After 35 distinguished years as an executive with Coca-Cola, Jim turned his passion for travel into a second career. Already well traveled, Jim founded Buckley's Escorted Tours of Distinction featuring personally-hosted escorted tours based on the concept of extraordinary service. Buckley's worked exclusively through Island Travel until Jim purchased the agency in 2001 in partnership with managing director Carlo Hilliard. Jim's management skills helped him assemble a staff of the finest and most knowledgeable travel professionals to be found. Jim is proud that his team's efforts have made Island Travel the Lowcountry's #1 Travel Agency (voted such in the Island Packet's 2006 through 2009 Readers' Poll!).
Jim and his wife Marty have traveled to every corner of the globe as travel guides and hosts. Jim is a Virtuoso Certified Travel Specialist.
Luxury Cruises, Escorted Tours, Independent tours and packages. Carlo is personally familiar with all of the deluxe cruise and tour operators including companies such as Silverseas, Regent Seven Seas, Crystal Cruises, Seabourn Cruises, Tauck Tours and others.
Our long-time part owner and Managing Partner of Island Travel, Carlo has 25 years of travel counseling experience including 20 of them on Hilton Head. Having traveled the world, Carlo's depth of knowledge and winning personality have made her both a favorite contact for Lowcountry travelers and a mentor to our staff.
Mediterranean cruise aboard the yacht Le Ponant with Tauck World Discovery; QE 2 transatlantic sailing with touring in London and environs; Princess Cruise in the Mediterranean with a transatlantic return; Walt Disney World and Carnival Cruise with her granddaughter.
Escorted and Independent Tours. Cruises. Family Travel. Spas and vacation packages including all-inclusive resorts.
Julie came to Island Travel in 1995 after a 14-year career in worldwide corporate travel with her family's agency in New Jersey. Leisure travel is now her specialty and she excels as a Travel Counselor. Julie's vivaciousness and caring dedication to her clients are invaluable assets to Island Travel.
Cruises, escorted groups and individual tours, independent travel. Linc has arranged a number of very successful fund-raising cruises.
Lincoln E "Linc" Tumey has recently relocated to Sun City from Ft Myers, Florida, after vacationing in the area for 27 years. Linc also entered the travel industry after a career in hospital administration. He served as manager of a full-service agency for seven years and has been an independent travel consultant for the last 5 years. Linc has designed individual tours and arranged escorted tours for many clients from all over the US to Europe, Israel, Australia/New Zealand and Hawaii. For the past three years, Linc has arranged fund-raiser cruises for his church, the most recent netting over $6,000. Each year these cruises have grown in the number of participants and the amount of money raised.
Europe, Israel, Australia/New Zealand, Hawaii, Germany, Switzerland and Scandinavia. The highlight of his travel to date was a 14-day river cruise from Amsterdam to Budapest in 2005.
Cruises and tours throughout the world. Extensive group travel history with blue water fishing and diving in North, Central and South America. Special interest in overseas real estate investment opportunities.
Tom's travel interests started young, accompanying his executive father throughout Europe and the Caribbean. As a podiatrist in Beaufort County for many years, the opportunity to acquire Beaufort Travel proved too tempting to pass up. He enjoyed operating the business for 13 years with his wife, Jane, and a great group of travel counselors. The changing travel environment and admiration for Jim Buckley's marketing expertise led to Island Travel's acquisition of Beaufort Travel in 2007.
Cruises, cruises, cruises... groups and independent travel. Ellen has arranged a number of very successful friends and neighbors groups cruises. Ellen and her husband Barry love to travel and have enjoyed many cruises to the Caribbean, the Mediterranean and Alaska. She uses her travel experience to help others get the vacation of their dreams.
Ellen has a varied background in real estate sales, bookkeeping and legal contract work and a passion for travel. Ellen and Barry moved to Bluffton four years ago and continues to turn her passion into helping others plan their next cruise or trip. Her strong organizational skills, her love of working with and meeting new people make her the perfect person or host your next vacation.
Ellen and Barry are members of Temple Oseh Shalom in Bluffton and Ellen is presently President of the Sisterhood of Temple Oseh Shalom and for the past three years she has been the Social Chairperson of her neighborhood on Sun City. Ellen is delighted to be joining the professional travel advisors at Island Travel.
Connie is a Georgia native, but calls Beaufort her true home. With over 25 years in the Tour and Travel Industry, her credentials include Certified Tour Professional of the National Tour Association. Connie's experiences as a travel advisor are a Manager as well as a Regional Travel Agent Trainer for a large travel consortium.
She was Tourism Sales Manager for the Atlanta Convention and Visitors Bureau for 7 years and worked with the Georgia Department of Tourism during the 1996 Atlanta Olympics. All inclusive Resorts, Escorted and Independent tours and Cruises are Connie's expertise.
Connie's first international trip, as a travel advisor, was a 12 day escorted tour throughout South America. During that trip she visited must see places as Machu Picchu, Iguassu Falls and Christ the Redeemer statute atop Corcovado Mountain in Rio. Since then, her travels have taken her to many places around the world. Her three most recent trips were a 10 day tour of Ireland, a 14 day Alaska Cruisetour and a 11 day land tour of Italy during the Christmas season.
Janice specializes in all-inclusive travel, corporate travel, individual travel and group travel. Personally, she loves Caribbean travel and anywhere with a beach!
Janice started with in the airline business in 1973, working for several commuter airlines and Braniff Airlines, as a Supervisor and a Station Manager. She started working in travel agencies in 1987, first as a corporate agent working with international business travelers and continuing into leisure. In November of 2012, she moved to the Lowcountry after retiring as Manager from an upstate New York travel agency.
Adventure and active travel including golf, bicycling, skiing, fishing and scuba diving. Cruises, independent travel and escorted tours.
Janelle joins us from Wisconsin where she is a past owner of a small full service travel agency.
With 18 years in the travel business and a true passion for travel, Janelle is looking forward to sharing her expertise in planning your next travel adventure.
Various Caribbean islands for scuba diving including Grand Turk, Barbados, Little Cayman and Bonaire. Numerous trips to Western Europe including Italy, France, Germany, Greece and the Netherlands.
Marion Payne was smitten to travel early in life and has always shared that passion with those around her. She and her husband, Bill, take great pride in traveling to 17 countries in a 3 year span while living and teaching in Isfahan, Iran. During her tenure as a principal and president, National Middle School Association, Marion toured schools in New Zealand, accompanied students to Japan and participated as a Hays-Fullbright scholar in Israel and Jordan. While her youngest son played professional football (soccer) in Europe, she attended games in several countries during his years of tenure. She and Bill love to travel and she looks forward to helping others experience the joy of traveling.
Janet creates exciting travel memories for your vacation anywhere in the world. She has extensive history in the travel industry and will provide tremendous customer service for both independent and group travelers. You choose the destination and let her take care of all the details!
She arrived at our Hilton Head Island office with a far-reaching career from owning her own agency to corporate, leisure and extensive group travel planning. Also, we find it helpful that she is a certified international and domestic tour manager. She brings with her an upbeat personality and positive attitude from the Buckeye State. "Can't do" is not a phrase you will hear her use - collaborating with you is what she does best. Travel memories are waiting to be made!
In the last 30 years I traveled extensively, taught a class in Russia, and became a motivational speaker, all while working in corporate America. But it wasn't until a travel agency owner said to me "Anyone who has been to all seven continents should have a career in the travel industry", that I realized my love for travel was matched perfectly with my desire to help others make their travel dreams comes true. So I became a travel advisor!
Traveling through the Galapagos photographing animals, relaxing in Bora Bora, and soaking in the history of Egypt are some of my favorite memories. These memories, and many more, fuel my passion to craft personalized travel experiences for others. I'll collaborate with you to make sure your travels tailor to your preferences, and then thoughtfully plan each detail. My bucket-list is as vast as the world and includes Machu Picchu, the Amazon, and experiencing the great migration in central Africa. What's on yours? I'll use my personal knowledge and insider tips to make your travel dreams a reality.
Visit the Virtuoso Blog to learn the latest travel industry happenings, trends, and insider news.
You have a doctor, lawyer, banker and accountant. They are your professional helpers and so is your travel counselor.
Vacations are an important part of your life. Planning one requires care and attention to detail. Doing it on your own can be as risky as self-diagnosing an illness. You deserve a great vacation without unpleasant surprises. If you're thinking about cruises, escorted tours, family groups, destination weddings/events or independent travel, Island Travel has the expertise you need.


text:
Teens killed in crashes in Hutchinson, Corcoran identified
The teens were killed in crashes that happened on July 29 and Aug. 2.
Melissa Turtinen
Kurt Bauschardt via Flickr
Two teenagers who were killed in separate crashes in Hutchinson and Corcoran have been identified.
Garrett Riley Car Damlow, 18, of Hutchinson, was a passenger in a vehicle that crashed on Yacht Avenue in Hutchinson just before 3 p.m. on July 29. He died from multiple blunt force injuries at the hospital at 4:26 p.m. on Aug. 1, the Hennepin County Medical Examiner said.
Damlow's obituary says he loved farming, being outside and had an interest in trucks. He liked to hunt, play X-Box, watch Netflix, eat Toaster Scramblers and enjoyed "just staying busy."
According to the McLeod County Sheriff's Office, the fatal crash involved two vehicles driven by a 19-year-old from Hutchinson and a 20-year-old from Glencoe. Two people were transported to the hospital, with the sheriff's office noting "most" of the people in the two vehicles were wearing their seatbelts.
Teen killed in Corcoran rollover crash
Ayriona Marie Derheim, 17, of Big Lake, died at the scene after the pickup truck she was driving went off the road and rolled on Oakdale Drive near Jefferey Lane in Corcoran at 6:35 p.m. on Aug. 2. The Hennepin County Medical Examiner says she died of penetrating and blunt force injuries sustained in the crash.
Derheim was driving with two other juveniles at the time of the crash, police said. They were both injured – one was treated at the scene, while the other was seriously injured and had to be taken by North Memorial Air Care to the hospital. Their condition is not known.
According to Derheim's obituary, she went to school at Wright Technical Center, loved animals – especially horses, and enjoyed fishing and hunting. Onna, as she was called, "will always be remembered as an outgoing, strong-willed, artistic, smart, confident, kind, loving, fearless" and a risk-taker.
In honor of her love of animals, her family is requesting memorials be given to the Minnesota Hooved Animal Rescue Foundation in Zimmerman.
Minnesotafatal crashBig LakeMedical ExaminerHutchinsonCorcoran
Teen driver killed in Minnetrista crash identified
The crash happened just after 8 p.m. on Friday, Nov. 29.
1 dead, 1 airlifted after crash involving youths in Corcoran
The crash happened around 6:35 p.m. Sunday.
Authorities investigating fatal motorcycle crash in Corcoran
The crash happened around 5:20 p.m. Friday.
3-year-old killed, driver arrested after car crashes into tree, splits in half
A 5-year-old girl was also injured in the crash.
Person killed in Bloomington crash identified as 103-year-old
The crash happened on October 1.
Both drivers die in head-on crash on Hwy. 55
The crash happened around 6:45 a.m. Thursday.
2 dead in head-on crash involving pickups near MN-WI border
The crash happened around 11:30 p.m. Thursday, July 2.
Family of 4 killed in Austin car crash identified
The single-car crash happened Thursday, March 19.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Update the limitation of memory/CPU for existing container in docker I do know that we can create a container with memory limitation like this
docker run -ti --memory-reservation 1G ubuntu:14.04 /bin/bash

but how to update the limitation of memory/CPU for existing container?

