-XX) 1 Numerical analysis (65-XX) 1 Computer science (68-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Operations research, mathematical programming (90-XX) 1 Biology and other natural sciences (92-XX)

### Citations contained in zbMATH Open

53 Publications have been cited 831 times in 521 Documents Cited by Year
On the existence of solutions for the Schrödinger-Poisson equations. Zbl 1159.35017
Zhao, Leiga; Zhao, Fukun
2008
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. Zbl 1286.35103
Zhao, Leiga; Liu, Haidong; Zhao, Fukun
2013
Positive solutions for Schrödinger-Poisson equations with a critical exponent. Zbl 1156.35374
Zhao, Leiga; Zhao, Fukun
2009
Periodic solutions for a class of nonautonomous second-order systems. Zbl 1050.34062
Zhao, Fukun; Wu, Xian
2004
On Hamiltonian elliptic systems with periodic or non-periodic potentials. Zbl 1205.35080
Zhao, Fukun; Ding, Yanheng
2010
Multiple solutions for a superlinear and periodic elliptic system on $${\mathbb{R}^N}$$. Zbl 1279.35038
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2011
Multiple solutions for asymptotically linear elliptic systems. Zbl 1170.35384
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2008
Infinitely many solutions for a class of nonlinear Dirac equations without symmetry. Zbl 1152.35501
Zhao, Fukun; Ding, Yanheng
2009
Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. Zbl 1189.35091
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2010
The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system. Zbl 1437.35699
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2017
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media. Zbl 1332.35345
Zhao, Leiga; Zhao, Fukun; Shi, Junping
2015
Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Zbl 1087.34022
Zhao, Fukun; Wu, Xian
2005
Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. Zbl 1345.35031
Zhang, Jian; Qin, Wenping; Zhao, Fukun
2013
Semiclassical limits of ground state solutions to Schrödinger systems. Zbl 1310.35104
Ding, Yanheng; Lee, Cheng; Zhao, Fukun
2014
Existence and concentration of ground state solutions for a subcubic quasilinear problem via Pohozaev manifold. Zbl 1308.35106
Wang, Wenbo; Yang, Xianyong; Zhao, Fukun
2015
Existence and multiplicity of solutions for a non-periodic Schrödinger equation. Zbl 1159.35431
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2008
Saddle point reduction method for some non-autonomous second order systems. Zbl 1053.34039
Zhao, Fukun; Wu, Xian
2004
Existence and concentration of ground states to a quasilinear problem with competing potentials. Zbl 1288.35412
Wang, Wenbo; Yang, Xianyong; Zhao, Fukun
2014
Ground state solutions for superlinear elliptic systems on $$\mathbb R^N$$. Zbl 1266.35051
Xia, Leran; Zhang, Jian; Zhao, Fukun
2013
Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method. Zbl 1400.35127
Yang, Xianyong; Zhang, Wei; Zhao, Fukun
2018
Infinitely many radial and non-radial solutions to a quasilinear Schrödinger equation. Zbl 1309.35031
Yang, Xianyong; Wang, Wenbo; Zhao, Fukun
2015
A periodic solution for a second-order asymptotically linear Hamiltonian system. Zbl 1167.34345
Zhao, Fukun; Chen, Jin; Yang, Minbo
2009
Multiple solutions for a class of nonperiodic Dirac equations with vector potentials. Zbl 1253.81059
Zhang, Jian; Qin, Wenping; Zhao, Fukun
2012
The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth. Zbl 1391.35403
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2018
Multiple solutions for superlinear elliptic systems of Hamiltonian type. Zbl 1223.35149
Zhang, Rumei; Chen, Jin; Zhao, Fukun
2011
On a diffusion system with bounded potential. Zbl 1173.35522
Ding, Yanheng; Zhao, Fukun
2009
A note on superlinear Hamiltonian elliptic systems. Zbl 1304.35266
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2009
Infinitely many sign-changing solutions for a nonlocal problem. Zbl 1405.35248
Gu, Guangze; Zhang, Wei; Zhao, Fukun
2018
On ground state solutions for superlinear Hamiltonian elliptic systems. Zbl 1276.35076
Zhao, Leiga; Zhao, Fukun
2013
Concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system involving critical exponent. Zbl 1428.35427
Yang, Zhipeng; Yu, Yuanyang; Zhao, Fukun
2019
Existence of infinitely many solutions for quasilinear equations perturbed from symmetry. Zbl 1300.35046
Liu, Xiangqing; Zhao, Fukun
2013
New existence and multiplicity theorems of periodic solutions for non-autonomous second order Hamiltonian systems. Zbl 1173.35436
Wu, Xian; Chen, Shao-Xiong; Zhao, Fukun
2007
Positive and sign-changing least energy solutions for a fractional Schrödinger-Poisson system with critical exponent. Zbl 1448.35425
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2020
Multiple solutions for asymptotically quadratic and superquadratic elliptic system of Hamiltonian type. Zbl 1410.35032
Zhang, Wen; Zhang, Jian; Zhao, Fukun
2015
Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation. Zbl 1401.35070
Zhang, Jian; Zhang, Wen; Zhao, Fukun
2018
The concentration behavior of ground state solutions for a critical fractional Schrödinger-Poisson system. Zbl 1426.35126
Yang, Zhipeng; Yu, Yuanyang; Zhao, Fukun
2019
Multiple solutions for a semiclassical Schrödinger equation. Zbl 1241.35193
Zhang, Jian; Zhao, Fukun
2012
A note on a superlinear and periodic elliptic system in the whole space. Zbl 1237.35047
He, Shuying; Zhang, Rumei; Zhao, Fukun
2011
Nonlinear maps preserving product $$X^{*}Y+Y^{*}X$$ on von Neumann algebras. Zbl 07040690
Li, C.; Zhao, F.; Chen, Q.
2018
Infinitely many positive solutions for a nonlocal problem. Zbl 06892639
Gu, Guangze; Zhang, Wei; Zhao, Fukun
2018
Numerical characteristics of word frequencies and their application to dissimilarity measure for sequence comparison. Zbl 1405.92213
Dai, Qi; Liu, Xiaoqing; Yao, Yuhua; Zhao, Fukun
2011
Homoclinic orbits for a class of nonperiodic Hamiltonian systems. Zbl 1244.34070
Qin, Wenping; Zhang, Jian; Zhao, Fukun
2012
On the existence of solutions for Schrödinger-Maxwell systems in $$R^3$$. Zbl 1253.35166
Yang, Minbo; Zhao, Fukun; Ding, Yanheng
2012
The least energy sign-changing solution for a nonlocal problem. Zbl 1365.35207
Gu, Guangze; Yu, Yuanyang; Zhao, Fukun
2017
Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Zbl 1473.35306
Yang, Minbo; Zhao, Fukun; Zhao, Shunneng
2021
Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry. Zbl 1191.39011
Yang, Minbo; Zhao, Fukun; Ding, Yanheng
2010
Three solutions for a fractional Schrödinger equation with vanishing potentials. Zbl 1375.35621
Yang, Zhipeng; Zhao, Fukun
2018
Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Zbl 1466.35304
Yang, Zhipeng; Zhao, Fukun
2021
Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems. Zbl 1079.34519
Zhao, Fukun; Wu, Xian
2005
On multiplicity and concentration of solutions for a gauged nonlinear Schrödinger equation. Zbl 1448.35133
Zhang, Jian; Tang, Xianhua; Zhao, Fukun
2020
Positive solutions for quasilinear systems with critical growth. Zbl 1296.35059
Guo, Yuxia; Liu, Xiangqing; Zhao, Fukun
2013
Infinitely many solutions to quasilinear elliptic equation with concave and convex terms. Zbl 1365.35042
Xia, Leran; Yang, Minbo; Zhao, Fukun
2014
Existence and concentration results for fractional Schrödinger-Poisson system via penalization method. Zbl 1464.35077
Yang, Zhipeng; Zhang, Wei; Zhao, Fukun
2021
Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Zbl 1473.35306
Yang, Minbo; Zhao, Fukun; Zhao, Shunneng
2021
Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Zbl 1466.35304
Yang, Zhipeng; Zhao, Fukun
2021
Existence and concentration results for fractional Schrödinger-Poisson system via penalization method. Zbl 1464.35077
Yang, Zhipeng; Zhang, Wei; Zhao, Fukun
2021
Positive and sign-changing least energy solutions for a fractional Schrödinger-Poisson system with critical exponent. Zbl 1448.35425
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2020
On multiplicity and concentration of solutions for a gauged nonlinear Schrödinger equation. Zbl 1448.35133
Zhang, Jian; Tang, Xianhua; Zhao, Fukun
2020
Concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system involving critical exponent. Zbl 1428.35427
Yang, Zhipeng; Yu, Yuanyang; Zhao, Fukun
2019
The concentration behavior of ground state solutions for a critical fractional Schrödinger-Poisson system. Zbl 1426.35126
Yang, Zhipeng; Yu, Yuanyang; Zhao, Fukun
2019
Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method. Zbl 1400.35127
Yang, Xianyong; Zhang, Wei; Zhao, Fukun
2018
The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth. Zbl 1391.35403
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2018
Infinitely many sign-changing solutions for a nonlocal problem. Zbl 1405.35248
Gu, Guangze; Zhang, Wei; Zhao, Fukun
2018
Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation. Zbl 1401.35070
Zhang, Jian; Zhang, Wen; Zhao, Fukun
2018
Nonlinear maps preserving product $$X^{*}Y+Y^{*}X$$ on von Neumann algebras. Zbl 07040690
Li, C.; Zhao, F.; Chen, Q.
2018
Infinitely many positive solutions for a nonlocal problem. Zbl 06892639
Gu, Guangze; Zhang, Wei; Zhao, Fukun
2018
Three solutions for a fractional Schrödinger equation with vanishing potentials. Zbl 1375.35621
Yang, Zhipeng; Zhao, Fukun
2018
The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system. Zbl 1437.35699
Yu, Yuanyang; Zhao, Fukun; Zhao, Leiga
2017
The least energy sign-changing solution for a nonlocal problem. Zbl 1365.35207
Gu, Guangze; Yu, Yuanyang; Zhao, Fukun
2017
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media. Zbl 1332.35345
Zhao, Leiga; Zhao, Fukun; Shi, Junping
2015
Existence and concentration of ground state solutions for a subcubic quasilinear problem via Pohozaev manifold. Zbl 1308.35106
Wang, Wenbo; Yang, Xianyong; Zhao, Fukun
2015
Infinitely many radial and non-radial solutions to a quasilinear Schrödinger equation. Zbl 1309.35031
Yang, Xianyong; Wang, Wenbo; Zhao, Fukun
2015
Multiple solutions for asymptotically quadratic and superquadratic elliptic system of Hamiltonian type. Zbl 1410.35032
Zhang, Wen; Zhang, Jian; Zhao, Fukun
2015
Semiclassical limits of ground state solutions to Schrödinger systems. Zbl 1310.35104
Ding, Yanheng; Lee, Cheng; Zhao, Fukun
2014
Existence and concentration of ground states to a quasilinear problem with competing potentials. Zbl 1288.35412
Wang, Wenbo; Yang, Xianyong; Zhao, Fukun
2014
Infinitely many solutions to quasilinear elliptic equation with concave and convex terms. Zbl 1365.35042
Xia, Leran; Yang, Minbo; Zhao, Fukun
2014
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. Zbl 1286.35103
Zhao, Leiga; Liu, Haidong; Zhao, Fukun
2013
Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. Zbl 1345.35031
Zhang, Jian; Qin, Wenping; Zhao, Fukun
2013
Ground state solutions for superlinear elliptic systems on $$\mathbb R^N$$. Zbl 1266.35051
Xia, Leran; Zhang, Jian; Zhao, Fukun
2013
On ground state solutions for superlinear Hamiltonian elliptic systems. Zbl 1276.35076
Zhao, Leiga; Zhao, Fukun
2013
Existence of infinitely many solutions for quasilinear equations perturbed from symmetry. Zbl 1300.35046
Liu, Xiangqing; Zhao, Fukun
2013
Positive solutions for quasilinear systems with critical growth. Zbl 1296.35059
Guo, Yuxia; Liu, Xiangqing; Zhao, Fukun
2013
Multiple solutions for a class of nonperiodic Dirac equations with vector potentials. Zbl 1253.81059
Zhang, Jian; Qin, Wenping; Zhao, Fukun
2012
Multiple solutions for a semiclassical Schrödinger equation. Zbl 1241.35193
Zhang, Jian; Zhao, Fukun
2012
Homoclinic orbits for a class of nonperiodic Hamiltonian systems. Zbl 1244.34070
Qin, Wenping; Zhang, Jian; Zhao, Fukun
2012
On the existence of solutions for Schrödinger-Maxwell systems in $$R^3$$. Zbl 1253.35166
Yang, Minbo; Zhao, Fukun; Ding, Yanheng
2012
Multiple solutions for a superlinear and periodic elliptic system on $${\mathbb{R}^N}$$. Zbl 1279.35038
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2011
Multiple solutions for superlinear elliptic systems of Hamiltonian type. Zbl 1223.35149
Zhang, Rumei; Chen, Jin; Zhao, Fukun
2011
A note on a superlinear and periodic elliptic system in the whole space. Zbl 1237.35047
He, Shuying; Zhang, Rumei; Zhao, Fukun
2011
Numerical characteristics of word frequencies and their application to dissimilarity measure for sequence comparison. Zbl 1405.92213
Dai, Qi; Liu, Xiaoqing; Yao, Yuhua; Zhao, Fukun
2011
On Hamiltonian elliptic systems with periodic or non-periodic potentials. Zbl 1205.35080
Zhao, Fukun; Ding, Yanheng
2010
Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. Zbl 1189.35091
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2010
Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry. Zbl 1191.39011
Yang, Minbo; Zhao, Fukun; Ding, Yanheng
2010
Positive solutions for Schrödinger-Poisson equations with a critical exponent. Zbl 1156.35374
Zhao, Leiga; Zhao, Fukun
2009
Infinitely many solutions for a class of nonlinear Dirac equations without symmetry. Zbl 1152.35501
Zhao, Fukun; Ding, Yanheng
2009
A periodic solution for a second-order asymptotically linear Hamiltonian system. Zbl 1167.34345
Zhao, Fukun; Chen, Jin; Yang, Minbo
2009
On a diffusion system with bounded potential. Zbl 1173.35522
Ding, Yanheng; Zhao, Fukun
2009
A note on superlinear Hamiltonian elliptic systems. Zbl 1304.35266
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2009
On the existence of solutions for the Schrödinger-Poisson equations. Zbl 1159.35017
Zhao, Leiga; Zhao, Fukun
2008
Multiple solutions for asymptotically linear elliptic systems. Zbl 1170.35384
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2008
Existence and multiplicity of solutions for a non-periodic Schrödinger equation. Zbl 1159.35431
Zhao, Fukun; Zhao, Leiga; Ding, Yanheng
2008
New existence and multiplicity theorems of periodic solutions for non-autonomous second order Hamiltonian systems. Zbl 1173.35436
Wu, Xian; Chen, Shao-Xiong; Zhao, Fukun
2007
Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Zbl 1087.34022
Zhao, Fukun; Wu, Xian
2005
Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems. Zbl 1079.34519
Zhao, Fukun; Wu, Xian
2005
Periodic solutions for a class of nonautonomous second-order systems. Zbl 1050.34062
Zhao, Fukun; Wu, Xian
2004
Saddle point reduction method for some non-autonomous second order systems. Zbl 1053.34039
Zhao, Fukun; Wu, Xian
2004
all top 5

