text:
Troubled Waters: Municipal Wastewater Pollution on the Atlantic Coast
Posted on August 31, 1996 by Environment Probe
Troubled Waters: Municipal Wastewater Pollution on the Atlantic Coast, by Martin Nantel, examines the environmental and socioeconomic effects caused by the daily discharge of 1.1 million cubic metres of treated and untreated sewage in the waters of the Atlantic region. The report also addresses governments' failure to enforce the legislation intended to regulate sewage treatment plants and proposes a solution to alleviate sewage pollution on the East Coast. Appendix A further illustrates the report's findings using Halifax harbour as a case study.
To read the PDF Version, click here.
Forty-four per cent of the wastewater generated by residents and businesses of the four Atlantic provinces ends up in septic systems, 29 per cent is directed to sewage treatment plants (STPs), and the remaining 27 per cent is spewed raw into our coastal waters.
On an annual basis, this means that our coastal waters receive 122 million cubic metres of "treated" wastewater, 32 million cubic metres of which only undergoes primary treatment before discharge. Over 600,000 people living in some 194 cities, towns, and municipali-ties of the Atlantic region discharge an additional 100 million cubic metres a year of raw sewage into the coastal waters.
Effluents containing raw or improperly treated sewage pose environmental and public health threats:
In addition to degrading and destroying aquatic habitat, sewage pollution causes both acute and chronic toxicity in aquatic organisms. Cancerous lesions and other symptoms associated with chemical mixtures have been reported in Halifax and St. John's harbours. Despite chlorine's proven harmful environmental effects, STPs in the Atlantic region use at least 692,200 kg/year of chlorine-based products to disinfect municipal wastewater effluents.
Sewage pollution is also a health hazard for people swimming at beaches that have been contaminated by sewage effluent. Pathogenic micro-organisms found in wastewater can cause serious diseases such as hepatitis and meningitis, as well as less serious conditions such as diarrhea and skin and ear infections.
In Atlantic Canada, most shellfish closures are caused by bacteriological pollution.
Indeed, the sole cause of approximately 20 per cent of all shellfish closures in the Maritimes, municipal wastewater is also implicated as a contributing pollution source in over 50 per cent of the contaminated shellfish areas.
As of April 1995, there were about 580 closures comprising a total surface area of over 2,000 square kilometres, and representing 35 per cent of the Atlantic region shellfish areas.
These closures represent a public health threat and cause significant economic loss to the shellfish industry, in addition to preventing recycling earnings into the local economy. For example, the Department of Fisheries and Oceans estimates that an income of over $8 million is forgone every year due to health-related closures of the soft shell clam fishery in south west New Brunswick alone.
Shellfish closures also increase pressures on the open areas, reduce employment, increase consumer prices, and impose severe demands on enforcement agencies to patrol the closed areas for illegal fishing.
The lax enforcement of many powerful regional, provincial, and federal laws have ensured that municipal sewage polluters go mostly unpunished and that unnecessary environmental degradation persists. Provincial Crowns and federal authorities have never filed any prosecution against municipal sewage offenders on the East Coast, despite the fact that many STP operators are chronically violating the laws.
As the population of the Atlantic region grows, as the volume of sewage increases, as STPs exceed their designated operating capacities, and as older system deteriorate, STPs run the risk of being further stressed, and so does the environment. If provincial Crowns cannot afford to cleanup our coastal waters, they should invite the private sector to do the job. Here's why:
Owing to operational advantages, tax benefits, and timing and construction costs efficiencies, the private sector can provide most municipal services at a 10 to 30 per cent lower cost than municipalities can.
More importantly, privatizing municipal wastewater utilities would eliminate the inherent conflict of interest that exists when a regulatory agency assumes financial responsibility for the same activity it regulates. The poor enforcement records of regulatory agencies are barely surprising when it is realized that these same agencies are financially responsible for up to 80 per cent of the capital costs of sewage infrastructure projects.
By limiting governments' interventions to the direct control of private wastewater utilities, i.e., by enacting strong and effective legislation preventing the erosion of environmental standards, pricing abuses, and services, privatization would allow governments to enforce long-ignored standards without fear of being asked to finance the improvements.
Governments should empower everybody affected by sewage pollution to sue polluters. Faced with the threat of private lawsuits and government sanctions, private entrepreneurs would find innovative ways to eliminate sewage pollution. Technological innovations, educational campaigns, financial incentives, water-pricing reforms, source control, and monitoring programs are only a few of the ways in which sewage pollution would improve.
From the "Home of Anne of Green Gables" to the beautiful beaches of "Canada's Ocean Playground," from Newfoundland's rocky shores to the picturesque coast of New Brunswick, the ocean relentlessly carves the physical and cultural landscape of the Atlantic provinces. In addition to nurturing a vibrant coastal-oriented tourism industry, the waters off Canada's East Coast also long-supported a vital fin and shell fishery. Unfortunately, these waters, on which so many rely, are increasingly becoming polluted. One of the culprits is municipal wastewater pollution.
Consider the following examples: In Nova Scotia, Halifax and Dartmouth have been discharging raw sewage into Halifax harbour for the past 250 years. Currently, about 54.8 million cubic metres a year, or 150,000 cubic metres a day, of untreated industrial and municipal sewage from the twin cities is discharged directly into the harbour from about 40 outfalls.[1] If spread over the city of Halifax, the total volume of raw sewage discharged annually would cover the city to a depth of .66 metres. (See Appendix A for more details.) Similarly, St. John's, Newfoundland's provincial capital, dumps 38.3 million cubic metres a year of raw sewage into its harbour.[2] In Saint John, New Brunswick, more than 8.5 million cubic metres of raw sewage—a volume equivalent to 204 Exxon Valdez oil spills—ends up annually in the St. John River or in the Bay of Fundy.[3]
These are by no means isolated cases. In fact, more than six hundred thousand people living in some 194 cities, towns, and municipalities of the Atlantic region dispose of their sewage in the same way.[4] Of the approximate 1.1 million cubic metres of wastewater generated daily by residents, businesses, and industries of the four Atlantic provinces, about a quarter is released raw into our coastal waters. Because these enormous quantities of wastewater are hazardous to aquatic life, pose a public health threat, and contribute to socioeconomic problems, this paper reviews the Atlantic province's ongoing municipal sewage pollution, and suggests an alternative to the status quo.
Sanitary Sewage
Various debris such as gravel, wood, plastic containers, condoms, tampons, and rags find their way into the sewerage system. Yet, the bulk of sanitary sewage consists of human excrement and water, and contributes mostly innocuous organic matter to the receiving environment—usually a body of water. Under normal circumstances, micro-organisms break down the organic matter that comes out of a municipal outfall.
Municipal wastewater causes problems, however, when the volumes discharged are greater than those the receiving waters can assimilate. Excess quantities of undissolved solids—also called suspended solids—will smother the habitat of bottom-dwelling organisms, degrade fish habitat, and fatally clog fish gills and abrade the exposed membranes of aquatic organisms. Excess organic matter may well kill fish and other aquatic species if the biological oxygen demand—the amount of oxygen used by micro-organisms breaking down the organic matter—reaches very high levels, i.e., if the dissolved oxygen level required by aquatic life becomes depressed.
Too much nutrients, by stimulating excessive and undesirable plant growth such as algal blooms—a phenomenon referred to as eutrophication—can put additional pressures on dissolved oxygen levels due to the increased number of oxygen-consuming micro-organisms required to decompose these aquatic plants. Certain species of non-tolerant organisms abandon these waters due to depressed oxygen levels. Unsightly growths of algae are also a nuisance to boaters and bathers.
In addition, pathogenic micro-organisms thrive in wastewater. These bacteria, parasites, and viruses found in human and animal stools, i.e., in wastewater, represent a latent threat to weaker fish that are more susceptible to bacterial invasion. If ingested by humans, these pathogens can cause many serious diseases such as hepatitis, myocardia, and meningitis and are implicated in infections such as chronic fatigue syndrome and diabetes. Less serious diseases such as diarrhea, skin and ear infections also ensue (See Appendix B).
Treated and untreated municipal wastewater released into Canadian waters also contains over 200 chemicals and other toxins that are dumped into sewers by households, businesses, and industries. Added to this witches' brew is urban runoff—a combination of oils, animal wastes, and poisonous substances that further contaminates our environment when inadequately treated. Appendix C summarizes the potential health effects associated with various toxins.
The purpose of wastewater treatment is to strip sewage of as much of its harmful elements as possible before releasing it to the receiving environment, especially when large volumes are involved. Municipal wastewater treatment can range from the most basic to more advanced. The difference lies in how much is taken out of the effluent, with each added level resulting in a cleaner effluent.
Primary treatment involves slowing down the wastewater and directing it to a sedimentation tank where larger suspended solids settle naturally due to gravitation. Secondary treatment consists of supplying oxygen to micro-organisms to enable them to grow and to eat organic matter in the sewage. This ensures that once released, the sewage effluent does not provide food for the micro-organisms that would consume excessive amounts of oxygen required by aquatic biota. Lastly, tertiary treatment uses a mechanical or a sand filtration to achieve a similar but more thorough treatment than secondary processes.
In certain cases, provincial regulators may decide to add extra steps to one of these three treatments in order to obtain an effluent of better quality. If an effluent is too rich in nutrients and adversely affects the receiving waters, the sewage treatment plant's (STP's) operators may have to further clean the effluent with a nutrient removal process. Similarly, to counter the potential risks associated with an effluent rich in pathogenic organisms that could harm recreational users, an STP's effluent may be disinfected and subsequently analysed for fecal coliforms—bacteria present in the intestines of all warm-blooded animals, including humans, that can function as an indicator of fecal pollution in water bodies.
Forty four per cent of the total population of the Atlantic Provinces, or over one million people, is serviced by septic systems.[5] The remaining 56 per cent is divided almost equally between those who are serviced by a municipal STP, and those who see their wastewater entering the environment directly, without any treatment. While approximately 32 million cubic metres of wastewater receives primary treatment before discharge, or 26 per cent of the total 122 million cubic metres treated every year in the region, an additional 100 million cubic metres a year is discharged raw into our coastal waters.[6]
Effects of Sewage on Fish
As mentioned above, sewage pollution harms fish and other aquatic life by degrading and destroying aquatic habitat. When an aquatic ecosystem reaches excessive pollution levels, some or all of its organisms may die. But fish mortality and acute toxicity are by no means the only indicators, although they are the most obvious ones, used to determine whether a substance is harmful to marine biota. A not so extreme and much more appropriate (but more expensive and more complicated) way to assess the health of aquatic life is to look at the multitude of sublethal effects that pollutants cause.[7]
The sublethal effects of water pollution on fish have been documented worldwide for a great many species.[8] They can take many forms—physiological, biochemical, pathological, and behavioural. Physiological abnormalities include reduction or inhibition of reproductive capacity, growth retardation, and reduced resistance to infection from pathogens. Biochemical disturbances cause alterations in metabolism, body fluids, and enzyme activities, leading to subtle organ impairments and physical abnormalities in developing young. Some common pathological disturbances include fin erosion, ulcerations, liver tumors, and skeletal anomalies caused by damaged genetic material. Also, by altering external surfaces, pollutants can facilitate invasion of pathogens. Behavioural changes such as altered feeding and migrating patterns are often due to chemical damage to fish sensory equipment and to their abilities to react to subtle chemical changes in water composition. As one investigator put it, the toxic effects of pollutants on sensory organs "are significant even if they do not cause permanent neurological damage, for [even] a temporary disability that prevents an organism from relating to a viable environment for only moments can be disastrous."[9]
Most of these sublethal effects have been associated with chemical mixtures from a variety of sources including municipal and industrial outfalls, urban runoff, pulp mills, and petroleum sites. Nonetheless, it is difficult to separate the effects of mill effluents from those of urban runoff and sewage pollution.[10] However, a recent fish health study in and around St. John's harbour revealed that many of the fish caught had been severely affected by pollution. In addition to evident fluid retention in fish gills, a general indication of a disturbed pathological condition, more than half the flounder caught inside the harbour had eroded fins.[11] Severe liver lesions were found in fish from the entire study area.[12] Although the funds available for the study did not allow for the establishment of direct links between pollutants in the harbour and the effects seen in fish, the fact that Saint John's does not have "heavy" industrial effluents entering its harbour suggest that the effects observed are likely linked to sewage and/or to traditional runoff chemicals.[13]
In the long-term, sublethal effects can adversely affect the community structure and dynamics of the fish population as well as ecosystem structures and functions.[14] Although the fish population may also be affected, individual responses to stress do not provide a basis for predicting population impacts.[15] Harm to individual fish does demonstrate, however, the presence of deleterious substances, and this in itself is a cause for concern. As they may produce subtle impacts over broader geographic areas, sublethal effects associated with chemical mixtures may be more significant than localized fish mortality.[16]
Since 1903, when Allen Hazen determined the role that chemically treated water could play in reducing the death rate of the human population, disinfection of potable and wastewater has become widespread. Owing to chlorine's powerful disinfecting capabilities, ease of application, and low cost, chlorination has become the most common method of disinfecting wastewater effluents in Canada and in the U.S., thus making water safer for recreational activities.[17] The use of chlorine-based bactericides in municipal water and sewage treatment is now the single largest use of pesticide in the Atlantic region.[18] In 1990, Environment Canada estimated STPs in the region used approximately 692,200 kg a year of chlorine-based products to disinfect municipal wastewater effluents.[19] These quantities are probably much higher today since over-chlorination is common, particularly at STPs that malfunction because of age, mechanical breakdown, overloading, or general neglect.[20]
However, Hazen's successors did not foresee that chlorinated wastewater effluents, unless they undergo a final dechlorination process prior to discharge into the receiving waters, adversely affect aquatic life. Indeed, chlorine itself is toxic, as evidenced by the kill in 1991 of thousands of fish in Nova Scotia's East River, due to the flushing of an industrial water line with a chlorine solution.[21] In addition, when added to water, chlorine and its various compounds do not remain in their original form. Instead, they react with various pollutants in the water, and can lead to the formation of numerous compounds in the effluent, many of which may be highly toxic and detrimentally affect the biota and public health.[22]
By "burning" fish tissues, especially gill structures, chlorine damages the biochemical ability of fish to uptake oxygen. To protect itself from the irritation, a fish secretes mucus that rapidly builds up and clogs its respiratory surface. Eventually, the fish dies of asphyxiation. At high doses, chlorinated effluents result in immediate fish kills.[23] Severe burning has also been observed to cause convulsions in fish, which die of a broken back.[24] Exposure to residual chlorine also increases gill permeability. In turn, this may lead to increased accumulation, and hence toxicity, of other chemical substances found in STP effluents. Chlorine is also thought to affect the nervous system of fish.[25]
In the 1970s, studies using caged sockeye and pink salmon fingerlings were conducted in three tributaries of British Columbia's Fraser River downstream of STPs discharging chlorinated effluents. Mortality rates of up to 100 per cent were observed at the farthest station, 277 metres downstream from the effluent outfalls. When chlorination was not performed, mortality did not occur, or it decreased significantly.[26]
In Nova Scotia, biological surveys conducted near three municipal STPs discharging chlorinated wastewater effluents revealed that the community structure of bottom-dwelling organisms was significantly altered at the furthest station sampled.[27] For example, although scientists could not prove a causal relation, the total number of invertebrates sampled 100 metres from a chlorinated effluent outfall discharging in Halifax harbour was significantly reduced when compared to a control station located 2.5 kilometres away from the outfall. The total number of several annelid, crustacea, and mollusc species frequenting the area was also significantly reduced.[28]
Although the average concentration of residual chlorine in STP effluents from the Atlantic provinces nears the recommended concentration of 0.5 mg/l, and although it is well below the levels known to have physiological effects in mammals and humans, it is still far above what aquatic invertebrates can tolerate. Acute toxicity tests have shown that concentrations as low as 0.002 mg/l can kill aquatic invertebrates.[29]
The available scientific data on the effects of chlorinated wastewater effluents led the federal Minister of Environment and the Minister of National Health and Welfare, in 1993, to declare chlorinated wastewater effluents "toxic" as defined under the Canadian Environmental Protection Act.[30] Unfortunately, until the federal government implements management strategies, chlorinated wastewater effluent will remain legal, and the more environmentally-friendly, but more expensive, disinfection techniques using ozone or ultraviolet irradiation will not gain popularity.
Effects of Sewage on Shellfish
Sewage pollution also harms bivalve molluscs and crustaceans, two classes of invertebrate fish that are referred to as shellfish. Bivalve molluscs are soft bodied aquatic invertebrates enclosed by two shells joined by a hinge, and crustaceans are mostly aquatic invertebrates with segmented bodies, jointed limbs, and a firm, crustlike shell. In the four Atlantic provinces, clams, mussels, oysters, and quahogs are the most common type of bivalve molluscs harvested from the miles of beaches and rocky coastline. The most common type of crustaceans harvested are lobsters and crabs.
Except for some crustaceans, shellfish are nonmigratory or only weakly migratory, so the environment in which they are found as adults is the one to which they have been exposed for much of their existence. Shellfish are, therefore, a good sentinel species for water pollution and sediment contamination; they are better markers and can more directly demonstrate cumulative or chronic effects than fish can. Toxic contaminants that exceed shellfish tolerances may result in their immediate or protracted deaths. And as with finfish, the body of literature documenting the varied subacute and chronic effects of pollution on shellfish fecundity, physiological processes, body structures, and behaviour is steadily growing.[31]
Cellular damage and deformities are two of the many pollution-associated diseases to afflict molluscan shellfish. Others include shell abnormalities and various cancers. Several studies using commercial shellfish species including oysters and periwinkles have disclosed a direct relationship between the developmental abnormalities in embryos and the degree of pollution.[32]
In California, black abalone in a sewage-polluted area were compared with those from a clean area. Abalone at the sewage outfall site demonstrated slow growth and high mortality rates. Starving abalones with eroded shells were also reported. Individuals relocated from the clean site to the polluted area failed to grow and eventually died from undetermined causes.[33] Studies also suggest that chronic exposure of molluscs to physiological stress, including toxic pollutants, decreases resistance to bacterial infection. Investigations conducted on mussels harvested from polluted areas of the northeast coast of the United States showed an increase in parasitism that correlated with centres of population density and industrial activity.[34]
Crustaceans have their own set of pollution- or stress-related diseases. Prominent among them is a condition associated with badly degraded estuarine and coastal waters known as "shell disease." Caused by bacterial or fungal destruction of portions of the protective outer shell, this condition often results in mortalities due to the deterioration of underlying tissues, secondary infections by opportunistic pathogens, erosion and loss of gill membranes, lost appendages, and increased vulnerability to predators.[35]
Another abnormality found to increase in prevalence in suboptimal or stressful growing conditions is "black gill disease," characterized by sediment accumulation in the gills and darkening of filaments in response to microbial growth and gill necrosis. Outer shell deterioration also ensues. Other stress-related pathology may include, but is not limited to, muscle and/or blood opacity, molt retardation, and abnormal or disoriented behaviour.[36]
As clearly stated by Carl J. Sindermann, a leading authority on parasitology, "Despite the relative lack of robustness of the data base, it is becoming apparent that many of the disorders of molluscs and crustaceans in degraded habitats are stress related and are often expressions of a stress syndrome, quite distinct from that of fish, yet similar in that heterostasis—a heightened, energy-demanding physiological response to continuing stress—is achieved."[37] And, although the Atlantic region has seen very little eco-toxicological shellfish research, reflecting the funding priorities of our government, there is no reason to believe that the elevated level of pollution encountered in some areas does not exert similar stress-related responses in exposed shellfish to those observed in other parts of the world.[38]
Shellfish Closures
Perhaps the most noticeable consequence of water contamination by municipal sewage, agricultural, and industrial wastes is the closure to shellfish harvesting of most of the productive estuaries of the Atlantic region.[39] Although the shellfish growing in contaminated areas may not themselves be harmed by wastewater pollution, the health-related closures of these areas to harvesting do cause socioeconomic hardships.
Because bivalve molluscs feed by filtering the water that surrounds them, they readily accumulate chemical and bacterial pollutants in addition to naturally occurring toxins enclosed in algae, even at a considerable distance from pollution sources. When waters are polluted by sewage, sewage-related bacteria and viruses are concentrated to high levels in the shellfish tissue. Since consumers prefer shellfish that are partially cooked, such as steamed clams, or raw, as in the case of oysters, there is the possibility of ingesting contaminated tissue and live pathogens. To ensure that bivalve molluscs are harvested from clean waters, Environment Canada closely monitors water quality in shellfish areas and prohibits harvesting when fecal coliforms reach levels that represent a risk to public health.
In Atlantic Canada, most shellfish closures are caused by bacteriological pollution. Indeed, the sole cause of approximately 20 per cent of all shellfish closures in the Maritime provinces, municipal wastewater is also implicated as a contributing pollution source in over 50 per cent of the contaminated shellfish areas.[40] As seen in Appendix D, other sources of bacteriological pollution include: industrial wastes; landwash from agricultural and urban areas; seepage from poorly operating septic tank and tile field disposal systems; direct discharges of raw sewage from pleasure crafts, fishing boats, and shore-side residences; and excrement of wildlife and domestic animals.
Since the 1940s, the number of closures has been escalating steadily, in large part reflecting increased monitoring. However, increased monitoring in turn reflects a higher demand for areas available for harvesting. Unfortunately, previously unmonitored shellfish grounds that become monitored in order to meet the increased demand are often found to be bacteriologically contaminated. In 1995, there were about 580 closures comprising a total surface area of approximately 2,009 square kilometres, and representing 35 per cent of the Atlantic region shellfish areas.[41]
Socioeconomic Effects of Sewage Pollution
The molluscan shellfish industry is important to the economy of many rural coastal communities in Atlantic Canada. In 1994 alone, the commercial landings of oysters, clams, quahogs, and scallops generated over $165.3 million to the economy of the Atlantic provinces.[42] Shellfish aquaculture, on the other hand, generated much less. The landed value of farmed bivalve molluscs for the same year barely reached $12.2 million.[43] The farmed molluscan shellfish industry is nonetheless growing rapidly and reflects a more general trend of the aquaculture sector as a whole.[44]
The economic repercussions of wild and farmed shellfish closures are felt throughout the region. In New Brunswick's Caraquet Bay and in Nova Scotia's Annapolis Basin, fecal pollution precludes almost 50 per cent of the potential shellfish harvesting activities. It is estimated that the shellfish resources in these closures are valued over $2 million.[45] The Department of Fisheries and Oceans (DFO) estimates that the clam fishery in the now closed areas in Yarmouth Sound and harbour, Nova Scotia, has a potential of generating $500,000 annually.[46]
In south west New Brunswick, the drop in landings of Mya arenaria, a soft shell clam that has brought record landings since 1880, has dramatically affected the local economy. Despite 1995 landings of 1,203 million tonnes that DFO values at $2.5 million, an estimated income of over $8 million is forgone every year due to health-related closures of the soft shell clam fishery.[47] This income loss affects not only the diggers themselves but also the local economy: The small overhead expenses related to harvesting (a bucket and a rake) ensure that the local economy, rather than the servicing of fishing-related debt, directly recycles the bulk of the earnings.[48]
Bacterial-related closures have a two-fold effect on the clam fishery: They remove the clams from the wild harvesting base, and they concentrate the diggers on the other open flats. In other words, digging areas are compressed. And while the digging rate on the most productive, but contaminated, beaches decreases, harvesting rate and pressures on the less productive beaches, which are usually the last ones to remain open, increase.[49]
In summary, shellfish closures not only result in the loss of millions of dollars to fisheries and in increased pressures on the environment, but also reduce employment and increase consumer prices. They also discourage tourists from digging clams, an exciting recreational opportunity. Shellfish closures also impose severe demands on enforcement agencies to patrol the closed areas for illegal fishing.[50]
Remedial Actions
Enough resources have already been spent documenting the ills caused by municipal wastewater pollution. Bookshelves full of literature documenting the adverse effects of sewage pollution in many different regions already exist. And although each city, town, or community has its own specific environmental and socioeconomic conditions, the available research points to a common conclusion: Sewage pollution is harming the environment, and it is affecting our health and our economy.
Provincial economic wizards argue that their governments' coffers are empty and that plans for municipal sewage treatment have to await flusher days. But clearly, as the volume of sewage and the pressures on the environment and on society increase, waiting for public funds to become available prior to initiating any action is a foolish strategy. To reduce sewage pollution in the Atlantic provinces, alternative methods of financing and operating major capital projects are needed. The private sector has proven again and again that this is what they know how to do best. Here's why:
First, privatizing municipal wastewater utilities would allow the construction of infrastructure that would do much towards eliminating national embarrassments such as Halifax and St. John's harbours. Indeed, owing to operational advantages, tax benefits, and timing and construction costs efficiencies, the private sector can provide most municipal services at a 10 to 30 per cent lower cost than municipalities can . . . without an investment of scarce tax revenues.[51]
This proposition is even more attractive when one considers that in Canada, experts estimate that the deterioration of wastewater assets, their deferred maintenance, unreliable water quality, inadequate and inefficient wastewater collection and treatment, underpricing in services, plus the cost of meeting increasing standards for water supply and wastewater treatment will require municipal wastewater utilities to almost double their investments in physical plants by the year 2015.[52] Securing private funds will become even more enticing as existing systems exceed their designed capacities and costly upgrades are needed.
Privatization also answers one basic question that public ownership, because of its nature, ignores: Quis custodiet ipsos custodes? Who will keep the keepers themselves? Indeed, as soon as a provincial regulator assumes financial responsibility for a certain activity, it automatically becomes entangled in a conflict of interest. Because prosecuting non-complying STPs may force an already financially strapped government agency to inject more money into cleaning up wastewater, bureaucratic reasoning holds it that it is far better to turn a blind eye than to prosecute.
In order to fulfill their mandate many provincial governments now attempt to remedy municipal wastewater pollution cooperatively with STP operators. Unfortunately, business-friendly cooperation seems to achieve little regarding STPs' performance. In Nova Scotia, despite many notices to non-complying STP operators, non-complying STPs make up the norm, and yet, none has ever been prosecuted for violating provincial and/or federal legislation.[53] Unsurprisingly, the province of Nova Scotia finances up to 50 per cent of the capital costs of municipal wastewater projects.[54] A similar situation exists in the other Atlantic provinces where the various provincial departments of the environment provide up to 80 per cent of the capital costs of STPs' upgrades and construction.[55] More examples can be found in Quebec, British Columbia, and most likely in the other provinces as well.[56]
Were the onus to build and operate STPs put in private hands, the above situation would be turned on its head. In the same way therapy relieves patients afflicted with Multiple Personality Disorder, privatization would liberate provincial Crowns from their financial personae and help them find their true regulatory selves. Thus liberated, governments could do what only they can do: regulate others and enforce the law. By limiting their interventions to the direct control of private wastewater utilities, i.e., by enacting strong and effective legislation preventing the erosion of environmental standards, pricing abuses, and services, governments, through regulatory boards, could finally enforce long ignored standards without fear of being asked to finance the improvements.
Conjointly, governments should empower everybody affected by sewage pollution to sue polluters.[57] Facing both the threat of private lawsuits and government sanctions, private entrepreneurs would find ways to eliminate sewage pollution: Technological innovation would spring up, education campaigns would become good investments, financial incentives not to discard harmful substances into sewers would become common, and comprehensive commercial and industrial source control and monitoring programs would be set up in order to exercise tight control over the substances discharged into the sewerage system. Moreover, by reforming the pricing system, privatization would increase consumers' awareness of the true costs of providing wastewater treatment services, and would encourage them to reduce their demands on the sewage system.
On the other hand, despite improved sewage treatment, or sewage treatment where none was previously available, customers are likely to object to any price hike. It is worth remembering, however, that we are already paying a high price for the costs of unabated sewage pollution. We pay in terms of forgone income to the fishery and to the tourism industry. We also pay in terms of a degraded environment, loss of aquatic habitat, and loss of enjoyment of public beaches and of other recreational pursuits. Privatization would ensure that individual, commercial, or industrial polluters, rather than their victims, pick up the tab for protecting and restoring our rivers, estuaries, and coastal waters from unabated sewage pollution.
Appendix A: Halifax Harbour
Appendix B: Diseases associated with pathogenic micro-organisms found in domestic sewage
Appendix C: Potential health and environmental effects of toxins found in STP effluents
Appendix D: Relative composition of contributing sources to shellfish closures in the four Atlantic provinces, 1993
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On July 4, 1861 during the fourth of July celebration townsfolk from Fremont raised the flag up the 150′ "liberty Pole." A southern sympathizer took a shot at the flag and a riot ensued. Soldiers who were getting ready to head off to join the Union Army broke up the conflagration. The riot was the first Civil War riot in New England and the fourth overall.
The marker is on the front lawn of the 1756 John Prescott Lovering's Inn.


