text:
# Use pgfplots and a custom length to define global width when inputing the plot

I'm trying to set the width for all my plots, that are defined in other files and imported into the main document, using a custom width, \sz. I'm trying to set this length because I want to change it for one or two columns. Like in this example, main.tex:

\documentclass{minimal}

\usepackage{standalone}
\usepackage{tikz}
\usepackage{pgfplots}

\newlength{\sz}
\setlength{\sz}{0.5\linewidth}

\begin{document}
\input{test}
\end{document}

test.tex:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
x tick label style={
/pgf/number format/1000 sep=},
ylabel=Population,
enlargelimits=0.05,
legend style={at={(0.5,-0.15)},
anchor=north,legend columns=-1},
ybar interval=0.7,
cycle list name=color list
]
coordinates {(1930,50e6) (1940,33e6)
(1950,40e6) (1960,50e6) (1970,70e6)};
coordinates {(1930,38e6) (1940,42e6)
(1950,43e6) (1960,45e6) (1970,65e6)};
coordinates {(1930,15e6) (1940,12e6)
(1950,13e6) (1960,25e6) (1970,35e6)};
coordinates {(1930,15e6) (1940,12e6)
(1950,13e6) (1960,25e6) (1970,35e6)};

\legend{Far,Near,Here,There}
\end{axis}
\end{tikzpicture}
\end{document}

However, when I'm compiling it, the length of \sz is reset to zero. And I get the error: Error: Plot width '0.0pt' is too small. Is there any way of setting the length so that the figure in the other file is able to 'see' it?

-

This happens because you use the minimal documentclass, which doesn't define \linewidth. If you use a proper class, like article, your setup works.


text:
### Re: [Rats] Age Claim in EAT

"Oliver, Ian (Nokia - FI/Espoo)" <ian.oliver@nokia-bell-labs.com> Thu, 30 July 2020 10:01 UTC

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From: "Oliver, Ian (Nokia - FI/Espoo)" <ian.oliver@nokia-bell-labs.com>
To: Hannes Tschofenig <Hannes.Tschofenig@arm.com>, Henk Birkholz <henk.birkholz@sit.fraunhofer.de>, "rats@ietf.org" <rats@ietf.org>
Thread-Topic: [Rats] Age Claim in EAT
Date: Thu, 30 Jul 2020 10:01:02 +0000
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Subject: Re: [Rats] Age Claim in EAT
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Interesting.

We've implemented something similar (or at least tried to be compliant with the spec as much as we can)

For each claim we have at minimum the time of the claim request (R) by the attestating authority, the time of claim creation (C) by the reporting authority and the time of receipt (T) by the attesting authority.

One of the attestation rules is thus: R < C < T

Another would then deal with timliness of the obtaining of the claim, eg:   T -R < \epsilon

I agree that some idea of the latches/stage-transitions would help.

Ian

--

Dr. Ian Oliver

Cybersecurity Research

Distinguished Member of Technical Staff

Nokia Bell Labs

+358 50 483 6237

________________________________
From: Hannes Tschofenig <Hannes.Tschofenig@arm.com>
Sent: 30 July 2020 11:28
To: Henk Birkholz <henk.birkholz@sit.fraunhofer.de>de>; rats@ietf.org <rats@ietf.org>
Subject: Re: [Rats] Age Claim in EAT

Thanks for the clarification, Henk.

Since I don't know what information consumers of Entity Attestation Tokens really need to make decisions I am not sure whether the current functionality is sufficient already or not. To me as a reader it appears underspecified.  At a minimum a reference to Appendix A of the RATS architecture document in the EAT spec would be useful.

Ciao
Hannes

-----Original Message-----
From: Henk Birkholz <henk.birkholz@sit.fraunhofer.de>
Sent: Wednesday, July 29, 2020 11:11 AM
To: Hannes Tschofenig <Hannes.Tschofenig@arm.com>om>; rats@ietf.org
Subject: Re: [Rats] Age Claim in EAT

Hi Hannes,

in the scope of Evidence:

Signing a CWT structure is creating an EAT. This event is defined as
time(EG) here:
https://tools.ietf.org/html/draft-ietf-rats-architecture-05#appendix-A

"Creating a token" would mean "Evidence Generation", for example.

Age also is a time interval based on a time unit (units that apply to Henk are years, I claim that Henk's age is 21, of course. Trust in Henk's Claims can be established via Endorsements). Age is a time interval expressed as a duration. That means that the beginning of that time interval is set as zero (an epoch) and then counts the time units.

The semantics of age in EAT are:

the epoch is time(EG) and the unit is seconds.
This only works, if time sources for relative time-counters are available, of course.

Collection of Claims - time(CC) was proposed to the architecture, but did not find consensus yet (we are working on time(AA), though, the time at which the Attester becomes aware of a changes value in the Target Environment right now).

But! there is time(VG), the Value Generation. This time can often only be inferred or approximated (in contrast to time(AA)), but it is the best we have defined right now in the context of:

> If that's the case, I wonder whether it would also make sense to take into account that different information may be collected at a different point in time and hence the age indication would better go (somehow) with specific claims where the time difference between the collection of the data and the signature generation matters.

As a note: We have not talked at all before about the common concept of "latches" or stage transitions... this is an important concept that does capture the "this event has happened" (before/after a defined event) and represents strong security implications that are not included in the RATS architecture today. Would that maybe help in your context?

Viele Grüße,

Henk

On 29.07.20 10:37, Hannes Tschofenig wrote:
> Hi Laurence, Hi all,
>
> How does the age claim work?
>
> The spec says "represents the number of seconds that have elapsed since the token was created".
>
> By creating a token you also protect the claims with a signature and hence you cannot change the content of the claims anymore (without breaking the signature).
>
> Hence, here "creating a token" must mean something different. I suspect it means when certain values have been collected from the device and before the token with the signature was put together. Correct?
>
> If that's the case, I wonder whether it would also make sense to take into account that different information may be collected at a different point in time and hence the age indication would better go (somehow) with specific claims where the time difference between the collection of the data and the signature generation matters. I also wonder whether the number of seconds matter and whether you really want to communicate something more abstract, such as "I created a hash over the firmware during boot time" rather than "I created the hash over the firmware in real-time when I was asked".
>
> Ciao
> Hannes
>
> IMPORTANT NOTICE: The contents of this email and any attachments are confidential and may also be privileged. If you are not the intended recipient, please notify the sender immediately and do not disclose the contents to any other person, use it for any purpose, or store or copy the information in any medium. Thank you.
>
> _______________________________________________
> RATS mailing list
> RATS@ietf.org
> https://www.ietf.org/mailman/listinfo/rats
>
IMPORTANT NOTICE: The contents of this email and any attachments are confidential and may also be privileged. If you are not the intended recipient, please notify the sender immediately and do not disclose the contents to any other person, use it for any purpose, or store or copy the information in any medium. Thank you.

format: plain


text:
# How to make “Show” function display images with original size?

h = ExampleData[{"TestImage", "House"}];
Grid[{{Show[h],
Show[h, Graphics[Line[{{0, 0}, ImageDimensions[h]*2}]],
PlotRange -> All]}}]

The second image is much smaller than the first. I want those two images to have same sizes.

Attempt 1

h=ExampleData[{"TestImage","House"}];
hDim=ImageDimensions[h];
a=Grid[{{Show[h,PlotRange->hDim],Show[h,Graphics[Line[{{0,0},hDim*2}]],PlotRange->All]}}]

Basically, it works. But the size of two images are still slightly different. Besides, it seems that I can not control the size any more, like use ImageSize.

• remove PlotRange-> All and check – Ali Hashmi May 27 '17 at 13:29
• But then I can not see the entire line. – Shen Zhou May 27 '17 at 13:39
• Grid[{{Show[h, PlotRange -> ImageDimensions[h]], Show[h, Graphics[Line[{{0, 0}, ImageDimensions[h]*2}]], PlotRange -> All]}}] – Ali Hashmi May 27 '17 at 13:50
• Thanks for your reply. But it still not perfect. And this problem looks like an easy one. But I still can not find a perfect solution after couples of hours struggling. – Shen Zhou May 27 '17 at 23:40
• Grid[{{Show[h, PlotRange -> hDim], Show[h, Graphics[Line[{{0, 0}, hDim*2}]], PlotRangePadding -> 0, PlotRange -> All]}}]? – kglr May 27 '17 at 23:53

You can make use of the new in M8 ImageSize syntax to do this:

h = ExampleData[{"TestImage","House"}];
hDim = ImageDimensions[h];
a = Grid[
{{
Show[h, ImageSize->hDim->200],
Show[h, Graphics[Line[{{0,0}, hDim*2}]], ImageSize->hDim->200,
]
}},
Alignment->Bottom
]

• Thanks a lot. Sorry for not voting due to my low reputation. – Shen Zhou May 28 '17 at 0:11

We can take control of the image size with ImagePad to resize the images and with the ImageSize -> option of the Show command. Here is some code that illustrates the approach.