### Cited by 480 Authors

 65 Tang, Xianhua 26 Zhang, Wen 25 Chen, Haibo 21 Wu, Tsungfang 21 Zhang, Jian 20 Zhang, Fubao 18 Sun, Juntao 18 Zhao, Fukun 17 Chen, Sitong 17 Wang, Jun 12 Wu, Xian 11 Ding, Yanheng 11 Xu, Junxiang 11 Yu, Yuanyang 10 Wang, Wenbo 10 Zhao, Leiga 8 Che, Guofeng 8 Gu, Guangze 8 Liu, Zhisu 7 Chen, Jianqing 7 Ma, Shiwang 7 Qin, Dongdong 7 Shi, Junping 7 Teng, Kaimin 7 Zhang, Hui 6 Chen, Yongpeng 6 Du, Miao 6 Li, Fuyi 6 Li, Quanqing 6 Li, Yongkun 6 Shen, Liejun 6 Wang, Dabin 6 Yang, Minbo 6 Yang, Zhipeng 6 Zhang, Jianjun 6 Zhang, Xingyong 5 Alves, Claudianor Oliveira 5 Chen, Jianhua 5 Chen, Shaowei 5 Feng, Xiaojing 5 Guan, Wen 5 Guo, Shangjiang 5 He, Xiaoming 5 Huang, Lirong 5 Liang, Zhanping 5 Liao, Fangfang 5 Mao, Anmin 5 Su, Jiabao 5 Tian, Lixin 5 Wu, Ke 5 Yang, Xian Yong 5 Zhang, Liang 4 Ambrosio, Vincenzo 4 Chen, Peng 4 Chen, Yi 4 d’Avenia, Pietro 4 Fang, Xiangdong 4 Guo, Fei 4 Huang, Wennian 4 Lei, Chunyu 4 Li, Yuhua 4 Lin, Xiaoyan 4 Liu, Hongliang 4 Liu, Xiaoying 4 Rocha, Eugénio A. M. 4 Shi, Hongxia 4 Tang, Chun-Lei 4 Wu, Yuanze 4 Zhang, Huabo 4 Zhou, Huan-Song 3 Agarwal, Ravi P. 3 Azzollini, Antonio 3 Benhassine, Abderrazek 3 Chen, Shangjie 3 Cheng, Bitao 3 Du, Yao 3 Duan, Shengzhong 3 Feng, Zhaosheng 3 Figueiredo, Giovany Malcher 3 Gao, Zu 3 Guo, Qi 3 Han, Zhiqing 3 He, Xiumei 3 He, Yubo 3 Huang, Yisheng 3 Jeanjean, Louis 3 Jia, Gao 3 Jia, HuiFang 3 Jiang, Yongsheng 3 Li, Gongbao 3 Liu, Zeng 3 Luo, Xiao 3 Meng, Qiong 3 Paşca, Daniel 3 Pomponio, Alessio 3 Qian, Aixia 3 Shan, Yuan 3 Suo, Hong-Min 3 Tang, Zhongwei 3 Wang, Lixia ...and 380 more Authors
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### Cited in 98 Serials

