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Home / Publications / Copyright and the Digital Economy (DP 79) / 16. Broadcasting / The definition of 'broadcast'
Copyright and the Digital Economy (DP 79)
Proposals and Questions
Matters outside the Terms of Reference
Principle 2: Maintaining incentives for creation of works and other subject matter
Principle 3: Promoting fair access to and wide dissemination of content
Principle 4: Providing rules that are flexible and adaptive to new technologies
Principle 5: Providing rules consistent with Australia's international obligations
3. Policy Context of the Inquiry
Trends in consumer use of copyright material
The implications of cultural policy for copyright reform
4. The Case for Fair Use in Australia
The changed environment
Arguments in favour of fair use for Australia
Arguments against fair use in Australia
ALRC's proposals for reform
The proposed fair use exception
Should third parties benefit?
Inhibiting innovation
Depends on the market
Fair dealing and fair use
Fair remuneration for rights holders
Repeal of statutory licences
Licensing uses covered by exceptions
When licences cannot be obtained voluntarily
7. Fair Dealing
The operation of fair dealing exceptions in the digital environment
Reducing complexity
Reform of fair dealing exceptions
8. Non-consumptive Use
Defining 'non-consumptive' use
Caching and indexing and network-related functions
Text and data mining
Non-consumptive uses and fair use
9. Private and Domestic Use
An alternative—a new confined exception
Back-up and data recovery
10. Transformative Use and Quotation
Defining 'transformative' use
Transformative use and fair use
Australian law
A stand-alone transformative use exception
11. Libraries, Archives and Digitisation
Is s 200AB working adequately in the digital environment?
Mass digitisation
Certain exceptions should be retained
Scope of the orphan works problem
Licensing orphan works
13. Educational Use
An illustrative purpose
Fair dealing for education
Fair dealing exceptions
Uses required by freedom of information laws
Use under an implied licence
Fair dealing for public administration
Use for judicial proceedings and for members of Parliament
Assumptions and options for reform
Option 1: Repeal of the retransmission scheme
Option 2: Retention of the retransmission scheme
Overseas models
The scope of amended exceptions
The remuneration caps
Voluntary licensing of sound recordings
Is there a need for reform?
Approaches to reform
Last modified on 4 June, 2013.
16.7 The Copyright Act defines the term 'broadcast' to mean 'a communication to the public delivered by a broadcasting service within the meaning of the Broadcasting Services Act'.[2]
16.8 The Broadcasting Services Act defines a 'broadcasting service' to mean 'a service that delivers television programs or radio programs to persons having equipment appropriate for receiving that service, whether the delivery uses the radiofrequency spectrum, cable, optical fibre, satellite or any other means or a combination of those means'. A broadcasting service does not include:
(a) a service (including a teletext service) that provides no more than data, or no more than text (with or without associated still images); or
(b) a service that makes programs available on demand on a point-to-point basis, including a dial-up service; or
(c) a service, or a class of services, that the Minister determines, by notice in the Gazette, not to fall within this definition.[3]
16.9 A ministerial determination, made in 2000 under the Broadcasting Services Act, excludes a 'service that makes available television and radio programs using the internet' from the definition of a broadcasting service.[4]
16.10 The primary reasons for the ministerial determination were to ensure that developing internet audio and video streaming services were not regulated as broadcasting services under the Broadcasting Services Act and to clarify the regulatory position of datacasting over broadcasting services bands.[5]
16.11 However, it also has a significant effect on the scope of the broadcast exceptions under the Copyright Act, as discussed below. Among other things, it means that while free-to-air and subscription cable and satellite television transmissions are covered, transmissions of television programs using the internet are not.[6]
[2]Copyright Act 1968 (Cth) s 10.
[3]Broadcasting Services Act 1992 (Cth) s 6.
[4]Commonwealth of Australia Gazette—Determination under Paragraph (c) of the Definition of 'Broadcasting Service', (No 1 of 2000), Commonwealth of Australia Gazette No GN 38, 27 September 2000.
[5] See Phonographic Performance Company of Australia Limited v Commercial Radio Australia Limited [2013] FCAFC 11, [52]–[46].
[6] While some forms of internet protocol television (IPTV) and internet radio are treated as broadcasting services under the Broadcasting Services Act, others are not—for example, where television-like content is delivered over an unmanaged network, such as broadband internet ('over the top'). This is discussed in more detail in Ch 15.


FORSUN ULTRA-HARD MATERIAL INDUSTRY CO., LTD., core bit, diamond drill bit, drilling core bit, on EUROPAGES.
EUROPAGES > > FORSUN ULTRA-HARD MATERIAL INDUSTRY CO., LTD.
Forsun Ultra-hard Material Industry Co., Ltd. (Forsun) is a professional manufacturer of drilling tools for geological and engineering prospecting, water well drilling, etc. Since the establishment in 2001, Forsun has committed in developing various products to meet the continuously-changing requirements of the drilling industry. So far, our products are sold all over the world. Forsun possesses excellent and professional employees, and advanced production technologies and equipment. Apart from standard products, Forsun can also design and manufacture required products as per customers' drawings or requirements. In November 2008, Forsun became officially the base for industry-university-research cooperation of Central South University. Welcome to our website to learn more about us and our products.


LIve streaming video of the Dubai World Cup Carnival.
Evasive (GB) – Stallion Elusive Quality – Canda / Storm Cat Sire of black type winners France, including G3 winner, placed in French Derby (G1).
Haydock Park Serious Sport…Serious Fun…Whether you are looking for an exciting racing atmosphere that cannot be matched or a unique leading conference centre in the heart of the North West of England, Haydock Park is the perfect choice for you.
Stutteri Hjortebo Stutteri Hjortebo is the leading stud in Scandinavia mangedobbelt champion i Danmark.
Turf & Dirt – On Horse Racing Horse Racing Opinions and Photos presented by EquiSport Photos. Kentucky Derby, Breeder's Cup, Photography, Other racing. Anything goes! Don't be left out of the discussion!
OH Racing Community an Online Horse Racing community, specialising in speed figures on the flat turf. Always up for talking horses though, no matter what code!
All Breed Pedigree Slå afstamingen op på heste af alle racer, og tilføj hvis du har info.
FEGENTRI – – International Federation of Gentlemen & Lady Riders It was founded in 1955 by a handful of enthusiastic amateur riders. The founding members were France, Germany, Italy, Sweden and Switzerland. Many nations have joined FEGENTRI since 1955 and today the membership consists of 25 different countries.
Nationaldagsgaloppen Den 6. juni 2013 er der galop midt i Stockholm, og denne side er til ære for dette event.
Thoroughbred Herritage Historic Sires/Dams, Portraits, Turf Hallsmarks, Breeders, Genetics etc.
Radioudsendelse og artikel – Kom glad i mål Om Galopsporten og om at hesten skal være glad – spændende!
Fotograf Christiane Slawik Hestefotos, kalendre, bøger m.m.
Bedford House Stables – Newmarket LUCA Cumani is one of British racing's leading trainers, having enjoyed top-class success all over the world. Cumani has successfully overseen the careers of many great racehorses, achieving multiple Classic triumphs at home and abroad.
Galoptræner Wido Neuroth Træner Wido & Jan-Erik Neuroth m.fl.
Wido Neuroth A Leading Stable – every single year!
Rennstall Wôhler Das Gestüt Ravensberg in Spexard bei Gütersloh ist seit Herbst 2004 die Heimat von Galopptrainer Andreas Wöhler und den rund 100 ihm anvertrauten Pferden.
Mark Usher Racing Racehorse trainer based in Upper Lambourn, who believes ownership doesn't have to be expensive!


Martin Louis Amis is an English novelist, the son of Kingsley Amis.
His best-known novels are Money (1984) and London Fields (1989).
He has been listed for the Booker Prize twice, once for Time's Arrow, a SF novel.
Amis served as the Professor of Creative Writing at the Centre for New Writing at the University of Manchester until 2011.


This is a bit overdue, I'm afraid. The first few weeks of the semester are busy enough that I haven't been able to keep up with Plover emails. But I'll get to them soon, I promise!
So Josh, our dauntless programmer, has released a new version of Plover.
* Fix for a bug that caused crash when adding, for example, an -ing suffix to words that end in a consonant followed by y, such as early -> earlier.
* Fix for a bug that caused crash when starting Plover configured to use a Gemini PR with a non-existent serial port.
* Update of the default dictionary to the latest from Mirabai.
* Refactor of the underlying config file logic such that addition of configuration options in future releases is invisible to the end user and less error prone.
I also was pleased to see in a recent post on Metafilter, someone asked a question about steno, and someone else -- who was not me -- answered it with a link to What Is Steno Good For. So that's exciting!
In other news, I discovered a few days ago that I can make Plover sort-of-kind-of work with my laptop's built-in keyboard, if I sort of roll or arpeggiate the keys. For instance, if I want to write my brief for "supercalifragilisticexpialidocious" (which, along with "antidisestablishmentarianism", is the first word people tend to say when they notice they're being transcribed in realtime), STPRAPBLG, I couldn't just press down all the keys at once the way I can with my SideWinder or Revolution Grand; my laptop's keyboard doesn't have n-key rollover, and recognizes only two or three keys at a time, maximum. But if I start out by pressing the S key (the A key on the qwerty keyboard), then the T and P, then while still holding onto the T and P, let go of the S, roll onto the A, keep the A pressed down and lift up the TP while going on to the PB, et cetera. When I get to the end, I release the last key, and the word pops up like magic. Plover is actually able to recognize all those keystrokes, as long as they're only pressed one or two keys at a time. And since it doesn't mark a stroke as complete until all keys have been lifted, you can essentially play this arpeggio of keys and come up with a correct steno translation for any word in your steno dictionary, even on a keyboard without proper n-key rollover. Now, it's nowhere near as quick as actual strokewise steno, and it's a bit tricky to do properly, but at the very least it can be useful for testing Plover when there isn't a SideWinder handy.
Very cool! I'm excited at the prospect of trying this out on my laptop keyboard. I was also thinking that Plover would probably be the best way to utilize my virtual piano via steno technique because, as you said, there's no delay between keystrokes and output. Can't wait to give it a whirl!
Hey dear Knight! I've just found Your blog and I'm confused, how could I miss it before?
I work on Polish site about shorthand and stenography and collect as much as possible informations. One of my interests is stenotype writing - as in Poland it's still science-fiction. Your invention is just great, I begin to dig thru Your work to know more about it.
Is there any way to get Plover working on Windows short of running it in a Linux virtual machine? Never really delved into the penguin too much myself, but I'm aware of the rabbit-hole effect.
Keep up the good work, man. I just tonight realized that the version of CaseCat which Stenograph hustledImeansoldme can only edit files, rather than transcribe. So it's spend several hundred dollars to upgrade or...cry about it on the internet.
And dear lord: going through their website and being bombarded with all the propaganda about why only being able to run CaseCat on 3 computers is more secure and convenient for you, the matrixpodpersonImeancustomer...just...ugh.
You have really nice Blog here and I am saving it in my favorite list. Keep posting similar stuff.
You've got to put the Stentura into TX Bolt mode.
You press the second and third buttons simultaneously (from the left, the leftmost being the power button) on the Protege. That should make the machine beep and a serial port icon appear on the little screen to indicate the mode change. Then configure Plover to be in "TX Bolt mode", connect your Protege, restart Plover, and press the red P so that it turns green. With luck, you should now be able to write using your Protege. Tell me how it goes, and if you run into any trouble, I'll try to figure out what's up. Keep in touch! I'm so happy that more steno people are starting to experiment with Plover.
Another tip for "arpeggiating" is to just hold down the leftmost key and type out all the remaining keys - that way you don't have to worry about accidentally letting go of all the keys before finishing.
On another note, I am an experienced developer and would like to add some velocity to the project, if there are things to be done.
We're always looking for developers on there! Thanks!


A guide to the ferment of ideas in the modern world.
By Modern Masters we mean the men who have changed and are changing the life and thought of our age. Everybody wants to know who they are and what they say, but hitherto it has often been very difficult to find out. This series makes it easy. Each volume is clear, concise and authoritative. Nothing else can offer in such an acceptable form, an assured grasp of these revolutionary thinkers. The authors are themselves masters, and they have written their books in the belief that general discussion of their subjects will henceforth be more informed and more exciting than ever before.
If this seemed somewhat over the top then the Times Literary Supplement offered a more balanced view, describing the series as 'just what is needed by the so-called "general reader" in search of a guide to intellectual currents'. This is still true of the books today, though now these monographs on the masters of modern literature, philosophy and science are celebrated not for their contents but their covers, which featured brightly-coloured abstract art and a rounded sans-serif typeface.
of covers for the first ten Fontana Modern Masters in 1970-71.
The covers were pocket-sized works of art, and it is as art that the story of the Fontana Modern Masters is presented here, for they did not just catch the eye, they hijacked it. This, then, is a cover story about books, art, and books as art, and the unique place that the Fontana Modern Masters occupy in publishing history. It is the story of a concept, from the 20th-century abstract art that inspired these covers to the 21st-century conceptual art that the covers have inspired in their turn.