ClearAll["Global*"]

h = ExampleData[{"TestImage", "House"}];
{wd, ht} = ImageDimensions[h];

imgLeft = ImagePad[h, {{0, 0}, {0, ht}}, White];
sizeLeft = ImageDimensions[imgLeft];

imgRight = ImagePad[h, {{0, wd}, {0, ht}}, White];
sizeRight = ImageDimensions[imgRight];

line = Graphics[{Line[{{0, 0}, ImageDimensions[imgRight]}]}];
m = 1/2;

Grid[{{
Show[imgLeft, ImageSize -> m*sizeLeft],
Show[{imgRight, line}, ImageSize -> m*sizeRight]}},
Alignment -> {{Left, Left}, {Bottom}},
Frame -> All]

We use the Frame -> All option initially to see the image spacing, borders, etc, and remove it later.

To create the images at the same scale, we first note that the diagonal line is the diagonal of an image that is twice as large as the original. Knowing that, we pad the original to be the same width and height as the line to obtain an image for the right panel. Padding does not distort the original. For the left image we pad the original to the same height as right image. The padding is only applied to the right and at the top of the original.

We can introduce a "magnification factor", $m$, to control the overall size of the images. The magnification is applied in the ImageSize -> option of the Show commands.

Note that the GraphicsGrid command is similar to Grid` command but seems to introduce additional, unwanted padding.


text:
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text:
# chemfig: order of atoms

I want to change the order of the "OH"-group at carbon 4: the "H" should be left of the "O".

\documentclass{standalone}
\usepackage{chemfig}

\begin{document}
\setatomsep{4em}\setcrambond{3pt}{}{}
\chemfig{
OH-[2,.7](-[3,0.15,,,draw=none]\scriptstyle\mbox{\small\textcolor{orange}{4}})
(
-[1,.7](-[2,0.5]-[3,0.5]OH)
-[0]{O}-[7,0.7]
)
<[7,0.7](-[2,0.5]HO)
-[,,,,line width=3pt](-[6,0.5]OH)
>[1,0.7](-[1,0.15,,,draw=none]\scriptstyle\mbox{\small\textcolor{orange}{1}})
-[2,.5]OH
}

\end{document}

• Welcome to TeX.sx! Change the beginning of your formula to HO-[2,.7,2]. You might want to check the chemfig manual for departure and arrival atoms and the optional arguments of bonds: <bond>[<angle>,<length factor>,<departure>,<arrival>,<tikz>] – cgnieder Jun 24 '15 at 18:56
• You should change the order on the sixth and third carbon atom as well, looks like hydrogen is attached to the carbon (and the oxygen) at the moment – user49901 Jun 24 '15 at 19:10

The order of atoms is the order you type them in: if you type OH then chemfig uses OH, if you type HO then chemfig uses HO. Ideally you also tell chemfig that the bond starts at the O. This can be done with the optional argument of a bond:

<bond>[<angle>,<length factor>,<departure>,<arrival>,<tikz>]

So starting your compound with HO-[2,.7,2] instead of OH-[2,.7] will give what you want. Applying this to other OH groups, too, gives:

\documentclass{standalone}
\usepackage{chemfig}

\begin{document}
\setatomsep{4em}\setcrambond{3pt}{}{}
\chemfig{
HO-[2,.7,2](-[3,0.15,,,draw=none]\scriptstyle\mbox{\small\textcolor{orange}{4}})
(
-[1,.7](-[2,0.5]-[3,0.5]HO)
-[0]{O}-[7,0.7]
)
<[7,0.7](-[2,0.5,,2]HO)
-[,,,,line width=3pt](-[6,0.5]OH)
>[1,0.7](-[1,0.15,,,draw=none]\scriptstyle\mbox{\small\textcolor{orange}{1}})
-[2,.5]OH
}

\end{document}


text:
Geometry Webinar

# Algebraic theory of ODEs depending on parameters

Transmissão através de Videoconferência

Por João Pedro dos Santos (Universidade de Paris-Sorbonne e Universidade de Montpellier).

Differential Galois theory has the objective to study linear ODEs with the help of algebraic groups. Roughly and explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbf C(x) )$ and a differential system $y'=Ay$, we associate a subgroup of ${\rm GL_n}(\mathbf C)$, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions (similar to standard Galois theory), monodromy representations (analytic approach)  and Tannakian categories (the categorical way).

If instead of working with complex coefficients we deal with a discrete valuation ring $R$, the construction of the differential Galois groups are less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry. In this talk, I shall explain how to associate to these differential equations certain flat $R$-group schemes, what properties these may have and, most importantly, how to compute them with the monodromy. In doing so, I shall explain our method to establish a certain "Deligne-Riemann-Hilbert" theorem with "parameters" (this is in connection to recent work of Fiorot-Monteiro-Sabbah), which is a way of finding extensions of differential equations having mild singularities. I shall emphasise the case of formal differential equations for the sake of clarity.

(The talk is a horizontal report on several works done in collaboration with P. H. Hai and his students N. D. Duong and P. T. Tam over the past years.)

Transmissão em direto via Zoom (Meeting ID: 799 297 2871).

15h00
CMAFcIO - Centro de Matemática, Aplicações Fundamentais e Investigação Operacional

Workshop promovido pelo GAPsi - Gabinete de Apoio Psicopedagógico.

Talks @DI, por Marcelo Inuzuka e Ruan Rodrigues.

Mathematical Logic Webinar, por Paulo Santos (Universidade Nova de Lisboa - CMA).

Bioecosys brings together expertise and experience in forest ecosystem management and economics to communicate recent findings and to accelerate the transfer of knowledge and best practices to stakeholders and forest actors.

"Desenvolvimento da Organocatálise Assimétrica", por Amélia Pilar Reuter.

Seminário no âmbito da Unidade Curricular de Introdução aos Recursos Naturais Vegetais, por Jorge Faria (INIAV - Instituto Nacional de Investigação Agrária e Veterinária / MED - Instituto Mediterrâneo para a Agricultura, Ambiente e Desenvolvimento).

Sessão de apresentação do livro coordenado por José Almeida Silva (IDL / Ciências ULisboa).

This Workshop will gather specialists in the field of Combinatorial Optimization, from several countries, to present and discuss recent research work around this subject. Luís Gouveia, coordinator of CMAFcIO and José Rui Figueira, president of CEG-IST will present the Opening Session.

Formação de Comunicação em Ciência(s), por Cristina Nobre Soares.

Concerto de Natal | Aula Magna da Reitoria da Universidade de Lisboa.

Conversa com Galopim de Carvalho.

Workshop promovido pelo GAPsi - Gabinete de Apoio Psicopedagógico.

The training program will provide industry and academia the technological and scientific pathways required for deployment of sustainable cost-effective large scale energy storage and conversion powered by renewable energies.

"Desenvolvimento da Organocatálise Assimétrica", por Amélia Pilar Reuter.

Candidaturas a decorrer até 15 de dezembro de 2021.

This meeting is a forum for scientists of all academic levels, (from master to principal investigators), to present their work and discuss, fostering new ideas and collaborations. We invite researchers from all over the world to participate and submit their work.

Conferência do Centro Interuniversitário de História das Ciências e da Tecnologia, por Céline Paillette (Université Paris 1 Panthéon-Sorbonne).

Desde o início de 2020 que somos bombardeados, através da comunicação social, com informações sobre o novo coronavírus humano - SARS-CoV-2 - que provoca a síndrome respiratória aguda grave. Nunca se comunicou tanto, em tempo real, um tema científico, mas será que já vimos todas as nossas perguntas sobre este tema respondidas?

Candidaturas online de 01 a 31 de dezembro de 2021.

At the end of this course, participants will be able to predict the three-dimensional structure of nucleic acid and proteins from their sequence and use the acquired knowledge to improve or create new methodologies in molecular biology research and diagnostics.

The course SoilEco aims at introducing attendants to an updated state of the art of diversity of the soil biota and the functional roles played by soil organisms in key ecological processes.

Understanding plant water relations at several scales under climate change is the theme of the XV Portuguese-Spanish Water relations Symposium: “Water relations under climate change: from genes to ecosystem”.

In this course, we promote a multidisciplinary approach presenting the most recent findings on the topic and challenging the traditional way of considering symbiotic associations as exceptions and not as the rule.

The goal of this course is to provide to the participants with current and practical knowledge on urban ecology, including ecological and social aspects.

This course aims to explore ways of communicating science to non-specialized audiences, such as policy makers, industry, general public (including students and teachers), through their engagement and participation in citizen science activities.


text:
Home NCERT Solutions NCERT Exemplar CBSE Sample Papers NCERT Books Class 12 Class 10
Question 7

# Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

fR → R is given by,

f(x) = 4x + 3

One-one:

Let f(x) = f(y).

∴ f is a one-one function.

Onto:

For y ∈ R, let y = 4x + 3.

Therefore, for any y ∈ R,   such that

∴ f is onto.

Thus, f is one-one and onto and therefore, f - 1 exists.

Let us define gR→ R by.

Hence, f is invertible and the inverse of f is given by

## Recently Viewed Questions of Class 12th mathematics

#### Write a Comment:

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## Algebra 2 (1st Edition)

$P(A$ and $B) = P(A)*P(B)$ P(A) = 0.3 P(B) = 0.4 $P(A$ and $B) = .3*.4$ or $.12$


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# nLab model structure on dg-modules

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

The category of dg-modules over a dg-algebra, or more generally a dg-category, admits dg-model structures which present the derived dg-category.