 48 Journal of Mathematical Analysis and Applications 42 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 29 ZAMP. Zeitschrift für angewandte Mathematik und Physik 28 Journal of Differential Equations 24 Boundary Value Problems 22 Journal of Mathematical Physics 19 Communications on Pure and Applied Analysis 18 Applied Mathematics Letters 16 Computers & Mathematics with Applications 13 Nonlinear Analysis. Real World Applications 13 Advances in Nonlinear Analysis 11 Applicable Analysis 11 Calculus of Variations and Partial Differential Equations 11 Complex Variables and Elliptic Equations 10 Mathematical Methods in the Applied Sciences 10 Communications in Contemporary Mathematics 10 Journal of Applied Mathematics and Computing 9 Applied Mathematics and Computation 9 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 8 Discrete and Continuous Dynamical Systems 7 Topological Methods in Nonlinear Analysis 7 Mediterranean Journal of Mathematics 6 Annali di Matematica Pura ed Applicata. Serie Quarta 6 Abstract and Applied Analysis 5 The Journal of Geometric Analysis 5 Advances in Mathematical Physics 4 Applications of Mathematics 4 NoDEA. Nonlinear Differential Equations and Applications 4 Science China. Mathematics 4 Journal of Applied Analysis and Computation 4 Journal of Function Spaces 4 Open Mathematics 4 AIMS Mathematics 4 Electronic Research Archive 3 Electronic Journal of Differential Equations (EJDE) 3 Advanced Nonlinear Studies 3 Advances in Difference Equations 3 Frontiers of Mathematics in China 3 Discrete and Continuous Dynamical Systems. Series S 3 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 3 Journal of Theoretical Biology 2 Indian Journal of Pure & Applied Mathematics 2 Nonlinearity 2 Chaos, Solitons and Fractals 2 Glasgow Mathematical Journal 2 Asymptotic Analysis 2 Communications in Partial Differential Equations 2 Taiwanese Journal of Mathematics 2 Discrete Dynamics in Nature and Society 2 Acta Mathematica Sinica. English Series 2 Discrete and Continuous Dynamical Systems. Series B 2 Acta Mathematica Scientia. Series B. (English Edition) 2 Asian-European Journal of Mathematics 2 Analysis and Mathematical Physics 2 SN Partial Differential Equations and Applications 1 Archive for Rational Mechanics and Analysis 1 Physica A 1 Rocky Mountain Journal of Mathematics 1 Reviews in Mathematical Physics 1 Applied Mathematics and Optimization 1 Archiv der Mathematik 1 Journal of the Korean Mathematical Society 1 Manuscripta Mathematica 1 Mathematische Nachrichten 1 Monatshefte für Mathematik 1 Pacific Journal of Mathematics 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Quaestiones Mathematicae 1 Results in Mathematics 1 Transactions of the American Mathematical Society 1 Stochastic Analysis and Applications 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Acta Mathematicae Applicatae Sinica. English Series 1 Optimization 1 Revista Matemática Iberoamericana 1 International Journal of Mathematics 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Potential Analysis 1 Turkish Journal of Mathematics 1 Opuscula Mathematica 1 Journal of Difference Equations and Applications 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Positivity 1 Annales Henri Poincaré 1 Qualitative Theory of Dynamical Systems 1 Differential Equations 1 Nonlinear Analysis. Modelling and Control 1 Journal of Applied Mathematics 1 Central European Journal of Mathematics 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Analysis and Applications (Singapore) 1 Communications in Mathematical Analysis 1 Journal of Fixed Point Theory and Applications 1 Mathematical Geosciences 1 Journal of Nonlinear Science and Applications 1 Journal of Pseudo-Differential Operators and Applications
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### Cited in 27 Fields

 454 Partial differential equations (35-XX) 57 Ordinary differential equations (34-XX) 53 Global analysis, analysis on manifolds (58-XX) 40 Dynamical systems and ergodic theory (37-XX) 29 Quantum theory (81-XX) 28 Calculus of variations and optimal control; optimization (49-XX) 27 Operator theory (47-XX) 9 Optics, electromagnetic theory (78-XX) 6 Difference and functional equations (39-XX) 6 Mechanics of particles and systems (70-XX) 5 Functional analysis (46-XX) 4 Real functions (26-XX) 4 Biology and other natural sciences (92-XX) 3 Differential geometry (53-XX) 3 Mechanics of deformable solids (74-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 1 Category theory; homological algebra (18-XX) 1 Integral equations (45-XX) 1 Algebraic topology (55-XX) 1 Probability theory and stochastic processes (60-XX) 1 Statistics (62-XX) 1 Numerical analysis (65-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX) 1 Geophysics (86-XX) 1 Systems theory; control (93-XX)


text:
Tag Info

10

First proof Suppose $\mathbb{Z}^n\cong\mathbb{Z}^m$, with $m\ne n$. Then you can find a set of $m$ free generators in $\mathbb{Z}^n$, say $\{f_1,f_2,\dots,f_m\}$. Then, for some integers $a_{ij}$, $b_{ij}$, you have \begin{align} f_j&=\sum_{i=1}^n a_{ij} e_i &&j=1,\dots, m\\ e_i&=\sum_{j=1}^m b_{ij} f_j &&i=1,\dots, n \end{align} ...

7

$K(\alpha, \beta)$ is Galois over $K$ and the corresponding Galois group is a subgroup of $\text{Gal}(K(\alpha)/K)\times \text{Gal}(K(\beta)/K)$. So $K(\alpha, \beta)/K$ is abelian. Thus, any intermediate Galois extension must be abelian since the corresponding Galois group would be a quotient of $\text{Gal}(K(\alpha,\beta)/K)$

6

$\mathbb{Z}^n \cong \mathbb{Z}^m$ implies $(\mathbb{Z}/2)^n \cong \mathbb{Z}^n / 2 \mathbb{Z}^n \cong \mathbb{Z}^m / 2 \mathbb{Z}^m \cong (\mathbb{Z}/2)^m$. By comparing the number of elements, we get $2^n=2^m$, i.e. $n=m$. (No linear algebra is necessary here!)

3

Whenever you have a condition $g^2=e$ in a group, it's equivalent to $g=g^{-1}$ (multiply both sides by $g^{-1}$). In this case, it applies to every element of the group, so you can add or remove inverses from any expression freely. So the proof is simply $$ab=(ab)^{-1}=b^{-1}a^{-1}=ba.$$

3

This is known as Smith normal form, and you can find lots of references on Google books. In particular, this textbook has exhaustive coverage of Smith normal form and all of its consequences.

3

You can simplify things a lot by noting that $|a|$ divides $K$ iff $a^K=1$. Using this, verifying that $H$ is a subgroup is easy. If you can use homomorphisms, then it is easier because $H=\ker\phi$, where $\phi(x)=x^K$. Since $G$ is Abelian, $\phi$ is a homomorphism.

2

A way that doesn't use group actions (which are awesome, by the way, so try to use them when you can): We want to know the order of the centralizer of $g,\ |C(g)|$. Now $x \in C(g)$ means that $xgx^{-1} = g$. Since $x(1\ 2\ 3)x^{-1} = (x(1)\ x(2)\ x(3))$ we can immediately see two things: $x$ must map $\{1,2,3\} \to \{1,2,3\}$ and thus $\{4,5\} \to ... 2 Let$G$acts on itself by conjugation. Consider the orbit of$g$, it's the conjugacy class of$g$in$S_{5}$, so the set of$3$cycles. Now,$H$is the stabilizer of$g$. If$x \in H$, then$xg=gx$so$xgx^{-1}=g$. And if$x \in Stab(g)$, then$xgx^{-1}=g$so$xg=gx$so$x \in H$. Now the orbit stabilizer theorem tells us that$|G|=|Orb(g)||Stab(g)|$, so ... 2 use the fundamental theorem of finitely generated Abelian groups: The invarint factor version. G is nothing but a k finite direct sum of Z_n's ordered in an ascending order of divisibility of the n's. Then certainly there is a summant Z_n. Now take a k-tuple with all entries 0 except one entry corresponding to Z_n in which it is 1. 2 The statement you want to prove is false in general. Let$k=1$for example, and let $$G = \langle x,y,z \mid x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx\rangle$$ be the group found in this answer. Then$G$is not abelian, but$G$has exponent three. It thus holds that$(ab)^3 = e = a^3 b^3$for all$a,b \in G$, and moreover it obviously holds that ... 1 Doesn't the first nontrivial case, with$k=2$,$m_1=2$, and$m_2=3$, give a counterexample? EDIT: User64726 has pointed out that this counterexample doesn't work if one allows$1$to be among the$d_i$'s. Under that convention, the problem becomes whether we can always take the number of nontrivial$d$'s to be$\leq k$, where$k$is the number of$m$'s; if ... 1 @Egreg's answer above is very complete and concise, but I thought I'd add another answer just for the heck of it. Suppose toward a contradiction that there exist distinct natural numbers$m,n\in\Bbb{N}$with$\Bbb{Z}^n=\Bbb{Z}^m$. Let$m\in\Bbb{N}$the least natural number for which there exists$n\in\Bbb{N}$with$m\neq n$such that ... 1 Every group homomorphism$\phi: \mathbb Z^n \to \mathbb Z^m$extends (uniquely) to a linear transformation$T: \mathbb Q^n \to \mathbb Q^m$of vector spaces over$\mathbb Q$. Moreover,$\phi$is injective iff$T$is injective. But$T$injective implies$n \le m$. Applying this to both$\phi$and$\phi^{-1}$, we conclude that$n \le m \le n$. 1 You want to prove that the map $$x \mapsto (\chi \mapsto \chi(x))$$ is injective. I assume that you have already proved that this is a homomorphism. So all you need to show is that the kernel of this map is trivial. That is, you want to show that $$\ker (x \mapsto (\chi \mapsto \chi(x))) = \{e\}$$ where$e$is the identity in$G$. Now the kernel is ... 1 Given that$(a\cdot b)^{n+2}=a^{n+2}\cdot b^{n+2}$we can multiply by$a^{-1}$and$b^{-1}$on the left and right to get $$(b\cdot a)^{n+1}=a^{n+1}\cdot b^{n+1}=(a\cdot b)^{n+1},$$ the latter equality holding by assumption. Repeating this trick we get $$(b\cdot a)^n=a^n\cdot b^n=(a\cdot b)^n,$$ and dividing these two left hand sides and right hand sides of ... 1 All you have done is correct. However it seems you made some claims which at first sight are correct, but in fact they could be based on some confusion. One of them is the following: when$G=\mathbb Z/p^{\alpha}\mathbb Z\times\mathbb Z/p^{\beta}\mathbb Z$you said that$\mathbb Z/p\mathbb Z\subset G$. This is right up to isomorphism: we identify$\mathbb ...

