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Generally chatting those will certainly be ideal served buying a tablet kind; Clen tablet computers are hardly ever counterfeited and also rather affordable. Many regularly located in 20mcg as well as 40mcg kind Clen tablet computer computers are rather easy to take care of as it is incredibly straightforward to control the application with efficiency. Did we review they are cheap? It is not uncommon to locate premium Clen in 40mcg tabs at a price of $10 for FIFTY tablet computer systems. Given, in most cases those which get Dianabol, Deca-Durobolin and also Clenbuterol are checking out need a reasonable little bit greater than 50 40mcg tabs nevertheless the expense to profit ratio in-terms of bucks and results is considerably in your favor.
Results are most properly accomplished if you stick to the advised patterns which are set out above in the Stacking Quick quick guide.
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The muscle building particular niche is massive and also, being a man dominated sport, as you would certainly expect, most of CrazyBulk consumers are males. However, women body building is preferred, so there is still a market for females (as well as there are female pleasant items to be discovered on CrazyBulk). Although mainly bodybuilding focussed, CrazyBulk products such as those made to boost toughness, energy as well as cardiovascular efficiency could equally as quickly be effectively targeted to basic strength trainers, athletes as well as anybody else looking to get fit, strong and construct muscle. People in below inquire about, which purchase Dianabol, Deca-Durobolin and also Clenbuterol generally obtain their item in both tablet computer system as well as fluid kind. Regardless of the kind, when you acquire Dianabol, Deca-Durobolin and also Clenbuterol in either style it is to be taken by mouth and specifically likewise; one type is merely solid as well as the various other in a fluid state.
Presuming both products are real, dosed correctly as well as of a sanitary design there is really no difference in either thing; despite type, when you purchase Dianabol, Deca-Durobolin as well as Clenbuterol in right here ask about. It will certainly accomplish within the physical body the same way tablet computer or otherwise. It applies, several of the liquid kinds, far more so compared to the tablet kinds are absolutely nothing short of rubbish or unsteady at finest as well as difficult to dose accurately. Due to this a tablet acquisition is always a far much better bet yet there are excellent quality liquid things about, you just should do a little research to guarantee you locate the very best ones.
You can purchase very normally numerous buy Dianabol, Deca-Durobolin as well as Clenbuterol in here inquire about as it is one of the most reliable. So trusted those that take many over-the-counter fat burners will certainly not enjoy or experience near the physical body fat burning results as those that shop Dianabol, Deca-Durobolin and also Clenbuterol in here ask about. No, this effective bronchodilator will definitely not change out sound diet regimen programs as well as exercise ideas nevertheless in additional fashion it can supply a much desired included side.
Why you should Acquire Dianabol, Deca-Durobolin and also Clenbuterol in below? Ask about Where to Shop Dianabol, Deca-Durobolin and also Clenbuterol in here inquire about Go to any sort of message board as well as there are all sort of stories as well as point of views in here ask about with regarding this bronchodilator; there are stories concerning merely just how much it assists the PCT procedure, how it develops fantastic harmony with an anabolic steroid pattern; the list goes on and on. It holds true, Dianabol, Deca-Durobolin as well as Clenbuterol Hydrochloride does have a moderate anabolic nature yet its hardly enough need to dietary supplement with it.
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Where Can I Get Dianabol Steroid In San Carlos De Bariloche Argentina. There are definitely only 2 better factors anyone would certainly ever purchase Dianabol, Deca-Durobolin as well as Clenbuterol in here ask about, for the treatment of persistent bronchial asthma and also for metabolic improving effects, i.e. body fat burning. Real, you could possibly buy Dianabol, Deca-Durobolin and also Clenbuterol in here inquire about in an effort to raise cardio efficiency and this could be helpful for some yet mostly the two after stated elements are the key concentration with body fat burning being the key concentration of the majority of any individual that finds themselves below.
Like all medications of any type of kind you have to have understanding, you have to know exactly just what you are doing so that you not simply experience top quality results nonetheless to make sure that you continue to be safe too; fortunately for you there are a number of overviews below on steroid.com that can offer to this end. It corrects here however we wish to discuss the type of Clen as it is generally known that you may definitely buy Of those in below ask about which make use of Clen most will definitely situate it can come either methods, as a dental tablet computer or in a liquid choice you simply eat. The greatest pointers you could acquire would certainly be making particular that you understand just how you can prepare.
Although this seems a little irrelevant, it is a capability which is not to be taken also gently. Although several locate it a lot simpler to merely buy a take-away, if you intend your physical body to respond positively to the medicine consuming right is the first thing you need to do. Among the perks of food preparation by yourself is that you comprehend exactly simply exactly what you are consuming and the quantity of calories you are consuming. This rejects the opportunities of over or under eating as a result of misdirected areas marketed on numerous take-away food choices!
Assurance that you consist of large quantities of fruits as well as vegetable in your fat burning. Eat bunches of citrus fruits which are high in vitamin C as well as superb for your health. Bananas are additionally fantastic when you are trying to shed fat and also develop muscular tissue. Eco-friendly Veggies are really high in Iron which furthermore great for your physical body as well as keepings you aware and also sharp. This will certainly place you at a considerable advantage as it will definitely enhance your concentration and maintain you urged. Ensure that you are additionally consuming foods which are high in calcium and also bountiful in vitamin D as this will aid to enhance your bone toughness which is fantastic for when you are constructing muscular tissue mass as your physical body will have the ability to sustain itself far much better. One more terrific idea is to set on your own little possible objectives. By setting your views too high you will certainly wind up disappointing on your own and coming to be demotivated. By establishing tiny goals you will definitely handle to document your progression conveniently and also you will going on your personal encouraged which as a result will definitely urge you to wage the process.
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Dianabol, Deca-Durobolin and also Clenbuterol Stacking is not an anabolic steroid however it does function effectively in combination. Normally any sort of anabolic steroid can be made use of nevertheless those that purchase Dianabol, Deca-Durobolin and Clenbuterol in here inquire about will usually doing this when anabolic steroids that advertise a leaner and also a lot more challenging body are in play. While Clen will absolutely offer to this end there are various other items that work well with it; as an example, lots of people that acquire Dianabol, Deca-Durobolin as well as Clenbuterol in here inquire about frequently financial investment thyroid gland bodily hormones such as Cytomel (T3) to accompany it as well as countless likewise supplement with Human Growth Hormonal agent. It holds true; a mix of Dianabol, Deca-Durobolin as well as Clenbuterol, Cytomel and Human Development Hormonal agent paired with a solid anabolic steroid pattern will certainly offer end results tough toacquire otherwise.
You might purchase Dianabol, Deca-Durobolin and Clenbuterol in here inquire about as well as definitely nothing else as well as get a moderate anabolic bump and also ramping of the metabolic procedure yet if you select this walloping it will be small. It is important to note the effects will certainly be far more pronounced when an anabolic steroid pattern is at hand however while essential it is hardly the most important factor. Those which acquire Dianabol, Deca-Durobolin as well as Clenbuterol in right here ask about yet fail to abide by audio diet regimen programs concepts will definitely not shed any type of physique fat; you can in fact take an enormous amount of Clen per day as well as if you are eating like a body fat pig after that assume merely what, youre getting fat. When we purchase. Dianabol, Deca-Durobolin as well as Clenbuterol there is really simply one approach to view it, as a supplement and the concept of a supplement is to consist of in the benefits, no Hydrochloride t create them. Clen can be part of the equation, a valuable gadget to your fat burning requirements; it is nevertheless not the response all alone.
Dianabol is among one of the most effective dental anabolic steroids on the market in Poltava Ukraine for the purpose of promptly raising both size as well as durability and also as by its nature is a lot more powerful than its normal relative Anadrol on a milligram for milligram basis. As sturdiness and dimension enhancing are its key ability most of those who purchase Dianabol will doing this for a bulking cycle, as have to come as no surprise yet it can be made use of properly throughout a decreasing pattern too.
Dr. John Ziegler did not have any type of kind of hand in developing D-bol, yet he contributed in the distribution and also very early usage of the steroid. He started by handling a wide range of his own professional people consisting of an appendectomy client and also shed victim. Dr. Ziegler tried evaluating on himself and also later on continued to teaming up with professional athletes searching for a way to raise their effectiveness.
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If you're a performance improving expert athlete or probably if you're merely getting ready to start there's a superb probability you'll buy Dianabol in Poltava Ukraine in the near future. The factors are simple; people in Poltava Ukraine acquisition Dianabol as it is amongst one of the most extremely effective regarding its purpose, it is really basic to control, it is typically really inexpensive and also for decades has actually been among one of one of the most extensively easily offered anabolic steroids of all time; in-fact, the little 5mg pink Anabol tabs, among one of one of the most popular brand names of all time have actually been seen in fitness centers as commonly as the weights themselves considered that before the Golden era of bodily structure.
Where Can I Get Dianabol Steroid In San Carlos De Bariloche Argentina. In order to test the new medicine as well as please a requested need, Dr. Ziegler worked with a set of three of American weight lifters, thinking about that they were having troubles competing with the Russian weightlifting group. The experiment situated mixed outcomes, yet in the end Dr. Ziegler believed it to be a failing, due to the fact that the only real outcome he situated in the weightlifters was improved horniness. However, evidence later on revealed the appropriate dosage neither pattern were being used.
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While this steroid can be trustworthy in a reducing pattern, generally most will identify numerous other choices to be a lot more efficient after that restriction this steroid to off-season durations of use. Those that supplement with this powerful C17-aa oral steroid typically report gains in size of around 20lbs in plain weeks, commonly a great deal more; it has to be remembered, the individual will certainly not multiply if appropriate nutrients are not in position. A normal misunderstanding is that you have to do is supplement with anabolic steroids as well as you'll increase; this is merely unreal. If you wish to permit you should feed your growth along with while those that obtain Dianabol will certainly find it efficient it is not magic. The concept behind this steroid as is with all steroids is to take merely what youre currently doing ideal nonetheless to do it with a far better degree of performance.
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Tyler Dunham is a polymath. With skills ranging from creative strategy to custom motorcycle design, he has spent his life learning, traveling and creating.
Tyler has had the fortune to work with the highly regarded branding firm Collins: in NYC, Senator Angus King in Maine, and spent 6 months driving a converted school bus around the country for good. He spends most days now producing projects and strategizing long-term for MAKEWILD.
Tyler likes maple syrup, powder days, and mezcal. He likes to meet people who exude enthusiasm, determination, optimism and a passion for exploring the natural world.