Central Region with its capital called Cape coast is where most of the slaves were sent before they were being send away from Ghana. The region was the first area in the country to make contact with the Europeans. Its capital, Cape Coast, was also the capital of the Gold Coast until 1877, when the capital was moved to Accra. It was in the castle of Cape Coast that the historic Bond of 1844 was signed between the British and the Fante Confederation.
The strategic location of Cape Coast having a sheltered beach in proximity to Elmina Castle made it a great attraction to the European nations. Central Region is blessed with its proximity to the country's capital in terms of tourism attraction. The region is more than just physical beauty, it provides an intimate view of the Ghanaian culture. It is within the Central Region that you will discover the true lifestyle of the Ghanaian people embedded in the bustling fishing communities, historic forts and castles, posuban shrines, vibrant markets and colorful festivals. Approximately two hours' drive west of Accra, Cape Coast, the capital of the Central Region is well prepared for your visit with classified accommodation and catering units, established tour operators, efficient transportation and a whole host of activities for you to enjoy.
Most tourists visit this region because of the numerous tourist centers. The region has tourist sites such as; Elmina castle, Fort Patience, Ft Good hope, Ft. saint Jogo, Kakum National Parks, Traditional Fishing Village and craft Village. The most frequents site tourists visit most often is the Elmina Castle and the Kakum National parks.
Elmina Castle: It was the first European building in Ghana (then Gold coast) built from materials imported from Portugal. It was used in 1482 as the Portuguese headquarters. UNESCO World Heritage Foundation has also identified it as world heritage monument. It gives rich information and represents an understanding and appreciation of the early age of European exploitation and interaction between Europeans and Africans. Cape Coast Castle is the largest of the castles, beside Elmina Castle, build by European powers in today Ghana, Known as Gold Coast.
The Kakum National Park is made up of mostly virgin rainforest and is one of the famous national parks in Ghana. It provides the easiest access to the rainforest along with its famous canopy walkway. Kakum National Park is situated about 33km hardwood trees up 65m in height. The Park – Open Monday through Saturday from 6 a.m. to 5pm but It is best to arrive at 7 a.m. to see as much wildlife as possible The park is clearly posted along Jukwa Road roughly 20 miles north of Cape Coast and Elmina. Walking is your only realistic option within the park, and you must have a guide with you at all times. Be advised that the walk to and from the canopy walkway can be difficult for some. There are several stairs and uneven trails during the journey on the canopy pathway.
Central region, your visit to Ghana wouldn't be complete without a visit to the region which is known to be the heartbeat of Ghana tourism.
Next Welcome to the land of the Volta….


