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A room box is a display box used for three-dimensional miniature scale environments, or scale models . Although the name would suggest room boxes generally only represent typical rooms such as those found in houses or other buildings (bedrooms, kitchens, offices, etc.), room boxes are used for all sorts of environments – exterior views as well as interior ones, realistic ones as well as fantastical ones. While some miniaturists concentrate their efforts specifically on room boxes, many use them to take a break from larger projects, such as dollhouses or miniature villages, to create a smaller environment on a different theme. A room box can be tailored to one’s interests or mirror an important step in life - for example, a bakery or restaurant scene might be created by or for a baker or cook, and a wedding dress storefront might be created for a bride to be or as a reminiscence of one's wedding. Making a room box is often a first step to learning new techniques in miniature making; such projects are popular at miniaturists' events where attendees have only 1–2 days to make and finish a project. Once techniques are perfected in these smaller settings, craftspersons and hobbyists often reapply them to larger projects.
Room boxes are a cost- and time-effective way to make miniature settings without attempting larger setups such as a dollhouse or train set . Commercially bought room boxes tend to be made of wood, pressed wood products or plywood, with the top and front window made of removable clear acrylic that lets in light and enables access and viewing from two perspectives. Dimensions usually meet standard dollhouse proportions ("1:12 scale" in dollhouse speak means that 1" in the dollhouse world represents 1' in the real world), but anyone can make a room box from a leftover shoebox, orange crate, etc. and adapt an idea to suit the box's scale. Since any material can be used, whether leftover or new, people of all economic classes express themselves through this craft.
One elaborate example of 1:12 scale miniature rooms are the 68 miniature Thorne Rooms , each with a different theme. They were designed by Narcissa Niblack Thorne and furniture for them was created by craftsmen in the 1930s and 1940s. They are now at the Art Institute of Chicago , Phoenix Art Museum . [ 1 ] As evidenced in the recent increase in craft book and magazine publishing on different types of miniatures, interest in making room-boxes for miniature settings has steadily grown since the 1990s. Room boxes have even found a place during prime-time television: the winter 2007 season of CSI: Crime Scene Investigation included a clever storyline recurring throughout the season, where a murderer named The Miniature Killer leaves clues for investigators in the form of intricately made 3-D room boxes showing scenes of the crimes she committed, reproduced in scale miniature. | https://en.wikipedia.org/wiki/Room_box |
Room temperature , colloquially, denotes the range of air temperatures most people find comfortable indoors while dressed in typical clothing. Comfortable temperatures can be extended beyond this range depending on humidity , air circulation , and other factors.
In certain fields, like science and engineering , and within a particular context, room temperature can mean different agreed-upon ranges. In contrast, ambient temperature is the actual temperature, as measured by a thermometer , of the air (or other medium and surroundings) in any particular place. The ambient temperature (e.g. an unheated room in winter) may be very different from an ideal room temperature .
Food and beverages may be served at "room temperature", meaning neither heated nor cooled.
Comfort temperature is interchangeable with neutral temperature in the scientific literature, which can be calculated through regression analysis between thermal sensation votes and indoor temperature. The neutral temperature is the solution of the resulting regression model by setting the thermal sensation vote as zero. The American Heritage Dictionary of the English Language identifies room temperature as around 20–22 °C (68–72 °F; 293–295 K), [ 1 ] while the Oxford English Dictionary states that it is "conventionally taken as about 20 °C (68 °F; 293 K)". [ 2 ]
Ideal room temperature varies vastly depending on the surrounding climate. Studies from Indonesia have shown that the range of comfortable temperature is 24–29 °C (75–84 °F) for local residents. [ 3 ] Studies from Nigeria show a comfortable temperature range of 26–28 °C (79–82 °F), comfortably cool 24–26 °C (75–79 °F) and comfortably warm 28–30 °C (82–86 °F). [ 4 ] A field study conducted in Hyderabad, India returned a comfort band of 26–32.45 °C (79–90 °F) with a mean of 29.23 °C (85 °F). [ 5 ] A study conducted in Jaipur, India among healthy young men showed that the neutral thermal comfort temperature was analyzed to be 30.15 °C (86 °F), although a range of 25.9–33.8 °C (79–93 °F) was found. [ 6 ]
People are highly sensitive to even small differences in environmental temperature. At 24 °C (75 °F), a difference of 0.38 °C (0.68 °F) can be detected between the temperature of two rooms. [ 7 ]
Owing to variations in humidity and (likely) clothing, recommendations for summer and winter may vary; a suggested [ by whom? ] typical range for summer is 23–25.5 °C (73–78 °F), with that for winter being 20–23.5 °C (68–74 °F). [ 8 ] Some studies have suggested that thermal comfort preferences of men and women may differ significantly, with women on average preferring higher ambient temperatures. [ 9 ] [ 10 ] [ 11 ]
In the recent past, it was common for house temperatures to be kept below the comfort level; a 1978 UK study found average indoor home temperatures to be 15.8 °C (60.4 °F) while Japan in 1980 had median home temperatures of 13 °C (55 °F) to 15 °C (59 °F). [ 12 ]
Rooms may be maintained at an ambient temperature above the comfort temperature in hot weather, or below it in cold weather, if required by cost considerations or practical issues (e.g. lack of air conditioning or relatively high expense of heating.)
In the UK, the Offices, Shops and Railway Premises Act 1963 provides for a minimum temperature in commercial premises, but not for a maximum temperature.
The World Health Organization in 1987 found that comfortable indoor temperatures of 18–24 °C (64–75 °F) were not associated with health risks for healthy adults with appropriate clothing, humidity, and other factors. For infants, elderly, and those with significant health problems, a minimum of 20 °C (68 °F) was recommended. Temperatures lower than 16 °C (61 °F) with humidity above 65% were associated with respiratory hazards including allergies. [ 13 ] [ 14 ]
The WHO's 2018 guidelines give a strong recommendation that a minimum of 18 °C (64 °F) is a "safe and well-balanced indoor temperature to protect the health of general populations during cold seasons". A higher minimum temperature may be necessary for vulnerable groups including children, the elderly, and people with cardiorespiratory disease and other chronic illnesses. However, the recommendation regarding risk of exposure to high indoor temperatures is only "conditional". Minimal-risk high temperatures range from about 21 to 30 °C (70 to 86 °F) depending on the region, with maximum acceptable temperatures between 25 and 32 °C (77 and 90 °F). [ 15 ] [ 16 ]
Temperature ranges are defined as room temperature for certain products and processes in industry, science, standards, and consumer goods. For instance, for the shipping and storage of pharmaceuticals , the United States Pharmacopeia -National Formulary (USP-NF) defines controlled room temperature as between 20 and 25 °C (68 and 77 °F), with excursions between 15 and 30 °C (59 and 86 °F) allowed, provided the mean kinetic temperature does not exceed 25 °C (77 °F). [ 17 ] The European Pharmacopoeia defines it as being simply 15 to 25 °C (59 to 77 °F), and the Japanese Pharmacopeia defines "ordinary temperature" as 15 to 25 °C (59 to 77 °F), with room temperature being 1 to 30 °C (34 to 86 °F). [ 18 ] [ 19 ] Merriam-Webster gives as a medical definition a range of 15 to 25 °C (59 to 77 °F) as being suitable for human occupancy, and at which laboratory experiments are usually performed. [ 20 ]
World Health Organization (2018). WHO Housing and Health Guidelines . ISBN 978-92-4-155037-6 . PMID 30566314 . Wikidata Q95379102 . Retrieved 2022-11-22 . | https://en.wikipedia.org/wiki/Room_temperature |
A rooster tail is a term used in fluid dynamics , automotive gear shifting, and meteorology . It is a region of commotion or turbulence within a fluid, caused by movement. In fluid dynamics, it lies directly in the wake of an object traveling within a fluid, and is accompanied by a vertical protrusion. If it occurs in a river, wise boaters upstream steer clear of its appearance. The degree of their formation [ clarification needed ] can indicate the efficiency of a boat's hull design . The magnitude of these features [ clarification needed ] in a boat increases with speed, while the relationship is inversely proportional for airplanes . Energetic volcanic eruptions can create rooster tail formations from their ejecta. They can form in relation to coronal loops near the Sun 's surface.
In gear shifting in motor vehicles, it is the relation between the coefficient of friction and the sliding speed of the clutch . Cars can throw rooster tails in their wake and loose materials under its wheels. In meteorology, a rooster tail satellite pattern can be applied [ clarification needed ] to either low or high level cloudiness, with the low cloud line seen in the wake of tropical cyclones and the high cloud pattern seen either within mare's tails or within the outflow jet of tropical cyclones.
Rooster tails are caused by constructive interference near and to the wake of objects within a flowing fluid. [ 1 ]
A fast current of water flowing over a rock near the surface of a stream or river can create a rooster tail—such commotions at the water's surface are avoided by boaters due to the near surface obstruction. [ clarification needed ] [ 1 ] Propellers on boats can produce a rooster tail of water in their wake, in the form of a fountain which shoots into the air behind the boat. [ 2 ] The faster a boat goes, the larger the rooster tails become. [ 3 ] The efficiency of a boat's hull design can be judged by the magnitude of the rooster tail—larger rooster tails indicate less efficient designs. [ 4 ] If a water skier is in tow, the skis also throw off a rooster tail. [ 5 ] Airplanes lifting off from a lake produce lengthening rooster tails behind their amphibious floats as their speed increases, until the plane lifts off the surface. [ 6 ]
An airplane leaves rooster tails in its wake in the form of two circulations at the tip of its wings. As the plane speeds up, the rooster tails become smaller. [ 3 ]
In low gravity and dusty environments, such as the Moon , they can be created by the wheels of moving vehicles . [ 7 ] A special energetic volcanic eruption known as a strombolian eruption produces bright arcs of ejecta, referred to as rooster tails, composed of basaltic cinders or volcanic ash . [ 8 ]
Coronal loops are the basic structures of the magnetic solar corona , the bright area seen around the Sun during solar eclipses . These loops are the closed-magnetic flux cousins of the open-magnetic flux that can be found in coronal hole (polar) regions and the solar wind . Loops of magnetic flux well up from the solar body and fill with hot solar plasma. [ 9 ] Due to the heightened magnetic activity in these coronal loop regions, coronal loops can often be the precursor to solar flares and coronal mass ejections (CMEs). Emerging magnetic flux within coronal loops can cause a rooster tail. [ 10 ]
The curve describing the relationship between the coefficient of friction and sliding speed of the clutch in manual transmission automobiles on a graph is known as a rooster tail characteristic. [ 11 ] Formations can occur when a car's motor revs up over puddles, loose soil, or mud.
Rooster tails have been mentioned in weather satellite interpretation since 2003 connected with tropical cyclones. In the low cloud field, it represents a convergence zone on the westward extent of the Saharan Air Layer seen at the back of tropical cyclones gaining latitude. If there are two systems, the one nearer the pole strengthens, while the system nearest the Equator weakens within an area with downward motion in the mid-levels of the troposphere. [ 12 ]
This description has also been used with high cloudiness spreading in a narrow channel towards the Equator within the outflow jet of a tropical cyclone, such as Hurricane Felix (1995) . [ 13 ] Mare's tail patterns within cirrus clouds are occasionally referred to by this term due to their appearance. | https://en.wikipedia.org/wiki/Rooster_tail |
RootMetrics (formerly Root Wireless ) offers scientifically collected and crowdsourced mobile network performance information to consumers and the industry. [ 1 ] The firm captures user information by testing network performance when consumers are using their mobile phone for voice or data communications. [ 2 ]
RootMetrics was acquired by analytics firm IHS Inc. in 2015.
On December 14, 2021, RootMetrics was acquired by the Seattle -based internet performance analyzer Ookla . [ 3 ]
RootScore Reports provide mobile network comparison information based on the company’s testing. In the United States, the firm publishes reports for the U.S. as a whole, the 50 U.S. states, the 125 most populous U.S. cities, and the 50 busiest U.S. airports. In the UK, RootMetrics publishes RootScore Reports for the four UK nations, 16 most populous metropolitan areas, and three London airports (Heathrow, Gatwick, and City).
The reports show network reliability and speed using two indexes: Network Reliability and Network Speed. [ 4 ] The Network Reliability Index is made up of results from data/mobile internet, call, and text testing; the Network Speed Index is compiled from data/mobile internet and text results.
The RootMetrics CoverageMap combines the company's scientifically-collected results with results crowdsourced from consumers. It is available both online and within the company’s mobile app for Android and iOS. It displays call performance, average call signal strength, download data speeds, and types of network technology available. [ 10 ]
Tests are conducted with unmodified Android-based smartphones purchased off the shelf at regular mobile phone stores. The company tests mobile networks at various locations and hours, both indoors and outdoors, and while driving, using a random sampling methodology to prevent bias. [ 11 ] [ 12 ]
Test locations are randomly selected in each state and each nation, and divided into groups by population size, with each population-based group given equal weighting. [ 13 ]
Complementing its professional testing, the firm also gathers crowdsourced network performance data from mobile users, [ 14 ] combining them to produce the CoverageMap comparison tool, available both online and within the mobile application.
RootScore Awards are assigned to the networks with the top score in data/mobile internet performance, call performance, text performance, and overall performance incorporating the three categories. [ 15 ] [ 16 ] For airports the awards are based on the data/mobile internet performance. [ 17 ] | https://en.wikipedia.org/wiki/RootMetrics |
Root hairs or absorbent hairs , are outgrowths of epidermal cells, specialized cells at the tip of a plant root . They are lateral extensions of a single cell and are only rarely branched. They are found in the region of maturation, of the root. Root hair cells improve plant water absorption by increasing root surface area to volume ratio which allows the root hair cell to take in more water. The large vacuole inside root hair cells makes this intake much more efficient. Root hairs are also important for nutrient uptake as they are main interface between plants and mycorrhizal fungi .
The function of all root hairs is to collect water and mineral nutrients in the soil to be sent throughout the plant. In roots, most water absorption happens through the root hairs. The length of root hairs allows them to penetrate between soil particles and prevents harmful bacterial organisms from entering the plant through the xylem vessels. [ 1 ] Increasing the surface area of these hairs makes plants more efficient in absorbing nutrients and interacting with microbes. [ 2 ] As root hair cells do not carry out photosynthesis , they do not contain chloroplasts .
Root hairs form an important surface as they are needed to absorb most of the water and nutrients needed for the plant. They are also directly involved in the formation of root nodules in legume plants. The root hairs curl around the bacteria, which allows for the formation of an infection thread into the dividing cortical cells to form the nodule. [ 3 ]
Having a large surface area, the active uptake of water and minerals through root hairs is highly efficient. Root hair cells also secrete acids (e.g., malic and citric acid), which solubilize minerals by changing their oxidation state , making the ions easier to absorb. [ 4 ]
Root hair cells vary between 15 and 17 micrometers in diameter, and 80 and 1,500 micrometers in length. [ 5 ] Root hairs are found only in the zone of maturation, also called the zone of differentiation. [ 6 ] They are not found in the zone of elongation, possibly because older root hairs are sheared off as the root elongates and moves through the soil. [ 7 ] Root hairs grow quickly, at least 1 μm/min, making them particularly useful for research on cell expansion. [ 8 ] Just prior to and during root hair cell development, there is elevated phosphorylase activity. [ 9 ]
Root hairs are essential for healthy plant nutrition, especially through their interactions with symbiotic fungi. Symbiotic fungi and root hairs produce mycorrhizal symbioses like arbuscular mycorrhiza , formed by AM fungi, and ectomycorrhiza , formed by EM fungi. [ 10 ] These are very common, [ 11 ] occurring in 90% of terrestrial plant species, [ 12 ] because of the benefits it brings to both the fungus and plant.
Formation of this relationship for EM fungi begins with the colonization of the root hairs. This process begins when the EM fungus adheres to the root hair from the soil. [ 13 ] The fungus then secretes diffusible factors, to which root hairs are highly sensitive, allowing the hyphae to penetrate into the epidermal cells and create a Hartig net in the first layers of the root cortex. [ 13 ] This highly branched structure serves as an interface between the two organisms as fungal cells adapt to the exchanges that occur between the plant and fungus. [ 14 ] This process is similar to how AM fungi colonize root hairs, but instead of diffusible factors, they secrete hydrolases to relax the cell wall, which allows hyphae to enter, and there is no Hartig net. [ 13 ]
Various effects of fungal colonization in root hairs show that this relationship is beneficial to both plant and fungal species, but the main effect is on root hair growth. Fungi actually affect the growth of root hairs if there is water or nutrient deficiency. [ 13 ] Since both of these organisms require nutrients and water, their cooperation is essential to their mutual survival. Upon detection of deficiency, the drought stress response of the plant is triggered, causing growth of the root hairs. [ 12 ] The mycorrhizae of the fungus then uses its extended system to help the plant find the correct area of nutrition, signaling the direction in which the roots should grow. [ 13 ] This makes root growth more efficient, preserving energy for other metabolic processes, which in turn benefits the fungus that feeds off those metabolic products.
When a new root hair cell grows, it excretes a hormone that inhibits the growth of root hairs in nearby cells. This ensures equal and efficient distribution of the actual hairs on these cells. [ citation needed ]
Repotting or transplanting a plant can result in root hair cells being pulled off, perhaps to a significant extent, which can cause wilting. [ 15 ] | https://en.wikipedia.org/wiki/Root_hair |
In bioinformatics , the root mean square deviation of atomic positions , or simply root mean square deviation (RMSD) , is the measure of the average distance between the atoms (usually the backbone atoms) of superimposed molecules. [ 1 ] In the study of globular protein conformations, one customarily measures the similarity in three-dimensional structure by the RMSD of the Cα atomic coordinates after optimal rigid body superposition.
When a dynamical system fluctuates about some well-defined average position, the RMSD from the average over time can be referred to as the RMSF or root mean square fluctuation . The size of this fluctuation can be measured, for example using Mössbauer spectroscopy or nuclear magnetic resonance , and can provide important physical information. The Lindemann index is a method of placing the RMSF in the context of the parameters of the system.
A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the RMSD. Coutsias, et al. presented a simple derivation, based on quaternions , for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors. [ 2 ] They proved that the quaternion method is equivalent to the well-known Kabsch algorithm . [ 3 ] The solution given by Kabsch is an instance of the solution of the d -dimensional problem, introduced by Hurley and Cattell. [ 4 ] The quaternion solution to compute the optimal rotation was published in the appendix of a paper of Petitjean. [ 5 ] This quaternion solution and the calculation of the optimal isometry in the d -dimensional case were both extended to infinite sets and to the continuous case in the appendix A of another paper of Petitjean. [ 6 ]
where δ i is the distance between atom i and either a reference structure or the mean position of the N equivalent atoms. This is often calculated for the backbone heavy atoms C , N , O , and C α or sometimes just the C α atoms.
Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of n {\displaystyle n} points v {\displaystyle \mathbf {v} } and w {\displaystyle \mathbf {w} } , the RMSD is defined as follows:
An RMSD value is expressed in length units. The most commonly used unit in structural biology is the Ångström (Å) which is equal to 10 −10 m.
Typically RMSD is used as a quantitative measure of similarity between two or more protein structures. For example, the CASP protein structure prediction competition uses RMSD as one of its assessments of how well a submitted structure matches the known, target structure. Thus the lower RMSD, the better the model is in comparison to the target structure.
Also some scientists who study protein folding by computer simulations use RMSD as a reaction coordinate to quantify where the protein is between the folded state and the unfolded state.
The study of RMSD for small organic molecules (commonly called ligands when they're binding to macromolecules, such as proteins, is studied) is common in the context of docking , [ 1 ] as well as in other methods to study the configuration of ligands when bound to macromolecules. Note that, for the case of ligands (contrary to proteins, as described above), their structures are most commonly not superimposed prior to the calculation of the RMSD.
RMSD is also one of several metrics that have been proposed for quantifying evolutionary similarity between proteins, as well as the quality of sequence alignments. [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Root_mean_square_deviation_of_atomic_positions |
The root microbiome (also called rhizosphere microbiome) is the dynamic community of microorganisms associated with plant roots . [ 1 ] Because they are rich in a variety of carbon compounds, plant roots provide unique environments for a diverse assemblage of soil microorganisms, including bacteria , fungi , and archaea . The microbial communities inside the root and in the rhizosphere are distinct from each other, [ 2 ] and from the microbial communities of bulk soil , [ 3 ] although there is some overlap in species composition.
Different microorganisms, both beneficial and harmful, affect the development and physiology of plants. Beneficial microorganisms include bacteria that fix nitrogen, various microbes that promote plant growth, mycorrhizal fungi, mycoparasitic fungi, protozoa, and certain biocontrol microorganisms. [ 1 ] Pathogenic microorganisms can also include certain bacteria, fungi, and nematodes that can colonize the rhizosphere. Pathogens are able to compete with protective microbes and break through innate plant defense mechanisms. [ 1 ] Some pathogenic bacteria that can be carried over to humans, such as Salmonella , enterohaemorhagic Escherichia coli , Burkholderia cenocepacia , Pseudomonas aeruginosa , and Stenotrophomonas maltophilia , can also be detected in root microbiomes and other plant tissues. [ 1 ]
Root microbiota affect plant host fitness and productivity in a variety of ways. Members of the root microbiome benefit from plant sugars or other carbon rich molecules. Individual members of the root microbiome may behave differently in association with different plant hosts, [ 4 ] or may change the nature of their interaction (along the mutualist-parasite continuum ) within a single host as environmental conditions or host health change. [ 5 ]
Despite the potential importance of the root microbiome for plants and ecosystems , our understanding of how root microbial communities are assembled is in its infancy. [ 6 ] [ 7 ] This is in part because, until recent advances in sequencing technologies, root microbes were difficult to study due to high species diversity , the large number of cryptic species , and the fact that most species have yet to be retrieved in culture . [ 8 ] Evidence suggests both biotic (such as host identity and plant neighbor) and abiotic (such as soil structure and nutrient availability) factors affect community composition. [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ]
Root associated microbes include fungi , bacteria , and archaea . In addition, other organisms such as viruses , algae , protozoa , nematodes , and arthropods are part of root microbiota. [ 1 ] Symbionts associated with plant roots subsist off of photosynthetic products (carbon rich molecules) from the plant host and can exist anywhere on the mutualist/parasite continuum.
Root symbionts may improve their host's access to nutrients , [ 14 ] [ 15 ] [ 16 ] produce plant-growth regulators , [ 17 ] improve environmental stress tolerance of their host, [ 18 ] [ 19 ] [ 20 ] induce host defenses and systemic resistance against pests or pathogens, [ 21 ] [ 22 ] [ 23 ] or be pathogenic . [ 24 ] Parasites consume carbon from the plant without providing any benefit or providing insufficient benefit relative to their carbon consumption, thereby compromising host fitness. Symbionts may be biotrophic (subsisting off of living tissue) or necrotrophic (subsisting off of dead tissue).
While some microbes may be purely mutualistic or parasitic , many may behave differently depending on the host species with which it is associated, environmental conditions, and host health. [ 4 ] A host's immune response controls symbiont infection and growth rates. [ 4 ] If a host's immune response is not able to control a particular microbial species, or if host immunity is compromised, the microbe-plant relationship will likely reside somewhere nearer the parasitic side of the mutualist-parasite continuum. Similarly, high nutrients can push some microbes into parasitic behavior, encouraging unchecked growth at a time when symbionts are no longer needed to aid with nutrient acquisition. [ 4 ]
Roots are colonized by fungi , bacteria , and archaea . Because they are multicellular , fungi can extend hyphae from nutrient exchange organs within host cells into the surrounding rhizosphere and bulk soil. Fungi that extend beyond the root surface and engage in nutrient-carbon exchange with the plant host are commonly considered to be mycorrhizal , but external hyphae can also include other endophytic fungi. Mycorrhizal fungi can extend a great distance into bulk soil, [ 5 ] thereby increasing the root system's reach and surface area, enabling mycorrhizal fungi to acquire a large percentage of its host plant's nutrients. In some ecosystems, up to 80% of plant nitrogen and 90% of plant phosphorus is acquired by mycorrhizal fungi . [ 14 ] In return, plants may allocate ~20–40% of their carbon to mycorrhizae. [ 25 ]
Mycorrhizal (from Greek) literally means "fungus roots" and defines symbiotic interaction between plants and fungi. Fungi are important for decomposing and recycling organic material. However, the boundaries between the pathogenic and symbiotic lifestyles of fungi are not always clear-cut. Most of the time, the association is symbiotic, with the fungus improving nutrient and water acquisition or increasing stress tolerance for the plant and benefiting from the carbohydrates produced by the plant in return. [ 26 ] Mycorrhizae include a wide variety of root-fungi interactions characterized by the mode of colonization. Essentially all plants form mycorrhizal associations, and there is evidence that some mycorrhizae transport carbon and other nutrients not only from soil to plant, but also between different plants in a landscape. [ 5 ] The main groups include ectomycorrhizae , arbuscular mycorrhizae , ericoid mycorrhizae , orchid mycorrhizae , and monotropoid mycorrhizae . Monotropoid mycorrhizae are associated with plants in the monotropaceae , which lack chlorophyll . Many Orchids are also achlorophyllous for at least part of their life cycle. Thus, these mycorrhizal-plant relationships are unique because the fungus provides the host with carbon and other nutrients, often by parasitizing other plants. [ 5 ] Achlorophyllous plants forming these types of mycorrhizal associations are called mycoheterotrophs .
Endophytes grow inside plant tissue—roots, stems, leaves—mostly symptomless. However, when plants age, they can become slightly pathogenic. [ 26 ] They may colonize inter-cellular spaces, the root cells themselves, or both. Rhizobia and dark septate endophytes (which produce melanin , an antioxidant that may provide resilience against a variety of environmental stresses [ 27 ] ) are examples.
The zone of soil surrounding the roots is rich in nutrients released by plants and is, therefore, an attractive growth medium for both beneficial and pathogenic bacteria. Root associated beneficial bacteria promote plant growth and provide protection from pathogens. They are mostly rhizobacteria that belong to Pseudomonadota and Bacillota , with many examples from Pseudomonas and Bacillus genera. [ 1 ] Rhizobium species colonize legume roots forming nodule structures. In response to root exudates, rhizobia produce Nod signalling factors that are recognized by legumes and induce the formation of nodules on plant roots. [ 28 ] Within these structures, Rhizobium fix atmospheric nitrogen into ammonia that is then used by the plant. In turn, plants provide the bacteria with a carbon source to energize the nitrogen fixation. [ 29 ] [ 30 ] In addition to nitrogen fixation, Azospirillum species promote plant growth through the production of growth phytohormones ( auxins , cytokinins , gibberellins ). Due to these phytohormones, root hairs expand to occupy a larger area and better acquire water and nutrients. [ 29 ] [ 31 ] Pathogenic bacteria that infect plants infect plant roots are most commonly from Pectobacterium , Ralstonia , Dickeya and Agrobacterium genera. Among the most notorious are Pectobacterium carotovorum , Pectobacterium atrosepticum , Ralstonia solanacearum , Dickeya dadanthi , Dickeya solani , and Agrobacterium tumefaciens .
