Canada-based fintech firm MOBI724 Global Solutions Inc. announced on Monday it has appointed Michael Schuck as Chief Sales Officer (CSO) and Johnny Hawa as Chief Operating Officer (COO).
The company reports that Schuck has more than 30 years of experience in both banking and financial services. He previously held the president position at consulting firm Omnico Solutions and served as senior executive vice president at MBNA. Hawa is co-founder of MOBI1724 Solutions and has over 25 years of experience in the sales and delivery industry.
The appointments come less than six months after MOBI1724 completed $553,000 private placement.
This entry was posted in Canada, Fintech and tagged appointments, canada, johnny hawa, marcel vienneau, michael schuck, mobi1724. Bookmark the permalink.


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PRESIDENT BUHARI TO AfDB PRESIDENT, ADESINA: I'LL STAND BY YOU
President Muhammadu Buhari says Nigeria will stand solidly behind Dr Akinwumi Adesina in his bid to get re-elected as President of the African Development Bank (AfDB)
The President spoke at State House, Abuja, Tuesday, while hosting Dr Adesina on a courtesy visit.
"In 2015, when you were to be elected for the first term, I wrote to all African leaders, recommending you for the position, president said.
The president, in a statement signed Femi Adesina Special Adviser to the President
(Media and Publicity) stated " I didn't say because you were a People's Democratic Party (PDP) Minister, and I belonged to the All Progressives Congress (APC), so I would withhold my support. I'll remain consistent with you, because no one has faulted the step I took on behalf of Nigeria," .
The President pledged that Nigeria would work with all other leaders and stakeholders in AfDB to ensure that Dr Adesina was elected for a second term built on the record of his achievements during his first term.
The African Union had already endorsed the incumbent AfDB President as sole candidate for the continent, but some other stakeholders are trying to ensure that Dr Adesina is re-investigated on some allegations, and rendered ineligible to run.
Giving a background to what was happening in the bank, Dr Adesina, a former Nigerian Minister for Agriculture, said the 16 allegations raised against him were trumped up, "and without facts, evidence, and documents, as required by the rules and regulations of the bank."
He added that the Ethics Committee of the bank cleared him of all the allegations, and calls for fresh investigation by the United States of America, were against the rules.
"My defense ran into 250 pages, and not a single line was faulted or questioned. The law says that report of the Ethics Committee should be transmitted to the Chairman of Governors of the bank. It was done, and the governors upheld the recommendations. That was the end of the matter, according to the rules. It was only if I was culpable that a fresh investigation could be launched. I was exonerated, and any other investigation would amount to bending the rules of the bank, to arrive at a predetermined conclusion," Dr Adesina said.
Stressing that the motive was to soil his name, and that of the bank, the AfDB President said he was proud to be Nigerian, and thanked President Buhari for his unflinching support.
"You helped me to get elected in the first place, and you have supported me robustly all along, and the African Union unanimously endorsed my re-election" he declared.
While commiserating with President Buhari on the death of the former Chief of Staff, Mallam Abba Kyari, Dr Adesina described Professor Ibrahim Gambari, new Chief of Staff as "a man of integrity, and of global standing."
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2020 Budget Review: National Assembly resumes Thursday
Contrary to its earlier resumption date of next Tuesday, June 2, 2020, both chambers of the National Assembly are expected…
Published: May 26, 20209:57 pm Updated: 9:58 pm
COAS Reiterates Commitment To Enhance Reforms Of Nigeria Army To Address Emerging Security Threats
Segun Awofadeji iñ Bauchi As part of efforts to enhance and strengthen the nation's security architecture,Chief of Army Staff(COAS), Lieutenant…

redpajama_set_name: RedPajamaCommonCrawl


We have listed all of the Niceville food pantries that we have found. With help from users like you we have compiled a list of some. If you know of a food pantry that is not included here, please submit new food pantries to our database by going to the ADD A LISTING link on the header of our website.
Food pantry serving Okaloosa county. Hours:Monday through Thursday,9:00am - 2:00pmFriday 9:00am - 12:00pmTo be eligible, must reside in Niceville or Valparaiso. Contact Lin Saber.


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Tennessee School Districts Now Have a Single Source for Print and Digital Content
May 30, 2017 09:00 AM Eastern Daylight Time
NASHVILLE, Tenn.--(BUSINESS WIRE)--Tennessee Book Company LLC, an Ingram Content Group company, has acquired assets of Thrivist, LLC; a teaching and learning platform to create a single, dynamic way for teachers to access both digital and print content for their classrooms.
Whether school districts have goals around personalized learning or providing easy access to digital resources, the platform provides solutions that help school districts make the transition. Thrivist will also help publishers create more connection points with Tennessee School Districts.
Currently, an estimated 54 percent of students and teachers have personal devices issued by the school. "Initiatives that drive student-to-device ratios will increase the integration of classroom technology, complimenting traditional print learning. In the K-12 school market, there has been a huge shift from primarily print to primarily digital over the last few years," said Todd Svec, vice president K-12 content and general manager for Tennessee Book Company.
Tennessee Book Company has served as Tennessee's state depository of printed textbooks since 1935. "K-12 schools are continuing to increase their use of digital course materials to help their students succeed. It only makes sense to help support our customers and streamline the process of making the transition of physical only course materials to a more diverse offering," said Svec.
As content continues to go digital, Tennessee Book Company intends to uphold its dedication to the advancement of Tennessee school districts. "We have every intention of walking side-by-side with our customers as course materials shift to digital. The addition of Thrivist to our company provides our customers a one-stop shop to implement their digital content needs, with seamless and easy access," added Svec.
"We are pleased to pair our capabilities together with Tennessee Book Company to empower teachers and better prepare K-12 students for the future," said Andrew McGarrity, Thrivist founder and new Director of Digital Solutions for Tennessee Book Company.
About Ingram | www.ingramcontent.com
Ingram Content Group ("Ingram") is comprised of several subsidiaries to Nashville-based Ingram Industries Inc. Ingram includes a comprehensive suite of publishing industry services companies that offer numerous solutions, including physical book distribution, print-on-demand and digital services. Committed to the success of its partners, Ingram works closely with publishers, retailers, librarians and schools around the world to provide them with the right products and services to help them succeed in the dynamic and increasingly complex world of content publishing and distribution. Ingram's operating units are Ingram Book Group LLC, Lightning Source LLC, VitalSource Technologies LLC, Ingram International LLC, Ingram Library Services LLC, Spring Arbor Distributors LLC, Ingram Publisher Services LLC, Tennessee Book Company LLC, and ICG Ventures LLC. Learn more about Ingram Content Group at www.ingramcontent.com.
About Thrivist
Thrivist is a learning and analytics company that enables K-12 schools to personalize learning and make better decisions based on data. Thrivist's platform offers districts a way to develop, deliver and manage online learning and correlate learning activities with student performance. Thrivist is based in Franklin, Tennessee. www.thrivist.com.
Ingram Content Group
Elizabeth Fielding, 615-812-8009
Tennessee Book Company LLC, an Ingram Content Group company, has acquired assets of Thrivist, LLC; a teaching and learning platform.


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H.Con.Res. 28:
Expressing the sense of Congress that the President's executive amnesty is illegal notwithstanding passage of H.R. 240, the Department of Homeland Security Appropriations Act of 2015.
Published: Wed, Mar 25th 2015 @ 12:39 pm EDT
Expressing the sense of Congress that the President's executive amnesty is illegal notwithstanding passage of H.R. 240, the Department of Homeland Security Appropriations Act of 2015.
Read more about Expressing the sense of Congress that the President's executive amnesty is illegal notwithstanding passage of H.R. 240, the Department of Homeland Security Appropriations Act of 2015.
S.Con.Res. 11:
S.Con.Res. 11
Published: Wed, Mar 25th 2015 @ 9:35 am EDT
An original concurrent resolution setting forth the congressional budget for the United States Government for fiscal year 2016 and setting forth the appropriate budgetary levels for fiscal years 2017 through 2025.
Read more about S.Con.Res. 11
IN STATE Act of 2015
Published: Mon, May 11th 2015 @ 10:32 am EDT
H.R. 1507, the IN STATE Act of 2015, would grant amnesty to illegal aliens who enlist in the military. The bill would also allow illegal aliens who meet the "Dreamer" requirements to receive in-state tuition rates.
Read more about IN STATE Act of 2015
S.Res. 104:
A resolution to express the sense of the Senate regarding the success of Operation Streamline and the importance of prosecuting first time illegal border crossers.
Published: Thu, Aug 27th 2015 @ 11:38 am EDT
Read more about A resolution to express the sense of the Senate regarding the success of Operation Streamline and the importance of prosecuting first time illegal border crossers.
Jobs Originated through Launching Travel Act of 2015
H.R. 1401, the Jobs Originated through Launching Travel Act of 2015, would allow the Secretary of Homeland Security to waive the maximum refusal rate to allow a certain country to qualify for the visa waiver program. The visa waiver program allows citizens from qualifying countries to legally enter the United States without a visa.
Read more about Jobs Originated through Launching Travel Act of 2015
Arizona Borderlands Protection and Preservation Act
H.R.1412, the Arizona Borderlands Protection and Preservation Act, would allow Border Patrol agents access to federal lands in Arizona, but would also weaken the definition of operational control from 100% situational awareness to only 90%.
Read more about Arizona Borderlands Protection and Preservation Act
Trade Act of 2015
Published: Fri, Jun 12th 2015 @ 10:24 am EDT
H.R. 1314, the Trade Promotion Authority Act, would set up fast-track procedures for the passage of trade deals negotiated by the president. Although Ambassador Froman, the U.S.
Read more about Trade Act of 2015
Child Tax Credit Integrity Preservation Act of 2015
H.R. 4044, the Child Tax Credit Integrity Preservation Act of 2015, would halt refugee resettlement of foreign nationals from Syria, Iraq, Libya, Yemen, Afghanistan or whose last known residence was in one of those countries.
Read more about Child Tax Credit Integrity Preservation Act of 2015
Published: Mon, Mar 2nd 2015 @ 1:12 pm EST
H.R. 1147, The Legal Workforce Act, would require most employers to use E-Verify within two years of enactment. Large employers, federal, state, and local agencies and federal and state contractors would need to comply within six months. The bill would also make the E-Verify program permanent.
Read more about Legal Workforce Act
Michael Davis, Jr. in Honor of State and Local Law Enforcement Act
Published: Tue, Mar 3rd 2015 @ 9:45 am EST
H.R. 1148, the Michael Davis, Jr. in Honor of State and Local Law Enforcement Act, would prevent cities from providing sanctuary to illegal aliens and would provide funding for the State Criminal Alien Assistance Program (SCAAP) that reimburses states for incarcerating illegal aliens. The legislation would help reduce rewards for illegal immigration by preventing cities from providing a safe harbor for illegal aliens.
Read more about Michael Davis, Jr. in Honor of State and Local Law Enforcement Act


