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The 1984 Summer Olympics in Los Angeles was the first time that massage therapy was televised as it was being performed on the athletes. And then, during the 1996 Summer Olympics in Atlanta massage therapy was finally offered as a core medical service to the US Olympic Team. Massage has been employed by businesses and ...
Massage
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"Known in Myanmar as Yoe Yar Nhake Nal Chin, meaning 'traditional massage', Burmese massage has its ancient origins from Thai, Chinese and Indian medicine. Currently, Burmese massage also includes the use of local natural ingredients such as Thanaka which helps to promote smooth skin and prevents sunburn." Burmese mas...
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In the traditional Hammam, massage involves not just vigorous muscle kneading, but also joint cracking, "not so much a tender working of the flesh as a pummeling, a cracking of joints, a twisting of limbs..." An 18th-century traveler reported: Lomilomi and indigenous massage of Oceania Lomilomi is the traditional ma...
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Massage used in the medical field includes decongestive therapy used for lymphedema which can be used in conjunction with the treatment of breast cancer. Light massage is also used in pain management and palliative care. Carotid sinus massage is used to diagnose carotid sinus syncope and is sometimes useful for differe...
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Sports massage is the use of specific massage therapy techniques in an athletic context to improve recovery time, enhance performance and reduce the risk of injury. This is accomplished using techniques that stimulate the flow of blood and lymph to and from muscles. Sports massage is often delivered before or after phy...
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Thai massage is a popular massage therapy that is used for the management of conditions such as musculoskeletal pain and fatigue. Thai massage involves a number of stretching movements that improve body flexibility, joint movement and also improve blood circulation throughout the body. In one study scientists found tha...
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Watsu, developed by Harold Dull at Harbin Hot Springs, California, is a type of aquatic bodywork performed in near-body-temperature water, and characterized by continuous support by the practitioner and gentle movement, including rocking, stretching of limbs, and massage. The technique combines hydrotherapy floating an...
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Vichy showers A Vichy shower is a form of hydrotherapy that uses a series of shower nozzles that spray large quantities of water over the client while they lie in a shallow wet bed, similar to a massage table, but with drainage for the water. The nozzles may usually be adjusted for height, direction, and temperature t...
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Medical and therapeutic use The main professionals that provide therapeutic massage are massage therapists, athletic trainers, physical therapists, and practitioners of many traditional Chinese and other eastern medicines. Massage practitioners work in a variety of medical settings and may travel to private residences...
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Pain relief: Relief from pain due to musculoskeletal injuries and other causes is cited as a major benefit of massage. A 2015 Cochrane Review concluded that there is very little evidence that massage is an effective treatment for lower back pain. A meta-analysis conducted by scientists at the University of Illinois Urb...
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Canada In regulated provinces massage therapists are known as Registered Massage Therapists, in Canada six provinces regulate massage therapy: British Columbia, Ontario, Newfoundland and Labrador, Prince Edward Island, Saskatchewan, and New Brunswick. Registered Massage Therapy in British Columbia is regulated by the...
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Germany In Germany, massage is regulated by the government on a federal and national level. Only someone who has completed 3,200 hours of training (theoretical and practical) can use the professional title "Masseur und Medizinischer Bademeister" 'Masseur and Medical Spa Therapist'. This person can prolong his training...
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Myanmar In Myanmar, massage is unregulated. However, it is necessary to apply for a spa license with the government to operate a massage parlor in major cities such as Yangon. Blind and visually impaired people can become masseurs, but they are not issued licenses. There are a few professional spa training schools in ...
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United Kingdom To practice commercial massage or massage therapy in the UK, an ITEC or VTCT certificate must be obtained through training which includes Beauty and Spa Therapy, Hairdressing, Complementary Therapies, Sports & Fitness Training and Customer Service. Therapists with appropriate paperwork and insurance ma...
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Forty-seven states, Puerto Rico, and the District of Columbia currently offer some type of credential to professionals in the massage and bodywork field—usually licensure, certification or registration. Forty-five states require some type of licensing for massage therapists. There are two nationally recognized tests ga...
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In 1997 there were an estimated 114 million visits to massage therapists in the US. Massage therapy is the most used type of alternative medicine in hospitals in the United States. Between July 2010 and July 2011 roughly 38 million adult Americans (18 percent) had a massage at least once. People state that they use mas...