text:
The Mixtape è un EP degli Anarbor, pubblicato l'8 marzo 2011 e reso disponibile in download gratuito sul sito ufficiale della band. L'EP contiene una nuova versione di Contagious (già presente nell'album di debutto The Words You Don't Swallow) in collaborazione con Sierra Kusterbeck dei VersaEmerge e Ryan Hunter degli Envy on the Coast, una versione rock di Cooler than Me di Mike Posner, una cover di Hash Pipe degli Weezer, una versione rock di Bulletproof dei La Roux, un inedito degli Anarbor (Fast, Cheap, and Out of Control), una telefonata da parte del manager della band, il remix di I Do What I Do e Let the Games Begin (due canzoni del primo album) e infine una versione acustica di quest'ultima canzone.

Tracce

Note

redpajama_set_name: RedPajamaWikipedia


I had the chance to fly to Bali last December just before new year, indeed a great experience to stay in Ubud.
Room is clean and bright with enough amenities, the rooftop pool with rice field view is amazing and it is located in a very strategic location in between Hanoman and Monkey forest road to ensure good and fast access to what Ubud has to offer. The hotel indeed offers good value for money and amenities, it costs usd 50/night - such a great deal compared to other hotels in the location. Hotel is new so the room properties are looking great, aircon + hot shower are all good.
It is clear that this hotel is so new, their system is not working properly yet.
The hotel has a fault in their system, so when I wanted to check in they say that my reservation is not valid, in fact, I use their website for reservation. thankfully, there is still room available. But then, they gave me the wrong room (asked for double bed and got the twin bed), yet need to wait for another room given. Staff look confused, does not show the standard level of hospitality that Bali has to offer, and seem not getting enough training. There is no room service provided yet, therefore, better have your own snack supply in the room. For this matter, its service level was lacking, unlike other hotels in Bali that I have been to. On the next day, breakfast was ordinary with very limited choice, but fret not good breakfast can be found in walking distance.


text:
Spatial analysis if prodution by macrophytes, phytoplankton and epiphyton in a large river system under different water-level conditions
Vis, C. et Hudon, C. et Carignan, R. et Gagnon, P. (2007). Spatial analysis if prodution by macrophytes, phytoplankton and epiphyton in a large river system under different water-level conditions. Ecosystems , 10 (2). p. 293-310. DOI: 10.1007/s10021-007-9021-3.
Relative contributions by macrophytes, epiphyton and phytoplankton to total primary production was estimated in a large (~300 km2) widening of the St. Lawrence River (Canada), over a 2-year period with contrasting flows and water levels. Spatially-explicit estimates of whole-system production were obtained by combining field measurements with remotely sensed data and empirical models using GIS. Primary production and relative contributions of each producer type differed markedly between open-water and wetland habitats. Spatial differences within each habitat arose from interactions between physical factors including light, water depth, water transit times and wind stress. At the whole-system level, annual primary production represented 105 gC m-2 y-1, divided roughly equally among phytoplankton (34 %), submerged macrophytes (27 %), emergent macrophytes (23 %) and epiphyton (16 %). A 10% decrease in annual flows and 1-m decline in water levels between 2000 and 2001 resulted in a 50 % loss of marsh habitat, a 60 % increase in phytoplankton production in the open-water zone, and in the appearance of conspicuous filamentous algal mats. Low water levels induced substantial shifts in the spatial configuration and relative importance of primary producers although total river primary production remained stable between years.
Article scientifique
Large rivers, Fluvial lake, Primary production, Phytoplankton, Epiphyton, Macrophytes, Low water levels
3. Végétation, milieux humides
http://belsp.uqtr.ca/id/eprint/574

redpajama_set_name: RedPajamaCommonCrawl


Every one when sending a Christmas gift desires it to appear as attractive as possible. When we receive a gift nicely wrapped with tissue paper and decorated with pretty Christmas seals, there is an added pleasure. This is especially so when we realize that the sender has spent time and cleverness to give this little touch of personal thoughtfulness.
Many varieties of Christmas seals may be purchased, but a little originality will not come amiss even in these small details.
There are seals [stickers] for the outside package and the heading to be used on the notepaper, conveying the good wishes of the giver, or on the little card of greeting. These two in each set match. This gives another touch of the unusual, which is always sought after.
Have a piece of carbon paper, place it over the notepaper, card or seal, and on top of this one of the designs. Carefully trace with a sharp pencil the entire outline. An exact duplicate of the design will be found on the notepaper or seal ready for tinting. This may be easily done with watercolors or wax crayons. Poinsettia should be colored a warm, bright red, with yellow centers and dark green stems.
Holly, the special emblem of this season, has berries of vivid, fiery red and glossy leaves of dark green.
A dainty basket of forget-me-nots would be pretty colored yellow, the flowers a faint light blue and the bow knot of pale pink. Now we have the mystic sign of the holidays borrowed from the Druids: the mistletoe. The berries of this plant are a pearly white and the leaves a yellowish green.
Spend a little extra time giving the Christmas gift this novel decoration, and the pleasure it affords will never cause you to regret having done so.
Economy is to be considered at Christmas time. Our purse-strings have been stretched to the breaking point purchasing gifts for relatives and friends. To put this fascinating bit of handiwork upon the Christmas package costs comparatively nothing, aside from the time you spend doing it. This fact counts very little when we add to the enjoyment of others.
These few suggestions will probably cause endless ideas for the Christmas package to present themselves to you. Make your gifts radiate the holiday spirit; give this little bit of personality and attractiveness to as small a thing as the Christmas seal.


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text:
Foreign ministers downplay shared Canada-U.K. embassies
Foreign ministers for Canada and the U.K. downplayed an agreement to share space in embassies around the world, amid criticism from opposition MPs in Ottawa.
Laura Payton · CBC News · Posted: Sep 24, 2012 2:19 PM ET | Last Updated: September 25, 2012
Foreign Affairs Minister John Baird and British Foreign Secretary William Hague hold a news conference on an agreement between Canada and the U.K. to share embassy space 17:07
Foreign ministers for Canada and the U.K. downplayed an agreement to share space in embassies around the world, amid criticism from opposition MPs in Ottawa on Monday.
Britain's Foreign Secretary William Hague was in Ottawa to announce an agreement between the two countries that Foreign Affairs Minister John Baird says will start with embassies in Haiti and Burma, and will allow civil servants to consider doing so in more countries as the need arises.
"We are not moving to merge all of our embassies and consulates around the world. We are not going to be sharing ambassadors or trade commissioners," Baird said.
"Each country will continue to have complete independence on policy and Canadian public servants will always protect and promote Canada's interests and Canada's values. In select locations, this simply allows Canadian diplomats to do their good work faster and at a lower cost to Canadian taxpayers."
Hague said the two countries were expanding on an agreement announced last fall when British Prime Minister David Cameron was in Ottawa.
"We are two countries with large diplomatic networks ... but we can't be everywhere," Hague said.
"This looks to increase our co-operation to maximize our reach and impact on how two vibrant foreign ministries with common goals in many respects can work together — common values, shared values ... It is about speed and flexibility, practicality, saving the taxpayer money in both countries."
'Sound practical sense'
Hague said the agreement doesn't change the two countries' independence from each other.
"It just makes sound practical sense," he said.
NDP foreign affairs critic Paul Dewar said it was confusing that the government argued the plan was an innovative approach to foreign affairs and announced it with much fanfare, but then said it's more about sharing administrative services.
In question period, Baird referred to the agreement as "small" and "administrative." The arrangement, he said, will allow a British diplomat to have an office at the Canadian Embassy in Haiti, where Britain now has no presence. At the same time, Canada can send a foreign service officer to Burma to work out of the British Embassy until the Canadian mission is established. Those are the only two countries included so far in the agreement.
NDP Leader Tom Mulcair accused the government of being unable to represent Canada abroad.
"If the Conservatives won't stand up for Canada in the world, why do they expect that the British will do it for us?" he said.
"It's all very nice to be nostalgic for the great British empire, but there are limits."
Interim Liberal Leader Bob Rae criticized Prime Minister Stephen Harper for avoiding the UN General Assembly this week.
"If we have such a wonderful independent foreign policy, why isn't the prime minister of Canada discussing that foreign policy in front of the United Nations this week like so many heads of state?" he said.
The NDP also raised the issue of French-language services and whether they would be available to francophones through the British missions.
Agreement includes crisis response, security
The British press broke the news yesterday that Canada and the U.K. would be announcing they will co-ordinate some embassies to share administrative costs.
"The U.K. and Canada have signed an agreement on closer working that will provide a framework for our missions overseas to co-operate more closely, sharing admin costs for example, where appropriate to promote U.K. and Canadian interests. This declaration sets out a number of areas to further co-operation, including co-location, crisis response, security and consular co-operation," the British Foreign and Commonwealth Office said in a statement to CBC News.
"This is part of the government's drive to deepen our relationships and improve our engagement with a number of Commonwealth allies across the full spectrum of foreign policy issues."
There is speculation in the U.K. that the agreement is meant to compete with efforts by the EU to co-ordinate its own diplomatic work. Hague denied that.
"Obviously we work very closely with other European foreign services ... and there are places where we sometimes share a compound with some of the other foreign ministries of the European Union," he said.
"It's not a zero sum world. And finding practical ways to co-operate successfully with other nations doesn't in any way diminish our relations within the European Union."
Hague's visit marks the first bilateral visit to Canada by a foreign secretary since 1996, he said this morning on Twitter.
Hague kicked off the visit with a meeting with Bank of Canada governor Mark Carney and the Canadian Council of Chief Executives, he wrote on Twitter.
This won't be Canada's first foray into sharing services. Canada and Australia have worked together in 26 locations around the world for years.
Since Canada has pulled its diplomats out of Iran, Italy has offered to handle consular matters in Tehran.
With files from James Cudmore
TRANSCRIPT | Peter Mansbridge talks to William Hague
Canada's consular co-operation
Should Canada and the U.K. share embassies abroad?
Canada, U.K. to cut costs by sharing embassies abroad