A: Docker Update Command
docker update --memory "1g" --cpuset-cpu "1" <RunningContainerNameOrID>

this will update the "RunningContainerNameOrId" to use 1g of memory and only use cpu core 1
To up date all running containers to use core 1 and 1g of memory:
docker update --cpuset-cpus "1" --memory "1g" $(docker ps | awk 'NR>1 {print $1}')

A: Dynamic resource allocation is currently not possible with only Docker. You would have to update the cpu/memory shares of the cgroup (control group). You must create a new container to change the resource limitations with Docker.
There is a Docker GitHub issue for dynamic resource configuration. This pull request suggests it will be added in Docker 1.10 with a docker set or docker update command. This command will allow you to update a container's configuration at runtime.
If you are running on a systemd enabled system, you can leverage that as well to change the cpu or memory shares. An example can be found in this blog post.

A: I'd have saved hours if even one person was running docker on WSL and mentioned that you need to do this from the .wslconfig file if you're running docker with wsl integration, on windows.  I'm writing it right now for if another ME comes along and now they know the trick.
https://itnext.io/wsl2-tips-limit-cpu-memory-when-using-docker-c022535faf6f
It's an old question, but if you come along and need this new answer, on that one platform, at least it'll get you thinking.

redpajama_set_name: RedPajamaStackExchange


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text:
Platform(s): PC, PlayStation 4, Xbox One
Publisher: THQ Nordic
Developer: Bugbear
Release Date: Aug. 27, 2019
'Wreckfest' (ALL) Gets PS4/XOne Date, Deluxe Edition, PC Content Update - Trailer
by Rainier on July 2, 2019 @ 11:54 a.m. PDT
Wreckfest is a demolition derby-themed racing game with special attention paid to soft-body damage modeling, sophisticated driving dynamics and in-depth vehicle upgrading, featuring both demolition derbies and more traditional track races.
Have you ever hoped you could get back to the glorious days of Demolition Derby and FlatOut, climb behind the wheel of your banged-up ride and once again wreck your way to the finish line and show who's the boss? You're not the only one. Over the years, we've been asked thousands of times when are we going to get back to our roots and make another racing game that would capture the same spirit of over-the-top action and reckless racing.
Wreckfest is going to bring those unforgettable moments up-to-date, throw in the mix vehicle upgrading and spice it all up with massive online races where players can battle it out with their tricked out bangers.
Bugbear's Wreckfest is all about fun, breakneck racing and over-the-top crashes. We don't shy away from contact, and indeed, our in-house engine is all about what happens when cars collide with each other and the world around them at reckless speeds.
It's motorsport, but not just any motorsport. It's motorsport with attitude.
Expect epic crashes and tight, neck-to-neck fights over the finish line on run-down racing tracks. Expect proper, big demolition derbies. Expect brand new ways for metal to bend and the defining once-in-a-lifetime moments that can only be achieved in Wreckfest, with its true-to-life physics simulation.
Delayed late last year for more polishing, Finnish game developer Bugbear Entertainment has completed the qualifying and stands on pole position for its release on August 27, 2019 to race onto PS4 and Xbox One.
"It's a great day! Bugbear Entertainment has been working hard on Wreckfest and we are happy to finally welcome the console players to the racing tracks. We've improved the PC version from 2018 tremendously, new tracks, cars, options, customizations and more - It's the best banger racing experience ever." says Janne Alanenpää, CEO and Creative Director of Bugbear Entertainment.
Wreckfest will be available in both standard and deluxe versions, while the digital deluxe edition offers a 24-hour head start into the game on August 26th 2019, both versions will include the exclusive Bandit Ripper V8 racer for everyone who chooses to preorder, digitally or retail**.
The Deluxe Edition also contains the Season Pass, which will grant access to 20 new cars, 20 special roof decorations and additional car customizations like armor, rims and more. This new content will be released step by step after the release, along with free content updates.
Do exisiting owners of Wreckfest (PC) get anything, when the console version launches? YES! On August 27 Wreckfest will receive a big free update, including:
3 entirely new tracks and 1 new derby arena
3 new crazy challenge vehicles
1 new reward car for finishing the carreer
New & improved career events
Several new paintjobs for existing vehicles
Additonal daytimes for all tracks
... and some more cool stuff!
Burn rubber and shred metal in the ultimate driving playground!
Awesome Cars – Old, banged-up, patched-together cars that simply ooze style! From old American heavy-hitters to agile Europeans and fun Asians, you won't find anything like this in other games.
Meaningful Customization – Change not only the look of your cars but also upgrade their body armor.
Multiplayer – Wreck your friends online and take racing to the limit while chasing for demolition dominance!
Already available on PC, Wreckfest is coming to PS4 and Xbox One on August 27, 2019.
More articles about Wreckfest


We carry decorator fabrics. The showroom has a large assortment meant to suit everyone's budget. Our fabric choices range from vinyls, leathers,and a variety of assorted textured, patterned or plain material. We also have great discounts on in-stock fabrics.
We are available with design and decor experience to help with your selections.


text:
Jeff Toyne Scoring 'Rogue'
Posted: March 16, 2013 by filmmusicreporter in Film Scoring Assignments, TV Scoring Assignments
Tags: Deadweigth, DirecTV, Jeff Toyne, Rogue
Jeff Toyne is scoring DIRECTV's first original series Rogue. The drama is created by Matthew Parkhill (The Caller) and stars Thandie Newton as a morally and emotionally-conflicted undercover detective who is tormented by the possibility that her own actions contributed to her son's death. Marton Csokas, Ian Hart, Claudia Ferri, Joshua Sasse, Jonathan Holmes, Sarah Jeffrey, Ian Tracey, Jarod Joseph, Matthew Beard and Martin Donovan are co-starring. Brian Kirk (Game of Thrones, Boardwalk Empire) has directed the first episode. Nick Hamm (The Hole, Godsend, Killing Bono) is executive producing the project for Greenroom Entertainment, alongside John Morayniss and Michael Rosenberg for eOne and Steven Marrs. Rogue marks Toyne's first major American TV show scoring assignment. The composer's other previous credits include the feature drama Dirty Girl and the documentary Magic Beyond Words: The JK Rowling Story. Rogue is set to debut on April 3, 2013 on DIRECTV's Audience network. To learn more about the series, visit the official show website.
Toyne's other upcoming projects also includes the thriller Deadweight. The film is directed by Leah Walker and stars Sam Trammell (True Blood) as a young partner in a prestigious law firm. Laura Harris, Josh Close and Lina Roessle are co-starring. Jordan Walker has co-written the script and is producing the film with Justin Kelly and David Anselmo. Toyne has previously collaborated with Walker on the feature The Third Eye and the short The Knockout. Deadweight is currently in post-production. No word yet on a release date. Also coming up for the composer is the indie feature Dear Jen directed by R. Stephen Suettinger and starring Brian Greenberg and Jamie Chung. The film marks Suettinger's directorial debut and is set to shoot this spring.


text:
New Portland Area Income Taxes Implemented
CCHTaxGroup
Portland, Oregon area voters approved two ballot measures imposing a new business profits tax and two high earner taxes in 2020. The new taxes took effect January 1, 2021.
The ballot measures were voted on in May and November. The May ballot measure applies to the Metro Area (Washington, Clackamas, and Multnomah Counties) and imposes a personal and business income tax. The November ballot measure applies only to Multnomah County and imposes a high earner tax.
May Ballot Measure
Metro Area voters approved the personal and business income tax to fund supportive housing services. The taxes are:
a marginal income tax of 1% on households with taxable income over $200,000 (or taxable income over $125,000 for individual tax filers); and
a business profits tax of 1%.
The Metro Council has been working on implementation plans since the ballot measure was approved in May. According to preliminary meeting minutes from a December council meeting, the Council voted to approve implementing ordinances.
The taxes take effect in 2021 and stay in effect through 2030 unless reauthorized by voters.
The administration ordinance requires all tax returns be filed with payment by the 15th day of the 4th month following the end of the tax year.
Every taxpayer with $1,000 of liability or more must estimate and pay the liability by:
making quarterly payments; or
employer supplied withholding from taxpayer's wages.
The first payment date for quarterly estimated payments is April 15th.
Beginning in 2021, withholding is voluntary, but employers must offer employees withholding from wages as soon as possible. In 2022, withholding will be mandatory for all employees that work in the Metro District and earn $200,000 or more.
The personal income tax will be assessed on residents of the Metro district and non-residents who have income earned in the Metro district. Only income above $200,000 ($125,000 individuals) will be taxed.
Business Income Tax
The business income tax has an exemption for small businesses with gross receipts under $5 million. Sole proprietorships and disregarded entities are not subject to the tax but are subject to the personal income tax.
November Ballot Measure
In November, Multnomah County voters approved an income tax on high earners to set up a tuition-free preschool program.
The tax will be on residents and on taxable income derived from sources in Multnomah County at the rate of:
1.5% on taxable income over $125,000 (single) and $200,000 (joint); and
an added 1.5% (3% total) on taxable income over $250,000 (single) and $400,000 (joint).
The rates will increase by 0.8% in 2026.
Households with taxable income below $200,000 (single filer below $125,000) will not pay the tax.
Tax for the year is due April 15th the following year.
Paying the Tax
Multnomah County has released a set of questions and answers regarding administration of the tax.
Employers will opt employees into withholding if they make over $200,000. Multnomah County employers must offer withholding in 2021, once their systems are updated to handle withholding. In 2022, withholding becomes mandatory.
The City of Portland will be accepting the remitted taxes. The planned launch is April 2021. The tax for 2021 is owed by April 15, 2022.
By Andrew Soubel, J.D.
Login to read more on CCHAnswerConnect.
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All stories by: CCHTaxGroup