In the case of dg-algebras in an abelian category $\mathcal{A}$, dg-modules are the same as modules over algebras over the associative operad in $Ch(\mathcal{A})$. These admit model structures as described in model structure on modules over an algebra over an operad, transferred from the model structures on chain complexes in $\mathcal{A}$.

More generally there are model structures on dg-modules over a dg-category, analogous to model structures on simplicial presheaves.

## Definition

###### Theorem

There is a cofibrantly generated model structure on the category of dg-modules, called the projective model structure, where the weak equivalences are object-wise quasi-isomorphisms of chain complexes, and the fibrations are object-wise epimorphisms. This is a dg-model structure.

###### Theorem

There is a model structure on the category of dg-modules, called the injective model structure, where the weak equivalences are object-wise quasi-isomorphisms of chain complexes, and the cofibrations are object-wise monomorphisms.

See (Toen 04, section 3) and (Keller 06, theorem 3.2). These model structures present the derived dg-category.

## References

Section 3 of

Paragraph 3.2 of

### For dg-algebras

For the case of dg-algebras, see the references below.

A general account is around section 11.2.5 of

• Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)

and in section 3 of

The homotopy category and triangulated category of dg-modules is discussed for instance also in

• Joseph Bernstein, DG-modules and equivariant cohomology (pdf).

See also

• Tobias Barthel, Peter May, Emily Riehl, Six model structures for DG-modules over DGAs: model category theory in homological action, New York J. Math. 20 (2014) 1077–1159 (pdf)

Revised on June 18, 2015 10:08:59 by Urs Schreiber (195.113.30.252)


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# Comet Calendar Event Details

Due to changes in campus operations, many events may be canceled, virtualized or postponed.
Mathematical Sciences Colloquium by Brendon Rhoades
Friday, Nov. 1, 2019
2 p.m. - 3 p.m. Location: CB1 1.104

University of California, San Diego

The algebra and geometry of ordered set partitions

An {\em ordered set partition} of size $n$ is a sequence $(B_1, \dots, B_k)$ of nonempty sets such that we have a disjoint union decomposition $B_1 \sqcup \cdots \sqcup B_k = \{1, 2, \dots, n \}$.  When $k = n$, we have a natural identification with permutations in the symmetric group $S_n$. We discuss algebraic and geometric incarnations of ordered set partitions, generalizing results of Chevalley, Artin, Steinberg, Borel, and others. We also discuss how ordered set partitions relate to the theory of Macdonald polynomials.

Persons with disabilities may submit a request for accommodations to participate in this event at UT Dallas' ADA website. You may also call (972) 883-5331 for assistance or send an email to [email protected]. All requests should be received no later than 2 business days prior to the event.
Contact Info:
Viswanath Ramakrishna, 972-883-6873
Questions? Email me.

Tagged as Lectures/Seminars, Professional Dev.
See more events from Natural Sciences & Mathematics
View other events on the Comet Calendar


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# Find eigenvalues & eigenvectors

#### rugerts

Problem Statement
Find eigenvalues & eigenvectors
Relevant Equations
det(A-r*I) = 0
Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at least for simple 2x2 cases like these) by simply taking a row (of the matrix after having plugged in the eigenvalue) and shaping it such that the leftmost value is the first value for the eigenvector (with opposite sign) and the second eigenvector value is the same as the rightmost value. I heard that this is valid for either row choice... but when I do this, the answers appear to be different (hence the question mark over the inequality). So, my question is, are these eigenvectors shown in the blue box equivalent to one another? Does this trick really work? It seems to.
Thanks for your time.

Related Calculus and Beyond Homework News on Phys.org

#### RPinPA

Sort of. Those are the same two eigenvectors up to a constant, but in opposite order.

Multiply your $\vec v$ by the constant (1 + i). As you know, any multiple of an eigenvector is still an eigenvector of the same eigenvalue.
$$(1 + i) {2 \choose {2 - 2i}} = {{2 + 2i} \choose {2(1 - i)(1+i)}} = {{2+2i} \choose 4}$$
and similarly multiply $\vec w$ by the constant (1 - i).
$$(1 - i) {2 \choose {2 + 2i}} = {{2 - 2i} \choose {2(1 + i)(1-i)}} = {{2-2i} \choose 4}$$
Since the order is reversed, so are the eigenvalues with which each is associated.

#### Mark44

Mentor
Also, you can verify that your eigenvalues and eigenvectors are correct or not by plugging them into the matrix equation and determining whether $(A - \lambda I) \vec x = \vec 0$.

#### rugerts

Sort of. Those are the same two eigenvectors up to a constant, but in opposite order.

Multiply your $\vec v$ by the constant (1 + i). As you know, any multiple of an eigenvector is still an eigenvector of the same eigenvalue.
$$(1 + i) {2 \choose {2 - 2i}} = {{2 + 2i} \choose {2(1 - i)(1+i)}} = {{2+2i} \choose 4}$$
and similarly multiply $\vec w$ by the constant (1 - i).
$$(1 - i) {2 \choose {2 + 2i}} = {{2 - 2i} \choose {2(1 + i)(1-i)}} = {{2-2i} \choose 4}$$
Since the order is reversed, so are the eigenvalues with which each is associated.
Ahh, I see. Thank you

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Browse Questions

# The point of contact the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2-2x+2y+1=0$ is

$\begin {array} {1 1} (a)\;(0,1) & \quad (b)\;(0,-1) \\ (c)\;(1,0) & \quad (d)\;(-1,0) \end {array}$

(b) (0,-1)


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# Group Actions

A group action of a group G on a set Ω (from the right) is defined by a map μ: Ω × G → Ω that satisfies the compatibility conditions μ(μ(x, g), h) = μ(x, g*h) and μ(x, one(G)) == x for all x ∈ Ω.

The maps μ are implemented as functions that take two arguments, an element x of Ω and a group element g, and return the image of x under g.

In many cases, a natural action is given by the types of the elements in Ω and in G. For example permutation groups act on positive integers by just applying the permutations. In such situations, the function ^ can be used as action function, and ^ is taken as the default whenever no other function is prescribed.

However, the action is not always determined by the types of the involved objects. For example, permutations can act on vectors of positive integers by applying the permutations pointwise, or by permuting the entries; matrices can act on vectors by multiplying the vector with the matrix, or by multiplying the inverse of the matrix with the vector; and of course one can construct new custom actions in situations where default actions are already available.

Thus it is in general necessary to specify the action function explicitly, see the following sections.

## Common actions of group elements

on_tuplesFunction
on_tuples(tuple::GAP.GapObj, x::GAPGroupElem)
on_tuples(tuple::Vector{T}, x::GAPGroupElem) where T
on_tuples(tuple::T, x::GAPGroupElem) where T <: Tuple

Return the image of tuple under x, where the action is given by applying ^ to the entries of tuple.

For Vector and Tuple objects, one can also call ^ instead of on_tuples.

Examples

julia> g = symmetric_group(3);  g[1]
(1,2,3)

julia> l = GAP.GapObj([1, 2, 4])
GAP: [ 1, 2, 4 ]

julia> on_tuples(l, g[1])
GAP: [ 2, 3, 4 ]

julia> on_tuples([1, 2, 4], g[1])
3-element Vector{Int64}:
2
3
4

julia> on_tuples((1, 2, 4), g[1])
(2, 3, 4)

julia> (1, 2, 4)^g[1]
(2, 3, 4)

source
on_setsFunction
on_sets(set::GAP.GapObj, x::GAPGroupElem)
on_sets(set::Vector, x::GAPGroupElem)
on_sets(set::Tuple, x::GAPGroupElem)
on_sets(set::AbstractSet, x::GAPGroupElem)

Return the image of set under x, where the action is given by applying ^ to the entries of set, and then turning the result into a sorted vector/tuple or a set, respectively.

For Set objects, one can also call ^ instead of on_sets.

Examples

julia> g = symmetric_group(3);  g[1]
(1,2,3)

julia> l = GAP.GapObj([1,3])
GAP: [ 1, 3 ]

julia> on_sets(l, g[1])
GAP: [ 1, 2 ]

julia> on_sets([1, 3], g[1])
2-element Vector{Int64}:
1
2

julia> on_sets((1, 3), g[1])
(1, 2)

julia> on_sets(Set([1, 3]), g[1])
Set{Int64} with 2 elements:
2
1

julia> BitSet([1, 3])^g[1]
BitSet with 2 elements:
1
2

source
permutedFunction
permuted(pnt::GAP.GapObj, x::PermGroupElem)
permuted(pnt::Vector, x::PermGroupElem)
permuted(pnt::Tuple, x::PermGroupElem)

Return the image of pnt under x, where the action is given by permuting the entries of pnt with x.