text:
The Observatorial
WorldWide News Outlet
In Antarctica, the glacier is retreating at an alarming rate in some places
Antarctica is warming faster than the global average due to climate change.
Thwaites, one of the largest glaciers on the continent, has become a particularly big concern, and large cracks were recently discovered in the ice shelf floating above the sea.
It is estimated that the ice shelf will crumble within 5–10 years.
The speed of the change also surprised the researchers, says the director of the Antarctic Institute in Chile Marcelo Leppe.
"The speed of change is surprising," researcher Marcelo Leppe says about the melting of the Antarctic glaciers.
Why are the observations made on the continent so significant?
The ice shelf has acted as a plug and prevented the Thwaites Glacier from draining into the sea. However, now the warmed sea water has started to melt the ice shelf quickly from below.
Researchers estimate that when the ice shelf breaks up, the glacier located on the continent can flow into the sea at an accelerating rate. That, in turn, would accelerate sea level rise in the coming decades.
Read more: What will happen if a giant glacier in Antarctica collapses? A cavity of tens of kilometers has appeared under the Thwaites glacier
How surprising were the results to the scientific community?
The speed of change is surprising. We are only now beginning to see how different types of events, from heat waves to changes in water salinity and ocean currents, affect each other. Predicting their mutual interaction is difficult.
What are some examples of such interactions?
In recent years, heat waves have increased large forest fires in Australia, for example. A lot of soot has traveled from there to Antarctica, which colors the snow dark. Dark snow absorbs more radiation and makes it melt more easily.
The reflectivity of the ice is also changed by different kinds of algae, which with warming can spread deeper into the continental glaciers.
Is the Antarctic ice shrinking all over?
No, some of the central parts of the continent have actually cooled. In the peripheral areas, however, the change is clear.
I have been visiting Antarctica for 20 years at various research stations, and in some areas the glaciers have retreated more than 200 meters during that time. Where there used to be the tip of the peninsula, there may now be an island.
In practice, we have to redraw maps, especially in the northern regions of the continent.
How is warming seen in plants and animals?
The zones of plant and animal species living on the edges of Antarctica are moving south.
For example, some penguin species have moved into areas that were previously too barren for them to nest.
In addition, alien species have spread to Antarctica from the surrounding continents. Close to the research stations, for example, sedges have started to grow.
How do you try to protect the continent?
Antarctica is no longer seen as a separate area from the rest of the world, the protection of which is locally related to, for example, the reduction of pollution and microplastics.
Ocean currents surrounding the continent affect the climate globally, just like the Gulf Stream. In order to protect it, the most important thing is to reduce emissions worldwide.
What gives you hope about climate action?
International cooperation is closer than ever, and Antarctica has come into the spotlight: researchers from the region were invited to participate, for example, in the work of the intergovernmental panel on climate change, the IPCC.
We are currently also witnessing the revolution of renewable forms of energy. In the 1980s, we achieved the Montreal Treaty on Propellant Gases, which has reduced ozone depletion – no small feat. We can do the same for carbon dioxide as well.
They discover the closest black hole to Earth; it is 10 times larger than the Sun, astronomers say
Oculus co-founder devises a VR headset that kills the player by destroying their brain if they lose in the game
Jan 30, 2023 Editor
Street Fighter 6, among the commentators Zelina Vega of WWE
OnePlus Pad adapts "the iconic design philosophy" of the brand
Tom Verlaine, a former front man for Television and a prominent member of the punk rock scene, has died
Capital gains case, De Siervo: "It's hard to think Juve is solely responsible"
The Swedish economy unexpectedly squatted at the end of the year: "The figure is really weak"
Finland announces that it wants to enter NATO together with Sweden
Leipzig: Zoo boss Jörg Junhold (58) makes 3000 euros in 20 minutes at Lidl


2018 was another eventful year for intellectual property litigation. In addition to developments in AIA post-grant trials before the PTAB, the Supreme Court and Federal Circuit issued several landmark decisions affecting both IP owners and accused infringers. By all indications, 2019 promises even more changes, as the Supreme Court already is hearing oral arguments in cases involving patents and copyrights.
Join Brooks Kushman's expert IP litigators as they discuss the developments from 2018 that have had the biggest impact on enforcement of IP rights, and their preview of the issues that the Supreme Court and Federal Circuit likely will consider in 2018.
Webinar is over, you can not register now. If you have any questions, please contact Webinar host: Brooks Kushman.
Hi there, You are invited to a Zoom webinar. When: Feb 6, 2019 2:00 PM Eastern Time (US and Canada) Topic: IP Litigation in 2019: Key Developments from 2018 and Predictions for the New Year Register in advance for this webinar: https://brookskushman.zoom.us/webinar/register/WN_8xxRL_tDSUOQhIfERq_gfg After registering, you will receive a confirmation email containing information about joining the webinar. ---------- Webinar Speakers Marc Lorelli (Co-Chair Litigation @Brooks Kushman) Marc is the Co-Chair of Brooks Kushman's intellectual property litigation practice. He has successfully tried patent, trademark, trade secret and copyright cases in Federal Courts nationwide, the Federal Circuit Court of Appeals, and the International Trade Commission, and is authorized to practice before the U.S. Patent and Trademark Office. Marc's litigation experience spans across various technologies and industries, including automotive, consumer electronics, medical device, and appliance industries. Chanille Carswell (Shareholder @Brooks Kushman) Chanille`s practice focuses on trademark, trade dress, copyright and patent litigation, unfair competition, and appeals in a wide range of industries such as automotive, banking, construction machinery, consumer products, e-commerce and insurance. She also represents clients in trademark opposition and cancellation proceedings before the United States Patent and Trademark Office.


text:
\section{Introduction}
The research into pulse power amplification in microwave
generators is gaining importance nowadays \cite{1,giri}. It has
been shown that even for short microwave pulses, the discharge
processes in the resonators of generators place limitations on the
radiation power amplification potential.  The possibilities of
coherent combining of fields generated by several microwave
generators  are actively studied in this connection as the means
of tackling these limitations. Particularly, phase-locking is a
subject of vigorous study \cite{giri,woo,whar,elch}.

It has also been demonstrated that two separate super-radiant
backward wave oscillators (BWO) connected to one and the same
voltage (power) supply of subnanosecond rise time can form two
 coherent waves, which causes a fourfold increase in the radiation power \cite{elch}.

 Thus, these experiments confirm that using $N$ number of BWOs in this way, two (or more)-channel
nanosecond relativistic microwave generators can be developed,
whose total power will be as high as $W\sim N^2$ ($N$ is the
number of BWOs \cite{whar}.

  Study of the application
potential of volume free electron lasers (VFEL) for the
development of relativistic microwave and optical generators with
increased power is another promising line of investigation in this
field \cite{bar12}.

Two(three)-dimensional distributed feedback (DFB) arising in VFEL
resonators with two-three dimensional spatially periodic
structures (now often called electromagnetic or photonic crystals)
enables producing coherent radiation from wide electron beams or
several beams (see Figs. \ref{volume}, \ref{multiwave}). A key
feature of the VFEL is that as a result of diffraction, the signal
is transferred from one point to another, thus linking the points
of the beam and making them generate coherently (see Figs.
\ref{volume}, \ref{multiwave}).
\begin{figure}[htbp]
\begin{center}
\epsfxsize = 10 cm \centerline{\epsfbox{volume.eps}} \caption{}
\label{volume}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.6]{multi3wave}
\quad \includegraphics[scale=0.6]{multi4wave}
\quad\includegraphics[scale=0.6]{multi6wave}
\end{center}
\caption{} \label{multiwave}
\end{figure}
This means that the VFEL is a
self-phase-locking generator of radiation (phase-locking inside).
When $N$ number of electron beams pass through electromagnetic
crystals, the radiated power
 will increase in proportion to $N^2$ as a result of  self-phase-locking  of the beams due to two(three)-dimensional
DFB formed in a VFEL resonator. When  beams are generated, there
is always a certain  spread in the  times when the produced
electron bunches enter a VFEL resonator, which results from the
instability of the current generation in a diode gap.

