Image Restoration and Enhancement
|
Introduction
This research project is mainly focusing on various applications of image resolution and enhancement, including image super-resolution, image denoising, image dehazing, image deblurring, and image enhancement. All of these works obtain competitive results. Until now, we have published many related papers on the flagship journals and conferences, e.g., IEEE TIP, TNNLS, IEEE TMM, IEEE TCSVT, ECCV, ICCV, ACMMM, IJCAI, etc. Our ultimate goal is to build a general image restoration model that does not need to be trained separately for different tasks. If you are interested in these works, welcome to join us.
Survey
#:Co-first authors *:Corresponding author †: First contributor, author names alphabetically.
Image Super-Resolution
#:Co-first authors *:Corresponding author †: First contributor, author names alphabetically.
-
Multi-scale Residual Network for Image Super-Resolution
Juncheng Li, Faming Fang*, Kangfu Mei, and Guixu Zhang.
European Conference on Computer Vision (ECCV), 2018.
-
An Effective Single-Image Super-Resolution Model Using Squeeze-and-Excitation Networks
Kangfu Mei, Aiwen Jiang*, Juncheng Li, Jihua Ye, Mingwen Wang
International Conference on Neural Information Processing (ICONIP), 2018.
-
Deep Residual Refining based Pseudo-Multi-Frame Network for Effective Single Image Super-Resolution
Kangfu Mei, Aiwen Jiang*, Juncheng Li, Bo Liu, Jihua Ye, Mingwen Wang
IET Image Processing, 2019.
-
Lightweight and Accurate Recursive Fractal Network for Image Super-Resolution [Oral]
Juncheng Li, Yiting Yuan, Kangfu Mei, Faming Fang*.
International Conference on Computer Vision Workshop (ICCV Workshop), 2019.
-
MDCN: Multi-scale Dense Cross Network for Image Super-Resolution
Juncheng Li, Faming Fang, Jiaqian Li, Kangfu Mei, and Guixu Zhang*.
IEEE Transactions on Circuits and Systems for Video Technology (IEEE TCSVT), 2020.
-
Soft-edge Assisted Network for Single Image Super-Resolution
Faming Fang, Juncheng Li†, and Tieyong Zeng*.
IEEE Transactions on Image Processing (IEEE TIP), 2020.
-
Lightweight Image Super-Resolution with Multi-scale Feature Interaction Network
Zhengxue Wang, Guangwei Gao*, Juncheng Li, Yi Yu, Huimin Lu
IEEE International Conference on Multimedia & Expo (ICME), 2021.
-
Feedback Network for Mutually Boosted Stereo Image Super-Resolution and Disparity Estimation
Qinyan Dai, Juncheng Li, Qiaosi Yi, Faming Fang*, and Guixu Zhang.
ACM International Conference on Multimedia (ACMMM), 2021.
-
Feature Distillation Interaction Weighting Network for Lightweight Image Super-Resolution
Guangwei Gao#, Wenjie Li#, Juncheng Li*, Fei Wu, Huimin Lu, and Yi Yu.
Association for the Advancement of Artificial Intelligence (AAAI), 2022.
-
Lightweight Bimodal Network for Single-Image Super-Resolution via Symmetric CNN and Recursive Transformer
Guangwei Gao#, Zhengxue Wang#, Juncheng Li*, Wenjie Li, Yi Yu, and Tieyong Zeng.
International Joint Conference on Artificial Intelligence (IJCAI), 2022.
-
Transformer for Single Image Super-Resolution
Zhisheng Lu#, Juncheng Li#, Hong Liu*, Chaoyan Huang, Linlin Zhang, and Tieyong Zeng
The Conference on Computer Vision and Pattern Recognition Workshop (CVPR Workshop), 2022.
-
Adjustable Super-Resolution Network via Deep Supervised Learning and Progressive Self-Distillation
Juncheng Li, Faming Fang*, Tieyong Zeng, Guixu Zhang, and Xizhao Wang.
Neurocomputing (NC), 2022.
Image Restoration under Extreme Condition
#:Co-first authors *:Corresponding author †: First contributor, author names alphabetically.
-
Higher-Resolution Network for Image Demosaicing and Enhancing
Kangfu Mei, Juncheng Li, Jiajie Zhang, Haoyu Wu, Jie Li, and Rui Huang*.
International Conference on Computer Vision Workshop (ICCV Workshop), 2019.
-
Multi-level Edge Features Guided Network for Image Denoising
Faming Fang, Juncheng Li†, Yiting Yuan, Tieyong Zeng*, and Guixu Zhang*.
IEEE Transactions on Neural Networks and Learning Systems (IEEE TNNLS), 2020.
-
Luminance-aware Pyramid Network for Low-light Image Enhancement
Jiaqian Li#, Juncheng Li#, Faming Fang*, Fang Li, and Guixu Zhang.
IEEE Transactions on Multimedia (IEEE TMM), 2020.
-
Multi-scale Grid Network for Image Deblurring with High-frequency Guidance
Yang Liu, Faming Fang*, Tingting Wang, Juncheng Li, Yun Sheng, and Guixu Zhang.
IEEE Transactions on Multimedia (IEEE TMM), 2021.
-
Multiple Degradation and Reconstruction Network for Single Image Denoising via Knowledge Distillation
Juncheng Li, Hanhui Yang, Qiaosi Yi, Faming Fang, Guangwei Gao, Tieyong Zeng*, and Guixu Zhang.
International Conference on Computer Vision Workshop (ICCV Workshop), 2022.
Image Restoration under Extreme Weather
#:Co-first authors *:Corresponding author †: First contributor, author names alphabetically.
-
Progressive Feature Fusion Network for Realistic Image Dehazing
Kangfu Mei, Aiwen Jiang*, Juncheng Li, Mingwen Wang.
Asian Conference on Computer Vision (ACCV), 2018.
-
Progressive Back-Traced Dehazing Network based on Multi-Resolution Recurrent Reconstruction
Qiaosi Yi, Aiwen Jiang*, Juncheng Li, Jianyi Wan, Mingwen Wang.
IEEE Access, 2020.
-
Efficient and Accurate Multi-scale Topological Network for Single Image Dehazing
Qiaosi Yi#, Juncheng Li#, Faming Fang*, Aiwen Jiang, Guixu Zhang
IEEE Transactions on Multimedia (TMM), 2021.
-
Structure-Preserving Deraining with Residue Channel Prior Guidance
Qiaosi Yi, Juncheng Li, Qinyan Dai, Faming Fang*, Guixu Zhang, and Tieyong Zeng.
International Conference on Computer Vision (ICCV), 2021.