Examples

julia> g = symmetric_group(3);  g[1]
(1,2,3)

julia> a = ["a", "b", "c"]
3-element Vector{String}:
"a"
"b"
"c"

julia> permuted(a, g[1])
3-element Vector{String}:
"c"
"a"
"b"

julia> permuted(("a", "b", "c"), g[1])
("c", "a", "b")

julia> l = GAP.GapObj(a, recursive = true)
GAP: [ "a", "b", "c" ]

julia> permuted(l, g[1])
GAP: [ "c", "a", "b" ]

source
on_indeterminatesFunction
on_indeterminates(f::GAP.GapObj, p::PermGroupElem)
on_indeterminates(f::Nemo.MPolyElem, p::PermGroupElem)
on_indeterminates(f::MPolyIdeal, p::PermGroupElem)
on_indeterminates(f::GAP.GapObj, p::MatrixGroupElem)
on_indeterminates(f::Nemo.MPolyElem{T}, p::MatrixGroupElem{T, S}) where T where S
on_indeterminates(f::MPolyIdeal, p::MatrixGroupElem)

Return the image of f under p. If p is a PermGroupElem then it acts via permuting the indeterminates, if p is a MatrixGroupElem then it acts via evaluating f at the vector obtained by multiplying p with the (column) vector of indeterminates.

For Nemo.MPolyElem and MPolyIdeal objects, one can also call ^ instead of on_indeterminates.

Examples

julia> g = symmetric_group(3);  p = g[1]
(1,2,3)

julia> R, x = PolynomialRing(QQ, ["x1", "x2", "x3"]);

julia> f = x[1]*x[2] + x[2]*x[3]
x1*x2 + x2*x3

julia> f^p
x1*x3 + x2*x3

julia> x = [GAP.Globals.X( GAP.Globals.Rationals, i ) for i in 1:3];

julia> f = x[1]*x[2] + x[2]*x[3]
GAP: x_1*x_2+x_2*x_3

julia> on_indeterminates(f, p)
GAP: x_1*x_3+x_2*x_3

julia> g = general_linear_group(2, 5);  m = g[2]
[4   1]
[4   0]

julia> R, x = PolynomialRing(base_ring(g), degree(g));

julia> f = x[1]*x[2] + x[1]
x1*x2 + x1

julia> f^m
x1^2 + 4*x1*x2 + 4*x1 + x2

source

## G-Sets

The idea behind G-sets is to have objects that encode the permutation action induced by a group (that need not be a permutation group) on a given set. A G-set provides an explicit bijection between the elements of the set and the corresponding set of positive integers on which the induced permutation group acts, see action_homomorphism(Omega::GSetByElements{T}) where T<:GAPGroup.

gsetMethod
gset(G::GAPGroup[, fun::Function], seeds, closed::Bool = false)

Return the G-set Omega that consists of the closure of the seeds seeds under the action of G defined by fun.

This means that Omega contains all elements fun(omega, g) for omega in seeds and g in G.

fun can be omitted if the element type of seeds implies a reasonable default, for example, if G is a PermGroup and seeds is a Vector{T} where T is one of Int, Set{Int}, Vector{Int}.

If closed is set to true then seeds is assumed to be closed under the action of G. In this case, collect(Omega) is guaranteed to be equal to collect(seeds); in particular, the ordering of points in seeds (if applicable) is kept. Note that the indexing of points in Omega is used by action_homomorphism.

Examples

julia> G = symmetric_group(4);

julia> length(gset(G, [[1]]))  # natural action
4

julia> length(gset(G, [[1, 2]]))  # action on ordered pairs
12

julia> length(gset(G, on_sets, [[1, 2]]))  # action on unordered pairs
6

source
permutationFunction
permutation(Omega::GSetByElements{T}, g::BasicGAPGroupElem{T}) where T<:GAPGroup

Return the element of the permutation group that describes the action of g on Omega, where g is an element of acting_group(Omega).

Examples

julia> G = symmetric_group(4);

julia> Omega = gset(G, [[1, 2]]);

julia> x = gen(G, 1)
(1,2,3,4)

julia> permutation(Omega, x)
(1,2,4,7)(3,6,9,12)(5,8,10,11)

source
action_homomorphismMethod
action_homomorphism(Omega::GSetByElements{T}) where T<:GAPGroup

Return the group homomorphism act with domain G = acting_group(Omega) and codomain symmetric_group(n) that describes the permutation action of G on Omega, where Omega has n elements.

This means that if an element g in G maps collect(Omega)[i] to collect(Omega)[j] then act(g) maps i to j.

Examples

julia> G = symmetric_group(6);

julia> Omega = gset(G, [Set([1, 2])]);  # action on unordered pairs

julia> acthom = action_homomorphism(Omega)
Group homomorphism from
Sym( [ 1 .. 6 ] )
to
Sym( [ 1 .. 15 ] )

julia> g = gens(G)[1]
(1,2,3,4,5,6)

julia> elms = collect(Omega);

julia> actg = acthom(g)
(1,2,3,5,7,10)(4,6,8,11,14,13)(9,12,15)

julia> elms[1]^g == elms[2]
true

julia> 1^actg == 2
true

source
orbitMethod
orbit(Omega::GSet, omega::T) where T

Return the G-set that consists of the elements fun(omega, g) where g is in the group of Omega and fun is the underlying action of Omega.

Examples

julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
Group([ (1,2), (3,4), (1,3)(2,4), (5,6) ])

julia> Omega = gset(G, [1, 5]);

julia> length(orbit(Omega, 1))
4

source
orbitMethod
orbit(G::GAPGroup[, fun::Function], omega)

Return the G-set that consists of the images of omega under the action of G defined by fun.

This means that the result contains all elements fun(omega, g) for g in G.

fun can be omitted if the type of Omega implies a reasonable default, for example, if G is a PermGroup and omega is one of Int, Set{Int}, Vector{Int}.

Examples

julia> G = symmetric_group(4);

julia> length(orbit(G, 1))
4

julia> length(orbit(G, [1, 2]))
12

julia> length(orbit(G, on_sets, [1, 2]))
6

source
orbitsMethod
orbits(Omega::GSet)

Return the vector of transitive G-sets in Omega.

Examples

julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
Group([ (1,2), (3,4), (1,3)(2,4), (5,6) ])

julia> orbs = orbits(gset(G));

julia> map(collect, orbs)
2-element Vector{Vector{Int64}}:
[1, 2, 3, 4]
[5, 6]

source

## Stabilizers

stabilizerMethod
stabilizer(G::Oscar.GAPGroup, pnt::Any[, actfun::Function])

Return the subgroup of G that consists of all those elements g that fix pnt under the action given by actfun, that is, actfun(pnt, g) == pnt holds.

The default for actfun depends on the types of G and pnt: If G is a PermGroup then the default actions on integers, Vectors of integers, and Sets of integers are given by ^, on_tuples, and on_sets, respectively. If G is a MatrixGroup then the default actions on FreeModuleElems, Vectors of them, and Sets of them are given by *, on_tuples, and on_sets, respectively.

Examples

julia> G = symmetric_group(5);

julia> S = stabilizer(G, 1);  order(S[1])
24

julia> S = stabilizer(G, [1, 2]);  order(S[1])
6

julia> S = stabilizer(G, Set([1, 2]));  order(S[1])
12

julia> S = stabilizer(G, [1,1,2,2,3], permuted);  order(S[1])
4

source


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#### IEEE-USA E-Books

• Abst ract We pres ent the rece nt adva nces mad e tow ards the reali zati on of elect ron and hole qua ntu m bit devi ces on a Silicon-On-Insulator (SOI) technology. Such devices are obtained by slightly modifying our standard process flow for nanowire transistor fabrication. Recent developments on electrical control of the qubits are reviewed.

• Quantum key distribution (QKD) uses the laws of quantum mechanics to distribute secret keys with information-theoretic security. To realize QKD for secure communications among multiple distant users, fiber-based QKD networks can be deployed with today's available technologies. Incorporating QKD into existing wavelength-division multiplexing (WDM) network infrastructure provides a practical way to reduce the difficulty and cost of QKD networking. Nonetheless, the costs of QKD nodes and QKD links are still considered as the major barriers to long-distance QKD network deployment in practice. In this work, we study how to minimize the cost of deploying QKD-over-WDM backbone networks. We introduce a general architecture of QKD-over-WDM backbone networks. A novel cost-oriented model is defined to elaborate the cost- minimized problem, in which various QKD network components including QKD transceivers, QKD auxiliary equipment for QKD backbone nodes and trusted repeater nodes (TRNs), and QKD links are considered. The physical-layer parameters such as secret-key rate, physical distance, and the layout of TRNs are also incorporated in our defined model. We develop an integer linear programming (ILP) model and a novel cost-efficient QKD networking (CEQN) heuristic algorithm to address the cost-minimized problem. The extensive simulations indicate that our proposed approaches are effective in reducing the cost of deploying QKD-over-WDM backbone networks, where the designed CEQN algorithm is efficient to demonstrate similar to the ILP model. Finally, some open issues in future work are discussed, given that this work only presents an early highlevel analysis as opposed to a definitive detailed model.

• We present an overview of Majorana qubits based on one-dimensional semiconducting nanowires partially covered with a conventional superconductor. Majorana zero modes emerge at the wire ends when this hybrid system transitions from a conventional superconducting phase to a topological phase, in general occurring on increasing a magnetic field. For sufficiently long wires different Majoranas are fully independent and Majorana-based qubit states become topologically protected, which make them insensitive to local sources of noise. We present qubit designs, materials and device development and ongoing experimental efforts.

• Secure keys in optical networks integrated with quantum key distribution (QKD) are virtualized to construct secure VONs (SVONs). The QKD-integrated optical network virtualization architecture and SVON embedding algorithm are proposed.