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Home Africa Development Bank of Nigeria –Everything You Need to Know about DBN
Development Bank of Nigeria –Everything You Need to Know about DBN
The idea of the Development Bank of Nigeria (DBN) was formed in 2014 and in 2017, President Buhari's government translated the idea into reality with the eventual establishment of the bank.
While the DBN is said to operate without interfering with Nigeria's existing development banks, the bank will instead assist with the provision of loans to sectors which aren't getting loans from the existing development banks such as Bank of Agriculture (BOA) and Bank of Industry (BOI).
This article contains essential information about DBN including the bank's audience, management team, vision statement and mode of recruitment.
Development Bank of Nigeria Recruitment
Application for entry into the Development Bank of Nigeria (DBN) is open to Nigerian graduates. While interested applicants are usually required to wait for the recruitment application form to be released online, DBN is said to have recorded large-scale yearly recruitment since its establishment. There can be specific application requirements but here are the general prerequisites the bank considers for recruitment:
The maximum age for application is 35 years –application isn't available to older people
Applicants must be able to operate computer
Applicants must hold authentic academic qualifications –being any of NCE, HND, OND, and B. Sc in any course of study –awarded by an authorized institution. The breakdown of the academic qualification for application is O' Level –any of General Certificate Examination (GCE) and National Examination Council (NECO) –with the minimum of 5 passes (including in English language) at no more than two sittings OR a minimum of 4 passes at one sitting. Alternatively, Senior Secondary School Certificate (SSCE) or West African School Certificate (WASCE) with 3 credits (including in English language) and the minimum of passes in 2 other subjects.
Alternatively again, applicants can submit (i) National Diploma (ND) awarded by an authorized tertiary institution, (ii) (Advanced Level) General Certificate of Education in 3 subjects based on two sittings or 2 subjects based on one sitting, (iii) National Certificate of Education (NCE) awarded by an authorized tertiary institution
Also Check e-Naira? EVERYTHING YOU NEED TO KNOW ABOUT ENAIRA
Development Bank of Nigeria's Management Team
In March 2017, the Nigerian government, through the Federal Ministry of Finance, officially declared DBN's management team, disclosing details of the bank's board members. Below are the names of DBN's board members and their official roles:
Tony Okpanachi –DBN's management leader and former Deputy Management Director of Ecobank Nigeria Limited
Shehu Yahaya –DBN's chairman and erstwhile Executive Director of AfDB
Uche Orji –Managing Director/Chief Executive of Nigeria Sovereign Investment Authority
Mohammed Kalif –Member of the African Development Bank
Also, the names of DBN's independent directors were declared as:
Oladimeji Alo –Managing Director, CEO Excel Professional Service Limited
Clare Omatseye –Founder/Managing Director of JNC International Limited
Bello Maccido –Chairman of FBN Merchant Bank
Andrew Alli –CEO and President of African Finance Corporation
Philips Oduoza –United Bank for Africa's (UBA) former Group Managing Director/CEO
Targets of Development Bank of Nigeria
The Development Bank of Nigeria (DBN) was initiated as a solution to the failure of Micro, Small and Medium Enterprises (MSMEs) to secure adequate access to loans from financial institutions. While proprietors of small and medium-scale businesses are considered the targets of the Development Bank of Nigeria, it is notable that DBN partners with qualified financial mediators often known as "participating financial institutions". These institutions serve as the mediums for MSMEs to secure better access to finances.
Vision of Development Bank of Nigeria
According to its Managing Director, Tony Okpanachi, the Development Bank of Nigeria maintains the vision of becoming the primary development finance institution in Nigeria, hence providing advisory services, Credit Guarantees and financial services aimed at promoting the growth and sustainability of the MSMEs.
Also Check "Only A Fool Wouldn't Look at Bitcoin" – Elon Musk
We hope the above information about the Development Bank of Nigeria is satisfactory. If otherwise, you may visit the bank's official website to learn more.
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Development Bank of Nigeria
Development Bank of Nigeria –Everything You Need to Know
Small and Medium Enterprises (MSMEs)
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redpajama_set_name: RedPajamaCommonCrawl


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Arguably the father of public housing in Hong Kong, Michael Wright was born and grew up in this city with many Chinese friends. He was interned in prison camp during the second world war. His background and personal experience were instrumental to his perseverance of building self-contained flats for the poor.
"It is all because of my background as an architect," was Michael Wright's answer to my question as to why he insisted on building self-contained flats for the grassroots people of Hong Kong. But for a journalist, his answer falls short of addressing the question. It took three years to figure out why Michael persevered in his quest for self-contained flats more than six decades ago.
Nowadays, self-contained flats are a universal feature of Hong Kong's public housing estates, as ordinary as the air that we breathe. However, it was not part of the plan when the administration committed to building resettlement estates to house the victims of the Shekkipmei Fire, and the poor living in squatters. But there are no "ifs" in history. There is no way to know what would have happened had Michael not intervened.
Shekkipmei fire on Christmas Eve, 1953
Michael Wright was Hong Kong's Director of Public Works between 1963 and 1969. He was born in the Victoria Hospital in the Peak in 1912, and joined the Public Works Department (PWD) in 1938. As a humanitarian, he believes it is the government's responsibility to provide decent accommodation for the poor.
The first generation resettlement blocks were built to house victims of the fatal fire in Shekkipmei on the Christmas Eve of 1953. But these were very crude abodes enclosed by four walls. The tenants had to share toilets and showers. Kitchens were makeshift structures along the public corridor. At that time, the PWD was the government's architectural office, and Michael was already the department's chief architect. He was charged with designing and building the resettlement blocks with the assigned budget. However, as a firm believer of decent accommodation for all, he was upset by the crude resettlement housings. He persistently pressed for self-contained flats for the poor.
Seven-storey resettlement blocks to house victims of the Shekkipmei fire
His firm commitment to building decent housing for the poor deeply impressed Douglas Clague, then chairman of the Housing Society. Both men fought in the Battle of Hong Kong and were heavily involved in the city's post-war reconstruction efforts. Clague invited Michael to sit in the society's building committee. The society adopted Michael's vision before the Government, and gave it the name of "the Wright Principle". In the obituary pieces, the press crowned him Father of Public Housing for his contribution in providing decent accommodation to the poor in the face of shoestring budget. But from where did this idea of decent accommodation originate? If Michael's architectural upbringing alone explains his commitment, there should have been a lot of Michael Wrights.
The answer slips through because both interviewee and the interviewer took public housing for granted. Brought up in a post-war value system that believes in shelter for all, the baby boomers, their children and grandchildren unconsciously take the concept of public housing for granted. It is only by re-tracing how radical the idea of public housing was could we begin to make sense of Michael's answer.
State provision of housing was such a radical idea that it took the progressives in Britain nearly a century to argue, experiment and implement. This eventually led to the birth of council housing. The long road to council housing included such milestones as the model dwelling movement, Octavia Hill management, and the Garden Cities in Letchworth and Welwyn. Industrialisation and urbanisation in Britain gave birth to new categories of workers in the manufacturing, transport and other industries. These workers lived in fast-growing industrial urban centres. Unfortunately, most of them were not able to afford decent accommodation in the urban area and ended up living in slums. The labourers and their slums were completely alien to the established system. The established value, belief and institution found that it had neither the tool nor language to deal with the alarming concentration of uncouth strangers in the cities.
In the early 19th-century, cities had no means to cope with the working mass. The existing system was obsolete. Individuals driven by religious love as well as secular empathy ventured in philanthropy. But, would philanthropy undermine the principle of self-help? Where should philanthropy end if assistance were to be made available? If the sick, aged and orphaned deserved kindness because they were victims of circumstance, what about the plights of the sinners? Should a line be drawn between the "deserved" and the "undeserved" when bestowing help? New arguments were put forth to challenge the convention. Poverty was not caused by moral decadence of the "undeserved". It was the result of a faulty system - low wages, lack of labour protection, economic fluctuation and the hopelessly unhealthy environment. It took the First World War to win the debate. The power-holders recognised that the soldiers turned war heroes should be treated with respect or they pose a threat to social stability. This provided the basis for decent housing for the poor.
When the colonial administration in Hong Kong under Alexander Grantham committed to building accommodations to the homeless after the fire in Shekkipmei, Britain's council housing had been in place for more than three decades. The socio-political factors that lead to Grantham's commitment to public housing, however, were quite different from the British context. In both cases, however, the government's commitment to public housing was only partly motivated by benevolence.
Built in 1975 and 1976, Lai Tak Tsuen was named after Michael Wright's Chinese name
Post-war Hong Kong was a total mess. Most of the buildings were destroyed, either by bombs, burglars, or people desperately in need of fuel for cooking and keeping warm. Producing electricity was the priority of the city's reconstruction. The city was so devastated that it needed to ferry food from the neighbouring countries and fly in cash from London. When the war ended, those who fled returned. Housing was in grave demand. The housing crisis deepened as China's civil war gradually tipped the balance in favour of the Communist Party. Refugees streamed across the border for political shelter.
Michael returned to the civil service after the war, where he continued his work in the PWD. One of his earliest projects was to design accommodation for the senior civil servants because the city was inhospitable for European women. The administration's logic was that the senior civil servants should concentrate fully on their work. This meant that they and their family must live in a secure and comfortable environment. However, even among the British elite, the decision to prioritise housing for senior civil servants over the rest of the community was met with severe criticism. It was argued that the government should first provide housing for the poor. One of the staunchest critics was Bishop Hall, head of the city's Anglican Church. When the government ignored his call, the bishop used donation from London and gathered a group of reform-minded community leaders to set up the Housing Society. The society's founding constitution was to provide housing for the labourer.
The fire in Shekkipmei was merely a trigger that sets the public housing programme into action. Behind the benevolence was the realisation that the refugee fleeing communist China would be here for good. The social cost of crude housing was much lower than the cost of leaving the people in squatters, not least because the land that was illegally occupied could be released for more productive uses. Within the remit set by the administration, the PWD architects, including Michael, were guided by their professionalism and designed homes for the poor. They equipped resettlement estates with the necessary facilities, including playground, school, market, shops and clinics.
Four of the eight blocks in Lai Tak Tsuen were built in unconventional cylindrical shape
But why did he insist on building self-contained flats for the working class? At the time Michael should receive formal education, Hongkong didn't have schools that could cater to children of British descent. The Wrights sent their children back to England for school. In 1921, at the age of eight, Michael left for school in England. The European War ended just three years ago. In 1918, the Tudor Walters Report recommended self-contained, two-storey family cottages with front and back gardens, front doors, water supply cooking and sanitary arrangements as the standard for council houses. Bedrooms were required for privacy, and three was the norm. The estate should include green spaces for children to play in. Distantly related to the topic but a landmark breakthrough in civil liberties, women in Britain aged 30 or over and with property won their right to vote in 1918. A decade later, women's voting age was lowered to 21. In the 1920s, progressive architects and planners were concerned with the design of council flats and management of the estates. Michael was brought up in an era that has already accepted it is the state's duty to provide decent accommodation to the poor.
In 1930, Michael became an article pupil of architecture. He also enrolled in the School of Building in the Regent Street Polytechnics (now University of Westminster). In the evening school, he studied the history of architecture, architecture design, building construction, drainage and sanitation, and everything an architect and surveyor needed for their job. By then, council housing had become integral to the British society. Michael matured alongside a new generation of architects who embraced modernism and humanism.
To an extent, Hong Kong resembled London. In London, industrialisation resulted in labours being congested in urban slums. In Hong Kong, colonisation brought fast track urbanisation. Since the colony's inception, labours have been pouring in from the surrounding rural communities. They had neither land nor property in their newfound home. During the155 years of British colonial rule, Hong Kong's population was highly correlated to political situation in mainland China. The British colony was a haven for those fleeing political instability across the border. Urbanisation, industrialisation and political stability attracted waves and waves of labourer and refugees. The Chinese population were congested in poor and overcrowded tenements. Thus, slums were common in both London and Hong Kong back when Michael was studying architecture.
It is easy to blame the colonisers' exploitation for the ills of Hong Kong society. One example is the notorious Peak Ordinance, under which the Chinese could only reside in the Peak with the Governor's permission. But inquisitive minds avoid generalisation and stereotypes. Social progression is built on compassion as well as critical analysis of the past and present. It is well-documented how early Chinese merchants seized racial prejudice to rebuff health regulators who wanted to mandate the building of alleyways between residential blocks. In 1879, a group of Chinese merchants petition against sanitary reforms under which they had to build alleys between buildings for air, sunlight and fire safety. Without alleys every inch of land could be sold. These merchants went as far as claiming that the Chinese were different from Westerners in their living standards and health practices. They argued that the Chinese community would not appreciate the sanitary provisions designed for a superior class of residence.
Michael Wright's father was also an architect who retired as the PWD's assistant director. In our interview, Michael shared with me one of the conversations he had with his father, on a subject both knew by heart. The conversation reveals the gap between these two generations of architects. The conversation took place in December 1945, where Michael was in his parent's home in Essex for his long overdue leave. The son asked his father why there were no amendments to the building ordinance to extend the frontage of the tenements.
Michael: The requirement for tenement frontages was always 14-foot. Why didn't you extend it to 20-foot?
Arthur: What's wrong with 14-foot? There was nothing wrong. It was rather good accommodation."
Michael: No, people were dying of tuberculosis. They were living in unhealthy, appalling living conditions."
Arthur: So what?
Michael was piqued by his father's attitude and never talked to him about Hong Kong again. Nevertheless, Michael went at lengths to explain that his father was not a bad person. It was merely a clash between the old world and the new.
Timeline of relevant events
1903 – Arthur Edgar Wright, Michael's father, came to Hong Kong to join the Public Works Department (PWD)
1912 – Michael was born in the Victoria Hospital in the Peak
1918 – The release of the Report of the Committee appointed by the President of the Local Government Board and the Secretary for Scotland to consider questions of building construction in connection with the provision of dwellings for the working classes in England and Wales, and Scotland, more commonly known as the Tudor Walters Report.
1919 – Mass provision of council housing in Britain began
1921 – The Wrights sent Michael and his elder brother Denis back to England for education
1929 – Le Corbusier's The City of Tomorrow and Its Planning published.
1929 – Ill health forced Arthur to early retirement. He returned to England in the same year, aged 49.
1930 – Michael became an article pupil of architecture and enrolled in evening school at the Regent Street Polytechnics.
1933 – Michael started working in an architecture studio in Regent Street. He moved to a bigger firm after being chartered.
1938 – Michael was offered the firm's partnership. Instead of accepting the partnership, he took up the PWD's offer, returning to Hong Kong the same year to join the building ordinance office.
Originally published at www.urbandiarist.com. Dr Chloe Lai is curator and storyteller of Urban Diary. She is also producer of the "Wright Chronicle" documentary.
#Urban Diary #community #city #Hong Kong #public housing