text:
\section*{Introduction}

Let $(W,S)$ be a finite Coxeter system whose length function is
denoted by $\ell: W \rightarrow {\mathbb{N}}=\{0,1,2,\dots\}$. In 1976, Solomon
introduced a remarkable subalgebra $\Sigma W$ of the group algebra
${\mathbb{Z}} W$, called the {\it Solomon descent algebra} \cite{solomon}.
Let us recall its definition. If $I \subset S$, let $W_I$ denote
the {\it standard parabolic subgroup} generated by $I$.
Then
$$
X_I =\set{w\in W \st \,\forall\,s \in I,\,\ell(ws)>\ell(w)}
$$
is a set of {\it minimal length coset representatives} of $W/W_I$.
Let $x_I=\sum_{w\in X_I} w \in {\mathbb{Z}} W$. Then $\Sigma W$ is defined
as the sub-${\mathbb{Z}}$-module of ${\mathbb{Z}} W$ spanned by $(x_I)_{I \subset
S}$. Moreover, $\Sigma W$ is endowed with a ${\mathbb{Z}}$-linear map $\theta:
\Sigma W \rightarrow {\mathbb{Z}}\mathop{\mathrm{Irr}}\nolimits W$ satisfying $\theta(x_I)=\mathop{\mathrm{Ind}}\nolimits_{W_I}^W
1_{W_I}$. This is an algebra homomorphism.

If $W$ is the symmetric group ${\mathfrak S}_n$, $\theta$ becomes an
epimorphism and the pair $(\Sigma {\mathfrak S}_n,\theta)$ provides a nice
construction of $\mathop{\mathrm{Irr}}\nolimits({\mathfrak S}_n)$ \cite{jollen}, which is the first
ingredient of several recent works, see for instance
\cite{Thibon,bless}. However, the morphism $\theta$ is surjective
if and only if $W$ is a product of symmetric groups.

In \cite{bonnafe-hohlweg}, we have constructed a subalgebra
$\Sigma'(W_n)$ of the group algebra ${\mathbb{Z}} W_n$ of the Coxeter group
$W_n$ of type $B_n$: it turns out that this subalgebra contains
$\Sigma W_n$, that it is also endowed with an algebra homomorphism
$\theta': \Sigma'(W_n) \rightarrow {\mathbb{Z}}\mathop{\mathrm{Irr}}\nolimits W_n$ extending $\theta$. Moreover,
$\theta'$ is surjective and ${\mathbb{Q}}
\otimes_{\mathbb{Z}} \mathop{\mathrm{Ker}}\nolimits \theta'$ is the radical of ${\mathbb{Q}} \otimes_{\mathbb{Z}}
\Sigma'(W_n)$. This lead to a construction of the irreducible
characters of $W_n$ following J\H ollenbeck's strategy. In fact,
$\Sigma'(W_n)$ is the {\it Mantaci-Reutenauer algebra}
\cite{mantaci-reutenauer}.

It is a natural question to ask whether this kind of construction
can be generalized to other groups.  However, the situation seems
to be much more complicated in the other types. Let us explain now
what kind of subalgebras we are looking for.

Let $T=\{wsw^{-1}~|~w \in W$ and $s \in S\}$ be the set of
reflections in $W$. Let $A$ be a fixed subset of $T$. If $I
\subset A$, we still denote
by $W_I$ the subgroup of $W$ generated by $I$ and we still set
$$
X_I =\set{w\in W \st \,\forall\,x \in W_I,\,\ell(wx) \ge \ell(w)}.
$$
Then $X_I$ is again a set of representatives for $W/W_I$. Now, let
$x_I=\sum_{w \in X_I} w \in {\mathbb{Z}} W$. Then
$$
\Sigma_A(W)=\sum_{I \subset A} {\mathbb{Z}} x_I
$$
is a sub-${\mathbb{Z}}$-module of ${\mathbb{Z}} W$. However, it is not in general a
subalgebra of ${\mathbb{Z}} W$.

Let us define another sub-${\mathbb{Z}}$-module of ${\mathbb{Z}} W$. If $w \in W$,
let
$$
D_A(w)=\{s \in A~|~\ell(ws) < \ell(w)\} \quad\subset A
$$
be the $A$-descent set of $w$. A subset $I$ of $A$ is said to be
{\it $A$-admissible} if there exists $w \in W$ such that
$D_A(w)=I$. Let ${\mathcal{P}}_{\textnormal{ad}}(A)$ denote the set of $A$-admissible
subsets of $A$. If $I \in {\mathcal{P}}_{\textnormal{ad}}(A)$, we set $D_I^A=\{w \in
W~|~D_A(w)=I\}$ and $d_I^A=\sum_{w \in D_I^A} w \in {\mathbb{Z}} W$. Now,
let
$$
{\mathcal{D}}_A(W)=\mathop{\oplus}_{I \in {\mathcal{P}}_{\textnormal{ad}}(A)} {\mathbb{Z}} d_I^A.
$$
As an example, ${\mathcal{D}}_S(W)=\Sigma_S(W)=\Sigma W$.
The main theorem of this paper is the following (here, $C(w)$
denotes the conjugacy class of $w$ in $W$):

\medskip

\noindent{\bf Theorem A.}
{\it If there exists two subsets $S_1$ and $S_2$ of $S$ such that
$A=S_1 \cup (\cup_{s \in S_2} C(s))$, then ${\mathcal{D}}_A(W)$ is a
subalgebra of ${\mathbb{Z}} W$.}

\medskip

This theorem contains the case of Solomon descent algebra (take
$A=S$), the Mantaci-Reutenauer algebra $\Sigma'(W_n)$ (take
$A=\{s_1,\dots,s_{n-1}\} \cup C(t)$, where
$S=\{t,s_1,\dots,s_{n-1}\}$ satisfies $C(t) \cap
\{s_1,\dots,s_{n-1}\}=\varnothing$), and
one of the subalgebras we have constructed in type $G_2$. But it
also gives a new algebra in type $F_4$ of ${\mathbb{Z}}$-rank $300$ (take
$A=\{s_1,s_2\} \cup C(s_3)$, where $S=\{s_1,s_2,s_3,s_4\}$
satisfies $\{s_1,s_2\} \cap C(s_3)=\varnothing$). Moreover, if
$A=T$, we get that ${\mathcal{D}}_A(W)={\mathbb{Z}} W$ (see Example \ref{group
algebra}). In the case of dihedral groups, we get another family
of algebras:

\medskip

\noindent{\bf Theorem B.}
{\it If $W$ is a dihedral group of order $4m$ ($m\geq 1$),
$S=\set{s,t}$ and $A=\set{s,t,sts}$ or $A=\set{t,sts}$, then
${\mathcal{D}}_A(W)$ is a subalgebra of ${\mathbb{Z}} W$.}

\medskip

It must be noted that the algebras constructed in Theorems A and B
are not necessarily unitary. More precisely $1 \in {\mathcal{D}}_A(W)$ if
and only if $S\subset A$. Moreover, if $S \subset A$, then $\Sigma
W \subset {\mathcal{D}}_A(W)$.