1

The map $f:G \to G$ given by \begin{align*} f(g_i) &= \begin{cases} g_i & \text{if}\;\; i \leq r; \\ 0 & \text{if}\;\; i > r \end{cases} \end{align*} has kernel $\langle{g_{r+1}, \dots, g_k\rangle} = H$ (since $G$ is free). Hence $G/H = f(G) = \langle g_1, \dots, g_r\rangle\subset G$.

1

You can take $v_1=(1,4)$, $k_1=1$, $v_2=(0,1)$, $k_2=7$.

1

Another way to see that $|zw| = |z||w|$: Let $z = a+ib$, and $w = c + id$. Then: $|zw| = |(a+ib)(c+id)| = |(ac-bd) + i(ad+bc)| = \sqrt{(ac-bd)^2 + (ad+bc)^2}$ $= \sqrt{a^2c^2 - 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2}$ $= \sqrt{a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2} = \sqrt{a^2(c^2 + d^2) + b^2(c^2 + d^2)}$ $= \sqrt{(a^2 + b^2)(c^2 + d^2)} = ... 1$\mathbb C^{\times}$is an Abelian group. The map$z \mapsto |z|$is a group homomorphism$\mathbb C^{\times} \to \mathbb R^{\times}$. The circle is the kernel of this map and so is a subgroup of$\mathbb C^{\times}$. The crucial point is that$|zw|=|z|\ |w|$and this can be proved as follows:$$|zw|^2=(zw)(\overline{zw})=z w \bar z \bar w=z \bar z w \bar ... 1 Hint: if$z\in G$, there exists$\varphi\in(-\pi,\pi]$with$z=e^{i\varphi}$. (Of course one can choose$\varphi \in [0,2\pi)$or$\varphi \in I$for any half closed intervall$I$of length$2\pi\$).

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Formatting chapter in memoir

I'm trying to reproduce this chapter heading (Styling my section/chapter) in memoir. This is more or less working with the following code:

\documentclass[11pt,oneside]{memoir}
\usepackage{blindtext}

\makechapterstyle{uno}{%
\setlength{\midchapskip}{5pt}
\renewcommand{\chapnumfont}{\Huge\bfseries}
\renewcommand{\printchaptername}{}%
\renewcommand{\printchapternum}{}%
\renewcommand{\chaptitlefont}{\Huge\bfseries\scshape}
\renewcommand{\printchaptertitle}[1]{%
\hspace*{-0.1\textwidth}\chaptitlefont ##1}
\renewcommand{\afterchaptertitle}{%
\vskip -2pt \hspace*{-\marginparwidth}\rule{0.7\textwidth}{5pt}}
}

\chapterstyle{uno}

\begin{document}
\chapter{Es war einmal ein Mann}
\blindtext[1]
\end{document}

But my question: Why does the following code:

\renewcommand{\afterchaptertitle}{%
\vskip\midchapskip\hspace*{-\marginparwidth}\rule{0.7\textwidth}{5pt}}
}

leave such a large separation to the rule? As far as I understand, \vskip is creating a new paragraph. But if I use

\renewcommand{\afterchaptertitle}{%
\vskip\midchapskip\hrule}
}

there is no separation at all (only the 5pt of \midchapskip). \rule and \hrule are behaving differently, but I can not understand why and I did not see a good explanation for this.

\hrule is a vertical mode command so in vertical mode stacks directly under the previous thing.

The LaTeX \rule starts with \leavevmode and so is a horizontal mode command and so if used in vertical mode you will get a vertical \parskip space and a horizontal \parindent.

You will see the same behaviour if you compare \hbox{hello} with \mbox[hello}

(Also in your example you have \hspace* in the \rule example which similarly is a horizontal command.)

Perhaps this is closer to your needs

 \renewcommand{\afterchaptertitle}{{%
\par\offinterlineskip\hbox{\hspace*{-\marginparwidth}\rule{0.7\textwidth}{5pt}}}}

• Many thanks for your answer. I'm reading right now doc about the two TEX modes in order to understand it. Where can I see the definition of \rule? – rogorido Apr 11 '12 at 17:06


text:
# tilde

0th

Percentile

##### Tilde Operator

Tilde is used to separate the left- and right-hand sides in a model formula.

Keywords
models
##### Usage
y ~ model
##### Arguments
y, model
symbolic expressions.
##### Details

The left-hand side is optional, and one-sided formulae are used in some contexts.

A formula has mode call. It can be subsetted by [[: the components are ~, the left-hand side (if present) and the right-hand side in that order.

##### References

Chambers, J. M. and Hastie, T. J. (1992) Statistical models. Chapter 2 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

formula


text:
# Hilbert transform

Where $z(\omega)$ is the complex coefficient (setting the amplitude and phase) of the positive-frequency complex sinusoid $\exp(j\omega t)$ at frequency. The hilbert transform is a time-domain to time-domain transformation which shifts the phase of a signal by 90 degrees positive frequency components are. In this video you will learn about the hilbert transform, which can be used to compute the analytic signal (a complex time series from which instantaneous. The hilbert transform is defined via the convolution with pv(1/x), namely before we proceed to the proof, we first give a few useful properties of the hilbert.

It is also shown that the fourier transform of the one-sided autocorrelogram of the response implicitly involves a hilbert transform the hilbert transform can be. Hilbert transform ang man shun 2012-11-13 reference hwei hsu , analog and digital communications 1 the hilbert transform ̂x(t) = h{x(t)} = x(t) ∗ 1 πt. In both hilbert transform (ht) and emd end effects end effects are problems for any data analysis method traditionally, window with tapered ends were used .

Fourier transform of the strong analytic signal gives us a one-sided spectrum in the frequency domain it is not hard to see that a function and its hilbert transform. Abstract: we study analogues of classical hilbert transforms as fourier multipliers on free groups we prove their complete boundedness on non. Hilbert transform (inttrans package) introduction the hilbert transform, sometimes called a quadrature filter , is useful in radar systems, single side-band . Computing the hilbert transform and its inverse sheehan olver abstract we construct a new method for approximating hilbert transforms and their inverse.

Compute the analytic signal, using the hilbert transform the transformation is scipyfftpackhilbert: return hilbert transform of a periodic sequence x notes. The hilbert transform of g(t) is the convolution of g(t) with the signal 1/πt it is the response to g(t) of a linear time-invariant filter (called a hilbert transformer). The integral is an extension of the riemann definition of integral, called the cauchy principal value as we will see in chapter 3, the hilbert transform arises from.

The hilbert transform of a real‐valued time‐domain signal x(t) is another real‐ valued time‐domain signal, denoted by x(t), such that z(t) = x(t) +. Guide: mathematics of the discrete fourier transform (dft) - julius o smith iii the analytic signal and hilbert transform filters. We study fourier multipliers of hilbert transform type on free groups we prove that they are completely bounded on noncommutative l p -spaces associated. The analytic signal produced by the hilbert transform is useful in many signal analysis applications if you bandpass filter the signal first, the analytic signal. In mathematics and in signal processing, the hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another.

## Hilbert transform

The hilbert transform of g(t) is the convolution of g(t) with the signal 1/πt it is the response to g(t) of a linear time-invariant filter (called a hilbert. The hilbert transform is a widely used transform in signal processing in this thesis we explore its use for three different applications:. Hilbert transform download mathematica notebook the hilbert transform ( and its inverse) are the integral transform. The hilbert transform (ht) and the analytic signal (as) are widely used in their one-dimensional version for various applications however, in the bi-dimensional .