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\section{Introduction}

For a couple of years superstring theories, and particularly the heterotic
string theory \cite {1',2'}, have provided an efficient way to construct the
Grand Unified
Superstring Theories ($GUST$) of all known interactions, despite the
fact that it is
still difficult to construct unique and fully realistic low energy models
resulting after decoupling of massive string modes.

In the fermionic formulation of the four-dimensional heterotic string theory
\cite {3',4'} in
addition to the two transverse bosonic coordinates $X_{\mu}$ ,$\bar{X}_{\mu}$
and their left-moving superpartners ${\psi}_{\mu}$, the internal sector
${\cal M}_{c_L;c_R}$ contains 44 right-moving ($c_R=22$) and 18 left-moving
($c_L=9$) real fermions (each real world-sheet fermion has
$c_f=1/2$).
String theories possess infinite dimensional symmetries that place many
specific constraints on the theory spectrum. These symmetries origin from
2 dimensional conformal invariance, modular invariance, and Virasoro and
Kac-Moody algebras.
Because of the presence of the affine Kac-Moody algebra (KMA)
$\hat g$ (which is a 2-dimensional manifestation of gauge symmetries of the
string itself) on the world sheet, string constructions yield definite
predictions concerning representation of the symmetry group,
especially for the rank 8 and greater,  that can be used
for low energy models building.

There are not so many GUSTs describing the observable sector of Standard
Model. They are well known: the $SU(3)\times SU(2)\times U(1)^n\times G_{hid}$
gauge group, the Pati-Salam ($SU(4)\times SU(2)\times SU(2)\times U(1)
\times G_{hid}$) gauge group,
the flipped $SU(5) \times U(1)\times G_{hid}$ gauge group
and $SO(10)\times G_{hid}$ gauge group \cite{20',20''}.
For the heterotic 10-dimensional  string the groups $E_8\otimes E_8$ and
$spin(32)/Z_2$ are characteristic. Hence it is interesting to consider GUSTs
in four dimension based on its various rank 8 and 16 subgroup \cite{25'}.
As the GUSTs originating from level one Kac-Moody algebra
(KMA) contain only low-dimensional representations, new types of GUSTs with
the $G \times G$ gauge groups can naturally
appear in consideration \cite {25',26'}.
Moreover for the observable gauge symmetry one can consider the diagonal
subgroups ${G'}^{sym}$ of the rank 16 group $G \times G \subset SO(16)
\times SO(16)$ or $\subset E(8) \times E(8)$.
Early \cite {25'} we considered the possible ways of breaking the "string"
gauge subgroups
$\subset E_8\otimes E_8$ down to low energy supersymmetric model that includes
Standard Model group and horizontal factor $SU(3)$.
There are good physical reasons for including the horizontal $SU(3)_H$ group
into the unification scheme. Firstly, this group naturally accommodates three
fermion families presently observed (explaining their origin) and, secondly,
can help to solve the flavour problem in SUSY GUTs and
can provide correct and economical description of the fermion mass spectrum
and mixing  without invoking high dimensional representation of conventional
$SU(5)$, $SO(10)$ or $E(6)$ gauge groups\cite{9'}.
Construction of a string model
(GUST) containing the horizontal gauge symmetry provides additional strong
motivation to this idea. Moreover, the fact that in GUSTs high dimensional
representations are forbidden by the KMA is a very welcome feature in
this context.
The constraints of horizontal model parameters followed from this approach
allow the existence of the interesting flavour-changing physics in the
TeV region. Also these models gives rise to a rather natural way of
the superweak-like CP-violation\cite {9'}.
All this leads us naturally to considering possible forms for horizontal
symmetry
$G_H$, and $G_H$ quantum number assignments for quarks (anti-quarks) and
leptons (anti-leptons) which can be realized within GUST's framework.

Here we present shortly the string models including Grand Unification group
$[SU(5)\times U(1) \times G_H]^{\otimes 2}$,
along with horizontal gauge symmetry $G_H=U(3)$.
Using the (2,0) world-sheet
 superconformal symmetry  we study the superpotential.
The form of obtained superpotential implies that 2
generations remain massless comparing with the $M_W$ scale.
Using the condition of  $SU(3)_H$ anomaly cancellation  the theory
predicts the existence of the Standard Model singlet
"sterile" particles that participate only in horizontal $SU(3)_H$
and/or $U(1)_H$
interactions.  As following from the form of the superpotential some of
them   could be light (much less than $M_W$) that will be very interesting in
sense of experimental accelerator and astrophysical searches.
In this model after anomalous $U(1)$ D-term  suppressing  the only surviving
horizontal gauge group is $SU(3)_H$.

We outline the perspective way of  including the symmetric subgroup on the
intermediate stage that does not involve higher level of Kac-Moody
algebra representations.
Starting from the rank 16 grand unified gauge group
of the form $G\times G$
\cite{25',26'} and making use of the KMA which select
the possible gauge group
representations we
discuss some  ways of breaking of string rank 16 gauge
group $[SU(5)\times U(1) \times G_H]^{\otimes 2}$
down to the  symmetric diagonal subgroups \cite {25'},\cite {Barbieri'}.
This model allows two ways of embedding
chiral matter (16 quarks, leptons and right neutrino) in  $\underline {1}$,
 $\underline {\bar 5}$
and $\underline {10}$ representation of $SU(5) \times U(1)$, which
 correspond to the flipped and non-flipped $SU(5) \times U(1)$
models respectively \cite {Barr'}.

 The main goal of our paper is to solve the problem of discrepancy between
the unification scales of
$SU(3^c)$, $SU(2)_{EW}$, $U(1)_Y$-gauge coupling constants ,
$M_G \sim 10^{16} GeV$,
and string scale in GUSTs, $M_{GUST}= M_{SU}=g_{string}\cdot 5\cdot 10^{17}
GeV$.

We consider two possibilities of the breaking of the primordial
$[U(5) \times G_H]^{\otimes 2}$ gauge symmetry with two variants of
$Q_{em}$ charge  quantization correspondingly.
For the various chains of gauge symmetry breaking in flipped and non-flipped
 cases of the $SU(5)$ model we carry out the RGE analysis of the behaviour of
the gauge coupling constants taking into account the  possible intermediate
thresholds (the additional Higgs doublets, color triplets,
SUSY threshold, massive fourth generation)
 and the threshold effects due to the  massive string states.
We show that only in
non-flipped case of $[U(5) \times G_H]^{\otimes 2}$ GUST it is possible
to make the unification scale of $g_{1,2,3}$-
coupling constants in supersymmetric standard model, $M_G$, to be
consistent  with the string
scale unification, $M_{GUST} = M_{SU}$ and
obtain estimation of the string coupling constant $g_{str}=O(1)$.
As an additional benefit,
the values of the $g_{str}$ and $M_{SU}$ allow us to estimate
the horizontal coupling constant $g_{3H}$ on the scale of $\sim$1 TeV.

\section{The features of the GUST spectrum with
$[SU(5)\times U(1)\times SU(3)\times U(1)]^{\otimes 2}.$
The ways of the gauge symmetry breaking.}\label{sbsec42}

Model~1 is defined by 6 basis vectors given in Table \ref{tabl1} which
generates the $Z_2\times Z_4\times Z_2\times Z_2\times Z_8\times Z_2$
group under addition.

\begin{table}[h]
\caption{\bf Basis of the boundary conditions for all world-sheet fermions.
Model~1.}
\label{tabl1}
\footnotesize
\begin{center}
\begin{tabular}{|c||c|ccc||c|cc|}
\hline
Vectors &${\psi}_{1,2} $ & ${\chi}_{1,..,6}$ & ${y}_{1,...,6}$ &
${\omega}_{1,...,6}$
& ${\bar \varphi}_{1,...,12}$ &
${\Psi}_{1,...,8} $ &
${\Phi}_{1,...,8}$ \\ \hline
\hline
$b_1$ & $1 1 $ & $1 1 1 1 1 1$ &
$1 1 1 1 1 1 $ & $1 1 1 1 1 1$ &  $1^{12} $ & $1^8 $ & $1^8$ \\
$b_2$ & $1 1$ & $1 1 1 1 1 1$ &
$0 0 0 0 0 0$ &  $0 0 0 0 0 0 $ &  $ 0^{12} $ &
${1/2}^8 $ & $0^8$  \\
$b_3$ & $1 1$ & $ 1 1 1 1 0 0 $ & $0 0 0 0 1 1 $ & $ 0 0 0 0 0 0 $ &
$0^4  1^8 $ & $ 0^8 $ &  $ 1^8$ \\
$b_4 = S $ & $1 1$ &
$1 1 0 0 0 0 $ & $ 0 0 1 1 0 0 $ & $ 0 0 0 0 1 1 $ &
$ 0^{12} $ & $ 0^8 $ & $ 0^8 $ \\
$  b_5 $ & $ 1 1 $ & $ 0 0  1 1 0 0 $ &
$0 0 0 0 0 0 $ &  $1 1  0 0 1 1$ &  $ 1^{12} $ &
${ 1/4}^5  {-3/4}^3 $ & $ {-1/4}^5\ {3/4}^3  $  \\
$  b_6 $ & $ 1 1 $ & $ 1 1  0 0  0 0 $ &
$ 0 0  0 0 1 1 $ &  $0 0  1 1 0 0$ &  $ 1^2 0^4 1^6 $ &
$ 1^8  $ & $ 0^8  $  \\
\hline \hline
\end {tabular}
\end{center}
\normalsize
\end{table}

The  model corresponds to the following chain of the gauge
symmetry breaking:
$\longrightarrow U(8)^2
\longrightarrow [U(5)\times U(3)]^2\ . $

Since the matter fields form the chiral multiplets of $SO(10)$, it is possible
to write down  $U(1)_{Y_5}$--hypercharges of massless states. In order to
construct the correct electromagnetic charges for matter fields we must define
the hypercharges operators for the observable $U(8)^{I}$ group as follows
\begin{equation}\label{eq16}
Y_5=\int^\pi_0 d\sigma\sum_a \Psi^{*a}\Psi^a ,\,\,\,\,\,
Y_3=\int^\pi_0 d\sigma\sum_i \Psi^{*i}\Psi^i
\end{equation}
and analogously for the $U(8)^{II}$ group.