In the present paper it is shown that despite this spread in the
entry times of the beams, a two(three)-dimensional feedback formed
in a VFEL resonator gives rise to the self-phase-locking process .
Equations are derived that describe the process of radiation
generation  by several electron bunches and enable studying the
process of superradiation produced by the bunches, depending on
the difference between the times of their entry into the
resonator. It is also shown that similar processes of
self-phase-locking, which lead to the power increase $W\sim N^2$
can also be observed when generation from electron bunches is
excited in a system of several relativistic BWOs, whose corrugated
waveguides form a single resonator by means of, e.g., slots (long
bridge corrugated waveguides or diffraction grating) made in a
cylinder along the axis of the waveguides.

\section{Lasing equations for
 the system with a photonic crystal (diffraction grating) with changing parameters}
\label{vp_sec:2}

In the general case the equations, which describe lasing process,
follow from the Maxwell equations:
\begin{eqnarray}
& \hspace{1cm}& \textrm{rot} \vec{H}=\frac{1}{c}\frac{\partial
\vec{D}}{\partial t}+\frac{4 \pi}{c} \vec{j}, ~\textrm{rot}
\vec{E}=-\frac{1}{c}\frac{\partial
\vec{H}}{\partial t}, \nonumber \\
&  &\textrm{div} \vec{D}=4 \pi \rho,~\frac{\partial \rho}{\partial
t}+ \textrm{div} \vec{j}=0, \label{vp_sys0}
\end{eqnarray}
here $\vec{E}$ and $\vec{H}$ are the electric and magnetic fields,
$\vec{j}$ and $\rho$ are the current and charge densities, the
electromagnetic induction $D_i(\vec{r},t^{\prime})=\int
\varepsilon_{il}
(\vec{r},t-t^{\prime})E_l(\vec{r},t^{\prime})dt^{\prime}$ and,
therefore, $D_i(\vec{r},\omega)=\varepsilon_{il}
(\vec{r},\omega)E_l(\vec{r},\omega)$, the indices $i,l=1,2,3$
correspond to the axes $x,y,z$, respectively.

The current and charge densities are respectively defined as:
\begin{equation}
\vec{j}(\vec{r},t)=e
\sum_{\alpha}\vec{v}_{\alpha}(t)\delta(\vec{r}-\vec{r}_{\alpha}(t)),~
\rho(\vec{r},t)=e
\sum_{\alpha}\delta(\vec{r}-\vec{r}_{\alpha}(t)), \label{eq2}
\end{equation}
where $e$ is the electron charge, $\vec{v}_{\alpha}$ is the
velocity of the particle $\alpha$ ($\alpha$ numerates the beam
particles),
\begin{equation}
 \frac{d
\vec{v}_{\alpha}}{dt}=\frac{e}{m \gamma_{\alpha}}\left\{
 \vec{E}(\vec{r}_{\alpha}(t),t)+\frac{1}{c} [
\vec{v}_{\alpha}(t) \times \vec{H}(\vec{r}_{\alpha}(t),t)
 ]-
 \frac{\vec{v}_{\alpha}}{c^2}(\vec{v}_{\alpha}(t)\vec{E}(\vec{r}_{\alpha}(t),t))
\right\},\label{vp_sys1}
\end{equation}
here $\gamma_{\alpha}=(1-\frac{v_{\alpha}^2}{c^2})^{-\frac{1}{2}}$
is the Lorentz-factor, $\vec{E}(\vec{r}_{\alpha}(t),t)$
($\vec{H}(\vec{r}_{\alpha}(t),t)$) is the electric (magnetic)
field at the point of location $\vec{r}_{\alpha}$ of  particle
$\alpha$.

It should be recalled that  (\ref{vp_sys1}) can also be written as
\begin{equation}
 \frac{d
\vec{p}_{\alpha}}{dt}=m\frac{d \gamma_{\alpha}
v_{\alpha}}{dt}={e}\left\{
 \vec{E}(\vec{r}_{\alpha}(t),t)+\frac{1}{c} [
\vec{v}_{\alpha}(t) \times \vec{H}(\vec{r}_{\alpha}(t),t)
 ]
\right\},\label{vp_sys11}
\end{equation}
where $p_{\alpha}$ is the particle momentum.

Let us recall here that the change in the particle energy through
the its interaction with electromagnetic fields is described by
the equation
\begin{equation}
\label{en} mc^2\frac{d\gamma_{\alpha}}{dt}=e\vec v_{\alpha}\vec
E(\vec r_{\alpha}(t),t).
\end{equation}

Combining the equations in (\ref{vp_sys0}), we obtain:
\begin{equation}
-\Delta
\vec{E}+\vec{\nabla}(\vec{\nabla}\vec{E})+\frac{1}{c^2}\frac{\partial^2
\vec{D}}{\partial t^2}=-\frac{4 \pi}{c^2} \frac{\partial
\vec{j}}{\partial t}. \label{vp_sys01}
\end{equation}

The dielectric permittivity tensor can be expressed as
$\hat{\varepsilon}(\vec{r})=1+\hat{\chi}(\vec{r})$, where
$\hat{\chi}(\vec{r})$ is the dielectric susceptibility.
When $\hat{\chi} \ll 1$,  (\ref{vp_sys01}) can be rewritten as:

\begin{equation}
\Delta \vec{E}(\vec{r},t)-\frac{1}{c^2}\frac{\partial^2}{\partial
t^2}
\int \hat{\varepsilon}(\vec{r},t-t^{\prime})
\vec{E}(\vec{r},t^{\prime}) dt^{\prime}
=4 \pi \left( \frac{1}{c^2} \frac{\partial
\vec{j}(\vec{r},t)}{\partial t} + \vec{\nabla} \rho (\vec{r},t)
\right). \label{vp_1}
\end{equation}

When the grating is ideal $\hat{\chi}(\vec{r}) = \sum_{\tau}
\hat{\chi}_{\tau} (\vec{r}) e^{i \vec{\tau} \vec{r}}$, where
$\vec{\tau}$ is the reciprocal lattice vector.

Let the photonic crystal (diffraction grating) period be smoothly
varied with distance, which is much greater then the diffraction
grating {(ptotonic crystal lattice)} period.
It is convenient in this case to present the susceptibility
$\hat{\chi}(\vec{r})$ in the form, typical of the theory of X-ray
diffraction in crystals with lattice distortion \cite{bar12}:
\begin{equation}
\hat{\chi}(\vec{r})=\sum_{\tau} e^{i \Phi_{\tau}(\vec{r})}
\hat{\chi}_{\tau} (\vec{r}), \label{vp_chi01}
\end{equation}
where $\Phi_{\tau}(\vec{r})=\int \vec{\tau} (\vec{r}^{\,\prime})
d\vec{l}^{\prime}$, $\vec{\tau} (\vec{r}^{\,\prime})$ is the
reciprocal lattice  vector in the vicinity of the point
$\vec{r}^{\,\prime}$.
In contrast to the theory of X-rays diffraction, in the case under
consideration $\hat{\chi}_{\tau}$ can also depend on $\vec{r}$.
Moreover, $\hat{\chi}_{\tau}$ depends on the volume of the lattice
unit cell $\Omega$, which can be significantly varied for
diffraction gratings (photonic crystals), as distinct from natural
crystals.
The volume of the unit cell $\Omega(\vec{r})$  depends on
coordinate and, for example, for a cubic lattice it is determined
as
$\Omega(\vec{r})=\frac{1}{d_1(\vec{r})d_2(\vec{r})d_3(\vec{r})}$,
where $d_i$ are the lattice periods.
If $\hat{\chi}_{\tau} (\vec{r})$ does not depend on $\vec{r}$, the
expression (\ref{vp_chi01}) converts to that usually used for
X-rays in crystals with lattice distortion.

Recall here that  for an ideal crystal without lattice
distortions, the wave, which propagates in the crystal can be
presented as a superposition of plane waves:
\begin{equation}
\vec{E}(\vec{r},t) = \sum_{\vec{\tau}=0}^{\infty}
\vec{A}_{\vec{\tau}} e^{i(\vec{k}_{\tau} \vec{r}-\omega t)},
\label{vp_field}
\end{equation}
where $\vec{k}_{\tau}=\vec{k}+\vec{\tau}$.

Let us now use the fact that in the case under consideration the
typical length for the change of the lattice parameters
significantly exceeds the lattice period. Then the field inside
the crystal with lattice distortion can be expressed similarly to
(\ref{vp_field}), but with $\vec{A}_{\vec{\tau}}$ depending on
$\vec{r}$  and $t$ and changing noticeably at the distances much
greater than the lattice period.

Similarly, the wave vector should be considered as a slowly
changing function of a coordinate.