The bad point in the above construction is that, in many of the
above cases, one has $\Sigma_A(W) \not= {\mathcal{D}}_A(W)$. Also, we are
not able to construct in general a morphism of algebras $\theta_A:
{\mathcal{D}}_A(W) \rightarrow {\mathbb{Z}} \mathop{\mathrm{Irr}}\nolimits W$ extending $\theta$ if $S \subset A$. It can
be proved that, in some cases, this morphism does not exist.

This paper is organized as follows. Section 1 is essentially
devoted to the proofs of Theorems A and B. One of the key step is
that the subsets $D_I^A$ are left-connected (recall that a subset
$E$ of $W$ is said to be {\it left-connected} if, for all $w$, $w'
\in E$, there exists a sequence $w=w_1$, $w_2$,\dots, $w_r=w'$
of elements of $E$ such that $w_{i+1} w_i^{-1} \in S$ for every $i
\in \{1,2,\dots,r-1\}$). In Section 2, we discuss more precisely
the case of dihedral groups.

\section{Descent sets}

Let $(W,S)$ be a finitely generated Coxeter system (not necessary
finite). If $s$, $s' \in S$, we denote by $m(s,s')$  the order of
$ss'\in W$. If $W$ is finite, we denote by $w_0$ its longest
element.

\subsection{Root system}
Let $V$ be an ${\mathbb{R}}$-vector space endowed with a basis indexed by $S$
denoted by $\Delta=\set{\alpha_s \st s\in S}$. Let $B: V \times V \rightarrow
{\mathbb{R}}$ be the symmetric bilinear form such that
$$
B(\alpha_s,\alpha_{s'}) = - \cos\left(\frac{\pi}{m(s,s')}  \right)
$$
for all $s$, $s' \in S$. If $s \in S$ and $v \in V$, we set
$$
s (v)=v-2B(\alpha_s , v)\alpha_s .
$$
Thus $s$ acts as the reflection in the hyperplane
orthogonal to $\alpha_s$ (for the bilinear form $B$). This extends
to an action of $W$ on $V$ as a group generated by reflections. It
stabilizes $B$.

We recall some basic terminology on root systems.
The {\it root system} of $(W,S)$ is the set $\Phi=\set{w(\alpha_s)\st w\in W,
s\in S}$ and the elements of
$\Delta$ are the {\it simple roots}.  The roots contained in
$$
 \Phi^+ = \big(\sum_{\alpha\in \Delta} {\mathbb{R}}^+ \alpha\big) \cap \Phi
$$
are said to be {\it positive}, while those contained in
$\Phi^-=-\Phi^+$ are said to be {\it negative}. Moreover, $\Phi$
is the disjoint union of $\Phi^+$ and $\Phi^-$. If $w\in W$,
$\ell(w)=\card{N(w)}$, where
$$
N(w)=\set{\alpha \in \Phi^+ \st \ w(\alpha)\in \Phi^-}.
$$

Let $\alpha = w(\alpha_s)\in \Phi$, then $s_\alpha = wsw^{-1}$ acts as the
reflection in the hyperplane orthogonal to $\alpha$
and $s_\alpha = s_{-\alpha}$.
Therefore, the {\it set of reflections of $W$}
$$
T=\bigcup_{w\in W} wSw^{-1}
$$
is in bijection with $\Phi^+$ (and thus $\Phi^-$).

Let us recall the following well-known result:

\begin{lem}\label{important}
Let $w \in W$. Then:
\begin{itemize}
\itemth{a} If $\alpha \in \Phi^+$, then $\ell(ws_\alpha)> \ell(w)$ if and only if
$w(\alpha) \in \Phi^+$.
\itemth{b} If $s \in S$, then
$$N(sw)=\begin{cases}
        N(w) \coprod \{w^{-1}(\alpha_s)\} & \text{if } \ell(sw) > \ell(w), \\
    N(w) \setminus \{-w^{-1}(\alpha_s)\} & \text{otherwise.}
    \end{cases}$$
\end{itemize}
\end{lem}

From now on, and until the end of this paper, we fix a subset $A$
of $T$. We start with easy observations.

As a consequence of Lemma \ref{important} (a), we get that
$$
D_A(w)=\set{s_\alpha \in A \st \alpha \in \Phi^+ \text{ and } w(\alpha)\in
\Phi^-}.
$$
We also set
$$
N_A(w)=\{\alpha \in \Phi^+~|~s_\alpha \in A \text{ and } w(\alpha) \in
\Phi^-\}.
$$
The map $N_A(w) \rightarrow D_A(w)$, $\alpha \mapsto s_\alpha$ is then a
bijection.

\subsection{Properties of the map ${\boldsymbol{D_A}}$}
First, using Lemma \ref{important} (b), we get:

\begin{cor}\label{trivial}
If $s \in S$ and if $w \in W$ is such that $w^{-1} s w=s_{w^{-1}(\alpha_s)} \not\in A$,
then $N_A(w)=N_A(sw)$ (and $D_A(w)=D_A(sw)$).
\end{cor}

\remarque{A1 in A2} If  $A_1\subset A_2 \subset T$, then $D_{A_1} (w)=D_{A_2}(w)\cap A_1$
for all $w\in W$. Therefore if $W$ is finite, ${\mathcal{D}}_{A_1}(W)
\subset {\mathcal{D}}_{A_2}(W)$.~$\square$

\begin{prop}\label{A-admissible}
We have:
\begin{enumerate}
\item[(a)] $\varnothing$ is $A$-admissible.
\item[(b)] $D_\varnothing^A = \set{1}$ if and
only if  $S\subset A$.
\end{enumerate}
\end{prop}

\begin{proof}
We have $D_A(1)=\varnothing$ so (a) follows. If $s \in S\setminus
A$, then $D_A(s)=\varnothing$. This shows (c).
\end{proof}

The notion of $A$-descent set is obviously compatible with direct
products:

\begin{prop}\label{decomposable}
Assume that $W=W_1 \times W_2$ where $W_1$ and $W_2$ are standard
parabolic subgroups of $W$. Then, for all $I \in {\mathcal{P}}_{\textnormal{ad}}(A)$, we
have
$$D_I^A = D_{I \cap W_1}^{A \cap W_1} \times D_{I \cap W_2}^{A \cap W_2}.$$
\end{prop}

\begin{cor}\label{produit algebres}
Assume that $W$ is finite and that $W=W_1 \times W_2$ where $W_1$
and $W_2$ are standard parabolic subgroups of $W$. Then
$${\mathcal{D}}_A(W) = {\mathcal{D}}_{A \cap W_1}(W_1) \otimes_{\mathbb{Z}} {\mathcal{D}}_{A \cap W_2}(W_2).$$
\end{cor}

\exemple{group algebra} Consider the case where $A=T$ (then $N_A(w)=N(w)$).
It is well-known \cite[Chapter VI, Exercise 16]{bourbaki} that the map
$w \mapsto N(w)$ from $W$ onto the set of subsets of $\Phi^+$ is
injective  (observe that if $\alpha\in N(w_1w_2^{-1})$ then $\pm
w_2^{-1}(\alpha)$ lives in the union, but not in the intersection,
of $N(w_1)$ and $N(w_2)$). Therefore, the map $W
\rightarrow
{\mathcal{P}}_{\textnormal{ad}}(T)$,
$w \mapsto D_T(w)$ is injective. In particular, if $W$ is finite,
then ${\mathcal{D}}_T(W)={\mathbb{Z}} W$.~$\scriptstyle{\square}$

\medskip

In the case of finite Coxeter groups, the multiplication on the
left by the longest element has the following easy property.

\begin{prop}\label{w0}
If $W$ is finite and if $w \in W$, then $D_A(w_0 w) = A \setminus
D_A(w)$.
\end{prop}

\begin{cor}\label{A-admissible-finite}
If $W$ is finite, then:
\begin{enumerate}
\item[(a)] $A$ is $A$-admissible;
\item[(b)]  $I\in {\mathcal{P}}_{\textnormal{ad}}(A)$ if and only if $A\setminus I\in
{\mathcal{P}}_{\textnormal{ad}}(A)$;
\item[(c)]  $D_A^A=\set{w_0}$ if and only
if $S\subset A$.
\end{enumerate}
\end{cor}

\begin{proof}
$D_A(w_0)=A$ so (a) follows. (b) follows from Proposition
\ref{w0}. (c) follows from Proposition \ref{w0} and Proposition
\ref{A-admissible} (b).
\end{proof}

\subsection{Left-connectedness}

In \cite{atkinson}, Atkinson gave a new proof of Solomon's result
by using an equivalence relation to describe descent sets. We
extend his result to $A$-descent sets. It shows in particular that
the subsets $D_I^A$ are left-connected.

Let $w$ and $w'$ be two elements of $W$. We say that $w$ is an
\textit{$A$-descent neighborhood} of $w'$, and we write $w
\smile_A w'$,
if $w' w^{-1} \in S$ and $w^{-1}w' \notin A$. It is easily seen
that $\smile_A$ is a symmetric relation. The reflexive and
transitive closure of the $A$-descent neighborhood relation is
called \textit{the $A$-descent equivalence}, and is denoted by
$\sim_A$. The next proposition characterizes this equivalence
relation in terms of $A$-descent sets.

\begin{prop}\label{descent_class}
Let $w$, $w' \in W$. Then
$$
w\edclass w' \Leftrightarrow D_A(w)=D_A(w') \Leftrightarrow N_A(w)=N_A(w').
$$
\end{prop}

\begin{proof}
The second equivalence is clear. If $w \smile_A w'$, then it
follows from Corollary \ref{trivial} that $N_A(w)=N_A(w')$. It
remains to show that, if $N_A(w)=N_A(w')$, then $w \sim_A w'$.

So, assume that $N_A(w)=N_A(w')$. Write $x=w' w^{-1}$ and let
$m=\ell(x)$. If $\ell(x)=0$, then $w=w'$ and we are done. Assume
that $m \ge 1$ and write $x=s_1 s_2 \dots s_m$ with $s_i \in S$.
We now want to prove on induction on $m$ that
$$w \smile_A s_m w \smile_A s_{m-1} s_m w \smile_A \dots \smile_A s_2 \dots s_m w
\smile_A s_1 s_2 \dots s_m w=w'.\leqno{(*)}$$
First, assume that $w \not\smile_A s_m w$. In other words, $w^{-1}
s_m w \in A$. For simplification, let $\alpha_i=\alpha_{s_i}$. By Lemma
\ref{important} (b), we have
$$N_A(s_m w) = N_A(w) \coprod \{w^{-1}(\alpha_m)\} \quad \text{or} \quad
N_A(w) \setminus \{-w^{-1}(\alpha_m)\}.
$$
In the first case, as $N_A(w)=N_A(w')$, and by applying again
Lemma \ref{important} (b), there exists a step $i \in
\{1,2,\dots,m-1\}$ between $N_A(w)$ to $N_A(w')$ where
$w^{-1}(\alpha_m)$ is removed from $N_A(s_i\dots s_m w)$, that is,
$(s_{i+1}\dots s_m w)^{-1}(\alpha_i)=-w^{-1}(\alpha_m)$. In the same way,
we get the same result in the second case. In other words, we have
proved that there exists $i \in \{1,2,\dots,m-1\}$ such that
$s_m\dots s_{i+1}(\alpha_i)=-\alpha_m$, so, by Lemma
\ref{important} (a), we have $\ell(s_m \dots s_{i+1} s_i) <
\ell(s_m \dots s_{i+1})$. This contradicts the fact that
$m=\ell(x)$. So $w \smile_A s_m w$, and  then $N_A(w)=N_A(s_m w)$.
Hence $N_A(s_m w)=N_A(w')$ and $w' (s_m w)^{-1} = s_1 s_2 \dots
s_{m-1}$.  We get then by induction that
$s_m w \smile_A s_{m-1} s_m w \smile_A \dots \smile_A s_1 s_2
\dots s_m w=w'$, which shows $(*)$.
\end{proof}

\begin{cor}
If $I \in {\mathcal{P}}_{\textnormal{ad}}(A)$, then $D_I^A$ is left-connected.
\end{cor}

\subsection{Nice subsets of ${\boldsymbol{T}}$}

We say that $A$ is {\it nice} if, for every $s \in A$ and $w \in
W$ such that $w^{-1} s w \not\in A$, we have $D_A(sw)=D_A(w)$.
Notice that every subset of $S$ is nice, by
Corollary~\ref{trivial}.