Bacteria attach to roots in a biphasic mechanism with two steps—first weak, non-specific binding, then a strong irreversible residence phase. Both beneficial and pathogenic bacteria attach in this fashion. Bacteria can stay attached to the outer surface or colonize the inner root. [ 29 ] Primary attachment is governed by chemical forces or extracellular structures such as pili or flagella . Secondary attachment is mainly characterized by the synthesis of cellulose , extracellular fibrils, and specific attachment factors such as surface proteins that help bacteria aggregate and form colonies. [ 29 ]
Though archaea are often thought of as extremophiles , microbes belonging to extreme environments, advances in metagenomics and gene sequencing have revealed that archaea are found in nearly any environment, including the root microbiome. [ 8 ] [ 32 ] [ 33 ] [ 34 ] [ 35 ] [ 36 ] For example, root-colonizing archaea have been discovered in maize , [ 33 ] rice , [ 37 ] wheat , [ 34 ] and mangroves . [ 38 ] Methanogen and ammonium-oxidizing archaea are prevalent members of the root microbiome, especially in anaerobic soils and wetlands. [ 32 ] [ 39 ] [ 40 ] [ 41 ] Archaeal phyla found in the root microbiome include Euryarchaeota , [ 32 ] [ 40 ] [ 42 ] Nitrososphaerota (formerly Thaumarchaeota), [ 32 ] [ 42 ] and Thermoproteota (formerly Crenarchaeota). [ 40 ]
The presence and relative abundance of archaea in various environments suggest that they likely play an important role in the root microbiome. [ 32 ] Archaea have been found to promote plant growth and development, provide stress tolerance, improve nutrient uptake, and protect against pathogens. [ 32 ] [ 36 ] [ 43 ] For example, Arabidopsis thaliana colonized with an ammonia-oxidizing soil archaea, Nitrosocosmicus oleophilius , exhibited increased shoot weight, photosynthetic activity, and immune response. [ 43 ]
Examination of microbial communities in soil and roots has identified archaeal organisms and genes with functions similar to that of bacteria and fungi , such as auxin synthesis, protection against abiotic stress , and nitrogen fixation . [ 36 ] [ 44 ] In some cases, key genes for plant growth and development, such as metabolism and cell wall synthesis, are more prevalent in archaea than bacteria. [ 36 ]
Archaeal presence in the root microbiome can also be affected by plant hosts, which can change the diversity, presence, and health of archaeal communities. [ 8 ] [ 38 ] [ 45 ]
Viruses also infect plants via the roots; however, to penetrate the root tissues, they typically use vectors such as nematodes or fungi. [ 1 ]
There is an ongoing debate regarding what mechanisms are responsible for assembling individual microbes into communities . There are two primary competing hypotheses. One is that "everything is everywhere, but the environment selects," meaning biotic and abiotic factors pose the only constraints, through natural selection , to which microbes colonize what environments . This is called the niche hypothesis. Its counterpart is the hypothesis that neutral processes, such as distance and geographic barriers to dispersal , control microbial community assembly when taxa are equally fit within an environment. In this hypothesis, differences between individual taxa in modes and reach of dispersal explain the differences in microbial communities of different environments. [ 7 ] Most likely, both natural selection and neutral processes affect microbial community assembly, though certain microbial taxa may be more restricted by one process or the other depending on their physiological restrictions and mode of dispersion. [ 7 ]
Microbial dispersal mechanisms include wind, water, and hitchhiking on more mobile macrobes. Microbial dispersion is difficult to study, and little is known about its effect on microbial community assembly relative to the effect of abiotic and biotic assembly mechanisms, [ 7 ] particularly in roots. For this reason, only assembly mechanisms that fit within the niche hypothesis are discussed below.
The taxa within root microbial communities seem to be partly drawn from the surrounding soil, though the relative abundance of various taxa may differ greatly from those found in bulk soil due to unique niches in the root and rhizosphere. [ 8 ]
Recent evidence shows that seed-transmitted bacteria contribute significantly to the composition of the root microbiome. In wheat, they can dominate over soil-derived microbes and structure the rhizosphere community through niche partitioning and facilitation. [ 46 ] These bacteria possess traits that allow them to degrade root-derived compounds and support the growth of other microbes, highlighting the importance of seed microbiota in microbial succession and community assembly within the rhizosphere.
Different parts of the root are associated with different microbial communities. For example, fine roots, root tips, and the main root are all associated with different communities, [ 8 ] [ 47 ] and the rhizosphere, root surface, and root tissue are all associated with different communities, [ 2 ] [ 3 ] likely due to the unique chemistry and nutrient status of each of these regions, which differ from those of the bulk soil. [ 48 ] For instance, root exudates provide specific carbon compounds that are enriched in the rhizosphere, selecting for microbial taxa with matching metabolic traits and shaping root-associated communities accordingly. [ 48 ] Additionally, different plant species, and even different cultivars, harbor different microbial communities, [ 9 ] [ 10 ] [ 47 ] probably due to host specific immune responses [ 4 ] and differences in carbon root exudates. [ 49 ] Host age affects root microbial community composition, likely for similar reasons as host identity. [ 8 ] The identity of neighboring vegetation has also been shown to impact a host plant's root microbial community composition. [ 9 ] [ 10 ] [ 50 ] [ 51 ]
Abiotic mechanisms also affect root microbial community assembly [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] because individual taxa have different optima along various environmental gradients , such as nutrient concentrations, pH, moisture, temperature, etc. In addition to chemical and climatic factors, soil structure and disturbance impact root biotic assembly. [ 8 ]
The root microbiome is dynamic and fluid within the constraints imposed by the biotic and abiotic environment. As in macroecological systems, the historical trajectory of the microbiotic community may partially determine the present and future community. Due to antagonistic and mutualistic interactions between microbial taxa, the taxa colonizing a root at any given moment could be expected to influence which new taxa are acquired, and therefore how the community responds to changes in the host or environment. [ 7 ] While the effect of initial community on microbial succession has been studied in various environmental samples, human microbiome , and laboratory settings, it has yet to be studied in roots. | https://en.wikipedia.org/wiki/Root_microbiome |
Root nodules are found on the roots of plants , primarily legumes , that form a symbiosis with nitrogen-fixing bacteria . [ 1 ] Under nitrogen -limiting conditions, capable plants form a symbiotic relationship with a host-specific strain of bacteria known as rhizobia . [ 2 ] This process has evolved multiple times within the legumes, as well as in other species found within the Rosid clade. [ 3 ] Legume crops include beans , peas , and soybeans .
Within legume root nodules, nitrogen gas (N 2 ) from the atmosphere is converted into ammonia (NH 3 ), which is then assimilated into amino acids (the building blocks of proteins), nucleotides (the building blocks of DNA and RNA as well as the important energy molecule ATP ), and other cellular constituents such as vitamins , flavones , and hormones . [ citation needed ] Their ability to fix gaseous nitrogen makes legumes an ideal agricultural organism as their requirement for nitrogen fertilizer is reduced. Indeed, high nitrogen content blocks nodule development as there is no benefit for the plant of forming the symbiosis. The energy for splitting the nitrogen gas in the nodule comes from sugar that is translocated from the leaf (a product of photosynthesis ). Malate as a breakdown product of sucrose is the direct carbon source for the bacteroid. Nitrogen fixation in the nodule is very oxygen sensitive. Legume nodules harbor an iron containing protein called leghaemoglobin , closely related to animal myoglobin , to facilitate the diffusion of oxygen gas used in respiration.
Plants that contribute to N2 fixation include the legume family – Fabaceae – with taxa such as kudzu , clovers , soybeans , alfalfa , lupines , peanuts , and rooibos . They contain symbiotic bacteria called rhizobia within the nodules, producing nitrogen compounds that help the plant to grow and compete with other plants. When the plant dies, the fixed nitrogen is released, making it available to other plants, and this helps to fertilize the soil . [ 4 ] [ 5 ] The great majority of legumes have this association, but a few genera (e.g., Styphnolobium ) do not. In many traditional farming practices, fields are rotated through various types of crops, which usually includes one consisting mainly or entirely of a leguminous crop such as clover, in order to take advantage of this. [ citation needed ]
Although by far the majority of plants able to form nitrogen-fixing root nodules are in the legume family Fabaceae , there are a few exceptions:
The ability to fix nitrogen is far from universally present in these families. For instance, of 122 genera in the Rosaceae , only 4 genera are capable of fixing nitrogen. All these families belong to the orders Cucurbitales , Fagales , and Rosales , which together with the Fabales form a nitrogen-fixing clade (NFC) of eurosids . In this clade, Fabales were the first lineage to branch off; thus, the ability to fix nitrogen may be plesiomorphic and subsequently lost in most descendants of the original nitrogen-fixing plant; however, it may be that the basic genetic and physiological requirements were present in an incipient state in the last common ancestors of all these plants, but only evolved to full function in some of them: [ citation needed ]
Betulaceae : Alnus (alders)
Cannabaceae : Trema
Casuarinaceae :
......
Coriariaceae : Coriaria
Datiscaceae : Datisca
Elaeagnaceae :
......
Myricaceae :
......
Rhamnaceae :
......
Rosaceae :
Two main types of nodule have been described in legumes: determinate and indeterminate. [ 9 ]
Determinate nodules are found on certain tribes of tropical legume such as those of the genera Glycine (soybean), Phaseolus (common bean), and Vigna . and on some temperate legumes such as Lotus . These determinate nodules lose meristematic activity shortly after initiation, thus growth is due to cell expansion resulting in mature nodules which are spherical in shape. Another type of determinate nodule is found in a wide range of herbs, shrubs and trees, such as Arachis ( peanut ). These are always associated with the axils of lateral or adventitious roots and are formed following infection via cracks where these roots emerge and not using root hairs . Their internal structure is quite different from those of the soybean type of nodule. [ 10 ]
Indeterminate nodules are found in the majority of legumes from all three sub-families, whether in temperate regions or in the tropics. They can be seen in Faboideae legumes such as Pisum (pea), Medicago (alfalfa), Trifolium (clover), and Vicia (vetch) and all mimosoid legumes such as acacia s, the few nodulated caesalpinioid legumes such as partridge pea . They earned the name "indeterminate" because they maintain an active apical meristem that produces new cells for growth over the life of the nodule. This results in the nodule having a generally cylindrical shape, which may be extensively branched. [ 10 ] Because they are actively growing, indeterminate nodules manifest zones which demarcate different stages of development/symbiosis: [ 11 ] [ 12 ] [ 13 ]
This is the most widely studied type of nodule, but the details are quite different in nodules of peanut and relatives and some other important crops such as lupins where the nodule is formed following direct infection of rhizobia through the epidermis and where infection threads are never formed. Nodules grow around the root, forming a collar-like structure. In these nodules and in the peanut type the central infected tissue is uniform, lacking the uninfected ells seen in nodules of soybean and many indeterminate types such as peas and clovers. [ citation needed ]
Actinorhizal-type nodules are markedly different structures found in non-legumes. In this type, cells derived from the root cortex form the infected tissue, and the prenodule becomes part of the mature nodule. Despite this seemingly major difference, it is possible to produce such nodules in legumes by a single homeotic mutation. [ 14 ]
Legumes release organic compounds as secondary metabolites called flavonoids from their roots, which attract the rhizobia to them and which also activate nod genes in the bacteria to produce nod factors and initiate nodule formation. [ 15 ] [ 16 ] These nod factors initiate root hair curling . The curling begins with the very tip of the root hair curling around the Rhizobium . Within the root tip, a small tube called the infection thread forms, which provides a pathway for the Rhizobium to travel into the root epidermal cells as the root hair continues to curl. [ 17 ]
Partial curling can even be achieved by nod factor alone. [ 16 ] This was demonstrated by the isolation of nod factors and their application to parts of the root hair. The root hairs curled in the direction of the application, demonstrating the action of a root hair attempting to curl around a bacterium. Even application on lateral roots caused curling. This demonstrated that it is the nod factor itself, not the bacterium that causes the stimulation of the curling. [ 16 ]
When the nod factor is sensed by the root, a number of biochemical and morphological changes happen: cell division is triggered in the root to create the nodule, and the root hair growth is redirected to curl around the bacteria multiple times until it fully encapsulates one or more bacteria. The bacteria encapsulated divide multiple times, forming a microcolony . From this microcolony, the bacteria enter the developing nodule through the infection thread, which grows through the root hair into the basal part of the epidermis cell, and onwards into the root cortex ; they are then surrounded by a plant-derived symbiosome membrane and differentiate into bacteroids that fix nitrogen . [ 18 ]
Effective nodulation takes place approximately four weeks after crop planting , with the size, and shape of the nodules dependent on the crop. Crops such as soybeans, or peanuts will have larger nodules than forage legumes such as red clover, or alfalfa, since their nitrogen needs are higher. The number of nodules, and their internal color, will indicate the status of nitrogen fixation in the plant. [ 19 ]
Nodulation is controlled by a variety of processes, both external (heat, acidic soils, drought, nitrate) and internal (autoregulation of nodulation, ethylene). Autoregulation of nodulation [ 20 ] controls nodule numbers per plant through a systemic process involving the leaf. Leaf tissue senses the early nodulation events in the root through an unknown chemical signal, then restricts further nodule development in newly developing root tissue. The Leucine rich repeat (LRR) receptor kinases (NARK in soybean ( Glycine max ); HAR1 in Lotus japonicus , SUNN in Medicago truncatula ) are essential for autoregulation of nodulation (AON). Mutation leading to loss of function in these AON receptor kinases leads to supernodulation or hypernodulation. Often root growth abnormalities accompany the loss of AON receptor kinase activity, suggesting that nodule growth and root development are functionally linked. Investigations into the mechanisms of nodule formation showed that the ENOD40 gene, coding for a 12–13 amino acid protein [41], is up-regulated during nodule formation [3].
Root nodules apparently have evolved three times within the Fabaceae but are rare outside that family. The propensity of these plants to develop root nodules seems to relate to their root structure. In particular, a tendency to develop lateral roots in response to abscisic acid may enable the later evolution of root nodules. [ 21 ]
Some fungi produce nodular structures known as tuberculate ectomycorrhizae on the roots of their plant hosts. Suillus tomentosus , for example, produces these structures with its plant host lodgepole pine ( Pinus contorta var. latifolia ). These structures have, in turn, been shown to host nitrogen fixing bacteria , which contribute a significant amount of nitrogen and allow the pines to colonize nutrient-poor sites. [ 22 ] | https://en.wikipedia.org/wiki/Root_nodule |
In mathematics , the root test is a criterion for the convergence (a convergence test ) of an infinite series . It depends on the quantity
where a n {\displaystyle a_{n}} are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series .
The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). [ 1 ] Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test . For a series
the root test uses the number
where "lim sup" denotes the limit superior , possibly +∞. Note that if
converges then it equals C and may be used in the root test instead.
The root test states that:
There are some series for which C = 1 and the series converges, e.g. ∑ 1 / n 2 {\displaystyle \textstyle \sum 1/{n^{2}}} , and there are others for which C = 1 and the series diverges, e.g. ∑ 1 / n {\displaystyle \textstyle \sum 1/n} .
This test can be used with a power series
where the coefficients c n , and the center p are complex numbers and the argument z is a complex variable.
The terms of this series would then be given by a n = c n ( z − p ) n . One then applies the root test to the a n as above. Note that sometimes a series like this is called a power series "around p ", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).
A corollary of the root test applied to a power series is the Cauchy–Hadamard theorem : the radius of convergence is exactly 1 / lim sup n → ∞ | c n | n , {\displaystyle 1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}},} taking care that we really mean ∞ if the denominator is 0.
The proof of the convergence of a series Σ a n is an application of the comparison test .
If for all n ≥ N ( N some fixed natural number ) we have | a n | n ≤ k < 1 {\displaystyle {\sqrt[{n}]{|a_{n}|}}\leq k<1} , then | a n | ≤ k n < 1 {\displaystyle |a_{n}|\leq k^{n}<1} . Since the geometric series ∑ n = N ∞ k n {\displaystyle \sum _{n=N}^{\infty }k^{n}} converges so does ∑ n = N ∞ | a n | {\displaystyle \sum _{n=N}^{\infty }|a_{n}|} by the comparison test. Hence Σ a n converges absolutely.
If | a n | n > 1 {\displaystyle {\sqrt[{n}]{|a_{n}|}}>1} for infinitely many n , then a n fails to converge to 0, hence the series is divergent.
Proof of corollary :
For a power series Σ a n = Σ c n ( z − p ) n , we see by the above that the series converges if there exists an N such that for all n ≥ N we have
equivalent to
for all n ≥ N , which implies that in order for the series to converge we must have | z − p | < 1 / | c n | n {\displaystyle |z-p|<1/{\sqrt[{n}]{|c_{n}|}}} for all sufficiently large n . This is equivalent to saying
so R ≤ 1 / lim sup n → ∞ | c n | n . {\displaystyle R\leq 1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}}.} Now the only other place where convergence is possible is when
(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so
Example 1:
Applying the root test and using the fact that lim n → ∞ n 1 / n = 1 , {\displaystyle \lim _{n\to \infty }n^{1/n}=1,}
Since C = 2 > 1 , {\displaystyle C=2>1,} the series diverges. [ 2 ]
Example 2:
The root test shows convergence because
This example shows how the root test is stronger than the ratio test . The ratio test is inconclusive for this series as if n {\displaystyle n} is even, a n + 1 / a n = 1 {\displaystyle a_{n+1}/a_{n}=1} while if n {\displaystyle n} is odd, a n + 1 / a n = 1 / 2 {\displaystyle a_{n+1}/a_{n}=1/2} , therefore the limit lim n → ∞ | a n + 1 / a n | {\displaystyle \lim _{n\to \infty }|a_{n+1}/a_{n}|} does not exist.
Root tests hierarchy [ 3 ] [ 4 ] is built similarly to the ratio tests hierarchy (see Section 4.1 of ratio test , and more specifically Subsection 4.1.4 there).
For a series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} with positive terms we have the following tests for convergence/divergence.
Let K ≥ 1 {\displaystyle K\geq 1} be an integer, and let ln ( K ) ( x ) {\displaystyle \ln _{(K)}(x)} denote the K {\displaystyle K} th iterate of natural logarithm , i.e. ln ( 1 ) ( x ) = ln ( x ) {\displaystyle \ln _{(1)}(x)=\ln(x)} and for any 2 ≤ k ≤ K {\displaystyle 2\leq k\leq K} , ln ( k ) ( x ) = ln ( k − 1 ) ( ln ( x ) ) {\displaystyle \ln _{(k)}(x)=\ln _{(k-1)}(\ln(x))} .
Suppose that a n − n {\displaystyle {\sqrt[{-n}]{a_{n}}}} , when n {\displaystyle n} is large, can be presented in the form
(The empty sum is assumed to be 0.)
Since a n − n = e − 1 n ln a n {\displaystyle {\sqrt[{-n}]{a_{n}}}=\mathrm {e} ^{-{\frac {1}{n}}\ln a_{n}}} , then we have
From this,
From Taylor's expansion applied to the right-hand side, we obtain:
Hence,
(The empty product is set to 1.)
The final result follows from the integral test for convergence .
This article incorporates material from Proof of Cauchy's root test on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Root_test |
The Rootare-Prenzlow Equation is named for Estonian-American scientist Hillar Rootare and American scientist Carl , first published in their 1967 paper, "Surface Areas from Mercury Porosimetry Measurements," Rootare, H.M., and Prenzlow, C.F., 71 J. Phys. Chem. p. 2733 (1967). The equation first formulated a means to calculate cumulative surface areas of porous solids based on data taken in mercury porosimetry testing. Rootare and Spencer later devised a computer program to carry out automated calculations, "A Computer Program for Pore Volume and Pore Area Distribution," Rootare & Spencer, Perspectives in Powder Metallurgy (Advanced Experimental Techniques in Powder Metallurgy) p. 225, Plenum Press (New York, London 1970).
The equation can be found at section 2.2.3 here: http://ethesis.helsinki.fi/julkaisut/mat/farma/vk/westermarck/ch2.html | https://en.wikipedia.org/wiki/Rootare–Prenzlow_equation |
The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type . It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals , respectively, are doubly occupied. This is generally called Restricted Hartree–Fock theory.
The method was developed independently by Clemens C. J. Roothaan and George G. Hall in 1951, and is thus sometimes called the Roothaan-Hall equations . [ 1 ] [ 2 ] [ 3 ] The Roothaan equations can be written in a form resembling generalized eigenvalue problem , although they are not a standard eigenvalue problem because they are nonlinear:
where F is the Fock matrix (which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and ϵ {\displaystyle \epsilon } is the (diagonal, by convention) matrix of orbital energies. In the case of an orthonormalised basis set the overlap matrix, S, reduces to the identity matrix. These equations are essentially a special case of a Galerkin method applied to the Hartree–Fock equation using a particular basis set.
In contrast to the Hartree–Fock equations - which are integro-differential equations - the Roothaan–Hall equations have a matrix-form. Therefore, they can be solved using standard techniques.
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Roothaan_equations |
RopB transcriptional regulator , also known as RopB/Rgg transcriptional regulator is a transcriptional regulator protein that regulates expression of the extracellularly secreted cysteine protease streptococcal pyrogenic exotoxin B (speB or streptopain ), which is an important virulence factor of Streptococcus pyogenes and is responsible for the dissemination of a host of infectious diseases including strep throat , impetigo , streptococcal toxic shock syndrome , necrotizing fasciitis , and scarlet fever . [ 2 ] Functional studies suggest that the ropB multigene regulon is responsible for not only global regulation of virulence [ 3 ] [ 4 ] but also a wide range of functions from stress response, metabolic function, and two-component signaling. [ 5 ] Structural studies implicate ropB's regulatory action being reliant on a complex interaction involving quorum sensing with the leaderless peptide signal speB-inducing peptide (SIP) [ 1 ] acting in conjunction with a pH sensitive histidine switch. [ 6 ]
Observations of an extracellularly secreted glucosyltransferase (gtfG) sequentially proximal to and activated by an rgg gene with inverted repeats in the intergenic region of Streptococcus gordonii served as a basis for studying its homology between Streptococcus pyogenes . It was discovered that S. pyogenes also shared an rgg/ropB gene located directly next to the subject of its transcriptional regulation, in this case speB protease , with intergenic inverted repeats. [ 7 ] Confirmation of linkage between rgg/ropB and speB secretion activation was achieved by means of ropB insertional disruption which resulted in decreased speB production. [ 8 ]
The location of the ropB gene is directly and sequentially proximal to the subject of its transcriptional regulation, speB, which lies downstream of a 941 bp intergenic region between the two. [ 9 ] Transcription of the ropB gene seems to necessitate a promoter within a series sequences between 238 and 480 bp and up to 800 bp upstream of the gene itself inside the highly repetitive intergenic region. [ 10 ]
The ropB protein binding location lies adjacent to speB promoter 1 [ 11 ] that is also located within the highly repetitive intergenic region, although the ropB gene and the speB gene are transcribed in opposite directions. The -10 and -35 regions of speB promoter 1 have poor consensus ; in order to ameliorate this, the ropB aids the RNA polymerase bondage with the help of a polyU polypyrimidine tract inside the palindromic inverted repeat region [ 10 ] in a fashion uncannily similar to intrinsic termination in E. coli .
The N-terminal domain consists of amino acids 1-56 and is an amino terminal responsible for DNA-binding and is a key mediator in the linkage between the C-terminal domain of the opposite dimer . The dimer interface II has its I255 side chain located in the N-terminal. [ 1 ]
The C-terminal domain, also known as ropB-CTD, is a carboxy terminal ligand-binding domain made of amino acids 56–280. RopB-CTD houses 5 TPR motifs and attaches to the SIP peptide in the innermost part of the SIP binding pocket in a sequence-specific manner without induction of polymerization . [ 6 ]
The tetratricopeptide repeat domain provides the concave surface necessitated for SIP recognition. RopB-CTD houses 5 stacked TPR motifs, each having sets of paired antiparallel helices that aid in the formation of a concave inner pathway and a convex exterior. The base of the recognition site is constructed by alpha helices α6 and α8, while the supporting walls are constructed from helices α2, and α12. The exterior portion of the recognition site is flanked by asparagines N152 and N192, thus providing a ridge of support for the peptide-protein complex. [ 1 ] [ 6 ]
The dimer interfaces of ropB are constructed by a union of the α8 - α12 helices of the N-terminal domain and the C-terminal domain. Additionally, there is an Interface I forged from three side chains (C22, Y224, and R226), an Interface II forged from one side chain (I255), and N-terminal domains that are all responsible for dimerizing ropB protein subunits together. [ 1 ]
The SIP peptide binding pocket is the docking station of the eight amino acid leaderless peptide signal, speB-inducing peptide (SIP). The binding pocket is a tripartite construction of the C-terminal's α12 helix which is a capping helix, TPR3's α6 helix that has a hydrophobic interplay with SIP sidechains, and TPR 4's α8 helix which electrostatically stabilizes SIP. Variations in pH level altered strength of adherence between SIP and the SIP binding pocket with acidic pH levels between 5.5 and 6.5 enhancing adherence and pH levels between 7 and 9 reducing adherence. [ 6 ]
Though the ropB protein has seven histidines (H12, H81, H93, H144, H265, H266, and H277) structurally present, the ropB histidine switch primarily operates with a single functionally involved histidine (H144) conveniently placed to associate with ropB sidechains (Y176 and E185) that near each other upon the addition of a hydrogen ion to H144 in acidic conditions. Only one histidine (H12) is located on the N-domain while the rest lie in the C-terminal domain. [ 6 ]
Streptococcus pyogenes has evolved an interwoven complex of gene regulatory mechanisms in the SIP signaling pathway by implanting a pH sensitive histidine switch onto the quorum-sensing ropB protein. During the neutral to basic pH conditions whether synthetically induced or naturally caused by low population density of S. pyogenes , the interaction between the unprotonated functionally involved histidine (H144) with relevant sidechains (Y176, Y182, E185) in the SIP binding pocket domain is impaired and speB protease expression is inhibited. On the other hand, as extracellular pH decreases to be more acidic in cases of high population density, S. pyogenes has no elaborate pH homeostatic capabilities relative to non-lactic bacteria, therefore intracellular cytosolic pH levels will more easily resemble extracellular levels. [ 12 ] Cytosolic acidification mobilizes the SIP pathway to allow for the SIP-ropB protein complex to form and increasing SIP production. Furthermore, increased cytosolic acidity enhances the maturation of speB zymogen (speBz) into mature speB protease (speBm) to dramatically increase its proteolytic activity and virulence. [ 6 ]
Rgg-like transcriptional regulators can be found in a variety of gram-positive bacteria . Where ropB regulates speB protease production in S. pyogenes , a roughly equivalent secretory control mechanism can be seen in Rgg's regulation of gtfG glucosyltransferase production in S. gordonii , [ 13 ] in the manner in which gadR regulates acid resistance in Lactococcus lactis , [ 8 ] how lasX regulates expression of lantibiotic lactocin S in Lactobacillus sakei , [ 10 ] and mutR's regulation of mutacin in S. mutans . [ 14 ] Sequentially, these genes are all localized contiguously to their respective subject of regulation and share promoters localized contiguously to inverted repeat regions.