Then the orthogonal combinations
\begin{equation}\label{eq17}
 \tilde Y_5 = {1\over 4}(Y_5 + 5Y_3), \,\,\,\,\,
 \tilde Y_3 = {1\over 4}(Y_3 - 3Y_5),
\end{equation}
play the role of the hypercharge operators of $U(1)_{Y_5}$ and
$U(1)_{Y_H}$ groups
respectively. In  Table \ref{tabl3'} we give the hypercharges
 $\tilde Y_5^{I},\tilde Y_3^{I}, \tilde Y_5^{II},\tilde Y_3^{II}$.

\begin{table}[t]
\caption{\bf The list of quantum numbers of the states. Model~1.}
\label{tabl3'}
\footnotesize
\noindent \begin{tabular}{|c|c||c|cccc|cccc|} \hline
N$^o$ &$ b_1 , b_2 , b_3 , b_4 , b_5 , b_6 $&
$ SO_{hid}$&$ U(5)^I $&$ U(3)^I $&$ U(5)^{II} $&$
U(3)^{II} $&$ {\tilde Y}_5^I $&$ {\tilde Y}_3^I $&$ {\tilde Y}_5^{II} $&$
{\tilde Y}_3^{II}$ \\ \hline \hline
1 & RNS &&5&$\bar 3$&1&1&--1&--1&0&0 \\
  &     &&1&1&5&$\bar 3$&0&0&--1&--1 \\
  &0\ 2\ 0\ 1\ 2(6)\ 0&&5&1&5&1&--1&0&--1&0 \\
${\hat \Phi }$
  &&&1&3&1&3&0&1&0&1 \\
  &&&5&1&1&3&--1&0&0&1 \\
  &&&1&3&5&1&0&1&--1&0 \\ \hline \hline
2 &0\ 1\ 0\ 0\ 0\ 0&&1&3&1&1&5/2&--1/2&0&0 \\
  &&&$\bar 5$&3&1&1&--3/2&--1/2&0&0 \\
  &&&10&1&1&1&1/2&3/2&0&0 \\
${\hat \Psi }$
  &0\ 3\ 0\ 0\ 0\ 0&&1&1&1&1&5/2&3/2&0&0 \\
  &&&$\bar 5$&1&1&1&--3/2&3/2&0&0 \\
  &&&10&3&1&1&1/2&--1/2&0&0 \\ \hline
3 &0\ 0\ 1\ 1\ 3\ 0&$-_1\ \pm_2$&1&1&1&3&0&--3/2&0&--1/2 \\
  &0\ 0\ 1\ 1\ 7\ 0&$-_1\ \pm_2$&1&$\bar 3$&1&1&0&1/2&0&3/2 \\
${\hat \Psi }^H$
  &0\ 2\ 1\ 1\ 3\ 0&$+_1\ \pm_2$&1&$\bar 3$&1&3&0&1/2&0&--1/2 \\
  &0\ 2\ 1\ 1\ 7\ 0&$+_1\ \pm_2$&1&1&1&1&0&--3/2&0& 3/2 \\ \hline
4 &1\ 1\ 1\ 0\ 1\ 1&$\mp_1\ \pm_3$&1&1&1&$\bar 3$&0&--3/2&0&1/2 \\
  &1\ 1\ 1\ 0\ 5\ 1&$\mp_1\ \pm_3$&1&$\bar 3$&1&1&0&1/2&0&--3/2 \\
${\hat \Phi }^H$
  &1\ 3\ 1\ 0\ 1\ 1&$\pm_1\ \pm_3$&1&$\bar 3$&1&$\bar 3$&0&1/2&0&1/2 \\
  &1\ 3\ 1\ 0\ 5\ 1&$\pm_1\ \pm_3$&1&1&1&1&0&--3/2&0&--3/2 \\ \hline
5 &0\ 1(3)\ 1\ 0\ 2(6)\ 1&$-_1\ \pm_3$&1&3($\bar 3$)&1&1&$\pm$5/4&$\pm$1/4
&$\pm$5/4&$\mp$3/4 \\
  &&$+_1\ \pm_3$&5($\bar 5$)&1&1&1&$\pm$1/4&$\mp$3/4&$\pm$5/4&$\mp$3/4 \\
${\hat \phi }$
  &0\ 1(3)\ 1\ 0\ 4\ 1&$-_1\ \pm_3$&1&1&1&3($\bar 3$)&$\pm$5/4&$\mp$3/4
&$\pm$5/4&$\pm$1/4 \\
  &&$+_1\ \pm_3$&1&1&5($\bar 5$)&1&$\pm$5/4&$\mp$3/4&$\pm$1/4&$\mp$3/4 \\
\hline
6 &1\ 2\ 0\ 0\ 3(5)\ 1&$\pm_1\ -_4$&1&1&1&1&$\pm$5/4&$\pm$3/4
&$\mp$5/4&$\mp$3/4 \\
  &1\ 1(3)\ 0\ 1\ 5(3)\ 1&$+_1\ \mp_4$&1&1&1&1&$\pm$5/4&$\pm$3/4
&$\pm$5/4&$\pm$3/4 \\
${\hat \sigma }$
  &0\ 0\ 1\ 0\ 2(6)\ 0&$\mp_3\ +_4$&1&1&1&1&$\pm$5/4&$\mp$3/4
&$\pm$5/4&$\mp$3/4 \\ \hline
\end{tabular}
\normalsize
\end{table}

With the chiral matter and "horizontal" Higgs fields available in Model~1
 the possible form of the renormalizable (trilinear)
part of the superpotential responsible for fermion mass matrices is
restricted not only by  the gauge symmetry.
Another strong constraint
 comes from the interesting observation that a modular invariant
N=1 space-time supersymmetric theory also extends to a global N=2
world sheet superconformal symmetry \cite{bank'} which now contains
 two distinct fermionic components of the energy-momentum tensor,
 $T_F^+$ and  $T_F^-$
 and there is also the  $U_J(1)$ current $J$. This conserved $U(1)$ current of
the N=2 superalgebra may play a key role in
constructing of realistic phenomenology. Thus all vertex operators
have the definite $U(1)$ charge.
Let us consider the contribution $R^2(NS)$ to the   three point
fermion-fermion-boson matter superpotential:

\begin{eqnarray}\label{eq32}
W_1&=& g\sqrt{2} \biggl[ {\hat \Psi}_{(1,3)}
{\hat \Psi}_{({\bar 5},1)}
{\hat\Phi}_{(5,{\bar 3})} +
 {\hat \Psi}_{(1,1)} {\hat \Psi}_{({\bar 5},3)}
{\hat\Phi}_{(5,{\bar 3})} + \nonumber\\
&+& {\hat \Psi}_{(10,3)} {\hat \Psi}_{({\bar 5},3)}
{\hat\Phi}_{({\bar5},3)} +
{\hat \Psi}_{(10,3)} {\hat \Psi}_{(10,1)}
{\hat\Phi}_{(5,{\bar 3})} \biggr]
\end{eqnarray}

From the above form of the Yukawa couplings it
follows that two (chiral) generations
have to be very light (comparing to $M_W$ scale).

The $SU(3)_H$ anomalies of the matter fields (row No 2) are
naturally canceled by
the chiral "horizontal" superfields  forming two sets:
${\hat \Psi}^H_{(1,N;1,N)}$ and ${\hat \Phi}^H_{(1, N;1, N)}$,
{}~$\Gamma = \underline 1$,~~$N = \underline 1, \,  \underline 3$,~
(with both ${SO(2)}$ chiralities, see Table \ref{tabl3'}, row No 3, 4
respectively).
The superpotential, $W_2$, consists of the following $R^2 NS$-terms:
\begin{eqnarray}\label{eq34}
W_2&=& g\sqrt{2} \biggl[
{\hat \Phi}^H_{(1,1;1,\bar3)}{\hat \Phi}^H_{(1,\bar 3;1,1)}
{\hat \Phi}_{(1,3;1,3)} +
{\hat \Phi}^H_{(1,1;1,1)}{\hat \Phi}^H_{(1,\bar 3;1,\bar 3)}
{\hat \Phi}_{(1,3;1,3)} + \nonumber\\
&+& {\hat \Phi}^H_{(1,\bar 3;1,\bar 3)}{\hat \Phi}^H_{(1,\bar 3;1,\bar 3)}
{\hat \Phi}_{(1,\bar 3;1,\bar 3)} +
 {\hat \Psi}^H_{(1,{\bar 3};1,1)}{\hat \Psi}^H_{(1,{\bar 3};1, 3)}
{\hat \Phi}_{(1,{\bar 3};1,{\bar 3})} +\nonumber\\
&+&{\hat \Psi}^H_{(1,1;1,3)}
{\hat \Psi}^H_{(1,{\bar 3};1,3)}
{\hat \Phi}_{(1, 3;1, 3)}
+{\hat \sigma }_{(-_1 -_4 )} {\hat \sigma }_{(+_1 +_4 )}
{\hat \Psi }_{(1,1;1,1)} \biggr]
\end{eqnarray}

From (\ref{eq34}) it follows that some of the horizontal fields in sectors
(No 3, 4) remain massless at the tree-level.
This is a remarkable prediction: "horizontal" fields are "sterile",
e.g. they interact with the ordinary
chiral
matter fields only through the $U(1)_H$ and $SU(3)_H$
gauge boson
and therefore this "sterile" matter
is of an  interest in the context of the
experimental searches on accelerators or in astrophysics.
The Higgs fields  could give the following
$(NS)^3$ contributions to the renormalizable superpotential:
\begin{eqnarray}\label{w3}
W_3=\sqrt{2}g\biggl\{
{\hat\Phi }_{ (5,1;1,3)}  {\hat\Phi }_{ (\bar 5,1;\bar 5,1)}
{\hat\Phi }_{ (1,1;5,\bar 3)} +
{\hat\Phi }_{ (5,1;1,3)}  {\hat\Phi }_{ (1,\bar 3;1,\bar 3)}
{\hat\Phi }_{ (\bar 5,3;1,1)} && \nonumber\\
+ {\hat\Phi }_{ (1,3;5,1)}  {\hat\Phi }_{ (\bar 5,1;\bar 5,1)}
{\hat\Phi }_{ (5,\bar 3;1,1)} +
{\hat\Phi }_{ (1,3;5,1)}  {\hat\Phi }_{ (1,\bar 3;1,\bar 3)}
{\hat\Phi }_{ (1,1;\bar 5,3)} +\mbox{conj.}
\biggr\}&&
\end{eqnarray}

So, $W_1 +W_2 +W_3$ is the most general renormalizable superpotential
which includes all nonzero three-point (F-type) expectation values
of the vertex operators for corresponding 2-dimensional conformal
model.
The only non-vanishing nonrenormalizable superpotential $W_4=R^4$
is as follows
\begin{eqnarray}
W_4\sim \frac{g^2}{M_{Pl}}{\hat\Psi}_{(1,3;1,1)}
{\hat\Phi}^H_{ (+_1,-_3)(1,\bar 3;1,1)}
{\hat\sigma}_{1\ (-_1,-_4)} {\hat\sigma}_{3\ (+_3,+_4)}.
\end{eqnarray}

As we can see from the states list for Model 1, the hidden group
$U(1)_1$ in this model appears to be anomalous: Tr $U(1)_1=12$.