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{% else %}{% trans %} This is a service for the Mozilla community, using the Mozilla OpenBadges standard and Open Source software. {% endtrans %}
{% endif %}{{ _("Click here to start:") }}
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View source
OpsSet
Enum class defining the sets of ops available to generate TFLite models.
tf.compat.v1.lite.OpsSet
tf.compat.v2.lite.OpsSet
WARNING: Experimental interface, subject to change.
SELECT_TF_OPS
TFLITE_BUILTINS
TFLITE_BUILTINS_INT8


»frag.jetzt« is funded by the Federal State of Hessen as part of the
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PyCon 2013 will be held on the Aalto University campus. The main sites will be in the Design Factory.
Otaniemi Campus map
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?TRACY
HERBARIUM,
TEXAS
AficM
UNIVERSITY
PLANTS
OF
GRENADA
Dactyloctenium
aegyptium
(L.)
Beauv.
Below
cliffs,
near
Victoria.
Alt.
near
sea
level.
Frank
W.
Gould
15188
28
March
1977
J
New
York
Botanical
Garden
lbD7107
1607107
Legend -
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val confirmationMethod: String?
One of automatic (default) or manual.
When confirmationMethod is automatic
, a PaymentIntent can be confirmed
using a publishable key. After next_action
s are handled, no additional
confirmation is required to complete the payment.
When confirmationMethod is manual
, all payment attempts must be made
using a secret key. The PaymentIntent returns to the
RequiresConfirmation
state after handling next_action
s, and requires your server to initiate each
payment attempt with an explicit confirmation.


MappedSchema |
A database schema that might be configured for this node. As well as a name and version for identifying the schema, also list the classes that may be used in the generated object graph in order to configure the ORM tool. open class MappedSchema |
UUIDConverter |
Used to convert to and from UUIDs when defining MappedSchema entity columns. object UUIDConverter : AttributeConverter<UUID, String> |
CordaPersistenceException |
Exception that encapsulates errors during Persistence. class CordaPersistenceException : CordaRuntimeException |


June 14
Except the Lord build the house, they labor in vain that build it. Psalm 127:1
What a time of monumental decisions and changes we're living in. If there ever was a day when you and I must know God's will and purpose for our lives, it's now. In fact, nothing else is going to work! God is changing and rearranging things, lining them up for the endtime move of the Spirit. If you're going to keep up, you must know God's perfect plan for your life--and how to carry it out in His power! How do you do that? Through prayer. In fact, God has provided us a prayer in His Word that we can use to receive the wisdom and understanding we need for this critical time. You'll find it in Colossians 1:9-11:
"For this cause we...do not cease to pray for you, and to desire that ye might be filled with the knowledge of [God's] will in all wisdom and spiritual understanding; That ye might walk worthy of the Lord unto all pleasing, being fruitful in every good work, and increasing in the knowledge of God; Strengthened with all might, according to his glorious power, unto all patience and longsuffering with joyfulness."
In recent months, God has instructed Gloria and me to pray that prayer for our ministry partners and for ourselves every day. It's a powerful, Holy Ghost-inspired prayer that will not only enable you to know what God's will is, but to have the wisdom and understanding to carry it out.
Once we get a glimpse of God's will for our lives in the Spirit, let's not mess things up by trying to carry it out in the flesh. Let's not try to build the house ourselves and end up doing it all in vain. Instead, let the Lord build the house in His power.
Pray this prayer that God has given us. It was written in the New Testament for you. Put your name in it. It's a prayer you can be sure God will answer. Put it to work in your life daily and God will fill you with His knowledge and understanding!
Scripture Reading: Colossians 1:9-22


The animation class can be applied to the progress tracker so each step has the same animation. Alternatively you can add the animation class to the progress step to have different animation on each step.
$ npm install progress-tracker --save-dev
After installation you can import it into your Sass files with the statement below.
@import "node_modules/progress-tracker/src/styles/progress-tracker.scss";
The JS that is part of this site is just for demonstration purposes, add your own JS as needed to toggle the classes for the step states.
Follow the HTML code example below for basic usage; each demo sets the first two steps as complete, the third step as active and the last two steps as inactive.
For additional styles add modifier classes and additional markup as needed in the examples below. You can add multiple modifier classes to achieve additional styles that those shown below.

Clone or download from Github.
$ npm install
$ gulp serve


fun toString(): String
Convert a currency Amount to a display string representation.
For example, with an amount with a quantity of "1234" GBP, returns "12.34 GBP". The result of fromDecimal is used to control the numerical formatting and the token specifier appended is taken from token.toString.
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Syria's Assad: anybody will be able to run at 2021 election
MOSCOW (Reuters) - Syrian President Bashar al-Assad said the Syrian presidential election in 2021 would be open to anybody who wants to run and that there would be numerous challengers for the presidency.
Assad, who made the comment in an interview broadcast on Monday on Russian state-funded television channel RT, faced two challengers at the 2014 election which he won by a landslide, but which his opponents dismissed as a charade.
“Last time we were three and this time of course we are going to have as much as they want to nominate. There are going to be numerous nominees,” Assad said.


Hello, I'm
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epic
website.
INTERalliance
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Worked at INTERalliance to implement a redesigned website.
Mason Student Activities
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Volunteered at my school to redesign the student activities website.
Brain Food
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Mar. 2022
Build and launch burgers in a post-apocalyptic diner.
GeoFare
Webpage
Mar. 2022
Food bank map made for HooHacks 2022.

Snake Maze
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Mar. 2020
Snake, but a maze! Get the gems to escape.
Luketris
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Mar. 2020
Tetris, but I’m the pieces!
Duck Souls
Game
Aug. 2021
A souls-like game featuring the amazing art of Gabe Salazar.
Cloud Keys
Repository
June 2019
A realtime streaming service for MIDI keyboard performances.
Tandem
Game
Sep. 2018
A cooperative rhythm game featuring original music.
Platformer Engine
Game
Dec. 2018
A custom platformer engine with camera interpolation and physics.
Conway's Game
Webpage
Dec. 2018
A small simulation of the Conway's Game Of Life simulation.