Online betting has gained immense popularity. You will find a large number of people consider betting online rather than traditional methods. Increasing popularity of online betting has led to an increase in betting websites. You will find that the internet is filled with many sites all claiming to offer the best betting experience. Though there are many, nothing can beat the services offered by Kalyan Mumbai SattaMatka. It is the best way to bet from the convenience of your home. It does not matter whether you are a novice or a professional bettor you can rely on these services. It is seen that people are investing a lot of money as the chances of earning huge profits is huge.
It is a simple lottery game. In this you are required to pick three numbers between 0-9. The three numbers chosen are further added in order to offer the final number. The last digit is later taken into consideration for the final sum. It is later added with the three numbers you have chosen. You can bet either on the entire set or seek expert help .
There are innumerable benefits associated with sattamatka. Mentioned below are some of the major benefits and suggestions to keep in mind if you plan on becoming the sattamatka.
It is of paramount importance that you check the reliability of the website before playing the game. There are a number of sites that are completely dedicated towards offering the best services. Ensure that the site you are playing from is legal.
The SattaMatka result is completely based on draws. All the draws are done with complete security. All the numbers displayed are true and appear only from the draw. A reliable website offers all the information and awards the winners accordingly.
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How to Modify Motor Scorpio became a Japstyle - Trending modification vintage kind certainly back booming later. In addition to the trend's La Honda CB's full of life in lots of areas, modification like Japstyle also was not able to claim the least. The thought of japstyle positioned ahead at the aspect of the foot legs that seem to be sturdy.
Elegant yet vintage seem to be can emanated from the kind changes that come from Japan. Of the numerous varieties of motorcycles, Yamaha Scorpio Z is probably one of the foremost extensively used. This is simply due to the fact Jenson button that helps as effectively as a large-capacity engine i.e. cc 215. with a big engine, Scorpio Japstyle be extra dashing.
As I stated above, japstyle is a modification of the stream that comes from the land of the growing sun. As the identify implies is a japstyle quick of Japannese Style. The stream of this modification already existed in Japan for the reason that 80 's. Then the trend's in Japan surely comply with taste motor ever released in Europe as properly as in such form of Amerikca Cafe Racer, Scambler or Flat Tracker. However, as the Japanese aren't the effects of modification known as allusion, and so present brat taste which originated from a workshop named Brat.
To swap the japstyle, so the primary factor to fix is the sector's order. The aft ingredient to the order wants to be dipapras after which trimmed back. To desist and truly ought to be introduced to the workshop of las. Revision of This wants to be achieved as the use of a shorter customized upholstery.
The thought of japstyle placed ahead sector ft considered sturdier. For that wheel rims and tires manufacturing facility default isn't used anymore. Instead aluminum rims measurement can elect 135-450 ring 17 at the again and 135-400 ring 17 to the front.
The use of a extra gambot tires, making the unique entrance shockbreaker can't be installed. So it might use with aftermarket segment airs extra width or can use the shockbreaker destiny belongs to fz16. Whereas for the tow, mono surprise brand was changed with a double surprise right the next with that arm. Swing arm ought to be changed extra width so that the ban might be entered as properly as for the set up of a double shock.
After completion of the Affairs of the feet, like this turn update "dress" the Scorpion. Starting from spatboar, the tank and edge disguise is changed with the product after the market comprised of galvanized plate. In order to be extra frugal can discover that one package deal deal and it's been painted. You don't need to do paint again.
Lastly, to adorn the appearance of the japstyle, all lights off lights default. Replace with the next spherical headlamp with a faded sein. In addition different accessories also want embedded like a rear view mirror and in addition customized exhaust.
How to make adjustments to the Motor So this Japstyle Scorpio. But to do a revision of the price range required isn't the least bit incredibly if desiring a exceptional aftermarket products. Cost levels among Rp 3 million to Rp 8 million depending on aftermarket merchandise which will you use.
That's How to Modify Motor Scorpio turned a Japstyle. Thank you.


Welcome to the artsBASICS resources page! On this page, we will provide you with useful resources, tools, websites and other information to help you integrate the arts into all areas of education, at all levels.
Resources are more than art supplies, musical instruments, or dance lessons. Below you will find different articles, links to supporting materials, and other resources you may find as helpful as we have. You have something to share, please drop us a line and we will review your submission and add it accordingly.
The following lessons are from participants from our 2018 artsBASICS Summer Arts Integration Seminar who also received credit from St. Ambrose University. Please feel free to adapt any of these lessons in your classes and join us next June for our 2019 Seminar at the new location of one of our schools, the Creative Arts Academy of the Quad Cities next to the Figge Art Museum!
ArtsEdge is the Kennedy Center's free digital resource for teaching and learning in, through and about the arts.
Developed by the Los Angeles County Office of Education in partnership with the Los Angeles Unified School District as part of the (CCSESA) Arts Initiative and the Curriculum and Instruction Steering Committee (CISC) Visual and Performing Arts Subcommittee.
In this two-day lesson, students look beyond the basics of haiku poetry and focus on the content of the haiku. Students reflect on their daily lives to find moments of peace and happiness. Using these moments and observations, they create a haiku and an accompanying photograph, which are combined into a digital visual class anthology.
In this lesson, students learn about the mythological 12-hour journey of the sun god Re and explore an online Smithsonian Museum exhibit. Students review locomotor and non-locomotor movements and the different qualities of movement, and choreograph a piece that represents one of the hours of Re's journey.
This video from NatGeo Channel's "StarTalk" program features musician David Byrne and Neil deGrasse Tyson discussing the vital importance of arts education.
The George Lucas Educational Foundation, a nonprofit operating foundation, was founded by filmmaker George Lucas in 1991. Growing up, Lucas was curious and creative – but at school, he often felt bored. Years later, after becoming a father, he once again found himself focused on schools' untapped potential to truly engage students and inspire them to become active, lifelong learners. He decided to invest in making a difference and created the Foundation to identify and spread innovative, replicable and evidence-based approaches to helping K-12 students learn better.
An interesting and thought provoking piece by Adam Grant, contributing Op-Ed writer for the Sunday Review supplement to the New York Times newspaper.
An article from District Administration, (DA), the most widely received and most regularly read publication for school district leaders nationwide in print and digital publishing. This article emphasizes developing artistic ideas, refining them, and following projects through to completion.
Art has long been recognized as an important part of a well-rounded education — but when it comes down to setting budget priorities, the arts rarely rise to the top. Many public schools….


At once the "2016 OTC Exhibition" will be hold on May 2-5 in Houston.
At the exhibition, the Centrifuge, Vertical Dryer and Shaker Screens that manufactured by GN Solids Control will be showed.
Today, in this article we give a introduction about the features of GN's centrifuge so that people could learn about its knowledge in advance.
3. GN's Centrifuge is the high-speed centrifuge. Drum and Helical Pusher inside the drum are the main working parts, their working speeds usually are above 3000RPM. The working principle of centrifuge is by the centrifugal force. Based on different densities between solid phase and liquid phase, the fluid in the centrifuge will form 2 rings: outside is the solid ring, inside is the liquid ring. The more higher rotation speed, the much better treatment effect.
Welcome you contact us or visit our booth at the exhibition for learning more about our centrifuge and other products.


Words can not describe how much I loved staying in this city. So pictures will just have to do.
One of my favorite cities . Working on another blog post about The 416! Thank you for liking my blog post on DC.


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Aleksander Wiktor Wiatrak (ur. 29 grudnia 1914 w Berlinie-Schlachtensee, zm. 12 lutego 1980 w Gdyni) – gdański przedsiębiorca i holenderski urzędnik konsularny.

Życiorys
Syn Antonio Wiatraka (1868–1959), przedsiębiorcy, kupca i konsula honorowego Gwatemali, Hondurasu i Kostaryki, i Josephine Frank. Ukończył Gimnazjum (Realgimnasium) w Sopocie, absolwent Politechnic-School of Commerce w Londynie (1932). Praktykował na plantacjach kawy w Meksyku i Gwatemali (1934), również w firmach Los Angeles, San Francisco i Nowym Jorku. Po powrocie do kraju prowadził firmę importu kawy w Gdyni oraz został konsulem honorowym Kostaryki w Gdańsku i wicekonsulem Kostaryki w Gdyni. Był członkiem i ekspertem do spraw kawy Izby Przemysłowo-Handlowej w Gdyni (1937).

Po wojnie był zatrudniony w UNRRA oraz pełnił funkcję kier. referatu cudzoziemskiego Urzędu Wojewódzkiego w Gdańsku, następnie pracował w konsulacie Holandii w Gdyni (od 1946), który p.o. konsula prowadził w latach 1952–1980. W trakcie swojej działalności Aleksander Wiatrak był poddawany działaniom służby bezpieczeństwa. W 1971 królowa Juliana odznaczyła go Medalem Rycerskim "Orde von Oranje Nassau". Od 1918 mieszkał, przy Cecilenstrasse 6 (obecnie ul. Czyżewskiego). Został pochowany na cmentarzu katolickim w Sopocie (kwatera H2-A-1).

Bibliografia
 Biogram

Przypisy

Ludzie urodzeni w Berlinie
Urzędnicy konsularni w Gdańsku
Holenderscy dyplomaci
Członkowie Izby Przemysłowo-Handlowej w Gdyni (II Rzeczpospolita)
Urodzeni w 1914
Zmarli w 1980
Pochowani na Cmentarzu katolickim w Sopocie

redpajama_set_name: RedPajamaWikipedia


* Spot the snowmen to progress.
* Click the cloth to shuffle.
* Clear some space with bombs.
Findin' glyphs in a sea of unicode - just like your gramps used to play.
Loved how the audio blended with the clicks!
The game graphics and effects are nicely well polished.
The game starts to be very hard after 40☃, cards expand on the whole screen and that's a real challenge to find the ☃ to progress to the next level.
Cool "Where is Waldo"-esque game. Could be great with a bit more polish.
Very fun game. It started to run very slow around 40.
Yeah, this is secretly just a browser-based stress test: how goods your 'puter/browser? I get up to 50+ on my mac before slowdowns - interestingly, Firefox handles things better than Chrome... those guys are gettin' good recently.
Very addictive game, I spent some time on it already. I love score games.
After 45 you start losing money, you must have luck. I got 50 snowmen with 2700 moneys.
Anyway I really enjoyed the game. I appreciate things the other commentators listed. Game is really polished and even though it's just "find a wally", you put inside many additional mechanism (cash, bombs, etc). Only one thing I don't understand: why standard cards gives me mainly +1$, but sometimes I loose 20$. Which one does it?
This a very nice game, simple and elegant. Nice soundtrack too. Goodjob!
I started playing without reading any instructions and I thought it was just going to be as simple as finding the snowman but there was a lot more too it then I thought there was going to be and it was a lot of fun. I'm looking forward to playing it again in a second now I know all the rules. I like the strategy element I like the fact that you don't actually ever lose and it's up to you to quit when the going is good. Just like real gambling. The graphics and sound were good, and the whole thing was polished. The performance got bad after level 40 for me. You can cheat by clicking the cash out button just to pause it and still continue to look for the snowman.


Showing results for tags 'adi-173'.
If you have a shape with many nodes like a letter converted to curves it's very difficult to select and move multiple nodes. There does not seem to be any logic to what node will move all of them. I get that the concept is to select the nodes and then pick one node and move all of them together. This does not work most of the time if there are many nodes close together. You have to select and find the random node that seem to move all of them. If you select the wrong one just the one node moves. It's a strange way to do this and to me looks like a bug. It's also too easy to deselect all the nodes you want to move forcing you do start over. This really needs to be fixed right away as this is a main feature in any vector program and Designer can not be used as is. If I'm misunderstanding how to do this please let me know.


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Sportsbook > Tennis > Betting on Tennis at French Open: Favorites Advance
Betting on Tennis at French Open: Favorites Advance
Posted by: Mike Davis Posted on: Mon, May 29th, 2017 5:31 PM
Paris is a moveable feast, but it did not follow Novak Djokovic – a favorite of those betting on tennis – anywhere he went. Au contraire, the Serb had to return to la scène du crime; Stade Roland Garros, where he won his career Grand Slam. Accompanied by new coach Andre Agassi, the No. 2 seed made short work of Marcel Granollers on Monday to move on to the second round of the tournament. Djokovic has not won any major titles since defeating Andy Murray in the 2016 French Open final, but the Scotsman had the last laugh when he bumped the Serb off the No. spot in the world rankings.
Nole lost in the third round at Wimbledon, which marked his earliest exit of in a Grand Slam in seven years. then he was eliminated in the first round at the Rio de Janeiro Olympics, made the final of the U.S. Open, and was again ousted early at the Australian Open. Djokovic hired Agassi as the Mr. Miyagi to his Daniel-san in an attempt to become the first racketman in the Open Era to win each of the four majors two times. What are the odds of that, wonder people who bet on tennis. Djokovic parted ways with coach Marian Vajda at the start of the month. "I've changed everyone, but I will not change my wife," Djokovic joked with a smile. Guess that puts the kibosh on a tennis-centric Wife Swap episode.
Meanwhile, Rafael Nadal is on a quest of his own; namely, to win his 10th French Open – a feat that no man or woman has ever achieved in the Open Era. The Nadster went through Benoit Paire 6-1, 6-4, 6-1. The Nadman put on a clinic with only 18 unforced errors. The favorite of people betting on tennis was in the second round in less than two hours. If you were a Richard Mille ambassador like Nads, you would have been able to time it with your $725,000 wristwatch – which kind looks like Ironman's watch if he wore one. And perhaps someday you'll be able to afford it too, if you bet on tennis wisely.
"For me, it's important to serve a little bit better than what I did today," El Naderino said. "I feel that some moments I was not serving that well. The rest of the things I am happy with." The Spaniard won his 10th Monte-Carlo Rolex Masters, 10th Barcelona Open Banc Sabadell, and fifth Mutua Madrid Open in the last two months. His win on Monday improved his record to 18-1 during this year's clay-court season and to 73-2 all-time at the French Open. The No. 4 seed Nadal will go up against Dutchman Robin Haase in the next round. "I know him very well and he knows me very well. He's a dangerous opponent," Nadal said of Haase. "He hits the ball very hard. So I need to be very focused on the serve because returning him can be tough." Nonetheless, Nadal will be the favorite of people betting on tennis in that matchup as well.