Characterization of the RRNPP family of quorum-sensing regulators (which stands for proteins Rap, NprR, PrgX, PlcRd) were used in comparisons with ropB to postulate its structural functions. The Rap protein derived from Bacilli regulates sporulation , the NprR protein in Bacillus thuringiensis regulates necrotrophism, the PrgX protein regulates conjugation in Enterococcus faecalis , and PlcR protein regulates transcription of virulence factors in both Bacillis thuringiensis and Bacillus cereus . [ 15 ] Similarities were observed in conserved asparagine residues on the TPR motifs of each of these proteins and in ropB. [ 1 ]
Quorum sensing regulates a menagerie of aspects in Bacillota including the production of ropB-like proteins in Streptococcus pneumoniae and S. pyogenes . [ 16 ] Similarities in the pH sensitivity of the cell signaling mechanisms were found in pneumococci , S. mutans , [ 6 ] and Staphylococcus aureus as well.
Amongst Rgg-like proteins, it has been observed that the pH sensitive histidine (particularly H144) and interacting amino acids (Y176, Y182, and E185) of ropB of Streptococcus pyogenes are conserved in S. porcinus , S. pseudoporcinus , S. salivarius , L. pentosus , L. aviaries , L. reuteri , and Enterococcus sp. including E. faecalis . [ 6 ] Thus, suggesting the usage of a pH sensitive histidine switch complex with gene-regulating effector molecules in a slew of other bacteria [See Also: allosteric regulation ].
RopB regulation speB is a key determinant in the expression of the speB proteinase which is a primary virulence factor and the most abundant extracellular protein in streptococcal secretions. [ 17 ] SpeB cleaves host serum proteins that make up the human extracellular matrix and bacterial proteins including other secreted streptococcal proteins. As previously mentioned, it is responsible for the dissemination of a host of infectious diseases including but not limited to pharyngitis , impetigo , streptococcal toxic shock syndrome , necrotizing fasciitis , and scarlet fever . [ 2 ] Therefore, study of the inactivation of speB's many functional pathways and regulators are of critical importance in developing potential novel therapeutics. | https://en.wikipedia.org/wiki/RopB_transcriptional_regulator |
In recreational mathematics , rope-burning puzzles are a class of mathematical puzzle in which one is given lengths of rope, fuse cord , or shoelace that each burn for a given amount of time, and matches to set them on fire, and must use them to measure a non-unit amount of time. The fusible numbers are defined as the amounts of time that can be measured in this way.
As well as being of recreational interest, these puzzles are sometimes posed at job interviews as a test of candidates' problem-solving ability, [ 1 ] and have been suggested as an activity for middle school mathematics students. [ 2 ]
A common and simple version of this problem asks to measure a time of 45 seconds using only two fuses that each burn for a minute. The assumptions of the problem are usually specified in a way that prevents measuring out 3/4 of the length of one fuse and burning it end-to-end, for instance by stating that the fuses burn unevenly along their length. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
One solution to this problem is to perform the following steps: [ 3 ]
Many other variations are possible, in some cases using fuses that burn for different amounts of time from each other. [ 5 ]
In common versions of the problem, each fuse lasts for a unit length of time, and the only operations used or allowed in the solution are to light one or both ends of a fuse at known times, determined either as the start of the solution or as the time that another fuse burns out. If only one end of a fuse is lit at time x {\displaystyle x} , it will burn out at time x + 1 {\displaystyle x+1} . If both ends of a fuse are lit at times x {\displaystyle x} and y {\displaystyle y} , it will burn out at time ( x + y + 1 ) / 2 {\displaystyle (x+y+1)/2} , because a portion of y − x {\displaystyle y-x} is burnt at the original rate, and the remaining portion of 1 − ( y − x ) {\displaystyle 1-(y-x)} is burnt at twice the original rate, hence the fuse burns out at
A number x {\displaystyle x} is a fusible number if it is possible to use unit-time fuses to measure out x {\displaystyle x} units of time using only these operations. For instance, by the solution to the example problem, 3 4 {\displaystyle {\tfrac {3}{4}}} is a fusible number. [ 7 ]
One may assume without loss of generality that every fuse is lit at both ends, by replacing a fuse that is lit only at one end at time x {\displaystyle x} by two fuses, the first one lit at both ends at time x {\displaystyle x} and the second one lit at both ends at time x + 1 / 2 {\displaystyle x+1/2} when the first fuse burns out.
In this way, the fusible numbers can be defined as the set of numbers that can be obtained from the number 0 {\displaystyle 0} by repeated application of the operation x , y ↦ ( x + y + 1 ) / 2 {\displaystyle x,y\mapsto (x+y+1)/2} , applied to pairs x , y {\displaystyle x,y} that have already been obtained and for which | x − y | < 1 {\displaystyle |x-y|<1} . [ 7 ]
The fusible numbers include all of the non-negative integers , and are a well-ordered subset of the dyadic rational numbers, the fractions whose denominators are powers of two . Being well-ordered means that, if one chooses a decreasing sequence of fusible numbers, the sequence must always be finite. Among the well-ordered sets, their ordering can be classified as ε 0 {\displaystyle \varepsilon _{0}} , an epsilon number (a special case of the infinite ordinal numbers ). Because they are well-ordered, for each integer n {\displaystyle n} there is a unique smallest fusible number among the fusible numbers larger than n {\displaystyle n} ; it has the form n + 1 / 2 k {\displaystyle n+1/2^{k}} for some k {\displaystyle k} . [ 7 ] This number k {\displaystyle k} grows very rapidly as a function of n {\displaystyle n} , so rapidly that for n = 3 {\displaystyle n=3} it is (in Knuth's up-arrow notation for large numbers) already larger than 2 ↑ 9 16 {\displaystyle 2\uparrow ^{9}16} . [ 8 ] The existence of this number k {\displaystyle k} , for each n {\displaystyle n} , cannot be proven in Peano arithmetic . [ 7 ]
If the rules of the fuse-burning puzzles are interpreted to allow fuses to be lit at more points than their ends, a larger set of amounts of time can be measured. For instance, if a fuse is lit in such a way that, while it burns, it always has three ends burning (for instance, by lighting one point in the middle and one end, and then lighting another end or another point in the middle whenever one or two of the current lit points burn out) then it will burn for 1/3 of a unit of time rather than a whole unit. By representing a given amount of time as a sum of unit fractions , and successively burning fuses with multiple lit points so that they last for each unit fraction amount of time, it is possible to measure any rational number of units of time. However, keeping the desired number of flames lit, even on a single fuse, may require an infinite number of re-lighting steps. [ 4 ]
The problem of representing a given rational number as a sum of unit fractions is closely related to the construction of Egyptian fractions , sums of distinct unit fractions; however, for fuse-burning problems there is no need for the fractions to be different from each other. Using known methods for Egyptian fractions one can prove that measuring a fractional amount of time x / y {\displaystyle x/y} , with x < y {\displaystyle x<y} , needs only O ( log y ) {\displaystyle O({\sqrt {\log y}})} fuses (expressed in big O notation ). [ 9 ] An unproven conjecture of Paul Erdős on Egyptian fractions suggests that fewer fuses, O ( log log y ) {\displaystyle O(\log \log y)} , may always be enough. [ 10 ]
In a booklet on these puzzles titled Shoelace Clock Puzzles , created by Dick Hess for a 1998 Gathering 4 Gardner conference, Hess credits Harvard statistician Carl Morris as his original source for these puzzles. [ 4 ] | https://en.wikipedia.org/wiki/Rope-burning_puzzle |
Rope caulk or caulking cord is a type of pliable putty or caulking formed into a rope-like shape. It is typically off-white in color, relatively odorless, and stays pliable for an extended period of time.
Rope caulk can be used as caulking or weatherstripping around conventional windows installed in conventional wooden or metal frames (see glazing ). It is also used as a form for epoxy work, since epoxy does not adhere to this material.
Rope caulk has also been applied to the metallic structure supporting the magnet for a dynamic speaker to cut unwanted resonance of the metal structure, leading to improved speaker performance. It has also been used as a sonic damping material in sensitive phonograph components.
Mortite brand rope caulk was introduced by the J.W. Mortell Co. of Kankakee, Illinois in the 1940s, and called "pliable plastic tape". [ 1 ] The trademark application was filed in March, 1943. [ 2 ] It was later marketed as "caulking cord". The company was later acquired by Thermwell Products .
Mortite putty is a brand of rope caulk marketed under the Frost King brand . Its primary ingredient is titanium dioxide ; it has a specific gravity of 1.34.
It is listed by the state of California as containing ingredients known to the state to cause cancer or adversely affect reproductive health (a "P65 Warning"). | https://en.wikipedia.org/wiki/Rope_caulk |
Rosa's rule , also known as Rosa's law of progressive reduction of variability , [ 1 ] is a biological rule that observes the tendency to go from character variation in more primitive representatives of a taxonomic group or clade to a fixed character state in more advanced members. An example of Rosa's rule is that the number of thoracic segments in adults (or holaspids ) may vary in Cambrian trilobite species, while from the Ordovician the number of thoracic segments is constant in entire genera, families, and even suborders. [ 2 ] Thus, a trend of decreasing trait variation between individuals of a taxon as the taxon develops across evolutionary time can be observed. The rule is named for Italian paleontologist Daniele Rosa . [ 3 ] | https://en.wikipedia.org/wiki/Rosa's_rule |
Rosa Vásquez Espinoza is a Peruvian chemical biologist and conservationist who founded Amazon Research Internacional, an organization focused on biodiversity research and conservation in the Amazon rainforest. Her work integrates traditional ecological knowledge with modern science, emphasizing sustainable practices and community collaboration.
Rosa Vásquez Espinoza was born in Peru, with family roots in both the Andes and the Amazon rainforest . [ 1 ] [ 2 ] [ 3 ] Her upbringing was influenced by her grandmother, a traditional healer in the Andes who relied on medicinal plants to create remedies. [ 2 ] [ 3 ] This early exposure to natural medicine and cultural traditions inspired Vásquez Espinoza and later influenced her scientific career. [ 2 ] [ 3 ]
Vásquez Espinoza attended school in Lima , but spent her summers in the Amazon and Andes, experiencing rural life and interacting with the local biodiversity . [ 2 ] She received a scholarship to attend Tennessee Tech , where she earned a B.S. in biology and chemistry in 2015. [ 4 ] [ 5 ] She completed a Ph.D. in chemical biology from the University of Michigan . [ 4 ]
From 2016 to 2022, Vásquez Espinoza worked in the life science laboratory at the University of Michigan under David H. Sherman , including as a postdoctoral researcher. [ 6 ] In 2019, she joined a scientific expedition to the shanay-timpishka to study its unique ecosystem . [ 7 ] During the trip, she collected samples of water, sediment, and microbes, documenting environmental conditions such as temperature and sunlight to better understand the microorganisms thriving in the river’s extreme heat. [ 7 ]
Vásquez Espinoza is a chemical biologist whose work centers on exploring biodiversity in extreme ecosystems, particularly the Amazon rainforest. Her research focuses on microorganisms and stingless bees , examining their ecological roles and potential applications in medicine and conservation. [ 1 ] [ 2 ] [ 3 ] She founded Amazon Research Internacional, an institute dedicated to collaborative research and the integration of traditional knowledge with modern science. [ 1 ]
Vásquez Espinoza's projects include mapping stingless bee populations and studying their honey's chemical properties, which have shown potential medicinal benefits. [ 1 ] [ 2 ] She collaborates with Indigenous communities, incorporating their knowledge and practices into conservation strategies. [ 1 ] [ 2 ] [ 3 ] Her partnerships extend to academic institutions and policymakers, focusing on biodiversity preservation and sustainability. [ 2 ]
Vásquez Espinoza has participated in initiatives to address the impacts of deforestation , climate change , and environmental pollutants on the Amazon's ecosystem. She has contributed to legislative efforts advocating for the protection of stingless bees in Peru, aiming to enhance conservation and support local economies . [ 1 ] [ 3 ] In 2024, she was listed on BBC 100 Women . [ 8 ] | https://en.wikipedia.org/wiki/Rosa_Vásquez_Espinoza |
Rosalind Audrey Clare Hudson ( née Latham ; 31 July 1926 – 7 July 2013) was a British codebreaker and architectural model maker.
Rosalind was born at Wirral, Cheshire and was educated at Adcote School , Shropshire . In her early years she developed a love for architecture. Being the granddaughter of Walter Aubrey Thomas , who designed the Royal Liver Building , Hudson also constructed architectural models. [ 1 ]
Hudson attempted to further her career in architecture at the Liverpool School of Art , but she eventually dropped out during World War II to join the Women's Royal Naval Service (WRNS). Hudson was stationed in Bletchley Park and Woburn Abbey for the duration of the war. [ 2 ]
Following training in the WRNS she was sent to Bletchley Park , home of the British government's the Government Code and Cypher School . Hudson worked in Hut 8 under cryptanalyst Alan Turing . [ 3 ] She never discussed her work at Bletchley Park following the war. [ 2 ]
After the war Hudson trained as a florist under Constance Spry , and arranged flowers at Claridge's and the Savoy hotels. [ 2 ] The Savoy gave her and her husband a suite overlooking the River Thames as a wedding present. Hudson worked as a florist for Somerset Maugham and his wife, the interior designer Syrie Maugham , and was an amateur pianist and watercolourist. [ 2 ]
Hudson was most notable for her architectural models, being particularly attracted to Georgian architecture . [ 2 ] Her works charted the development of the urban façade during the Georgian period. [ 4 ]
Bath's Building of Bath Collection and Bath's Pump Room contain models made by Hudson. A scale model of Dulwich Picture Gallery made by Hudson stands in the gallery's foyer. [ citation needed ] Shortly after her death, a special exhibition of her work was held in 2014 at the Bath Museum. [ 5 ]
Hudson made a model of Highgrove , the country house of Charles, Prince of Wales as a present for the Prince and Princess of Wales's wedding in 1981. [ 2 ] Hudson was later commissioned by Charles to alter the model when he added a porch to the house. Hudson made models of other private houses, and was reluctant to accept payment for her models. [ 2 ]
She met her husband, Richard Hudson in 1945. Hudson was serving in the Royal Marines , they married in 1949. They had three sons and two daughters and lived on a farm near Bath . [ 2 ]
She died on 7 July 2013, shortly before her 87th birthday. [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Rosalind_Hudson |
Rosana Sonia Chirinos Gallardo (born 1970, Lima) is a engineer, professor, and researcher specializing in food biotechnology . Her work focuses on bioactive compounds derived from native biodiversity, functional foods, and sustainable bioprocesses. She is a professor at the Faculty of Food Industries at the National Agrarian University La Molina (UNALM) and main researcher at the Biotechnology Institute – Industrial Biotechnology & Bioprocesses Area of UNALM. [ 1 ] [ 2 ]
Rosana Sonia Chirinos Gallardo was born1970, in Lima . [ 3 ] Her interest in science began from curiosity about the characteristics of plants and their benefits according to their composition. [ 4 ]
In 1994, she realized her studies in Food Industries Engineering and obtain a Master's in Food Technology at the National Agrarian University La Molina (UNALM). Subsequently, she specialized in biotechnology and biological engineering, obtaining a Ph.D. in Agricultural Sciences and Biological Engineering at the Université catholique de Louvain in Belgium . [ 2 ] During her doctoral studies, she focused her research on enzymatic technologies applied to food processing and bioproduct development.
Her early research focused on enzymatic transformations in plant-derived compounds, contributing to the development of innovative bioproducts with high nutritional and functional value. [ 3 ]
Her doctoral research at the Université catholique de Louvain explored enzymatic hydrolysis for the extraction and stabilization of bioactive compounds from native Andean crops. [ 5 ] This work laid the foundation for her later studies on functional foods and nutraceuticals.
Together with other researchers work at the Industrial Biotechnology & Bioprocesses area of the Biotechnology Institute at National Agrarian University La Molina. This research group focuses on various aspects of food biotechnology and bioprocessing. [ 1 ] [ 6 ]
Her expertise includes the processing, extraction, purification, and characterization of bioactive compounds such as prebiotics , probiotics , antioxidants , polyphenols , glucosinolates , phytosterols , tocopherols , and high-intensity sweeteners. [ 2 ]
Throughout her career, she has investigated the antioxidant properties and potencial applications in pharmaceuticals and food industries of native Peruvian plants, including sacha inchi , mashua , cañihua , kiwicha , and muña . [ 4 ]
Since 2007, she has been a professor at UNALM's Faculty of Food Industries, advising over 50 undergraduate, master's, and doctoral theses. [ 2 ]
In 2003, she conducted her first research studies together other collaborators, publishing a study about the nutritional benefits of yacón and it was important to foundation for her subsequent studies on native Peruvian plants. [ 7 ]
In 2017, she contributed on the book Advances in Food and Nutrition Research co-authoring the chapter "Bioactive Potential of Andean Fruits, Seeds, and Tubers". The chapter provides a comprehensive analysis of major Andean crops, including fruits such as lúcuma , cherimoya , pepino melon , and elderberry ; roots and tubers such as potato , sweet potato , yacón , chicuru, mashua , and olluco ; and seeds such as quinoa , amaranth , and tarwi . It explores their nutritional and functional properties. [ 8 ]
Her most recent publication, released in 2024, is titled "Conventional and ultrasound-assisted extractions of protein from sacha inchi (Plukenetia volubilis) and their impact on the physicochemical and structural characteristics". This study was conducted in collaboration with her research team at the Universidad Nacional Agraria La Molina (UNALM). [ 9 ] | https://en.wikipedia.org/wiki/Rosana_Chirinos_Gallardo |
Rose's law is the observation that the number of qubits on chips doubles about every 18 months. [ 1 ] [ 2 ] It is the quantum computing equivalent of Moore's law . [ 3 ] [ 4 ]
The term was coined by Steve Jurvetson when he met Geordie Rose, the founder of D-Wave Systems and the law's namesake. [ 5 ] | https://en.wikipedia.org/wiki/Rose's_law |
Rose's metal , Rose metal or Rose's alloy is a fusible alloy with a low melting point .
Rose's metal consists of 50% bismuth , 25–28% lead and 22–25% tin . Its melting point is between 94 and 98 °C (201 and 208 °F). The alloy does not appreciably contract or expand on solidification, this characteristic being a function of its bismuth percentage, but does slightly contract on cooling. [ 1 ]
Rose's metal has several common uses:
It was discovered by the German chemist Valentin Rose the Elder , the grandfather of Heinrich Rose .
This alloy-related article is a stub . You can help Wikipedia by expanding it .
This article about a specific mineral or mineraloid is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rose's_metal |
Rosemary A. Bailey FRSE (born 1947) is a British statistician who works in the design of experiments and the analysis of variance and in related areas of combinatorial design , especially in association schemes . She has written books on the design of experiments , on association schemes , and on linear models in statistics.
Bailey's first degree and Ph.D. were in mathematics at the University of Oxford . She was awarded her doctorate in 1974 for a dissertation on permutation groups , Finite Permutation Groups supervised by Graham Higman . [ 1 ] Bailey's career has not been in pure mathematics but in statistics where she has specialised in the algebraic problems associated with the design of experiments.
Bailey worked at the University of Edinburgh with David Finney and at The Open University . She spent 1981–91 in the Statistics Department of Rothamsted Experimental Station . In 1991 Bailey became Professor of Mathematical Sciences at Goldsmiths College in the University of London and then Professor of Statistics at Queen Mary, University of London where she is Professor Emerita of Statistics. She is currently Professor of Mathematics and Statistics in the School of Mathematics and Statistics at the University of St Andrews , Scotland.
Bailey is a Fellow of the Institute of Mathematical Statistics [ 2 ] and in 2015 was elected a Fellow of the Royal Society of Edinburgh . [ 3 ] | https://en.wikipedia.org/wiki/Rosemary_A._Bailey |
In mathematical optimization , the Rosenbrock function is a non- convex function , introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms . [ 1 ] It is also known as Rosenbrock's valley or Rosenbrock's banana function .
The global minimum is inside a long, narrow, parabolic -shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult.
The function is defined by
It has a global minimum at ( x , y ) = ( a , a 2 ) {\displaystyle (x,y)=(a,a^{2})} , where f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Usually, these parameters are set such that a = 1 {\displaystyle a=1} and b = 100 {\displaystyle b=100} . Only in the trivial case where a = 0 {\displaystyle a=0} the function is symmetric and the minimum is at the origin.
Two variants are commonly encountered.
One is the sum of N / 2 {\displaystyle N/2} uncoupled 2D Rosenbrock problems, and is defined only for even N {\displaystyle N} s:
This variant has predictably simple solutions.
A second, more involved variant is
has exactly one minimum for N = 3 {\displaystyle N=3} (at ( 1 , 1 , 1 ) {\displaystyle (1,1,1)} ) and exactly two minima for 4 ≤ N ≤ 7 {\displaystyle 4\leq N\leq 7} —the global minimum at ( 1 , 1 , . . . , 1 ) {\displaystyle (1,1,...,1)} and a local minimum near x ^ = ( − 1 , 1 , … , 1 ) {\displaystyle {\hat {\mathbf {x} }}=(-1,1,\dots ,1)} . This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of x {\displaystyle x} . For small N {\displaystyle N} the polynomials can be determined exactly and Sturm's theorem can be used to determine the number of real roots , while the roots can be bounded in the region of | x i | < 2.4 {\displaystyle |x_{i}|<2.4} . [ 5 ] For larger N {\displaystyle N} this method breaks down due to the size of the coefficients involved.
Many of the stationary points of the function exhibit a regular pattern when plotted. [ 5 ] This structure can be exploited to locate them.
The Rosenbrock function can be efficiently optimized by adapting appropriate coordinate system without using any gradient information and without building local approximation models (in contrast to many derivate-free optimizers). The following figure illustrates an example of 2-dimensional Rosenbrock function optimization by adaptive coordinate descent from starting point x 0 = ( − 3 , − 4 ) {\displaystyle x_{0}=(-3,-4)} . The solution with the function value 10 − 10 {\displaystyle 10^{-10}} can be found after 325 function evaluations.
Using the Nelder–Mead method from starting point x 0 = ( − 1 , 1 ) {\displaystyle x_{0}=(-1,1)} with a regular initial simplex a minimum is found with function value 1.36 ⋅ 10 − 10 {\displaystyle 1.36\cdot 10^{-10}} after 185 function evaluations. The figure below visualizes the evolution of the algorithm. | https://en.wikipedia.org/wiki/Rosenbrock_function |
The Rosenmund reduction is a hydrogenation process in which an acyl chloride is selectively reduced to an aldehyde . The reaction was named after Karl Wilhelm Rosenmund , who first reported it in 1918. [ 1 ]
The reaction, a hydrogenolysis , is catalysed by palladium on barium sulfate , which is sometimes called the Rosenmund catalyst . Barium sulfate has a low surface area which reduces the activity of the palladium, preventing over-reduction. However, for certain reactive acyl chlorides the activity must be reduced further, by the addition of a poison . Originally this was thioquinanthrene although thiourea [ 2 ] has also been used. [ 3 ] [ 4 ] Deactivation is required because the system must reduce the acyl chloride but not the subsequent aldehyde. If further reduction does take place, it will create a primary alcohol which would then react with the remaining acyl chloride to form an ester .
Rosenmund catalyst can be prepared by reduction of palladium(II) chloride solution in the presence of BaSO 4 . Typical reducing agent is formaldehyde. [ 5 ]
While Rosenmund reduction method can be used to prepare several aldehydes, formaldehyde cannot be prepared, as formyl chloride is unstable at room temperatures. [ 6 ] | https://en.wikipedia.org/wiki/Rosenmund_reduction |
The Rosenmund–von Braun synthesis is an organic reaction in which an aryl halide reacts with cuprous cyanide to yield an aryl nitrile . [ 1 ] [ 2 ] [ 3 ]
The reaction was named after Karl Wilhelm Rosenmund who together with his Ph.D. student Erich Struck discovered in 1914 that aryl halide reacts with alcohol water solution of potassium cyanide and catalytic amounts of cuprous cyanide at 200 °C. The reaction yields the carboxylic acid , not the nitrile, but Rosenmund speculated that the intermediate should be the nitrile, since nitriles on aromatic rings can react to form carboxylic acids. [ 1 ] Independently Alfred Pongratz [ 4 ] and Julius von Braun [ de ] [ 2 ] improved the reaction by changing the reaction conditions to higher temperatures and used no solvent for the reaction. Further improvement of the reaction was done in the following years, [ 5 ] for example the use of ionic liquids as solvent for the reaction. [ 6 ] | https://en.wikipedia.org/wiki/Rosenmund–von_Braun_reaction |
Rosenthal's reagent is a metallocene bis(trimethylsilyl)acetylene complex with zirconium ( Cp 2 Zr ) or titanium (Cp 2 Ti) used as central atom of the metallocene fragment Cp 2 M. Additional ligands such as pyridine or THF are commonly used as well. With zirconium as central atom and pyridine as ligand ( Zirconocene bis(trimethylsilyl)acetylene pyridine ), a dark purple to black solid with a melting point of 125–126 °C is obtained. [ 1 ] Synthesizing Rosenthal's reagent of a titanocene source yields golden-yellow crystals of the titanocene bis(trimethylsilyl)acetylene complex with a melting point of 81–82 °C. [ 2 ] [ 3 ] This reagent enables the generation of the themselves unstable titanocene and zirconocene under mild conditions. [ 4 ]
The reagent is named after the German chemist Uwe Rosenthal [ de ] (born 1950) and was first synthesized by him and his co-workers in 1995. [ 5 ]
Rosenthal's reagent can be prepared by reduction of titanocene or zirconocene dichloride with magnesium in the presence of bis(trimethylsilyl)acetylene in THF. The illustrated product for a titanocene complex can be represented by the resonance structures A and B . If zirconium is used as central atom, additional ligands (e.g. pyridine) are necessary for stabilization. [ 6 ]
The first successful synthesis of titanocene bis(trimethylsilyl)acetylene was accomplished by Uwe Rosenthal in 1988, via the reduction of Cp 2 TiCl 2 with magnesium and the alkyne Me 3 SiC 2 SiMe 3 , in THF. [ 7 ]
This synthesis was immediately used to make other similar titanocene and zirconocene alkyne complexes. [ 8 ] Under the same conditions, various zirconium complexes were synthesized, most utilizing other stabilizing ligands, including pyridine and THF. Notably, this synthesis also enabled the subsequent synthesis and characterization of the first zirconocene-alkyne complex without addition stabilizing ligands. This was accomplished with the reduction of racemic (EBTHI)ZrCl 2 [EBTHI = 1,2-ethylene-1,1‘-bis(η5-tetrahydroindenyl)]. [ 9 ]
Zirconocene bis(trimethylsilyl)acetylene pyridine was originally synthesized by Rosenthal’s group in 1994 after they exchanged the coordinating solvent from tetrahydrofuran (THF) to pyridine. [ 10 ] The exchange of the THF for the pyridine ligand provides extra stability in organic solvents preventing dimerization. [ 1 ] The original synthesis involved the reduction of zirconocene dichloride and the addition of bis(trimethylsilyl)acetylene in THF before transferring to pyridine as mentioned above. More recently, Tilley and coworkers demonstrated a simpler synthesis with a higher yield bypassing the isolation of the less stable THF adduct. This newer method reacts zirconocene dichloride with 2 equivalents of n-Butyllithium in THF to form a metallacyclopropane which is subsequently substituted by bis(trimethylsilyl)acetylene and pyridine. [ 11 ]
For much of the history of titanocene bis(trimethylsilyl)acetylene, there has been no X-ray crystal structure . Many attempts to obtain crystals failed, due to the complex’s extremely high solubility in all suitable solvents. However, researchers obtained many crystal structures of similar compounds of the type Cp 2 Ti(η 2 R 3 SiC 2 SiR 3 ), such as Cp=Cp*, R=tBu, and R=Ph. [ 8 ] The crystal structure of the parent complex was not obtained until suitable crystals were serendipitously recovered from reaction mixtures. [ 13 ] Once successfully obtained, the crystal structure displayed a bent titanocene with the coordinated alkyne ligand located between the Cp ligand planes. The angles between the titanium-coordinated alkyne ligand and each Cp ligand plane are 21.5° and 25.2°, respectively. To themselves, the Cp ligands form an angle of 46.6°. The Si atoms bonded to the alkyne carbons are almost perfectly in plane, with a torsion angle of 6.5°.