• Molecular nanomagnets can be considered as serious candidates for the definition of a scalable and reliable quantum computing technology: information is encoded on their spins and they ensure relaxation and decoherence times sufficiently long for the execution of tens of quantum operations. Among these devices, $\text{Cr}_7\text{Ni}$ supramolecular complexes are extremely of interest since they implement a universal set of quantum gates, made of single-qubit gates and the two-qubit Controlled-phase gate. A model for analyzing $\text{Cr}_7\text{Ni}$ molecules and a potential quantum computer architecture are proposed: starting from the energy parameters required for spin manipulations, these systems have been proved to implement elementary quantum algorithms. Within this paper, two, three and four-qubit systems have been analyzed: the simulation of quantum operations and the analysis of dynamical non-idealities have been done in parallel. Our approach enables the definition of an operating point - in terms of magnetic driving fields and latency - in which the execution of elementary quantum algorithms is characterized by negligible errors. The proposed model has been entirely implemented in MATLAB, thus obtaining a software infrastructure for the analysis of $\text{Cr}_7\text{Ni}$ molecules that can be extended to quantum systems with analogous Hamiltonian and behavior.

• In this paper, a hybrid quantum genetic algorithm (QGA) is proposed to solve a new job shop scheduling problem. The improvement of quantum genetic algorithm is as follows: the coding of job shop scheduling problem (JSSP) is analyzed; a new quantum rotation gate is introduced to update quantum chromosome. Finally, the simulation examples show that the new quantum rotation gate improves the traditional genetic algorithm and provides powerful support for the scheduling job shop.

• In this paper, we propose a simple neural net that requires only O(n log_2k) number of qubits and O(nk) quantum gates: Here, N is the number of input parameters, and k is the number of weights applied to these parameters in the proposed neural net. We describe the network in terms of a quantum circuit, and then draw its equivalent classical neural net which involves O(k^n) nodes in the hidden layer. Then, we show that the network uses a periodic activation function of cosine values of the linear combinations of the inputs and weights. The backpropagation is described through the gradient descent, and then iris and breast cancer datasets are used for the simulations. The numerical results indicate the network can be used in machine learning problems and it may provide exponential speedup over the same structured classical neural net.

• Superconducting parametric amplifier, originally developed more than half a century ago, gained renewed interests in some experiments of superconducting quantum electronics about a decade ago, and now has become an indispensable tool in the field of superconducting quantum information processing. More recently, there are several proposals, where a parametric oscillator, which is a parametric amplifier pumped above the threshold, is used as a quantum bit and the network of the parametric oscillators solve some computational tasks. Here, we briefly introduce the research activity on the development of the superconducting parametric devices, including our results, with some historical backgrounds.

• Firefly algorithm is a new optimization technique based on swarm intelligence. It simulates the social behavior of fireflies. The search pattern of firefly algorithm is determined by the attractions among fireflies, whereby a less bright firefly moves toward a brighter firefly. In firefly algorithm, each firefly can be attracted by all other brighter fireflies in the population. But firefly algorithm is similar to other swarm intelligence algorithms; the performance of firefly algorithm is poor in high dimensional problems. It has low local search accuracy and is easy to fall into local extremum in some case. To overcome these problems, the quantum theory and mutation operation was used to improve firefly algorithm, a quantum-inspired hybrid firefly algorithm was proposed. In proposed algorithm, each quantum firefly can express two position of solution space, location update is implemented by quantum gate calculation, the mutation operation is used to jump out of the local extremum. Optimization Experiments are conducted using well-known benchmark functions. The results show that the proposed algorithm can efficiently improve the global search capability and the accuracy of solutions.

• A programmable quantum computer based on trapped ions interacting via magnetic gradient induced coupling (MAGIC) together with elements for scaling quantum computing - transport of ions and a novel trap for 2D ion arrays - are reported. © 2019 The Author(s).


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# How can I claim a one-sided limit doesn't exist?

I have to find the limit $$\lim_{x\to 0^+} \frac{\ln(1+2x)\sin x}{\sqrt {x^3}}$$

Now, I tried using L'Hôpital's rule, but it doesn't lead anywhere.

Manually trying to convert the functions to another form doesn't seem to go anywhere either, so I determined that the limit must be undefined.

However, I cannot prove it. What can I do?

• Using Taylor, the numerator is $2x\cdot x+o(x^3)$ and the limit is $0$. – Yves Daoust Jan 30 at 16:37
• We haven't been taught about Taylor yet, I should add this to my question. However, thanks for the answer, i'll research Taylor on my own. – sdds Jan 30 at 16:40
• To answer the question in the title, see for example here – BlueRaja - Danny Pflughoeft Jan 30 at 21:37
• After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $\checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?. – Clement C. Feb 3 at 23:26

Hint:

Use the facts that (1) $$\lim_{x\to 0} \frac{\ln(1+2x)}{2x} = 1$$, (2) $$\lim_{x\to 0} \frac{\sin x}{x} = 1$$, and (3) $$\lim_{x\to 0^+} \frac{x^2}{\sqrt{x^3}} = 0$$ to show the limit is $$2\cdot 1\cdot 0 = 0$$.

To do so, rewrite, for $$x>0$$, $$\frac{\ln(1+2x)\sin x}{\sqrt{x^3}}=2\cdot \frac{\ln(1+2x)}{2x}\cdot\frac{\sin x}{x}\cdot\frac{x^2}{\sqrt{x^3}}$$

• Thank you, this conversion never occurred to me – sdds Jan 30 at 16:37
• @sdds You're welcome! – Clement C. Jan 30 at 16:40

$$\frac{\ln(1+2x)\sin{x}}{\sqrt{x^3}}=\frac{\ln(1+2x)}{2x}\cdot\frac{\sin{x}}{x}\cdot2\sqrt{x}\rightarrow0.$$

It is known that

$$\lim_{x\to 0^+}\frac{\sin x}x=1$$ and $$\lim_{x\to 0^+}\frac{\log(1+x)}x=1=\lim_{x\to 0^+}\frac{\log(1+2x)}{2x}.$$

Then

$$\lim_{x\to 0^+} \frac{\ln(1+2x)\sin x}{\sqrt {x^3}}=1\cdot2\cdot\lim_{x\to 0^+} \sqrt x.$$

• It's true, but how is that different from the current two answers? – Clement C. Jan 30 at 16:45

If you want to, you should apply L'Hospital's rule twice: $$\lim_{x\to 0^+} \frac{\ln(1+2x)\sin x}{\sqrt {x^3}}=\\ \lim_{x\to 0^+} \frac{\frac{2}{1+2x}\sin x+\ln(1+2x)\cos x}{1.5\sqrt {x}}=\\ \lim_{x\to 0^+} \frac{\frac{-4}{(1+2x)^2}\sin x+\frac{2}{1+2x}\cos x+\frac{2}{1+2x}\cos x-\ln(1+2x)\sin x}{\frac{0.75}{\sqrt {x}}}=0.$$


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# Search for direct production of electroweakinos in final states with missing transverse momentum and a Higgs boson decaying into photons in pp collisions at root s=13 TeV with the ATLAS detector

Aad, G, Abbott, B, Abbott, DC, Abud, A, Abeling, K, Abhayasinghe, DK, Abidi, SH, AbouZeid, OS, Abraham, NL, Abramowicz, H
et al (show 2946 more authors) (2020) Search for direct production of electroweakinos in final states with missing transverse momentum and a Higgs boson decaying into photons in pp collisions at root s=13 TeV with the ATLAS detector. JOURNAL OF HIGH ENERGY PHYSICS (10).

Access the full-text of this item by clicking on the Open Access link.


text:
• Role of MnO in manganese–borate binary glass systems: a study on structure and thermal properties

• # Fulltext

Permanent link:
https://www.ias.ac.in/article/fulltext/boms/040/05/0933-0938

• # Keywords

Mn; infrared (IR) spectroscopy; glass; thermal properties.

• # Abstract

Structural and thermal properties of $x$MnO−($100−x$)B$_2$O$_3$ (where $x = 40, 50$ and 60 mol%) glass samples have been investigated with the employment of various techniques. Fourier transform infrared spectroscopy results revealed the influence of MnO on glass matrix. Decrease of B–O bond-related band intensities has been observed. MnO addition was found to introduce broken [BO$_2$O$^−$]$_n$ chains. Differential scanning calorimetry (DSC) measurements presented decreasing $T_g$ that indicates depolymerization of glass matrix in the considered compositional range. Moreover, thermal stability (TS) parameter has been evaluated using the DSC technique. It slightly decreased with MnO content. X-ray photoelectron spectroscopy results provided the evidence for Mn$^{2+}$ and Mn$^{3+}$ presence. Multiplet splitting, close to that of MnO, has been observed. It has been concluded that most of the manganese ions existed in the divalent state. Photoluminescence study revealed that manganese ions are tetragonally co-ordinated in a glassy matrix.

• # Author Affiliations

1. Department of Solid State Physics, Gdansk University of Technology, 80-233 Gdansk, Poland

• # Bulletin of Materials Science

Current Issue
Volume 42 | Issue 5
October 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019

Click here for Editorial Note on CAP Mode

© 2017-2019 Indian Academy of Sciences, Bengaluru.