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\section{Introduction}Casimir effect \cite{16} has wide applications in different areas of physics such as quantum field theory, gravitation and cosmology, atomic physics, condensed matter, nanotechnology and mathematical physics (see e.g. \cite{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}). According to Casimir, there is an attractive force
 \begin{equation}\label{eq8_26_1}F_{\text{Cas}}= -\frac{\pi^2 \hbar c A}{240 a^4}\end{equation} acting between two perfectly conducting plates with area $A$ and separation distance $a$. Naively the Casimir energy due to a quantum field is the sum of the zero point energies of the quantum field:
\begin{equation}\label{eq8_26_2}
E_{\text{Cas}} =\frac{\hbar}{2}\sum \omega,
\end{equation}where  the sum runs through all eigenfrequencies $\omega$ of the field. Generically, this sum is an infinite sum, and Casimir is the first one to extract a finite force \eqref{eq8_26_1} acting between two parallel perfectly conducting plates. Since then, various methods have been proposed for the regularization of the Casimir energy \eqref{eq8_26_2}.

 Casimir force is a pure quantum effect which can be considered as the macroscopic manifestation of the London-van der Waals force between atoms and molecules. Lifshitz \cite{3,28, 29} has laid a theoretical foundation for the description of both the London-van der Waals force and the Casimir force. The celebrated Lifshitz formula expresses the finite temperature Casimir force between two plane parallel plates as some functionals of the electric permittivities of the plate materials. This formula has been re-derived using different approaches \cite{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, and is the main tool for the theoretical investigation of the Casimir effect on real materials at nonzero temperature. The original Lifshitz formula models the two parallel plates as two semi-infinite slabs which have infinite thickness. This approach is justified if the skin-depth of the material of the two plates is much smaller than the thickness of the plates. A refinement of the Lifshitz formula where the finite thickness of the plates is taken into consideration has been derived and usually related to the consideration of the multi-layer models \cite{new1, 5,9,10,11,12,13,54}.

Recently the issue of   divergences in Casimir effect was discussed extensively in \cite{new2,new3, new4, new5}. In connection to the application of Casimir effect in quantum field theory, this has been pointed out in \cite{30,31}. Recently, this issue was reconsidered in \cite{32,33,34,35} where it was argued that surface divergence terms in the Casimir energy cannot be removed by any renormalization of the physical parameters. Although the  arguments of \cite{32, 33, 34, 35}  have been refuted in the works \cite{36, 37}, one still need to be cautious in regularizing the Casimir energy. In \cite{38}, Cavalcanti introduced a geometric setup called piston and showed that the Casimir force acting on a piston is free of divergences, due to the cancelations of the divergences from the two regions separated by the piston. By taking certain limits, one can recover the Casimir force acting on a pair of parallel plates. Therefore, the piston approach provides an interpretation for the formula of the finite Casimir force acting between parallel plates. Since then, Casimir effect in the piston setup has attracted considerable interest \cite{39,40,7,41,42,43,6,8,44}. It has also been used to explore the Casimir effect in the presence of extra dimensions \cite{45,46,47,48,49,50,51,52}. Nevertheless, these works either considered scalar fields with Dirichlet or Neumann boundary conditions or more general Robin boundary conditions, or electromagnetic fields with perfect electric conductor conditions or perfect magnetic conductor conditions. In these cases, the piston becomes a barrier for the quantum fields to penetrate from one region to the other, and therefore one can compute the Casimir energies in the two regions separated by the piston independently. The divergent part of the Casimir energies in the two regions would give rise to Casimir force of opposite direction on the piston and therefore cancel each other.

Due to the success of the piston approach in the parallel plate scenario, one would expect a similar approach can be used to derive the Lifshitz formula for the Casimir effect on real materials. As a matter of fact, piston type approach has been employed by Schram \cite{53} in 1973 as a regularization scheme to derive the Lifshitz formula at zero temperature. The generalization to include the thermal effect is still lacking. Moreover, in the work of Schram, some subtraction scheme was employed to get rid of the divergences, instead of observing an automatic cancelation of divergence as in the recent works on pistons. In this work, we re-derive the Lifshitz formula using the piston approach. In fact, we consider a more general setup. First we derive the Casimir energy for a three-layer model, i.e., three parallel layers of different materials    with electric permittivities $\varepsilon_1,\varepsilon_2,\varepsilon_3$ and magnetic permeabilities $\mu_1,\mu_2,\mu_3$. Using the piston approach, we first embed the layers   inside a large rectangular box which has perfectly conducting walls  serving as impenetrable barriers to  the electromagnetic field. The finite temperature Casimir energy for the electromagnetic field inside the box is computed using mode sum approach. An exponential cut-off is used to regularize the Casimir energy. The result is written as a divergent part (which goes to infinity if the cut-off parameter approaches zero) plus a finite interaction term (which is independent of the cut-off parameter). Treating the second layer as a non-deformable piston, we compute the Casimir force acting on a piston made of real material by differentiating with respect to the distance between the piston and one end of the rectangular box. The limit where the other end of the rectangular box is brought to infinity gives the Casimir force acting between a perfectly conducting plate and a plate made of real material. It is shown that the divergent part of the Casimir force will only be canceled under certain conditions, for example, when the regions separated by the piston or the plates are filled with   media  of the same refractive index $n=c\sqrt{\varepsilon\mu}$. In general, one cannot ignore the cut-off parameter.

To derive  the Casimir force acting on two parallel plates both made with real materials, it is necessary to consider the five-layer model. Using the same approach as the three-layer model, we derive the Casimir energy and the Casimir force for the five-layer model. In the limit where both ends of the confining rectangular box go to infinity, we obtain  the Casimir force acting on two parallel plates, both made with real materials. The limit where the thickness of the two plates are infinite recovers the celebrated Lifshitz formula. As in the case of the three-layer model, we observe that the divergences of the Casimir force would cancel each other if the three regions separated by the two plates are filled with   media of the same refractive index.

Finally, we would like to remark  that in this work, the piston approach does not really refer to the piston, but rather the finite rectangular box with perfectly conducting walls that serve to confine the field into a finite region. The advantage of this approach is that in solving for the field modes, one do not have to use imaginary energy modes as in some of the previous works. This bridge the gap between the derivation of the Casimir force for real materials and the derivation of the Casimir force for ideal metals.

\section{Casimir energy inside a piston system}\label{s2}

 \begin{figure}
\epsfxsize=0.4\linewidth \epsffile{piston_1.eps} \caption{\label{f1}A   piston with electric permittivity $\varepsilon_2$ and magnetic permeability $\mu_2$ diving the rectangular box into two regions, which are filled with materials having electric permittivities $\varepsilon_1,\varepsilon_3$ and magnetic permeabilities $\mu_1,\mu_3$. }\end{figure}
We consider a  piston with thickness $d_2=b-a$ dividing a rectangular box of dimension $L\times L_2\times L_3$ (see FIG. \ref{f1})   into two regions filled with materials with electric permittivities and magnetic permeabilities $\varepsilon_1,   \mu_1$ and $  \varepsilon_3, \mu_3$ respectively. The  piston is assumed to have electric permittivity  $\varepsilon_2$ and magnetic permeability   $\mu_2$   and the walls of the rectangular box is assumed to be perfectly conducting.   Maxwell's equations take the usual form
\begin{equation}\label{eq3_26_1}\begin{split}
\nabla . \textbf{D}  =&\rho_f,\hspace{1cm}\nabla\times \textbf{E} +\frac{\partial \textbf{B}}{\partial t} =0,\\
\nabla . \textbf{B} =& 0,\hspace{1.2cm}
\nabla\times \textbf{H}-\frac{\partial\textbf{D}}{\partial t}=\textbf{J}_f,
\end{split}
\end{equation}where $\textbf{D}=\varepsilon\textbf{E}$ and $\textbf{B}=\mu\textbf{H}$. As usual, we introduce the potential $\Phi$ and $\textbf{A}$ such that
\begin{equation*}\begin{split}
\textbf{E}=& -\frac{\partial \textbf{A}}{\partial t}-\nabla \Phi,\hspace{1cm}
\textbf{B}=\nabla\times\textbf{A},
\end{split}
\end{equation*}and impose the radiation gauge
\begin{equation*}
 \nabla .\textbf{A}=0.
\end{equation*}In the absence of free charges and free current, i.e., $\rho_f=0$ and $\textbf{J}_f=\textbf{0}$, we can impose the additional condition $\Phi=0$. Then the first three equations in \eqref{eq3_26_1} are automatically satisfied and the fourth equation give
\begin{equation}\label{eq3_26_2}
\left(\varepsilon\mu\frac{\partial^2}{\partial t^2}-\nabla^2 \right)\textbf{A}=0.
\end{equation}For the boundary conditions, the continuity of $\textbf{D}\,.\,\textbf{n}, \textbf{E}\times \mathbf{n}, \textbf{B}\,.\,\mathbf{n}$ and $\textbf{H}\times\mathbf{n}$   across the boundaries implies that    $\varepsilon E_{1}$, $E_2$, $E_3$ and $B_1$, $\frac{1}{\mu}B_2$, $\frac{1}{\mu}B_3$ must be continuous in the $x_1$ direction. On the other hand, the perfectly conducting conditions on the boundary of the rectangular box imply that on the boundaries $x_1=0$ and $x_1=L$, $E_2=E_3=0$ and $B_1=0$. On the boundaries $x_2=0$ and $x_2=L_2$, $E_1=E_3=0$ and $B_2=0$. On the boundaries $x_3=0$ and $x_3=L_3$, $E_1=E_2=0$ and $B_3=0$.