With thousands of years history and your own experience in using or benefiting from meridian-based medical practices, you many not feel that you need physical proof of the existence of meridian lines and points.  However, it is the lack of visual proof that has kept the Western health community, including the insurance companies, from fully accepting the practices of the East like acupuncture and acupressure. Those who understand meridian theory know that it is not just the ability to help heal that is important in Eastern practice, but the focus on health maintenance through maintaining the flow of energy throughout the body.  There are many people trying to establish the proof of meridian theory and this book contributes to this goal by presenting a sample of the work of Dr. Shui Yin Lo, who uses infrared photography in his research and has discovered its ability to reveal easily the 14 major meridians in Eastern meridian theory. We all know that we need a more efficient and cost effect approach to medical care and this book will provide a major steppingstone towards that goal.
This is the page of Shui Yin Lo on 24symbols. Here you can see and read his/her books.


text:
Gimbrett  est un village français du Bas-Rhin, à vocation agricole, situé dans une cuvette au pied du Galgenberg (). Gimbrett est une ancienne commune indépendante, associée à Berstett depuis le .

Histoire
 Gimbrett est mentionné pour la première fois en 929 dans les archives. L'archevêque de Trèves cède les biens situés à Genebredde.
 Au , le Domstift de Strasbourg acquiert des biens de Jenebreten.
 À partir de 1295, les comtes de Lichtenberg prélèvent les impôts de l'église de Gynebreten.
 En 1840, le village fait partie du Comté de Hanau-Lichtenberg.
 Avant la réforme, Gimbrett était une paroisse autonome, à partir de 1545, la paroisse est affiliée à celle de Reitwiller et les catholiques font partie de celle de Rumersheim.
 En 1927, la foudre a frappé le clocher de style roman, qui date du , il a été reconstruit en 1928.
 Au début du , 1903-1904, une épidémie de typhus a sévi dans le village.
 En 1928, Gimbrett comptait 28 exploitations agricoles d'une superficie moyenne de 5 hectares qu'occupaient 17 artisans, 32 ouvriers et 16 journaliers. L'effectif du cheptel était alors de 77 chevaux et 279 bovins.
 Le , la commune de Gimbrett est rattachée à celle de Berstett sous le régime de la fusion-association.
 En 2007, fonctionnent encore 13 exploitations de 25 hectares de superficie moyenne.

Héraldique

Politique et administration

Démographie

Lieux et monuments
L'église de Gimbrett, affectée au culte luthérien, a été construite au . Cependant, il ne reste de l'église primitive que l'abside et la base de la tour. La nef et la partie supérieure du clocher datent du . Jusqu'en 1882, cet édifice était entouré d'un cimetière qui a été déplacé à cette date. À l'intérieur, on trouve un orgue construit par le facteur Stiehr-Mocker en 1854 qui fait l'objet d'un projet de restauration.

Notes et références

Ancienne commune dans le Bas-Rhin
Bailliage de Bouxwiller
Commune associée dans le Bas-Rhin
Communauté de communes du Kochersberg

redpajama_set_name: RedPajamaWikipedia


attitude, focused and capable of working under pressure with target.
There is no service bond or security required.
with suitable salary as per the performance of the candidate during the training period.
There will be skills which the candidate should attain during the training.
The candidates completing the training will be given certificate of proficiency.
There is no bond or of any kind.


Due to the high volume of drivers in Las Vegas, Nevada, there is a state-wide ban on texting and calling while driving due to the amount of accidents caused by either texting or calling in the car. This state-wide law is also enforced in 39 other states in the U.S. Both texting and calling while driving still occur at an extreme volume and there have been over 1,000 accidents in Nevada as of 2010. The Nevada Highway patrol issue traffic alerts via social media to decrease the risk of accidents in the state. Almost all accidents in Las Vegas, Nevada have been due to driver inattention. Fines are issued to people who fail to obey the law, the first fine is $50, the second is $100, and third offense is $250 and eventually other consequences can and will be issued, these fines are doubled in work zones in order to keep workers concentrated during driving and work hours. These laws will crack down on offenders, so keep them in mind and pay attention on the road.


text:
\section{Introduction}

A fundamental role of p-branes in M-theory \cite{M1} attracts much of interest
to studying  the membrane ($p=2$) nonlinear equations. However, not so much is known
about their exact classical solutions
(see e.g. \cite{tucker, hoppe1, Z_0, BZ_0, hoppe2,hoppe4,UZ, hoppe6, JU1})
and a membrane realization of the string program \cite{Reb}.
Recently Hoppe in \cite{JU1} observed that
adding the $U(1)$ global symmetry to the problem
transforms the original 3-dim membrane equations into a 2-dim string-like nonlinear problem
still capturing characteristic nonlinearities of the membrane dynamics.

Here the solvability of Hoppe's
non-linear equations  \cite{JU1}  and their  Hamiltonian formulation are  investigated
\footnote{We thank Jens Hoppe for bringing to our attention this interesting problem.}.
We prove the exact solvability of the non-linear
problem for the $U(1)$ static membrane in
$D=2N+1$-dimensional Minkowski space, including the M-theory case $D=11$,
and present the general solution. Then the time-dependent solutions for  $D=5$
are analyzed and two physically interesting solutions are found. The
first solution is given  by the family of the Jacobi elliptic cosine
functions parametrized by the discrete integer number $n$ and describing a
contracting tori. The second solution describes a spinning torus characterized
by the upper boundary $\Omega_{max}=\frac{T^{1/3}}{\sqrt{\pi}}$ for
the rotation frequency  $\Omega$, where $T$ is the membrane  tension.

\section{The membrane dynamics}

The action for a p-brane without boundaries
is given by the integral in the world-volume
parameters $\xi^{\alpha}$ ($\alpha=0,\ldots,p$)
\footnote {Here the D-dimensional Minkowski space has
the signature $\eta_{\mu\nu}=(+,-,\ldots,-)$.}
\[
S=\int \sqrt{|G|}d^{p+1}\xi, \label{1}
\]
where $G$ is determinant of the induced metric $G_{\alpha
\beta}:=\partial_{\alpha} x_{\mu}\partial_{\beta} x^{\mu}.
\label{2}$ After splitting of the embedding
$x^{\mu}=(x^0,x^i)=(t,\vec{x})$ and internal coordinates
$\xi^{\alpha}=(\tau,\sigma^r)$, the Euler-Lagrange equations and
$p+1$ primary constraints generated by $S$ take the  form \footnote{
We  have chosen  the coordinates $x^{\mu}$  to be dimensionless
(similarly $\xi^{\alpha}$)  by rescaling  $x^{\mu}  \to x^{\mu} /
\sqrt{\alpha'}$,  where $\alpha'$ parameterizes the p-brane tension
$T \sim 1/{{\alpha'}^{\frac{p+1}{2}}}$.}
\begin{equation}\label{5}
\dot{\mathcal{P}}^{\mu}=-\partial_r \sqrt{|G|}G^{r\alpha}\partial_{\alpha}x^{\mu}, \ \ \
\mathcal{P}^{\mu}=\sqrt{|G|}G^{\tau \beta}\partial_{\beta} x^{\mu},
\end{equation}
\begin{equation}
\tilde{T}_{r}:=\mathcal{P}^{\mu} \partial_{r} x_{\mu} \approx 0, \ \ \ \
\tilde{U}:=\mathcal{P}^{\mu}\mathcal{P}_{\mu}-|\det G_{rs}| \approx  0, \label{6}
\end{equation}
where $\mathcal{P}^{\mu}$ is the energy-momentum density.
Next we follow \cite{hoppe4} and use  the orthogonal gauge
\begin{eqnarray} \label{7}
\tau=x^0, \ \ \ \ G_{\tau r}= -(\dot{\vec{x}} \cdot \partial_r \vec{x})=0, \\
 g_{rs}:=\partial_r \vec{x} \cdot \partial_s \vec{x}, \ \ \ \
G_{\alpha\beta}=\left( \begin{array}{cc}
                       1- {\dot{\vec{x}}}^2 & 0    \\
                         0 & -g_{rs}
                              \end{array} \right)
                               \nonumber
\end{eqnarray}
to simplify the metric $G_{\alpha\beta}$.
The solution of the constraint $\tilde{U}$ (\ref{6})
takes the form
\begin{equation}
\mathcal{P}_0=\sqrt{\vec{\mathcal{P}}^2+g}, \ \ \ \
g=\det g_{rs} \label{9}
\end{equation}
and becomes the  Hamiltonian density $\mathcal{H}_0$ of the p-brane,
because $\dot{\mathcal{P}}_0=0$ in view of  Eqn. (\ref{5}).
Respectively the evolution of $\vec{\mathcal{P}}$
is described as follows
\begin{equation}\label{13}
\vec{\mathcal{P}}=\mathcal{P}_0 \dot{\vec{x}}=\sqrt{\frac{g}{1-\dot{\vec{x}}^2}}\dot{\vec{x}}, \ \ \ \
\dot{\vec{\mathcal{P}}}=  \partial_r \left( \frac{g}{\mathcal{P}_0} g^{rs}\partial_s \vec{x}\right)
\end{equation}
and  Eqns. (\ref{13}) yield the second order PDE for $\vec{x}$
\begin{equation}
 \ddot{\vec{x}}=\frac{1}{\mathcal{P}_0}\partial_r \left( \frac{g}{\mathcal{P}_0} g^{rs}\partial_s \vec{x}\right).
\end{equation}
The constraints $\tilde{T}_r$ take the form
$T_r:=\vec{\mathcal{P}}\partial_r \vec{x}=0 $
and are satisfied on the surface of the gauge conditions (\ref{7}).
The gauge (\ref{7}) has a residual symmetry
\begin{equation}
\tilde{\tau}=\tau, \ \ \ \ \tilde{\sigma}^r=f^r(\sigma^s) \label{diff}
\end{equation}
which allows (see \cite{Mos})
to simplify  Eqns. (\ref{13}) by choosing
the additional gauge condition  $\mathcal{H}_0(=\mathcal{P}_0)=const.=C>0 $ resulting in
\begin{eqnarray}
U:=\vec{\mathcal{P}}^2+g-C^2=0,  \nonumber \\
\dot{\vec{x}}=\frac{1}{C}\vec{\mathcal{P}}, \ \ \ \ \dot{\vec{\mathcal{P}}}
=\frac{1}{C}\partial_r (g g^{rs} \partial_s \vec{x}).
\nonumber
\end{eqnarray}
Then the equations of motion admit the canonical formulation
\[
\dot{\vec{x}}=\{H,\vec{x}\}, \ \ \ \ \dot{\vec{\mathcal{P}}}=\{H,\vec{\mathcal{P}}\}, \ \ \ \
\{\mathcal{P}_i(\sigma), x_j(\tilde{\sigma})  \}=
\delta_{ij}\delta^{(2)}(\sigma^r-\tilde{\sigma}^r)
\]
with the Hamiltonian given by
\begin{eqnarray}\label{hampbr}
H=\frac{1}{2C}\int d^p \sigma (\vec{\mathcal{P}}^2+g).
\end{eqnarray}
Next we focus our analysis on the case of membranes ($p=2$) with
additional $U(1)$ symmetry.