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A biofilm is a syntrophic community of microorganisms in which cells stick to each other and often also to a surface. These adherent cells become embedded within a slimy extracellular matrix that is composed of extracellular polymeric substances (EPSs). The cells within the biofilm produce the EPS components, which are...
Biofilm
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Origin of biofilms Biofilms are thought to have arisen during primitive Earth as a defense mechanism for prokaryotes, as the conditions at that time were too harsh for their survival. They can be found very early in Earth's fossil records (about 3.25 billion years ago) as both Archaea and Bacteria, and commonly protec...
Biofilm
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During surface colonization bacteria cells are able to communicate using quorum sensing (QS) products such as N-acyl homoserine lactone (AHL). Once colonization has begun, the biofilm grows by a combination of cell division and recruitment. Polysaccharide matrices typically enclose bacterial biofilms. The matrix exopol...
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It was generally assumed that cells dispersed from biofilms immediately go into the planktonic growth phase. However, studies have shown that the physiology of dispersed cells from Pseudomonas aeruginosa biofilms is highly different from that of planktonic and biofilm cells. Hence, the dispersal process is a unique sta...
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The EPS matrix consists of exopolysaccharides, proteins and nucleic acids. A large proportion of the EPS is more or less strongly hydrated, however, hydrophobic EPS also occur; one example is cellulose which is produced by a range of microorganisms. This matrix encases the cells within it and facilitates communication ...
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Biofilms are ubiquitous in organic life. Nearly every species of microorganism have mechanisms by which they can adhere to surfaces and to each other. Biofilms will form on virtually every non-shedding surface in non-sterile aqueous or humid environments. Biofilms can grow in the most extreme environments: from, for ex...
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Slow sand filter Slow sand filters are used in water purification for treating raw water to produce a potable product. They work through the formation of a biofilm called the hypogeal layer or Schmutzdecke in the top few millimetres of the fine sand layer. The Schmutzdecke is formed in the first 10–20 days of operation...
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Rhizosphere Plant-beneficial microbes can be categorized as plant growth-promoting rhizobacteria. These plant growth-promoters colonize the roots of plants, and provide a wide range of beneficial functions for their host including nitrogen fixation, pathogen suppression, anti-fungal properties, and the breakdown of or...
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Plants that have been colonized by PGPR forming a biofilm have gained systemic resistances and are primed for defense against pathogens. This means that the genes necessary for the production of proteins that work towards defending the plant against pathogens have been expressed, and the plant has a "stockpile" of comp...
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Human environment In the human environment, biofilms can grow in showers very easily since they provide a moist and warm environment for them to thrive. Mold biofilms on ceilings may form due to roof leaks. They can form inside water and sewage pipes and cause clogging and corrosion. On floors and counters, they can ma...
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Dental plaque is an oral biofilm that adheres to the teeth and consists of many species of both bacteria and fungi (such as Streptococcus mutans and Candida albicans), embedded in salivary polymers and microbial extracellular products. The accumulation of microorganisms subjects the teeth and gingival tissues to high c...
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A peptide pheromone quorum sensing signaling system in S. mutans includes the competence stimulating peptide (CSP) that controls genetic competence. Genetic competence is the ability of a cell to take up DNA released by another cell. Competence can lead to genetic transformation, a form of sexual interaction, favored u...
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Taxonomic diversity Many different bacteria form biofilms, including gram-positive (e.g. Bacillus spp, Listeria monocytogenes, Staphylococcus spp, and lactic acid bacteria, including Lactobacillus plantarum and Lactococcus lactis) and gram-negative species (e.g. Escherichia coli, or Pseudomonas aeruginosa). Cyanobacte...
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Infectious diseases Biofilms have been found to be involved in a wide variety of microbial infections in the body, by one estimate 80% of all infections. Infectious processes in which biofilms have been implicated include common problems such as bacterial vaginosis, urinary tract infections, catheter infections, middl...
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Research has shown that sub-therapeutic levels of β-lactam antibiotics induce biofilm formation in Staphylococcus aureus. This sub-therapeutic level of antibiotic may result from the use of antibiotics as growth promoters in agriculture, or during the normal course of antibiotic therapy. The biofilm formation induced b...