This means that at one-loop string level there exists Fayet-Iliopoulos
D-term determined by VEV of the dilaton and it is proportional to
Tr $U(1)_1=12$.
Potentially this term could break SUSY at the high scale
and destabilize the vacua \cite{Dine'}. This could be avoided
if the potential has
D-flat direction on $U(1)_1$-charged fields which have VEVs that break
anomalous group (and may be some other groups), compensate D-term, and
restore SUSY. Those fields have to have appropriate charges on the
remaining groups in order that  cause no  SUSY breaking via their D-terms.
In our case we can avoid the SUSY breaking caused by D-term by using the
pairs of fields ${\hat\phi}_1$ and/or ${\hat\phi}_3$ from the sector 5.
In addition to anomalous group breaking we also obtain breaking of the
groups $U(1)_3^{hid}$ (hidden),
$U(1)^I_H\times U(1)^{II}_H\rightarrow U(1)_H^{'} $
(in the case of using only one of ${\hat\phi}_1$, ${\hat\phi}_3$,
otherwise the group is broken totally)
and
$U(1)^I_5\times
U(1)^{II}_5\rightarrow U(1)^{sym}_5$, and also horizontal
group $SU(3)_H^{I}\rightarrow SU(2)_H^I$ and/or
$SU(3)_H^{II}\rightarrow SU(2)_H^{II}$. For complete breaking
the initial horizontal symmetry $SU(3)_H^I \times SU(3)_H^{II}$
we also need to use the VEVs of the ${\hat{\Phi}}_{(1,3;1,3)}$- Higgs fields.
Besides that fields potentially the fields ${\hat\sigma}_2$
can also obtain VEVs,
that  additionally breaks down
the hidden group $SO(6)_4$ and also breaks  $U(1)_H^{'}$.
Note that
since the  superfields mentioned above besides the $\hat{\sigma}_2$-field
do not participate in construction
of the renormalizable superpotential $W_1+W_2+W_3$
in this scheme we have no problem with the F-flat directions.
Finally, note that in non-flipped $SU(5)\times U(1)$ model
we can give the VEVs to the fields ${\hat\sigma}_1$
for breaking $U(1)^{sym}_5$. In this case for the choice of the $D$-flat
direction we also need to use VEVs of the fields ${\hat\sigma}_3$.
From the form of the ${\hat\sigma}$-depended contribution to the
superpotential $W_2$  (\ref{eq34}) it follows that the field
of the fourth generation ${\hat\Psi}_{(1,1;1,1)}$ (the fourth neutrino
in non-flipped scheme)  obtains a heavy mass.

 Further we shortly discuss the problem of
  gauge symmetry breaking in Model~1.  The most
important point  is that the Higgs fields
$(\underline{10}_{1/2}+\underline{\bar{10}}_{-1/2})$ do not appear.
However there exists some
possibilities to break the GUST group $[(U(5)\times U(3))^I]^{\times 2}$
down to the symmetric subgroups using the following VEVs of the Higgs fields
$(\underline 5,\underline 1;\underline 5,\underline 1)_{(-1,0;-1,0)}$:
\begin{description}
\itemsep=0cm
\item[a)] $\qquad\qquad <(\underline 5,\underline 5)> =
a\cdot \mbox{diag}(1,1,1,1,1)$
\item[b)] $\qquad\qquad <(\underline 5,\underline 5)>_0 =
 \mbox{diag}(x,x,x,y,y)$,
\item[c)] $\qquad\qquad <(\underline 5,\underline 5)>_0 =
a \cdot \mbox{diag}(0,0,0,1,1)$,
\item[d)] $\qquad\qquad <(\underline 5,\underline 5)>_0 =
a \cdot \mbox{diag}(1,1,1,0,0)$.
\end{description}
With the VEV a) the GUST group $[(U(5)\times U(3))^I]^{\otimes 2}$
 breaks to symmetric group:
$$ \mbox{\bf a)}\qquad\qquad U(5)^I\times U(5)^{II} \rightarrow {U(5)}^{sym}
\rightarrow ...$$
With such a breaking tensor Higgs fields transform
under the $(SU(5)\times U(1))^{sym}\times G_H$ group in the
following way:
\begin{eqnarray}\label{eq19}
(\underline 5,\underline 1;\underline 5,\underline 1)_{(-1,0;-1,0)}
{}~&\rightarrow~~& {(\underline {24},\underline 1) }_{(0,0)}~~
+~~(\underline 1,\underline 1) _{0,0}.
\end{eqnarray}

The diagonal VEVs
of the Higgs fields break the GUST with $G \times G$ down to
the "skew"-symmetric group with the generators $\triangle^{sym}$ of the form:
\begin{equation}\label{eq21}
\triangle^{sym} (t) = -t^* \times 1 + 1\times t,
\end{equation}
The corresponding  hypercharge of the symmetric group reads:
\begin{equation}\label{eq22}
\bar Y = \tilde {Y}^{II} - \tilde {Y}^{I}.
\end{equation}
Adjoint  representations which appear on the $rhs$ of (\ref{eq19}) can be used
for
further breaking of the symmetric group. This can lead to the final physical
symmetry
$$ \qquad SU(3^c)\times SU(2_{EW})\times U(1)_5 \times U(1)^{sym}
\times G_H^{'}$$
with the low-energy gauge symmetry of the quark--lepton generations with an
additional $U(1)^{sym}$--factor. As we already discussed above
the form of the $G_H^{'}$ depends on the way of the U(1)
anomaly group cancellation and the complete breaking of this group is
realized by the VEVs of the ${\hat \Phi}_{(1,3;1,3)}$ Higgs fields
and/or ${\hat \Phi}_{(1,1;5,{\bar 3})}$,
${\hat \Phi}_{(5,{\bar 3};1,1)}$ Higgs fields.

Note, that when we use the VEVs b),c),d) there exist also
the others interesting ways of breaking the $G^I \times G^{II}$ gauge symmetry
down to
\begin{description}
\itemsep=0cm
\leftmargin=5mm
\rightmargin=0mm
\item[b)] $\quad SU(3^c) \times
 SU(2)_{EW} \times U(1)_5 \times U(1)^{sym} \times G_H^{'} \rightarrow \ldots$,
\item[c)] $\quad  SU(3^c) \times SU(2)^I_{EW}
\times SU(2)^{II}_{EW} \times U(1)_5\times U(1)^{sym}
\times G_H^{'} \rightarrow \ldots $ ,
\item[d)] $\quad SU(3^c)_I \times SU(3^c)_{II} \times SU(2)_{EW} \times U(1)_5
\times U(1)^{sym} \times G_H^{'} \rightarrow \ldots$ .
\end{description}
 We could consider the GUST construction involving
$[SO(10)  \times G_H]^{\otimes 2}\times G_{hidden}$
 as the  gauge group \cite{25'}.
In that model the only Higgs fields appeared are $(10,10)$ of
$SO(10) \times SO(10) $ and the hidden group $G_{hidden}=U(1)\times SO(6)$
 is anomaly free.
As an illustration we would like to remark that for the
$SO(10) \times SO(10) \times G_H \times G_H$ GUST we can consider similarly
the following VEVs of the Higgs fields $(10,10)$:
\begin{description}
\itemsep=0cm
\item[a')]$\quad <(10,10)>_0 = a\cdot \mbox{diag} (1,1,1,1,1,1,1,1,1,1)$,
\item[b')]$\quad <(10,10)>_0 = a \cdot \mbox{diag} (1,1,1,1,1,1,0,0,0,0)$,
\item[c')]$\quad <(10,10)>_0 = a \cdot \mbox{diag} (1,1,1,1,1,1,x,x,x,x)$.
\end{description}
These cases
lead  correspondingly to the following chains of
$[SO(10)]^{\otimes 2}$ breaking ways:
\begin{description}
\itemsep=0cm
\item[a')]$\quad [SO(10)]^{sym}$,
\item[b')]$\quad SU(4)\times SU(2)_{I1}\times SU(2)_{I2}\times
 SU(2)_{II1} \times SU(2)_{II2}$,
\item[c')]$\quad SU(4)\times SU(2)_{I}\times SU(2)_{II}$.
\end{description}

\section{The GUT and string unification scale. The paths of unification in
flipped and non-flipped GUST models.}

Indeed the estimates on the $M_{H_0}$ scale depend on the
value of the family gauge coupling.
String theories imply a natural unification of the gauge
and gravitational
couplings, $g_i$ and $G_N$ respectively. For example, it turns out that
these couplings unify
at tree level to form one coupling constant $g_{string}$ \cite{Ginsparg'}:
\begin{eqnarray}
8\pi \frac{G_N}{\alpha'}=g_ik_i=g^2_{str}.
\end{eqnarray}
\medskip
Here $\alpha'$ is the Regge slope, the coupling constants $g_i$
correspond to the gauge group $G_i$ with the Kac-Moody levels $k_i$.
In string theory the scale of unification is fixed by the Planck scale
$M_{Pl}\approx 10^{19}$.
In one-loop string calculations
the value of the unification scale could be divided into moduli independent
part and a part that depends on the VEVs of the moduli fields. The latter
part considered as the correction to the former and obviously it is
different for the various models. Like in the gauge fields theory
this correction is called the string threshold correction of the massive
string states. Moduli independent contribution depends on the renormalizing
scheme used,
so in $\overline{DR}$ renormalization scheme the scale of string
unification is shifted to \cite{Kaplunovsky'}a :
\begin{equation}
M_{SU} = \frac{e^{(1-\gamma)/2} 3^{-3/4}}{4\pi} g_{str} M_{Pl
 \approx 5\,g_{str} \times {10}^{17} GeV. \label{Mstring}
\end{equation}

There  exists the most important difference between the unification
scales of gauge coupling
unification in string theory and  in field theory. In field theory this scale
is determined via extrapolation of data within the Supersymmetric
$SU(3) \times SU(2) \times U(1)$ Standard Model using
 the renormalization group equations (RGE) for the gauge couplings
$${\alpha}_{3}^{-1}(M_Z) = 8.93\ ,$$
$${\alpha}_{2}^{-1}= ({\alpha}_{em}/
{\sin}^{2}{\theta}_{W})^{-1}|_{M_Z}=29.609\ ,$$
$${\alpha}_{1}^{-1}= (5{\alpha}_{em}/ 3{\cos}^{2}
{\theta}_{W})^{-1}|_{M_Z}=58.975\ .$$
The factor 5/3 in the definition of ${\alpha}_1$ has been included for the
normalization at the unification scale $M_{G}$.

The one loop   renormalization group equations for these gauge couplings
 are given by

\begin{equation}
\frac {d{\alpha}_{i}}{d \ln \mu}= \frac{{\alpha}_{i}^2}{2\pi} b_i
\label{RGEi}
\end{equation}

 Beta functions coefficients
for the $SU(3)\times SU(2) \times U(1)$ coupling constants
in SUSY models are given by the following:

\begin{eqnarray}
b_3&=& -9 +2N_g + 0\cdot N_h + \frac{1}{2}\cdot N_3,\\
b_2&=& -6 +2N_g + \frac{1}{2}\cdot N_h +  0\cdot N_3,\\
b_1&=&\,0 +2N_g + \frac{3}{10}\cdot N_h + \frac{1}{5}\cdot N_3,
\label{bi}
\end{eqnarray}
where $N_g$ is the number of generation and $N_h$ is the number of Higgs
doublets and we also include some possible intermediate thresholds for heavy
Higgs doublets $(N_h - 2)$ and color triplets, $N_3$,
which exist in the spectrum of Model 1 and can be take into account for
RGE.

The RGE are integrated from $M_Z$-mass to the unification scale $M_{G}$.
In the presence of various intermediate scale, $M_I$, $I=1,2,3,...$,
the RGE are given by:

\begin{eqnarray}
{\alpha}_i^{-1}(M_Z)=
{\alpha}_i^{-1}(M_{G}) + \frac{b_i}{2\pi}\ln{\frac{M_{G}}{M_Z}}
-\frac{b_{i1}}{2\pi}\ln{\frac{M_1}{M_Z}}
-\frac{b_{i2}}{2\pi}\ln{\frac{M_2}{M_Z}}-...
\end{eqnarray}

where $b_{iI}$ are the additional corresponding contributions of the new
thresholds to the beta functions. At the Z-mass
scale we have:

\begin{eqnarray}
{\sin}^2{\theta}_{W}(M_Z) &\approx & 0.2315 \pm 0.001\ ,\nonumber\\
M_Z&=&91.161\pm 0.031 GeV, \label{experiment}
\end{eqnarray}

Note, that for flipped models
\begin{equation}
\sin^2 \theta_W(M_{G}) =\frac{15k^2}{16k^2 +24}
\end{equation}
where $k^2=g_1^2/g_5^2$.
In the SO(10) limit (or for non-flipped case) we have
$k^2=1$ and $\sin^2 \theta_W = 3/8$.

The string unification scale could be contrasted with
MSSM, $SU(3^c)\times SU(2) \times U(1)$
naive unification scale,
\begin{equation}
M_{G} \approx 2\times {10}^{16} GeV
\end{equation}
obtained by running the SM particles and their SUSY-partners to high energies.