Cop 'runs' to protest transfer
by Sanjay PandeySome people run for their health while some others do it for prizes. In Uttar Pradesh, however, a cop almost ran a Marathon race to protest his transfer.
According to the reports, 30-year-old Vijay Pratap, who was a sub-inspector (SI) in the UP police, was sore at senior officials for transferring him to a remote police station in Etawah district, about 250 km from here.
Vijay Pratap, who reported for duty on Friday after remaining absent for several days without permission, was asked to join his duty at Bithauli police station, which was about 60 km away from the district headquarters.
Vijay asked the officials to cancel his transfer and post him at some other police station but they declined, reports said.
The SI was so upset with the senior officials at his transfer that he decided to run all the way to the police station to "register" his "protest".
He started his protest run at around nine in the morning, aiming to reach Bithauli by evening but he fell after completing a distance of around 30 km near Hanumantpura village and fainted.
He was rushed to the hospital by the cops at Hanumantpura police outpost, where his condition was stated to be "out of danger".
Taking his run as an act of ''indiscipline'', the district police chief has suspended Vijay. "It is an act of indiscipline... a probe has been ordered... we will take stern action against the SI," said a senior police official in Etawah.


AppBase
Class
Namespace: Imagini
Base types:
Declared in: Imagini
Base app class which instantiates a window and event loop.
Derive from this if you want to provide your own renderer for the
game loop.
Members
Property
Window
Events
TotalTime
ElapsedAppTime
IsMouseVisible
CaptureMouse
MousePosition
IsActive
TargetElapsedTime
InactiveSleepTime
MaxElapsedTime
IsFixedTimeStep
IsRunningSlowly
IsDisposed
Constructor
AppBase(WindowSettings)
AppBase(Window)
Method
CancelExitRequest()
Dispose()
RequestExit()
ResetElapsedTime()
Run()
SetupEvents()
SuppressDraw()
Terminate()
Tick()
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Meghan Duffy is an ecologist at the
University of Michigan, where she is an Associate Professor in the
Department of Ecology and Evolutionary Biology. Her research
focuses on disease ecology and aquatic ecology. She also writes
for the Dynamic
Ecology blog.


Sleep: {{avg_sleep|floatformat:1}} hrs/day
Work: {{avg_work|floatformat:1}} hrs/day ({{avg_work_per_workday|floatformat:1}} hrs/workday)
Alone: {{avg_alone|floatformat:1}} hrs/day
Friend: {{avg_friend|floatformat:1}} hrs/day
Public: {{avg_public|floatformat:1}} hrs/day
Relationship: {{avg_relationship|floatformat:1}} hrs/day
Presence: {{avg_presence|floatformat:0}}/10
Happiness: {{avg_happiness|floatformat:0}}/10
Creativity: {{avg_creativity|floatformat:0}}/10
Morning mood: {{avg_morning_mood|floatformat:0}}/10
Unbusy: {{avg_unbusy|floatformat:0}}/10
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There are a few key points for apfell that will be explained in this section:


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--- layout: documentation title: KerberosDelegationType ---
The Delegation field specifies the Kerberos delegation used for the Windows computer account.
| Field Name | Type | Description |
|---|---|---|
| Bitmask0..1 | HexBinaryObjectPropertyType |
The Bitmask field specifies the bitmask used in the Kerberos delegation for the Windows computer account. |
| Service0..1 | KerberosServiceType |
The Service field specifies the properties of the Kerberos delegation service for the Windows computer account. |


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Ant and Dec tell Piers Morgan to 'grow up' as fiery Twitter argument breaks out
Geordie pair Ant and Dec put controversial Piers Morgan in his place as they told him 'grow up you big wally' during heated series of tweets
by Brogan-Leigh HurstThe nation's favourite telly presenters Ant McPartlin and Dec Donnelly told controversial Piers Morgan to "grow up" during a huge social media argument.
Ant, 43, and Dec, 44, made a joke about Good Morning Britain's Piers on last night's first I'm A Celeb episode of the 2019 series.
The pair branded Piers, 54, a "snowflake" on their shared Twitter account after a story emerged about the hilarious moment, and it turns out the GMB presenter really wasn't pleased.
Piers himself then tweeted the Geordie lads back with a sly dig. He said: "At least I have my own Twitter account & don't have to share one with my mate."
And the boys lashed out with a savage response, saying: "Oh yeah? Well our dads are bigger than your dad.
Ant and Dec humiliated the presenter in front of their joint 6.8million followers for his childish response, and it seems they put him in place as Piers has yet to respond.
Their feisty argument has enlightened fans on Twitter who have also shared their comments.
One fascinated user penned: "Does he really think he can take on two national treasures?!?!"
While another joked: "This is the twitter war we’ve all been waiting for!!! On behalf of Britain, thank you".
And a third added: "As if the British public would ever side with Piers Morgan over Ant & Dec... this is brilliant".
One person even joked there should be a charity boxing match to settle the squabble. They wrote: "Now now boys charity boxing match will sort this #boxing works".
It comes after Piers named his price to appear on ITV's I'm a Celebrity...Get Me Out Of Here!
He responded to the Daily Mirror's article calling for him to enter the jungle this year.
He tweeted: "Like I said, £5 million and I'll come and munch kangaroo testicles. Not a penny less."
But Piers wasn't selected as part of this year's line-up and instead his co-star Kate Garraway stepped foot into the Jungle.
*I'm A Celebrity continues tomorrow at 9pm on ITV
Do you have a story to sell? Get in touch with us at<EMAIL_ADDRESS>or call us direct 0207 29 33033