The triple bond of the alkyne has a length of 1.283(6) Å. This value is longer than that of the free alkyne (1.208 Å), and closer to that of a double bond (1.331 Å). Furthermore, the distances between the titanium center and the carbon atoms of the coordinated alkyne are 2.136(5) Å and 2.139(4) Å. These values fall within the range of reported endocyclic Ti-C(sp 2 ) σ-bonds. [ 14 ]
Researchers have calculated the bonding nature of various metallocene acetylene complexes. Cp 2 Ti(η 2 -Me 3 SiC 2 SiMe 3 ) was modeled using a B3LYP density functional theory (DFT) computation. This revealed the metallocyclopropane group is composed of two in-plane σ-bonds from the carbons to the metal, and one out-of-plane π-bond that also interacts with the metal. This type of interaction is a 3-center, 2-electron bond . Although the aromatic stabilization is the lowest for titanium of the Group 4 metals, the complex is aromatic . These computational results were in agreement with the X-ray structural data. [ 15 ]
Further DFT calculations were carried out using the PBE0 D3BJ/ def2-TZVP functional and visualization in IBOview . These illustrate the nature of the frontier orbitals in titanocene bis(trimethylsilyl)acetylene. The blue and purple orbitals display the highest occupied molecular orbital (HOMO) found on the complex. These are located between the coordinated alkyne and the metal, in the 2-electron, 3-center system. The green and yellow orbitals display the lowest occupied molecular orbital (LUMO), found on titanium.
The two main resonance structures of Zirconocene bis(trimethylsilyl)acetylene pyridine include a variation where the C-C triple bond binds side on to the metal and another with the 1-metallacyclopropene configuration. Density functional theory (DFT) calculations showed metal-carbon sigma bonds in addition to an out of plane pi bond corresponding to the 1-metallacyclopropene depiction being a major resonance form. [ 16 ] However, a subsequent series of calculations by Leites and colleagues using a higher level of theory showed molecular orbitals more consistent with the triple bond description. [ 17 ]
The original creation of zirconocene bis(trimethylsilyl)acetylene pyridine was accompanied by reactivity studies of the complex with common small molecules in the form of carbon dioxide and water . Both reactions involved the loss of the pyridine ligand and creation of bimetallic complexes containing bridging-oxo substituents, with the carbon dioxide inserting to create a series of fused metallacycles and the water’s hydrogen atoms breaking up the metallacyclopropenes. [ 1 ]
More generally, the main reactivity for this version of Rosenthal’s reagent is its reaction with alkynes to replace the zirconacyclopropene with a larger zirconacyclopentadiene rings. [ 18 ] T. Don Tilley and colleagues have extensively utilized this functionality to create zirconocene based macrocycles with considerable tunability based on the alkyne used. [ 19 ] These large macrocycles can subsequently be reacted with hydrochloric acid to lose the zirconocene dichloride leaving behind new carbon-carbon bonds. [ 20 ] Following these macrocycles, the Tilley group also showed that the zirconocene bis(trimethylsilyl)acetylene pyridine could aid in the creation of various polycyclic aromatic hydrocarbons via [2+2+n] cycloaddition reactions. [ 21 ] [ 22 ] As seen with the acidic conditions above, the zirconocene fragment is easily displaced and Tilley demonstrated the ability to insert selenium into the framework. [ 22 ] Rivard's group also showed an analogous transmetalation process allowing for the replacement of zirconium with tellurium . [ 23 ]
Rosenthal also continued exploring the reactivity of the zirconocene bis(trimethylsilyl)acetylene pyridine showing the ability to functionalize the zirconacyclopentadienes in addition to modifying the ring itself. [ 24 ] This latter study involved the reaction of substrates like tertbutyl substituted 1,3 butadiyne to create novel zirconacyclocumulene complexes. [ 25 ]
In 2018, Staubitz and coworkers used the pyridine complex in combination with dialkyne complexes to form the zirconacyclopentadiene after loss of the pyridine and the bis(trimethylsilyl)acetylene. These zirconium metallacycles can then be transmetalated to create functionalized stannoles which Staubitz later used in Stille cross coupling reactions to form polymers with thiophene groups. [ 26 ] [ 27 ] [ 28 ]
Staubitz’s group followed this with a reactivity comparison between Cp 2 Zr(btmsa)(py) and Negishi’s reagent with respect to forming zirconacyclopentadienes. [ 29 ] They found that this reaction took place quicker and more efficiently than with Negishi’s reagent.
In 2019, Ye and coworkers further extended the scope of the pyridine Rosenthal reagent reactivity, demonstrating its reaction with bis(alkylnyl)boranes in an attempt to create compounds capable of activating small molecules. The product of this reaction has resonance structures including a boron zirconium(IV) 6-member heterocycle and a zirconium(II) donating into the boron stabilized by the two alkynes. [ 30 ]
Zirconocene bis(trimethylsilyl)acetylene pyridine was also shown to react with other zirconocene derivatives containing alkyne substituents with Lindenau et. al. showing the creation of a bimetallic transition metal hydride . This was achieved by the reaction of Rosenthal’s reagent with Zr(Cp) 2 (CH 3 )(CCSiMe 3 ) to create a methyl bridged complex which could be converted to the hydride upon the addition of BH 3 •NHMe 2 . [ 31 ]
Tonks and colleagues looked into the reactivity of this Rosenthal reagent as a potential ring opening complex, but instead formed new zirconocene heterocycles. Upon addition of the zirconocene bis(trimethylsilyl) acetylene pyridine to cyclopropyl methyl ketone, a zirconium oxygen bond formed simultaneously forming a new carbon-carbon bond from the cyclopropene and the carbonyl carbon. [ 32 ]
There exist 2 resonance forms of this complex, the acetylenic pi-complex, and the metallocyclopropene complex. The major type of interaction dictates the reaction pathway the complex will follow. The insertion pathway involves insertion of the substrate to form a metallocycloprane ring, followed by loss of the alkyne. The dissociation pathway involves dissociation of the alkyne to generate the reactive Cp 2 Ti intermediate, which is then trapped by reaction with the substrate. The interaction between the metal and alkyne can be controlled by changing the metal (Ti or Zr) and the ligands, including the type of Cp ligand and the substitution on the alkyne. The Cp 2 Ti species is an unstable Ti(II), d 2 complex with 14 total electrons. Because it contains a lone electron pair held in 2 valence orbitals, its reactivity can be compared to carbenes. This form often undergoes reactions with a variety of olefins to yield metallacycles. [ 14 ]
A special feature of titanocene bis(trimethylsilyl) and its zirconium analogues is the ability it derives from coordination of the alkyne to stabilize the metallocene fragment. This alkyne can be released under relatively mild conditions to yield the reactive and unstable Cp 2 Ti intermediate. This reactivity manifests in a variety of reactions, some of which are detailed below. For a comprehensive review, visit "Recent Synthetic and Catalytic Applications of Group 4 Metallocene Bis(trimethylsilyl)acetylene Complexes".
Titanocene bis(trimethylsilyl)acetylene reacts with carbonyl compounds to generate metallacyclic titanium-dihydrofuran complexes. The constitution of the products depends on the steric bulk of the groups on the carbonyl compound, with the metallocyclopropane product only being obtained with sufficiently sterically bulky groups, such as R/R' = phenyl. [ 33 ]
Heterocyclic systems containing C=N bonds undergo a ring enlargement via a coupling reaction. [ 33 ]
Polymerization of acetylene was achieved at 20-60 °C when titanocene bis(trimethylsilyl)acetylene was utilized as a precatalyst. Yield and properties of the resulting polyacetylene could be modulated by the solvent used. 100% trans-polyacetylene could be obtained in pyridine. [ 34 ]
Titanocene bis(trimethylsilyl)acetylene afforded the linear polymerization of 1-alkenes with a selectivity over 98%. This reaction also accomplished a turnover number of 1200-1500. [ 35 ]
The main area of application is the synthesis of synthetically challenging organic structures such as macrocycles and heterometallacycles. Rosenthal's reagent allows the selective preparation of these compounds with high yields . [ 36 ] [ 37 ]
Currently, Rosenthal's reagent is often used instead of Negishi 's reagent (1-butene)zirconocene to generate zirconocene fragments as it offers a number of compelling advantages. Unlike Negishi's reagent, Rosenthal's reagent is stable at room temperature and can be stored indefinitely under an inert atmosphere. A much more precise control over the stoichiometry of reactions is possible, especially because the instable (1-butene)zirconocene cannot be formed quantitatively. [ 37 ] Stoichiometric and catalytic reactions can be performed and influenced by the use of different ligands, metals and substrate substituents . While for titanium complexes, a dissociative reaction mechanism has been observed, zirconium complexes favor an associative pathway. [ 36 ] The combination of these organometallic complexes with different suitable substrates (e.g. carbonyl compounds , acetylenes , imines , azoles , etc.) often leads to novel bond types and reactivities. [ 4 ] [ 38 ] A particularly interesting aspect is the novel C–C coupling reaction of nitriles to form precursors for the realization of so far unknown heterometallacycles. [ 36 ] As main side products of coupling reactions with Rosenthal's reagent, pyridine and bis(trimethylsilyl)acetylene are obtained. These compounds are soluble and volatile , and therefore easy to remove from the product mixture. [ 37 ]
Past synthesis, including those mentioned previously, have been straightforward, but require extreme caution in the exclusion of water and air to obtain a pure, catalytically useful complex. The success of the synthesis is also heavily dependent on the quality of Mg(0) used. In 2020, Beckhaus and coworkers reported a more robust synthesis of titanocene bis(trimethylsilyl)acetylene from Cp 2 TiCl 2 and EtMgBr. [ 39 ] This synthesis is predicted to have a positive impact on the growth of investigations into applications of the complex. [ 40 ]
Similarly, the synthesis of other titanocene bis(trimethylsilyl) acetylene complexes have been reported, such as the low-valent ansa -dimethylsilylene, dimethylmethylene–bis(cyclopentadienyl)titanium. [ 41 ]
Titanocene bis(trimethylsilyl)acetylene complexes were first mentioned by the group of Vol’pin in Moscow in 1961. Using the isolobal analogy , the group argued that silacyclopropanes would be a stable group of compounds, due to their similarities to the cyclopropenyl cation . [ 42 ] However, true three-membered rings containing a silicon atom and a carbon-carbon double bond, silirenes, were not reported until 1971. Seyferth and coworkers were the first to synthesize these molecules. [ 43 ] Later, Vol’pin again utilized the isolobal analogy to react diphenylacetylene with titanocene (Cp 2 Ti, where Cp = cyclopentadienyl, rather than dialkylsilene) in an attempt to synthesize unsaturated 1-heterocyclopropanes. Although this was unsuccessful, titanocyclopropane (Cp 2 Ti(η 2 -PhC 2 Ph)) was isolated. [ 44 ] In 1988, Vol’pin selected the alkyne bis(trimethylsilyl)acetylene as the most likely reactant for the synthesis of a stable titanocene-alkyl complex. The group, led by the postdoctoral associate Rosenthal, successfully obtained Cp 2 Ti(η 2 -Me 3 SiC 2 SiMe 3 ) in high yield, as a yellow-orange substance. [ 45 ] | https://en.wikipedia.org/wiki/Rosenthal's_reagent |
Rosetta Mini: 3.78 / October 3, 2017 ; 7 years ago ( 2017-10-03 )
Rosetta@home is a volunteer computing project researching protein structure prediction on the Berkeley Open Infrastructure for Network Computing (BOINC) platform, run by the Baker lab. Rosetta@home aims to predict protein–protein docking and design new proteins with the help of about fifty-five thousand active volunteered computers processing at over 487,946 Giga FLOPS on average as of September 19, 2020. [ 4 ] Foldit , a Rosetta@home videogame, aims to reach these goals with a crowdsourcing approach. Though much of the project is oriented toward basic research to improve the accuracy and robustness of proteomics methods, Rosetta@home also does applied research on malaria , Alzheimer's disease , and other pathologies. [ 5 ]
Like all BOINC projects, Rosetta@home uses idle computer processing resources from volunteers' computers to perform calculations on individual workunits . Completed results are sent to a central project server where they are validated and assimilated into project databases . The project is cross-platform , and runs on a wide variety of hardware configurations. Users can view the progress of their individual protein structure prediction on the Rosetta@home screensaver.
In addition to disease-related research, the Rosetta@home network serves as a testing framework for new methods in structural bioinformatics . Such methods are then used in other Rosetta-based applications, like RosettaDock or the Human Proteome Folding Project and the Microbiome Immunity Project , after being sufficiently developed and proven stable on Rosetta@home's large and diverse set of volunteer computers. Two especially important tests for the new methods developed in Rosetta@home are the Critical Assessment of Techniques for Protein Structure Prediction (CASP) and Critical Assessment of Prediction of Interactions (CAPRI) experiments, biennial experiments which evaluate the state of the art in protein structure prediction and protein–protein docking prediction, respectively. Rosetta consistently ranks among the foremost docking predictors, and is one of the best tertiary structure predictors available. [ 6 ]
With an influx of new users looking to participate in the fight against the COVID-19 pandemic , caused by SARS-CoV-2 , Rosetta@home increased its computing power up to 1.7 PetaFlops as of March 28, 2020. [ 7 ] [ 8 ] On September 9, 2020, Rosetta@home researchers published a paper describing 10 potent antiviral candidates against SARS-CoV-2. Rosetta@home contributed to this research and these antiviral candidates are heading towards Phase 1 clinical trials, which may begin in early 2022. [ 9 ] [ 10 ] [ 11 ] [ 12 ] According to the Rosetta@home team, Rosetta volunteers contributed to the development of a nanoparticle vaccine. [ 9 ] This vaccine has been licensed and is known as the IVX-411 by Icosavax, which began a Phase I/II clinical trial in June 2021, [ 13 ] and GBP510 which is being developed by SK Bioscience and is already approved for a Phase III clinical trial in South Korea. [ 14 ] [ 15 ]
NL-201 , a cancer drug candidate that was first created at the Institute of Protein Design (IPD) and published in a January 2019 paper, [ 16 ] began a Phase 1 Human clinical trial in May 2021 with the support of Neoleukin Therapeutics, itself a spin-off from the IPD. [ 17 ] Rosetta@home played a role in the development of NL-201 and contributed with "forward folding" experiments that helped validate protein designs. [ 18 ]
The Rosetta@home application and the BOINC volunteer computing platform are available for the operating systems Windows , Linux , and macOS ; BOINC also runs on several others, e.g., FreeBSD. [ 19 ] Participation in Rosetta@home requires a central processing unit (CPU) with a clock speed of at least 500 MHz , 200 megabytes of free disk space , 512 megabytes of physical memory , and Internet connectivity. [ 20 ] As of July 20, 2016, the current version of the Rosetta Mini application is 3.73. [ 21 ] The current recommended BOINC program version is 7.6.22. [ 19 ] Standard Hypertext Transfer Protocol (HTTP) ( port 80) is used for communication between the user's BOINC client and the Rosetta@home servers at the University of Washington; HTTPS (port 443) is used during password exchange. Remote and local control of the BOINC client use port 31416 and port 1043, which might need to be specifically unblocked if they are behind a firewall . [ 22 ] Workunits containing data on individual proteins are distributed from servers located in the Baker lab at the University of Washington to volunteers' computers, which then calculate a structure prediction for the assigned protein. To avoid duplicate structure predictions on a given protein, each workunit is initialized with a random seed number. This gives each prediction a unique trajectory of descent along the protein's energy landscape . [ 23 ] Protein structure predictions from Rosetta@home are approximations of a global minimum in a given protein's energy landscape. That global minimum represents the most energetically favorable conformation of the protein, i.e., its native state .
A primary feature of the Rosetta@home graphical user interface (GUI) is a screensaver which shows a current workunit 's progress during the simulated protein folding process. In the upper-left of the current screensaver, the target protein is shown adopting different shapes (conformations) in its search for the lowest energy structure. Depicted immediately to the right is the structure of the most recently accepted. On the upper right the lowest energy conformation of the current decoy is shown; below that is the true, or native, structure of the protein if it has already been determined. Three graphs are included in the screensaver. Near the middle, a graph for the accepted model's thermodynamic free energy is displayed, which fluctuates as the accepted model changes. A graph of the accepted model's root-mean-square deviation (RMSD), which measures how structurally similar the accepted model is to the native model, is shown far right. On the right of the accepted energy graph and below the RMSD graph, the results from these two functions are used to produce an energy vs. RMSD plot as the model is progressively refined. [ 24 ]
Like all BOINC projects, Rosetta@home runs in the background of the user's computer, using idle computer power, either at or before logging into an account on the host operating system . The program frees resources from the CPU as they are needed by other applications so that normal computer use is unaffected. Many program settings can be specified via user account preferences, including: the maximum percentage of CPU resources the program can use (to control power consumption or heat production from a computer running at sustained capacity), the times of day during which the program can run, and many more. [ citation needed ]
With the proliferation of genome sequencing projects , scientists can infer the amino acid sequence, or primary structure , of many proteins that carry out functions within the cell. To better understand a protein's function and aid in rational drug design , scientists need to know the protein's three-dimensional tertiary structure .
Protein 3D structures are currently determined experimentally via X-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy. The process is slow (it can take weeks or even months to figure out how to crystallize a protein for the first time) and costly (around US$100,000 per protein). [ 25 ] Unfortunately, the rate at which new sequences are discovered far exceeds the rate of structure determination – out of more than 7,400,000 protein sequences available in the National Center for Biotechnology Information (NCBI) nonredundant (nr) protein database, fewer than 52,000 proteins' 3D structures have been solved and deposited in the Protein Data Bank , the main repository for structural information on proteins. [ 26 ] One of the main goals of Rosetta@home is to predict protein structures with the same accuracy as existing methods, but in a way that requires significantly less time and money. Rosetta@home also develops methods to determine the structure and docking of membrane proteins (e.g., G protein–coupled receptors (GPCRs)), [ 27 ] which are exceptionally difficult to analyze with traditional techniques like X-ray crystallography and NMR spectroscopy, yet represent the majority of targets for modern drugs. [ 28 ]
Progress in protein structure prediction is evaluated in the biannual Critical Assessment of Techniques for Protein Structure Prediction (CASP) experiment, in which researchers from around the world attempt to derive a protein's structure from the protein's amino acid sequence. High scoring groups in this sometimes competitive experiment are considered the de facto standard-bearers for what is the state of the art in protein structure prediction. Rosetta, the program on which Rosetta@home is based, has been used since CASP5 in 2002. In the 2004 CASP6 experiment, Rosetta made history by being the first to produce a close to atomic-level resolution, ab initio protein structure prediction in its submitted model for CASP target T0281. [ 29 ] Ab initio modeling is considered an especially difficult category of protein structure prediction, as it does not use information from structural homology and must rely on information from sequence homology and modeling physical interactions within the protein. Rosetta@home has been used in CASP since 2006, where it was among the top predictors in every category of structure prediction in CASP7. [ 30 ] [ 31 ] [ 32 ] These high quality predictions were enabled by the computing power made available by Rosetta@home volunteers. [ 33 ] Increasing computing power allows Rosetta@home to sample more regions of conformation space (the possible shapes a protein can assume), which, according to Levinthal's paradox , is predicted to increase exponentially with protein length. [ citation needed ]
Rosetta is also used in protein–protein docking prediction, which determines the structure of multiple complexed proteins , or quaternary structure . This type of protein interaction affects many cellular functions, including antigen–antibody and enzyme–inhibitor binding and cellular import and export. Determining these interactions is critical for drug design . Rosetta is used in the Critical Assessment of Prediction of Interactions (CAPRI) experiment, which evaluates the state of the protein docking field similar to how CASP gauges progress in protein structure prediction. The computing power made available by Rosetta@home's project volunteers has been cited as a major factor in Rosetta's performance in CAPRI 2007, where its docking predictions have been among the most accurate and complete. [ 34 ]
In early 2008, Rosetta was used to computationally design a protein with a function never before observed in nature. [ 35 ] This was inspired in part by the retraction of a high-profile paper from 2004 which originally described the computational design of a protein with improved enzymatic activity relative to its natural form. [ 36 ] The 2008 research paper from David Baker's group describing how the protein was made, which cited Rosetta@home for the computing resources it made available, represented an important proof of concept for this protein design method. [ 35 ] This type of protein design could have future applications in drug discovery, green chemistry , and bioremediation . [ 35 ]
The Rosetta computer program was cited in the 2024 Scientific Background to the Nobel Prize in Chemistry . [ 37 ]
In addition to basic research in predicting protein structure, docking and design, Rosetta@home is also used in immediate disease-related research. [ 38 ] Numerous minor research projects are described in David Baker's Rosetta@home journal. [ 39 ] As of February 2014, information on recent publications and a short description of the work are being updated on the forum. [ 40 ] The forum thread is no longer used since 2016, and news on the research can be found on the general news section of the project. [ 41 ]
A component of the Rosetta software suite, RosettaDesign, was used to accurately predict which regions of amyloidogenic proteins were most likely to make amyloid -like fibrils. [ 42 ] By taking hexapeptides (six amino acid-long fragments) of a protein of interest and selecting the lowest energy match to a structure similar to that of a known fibril forming hexapeptide, RosettaDesign was able to identify peptides twice as likely to form fibrils as are random proteins. [ 43 ] Rosetta@home was used in the same study to predict structures for amyloid beta , a fibril-forming protein that has been postulated to cause Alzheimer's disease . [ 44 ] Preliminary but as yet unpublished results have been produced on Rosetta-designed proteins that may prevent fibrils from forming, although it is unknown whether it can prevent the disease. [ 45 ]
Another component of Rosetta, RosettaDock, [ 46 ] [ 47 ] [ 48 ] was used in conjunction with experimental methods to model interactions between three proteins—lethal factor (LF), edema factor (EF) and protective antigen (PA)—that make up anthrax toxin . The computer model accurately predicted docking between LF and PA, helping to establish which domains of the respective proteins are involved in the LF–PA complex. This insight was eventually used in research resulting in improved anthrax vaccines. [ 49 ] [ 50 ]
RosettaDock was used to model docking between an antibody ( immunoglobulin G ) and a surface protein expressed by the cold sore virus, herpes simplex virus 1 (HSV-1) which serves to degrade the antiviral antibody. The protein complex predicted by RosettaDock closely agreed with the especially difficult-to-obtain experimental models, leading researchers to conclude that the docking method has potential to address some of the problems that X-ray crystallography has with modelling protein–protein interfaces. [ 51 ]
As part of research funded by a $19.4 million grant by the Bill & Melinda Gates Foundation , [ 52 ] Rosetta@home has been used in designing multiple possible vaccines for human immunodeficiency virus ( HIV ). [ 53 ] [ 54 ]
In research involved with the Grand Challenges in Global Health initiative, [ 55 ] Rosetta has been used to computationally design novel homing endonuclease proteins, which could eradicate Anopheles gambiae or otherwise render the mosquito unable to transmit malaria . [ 56 ] Being able to model and alter protein–DNA interactions specifically, like those of homing endonucleases, gives computational protein design methods like Rosetta an important role in gene therapy (which includes possible cancer treatments). [ 38 ] [ 57 ]
In 2020, the Rosetta molecular modelling suite was used to accurately predict the atomic-scale structure of the SARS-CoV-2 spike protein weeks before it could be measured in the lab. [ 58 ] On June 26 of 2020, the project announced it had succeeded in creating antiviral proteins that neutralize SARS-CoV-2 virions in the lab and that these experimental antiviral drugs are being optimized for animal testing trials. [ 59 ]
In a follow-up, a paper describing 10 SARS-CoV-2 miniprotein inhibitors was published in Science on September 9. Two of these inhibitors, LCB1 and LCB3, are several times more potent than the best monoclonal antibodies being developed against SARS-CoV-2, both on a molar and mass basis. In addition, the research suggests that these inhibitors retain their activity at elevated temperatures, are 20-fold smaller than an antibody and thus, have 20-fold more potential neutralizing sites, increasing the potential efficacy of a locally administered drug. The small size and high stability of the inhibitors is expected to make them adequate to a gel formulation that can be nasally applied or as a powder to be administered directly onto the respiratory system. The researchers will work on developing these inhibitors into therapeutics and prophylactics in the months ahead. [ 10 ] As of July 2021, these antiviral candidates were forecasted to begin clinical trials in early 2022 and had received funding from the Bill & Melinda Gates Foundation for preclinical and early clinical trials. [ 12 ] In animal testing trials, these antiviral candidates were effective against variants of concern including Alpha, Beta and Gamma. [ 12 ] [ 60 ] [ 61 ]
Rosetta@home was used to help screen the over 2 million SARS-CoV-2 Spike-binding proteins that were computationally designed, and thus, contributed to this research. [ 62 ] [ 63 ]
Per the Rosetta@home team at the Institute of Protein Design, Rosetta@home volunteers contributed to the development of antiviral drug candidates [ 10 ] and to a protein nanoparticle vaccine. [ 64 ] The IVX-411 vaccine is already on a Phase 1 clinical trial run by Icosavax [ 13 ] while the same vaccine, licensed to another manufacturer and under the name GBP510, has been approved in South Korea for a Phase III trial run by SK Bioscience . [ 15 ] [ 14 ] The candidate antivirals are also going towards Phase 1 clinical trials. [ 9 ]
Rosetta@home researchers have designed an IL-2 receptor agonist called Neoleukin-2/15 that does not interact with the alpha subunit of the receptor. Such immunity signal molecules are useful in cancer treatment. While the natural IL-2 suffers from toxicity due to an interaction with the alpha subunit, the designed protein is much safer, at least in animal models. [ 16 ] Rosetta@home contributed in "forward folding experiments" which helped validate designs. [ 18 ]
In a September 2020 feature in the New Yorker , David Baker stated that Neoleukin-2/15 would begin human clinical trials "later this year". Neoleukin-2/15 is being developed by Neoleukin , a spin-off company from the Baker lab. [ 65 ] In December 2020, Neoleukin announced it would be submitting an Investigational New Drug application with the Food and Drug Administration in order to begin a Phase 1 clinical trial of NL-201, which is a further development of Neoleukin-2/15. A similar application was submitted in Australia and Neoleukin hopes to enrol up 120 participants on the Phase 1 clinical trial. [ 66 ] The Phase 1 human clinical trial began on May 5, 2021. [ 17 ]
Rosetta is the software responsible for performing structure prediction in Rosetta@home. Besides a BOINC cluster, Rosetta can run on a single local computer, or on a local supercomputer. Similar to other bioinformatic programs, there are online public servers offering to run Rosetta from a web interface. [ 71 ] The software is freely licensed to the academic community and available to pharmaceutical companies for a fee. [ 72 ]
Originally introduced by the Baker laboratory at the University of Washington in 1998 as an ab initio approach to structure prediction, Rosetta has since branched into several development streams and distinct services, providing features such as macromolecular docking and protein design . [ 67 ] Many of the graduate students and other researchers involved in Rosetta's initial development have since moved to other universities and research institutions, and subsequently enhanced different parts of the Rosetta project.