The hilbert transform of a function f(x) is defined by: f(t)=1/pi integral -inf to +inf f theoretically, the integral is evaluated as a cauchy principal value. We have generalized the hilbert transform by defining the fractional hilbert transform (fht) operation in the first stage, two different approaches for defining the. Fast hilbert transform details the hilbert transform of a function x(t) is defined as using fourier identities, you can show the fourier transform of the hilbert. There, the reconstruction requires recovering a family of one-dimensional functions $f$ supported on compact intervals $[a_2,a_4]$ from its hilbert transform.

Asymptotic expansions are obtained for the hilbert transform $h_f (x) = \frac{1}{\pi }\int_{ - \infty }^\infty {\frac{{f(t)}}{{t - x}}dt\quad (x{\text{ real}}),}$ where the bar. In the paper, an accuracy of the discrete hilbert transform method in a detection of amplitude and frequency of vibration signals has been examined this issue. The hilbert transform let us now turn to the hilbert transform of an integrable function g hg(x) = 1 π ∫ r g(t) t − x dt elijah liflyand (bar-ilan university. [APSNIP--]

Hilbert transform
Rated 5/5 based on 38 review

2018.


text:
## Stats: proving an equality with summations

Spent several hours trying to prove this, and now my eyes are swimming :/

I need to show that: $ns^2_{n+1} = (n-1)s^2_n + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2$

where $\bar{x}$ is the mean, and $s^2_n$ is the variance for a set of data with n elements, and $s^2_{n+1}$ is the variance for the same set of data, with 1 additional element ( $x_{n+1}$) added to it. The goal being, I assume, to calculate the new variance based on the old variance without having to redo the grunt work.

edit:
some formulas
$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$

$s^2 = \frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}$

or, alternatively

$s^2 = \frac{\sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n}}{n-1}$

In a different portion of question, they also mentioned that the mean of the set with $x_{i+1}$ added to it can be derived from the current mean. I determined that this formula is $\bar{x}_{i+1} = \frac{n\bar{x}_n + x_{n+1}}{n+1}$


text:
Volume 374 - Light Cone 2019 - QCD on the light cone: from hadrons to heavy ions (LC2019) - Contributed
High energy scattering in QCD: from low to high Bjorken $x$
J. Jalilian-Marian
Full text: pdf
Pre-published on: 2020 May 06
Published on: 2020 May 26
Abstract
We generalize the Color Class Condensate formalism for particle production in the small Bjorken $x$ kinematics to include contribution of large $x$ gluons. We consider scattering of a quark from the color fields of small and large $x$ gluons of a target proton or nucleus and demonstrate that this leads to spin and angular asymmetries in high energy collisions as well as rapidity loss of the projectile quark.
DOI: https://doi.org/10.22323/1.374.0047
Open Access
Copyright owned by the author(s) under the term of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.


text:
# Construction of Alternator

• The AC generator (alternator) or synchronous generator is a machine which converts the mechanical power or energy into electrical power.
• The construction of an alternator is very similar to the DC generator but the main difference between them in DC generator the armature winding is the rotating part and field winding is the stationary part whereas in an alternator the armature winding is stationary and field winding is the rotary part.

## Stator

• As the name suggests it is the stationary part of the machine and it is made up of special magnetic material which can allow high magnetic permeability and low magnetic hysteresis such as fabricated steel.

• The stator core is laminated to minimize the effect of eddy current losses.The lamination is insulated from each other by a thin coating of an oxide and has space between them to allow passage of cool air flow.
• For the small machine, the laminations are stamped out in the complete ring structure and for the large machine, the laminations are divided into the number of segments.
• The slots are provided in the inner periphery of the core and the armature conductors or coils are assembled in it.
• Generally, open slots are used permitting easy installation or removal of the stator coil.
• The fractional number of slots per pole is used in order to eliminate the harmonic in the waveform.
• The armature winding of an alternator is usually connected in star and its neutral is connected to the ground.

## Why is the Armature winding of an Alternator connected in Star?

• The phase voltages in star connection are 57.7 % of the line voltages, i.e. the armature winding in star connection is less exposed to voltage as compared to the delta connection which in turn prove more economic if we consider insulation, breakdown strength, the requirement of conductor material etc.
• In star connection, if the neutral is grounded then it also provides a path for the Zero-Sequence currents during faults, whereas in the delta connection the zero sequence currents flow within the delta circuit and hence increasing the load on the winding.

## Rotor

• The revolving field structure of the electrical machine is called as the rotor. In a synchronous generator, the rotor carries a field winding which is supplied by the DC source.
• The DC source is also called an exciter which is generally a small d.c shunt or compounded generator mounted on the shaft of the alternator.

There are two types of rotor construction

1. Salient Pole Type
2. Cylindrical Type (non-salient pole)

## Salient (or projecting Pole) Type

• The poles are made up of thick steel lamination to reduce eddy current heating loss and it is attached to a rotor by means of the dovetail joint.
• In salient pole rotor, the poles are always projected in the outward direction as shown in the figure.
• The field winding in the salient pole type is connected in series in such a way that when the field winding is energised by the exciter, then adjacent poles will have opposite polarities. The number of poles does not affect the number of phases in the alternator output.
• To reduce the effect of haunting damper winding is provided in the pole faces. They don’t let the motor to oscillate abruptly, they damp the oscillations thus increasing the stability of the machine.
• Salient pole rotor found application for diesel engine and water turbine because they both required medium speed (120-1000 rpm).
• The pole and Pole shoe cover 2/3 of the pole pitch.
• The rating of salient pole rotor is less than 500 kW.

### Disadvantage of Salient Pole Rotor

The salient pole rotor has following disadvantages

1. The salient pole rotor cause excessive windage looses if they are driven at high speed and it also increases the noise produced by an alternator.
2. The construction of salient pole rotor cannot withstand high mechanical stress.
3. The speed of an alternator is inversely proportional to the numbers of pole required (Ns = 120f/p) so to operate a salient pole type alternator, a large number of poles are required which increases the diameter of the generator thus increasing space requirement for installation and initial cost due to extra material used.

## Smooth Cylindrical Type | Non-Salient Pole Alternator

• This type of Rotor is used for steam driven alternator i.e turbo alternator which runs at very high speed.
• The Rotor is made up of smooth solid forgings of alloy steel cylinder having the number of slots along the outer periphery.
• The field windings of cylindrical type rotor are connected in series to the slip rings through which they are excited by the DC exciter.
• The top portion of the slot is covered with the help of steel or manganese wedges and the unslotted portion of the cylinder acts as the poles of an alternator.
• The field windings are arranged in such a way that its flux density is maximum on the polar central line.
• In cylindrical rotor, the pole doesn’t project out from the smooth surface of the rotor hence they maintain the uniform air gap between stator and rotor.
• Since steam turbine runs at very high speed, therefore, they required less number of poles hence the diameter of the rotor is small and axial or rotor length is large.

### Advantages of Cylindrical Rotor type Alternator

• The main advantages of the cylindrical rotor are that their construction has mechanical robustness and gives noiseless operation at very high speed (1500-3000 rpm).
• The flux distribution is nearly uniform sine wave hence better waveform is obtained.
• The hunting effect is very rare in the cylindrical rotor, therefore, there is no need to provide damper winding except in case of assisting the alternator for synchronising purpose.

## Types Of Alternator

The alternator can be divided into different types based on their application, prime mover, design, output power, and cooling.

### Alternator Based on their Output Power

1. Single Phase Alternator
2. Two-Phase Alternator
3. Three Phase Alternator

## Single Phase Alternator

The single phase alternator produces a continuous single alternating voltage. The armature coils are connected in series forming a Single circuit in which output voltage is generated.

In the above figure, the stator has four poles which are evenly spaced around the stator frame. The rotor also consists 4 poles and each pole has opposite polarity to its neighbours which are angled at 90 degrees. Each coil also has opposite winding to its neighbours. This configuration allows the lines of force at 4 poles to be cut by 4 coils at the same amount at a given time. At each 90-degree rotation, the voltage output polarity is switched from one direction to the other. Therefore, there are 4 cycles of the AC output in one rotation.

Single-phase generators are used as standby generators in case of the main power supply is interrupted and for supplying temporary power on construction sites.

## Two-Phase Alternator

In a two-phase alternator, there are two single-phase windings spaced physically so that the ac voltage induced in one is 90° out of phase with the voltage induced in the other. The windings are electrically separate from each other. Suppose in the first quarter first winding produce maximum flux, then the second winding generates zero flux and in the second quarter the second winding generates maximum flux and first winding generate zero flux. This condition establishes a 90° relation between the two phases.

## Three Phase Alternator

A three-phase alternator has 3 sets of single-phase windings arrangement so that the voltage induced in each winding is 120° out of phase with the voltages in the other two windings. These windings are connected in the star to provide a three-phase output.

1. The three-phase alternator gives the most constant output than the single phase alternator.
2. Three phase power supply is more economical than the other two phases because three separate single-phase voltage can be delivered at the same time from the power system.

### Alternator based on their applications

According to their application usage, the alternator can be divided into 5 main part.