Every Flink application depends on a set of Flink libraries. At the bare minimum, the application depends on the Flink APIs. Many applications depend in addition on certain connector libraries (like Kafka, Cassandra, etc.). When running Flink applications (either in a distributed deployment, or in the IDE for testing), the Flink runtime library must be available as well.
As with most systems that run user-defined applications, there are two broad categories of dependencies and libraries in Flink:
Flink Core Dependencies: Flink itself consists of a set of classes and dependencies that are needed to run the system, for example coordination, networking, checkpoints, failover, APIs, operations (such as windowing), resource management, etc. The set of all these classes and dependencies forms the core of Flink’s runtime and must be present when a Flink application is started.
These core classes and dependencies are packaged in the flink-dist
jar. They are part of Flink’s lib
folder and
part of the basic Flink container images. Think of these dependencies as similar to Java’s core library (rt.jar
, charsets.jar
, etc.),
which contains the classes like String
and List
.
The Flink Core Dependencies do not contain any connectors or libraries (CEP, SQL, ML, etc.) in order to avoid having an excessive number of dependencies and classes in the classpath by default. In fact, we try to keep the core dependencies as slim as possible to keep the default classpath small and avoid dependency clashes.
The User Application Dependencies are all connectors, formats, or libraries that a specific user application needs.
The user application is typically packaged into an application jar, which contains the application code and the required connector and library dependencies.
The user application dependencies explicitly do not include the Flink DataSet / DataStream APIs and runtime dependencies, because those are already part of Flink’s Core Dependencies.
Every Flink application needs as the bare minimum the API dependencies, to develop against. For Maven, you can use the Java Project Template or Scala Project Template to create a program skeleton with these initial dependencies.
When setting up a project manually, you need to add the following dependencies for the Java/Scala API (here presented in Maven syntax, but the same dependencies apply to other build tools (Gradle, SBT, etc.) as well.
Important: Please note that all these dependencies have their scope set to provided. That means that they are needed to compile against, but that they should not be packaged into the project’s resulting application jar file - these dependencies are Flink Core Dependencies, which are already available in any setup.
It is highly recommended to keep the dependencies in scope provided. If they are not set to provided, the best case is that the resulting JAR becomes excessively large, because it also contains all Flink core dependencies. The worst case is that the Flink core dependencies that are added to the application’s jar file clash with some of your own dependency versions (which is normally avoided through inverted classloading).
Note on IntelliJ: To make the applications run within IntelliJ IDEA, the Flink dependencies need
to be declared in scope compile rather than provided. Otherwise IntelliJ will not add them to the classpath and
the in-IDE execution will fail with a NoClassDefFountError
. To avoid having to declare the
dependency scope as compile (which is not recommended, see above), the above linked Java- and Scala
project templates use a trick: They add a profile that selectively activates when the application
is run in IntelliJ and only then promotes the dependencies to scope compile, without affecting
the packaging of the JAR files.
Most applications need specific connectors or libraries to run, for example a connector to Kafka, Cassandra, etc. These connectors are not part of Flink’s core dependencies and must hence be added as dependencies to the application
Below is an example adding the connector for Kafka 0.10 as a dependency (Maven syntax):
We recommend to package the application code and all its required dependencies into one jar-with-dependencies which we refer to as the application jar. The application jar can be submitted to an already running Flink cluster, or added to a Flink application container image.
Projects created from the Java Project Template or
Scala Project Template are configured to automatically include
the application dependencies into the application jar when running mvn clean package
. For projects that are
not set up from those templates, we recommend to add the Maven Shade Plugin (as listed in the Appendix below)
to build the application jar with all required dependencies.
Important: For Maven (and other build tools) to correctly package the dependencies into the application jar, these application dependencies must be specified in scope compile (unlike the core dependencies, which must be specified in scope provided).
Scala versions (2.10, 2.11, 2.12, etc.) are not binary compatible with one another. For that reason, Flink for Scala 2.11 cannot be used with an application that uses Scala 2.12.
All Flink dependencies that (transitively) depend on Scala are suffixed with the
Scala version that they are built for, for example flink-streaming-scala_2.11
.
Developers that only use Java can pick any Scala version, Scala developers need to pick the Scala version that matches their application’s Scala version.
Please refer to the build guide for details on how to build Flink for a specific Scala version.
Note: Because of major breaking changes in Scala 2.12, Flink 1.5 currently builds only for Scala 2.11. We aim to add support for Scala 2.12 in the next versions.
General rule: It should never be necessary to add Hadoop dependencies directly to your application. (The only exception being when using existing Hadoop input-/output formats with Flink’s Hadoop compatibility wrappers)
If you want to use Flink with Hadoop, you need to have a Flink setup that includes the Hadoop dependencies, rather than adding Hadoop as an application dependency. Please refer to the Hadoop Setup Guide for details.
There are two main reasons for that design:
Some Hadoop interaction happens in Flink’s core, possibly before the user application is started, for example setting up HDFS for checkpoints, authenticating via Hadoop’s Kerberos tokens, or deployment on YARN.
Flink’s inverted classloading approach hides many transitive dependencies from the core dependencies. That applies not only to Flink’s own core dependencies, but also to Hadoop’s dependencies when present in the setup. That way, applications can use different versions of the same dependencies without running into dependency conflicts (and trust us, that’s a big deal, because Hadoops dependency tree is huge.)
If you need Hadoop dependencies during testing or development inside the IDE (for example for HDFS access), please configure these dependencies similar to the scope of the dependencies to test or to provided.
To build an application JAR that contains all dependencies required for declared connectors and libraries, you can use the following shade plugin definition:


SID AND NANCY A film review by Billy Green Copyright 1988 Billy Green
When I heard that a film was being made about Sid Vicious and Nancy Spungen, I figured, "Well, I guess it was inevitable that someone would wind up doing a Sex Pistols retrospective. Let's go see how badly they messed it up."
I was amazed that Alex Cox managed to make a movie that was both true to the characters *and* comercially viable. SID AND NANCY is a very slick movie about a very dirty, grimy subject. The slickness of the photography and the choreography of the scenes manages to balance out the coarseness of the subject matter.
Gary Oldman plays Sid. More to the point, Gary Oldman *is* Sid Vicious (or the closest we can hope to see since Sid's death). He even looks like Sid. It's almost scary. Give the man a cookie. He's earned it.
Chloe Webb is made to look a whole lot like Nancy. The most uncannily accurate aspect of her portrayal is the voice--she sounds 99% like Nancy. The only real failure I saw was that Chloe didn't affect a British accent every thirtieth word (Nancy Spungen lived a long time in London, and her heavy Bronx accent would give way to an altered British pronunciation every once in a while).
The characters of Sid and Nancy are so faithfully portrayed that it's almost scary. However, some of the lesser characters are not quite to faithfully portrayed. John Lydon's character seems to have been cast for the singing voice alone. The actor (whose name escapes me) doesn't quite manage to capture Johnny Rotten (except for the singing voice). His mannerisms are not right, his composure is far off base--but the overall impression is right on target. The same holds true for the characterization of Malcolm McLaren--the character in the movie is an exaggeration of Malcolm, not a realistic portrayal. In both characters, we do not see a realistic portrayal. Neither actor looks or acts like the person he portrays. Instead, we are given an impressionistic view--both characters represent the *effects* that the people had on Sid and Nancy, not the actual people they are based on. (And, don't forget, this is SID AND NANCY, not SID AND NANCY AND JOHN AND MALCOLM. :-))
Apart from these portrayals, Mr. Cox has gone to some pains to set his scenes in a world controlled subtly by the characters. For example, the first view we have of Sid and John is of them as they demolish a Rolls Royce. A bit later, we see Sid and friends walking along the street. On the surface, you hear a fairly mundane conversation. However, if you pay attention to the background, you will see a bunch of school kids (in uniform) running up the street and making a ruckus. They are beating cars with their lacrosse sticks. Shades of Sid and John's influence on the kids? Maybe. Pay closer attention. The kids are beating only on the old, beat-up cars. They don't touch the Rolls Royces or BMW's. This is actually a much better image of how the mainstream kids of England took the Sex Pistols and their anarchy.
As a whole, I give extreme kudos to Alex Cox for making SID AND NANCY. In this movie, he has given much more information about Sid Vicious, the Sex Pistols, Nancy Spungen and the whole musical/social revolution in the UK than anyone else has dared to try before. The movie is far more impressionistic (and, at times, surrealistic) than anyone tends to give it credit for.
I also would like to suggest D.O.A. as a companion piece for SID AND NANCY. D.O.A. chronicles the Sex Pistols' American tour (during which they disbanded--pardon the pun). D.O.A. is a documentary (including interviews with Sid and Nancy). SID AND NANCY is a grand piece of impressionistic fictionalization. Watch them both. Turn the sound up REALLY LOUD. Bang your head. Slam dance with whoever is closest to you, even if you've never slammed before. Spit at the TV. Find the punk that you didn't think resided in you.
(The last time that D.O.A. played at UNC, the audience did several hundred dollars worth of damage to the Student Union. No real point--I'd just thought I'd say that. :-) )
Billy Green Carrboro, NC {seismo,gatech,rutgers,uunet}!mcnc!briarpatch!billy
[Moderator's note: Do not confuse the D.O.A. that Green describes with the 1949 Edmond O'Brien film that shows up all the time on television because it's in the public domain, not the remake with Dennis Quaid soon to be released. They're interesting in their own way, of course, but if you're expecting the Sex Pistols and get Edmond O'Brien instead, you'll probably be disappointed. :-) ]
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Descent Algebras#
AUTHORS:
Travis Scrimshaw (2013-07-28): Initial version
- class sage.combinat.descent_algebra.DescentAlgebra(R, n)#
Bases:
UniqueRepresentation
,Parent
Solomon’s descent algebra.
The descent algebra \(\Sigma_n\) over a ring \(R\) is a subalgebra of the symmetric group algebra \(R S_n\). (The product in the latter algebra is defined by \((pq)(i) = q(p(i))\) for any two permutations \(p\) and \(q\) in \(S_n\) and every \(i \in \{ 1, 2, \ldots, n \}\). The algebra \(\Sigma_n\) inherits this product.)
There are three bases currently implemented for \(\Sigma_n\):
the standard basis \(D_S\) of (sums of) descent classes, indexed by subsets \(S\) of \(\{1, 2, \ldots, n-1\}\),
the subset basis \(B_p\), indexed by compositions \(p\) of \(n\),
the idempotent basis \(I_p\), indexed by compositions \(p\) of \(n\), which is used to construct the mutually orthogonal idempotents of the symmetric group algebra.
The idempotent basis is only defined when \(R\) is a \(\QQ\)-algebra.
We follow the notations and conventions in [GR1989], apart from the order of multiplication being different from the one used in that article. Schocker’s exposition [Sch2004], in turn, uses the same order of multiplication as we are, but has different notations for the bases.
INPUT:
R
– the base ringn
– a nonnegative integer
REFERENCES:
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: D = DA.D(); D Descent algebra of 4 over Rational Field in the standard basis sage: B = DA.B(); B Descent algebra of 4 over Rational Field in the subset basis sage: I = DA.I(); I Descent algebra of 4 over Rational Field in the idempotent basis sage: basis_B = B.basis() sage: elt = basis_B[Composition([1,2,1])] + 4*basis_B[Composition([1,3])]; elt B[1, 2, 1] + 4*B[1, 3] sage: D(elt) 5*D{} + 5*D{1} + D{1, 3} + D{3} sage: I(elt) 7/6*I[1, 1, 1, 1] + 2*I[1, 1, 2] + 3*I[1, 2, 1] + 4*I[1, 3]
As syntactic sugar, one can use the notation
D[i,...,l]
to construct elements of the basis; note that for the empty set one must useD[[]]
due to Python’s syntax:sage: D[[]] + D[2] + 2*D[1,2] D{} + 2*D{1, 2} + D{2}
The same syntax works for the other bases:
sage: I[1,2,1] + 3*I[4] + 2*I[3,1] I[1, 2, 1] + 2*I[3, 1] + 3*I[4]