According to the above, let us find the solution of (\ref{vp_1})
in the form:
\begin{equation}
\vec{E}(\vec{r},t)=\textrm{Re} \left\{
\sum_{\vec{\tau}=0}^{\infty}  \vec{A}_{\vec{\tau}}
e^{i(\phi_{\tau} (\vec{r})-\omega t)} \right\}, \label{vp_*}
\end{equation}
where $\phi_{\tau} (\vec{r})= \int_0^{\vec{r}} k(\vec{r}) d
\vec{l}+ \Phi_{\tau}(\vec{r})$, where $k(\vec{r})$ can be found as
a solution of the dispersion equation in the vicinity of the point
with the coordinate vector $\vec{r}$, integration is made over the
quasiclassical trajectory, which describes motion of the
wavepacket in the crystal with lattice distortion.

Now let us consider the case when all the waves participating in
the diffraction process lie in a plane (coupled wave diffraction,
multiple-wave diffraction), i.e., all the reciprocal lattice
vectors $\vec{\tau}$ lie in one plane.
Suppose the wave polarization vector is orthogonal to the plane of
diffraction.

Let us rewrite (\ref{vp_*}) in the form
\begin{equation}
\vec{E}(\vec{r},t)=\vec{e}\,E(\vec{r},t)=\vec{e} \, \textrm{Re}
\left\{
 {A}_1
e^{i(\phi_{1} (\vec{r})-\omega t)}+ {A}_2 e^{i(\phi_{2}
(\vec{r})-\omega t)} + ... \right\}, \label{vp_*1}
\end{equation}
where
\begin{equation}
\phi_1(\vec{r})=\int_0^{\vec{r}} \vec{k}_1(\vec{r}^{\, \prime}) d
\vec{l}, \label{vp_phi1}
\end{equation}
\begin{equation}
\phi_2(\vec{r})=\int_0^{\vec{r}} \vec{k}_1(\vec{r}^{\,\prime}) d
\vec{l} + \int_0^{\vec{r}} \vec{\tau}(\vec{r}^{\,\prime}) d
\vec{l}. \label{vp_phi2}
\end{equation}

Then multiplying (\ref{vp_1}) by $\vec{e}$, one can get:
\begin{equation}
\Delta {E}(\vec{r},t)-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}
\int
\hat{\varepsilon}(\vec{r},t-t^{\prime}){E}(\vec{r},t^{\prime})
dt^{\prime}
=4 \pi \vec{e} \left( \frac{1}{c^2} \frac{\partial
\vec{j}(\vec{r},t)}{\partial t} + \vec{\nabla} \rho (\vec{r},t)
\right). \label{vp_2}
\end{equation}
Applying the equality $\Delta {E}(\vec{r},t)=\vec{\nabla}
(\vec{\nabla} E)$ and using (\ref{vp_*1}), we obtain
\begin{eqnarray}
\hspace{-1 cm} \Delta (A_1 e^{i(\phi_{1} (\vec{r})-\omega t)})=
e^{i(\phi_{1} (\vec{r})-\omega t)}
[2i  \vec{\nabla} \phi_1 \vec{\nabla} A_1 +i \vec{\nabla}
\vec{k}_1 (\vec{r}) A_1  - k_1^2(\vec{r})  A_1], \label{vp_3a}
\end{eqnarray}

Therefore, substitution of the above expression into (\ref{vp_2})
gives the following system:
\begin{eqnarray}
& & \frac{1}{2} e^{i(\phi_{1} (\vec{r})-\omega t)} \left[ 2i
\vec{k}_1(\vec{r}) \vec{\nabla} A_1 +i \vec{\nabla}
\vec{k}_1 (\vec{r}) A_1  - k_1^2(\vec{r}) A_1  \right.\nonumber \\
& & \left.+ \frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_1
+ i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_1}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{-\tau}(\omega,\vec{r}) A_2 \right.\nonumber\\
& &+ \left.i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_{-\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_2}{\partial t}
\right] \nonumber \\
& & + \textrm{~conjugated~terms~}
=4 \pi \vec{e} \left( \frac{1}{c^2} \frac{\partial
\vec{j}(\vec{r},t)}{\partial t} + \vec{\nabla} \rho (\vec{r},t)
\right), \nonumber \\
& & \frac{1}{2} e^{i(\phi_{2} (\vec{r})-\omega t)} [ 2i
\vec{k}_2(\vec{r}) \vec{\nabla} A_2 +i \vec{\nabla}
\vec{k}_2 (\vec{r}) A_2  - k_2^2(\vec{r}) A_2  \nonumber \\
& & + \frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_2 + i
\frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_2}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{\tau}(\omega,\vec{r}) A_1 \nonumber\\
& &+ i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_{\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_1}{\partial t}
] \nonumber \\
& & + \textrm{~conjugated~terms~}
=4 \pi \vec{e} \left( \frac{1}{c^2} \frac{\partial
\vec{j}(\vec{r},t)}{\partial t} + \vec{\nabla} \rho (\vec{r},t)
\right),  \label{vp_3}
\end{eqnarray}
where vector $\vec{k}_2 (\vec{r})=\vec{k}_1 (\vec{r})+
\vec{\tau}$,
$\varepsilon_0(\omega,\vec{r})=1 + {\chi}_{0}(\vec{r})$, here the
notation ${\chi}_{0} (\vec{r})={\chi}_{\tau=0} (\vec{r})$ is used,
$\varepsilon_{\tau}(\omega,\vec{r})={\chi}_{\tau} (\vec{r})$.
Note here that for a numerical analysis of (\ref{vp_3}), if
${\chi}_{0} \ll 0$, it is convenient to take  vector $\vec{k}_1
(\vec{r})$ in the form $\vec{k}_1
(\vec{r})=\vec{n}\sqrt{k^2+\frac{\omega^2}{c^2} \chi_0(\vec{r})}$.

Let us multiply the first equation by $e^{-i(\phi_{1}(\vec
r)-\omega t)}$ and the second by $e^{-i(\phi_{2}(\vec r)-\omega
t)}$.
This procedure enables neglecting the conjugated terms, which
appear fast oscillating (when averaging over the oscillation
period they become zero).

Considering the right-hand side of (\ref{vp_3}), let us take into
account that microscopic currents and densities are the sums of
terms, containing delta-functions, therefore, the right-hand side
can be rewritten as:
\begin{eqnarray}
\label{fun} & & e^{-i(\phi_{1}(\vec r)-\omega t)}4 \pi \vec{e}
\left( \frac{1}{c^2} \frac{\partial \vec{j}(\vec{r},t)}{\partial
t} + \vec{\nabla} \rho (\vec{r},t)
\right) \\
& &= - \frac{4 \pi i \omega e}{c^2}  \vec{e} \sum_{\alpha} \vec
{v}_{\alpha} (t) \delta(\vec r -\vec r_{\alpha}(t))
e^{-i(\phi_{1}(\vec r)-\omega t)} \, \theta (t-t_{\alpha}) \,
\theta (T_{\alpha}-t). \nonumber
\end{eqnarray}

Here $t_\alpha$ is the time of entrance of particle $\alpha$  to
the resonator, $T_\alpha$ is the time of particle leaving the
resonator, $\theta-$functions in (\ref{fun}) indicate that for the
time moments preceding $t_{\alpha}$ and following $T_{\alpha}$,
the particle  ${\alpha}$ does not contribute to the process.

Upon averaging the system of equations over the oscillation period
$\frac{2\pi}{\omega}$, we can write:
\begin{eqnarray}
\label{5mod} & &\left[2i \vec{k}_1(\vec{r}) \vec{\nabla} A_1 +i
\vec{\nabla} \vec{k}_1 (\vec{r}) A_1  - k_1^2(\vec{r}) A_1 +
\frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_1
\right.\nonumber
\\
& & \left. + i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_1}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{-\tau}(\omega,\vec{r}) A_2 + i \frac{1}{c^2}
\frac{\partial \omega^2
\varepsilon_{-\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_2}{\partial t}\right]
\\
 & &= - \frac{8\pi i\omega
e}{c^2}\sum_{\alpha}\int^{t+\frac{2\pi}{\omega}}_{t}\vec e\,\vec
v_{\alpha}(t')\delta(\vec r-\vec r_{\alpha}(t',t_{\alpha}, \vec
r_{\alpha 0})e^{-i\varphi_1[\vec r_{\alpha}(t',t_{\alpha},\vec
r_{\alpha\,
0}),t']}\theta(t'-t_{\alpha})\theta(t_{\alpha}-t'),\nonumber
\end{eqnarray}
where the phase $\varphi_1[\vec r_{\alpha}(t' ,t _{\alpha},\vec
r_{\alpha\, 0}),t']=\phi_1(\vec r_{\alpha}(t',t_{\alpha}, \vec
r_{\alpha\,0})-\omega t'$.
\begin{eqnarray}
\label{5mod2} & &\left[2i \vec{k}_2(\vec{r}) \vec{\nabla} A_2 +i
\vec{\nabla} \vec{k}_2 (\vec{r}) A_2  - k_2^2(\vec{r}) A_2 +
\frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_2
\right.\nonumber
\\
& & \left. + i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_2}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{\tau}(\omega,\vec{r}) A_1 + i \frac{1}{c^2}
\frac{\partial \omega^2
\varepsilon_{\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_1}{\partial t}\right]
\\
 & &= -\frac{8\pi i\omega
e}{c^2}\sum_{\alpha}\int^{t+\frac{2\pi}{\omega}}_{t}\vec e\,\vec
v_{\alpha}(t')\delta(\vec r-\vec r_{\alpha}(t',t_{\alpha}, \vec
r_{\alpha 0})e^{-i\varphi_2[\vec r_{\alpha}(t',t_{\alpha},\vec
r_{\alpha\,
0}),t']}\theta(t'-t_{\alpha})\theta(t_{\alpha}-t'),\nonumber
\end{eqnarray}
where the phase $\varphi_2[\vec r_{\alpha}(t',t_{\alpha},\vec
r_{\alpha\, 0}),t']=\phi_2(\vec r_{\alpha}(t',t_{\alpha}, \vec
r_{\alpha\,0})-\omega t'$ and $\varepsilon_0=1+\chi_0$,
$\varepsilon_{\tau}=\chi_{\tau}$.