A solution of equation \eqref{eq3_26_2}  satisfying the gauge and the boundary conditions can be written in the form
\begin{equation*}
\begin{split}
A_1=&\pi \left(\left[\frac{k_2}{L_2}\right]^2+\left[\frac{k_3}{L_3}\right]^2\right) \phi(x_1) \sin \frac{\pi k_2x_2}{L_2}\sin\frac{\pi k_3x_3}{L_3}e^{-i\omega t},\\
A_2=&\frac{k_2}{L_2}\frac{\partial\phi(x_1)}{\partial x_1} \cos\frac{\pi k_2x_2}{L_2}\sin\frac{\pi k_3x_3}{L_3}e^{-i\omega t},\hspace{2cm}(k_2, k_3)\in \mathcal{S}_{\text{TM}}:= \mathbb{N}^2,\\
A_3=&\frac{k_3}{L_3}\frac{\partial\phi(x_1)}{\partial x_1} \sin\frac{\pi k_2x_2}{L_2}\cos\frac{\pi k_3x_3}{L_3}e^{-i\omega t},
\end{split}
\end{equation*}for TM modes, and
\begin{equation*}
\begin{split}
A_1=&0,\\
A_2=& \frac{k_3}{L_3} \psi(x_1) \cos\frac{\pi k_2x_2}{L_2}\sin\frac{\pi k_3x_3}{L_3}e^{-i\omega t},\hspace{2cm}(k_2, k_3)\in \mathcal{S}_{\text{TE}}: \mathbb{N}_0^2\setminus\{(0,0)\},\\
A_3=&-\frac{k_2}{L_2}\psi(x_1)\sin\frac{\pi k_2x_2}{L_2}\cos\frac{\pi k_3x_3}{L_3}e^{-i\omega t},
\end{split}
\end{equation*}for TE modes. Here $ \mathbb{N}_0=\mathbb{N}\cup\{0\}$. The functions $\varepsilon \phi (x_1)$ and $\frac{\partial \phi(x_1)}{\partial x_1}$ are continuous, and $\phi(x_1)$ satisfies
\begin{equation*}
\frac{\partial^2\phi(x_1)}{\partial x_1^2}=-p(x_1)^2\phi(x_1),\hspace{1cm}\left.\frac{\partial \phi(x_1)}{\partial x_1}\right|_{x_1=0,L}=0,
\end{equation*}where $p(x_1)=p_j$ for $x_1$ in region $j$ whose  electric permittivity and magnetic permeability are $\varepsilon_j(\omega)$ and $\mu_j(\omega)$ respectively. The functions $\psi(x_1)$, $\frac{1}{\mu}\frac{\partial\psi(x_1)}{\partial x_1}$ are continuous, and $\psi(x_1)$ satisfies
\begin{equation*}
\frac{\partial^2\psi(x_1)}{\partial x_1^2}=-p(x_1)^2\psi(x_1), \hspace{1cm} \psi(0)=\psi(  L)=0.
\end{equation*}Finally, for $j=1,2,3$,
\begin{equation*}
\varepsilon_j(\omega)\mu_j(\omega)\omega^2=p_j^2+\left[\frac{\pi k_2}{L_2}\right]^2+\left[\frac{\pi k_3}{L_3}\right]^2=p_j^2+\lambda_{\boldsymbol{k}}^2.
\end{equation*}
For the function $\phi(x_1)$, assume that
\begin{equation*}
\begin{split}
\phi(x_1)=\begin{cases}
\Lambda_1 e^{ip_1x_1}+\Lambda_2 e^{-ip_1x_1}, \hspace{1cm}&0\leq x_1\leq a,\\
\Lambda_3 e^{ip_2x_1}+\Lambda_4e^{-ip_2x_1}, & a\leq x_1\leq b,\\
\Lambda_5e^{ip_3x_1}+\Lambda_6e^{-ip_3x_1}, & b\leq x_1\leq L.
\end{cases}
\end{split}
\end{equation*}Then the boundary conditions give six equations for the six variables $\Lambda_1,\ldots,\Lambda_{6}$:
\begin{equation}\label{eq8_4_1}
\begin{split}
&\Lambda_1  -\Lambda_2  =0,\\
&\varepsilon_1\left(\Lambda_1 e^{ip_1a}+\Lambda_2e^{-ip_1a}\right)=\varepsilon_2\left( \Lambda_3e^{ip_2a}+\Lambda_4e^{-ip_2a}\right),\\
&ip_1\left(\Lambda_1 e^{ip_1a}-\Lambda_2e^{-ip_1a}\right)=ip_2\left( \Lambda_3e^{ip_2a}-\Lambda_4e^{-ip_2a}\right),\\
&\varepsilon_2\left(\Lambda_3 e^{ip_2b}+\Lambda_4e^{-ip_2b}\right)=\varepsilon_3\left( \Lambda_5e^{ip_3b}+\Lambda_6e^{-ip_3b}\right),\\
&ip_2\left(\Lambda_3 e^{ip_2b}-\Lambda_4e^{-ip_2b}\right)=ip_3\left( \Lambda_5e^{ip_3b}-\Lambda_6e^{-ip_3b}\right)\\
&\Lambda_5e^{ip_3L}-\Lambda_6e^{-ip_3L}=0.
\end{split}
\end{equation}These homogeneous system in $(\Lambda_1,\ldots, \Lambda_{6})^T$ has a nontrivial solution if and only if the determinant of the corresponding matrix is zero. However, we can also solve the system in the following way. The first equation and the last equation give
\begin{equation}\label{eq3_27_1}
\Lambda_2=\Lambda_1, \hspace{1cm} \Lambda_5 = \Lambda_{6}e^{-2ip_3L}.
\end{equation}The second and third, fourth and fifth equations give respectively
\begin{equation}\label{eq8_3_1}
\begin{split}
\begin{pmatrix} \Lambda_3\\ \Lambda_{4}\end{pmatrix} =&\frac{1}{2}\begin{pmatrix} r_{12 }^+ e^{i(p_1-p_2)a} & -r_{12 }^- e^{-i(p_1+p_2)a}\\
-r_{12 }^-e^{i(p_1+p_2)a}& r_{12 }^+ e^{-i(p_1-p_2)a}\end{pmatrix}\begin{pmatrix} \Lambda_1\\ \Lambda_2\end{pmatrix}=\frac{1}{2}\begin{pmatrix} r_{12 }^+ e^{i(p_1-p_2)a} - r_{12 }^- e^{-i(p_1+p_2)a}\\
-r_{12 }^-e^{i(p_1+p_2)a}+ r_{12 }^+ e^{-i(p_1-p_2)a}\end{pmatrix}\Lambda_1,
\\\begin{pmatrix} \Lambda_3\\ \Lambda_{4}\end{pmatrix} =&\frac{1}{2}\begin{pmatrix} r_{32}^+ e^{i(p_3-p_2)b} & -r_{32}^- e^{-i(p_3+p_2)b}\\
-r_{32}^-e^{i(p_3+p_2)b}& r_{32}^+ e^{-i(p_3-p_2)b}\end{pmatrix}\begin{pmatrix} \Lambda_5\\ \Lambda_6\end{pmatrix}=\frac{1}{2}\begin{pmatrix} r_{32}^+ e^{i(p_3-p_2)b} e^{-2ip_3L}- r_{32}^- e^{-i(p_3+p_2)b}\\
-r_{32}^-e^{i(p_3+p_2)b}e^{-2ip_3L}+ r_{32}^+ e^{-i(p_3-p_2)b}\end{pmatrix}\Lambda_6,
\end{split}
\end{equation}where $$r_{jk}^{\pm}=\frac{p_j}{p_k}\pm\frac{\varepsilon_j}{\varepsilon_k}.$$
Equating the  two equations in \eqref{eq8_3_1} give an equation of the form
\begin{equation*}
\begin{pmatrix} \alpha\\\beta\end{pmatrix}\Lambda_1=\begin{pmatrix} \gamma\\\delta \end{pmatrix}\Lambda_{6}
\end{equation*}which has a nontrivial solution if and only if $\alpha\delta -\beta\gamma=0$. In other words, the system \eqref{eq8_4_1} has a nontrivial solution if and only if
\begin{equation*}
\begin{split}
e^{-ip_3L}\Biggl\{ &\left(r_{12}^+e^{ip_1d_1}-r_{12}^-e^{-ip_1d_1}\right)\left(r_{32}^+e^{ip_3d_3}-r_{32}^-e^{-ip_3d_3}\right)e^{ip_2d_2}\\&-\left( r_{12}^+e^{-ip_1d_1}-r_{12}^-e^{ip_1d_1}\right)\left(r_{32}^+e^{-ip_3d_3}-r_{32}^-e^{ip_3d_3}\right)e^{-ip_2d_2}\Biggr\}=0,
\end{split}
\end{equation*}where $d_1=a$, $d_2=b-a$ and $d_3=L-b$.
Discarding the nonzero factor $e^{-ip_3L}$ and define
\begin{equation}\label{eq8_3_9}\begin{split}
F_{0;\text{TM}}(\omega, \boldsymbol{k}) =&   \left(r_{12}^+e^{ip_1d_1}-r_{12}^-e^{-ip_1d_1}\right)\left(r_{32}^+e^{ip_3d_3}-r_{32}^-e^{-ip_3d_3}\right)e^{ip_2d_2}\\&-\left( r_{12}^+e^{-ip_1d_1}-r_{12}^-e^{ip_1d_1}\right)\left(r_{32}^+e^{-ip_3d_3}-r_{32}^-e^{ip_3d_3}\right)e^{-ip_2d_2}, \hspace{0.5cm}\boldsymbol{k}=(k_2,k_3)\in \mathcal{S}_{\text{TM}}\end{split}
\end{equation} where for $j=1,2,3$,
\begin{equation*}
p_j(\omega,\boldsymbol{k}) =\sqrt{ \varepsilon_j(\omega)\mu_j(\omega)\omega^2-\lambda_{\boldsymbol{k}}^2}.
\end{equation*}The set of TM energy eigenmodes   is the union of the sets of real nonnegative zeros of $F_{0;\text{TM}}(\omega, \boldsymbol{k}), \boldsymbol{k}=(k_2,k_3)\in \mathcal{S}_{\text{TM}}$. For the TE modes, one can show analogously that the set of TE energy eigenmodes is the union of the sets of real nonnegative zeros of
\begin{equation}\label{eq8_3_10}\begin{split}
F_{0;\text{TE}}(\omega, \boldsymbol{k}) =&\left(s_{12}^+e^{ip_1d_1}-s_{12}^-e^{-ip_1d_1}\right)\left(s_{32}^+e^{ip_3d_3}-s_{32}^-e^{-ip_3d_3}\right)e^{ip_2d_2}\\&-\left( s_{12}^+e^{-ip_1d_1}-s_{12}^-e^{ip_1d_1}\right)\left(s_{32}^+e^{-ip_3d_3}-s_{32}^-e^{ip_3d_3}\right)e^{-ip_2d_2}, \hspace{1cm}\boldsymbol{k}=(k_2,k_3)\in \mathcal{S}_{\text{TE}},
\end{split}\end{equation} where
$$s_{jk}^{\pm}= 1\pm\frac{p_j\mu_k}{p_k\mu_j}.$$Compare \eqref{eq8_3_10} to \eqref{eq8_3_9}, we find that $F_{0;\text{TE}}(\omega, \boldsymbol{k}) $ is obtained from $F_{0;\text{TM}}(\omega, \boldsymbol{k})$ by replacing $r_{jk}^{\pm}$ by $s_{jk}^{\pm}$. In the following, we assume that all the zeros of $F_{0;\text{TM}}(\omega,\boldsymbol{k}),\boldsymbol{k}=(k_2,k_3)\in \mathcal{S}_{\text{TM}},$ and $F_{0;\text{TE}}(\omega,\boldsymbol{k}), \boldsymbol{k}=(k_2,k_3)\in \mathcal{S}_{\text{TE}},$ are real.

The finite temperature Casimir energy   inside the piston  is given by
\begin{equation}\label{eq8_3_4}
E_{\text{Cas}}( t_c) =\frac{\hbar}{2}\sum_{\text{modes}}\omega e^{-t_c\omega}+k_BT\sum_{\text{modes}}\log\left(1-\exp\left(-\frac{\hbar \omega}{k_B T}\right)\right),
\end{equation}where we have introduced a cut-off parameter $t_c$ to render the first sum finite. Using the generalized Abel-Plana summation formula \cite{1,2} (see Appendix \ref{a1}), we compute this finite temperature Casimir energy in Appendix \ref{a2}. Up to the terms that give non-trivial limits when $t_c\rightarrow 0^+$, the result is
\begin{equation}\label{eq8_4_5}
\begin{split}
E_{\text{Cas}}(t_c) =& \sum_{j=1}^3d_j\Xi_{j}(t_c)+\Delta E_{\text{Cas}},\end{split}
\end{equation}where\begin{equation}\label{eq8_3_13}
\begin{split}
\Xi_j(t_c) =\frac{1}{\pi}\left(\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}+\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TE}}}\right)\int_{p_j(\omega,\boldsymbol{k})\geq 0}\left\{
\frac{\hbar\omega}{2} e^{-t_c\omega}+k_BT\log\left(1-\exp\left(-\frac{\hbar\omega}{k_BT}\right)\right)\right\}dp_j(\omega,\boldsymbol{k}),
\end{split}
\end{equation}and
\begin{equation}\label{eq8_3_12}
\begin{split}
\Delta E_{\text{Cas}}=& \frac{k_BT}{2}\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty}\log\Biggl\{\left(1-\Delta_{12}^{\text{TM}}(i\xi_l,\boldsymbol{k}) e^{-2q_1( \xi_l,\boldsymbol{k})d_1}\right)
\left(1-\Delta_{32}^{\text{TM}}(i\xi_l,\boldsymbol{k})e^{-2q_3( \xi_l,\boldsymbol{k})d_3}\right) \\& -\left(  e^{-2q_1( \xi_l,\boldsymbol{k})d_1}-\Delta_{12}^{\text{TM}}(i\xi_l,\boldsymbol{k}) \right)\left( e^{-2q_3( \xi_l,\boldsymbol{k})d_3} -\Delta_{32}^{\text{TM}}(i\xi_l,\boldsymbol{k})\right)e^{-2q_2( \xi_l,\boldsymbol{k})d_2}\Biggr\} +\left(\text{TM}\longrightarrow \text{TE}\right),
\end{split}
\end{equation}with
\begin{equation}\label{eq8_5_2}\begin{split}q_j(\xi,\boldsymbol{k})=&\sqrt{\varepsilon_j(i\xi)\mu_j(i\xi)\xi^2+\lambda_{\boldsymbol{k}}^2},\hspace{1cm}\xi_l=\frac{2\pi |l| k_BT}{\hbar},\\\Delta_{jk}^{\text{TM}}(i\xi,\boldsymbol{k})=&\frac{\varepsilon_k(i\xi)q_j(\xi,\boldsymbol{k})-\varepsilon_j(i\xi)q_k(\xi,\boldsymbol{k})}
{\varepsilon_k(i\xi)q_j(\xi,\boldsymbol{k})+\varepsilon_j(i\xi)q_k(\xi,\boldsymbol{k})},\hspace{0.5cm}\Delta_{jk}^{\text{TE}}(i\xi,\boldsymbol{k})=
\frac{\mu_j(i\xi)q_k(\xi,\boldsymbol{k})-\mu_k(i\xi)q_j(\xi,\boldsymbol{k})}{
\mu_j(i\xi)q_k(\xi,\boldsymbol{k})+\mu_k(i\xi)q_j(\xi,\boldsymbol{k})}.\end{split}\end{equation}
Notice that $\sum_{j=1}^3d_j\Xi_{j}(t_c)$ contains all the $t_c\rightarrow 0^+$ divergences. The term  $d_1\Xi_1(t_c)$ is independent of $  d_2, d_3$ and the parameters of media 2 and 3. It represents the self energy of media 1.  The term $\Xi_1(t_c)$ can be interpreted as the Casimir energy per unit length that would exists in the region between $x_1=0$ and $x_1=a$ if the boundaries at $x_1=0$ and $x_1=a$ were absent. Similar interpretation can be given to the terms $d_2\Xi_2(t_c)$ and $d_3\Xi_3(t_c)$. The term $\Delta E_{\text{Cas}}$ is called the interaction term. In the literatures on Casimir effect,   $\Delta E_{\text{Cas}}$ is usually regarded as the regularized Casimir energy by claiming that the term  $\sum_{j=1}^3d_j\Xi_{j}(t_c)$ should be subtracted away as the energy in the absence of boundary. This regularization scheme is a little superficial, since we have to consider the three regions that have different   properties separately. As explained in \cite{new2, new3, new4, new5}, one should not ignore the divergent terms, but should instead give a physical interpretation to the cut-off parameter $t_c$.  Nevertheless, in the special case $\Xi_1(t_c)=\Xi_2(t_c)=\Xi_3(t_c)$, then the term $d_1\Xi_1(t_c)+d_2\Xi_2(t_c)+d_3\Xi_3(t_c)=L\Xi_1(t_c)$ is independent of $d_1, d_2, d_3$, and the subtraction of this term  from the Casimir energy becomes  natural since it would not contribute to the Casimir force (see next section). $\Xi_1(t_c)=\Xi_2(t_c)=\Xi_3(t_c)$ holds when for example, the system is isorefractive, i.e., the refractive index $n(\omega)=c\sqrt{\varepsilon(\omega)\mu(\omega)}$ is the same for the three materials.