If $w\in W$ and $I$, $J\in{\mathcal{P}}_{\textnormal{ad}} (A)$, we set
$$
D_A(I,J,w) = \set{(u,v) \in D_I^A \times D_J^A\st uv=w }.
$$
The next lemma gives an obvious characterization of the fact
that ${\mathcal{D}}_A(W)$ is an algebra in terms of these sets.

\begin{lem}\label{key}
Assume that $W$ is finite. Then the following are equivalent:
\begin{itemize}
\itemth{1} ${\mathcal{D}}_A(W)$ is a subalgebra of ${\mathbb{Z}} W$.
\itemth{2} For all $I$, $J \in {\mathcal{P}}_{\textnormal{ad}}(A)$ and for all $w$, $w' \in W$
such that $D_A(w)=D_A(w')$, we have $|D_A(I,J,w)|=|D_A(I,J,w')|$.
\end{itemize}
If these conditions are fulfilled,
we choose for any $I\in{\mathcal{P}}_{\textnormal{ad}} (A)$ an element $z_I$ in $D_I^A$.
Then
$$d_I^A d_J^A = \sum_{K \in {\mathcal{P}}_{\textnormal{ad}}(A)} |D_A(I,J,z_K)| d_K^A.$$
\end{lem}

\def{\mathrm{Id}}{{\mathrm{Id}}}

Let us fix now $s \in S$ and  let $(u,v) \in W \times W$. If $u
\smile_A su$, we set $\psi_s^A(u,v)=(su,v)$. If
$u \not\smile_A su$, then we set $\psi_s^A(u,v)=(u,u^{-1}su v)$.
Note that, in the last case, $u^{-1}su \in A$. We have
$(\psi_s^A)^2 = {\mathrm{Id}}_{W \times W}$. In particular, $\psi_s^A$ is a
bijection. Using $\psi_s^A$, one can relate the notion of nice
subsets to the property (2) stated in Lemma \ref{key}.

\begin{prop}\label{key 2}
Assume that $A$ is nice. Let $I$, $J \in {\mathcal{P}}_{\textnormal{ad}}(W)$, let $w \in
W$ and let $s \in S$ be such that $w \smile_A s w$. Then
$\psi_s^A(D_A(I,J,w))=D_A(I,J,sw)$.
\end{prop}

\begin{proof}
Let $(u,v)$ be an element of $D_A(I,J,w)$. By symmetry, we only need to prove
that $\psi_s^A(u,v) \in D_A(I,J,sw)$. If $u \smile_A su$, then
$D_A(su)=D_A(u)=I$ by Proposition \ref{descent_class}, so
$\psi_s^A(u,v)=(su,v) \in D_A(I,J,sw)$. So, we may, and we will,
assume that $u \not\smile_A su$. Let $s'=u^{-1} s u \in A$. Note
that $w^{-1} s w =v^{-1}s'v \not\in A$. Then,
$\psi_s^A(u,v)=(u,s'v)$ and $us'v=sw$. So, we only need to prove
that $D_A(s'v)=D_A(v)$. But this just follows from the definition
of nice subset of $T$.
\end{proof}

\begin{cor}\label{voila}
If $W$ is finite and if $A$ is nice, then ${\mathcal{D}}_A(W)$ is a
subalgebra of ${\mathbb{Z}} W$. It is unitary if and only if $S \subset A$.
\end{cor}

\begin{proof}
This follows from Lemma \ref{key} and from Propositions
\ref{descent_class} and \ref{key 2}.
\end{proof}

\remarque{question nice} It would be interesting to know if the
converse of Corollary \ref{voila} is true.~$\scriptstyle{\square}$

\subsection{Proof of Theorems A and B}

Using Corollary \ref{voila}, we see that Theorems A and B are
direct consequences of the following theorem (which holds also for
infinite Coxeter groups):

\begin{thm}\label{le theoreme}
Assume that one of the following holds:
\begin{itemize}
\itemth{1} There exists two subsets $S_1$ and $S_2$ of $S$ such that
$A=S_1 \cup (\cup_{s \in S_2} C(s))$.
\itemth{2} $S=\{s,t\}$, $m(s,t)$ is even or $\infty$,
and $A=\set{s,t,sts}$ or $A=\set{t,sts}$.
\end{itemize}
Then $A$ is nice.
\end{thm}

\begin{proof}
Assume that (1) or (2) holds. Let $r \in A$ and let $w \in W$ be
such that $w^{-1} r w \not\in A$. We want to prove that
$D_A(rw)=D_A(w)$. By symmetry, we only need to show that $D_A(rw)
\subset D_A(w)$. If $r \in S$, then this follows from Corollary
\ref{trivial}. So we may, and we will, assume that $r \notin S$.

$\bullet$ Assume that (1) holds. Write $A'=\cup_{s \in S_2} C(s)$
and $S' = A \setminus A'$. Then $S' \subset S$, $A=A' \coprod S'$
and $A'$ is stable under conjugacy. Then $r\in A'$ and $w^{-1} r w
\in A'\subset A$, which contradicts our hypothesis.

$\bullet$ Assume that (2) holds. If $m(s,t)=2$, then $A$ is
contained in $S$ and therefore is nice by Corollary~\ref{trivial}.
So we may, and we will, assume that
$m(s,t)\ge 4$. Since $r \not\in S$, we have $r=sts$.
Assume that $D_A(rw)\not\subset D_A(w)$. Let
$$\Phi_A=\{\alpha \in \Phi^+~|~s_\alpha \in A\} \subset \{\alpha_s,\alpha_t,s(\alpha_t)\}.$$
There exists $\alpha \in \Phi_A$ such that $rw(\alpha) \in \Phi^-$ and
$w(\alpha) \in
\Phi^+$. So $w(\alpha) \in N(r)=\{\alpha_s,s(\alpha_t),st(\alpha_s)\}$. Since
$w^{-1} r w \not\in A$, we have that $w s_\alpha w^{-1} \not= r$, so
$w(\alpha) \not= s(\alpha_t)$. So $w(\alpha)=\alpha_s$ or $st(\alpha_s)$. But the
roots $\alpha_s$ and $\alpha_t$ lie in different $W$-orbits. So $\alpha=\alpha_s$
and $w(\alpha_s) \in \{\alpha_s,st(\alpha_s)\}$. If $W$ is infinite, this
gives that $w \in \{1,st\}$. This contradicts the fact that
$w^{-1} rw \not\in A$. If $W$ is finite, this gives that $w \in
\{1,st,w_0s,sts w_0\}$. But again, this contradicts the fact that
$w^{-1} rw \not\in A$.
\end{proof}

\exemple{f4}
Assume here that $W$ is of type $F_4$, that
$S=\{s_1,s_2,s_3,s_4\}$ and that $A=C(s_1) \cup S$. Then, using
{\tt GAP}, one can see that $\mathop{\mathrm{rank}}\nolimits_{\mathbb{Z}} {\mathcal{D}}_A(W)=300$ and
$\mathop{\mathrm{rank}}\nolimits_{\mathbb{Z}} \Sigma_A(W)=149$. Moreover, $\Sigma_A(W)$ is not a
subalgebra of ${\mathcal{D}}_A(W)$.~$\scriptstyle{\square}$

\medskip

Some computations with {\tt GAP} suggest that the following
question has a positive answer:

\smallskip

\begin{quotation}
\noindent{\bf Question. } {\it Let $A$ be a nice subset of $T$
containing $S$. Is it true that $A$ is one of the subsets
mentioned in Theorem \ref{le theoreme}?}
\end{quotation}

\def{\bar{s}}{{\bar{s}}}
\def{\bar{t}}{{\bar{t}}}

\section{The dihedral groups}\label{section 2}

The aim of this section is to study the unitary subalgebras
${\mathcal{D}}_A(W)$ constructed in Theorems A and B whenever $W$ is finite
and dihedral.

\smallskip

\begin{quotation}
\noindent{\bf Hypothesis:} {\it From now on, and until the end of
this paper, we assume that $S=\{s,t\}$ with $s \not= t$ and that
$m(s,t)=2m$, with $2 \le m < \infty$.}
\end{quotation}

\smallskip

Note that $w_0=(st)^m$ is central. We denote by ${\mathcal{P}}_0(A)$ the set
of subsets $I$ of $A$ such that $W_I \cap A = I$. With this
notation, we have
$$\Sigma_A(W)=\sum_{I \in {\mathcal{P}}_0(A)} {\mathbb{Z}} x_I.$$
We now define an equivalence relation $\equiv$ on ${\mathcal{P}}_0(A)$: if
$I$, $J \in {\mathcal{P}}_0(A)$, we write $I \equiv J$ if $W_I$ and $W_J$
are conjugate in $W$. We set
$$\Sigma^{(1)}_A(W)=
\sum_{\scriptstyle{I,J \in {\mathcal{P}}_0(A)} \atop \scriptstyle{I \equiv J}} {\mathbb{Z}}(x_I-x_J).$$

In what follows, we will also need some facts on the character
table of $W$. Let us recall here the construction of $\mathop{\mathrm{Irr}}\nolimits W$.
First, let $H$ be the subgroup of $W$ generated by $st$. It is
normal in $W$, of order $2m$ (in other words, of index $2$). We
choose the primitive $(2m)$-th root of unity $\zeta \in {\mathbb{C}}$ of
argument $\frac{\pi}{m}$. If $i \in
{\mathbb{Z}}$, we denote by $\xi_i: H \rightarrow {\mathbb{C}}^\times$ the unique linear
character such that $\xi_i(st)=\zeta^i$. Then $\mathop{\mathrm{Irr}}\nolimits H = \{\xi_i~|~0
\le i \le 2m-1\}$. Now, let
$$
\chi_i=\mathop{\mathrm{Ind}}\nolimits_H^W \xi_i.
$$
Then $\chi_i=\chi_{2m-i}$ and, if $1 \le i \le m-1$, $\chi_i \in
\mathop{\mathrm{Irr}}\nolimits W$. Also, $\chi_i$ has values in ${\mathbb{R}}$. More precisely, for
$1 \le i \le m-1$ and $j\in{\mathbb{Z}}$
$$
\chi_i\big((ts)^j\big)=\zeta^j + \zeta^{-j}=2\cos\left(\frac{ij\pi}{m}\right)
\quad\textrm{and}\quad\chi_i\big(s(ts)^j\big)=0.
$$
Let $1$ denote the trivial character of $W$, let $\varepsilon$ denote the
sign character and let $\gamma: W \rightarrow \{1,-1\}$ be the unique linear
character such that $\gamma(s)=-\gamma(t)=1$. Then
\refstepcounter{prop}$$~\label{irreductibles W} \mathop{\mathrm{Irr}}\nolimits W = \{1,\varepsilon,\gamma,\varepsilon\gamma\} \cup
\{\chi_i~|~1 \le i \le m-1\}.
\leqno{\boldsymbol{(\arabic{section}.\arabic{prop})}}~$$
In particular, $|\mathop{\mathrm{Irr}}\nolimits W|=m+3$.