1. Automotive Type Alternator
2. Diseal electric locomotive Alternator
3. Brushless type Alternator
4. Marine Type Alternator


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# MATLAB model for equalizer using LMS algorithm adaptive filter and unexpected output curve

The MATLAB code below is for equalizer using lms algorithm adaptive filter and then plotting MSE (Mean Square Error) Vs Iteration numbers

%% Channel Equalization using Least Mean Square (LMS) algorithm
% Author: SHUJAAT KHAN
clc;clear all;close all;
%% Channel and noise level
h = [0.9 0.3 0.5 -0.1]; % Channel
SNRr = 10;              % Noise Level
%% Input/Output data
N = 1000;               % Number of samples
Bits = 2;               % Number of bits for modulation (2-bit for Binary modulation)
data = randi([0 1],1,N);        % Random signal
d = real(qammod(data,Bits));    % BPSK Modulated signal (desired/output)
r = filter(h,1,d);              % Signal after passing through channel
x = awgn(r, SNRr);              % Noisy Signal after channel (given/input)
%% LMS parameters
epoch = 50;        % Number of epochs (training repmiotion)
mio = 1e-3;         % Learning rate / step size
order=12;           % Order of the equalizer
U = zeros(1,order); % Input frame
W = zeros(1,order); % Initial Weigths
%% LMS Algorithm
for k = 1 : epoch
for n = 1 : N
U(1,2:end) = U(1,1:end-1);  % Sliding window
U(1,1) = x(n);              % Present Input

y = (W)*U';             % Calculating output of LMS
e(n) = d(n) - y;           % Instantaneous error
W = W +  mio * e(n) * U ;  % Weight update rule of LMS
J(k,n) = e(n) * e(n)';        % Instantaneous square error
end
end
%% Calculation of performance parameters
MJ = mean(J,2);     % Mean square error
%% Plots
figure % MSE
plot(10*log10(MJ),'linewidth',lw)
hg=legend('MSE','Location','Best');
grid minor
xlabel('Epochs iterations');
ylabel('Mean squared error (dB)');
title('Cost function');

But the curve plotted is unexpected, as the MSE should have fluctuations across iterations and to be very smooth like that as it LMS algorithm is used

That's because it doesn't take the Expectation in the adaptive equation.

The curve plotted is as the following:

[

Where the expected plot is as the following:

• Looks plausible to me. What exactly did you expect and how is it different from what you see ? – Hilmar Dec 17 '20 at 13:48
• @Hilmar The curve plotted is very smooth and it should contain fluctuations as the algorithm used is LMS algorithm. – Aren dg Dec 18 '20 at 0:30
• Isn't it because of the added noise? – megasplash Dec 18 '20 at 8:45
• @megasplash even without the noise, the same. But my question the curve is so smooth while the expected output cost function to have many fluctuations. – Aren dg Dec 18 '20 at 12:02


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# 2.3. Plotting a scalar field in cylindrical coordinates¶

This example shows how to initialize and visualize the scalar field $$u = \sqrt{z} \, \exp(-r^2)$$ in cylindrical coordinates.

from pde import CylindricalSymGrid, ScalarField

grid = CylindricalSymGrid(radius=3, bounds_z=[0, 4], shape=16)
field = ScalarField.from_expression(grid, "sqrt(z) * exp(-r**2)")
field.plot(title="Scalar field in cylindrical coordinates")

Total running time of the script: ( 0 minutes 0.490 seconds)


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Question

# A pack of cards consists of the Ace, King, Queen, Jack and ten of all the four suits. Find the probability of selecting an Ace.

Verified
129.3k+ views
Hint: Figure out the total number of cards in a deck before moving forward with the solution of the problem. There is an Ace in each suit and there are four suits in a deck. Take the ratio of the number of aces and the total number of cards in a deck to calculate probability.

Let us first understand the question properly. In a deck, there are total $52$ cards in which $26$ are of black colour and $26$ red colour. Out of which in each of the four suits of Spades, Hearts, Diamonds, and Clubs. (Diamonds and Hearts are of red colour cards while Spades and Clubs are of black colours.)
Each suit contains $13$ cards: $2,3,4,5,6,7,8,9,10$, Jack, Queen, King and Ace.
$\Rightarrow \operatorname{Probability} = \dfrac{Favourable outcomes}{Total outcomes}$
Since we already know that, there are a total of four Ace in a pack of cards and the total number of cards is $52$.
$\Rightarrow$ Required Probability$= \dfrac{4}{{52}} = \dfrac{1}{{13}}$
Therefore, the probability of pick an Ace from the deck is $\dfrac{1}{{13}}$


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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 Asymptotic evaluation of $\int_0^\infty(\sin x/x)^ndx$ Commun. Korean Math. Soc.Published online August 4, 2020 Jan-Christoph Schlage-Puchta University of Rostock Abstract : We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with an explicit error bound. Keywords : sine integral, asymptotic expansion MSC numbers : 26D15, 33F05 Full-Text :


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# Activation energy

### Activation energy

#### Lessons

In this lesson, we will learn:

• To use the potential energy curve to describe reactant to product transition in chemical reactions.
• The difference between a transition state and an intermediate in chemical reactions.
• The definition of successful collision and the conditions for it to occur in a chemical reaction.
Notes:

• In the 'Introduction to kinetics' (C12.1.1) lesson we introduced activation energy. It is the minimum energy that reactant molecules need to successfully collide and start changing from reactants to products. It is an obstacle that must be overcome for any reaction and rate of reaction to be measured.
Here, we will talk about what happens to molecules when that obstacle is overcome.
• The chemical reaction actually occurs at the peak of the curve marked Ea on an energy diagram – when the reactant molecules have enough potential energy to form an activated complex. This can either be a specific intermediate or just a transition state.
• The transition state is a temporary high energy state seen during the breaking of reactant bonds and forming of product bonds – these are extremely short lived and can't be isolated as chemical substances on their own.
• An intermediate is an unstable short-lived state the reactants take when reactant bonds have broken, but product bonds have not formed. Not all reactions have intermediates. Unlike transition states, intermediates can sometimes be isolated as they have a specific lifetime, though it may be extremely short, before the products are formed. An intermediate is shown by a small 'well' cut into the peak of the potential energy curve.
• Potential energy diagrams during reactions can be explained in the following way. Events marked as points on the potential energy curve are highlighted in red:
• Moving molecules have energy in the form of motion (AKA kinetic energy), so they will collide with each other. Remember that molecules at room temperature and pressure generally collide a few billion times per second!
• Before they collide, when they get near each other repulsion between electrons in the separate molecules will start happening (in a similar way to when London forces start to form). This reduces the energy of motion in the molecules; this reduced 'kinetic energy' is stored as potential energy in the electrons. This is where an increase in potential energy begins in reactant molecules.
• The faster the motion of the molecules originally was, the greater the increase in potential energy this will cause. If this rise in potential energy does not match the activation energy, nothing will happen. The molecules will eventually repel back, the excited electrons will repel each other again and send the unchanged reactant molecules back in motion in opposing directions.
• If the molecules do reach a sufficiently high potential energy then the molecules, still close together, can form an activated complex or transition state. This is where the curve flattens out at the top of the potential energy 'hill'.
• Once this occurs, the chemical bonds get rearranged. The reactant bonds are broken and the bonds of the product made, in a relatively short time period.
• The newly made products begin repelling each other by their electrons, moving away from each other. This reduces potential energy in the electrons, which is now being used again as kinetic energy. This is where the curve begins dropping again, as PE decreases and kinetic energy increases.
• As the molecules get further away, the remaining potential energy from the electron repulsion is 'converted' to kinetic energy until the products reach their 'ground state', their lowest energy state available, like in the reactants before the molecules collided.
• Molecules that go through the reactant to product process described above have undergone a successful collision. There are two conditions for a successful collision. A molecule must have:
• Sufficient energy – the activation energy required as explained above.
• Correct orientation – the molecules have to be arranged in space in an ideal way that the reactant bonds will break and product bonds form when colliding. If they are not, the activation energy for the reaction will be significantly higher. Activation energy on PE curves is shown assuming there is perfect orientation – its lowest possible value.
• In the lesson on the Boltzmann Distribution (C12.1.5) we saw the distribution of molecules and the kinetic energy they possess. The activation energy barrier is directly related to the rate of reaction through this – the lower the activation energy, the more likely any given molecule will possess enough kinetic energy that can be stored as potential energy when molecules approach and collide. Therefore, the lower the activation energy 'hill', the faster the reaction.
• Introduction
What happens when molecules have activation energy?
a)
Potential energy changes in a chemical reaction

b)
The transition state / activated complex.

c)
Conditions of a successful collision.

• 1.
a)
Explain using activation energy why incorrect orientation does not lead to successful collisions. Use the potential energy diagram to illustrate your answer.

b)
The reaction between chemicals A and B is exothermic A and is shown by the equation:
2A $+$ B → A$_2$B

It is known that the activated complex forms an intermediate structure. Draw a potential energy curve for the reaction showing reactants, products and the intermediate structure.