When several ($N$ number) electron beams move through a spatially
periodic medium, the sum $\sum_{\alpha}$ over the particles can be
represented as a sum of contributions coming from individual
electron beams to the total current:
\[
\sum_{\alpha}=\sum_{\alpha_1}+\sum_{\alpha_2}+...\sum_{\alpha\,N},
\]
where $N$ is the number of electron beams.

Using the definitions of $\phi_m$ (see (\ref{vp_phi1}),
(\ref{vp_phi2}) we can obtain the following relationship for the
phases $\varphi_m$:
\begin{equation}
\label{ph1}
\frac{d\varphi_m}{dt}=\vec k_m(\vec r_{\alpha}(t, t_{\alpha},
r_{\alpha\,0}))\vec v_{\alpha}(t)-\omega
\end{equation}
and
\begin{equation}
\label{ph2} \frac{d^2\varphi_m}{dt^2}=\vec
v_{\alpha}(t)\frac{d\vec k_m(r_{\alpha}(t))}{dt}+\vec
k_m(r_{\alpha}(t))\frac{d\vec v_{\alpha}}{dt}.
\end{equation}
Equations (\ref{vp_sys1})--(\ref{en}), describing particle motion
in electromagnetic fields, and equations
(\ref{5mod})--(\ref{5mod2}) for the fields are written using
slowly changing amplitudes $A$ and phases $\varphi_m=\phi_m-\omega
t$. They give a closed, nonlinear set of equations that defines
the amplitude $A_m$ and  phases $\varphi_m$ (as well as the change
in the energy of particles interacting with the fields) and can be
numerically analyzed using, say, the large-particle method.

Because of random distribution of particles in the bunches
incident on a resonator (electromagnetic, photonic crystal), the
times $t_{\alpha}$ of particle entry  into the resonator as well
as the distribution of the entry point coordinates $\vec r_s$ of
the bunch particles over the entire surface of the resonator are
random. Each  bunch  also has a certain distribution of initial
velocities $\vec v_{\alpha}^{(0)}$. This enables one to average
(\ref{5mod}) and (\ref{5mod2}) over the distribution of the
quantities $t_{\alpha}$$\vec r_s$, and $\vec v_{\alpha}^{(0)}$.
Such averaging can be made by generalizing the averaging method
developed for the case of one-dimensional generators like TWT,
BWO, FEL  to the case of a non-one-dimensional distributed
feedback (DFB) The equations obtained as a result of such
averaging in a stationary  case when one beam moves in a VFEL
resonator are given in \cite{bar12}.

We shall further consider self-phase-locking arising when photons
are emitted by several electron beams in a spatially periodic VFEL
resonator in the case of quasi-Cherenkov (diffraction) spontaneous
radiation mechanism (recall here that this radiation  mechanism
underlies the operation of conventional one-dimensional TWTs and
BWOs).

For better understanding, let us suppose now that a strong
magnetic field is applied for beam guiding through the generation
area. Electron beams move along the direction of this field. Let
us choose the direction of beam motion as the $z$-axis. We shall
also consider the case when the period of the resonator's
diffraction grating changes along the direction of the $z$-axis.
In this case, equations (\ref{5mod}) and (\ref{5mod2}) can be
presented in the form:
\begin{eqnarray}
\label{ins1}
& & \left[2i \vec{k}_1(\vec{r}) \vec{\nabla} A_1 +i
\vec{\nabla} \vec{k}_1 (\vec{r}) A_1  - k_1^2(\vec{r}) A_1
\frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_1
\right.\nonumber
\\
& & \left. + i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_1}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{-\tau}(\omega,\vec{r}) A_2 + i \frac{1}{c^2}
\frac{\partial \omega^2
\varepsilon_{-\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_2}{\partial t}\right]\nonumber
\\
& & =-\frac{8\pi i \omega e\vartheta_1}{c^2}g_1(r_{\perp}, z, t),
\end{eqnarray}
where
\begin{eqnarray*}
g_1(r_{\perp}, z, t) &= &
\sum_{N}\langle\langle\int^{t+\frac{2\pi}{\omega}}_{t}
\sum_{\alpha_N}u_{\alpha_N}(t)\delta(r_{\perp}-r_{\alpha_N\perp})\delta(z-z_{\alpha_N}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))
\\
&  \times & e^{-i\vec k_{\perp}\vec
r_{\alpha_N\perp}}e^{-i[\phi_1(z_{\alpha}(t',t_{\alpha_N},u_{\alpha_N}^{(0)}))-\omega
t']} \theta(t'-t_{\alpha}) \theta(T_{\alpha-t'})\rangle\rangle d
t',
\end{eqnarray*}
\begin{eqnarray}
\label{ins2} & &\left[2i \vec{k}_2(\vec{r}) \vec{\nabla} A_2 +i
\vec{\nabla} \vec{k}_2 (\vec{r}) A_2  - k_2^2(\vec{r}) A_2 +
\frac{\omega^2}{c^2} \varepsilon_0(\omega,\vec{r}) A_2
\right.\nonumber
\\
& & \left. + i \frac{1}{c^2} \frac{\partial \omega^2
\varepsilon_0(\omega,\vec{r})}{\partial \omega} \frac{\partial
A_2}{\partial t} + \frac{\omega^2}{c^2}
\varepsilon_{\tau}(\omega,\vec{r}) A_1 + i \frac{1}{c^2}
\frac{\partial \omega^2
\varepsilon_{\tau}(\omega,\vec{r})}{\partial \omega}
\frac{\partial A_1}{\partial t}\right]\nonumber
\\
& &=-\frac{8\pi i \omega e\vartheta_2}{c^2}g_2(r_{\perp}, z, t),
\end{eqnarray}
where
\begin{eqnarray*}
g_2(r_{\perp}, z, t) &= &
\sum_{N}\langle\langle\int^{t+\frac{2\pi}{\omega}}_{t}
\sum_{\alpha_N}u_{\alpha_N}(t)\delta(r_{\perp}-r_{\alpha_N\perp})\delta(z-z_{\alpha_N}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))
\\
&  \times & e^{-i\vec k_{\perp}\vec
r_{\alpha_N\perp}}e^{-i[\phi_2(z_{\alpha}(t',t_{\alpha_N},u_{\alpha_N}^{(0)}))-\omega
t']} \theta(t'-t_{\alpha}) \theta(T_{\alpha-t'})\rangle\rangle d
t'.
\end{eqnarray*}
Here $\phi_1(z_{\alpha}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))=\int_0^{z_{\alpha}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))}k_{1z}(z')dz'$ and
$\phi_2(z_{\alpha}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))=\int_0^{z_{\alpha}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))}k_{2z}(z')dz'$. If the period of the
diffraction grating is constant along the $z$-axis, then
$\phi_1=k_1z_{\alpha}(t', t_{\alpha_N}, u_{\alpha_N}^{(0)})$. The
sign $\langle\langle...\rangle\rangle$ means averaging over the
distribution of beam particles over the transverse coordinate of
the entry points, the entry time, and over the velocity
distribution of the beams entering the resonator;
\[
\vartheta_m=\sqrt{1-\frac{\omega^2}{\beta^2 k_m^2 c^2}},\quad \beta^2=1-\frac{1}{\gamma^2},
\quad \vec k_1=\vec k_{\tau=0}, \quad \vec k_2=\vec k_1+\vec{\tau}.
\]

Let $\rho(\vec r_{\perp})$ denote the particle density in the plane transverse relative to the particle velocity and $\dot{n}$
denote the number of particles traversing the  inner surface of the resonator per unit time.
We also make use of the fact that
\[
\delta(z-z_{\alpha_N}(t',t_{\alpha_N},
u_{\alpha_N}^{(0)}))=\delta(t'-\tau(z,t_{\alpha_N},
u_{\alpha_N}^{(0)}))/\left|\frac{\partial
z_{\alpha_N}(t',t_{\alpha_N}, u_{\alpha_N}^{(0)})}{\partial
t'}\right|,
\]
where
\[
\tau(z,t_{\alpha_N}, u_{\alpha_N}^{(0)})=
t_{\alpha_N}+\int_0^z\frac{d
z'}{u_N(z',t_{\alpha_N}u_{\alpha_N}^{(0)})}.
\]
Here $\tau $ is the time required for the particle entering the
interaction area $z=0$ at time $t_{\alpha}$ at initial  speed
$u_{\alpha_N}^{(0)}$ to reach the point $z$ and
$u_N(z,t_{\alpha_N},u_{\alpha_N}^{(0)})$ is the speed at point $z$
for the particle whose initial speed  at $z=0$ and time
$t_{\alpha_N}$ equaled $u_{\alpha_N}^{(0)}$, while $\frac{\partial
z_{\alpha_N}(t',t_{\alpha_N}, u_{\alpha_N}^{(0)})}{\partial t'}$
is the particle speed at time $t$ if at $z=0$ and $t_{\alpha_N}$
its speed was $u_{\alpha_N}^{(0)}$. Obviously, for the expression
to be finite, this speed should not vanish; otherwise, such
transformation of the $\delta$-function becomes invalid (this is
possible, for example, in the case when the beam's current exceeds
the so-called limiting current and the virtual cathode is formed).