\subsection{The subset ${\boldsymbol{A=\{s,t,sts\}}}$}\label{A}

From now on, we assume that $A=\{s,t,sts\}$.  We set
${\bar{s}}=A \setminus \{s\}=\{t,sts\}$ and ${\bar{t}}=A
\setminus\{t\}=\{s,sts\}$. It is easy to see that
$${\mathcal{P}}_{\textnormal{ad}}(A)=\{\varnothing,\{s\},\{t\},{\bar{s}},{\bar{t}},A\}.$$
For simplification,  we will denote by $d_I$ the element $d_I^A$ of
${\mathbb{Z}} W$ (for $I \in {\mathcal{P}}_{\textnormal{ad}}(W)$) and we set $d_s=d_{\{s\}}$ and
$d_t=d_{\{t\}}$. We have
$$
\begin{array}{rcl@{\qquad}rcl}
d_\varnothing &=& 1, & d_{\bar{s}}&=& w_0 s, \\
d_s   &=& s, &d_A   &=& w_0, \\
d_t   &=&\displaystyle{\sum_{i=1}^{m-1}} \Bigl((st)^i + (ts)^{i-1}t\Bigr), &
d_{\bar{t}}&=&\displaystyle{\sum_{i=1}^{m-1}} \Bigl((st)^is + (ts)^i\Bigr). \\
\end{array}
$$
The multiplication table of ${\mathcal{D}}_A(W)$ is given by

\medskip
{\small
\begin{centerline}
{\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$         & $1$      & $d_s$    & $d_{\bar{s}}$ & $d_A$    & $d_t$    & $d_{\bar{t}}$ \\
\hline
\hline
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $1$      & $1$      & $d_s$    & $d_{\bar{s}}$ & $d_A$    & $d_t$    & $d_{\bar{t}}$ \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $d_s$    & $d_s$    & $1$      & $d_A$    & $d_{\bar{s}}$ & $d_t$    & $d_{\bar{t}}$ \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $d_{\bar{s}}$ & $d_{\bar{s}}$ & $d_A$    & $1$      & $d_s$    & $d_{\bar{t}}$ & $d_t$    \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $d_A$    & $d_A$    & $d_{\bar{s}}$ & $d_s$    & $1$      & $d_{\bar{t}}$ & $d_t$    \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $d_t$    & $d_t$    & $d_{\bar{t}}$ & $d_t$    & $d_{\bar{t}}$ & $z_A$    & $z_A$    \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$ $d_{\bar{t}}$ & $d_{\bar{t}}$ & $d_t$   & $d_{\bar{t}}$ & $d_t$    & $z_A$    & $z_A$    \\
\hline
\end{tabular}}
\end{centerline}
}

\medskip

\noindent where
$z_A=(m-1)\big(1+d_A+d_s+d_{\bar{s}}\big)+(m-2)\big(d_t+d_{\bar{t}}\big)$.
We now study the sub-${\mathbb{Z}}$-module $\Sigma_A(W)$: we will show that
it coincides with ${\mathcal{D}}_A(W)$. First, it is easily seen that
$$
{\mathcal{P}}_0 (A)=\set{\varnothing,\, \set{s},\, \set{t},\, \set{sts},\, {\bar{s}},\, A}
$$
and that
$$\begin{array}{ccclllll}
x_A    & = & 1 &&&&& \\
x_{\bar{s}} & = & 1 & + d_s &&&& \\
x_{sts} & = & 1 & + d_s & + d_t &&& \\
x_t    & = & 1 & + d_s &       & +d_{\bar{t}} && \\
x_s    & = & 1 &       & + d_t &         & + d_{\bar{s}} & \\
x_\varnothing&=&1&+d_s & + d_t & +d_{\bar{t}} & + d_{\bar{s}} & + d_A \\
\end{array}$$
Therefore, $\Sigma_A(W)={\mathcal{D}}_A(W)=\oplus_{I \in {\mathcal{P}}_0(A)} {\mathbb{Z}} x_I$.
So we can define a map $\theta_A: \Sigma_A(W) \rightarrow {\mathbb{Z}} \mathop{\mathrm{Irr}}\nolimits W$ by
$\theta_A(x_I)=\mathop{\mathrm{Ind}}\nolimits_{W_I}^W 1_{W_I}$.

\begin{prop}\label{s,t,sts}
Assume that $S=\{s,t\}$ with $s \not= t$, that $m(s,t)=2m$ with $m
\ge 2$ and that $A=\{s,t,sts\}$. Then:
\begin{itemize}
\itemth{a} $\Sigma_A(W)={\mathcal{D}}_A(W)$ is a subalgebra of ${\mathbb{Z}} W$ of ${\mathbb{Z}}$-rank $6$.
\itemth{b} $\theta_A$ is a morphism of algebras.
\itemth{c} $\mathop{\mathrm{Ker}}\nolimits \theta_A = {\mathbb{Z}} (x_t - x_{sts}) = \Sigma_A^{(1)}(W)$.
\itemth{d} ${\mathbb{Q}} \otimes_{\mathbb{Z}} \mathop{\mathrm{Ker}}\nolimits \theta_A$ is the
radical of ${\mathbb{Q}} \otimes_{\mathbb{Z}} \Sigma_A(W)$.
\itemth{e} $\theta_A$ is surjective if and only if $m=2$ that is, if and only if
$W$ is of type $B_2$.
\end{itemize}
\end{prop}

\begin{proof}
(a) has already been proved.
For proving the other assertions, we need to compute explicitly
the map $\theta_A$. It is given by the following table:
$$\begin{array}{|c||c|c|c|c|c|c|}
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} d_I & 1 & d_s & d_{\bar{s}} & d_A & d_t & d_{\bar{t}} \\
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} \theta_A(d_I)&1&\varepsilon\gamma&\gamma&\varepsilon&\displaystyle{\sum_{i=1}^{m-1}}\chi_i&\displaystyle{\sum_{i=1}^{m-1}}\chi_i\\
\hline
\end{array}$$

(c) This shows that $\mathop{\mathrm{Ker}}\nolimits \theta_A
={\mathbb{Z}}(d_{\bar{t}}-d_t)={\mathbb{Z}}(x_t-x_{sts})=\Sigma_A^{(1)}(W)$, so (c)
holds.

(b) To prove that $\theta_A$ is a morphism of algebras, the only
difficult point is to prove that $\theta_A(d_t^2)=\theta_A(d_t)^2$. Let
$\rho$ denote the regular character of $W$. Then
$$\theta_A(d_t)=\frac{1}{2}\big(\rho-1-\gamma-\varepsilon\gamma-\varepsilon).$$
Therefore,
$$\theta_A(d_t)^2= (m-2) \rho + 1+\gamma+\varepsilon\gamma+\varepsilon.$$
But, $d_t^2=z_A=(m-2) x_\varnothing + 1 + d_s + d_{\bar{s}} + d_A$.
This shows that $\theta_A(d_t^2)=\theta_A(d_t)^2$.

(d) Let $R$ denote the radical of ${\mathbb{Q}} \otimes_{\mathbb{Z}} \Sigma_A(W)$.
We only need to prove that ${\mathbb{C}}
\otimes_{\mathbb{Q}} R = {\mathbb{C}} \otimes_{\mathbb{Z}} \Sigma_A(W)$. Since ${\mathbb{C}}\mathop{\mathrm{Irr}}\nolimits W$ is
a split semisimple commutative algebra, every subalgebra of
${\mathbb{C}}\mathop{\mathrm{Irr}}\nolimits W$ is semisimple. So $({\mathbb{C}}
\otimes_{\mathbb{Z}} \Sigma_A(W))/({\mathbb{C}} \otimes_{\mathbb{Z}} \mathop{\mathrm{Ker}}\nolimits \theta_A)$ is a
semisimple algebra. This shows that $R$ is contained in ${\mathbb{C}}
\otimes_{\mathbb{Z}} \mathop{\mathrm{Ker}}\nolimits \theta_A$. Moreover, since
$(x_t-s_{sts})^2=(d_t-d_{\bar{t}})^2=0$, $\mathop{\mathrm{Ker}}\nolimits \theta_A$ is a nilpotent
two-sided ideal of $\Sigma_A(W)$. So ${\mathbb{C}} \otimes_{\mathbb{Z}} \mathop{\mathrm{Ker}}\nolimits \theta_A$
is contained in ${\mathbb{C}} \otimes_{\mathbb{Z}} R$. This shows (d).

(e) If $m=2$, then $\mathop{\mathrm{Irr}}\nolimits W=\{1,\gamma,\varepsilon\gamma,\varepsilon,\chi_1\}
=\theta_A(\{1,d_s,d_{\bar{s}},d_A,d_t\})$ so $\theta_A$ is surjective.
Conversely, if $\theta_A$ is surjective, then $|\mathop{\mathrm{Irr}}\nolimits W|=5$ (by (a)
and (c)). Since $|\mathop{\mathrm{Irr}}\nolimits W|=m+3$, this gives $m=2$.
\end{proof}

We close this subsection by giving, for $\Sigma_A(W)$, a complete set
of orthogonal primitive idempotents,  extending to our case those
given in \cite{bbht}, and the corresponding  irreducible
representations:
$$
E_\varnothing=\frac{1}{4m}x_\varnothing, \qquad
E_s=\frac{1}{2}\left(x_s-\frac{1}{2}x_\varnothing\right), \qquad
E_t=\frac{1}{2}\left(x_t-\frac{1}{2}x_\varnothing\right),
$$
\begin{eqnarray*}
E_{\bar s}&=&\frac{1}{2}\left(x_{\bar s}-\frac{1}{2}x_t-\frac{1}{2}x_{sts}+\frac{m-1}{2m}x_\varnothing\right)\\
E_A&=&1-\frac{1}{2}x_s-\frac{1}{4}x_t+\frac{1}{4}x_{sts}-\frac{1}{2}x_{\bar{s}}+\frac{1}{4}x_\varnothing
\end{eqnarray*}

\def{\mathrm{ev}}{{\mathrm{ev}}}

For $i \in \mathcal  \{\varnothing, s,t,{\bar{s}},A\}$, we denote by
$P_i$ the indecomposable projective module ${\mathbb{Q}}\Sigma_A(W) E_i$.
Let ${\mathbb{Q}}_i^A$ denote the unique simple ${\mathbb{Q}}\Sigma_A(W)$-module
lying in the head of $P_i$. It is easily seen, using the previous
multiplication table, that
$$\dim P_\varnothing=\dim P_t = \dim P_{\bar{s}} = \dim P_A=1$$
$$\dim P_s =2\leqno{\text{and that}}$$
(note that $x_{\bar{s}} E_s = -(x_t-x_{sts})$). In other words,
$$P_\varnothing={\mathbb{Q}}_\varnothing^A,\quad P_t = {\mathbb{Q}}_t^A, \quad P_{\bar{s}}={\mathbb{Q}}_{\bar{s}}^A,\quad
P_A={\mathbb{Q}}_A^A$$
$${\textnormal{Rad}}(P_s)\simeq {\mathbb{Q}}_t^A.\leqno{\text{and}}$$

For $w \in W$, we denote by ${\mathrm{ev}}_w$ the morphism of algebras
${\mathbb{Z}}\mathop{\mathrm{Irr}}\nolimits W \rightarrow {\mathbb{Z}}$, $\chi \mapsto
\chi(w)$ and we set ${\mathrm{ev}}_w^A = {\mathrm{ev}}_w \circ \theta_A$. Then
the morphism of algebras
$\Sigma_A(W) \rightarrow {\mathbb{Z}}$ associated to the
simple module ${\mathbb{Q}}_i^A$ is ${\mathrm{ev}}_{f(i)}^A$, where
$$f(\varnothing)=1,\quad f(s)=s,\quad f(t)=t, \quad f({\bar{s}})= (st)^m = w_0
\quad{\text{and}}\quad f(A)=st.$$

\subsection{The subset $\boldsymbol{B=\set{s}\cup C(t)}$}

Let $B=\set{s}\cup C(t)$ (so that $|B|=m+1$).
It is easy to see that
${\mathcal{P}}_{\textnormal{ad}}(B)$ consists of the sets
$$
\begin{array}{ccl}
\varnothing                &=& D_B(1), \\
B                        &=& D_B(w_0), \\
\set{s}                  &=& D_B(s), \\
C(t)                     &=& D_B (w_0s), \\
D_B\big((ts)^i \big)     &=& D_B\big( s(t s)^i\big), \quad 1\leq i\leq m-1,\\
D_B\big((st)^{j}\big) &=& D_B\big( (t s)^{j-1} t\big), \quad 1\leq j\leq m-1. \\
\end{array}
$$
Therefore ${\mathcal{D}}_B(W)$ is a subalgebra of ${\mathbb{Z}} W$ of ${\mathbb{Z}}$-rank
$(2m+2)$.