The interaction term of the energy $\Delta E_{\text{Cas}}$ shows that in the high temperature limit, the leading term of the energy is linear in $T$, given by the sum of the $l=0$ terms. The zero temperature Casimir energy  is obtained by taking the limit $T\rightarrow 0$ in \eqref{eq8_3_13} and \eqref{eq8_3_12}, which gives
\begin{equation*}
E_{\text{Cas}}^{T=0}( t_c) =\frac{\hbar}{2\pi}\left(\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}+\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TE}}}\right)\int_{p_j(\omega,\boldsymbol{k})\geq 0}
 \omega  e^{-t_c\omega}dp_j(\omega,\boldsymbol{k})+\Delta E_{\text{Cas}}^{T=0},
\end{equation*}where the interaction term is
\begin{equation*}
\begin{split}
\Delta E_{\text{Cas}}^{T=0}=&\frac{\hbar}{2\pi}\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\int_0^{\infty}
\log\Biggl\{\left(1-\Delta_{12}^{\text{TM}}(i\xi,\boldsymbol{k})e^{-2q_1(\xi,\boldsymbol{k})d_1}\right)
\left(1-\Delta_{32}^{\text{TM}}(i\xi,\boldsymbol{k})e^{-2q_3(\xi,\boldsymbol{k})d_3}\right) \\&-\left(  e^{-2q_1(\xi,\boldsymbol{k})d_1}-\Delta_{12}^{\text{TM}}(i\xi,\boldsymbol{k})\right)\left( e^{-2q_3(\xi,\boldsymbol{k})d_3} -\Delta_{32}^{\text{TM}}(i\xi,\boldsymbol{k})\right)e^{-2q_2(\xi,\boldsymbol{k})d_2}\Biggr\}d\xi +\left( \text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation*}

 In the    limit where the two ends of the rectangular box $x_1=0$ and $x_1=L$ are brought to infinity, i.e., $d_1\rightarrow \infty$ and $d_3\rightarrow \infty$, the interaction term of the Casimir energy  \eqref{eq8_3_12}  gives
\begin{equation*}
\begin{split}
\Delta E_{\text{Cas}}=& \frac{k_BT}{2}\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty}\log \left(1 -  \Delta_{12}^{\text{TM}}(i\xi_l,\boldsymbol{k}) \Delta_{32}^{\text{TM}}(i\xi_l,\boldsymbol{k}) e^{-2q_2(\xi_l,\boldsymbol{k})d_2}\right) +\left( \text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation*}In addition, if  the media are infinite in the transversal direction, i.e., $L_2,L_3\rightarrow \infty$, then  $\lambda_{\boldsymbol{k}}^2$ has to be replaced by $k^2$ and  the summation over $\boldsymbol{k}$ has to be changed to an appropriate integral. More precisely, in the limit $L_2,L_3\rightarrow \infty$, the interaction term of the Casimir energy density is
\begin{equation}\label{eq8_11_8}
\begin{split}
\Delta \mathcal{E}_{\text{Cas}}= \frac{k_BT}{4\pi}\sum_{l=-\infty}^{\infty}\int_0^{\infty} \left\{\log \left(1 -  \Delta_{12}^{\text{TM}}(i\xi_l,k) \Delta_{32}^{\text{TM}}(i\xi_l,k) e^{-2q_2(\xi_l,k)d_2}\right)+\left(\text{TM}\longrightarrow \text{TE}\right)\right\}kdk,
\end{split}
\end{equation}where $q_j(\xi, k)=\sqrt{\varepsilon_j(i\xi)\mu_j(i\xi)\xi^2+k^2}$.
This is precisely the Lifshitz formula \cite{3} for the Casimir energy density when two semi-infinite media with permittivities and permeabilities $\varepsilon_1,\mu_1$ and $\varepsilon_3,\mu_3$ are separated by a medium of permittivity $\varepsilon_2$ and permeability $\mu_2$. Notice that we have   ignored (or subtracted away) the divergent part of the Casimir energy $d_1\Xi_1(t_c)+d_2\Xi_2(t_c)+d_3\Xi_3(t_c)$ and only take the limit $d_1, d_3\rightarrow \infty$ on the interaction term. Later on, we are going to show some situations where the divergences would naturally disappear when we consider the Casimir force.

\section{Casimir force acting on the piston}\label{s3}

We assume that the piston   is allowed to move freely inside the rectangular box, but it is non-deformable, i.e., its thickness $d_2=b-a$ is fixed. The position of the piston can be described by the variable $a=d_1$. As a function of $a$, $d_1=a$, $d_2$ is fixed and $d_3=L-a-d_2$. The Casimir force acting on the piston is given by
\begin{equation*}
\begin{split}
F_{\text{Cas}}^{\text{piston}}(a) = -\frac{\partial E_{\text{Cas}}}{\partial a}=-\frac{\partial E_{\text{Cas}}}{\partial d_1}+\frac{\partial E_{\text{Cas}}}{\partial d_3}.
\end{split}
\end{equation*}From \eqref{eq8_4_5}, \eqref{eq8_3_13} and  \eqref{eq8_3_12}, we find that
\begin{equation*}
F_{\text{Cas}}^{\text{piston}}(a)= \Xi_3(t_c)-\Xi_1(t_c) +\Delta F_{\text{Cas}}^{\text{piston}}(a),
\end{equation*}where $\Xi_1(t_c)$ and $\Xi_3(t_c)$ are defined by \eqref{eq8_3_13}, and the interaction term is
\begin{equation}\label{eq8_26_3}
\begin{split}
\Delta F_{\text{Cas}}^{\text{piston}}(a)=\Delta F_{\text{Cas};\text{TM}}^{  L}+\Delta F_{\text{Cas};\text{TE}}^{ L}-\Delta F_{\text{Cas};\text{TM}}^{ R}-\Delta F_{\text{Cas};\text{TE}}^{ R},
\end{split}
\end{equation}with
\begin{equation*}
\begin{split}
&\Delta F_{\text{Cas};\text{TM}}^{  L}=-k_BT\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty} \frac{q_1e^{-2q_1 d_1} \Bigl(\Delta_{12}^{\text{TM}}
\left(1-\Delta_{32}^{\text{TM}} e^{-2q_3 d_3}\right)-\left( \Delta_{32}^{\text{TM}} -e^{-2q_3 d_3} \right)e^{-2q_2d_2}\Bigr)}{ \left(1-\Delta_{12}^{\text{TM}}
e^{-2q_1 d_1}\right) \left(1-\Delta_{32}^{\text{TM}} e^{-2q_3 d_3}\right)
  -\left( \Delta_{12}^{\text{TM}}  -e^{-2q_1 d_1}\right)\left( \Delta_{32}^{\text{TM}} -e^{-2q_3 d_3} \right)e^{-2q_2 d_2}},
\end{split}
\end{equation*} $\Delta F_{\text{Cas};\text{TM}}^{  R}$ is obtained from $\Delta F_{\text{Cas};\text{TM}}^{  L}$ by interchanging the index $1$ with $3$, and  $\Delta F_{\text{Cas};\text{TE}}^{  L/R}$ is obtained from $\Delta F_{\text{Cas};\text{TM}}^{  L/R}$ by changing $\Delta_{jk}^{\text{TM}}$ to $\Delta_{jk}^{\text{TE}}$. Unlike the piston cases that have been considered so far, there are still divergences $\Xi_3(t_c)-\Xi_1(t_c) $ in the Casimir force coming from the self energy terms. This term will vanish if and only if $\Xi_1(t_c)=\Xi_3(t_c)$, which would happen for example when the two media separated by the piston is isorefractive.  In particular, if the two media 1 and 3 are both vacuum,    which is the case in the previous works that have  considered the piston scenarios, then there are no divergences, and regularization is not required for the Casimir force.

For the particular case where  $\varepsilon_1=\varepsilon_3$, $\mu=\mu_3$, the self-energy terms   $\Xi_1(t_c)$ and $\Xi_3(t_c) $ cancel and we can set the cut-off parameter $t_c$ to zero. Let $q=q_1=q_3$ and $\Delta^{\text{TM}/\text{TE}}=\Delta_{12}^{\text{TM}/\text{TE}}=\Delta_{32}^{\text{TM}/\text{TE}}$. Then     the Casimir force acting on the piston is given by \begin{equation}\label{eq8_24_1}
\begin{split}
  F_{\text{Cas}}^{  \text{piston}}=&-k_BT\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty} \frac{q\Delta^{\text{TM}}\left(e^{-2qd_1}-e^{-2qd_3}\right) \left(1-e^{-2q_2d_2}\right)}{ \left(1-\Delta^{\text{TM}}e^{-2q  d_1}
\right) \left(1-\Delta^{\text{TM}} e^{-2q  d_3}\right)
  -\left( \Delta^{\text{TM}} - e^{-2q  d_1}\right)\left( \Delta^{\text{TM}} -e^{-2q  d_3} \right)e^{-2q_2 d_2}}\\&+\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation} \begin{figure}
\epsfxsize=0.49\linewidth \epsffile{Lipshitz_6.eps} \epsfxsize=0.49\linewidth \epsffile{Lipshitz_7.eps} \caption{\label{f3}The Casimir force $F_{\text{Cas}}^{\text{piston}}(a)$ as a function of the plate separation $a$ for different values of $\varepsilon_{2,r}=\varepsilon_2/\varepsilon_0$ and $\mu_{2,r}=\mu_2/\mu_0$.   Here   $L=2$cm, $L_2=L_3=1$cm, $d_2=1$mm and $T=1$K. The two regions separated by the piston are vacuum. For the graph on the left, $\mu_{2}=\mu_0$. For the graph on the right, $\varepsilon_2=\varepsilon_0$. }\end{figure}
\begin{figure}
\epsfxsize=0.49\linewidth \epsffile{Lipshitz_8.eps} \epsfxsize=0.49\linewidth \epsffile{Lipshitz_9.eps} \caption{\label{f4}The Casimir force $F_{\text{Cas}}^{\text{piston}}(a)$ as a function of the piston thickness $d_2$ for different values of $\varepsilon_{2,r}=\varepsilon_2/\varepsilon_0$ and $\mu_{2,r}=\mu_2/\mu_0$.   Here   $L=5$cm, $L_2=L_3=1$cm, $d_1=5$mm, $T=1$K. The two regions separated by the piston are vacuum. For the graph on the left, $\mu_{2}=\mu_0$. For the graph on the right, $\varepsilon_2=\varepsilon_0$. }\end{figure}From this, we find that  the sign of the Casimir force is determined by the sign of $\Delta^{\text{TM}}, \Delta^{\text{TE}}$, and the sign of $d_1-d_3$. Notice that in general the Casimir force depends on the thickness of the piston, and it vanishes as the thickness of the piston goes to zero, unless $\Delta^{\text{TM}}=\pm 1$ and $\Delta^{\text{TE}}=\pm1$. FIG. \ref{f3} and FIG. \ref{f4} show the special case where the regions separated by the piston are vacuum, i.e., $\varepsilon_1=\varepsilon_3=\varepsilon_0$, $\mu_1= \mu_3=\mu_0$ and the piston is made of  material with constant permittivity $\varepsilon_2$ and permeability $\mu_2$.  We find that when $\varepsilon_2>\varepsilon_0$, the Casimir force attracts the piston to the closer wall, and when $\mu_2>\mu_0$, the Casimir force push the piston away towards the middle. This can also be directly verified from \eqref{eq8_24_1}.