We shall further assume that  the particles are not retarded
significantly  during the interaction and write the quantity in
the denominator of $\frac{\partial z_{\alpha_N}}{\partial
t}=u_{\alpha_N}^{(0)}$. As a result, we have
$\frac{\dot{n}}{u_\alpha}=\rho_z(t_{\alpha})$, where
$\rho_z(t_{\alpha})$ is the  beam's time-dependent density
distribution along the $z$-axis.

Let us also suppose that the duration of the bunches injected into
the resonator is larger than the period $\frac{2\pi}{\omega}$ of
excited oscillations of the electromagnetic wave. Taking into
account the relationship for
$\tau=t_{\alpha_N}+\frac{z}{u_{\alpha}}$, which follows in this
case from the $\delta$-function, the distribution density
$\rho_z(t'_{\alpha})$ in this case  can be removed from the sign
of integration  over the  entry time $t_{\alpha_N}$ at point
$t'-\frac{z}{u_{\alpha}}$, and so we have
\[\rho_z(t_{\alpha})=\rho_z(t'-\frac{z}{u_{\alpha}}).
\]
In a real situation, the distribution of the beam's longitudinal
velocity is much less than the average longitudinal velocity of
the bunch, so we can remove the density $\rho_z$ from the sign of
integration describing averaging over the velocity distribution.
However, analyzing the phase dependence on the velocity
distribution of particles in a bunch, one should bear in mind that
the velocity distribution can appreciably affect the phase.

As a result, we can obtain the following expression for $g_n$:
\begin{eqnarray}
\label{eq_g} g_n(r_{\perp}, z, t) & = & \sum _{N}e^{-i\vec
k_{\perp}\vec r_{\perp}}\rho_N(r_{\perp}, z- u_N t)\nonumber
\\
& \times & u_N
\int_{t-\frac{z}{u_N}}^{t-\frac{z}{u_N}+\frac{2\pi}{\omega}}e^{-i[\phi_n(z(\tau,
t_0, u^{(0)}))-\omega\tau(t_{\alpha},u^{(0)})]}
f(u_N^{(0)})du_N^{(0)}dt_0.
\end{eqnarray}
Here $d t_0$ means integration over the initial times,
$f(u^{(0)}_N)$ is the velocity distribution function in bunch $N$,
and $u_N$ is the average velocity of the $N$-th bunch; the
right-hand part of equation differs from zero at times $t>0$ from
the initial moment defined as the instant of time when  the first
particle of the first bunch enters the resonator.

The derived set of equations enables describing the process of
radiation from several beams in a spatially periodic system
(photonic crystal), including the case when the beams move
opposite to one another. The geometry when the beams move opposite
to one another can be used for beam diagnosing in the bunch-bunch
collision region in colliding-beam storage rings. For short
bunches, this set of equations describes the phenomenon of
super-radiance produced when several bunches of relativistic
particles pass through a VFEL resonator. Particularly, it is
possible to investigate radiation as a function of the difference
between the times of electron bunches entry into the photonic
crystal and as a function of the transverse distance between the
bunches moving in the electromagnetic (photonic) crystal.

According to the equations derived here, the electromagnetic field
induced in the crystal by different beams does not contain random
phases $r_{\alpha_N}$ and $t_{\alpha}$ any longer, and the total
field, as a result, is a coherent sum of the induced fields, which
means that the radiation power increases as ${W\sim E^2=(\sum_N
E_n)^2\simeq N^2 E_1}$. It should be noted that when the
parameters $\chi_{\tau}$ grow, $|\chi_{\tau}|\geq 1$, the
plane-wave expansion of the solution to Maxwell'equations, which
is used in the dynamical diffraction of waves in crystals,
requires that for accurate description of the radiation generation
process a larger number of waves should be considered. However, in
this case one can expand the electromagnetic field into the
analogue of the Wannier functions, which are used for describing
the band structure of electrons in crystals in the case of tight
binding.

Let us consider the following example.  Let a spatially periodic
resonator be formed by  axially corrugated cylindrical waveguides.
we shall choose the direction of the waguides' axis as the
$z$-axis. Coupled through the slots in their walls (long bridge
corrugated waveguides or diffraction grating), the waveguide form
a single spatially periodic electrodynamical system. In the
general case, we have a 2D periodic system in the ($x,y$) plane,
orthogonal to the $z$-axis. The beams move along the $z$-axis.
 Depending on the position of the
cylinders in the transverse plane, square gratings or more
complicated structures  can be formed, e.g the cylinders can be
arranged in a circle (as it occurs in the magnetron).

To describe generation in this system, it is convenient start with
the expansion Maxwell's equations in terms of the eigenfuctions
$\vec Y_n(\vec r_{\perp})$ of this transverse grating (see a
similar approach used for describing the motion of fast electrons
in crystals in channeling regime (mode)) \cite{barbook}.
In this case the eigenfunction $\vec Y_n(\vec r_{\perp})$ is a sum
of the localized Wannier functions $W_n$
\[
\vec Y_{n\vec \kappa}(\vec r)=\sum_m W_n (\vec r_{\perp}-\vec
R_{\perp m})e^{i\vec \kappa\vec R_{\perp m}},
\]
where $\vec \kappa$ is the reduced wave vector, $n$ is the set of
indices defining stationary wave functions , e.g., the wave
function used for the formation of the structures periodic in the
transverse plane, and $\vec R_m$ is the coordinate of the centre
of the elementary cell $m$ of the structure periodic in the
transverse plane.

As a result, for the analysis of generation of radiation we obtain
one dimensional along the $z$-axis equations, where the excitation
current is the total current $I$ produced by the beams moving in
the system. Let us average the current $I$ over the electron entry
times, the distribution of the initial velocities,   and  the
distribution of electrons in the transverse plane (which in this
case have the peaks in the regions where the electrons from each
beam producing the current $I$ move).

As a result we obtain the equations similar in form to those used
for the analysis of the generation process induced by one beam
moving in a waveguide that is spatially periodic along the
$z$-axis axis (formed by , e.g., a corrugated waveguide of a
relativistic BWO.)
Hence, we can conclude that the considered  system,  excited by
several beams, generates common coherent radiation. Now, let us
give a more detailed consideration of the case when the resonator
is formed by the two elements of the grating.

When the resonator period is formed by corrugation of the
waveguide surface, we obtain a system consisting of two corrugated
waveguides coupled through, say,  a slot. For the BWO in the
stationary case when two stationary electron  beams move through
circular waveguides, this system was analyzed neglecting the
influence of  the  wave moving in the same direction \cite{balak}.
Using numerical analysis, the authors of \cite{balak} showed that
at certain parameters, a single-frequency oscillation mode is stet
in the system, i.e., in fact, coherent summation of the amplitudes
of the fields induced by two separate beams is possible. As
follows from the above analysis, such coherent summation is also
possible in the case non-stationary excitation of the system by
two pulses of electron beams.

Note here that the equations derived in this paper enable taking
account of the influence of a coherent wave on the generation
process in a system of several BWOs. Moreover, according to
\cite{bar12}, just in the range of parameters where the amplitudes
of the incident and diffracted waves are comparable, in a
two-three dimensional periodic system the increment of radiative
instability increases sharply and the threshold for the generation
start drops dramatically.