Using {\tt GAP}, we can see that, in general, $\Sigma_B
(W)\not={\mathcal{D}}_B(W)$. First examples are given in the following table
$$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|}
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}  m                                   & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11    \\
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} {\mathbb{Z}}\textnormal{-rank of }{\mathcal{D}}_B(W)    & 6 & 8 & 10& 12& 14& 16& 18& 20& 22 & 24    \\
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} {\mathbb{Z}}\textnormal{-rank of }\Sigma_B(W) & 6 & 8 & 10& 10& 14& 12& 18& 18& 22 & 16 \\
\hline
\end{array}$$

\remarque{surjectif}
The linear map $\theta_B:\Sigma_B(W)\rightarrow {\mathbb{Z}}\mathop{\mathrm{Irr}}\nolimits
W$, $x_I\mapsto \mathop{\mathrm{Ind}}\nolimits_{W_I}^W 1_{W_I}$, is well-defined and
surjective if and only if $m \in \{2,3\}$ (recall that $m \ge 2$).
Indeed, the image of $\theta_B$ can not contain a non-rational
character. But all characters of $W$ are rational if and only if $W$
is a Weyl group, that is, if and only if $2m\in\set{2,3,4,6}$.

Moreover, if $m=2$, then $A=B$ and $\theta_B=\theta_A$ is surjective by
Proposition~\ref{s,t,sts}. If $m=3$, then this follows from
Proposition \ref{s,t,sts,tstst} below.~$\scriptstyle{\square}$

\subsection{The algebra $\boldsymbol{{\mathcal{D}}_B(G_2)}$}

From now on, and until the end, we assume that $m=3$. That is $W$
is of type $I_2(6)=G_2$. For convenience, we keep the same
notation as in \S\ref{A}. We have
$$\begin{array}{rcl}
d_\varnothing &=& 1,\\
d_s         &=& s,\\
d_{\bar{s}}      &=& d^B_{C(t)}            = w_0 s,\\
d_A         &=& d^B_B                 = w_0,
\end{array}
\qquad
\begin{array}{rcl}
d_1         &=& d^B_\set{t}           =t+st,\\
d_2         &=& d^B_\set{s,sts}       = ts+sts,\\
d_3         &=& d^B_\set{s,sts,tstst} = tsts+ststs,\\
d_4         &=& d^B_\set{t,tstst}     = tst+stst.
\end{array}
$$

Let us now show that $\Sigma_B(W)={\mathcal{D}}_B(W)$. First, it is easily seen that
$$
{\mathcal{P}}_0 (B)=\set{\varnothing,\, \set{s},\, \set{t},\, \set{sts},\,
{\bar{s}},\, \set{tstst},\set{s,tstst},B}
$$
and that
$$\begin{array}{ccclllllll}
x_A = x_B            & = & 1 &&&&& &&\\
x_{\bar{s}}              & = & 1 & + d_s &&&&&& \\
x_\set{s,tstst} & = & 1 &       &+d_1 &&&&&\\
x_{tstst} & = & 1 &  + d_s&+d_1 & + d_2  &&&&     \\
x_t                 & = & 1 & +d_s  &     & +d_2       & +d_3&&&\\
x_{sts}             & = & 1 & + d_s & + d_1 &&&+d_4&& \\
x_s                 & = & 1 &       & + d_1 &         &  &+d_4&+d_{\bar{s}}& \\
x_\varnothing       & = &1&+d_s & + d_1 &+d_2& +d_3 & + d_4 & + d_{\bar{s}} &+d_A\\
\end{array}$$
Therefore, $\Sigma_B(W)={\mathcal{D}}_B(W)=\oplus_{I \in {\mathcal{P}}_0(B)} {\mathbb{Z}}
x_I$. So we can define a map $\theta_B: \Sigma_B(W) \rightarrow {\mathbb{Z}} \mathop{\mathrm{Irr}}\nolimits W$
by $\theta_B(x_I)=\mathop{\mathrm{Ind}}\nolimits_{W_I}^W 1_{W_I}$.

\begin{prop}\label{s,t,sts,tstst}
Assume that $S=\{s,t\}$ with $s \not= t$, that $m(s,t)=6$ and that
$B=\{s,t,sts,tstst\}$. Then:
\begin{itemize}
\itemth{a} $\Sigma_B(W)={\mathcal{D}}_B(W)$ is a subalgebra of ${\mathbb{Z}} W$ of
${\mathbb{Z}}$-rank $8$.
\itemth{b} $\theta_B$ is a surjective  linear map (and not a morphism
of algebras).
\itemth{c} $\mathop{\mathrm{Ker}}\nolimits \theta_B = {\mathbb{Z}} (x_{tstst} -x_t)\oplus{\mathbb{Z}}(x_{tstst}-
x_{sts}) = \Sigma_B^{(1)}(W)$.
\itemth{d} $\mathop{\mathrm{Irr}}\nolimits W=\theta_B\left(\set{1,d_s,d_{\bar{s}},d_A,d_1,d_2}\right)$.
\end{itemize}
\end{prop}

\begin{proof}
The map $\theta_B$ is given by the following table:
$$\begin{array}{|c||c|c|c|c|c|c|c|c|}
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} d^B_I & 1 & d_s & d_{\bar{s}} & d_A & d_1 & d_2 & d_3&d_4 \\
\hline
\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}} \theta_B(d^B_I)&1&\varepsilon\gamma&\gamma&\varepsilon&\chi_2&\chi_1&\chi_2 &\chi_1 \\
\hline
\end{array}$$

This shows (b) and (d). As $d_1-d_3=x_\set{tstst}-x_t$ and
$d_2-d_4=x_{tstst}-x_{sts}$, (c) is proved. Finally, the fact that
$\theta_B$ is not a morphism of algebras follows from
$\theta_B(d_1^2)(w_0)=(1+\varepsilon\gamma+\chi_1)(w_0)=-2\not=
\theta_B(d_1)^2(w_0)=\chi_2(w_0)^2=4.$
\end{proof}

For information,
the multiplication table in the basis $(d_I^B)_{I \in {\mathcal{P}}_{\textnormal{ad}}(B)}$ is given by

\medskip

{\small
\begin{centerline}
{\begin{tabular}{c}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$             & $1$        & $d_s$      & $d_1$         & $d_2$         & $d_3$                  & $d_4$                   & $d_{{\bar{s}}}$ & $d_A$      \\
\hline
\hline
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $1$        & $1$        & $d_s$      & $d_{1}$       & $d_2$         & $d_3$                  & $d_4$                   & $d_{{\bar{s}}}$ & $d_A$      \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_s$      & $d_s$      & $1$        & $d_{1}$       & $d_2$         & $d_3$                  & $d_4$                   & $d_A$      & $d_{{\bar{s}}}$ \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_1$      & $d_{1}$    & $d_2$      & $1 + d_s + d_4$ & $1 + d_s + d_3$ & $d_2 + d_{{\bar{s}}} + d_A$ & $d_1 + d_{{\bar{s}}} + d_A$  & $d_4$  & $d_3$      \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_2$      & $d_2$      & $d_{1}$    & $1 + d_s + d_4$ & $1 + d_s + d_3$ & $d_2 + d_{{\bar{s}}} + d_A$ & $d_1 + d_{{\bar{s}}} + d_A$  & $d_3$  & $d_4$      \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_3$      & $d_3$      & $d_4$      & $d_2 + d_{{\bar{s}}} + d_A$ & $d_1 + d_{{\bar{s}}} + d_A$ & $1 + d_s + d_4$ & $1 + d_s + d_3$  &  $d_2$   & $d_{1}$    \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_4$      & $d_4$      & $d_3$      & $d_2 + d_{{\bar{s}}} + d_A$ & $d_1 + d_{{\bar{s}}} + d_A$ & $1 + d_s + d_4$ & $1 + d_s + d_3$  & $d_1$    & $d_2$      \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_{{\bar{s}}}$ & $d_{{\bar{s}}}$ & $d_A$      & $d_3$         & $d_4$         &  $d_{1}$               & $d_2$                   & $1$      &  $d_s$       \\
$\vphantom{\frac{\displaystyle{A}}{\displaystyle{A}}}$  $d_A$      & $d_A$      & $d_{{\bar{s}}}$ & $d_3$         & $d_4$         &  $d_{1}$               & $d_2$                   & $d_s$        &  $1$     \\
\hline
\end{tabular}\\
\end{tabular}}
\end{centerline}
}

\medskip

redpajama_set_name: RedPajamaArXiv


This panel was created as a direct reaction to the news of the great European catastrophe. Herman Wald's took a trunk of shamfuti wood and submitted himself to its material and the line of its grain, carving out these three tortured human figures, their arms upstretched like burning flames and their feet and legs consumed by fire. With this panel Herman Wald fuses Jewish martyrdom throughout the ages, from the Medieval period to the Inquisition with the horrors of the Holocaust.


text:
Manchin Pays Tribute to American Flag
Senator Manchin and staff members participated in flag presentations across West Virginia to honor Flag Day
Washington, D.C. – U.S. Senator Joe Manchin (D-W.Va.) today celebrated Flag Day by honoring veterans and meeting with seniors, community leaders and area residents in Greenbrier County. He spent the morning at Fairlea Senior Center and then paid tribute to the American flag with a community flag presentation at Greenbrier County Community Based Outpatient Clinic.
Senator Manchin also submitted into the official record a congressional record statement to commemorate Flag Day.
Representatives from Senator Manchin's office also spent Friday recognizing Flag Day in West Virginia by making stops in Logan, Braxton, Ohio, Jefferson, Wood, Hardy and Barbour counties to present flags to the local communities.
"Like all West Virginians, I feel a special surge of emotion every time I see the American flag," Senator Manchin said. "Every day, schoolchildren, Scouts, veterans, Senators – in fact, Americans all across this great land – pledge their allegiance to the Stars and Stripes. But one day a year, we pay special honor to our flag. I am thankful to celebrate Flag Day with fellow West Virginians, especially during our state's 150th birthday. Displaying the flag, our most enduring symbol, represents the unity of our people and the cause of liberty and justice for all."
To see photos from the Flag Day events, please click here.
Below is the full text of Senator Manchin's congressional record statement:
Mr. President, like all West Virginians, I feel a special surge of emotion every time I see the American flag. After all, Old Glory is the most enduring symbol of our country, representing the unity of our people and the cause of liberty and justice for all.
But the Star Spangled Banner is also the most recognized symbol of freedom wherever it flies in the world, a powerful inspiration to people everywhere who are "yearning to breathe free," as it is inscribed on the Statue of Liberty.
Every day, Americans all across this great land pledge their allegiance to flag of the United States. We salute it; we fight for it; we cherish it; we honor it.
But one day a year, we pay special honor to our flag. We set aside every June 14th as Flag Day, commemorating the date in 1777 when the Continental Congress officially made the Stars and Stripes the symbol of America.
Tomorrow, my office is planning special events in West Virginia commemorating Flag Day. Members of my staff will be presenting American flags to selected organizations all across the state that have requested flags:
To veterans in Logan at the "Spirit of the Doughboy" statue, which honors the victorious American soldiers of World War One.
To the Veterans Museum of Mid-Ohio Valley in Parkersburg, which pays tribute to West Virginians who have fought to preserve this country's freedom.
To Shepherd University in Shepherdstown, in conjunction with its Team River Runner program which includes kayaking programs for wounded warriors and their families.
To American Legion Post 33, in Sutton, honoring them for conducting memorial services for veterans in Braxton County.
To the City Council of Wardensville, to be displayed at the Wardensville Town Office.
To the "Here And There" Transit of Philippi, as part of the dedication of its new operations facilities.
And to the West Virginia Northern Community College in Wheeling, which only last month opened its Applied Technology Center to veterans and other students.
Mr. President, Flag Day has a special significance to West Virginia. Our state was born out of the fiery conflict of the Civil War, and next week we will celebrate our 150th birthday.
In that terrible war, West Virginians had a choice of two flags. We chose to follow the Stars and Stripes – and in doing so, West Virginia became the 35th Star on that Grand Old Flag.
So as we prepare for our state's 150th birthday celebration, I urge all West Virginians to join me in celebrating Flag Day – by displaying the flag that from the first days of America came to symbolize a "new constellation" of hope and freedom and from the first days of West
Virginia came to represent an allegiance to our remarkable Constitution.
In doing so, we honor not only our flag, but also the ideals on which America was founded as well as the generations of Americans who have defended those ideals in battle, always ensuring at the end of the fight that "our flag was still there."
The Star Spangled Banner is a symbol of their sacrifice and our faith.
Mr. President, not long after Congress officially adopted the Stars and Stripes as the flag of the United States, George Washington said, "We take the stars from Heaven, the red from our mother country, separating it by white stripes, thus showing that we have separated from her, and the white stripes shall go down to posterity representing liberty."
But a little poem I learned as a child from my Uncle Jimmy perfectly captures how I feel about the American flag even now:
It's only some stripes of red and white.
It's only some stars on a field of blue.
It's only a little cotton flag.
Does it mean anything to you?
Oh yes it does,
For beneath its folds
Our people are safe at land and sea.
It stands for a land where God is still king,
And His truth and His freedoms are free.
So let us love it well
And keep it pure as our banner of liberty.
Mr. President, this "little cotton flag" is displayed proudly in our homes, in our schools, in our businesses, over the Capitol and the White House, in parades and ballparks, on the field of battle, and on the graves of the heroes who fought in those battles.
It has flown from the tops of mountains, from the 9/11 rubble of Ground Zero, over the scarred wall of the Pentagon and from the surface of the moon – not once, not twice, but six times.
May our beautiful flag ever wave, and may God ever bless the country for which it stands.