In the special case where the regions separated by the piston are vacuum, i.e., $\varepsilon_1=\varepsilon_3=\varepsilon_0$, $\mu_1= \mu_3=\mu_0$,  and the piston is perfectly conducting,  achieved by the limit $\varepsilon_2\rightarrow \infty$, we find that
\begin{equation}\label{eq8_4_6}\begin{split}
\Delta_{12}^{\text{TM}}(i\xi_l, \boldsymbol{k})=\Delta_{32}^{\text{TM}}(i\xi_l, \boldsymbol{k})=\frac{\varepsilon_2\sqrt{\frac{\xi_l^2}{c^2}+\lambda_{\boldsymbol{k}}^2}-\varepsilon_0 \sqrt{\varepsilon_2\mu_2\xi_l^2+\lambda_{\boldsymbol{k}}^2}}{\varepsilon_2\sqrt{\frac{\xi_l^2}{c^2}+\lambda_{\boldsymbol{k}}^2}+\varepsilon_0 \sqrt{\varepsilon_2\mu_2\xi_l^2+\lambda_{\boldsymbol{k}}^2}}\longrightarrow & 1,\\\Delta_{12}^{\text{TE}}(i\xi_l, \boldsymbol{k})=\Delta_{32}^{\text{TE}}(i\xi_l, \boldsymbol{k})=\frac{\mu_0 \sqrt{\varepsilon_2\mu_2\xi_l^2+\lambda_{\boldsymbol{k}}^2}-\mu_2\sqrt{\frac{\xi_l^2}{c^2}+\lambda_{\boldsymbol{k}}^2}}{\mu_0 \sqrt{\varepsilon_2\mu_2\xi_l^2+\lambda_{\boldsymbol{k}}^2}+\mu_2\sqrt{\frac{\xi_l^2}{c^2}+\lambda_{\boldsymbol{k}}^2}}\longrightarrow &1.
\end{split}\end{equation}When $l=0$,   the limit  of the  second term has to be taken using the prescription of Schwinger, DeRaad and Milton \cite{4} where  one first takes the limit $\varepsilon_2\rightarrow \infty$ before setting $l=0$. Alternatively, one can also take the limit $\mu_2\rightarrow 0$. In fact,   superconductors can be considered as perfectly conducting and it is well-known that for superconductors, the magnetic permeability is zero.  Substituting the limits \eqref{eq8_4_6} into the Casimir force \eqref{eq8_24_1}, we find that
\begin{equation}\label{eq8_4_7}
F_{\text{Cas}}^{  \text{piston}}=-k_BT\left(\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}+\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TE}}}\right)\sum_{l=-\infty}^{\infty} \left\{\frac{\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}}{ \exp\left(2d_1\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}\right)-1}-\frac{\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}}{ \exp\left(2d_3\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}\right)-1} \right\}.
\end{equation}The two terms in the brackets represent the Casimir energies from the left region (with thickness $d_1=a$) and from the right region (with thickness $d_2=L-a-d_2$) respectively. This formula is exactly the same as the formula  for a perfectly conducting piston moving freely inside a perfectly conducting rectangular cavity derived in \cite{7, 6}. Notice that this formula is independent of the thickness of the piston $d_2$.

In the case the piston is infinitely permeable,  the limit is obtained by $\mu_2\rightarrow\infty$. As dual to the perfectly conducting case, we stipulate that $\varepsilon_2\rightarrow 0$. Then
\begin{equation*}
\Delta_{12}^{\text{TM}}(i\xi_l, \boldsymbol{k})\longrightarrow -1, \hspace{1cm}\Delta_{12}^{\text{TE}}(i\xi_l, \boldsymbol{k})\longrightarrow -1.
\end{equation*} In this limit,   the Casimir force \eqref{eq8_24_1} becomes
\begin{equation}\label{eq8_4_8}
F_{\text{Cas}}^{  \text{piston}}=k_BT\left(\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}+\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TE}}}\right)\sum_{l=-\infty}^{\infty} \left\{\frac{\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}}{ \exp\left(2d_1\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}\right)+1}-\frac{\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}}{ \exp\left(2d_3\sqrt{\left(\frac{2\pi l k_BT}{\hbar c}\right)^2+\lambda_{\boldsymbol{k}}^2}\right)+1} \right\},
\end{equation}which coincides with the result for an infinitely permeable piston moving freely inside a perfectly conducting piston we derived in \cite{8}. Notice again that the Casimir force is independent of the thickness of the piston $d_2$.

As one can see, in general the nature and the strength of the interaction term of the Casimir force acting on the piston \eqref{eq8_26_3} depend on the properties of the materials and the thickness of each material. It does not decouple into the Casimir force acting from the left region and the Casimir force acting from the right region.
In the limit the right end of the rectangular box is moved to infinity, i.e., $d_3\rightarrow \infty$,  we find that $\Delta F_{\text{Cas};\text{TM}}^{ R}\rightarrow 0, \Delta F_{\text{Cas};\text{TE}}^{ R}\rightarrow 0$, and
\begin{equation}\label{eq8_5_1}\begin{split}
\Delta F_{\text{Cas};\text{TM}}^{  \text{piston}, d_3\rightarrow\infty}=-k_BT\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty} q_1
 \left(\frac{\left(1-
\Delta_{12}^{\text{TM}}  \Delta_{32}^{\text{TM}}  e^{-2q_2d_2}\right)}{\left(\Delta_{12}^{\text{TM}} -\Delta_{32}^{\text{TM}} e^{-2q_2 d_2}\right)}e^{2q_1 d_1}-1\right)^{-1} +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}\end{equation}This formula can be interpreted as the interaction term of the Casimir force acting on a perfectly conducting plate and   a plate made  of real material (with permittivity $\varepsilon_2$ and permeability $\mu_2$) which are   embedded  orthogonally   in an infinitely long rectangular cylinder, with the medium between them having permittivity $\varepsilon_1$ and permeability $ \mu_1$ and the medium outside them   having permittivity $\varepsilon_3$ and permeability $\mu_3$. \eqref{eq8_5_1} shows that when the separation between the plates $d_1$ gets large, the magnitude of the interaction term of the Casimir force decreases exponentially.

\section{Casimir force between two perfectly conducting plates separated by medium with constant refractive index}
If  both plates are perfectly conducting so that $\Delta_{12}^{\text{TM}}=\Delta_{32}^{\text{TM}}=\Delta_{12}^{\text{TE}}=\Delta_{32}^{\text{TE}}=1$, and the medium 1 between the plates has constant  refractive index $n_1=c\sqrt{\varepsilon_1\mu_1}$, then we find that the interaction term of the Casimir force \eqref{eq8_5_1} becomes
\begin{equation}\label{eq8_11_2}
\begin{split}
&\Delta F_{\text{Cas}} =-\frac{k_BT}{2}\sum_{(k_2,k_3)\in {\mathbb{Z}}^2\setminus\{\boldsymbol{0}\}}\sum_{l=-\infty}^{\infty} \frac{\sqrt{ n_1^2\left(\frac{2\pi lk_BT}{\hbar c}\right)^2+ \lambda_{\boldsymbol{k}}^2}}{
  \exp\left(2d_1\sqrt{ n_1^2\left(\frac{2\pi lk_BT}{\hbar c}\right)^2+ \lambda_{\boldsymbol{k}}^2}\right)-1}.
\end{split}
\end{equation}This term is always attractive, and its magnitude   gets smaller if the medium between the perfectly conducting plates has higher refractive index.
In the zero temperature limit,  the interaction term of the  Casimir force \eqref{eq8_11_2} is
\begin{equation*}
\begin{split}
\Delta F_{\text{Cas}}^{ T=0}=&-\frac{\hbar c}{n_1}\Biggl(\frac{1}{4  d_1}\sum_{k_1=1}^{\infty}  \sum_{(k_2,k_3)\in {\mathbb{Z}}^2\setminus\{\boldsymbol{0}\}}\frac{\sqrt{\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2}}{k_1}K_1\left( 2\pi k_1d_1\sqrt{\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2}\right)\\&+\frac{1}{2}\sum_{k_1=1}^{\infty}  \sum_{(k_2,k_3)\in {\mathbb{Z}}^2\setminus\{\boldsymbol{0}\}}\left(\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2\right)K_0\left( 2\pi k_1d_1\sqrt{\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2}\right)\Biggr),
\end{split}
\end{equation*}which is $n_1$ times smaller than the corresponding Casimir force on perfectly conducting plates separated by vacuum. When the plate separation $d_1$ is much smaller than the size of the cross section, i.e., $d_1\ll L_2, L_3$, the leading terms of the interaction term of the Casimir force is
\begin{equation*}\begin{split}
\Delta F_{\text{Cas}} =&-\frac{\hbar c \pi^2 L_2L_3}{240 n_1d_1^4},\hspace{2cm} \text{if}\;\; T=0,\\
\Delta F_{\text{Cas}} =&-\frac{\zeta_R(3)L_2L_3k_BT}{4\pi d_1^3},\hspace{1cm}\text{if}\;\; d_1T\gg 1.\end{split}
\end{equation*}Notice that when $T=0$, the leading term is $n_1$ times smaller than the corresponding term for perfectly conducting plates separated by vacuum, but when the temperature is large enough, the leading term becomes independent of the refractive index of the medium between the plates.

The interaction term \eqref{eq8_11_2} has not taken into full account the influence of the material between the plates. There is an additional contribution to the Casimir force arising from the difference of materials in the region between the plates and the region outside the plates, given by
\begin{equation*}
\Xi_3(t_c)-\Xi_1(t_c)=\frac{1}{\pi}\left(\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TM}}}+\sum_{\boldsymbol{k}\in\mathcal{S}_{\text{TE}}}\right)\int_{p_3(\omega,\boldsymbol{k})\geq 0}\left\{
\frac{\hbar\omega}{2} e^{-t_c\omega}+k_BT\log\left(1-\exp\left(-\frac{\hbar\omega}{k_BT}\right)\right)\right\}dp_3(\omega,\boldsymbol{k})-\left(p_3\longrightarrow p_1\right).
\end{equation*}
Assume that the permittivity $\varepsilon_3$ and the permeability $\mu_3$ of the medium in region 3 are also constants with refractive index $n_3$. Then  up to the term constant in $t_c$,
\begin{equation}\label{eq12_07_3}\begin{split}
 \Xi_1(t_c)=&\frac{3\hbar L_2L_3}{\pi^2c^3}n_1^3t_c^{-4} -\frac{\hbar n_1 t_c^{-2}}{4\pi c} -\frac{\hbar c}{n_1}\left(\frac{\pi^2}{720}\frac{L_3}{L_2^3}+\frac{\zeta_R(3)}{16\pi L_3^2}+\frac{1}{2L_2^{\frac{3}{2}}L_3^{\frac{1}{2}}}\sum_{k_2=1}^{\infty}\sum_{k_3=1}^{\infty}\left(\frac{k_2}{k_3}\right)^{\frac{3}{2}}K_{\frac{3}{2}}\left(\frac{2\pi k_2k_3L_3}{L_2}\right)\right)\\&-\frac{k_BT}{2}\sum_{l=1}^{\infty}\sum_{(k_2,k_3)\in {\mathbb{Z}}^2\setminus\{\boldsymbol{0}\}}\frac{\sqrt{\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2}}{l}K_1\left( \frac{\pi \hbar c l}{n_1k_BT}\sqrt{\left(\frac{k_2}{L_2}\right)^2+\left(\frac{k_3}{L_3}\right)^2}\right),
\end{split}\end{equation}and $\Xi_3(t_c)$ is obtained from this formula by replacing $n_1$ with $n_3$.  It is obvious that if $n_1\neq n_3$, the terms  $\Xi_1(t_c)$ and $\Xi_3(t_c)$ due to the self energy of the system are not equal.  As explained in \cite{new2, new3, new4, new5}, in this case, the parameter $t_c$ has to be present in the expressions for Casimir energy and Casimir force.
\section{The five-layer model}
\begin{figure}
\epsfxsize=0.5\linewidth \epsffile{piston_2.eps} \caption{\label{f2}A five-layer model }\end{figure}
The piston system we consider in Sections \ref{s2} and \ref{s3} can be considered as a regularization device for a three-layer model. We have asserted that in the limit where one end of the rectangular box is brought to infinity,  we obtain the Casimir force acting on a pair of parallel plates, one being perfectly conducting and the other made of real material.  In this section, we consider the more general case of the five-layer model (FIG. \ref{f2}) which is the more appropriate model for two parallel plates both made of real materials.