- class B(alg, prefix='B')#
Bases:
CombinatorialFreeModule
,BindableClass
The subset basis of a descent algebra (indexed by compositions).
The subset basis \((B_S)_{S \subseteq \{1, 2, \ldots, n-1\}}\) of \(\Sigma_n\) is formed by
\[B_S = \sum_{T \subseteq S} D_T,\]where \((D_S)_{S \subseteq \{1, 2, \ldots, n-1\}}\) is the
standard basis
. However it is more natural to index the subset basis by compositions of \(n\) under the bijection \(\{i_1, i_2, \ldots, i_k\} \mapsto (i_1, i_2 - i_1, i_3 - i_2, \ldots, i_k - i_{k-1}, n - i_k)\) (where \(i_1 < i_2 < \cdots < i_k\)), which is what Sage uses to index the basis.The basis element \(B_p\) is denoted \(\Xi^p\) in [Sch2004].
By using compositions of \(n\), the product \(B_p B_q\) becomes a sum over the non-negative-integer matrices \(M\) with row sum \(p\) and column sum \(q\). The summand corresponding to \(M\) is \(B_c\), where \(c\) is the composition obtained by reading \(M\) row-by-row from left-to-right and top-to-bottom and removing all zeroes. This multiplication rule is commonly called “Solomon’s Mackey formula”.
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: B = DA.B() sage: list(B.basis()) [B[1, 1, 1, 1], B[1, 1, 2], B[1, 2, 1], B[1, 3], B[2, 1, 1], B[2, 2], B[3, 1], B[4]]
- one_basis()#
Return the identity element which is the composition \([n]\), as per
AlgebrasWithBasis.ParentMethods.one_basis
.EXAMPLES:
sage: DescentAlgebra(QQ, 4).B().one_basis() [4] sage: DescentAlgebra(QQ, 0).B().one_basis() [] sage: all( U * DescentAlgebra(QQ, 3).B().one() == U ....: for U in DescentAlgebra(QQ, 3).B().basis() ) True
- product_on_basis(p, q)#
Return \(B_p B_q\), where \(p\) and \(q\) are compositions of \(n\).
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: B = DA.B() sage: p = Composition([1,2,1]) sage: q = Composition([3,1]) sage: B.product_on_basis(p, q) B[1, 1, 1, 1] + 2*B[1, 2, 1]
- to_D_basis(p)#
Return \(B_p\) as a linear combination of \(D\)-basis elements.
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: B = DA.B() sage: D = DA.D() sage: list(map(D, B.basis())) # indirect doctest [D{} + D{1} + D{1, 2} + D{1, 2, 3} + D{1, 3} + D{2} + D{2, 3} + D{3}, D{} + D{1} + D{1, 2} + D{2}, D{} + D{1} + D{1, 3} + D{3}, D{} + D{1}, D{} + D{2} + D{2, 3} + D{3}, D{} + D{2}, D{} + D{3}, D{}]
- to_I_basis(p)#
Return \(B_p\) as a linear combination of \(I\)-basis elements.
This is done using the formula
\[B_p = \sum_{q \leq p} \frac{1}{\mathbf{k}!(q,p)} I_q,\]where \(\leq\) is the refinement order and \(\mathbf{k}!(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}!(q,p)\) to be \(l(q_{(1)})! l(q_{(2)})! \cdots l(q_{(k)})!\), where \(l(r)\) denotes the number of parts of any composition \(r\).
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: B = DA.B() sage: I = DA.I() sage: list(map(I, B.basis())) # indirect doctest [I[1, 1, 1, 1], 1/2*I[1, 1, 1, 1] + I[1, 1, 2], 1/2*I[1, 1, 1, 1] + I[1, 2, 1], 1/6*I[1, 1, 1, 1] + 1/2*I[1, 1, 2] + 1/2*I[1, 2, 1] + I[1, 3], 1/2*I[1, 1, 1, 1] + I[2, 1, 1], 1/4*I[1, 1, 1, 1] + 1/2*I[1, 1, 2] + 1/2*I[2, 1, 1] + I[2, 2], 1/6*I[1, 1, 1, 1] + 1/2*I[1, 2, 1] + 1/2*I[2, 1, 1] + I[3, 1], 1/24*I[1, 1, 1, 1] + 1/6*I[1, 1, 2] + 1/6*I[1, 2, 1] + 1/2*I[1, 3] + 1/6*I[2, 1, 1] + 1/2*I[2, 2] + 1/2*I[3, 1] + I[4]]
- to_nsym(p)#
Return \(B_p\) as an element in \(NSym\), the non-commutative symmetric functions.
This maps \(B_p\) to \(S_p\) where \(S\) denotes the Complete basis of \(NSym\).
EXAMPLES:
sage: B = DescentAlgebra(QQ, 4).B() sage: S = NonCommutativeSymmetricFunctions(QQ).Complete() sage: list(map(S, B.basis())) # indirect doctest [S[1, 1, 1, 1], S[1, 1, 2], S[1, 2, 1], S[1, 3], S[2, 1, 1], S[2, 2], S[3, 1], S[4]]
- class D(alg, prefix='D')#
Bases:
CombinatorialFreeModule
,BindableClass
The standard basis of a descent algebra.
This basis is indexed by \(S \subseteq \{1, 2, \ldots, n-1\}\), and the basis vector indexed by \(S\) is the sum of all permutations, taken in the symmetric group algebra \(R S_n\), whose descent set is \(S\). We denote this basis vector by \(D_S\).
Occasionally this basis appears in literature but indexed by compositions of \(n\) rather than subsets of \(\{1, 2, \ldots, n-1\}\). The equivalence between these two indexings is owed to the bijection from the power set of \(\{1, 2, \ldots, n-1\}\) to the set of all compositions of \(n\) which sends every subset \(\{i_1, i_2, \ldots, i_k\}\) of \(\{1, 2, \ldots, n-1\}\) (with \(i_1 < i_2 < \cdots < i_k\)) to the composition \((i_1, i_2-i_1, \ldots, i_k-i_{k-1}, n-i_k)\).
The basis element corresponding to a composition \(p\) (or to the subset of \(\{1, 2, \ldots, n-1\}\)) is denoted \(\Delta^p\) in [Sch2004].
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: D = DA.D() sage: list(D.basis()) [D{}, D{1}, D{2}, D{3}, D{1, 2}, D{1, 3}, D{2, 3}, D{1, 2, 3}] sage: DA = DescentAlgebra(QQ, 0) sage: D = DA.D() sage: list(D.basis()) [D{}]
- one_basis()#
Return the identity element, as per
AlgebrasWithBasis.ParentMethods.one_basis
.EXAMPLES:
sage: DescentAlgebra(QQ, 4).D().one_basis() () sage: DescentAlgebra(QQ, 0).D().one_basis() () sage: all( U * DescentAlgebra(QQ, 3).D().one() == U ....: for U in DescentAlgebra(QQ, 3).D().basis() ) True
- product_on_basis(S, T)#
Return \(D_S D_T\), where \(S\) and \(T\) are subsets of \([n-1]\).
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: D = DA.D() sage: D.product_on_basis((1, 3), (2,)) D{} + D{1} + D{1, 2} + 2*D{1, 2, 3} + D{1, 3} + D{2} + D{2, 3} + D{3}
- to_B_basis(S)#
Return \(D_S\) as a linear combination of \(B_p\)-basis elements.
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: D = DA.D() sage: B = DA.B() sage: list(map(B, D.basis())) # indirect doctest [B[4], B[1, 3] - B[4], B[2, 2] - B[4], B[3, 1] - B[4], B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4], B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4], B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4], B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3] - B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]]
- to_symmetric_group_algebra_on_basis(S)#
Return \(D_S\) as a linear combination of basis elements in the symmetric group algebra.
EXAMPLES:
sage: D = DescentAlgebra(QQ, 4).D() sage: [D.to_symmetric_group_algebra_on_basis(tuple(b)) ....: for b in Subsets(3)] [[1, 2, 3, 4], [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3], [1, 3, 2, 4] + [1, 4, 2, 3] + [2, 3, 1, 4] + [2, 4, 1, 3] + [3, 4, 1, 2], [1, 2, 4, 3] + [1, 3, 4, 2] + [2, 3, 4, 1], [3, 2, 1, 4] + [4, 2, 1, 3] + [4, 3, 1, 2], [2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1] + [4, 1, 3, 2] + [4, 2, 3, 1], [1, 4, 3, 2] + [2, 4, 3, 1] + [3, 4, 2, 1], [4, 3, 2, 1]]
- class I(alg, prefix='I')#
Bases:
CombinatorialFreeModule