redpajama_set_name: RedPajamaCommonCrawl


When it comes to Christmas gift baskets, this one takes the cake. Standing two feet tall and over two feet wide, this GORGEOUS gift will take their breath away! They will be delighted as they discover all of the amazing treats tucked inside.Assortment of Decadent Chocolate TrufflesMilk Chocolate Coconut ClustersCollector's Holiday Tin Filled with Assorted Danish Butter CookiesA Holiday Canister of Virginia Peanuts and Chocolate Covered Cranberry Nut MixMilk Chocolate Covered CherriesCookies and Cream BarkChocolate Peppermint Filled Wafer CookiesAssorted Chocolate CookiesHoliday Spice CookiesVanilla Cream Filled Wafer CookiesChocolate Truffle CookiesTriple Chocolate Hot Fudge BrownieAssorted Fruit Flavored Hard CandyAlmond Roca (Buttercrunch Toffee with Chocolate and Almonds)Assorted Belgian ChocolatesCranberry Bog FrogsCreamy Peppermint FudgeCreamy Chocolate Fudge"Smooth N' Melty" Petite MintsYogurt Frosted PretzelsCaramel and Cheddar Gourmet Popcorn MixHoliday Gift Box of TeaChocolate Caramel Peanut ClustersFestive Sleigh Bell Ornament All of this arrives artfully arranged in a hand-painted, metal embossed, old-fashioned, keepsake sleigh, which is decorated with festive holiday picks and a designer bow! Just imagine their excitement when this gift basket comes dashing through the snow right to their front doorstep! This impressive gift basket is an absolutely stunning way to help create a Christmas to Remember!


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Striking writers to work on Grammys
metrowebukmetroTuesday 29 Jan 2008 9:30 am
Striking writers have been given the go-ahead by their guild to work on the Grammy Awards show next month.
The Writers Guild of America gave its blessing last week to a picket-free Grammys.
With the board's decision to sign an interim agreement for the February 10 ceremony, the Grammys will escape the fate that befell this month's Golden Globes.
The Globes were stripped of stars and pomp when the guild would not agree to an interim deal and the Screen Actors Guild encouraged its members to boycott the ceremony, which was reduced to a news conference.
The agreement allowing guild-covered writing for the Grammys is in support of union musicians and will also help advance writers' own quest for "a fair contract", the guild said in a statement.
"Professional musicians face many of the same issues that we do concerning fair compensation for the use of their work in new media," Patric Verrone, president of the Writers Guild of America, West, said in the statement.
Payment for projects distributed via the internet is a central issue in the contract dispute between the writers union and the alliance that represents studios.
Informal talks began last week between the union and several studio chiefs in an effort to resolve the nearly three-month-old strike that has disrupted movie and TV production. Formal negotiations had broken down early in December.
During the impasse, the Directors Guild of America reached a tentative deal with the alliance that addressed new-media issues.
Directors Guild of AmericaScreen Actors Guild AwardsWriters Guild of America
SAG Awards 2020: Millie Bobby Brown slays androgynous look
SAG Awards 2020: Pregnant Michelle Williams hits red carpet with 'fiance'
SAG Awards 2020: Brad Pitt jokingly flirts with female presenter
Welcome to HMP Belmarsh with Ross Kemp Trailer


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Kal Penn Will Return for A Very Harold & Kumar Christmas
A little while back we received word from John Cho that A Very Harold & Kumar Christmas, the third installment of the stoner comedy series that all started with White Castle, was definitely on the way, and it might be filming as early as this summer for a hopeful holiday release this year. However, at the time Cho (Harold) wasn't sure if his counterpart Kal Penn (Kumar) would be returning since his job at The White House was keeping him busy. But now Deadline reports, though not necessarily definitively, that Penn will indeed return for the sequel, which shoots in summer and, get this, is being seriously considered for 3-D.
Deadline simply says they're "doing the film in 3-D," so I'm not entirely sure if that just means they want to shoot it in 3-D or convert it in post-production. Obviously the former would be the better choice, but either way, a 3-D flick geared towards the stoner crowd will definitely hit the mark. Though Kal Penn infamously left his regular series role on "House" for a job with President Obama's administration as associate director of the White House Office Of Public Engagement, apparently his recruitment for that position came with the agreement that it would only be for one year. And since it's getting close to the time when he'll be vacating that position, so it's pretty much certain that we'll see Cho and Penn back together on-screen for one smoky R-rated comedy that will give a whole new meaning to lighting up the Christmas Tree. Sound good fans?
Find more posts: Casting, Movie News, Opinions
Wow a comedy in 3-D just sounds fucking stupid, I'll see this in 2-D for damn sure.
Xerxex on Mar 31, 2010
Hell yes, it does! Sign me up!
Finch on Mar 31, 2010
Sweeeet...but yeah be seeing it 2-d....the 3-d idea doesnt surprise me one bit though.
Cody w on Mar 31, 2010
Where they gonna do it without him. Dont think he's too busy. Dont really care for this movies, laugh here and there, that's about it.
People's Champ on Mar 31, 2010
as long as it's better than the second movie... i mean a lot better
Kenji on Mar 31, 2010
Fucking 3d...it's everywhere, not unlike herpes.
Colt on Mar 31, 2010
DOOOOO IT! Yea, so excited.
jman571 on Mar 31, 2010
kool :), i dont know about the 3d tho :/
DaftPUNKFAN on Apr 1, 2010
FUCK 3D! im sick and tired of this 3d trend, theres nothing cool about it...maybe cept for avatar, but other than that its just gay!! if im high and im watching harold and kumar3 and im out of munchies im probally gonna eat the damn glasses and realise 10 minutes later that i needed the glasses to watch the damn movie...fuck...just let 3d go it works for some movies not ALL!
cesking on Apr 1, 2010
Why would Penn leave that cush gig on house, where he made in one episode what he made all year working for Obama. And it was only a one year position.
WB Exec on Apr 1, 2010


The Backwoods Engineer: Coming Soon to Your Neighborhood Gun Store: 4473 Burn Parties?
Coming Soon to Your Neighborhood Gun Store: 4473 Burn Parties?
Ol' Backwoods will never forget when I was a kid in the mid 80's, watching the scene in Red Dawn where the Commie major tells his minions to go to the gun store and get all the Forms 4473. Those are the yellow forms, kept in every gun store for up to 20 years, with a record of who bought what.
Look, Ol' Backwoods agrees with many in our 2nd Amendment community that eventually, there will be a confiscation order given by Obama or whatever Progressive succeeds him. They cannot fully implement communism with an armed citizenry, especially one that is continuing to arm itself at a rate of over 16.8 MILLION firearms per year, a rate which is accelerating. That has got to be scaring their tyrannical pants off.
They need registration, as a first step. We know what always follows: confiscation, at gunpoint.
This burn barrel day -- likely to be held the day after the President signs any new draconian legislation -- will include speeches, public prayers, and the blessing of those who have gathered by ministers, rabbis, and priests.
When Ol' Backwoods was a kid, that scene in Red Dawn didn't cause me to fear as much for myself as for others, because I knew my rifle and shotgun were not bought from a gun store, but handed down from my Daddny. Still, I feared for the others whose where.
Now, I have 4473's scattered in gun stores here and yon. So, probably, do you. If a registration order comes down, they must be destroyed. This will save lives.
In that day, I hope to stand and make a decent account of myself, so that I may not be ashamed in the company of my Revolutionary War ancestors in Glory.


The EB-5 investor immigration visa program has been subject to an overabundance of criticism disproportionate to the beneficial effect that the program has for investors, American jobs, and the U.S. economy.
Even advocates of EB-5 complain that the process moves too slowly. The typical response to that complaint acknowledges the sluggish nature of the process, but often lays the blame on applications with incomplete information.
The generic problem with the USCIS visa program seems to be that, for whatever reason, individuals within the bureaucracy appear to have broad discretionary power over what is processed and what is not, or which cases have priority over others. This certainly explains how Senator Harry Reid was able to use his influence to get USCIS bureaucrats to put a Nevada EB-5 project on a fast track.
The EB-5 program is unfairly criticized.
EB-5 applicants may be unfairly, overly scrutinized compared to applicants for other types of visas.
If one part of the system is broken, in effect, the whole system is broken. Try driving a car with just on broken piston rod.
We advocate for reasonable equity across all visa application types. Every individual needs to be adequately vetted, but, other than verifying the source of investment funds, why should the EB-5 be any more difficult than any other?


Franz Schubert's Impromptus are a series of eight pieces for solo piano. They were published in two sets of four impromptus each: the first was published in the composer's lifetime as Op. 90, and the second was published posthumously as Op. posth. 142. They are now catalogued as D. 899 and D. 935 respectively. They are considered to be among the most important examples of this popular early 19th-century genre.


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March of Dimes: Workers ready for OPSEU's first e-picket
The locked-out workers from Local 262 March of Dimes in Oakville may be a small unit of women, but they are gigantic in their efforts to get a decent contract. They are now reaching out in OPSEU's first electronic picket.
The e-picket is the newest campaign strategy for this enthusiastic group. Some workers will dedicate their time sending out the message "Thinking about giving to the March of Dimes? Think again!" through various social media channels.
"The March of Dimes relies on donations to help build its assets," says bargaining team member Jolene Schotsman. "That money should be used for services, not making profits. We want to encourage the employer to get back to the table to negotiate a decent contract."
The workers still maintain a picket line at the worksite as well as canvassing door-to-door in their own communities.
The March of Dimes in Oakville locked out its workers on October 28 when the members refused to accept wage concessions.


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Everything is hypnosis. Nothing is hypnosis. Milton Ericons answers the question. What is indirect suggestion? What is hypnosis? If any human being was able to truly answer this question, we'd all have discovered the mystical secrets of the Universe.


Mohammad Haskali gained his Bachelor's degree in analytical chemistry in 2008 from Victoria University (Australia). In 2014, Mohammad has submitted his PhD thesis from his research at the University of Melbourne in collaboration with the Peter MacCallum Cancer Centre. In his graduate studies, Mohammad synthesized novel peptide and improved their radiolabelling procedures. Furthermore, he succeeded in automating his procedures with either the TRACERlab FXFN module or the newly developed iPHASE Flexlab dual reactor. From January 2013 to October 2014, Mohammad worked at Cyclotek Pty. Ltd., Australia's leading production facility for 18F-labeled radiopharmaceuticals. His major role in the company was in GMP validation of new 18F-labeled investigational medicinal products. His current role in the PET Radiopharmaceutical Sciences Section is to perform regular productions of 18F- and 11C-labeled radiotracers for clinical studies, develop processes and documents in support of Investigational New Drug applications, and also to develop fluorine-18 and carbon-11 radiochemistry.


Sep. 28, 2018, 9:29 p.m.
Behind an exceptional performance from quarterback Steven Montez, Colorado defeated UCLA 38-16 on Friday night.
The Bruins trailed the Buffaloes 14-13 at the half and took the lead early in the third quarter with a field goal, but Colorado dominated the second half. Montez completed 22 of 26 pass attempts to gain 237 yards and score a touchdown, along with 81 rushing yards and two rushing touchdowns.
Montez's primary target, Laviska Shenault, tallied 126 receiving yards and a touchdown, while Travon McMillian ran for 102 yards and a touchdown.