\subsection{$U(1)$ invariant membranes}

The 2-dim surface  $\Sigma$ of a  $U(1)$ invariant membrane
 has the group  $U(1)$ as its isometry
with the Killing vector $\frac{\partial}{\partial \sigma^2}$.
Thus, the metric tensor $g_{rs}$ on $\Sigma$
is independent of $\sigma^2$, i.e. $\partial_{\sigma^2}g_{rs}=0$.
 Our further analysis will be narrowed to the case of $U(1)$ invariant membranes without boundaries.
 The $U(1)$ membranes with boundaries are treated similarly
taking into account additional boundary terms.
The $U(1)$  membranes may be compact or non-compact. For the latter
there are no conditions on the world-volume parameter $\sigma^1$.

The $U(1)$ membrane vector $\vec{x}$  may be chosen in the form \cite{JU1}
\begin{equation} \label{anzats}
\vec{x}^T=(m_1\cos\sigma^2,m_1\sin\sigma^2,\ldots,m_N\cos\sigma^2,m_N\sin\sigma^2), \ \
m_a=m_a(\tau,\sigma^1).
\end{equation}
It belongs to the $D=2N+1$ dimensional Minkowski space and generates the
metric  $g_{rs}=\partial_r \vec{x}\partial_s \vec{x}$ independent on $\sigma^{2}$.
The geometric restrictions associated with the representation (\ref{anzats}) are clarified from the
next observation. Because any $2N$-dim vector $\vec{x}$ is fixed
by $N$ pairs of its polar coordinates,
the space vector $\vec{x}$
of  any membrane may be presented in the form
\[
\vec{x}^T(\tau,\sigma^{1}, \sigma^{2})=
(m_1\cos\theta^1,m_1\sin\theta^1,\ldots,m_N\cos\theta^N,m_N\sin\theta^N)
\]
(where $T$ means the transposition)
with $m_a=m_a(\tau,\sigma^{r}),\  \theta^a=\theta^a(\tau,\sigma^{r})$
parametrized by its world volume coordinates $(\tau,\sigma^{1}, \sigma^{2})$.
The ansatz (\ref{anzats})  is obtained  from the above representation
by equaling all
the polar angles $\theta_a$ to $\sigma^2$ and the manifestation
of the radial coordinate $m_a=m_a(\tau,\sigma^1)$ independence on the parameter $\sigma^{2}$.
It means  geometrically that
(\ref{anzats}) describes a 2-dim surface swept by global rotations of a subgroup
$O(2)\in SO(2N)$,  parametrized by the angle $\sigma^{2}$, of a plane
closed  ${\bold m}$-curve given by its radial
coordinates $m_a=m_a(\sigma^1)$ at any fixed moment $\tau$.
 Some of these $O(2)$ rotation subgroups
 will create their own  $U(1)$ invariant closed 2dim surfaces.
 At least, it concerns of the $O(2)$ rotations with their axices lying in the plane of
the ${\bold m}$-curve.
It is easy to see that the ansatz (\ref{anzats}), e.g. in  $D=2N+1=5$
\begin{equation}\label{anz}
\vec{x}^T=(m_1\cos\sigma^2,m_1\sin\sigma^2,m_2\cos\sigma^2,m_2\sin\sigma^2),
\end{equation}
is created by the following one-parametric subgroup of $SO(4)$ rotations
\begin{equation}\label{4rot}
S=\left( \begin{array}{cccc}
                       \cos \sigma^2 & -\sin \sigma^2 & 0 & 0   \\
                         \sin \sigma^2 & \cos \sigma^2 & 0 & 0   \\
                        0 & 0   &   \cos \sigma^2 & -\sin \sigma^2  \\
                        0 & 0     &  \sin \sigma^2 & \cos \sigma^2
                              \end{array} \right),
\ \ \ S\in SO(4)
\end{equation}
applied to the vector $\vec{x_0}^T=(m_1,0,m_2,0)$ (lying in the $x_1x_3$ plane)
simultaneously in the $x_1x_2$ and $x_3x_4$ planes. The similar block structure of the
 rotation matrix preserves for higher N.
Because of the arbitrariness of
the  ${\bold m}$-curve other global symmetries of the membrane  surface, except of the
rotational  $U(1)$ symmetry, are not assumed.

So, by construction the ansatz (\ref{anzats}) desribes one of the representatives
of the family of the $U(1)$ invariant surfaces created by various  rotations
of an closed curve in the $2N$ dimensional Euclidean space.
Each of the members of the family has a $U(1)$ symmetry as its inherent global symmetry.

The membrane world-volume metric $G_{\alpha \beta}$ corresponding to (\ref{anzats}) is
 \[
 G_{\alpha\beta}=diag(1- \dot{\bold{m}}^2, -\bold{m}^{' 2},-\bold{m}^2 ),
\  \ \  \bold{m}:=(m_1,..,m_N).
 \]
The canonical momentum
$\boldsymbol{\pi}:=(\pi_1,..., \pi_N)$ conjugate  to
$\bold{m}$ and defined as
$\pi_a=\frac{\partial \mathcal{L}}{\partial \dot{m_a}}=
\vec{\mathcal{P}}\frac{\partial{\dot{\vec{x}}}}{\partial \dot{m_a}},
\ \ (a=1,2,.., N)$,
is presented in the explicit  form as
\begin{equation}\label{Imp}
\pi_a = \mathcal{P}_0 \dot{m}_a, \ \ \
\mathcal{P}_0 =\sqrt{\frac{\bold{m}^2\bold{m}^{' 2}}{1-\dot{\bold{m}}^2}} \ \ ,
\end{equation}
 where $\bold{m}^{'}$ and $\dot{\bold{m}}$ are partial derivatives
with respect to $\sigma^1$ and $t$, after using  (\ref{13}) and the relations: \,
$   \vec{x}^2=\bold{m}^2, \ \ \ \dot{\vec{x}}^2=\dot{\bold{m}}^2, \ \ \
\vec{x}^{' 2}=\bold{m}^{' 2},
\ \ \ g=\bold{m}^2\bold{m}^{' 2}  $.
 Then the Hamiltonian density (\ref{9}) and the constraints become
\[
\mathcal{H}_0=\mathcal{P}_0=\sqrt{\boldsymbol{\pi}^2+\bold{m}^2\bold{m}^{' 2}}, \ \ \
\dot{\mathcal{P}_0}=0,
\]
\begin{equation}\label{T1}
T:=T_1=\boldsymbol{\pi}\bold{m}^{'}=0.
\end{equation}
The corresponding Hamiltonian
equations of motion are transformed in Eqns.
\begin{equation} \label{hameq}
\dot{\bold{m}}=\{H_0,\bold{m}\}=\frac{1}{\mathcal{P}_0}\boldsymbol{\pi}, \ \ \ \
\dot{\boldsymbol{\pi}}=\{H_0,\boldsymbol{\pi}\}=
\left(\frac{\bold{m}^2 \bold{m}^{'}}{\mathcal{P}_0} \right)^{'},
\end{equation}
where the canonical  Poisson bracket and the Hamiltonian are defined by
\[
\{\pi_a(\sigma^1),m_b(\tilde{\sigma^1})\}=\delta_{ab}\delta(\sigma^1-\tilde{\sigma^1}), \ \ \ \
H_0=\int d\sigma^1 \sqrt{\boldsymbol{\pi}^2+\bold{m}^2\bold{m}^{' 2}}\ \  . \label{hameq1}
\]
The  Hamiltonian Eqns. (\ref{hameq}) yield the
following
system of PDE's
\begin{equation} \label{pde}
\ddot{\bold{m}}=\frac{1}{\mathcal{P}_0}\left( \frac{\bold{m}^2\bold{m}^{'} }{\mathcal{P}_0 }
\right)^{'}- \left( \frac{1}{\mathcal{P}_0} \bold{m}^{'} \right)^2 \bold{m} ,
\end{equation}
which coincides with equations (\ref{13}) after the substitution of (\ref{anzats}).

The gauge condition $\mathcal{P}_0=C$
transforms the constraint  $U$  to the following
\[
C^2\dot{\bold{m}}^2+\bold{m}^2\bold{m}^{' 2}-C^2=0,
\]
and the membrane dynamics is now completely determined by the
simplified  equations and constraints
\begin{eqnarray}
C^2\ddot{\bold{m}}=(\bold{m}^2\bold{m}^{'})^{'}-\bold{m}^{' 2}\bold{m} \label{eq11},\\
C^2\dot{\bold{m}}^2+\bold{m}^2\bold{m}^{' 2}-C^2=0, \ \ \ \ \dot{\bold{m}}\bold{m}^{'}=0. \label{eq2}
\end{eqnarray}
Eqns. (\ref{eq11}) and (\ref{eq2}) form the system of dynamical equations of
 the U(1)-invariant membrane (\ref{anzats})
and  their  study is the main goal of this paper.