The Rosetta platform derives its name from the Rosetta Stone , as it attempts to decipher the structural "meaning" of proteins' amino acid sequences. [ 73 ] Development of the Rosetta code is done by Rosetta Commons. [ 72 ] Rosetta participates in CASP and CAPRI .
Rosetta was rewritten in C++ to allow easier development than that allowed by its original version, which was written in Fortran . This new version is object-oriented , and was released to Rosetta@Home February 8, 2008. [ 21 ] [ 74 ]
RosettaDesign, a computing approach to protein design based on Rosetta, began in 2000 with a study in redesigning the folding pathway of Protein G . [ 75 ] In 2002 RosettaDesign was used to design Top7 , a 93-amino acid long α/β protein that had an overall fold never before recorded in nature. This new conformation was predicted by Rosetta to within 1.2 Å RMSD of the structure determined by X-ray crystallography , representing an unusually accurate structure prediction. [ 76 ] Rosetta and RosettaDesign earned widespread recognition by being the first to design and accurately predict the structure of a novel protein of such length, as reflected by the 2002 paper describing the dual approach prompting two positive letters in the journal Science , [ 77 ] [ 78 ] and being cited by more than 240 other scientific articles. [ 79 ] The visible product of that research, Top7 , was featured as the RCSB PDB's 'Molecule of the Month' in October 2006; [ 80 ] a superposition of the respective cores (residues 60–79) of its predicted and X-ray crystal structures are featured in the Rosetta@home logo. [ 29 ]
Brian Kuhlman, a former postdoctoral associate in David Baker's lab and now an associate professor at the University of North Carolina, Chapel Hill , [ 81 ] offers RosettaDesign as an online service. [ 82 ]
RosettaDock was added to the Rosetta software suite during the first CAPRI experiment in 2002 as the Baker laboratory's algorithm for protein–protein docking prediction. [ 83 ] In that experiment, RosettaDock made a high-accuracy prediction for the docking between streptococcal pyogenic exotoxin A and a T cell-receptor β-chain , and a medium accuracy prediction for a complex between porcine α-amylase and a camelid antibody . While the RosettaDock method only made two acceptably accurate predictions out of seven possible, this was enough to rank it seventh out of nineteen prediction methods in the first CAPRI assessment. [ 83 ]
Development of RosettaDock diverged into two branches for subsequent CAPRI rounds as Jeffrey Gray, who laid the groundwork for RosettaDock while at the University of Washington , continued working on the method in his new position at Johns Hopkins University . Members of the Baker laboratory further developed RosettaDock in Gray's absence. The two versions differed slightly in side-chain modeling, decoy selection and other areas. [ 48 ] [ 84 ] Despite these differences, both the Baker and Gray methods performed well in the second CAPRI assessment, placing fifth and seventh respectively out of 30 predictor groups. [ 85 ] Jeffrey Gray's RosettaDock server is available as a free docking prediction service for non-commercial use. [ 86 ]
In October 2006, RosettaDock was integrated into Rosetta@home. The method used a fast, crude docking model phase using only the protein backbone . This was followed by a slow full-atom refinement phase in which the orientation of the two interacting proteins relative to each other, and side-chain interactions at the protein–protein interface, were simultaneously optimized to find the lowest energy conformation. [ 87 ] The vastly increased computing power afforded by the Rosetta@home network, combined with revised fold-tree representations for backbone flexibility and loop modeling , made RosettaDock sixth out of 63 prediction groups in the third CAPRI assessment. [ 6 ] [ 34 ]
The Robetta (Rosetta Beta) server is an automated protein structure prediction service offered by the Baker laboratory for non-commercial ab initio and comparative modeling. [ 88 ] It has participated as an automated prediction server in the biannual CASP experiments since CASP5 in 2002, performing among the best in the automated server prediction category. [ 89 ] Robetta has since competed in CASP6 and 7, where it did better than average among both automated server and human predictor groups. [ 32 ] [ 90 ] [ 91 ] It also participates in the CAMEO3D continuous evaluation. Robetta tasks run on Baker lab servers, Janelia Research Campus machines, and Rosetta@home participant computers. [ 88 ]
In modeling protein structure as of CASP6, Robetta first searches for structural homologs using BLAST , PSI-BLAST , and 3D-Jury , then parses the target sequence into its individual domains , or independently folding units of proteins, by matching the sequence to structural families in the Pfam database . Domains with structural homologs then follow a "template-based model" (i.e., homology modeling ) protocol. Here, the Baker laboratory's in-house alignment program, K*sync, produces a group of sequence homologs, and each of these is modeled by the Rosetta de novo method to produce a decoy (possible structure). The final structure prediction is selected by taking the lowest energy model as determined by a low-resolution Rosetta energy function. For domains that have no detected structural homologs, a de novo protocol is followed in which the lowest energy model from a set of generated decoys is selected as the final prediction. These domain predictions are then connected together to investigate inter-domain, tertiary-level interactions within the protein. Finally, side-chain contributions are modeled using a protocol for Monte Carlo conformational search. [ 92 ]
In CASP8, Robetta was augmented to use Rosetta's high resolution all-atom refinement method, [ 93 ] the absence of which was cited as the main cause for Robetta being less accurate than the Rosetta@home network in CASP7. [ 33 ] In CASP11, a way to predict the protein contact map by co-evolution of residues in related proteins called GREMLIN was added, allowing for more de novo fold successes. [ 94 ]
Rosetta is available as an online service from a number of other public servers. ROSIE offers a variety of functions from RNA structure prediction and design to ligand docking and antibody modeling. [ 95 ]
On May 9, 2008, after Rosetta@home users suggested an interactive version of the volunteer computing program, the Baker lab publicly released Foldit , an online protein structure prediction game based on the Rosetta platform. [ 96 ] As of September 25, 2008 [update] , Foldit had over 59,000 registered users. [ 97 ] The game gives users a set of controls (for example, shake, wiggle, rebuild) to manipulate the backbone and amino acid side chains of the target protein into more energetically favorable conformations. Users can work on solutions individually as soloists or collectively as evolvers , accruing points under either category as they improve their structure predictions. [ 98 ]
Foldit can work as a GUI frontend to Rosetta under a tailored "professional mode". [ 71 ]
RoseTTAFold, which is inspired by AlphaFold , uses a neural network to predict the distance and orientation between residues. These predictions guide Rosetta software in producing a structure. RoseTTAFold is open source under the MIT license . [ 99 ]
The Jianyi Yang lab in China offers a modified version of Rosetta termed tr-RosettaX2 (transform-restrained Rosetta). [ 100 ] It uses a deep learning-based contact prediction method different from RoseTTAFold to guide the usual Rosetta folding algorithm. trRosetta predates RoseTTAFold. [ 101 ]
There are several volunteer computed projects which have study areas similar to those of Rosetta@home, but differ in their research approach:
Of all the major volunteer computing projects involved in protein research, Folding@home is the only one not using the BOINC platform. [ 102 ] [ 103 ] [ 104 ] Both Rosetta@home and Folding@home study protein misfolding diseases such as Alzheimer's disease , but Folding@home does so much more exclusively. [ 105 ] [ 106 ] Folding@home almost exclusively uses all-atom molecular dynamics models to understand how and why proteins fold (or potentially misfold, and subsequently aggregate to cause diseases). [ 107 ] [ 108 ] In other words, Folding@home's strength is modeling the process of protein folding, while Rosetta@home's strength is computing protein design and predicting protein structure and docking.
Some of Rosetta@home's results are used as the basis for some Folding@home projects. Rosetta provides the most likely structure, but it is not definite if that is the form the molecule takes or whether or not it is viable. Folding@home can then be used to verify Rosetta@home's results, and can provide added atomic-level information, and details of how the molecule changes shape. [ 108 ] [ 109 ]
The two projects also differ significantly in their computing power and host diversity. Averaging about 6,650 tera FLOPS from a host base of central processing units (CPUs), graphics processing units (GPUs), and (formerly) PS3s , [ 110 ] Folding@home has nearly 108 times more computing power than Rosetta@home. [ 111 ]
Both Phase I and Phase II of the Human Proteome Folding Project (HPF), a subproject of World Community Grid , have used the Rosetta program to make structural and functional annotations of various genomes . [ 112 ] [ 113 ] Although he now uses it to create databases for biologists, Richard Bonneau , head scientist of the Human Proteome Folding Project, was active in the original development of Rosetta at David Baker's laboratory while obtaining his PhD. [ 114 ] More information on the relationship between the HPF1, HPF2 and Rosetta@home can be found on Richard Bonneau's website. [ 115 ]
Like Rosetta@home, Predictor@home specialized in protein structure prediction. [ 116 ] While Rosetta@home uses the Rosetta program for its structure prediction, Predictor@home used the dTASSER methodology. [ 117 ] In 2009, Predictor@home shut down.
Other protein related volunteer computing projects on BOINC include QMC@home , Docking@home , POEM@home , SIMAP , and TANPAKU . RALPH@home, the Rosetta@home alpha project which tests new application versions, work units, and updates before they move on to Rosetta@home, runs on BOINC also. [ 118 ]
Rosetta@home depends on computing power donated by individual project members for its research. As of March 28, 2020 [update] , about 53,000 users from 150 countries were active members of Rosetta@home, together contributing idle processor time from about 54,800 computers for a combined average performance of over 1.7 Peta FLOPS . [ 111 ] [ 119 ]
Users are granted BOINC credits as a measure of their contribution. The credit granted for each workunit is the number of decoys produced for that workunit multiplied by the average claimed credit for the decoys submitted by all computer hosts for that workunit. This custom system was designed to address significant differences between credit granted to users with the standard BOINC client and an optimized BOINC client, and credit differences between users running Rosetta@home on Windows and Linux operating systems. [ 120 ] The amount of credit granted per second of CPU work is lower for Rosetta@home than most other BOINC projects. [ 121 ] Rosetta@home is thirteenth out of over 40 BOINC projects in terms of total credit. [ 122 ]
Rosetta@home users who predict protein structures submitted for the CASP experiment are acknowledged in scientific publications regarding their results. [ 33 ] Users who predict the lowest energy structure for a given workunit are featured on the Rosetta@home homepage as Predictor of the Day , along with any team of which they are a member. [ 123 ] A User of the Day is chosen randomly each day to be on the homepage also, from among users who have made a Rosetta@home profile. [ 124 ]
Online Rosetta services | https://en.wikipedia.org/wiki/Rosetta@home |
A rosette sampler (also known as a CTD-rosette or carousel ) is a device used for water sampling in deep water. Rosette samplers are used in the ocean and large inland water bodies such as the North American Great Lakes in order to investigate quality . Rosette samplers are a key piece of equipment in oceanography and have been used to collect information over many years in repeat hydrographic surveys .
A rosette sampler is made of an assembly of 12 to 36 sampling bottles. [ 1 ] Each bottle is a volume that range from a minimum value of 1.2 L to a maximum value of 30 L. [ 1 ] The bottles are clustered around a cylinder situated in the center of the assembly, [ 1 ] where there is a sensing system called Sea-Bird or CTD , that stands for "Conductivity, Temperature and Depth", although other variables can be measured by modern CTDs (e.g. water turbidity , dissolved oxygen concentration , chlorophyll concentration and pH ). [ 2 ]
The apparatus is attached to a wire rope. A winch on board of the boat unrolls the rope during descent and rolls up it during the ascent (i.e. at the end of the samples collection). During operations in the ocean, a rosette sampler can approach the seabed at a distance from 1 to 5 m, depending on the particular sea conditions. [ 3 ]
The opening of each sampling bottle can be automatic (by reaching a certain depth) or manual (by operator, remotely). [ 2 ]
Water sampling is used in general for chemical analysis and ecotoxicological assessment. [ 3 ]
A rosette sampler is preferred to Winchester sampler for collection of water sampling at depths greater than 50 m . [ 3 ]
This water supply –related article is a stub . You can help Wikipedia by expanding it .
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rosette_sampler |
In fluid mechanics , the Roshko number ( Ro ) is a dimensionless number describing oscillating flow mechanisms. It is named after the Canadian Professor of Aeronautics Anatol Roshko . It is defined as
where
Roshko determined the correlation [ 1 ] below from experiments on the flow of air around circular cylinders over range Re=50 to Re=2000:
Ormières and Provansal [ 2 ] investigated vortex shedding in the wake of a sphere and found a relationship between Re and Ro in the range 280 < Re < 360. | https://en.wikipedia.org/wiki/Roshko_number |
Rosickyite is a rare native element mineral that is a polymorph of sulfur . It crystallizes in the monoclinic crystal system and is a high temperature, high density polymorph. It occurs as soft, colorless to pale yellow crystals and efflorescences. [ 2 ] [ 3 ]
It was first described in 1930 for an occurrence in Havirna, near Letovice , Moravia , Czech Republic . It was named for Vojtĕch Rosický (1880–1942), of Masaryk University , Brno . [ 2 ] [ 3 ]
Rosickyite occurs as in Death Valley within an evaporite layer produced by a microbial community. The otherwise unstable polymorph was produced and stabilized within a cyanobacteria dominated layer. [ 5 ] | https://en.wikipedia.org/wiki/Rosickýite |
Rosin is a glycoside ester of cinnamyl alcohol and a constituent of Rhodiola rosea .
The three cinnamyl alcohol-vicianosides of Rhodiola rosea, commonly referred to as "rosavins," are rosin , and the structurally related disaccharide rosavin , which is the arabinose ester of rosin, and rosarin , the arabinofuranose ester of rosin. Salidroside , common in Rhodiola spp. and occurring in Rhodiola rosea is not a cinnamyl alcohol glycoside , but a glycoside of tyrosol . [ 1 ]
The cinnamyl alcohol glycosides rosin, rosavin and rosarin occur in the context of rhodiola species, only in Rhodiola rosea. [ 1 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rosin_(chemical) |
The Rosiwal scale is a hardness scale in mineralogy , with its name given in memory of the Austrian geologist August Karl Rosiwal . The Rosiwal scale attempts to give more quantitative values of scratch hardness, unlike the Mohs scale which is a qualitative measurement with relative values.
The Rosiwal method (also called the Delesse-Rosiwal method [ 1 ] ) is a method of petrographic analysis and is performed by scratching a polished surface under a known load using a scratch-tip with a known geometry. The hardness is calculated by finding the volume of removed material, but this measurement can be difficult and must sample a large enough number of grain in order to have statistical significance. [ 2 ]
Measures the scratch hardness of a mineral expressed on a quantitative scale. These measurements must be performed in a laboratory , since the surfaces must be flat and smooth. The base value of the Rosiwal scale is defined as corundum set to 1000 (unitless). | https://en.wikipedia.org/wiki/Rosiwal_scale |
In organic chemistry , the Roskamp reaction is a name reaction describing the reaction between α-diazoesters (such as ethyl diazoacetate ) and aldehydes to form β-ketoesters, often utilizing various Lewis acids (such as BF 3 , SnCl 2 , and GeCl 2 ) as catalysts. [ 1 ] [ 2 ] [ 3 ] The reaction is notable for its mild reaction conditions and selectivity.
The Roskamp reaction was published seminally in 1989 by Roskamp and co-workers. The authors initially proposed that the reaction would convert aldehydes to alkenes via a pseudo- Wittig type reaction ; however, β-ketoesters were the only products to be observed. The authors also noted that aliphatic aldehyde gave higher yield than aromatic aldehydes due to enolization . Additionally, the mild reaction conditions shows advantages in preventing side reactions and increasing functional group tolerance.
In 1992, Roskamp and co-workers expanded the scope of diazoacetate to diazo sulfones , diazo phosphonates and diazo phosphine oxides. [ 4 ]
Diazo compounds are ambiphilic reagents. According to its resonance structure, the carbon adjacent to the diazo group has partial negative charge. [ 5 ] If R’ = H, this can be regarded as a hydride-transfer process.
Aldehydes containing protected amines are tolerated under Roskamp conditions. [ 6 ] Olefins can be used to generate the desired aldehyde in situ through ozonolysis , where tin(II) chloride would serve as both the reducing agent in the ozonolysis step as well as the Lewis acid catalyst in the Roskamp reaction step. [ 7 ]
In the Roskamp reaction, the alpha position of the formed β-ketoester can potentially be a chiral center, so asymmetric variants of the Roskamp reaction were investigated. In 2009, Maruoka and co-workers reported a Lewis acid-catalyzed asymmetric Roskamp reaction. [ 8 ] The chiral information is introduced by chiral auxiliaries from the diazo compound.
In 2010, the Feng group developed a chiral Sc-catalyzed enantioselective Roskamp reaction, sometimes referred to as the Roskamp–Feng reaction , the first case of an asymmetric and catalytic Roskamp reaction. [ 9 ] N,N’-dioxide-Sc(OTF) 3 chiral ligands were used, an emerging class of privileged ligand from the Feng group. [ 10 ]
A disadvantage of the Roskamp–Feng reaction is that the substrate is limited to aromatic aldehydes. The authors also demonstrated that the ketoester products can be further reduced to access chiral 1,3-diols, a useful class of building blocks in natural product synthesis.
In 2012, Do Hyun Ryu from Sungkyunkwan University developed a catalytic, asymmetric Roskamp reaction with broad applicability. [ 11 ] They utilized oxazaborolidinium ion Lewis acid catalysts, which are generated from the corresponding oxazaborolidines by protonation with triflic acid .
Compared to the Roskamp–Feng reaction, this method has a broader scope of aldehydes, notably being compatible with aliphatic aldehydes. In 2015, the same group reported asymmetric Roskamp reaction of the α-aryl diazo Weinreb amide , using the same chiral oxazaborolidine catalyst. [ 12 ]
The Roskamp reaction was utilized in the total synthesis of (+)-Galbulimima Alkaloid 13 and (+)-Himgaline. [ 13 ] | https://en.wikipedia.org/wiki/Roskamp_reaction |
Rosocyanine and rubrocurcumin are two red colored materials, which are formed by the reaction between curcumin and borates .
The color reaction between borates and curcumin is used for the spectrophotometric determination and quantification of boron present in food or materials. Curcumin is a yellow coloring natural pigment found in the root stocks of some Curcuma species, especially Curcuma longa (turmeric), in concentrations up to 3%. In the so-called curcumin method for boron quantification it serves as reaction partner for boric acid . The reaction is very sensitive and so the smallest quantities of boron can be detected. The maximum absorbance at 540 nm for rosocyanine is used in this colorimetric method. The formation of rosocyanine depends on the reaction conditions. The reaction is carried out preferentially in acidic solutions containing hydrochloric or sulfuric acid. The color reaction also takes place under different conditions; however, in alkaline solution, gradual decomposition is observed. The reaction might be disturbed at higher pH values, interfering with other compounds. [ citation needed ]
Rosocyanine is formed as a 2:1 complex from curcumin and boric acid in acidic solutions. The boron complexes formed with rosocyanine are dioxaborines (here a 1,3,2-dioxaborine). Curcumin possesses a 1,3- diketone structure and can therefore be considered as a chelating agent . Unlike the simpler 1,3-diketone–containing compound acetylacetone (which forms acetylacetonate complexes with metals), the entire skeleton of curcumin is in resonance with the 1,3-dicarbonyl section, making the backbone an extended conjugated system . Investigations of the structure have shown that the positive charge is distributed throughout the molecule. In rosocyanine, the two curcumin moieties are not coplanar but rather perpendicular relative to one another (as seen in the 3D model), as a result of the tetrahedral geometry of tetracoordinate boron. The same applies to rubrocurcumin.
In order to exclude the presence of other materials during the boron quantification using the curcumin method, a variant of the process was developed. In this process, 2,2-dimethyl-1,3-hexanediol or 2-ethyl-1,3-hexanediol are added, in addition to curcumin, to a neutral solution of the boron-containing solution. The complex formed between boron and the 1,3-hexanediol derivative is removed from the aqueous solution by extraction in an organic solvent. Acidification of the organic phase yields rubrocyanine, which can be detected by colorimetric methods. The reaction of curcumin with borates in presence of oxalic acid produces the coloring compound rubrocurcumin .
Rosocyanine is a dark green solid with a glossy, metallic shine that forms red colored solutions. It is almost insoluble in water and some organic solvents, very slightly soluble (up to 0.01%) in ethanol , and somewhat soluble (approximately 1%) in pyridine , sulfuric acid , and acetic acid . An alcoholic solution of rosocyanine temporarily turns deeply blue on treatment with alkali .
In rubrocurcumin one molecule of curcumin is replaced with oxalic acid . Rubrocurcumin produces a similar red colored solution. Rosocyanine is an ionic compound, while rubrocurcumin is a neutral complex. | https://en.wikipedia.org/wiki/Rosocyanine |
Ross Baldick is an American professor emeritus of electrical and computer engineering at the University of Texas at Austin . He is an Institute of Electrical and Electronics Engineers (IEEE) fellow of power and energy society . He is the chairman of the System Economics Sub-Committee of the IEEE Power Engineering and an associate editor of IEEE Transactions on Power Systems.
His research interests are optimization and economic theory application to electric power system operations, public policy, and technical issues related to electric transmission under deregulation.
He received his bachelor of science in mathematics and physics and bachelor of engineering in electrical engineering from the University of Sydney , Australia in 1983 and 1985, respectively. He received his Master of Science and Doctor of Philosophy in electrical engineering and computer sciences from University of California , Berkeley in 1988 and 1990, respectively. From 1991-1992, after completing his doctoral studies, he worked as a post-doctoral fellow at the Lawrence Berkeley National Laboratory . From 1992 to 1993, he was an assistant professor at Worcester Polytechnic Institute , Worcester, MA. In 1993, Baldick joined the University of Texas at Austin faculty, where he remained until his retirement in 2021. [ 1 ] [ 2 ]
Baldick's research interests in electric power span across multiple areas, and he has contributed to over one hundred peer-reviewed journal articles. [ 3 ]
Baldick's research focuses on optimization and economic theory applied to electric power system operations and the public policy and technical issues associated with electric transmission under deregulation. He is the author of the textbook "Applied Optimization: Formulation and Algorithms for Engineering Systems." [ 4 ] [ 5 ]
In 2008, Baldick was named an IEEE Fellow for his contributions to analyzing and optimizing electric power systems. [ 6 ]
Baldick has received the 2014 IEEE Power and Energy Society Outstanding Engineering Educator Award. [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Ross_Baldick |
Rossby Wave Instability ( RWI ) is a concept related to astrophysical accretion discs . In non-self-gravitating discs, for example around newly forming stars, the instability can be triggered by an axisymmetric bump, at some radius r 0 {\displaystyle r_{0}} , in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation in the vicinity of r 0 {\displaystyle r_{0}} consisting of anticyclonic vortices. These vortices are regions of high pressure and consequently act to trap dust particles which in turn can facilitate planetesimal growth in proto-planetary discs. [ 2 ] The Rossby vortices in the discs around stars and black holes may cause the observed quasi-periodic modulations of the disc's thermal emission.
Rossby waves , named after Carl-Gustaf Arvid Rossby , are important in planetary atmospheres and oceans and are also known as planetary waves . [ 3 ] [ 4 ] [ 5 ] [ 6 ] These waves have a significant role in the transport of heat from equatorial to polar regions of the Earth. They may have a role in the formation of the long-lived ( > 300 {\displaystyle >300} yr) Great Red Spot on Jupiter which is an anticyclonic vortex. [ 7 ] The Rossby waves have the notable property of having the phase velocity opposite to the direction of motion of the atmosphere or disc in the comoving frame of the fluid. [ 2 ] [ 3 ]
The theory of the Rossby wave instability in accretion discs was developed by Lovelace et al. [ 8 ] and Li et al. [ 9 ] for thin Keplerian discs with negligible self-gravity and earlier by Lovelace and Hohlfeld [ 10 ] for thin disc galaxies where the self-gravity may or may not be important and where the rotation is in general non-Keplerian.
The Rossby wave instability occurs because of the local wave trapping in a disc. It is related to the Papaloizou and Pringle instability; [ 11 ] [ 12 ] where the wave is trapped between the inner and outer radii of a disc or torus. | https://en.wikipedia.org/wiki/Rossby_wave_instability |
In economics , Rosser's equation (named after J. Barkley Rosser, Jr. ) calculates future US Social Security Administration Trust Fund balances and payments as the ratio of benefit payments in real terms for a given income level to be received the year after the Trust Fund would be exhausted, to those of the same income level for an initial year.
( FRA ij ( T )/ FRA ij ( t ))·100
where:
Rosser's equation was used in Rosser (2005) [ 1 ] to make calculations based on given reports and projections. The label was coined by Bruce Webb in 2010, picked up by others, [ 2 ] with Webb declaring it as "something between an inside joke and a tribute to Prof. Barkley Rosser, Jr. of James Madison University , an economist friend of mine who pointed out a surprising result: real payable benefits after projected Trust Fund depletion and subsequent 25% cut will still be higher in actual basket of goods terms than those of current retirees,". [ 3 ] The most important inputs to the equation are the projections from the SSA Trust Fund reports, which depend on demographic and economic assumptions. [ 4 ] In his original discussion in a letter to The Breeze, published 2/14/05, Rosser discussed an informal survey of students in economics classes made by himself and three other professors at JMU regarding their knowledge of what was projected by the SSA to happen after it ran out of "accumulated assets, thereby becoming 'bankrupt.'" They were offered four possible options in terms of the equation, to which they responded by raising their hands, reporting the majority outcomes for the seven classes.
"In one class everyone said a), zero. In five classes, a majority said b), between zero and 50%. In one class a majority said c), between 50% and 100%. Among the roughly 250 students not a single one said d), above 100%, the correct answer." | https://en.wikipedia.org/wiki/Rosser's_equation |
In mathematical logic , Rosser's trick is a method for proving a variant of Gödel's incompleteness theorems not relying on the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160). This method was introduced by J. Barkley Rosser in 1936, as an improvement of Gödel's original proof of the incompleteness theorems that was published in 1931.
While Gödel's original proof uses a sentence that says (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".
Rosser's trick begins with the assumptions of Gödel's incompleteness theorem. A theory T {\displaystyle T} is selected which is effective, consistent, and includes a sufficient fragment of elementary arithmetic.
Gödel's proof shows that for any such theory there is a formula Proof T ( x , y ) {\displaystyle \operatorname {Proof} _{T}(x,y)} which has the intended meaning that y {\displaystyle y} is a natural number code (a Gödel number) for a formula and x {\displaystyle x} is the Gödel number for a proof, from the axioms of T {\displaystyle T} , of the formula encoded by y {\displaystyle y} . (In the remainder of this article, no distinction is made between the number y {\displaystyle y} and the formula encoded by y {\displaystyle y} , and the number coding a formula ϕ {\displaystyle \phi } is denoted # ϕ {\displaystyle \#\phi } .) Furthermore, the formula Pvbl T ( y ) {\displaystyle \operatorname {Pvbl} _{T}(y)} is defined as ∃ x Proof T ( x , y ) {\displaystyle \exists x\operatorname {Proof} _{T}(x,y)} . It is intended to define the set of formulas provable from T {\displaystyle T} .