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## The charged $Z_c$ and $Z_b$ structures in a constituent quark model approach

Pablo Garcia Ortega, Jorge Segovia, David Rodriguez Entem and Francisco Fernández

SciPost Phys. Proc. 3, 013 (2020) · published 25 February 2020

### Proceedings event

24th European Few Body Conference

### Abstract

The nature of the recently discovered $Z_c$ and $Z_b$ structures is intriguing. Their charge forces its minimal quark content to be $Q\bar Q q\bar q$ (where $Q=\{c,b\}$ and $q=\{u,d\}$). In this work we perform a molecular coupled-channels calculation of the $I^G(J^{PC})=1^+(1^{+-})$ charm and bottom sectors in the framework of a constituent quark model which satisfactorily describes a wide range of properties of (non-)conventional hadrons containing heavy quarks. All the relevant channels are included for each sector, i.e.: The $D^{(\ast)}\bar D^{\ast}+h.c.$, $\pi J/\psi$ and $\rho\eta_c$ channels for the $Z_c$ and $B^{(\ast)}B^{\ast}$ and $\Upsilon(nS)\pi$ ($n=1,2,3$) channels for the $Z_b$ analysis. Possible structures of these resonances will be discussed

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.
• 1 Pablo G. Ortega,
• 2 Jorge Segovia,
• 1 David Rodriguez Entem,
• 1 Francisco Fernández
Funders for the research work leading to this publication


text:
Literature

### Derivation of the time-independent Landau-Lifshitz equation (Video)

Static configurations of the magnetisation $\mathbf{m}(\mathbf{x})$ minimize the magnetic energy $E(\mathbf{m})$.

We will consider only the exchange energy. We have the Lagrangian density

$$\mathcal{L} = \frac{1}{2}\, \partial_x\mathbf{m}\cdot\partial_x\mathbf{m}.$$

We have to consider the constraint $\mathbf{m}^2 = 1$. The constraint is imposed via a Lagrange multiplier $\lambda(x)$ [Raj, Sec. 3.3][Yan, Sec. 2.1.1][FG, Sec. 12.2].

We have to extremize the functional

$$L[\mathbf{m}] = \int dx \underbrace{\left[ \frac{1}{2}\,\partial_x\mathbf{m}\cdot\partial_x\mathbf{m} + \frac{\lambda(x)}{2} (1-\mathbf{m}^2) \right]}_{\mathcal{L}}.$$

The Euler-Lagrange equation

The functional $L$ is minimized for $\mathbf{m}(x)$ that satisfies the Euler-Lagrange equation

$$-\frac{\delta L}{\delta \mathbf{m}} = 0 \Rightarrow \frac{d}{dx}\left(\frac{\partial \mathcal{L}}{\partial_x\mathbf{m}}\right) - \frac{\partial\mathcal{L}}{\partial\mathbf{m}} = 0.$$

We calculate

$$-\frac{\delta L}{\delta \mathbf{m}} = \frac{d}{dx}\left( \partial_x\mathbf{m} \right) + \lambda \mathbf{m} = 0$$

or

$$\partial_x^2 \mathbf{m} + \lambda \mathbf{m} = 0.$$


text:
# The Probability of a Computable Output from a Random Oracle

@article{Barmpalias2017ThePO,
title={The Probability of a Computable Output from a Random Oracle},
author={George Barmpalias and Douglas A. Cenzer and Christopher P. Porter},
journal={ACM Transactions on Computational Logic (TOCL)},
year={2017},
volume={18},
pages={1 - 15}
}
• Published 27 December 2016
• Computer Science, Mathematics
• ACM Transactions on Computational Logic (TOCL)
Consider a universal oracle Turing machine that prints a finite or an infinite binary sequence, based on the answers to the binary queries that it makes during the computation. We study the probability that this output is infinite and computable when the machine is given a random (in the probabilistic sense) stream of bits as the answers to its queries during an infinitary computation. Surprisingly, we find that these probabilities are the entire class of real numbers in (0,1) that can be…
2 Citations
Aspects of Chaitin's Omega
The purpose of this survey is to expose developments and tell a story about Omega, which outlines its multifaceted mathematical properties and roles in algorithmic randomness.
CHAITIN’S Ω AS A CONTINUOUS FUNCTION
• Mathematics
The Journal of Symbolic Logic
• 2020
It is proved that the continuous function $X:2^\omega \to$ for all \$X is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic.

## References

SHOWING 1-10 OF 63 REFERENCES
Random numbers as probabilities of machine behavior
• Computer Science
Theor. Comput. Sci.
• 2017
A Highly Random Number
• Computer Science, Mathematics
DMTCS
• 2001
It is proved that α is a random number that goes beyond Ω, the probability that a universal self delimiting machine halts, and similar to the algorithmic complexity of Ω', the halting probability of an oracle machine.
Differences of halting probabilities
• Mathematics
J. Comput. Syst. Sci.
• 2017
Randomness and halting probabilities
• Mathematics
Journal of Symbolic Logic
• 2006
It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2≤ω recursive in ∅′ ⊕ r, such that ΩU[X] = r.
From index sets to randomness in ∅n: random reals and possibly infinite computations part II
• Mathematics, Computer Science
The Journal of Symbolic Logic
• 2009
A large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin are obtained and methods to transfer many-one completeness results of index sets to n- randomness of associated probabilities are developed.
Another Example of Higher Order Randomness
• Mathematics, Computer Science
Fundam. Informaticae
• 2002
It is proved that the probability β that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem.
Series Recursively Enumerable Reals and Chaitin Numbers
• Mathematics
• 1997
A real is called recursively enumerable if it can be approximated by an increasing, recursive sequence of rationals. The halting probability of a universal selfdelimiting Turing machine (Chaitin's
Randomness and Computability
The two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent.
The Definition of Random Sequences
Using random sets as oracles
• Computer Science, Mathematics
• 2007
It is shown that the bases for 1‐randomness are exactly the K‐trivial sets, and several consequences of this result are discussed.


text:
# Term independent of x and y in this expansion

Find the term independent of $x$ and $y$ in
$$\left(x+y+\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}+2\right)^{20}$$

My attempt

factorised it as$$\frac{(x+y+2)(xy+1)-1)}{xy}$$ but still the method is too lengthy

#### Solutions Collecting From Web of "Term independent of x and y in this expansion"

$$\left(x+y+\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}+2\right)^{20} = \sum\binom{20}{a,b,c,d,e,f}2^f x^{a-c-e}y^{b-d-e}$$
where the sum is performed over the tuples $(a,b,c,d,e,f)\in\mathbb{N}^6$ such that $a+b+c+d+e+f=20$. If we are just interested in the constant term, we are just interested in the tuples for which $a=c+e$ and $b=d+e$, i.e. the triples $(c,d,e)\in\mathbb{N}^3$ such that $2c+2d+3e\leq 20$. The constant term is so given by
$$\sum_{\substack{(c,d,e)\in\mathbb{N}^3\\2c+2d+3e\leq 20}}\frac{20!}{(c+e)!(d+e)!c!d!e!f!}2^f\quad \text{where}\;f=20-2c-2d-3d$$
that is trivially greater than $2^{20}+760\cdot 2^{18}$ (by restricting the sum just to $f=20$ and $f=18$, since $f=19$ is impossible). There are exactly $192$ triples contributing to the last sum. Thanks to Mathematica, the constant term is actually $\color{red}{737880155312536}$, between $2^{49}$ and $2^{50}$ and with a huge prime factor ($5957949707$).


text:
# Reducing Fractions (Simplest Form)

A fraction is in simplest form if the greatest common factor (GCF) of its numerator and denominator is $1$ -- that is, if they share no common factors other than $1$ .

For example,

$\frac{3}{11}$

is in simplest form, since $3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}11$ have no common factors other than $1$ . However,

$\frac{6}{21}$

is not in simplest form, since $6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}21$ share a common factor of $3$ .

To reduce a fraction, divide both the numerator and denominator by the GCF. (This is also known as "writing a fraction in lowest terms".)

This is sometimes shown as "canceling" the common factors.

$\frac{6}{21}=\frac{\overline{)3}\cdot 2}{\overline{)3}\cdot 7}=\frac{2}{7}$

Note that the result is an equivalent fraction in simplest form.


text:
# $ab$ and $ba$ invertible elements imply $a$ and $b$ invertible in a Banach algebra

I encountered the following question as a question about linear operators. If $AB$ and $BA$ are invertible for some $A, B \in \frak{L}$$(E)$ where $E$ is a Banach space, then show that $A$ and $B$ are also invertible.

I worked this out by showing that $A$ is bijective and its inverse is linear and bounded. But that required making explicit the action of $A$ and $B$ as operators on an underlying space. So my question is whether there is an 'algebraic' way of showing the statement is true, i.e. whether if $ab$ and $ba$ are invertible elements in a Banach algebra imply $a$ and $b$ invertible.

Let $C$ be the inverse of $AB$ and $D$ be the inverse of $BA$.
Then: $I=C(AB)=(CA)B$ and $I=(BA)D=B(AD)$. With $X=CA$ and $Y=AD$ we therefore have
$XB=I=BY$.
This shows that $B$ is invertible. A similar proof shows that $A$ is invertible.


text:
# Circle Geometry Question

1) In triangle $ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Let $P$ be the point on the circumcircle of triangle $ABC$ so that $\angle PCA = 45^\circ$. Find $CP$. Diagram(1)

2) Let $B$, $C$, and $D$ be points on a circle. Let $\overline{BC}$ and the tangent to the circle at $D$ intersect at $A$. If $AB = 4$, $AD = 8$, and $\overline{AC} \perp \overline{AD}$, then find $CD$. Diagram(2)

I think both of them have to do with power of a point

• In the first one, $AB$ is the diameter and $\angle PBA =\angle PAB = 45 ^\circ$. Then you can use Ptolemy's theorem. Jul 9 '16 at 20:38
• In the second one, power of point, yes. $4AC=8^2$, and $AC^2+8^2=CD^2$ by Pythagoras. Jul 9 '16 at 20:43

Analytical solution for question (1):

Take coordinate axes centered in $C$ with natural coordinate axes directed by $\vec{CA}$ and $\vec{CB}$ giving the following coordinates: $A(8,0)$ and $B(0,6)$.