To this end  we  note that Eqns. (\ref{eq11}) are invariant under the
scaling transformations: $\tau \to \tilde \tau=\frac{1}{ab} \tau$,
 $\sigma\to \tilde\sigma =\frac{1}{b} \sigma$, $\bold{m}\to
\tilde{\bold{m}}= a\bold{m}$, so that $\tilde{\bold{m}}(\tilde \tau, \tilde\sigma^1)=a\bold{m}(\tau,\sigma^1)$.
After performing the rescaling and fixing $a$ and $b$ by the condition
 $a^4b^2C^2=1$, the constraints (\ref{eq2}) transform into
\begin{equation}
C^2\dot{\bold{m}}^2+\bold{m}^2\bold{m}^{' 2}-1=0,
\ \ \ \  \dot{\bold{m}}\bold{m}^{'}=0 \label{eq21}.
\end{equation}
Therefore making an additional change of the time variable into $\tau^*=\tau/C$ we find
that the Eqns. (\ref{eq11},\ref{eq2}) transform into
\begin{eqnarray}
\ddot{\bold{m}}=(\bold{m}^2\bold{m}^{'})^{'}-\bold{m}^{' 2}\bold{m} \label{eq31},\\
\dot{\bold{m}}^2+\bold{m}^2\bold{m}^{' 2}-1=0, \ \ \ \ \dot{\bold{m}}\bold{m}^{'}=0. \label{eq32}
\end{eqnarray}
The above equations are  in agreement with \cite{JU1}, where they were derived by
a change of the variable $\sigma^1$. Moreover, it was observed  in \cite{JU1} that for $D=5$
(i.e. for $N=2$) the  constraints (\ref{eq32}) imply  Eqns. (\ref{eq31})  provided
 that $\bold{m}^{'}$ and $\dot{\bold{m}}$ are independent
\footnote {We observe that the static membrane characterized by Eqns. $\dot{\bold{m}}=0$,
 has to be treated
on a different footing.}. In the reduced phase space with $\mathcal{P}_0=C=1$
the Hamiltonian $H$ and the constraints take the form
\begin{eqnarray}
 H=\int d\sigma^1 \mathcal{H}= \int d\sigma^1 (\boldsymbol{\pi}^2+\bold{m}^2\bold{m}^{' 2}),
\label{Gam}\\
U=\boldsymbol{\pi}^2+\bold{m}^2\bold{m}^{' 2}-1=0, \ \ \ T=\boldsymbol{\pi}\bold{m}^{'}=0
\label{Svz}
\end{eqnarray}
with  the Hamiltonian density $\mathcal{H}|_U=\mathcal{P}_0=1$ on the constraint surface.

\subsection{General static solution}

The static membranes play a great role in M/string theory and it is important to understand
 the role
of the global $U(1)$ symmetry there. The obtained  answer is surprising: the dynamics of
static $U(1)$-invariant membrane
occurs to be exactly solvable  in any odd dimension $D=2N+1$, including $D=11$.

To prove this we  put  $\dot{\bold{m}}=0$ in  Eqns. (\ref{eq11}-\ref{eq2}) and obtain the equations
\begin{eqnarray}\label{sta1}
(\bold{m}^2\bold{m}^{'})^{'} -\bold{m}^{'2}\bold{m}=0,\\
\bold{m}^{'2}\bold{m}^2 -1 =0     \label{sta2}.
\end{eqnarray}
Using  (\ref{sta2}) we express  $\bold{m}^{'2}=1/\bold{m}^{2}$
and transform Eqns. (\ref{sta1})  to the  form
\begin{equation} \label{sta1'}
 \bold{m}^{2}(\bold{m}^2\bold{m}^{'})^{'} - \bold{m}=0,
\end{equation}
where the derivative $\partial_{\sigma^1}$ appears only in the
combination  $\bold{m}^{2}$$\partial_{\sigma^1}$.
This observation proposes a natural change of
the variable $\sigma^1$ by a new one $\xi(\sigma^1)$ defined by the differential relation
\begin{equation}\label{ksi}
d\xi= \frac{1}{\bold{m}^{2}(\sigma^1)}d\sigma^1,\ \ \ \ \ \ \  \bold{q}(\xi)=\bold{m}(\sigma^1).
\end{equation}
Using the variable $\xi$ one can present Eqns. (\ref{sta1}) and (\ref{sta2})
in the form
\begin{equation}
\partial_{\xi\xi}\bold{q}-\bold{q}=0, \ \ \ \ (\partial_{\xi}\bold{q})^2=\bold{q}^{2}  \label{lin1}
\end{equation}
of the $N$ linear differential equations with a simple nonlinear constraint.

The $\bold{q}$-system (\ref{lin1}) is equivalent to the original
$\bold{m}$-system (\ref{sta1}-\ref{sta2}) and the solution of one gives a solution of the other.
 The general solution  of the $\bold{q}$-system may be presented in the form
\begin{equation}\label{lnsol}
\bold{q}=\bold{C}_1 \cosh\xi+ \bold{C}_2 \sinh \xi, \ \ \ \ \ \
\bold{C}_1^2=\bold{C}_2^2=c^2,
\end{equation}
where $\bold{C}_1,\,\bold{C}_2$ are the integration constants.
 It proves the claimed exact solvability of the static $U(1)$-membrane equations.

The next step is to express the general solution (\ref{lnsol})
 in the  $\bold{m}$-representation. To this end
we need to integrate  Eq. (\ref{ksi}) to obtain $\xi$ as a function
of $\sigma^1$. The subsequent substitution of this
solution $\xi(\sigma^1)$ to (\ref{lnsol})
will create the desired general solution in the $\bold{m}$-representation.
Let us realize this program.

The integration of (\ref{ksi}) gives $\sigma^1$  as an explicit  function of $\xi$
\begin{equation}\label{implic}
\sinh 2\xi +\delta \cosh 2\xi=z(\sigma^1),
\ \ \
z:=\frac{2(\sigma^1 + \sigma_0^1)}{c^2},
\ \ \
\delta:=\frac{1}{c^2}(\bold{C}_1\bold{C}_2),
\end{equation}
where  $ \sigma_0^1$ and  $\delta$ are the integration constants, and
 $\delta \ \ (0\leq\delta\leq1)$ coincides with the cosine of
the angle between  $\bold{C}_1$ and $\bold{C}_2$.
On the other hand taking into account the relations following from
 (\ref{ksi}-\ref{lnsol})
\begin{equation}\label{scalprod}
(\bold{m}^4)^{'}/4=\bold{q}\partial_{\xi}\bold{q}=c^2(\sinh 2\xi +\delta \cosh 2\xi)
\end{equation}
and combining  (\ref{scalprod}) with (\ref{implic}) we  find
$\bold{m}^2$ as the explicit
 function of $\sigma^1$
\begin{equation}\label{msquar}
\bold{m}^2=c^2\sqrt{z^2 +4\gamma/c^4},
\end{equation}
where $\gamma$ is the integration constant.
The substitution of $\bold{m}^2(\sigma^1)$ (\ref{msquar}) in Eqn. (\ref{ksi})
and its integration in $\sigma^1$ gives the
desired  explicit presentation of $\xi(\sigma^1)$
\begin{equation}\label{ksisolut}
2\xi(\sigma^1)=  {\rm arcsinh}(\mu z) + \tilde\gamma,
\ \ \
\mu:= c^2/(2\sqrt{\gamma})=1-\delta^2,
\ \
\tilde\gamma=-{\rm arctanh\delta}
\end{equation}
The substitution of $\xi(\sigma^1)$ (\ref{ksisolut}) into the general
solution (\ref{lnsol}) yields the desired general solution in the $\bold{m}$-representation.

To discuss this general solution we note that because of the global
invariance of Eqns. (\ref{sta1}-\ref{sta2}) of the $\bold{m}$-system under the
scaling transformations
$\tilde\sigma^1 =b \sigma^1, \ \ \   \tilde{\bold{m}}= \sqrt{b}\bold{m}$ one can put
$\mu=1$ in (\ref{ksisolut}) without loss of generality.
This choice of $\mu$ is equivalent to $\delta=\tilde\gamma=(\bold{C}_1\bold{C}_2)=0$
and results in
\begin{equation}\label{ro}
2\xi(\sigma^1)= {\rm arcsinh}z,
\ \ \
\bold{m}(\sigma^1)^2= c^2\sqrt{z^2 + 1},
\ \ \
z:=2(\sigma^1 + \sigma_0^1)/c^2
\end{equation}
producing a simplified representation of the general solution of Eqns. (\ref{sta1}-\ref{sta2})
\begin{eqnarray}\label{sol2}
\bold{m}(\sigma^1)=\bold{C}_1 \sqrt{\frac{\cosh(2\xi) +1}{2}}+
\bold{C}_2 \sqrt{\frac{\cosh(2\xi) -1}{2}} = \\ \nonumber
\bold{C}_1\sqrt{\frac{\sqrt{1+z^2}+1}{2}}+
\bold{C}_2\sqrt{\frac{\sqrt{1+z^2}-1}{2}}.
\end{eqnarray}

If the  parameter $\sigma^1$ was bounded, e.g. $\sigma^1
\in [0,2\pi]$ - the case discussed  in more details in
Section 3 -  then the solution (\ref{sol2}) would be bounded too.
It corresponds to a membrane with the boundary, since the periodicity
condition $\bold{m}(0)=\bold{m}(2\pi)$ can not be
satisfied. On the other hand, if  we let $\sigma^1\in[0, \infty)$ then the general
 solution describes a
 $U(1)$-invariant  membrane without a boundary
\footnote{Since equations (\ref{eq11}) and
(\ref{eq2}) can be derived without referring to the Moser
theorem \cite{Mos}, the assumptions of the theorem (in particular the
compactness of the membrane) can be omitted allowing one to consider noncompact
membrane solutions.}.