The assumptions on T {\displaystyle T} also show that it is able to define a negation function neg ( y ) {\displaystyle {\text{neg}}(y)} , with the property that if y {\displaystyle y} is a code for a formula ϕ {\displaystyle \phi } then neg ( y ) {\displaystyle {\text{neg}}(y)} is a code for the formula ¬ ϕ {\displaystyle \neg \phi } . The negation function may take any value whatsoever for inputs that are not codes of formulas.
The Gödel sentence of the theory T {\displaystyle T} is a formula ϕ {\displaystyle \phi } , sometimes denoted G T {\displaystyle G_{T}} , such that T {\displaystyle T} proves ϕ {\displaystyle \phi } ↔ ¬ Pvbl T ( # ϕ ) {\displaystyle \neg \operatorname {Pvbl} _{T}(\#\phi )} . Gödel's proof shows that if T {\displaystyle T} is consistent then it cannot prove its Gödel sentence; but in order to show that the negation of the Gödel sentence is also not provable, it is necessary to add a stronger assumption that the theory is ω-consistent , not merely consistent. For example, the theory T = PA + ¬ G P A {\displaystyle T={\text{PA}}+\neg {\text{G}}_{PA}} , in which PA is Peano axioms , proves ¬ G T {\displaystyle \neg G_{T}} . Rosser (1936) constructed a different self-referential sentence that can be used to replace the Gödel sentence in Gödel's proof, removing the need to assume ω-consistency.
For a fixed arithmetical theory T {\displaystyle T} , let Proof T ( x , y ) {\displaystyle \operatorname {Proof} _{T}(x,y)} and neg ( x ) {\displaystyle {\text{neg}}(x)} be the associated proof predicate and negation function.
A modified proof predicate Proof T R ( x , y ) {\displaystyle \operatorname {Proof} _{T}^{R}(x,y)} is defined as:
Proof T R ( x , y ) ≡ Proof T ( x , y ) ∧ ¬ ∃ z ≤ x [ Proof T ( z , neg ( y ) ) ] , {\displaystyle \operatorname {Proof} _{T}^{R}(x,y)\equiv \operatorname {Proof} _{T}(x,y)\land \lnot \exists z\leq x[\operatorname {Proof} _{T}(z,\operatorname {neg} (y))],}
which means that
¬ Proof T R ( x , y ) ≡ Proof T ( x , y ) → ∃ z ≤ x [ Proof T ( z , neg ( y ) ) ] . {\displaystyle \lnot \operatorname {Proof} _{T}^{R}(x,y)\equiv \operatorname {Proof} _{T}(x,y)\to \exists z\leq x[\operatorname {Proof} _{T}(z,\operatorname {neg} (y))].}
This modified proof predicate is used to define a modified provability predicate Pvbl T R ( y ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(y)} :
Pvbl T R ( y ) ≡ ∃ x Proof T R ( x , y ) . {\displaystyle \operatorname {Pvbl} _{T}^{R}(y)\equiv \exists x\operatorname {Proof} _{T}^{R}(x,y).}
Informally, Pvbl T R ( y ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(y)} is the claim that y {\displaystyle y} is provable via some coded proof x {\displaystyle x} such that there is no smaller coded proof of the negation of y {\displaystyle y} . Under the assumption that T {\displaystyle T} is consistent, for each formula ϕ {\displaystyle \phi } the formula Pvbl T R ( # ϕ ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(\#\phi )} will hold if and only if Pvbl T ( # ϕ ) {\displaystyle \operatorname {Pvbl} _{T}(\#\phi )} holds, because if there is a code for the proof of ϕ {\displaystyle \phi } , then (following the consistency of T {\displaystyle T} ) there is no code for the proof of ¬ ϕ {\displaystyle \neg \phi } . However, Pvbl T ( # ϕ ) {\displaystyle \operatorname {Pvbl} _{T}(\#\phi )} and Pvbl T R ( # ϕ ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(\#\phi )} have different properties from the point of view of provability in T {\displaystyle T} .
An immediate consequence of the definition is that if T {\displaystyle T} includes enough arithmetic, then it can prove that for every formula ϕ {\displaystyle \phi } , Pvbl T R ( ϕ ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(\phi )} implies ¬ Pvbl T R ( neg ( ϕ ) ) {\displaystyle \neg \operatorname {Pvbl} _{T}^{R}({\text{neg}}(\phi ))} . This is because otherwise, there are two numbers n , m {\displaystyle n,m} , coding for the proofs of ϕ {\displaystyle \phi } and ¬ ϕ {\displaystyle \neg \phi } , respectively, satisfying both n < m {\displaystyle n<m} and m < n {\displaystyle m<n} . (In fact T {\displaystyle T} only needs to prove that such a situation cannot hold for any two numbers, as well as to include some first-order logic.)
Using the diagonal lemma , let ρ {\displaystyle \rho } be a formula such that T {\displaystyle T} proves ρ ⟺ ¬ Pvbl T R ( # ρ ) {\displaystyle \rho \iff \neg \operatorname {Pvbl} _{T}^{R}(\#\rho )} . The formula ρ {\displaystyle \rho } is the Rosser sentence of the theory T {\displaystyle T} .
Let T {\displaystyle T} be an effective, consistent theory including a sufficient amount of arithmetic, with Rosser sentence ρ {\displaystyle \rho } . Then the following hold (Mendelson 1977, p. 160):
In order to prove this, one first shows that for a formula y {\displaystyle y} and a number e {\displaystyle e} , if Proof T R ( e , y ) {\displaystyle \operatorname {Proof} _{T}^{R}(e,y)} holds, then T {\displaystyle T} proves Proof T R ( e , y ) {\displaystyle \operatorname {Proof} _{T}^{R}(e,y)} . This is shown in a similar manner to what is done in Gödel's proof of the first incompleteness theorem: T {\displaystyle T} proves Proof T ( e , y ) {\displaystyle \operatorname {Proof} _{T}(e,y)} , a relation between two concrete natural numbers; one then goes over all the natural numbers z {\displaystyle z} smaller than e {\displaystyle e} one by one, and for each z {\displaystyle z} , T {\displaystyle T} proves ¬ Proof T ( z , (neg ( y ) ) {\displaystyle \neg \operatorname {Proof} _{T}(z,{\text{(neg}}(y))} , again, a relation between two concrete numbers.
The assumption that T {\displaystyle T} includes enough arithmetic (in fact, what is required is basic first-order logic) ensures that T {\displaystyle T} also proves Pvbl T R ( y ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(y)} in that case.
Furthermore, if T {\displaystyle T} is consistent and proves ϕ {\displaystyle \phi } , then there is a number e {\displaystyle e} coding for its proof in T {\displaystyle T} , and there is no number coding for the proof of the negation of ϕ {\displaystyle \phi } in T {\displaystyle T} . Therefore Proof T R ( e , y ) {\displaystyle \operatorname {Proof} _{T}^{R}(e,y)} holds, and thus T {\displaystyle T} proves Pvbl T R ( # ϕ ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(\#\phi )} .
The proof of (1) is similar to that in Gödel's proof of the first incompleteness theorem: Assume T {\displaystyle T} proves ρ {\displaystyle \rho } ; then it follows, by the previous elaboration, that T {\displaystyle T} proves Pvbl T R ( # ρ ) {\displaystyle \operatorname {Pvbl} _{T}^{R}(\#\rho )} . Thus T {\displaystyle T} also proves ¬ ρ {\displaystyle \neg \rho } . But we assumed T {\displaystyle T} proves ρ {\displaystyle \rho } , and this is impossible if T {\displaystyle T} is consistent. We are forced to conclude that T {\displaystyle T} does not prove ρ {\displaystyle \rho } .
The proof of (2) also uses the particular form of Proof T R {\displaystyle \operatorname {Proof} _{T}^{R}} . Assume T {\displaystyle T} proves ¬ ρ {\displaystyle \neg \rho } ; then it follows, by the previous elaboration, that T {\displaystyle T} proves Pvbl T R ( neg # ( ρ ) ) {\displaystyle \operatorname {Pvbl} _{T}^{R}({\text{neg}}\#(\rho ))} . But by the immediate consequence of the definition of Rosser's provability predicate, mentioned in the previous section, it follows that T {\displaystyle T} proves ¬ Pvbl T R ( # ρ ) {\displaystyle \neg \operatorname {Pvbl} _{T}^{R}(\#\rho )} . Thus T {\displaystyle T} also proves ρ {\displaystyle \rho } . But we assumed T {\displaystyle T} proves ¬ ρ {\displaystyle \neg \rho } , and this is impossible if T {\displaystyle T} is consistent. We are forced to conclude that T {\displaystyle T} does not prove ¬ ρ {\displaystyle \neg \rho } . | https://en.wikipedia.org/wiki/Rosser's_trick |
The Rossi X-ray Timing Explorer ( RXTE ) was a NASA satellite that observed the time variation of astronomical X-ray sources, named after physicist Bruno Rossi . The RXTE had three instruments — an All-Sky Monitor, the High-Energy X-ray Timing Experiment (HEXTE) and the Proportional Counter Array. The RXTE observed X-rays from black holes , neutron stars , X-ray pulsars and X-ray bursts . It was funded as part of the Explorer program and was also called Explorer 69 .
RXTE had a mass of 3,200 kg (7,100 lb) and was launched from Cape Canaveral on 30 December 1995, at 13:48:00 UTC , on a Delta II launch vehicle . Its International Designator is 1995-074A. [ 3 ]
The X-Ray Timing Explorer (XTE) mission has the primary objective to study the temporal and broad-band spectral phenomena associated with stellar and galactic systems containing compact objects in the energy range 2--200 KeV and in time scales from microseconds to years. The scientific instruments consists of two pointed instruments, the Proportional Counter Array (PCA) and the High-Energy X-ray Timing Experiment (HEXTE), and the All Sky Monitor (ASM), which scans over 70% of the sky each orbit. All of the XTE observing time were available to the international scientific community through a peer review of submitted proposals. XTE used a new spacecraft design that allows flexible operations through rapid pointing, high data rates, and nearly continuous receipt of data at the Science Operations Center (SOC) at Goddard Space Flight Center via a Multiple Access link to the Tracking and Data Relay Satellite System (TDRSS). XTE was highly maneuverable with a slew rate of greater than 6° per minute. The PCA/HEXTE could be pointed anywhere in the sky to an accuracy of less than 0.1°, with an aspect knowledge of around 1 arcminute . Rotatable solar panels enable anti-sunward pointing to coordinate with ground-based night-time observations. Two pointable high-gain antennas maintain nearly continuous communication with the TDRSS. This, together with 1 GB (approximately four orbits) of on-board solid-state data storage, give added flexibility in scheduling observations. [ 3 ]
The All-Sky Monitor (ASM) provided all-sky X-ray coverage, to a sensitivity of a few percent of the Crab Nebula intensity in one day, in order to provide both flare alarms and long-term intensity records of celestial X-ray sources. [ 5 ] The ASM consisted of three wide-angle shadow cameras equipped with proportional counters with a total collecting area of 90 cm 2 (14 sq in). The instrumental properties were: [ 6 ] [ 7 ]
It was built by the CSR at Massachusetts Institute of Technology . The principal investigator was Dr. Hale Bradt .
The High-Energy X-ray Timing Experiment (HEXTE) is a scintillator array for the study of temporal and temporal/spectral effects of the hard X-ray (20 to 200 keV) emission from galactic and extragalactic sources. [ 8 ] The HEXTE consisted of two clusters each containing four phoswich scintillation detectors . Each cluster could "rock" (beam switch) along mutually orthogonal directions to provide background measurements 1.5° or 3.0° away from the source every 16 to 128 seconds. In addition, the input was sampled at 8 microseconds so as to detect time-varying phenomena. Automatic gain control was provided by using an 241 Am radioactive source mounted in each detector's field of view. The HEXTE's basic properties were: [ 9 ]
The HEXTE was designed and built by the Center for Astrophysics & Space Sciences (CASS) at the University of California, San Diego . The HEXTE principal investigator was Dr. Richard E. Rothschild .
The Proportional Counter Array (PCA) provides approximately 6,500 cm 2 (1,010 sq in) of X-ray detector area, in the energy range 2 to 60 keV, for the study of temporal/spectral effects in the X-ray emission from galactic and extragalactic sources. [ 10 ] The PCA was an array of five proportional counters with a total collecting area of 6,500 cm 2 (1,010 sq in). The instrumental properties were: [ 11 ]
The PCA is being built by the Laboratory for High Energy Astrophysics (LHEA) at Goddard Space Flight Center . The principal investigator was Jean Swank . [ 11 ]
Observations from the Rossi X-ray Timing Explorer have been used as evidence for the existence of the frame-dragging effect predicted by the theory of general relativity of Einstein . RXTE results have, as of late 2007, been used in more than 1400 scientific papers.
In January 2006, it was announced that Rossi had been used to locate a candidate intermediate-mass black hole named M82 X-1 . [ 12 ] In February 2006, data from RXTE was used to prove that the diffuse background X-ray glow in our galaxy comes from innumerable, previously undetected white dwarfs and from other stars' coronae . [ 13 ] In April 2008, RXTE data was used to infer the size of the smallest known black hole. [ 14 ]
RXTE ceased science operations on 12 January 2012. [ 15 ]
NASA scientists said that the decommissioned RXTE would re-enter the Earth's atmosphere "between 2014 and 2023" (30 April 2018). [ 16 ] Later, it became clear that the satellite would re-enter in late April or early May 2018, [ 17 ] and the spacecraft fell out of orbit on 30 April 2018. [ 18 ] | https://en.wikipedia.org/wiki/Rossi_X-ray_Timing_Explorer |
Rostislav Ivanovich Grigorchuk ( Russian : Ростислав Иванович Григорчук ; Ukrainian : Ростисла́в Iва́нович Григорчу́к ; b. February 23, 1953) is a mathematician working in different areas of mathematics including group theory , dynamical systems , geometry and computer science . He holds the rank of Distinguished Professor in the Mathematics Department of Texas A&M University . Grigorchuk is particularly well known for having constructed, in a 1984 paper, [ 1 ] the first example of a finitely generated group of intermediate growth , thus answering an important problem posed by John Milnor in 1968. This group is now known as the Grigorchuk group [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] and it is one of the important objects studied in geometric group theory , particularly in the study of branch groups, automaton groups and iterated monodromy groups . Grigorchuk is one of the pioneers of asymptotic group theory as well as of the theory of dynamically defined groups. He introduced the notion of branch groups [ 7 ] [ 8 ] [ 9 ] [ 10 ] and developed the foundations of the related theory. Grigorchuk, together with his collaborators and students, initiated the theory of groups generated by finite Mealy type automata, [ 11 ] [ 12 ] [ 13 ] interpreted them as groups of fractal type, [ 14 ] [ 15 ] developed the theory of groups acting on rooted trees, [ 16 ] and found numerous applications [ 17 ] [ 18 ] [ 19 ] of these groups in various fields of mathematics including functional analysis , topology , spectral graph theory , dynamical systems and ergodic theory .
Grigorchuk was born on February 23, 1953, in Ternopil Oblast , now Ukraine (in 1953 part of the USSR ). [ 20 ] He received his undergraduate degree in 1975 from Moscow State University .
He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from Moscow State University , where his thesis advisor was Anatoly M. Stepin . Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics in Moscow . [ 20 ] During the 1980s and 1990s, Rostislav Grigorchuk held positions at the Moscow State University of Transportation , and subsequently at the Steklov Institute of Mathematics and Moscow State University . [ 20 ] In 2002 Grigorchuk joined the faculty of Texas A&M University as a Professor of Mathematics, and he was promoted to the rank of Distinguished Professor in 2008. [ 21 ]
Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians in Kyoto [ 22 ] an AMS Invited Address at the March 2004 meeting of the American Mathematical Society in Athens, Ohio [ 23 ] and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society . [ 24 ]
Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics" , [ 25 ] published by the European Mathematical Society , and is or was a member of the editorial boards of the journals "Mathematical Notes" , [ 26 ] "International Journal of Algebra and Computation" , [ 27 ] "Journal of Modern Dynamics" , [ 28 ] "Geometriae Dedicata" , [ 29 ] "Ukrainian Mathematical Journal" , [ 30 ] "Algebra and Discrete Mathematics" , [ 31 ] "Carpathian Mathematical Publications" , [ 32 ] "Bukovinian Mathematical Journal" , [ 33 ] and "Matematychni Studii" . [ 34 ]
Grigorchuk is most well known for having constructed the first example of a finitely generated group of intermediate growth which now bears his name and is called the Grigorchuk group (sometimes it is also called the first Grigorchuk group since Grigorchuk constructed several other groups that are also commonly studied). This group has growth that is faster than polynomial but slower than exponential. Grigorchuk constructed this group in a 1980 paper [ 35 ] and proved that it has intermediate growth in a 1984 article. [ 1 ] This result answered a long-standing open problem posed by John Milnor in 1968 about the existence of finitely generated groups of intermediate growth. Grigorchuk's group has a number of other remarkable mathematical properties. It is a finitely generated infinite residually finite 2-group (that is, every element of the group has a finite order which is a power of 2). It is also the first example of a finitely generated group that is amenable but not elementary amenable , thus providing an answer to another long-standing problem, posed by Mahlon Day in 1957. [ 36 ] Also Grigorchuk's group is "just infinite" : that is, it is infinite but every proper quotient of this group is finite. [ 2 ]
Grigorchuk's group is a central object in the study of the so-called branch groups and automata groups. These are finitely generated groups of automorphisms of rooted trees that are given by particularly nice recursive descriptions and that have remarkable self-similar properties. The study of branch, automata and self-similar groups has been particularly active in the 1990s and 2000s and a number of unexpected connections with other areas of mathematics have been discovered there, including dynamical systems , differential geometry , Galois theory , ergodic theory , random walks , fractals , Hecke algebras , bounded cohomology, functional analysis , and others. In particular, many of these self-similar groups arise as iterated monodromy groups of complex polynomials. Important connections have been discovered between the algebraic structure of self-similar groups and the dynamical properties of the polynomials in question, including encoding their Julia sets . [ 37 ]
Much of Grigorchuk's work in the 1990s and 2000s has been on developing the theory of branch, automata and self-similar groups and on exploring these connections. For example, Grigorchuk, with co-authors, obtained a counter-example to the conjecture of Michael Atiyah about L 2 -betti numbers of closed manifolds. [ 38 ] [ 39 ]
Grigorchuk is also known for his contributions to the general theory of random walks on groups and the theory of amenable groups , particularly for obtaining in 1980 [ 40 ] what is commonly known (see for example [ 41 ] [ 42 ] [ 43 ] ) as Grigorchuk's co-growth criterion of amenability for finitely generated groups .
In 1979 Rostislav Grigorchuk was awarded the Moscow Mathematical Society . [ 44 ]
In 1991 he obtained Fulbright Senior Scholarship , [ 45 ] Columbia University , New York.
In 2003 an international group theory conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy . [ 46 ] Special anniversary issues of the "International Journal of Algebra and Computation" , [ 47 ] the journal "Algebra and Discrete Mathematics" [ 20 ] and the book "Infinite Groups: Geometric, Combinatorial and Dynamical Aspects" [ 48 ] were dedicated to Grigorchuk's 50th birthday.
In 2009 Grigorchuk R.I. was awarded the Association of Former Students Distinguished Achievement in Research, [ 49 ] Texas A&M University .
In 2012 he became a fellow of the American Mathematical Society . [ 50 ]
In 2015 Rostislav Grigorchuk was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research. [ 51 ]
In 2020 Grigorchuk R.I. received the Humboldt Research Award by Germany’s Alexander von Humboldt Foundation . [ 52 ] | https://en.wikipedia.org/wiki/Rostislav_Grigorchuk |
Rot-proof or rot resistant is a condition of preservation or protection , by a process or treatment of materials used in industrial manufacturing or production to prevent biodegradation and chemical decomposition . Decomposition is a factor in which organic matter breaks down over time. It is commonly caused by fungus , mold or mildew .
There are natural conditions where the environment is inhospitable to animals, bacteria and fungus, for example in high altitude and the freezing subzero temperatures of the Arctic and Antarctic, which creates a similar suspension. The proofing of materials may also prevent dry rot and wet rot . | https://en.wikipedia.org/wiki/Rot-proof |
In linear algebra and matroid theory , Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases , named after Gian-Carlo Rota . It states that, if X is either a vector space of dimension n or more generally a matroid of rank n , with n disjoint bases B i , then it is possible to arrange the elements of these bases into an n × n matrix in such a way that the rows of the matrix are exactly the given bases and the columns of the matrix are also bases. That is, it should be possible to find a second set of n disjoint bases C i , each of which consists of one element from each of the bases B i .
Rota's basis conjecture has a simple formulation for points in the Euclidean plane : it states that, given three triangles with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three "rainbow" triangles having one vertex of each color. The triangles are all required to be non-degenerate, meaning that they do not have all three vertices on a line.
To see this as an instance of the basis conjecture, one may use either linear independence of the vectors ( x i , y i , 1 {\displaystyle x_{i},y_{i},1} ) in a three-dimensional real vector space (where ( x i , y i {\displaystyle x_{i},y_{i}} ) are the Cartesian coordinates of the triangle vertices) or equivalently one may use a matroid of rank three in which a set S of points is independent if either | S | ≤ 2 or S forms the three vertices of a non-degenerate triangle. For this linear algebra and this matroid, the bases are exactly the non-degenerate triangles. Given the three input triangles and the three rainbow triangles, it is possible to arrange the nine vertices into a 3 × 3 matrix in which each row contains the vertices of one of the single-color triangles and each column contains the vertices of one of the rainbow triangles.
Analogously, for points in three-dimensional Euclidean space, the conjecture states that the sixteen vertices of four non-degenerate tetrahedra of four different colors may be regrouped into four rainbow tetrahedra.
The statement of Rota's basis conjecture was first published by Huang & Rota (1994) , crediting it (without citation) to Rota in 1989. [ 1 ] The basis conjecture has been proven for paving matroids (for all n ) [ 2 ] and for the case n ≤ 3 (for all types of matroid). [ 3 ] For arbitrary matroids, it is possible to arrange the basis elements into a matrix the first Ω( √ n ) columns of which are bases. [ 4 ] The basis conjecture for linear algebras over fields of characteristic zero and for even values of n would follow from another conjecture on Latin squares by Alon and Tarsi. [ 1 ] [ 5 ] Based on this implication, the conjecture is known to be true for linear algebras over the real numbers for infinitely many values of n . [ 6 ]
In connection with Tverberg's theorem , Bárány & Larman (1992) conjectured that, for every set of r ( d + 1) points in d -dimensional Euclidean space , colored with d + 1 colors in such a way that there are r points of each color, there is a way to partition the points into rainbow simplices (sets of d + 1 points with one point of each color) in such a way that the convex hulls of these sets have a nonempty intersection. [ 7 ] For instance, the two-dimensional case (proven by Bárány and Larman) with r = 3 states that, for every set of nine points in the plane, colored with three colors and three points of each color, it is possible to partition the points into three intersecting rainbow triangles, a statement similar to Rota's basis conjecture which states that it is possible to partition the points into three non-degenerate rainbow triangles. The conjecture of Bárány and Larman allows a collinear triple of points to be considered as a rainbow triangle, whereas Rota's basis conjecture disallows this; on the other hand, Rota's basis conjecture does not require the triangles to have a common intersection. Substantial progress on the conjecture of Bárány and Larman was made by Blagojević, Matschke & Ziegler (2009) . [ 8 ] | https://en.wikipedia.org/wiki/Rota's_basis_conjecture |
In chemistry , rotamers are chemical species that differ from one another primarily due to rotations about one or more single bonds . Various arrangements of atoms in a molecule that differ by rotation about single bonds can also be referred to as conformations . Conformers/rotamers differ little in their energies, so they are almost never separable in a practical sense. Rotations about single bonds are subject to small energy barriers. [ 1 ] When the time scale for interconversion is long enough for isolation of individual rotamers (usually arbitrarily defined as a half-life of interconversion of 1000 seconds or longer), the species are termed atropisomers ( see: atropisomerism ). [ 2 ] [ 3 ] [ 4 ] The ring-flip of substituted cyclohexanes constitutes a common form of conformers. [ 5 ]
The study of the energetics of bond rotation is referred to as conformational analysis . [ 6 ] In some cases, conformational analysis can be used to predict and explain product selectivity, mechanisms, and rates of reactions. [ 7 ] Conformational analysis also plays an important role in rational, structure-based drug design .
rotamer : One of a set of conformers arising from restricted rotation about one single bond. [ 8 ]
Rotating their carbon–carbon bonds, the molecules ethane and propane have three local energy minima. They are structurally and energetically equivalent, and are called the staggered conformers . For each molecule, the three substituents emanating from each carbon–carbon bond are staggered, with each H–C–C–H dihedral angle (and H–C–C–CH 3 dihedral angle in the case of propane) equal to 60° (or approximately equal to 60° in the case of propane). The three eclipsed conformations, in which the dihedral angles are zero, are transition states (energy maxima) connecting two equivalent energy minima, the staggered conformers. [ citation needed ]
The butane molecule is the simplest molecule for which single bond rotations result in two types of nonequivalent structures, known as the anti - and gauche- conformers (see figure).
For example, butane has three conformers relating to its two methyl (CH 3 ) groups: two gauche conformers, which have the methyls ±60° apart and are enantiomeric , and an anti conformer, where the four carbon centres are coplanar and the substituents are 180° apart (refer to free energy diagram of butane). The energy separation between gauche and anti is 0.9 kcal/mol associated with the strain energy of the gauche conformer. The anti conformer is, therefore, the most stable (≈ 0 kcal/mol). The three eclipsed conformations with dihedral angles of 0°, 120°, and 240° are transition states between conformers. [ 6 ] Note that the two eclipsed conformations have distinct energies: at 0° the two methyl groups are eclipsed, resulting in higher energy (≈ 5 kcal/mol) than at 120°, where the methyl groups are eclipsed with hydrogens (≈ 3.5 kcal/mol). [ 10 ]
A rough approximate function can illustrate the main features of the conformational analysis for unbranched linear alkanes with rotation around a central C–C bond (C1–C2 in ethane, C2–C3 in butane, C3–C4 in hexane, etc.). [ 11 ] The members of this series have the general formula C 2n H 4n+2 with the index n = 1, 2, 3, etc. It can be assumed that the angle strain is negligible in alkanes since the bond angles are all near the tetrahedral ideal. The energy profile is thus periodic with 2 π / 3 {\displaystyle 2\pi /3} (120°) periodicity due to the threefold symmetry of sp 3 -hybridized carbon atoms. This suggests a sinusoidal potential energy function V ( θ , k ) {\displaystyle V(\theta ,k)} , typically modelled using a Fourier series truncated to the dominant terms: [ 12 ]
V ( θ , k ) = ∑ k = 0 ∞ V k ( n ) 2 [ 1 − cos ( k θ ) ] {\displaystyle V(\theta ,k)=\sum _{k=0}^{\infty }{\frac {V_{k}(n)}{2}}[1-\cos(k\theta )]}
Here:
For alkanes, the dominant term is usually k = 3 {\displaystyle k=3} , reflecting the threefold rotational symmetry. Higher terms may be included for precision where steric effects vary. The primary contribution comes from torsional strain due to alkyl groups eclipsing, captured by the cos ( 3 θ ) {\displaystyle \cos(3\theta )} term. Steric interactions rise with the size of substituents (H– for ethane, CH 3 – for butane, C 2 H 5 – for hexane, etc.), taken into account by the cos ( θ ) {\displaystyle \cos(\theta )} term ( k = 1 ) {\displaystyle (k=1)} . The number of carbon atoms clearly influences the size of substituents on the central C–C bond. In general, for unbranched linear alkanes with even-numbered chains, there will be two C n-1 H 2n-1 group substituents.