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# Tag Info

6

This is well covered in the corresponding chapter of Russell & Norvig (chapter 3.5, pages 93 to 99 (Third Edition)). Check that out for more details. First, let's review the definitions: Your definitions of admissible and consistent are correct. An admissible heuristic is basically just "optimistic". It never overestimates a distance. A consistent ...

5

Yes. If you leave A* running (i.e. do not impose a goal condition on a newly-encountered state), all states will be explored, just as they would be in breadth- or depth- first search.

4

A* is a best-first search algorithm, which means that it is an algorithm that uses both "past knowledge", gathered while exploring the search space, denoted by $g(n)$, and an admissible heuristic function, denoted by $h(n)$, which estimates the distance to the goal node, for each node $n$. There are other best-first search algorithms, which differ only in ...

4

Yes, UCS is a special case of A*. UCS uses the evaluation function $f(n) = g(n)$, where $g(n)$ is the length of the path from the starting node to $n$, whereas A* uses the evaluation function $f(n) = g(n) + h(n)$, where $g(n)$ means the same thing as in UCS and $h(n)$, called the "heuristic" function, is an estimate of the distance from $n$ to the goal ...

3

The evaluation function in A* is $f(n) = g(n) + h(n)$, where $g(n)$ is the cost of the path from the starting node to $n$ and $h(n)$ is an estimate of the distance from $n$ to the goal node. To compute $g(n)$, you simply do $g(n) = g(p) + c(p, n)$, where $p$ is the parent node, and $c(p, n)$ is the $c$ost of the edge between $p$ and $n$. So, yes, to compute $... 3 Both algorithms fall into the category of "best-first search" algorithms, which are algorithms that can use both the knowledge acquired so far while exploring the search space, denoted by$g(n)$, and a heuristic function, denoted by$h(n)$, which estimates the distance to the goal node, for each node$n$in the search space (often represented as a graph). ... 2 According to the book Artificial Intelligence: A Modern Approach (3rd edition), by Stuart Russel and Peter Norvig, specifically, section 3.5.1 Greedy best-first search (p. 92) Greedy best-first search tries to expand the node that is closest to the goal, on the grounds that this is likely to lead to a solution quickly. Thus, it evaluates nodes by using ... 2 You forgot to calculate and take into account the costs of the actual paths. You forgot to accumulate the cost of the edges for going forward and backward multiple times! The evaluation function of uniform-cost search (UCS) is$f(n) = g(n)$, where$g(n)$represents the cost of the path from the start node to$n$. The evaluation function of A* is$f(n) = g(n)...

2

The key phrase here is because heuristics are admissible In other words, the heuristics never overestimate the path length: $$cost(n) + heuristic(n) \le cost(\text{any path going through n})$$ And since the frontier is ordered by $\textbf{cost + heuristic}$, when a completed path $p$ is dequeued from the frontier, we know that it must necessarily be $\... 1 Check below reference url for A* algorithms ... https://takinginitiative.wordpress.com/2011/05/02/optimizing-the-a-algorithm/ https://en.wikipedia.org/wiki/Heap_%28data_structure%29 1 The first step of optimisation is to measure where inside the implementation most time is spent -- you don't actually optimise the algorithm itself, but a specific implementation of it. This step should give you an overview of where you can make improvements. Speculative changes usually don't do much. There are several aspects of A* which would be ... 1 I will use the 8-puzzle game to show you why Nilson's sequence score heuristic function is not admissible. In the 8-puzzle game, you have a$3 \times 3\$ board of (numbered) squares as follows. +---+---+---+ | 0 | 1 | 2 | +---+---+---+ | 7 | 8 | 3 | +---+---+---+ | 6 | 5 | 4 | +---+---+---+ The numbers in these squares are just used to refer to the ...

1

A* is an informed search algorithm. A* informed because it is based on the use of a heuristic function, which estimates the distance of each node to the goal, that is, the heuristic function provides information about the distance from any node to the goal node. The heuristic function can e.g. be the Euclidean distance (in the case this can be defined). ...

1

What you said isn't totally wrong, but the A* algorithm becomes optimal and complete if the heuristic function h is admissible, which means that this function never overestimates the cost of reaching the goal. In that case, the A* algorithm is way better than the greedy search algorithm.

1

Question 1: First of all, you state that that the goal G2 will be found first by relying on the expansion order R, B, D, G2. This is wrong. It is extremely easy to see that this is wrong, because A* is a search algorithm that guarantees to find an optimal solution given that only admissible heuristics are used. (A heuristic is being admissible if it never ...

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


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# The Ksp for Zn(OH)2 is 5.0x10^-17 . Determine the molar solubility of zn(OH)2 in a buffer solution with pH of 11.5 ?

You have the K_(sp) expression, which was explained in the other thread http://socratic.org/questions/in-which-of-the-following-aqueous-solutions-would-you-expect-to-agbr-to-have-the168330.

Here we know the [OH^-] value (how; what is the pH?). Therefore you have the tools to solve for [Zn^(2+)], whose concentration represents the solubility of zinc hydroxide. Remember that hydroxide ion concentration is squared in the K_(sp) expression.

As an extension, it may be speculated that we could reduce the [Zn^(2+)] value to any degree, simply by increasing [OH^-]. Note that at very high levels of [OH^-], we are likely to form complex ions such as, Zn(OH)_4^(2-) (I assure you this species is real). This is a competing equilibrium that can occur.


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# LOGICAL FOUNDATIONS OF PHYSICS. PART I. ZENO’S DICHOTOMY PARADOX,SUPERTASKS AND THE FAILURE OF CLASSICAL INFINITARY REASONING

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-09 3:56am GMT

Srinivasan, Radhakrishnan (2019) LOGICAL FOUNDATIONS OF PHYSICS. PART I. ZENO’S DICHOTOMY PARADOX,SUPERTASKS AND THE FAILURE OF CLASSICAL INFINITARY REASONING. [Preprint]

# How dark matter came to matter

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-09 3:54am GMT

de Swart, J. G. and Bertone, G. and van Dongen, J. (2017) How dark matter came to matter. Nature Astronomy, 1 (3). 0059. ISSN 2397-3366

# Proton Decay and the Quantum Structure of Spacetime. (arXiv:1903.02940v1 [hep-th])

## hep-th updates on arXiv.org

on 2019-3-09 1:23am GMT

Virtual black holes in noncommutative spacetime are investigated using coordinate coherent state formalism such that the event horizon of black hole is manipulated by smearing it with a Gaussian of width $\sqrt \theta$, where $\theta$ is the noncommutativity parameter. Proton lifetime, the main associated phenomenology of the noncommutative virtual black holes, has been studied: first in $4$ dimensional spacetime and then generalized to $D$ dimensions. The lifetime depends on $\theta$ and the number of spacetime dimensions such that it emphasizes on the measurement of proton lifetime as a potential probe for the micro-structure of spacetime.

# Higher Structures in Algebraic Quantum Field Theory. (arXiv:1903.02878v1 [hep-th])

## hep-th updates on arXiv.org

on 2019-3-09 1:23am GMT

Authors: Marco BeniniAlexander Schenkel

A brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.

# Born rule and Schr\”odinger equation from first principles. (arXiv:1903.03077v1 [quant-ph])

## quant-ph updates on arXiv.org

on 2019-3-09 1:23am GMT

Authors: Robert Oeckl (CCM-UNAM)

A concise review of the derivation of the Born rule and Schr\”odinger equation from first principles is provided. The starting point is a formalization of fundamental notions of measurement and composition, leading to a general framework for physical theories known as the positive formalism. Consecutively adding notions of spacetime, locality, absolute time and causality recovers the well established convex operational framework for classical and quantum theory. Requiring the partially ordered vector space of states to be an anti-lattice one obtains quantum theory in its standard formulation. This includes Born rule and Schr\”odinger equation.

# The best of many worlds, or, is quantum decoherence the manifestation of a disposition?

## ScienceDirect Publication: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

on 2019-3-09 1:21am GMT

Publication date: Available online 4 March 2019

Source: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Author(s): Florian J. Boge

##### Abstract

In this paper I investigate whether the phenomenon of quantum decoherence, the vanishing of interference and detectable entanglement on quantum systems in virtue of interactions with the environment, can be understood as the manifestation of a disposition. I will highlight the advantages of this approach as a realist interpretation of the quantum formalism, and demonstrate how such an approach can benefit from advances in the metaphysics of dispositions. I will also confront some commonalities with and differences to the many worlds interpretation, and address the difficulties induced by quantum non-locality. I conclude that there are ways to deal with these issues and that the proposal hence is an avenue worth pursuing.

# Going quantum with radio waves

## Science current issue

on 2019-3-07 6:41pm GMT

# Schematizing the Observer and the Epistemic Content of Theories

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-07 9:57am GMT

Curiel, Erik (2019) Schematizing the Observer and the Epistemic Content of Theories. [Preprint]

# No Chances in a Deterministic World

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-07 9:53am GMT

McCoy, C.D. (2019) No Chances in a Deterministic World. [Preprint]

# Quantum Monism Could Save the Soul of Physics

## Scientific American Content: Global

on 2019-3-05 5:00pm GMT

The multiverse may be an artifact of a deeper reality that is comprehensible and unique

— Read more on ScientificAmerican.com

show enclosure

# The Universe’s Ultimate Complexity Revealed by Simple Quantum Games

## Quanta Magazine

on 2019-3-05 3:40pm GMT

One of the biggest and most basic questions in physics involves the number of ways to configure the matter in the universe. If you took all that matter and rearranged it, then rearranged it again, then rearranged it again, would you ever exhaust the possible configurations, or could you go on reconfiguring forever?