When applied to this case, general equations for describing the
excitation of two relativistic BWOs by two pulses of electron
beams can be written in the form:

Neglecting dispersion in considering the generation process in a
system of two BWOs with a constant grating period (corrugation
period), one can write these equations in the form:
\begin{eqnarray}\label{2bwt1}
\left\{\begin{array}{c}
   2ik_{1 z}\frac{\partial
A^a_1}{\partial z}  +  2 i\frac{\omega}{c^2}\frac{\partial
A^a_1}{\partial
t}+\left[\frac{\omega^2}{c^2}\varepsilon_0-k^2_{mn}-k^2_{1z}\right]
A_1^a\\
+\frac{\omega^2}{c^2}\varepsilon_{\tau}A_2^a+\frac{\omega^2}{c^2}\chi_{ab}^1
A^b_1+\frac{\omega^2}{c^2}\chi^2_{ab}A^b_2=\vec Y_{mn}\vec
j_1,\\
\\
 2ik_{2 z}\frac{\partial A^a_2}{\partial z}  +  2
i\frac{\omega}{c^2}\frac{\partial A^a_2}{\partial
t}+\left[\frac{\omega^2}{c^2}\varepsilon_0-k^2_{mn}-k^2_{2z}\right]
A_2^a \\
 +\frac{\omega^2}{c^2}\varepsilon_{\tau}A_2^a+\frac{\omega^2}{c^2}\chi_{ab}^1
A^b_1+\frac{\omega^2}{c^2}\chi^2_{ab}A^b_2=\vec Y_{mn}\vec
j_{1\tau},
\end{array}\right.
\end{eqnarray}

\begin{eqnarray}\label{2bwt2}
\left\{\begin{array}{c}
   2ik_{1 z}\frac{\partial
A^b_1}{\partial z}  +  2 i\frac{\omega}{c^2}\frac{\partial
A^b_1}{\partial
t}+\left[\frac{\omega^2}{c^2}\varepsilon_0-k^2_{mn}-k^2_{1z}\right]
A_1^b\\
+\frac{\omega^2}{c^2}\varepsilon_{\tau}A_2^b+\frac{\omega^2}{c^2}\chi_{ab}^1
A^b_1+\frac{\omega^2}{c^2}\chi^2_{ab}A^b_2=\vec Y_{mn}\vec
j_2,\\
\\
 2ik_{2 z}\frac{\partial A^b_2}{\partial z}  +  2
i\frac{\omega}{c^2}\frac{\partial A^b_2}{\partial
t}+\left[\frac{\omega^2}{c^2}\varepsilon_0-k^2_{mn}-k^2_{2z}\right]
A_2^b \\
 +\frac{\omega^2}{c^2}\varepsilon_{\tau}A_2^b+\frac{\omega^2}{c^2}\chi_{ab}^1
A^b_1+\frac{\omega^2}{c^2}\chi^2_{ab}A^b_2=\vec Y_{mn}\vec
j_{2\tau},
\end{array}\right.
\end{eqnarray}

We have for the  BWO mode
\[
A_1^{a(b)}(z=0)=0,\qquad A_2^{a(b)}(z=L)=0,
\]
where $L$ is the resonator length.

 For the case when more  than
two waveguides are involved, say, $N$ number - $N$ number of pairs
of equation sets are required, instead of  the two sets given
above, and the terms describing the waves produced by other
waveguides that are similar to $\chi_{ab}^1 A_1^b$ and
$\chi_{ab}^2 A_2^b$ in (\ref{2bwt1}), (\ref{2bwt2})  should be
added to each pair of equations.

It is worth noting that in a real case of arbitrary $\chi_{\tau}$,
the coefficients appearing in these equations should be considered
as phenomenological coefficients determined from the experiment on
the passage of an electromagnetic wave through such structures.

The derived system of equations enables describing the process of
generation excited by the combined pulses of electron bunches in a
periodic system of coupled periodic waveguides (artificial
electromagnetic crystal, VFEL resonator), which forms
self-phase-locking of coherent oscillations. The derived set of
equations enabled studying the dependence of the radiation power
on the difference between the times of the bunches'entry into the
resonator of such a periodic system. This equation set is
applicable to describing radiation produced by bunches with
various duration, and consequently in the case of  short bunches
it allows one to describe the phenomenon of superradiation and
phase-locking  in the system of several relativistic BWOs coupled
into the grating.

\section{Conclusion}

The above analysis shows that when a spatially periodic system of
a VFEL resonator is excited by several ($N$) pulsed electron beams
that enter the resonator at different times, the
two-(three)-dimensional distributed feedback, formed in the
resonator, gives rise to self-phase-locking  of the radiation
process and coherent collective oscillations, which result into a
square in $N$ increase of the radiation power $W$ with growing
number of beams: $W\sim N^2$. Such self-phase locking occurring in
a VFEL resonator makes it possible for us to consider the VFEL as
the self-phase-locking system. Using this equation set, one can
describe generation of generation  by several beams in different
modes: as superradiation from several electron beams and as
radiation from long beams. These equations also make it possible
to study the process of generation of superadiation as a function
of the difference between the times of bunches' entry into the
resonator.

redpajama_set_name: RedPajamaArXiv


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text:
College Colloquium Guides
Posted on November 1, 2019 at 4:29 pm.
Written by John Repplinger
Each fall our librarians put together amazing web pages for each College Colloquium freshman class to customize library resources and services. The topics vary from class to class, and represents the interests of the faculty instructor who teaches the course. We'd like to highlight some of these library guides (click here to view all 27 of our library college colloquium guides).
Harry Potter and the Ethics of Difference
Harry Potter is undoubtedly a cultural phenomenon which all stemmed from that original magical series. Part of what makes this book series so interesting is that it addresses social (un)justice through the eyes of the main character Harry Potter (HP), through his interactions with the cruel Dursley family in the Muggle world, to his dealings with classroom bullying, and a corrupt wizarding government. This Harry Potter-themed course investigates the social hierarchy of the wizarding world and how it overlaps with the real world.
Our librarians had a lot of fun pulling together useful and fun resources to support this course. References to HP are woven throughout this resource, such as a Harry Potter name generator, unofficial HP recipes and cookbooks, popular mobile games (e.g. Wizards Unite and Hogwarts Mysteries), related books (e.g. The Psychology of Harry Potter, and Legilimens–the spell that allows wizards to see into another person's mind), quotes from the characters ("It matters not what someone is born, but what they grow to be," Albus Dumbledore, Goblet of Fire), HP-inspired conceptual art (from Van Gogh styles to stain glass and anime art styles), and ideas for pumpkin carving. Even the tabbed pages are labelled with well-known spells from this classic series!
Asia in Oregon
This colloquium explores the experiences of people of Asian heritage who call Oregon home and examines Asian influences on the land and culture of the state. One of the fun aspects of this class are the field trips to local archives to experience first-hand (primary) sources in Archives and Special Collections.
The library guide for this course highlights a nice selection of Asian-related books (e.g. Encyclopedia of Japanese American History, and Roots and Reflections: South Asians in the Pacific Northwest) and films, including feature films (e.g. Snow Falling on Cedars), Asian-American documentaries (e.g. Between Two Worlds: The Hmong Shaman in America), and Oregon-based documentaries (e.g. 7,500 Miles to Redemption). The guide also features resources from our Archives including Willamette's newspaper The Collegian, The Campus Photograph Collection (2,700+ photos of Willamette's campus), The Willamette Scene (alumni publication), the Wallulah (Willamette yearbook), and the Germaine Louise Fuller Papers which offers the designs for Willamette's Japanese Garden. Willamette also has a long and rich history with the Tokyo International University of America (TIUA), and the library has the archival collections for TIUA (1973-2016) and the Barry Duell TIU collection (1975-2015).
Games: Design, Strategy, Philosophy, & Society
Games come in an almost endless array of strategies, goals, designs, etc. In this colloquium, students create their own games as a final project. During the semester, they learn to play a variety of games, some classic games and many that students probably haven't even heard of that push boundaries of design and play experience.
The guide for this course features a small selection of cool, board game-related books from our collection, such as Rules of Play: Game Design Fundamentals, Games and Decision Making, and The Mathematics of Games and Gambling. Some articles from the Academic Search Premier database have been highlighted (e.g. Manipulation in Board Game Interactions: Being a Sporting Player; and Using the Board Game Borel to illustrate probability calculations). The "Trivial Pursuit Fun Stuff" page highlights a website that lists Board Games by Decade as well as Collectible Games between 1900-1950.
Hidden Bodies of Art
Art of the human body goes back several millennia, even prehistorically such as the caves of Lascaux, France. Each culture expresses their values, beliefs and conventions through their artistic renditions of the human body, such as the Renaissance depictions of the highly idealized body or sometimes the complete absence of the body as is often the case in abstract art.
Since the human body is a classic area of study, there are many outstanding books and encyclopedias on or related to this topic, some of which are included in the library guide for this college colloquium and include The Grove Encyclopedia of Classical Art and Architecture, Michelangelo and the Reinvention of the Human Body, and Despotic Bodies and Transgressive Bodies. There is a useful list of Library of Congress general call numbers (classifications) which allows users to browse our book collection. This guide also lists films from our collection: Feature films (e.g. Warm Bodies), Documentaries (e.g. Cave of Forgotten Dreams), Related Arts films (e.g. Against the Odds: the Artists of the Harlem Renaissance), and Foreign Language films (e.g. The Skin I Live In). All of our colloquium guides include a list of helpful databases to use when looking for articles, a page for interlibrary loan explaining how to request materials not available at Willamette, and a citation style guide appropriate to the class.
Knitting Culture
There is a lot that goes into knitting, from the procurement of materials to the end product. Students really learn how to knit in this class as well as examine the impact and value of knitting through a multidisciplinary approach of history, fashion, politics, science, psychology, and philosophy.
This library guide has cleverly used a knitting theme; its tabbed pages use knitting terminology (e.g. "purlwise" for books, "whipstick" for articles, and "frogging" for fun stuff). Also, instead of the standard Facebook logo, a knitted blue letter "F" is used, and a knitted stocking cap with an American flag is included with the course description. The page for books offers some wonderful titles from our collection such as Knitting School: the Complete Guide to Becoming a Confident Knitter, Crochet One-Skein Wonders: 101 Projects from Crocheters Around the World, and The Close Knit Circle: American Knitters Today. Some of the fun things listed on the Frogging page are YouTube videos of how to turn wool into yarn by hand, and how to change yarn colors. Local yarn and fabric shops are lists (Teaselwick Wools), as well as popular thread-themed games (Unravel and Run Sackboy! Run!), and the highly rated Ravelry.com website for knitters and crocheters which provides free patterns and tips.