A parameterization using energy values derived from rotational spectroscopy data and theoretical calculations [ 13 ] gives the following simplified equation:
V ( θ , n ) = 0.25 ( n − 1 ) [ 1 − cos ( θ ) ] + [ 1.45 + 0.05 ( n − 1 ) ] [ 1 − cos ( 3 θ ) ] {\displaystyle V(\theta ,n)=0.25(n-1)[1-\cos(\theta )]+[1.45+0.05(n-1)][1-\cos(3\theta )]}
Here V ( θ , n ) {\displaystyle V(\theta ,n)} is given in kcal/mol and k = 1 , 3 {\displaystyle k=1,3} . This function largely neglects angle strain and long-range interactions for the n {\displaystyle n} members of the series.
While simple molecules can be described by these types of conformations, more complex molecules require the use of the Klyne–Prelog system to describe the conformers. [ 6 ]
More specific examples of conformations are detailed elsewhere:
Conformers generally exist in a dynamic equilibrium [ 15 ]
Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at various temperatures. At a free energy difference of 0 kcal/mol, this analysis gives an equilibrium constant of 1, meaning that two conformers exist in a 1:1 ratio. The two have equal free energy; neither is more stable, so neither predominates compared to the other. A negative difference in free energy means that a conformer interconverts to a thermodynamically more stable conformation, thus the equilibrium constant will always be greater than 1. For example, the Δ G° for the transformation of butane from the gauche conformer to the anti conformer is −0.47 kcal/mol at 298 K. [ 16 ] This gives an equilibrium constant is about 2.2 in favor of the anti conformer, or a 31:69 mixture of gauche : anti conformers at equilibrium. Conversely, a positive difference in free energy means the conformer already is the more stable one, so the interconversion is an unfavorable equilibrium ( K < 1).
The fractional population distribution of various conformers follows a Boltzmann distribution : [ 17 ]
The left hand side is the proportion of conformer i in an equilibrating mixture of M conformers in thermodynamic equilibrium. On the right side, E k ( k = 1, 2, ..., M ) is the energy of conformer k , R is the molar ideal gas constant (approximately equal to 8.314 J/(mol·K) or 1.987 cal/(mol·K)), and T is the absolute temperature . The denominator of the right side is the partition function.
The effects of electrostatic and steric interactions of the substituents as well as orbital interactions such as hyperconjugation are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier. Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions. [ 18 ] [ 19 ]
In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by A values , which measure the energy difference when a substituent on cyclohexane in the axial as compared to the equatorial position. In large (>14 atom) rings, there are many accessible low-energy conformations which correspond to the strain-free diamond lattice. [ 20 ]
The short timescale of interconversion precludes the separation of conformer in most cases. Atropisomers are conformational isomers which can be separated due to restricted rotation. [ 21 ] The equilibrium between conformational isomers can be observed using a variety of spectroscopic techniques . [ 22 ]
Protein folding also generates conformers which can be observed. The Karplus equation relates the dihedral angle of vicinal protons to their J-coupling constants as measured by NMR. The equation aids in the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. [ 23 ] Protein side chains exhibit rotamers, whose distribution is determined by their steric interaction with different conformations of the backbone. This effect is evident from statistical analysis of the conformations of protein side chains in the Backbone-dependent rotamer library . [ 24 ]
Conformational dynamics can be monitored by variable temperature NMR spectroscopy. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called " fluxional ". For example, in cyclohexane derivatives , the two chair conformers interconvert rapidly at room temperature. The ring-flip proceeds at a rates of approximately 10 5 ring-flips/sec, with an overall energy barrier of 10 kcal/mol (42 kJ/mol). This barrier precludes separation at ambient temperatures. [ 25 ] However, at low temperatures below the coalescence point one can directly monitor the equilibrium by NMR spectroscopy and by dynamic, temperature dependent NMR spectroscopy the barrier interconversion. [ 26 ]
Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, ν CBr differs by almost 50 cm −1 . [ 25 ]
Reaction rates are highly dependent on the conformation of the reactants. In many cases the dominant product arises from the reaction of the less prevalent conformer, by virtue of the Curtin-Hammett principle . This is typical for situations where the conformational equilibration is much faster than reaction to form the product. The dependence of a reaction on the stereochemical orientation is therefore usually only visible in Configurational analysis , in which a particular conformation is locked by substituents. Prediction of rates of many reactions involving the transition between sp2 and sp3 states, such as ketone reduction, alcohol oxidation or nucleophilic substitution is possible if all conformers and their relative stability ruled by their strain is taken into account. [ 27 ]
One example where the rotamers become significant is elimination reactions , which involve the simultaneous removal of a proton and a leaving group from vicinal or anti periplanar positions under the influence of a base.
The mechanism requires that the departing atoms or groups follow antiparallel trajectories. For open chain substrates this geometric prerequisite is met by at least one of the three staggered conformers. For some cyclic substrates such as cyclohexane, however, an antiparallel arrangement may not be attainable depending on the substituents which might set a conformational lock. [ 28 ] Adjacent substituents on a cyclohexane ring can achieve antiperiplanarity only when they occupy trans diaxial positions (that is, both are in axial position, one going up and one going down). [ citation needed ]
One consequence of this analysis is that trans -4- tert -butylcyclohexyl chloride cannot easily eliminate but instead undergoes substitution (see diagram below) because the most stable conformation has the bulky t -Bu group in the equatorial position, therefore the chloride group is not antiperiplanar with any vicinal hydrogen (it is gauche to all four). The thermodynamically unfavored conformation has the t -Bu group in the axial position, which is higher in energy by more than 5 kcal/mol (see A value ). [ 29 ] As a result, the t -Bu group "locks" the ring in the conformation where it is in the equatorial position and substitution reaction is observed. On the other hand, cis -4- tert -butylcyclohexyl chloride undergoes elimination because antiperiplanarity of Cl and H can be achieved when the t -Bu group is in the favorable equatorial position.
The repulsion between an axial t -butyl group and hydrogen atoms in the 1,3-diaxial position is so strong that the cyclohexane ring will revert to a twisted boat conformation. The strain in cyclic structures is usually characterized by deviations from ideal bond angles ( Baeyer strain ), ideal torsional angles ( Pitzer strain ) or transannular (Prelog) interactions.
Alkane conformers arise from rotation around sp 3 hybridised carbon–carbon sigma bonds . The smallest alkane with such a chemical bond, ethane , exists as an infinite number of conformations with respect to rotation around the C–C bond. Two of these are recognised as energy minimum ( staggered conformation ) and energy maximum ( eclipsed conformation ) forms. The existence of specific conformations is due to hindered rotation around sigma bonds, although a role for hyperconjugation is proposed by a competing theory. [ citation needed ]
The importance of energy minima and energy maxima is seen by extension of these concepts to more complex molecules for which stable conformations may be predicted as minimum-energy forms. The determination of stable conformations has also played a large role in the establishment of the concept of asymmetric induction and the ability to predict the stereochemistry of reactions controlled by steric effects. [ citation needed ]
In the example of staggered ethane in Newman projection , a hydrogen atom on one carbon atom has a 60° torsional angle or torsion angle [ 30 ] with respect to the nearest hydrogen atom on the other carbon so that steric hindrance is minimised. The staggered conformation is more stable by 12.5 kJ / mol than the eclipsed conformation, which is the energy maximum for ethane. In the eclipsed conformation the torsional angle is minimised.
In butane , the two staggered conformations are no longer equivalent and represent two distinct conformers:the anti-conformation (left-most, below) and the gauche conformation (right-most, below).
Both conformations are free of torsional strain, but, in the gauche conformation, the two methyl groups are in closer proximity than the sum of their van der Waals radii. The interaction between the two methyl groups is repulsive ( van der Waals strain ), and an energy barrier results.
A measure of the potential energy stored in butane conformers with greater steric hindrance than the 'anti'-conformer ground state is given by these values: [ 31 ]
The eclipsed methyl groups exert a greater steric strain because of their greater electron density compared to lone hydrogen atoms.
The textbook explanation for the existence of the energy maximum for an eclipsed conformation in ethane is steric hindrance , but, with a C-C bond length of 154 pm and a Van der Waals radius for hydrogen of 120 pm, the hydrogen atoms in ethane are never in each other's way. The question of whether steric hindrance is responsible for the eclipsed energy maximum is a topic of debate to this day. One alternative to the steric hindrance explanation is based on hyperconjugation as analyzed within the Natural Bond Orbital framework. [ 32 ] [ 33 ] [ 34 ] In the staggered conformation, one C-H sigma bonding orbital donates electron density to the antibonding orbital of the other C-H bond. The energetic stabilization of this effect is maximized when the two orbitals have maximal overlap, occurring in the staggered conformation. There is no overlap in the eclipsed conformation, leading to a disfavored energy maximum. On the other hand, an analysis within quantitative molecular orbital theory shows that 2-orbital-4-electron (steric) repulsions are dominant over hyperconjugation. [ 35 ] A valence bond theory study also emphasizes the importance of steric effects. [ 36 ]
Naming alkanes per standards listed in the IUPAC Gold Book is done according to the Klyne–Prelog system for specifying angles (called either torsional or dihedral angles ) between substituents around a single bond: [ 30 ]
Torsional strain or "Pitzer strain" refers to resistance to twisting about a bond.
In n -pentane , the terminal methyl groups experience additional pentane interference . [ citation needed ]
Replacing hydrogen by fluorine in polytetrafluoroethylene changes the stereochemistry from the zigzag geometry to that of a helix due to electrostatic repulsion of the fluorine atoms in the 1,3 positions. Evidence for the helix structure in the crystalline state is derived from X-ray crystallography and from NMR spectroscopy and circular dichroism in solution. [ 38 ] | https://en.wikipedia.org/wiki/Rotamer |
A rotameter is a device that measures the volumetric flow rate of fluid in a closed tube. [ 1 ]
It belongs to a class of meters called variable-area flowmeters , which measure flow rate by allowing the cross-sectional area the fluid travels through to vary, causing a measurable effect. [ 2 ]
The first variable area meter with rotating float was invented by Karl Kueppers (1874–1933) in Aachen in 1908. This is described in the German patent 215225. Felix Meyer founded the company " Deutsche Rotawerke GmbH " in Aachen recognizing the fundamental importance of this invention. They improved this invention with new shapes of the float and of the glass tube. Kueppers invented the special shape for the inside of the glass tube that realized a symmetrical flow scale.
The brand name Rotameter was registered by the British company GEC Elliot automation, Rotameter Co. In many other countries the brand name Rotameter is registered by Rota Yokogawa GmbH & Co. KG in Germany which is now owned by Yokogawa Electric Corp.
A rotameter consists of a tapered tube, typically made of glass with a 'float' (a shaped weight, made either of anodized aluminum or a ceramic), inside that is pushed up by the drag force of the flow and pulled down by gravity. The drag force for a given fluid and float cross section is a function of flow speed squared only, see drag equation . [ 3 ]
A higher volumetric flow rate through a given area increases flow speed and drag force, so the float will be pushed upwards. However, as the inside of the rotameter is cone shaped (widens), the area around the float through which the medium flows increases, the flow speed and drag force decrease until there is mechanical equilibrium with the float's weight.
Floats are made in many different shapes, with spheres and ellipsoids being the most common. The float may be diagonally grooved and partially colored so that it rotates axially as the fluid passes. This shows if the float is stuck since it will only rotate if it is free. Readings are usually taken at the top of the widest part of the float; the center for an ellipsoid, or the top for a cylinder. Some manufacturers use a different standard. [ 3 ]
The "float" must not float in the fluid: it has to have a higher density than the fluid, otherwise it will float to the top even if there is no flow.
The mechanical nature of the measuring principle provides a flow measurement device that does not require any electrical power. If the tube is made of metal, the float position is transferred to an external indicator via a magnetic coupling. This capability has considerably expanded the range of applications for the variable area flowmeter, since the measurement can observed remotely from the process or used for automatic control. [ 3 ] | https://en.wikipedia.org/wiki/Rotameter |
The New South Wales Government Railways constructed in 1903 a device for measuring the length of its lines of railway. That authority named the machine a Rotameter . It consisted of a four-wheel trolley with an additional large fifth wheel which traveled along the running surface of the rail. Its last recorded use was in the 1920s. [ 1 ]
This standards - or measurement -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotameter_(railway) |
The rotarod performance test is a performance test based on a rotating rod with forced motor activity being applied, usually by a rodent . The test measures parameters such as riding time (seconds) or endurance . Some of the functions of the test include evaluating balance, grip strength and motor coordination of the subjects; especially in testing the effect of experimental drugs [ 1 ] or after traumatic brain injury . [ 2 ]
In the test, a rodent is placed on a horizontally oriented, rotating cylinder (rod) suspended above a cage floor, which is low enough not to injure the animal, but high enough to induce avoidance of fall. Rodents naturally try to stay on the rotating cylinder, or rotarod, and avoid falling to the ground. The length of time that a given animal stays on this rotating rod is a measure of their balance , coordination , physical condition, and motor-planning . The speed of the rotarod is mechanically driven, and may either be held constant, or accelerated. [ 3 ]
A human analog to rotarod test might be treadmill running. Hamster , gerbil , and mouse owners can observe the principle in action when an animal climbs on the outside of its wheel , instead of inside of it. In the rotarod test, however, the rotation of the cylinder in experiments is mechanically driven.
The advantage of this test is that it creates a discretely measurable, continuous variable (length of time) that can be used for statistical purposes to quantify the effects of different drugs, conditions, and procedures. This test does not use subjective judgments of ability, and inter-rater reliability will be virtually perfect. Inter-laboratory reliability will only be achieved if the various parameters (size of cylinder, speed of cylinder, composition material of surface, and amount of practice/training given the animal) are also replicated. [ 4 ] The experiment is also very replicable from lab to lab (ibid). Moreover, these parameters may be adjusted variously to optimize the statistical separation of different conditions. For instance, alcohol effects on mice become less apparent when the speed is increased. [ 5 ]
Because of concern for impairment in human motor behavior from the use of prescription medications , the rotarod test is frequently used in early stages of drug development [ 6 ] to screen out drugs that might later cause subtle impairments, which might not be detected epidemiologically in a human population for a very long time.
The test may be useful as a sensitive indicator of trauma induced by brain injury to laboratory rats . [ 7 ] Alcohol markedly impairs mouse performance in the rotarod test. [ 8 ] Research using the rotarod test with various chemical agonists and antagonists may help scientists determine which components of neurons mediate the effects of chemicals. [ 9 ] Testing of genetic knockout animals may help determine the genes most responsible for maintaining mammalian balance and coordination. [ 10 ] Comparing the performance of different animals with specific brain lesions helps scientists map which structures are critical for maintaining balance. [ 11 ] | https://en.wikipedia.org/wiki/Rotarod_performance_test |
Rotary atomizers use a high speed rotating disk, cup or wheel to discharge liquid at high speed to the perimeter, forming a hollow cone spray. The rotational speed controls the drop size. Spray drying and spray painting are the most important and common uses of this technology.
Many industries need to convert a large mass of liquid into a dispersion of small (micron-size) droplets (generate a spray). Some examples of this need are evaporative cooling , meteorology , printing, medical applications, spray combustion, coating, and drying. Various devices exist to generate sprays, such as atomizers, sprayers , nozzles , and applicators. Sprays are typically generated by producing a high speed difference between the phase of gases and the liquid to be atomized. These devices achieve this atomization by releasing the liquid at very high speed into the unagitated air. The liquid can also be atomized by using a reverse process, instead of accelerating the liquid, gas can be accelerated to achieve a relatively higher speed than the liquid. The devices using this method to achieve atomization are called as airblast, air-assist, or popularly twin-fluid atomizers. In a Rotary Atomizer , the rotating cup or disc forces the liquid to come out at a very high speed through its rim. [ 1 ] [ 2 ]
The Rotary, Pressure-swirl [ 3 ] or Twin-fluid Atomizers [ 4 ] are the most common methods for spray generation. For special applications, alternative atomizer types exist such as the electrostatic atomizer [ 5 ] in which electrical pressure is used to drive the atomization, and the ultrasonic atomizing device [ 6 ] in which the liquid is passed through a transducer vibrating at ultrasonic frequencies to generate shorter wavelengths which convert the fluid into smaller droplets. Since the flow rate of liquid is low in both of these devices, their applications are limited.
Rotary atomizers [ 7 ] work on the principle of centrifugal energy; this energy is used to produce a high relative speed between the fluid and air which is essential for atomization. A rotary atomizer comprises a rotating surface. This surface can be in the form of a flat or a vaned disc, a cup, or a slotted wheel. A basic rotary atomizer is displayed in the figure.
The liquid first flows radially outwards in the disc and is then released from the disc's outer limits at a relatively very high speed. The atomization relies on the liquid's flow rate and the disc's rotational speed. The fluid is released from the disc's outer limits as uniform-sized droplets at low flow rates. At a comparatively high flow rate, ligaments are generated along the disc's outer limits which later on break into smaller droplets. When the flow rate is further increased, the ligaments become unable to fit in with the liquid flow, and hence a fine sheet of liquid is produced which expands past the disc's rim. This sheet, later on, disintegrates into ligaments and finally, drops are formed. The transition from ligament to sheet formation can be delayed by ragging the disc's edges. Rotary atomizers belong to the mechanical atomizers; hence, neither a high-pressure liquid nor a pressurized gas is required for atomization. The energy required for atomization is transferred directly from the atomizer body to the liquid. This gives us an advantage that the energy required for atomizing the liquid is directly supplied mechanically and energetically. The complicated and costly production of compressed gas, for example, compressed air, is no longer necessary. It is sufficient to feed the liquid to be atomized to the atomizer under low pressure. Sometimes a low hydrostatic pressure is adequate.
The spray generated by a device like a rotary atomizer can be viewed as liquid droplets submerged in a continuous phase of gases. The size of droplet formed by atomizer depends on various properties of the fluid (both liquid and gaseous fluid) such as density, viscosity and surface tension between fluids.
Generally, small gas turbines operate under high rotational speed of more than 100,000 rpm. Even small-sized atomizer of 10 cm diameter revolving at 30,000 rpm can impart an acceleration of 490, 000 m/s 2 (which is fifty thousand times of gravity) on the liquid fuel. Eventually, such fuel atomizers create very tiny droplets. [ 8 ] The rotary atomizer in which liquid is revolving along with it at the rate of ω and has radial channels at nominal radius R=(R 1 +R 2 )/2 in the edge from which high-speed liquid interacts with gas to form droplets.
Considering the nominal radius of the channel and thus of mass of liquid inside channel equal to R, the liquid inside channel will experience the centrifugal acceleration of Rω 2 , which causes the liquid to form a thin layer of thickness t on both walls of the channel. At very high acceleration thickness of the liquid layer (film) is very small in order μm. The shape of the channel also decides the effectiveness of atomization and the size of droplets. That is one aspect of determining the size of the droplet is the velocity of liquid in the channel (v=Rω).
So, we have four dimensionless terms derived from the above properties which determine the performance of atomization. [ 9 ]
1. Liquid-gas density ratio
r = [ρ L / ρ G ] where ρ L and ρ G are densities of liquid and gas respectively
2. Viscosity ratio
m = [μ L / μ G ] where, μ L and μ G are viscosities of liquid and gas respectively
3. Weber Number
We t = [ρ G V c 2 t/σ s ] where σ s is surface tension between liquid and gas contact surface. It is the ratio of the force applied by the gas on the liquid layer to the surface tension force acting on liquid.
4. Ohnesorge Number
Oh t = [μ L / (ρ L σ s t) 1/2 ]
It is the ratio of viscous force inside the layer to the surface tension force acting on liquid. Altogether, all these terms describe three main phenomena of atomization viz., inertia, viscous diffusion and surface tension. For practical fuel atomizer, Ohnesorge number is limited to Oh t <<1 and the size of the droplet are not much affected by Ohnesorge number. So, viscous effects can be neglected. But Weber number can't be neglected since surface tension and inertia are the major phenomena of the atomization process.
For small values of We, surface tension is dominant, and this force pulls the liquid towards the wall of the channel, making a single column that eventually breaks after meeting air resulting in comparatively larger droplets. This is known as the subcritical breakup of liquid. Whereas, for the supercritical breakup of liquid (more significant values of We), force applied by gas is dominant for breaking of liquid which results in fine small size of droplets.
[ 10 ] Rotary atomizer technology is found often on paint lines in the industrial finishing industry. A rotary atomizer is mounted to a paint robot or a reciprocator. Often call a Rotary bell atomizer, this paint applicator is often paired with electrostatic technology in order to maximize transfer efficiency of the paint. Rotary atomizers spin at extremely high speeds to break up the paint into fine, even particle sizes. Leading to a very high quality, consistent finish. This technology is used to paint a variety of industries including [ 11 ] automotive, [ 12 ] aerospace, [ 13 ] aluminum extrusion, [ 14 ] agricultural equipment, [ 15 ] cosmetics, [ 16 ] household cookware, [ 17 ] electronics and more. | https://en.wikipedia.org/wiki/Rotary_atomizers |
A rotary dryer is a type of industrial dryer used to reduce or minimize the moisture content of the material it is handling by bringing it into contact with a heated gas. [ 1 ] The dryer is made up of a rotating cylinder ("drum" or "shell"), a drive mechanism, and a support structure (commonly concrete posts or a steel frame). The cylinder is inclined slightly with the discharge end is lower than the material feed end so that material moves through the dryer under the influence of gravity . Material to be dried enters the dryer and, as the dryer rotates, the material is lifted up by a series of fins (known as flights ) lining the inner wall of the dryer. When the material gets high enough, it falls back down to the bottom of the dryer, passing through the hot gas stream as it falls.
Rotary Dryers have many applications but are most commonly seen in the mineral industry for drying sand, stone, soil, and ore . [ 2 ] They are also used in the food industry for granular material such as grains, cereals, pulses , and coffee beans.
A wide variety of rotary dryer designs are available for different applications. Gas flow, heat source, and drum design all affect the efficiency and suitability of a dryer for different materials.
The stream of hot gas can either be moving toward the discharge end from the feed end (known as co-current flow), or toward the feed end from the discharge end (known as counter-current flow ). The direction of gas flow combined with the inclination of the drum determine how quickly material moves through the dryer.
The gas stream is most commonly heated with a burner using gas, coal or oil . If the hot gas stream is made up of a mixture of air and combustion gases from a burner, the dryer is known as "directly heated". Alternatively, the gas stream may consist of air or another (sometimes inert) gas that is preheated. Where burner combustion gases do not enter the dryer, the dryer is known as "indirectly-heated". [ 3 ] Often, indirectly heated dryers are used when product contamination is a concern. In some cases, a combination of direct-indirect heated rotary dryers are also used to improve the overall efficiency.
A rotary dryer can consist of a single shell or several concentric shells, though any more than three shells is not usually necessary. Multiple drums can reduce the amount of space that the equipment requires to achieve the same throughput . Multi-drum dryers are often heated directly by oil or gas burners. The addition of a combustion chamber at the feed end helps ensure efficient fuel usage, and homogenous drying air temperatures.
Some rotary dryers have the ability to combine other processes with drying. Other processes that can be combined with drying include cooling, cleaning, shredding and separating. | https://en.wikipedia.org/wiki/Rotary_dryer |
A rotary evaporator [ 1 ] ( rotovap ) is a device used in chemical laboratories for the efficient and gentle removal of solvents from samples by evaporation . When referenced in the chemistry research literature, description of the use of this technique and equipment may include the phrase "rotary evaporator", though use is often rather signaled by other language (e.g., "the sample was evaporated under reduced pressure").
Rotary evaporators are also used in molecular cooking for the preparation of distillates and extracts.
A simple rotary evaporator system was invented by Lyman C. Craig . [ 2 ] It was first commercialized by the Swiss company Büchi in 1957. [ 3 ] The device separates substances with different boiling points, and greatly simplifies work in chemistry laboratories. In research the most common size accommodates round-bottom flasks of a few liters, whereas large scale (e.g., 20L-50L) versions are used in pilot plants in commercial chemical operations. [ citation needed ]
The main components of a rotary evaporator are:
The vacuum system used with rotary evaporators can be as simple as a water aspirator with a trap immersed in a cold bath (for non-toxic solvents), or as complex as a regulated mechanical vacuum pump with refrigerated trap. Glassware used in the vapor stream and condenser can be simple or complex, depending upon the goals of the evaporation, and any propensities the dissolved compounds might give to the mixture (e.g., to foam or "bump"). Commercial instruments are available that include the basic features, and various traps are manufactured to insert between the evaporation flask and the vapor duct. Modern equipment often adds features such as digital control of vacuum, digital display of temperature and rotational speed, and vapor temperature sensing.
Vacuum evaporators as a class function because lowering the pressure above a bulk liquid lowers the boiling points of the component liquids in it. Generally, the component liquids of interest in applications of rotary evaporation are research solvents that one desires to remove from a sample after an extraction, such as following a natural product isolation or a step in an organic synthesis. Liquid solvents can be removed without excessive heating of what are often complex and sensitive solvent-solute combinations.
Rotary evaporation is most often and conveniently applied to separate "low boiling" solvents such a n-hexane or ethyl acetate from compounds which are solid at room temperature and pressure. However, careful application also allows removal of a solvent from a sample containing a liquid compound if there is minimal co-evaporation ( azeotropic behavior), and a sufficient difference in boiling points at the chosen temperature and reduced pressure.
Solvents with higher boiling points such as water (100 °C at standard atmospheric pressure, 760 torr or 1 bar), dimethylformamide (DMF, 153 °C at the same), or dimethyl sulfoxide (DMSO, 189 °C at the same), can also be evaporated if the unit's vacuum system is capable of sufficiently low pressure. (For instance, both DMF and DMSO will boil below 50 °C if the vacuum is reduced from 760 torr to 5 torr [from 1 bar to 6.6 mbar]) However, more recent developments are often applied in these cases (e.g., evaporation while centrifuging or vortexing at high speeds). Rotary evaporation for high boiling hydrogen bond-forming solvents such as water is often a last recourse, as other evaporation methods or freeze-drying ( lyophilization ) are available. This is partly due to the fact that in such solvents, the tendency to "bump" is accentuated. The modern centrifugal evaporation technologies are particularly useful when one has many samples to do in parallel, as in medium- to high-throughput synthesis now expanding in industry and academia.
Evaporation under vacuum can also, in principle, be performed using standard organic distillation glassware — i.e., without rotation of the sample. The key advantages in use of a rotary evaporator are
A key disadvantage in rotary evaporations, besides its single sample nature, is the potential of some sample types to bump, e.g. ethanol and water, which can result in loss of a portion of the material intended to be retained. Even professionals experience periodic mishaps during evaporation, especially bumping, though experienced users become aware of the propensity of some mixtures to bump or foam, and apply precautions that help to avoid most such events. In particular, bumping can often be prevented by taking homogeneous phases into the evaporation, by carefully regulating the strength of the vacuum (or the bath temperature) to provide for an even rate of evaporation, or, in rare cases, through use of added agents such as boiling chips (to make the nucleation step of evaporation more uniform). Rotary evaporators can also be equipped with further special traps and condenser arrays that are best suited to particular difficult sample types, including those with the tendency to foam or bump.