The Casimir energy and Casimir force of a five-layer model have been derived by some authors using different methods \cite{new1, 5,9, 10, 11, 12, 13, 54}. We are going to use the piston approach. Namely, first we assume that the whole system is inside a large perfectly conducting rectangular cavity $[0,L]\times[0,L_2]\times [0,L_3]$. As depicted in FIG. \ref{f2}, there are five layers of materials with electric permittivities $\varepsilon_1,\varepsilon_2, \varepsilon_3,\varepsilon_4, \varepsilon_5$ and magnetic permeabilities $\mu_1,\mu_2,\mu_3,\mu_4, \mu_5$ from left to right. The first medium extends from $x_1=0$ to $x_1=a_1$, the second from $x_1=a_1$ to $x_1=a_2$, the third from $x_1=a_2$ to $x_1=a_3$, the fourth from $x_1=a_3$ to $x_1=a_4$ and the last one from $x_1=a_4$ to $x_1=L$.

Let $d_1=a_1, d_2=a_2-a_1, d_3=a_3-a_2, d_4=a_4-a_3$ and $d_5=L-a_4$. Then using the same method as Section \ref{s2}, we find that the Casimir energy of the system is
\begin{equation}\label{eq8_26_4}
E_{\text{Cas}}(t_c)=\sum_{j=1}^5 d_j\Xi_j(t_c)+\Delta E_{\text{Cas}},
\end{equation}where $\Xi_j(t_c)$ is as in \eqref{eq8_3_13}, and the interaction term of the Casimir energy is
\begin{equation*}\begin{split}
&\Delta E_{\text{Cas}}=\frac{k_BT}{2}\sum_{\boldsymbol{k}\in \mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty} \log\Biggl\{\Bigl(\left[1+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}e^{-2q_2d_2}\right]
-\left[\Delta^{\text{TM}}_{12}+\Delta^{\text{TM}}_{23}
e^{-2q_2d_2}\right]e^{-2q_1d_1}\Bigr)\\&\times\Bigl(\left[1+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}e^{-2q_4d_4}\right]-\left[\Delta^{\text{TM}}_{54}+\Delta^{\text{TM}}_{43}
e^{-2q_4d_4}\right]e^{-2q_5d_5}\Bigr)-\Bigl(\left[e^{-2q_2d_2}+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}\right]e^{-2q_1d_1}-\left[\Delta^{\text{TM}}_{12}e^{-2q_2d_2}
+\Delta^{\text{TM}}_{23}
\right]\Bigr)\\&\times\Bigl(\left[e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}\right]e^{-2q_5d_5}-\left[\Delta^{\text{TM}}_{54}e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}
\right]\Bigr)e^{-2q_3d_3}\Biggr\} +\left(\text{TM}\longrightarrow\text{TE}\right), \\& \text{where}\;\; q_j=q_j(\xi_l,\boldsymbol{k}), \;\; \Delta^{\text{TM}}_{jk}=\Delta^{\text{TM}}_{jk}(i\xi_l,\boldsymbol{k}),\;\; \Delta^{\text{TE}}_{jk}=\Delta^{\text{TE}}_{jk}(i\xi_l,\boldsymbol{k}).
\end{split}\end{equation*}The functions $q_j(\xi_l,\boldsymbol{k})$, $\Delta^{\text{TM}}_{jk}=\Delta^{\text{TM}}_{jk}(i\xi_l,\boldsymbol{k})$ and $\Delta^{\text{TE}}_{jk}=\Delta^{\text{TE}}_{jk}(i\xi_l,\boldsymbol{k})$ are defined as in \eqref{eq8_5_2}.

 Now assume that   medium 2 and medium 4 are two non-deformable pistons/plates embedded orthogonally inside  a rectangular box, dividing it into three regions filled with different materials. The Casimir forces acting on the plate on the left (medium 2) and the plate on the right (medium 4) are given respectively by
\begin{equation*}\begin{split}
 F_{\text{Cas}}^L=&-\frac{\partial   E_{\text{Cas}}(t_c)}{\partial d_1}+\frac{\partial  E_{\text{Cas}}(t_c)}{\partial d_3}=-\Xi_1(t_c)+\Xi_3(t_c)-\frac{\partial  \Delta E_{\text{Cas}} }{\partial d_1}+\frac{\partial \Delta  E_{\text{Cas}} }{\partial d_3},\\
 F_{\text{Cas}}^R=&-\frac{\partial   E_{\text{Cas}}(t_c)}{\partial d_3}+\frac{\partial  E_{\text{Cas}}(t_c)}{\partial d_5}=-\Xi_3(t_c)+\Xi_5(t_c)-\frac{\partial  \Delta E_{\text{Cas}} }{\partial d_3}+\frac{\partial \Delta  E_{\text{Cas}} }{\partial d_5}.
\end{split}\end{equation*}
By moving the two auxiliary ends of the rectangular box to infinity, i.e., $d_1,d_5\rightarrow \infty$, since $$\frac{\partial \Delta  E_{\text{Cas}} }{\partial d_1}\xrightarrow{d_1\rightarrow \infty} 0, \;\;\frac{\partial \Delta  E_{\text{Cas}} }{\partial d_5}\xrightarrow{d_5\rightarrow \infty} 0,$$
we find that\begin{equation*}\begin{split}
 F_{\text{Cas}}^L=& -\Xi_1(t_c)+\Xi_3(t_c) +\Delta F_{\text{Cas}},\\
 F_{\text{Cas}}^R=& -\Xi_3(t_c)+\Xi_5(t_c)-\Delta F_{\text{Cas}},
\end{split}\end{equation*}  where the interaction term of the Casimir force acting on the two plates is
\begin{equation}\label{eq8_5_4}
\begin{split}
&\Delta F_{\text{Cas}}=-\frac{\partial  \Delta E_{\text{Cas}} }{\partial d_3}(d_1, d_5\rightarrow \infty)\\=&- k_BT \sum_{\boldsymbol{k}\in \mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty}  q_3\left(\frac{\left( 1+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}e^{-2q_2d_2} \right)\left( 1+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}e^{-2q_4d_4} \right) }{\left(\Delta^{\text{TM}}_{12}e^{-2q_2d_2}+\Delta^{\text{TM}}_{23}
 \right)\left(  \Delta^{\text{TM}}_{54}e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}
 \right)}e^{2q_3d_3} -1\right)^{-1} +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation} The divergences $-\Xi_1(t_c)+\Xi_3(t_c)$ and $-\Xi_3(t_c)+\Xi_5(t_c)$   come from the self energy of the system and in general do not vanish. A special case they would vanish is that the media 1, 3 and 5 are isorefractive. In the symmetric setup, i.e.,  media 1 and 5 are isorefractive, then the force $-\Xi_1(t_c)+\Xi_3(t_c)$ acting on the plate on the left  and the force $-\Xi_3(t_c)+\Xi_5(t_c)$ acting on the plate on the right have opposite signs.
In the limit the plate on the left is perfectly conducting, i.e., $\varepsilon_2\rightarrow \infty$, then $\Delta^{\text{TM}}_{12}, \Delta^{\text{TE}}_{12}\rightarrow 1$ and $\Delta^{\text{TM}}_{23}, \Delta^{\text{TE}}_{23}\rightarrow -1$, we find that the formula \eqref{eq8_5_4} coincides with the formula \eqref{eq8_5_1} after the re-indexing $3\rightarrow 1, 4\rightarrow 2$ and $5\rightarrow 3$. This is consistent with our interpretation of formula \eqref{eq8_5_1} as the Casimir force acting between a perfectly conducting plate and a plate made of real material.

Usually in the works of Casimir effect on parallel plates, it is assumed that the size of the cross section of the plates is much larger than the separation between the plates. In this limit, the interaction term of the Casimir force density acting on the plates is
\begin{equation}\label{eq8_26_5}
\begin{split}
\Delta \mathcal{F}_{\text{Cas}}=&- \frac{k_BT }{2\pi} \sum_{l=-\infty}^{\infty} \int_0^{\infty} q_3\left(\frac{\left( 1+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}e^{-2q_2d_2} \right)\left( 1+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}e^{-2q_4d_4} \right) }{\left(\Delta^{\text{TM}}_{12}e^{-2q_2d_2}+\Delta^{\text{TM}}_{23}
 \right)\left(  \Delta^{\text{TM}}_{54}e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}
 \right)}e^{2q_3d_3} -1\right)^{-1}kdk +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation}
In the zero temperature limit, \eqref{eq8_5_4} gives
\begin{equation}\label{eq8_11_10}
\begin{split}
\Delta F_{\text{Cas}}^{T=0}= &- \frac{\hbar}{\pi}\sum_{\boldsymbol{k}\in \mathcal{S}_{\text{TM}}}\int_0^{\infty}  q_3\left(\frac{\left( 1+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}e^{-2q_2d_2} \right)\left( 1+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}e^{-2q_4d_4} \right) }{\left(\Delta^{\text{TM}}_{12}e^{-2q_2d_2}+\Delta^{\text{TM}}_{23}
 \right)\left(  \Delta^{\text{TM}}_{54}e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}
 \right)}e^{2q_3d_3} -1\right)^{-1}d\xi +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation}In addition, if $d_3\ll L_2, L_3$, then the zero temperature Casimir force density acting on the plates is
\begin{equation}\label{eq8_11_12}
\begin{split}
\Delta \mathcal{F}_{\text{Cas}}^{T=0}= &- \frac{\hbar}{2\pi^2}\int_0^{\infty}\int_0^{\infty}  q_3\left\{\left(\frac{\left( 1+\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{12}e^{-2q_2d_2} \right)\left( 1+\Delta^{\text{TM}}_{43}\Delta^{\text{TM}}_{54}e^{-2q_4d_4} \right) }{\left(\Delta^{\text{TM}}_{12}e^{-2q_2d_2}+\Delta^{\text{TM}}_{23}
 \right)\left(  \Delta^{\text{TM}}_{54}e^{-2q_4d_4}+\Delta^{\text{TM}}_{43}
 \right)}e^{2q_3d_3} -1\right)^{-1}+\left(\text{TM}\longrightarrow \text{TE}\right)\right\}d\xi kdk.
\end{split}
\end{equation}

Formulas \eqref{eq8_5_4}, \eqref{eq8_26_5}, \eqref{eq8_11_10} and \eqref{eq8_11_12} can be used to study the nature and properties of the Casimir force acting on two plates made of real materials. We would like to emphasize that in the case the materials in the regions 1, 3, 5 separated by the plates are not isorefractive, one   also has to take into account the forces coming from the terms $\Xi_1, \Xi_3$ and $\Xi_5$ and the cut-off parameter $t_c$ cannot be set to zero.

In the case the three media 1, 3, 5 are the same, i.e., $\varepsilon_1=\varepsilon_3=\varepsilon_5$, and $\mu_1=\mu_3=\mu_5$, we find that the self energy terms cancel and the Casimir force acting on the plates is \eqref{eq8_5_4}:
\begin{equation}\label{eq8_26_6}
\begin{split}
 F_{\text{Cas}} =&- k_BT \sum_{\boldsymbol{k}\in \mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty}  q_3\left(\frac{\left( 1-\left[\Delta^{\text{TM}}_{23}\right]^2e^{-2q_2d_2} \right)\left( 1-\left[\Delta^{\text{TM}}_{43}\right]^2e^{-2q_4d_4} \right) }{\Delta^{\text{TM}}_{23}\Delta^{\text{TM}}_{43}\left(1-e^{-2q_2d_2}
 \right)\left(  1-e^{-2q_4d_4}
 \right)}e^{2q_3d_3} -1\right)^{-1} +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation}In addition, if the two plates are made of the same material, i.e., $\varepsilon_2=\varepsilon_4$ and $\mu_2=\mu_4$, \eqref{eq8_26_6} becomes  \begin{equation}
\begin{split}
 F_{\text{Cas}} =&- k_BT \sum_{\boldsymbol{k}\in \mathcal{S}_{\text{TM}}}\sum_{l=-\infty}^{\infty}  q_3\left(\frac{\left( 1-\left[\Delta^{\text{TM}}_{23}\right]^2e^{-2q_2d_2} \right)\left( 1-\left[\Delta^{\text{TM}}_{23}\right]^2e^{-2q_2d_4} \right) }{\left[\Delta^{\text{TM}}_{23}\right]^2\left(1-e^{-2q_2d_2}
 \right)\left(  1-e^{-2q_2d_4}
 \right)}e^{2q_3d_3} -1\right)^{-1} +\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation}It is easy to see that in this case, the Casimir force is always attractive. This is a special case of the theorem \cite{14, 15} which asserts that the Casimir force between   two bodies with the same property is attractive. In general \eqref{eq8_26_6} shows that the dependence of the Casimir force on the thickness of the plates and the distance between the plates can be quite complicated. In FIG. \ref{f6}, we plot the dependence of the Casimir force on the distance between the plates and the thickness of the plates. The graphs show that with $\varepsilon_2=2\varepsilon_0$, $\varepsilon_4=4\varepsilon_0$, $\mu_4=\mu_0$ and $\mu_2=3.6\mu_0, 3.8\mu_0, 4.0\mu_0$,  the Casimir force can be attractive or repulsive depending on the distance between the plates and thickness of the plates.