Physicists don’t know, but in the absence of certain knowledge, they make assumptions. And those assumptions differ depending on the area of physics they happen to be in. In one area they assume the number of configurations is finite. In another they assume it’s infinite. For now, at least, there’s no way to tell who’s right.

But over the last couple years, a select group of mathematicians and computer scientists has been busy creating games that could theoretically settle the question. The games involve two players placed in isolation from each other. The players are asked questions, and they win if their answers are coordinated in a certain way. In all of these games, the rate at which players win has implications for the number of different ways the universe can be configured.

“There’s this philosophical question: Is the universe finite or infinite-dimensional?” said Henry Yuen, a theoretical computer scientist at the University of Toronto. “People will think this is something you can never test, but one possible way of resolving this is with a game like what William came up with.”

Yuen was referring to William Slofstra, a mathematician at the University of Waterloo. In 2016 Slofstra invented a game that involves two players who assign values to variables in hundreds of simple equations. Under normal circumstances even the most cunning players will sometimes lose. But Slofstra proved that if you give them access to an infinite amount of an unorthodox resource — entangled quantum particles — it becomes possible for the players to win this game all the time.

Other researchers have since refined Slofstra’s result. They’ve proved that you don’t need a game with hundreds of questions to reach the same conclusion Slofstra did. In 2017 three researchers proved that there are games with just five questions that can be won 100 percent of the time if the players have access to an unlimited number of entangled particles.

These games are all modeled on games invented more than 50 years ago by the physicist John Stewart Bell. Bell developed the games to test one of the strangest propositions about the physical world made by the theory of quantum mechanics. A half-century later, his ideas may turn out to be useful for much more than that.

## Magic Squares

Bell came up with “nonlocal” games, which require players to be at a distance from each other with no way to communicate. Each player answers a question. The players win or lose based on the compatibility of their answers.

One such game is the magic square game. There are two players, Alice and Bob, each with a 3-by-3 grid. A referee tells Alice to fill out one particular row in the grid — say the second row — by putting either a 1 or a 0 in each box, such that the sum of the numbers in that row is odd. The referee tells Bob to fill out one column in the grid — say the first column — by putting either a 1 or a 0 in each box, such that the sum of the numbers in that column is even. Alice and Bob win the game if Alice’s numbers give an odd sum, Bob’s give an even sum, and — most important — they’ve each written down the same number in the one square where their row and column intersect.

Here’s the catch: Alice and Bob don’t know which row or column the other has been asked to fill out. “It’s a game that would be trivial for the two players if they could communicate,” said Richard Cleve, who studies quantum computing at the University of Waterloo. “But the fact that Alice doesn’t know what question Bob was asked and vice versa means it’s a little tricky.”

In the magic square game, and other games like it, there doesn’t seem to be a way for the players to win 100 percent of the time. And indeed, in a world completely explained by classical physics, 89 percent is the best Alice and Bob could do.

But quantum mechanics — specifically, the bizarre quantum phenomenon of “entanglement” — allows Alice and Bob to do better.

In quantum mechanics, the properties of fundamental particles like electrons don’t exist until the moment you measure them. Imagine, for example, an electron moving rapidly around the circumference of a circle. To find its position you perform a measurement. But prior to the measurement, the electron has no definite position at all. Instead, the electron is characterized by a mathematical formula expressing the likelihood that it’s in any given position.

When two particles are entangled, the complex probability amplitudes that describe their properties are intertwined. Imagine two electrons that were entangled such that if a measurement identifies the first electron in one position around the circle, the other must occupy a position directly across the circle from it. This relationship between the two electrons holds when they’re right next to each other and when they’re light-years apart: Even at that distance, if you measure the position of one electron, the position of the other becomes instantly determined, even though no causal event has passed between them.

The phenomenon seems preposterous because there’s nothing about our non-quantum-scale experience to suggest such a thing is possible. Albert Einstein famously derided entanglement as “spooky action at a distance” and argued for years that it couldn’t be true.

To implement a quantum strategy in the magic square game, Alice and Bob each take one of a pair of entangled particles. To determine which numbers to write down, they measure properties of their particles — almost as if they were rolling correlated dice to guide their choice of answers.

What Bell calculated, and what many subsequent experiments have shown, is that by exploiting the strange quantum correlations found in entanglement, players of games like the magic square game can coordinate their answers with greater exactness and win the game more than 89 percent of the time.

Bell came up with nonlocal games as a way to show that entanglement was real, and that our classical view of the world was incomplete — a conclusion that was very much up for grabs in Bell’s time. “Bell came up with this experiment you could do in a laboratory,” Cleve said. If you recorded higher-than-expected success rates in these experimental games, you’d know the players had to be exploiting some feature of the physical world not explained by classical physics.

What Slofstra and others have done since then is similar in strategy, but different in scope. They’ve shown that not only do Bell’s games imply the reality of entanglement, but some games have the power to imply a whole lot more — like whether there is any limit to the number of configurations the universe can take.

## More Entanglement, Please

In his 2016 paper Slofstra proposed a kind of nonlocal game involving two players who provide answers to simple questions. To win, they have to give responses that are coordinated in a certain way, as in the magic square game.

Imagine, for example, a game that involves two players, Alice and Bob, who have to match socks from their respective sock drawers. Each player has to choose a single sock, without any knowledge of the sock the other has chosen. The players can’t coordinate ahead of time. If their sock choices form a matching pair, they win.

Given these uncertainties it’s unclear which socks Alice and Bob should pick in the morning — at least in a classical world. But if they can employ entangled particles they have a better chance of matching. By basing their color choice on the results of measurements of a single pair of entangled particles they could coordinate along that one attribute of their socks.

Yet they’d still be guessing blindly about all the other attributes — whether they were wool or cotton, ankle-height or crew. But with additional entangled particles they’d get access to more measurements. They could use one set to correlate their choice of material and another to correlate their choice of sock height. In the end, because they were able to coordinate their choices for many attributes, they’d be more likely to end up with a matching pair than if they’d only been able to coordinate for one.

“More complicated systems allow for more correlated measurements, which enable coordination at more complicated tasks,” Slofstra said.

The questions in Slofstra’s game aren’t really about socks. They involve equations such as a + b + c and b + c + d. Alice can make the value of each variable either 1 or 0 (and the values have to remain consistent across the equations — b has to have the same value in every equation where it appears). And her equations have to sum to various numbers.

Bob is given just one of Alice’s variables, say b, and asked to assign a value to it: 0 or 1. The players win if they both assign the same value to whichever variable Bob is given.

If you and a friend were to play this game, there’s no way you could win it all the time. But with the aid of a pair of entangled particles, you could win more consistently, as in the sock game.

Slofstra was interested in understanding whether there is an amount of entanglement past which a team’s winning probability stops increasing. Perhaps players could achieve an optimal strategy if they shared five pairs of entangled particles, or 500. “We’d hoped you could say, ‘You need this much entanglement to play it optimally,’” Slofstra said. “That’s not what is true.”

He found that adding more pairs of entangled particles always increased the winning percentage. Moreover, if you could somehow exploit an infinite number of entangled particles, you would be able to play the game perfectly, winning 100 percent of the time. This clearly can’t be done in a game with socks — ultimately you’d run out of sock features to coordinate. But as Slofstra’s game has made clear, the universe can be far knottier than a sock drawer.

## Is the Universe Infinite?

Slofstra’s result came as a shock. Eleven days after his paper appeared, the computer scientist Scott Aaronsonwrote that Slofstra’s result touches “on a question of almost metaphysical significance: namely, what sorts of experimental evidence could possibly bear on whether the universe was discrete or continuous?”

Aaronson was referring to the different states the universe can take — where a state is a particular configuration of all the matter within it. Every physical system has its own state space, which is an index of all the different states it can take.

Researchers talk about a state space as having a certain number of dimensions, reflecting the number of independent characteristics you can adjust in the underlying system.

For example, even a sock drawer has a state space. Any sock might be described by its color, its length, its material, and how raggedy and worn it is. In this case, the dimension of the sock drawer’s state space is four.

A deep question about the physical world is whether there’s a limit to the size of the state space of the universe (or of any physical system). If there is a limit, it means that no matter how large and complicated your physical system is, there are still only so many ways it can be configured. “The question is whether physics allows there to be physical systems that have an infinite number of properties that are independent of each other that you could in principle observe,” said Thomas Vidick, a computer scientist at the California Institute of Technology.

The field of physics is undecided on this point. In fact, it maintains two contradictory views.

On the one hand, students in an introductory quantum mechanics course are taught to think in terms of infinite-dimensional state spaces. If they model the position of an electron moving around a circle, for instance, they’ll assign a probability to each point on the circle. Because there are infinite points, the state space describing the electron’s position will be infinite-dimensional.