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Streptococcus pneumoniae Streptococcus pneumoniae is the main cause of community-acquired pneumonia and meningitis in children and the elderly, and of sepsis in HIV-infected persons. When S. pneumoniae grows in biofilms, genes are specifically expressed that respond to oxidative stress and induce competence. Formation...
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Escherichia coli Escherichia coli biofilms are responsible for many intestinal infectious diseases. The Extraintestinal group of E. coli (ExPEC) is the dominant bacterial group that attacks the urinary system, which leads to urinary tract infections. The biofilm formation of these pathogenic E. coli is hard to eradica...
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S. marcescens creates biofilms that ultimately develop into a highly porous, thread-like structure composed of chains of cells, filaments, and cell clusters. Research has shown that S. marcescens biofilms exhibit complex structural organization, including the formation of microcolonies and channels that facilitate nutr...
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The level of antibiotic resistance in a biofilm is much greater than that of non-biofilm bacteria, and can be as much as 5,000 times greater. The extracellular matrix of biofilm is considered one of the leading factors that can reduce the penetration of antibiotics into a biofilm structure and contributes to antibiotic...
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Food industry Biofilms have become problematic in several food industries due to the ability to form on plants and during industrial processes. Bacteria can survive long periods of time in water, animal manure, and soil, causing biofilm formation on plants or in the processing equipment. The buildup of biofilms can af...
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New forms of cleaning procedures are being tested to reduce biofilm formation in these processes which will lead to safer and more productive food processing industries. These new forms of cleaning procedures also have a profound effect on the environment, often releasing toxic gases into the groundwater reservoirs. As...
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Biofilms in aquatic environments are often founded by diatoms. The exact purpose of these biofilms is unknown, however there is evidence that the EPS produced by diatoms facilitates both cold and salinity stress. These eukaryotes interact with a diverse range of other organisms within a region known as the phycosphere,...
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Horizontal gene transfer Horizontal gene transfer is the lateral transfer of genetic material between cellular organisms. It happens frequently in prokaryotes, and less frequently in eukaryotes. In bacteria, horizontal gene transfer can occur through transformation (uptake of free floating DNA in the environment), tra...
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Examples Conjugative plasmids may encode biofilm-associated proteins, such as PtgA, PrgB, or PrgC which promote cell adhesion (required for early biofilm formation). Genes encoding type III fimbriae are found in pOLA52 (Klebsiella pneumoniae plasmid) which promote conjugative-pilus-dependent biofilm formation. Transf...
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Recent research has shown that the archaeal species H. volcanii has some biofilm phenotypes similar to bacterial biofilms such as differentiation and HGT, which required cell-cell contact and involved formation of cytosolic bridges and cellular fusion events. Cultivation devices There is a wide variety of biofilm cul...
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Star formation is the process by which dense regions within molecular clouds in interstellar space, sometimes referred to as "stellar nurseries" or "star-forming regions", collapse and form stars. As a branch of astronomy, star formation includes the study of the interstellar medium (ISM) and giant molecular clouds (GM...
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Observations indicate that the coldest clouds tend to form low-mass stars, which are first observed via the infrared light they emit inside the clouds, and then as visible light when the clouds dissipate. Giant molecular clouds, which are generally warmer, produce stars of all masses. These giant molecular clouds have ...
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In triggered star formation, one of several events might occur to compress a molecular cloud and initiate its gravitational collapse. Molecular clouds may collide with each other, or a nearby supernova explosion can be a trigger, sending shocked matter into the cloud at very high speeds. (The resulting new stars may th...
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During the collapse, the density of the cloud increases towards the center and thus the middle region becomes optically opaque first. This occurs when the density is about . A core region, called the first hydrostatic core, forms where the collapse is essentially halted. It continues to increase in temperature as deter...
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When the surrounding gas and dust envelope disperses and accretion process stops, the star is considered a pre-main-sequence star (PMS star). The energy source of these objects is (gravitational contraction)Kelvin–Helmholtz mechanism, as opposed to hydrogen burning in main sequence stars. The PMS star follows a Hayashi...
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The structure of the molecular cloud and the effects of the protostar can be observed in near-IR extinction maps (where the number of stars are counted per unit area and compared to a nearby zero extinction area of sky), continuum dust emission and rotational transitions of CO and other molecules; these last two are ob...
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On February 21, 2014, NASA announced a greatly upgraded database for tracking polycyclic aromatic hydrocarbons (PAHs) in the universe. According to scientists, more than 20% of the carbon in the universe may be associated with PAHs, possible starting materials for the formation of life. PAHs seem to have been formed sh...