The solution (\ref{sol2}) shows that the angle $\theta(\sigma^1)$
between $\bold{m}$ and $\bold{m}^{'}$
at any point $\sigma^1$ is given by the relation
\[
\bold{m}\bold{m}^{'}=\cos\theta= (1+z^{-2})^{-1/2},
\]
and goes to zero when $\sigma^1$ increases. The asymptotic
behavior of $\bold{m}(\sigma^1)$ when $\sigma^1 \rightarrow \infty $
is described by the vector function $\bold{m_{\infty}}(\sigma^1)$ is
\begin{equation}\label{asymt}
\bold{m(\sigma^1)} \rightarrow
\bold{m_{\infty}}=\bold{C}\sqrt{2\sigma^1},  \ \ \
\bold{C}:= (\bold{C}_1 + \bold{C}_2)/\sqrt2, \ \ \  \bold{C}^2=c^2.
\end{equation}
The comparison  of the product  $\bold{m_{\infty}}^2\bold{m_{\infty}^{'}}^2= c^4$
 with the constraint (\ref{sta2}) hints that $\bold{m_{\infty}}$ could be a special, exact
solution $\bold{m^{(pl)}}$ of Eqns. (\ref{sta1}-\ref{sta2})
provided that $\bold{C}$  satisfies the additional constraint  $\bold{C}^2=c^2=1$ that implies
\begin{equation}\label{plsol}
\bold{m^{(pl)}}(\sigma^1)= \pm\bold{n}\sqrt{2\sigma^1}, \ \ \
\end{equation}
where $\bold{n}$ is a unit constant vector with $N$ components.
The substitution of $\bold{m^{(pl)}}(\sigma^1)$ (\ref{plsol}) into
Eqns. (\ref{sta1}) proves that $\bold{m^{(pl)}}(\sigma^1)$ actually is
a special solution of (\ref{sta1}) characterized  by the additional
collinearity constraint
\begin{equation}\label{colin}
\bold{m^{(pl)}}(\sigma^1)\bold{m^{'(pl)}}(\sigma^1)=1.
\end{equation}
The constraint (\ref{colin}) means that the vectors
$\bold{m^{(pl)}}$ and $\bold{m^{'(pl)}}$ are parallel, but their
lengths $\rho^{(pl)}(\sigma^1)= \sqrt{2\sigma^1}$ and
$\rho^{'(pl)}(\sigma^1)= \frac{1}{\sqrt{2\sigma^1}}$ are inversely
proportional. So, the  gradient $\rho^{'(pl)}$ is equal to the
infinity at the end point $\sigma^1=0$ of the interval $[0,
\infty)$, and respectively to zero at $\sigma^1=\infty$.

For the first nontrivial case $N=2$, corresponding to $ D=5$,  the constraints (\ref{lnsol})
 for  the 2-vectors $\bold{C}_1$ and $\bold{C}_2$ of the $\bold{m}$-plane
 are  explicitly solved by introducing two arbitrary constants $C$ and $D$
\[
\bold{C}_1=(C,D), \ \ \ \ \bold{C}_2=(D,-C), \ \ \ C,D \in \mathbb{R}.
\]
In this  case the  solution (\ref{sol2}) yields a curve in the $\bold{q}$-plane
parametrized by $\xi$
\begin{eqnarray}\label{crvq}
 {q_{1}}^2 - {q_{2}}^2 = (C^2- D^2) + 2CD\sinh(2\xi)
\end{eqnarray}
which is obtained  from the  rectangular hyperbola: $\tilde q_{1}=c\,\cosh\xi,\ \ \tilde q_{2}=c\, \sinh\xi$
\begin{equation}\label{hyp1}
 {\tilde q_{1}}^2 - {\tilde q_{2}}^2 = c^2,\ \ \ c^2=C^2+D^2
\end{equation}
rotated in the $\bold{q}$-plane:
$q_{1}=\tilde q_{1}\cos\alpha  - \tilde q_{2}\sin\alpha, \ \ \ q_{2}= \tilde q_{1}\cos\alpha + \tilde q_{1}\sin\alpha$
with $\cos 2\alpha = (C^2-D^2)c^{-2}, \ \ \ \ \sin 2\alpha=-2CDc^{-2}$.
 Respectively,  the equation of the rotated hyperbola parametrized by $\sigma^1$
has the form
\begin{eqnarray}\label{crv2}
{m_{1}}^2 - {m_{2}}^2=C^2- D^2 +4CD(\sigma^1 + \sigma_0^1)/c^2.
\end{eqnarray}
On the other hand the asymptotic solution (\ref{plsol}) for $D=5$
\begin{equation}\label{plsol'}
\bold{m^{(pl)}}(\sigma^1)= \pm\bold{n}\sqrt{2\sigma^1}, \ \ \
\bold{n}=(\cos\psi_0, \sin\psi_0),
\end{equation}
parametrized by $\psi_0$, yields  a straight line in the
$\bold{m}$-plane presented by the equation
\begin{equation}\label{line}
m_{2}^{(pl)}=\lambda m_{1}^{(pl)}, \ \ \     \lambda = \tan \psi_{_0}.
\end{equation}
The asymptotic solution defines a membrane whose surface is a plane  going through the origin
of  the 4-dimensional  Euclidean  $\vec{x}$-space (\ref{anz})
\begin{eqnarray}\label{pleqn}
\vec{N}\vec{x}=0, \ \ \  \vec{N}=(-\sin\psi_0, -\sin\psi_0, \cos\psi_0, \cos\psi_0),
\end{eqnarray}
where $ \vec{x}^T=
\pm\sqrt{2\sigma^1}(\cos\psi_0\cos\sigma^2,\cos\psi_0\sin\sigma^2,\sin\psi_0\cos\sigma^2,\sin\psi_0\sin\sigma^2)$
is the world-volume vector of the $U(1)$ invariant membrane embedded in the five dimensional
Minkowski space-time.
Thus, we established that equations (\ref{sta1}),
(\ref{sta2}) are exactly solvable and their solutions (\ref{sol2}),
(\ref{plsol}) correspond to non-compact membranes having the above described  fixed shapes.

\subsection{The D=5 static solution in polar coordinates}

Here we study time dependent membrane solutions in D=5 using the
polar representation of the 2-dimensional vector $\bold{m}$ proposed
in \cite{JU1}. To this end we analyse how the polar representation
works in the discussed static case.

For $D=5$, the two-dimensional vector  $\bold{m}$ and
the constraint (\ref{sta2}) are
\begin{equation}\label{sanz}
\bold{m}=\rho(\sigma^1)\left(\cos \psi(\sigma^1),\sin \psi(\sigma^1) \right), \ \ \
\end{equation}
\begin{equation}\label{static1}
\rho^2(\rho^{' 2}+\psi^{' 2}\rho^2)=1 ,
\end{equation}
hence the  two components of a vector equation (\ref{sta1}) take  the form
\[
f_1 \rho \cos\psi+f_2\rho^2\sin\psi=0, \ \ \ \
\ \ \ \ \  f_1 \rho \sin\psi-f_2\rho^2\cos\psi=0,
\]
\[
f_1=2 \rho^2 \psi^{'2} - \rho^{'2} - \rho \rho^{''}, \ \ \ \   f_2=4\rho^{'}\psi^{'}+\rho\psi^{''}
\]
which generically imply that
\begin{equation}\label{static_5}
f_1=f_2=0.
\end{equation}
Because of the presence of the $\psi^{'}$ and $\psi^{''}$ in $f_2$, we
first consider the case when $\psi$ is a linear function $\psi=k\sigma^1+\psi_0$.

For $k=0$ the solution is $\rho=\pm\sqrt{2\sigma^1}$
and describes the non-compact membrane (\ref{plsol'}). The curve  $m_2 =F(m_1)$ in  the pane  $x_1x_3$
is a straight line and its  rotation (\ref{4rot})
yields a two parameter family of planes in $\mathbb{R}^4$ given by
\[
A(x_1-x_3\tan \psi_0) +B(x_2- x_4\tan \psi_0)=0 , \ \ \ A,B \in \mathbb{R}.
\]

For the case of  $k \ne 0$ we find that the solution of (\ref{static1}) is
\[
\rho(\sigma^1)=\pm \sqrt{\sin(2k\sigma^1+c)/k},
\]
where $k$ and $c$ are such that $\frac{1}{k}\sin(2k\sigma^1+c) \ge 0$.
 However, it turns out that this solution
does not satisfy (\ref{static_5}) or equivalently Eqns. (\ref{eq11}),
because
\[
\ddot{\bold{m}}-(\bold{m}^2\bold{m}^{'})^{'}+\bold{m}^{' 2}\bold{m} =
4k \rho(\sigma^1) (\sin(3k\sigma^1+c),-\cos(3k\sigma^1+c))\ne0.
\]
It shows that the ansatz $\psi=k\sigma^1+\psi_0$, $k\ne 0$, is a
solution of the constraint (\ref{static1}) but not of the
static-membrane equations.

Next, assuming that  $\rho \ne 0$ and that $\psi$ is not linear in $\sigma^1$, we find
from Eqn. (\ref{static_5})
 (i.e. from $f_2=0$) that $\rho^4\psi^{'}=1$ which substituted to $(\ref{static1})$
and (\ref{static_5}) (i.e. to $f_1=0$) gives
\begin{equation}
\rho=\pm\sqrt{2/b}\sqrt[4]{(a+b\sigma^1)^2+1}, \ \ \ \ \psi=\pm\arctan(a+b\sigma^1)/2+\phi, \ \ \
b>0.  \label{sol1}
\end{equation}
Note that the parameters $a$ and $b$ could be argued from the scaling invariance of the
equations: $\sigma^1 \to a+ b \sigma^1$, $\bold{m}\to \sqrt{b}\bold{m}$.