One of the first way to explain the difference between these two mass
scales, $M_{SU}$ and $M_{MSSM}=M_G$, was the  attempts to take into account
the  string thresholds corrections of the massive string states
\cite {Kaplunovsky'}:
\begin{equation}
\frac{1}{g_i^2(\mu)} = k_i\frac{1}{g_{string}^2}
+2b_i\ln(\frac{\mu}{ M_{SU}}) + {\tilde\Delta}_{G_i}.
\end{equation}
where the index $i$ runs over gauge coupling and  $\mu$ is
some phenomenological scale such as $M_Z$ or $M_{G}$.
The coefficients $k_i$ are the Kac-Moody levels (e.g.
for SU(5) $k_2=k_3=1,k_1=5/3$).
The quantities $ {\tilde\Delta}_{G_i}$ represent
the heavy thresholds corrections, which are the corrections arising
from the infinite towers of massive string states.
Although these states have the Planck mass scale, there are infinite
number of them, so they together could have the considerable effect.
In general the full string thresholds corrections are  of the form
\begin{equation}
{\tilde {\Delta}}_i={\Delta}_i +k_i Y,
\end{equation}
where Y is independent of the gauge
group factor. Moreover, the low energy predictions for
${\sin}^2{\theta}_W(M_Z)$ and ${\alpha}_3(M_Z)$ depend only on the differences
$
({\tilde{\Delta}}_i -{\tilde{\Delta}}_j)=(\Delta_i -\Delta_j)
$
for the different gauge groups.

However the $Y$ factor makes influence on the estimation of the value of
$g_{str}$ if we base on low energy gauge constants and use RGE. Note that
in general $g_{str}$ is defined by VEV of the dilaton moduli field but
because of degeneracy of the classic potential of the moduli fields
we do not know the $g_{str}$ a priori. If we could have the value of the
$g_{str}$ then the its coincidence with our estimates will show the
correctness of our model.

It is supposed that the value of the $Y$ factor is small enough
\cite{KalaraLopez} so we neglect it in our calculations of $g_{str}$
via RGE.

In the  GUSTs examples considered the threshold corrections
of the massive string states are not large enough to
compensate the difference between the scales of unification of
the string and the MSSM (GUT) models.
Later we will discuss the possible effects of them in $G \times G$ models,
for example in Model 1.

In our calculations for Model 1
we will follow the way suggested in \cite{Kaplunovsky'}c,\cite {Dienes'}.
According to it
we have to calculate an integral of the modified partition function
over the fundamental domain $\Gamma$ of modular group.
$$
\delta\Delta=\int_\Gamma\frac{\mbox{d}^2\tau}{\tau_2}
\left(|\eta(\tau)|^{-4}\hat{Z}(\tau)-b_G\right)
$$
where $b_G$ is the beta function coefficient,
$\hat Z(\tau)$ is modified partition function.  Modified in this
context means that the charge operators $Q$ are inserted in the trace
in partition function in the following way:
$$ \hat
Z(\tau)=-\mbox{Tr}Q^2_sQ^2_Iq^Hq^{\bar H}, $$
where $Q_s$ is helicity operator and
$Q_i$ is a generator of a gauge group.  We are interesting in particular
in the difference between groups $SU(5)_I$ and $U(1)_I$ in Model~1.
For this case charge polynomial is $5Q_1Q_2$  (see \cite{Kaplunovsky'}c).
Rewritten via well known theta
function, the modified function $\hat Z(\tau)$ for our case reads:
$$ \hat{Z}(\tau)=-\frac5{512}\sum_{{\bf a,b}}c\left({\bf a\atop b}\right)
\eta^{-1}\theta'\left( a_\Psi\atop b_\Psi\right)\bar\eta^{-2}
 \hat{\bar\theta}^2\left( a_I\atop b_I\right)
 \prod\eta^{-1}\theta\left(a_j\atop b_j\right)
 \prod\bar\eta^{-1}\bar\theta\left(a_k\atop b_k\right)
 $$
where  512 is normalizing factor
and products calculated over all fermions excluding fermions
 which $Q$ operators apply to. Namely $\theta'$ and $\hat{\bar\theta}$
 denote action of helicity and gauge group operators respectively.
 The sum is taken over all pairs of boundary condition vectors that
 appear on the model.
According to \cite{Dienes'} we expand the resulting expression with
$\theta$-functions via $q$ and $\bar q$ in order to achieve appropriate
precision.

The final results for Model 1 are as follows ($\delta$ denotes the difference
between corresponding quantities for $U(1)_I$ and $SU(5)_I$).
$\delta b=26.875; \qquad \delta\Delta=5.97 $
Given this relative threshold corrections we can compute its effect on
string unification scale $M_{SU}$.
We find that the correction unification scale is:
\begin{equation}\label{Mstrcor}
M_{SU}^{corr}=M_{SU} \,\exp\frac{({\Delta}_5-{\Delta}_1)}{2(b_5-b_1)}
\approx g_{str.}\cdot 5.6\cdot 10^{17} GeV\ .
\end{equation}

However there are some  ways to explain the difference between
scales of string ($M_{SU}$) and ordinary ($M_{G}=M_{SU}$) unifications
(without additional intermediate exotic vector matter fields
that does not fit into $5$ or $\bar 5$ representations of SU(5).
\cite {Dienes'})
Perhaps the most natural  way is related to  the $G^I \times G^{II}$
String GUT.
If one uses the breaking scheme $G^I\times G^{II}\,\rightarrow G^{sym}$
( where $G^{I,II}=U(5)\times U(3)_H \subset E_8$ ) on the $M_{sym}$-scale,
then unification scale $M_{MSSM}= M_{G} \sim 2\cdot 10^{16}\,$GeV is
the scale of breaking
the $G_{sym}$ group, and string unification do supply the equality
of coupling constant $G\times G$ on the string scale
$M_{SU}\sim g \cdot 5\cdot10^{17}\,$GeV.
Otherwise, we can have an addition scale of the symmetry breaking
$M_{sym} > M_{G}$.
In any case on the scale of breaking
$U(5)^I\times U(5)^{II}\,\rightarrow U(5)^{sym}$
the gauge coupling constants satisfy the equation
\begin{equation}
({\alpha}^{Sym})^{-1}= ({\alpha}^I)^{-1} + ({\alpha}^{II})^{-1} \ .
\label{symscale}
\end{equation}
Thus in this scheme the knowledge of scales $M_{SU}$ and $M_{Sym}$ gives
us a principal possibility to trace the evolution of coupling
constant of the original  group
$SU(5)^I \times U(1)^I \times SU(5)^{II} \times U(1)^{II}$  through
the $SU(5)^{Sym} \times U(1)^{Sym}$ to the low energies
and estimate the values of all coupling constants including
the horizontal gauge constant $g_{3H}$.

The coincidence of $\sin^2 \theta_W$ and ${\alpha}_3$ with experiment will
show how realistic this model is.
The evolution of the gauge constant from the string constant $g_{str}$
to the scale of $M_G$ is described by the equation:
\begin{equation}
({\alpha}_{5,1}^{Sym})^{-1}(M_{G})
=2({\alpha}_{5,1}^{Str})^{-1}(M_{SU}) +
\frac{(b_{5,1}^{I}+ b_{5,1}^{II})}{2 \pi}\ln(M_{SU}/M_{Sym})
+\frac{b_{5,1}^{Sym}}{2 \pi}\ln(M_{Sym}/M_{G})
\label{RGES}
\end{equation}

where ${\alpha}_{5}^{Str}(M_{SU})=g_{str}^2/4 \pi$
and $g_{str}$ is the string coupling.
\bigskip
Now we can get the relation between $g_{str}=g$ and $M_{Sym}$ from RGE's
for gauge running constants $g_5^{Sym}=g_5$, $g_5^{I}$ and , $g_5^{II}$
on the $M_{G}-M_{SU}$ scale.
For example in  Model 1 for the breaking scheme a) one can get:
\begin{equation} \label{b5values}
b_5^{Sym}=12,\,\,\,
b_5^{I}=5,\,\,\,
b_5^{II}=-3
\end{equation}
and
\begin{equation} \label{b1values}
b_1^{Sym}=27,\,\,\,
b_1^{I}=\frac{105}{4},\,\,\,
b_1^{II}=\frac{73}{4}.
\end{equation}

Let us try to make the behaviours of
these coupling constants consistent above and below $M_{G}$ scale.
To do this we have to remember that
there are two possibilities to embed quark-lepton matter in
$SU(5)\times U(1)$ of SO(10) multiplets,
$\underline {1}_{5/2} $,
$ \underline {\bar 5}_{-3/2} $, $\underline {10}_{1/2}$.
For the electromagnetic charge we get:
\begin{eqnarray}\label{eq23}
Q_{em}&=& Q^{II} - Q^{I} = {\bar T}_3 + \frac{1}{2}{\bar y},\nonumber\\
{\bar T}_3&=&T^{II}-T^{I};\,\,\,\,\quad
{\bar y}= y^{II} - y^{I}.
\end{eqnarray}
where
\begin{equation} \label {23'''}
\frac{1}{2}{y^{I,II}}
=\alpha {T^{I,II}}_{5z} + \beta {Y^{I,II}}_5\ , \quad
{T^{I,II}}_{5z} = \mbox{diag} (-1/3,-1/3,-1/3,1/2,1/2).
\end{equation}
In usual non-flipped Georgi-Glashow  SU(5)
model the $Q_{e.m.}$ is expressed
via SU(5) generators only: $\alpha= 1;\,\,\, \beta= 0$. For flipped
$SU(5)\times U(1)$ we have
$\alpha=- \frac{1}{5};\,\,\, \beta= \frac{2}{5}.$
Note, that this charge quantization does not lead to exotic
states with fractional electromagnetic charges (e.g. $Q_{em} =\pm 1/2,
\pm 1/6$)\cite{27',25'}. Also in non-flipped
$SU(5) \times U(1)$ gauge symmetry breaking
scheme  there are  no  SU(3) color triplets and SU(2) doublets with
exotic hypercharges.

For example in flipped (non-flipped) case of Model 1  we can
use the Higgs doublets from
$(5,1;1,1) + ({\bar 5},1;1,1)$,
(the fields $\hat{\phi}_2+\bar{\hat{\phi}}_2$ from sector 5)
for breaking the SM symmetry and low energy $U(1)_5^{sym}$-symmetry.
Below the $M_{G}$  scale in non-horizontal sector
the evolution of gauge coupling constants is described by equations
\begin{eqnarray}
&&\alpha^{-1}_S(\mu )=\alpha^{-1}_5(M_{G} ) + \frac{b_3}{2\pi}
\ln (M_{G}/\mu) \\
&&\alpha^{-1}(\mu )\sin^2\theta_W =\alpha^{-1}_5(M_{G} ) +
\frac{b_2}{2 \pi}\ln (M_{G}/\mu) \\
&&\frac{15k^2}{k^2+24}{\alpha}^{-1}(\mu )\cos^2\theta_W =
{\alpha}_5^{-1}(M_{G}) +\frac{{\bar b}_1}{2 \pi}\ln (M_{G}/\mu)  ,
\end{eqnarray}
where for $N_g=4$ generations and for the minimal set of Higgs fields we have:
$$b_3=-1\ ,\quad b_2=3\ ,\quad
{\bar b}_1=\frac{25k^2}{k^2+24}\cdot b_1 |_{4\ gen.}
=\frac{15k^2}{(k^2+24)}\frac{43}{3} .$$

From these equations and from the experimental data we can find for $N_g=3,4$,
respectively:
\begin{eqnarray}
M_{G}&=&1.2\cdot 10^{16}\mbox{GeV}\ ,\qquad \alpha_5^{-1}=24.4,
\qquad k^2=0.98\ ,\nonumber\\
M_{G}&=&1.17\cdot 10^{16}\mbox{GeV}\ ,\qquad \alpha_5^{-1}=14.1,
\qquad k^2=0.97.
\end{eqnarray}

Here we assume that additional Higgs doublets and triplets appeared
in the theory are heavy ($>M_G$).

From the other hand for the Model 1 and with the mass of the fourth
generation  sufficiently heavy to be invisible but less than $M_G$
the equations
for ${\alpha}_{5,1}^{I.II.Sym}$ for all $M_{sym}$-scale in the range,
$M_G <M_{sym}< M_{SU}$,
give the contradicting value for $k^2$ that is considerably less than 1.
For example, for $M_{Sym}=1.6 \cdot10^{17}GeV$, we get:
\begin{eqnarray}
 M_{SU} = 9.6\cdot10^{17}GeV,\,\,\,
g_{str}=1.7\,\,\rightarrow k^2=0.44.
\end{eqnarray}

In the non-flipped case in Model 1  we have an additional neutral singlet
$\hat{\sigma}^1$ field, which could be used for breaking $U(1)^{Sym}$ group
(of  $U(1)_5^I \times U(1)_5^{II}$) at any high scale, independently on
the $M_{Sym}$ scale, where the
$$SU(5)^I\times SU(5)^{II}\longrightarrow SU(5)^{Sym}$$
 Therefore we have no
constraints on $k^2$ parameter in the range from string scale down to $M_G$.
As a result in  non-flipped case of the
$G^I \times G^{II}$ ($G=SU(5) \times U(1)\times G_H$) GUST the string
unification scale, $M_{SU}$, can be consistent with the
$M_{G}$ ($M_{MSSM}$) scale, i.e using low energy values of the
$g_{1,2,3}$- coupling constants and their RGE (\ref {RGEi}, \ref {bi},
\ref {symscale},\ref {RGES})
 we get for the GUST scale $M_{SU}$  the expression
(\ref {Mstrcor} )
with the corresponding value of the string coupling constant.