\begin{figure}
\epsfxsize=0.49\linewidth \epsffile{Lipshitz_12.eps} \epsfxsize=0.49\linewidth \epsffile{Lipshitz_13.eps} \caption{\label{f6}The Casimir force $F_{\text{Cas}}$ as a function of the plate separation $d_3$ (left) and the plate thickness $d_2=d_4$ (right). Here $L_2=L_3=1$cm, $T=0.1$K, the regions separated by the plates are vacuum. For the plate on the left $\varepsilon_{2}=2\varepsilon_0$ and $\mu_2=\mu_{2,r}\mu_0$, where $\mu_{2,r}=3.6,3.8,4.0$.  For the plate on the right, $\varepsilon_4=4\varepsilon_0$, $\mu_4=\mu_0$. For the graph on the left, the thickness of the plates are $d_2=d_4=0.5$mm. For the graph on the right, the separation between the plates is $d_3=1$mm.  }\end{figure}

Finally we see how we can recover the Lifshitz formula from \eqref{eq8_26_5}.
As mentioned before, when the regions separated by the two plates are filled with isorefractive media, then the divergences of the Casimir force cancel. If in addition, we assume that both plates have infinite thickness,  i.e., $d_2, d_4\rightarrow \infty$, \eqref{eq8_26_5} gives
\begin{equation*}
\begin{split}
F_{\text{Cas}}=&- \frac{k_BT }{2\pi} \sum_{l=-\infty}^{\infty} \int_0^{\infty} q_3(\xi_l, k)\left( \left[\Delta^{\text{TM}}_{23}(i\xi_l, k)\right]^{-1}
 \left[\Delta^{\text{TM}}_{43}(i\xi_l, k)\right]^{-1}
  e^{2q_3(\xi_l,k)d_3} -1\right)^{-1} kdk+\left(\text{TM}\longrightarrow \text{TE}\right).
\end{split}
\end{equation*}
This is precisely the Lifshitz formula for the Casimir force acting on two parallel plates made of real materials. Thus we see that the piston approach can be used to derive the Lifshitz formula. An advantage of this approach is that the divergence of the Casimir force is not ignored by ad hoc subtraction, but is seen to vanish due to pairwise cancelation.

\section{Conclusion} In this article, we derive the Casimir energy and Casimir force acting on a piston made of real material that moves freely inside a perfectly conducting rectangular box. We also  use the piston approach to derive the Casimir energy and Casimir force acting on two parallel plates made of real materials. We use an exponential cut-off to regularize the Casimir energy. When the regions separated by the piston or the plates are filled with isorefractive media, we show that the divergences of the Casimir force cancel each other and therefore the cut-off parameter can be set to zero. In general, the Casimir energy contains divergent terms that depend on the cut-off parameter that cannot be ignored. Although we have restricted our consideration to three-layer model (for the piston) and five-layer model (for the parallel plates), our method can be easily   generalized to Casimir effect on multi-layer models. Moreover, it is easy to see that there is nothing special in choosing the rectangular cavity as a space cut-off for the electromagnetic field. We can also use a cylinder with arbitrary cross section as a substitute. For doing so, one just has to replace  the $\lambda_{\boldsymbol{k}}^2$ by an appropriate spectrum.

For simplicity, we have illustrated the behavior of the Casimir force using non-realistic examples where the electric permittivities and magnetic permeabilities of the materials are assumed to be constants. In reality, the electric permittivities and magnetic permeabilities depend on temperature and also the frequency of the field. There are some models that have been used for the electric permittivities such as the plasma model and the Drude model. An issue that we do not discuss in this article is the leading term of the thermal correction to the Casimir force at low and high temperatures. This has been an active topic of research and the results often depend on the particular model chosen for the electric permittivities and magnetic permeabilities. In the case where the self energy terms cancel, we have shown that the high temperature leading term of the Casimir force is linear in temperature.  In case the self energy terms do not cancel, further regularization may be needed to cancel the high temperature leading terms in the self energy that are higher than linear orders \cite{new6}. This issue will be discussed in more detail elsewhere.

\begin{acknowledgments}
This project is   funded by Ministry of Science, Technology and Innovation, Malaysia under e-Science fund 06-02-01-SF0080. We would like to thank the anonymous referee for the helpful comments.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


"Exceptional support. I am not from the USA but I have always wanted to attend an American university. I feel like I have family here. My family back home is happy I have support here as well."
We will help you create the perfect tour depending on your interest, budget and travel dates. All customized tours include a tour manager AND a performing arts college specialist as your guide.


On the 25th June 1904 Albert Philip Galienne was born. As far as I know, he was the first of four children to Albert Gallienne (b.1870) and Florence Sophia Hamon (b. 1883). He was my great grandmother's older brother.
1904 – Albert Philip Gallienne is born on the 25th June in Union Street, Jersey, Channel Islands.
1911 – Aged 7 Albert is living at 1 Gregory's Cottages, James Street, St. Helier, Jersey with his father, mother, sister Florence and brother Henry.
I have no further details about Albert's life. Interestingly enough I have just looked on the Jersey archives website to double check his address. On the identity card Albert is living at 120 Halkett Place. On the list of addresses on the website there is no such place as 120 Halkett Place. Under his name on the website he is listed as living at 20 Halkett Place.
It is still a mystery as to whether Albert lived at 20 or 120 Halkett Place and I will have to try and solve it.
I would love to hear from anyone who can confirm any of the above details or even better add to Albert's currently very short timeline.


CPU is used at 30 to 50%.
RB IDE as front-most application, my CPU is used at 30 to 50%.
> CPU is used at 30 to 50%.
> RB IDE as front-most application, my CPU is used at 30 to 50%.
> How is it possible???
in total take up 1/4 as much CPU time as your Mandelbrot app).
Perhaps you give your CPU something better to do?
Play music with iTunes or watch a DVD.
"Diese Brcke h?lt schon 30 Jahre"
That's not normal I think! No?
> in total take up 1/4 as much CPU time as your Mandelbrot app).
by because it waits after the display?
4. NOTHING, NOTHING AT ALL!!
7. Using API NetMessageBufferSend: Nothing!
11. using evens in define..


High-performance drives are generating ever higher motor speeds and drive torques. Although performance requirements are increasing, less and less sp ace is available for the brake. The INTORQ spring-operated brake for high-performance applications is the BFK468. The innovative multipole technology allows it to be released through a large air gap, even when forces are high.


From the creators of the award-winning An Egg Is Quiet, A Seed Is Sleepy and A Rock Is Lively comes this gorgeous and informative introduction to the world of butterflies. An incredible variety of butterflies are celebrated here in all of their beauty and wonder, from the tiny Arian Small Blue to the grand Queen Alexandra's Birdwing. Perfect for a child's bedroom bookshelf or for the classroom!
Dianna Hutts Aston is the author of many bestselling books for children. She is also the founder of The Oz Project, a nonprofit foundation which provides inspirational hot air balloon experiences to disadvantaged children.
Sylvia Long is the award-wining illustrator of many bestselling books for children. Her detailed paintings are inspired by her love of animals and the outdoors. She lives in Scottsdale, Arizona, with her husband and their dogs, Jackson and Truman.


Old Boys Gallery is really a rogues gallery when it comes to international craft beers and delectable tapas. When it comes to beer, there's literally walls of the stuff at this bar and restaurant in the Kampong Bahru area. Located in a former art space along Kampong Bahru Road, Old Boys Gallery has created a temple to the aesthetics and flavours of our hoppy favourites, matched perfectly with the Spanish small plate phenomenon (and a range of Asian fusion delicacies) that has taken Singapore by storm.
Prominent on Old Boys Gallery's wall of beer – yes, there's a whole wall to choose from, perfect for Singapore's thirsty hopheads – is the famous family of Japanese Hitachino Nest beer. Dai Dai IPA, Espresso Stout, Red Rice, and the brilliantly named Extra High are just four of the varieties in stock at Kampong Bahru's peerlessly geeky craft beer emporium. While the art of the bottled microbrew is the chief commodity here, Old Boys Gallery's Kampong Bahru Road premises stay true to the space's former function as an art space, offering art by emerging Singaporean artists for your visual pleasure, and novice art collector impulses.
Alongside Old Boys Gallery's fantastic selection of beer – surely among the most extensive in Singapore – is an equally superlative menu of small plates and tapas, taking cues from both Spanish traditions and futurethinking Asian fusion cuisine. So, expect inventive takes on the small plate like cremini in disguise and classic roast pork aglio olio to be found rubbing shoulders with burgers – huge, delicious burgers like Old Boys Gallery's signature Manster Burger, big enough for the heartiest of appetites. Find Old Boys Gallery down Kampong Bahru Road, and settle in with a cold one for a feast that satisfies both the eyes and the belly.


On Monday 17th December, a fantastic turnout saw 16 players competing for the Turkey tournament title. A handicap tournament against the clock, the Turkey tournament ensured lots of games for everyone and gave everyone a chance of competing, regardless of ability. A tremendous final saw Danny Jnr beat Sam 32-31 with the last shot of the match. Well done Dan. Thanks Alistair for the prizes.
Alistair won a close battle with Roger in the final. Thanks to Alistair for the prizes and Danny for organising.
Thanks to Matt for organising and Stephens' butchers for sponsoring. Report.
On Sunday 4th March 2012 St Ivo Racquets club resurrected the infamous 'Turkey' tournament. There was a very good turnout with 14 players of all standards battling it out for some Karakal squash goodies, and the ultimate prize of Turkey Tournament champion!
Players were each given a handicap based on their ability, and games were timed at fifteen minutes. The player with the higher handicap was given a points start equal to the difference in handicaps, which ensured many closely fought matches.
In the early rounds the tournament took on a round robin format. Danny Caulkett Junior progressed easily from his group, despite a scratch handicap his fierce shots were too much for his opposition but he survived a very close match with Gayle Jackson to win his group. He was followed into the semi finals by his father, Dan Caulkett Snr whose tireless running game was too much for his group opponents. Alex Curtis won his group with relative ease, seeing off a potentially tricky final opponent in Nick Bach to go through. The final semi final place was claimed by the returning Matt Dixon who went through after an epic tussle with Ian Pask saw him win the deciding group match 39-38.
The first semi final between Dan Caulkett senior and Alex Curtis was always going to be a big ask for the older Caulkett. Giving away 13 shots to the younger player meant Dan was always playing catch up, and Alex saw the match through with a thumping 61-19 victory.
The second semi final was an eagerly anticipated match between the competitions two best players Danny jnr and Matt. Matt had a five point advantage, and started well but Dan reeled the score back and smashed his way to a 34-26 victory.
The final was an epic encounter. Alex had a 28 point advantage, and at at the halfway stage it looked like he could hold on for victory. Danny had a very good run of points and took the lead with two minutes remaining. However Alex was hitting some unstoppable forehand drives and got the score back to level with only 30 seconds remaining. Danny composed himself and managed to win the last two points to claim the title 47-45. Well done Danny and Alex for a great final.
Every year before the AGM the club hold a team tournament. Players of all ages and ability are encouraged to enter. Teams are selected to include a player from each playing category (Advanced through to beginners), and players play against their opposite numbers from other teams.
It is a great opportunity for all the club, team players, social players, and juniors to come together and play in a friendly, fun competition, where everybody gets to play players of a similar standard. Make your entry on the noticeboard.
The final of the annual handicap turkey tournament took place on Monday 1st February. A large crowd was at the Ivo centre to witness a tense and exciting finale to the competition as the hard hitting Danny Caulkett Junior took on the skillful Steve Mason.
Mason was awarded a 12 point handicap start by match referee Toby Reeves. Under the rules of the Turkey, Caulkett junior had 20 minutes to overturn the deficit in a continuous game of point-a-rally squash.
Caulkett started well and reduced Mason's lead to eight points with ten minutes remaining. The two players slugged it out over the next five minutes in a game of high quality volleys, fierce drives, and exquisite drops. As the game progressed into the final five minutes, Caulkett started to find a good length with his drives and with two minutes remaining Mason held onto a slender lead of two points. Both players refused to take a step back and an epic rally ended with Caulkett tinning an easy drop. The roar of frustration Caulkett emitted still echoes around the Ivo centre to this day.
With victory in his sight Mason could have sat back and played for time, but to his credit he took the next point with a sublime dropshot to claim the title. Well done Steve, and a big thank you to Toby Reeves for organising the tournament.
Mason(right) is presented with a cheque shaped turkey for the winners prize.


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