The curve $m_2=F(m_1)$ in the $x_1x_3$ plane  is now given by
\begin{equation}\label{curve}
m_1^2-m_2^2= 2\cos(2\phi)/b-2(a+b\sigma^1)\sin(2\phi)/b,
\end{equation}
 which is recognized as the hyperbola $m_1^2-m_2^2= \frac{2}{b}$ rotated in that plane
through the angle $\phi$ (cp. (\ref{sanz})).
Therefore, this  solution is equivalent to the non-compact membrane (\ref{crv2}) that
 can be seen by writing
\[
a=\frac{2\sigma_0}{C^2+D^2}, \ \ \ \ b=\frac{2}{C^2+D^2}, \ \ \
 \sin(2\phi)=-\frac{2CD}{C^2+D^2}, \ \ \ \ \cos(2\phi)=\frac{C^2-D^2}{C^2+D^2}
\]
and next identification of  $\phi$ with $\alpha$.
 The corresponding implicit equation of the membrane surface is given
by (cp. (\ref{anz}))
\[
 \left(x_1^2+x_2^2-x_3^2-x_4^2-2/b\cos(2\phi)\right)^2=b^2\sin(2\phi)^2
\left  ((x_1^2+x_2^2+x_3^2+x_4^2)^2 -4/b^2 \right),
\]
\[
 x_1x_4=x_2x_3.
\]

\section{Time dependent solutions}

Here we consider a class of solutions corresponding to  the following ansatz
\begin{equation}
\bold{m}=\rho(\xi)\left(\cos \psi(\eta),\sin \psi(\eta) \right), \ \ \ \xi=a_1\sigma +a_2 \tau, \ \ \
\eta=a_3\sigma + a_4 \tau. \label{ganz}
\end{equation}
In that case the  constraints (\ref{eq2}) become
\[
 a_1a_2 \rho_{\xi}^2+a_3a_4\psi_{\eta}^2=0,\ \ \
a_2^2 \rho_{\xi}^2+ a_4^2\rho^2\psi_{\eta}^2+\rho^2(a_1^2 \rho_{\xi}^2+ a_3^2\rho^2\psi_{\eta}^2) =1
\]
and one can see that if $a_i=0$ for at least one of $i$ then the solutions are either trivial
or correspond to
\[
\begin{array}{cccc}
a) & \bold{m} & = & \rho(\tau)\left(\cos \psi(\sigma^1),\sin \psi(\sigma^1) \right),  \\
b) & \bold{m} & = & \rho(\sigma^1)\left(\cos \psi(\tau),\sin \psi(\tau)\right),  \\
c) & \bold{m} & = & \rho(\sigma^1)\left(\cos \psi(\sigma^1),\sin \psi(\sigma^1)\right), \\
d) & \bold{m}&=&\rho(\tau)\left(\cos \psi(\tau),\sin \psi(\tau) \right),
\end{array}
\]
from which one can exclude the previously studied static case c).
The purely time dependent case d) yields a degenerate world-volume
metric, as $\det g=0$ everywhere, and describes the closed (in
$\sigma^2$) $U(1)$ invariant null string moving with the velocity of
light ($\dot{\bold{m}}^2=1$).

On the other hand if $a_i \ne 0$ for all $i$ then the first constraint is solved by
\[
\rho=\rho_0 e^{\pm \frac{a_3a_4}{a_1a_2}\psi_0^2 \xi },
\ \ \ \psi=\psi_0 \eta + \psi_1, \ \ \ a_1a_2a_3a_4>0
\]
in contradiction with the second constraint hence we can narrow the
discussion to a) and b) that correspond to compact membranes.

Since there are two possible topologies corresponding to a U(1) invariant,
compact, orientable membrane surface $\Sigma$ without boundaries - the sphere  $S^2$ with the genus
 $\bold{g}=0$ and the torus  $T^2$ with  $\bold{g}=1$ -
we shall take advantage of using the Gauss-Bonnet theorem
\begin{equation}\label{tgb}
\frac{1}{2\pi}\int_{\Sigma}d^2\sigma\sqrt{g} K=\chi(\Sigma)=2-2\bold{g},
\end{equation}
where $\chi(\Sigma)$ is  the
 Euler characteristic and $K$ is the Gauss curvature
 \begin{equation}  \label{K}
K=-\frac{1}{2}\frac{1}{\sqrt{\bold{m}^2\bold{m}^{'2}}}\left(
\frac{1}{\sqrt{\bold{m}^2\bold{m}^{'2}}} \bold{m}^{2'}  \right)^{'}
\end{equation}
given by $\vec{x}$ (\ref{anz}).
The integration range of $\sigma^1$ is $[0,\pi]$ or
$[0,2\pi]$ for $S^2$ or $T^2$ respectively.
The integration  over $\sigma^2 \in [0,2\pi]$
in  (\ref{tgb}) gives the equation
\begin{equation}\label{gb}
\int d\sigma^1
\left( \frac{1}{\sqrt{\bold{m}^2\bold{m}^{'2}}} \bold{m}^{2'}
\right)^{'} = 4\bold{g}-4
\end{equation}
with the  vanishing  integrand for the cases  a) and b). It  results in $\bold{g}=1$ and proves  that the a) and b)  cases  describe the surface $\Sigma$ which is a torus.

\subsection{Contracting  tori}

Let us at first  consider the ansatz a). The constraints (\ref{eq2}) are satisfied provided
$\dot{\rho}^2+\psi^{' 2}\rho^4=1 $ which implies that $\psi^{' 2}=\omega^2=const.$ and
\[
F\left(\arccos(\sqrt{\omega}\rho);1/\sqrt{2}\right)=\mp\sqrt{2\omega}(\tau+\tau_0),
\ \ \ \tau_0\in \mathbb{R},
\]
where $F(\phi;k)$ is the elliptic integral of the first kind.
It follows that for fixed $\tau$,
$\bold{m}(\sigma^1,\tau)$ is bounded, so that the membrane is compact
and in view of $K=0$ corresponds to a torus.  That
implies that $\omega=n$ $(n\in \mathbb{Z})$ due to the periodicity
condition $\bold{m}(\sigma^1,\tau)=\bold{m}(\sigma^1+2\pi,\tau)$.
Thus the solution reads
\[
\rho(\tau;n)=\frac{1}{\sqrt{n}}cn\left(\sqrt{2n}
(\tau+\tau_0),\frac{1}{\sqrt{2}}\right), \ \ \ \ \psi(\sigma^1)=n\sigma^1
+\psi_0, \ \ \ \tau_0,\psi_0\in\mathbb{R},  \ \ \ \ n \in \mathbb{Z},
\]
where $cn(\phi,k)$ is the Jacobi elliptic cosine function.
If the initial data are such that
with $\dot{\rho}(\tau_0)>0$ then
the solution describes an expanding torus which at
some point reaches the maximal size $\rho_{max}=1/\sqrt{n}$
and then shrinks to a point after a finite time
$\bold{ K}(1/\sqrt{2})/\sqrt{2n}$
(where $\bold{ K}(1/\sqrt{2})=1,8451$ is the quarter period of  the elliptic
cosine function).

The dynamical equation (\ref{eq11}) is automatically satisfied because $\bold{m}^{'}$
and $\dot{\bold{m}}$
are independent in this case.
The implicit presentation of time dependent surface $\Sigma(\tau)$
corresponding to the solution is given by
\[
 x_1^2+x_2^2+x_3^2+x_4^2=\frac{1}{n}cn\left(\sqrt{2n}(\tau+\tau_0),\frac{1}{\sqrt{2}}\right)^2,
\ \  x_1x_4=x_2x_3.
\]

\subsection{ Spinning torus}

Finally we study the torus ansatz b) characterized by $\bold{m}^2=\rho^2$ and
$\bold{m}^{' 2}=\rho^{' 2}$ that results in $K=0$
and the following solutions of the constraints (\ref{eq2})
\[
\rho(\sigma^1)^2=-\left(\omega \sigma^1\pm
\tilde{B}/\omega\right)^2+\omega^{-2},  \ \ \ \psi(\tau)=\omega \tau
+\psi_0,   \ \ \ \ \omega,\psi_0,\tilde{B} \in \mathbb{R}.
\]
Note that $\sigma^1$ is bounded since $\rho(\sigma^1)^2>0$
and the membrane is  compact similarly to the a) case.
The periodicity condition
$\rho(0)^2=\rho(2\pi)^2$  fixes the integration constant $\tilde{B}:
\omega^2\pi=\mp \tilde{B}$
representing the solution in the form
\begin{equation}
\rho(\sigma^1)^2=\omega^{-2}-(\pi-\sigma^1)^2\omega^2 , \ \ \ \psi(\tau)=\omega \tau +\psi_0  \label{spanz}
\end{equation}
describing a rotating torus with the preserved shape.
Here $\bold{m}^{'}$ and $\dot{\bold{m}}$
stay independent hence the ansatz b) with (\ref{spanz}) is the solution of
(\ref{eq11}). The function $\rho(\sigma^1)^2$ has a maximum at
$\sigma^1=\pi$ with $\rho(\pi)=1/\omega$ and a minimum at
$\sigma^1=0$ with
\[
\rho(0)=\rho(2\pi)=\frac{1}{\omega}\sqrt{1-\pi^2\omega^4}
\]
which shows  that  $\omega$ has the  upper limit
$\omega\le \omega_{max}=1/\sqrt{\pi}$.
Restoring the physical units ($[\Omega]=L^{-1}$) for  $\omega$ with help of
the membrane tension $T$,
one can  present the critical frequency in the form
\[
\Omega_{max}=T^{1/3}\omega_{max}=T^{1/3}/\sqrt{\pi} \quad .
\]
This time the implicit description of $\Sigma(t)$ is given by
\[
x_1^2+x_2^2+x_3^2+x_4^2=\omega^{-2}-\left(\pi-\arctan x_2/x_1 \right)^2\omega^2,
 \ \  x_1x_4=x_2x_3.
\]

\section{Summary}

The $U(1)$ globally symmetric membranes realizing the idea of
 reduction of the 3-dim nonlinear problem to a 2-dim string-like one were studied.
 Such type of reductions could sharpen the role of the methods developed
in the theory of 2D chiral models \cite{FaTa} for the membrane quantization problem.
To this end an attempt to integrate the  $U(1)$ membrane equations was undertaken here.
We established the exact solvability of these  nonlinear equations
for the static membranes in the odd dimensions $D= 5, 7, 9, 11,.., 2N+1$
of the Minkowski space, and found their general solution.
The general solution  for $D= 5$ includes
the infinitely extended membranes whose surface is a 2-dimensional plane.
We also found the time  dependent solutions  for
$D=5$ that show a deep connection of the elliptic functions with
 the membrane  dynamics (see also \cite{hoppe7}).
These functions encode peculiarities of the
hidden structure of the  membrane dynamics different from the string dynamics.
We hope that the solutions presented here will prove useful in further study of the integrability
of the  membrane dynamics along the line studied in \cite{hoppe2,hoppe7}.