,BindableClass
The idempotent basis of a descent algebra.
The idempotent basis \((I_p)_{p \models n}\) is a basis for \(\Sigma_n\) whenever the ground ring is a \(\QQ\)-algebra. One way to compute it is using the formula (Theorem 3.3 in [GR1989])
\[I_p = \sum_{q \leq p} \frac{(-1)^{l(q)-l(p)}}{\mathbf{k}(q,p)} B_q,\]where \(\leq\) is the refinement order and \(l(r)\) denotes the number of parts of any composition \(r\), and where \(\mathbf{k}(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}(q,p)\) to be the product \(l(q_{(1)}) l(q_{(2)}) \cdots l(q_{(k)})\).
Let \(\lambda(p)\) denote the partition obtained from a composition \(p\) by sorting. This basis is called the idempotent basis since for any \(q\) such that \(\lambda(p) = \lambda(q)\), we have:
\[I_p I_q = s(\lambda) I_p\]where \(\lambda\) denotes \(\lambda(p) = \lambda(q)\), and where \(s(\lambda)\) is the stabilizer of \(\lambda\) in \(S_n\). (This is part of Theorem 4.2 in [GR1989].)
It is also straightforward to compute the idempotents \(E_{\lambda}\) for the symmetric group algebra by the formula (Theorem 3.2 in [GR1989]):
\[E_{\lambda} = \frac{1}{k!} \sum_{\lambda(p) = \lambda} I_p.\]Note
The basis elements are not orthogonal idempotents.
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: I = DA.I() sage: list(I.basis()) [I[1, 1, 1, 1], I[1, 1, 2], I[1, 2, 1], I[1, 3], I[2, 1, 1], I[2, 2], I[3, 1], I[4]]
- idempotent(la)#
Return the idempotent corresponding to the partition
la
of \(n\).EXAMPLES:
sage: I = DescentAlgebra(QQ, 4).I() sage: E = I.idempotent([3,1]); E 1/2*I[1, 3] + 1/2*I[3, 1] sage: E*E == E True sage: E2 = I.idempotent([2,1,1]); E2 1/6*I[1, 1, 2] + 1/6*I[1, 2, 1] + 1/6*I[2, 1, 1] sage: E2*E2 == E2 True sage: E*E2 == I.zero() True
- one()#
Return the identity element, which is \(B_{[n]}\), in the \(I\) basis.
EXAMPLES:
sage: DescentAlgebra(QQ, 4).I().one() 1/24*I[1, 1, 1, 1] + 1/6*I[1, 1, 2] + 1/6*I[1, 2, 1] + 1/2*I[1, 3] + 1/6*I[2, 1, 1] + 1/2*I[2, 2] + 1/2*I[3, 1] + I[4] sage: DescentAlgebra(QQ, 0).I().one() I[]
- one_basis()#
The element \(1\) is not (generally) a basis vector in the \(I\) basis, thus this returns a
TypeError
.EXAMPLES:
sage: DescentAlgebra(QQ, 4).I().one_basis() Traceback (most recent call last): ... TypeError: 1 is not a basis element in the I basis
- product_on_basis(p, q)#
Return \(I_p I_q\), where \(p\) and \(q\) are compositions of \(n\).
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: I = DA.I() sage: p = Composition([1,2,1]) sage: q = Composition([3,1]) sage: I.product_on_basis(p, q) 0 sage: I.product_on_basis(p, p) 2*I[1, 2, 1]
- to_B_basis(p)#
Return \(I_p\) as a linear combination of \(B\)-basis elements.
This is computed using the formula (Theorem 3.3 in [GR1989])
\[I_p = \sum_{q \leq p} \frac{(-1)^{l(q)-l(p)}}{\mathbf{k}(q,p)} B_q,\]where \(\leq\) is the refinement order and \(l(r)\) denotes the number of parts of any composition \(r\), and where \(\mathbf{k}(q,p)\) is defined as follows: When \(q \leq p\), we can write \(q\) as a concatenation \(q_{(1)} q_{(2)} \cdots q_{(k)}\) with each \(q_{(i)}\) being a composition of the \(i\)-th entry of \(p\), and then we set \(\mathbf{k}(q,p)\) to be \(l(q_{(1)}) l(q_{(2)}) \cdots l(q_{(k)})\).
EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: B = DA.B() sage: I = DA.I() sage: list(map(B, I.basis())) # indirect doctest [B[1, 1, 1, 1], -1/2*B[1, 1, 1, 1] + B[1, 1, 2], -1/2*B[1, 1, 1, 1] + B[1, 2, 1], 1/3*B[1, 1, 1, 1] - 1/2*B[1, 1, 2] - 1/2*B[1, 2, 1] + B[1, 3], -1/2*B[1, 1, 1, 1] + B[2, 1, 1], 1/4*B[1, 1, 1, 1] - 1/2*B[1, 1, 2] - 1/2*B[2, 1, 1] + B[2, 2], 1/3*B[1, 1, 1, 1] - 1/2*B[1, 2, 1] - 1/2*B[2, 1, 1] + B[3, 1], -1/4*B[1, 1, 1, 1] + 1/3*B[1, 1, 2] + 1/3*B[1, 2, 1] - 1/2*B[1, 3] + 1/3*B[2, 1, 1] - 1/2*B[2, 2] - 1/2*B[3, 1] + B[4]]
- a_realization()#
Return a particular realization of
self
(the \(B\)-basis).EXAMPLES:
sage: DA = DescentAlgebra(QQ, 4) sage: DA.a_realization() Descent algebra of 4 over Rational Field in the subset basis
- class sage.combinat.descent_algebra.DescentAlgebraBases(base)#
Bases:
Category_realization_of_parent
The category of bases of a descent algebra.
- class ElementMethods#
Bases:
object
- to_symmetric_group_algebra()#
Return
self
in the symmetric group algebra.EXAMPLES:
sage: B = DescentAlgebra(QQ, 4).B() sage: B[1,3].to_symmetric_group_algebra() [1, 2, 3, 4] + [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3] sage: I = DescentAlgebra(QQ, 4).I() sage: elt = I(B[1,3]) sage: elt.to_symmetric_group_algebra() [1, 2, 3, 4] + [2, 1, 3, 4] + [3, 1, 2, 4] + [4, 1, 2, 3]
- class ParentMethods#
Bases:
object
- is_commutative()#
Return whether this descent algebra is commutative.
EXAMPLES:
sage: B = DescentAlgebra(QQ, 4).B() sage: B.is_commutative() False sage: B = DescentAlgebra(QQ, 1).B() sage: B.is_commutative() True
- is_field(proof=True)#
Return whether this descent algebra is a field.
EXAMPLES:
sage: B = DescentAlgebra(QQ, 4).B() sage: B.is_field() False sage: B = DescentAlgebra(QQ, 1).B() sage: B.is_field() True
- to_symmetric_group_algebra()#
Morphism from
self
to the symmetric group algebra.EXAMPLES:
sage: D = DescentAlgebra(QQ, 4).D() sage: D.to_symmetric_group_algebra(D[1,3]) [2, 1, 4, 3] + [3, 1, 4, 2] + [3, 2, 4, 1] + [4, 1, 3, 2] + [4, 2, 3, 1] sage: B = DescentAlgebra(QQ, 4).B() sage: B.to_symmetric_group_algebra(B[1,2,1]) [1, 2, 3, 4] + [1, 2, 4, 3] + [1, 3, 4, 2] + [2, 1, 3, 4] + [2, 1, 4, 3] + [2, 3, 4, 1] + [3, 1, 2, 4] + [3, 1, 4, 2] + [3, 2, 4, 1] + [4, 1, 2, 3] + [4, 1, 3, 2] + [4, 2, 3, 1]
- to_symmetric_group_algebra_on_basis(S)#
Return the basis element index by
S
as a linear combination of basis elements in the symmetric group algebra.EXAMPLES:
sage: B = DescentAlgebra(QQ, 3).B() sage: [B.to_symmetric_group_algebra_on_basis(c) ....: for c in Compositions(3)] [[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1], [1, 2, 3] + [2, 1, 3] + [3, 1, 2], [1, 2, 3] + [1, 3, 2] + [2, 3, 1], [1, 2, 3]] sage: I = DescentAlgebra(QQ, 3).I() sage: [I.to_symmetric_group_algebra_on_basis(c) ....: for c in Compositions(3)] [[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1], 1/2*[1, 2, 3] - 1/2*[1, 3, 2] + 1/2*[2, 1, 3] - 1/2*[2, 3, 1] + 1/2*[3, 1, 2] - 1/2*[3, 2, 1], 1/2*[1, 2, 3] + 1/2*[1, 3, 2] - 1/2*[2, 1, 3] + 1/2*[2, 3, 1] - 1/2*[3, 1, 2] - 1/2*[3, 2, 1], 1/3*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] + 1/3*[3, 2, 1]]
- super_categories()#
The super categories of
self
.EXAMPLES:
sage: from sage.combinat.descent_algebra import DescentAlgebraBases sage: DA = DescentAlgebra(QQ, 4) sage: bases = DescentAlgebraBases(DA) sage: bases.super_categories() [Category of finite dimensional algebras with basis over Rational Field, Category of realizations of Descent algebra of 4 over Rational Field]