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Massive stars emit copious quantities of radiation which pushes against infalling material. In the past, it was thought that this radiation pressure might be substantial enough to halt accretion onto the massive protostar and prevent the formation of stars with masses more than a few tens of solar masses. Recent theor...
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Both the core mass function (CMF) and filament line mass function (FLMF) observed in the California GMC follow power-law distributions at the high-mass end, consistent with the Salpeter initial mass function (IMF). Current results strongly support the existence of a connection between the FLMF and the CMF/IMF, demonstr...
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Laboratory glassware is a variety of equipment used in scientific work, traditionally made of glass. Glass may be blown, bent, cut, molded, or formed into many sizes and shapes. It is commonly used in chemistry, biology, and analytical laboratories. Many laboratories have training programs to demonstrate how glassware...
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Modern era Many glasses that were produced in bulk in the 1830s would quickly become unclear and dirty because of the low quality glass being used. During the 19th century, more chemists began to recognize the importance of glassware due to its transparency, and the ability to control the conditions of experiments. Jö...
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Laboratory glassware selection Laboratory glassware is typically selected by a person in charge of a particular laboratory analysis to match the needs of a given task. The task may require a piece of glassware made with a specific type of glass. The task may be readily performed using low cost, mass-produced glasswar...
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Quality assurance Metrology Laboratory glassware can be used for high precision volumetric measurements. With high precision measurements, such as those made in a testing laboratory, the metrological grade of the glassware becomes important. The metrological grade then can be determined by both the confidence interva...
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Examples of glassware containers include: Beakers are simple cylindrical shaped containers used to hold reagents or samples. Flasks are narrow-necked glass containers, typically conical or spherical, used in a laboratory to hold reagents or samples. Examples flasks include the Erlenmeyer flask, Florence flask, and Sc...
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A calorimeter is a device used for calorimetry, or the process of measuring the heat of chemical reactions or physical changes as well as heat capacity. Differential scanning calorimeters, isothermal micro calorimeters, titration calorimeters and accelerated rate calorimeters are among the most common types. A simple c...
Calorimeter
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No adiabatic calorimeter is fully adiabatic - some heat will be lost by the sample to the sample holder. A mathematical correction factor, known as the phi-factor, can be used to adjust the calorimetric result to account for these heat losses. The phi-factor is the ratio of the thermal mass of the sample and sample hol...
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A bomb calorimeter is a type of constant-volume calorimeter used in measuring the heat of combustion of a particular reaction. Bomb calorimeters have to withstand the large pressure within the calorimeter as the reaction is being measured. Electrical energy is used to ignite the fuel; as the fuel is burning, it will he...
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Since there is no heat exchange between the calorimeter and surroundings (Q = 0) (adiabatic), no work is performed (W = 0) Thus, the total internal energy change Also, total internal energy change (constant volume ) where is heat capacity of the bomb Before the bomb can be used to determine heat of combus...
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Calvet-type calorimeters The detection is based on a three-dimensional fluxmeter sensor. The fluxmeter element consists of a ring of several thermocouples in series. The corresponding thermopile of high thermal conductivity surrounds the experimental space within the calorimetric block. The radial arrangement of the th...
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An example is a coffee-cup calorimeter, which is constructed from two nested Styrofoam cups, providing insulation from the surroundings, and a lid with two holes, allowing insertion of a thermometer and a stirring rod. The inner cup holds a known amount of a solvent, usually water, that absorbs the heat from the reacti...
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In a heat flux DSC, both pans sit on a small slab of material with a known (calibrated) heat resistance K. The temperature of the calorimeter is raised linearly with time (scanned), i.e., the heating rate dT/dt = β is kept constant. This time linearity requires good design and good (computerized) temperature control....
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A DSC may also be used as an initial safety screening tool. In this mode the sample will be housed in a non-reactive crucible (often gold, or gold-plated steel), and which will be able to withstand pressure (typically up to 100 bar). The presence of an exothermic event can then be used to assess the stability of a sub...
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In traditional heat flow calorimeters, one reactant is added continuously in small amounts, similar to a semi-batch process, in order to obtain a complete conversion of the reaction. In contrast to the tubular reactor, this leads to longer residence times, different substance concentrations and flatter temperature prof...