\noindent{\bf Acknowledgments}

We are grateful to J. Hoppe for valuable collaboration, kind support
and critical remarks. AZ would like to thank  J.
{\AA}man, I. Bengtsson, R. Myers, N. Pidokrajt and N. Turok for useful discussions,
 and Fysikum at
Stockholm University, Department of Mathematics of Royal Inst. of
Technology KTH Stockholm, Nordic Inst. for Theoretical Physics
Nordita and Perimeter Inst. for Theoretical Physics for kind hospitality.
 MT would like to thank Albert Einstein Institute for hospitality.
This research was supported in part by
Knut and Alice Wallenberg Foundation, Nordita, Marie Curie Research Training Network
ENIGMA (contract MRNT-CT-2004-5652)  and  Perimeter Institute for Theoretical Physics .

redpajama_set_name: RedPajamaArXiv


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text:
Snapped cable on cross-border Shenzhen Bay Bridge results in closure of two lanes but Hong Kong's highways chief says there are no structural safety issues
Cannix Yau
16 February 2019, 5:07 am
Part of the Hong Kong side of the Shenzhen Bay Bridge will be closed to traffic for at least three weeks as emergency repairs are being undertaken after a steel cable snapped, even as authorities stressed the cross-border connection remained safe.
As a safety precaution, the Highways Department would also check all bridges with a similar design across the city, Director of Highways Jimmy Chan Pai-ming said on Saturday evening.
The revelation was the latest episode of infrastructure problems the government had to tackle following construction scandals involving the Sha Tin-Central rail link and Hong Kong-Zhuhai-Macau Bridge, another boundary crossing.
Lawmakers and experts raised concerns about the standard of maintenance at the Shenzhen Bay Bridge, with a veteran engineer saying the incident set off alarm bells.
Chan, describing the broken cable as an "unusual" occurrence that was discovered during routine maintenance checks, sought to reassure motorists about the structural safety of the bridge.
Highways Department knew about late mega bridge documents as early as 2016
"When the bridge was designed, we took into account the need for cable replacement from time to time. According to our experts, even without one cable, it won't affect the safety of vehicles on the bridge," he said.
Chan said maintenance was done by Hong Kong contractors, but he did not name them.
Checks on other bridges of similar designs would be carried out over the next three months, he added.
When asked which bridges used that design, Chan said "there are quite a lot" and identified only two areas: the Tsing Yi South Bridge and Route 8, a road between Sha Tin and North Lantau. Route 8 includes the Lantau Link – a section made up of the Tsing Ma Bridge, Ma Wan Viaduct and Kap Shui Mun Bridge.
The 5.5km Shenzhen Bay Bridge, which took four years to build and opened on July 1, 2007, is the fourth of five – and the second busiest in 2017 – road-based boundary crossings linking Hong Kong and mainland China.
It is a dual three-lane highway spanning Deep Bay from Lau Fau Shan in Hong Kong's New Territories to reclaimed land in the Shekou area of Shenzhen. In the first 11 months of last year, there were 4.02 million vehicular trips through the Shenzhen Bay port.
Chan said the snapped cable, with a diameter of around 160mm, was one of six "external prestressed tendons" at the affected area. Another 40 internal prestressed tendons also support the section.
He said following a preliminary study, it was believed the cable, comprised of many small steel bars, had broken due to components rusting. But it would take some time to find out why corrosion occurred after testing the material and strength.
Workers only spotted the problem at noon on Friday, Chan said. The last routine cable check in September and an inspection around two weeks ago found nothing amiss.
A grouting process had been carried out to protect the whole cable so it was difficult to spot potential corrosion problems from outside, he said.
Two lanes out of three on the Hong Kong-bound carriageway were closed to allow the cable to be replaced, which was expected to take around three weeks.
Veteran structural engineer Ngai Hok-yan said the incident set off alarm bells that other parts of the bridge were not properly maintained. He said if the structure was adequately maintained, rusting of components and cables snapping would not happen.
"This incident can be said to be unusual and rare," Ngai said.
"If the real cause of the broken cable is rusting, it shows the structure has not been inspected for a long time or its routine inspection has not been sufficiently done," Ngai said.
He warned of potential hazards in other parts of the bridge if maintenance procedures were flawed.
"The government needs to find out what went wrong with the maintenance and to thoroughly inspect other parts of the structure," he said.
Jeremy Tam Man-ho, deputy chairman of the Legislative Council's transport panel, said the incident was outrageous and he questioned whether anti-rusting measures and other maintenance works were properly conducted.
This incident will undermine public confidence over large infrastructure projects in Hong Kong
Jeremy Tam, lawmaker
"This bridge is relatively new compared to the others before the handover era such as the Tsing Ma Bridge. This incident will undermine public confidence over large infrastructure projects in Hong Kong especially after the handover," Tam said.
Stanley Chiang Chi-wai, spokesman for truck drivers' group the Land Transportation Alliance, said the partial closure of the bridge would not affect them.
"The Shenzhen Bay Bridge is not a popular choice for cross-border trucks so we aren't worried there will be congestion after the closure of the two lanes," he said.
The main contractor of the Hong Kong section was Gammon-Skanska-MBEC Joint Venture. The government expected the project, costing HK$3.2 billion (US$410.3 million) back then, would bring a net economic benefit of HK$175 billion from 2000 to 2020. Then-president Hu Jintao officiated at the opening.
This article Snapped cable on cross-border Shenzhen Bay Bridge results in closure of two lanes but Hong Kong's highways chief says there are no structural safety issues first appeared on South China Morning Post
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Japan shares fall on pandemic worries as rest of Asia rises
Asian shares were mostly higher Wednesday, ahead of Joe Biden's inauguration as U.S. president. Japan's benchmark lost early gains as worries grew about the surge in coronavirus cases. Japan's benchmark Nikkei 225 lost early gains to slip 0.4% in morning trading to 28,515.21.
US inaugurations: High drama and sore losers
Outgoing US President Donald Trump will be far from the first to boycott his successor Joe Biden's inauguration on Wednesday -- but his absence will be the first since 1869.
NY gov explores buying virus vaccine directly from maker
Frustrated by the flow of coronavirus vaccine from the federal government, Gov. Andrew Cuomo on Monday floated the idea of buying shots for New Yorkers directly from one of the vaccine makers, Pfizer. Regardless, Cuomo said he felt compelled to broach the idea as his state, like many others, faces tough vaccine math. At the current pace of federal vaccine shipments to New York, it could take six months or more to get shots to the 7 million residents already eligible under federal guidelines, let alone the roughly 12 million other New Yorkers.


text:
International Business Degree Guide
A Guide To Earning Your International Business Degree
Comparing MBA Rankings
International Business Degree Program Rankings
Personality Traits, Employment Requirements, and Career Opportunities For Success In International Business
With the spread of the Internet and life in the "digital age," the world has effectively become smaller in a way that everyone is competing in a global marketplace. Many find that obtaining a job in international business will allow a person to benefit from the affects of the changing global trends combined with advancing technology. The most obvious way to pursue this goal would be to earn an international business degree. Since business has traditionally been one of the most popular fields of study for students looking to earn a healthy salary after graduation, it follows that international business has been rapidly expanding and may allow for even greater opportunity. Most large colleges and universities have international business programs, so finding the program which best suits your educational needs, budget, and professional aspirations is vital to propelling goals.
Once enrolled, a student will have traditional business courses as well as ones that focus on things like international financing and markets, and cross-cultural management. The curriculum in these classes will cover things like how to thrive in a sometimes volatile business setting, often in different countries, as well as acquire knowledge regarding currency exchanges, international laws, and tariffs.
Although an international business degree is helpful, a person does not necessarily have to have one to be employed and successful in international business. Pairing a marketing degree with a fluency in a foreign language could turn a person into a great asset for a firm. Similarly, a solid upper level educational foundation, the mastery of sales, and an extensive record of traveling abroad might be enough to get a person started in the field.
Aside from degrees, there are certain traits that companies will look for when hiring employees to work for them internationally. Many of these are applicable to all jobs, but they become especially important in an international setting. Businesses want people who are, curious, flexible, and resourceful since they may find themselves in unfamiliar situations and need to improvise. Additionally, it is key to have strong communication skills, the ability to negotiate, cultural sensitivity, and an interest in travel.
The jobs that can be included under the umbrella of international business are wide ranging, but unified by the fact that they are fairly to tremendously lucrative. Three examples of these jobs involve positions in international sales, customs compliance, and international banking. International sales representatives work at multinational corporations, and they focus on sales outside of the home country. The salaries for this position are often tied to commission, which makes them hard to gauge, but a rough average puts them around $60,000.
A person working in customs compliance concerns themselves with goods traveling across an international border. Each nation has different laws about not only the taxes on items entering their country but also about what's allowed in at all. Certain weaponry or vegetation may not be allowed to cross a border even if it is only passing through on its way to a separate final destination. Failure to comply with these regulations may lead to a business being fined and possible harm to existing relationships. The median salary for someone working in customs compliance is $58,000.
Of all the opportunities for a person engaged in international business, international banking may be the most lucrative. The compensation for these jobs is easily six figures, and with experience the pay can creep over a million dollar a year. The value these individuals provide is that they have the expertise that can allow corporations and high worth individuals to do things like structure transactions in a way that greatly reduces their taxes or facilitate international investments.
Whether a person wants to become an international banker or simply translate business documents, international business is a great field to get into. All it takes is the right degree and an attitude built for success.
This entry was posted in Careers In International Business on August 22, 2013 by Staff Editor.
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What is International Business?
What is an International Business Degree?
What is Studying for an International Business Degree Like?
What Is An International Business Degree Program Like?
What Determines the Quality of Education at an International Business School?
Can I get my MBA in International Business
Will Getting A Graduate Degree In International Business Help My Career?
What is the Highest Paying Job in International Business?
What Can A Degree in International Business Earn You Beyond A Salary?
How Long Will It Take To Complete My International Business Degree?
Will Completing The Top International Business MBA Programs Help Me Find A Job?
What Jobs Can I Do With An Online International Business Degree?
What Is The Difference Between International Business Degrees and International Relations Degrees?
Hult International Business School's Programs
Sources And Tips For International Business Degree Program Scholarships
Applying For and Getting Accepted Into Accredited International Business Degree Programs
The Advantages and Disadvantages to Online International Business Degree Courses
The Top 10 Free Classes Available Online For International Business Majors
International Business Career Information
10 International Business Degree Career Options For MBA Graduates
Jobs for International Business Degree Majors and Graduates
The 10 Best Places To Find International Business Jobs
The Highest Paying Jobs In International Business
International Business Careers That Involve Global Travel
How The Global Economy Affects International Business Careers