In this case while $M_{sym}$ changes in the range from
$M_G=1.17\cdot 10^{16}$ GeV to $10^{18}$ GeV we could
expect the string constant and string scale (\ref{Mstrcor}) to be in the range
$$g\sim (1.40\div 2.13)\ ,
\quad M_{SU}\sim (0.79\div1.20) \times 10^{18} GeV\ . $$

It is interesting to estimate the value of horizontal coupling constant.
The analysis of RG-equations allows to
state that the horizontal coupling constant $g_{3H}$ does not exceed
the electro-weak one $g_2$.

In Model~1 after cancellation of the U(1) anomaly by VEVs
of the fields ${\hat\phi}_1$ or ${\hat\phi}_3$
corresponding $SU(3)^{II}$ or $SU(3)^{I}$ horizontal gauge
symmetry group survives.

Using RG equations for the running constant $g_{3H}^{I,II}$
and the value of the string coupling constant $g_{str}$ at $M_{SU}$
we can estimate a value of
the horizontal coupling constant at low energies.
For Model~1 we have
$$
b_{3H}^{I}= 21\ ,\qquad
b_{3H}^{II}= 13\ ,$$
and
we find from RGE for $g_{3H}$ that
$$ g^I_H\sim 0.3\ ,\qquad g^{II}_H\sim 0.4\ .$$
Below we consider in details the gauge symmetry breaking by VEV's
b), c), d) applied to Model 1 both in flipped and non-flipped cases.
We assume that some additional Higgs doublets and triplets originating
from the group $G^{II}$ (see sector 1 in table \ref{tabl3'}) could
be lighter than $M_G$. The fact that this Higgs fields were initially
(i.e. before breaking $G^I\times G^{II}\rightarrow G$) related to
the group $G^{II}$ excludes their dangerous interaction with matter
that could lead to proton decay.

Also we investigate the dependency on the fourth generation mass $M_4$
and take into account the SUSY threshold.

In general the RGE with thresholds between $M_Z$ and $M_{G}$ are as follows
($k^2\equiv 1$ for non-flipped $SU(5)$):
\begin{eqnarray}\label{rgeeq}
\alpha_3^{-1}&=&{\alpha_5^{-1}} +\frac{1}{2\pi}\ (z + b_3 \ln M_{G}
-b_3^0 \ln M_Z)
\nonumber \\
\alpha_2^{-1}&=&{\alpha_5^{-1}} +\frac{1}{2\pi}\ (y + b_2 \ln
M_{G}-b_2^0 \ln M_Z) \\
\frac{25k^2}{k^2+24}\ \alpha_1^{-1}&=& \alpha_5^{-1}
+\frac{1}{2\pi}\ (x + {\bar b}_1 \ln M_{G}-{\bar b}_1^0 \ln M_Z)\nonumber
\end{eqnarray}
${\bar b}_1,\ b_2, \ b_3$ denote beta function coefficients for the corresponding
coupling constants that take into account all fields below $M_{G}$ scale.
Similarly ${\bar b}^0_1,\ b^0_2,\ b^0_3$ are beta function coefficients that
take all fields of the MSSM excluding superpartners.
We also introduce thresholds factors $x,\  y,\ z$ that depend on various
thresholds $M_I$:
$$ x,y,z= -\sum_I \Delta b^I_{1,2,3}\ \ln M_I $$
In particular we are going to consider SUSY threshold $M_{SUSY}$,
4th generation masses  $M_4$ and effective masses of addition doublets
and triplets $M_{2,3}$.

 In the context of Model 1 we  have
$$ {\alpha^I_5}=\alpha_5(1+q^2),\quad q=g^I_5/g^{II}_5,\quad
{g^I_5}^2=\frac{16\pi^2 g^2_{str}}{16\pi^2 -g^2_{str}b^I_5 \ln\left(\frac{M^2_{G}}{M^2_{str}}
\right)}
$$
and the similar formulae for $g^{II}_5,\ g^I_1,\ g^{II}_1$.

In this scheme the representations like $({\bf 5, 3})$ of the Model 1
break into
the equal number of vector-like triplets and doublets  ( $({\bf 3,1})$ and
$({\bf 1,2})$ under the $SU(3)\times SU(2)$ group). We consider the case
when the fields in one $({\bf 5, 3})$ representation  with the masses below
$M_{G}$ are from the second group $U(5) \times U(3)$.
(In this case we have no problems with the dimension four, five,
six operators of proton decay in Higgs sector.)
 Hence in addition to the MSSM Higgs  vector-like
doublet we have 2 doublets and 3 triplets.

Below are the values of the  $b$ coefficients for our case:
$$
{\bar b}_1=\frac{275k^2}{24+k^2}\ ,\ b_2=5\ ,\ b_3=2\ ,\
{\bar b}^0_1=\frac{105k^2}{24+k^2}\ ,\ b^0_2=-3\ ,\ b^0_3=-7\ .
$$

 Considering  flipped $SU(5)$ case we have to pay attention to the
consistency of the value of $k^2$ derived from (\ref{rgeeq}) and from
RGE of  the string coupling $g_{str}$ above the $M_{G}$ scale.
We use $b$
  coefficients from (\ref{b5values}, \ref{b1values}).

From the system (\ref{rgeeq}) we can get a set of appropriate masses
in the range of $M_Z-M_{G}$ and values of $\alpha_5,\ k^2$ and $q^2$
as well. But then we should apply RGE between the string scale
and $M_G$ scale
to check out whether
this values are consistent. This equations give us $k^2 < 1$.
 Our calculations show
that with  $k^2\leq 1$ for flipped $SU(5)$ in the Model 1
 one cannot get appropriate values for
$M_{SUSY},\, M_{2,3,4},\,M_{G}$ (i.e. that are within range
$M_{SUSY}-M_{G}$) that are consistent with string RGE.

For non-flipped case we apparently obtain the demand that
constants $\alpha_{1,2,3}$ converge to one point (that is equivalent
to $k^2\equiv 1$) which is consistent with RGE in the framework
of the MSSM-like models.

For this case we consider the b)  breaking way of
$ [SU(5) \times U(1)]^{\otimes 2}$. We can consider the cases c) and d)
as a limits ($x \ll y$ and $x \gg y$). As it follows from our analysis
there exists a range of parameters values (threshold masses) that make
system (\ref{rgeeq}) consistent and we have an appropriate hierarchy
of the scales.

The maximum value of $M_G$ one can obtain in this case is
$M_G\sim 1.3\cdot 10^{16}$ GeV. The mutual dependencies of the threshold masses
are shown on Fig. 1 where $M_{2,3}$ are effective masses because the equation
(\ref{rgeeq}) depends on them only ($M_2=\sqrt{M_2^{(1)} M_2^{(2)}}$;
$M_3=\sqrt[3]{M_2^{(1)} M_2^{(2)} M_2^{(3)}}$).

Note that the Higgs triplets and doublets considered obtain their masses
via $F^2$-term of the field ${\hat\Phi}_{(5,1;1,3)}$ (see the first term
in the $W_3$ (\ref{w3})). Hence $M^2_3\sim |x|^2$; $M^2_2\sim |y|^2$.
At the same time  squared masses of the vector bosons of the broken
groups $SU(3)$ and $SU(2)$ are proportional to $(g^2_I+g^2_{II})|x|^2$;
$(g^2_I+g^2_{II})|y|^2$. This means that  in general above the thresholds
$M_{2,3}$ we should take into account the restoration of the symmetry
$SU(n)\longrightarrow SU(n)_I\times SU(n)_{II},\ \ \ n=2,3$.
I.e. our plots are correct only in the region with $M_2$ close to $M_3$.
The other cases demand more careful accounting of the symmetry
restoration thresholds. This question is currently under
consideration and we will present the results in the future.

The horizontal gauge constant on the scale of 1 TeV for first
or second $SU(3)_H$ group (depending on which of them will survive
after suppression of the $U(1)$ anomaly) appears to be of the order

$${g_{3H}^I}^2 \biggl( O(1\mbox{TeV})\biggr)\approx 0.10\div 0.11\ , \quad
{g_{3H}^{II}}^2 \biggl( O(1\mbox{TeV})\biggr)\approx 0.15\div 0.17\ .
$$

The calculations for our model for different breaking chains
show that for evaluation of intensity
of a processes with a gauge horizontal bosons at low energies
we can use inequality
${{\alpha}_{3H}(\mu)}\,\leq \,{{\alpha}_2(\mu)}\ .$

This work was supported in part by Russian Foundation for Basic Researches,
grant No 95-02-06121a. One of us (G.G.V.) would like to thank INFN
for financial support, especially Prof. G. Puglierin and Prof. C. Voci,
for hospitality during his stay in Padova. It is a great pleasure for him
to thank
Profs. G. Costa, and A. Masiero for useful discussions, for help and support.
Many  helpful advises and  interesting discussions with R. Barbieri,
J. Ellis, H. Dahmen, K. Dienes, J. Gasser,
L. Ibanez, H. Leutwyler, P. Minkowsky, N.Paver, G.Ross and F. Zwirner during
this work are also acknowledged.

\begin{figure}[h]
\label{grafik2}
\vskip -7cm
\vbox to 23cm {\hbox to 14cm {\epsfxsize=14cm\epsffile{plot3.ps}}%
\vskip -9cm
\hbox to 14cm {\epsfxsize=14cm\epsffile{plot4.ps}}}
\caption{$M_{2,3}$ behavior.
 {\tt A5}$=\alpha_5^{-1}$, $M_G=1.26\cdot 10^{16}$, {\tt g} means
 $g_{str}$, $g^I_3=0.32,\ \ g_3^{II}=0.40$.}
\end{figure}