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In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture...
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Ideal gas mixtures Ideally the ratio of partial pressures equals the ratio of the number of molecules. That is, the mole fraction of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component: and the partial pressure of an individua...
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For example, at any given temperature, methyl chloride has the highest vapor pressure of any of the liquids in the chart. It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of ...
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where is also referred to as the Henry's law constant. As can be seen by comparing equations () and () above, is the reciprocal of . Since both may be referred to as the Henry's law constant, readers of the technical literature must be quite careful to note which version of the Henry's law equation is being used. He...
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The effect of a toxic contaminant such as carbon monoxide in breathing gas is also related to the partial pressure when breathed. A mixture which may be relatively safe at the surface could be dangerously toxic at the maximum depth of a dive, or a tolerable level of carbon dioxide in the breathing loop of a diving rebr...
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Plumbing is any system that conveys fluids for a wide range of applications. Plumbing uses pipes, valves, plumbing fixtures, tanks, and other apparatuses to convey fluids. Heating and cooling (HVAC), waste removal, and potable water delivery are among the most common uses for plumbing, but it is not limited to these a...
Plumbing
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Plumbing reached its early apex in ancient Rome, which saw the introduction of expansive systems of aqueducts, tile wastewater removal, and widespread use of lead pipes. The Romans used lead pipe inscriptions to prevent water theft. With the Fall of Rome both water supply and sanitation stagnated—or regressed—for well ...
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Lead was the favoured material for water pipes for many centuries because its malleability made it practical to work into the desired shape. Such use was so common that the word "plumbing" derives from plumbum, the Latin word for lead. This was a source of lead-related health problems in the years before the health haz...
Plumbing
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Bronze fittings and short pipe segments are commonly used in combination with various materials. Difference between pipes and tubes The difference between pipes and tubes is a matter of sizing. For instance, PVC pipe for plumbing applications and galvanized steel pipe are measured in iron pipe size (IPS). Copper tube...
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Wall thickness does not affect pipe or tubing size. 1/2" L copper has the same outer diameter as 1/2" K or M copper. The same applies to pipe schedules. As a result, a slight increase in pressure losses is realized due to a decrease in flowpath as wall thickness is increased. In other words, 1 foot of 1/2" L copper has...
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Most typical single family home systems will not require supply piping larger than due to expense as well as steel piping's tendency to become obstructed from internal rusting and mineral deposits forming on the inside of the pipe over time once the internal galvanizing zinc coating has degraded. In potable water dist...
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Plastic pipe is in wide use for domestic water supply and drain-waste-vent (DWV) pipe. Principal types include: Polyvinyl chloride (PVC) was produced experimentally in the 19th century but did not become practical to manufacture until 1926, when Waldo Semon of BF Goodrich Co. developed a method to plasticize PVC, makin...
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Aqua – known as PEX-Al-PEX, for its PEX/aluminum sandwich, consisting of aluminum pipe sandwiched between layers of PEX, and connected with modified brass compression fittings. In 2005, many of these fittings were recalled.
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Present-day water-supply systems use a network of high-pressure pumps, and pipes in buildings are now made of copper, brass, plastic (particularly cross-linked polyethylene called PEX, which is estimated to be used in 60% of single-family homes), or other nontoxic material. Due to its toxicity, most cities moved away f...
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Specialized plumbing tools include pipe wrenches, flaring pliers, pipe vise, pipe bending machine, pipe cutter, dies, and joining tools such as soldering torches and crimp tools. New tools have been developed to help plumbers fix problems more efficiently. For example, plumbers use video cameras for inspections of hidd...
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Each Government at the state level has their own Authority and regulations in place for licensing plumbers. They are also responsible for the interpretation, administration and enforcement of the regulations outlined in the NCC. These Authorities are usually established for the sole purpose of regulating plumbing activ...
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Some widely used Standards in the United States are: ASME A112.6.3 – Floor and Trench Drains ASME A112.6.4 – Roof, Deck, and Balcony Drains ASME A112.18.1/CSA B125.1 – Plumbing Supply Fittings ASME A112.19.1/CSA B45.2 – Enameled Cast Iron and Enameled Steel Plumbing Fixtures ASME A112.19.2/CSA B45.1 – Ceramic Plum...