Nicola Adams, two-time Olympic champion and boxing pioneer, retires – video report
by Nicholas WilliamsTwo-time olympic boxing champion Nicola Adams has retired. The current WBO flyweight world champion has been advised to stop due to eyesight issues. During a glittering career Adams broke new grounds, becoming the first woman to win Olympic gold in her sport as well as being the first British woman to win European and world championship medals. She will go down as one of the British Boxing’s true pioneers.


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Layouts Basics
To achieve the best results and ensure your page flexibility for the future and good support across different devices, please use the following basic layout structure.

Header content

Main text content

Sidebar text

Footer content

Once you have the structure, then you can inject your one the layouts style in the BODY tag in your application.less file.
body {
.fixed-layout; // if you want a fixed-width layout
}
There are couple of layout styles available
- .fixed-layout - the fixed width layout
- .basic-layout - the basic full page width layout
Sidebars Position
Sidebars are done by using the ASIDE tag and position of the sidebar depends on where it's placed in the HTML structure.

Main text content

Sidebar is on the right

If you want your sidebar to be on the left side, just move it before the ARTICLE section.

Sidebar is on the left

Main text content

Another way, you can change the flex-box order for the ASIDE tag
You also can have sidebars on the both sides.
Grid System
The grid system in pain.less.css is dead simple, add the .grid class to an element and that's it. All the child elements will be considered grid cells, and you can change their size by adding the .half, .third or .quoter css classes to them.

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Our grid system is based on the flex-box settings, so if you amend the size classes on any of the elements, they will try to take all the rest of the space.

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You cant create tight grids, the ones that don't have any paddings between the elements, simply by adding the .tight class to the main grid element.

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virus {border-style px;} #ffffff;} disease SARS-CoV- px; spread {font-family SARS epidemic font-size public University double; health host need new novel transmission viruses Elsevier available border-color border-width efforts outbreak rapidly respiratory solid; {font-style {font-weight COVID- bat bold centre coronaviruses courier; diagnostic em; epidemic, especially human humans. including information molecular pt;} questions receptor research resource