“In order to describe the system we need a parameter for every possible position the electron can be in,” Yuen said. “There are infinitely many positions, so you need infinitely many parameters. Even in one-dimensional space , the state space of the particle is infinite-dimensional.”

But perhaps the idea of infinite-dimensional state spaces is nonsense. In the 1970s, the physicists Jacob Bekenstein and Stephen Hawking calculated that a black hole is the most complicated physical system in the universe, yet even its state can be specified by a huge but finite number of parameters — approximately 1069 bits of information per square meter of the black hole’s event horizon. This number — the “Bekenstein bound” — suggests that if a black hole doesn’t require an infinite-dimensional state space, then nothing does.

These competing perspectives on state spaces reflect fundamentally different views about the nature of physical reality. If state spaces are truly finite-dimensional, this means that at the smallest scale, nature is pixelated. But if electrons require infinite-dimensional state spaces, physical reality is fundamentally continuous — an unbroken sheet even at the finest resolution.

So which is it? Physics hasn’t devised an answer, but games like Slofstra’s could, in principle, provide one. Slofstra’s work suggests a way to test the distinction: Play a game that can only be won 100 percent of the time if the universe allows for infinite-dimensional state spaces. If you observe players winning every time they play, it means they’re taking advantage of the kinds of correlations that can only be generated through measurements on a physical system with an infinite number of independently tunable parameters.

“He gives an experiment such that, if it can be realized, then we conclude the system that produced the statistics that were observed must have infinite degrees of freedom,” Vidick said.

There are barriers to actually carrying out Slofstra’s experiment. For one thing, it’s impossible to certify any laboratory result as occurring 100 percent of the time.

“In the real world you’re limited by your experimental setup,” Yuen said. “How do you distinguish between 100 percent and 99.9999 percent?”

But practical considerations aside, Slofstra has shown that there is, mathematically at least, a way of assessing a fundamental feature of the universe that might otherwise have seemed beyond our ken. When Bell first came up with nonlocal games, he hoped that they’d be useful for probing one of the most beguiling phenomena in the universe. Fifty years later, his invention has proved to have even more depth than that.

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# The Problem of Time

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-05 9:36am GMT

Thebault, Karim P Y (2019) The Problem of Time. [Preprint]

# Did Universe Have a Chance?

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-05 9:33am GMT

McCoy, C.D. (2018) Did Universe Have a Chance? In: UNSPECIFIED.

# The Physics Still Hiding in the Higgs Boson

## Quanta Magazine

on 2019-3-04 3:11pm GMT

In 2012, particles smashed together in the Large Hadron Collider’s 27-kilometer circular tunnel conjured up the Higgs boson — the last missing particle predicted by the Standard Model of particle physics, and the linchpin that holds that decades-old set of equations together.

But no other new particles have materialized at the LHC, leaving open many mysteries about the universe that the Standard Model doesn’t address. A debate has ensued over whether to build an even more enormous successor to the LHC — a proposed machine 100 kilometers in circumference, possibly in Switzerland or China — to continue the search for new physics.

Physicists say there’s much we can still learn from the Higgs boson itself. What’s known is that the particle’s existence confirms a 55-year-old theory about the origin of mass in the universe. Its discovery won the 2013 Nobel Prize for Peter Higgs and François Englert, two of six theorists who proposed this mass-generating mechanism in the 1960s. The mechanism involves a field permeating all of space. The Higgs particle is a ripple, or quantum fluctuation, in this Higgs field. Because quantum mechanics tangles up the particles and fields of nature, the presence of the Higgs field spills over into other quantum fields; it’s this coupling that gives their associated particles mass.

But physicists understand little about the omnipresent Higgs field, or the fateful moment in the early universe when it suddenly shifted from having zero value everywhere (or in other words, not existing) into its current, uniformly valued state. That shift, or “symmetry-breaking” event, instantly rendered quarks, electrons and many other fundamental particles massive, which led them to form atoms and all the other structures seen in the cosmos.

But why? “Why should the universe decide to have this Higgs presence all over? That is a big, big question,” said Michelangelo Mangano, a particle theorist at CERN, the laboratory that houses the LHC.

Physicists wonder whether the Higgs symmetry-breaking event had a role in creating the universe’s matter-antimatter asymmetry — the unexplained fact that so much more matter exists than antimatter. Another question is whether the Higgs field’s current value is stable or could suddenly change again — an unsettling prospect known as “vacuum decay.” The value of the Higgs field can be thought of as a ball settled at the bottom of a valley. The question is, are there yet deeper valleys in the mathematical curve that defines the field’s possible values? If so, the ball will eventually tunnel to the lower, more stable valley, corresponding to a drop in the energy of the Higgs field. A bubble of the more stable “true vacuum” would grow and encompass the “false vacuum” that we’ve been living in, obliterating everything.

Not only is the Higgs field tied to the origin and fate of the universe, but the Higgs particle’s behavior can also reveal hidden or otherwise unknown particles that it interacts with — perhaps those that make up the cosmos’s missing dark matter. In a particle collider, when particles smash together at nearly light speed, their kinetic energy converts into matter, occasionally forming heavy particles such as the Higgs boson. This Higgs then quickly morphs into other particles, such as a pair of top quarks or W bosons, where the probability of each outcome depends on the strength of the Higgs’ coupling to each type of particle. Precisely measuring the probabilities of these different Higgs decays and comparing the numbers to Standard Model predictions reveal if anything is missing, since the probabilities must add up to one.

“The more we study , the more we may find that this whole story might not fit together exactly as we expect, which would lead to new physics,” said Melissa Franklin, a particle physicist at Harvard University. “From an experimental point of view, we just want to make a bunch of them and see what happens.”

That’s one reason she and many of her colleagues want to build a bigger, better machine. The first phase of the proposed supercollider has been nicknamed the “Higgs factory,” because the machine would collide electrons and positrons with energies precisely tuned to maximize their chance of yielding Higgs bosons, whose subsequent decays could be measured in detail. In phase two, the giant machine would slam together protons, resulting in messier but much more energetic collisions.

With the LHC, most of the Higgs boson’s couplings with other Standard Model particles have been measured with roughly 20 percent precision, but a future collider, by producing many more Higgs bosons, could pin the numbers down with an accuracy of 1 percent. This would give physicists a much better sense of whether the probabilities add to one, or whether Higgs bosons are occasionally decaying into hidden particles. Extra particles coupled to the Higgs appear in many theories of physics beyond the Standard Model, including the “twin Higgs” and “relaxion”models. “Unfortunately, there are so many models and so many parameters that there is no hope of a no-lose theorem,” said the particle physicist Matt Strassler — “just a might-win opportunity.”

Perhaps the most important coupling that physicists want to nail down is called the triple Higgs coupling — essentially the strength of the Higgs boson’s interaction with itself. This number is measured by counting rare events, not yet seen at the LHC, in which a Higgs boson decays into two of itself. The Standard Model makes a prediction for the value of the triple Higgs coupling, so any measured deviations from this prediction would signify the existence of new particles not included in the Standard Model that affect the Higgs.

Measuring the triple Higgs coupling would also reveal the shape of the mathematical curve that defines the Higgs field’s different possible values, helping to determine whether the vacuum of our universe is stable or only metastable — settled in a local rather than a global minimum of the curve. If the Standard Model’s prediction for the coupling is correct, then the universe is metastable, destined to decay billions or trillions of years from now. This is nothing to worry about, but rather an important clue about the larger story of our cosmos. The ability to reveal the universe’s fate is why the triple Higgs coupling “is at the heart of the experimental program at the future colliders,” said Cédric Weiland, a particle physicist at the University of Pittsburgh who has studied this coupling.

With a Higgs factory, Weiland said, physicists could measure the triple Higgs coupling with a precision of 44 percent. The second-phase proton-proton collider could nail its value to within 5 percent.

The baseline expectation is that measurements at a future collider will simply confirm the Standard Model, which seems frustratingly unbreakable even as it gives an incomplete account of the physical universe. Some physicists balk at the prospect of investing billions of dollars in a machine that might simply add more decimal places of precision to our knowledge of an existing set of equations.

Physicists and funding agencies will actively debate the value of an LHC successor over the next few years. Whether to spend 20 years and as many billions of dollars constructing a 100-kilometer-circumference collider hinges on its discovery potential. Past colliders struck upon the puzzle pieces of the Standard Model one by one. But with that puzzle complete, there’s no guarantee that a future machine will find anything new, leaving physicists with a dilemma: to build or not to build?

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# On the measurement process in Bohmian mechanics (reply to Gao)

## Philsci-Archive: No conditions. Results ordered -Date Deposited.

on 2019-3-04 7:32am GMT

Lazarovici, Dustin (2019) On the measurement process in Bohmian mechanics (reply to Gao). [Preprint]

# Aharonov–Bohm interference of fractional quantum Hall edge modes

## Nature Physics – Issue – nature.com science feeds

on 2019-3-04 12:00am GMT

Aharonov–Bohm interference of fractional quantum Hall edge modes

Aharonov–Bohm interference of fractional quantum Hall edge modes, Published online: 04 March 2019; doi:10.1038/s41567-019-0441-8

An interferometer device demonstrates the interference of fractional quantum Hall effect edge states. This is a big step towards braiding non-Abelian anyons.