Let $(C)$ be the circle. Its center $\Omega$ clearly has coordinates $(4,3)$.

Thus the equation of $(C)$ has the form

$$(x-4)^2+(y-3)^2=R^2 \ \ \ (*)$$

Knowing that $(x,y)=(0,0)$ is a solution of (*), $R^2=25$. Therefore, the equation of (C) is:

$$x^2+y^2-8x-6y=0 \ \ \ (1)$$

The abscissas of the intersection points of $(C)$ with straight line

$$y=x \ \ \ (2)$$

($\pi/4$ angle means slope 1).

are obtained by plugging (2) into (1), giving $x(x-7)=0$. Excluding $x=0$ that corresponds to $C$, it remains $x=7$ for the abscissa of $P$. Thus $P(7,7)$.

Therefore $CP=7\sqrt{2}$.

1) Since the angle $$\angle PCA = 45^\circ$$ and the $$\triangle ACB$$ is clearly Pythagorean (double of the minimal of sides $$3,4,5$$) the diameter of the circle is equal to $$\overline{AB}$$ and because the angle $$\angle APB$$ subtends an arc of $$90^\circ$$ the quadrilatere $$ACBP$$ is a rectangle. Hence $$\overline{CP}=\overline{AB}=\color{red}{10}$$ 2) By the power of the point $$A$$ respect to the given circle we have $$4(4+x)=8^2\iff x=12$$ It follows $$\overline{CD}=\sqrt{8^2+16^2}=\color{red}{8\sqrt 5}$$


text:
218 articles – 619 Notices  [english version]
 HAL : hal-00001338, version 2
 arXiv : astro-ph/0403543
 Astronomy and Astrophysics 426 (2004) 65-73
 Versions disponibles : v1 (23-03-2004) v2 (02-06-2004)
 Dark halo baryons not in ancient halo white dwarfs
 M. Crézé1, 2, V. Mohan3, A.C. Robin4, C. Reylé4, H. J. Mc Cracken5, J.-C. Cuillandre6, O. Le Fèvre7, Y. Mellier5, 8
 (2004)
 Having ruled out the possibility that stellar objects are the main contributor of the dark matter embedding galaxies, microlensing experiments cannot exclude the hypothesis that a significant fraction of the Milky Way dark halo might be made of MACHOs with masses in the range $0.5-0.8 \msun$. Ancient white dwarfs are generally considered the most plausible candidates for such MACHOs. We report the results of a search for such white dwarfs in a proper motion survey covering a 0.16 sqd field at three epochs at high galactic latitude, and 0.938 sqd at two epochs at intermediate galactic latitude (VIRMOS survey), using the CFH telescope. Both surveys are complete to I = 23, with detection efficiency fading to 0 at I = 24.2. Proper motion data are suitable to separate unambiguously halo white dwarfs identified by belonging to a non rotating system. No candidates were found within the colour-magnitude-proper motion volume where such objects can be safely discriminated from any standard population as well as from possible artefacts. In the same volume, we estimate the maximum white dwarf halo fraction compatible with this observation at different significance levels if the halo is at least 14 gigayears old and under different ad hoc initial mass functions. Our data alone rules out a halo fraction greater than 14% at 95% confidence level. Combined with two previous investigations exploring comparable volumes pushes the limit below 4 % (95% confidence level) or below 1.3% ( 64% confidence), this implies that if baryonic dark matter is present in galaxy halos, it is not, or it is only marginally in the form of faint hydrogen white dwarfs.
 Domaine : Physique/Astrophysique/Cosmologie et astrophysique extra-galactiquePlanète et Univers/Astrophysique
 Mots Clés : dark matter – galaxy halo – white dwarf
Liste des fichiers attachés à ce document :
 PDF
 aa20040491_darkbaryons.pdf(606.7 KB)
 PS
 aa20040491_darkbaryons.ps(2.7 MB)
 hal-00001338, version 2 http://hal.archives-ouvertes.fr/hal-00001338 oai:hal.archives-ouvertes.fr:hal-00001338 Contributeur : Michel Crézé <> Soumis le : Mercredi 2 Juin 2004, 13:54:30 Dernière modification le : Mercredi 13 Septembre 2006, 13:47:22


text:
# [NTG-context] math: too big space between function and argument

Tue May 24 15:56:34 CEST 2011

On Tue, 24 May 2011, Paul Menzel wrote:

> Dear ConTeXt folks,
>
>
> to markup the probability measure and the parenthesis around the
> argument, I defined the following command.
>
> 	\define[1]\P{{\mathbf P}\left( #1 \right)}
>
> Unfortunately the space between the P and the left ( is a little big in
> my opinion. Is that correct or should/can I fix that somehow?

Getting these spaces is tricky; especially if you consider spaces before
and after the definition. Choose your pick:

\starttext
\startformula
\stopformula
\stoptext

Some information from the mathsets module (which does not work with MkIV).
The entry is the latex bug database is not viewable now.

%D The \type{\left} and \type{\right} generate a math atom of type inner,
%D while for math sets, we want a math open atom. To see the difference,
%D consider
%D
%D \startbuffer
%D \startformula
%D  2\biggl( \frac {3}{4} \biggr)
%D \stopformula
%D
%D and
%D
%D \startformula
%D  \Pr\biggl( \frac {3}{4} \biggr)
%D \stopformula
%D \stopbuffer
%D \typebuffer
%D
%D which gives (notice the spacing before the parenthesis)
%D
%D \getbuffer
%D
%D I will assume that if \type{text} is something, then
%D the default behaviour is desirable, if \type{text} is empty, then I add
%D \type{\mathopen} and \type{\mathclose}.  Using \type{\mathopen} to correct
%D the spacing is due to Frank Mittelbach, see
%D \hyphenatedurl{http://www.latex-project.org/cgi-bin/ltxbugs2html?pr=latex/3853}
%D
%D Mathset module ensures that we get the correct spacing in both cases
%D \startbuffer
%D \definemathset[SET][left=(,right=)]
%D \startformula
%D  \PR{ \frac{3}{4} }
%D \stopformula
%D \stopbuffer
%D \getbuffer[PR] \getbuffer which was typed as \typebuffer
%D
%D Also, if its argument is a single character, \type{\mathop} centers it
to
%D with respect to the math||axis. Compare the outputs of
%D
%D \startbuffer
%D \ruledhbox{$\mathop{y}\nolimits_x\left\{A\,\middle|\,B\right\}$}
%D \ruledhbox{$\mathop{\kern\zeropoint y}\nolimits_x\left\{A\,\middle|\,B\right\}$}
%D \stopbuffer
%D
%D \typebuffer
%D \getbuffer
%D
%D I have added a \type{\kern\zeropoint} to prevent that.


text:
Lemma 66.8.1. Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. There exist reduced closed subspaces

$\emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_ n = X$

such that with $X_ r = Z_ r \setminus Z_{r - 1}$ the stratification $X = \coprod _{r = 0, \ldots , n} X_ r$ is characterized by the following universal property: Given $g : T \to X$ the projection $W \times _ X T \to T$ is finite locally free of degree $r$ if and only if $g(|T|) \subset |X_ r|$.

Proof. Let $n$ be an integer bounding the degrees of the fibres of $W \to X$. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Apply More on Morphisms, Lemma 37.40.3 to $W \times _ X U \to U$. We obtain closed subsets

$\emptyset = Y_{-1} \subset Y_0 \subset Y_1 \subset Y_2 \subset \ldots \subset Y_ n = U$

characterized by the property stated in the lemma for the morphism $W \times _ X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times _ X U$ with projection maps $s, t : R \to U$ we conclude that

$s^{-1}(Y_ r) = t^{-1}(Y_ r)$

as closed subsets of $R$. In other words the closed subsets $Y_ r \subset U$ are $R$-invariant. This means that $|Y_ r|$ is the inverse image of a closed subset $Z_ r \subset |X|$. Denote $Z_ r \subset X$ also the reduced induced algebraic space structure, see Properties of Spaces, Definition 64.12.5.

Let $g : T \to X$ be a morphism of algebraic spaces. Choose a scheme $V$ and a surjective étale morphism $V \to T$. To prove the final assertion of the lemma it suffices to prove the assertion for the composition $V \to X$ (by our definition of finite locally free morphisms, see Morphisms of Spaces, Section 65.46). Similarly, the morphism of schemes $W \times _ X V \to V$ is finite locally free of degree $r$ if and only if the morphism of schemes

$W \times _ X (U \times _ X V) \longrightarrow U \times _ X V$

is finite locally free of degree $r$ (see Descent, Lemma 35.20.30). By construction this happens if and only if $|U \times _ X V| \to |U|$ maps into $|Y_ r|$, which is true if and only if $|V| \to |X|$ maps into $|Z_ r|$. $\square$

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