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A candle is an ignitable wick embedded in wax, or another flammable solid substance such as tallow, that provides light, and in some cases, a fragrance. A candle can also provide heat or a method of keeping time. Candles have been used for over two millennia around the world, and were a significant form of indoor light...
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Prior to the invention of candles, ancient people used open fire, torches, splinters of resinous wood, and lamps to provide artificial illumination at night. Primitive oil lamps in which a lit wick rested in a pool of oil or fat were used from the Paleolithic period, and pottery and stone lamps from the Neolithic perio...
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In medieval Europe, candles were initially used primarily in Christian churches. Its use spread later to the households of the wealthy as a luxury item. In northern Europe, rushlight made of greased rushes were commonly used especially in England, but tallow candles were used during the Middle Ages, with a mention of ...
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In the Middle East, during the Abbasid and Fatimid Caliphates, beeswax was the dominant material used for candle making. Beeswax was often imported from long distances; for example, candle makers from Egypt used beeswax from Tunis. As in Europe, these candles were expensive and limited to the elite, and most commoners ...
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The manufacture of candles became an industrialized mass market in the mid 19th century. In 1834, Joseph Morgan, a pewterer from Manchester, England, patented a machine that revolutionised candle making. It allowed for continuous production of molded candles by using a cylinder with a moveable piston to eject candles a...
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Symbolic use of candles has extended from the religious to the secular, for example, a candlelight vigil may be held in remembrance for a person, for a cause or an event, or as a form of political action or protest. In a social setting, candles are commonly used on birthday cakes. In the 21st century, there has been a...
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A candle wick works by capillary action, drawing ("wicking") the melted wax or fuel up to the flame. When the liquid fuel reaches the flame, it vaporizes and combusts. The candle wick influences how the candle burns. Important characteristics of the wick include diameter, stiffness, fire resistance, and tethering. A c...
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The hottest part of a candle flame is just above the very dull blue part to one side of the flame, at the base. At this point, the flame is about . However, this part of the flame is very small and releases little heat energy. The blue color is due to chemiluminescence, while the visible yellow color is due to radiativ...
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A candle flame is formed because wax vaporizes on burning. A candle flame is widely recognized as having between three and five regions or "zones": Zone I – this is the non-luminous, lowest, and coolest part of the candle flame. It is located around the base of the wick where there is insufficient oxygen for fuel to b...
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The liquid wax is hot and can cause skin burns, but the amount and temperature are generally rather limited and the burns are seldom serious. The best way to avoid getting burned from splashed wax is to use a candle snuffer instead of blowing on the flame. A candle snuffer is usually a small metal cup on the end of a l...
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Regulation International markets have developed a range of standards and regulations to ensure compliance, while maintaining and improving safety, including: Europe: GPSD, EN 15493, EN 15494, EN 15426, EN 14059, REACH, RAL-GZ 041 Candles (Germany), French Decree 91-1175 United States: ASTM F2058, ASTM F2179, ASTM F24...
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For tea light candles, there is a variety of candle holders, including small glass holders and elaborate multi-candle stands. The same is true for votives. Wall sconces are available for tea light and votive candles. For pillar-type candles, the assortment of candle holders is broad. A fireproof plate, such as a glass ...
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Candle followers are often found in churches on altar candles. Gallery
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In mathematics, a permutation of a set can mean one of two different things: an arrangement of its members in a sequence or linear order, or the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set {1, 2, 3}: written as tuples, ...
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History Permutation-like objects called hexagrams were used in China in the I Ching (Pinyin: Yi Jing) as early as 1000 BC. In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to...
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A first case in which seemingly unrelated mathematical questions were studied with the help of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that properties of the permutations of the roots of an equation are related to the possibilities to solve it. This ...
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As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, termed an active permutation or substitution. An older viewpoint sees a permutation as an ordered arrangement or list of all the elements of S, called a passive permutation. According to this definition, all permuta...
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Under this assumption, one may omit the first row and write the permutation in one-line notation as , that is, as an ordered arrangement of the elements of S. Care must be taken to distinguish one-line notation from the cycle notation described below: a common usage is to omit parentheses or other enclosing marks for ...
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Following the convention of omitting 1-cycles, one may interpret an individual cycle as a permutation which fixes all the elements not in the cycle (a cyclic permutation having only one cycle of length greater than 1). Then the list of disjoint cycles can be seen as the composition of these cyclic permutations. For exa...
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