redpajama_set_name: RedPajamaArXiv


Copyright © 2014 A. Jayachitra and R. Vinodha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Genetic algorithm (GA) based PID (proportional integral derivative) controller has been proposed for tuning optimized PID parameters in a continuous stirred tank reactor (CSTR) process using a weighted combination of objective functions, namely, integral square error (ISE), integral absolute error (IAE), and integrated time absolute error (ITAE). Optimization of PID controller parameters is the key goal in chemical and biochemical industries. PID controllers have narrowed down the operating range of processes with dynamic nonlinearity. In our proposed work, globally optimized PID parameters tend to operate the CSTR process in its entire operating range to overcome the limitations of the linear PID controller. The simulation study reveals that the GA based PID controller tuned with fixed PID parameters provides satisfactory performance in terms of set point tracking and disturbance rejection.
PID controllers are still widely used in 90% of industries, since no other advanced control schemes such as model predictive control, internal model control (IMC), and sliding mode control (SMC) match the simplicity, clear functionality, applicability, and ease of use provided by this controller . PID controller tuned at a particular operating point will not provide a satisfying response when there exists deviation in the process operating range . Hence, soft computing based PID controller tuning is widely proposed by the researchers during the last few decades [3–7].
CSTR process exhibits typical nonlinear and time varying behaviour where control of the parameter reactor concentration with few computation steps is a challenge to the researchers. Therefore, it becomes essential to use powerful heuristic algorithms like GA to tune the PID parameters, thereby achieving good set point tracking and disturbance rejection control in the chemical system CSTR. Vinodha et al. have designed three control schemes for nonlinear CSTR process such as extraction of PID controller parameters based on artificial intelligence, model predictive controllers using the weighted sum of the output from local predictive controllers, and multiple model PID controllers . The presence of parameter uncertainty in nonlinear systems like CSTR has been controlled by designing the nonlinear predictive adaptive controller (NPAC) and a nonlinear disturbance observer (NDO) in literature  and the proposed compensation method improves system tracking accuracy and robustness. Prakash and Senthil formulated a nonlinear observer based model predictive controller (NMPC) based on fuzzy Kalman filter (FKF) and augmented state fuzzy Kalman filter (ASFKF) for CSTR process . A model based control scheme (F-NMPC) reduces the number of computations needed and is considered as an alternative to analytical model based control scheme as discussed in [11, 12].
A partial state feedback controller has been proposed to achieve the global set point tracking control of CSTR . Indirect adaptive control based pole placement and adaptive general predictive control (GPC) methodologies are used to estimate and control parameters of CSTR which has been reported by Delbari et al. . Shopova and Vaklieva-Bancheva have introduced in detail a genetic algorithm called BASIC, designed to deal with various engineering optimization problems . The self-organizing GA optimization method has been used for evolving PID controller parameters to avoid premature convergence and to achieve good optimization performance . Krohling and Rey have discussed a method to design an optimal disturbance rejection PID controller based on genetic algorithms for solving the constrained optimization problem in a servo motor system . Kumar et al. have described the design of GA based controller for a bioreactor model that outperformed Ziegler Nichols and Skogestad's tuned controller in terms of overshoot and undershoot as well as disturbance rejection and set point tracking . A modified genetic algorithm has been reported  to obtain the optimal PID controller parameters for different process types. Fuzzy gain scheduling has been used to determine the controller parameter transition policy for the dynamic plant .
Nagaraj and Murugananth have discussed PID tuning using soft computing based controller, improving the performance of process in terms of time domain specifications, set point tracking, and regulatory changes . In the work of Chang, PID controller has been designed using an artificial bee colony algorithm for an unstable and nonlinear CSTR chemical system . Wang and Jin have determined state feedback linearization control method for nonminimum phase characteristics of CSTR system . Singh and Sharma have proposed a fractional order PID controller with a PSO approach, providing better servo and regulatory response . The combined PID control tuning by coupling the gain-phase margin method with genetic algorithm has been applied for position control in a servo system . A virtual feedback control has been implemented to control the state variables using extended Kalman filter (EKF) in the feedback path using particle swarm optimization technique for CSTR process . In our proposed work to depict the nonlinearity of CSTR process, the differential equation model is used. The methodology proposed is the design of global GA tuned PID values constrained at individual optimum values ISE, IAE, and ITAE and weighted combination of objective functions considered to operate the plant in the entire operating range.
Organization of the paper is presented as follows. Section 2 presents the proposed control approach and Section 3 describes the CSTR plant. Introduction to genetic algorithm is provided in Section 4 and Section 5 includes the simulation results of CSTR process for both servo and regulatory cases. Finally, Section 6 concludes how GA based PID controller gives better performance with minimum computation steps to achieve optimized PID control of the CSTR plant in a wide range.
A PID controller is a combination of a proportional, an integral, and a derivative controller, integrating the main features of all three. A simplified block diagram of control approach followed in the work is shown in Figure 1.
Figure 1: Structure of PID controller.
From Figure 1, it is inferred that the objective of the work is to tune the PID gains with minimum error between the set value and actual concentration value, where is a difference between the desired output and actual output , is the PID control law, and parameters , , and are the proportional gain and integral and derivative times, respectively. Commonly employed error criteria to achieve optimized PID tuning values are integral square error (ISE), integral absolute error (IAE), and integrated time absolute error (ITAE), respectively, and are formulated as given in (1), (2), and (3), respectively. From the block diagram, it is clear that a weighted combination of ISE, IAE, and ITAE  is used as an objective function to obtain PID parameters through genetic algorithm. Consider The objective function used in genetic algorithm is defined as The PID control law utilized in velocity form is given in One or a weighted combination of these criteria is employed to achieve optimized parameter tuning of PID controller. It is generally accepted that weighted indices are more appropriate than individual performance indices as reported in [3–6].
The first principles model of the continuous stirred tank reactor and the operating point data as specified by Pottman and Seborg  and design procedure described  has been used in the simulation studies. In the process considered for simulation study (shown in Figure 2), an irreversible, exothermic reaction → (Reactant of Concentration is converted to Product of Concentration ) occurs in a constant volume reactor that is cooled by a single coolant stream . The steady state operating specifications are given in Table 1.
Table 1: Steady state operating data of CSTR process.
Figure 2: Schematic of CSTR process.
The fundamental dependent quantities for the reactor are the total mass of the reacting mixture in the tank, mass of chemical in the reacting mixture, and total energy of the reacting mixture in the tank.
The mass balance on component is The reactor energy balance, assuming constant volume, heat capacity , density and neglecting changes in potential and kinetic energy is given in equation (7). Consider The linearized transfer function model of the CSTR process formulating as a control variable takes the form specified in (8) with and being damping factor and undamped natural frequency, respectively. Consider The model gives a drastic variation in as the operating concentration varies. The variations are mentioned in Table 2. Hence, it is observed that PID parameters , designed for a particular operating point, are not suitable to control the process globally. Therefore, there is a need to motivate an insight into the heuristic algorithm that satisfies with the fixed PID parameters, constrained to certain objectives.
Table 2: Variation in and with respect to concentration.
Genetic algorithm is a random search method that can be used to solve nonlinear system of equations and optimize complex problems. GA uses probabilistic transition rules instead of deterministic rules and handles a population of potential solutions known as individuals or chromosomes that evolve iteratively. Each iteration of the algorithm is termed a generation. The evolution of solutions is simulated through a fitness function and genetic operators such as reproduction, crossover, and mutation . Genetic algorithm as illustrated in Figure 3 is typically initialized with a random population. This population (mating pool) is usually represented by a real-valued number or a binary string called a chromosome. The performance of the individual is measured and assessed by the objective function, which assigns each individual a corresponding number called its fitness.
Figure 3: Flow diagram of genetic algorithm.
The fitness of each chromosome is assessed and a survival of the fittest strategy is applied. In this work, the error value is used to assess the fitness of each chromosome. There are three main operations in a genetic algorithm: reproduction, crossover, and mutation. The sequences of operations involved in GA are described in Figure 3.
Step 1. Initialize the parameter with a population of random solutions, such as crossover rate, mutation rate, number of clusters, and number of generations. Determine the coding mode.
Step 2. Compute and evaluate the value of the fitness function.
Step 3. Proceed with crossover and mutation operation and make up the new cluster.
Step 4. Repeat Step 2, till the best value is obtained.
Genetic algorithms work with a population of strings or chromosomes, instead of considering parameters directly. Hence, to solve our problem, the controller parameter vector should be coded to a string called chromosome. For convenience and simplicity, the binary coding method is chosen. Based on the binary coding method, every element of the parameter vector is coded as a string of length, which consists of zeros and ones for the desired resolution.
Fitness is a measure to evaluate the suitability of a chromosome. By the principle of survival of the fittest, a chromosome with higher fitness value has a higher probability of contributing one or more offspring in the next generation. By employing genetic algorithm, the performance criterion is related to fitness function and optimal PID parameters are derived by minimizing an objective, which inculcates a weighted combination of IAE, ISE, and ITAE.
Reproduction is a basic operator of genetic algorithm. It is operated on the basis of the survival of the fittest. In each generation, the chromosome of the current population is reproduced or copied into the next generation, according to the reproduction probability , which is defined in where is the population size.
Reproduction directs the search of genetic algorithm towards the best individuals. Crossover operation is performed to exchange the information between any two chromosomes via probabilistic decision in the mating pool and to provide a mechanism to mix chromosomes with the splice point.
In genetic algorithms, however, the gene pool tends to become more and more homogeneous as one better gene begins to dominate after several generations and leads to premature convergence of nonoptimal solution. To overcome this undesirable convergence, the third genetic operator mutation is introduced in genetic algorithm with appropriate probability. Mutation is an occasional alternation of the gene from zero to one or from one to zero with the mutation point determined uniformly at random.
Genetic Algorithm Based PID Tuning. It is summarized as follows. GA initially generates a random population, which is implemented with small population size in order to allow the controller to be optimized and converge at a faster rate. The initial population is set by encoding the PID parameters , , and into binary strings known as a chromosome. The fitness of each chromosome is calculated by converting its binary string into real value which represents the PID parameters. Each set of PID parameters is passed to PID controller. The complete response of the system for each PID parameter value and its initial fitness value is computed using individual cost functions like ISE, IAE, and ITAE a weighted combination of these three cost functions. This process will go through Steps 2 and 3 until the end of the generations where the best fitness value is achieved. The ultimate aim of GA is to seek global PID values (, , and ) with minimum fitness value to operate the CSTR plant in the entire range. The details of GA parameters used in the simulation are shown in Table 3.
Table 3: Parameters used in genetic algorithm.
Simulation using genetic algorithm based PID controller for a CSTR plant, including different performance indices such as ISE, IAE, and ITAE separately and a weighted combination of these three functions, is carried out for both servo and servo regulatory cases. In all simulation runs, the CSTR process is simulated using the nonlinear first principles model given by (6) and (7) and the true state variable (concentration) is computed by solving the nonlinear differential equation using Matlab 7.0. The controller saturation limit between 97 and 109 lit/min is considered with initial conditions given by lit/min, mol/lit, and K and the sampling time of about 0.083 sec is selected for all the simulation studies. The range of PID parameters considered for the simulation is given in Table 4.
Table 4: PID parameter range.
The set point variations as shown in Figure 4(a) have been introduced for assessing the tracking capability of the proposed PID controllers utilizing ISE, IAE, and ITAE separately and a weighted combination of these three functions as objective functions. The weights associated with combined objective function are , , and . From the response, it is observed that all the controllers are able to track the variation in the reactor concentration when the set point concentration is changed from 0.0989 mol/L to 0.1245 mol/L, 0.1245 mol/L to 0.0885 mol/L, and 0.0885 mol/L to 0.0989 mol/L. Weighted GA based PID controller settles faster in all set point transitions. The variation in the controller outputs is presented in Figure 4(b). The performance measures like percentage peak overshoot and settling time are calculated for various objective functions as shown in Table 5. Further, it is evident that the coolant flow rate variation is found to be smooth in all the controllers (refer to Figures 4 and 5).
Table 5: Performance measures under various cost functions.
Figure 4: Servo response of CSTR process. (a) Process output. (b) Controller output.
Figure 5: Servo regulatory response of CSTR process. (a) Feed temperature. (b) Process output. (c) Controller output.
Table 5 implies that a weighted combination of cost functions like ISE, IAE, and ITAE gives a better reduction in percentage peak overshoot and settling time compared to using ISE, IAE, and ITAE separately. Overshoot is negligible, both for ISE and for combined cost functions when concentration set point is changed from 0.1245 mol/L to 0.0885 mol/L and 0.0885 mol/L to 0.0989 mol/L.
Simulation studies have been carried out to demonstrate the disturbance rejection capability of the CSTR process at nominal and shifted operating point in the presence of change in the feed temperature. A step change in the inlet temperature of magnitude 2 K (from 350 K to 348 K) has been introduced at the 100th sampling instant and maintained at up to 200th sampling instant (refer to Figure 5(a)).
From 100th sampling instants to 150th sampling instants of Figure 5(b), it can be inferred that the controllers utilizing ISE, IAE, and ITAE separately and a weighted combination of these three functions as objective functions are able to reject the disturbance quickly and bring the reactor concentration back to the nominal value of the set point. The flow rate variation of the controllers is shown in Figure 5(c).
When disturbance persists, a step change in the set point has been introduced at the 150th sampling instant. It can be inferred that all the controllers are able to maintain the concentration at the specific set point, as shown in Figure 5(b). This part of the simulation shows that the controllers are capable of rejecting the disturbance, also at shifted operating points.
Similar observation has been obtained by disturbing the process and by varying the feed concentration as shown in Figure 6(a). The controllers utilizing ISE, IAE, and ITAE separately and a weighted combination of these three functions as objective functions are able to reject the disturbance quickly and bring the reactor concentration back to the nominal value of the set point as shown in Figure 6(b). The flow rate variation of the controllers is shown in Figure 6(c).
Figure 6: Servo regulatory response of CSTR process. (a) Feed concentration. (b) Process output. (c) Controller output.
Simulation study has been carried out by providing the disturbances feed temperature and feed concentration simultaneously as shown in Figures 7(a) and 7(b). For combined disturbance also, GA based PID controllers with various error goals have given satisfactory performance as shown in Figure 7(c). The respective flow rate variation of the controllers is shown in Figure 7(d).
Figure 7: Servo regulatory response of CSTR process. (a) Feed temperature. (b) Feed concentration. (c) Process output. (d) Controller output.
In this paper, GA based PID controller tuning has been designed and applied to CSTR process. From the simulation studies, we infer that optimized controller parameters obtained by implementing this algorithm with a weighted combination of ISE, IAE, and ITAE as cost indices have achieved satisfactory set point tracking and disturbance rejection in the entire operating range of the CSTR process.
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