Possible hazards include implosions resulting from use of glassware that contains flaws, such as star-cracks . Explosions may occur from concentrating unstable impurities during evaporation, for example when rotavapping an ethereal solution containing peroxides . This can also occur when taking certain unstable compounds, such as organic azides and acetylides , nitro-containing compounds, molecules with strain energy , etc. to dryness.
Users of rotary evaporation equipment must take precautions to avoid contact with rotating parts, particularly entanglement of loose clothing, hair, or necklaces. Under these circumstances, the winding action of the rotating parts can draw the users into the apparatus resulting in breakage of glassware, burns, and chemical exposure. Extra caution must also be applied to operations with air reactive materials, especially when under vacuum. A leak can draw air into the apparatus and a violent reaction can occur. | https://en.wikipedia.org/wiki/Rotary_evaporator |
Rotary feeders , also known as rotary airlocks or rotary valves , are commonly used in industrial and agricultural applications as a component in a bulk or specialty material handling system. Rotary feeders are primarily used for discharge of bulk solid material from hoppers /bins, receivers, and cyclones into a pressure or vacuum-driven pneumatic conveying system. Components of a rotary feeder include a rotor shaft, housing, head plates, and packing seals and bearings .
Rotors have large vanes cast or welded on and are typically driven by small internal combustion engines or electric motors .
Rotary airlock feeders have wide application in industry wherever dry free-flowing powders, granules, crystals, or pellets are used. Typical materials include: cement, ore, sugar, minerals, grains, plastics, dust, fly ash , flour, gypsum , lime , coffee, cereals, pharmaceuticals, etc.
Industries that utilize this type include cement, asphalt, chemicals, mining, plastics, and food processing, among others.
Rotary feeders are ideal for pollution control applications in wood, grain, food, textile, paper, tobacco, rubber, and paint industries, the Standard Series works beneath dust collectors and cyclone separators even with high temperatures and different pressure differentials.
Rotary valves are available with square or round inlet and outlet flanges. Housing can be fabricated out of sheet material or cast. Common materials are cast iron , carbon steel , 304 SS, 316 SS, and other materials.
Rotary airlock feeders are often available in standard and heavy duty models, the difference being the head plate and bearing configuration. Heavy duty models use an outboard bearing in which the bearings are moved out away from the head plate.
Housing inlet and discharge configurations are termed drop-thru or side entry.
Various wear protections are available such as hard chrome or ceramic plating applied to the inner housing surfaces.
Grease and air purge fittings are often provided to prevent contaminants from entering the packing seals.
Airlock-type rotary valves function as isolation devices in the event of a fire or deflagration inside the processing facility. They do not put out fires, but by sealing off the flow of air and material, they may slow down or prevent flames from spreading further down the conveying line. [ 2 ]
In order to function properly as isolation devices, the NFPA requires certain features in rotary airlock valves, including a 0.0079’’ clearance between the rotor and housing, specific materials of construction, and eight or more rotor vanes. [ 3 ]
The basic use of the rotary airlock feeder is as an airlock transition point, sealing pressurized systems against loss of air or gas while maintaining a flow of material between components with different pressure and suitable for air lock applications ranging from gravity discharge of filters, rotary valves, cyclone dust collectors, and rotary airlock storage devices to precision feeders for dilute phase and continuous dense phase pneumatic convey systems. (More on this in “How airlocks work in a conveying system.”)
Rotary airlock feeders and valves are utilized in pneumatic conveying systems, dust control devices, and for regulating volumetric feed.
Rotary airlock valves are commonly used as volumetric feeders to control the precise flow of materials from bins, hoppers, or silos to conveying or processing systems.
Airlock applications include gravity discharge of filters, rotary valves, cyclone dust collectors, and rotary airlock storage devices, as well as precision feeders for dilute phase and continuous dense phase pneumatic conveying systems.
ARF is a variable speed rotary feeder designed to handle bulk materials, working smoothly with the material's natural flow. The ARF, using a rotating member and articulable gate, controls the volume of material flow based on the material's inherent angle of repose. The speed of rotation and the positioning of the articulable gate provides a consistent feed of your material. Drum rotation allows the material to be gently fed without retarding material flow, while minimizing surge loading, product degradation, dust generation and maintaining uniformity of material flow.
Drop through rotary airlock feeders are designed for rugged applications that require an outboard bearing style unit where contamination and /or an abrasive product cannot be handled with an inboard bearing style.
The outboard bearing feeders is engineered for use in high pressure pneumatic conveying systems, with high temperatures where more of an effective seal is required due to high or excessive wear that is experienced with a simple dust collector .
The blow-thru rotary airlock feeder is ideal for pneumatic conveying applications in food, grain, chemical, milling, baking, plastics and pharmaceutical industries. The blow-thru airlocks feature a low profile with large capacity. High pressure differentials integral mounting feet, and retrofit competitive units. The blow-thru valves are available with 10-vane open-end rotor; outboard bearings and replaceable shaft seals.
Rotary airlock feeders are very often installed to dose material in a pneumatic conveying line. As a consequence, a lot of gas coming from the conveying line is brought upstream by the rotation of the valve and the leakages in between the rotor and the stator. To ensure a smooth material flow and a well-functioning system, a small hopper with a filter is often connected to the feeder. This setup allows the air to be vented, while the product is efficiently fed into the valve. [ 4 ]
The easy-clean series rotary feeders can be fast and simply disassembled, thoroughly quick cleaned, sanitized and inspected or maintenance in a minimum amount of time without the use of tools or removal from service, thereby reducing downtime and increasing system production. Reassembly without tools is accomplished in minutes. Internal clearances are automatically re-established every time.
The Clean-in-place rotary feeder is a special purpose valve designed for where cross-contamination is a major concern and lengthy shut-downs for clean-out are cost-prohibitive, suited for Dairy, Pharmaceutical industries, Food, Baking, Chemical, Plastics, Paint, and Powder Coating plants.
It is ideal for batch mixing systems such as those handling different colored resins which demand regular cleaning between cycles.
The filter valve is a low-cost solution designed for light duty dust collector applications.
This type of feeder is used for discharge of secondary fuel as for example: plastics or wood. The knife is cutting the oversize material and is preventing the rotor from blockage. | https://en.wikipedia.org/wiki/Rotary_feeder |
A rotary kiln is a pyroprocessing device used to raise materials to a high temperature ( calcination ) in a continuous process. Materials produced using rotary kilns include:
They are also used for roasting a wide variety of sulfide ores prior to metal extraction.
The kiln is a cylindrical vessel, inclined slightly from the horizontal, which is rotated slowly about its longitudinal axis. The process feedstock is fed into the upper end of the cylinder. As the kiln rotates, material gradually moves down toward the lower end, and may undergo a certain amount of stirring and mixing. Hot gases pass along the kiln, sometimes in the same direction as the process material (co-current), but usually in the opposite direction (counter-current). The hot gases may be generated in an external furnace, or may be generated by a flame inside the kiln. Such a flame is projected from a burner-pipe (or "firing pipe") which acts like a large bunsen burner . The fuel for this may be gas, oil, pulverized petroleum coke or pulverized coal.
The basic components of a rotary kiln are the shell, the refractory lining, support tyres (riding rings) and rollers, drive gear and internal heat exchangers.
The rotary kiln was invented in 1873 by Frederick Ransome . [ 1 ] He filed several patents in 1885-1887, but his experiments with the idea were not a commercial success. Nevertheless, his designs provided the basis for successful kilns in the US from 1891, subsequently emulated worldwide.
This is made from rolled mild steel plate, usually between 15 and 30 mm (0.6 and 1.2 in), welded to form a cylinder which may be up to 230 m (750 ft) in length and up to 6 m (20 ft) in diameter.
Upper limits on diameter are set by the tendency of the shell to deform under its own weight to an oval cross section, with consequent flexure during rotation. Length is not necessarily limited, but it becomes difficult to cope with changes in length on heating and cooling (typically around 0.1 to 0.5% of the length) if the kiln is very long.
The purpose of the refractory lining is to insulate the steel shell from the high temperatures inside the kiln, and to protect it from the corrosive properties of the process material. It may consist of refractory bricks or cast refractory concrete, or may be absent in zones of the kiln that are below approximately 250 °C (482 °F). The refractory selected depends upon the temperature inside the kiln and the chemical nature of the material being processed. In some processes, such as cement, the refractory life is prolonged by maintaining a coating of the processed material on the refractory surface. The thickness of the lining is generally in the range 80 to 300 mm (3 to 12 in). A typical refractory will be capable of maintaining a temperature drop of 1000 °C (1,800 °F) or more between its hot and cold faces. The shell temperature needs to be maintained below around 350 °C (662 °F) to protect the steel from damage, and continuous infrared scanners are used to give early warning of "hot-spots" indicative of refractory failure.
Tyres, sometimes called riding rings, usually consist of a single annular steel casting, machined to a smooth cylindrical surface, which attach loosely to the kiln shell through a variety of "chair" arrangements. These require some ingenuity of design, since the tyre must fit the shell snugly, but also allow thermal movement. The tyre rides on pairs of steel rollers, also machined to a smooth cylindrical surface, and set about half a kiln-diameter apart. The rollers must support the kiln, and allow rotation that is as nearly frictionless as possible. A well-engineered kiln, when the power is cut off, will swing pendulum-like many times before coming to rest. The mass of a typical 6 m × 60 m (20 ft × 197 ft) kiln, including refractories and feed, is around 1,100 t (2,400,000 lb), and would be carried on three tyres and sets of rollers, spaced along the length of the kiln. The longest kilns may have 8 sets of rollers, while very short and small kilns may have none. Kilns usually rotate at 0.5 to 2 rpm. The Kilns of modern cement plants are running at 4 to 5 rpm. The bearings of the rollers must be capable of withstanding the large static and live loads involved and must be carefully protected from the heat of the kiln and the ingress of dust. Since the kiln is at an angle, it also needs support to prevent it from walking off the support rollers. Usually upper and lower "retaining (or thrust) rollers" bearing against the side of tyres prevent the kiln from walking off the support rollers.
The kiln is usually turned by means of a single Girth Gear surrounding a cooler part of the kiln tube, but sometimes it is turned by driven rollers. The gear is connected through a gear train to a variable-speed electric motor . This must have high starting torque to start the kiln with a large eccentric load. A 6 m × 60 m (20 ft × 197 ft) kiln requires around 800 kW to turn at 3 rpm. The speed of material flow through the kiln is proportional to rotation speed; a variable-speed drive is needed to control this. When driving through rollers, hydraulic drives may be used. These have the advantage of developing extremely high torque. In many processes, it is dangerous to allow a hot kiln to stand still if the drive power fails. Temperature differences between the top and bottom of the kiln may cause the kiln to warp, and refractory is damaged. Hence, normal practice is to provide an auxiliary drive for use during power cuts. This may be a small electric motor with an independent power supply, or a diesel engine . This turns the kiln very slowly, but enough to prevent damage.
Heat exchange in a rotary kiln may be by conduction , convection and radiation , in descending order of efficiency. In low-temperature processes, and in the cooler parts of long kilns lacking preheaters, the kiln is often furnished with internal heat exchangers to encourage heat exchange between the gas and the feed. These may consist of scoops or "lifters" that cascade the feed through the gas stream, or may be metallic inserts that heat up in the upper part of the kiln, and impart the heat to the feed as they dip below the feed surface as the kiln rotates. The latter are favoured where lifters would cause excessive dust pick-up. The most common heat exchanger consists of chains hanging in curtains across the gas stream.
The kiln connects with a material exit hood at the lower end and ducts for waste gases. This requires gas-tight seals at either end of the kiln. The exhaust gas may go to waste, or may enter a preheater, which further exchanges heat with the entering feed. The gases must be drawn through the kiln, and the preheater if fitted, by a fan situated at the exhaust end. In preheater installations which may have a high pressure-drop, considerable fan power may be needed, and the fan drive is often the largest drive in the kiln system. Exhaust gases contain dust, and there may be undesirable constituents, such as sulfur dioxide or hydrogen chloride . Equipment is installed to scrub these from the gas stream before passing to the atmosphere. | https://en.wikipedia.org/wiki/Rotary_kiln |
A rotary phase converter , abbreviated RPC , is an electrical machine that converts power from one polyphase system to another, converting through rotary motion. Typically, single-phase electric power is used to produce three-phase electric power locally to run three-phase loads in premises where only single-phase is available.
A basic three-phase induction motor will have three windings, each end connected to terminals typically numbered (arbitrarily) as L1, L2, and L3 and sometimes T1, T2, T3.
A three-phase induction motor can be run at two-thirds of its rated horsepower on single-phase power applied to a single winding, once spun up by some means. A three-phase motor running on a single phase cannot start itself because it lacks the other phases to create a rotation on its own, much like a crank that is at dead center.
A three-phase induction motor that is spinning under single-phase power applied to terminals L1 and L2 will generate an electric potential (voltage) across terminal L3 in respect with L1 and L2. However, L1 to L3 and L2 to L3 will be 120 degrees out of phase with the input voltage, thus creating three-phase power. However, without current injection, special idler windings, or other means of regulation, the voltage will sag when a load is applied.
Power-factor correction is a very important consideration when building or choosing an RPC. This is desirable because an RPC that has power-factor correction will consume less current from the single-phase service supplying power to the phase converter and its loads.
A major concern with three phase power is that each phase be at similar voltages. A discrepancy between phases is known as phase imbalance. As a general guideline, unbalanced three-phase power that exceeds 4% in voltage variation can damage the equipment that it is meant to operate.
At the beginning of the 20th century, there were two main principles of electric railway traction current systems:
These systems used series-wound traction motors . All of them needed a separated supply system or converters to take power from the standard 50 Hz electric network.
Kálmán Kandó recognized that the electric traction system must be supplied by single-phase 50 Hz power from the standard electric network, and it must be converted in the locomotive to three-phase power for traction motors.
He created an electric machine called a synchronous phase converter, which was a single-phase synchronous motor and a three-phase synchronous generator with common stator and rotor.
It had two independent windings:
The direct feed from a standard electric network makes the system less complicated than the earlier systems and makes possible simple recuperation.
The single-phase feed makes it possible to use a single overhead line. More overhead lines increase the costs, and restrict the maximum speed of the trains.
The asynchronous traction motor can run on a single RPM determined by the frequency of the feeding current and the loading torque.
The solution was to use more secondary windings on phase converter, and more windings on motor different number of magnetic poles.
A rotary phase converter (RPC) may be built as a motor-generator set. These completely isolate the load from the single-phase supply and produce balanced three-phase output. However, due to weight, cost, and efficiency concerns, most RPCs are not built this way.
Instead, they are built out of a three-phase induction motor or generator, called an idler, on which two of the terminals (the idler inputs) are powered from the single-phase line. The rotating flux in the motor produces a voltage on the third terminal. A voltage is induced in the third terminal that is phase shifted from the voltage between the first two terminals. In a three-winding motor, two of the windings are acting as a motor, and the third winding is acting as a generator. Since the third, synthesized phase is driven differently from the other two, its response to load changes may be different causing this phase to sag more under load. Since induction motors are sensitive to voltage imbalance, this is another factor in de-rating of motors driven by this type of phase converter. For example, a small 5% imbalance in phase voltage requires a much larger 24% reduction of motor rated power. [ 1 ] Thus tuning a rotary phase converter circuit for equal phase voltages under maximum load may be quite important.
A common measure of the quality of the power produced by an RPC or any phase converter is the voltage balance, which may be measured while the RPC is driving a balanced load such as a three-phase motor. Other quality measures include the harmonic content of the power produced and the power factor of the RPC motor combination as seen by the utility. Selection of the best phase converter for any application depends on the sensitivity of the load to these factors. Three-phase induction motors are very sensitive to voltage imbalances.
The quality of three-phase power generated by such a phase converter depends upon a number of factors including:
RPC manufacturers use a variety of techniques to deal with these problems. Some of the techniques include,
Demand exists for phase converters due to the use of three-phase motors. With increasing power output, three-phase motors have preferable characteristics to single-phase motors; the latter not being available in sizes over 15 hp (11 kW) and, though available, rarely seen larger than 5 hp (3.7 kW). (Three-phase motors have higher efficiency, reduced complexity, with regards to starting, and three-phase power is significantly available where they are used.)
Rotary phase converters are used to produce a single-phase for the single overhead conductor in electric railways . [ citation needed ] Five European countries ( Germany , Austria , Switzerland , Norway , and Sweden ), where electricity is three-phase AC at 50 Hz , have standardised on single-phase AC at 15 kV 16 + 2 ⁄ 3 Hz for railway electrification; phase converters are, therefore, used to change both phases and frequency . In the Soviet Union , they were used on AC locomotives to convert single phase, 50 Hz to 3-phase for driving induction motors for traction motor cooling blowers, etc. [ 2 ]
Alternatives exist to rotary phase converters for operation of three-phase equipment on a single-phase power supply.
These may be an alternative where the issue at hand is starting a motor, rather than polyphase power itself. The static phase converter is used to start a three-phase motor. The motor then runs on a single phase with a synthesised third pole. However, this makes the power balance, and thus motor efficiency, extremely poor, requiring de-rating the motor (typically to 60% or less). Overheating, and quite often destruction of the motor, will result from failing to do so. (Many manufacturers and dealers specifically state that using a static converter will void any warranty.) An oversized static converter can remove the need to de-rate the motor, but at an increased cost.
The popularity of the Variable-frequency drive (VFD) has increased in the last decade, especially in the home-shop market. This is because of their relative low cost and ability to generate three-phase output from single phase input. A VFD converts AC power to DC and then converts it back to AC through a transistor bridge , a technology that is somewhat analogous to that of a switch-mode power supply . As the VFD generates its AC output from the DC bus, it is possible to power a three-phase motor from a single-phase source. Nevertheless, commercial-grade VFDs are produced that require three-phase input, as there are some efficiency gains to be had with such an arrangement.
A typical VFD functions by rapidly switching transistors on and off to "chop" the voltage on the DC bus through what is known as pulse-width modulation (PWM). Proper use of PWM will result in an AC output whose voltage and frequency can be varied over a fairly wide range. As an induction motor 's rotational speed is proportional to input frequency, a change in the VFD's output frequency will cause the motor to change speed. Voltage is also changed in a way that results in the motor producing a relatively constant torque over the useful speed range.
The output of a quality VFD is an approximation of a sine wave , with some high frequency harmonic content . Harmonic content will elevate motor temperature and may produce some whistling or whining noise that could be objectionable. The effects of unwanted harmonics can be mitigated by the use of reactive output filtering , which is incorporated into better quality VFDs. Reactive filtration impedes the high frequency harmonic content but has little effect on the fundamental frequency that determines motor speed. The result is an output to the motor that is closer to an ideal sine wave.
In the past, VFDs that have a capacity greater than 3 hp (2.2 kW) were costly, thus making the rotary phase converter (RPC) an attractive alternative. However, modern VFDs have dropped considerably in cost, making them more affordable than comparable RPCs. Also working in the VFD's favor is its more compact size relative to its electrical capacity. A plus is many VFDs can produce a "soft start" effect (in which power is gradually applied to the motor), which reduces the amount of current that must be delivered at machine start-up.
Use of a VFD may result in motor damage if the motor is not rated for such an application. This is primarily because most induction motors are forced-air cooled by a fan or blower driven by the motor itself. Operating such a motor at a lower-than-normal speed will substantially reduce the cooling airflow, increasing the likelihood of overheating and winding damage or failure, especially while operating at full load. A manufacturer may void the warranty on a motor powered by a VFD unless the motor is "inverter-rated." As VFDs are the most popular method of powering motors in new commercial installations, most three-phase motors sold today are, in fact, inverter-rated. | https://en.wikipedia.org/wiki/Rotary_phase_converter |
A rotary transfer machine is a machine tool , typically for metal working by machining, comprising a large indexing table with machining stations surrounding the table. Such rotary transfer machines are used for producing a large number of parts in fairly short cycle times. [ 1 ] [ 2 ] [ 3 ]
In rotary transfer machines, the workpieces are located and clamped in pallet type fixtures that are indexed in a circular path. During one cycle, sequential machining operations are performed simultaneously on the workpieces. The indexed table turns vertically or horizontally, and its movement could be continuous or intermittent. As the indexing table turns, the subsequent machining operation is repeated on the workpiece which was just machined by the previous station. This design combines automated part feed with simultaneous operations, enabling rapid completion of parts.
Rotary transfer machines are commonly used for the mass-production of metal parts in the automotive industry and for pneumatic and hydraulic fittings. The parts can range from simple to complex, depending on the layout of the machining tool, which is often custom-designed for the manufacturing of a single part or family of parts. Rotary arrangement presents a compact arrangement that saves floor space. The annual production capacity of one rotary transfer machine can range from 100'000 units to tens of millions of units.
Rotary transfer machines can generally cope with all standard machining operations like turning, milling, drilling, reaming, threading, recessing, marking, deburring, etc... for sizes ranging more or less from a fingernail up to a backpack.
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotary_transfer_machine |
A rotary union is a union that allows for rotation of the united parts. It is thus a device that provides a seal between a stationary supply passage (such as pipe or tubing) and a rotating part (such as a drum, cylinder, or spindle ) to permit the flow of a fluid into and/or out of the rotating part. Fluids typically used with rotary joints and rotating unions include various heat transfer media and fluid power media such as steam , water , thermal oil , hydraulic fluid , and coolants . [ 2 ] A rotary union is sometimes referred to as a rotating union , rotary valve , swivel union, rotorseal , [ 3 ] rotary couplings , rotary joint , rotating joints , hydraulic coupling , pneumatic rotary union , through bore rotary union , air rotary union , electrical rotary union , or vacuum rotary union [ 4 ]
A rotary union will lock onto an input valve while rotating to meet an outlet. During this time the liquid and/or gas will flow into the rotary union from its source and will be held within the device during its movement. This liquid and/or gas will leave the union when the valve openings meet during rotation, and more liquid and/or gas will flow into the union again for the next rotation.
Often functioning under high pressure and constant movement, a rotary union is designed to rotate around an axis. A rotary union's design can be altered to change this or to increase the psi or rpm it needs to withstand as well as the number of valves required.
While rotary unions come in many shapes, sizes, and configurations, they always have the same four basic components: a housing unit, a shaft, a bearing (mechanical) (or bearings), and a seal. Rotary unions typically are constructed from stainless steel to resist rust and corrosion, but many other metals can be involved, like aluminum .
The housing is the component that holds all of the other elements of the rotary union together. The housing has an inlet port, which is a threaded port to which the hose supplying the medium will be attached. The rotary union may also have an outlet port if the same joint is being used both to supply fluid to a roll and to remove fluid from the roll. In smaller rotary unions the housing is stationary. In larger rotary unions the housing is usually bolted to the drum or roll using a flange. In these cases, the housing rotates at the same speed as the drum [ 5 ]
The shaft is the component that carries the medium through the rotary union into the drum or roll. In many cases, the shaft will turn with the drum or roll. In some cases, like in larger flanged rotary unions, the shaft may be stationary while the housing rotates. The bearings and seal are typically assembled around the shaft.
The second most important part of the rotary union is the bearing. A rotary union may have only one bearing, but multiple bearings are much more common. Roller bearings; such as ball bearings and tapered roller bearings; or non-roller bearings, like graphite bearings and bronze bushings, may be used in a rotary union. The bearings are always used to allow a part of the joint, either the shaft or the housing, to rotate.
The heart of the rotary union is the seal. The seal prevents the medium from leaking outside the rotary union while in operation. Seal types can vary from pusher-type end-face mechanical seals, non-pusher-type end-face mechanical seals, lip seals, and o-ring seals. Most rotary unions have more than one seal. [ 6 ]
Many rotary unions incorporate multiple ports, some of which are designed to handle different types of material simultaneously.
A rotary union with a straight port transfers the substance directly through the rotary union. Other designs include an elbow port, which causes the material to flow out at an angle, and multiple ports. A multiple-port rotary union looks like a perforated cylinder. At the end of the cylinder is a threaded screw with a seal or seals that lock onto it. The transferred material flows into the cylinder and out of the input holes.
In the case of a rotary union with multiple inputs, chambers separated by seals keep the materials from inadvertently mixing. This type of rotary union is often used in the manufacture of plastics and other petroleum products, for which multiple inputs may need to be streamlined, but kept separate. [ citation needed ]
Rotary unions can be classified according to the media they transfer. For example, if they carry coolant , they are referred to as coolant unions. The same is also true for other media like water, air, steam, etc.
Many assembly lines incorporate multiple rotary unions because they are highly versatile and take up less space than other devices designed for a similar purpose. Rotary unions also appear in automobiles and other machines that require constant supplies of lubrication, air, or other liquids for moving parts to run smoothly. Brakes, for example, use rotary unions to maintain a constant supply of pressurized brake fluid. Rotary unions are also heavily used in crude oil processing, the chemical industry, commercial food production, and pharmaceutical applications.
Equipment used in grain harvesting including combines, tractors, grain carts, and threshers employ rotary unions. Once harvested, many crops will be processed with equipment that uses rotary unions. Food processing equipment that uses rotary unions includes cooling conveyors, flaking mills, shredders, steam cookers, starch dryers, rotary cutters , and roll-forming.
Auto manufacturing is a diverse user of rotary unions for a broad range of parts or components and materials, whether machined steel, iron or aluminum, stampings, plastics, glass, or paperboard. Rotary unions are used for operations that require coolant, lubricant, or hydraulics.
There are two kinds of car wash facilities that use unions: the automatic and the hand-operated. Most manufacturers of automatic systems have several revolving brushes that use 55 series to introduce low-pressure detergent water through the supporting shaft to the brushes. In addition, automatic car washes have spinners that require rotary unions to transmit high-pressure water into the spinning mechanisms.
Downstream processing of paper, plastic film, foil, and related substrate materials into finished, printed packaging such as bags, pouches, labels, tags, folding cartons, and corrugated shipping cases is called converting . Rotary unions are used in all types of converting for water, steam, thermal oil, air, or hydraulics.
Rotary unions may be used to transmit coolant, cutting oil, MQL, and pressurized air in a bearingless or bearing-supported configuration. Besides coolant delivery, rotary unions are used for chucking, tool sensing, rotary index tables, and other machine tool applications.
Electro-hydraulic equipment used in mining operations can employ rotary unions including shuttle cars and coal cars, drill heads, backhoes, clamshell cranes, and draglines. In addition, boom hoists, retrieving drums, and bucket drum clutches each require rotary unions.
Drilling rigs (oil or gas) use air clutches and brakes that require rotary unions. Water unions are used to flush mud from the drill tip, and must withstand shock and vibration in this severe application. Oil and petrochemical refineries use batch mixers, flaking mills, blenders, and drying rolls that each require rotary unions. The development of subsea oil and gas fields requires specialized equipment. Subsea swivels, manufactured by Dynamic Sealing Technologies, Inc. , are designed for deepwater oil production systems that provide equipment operations added flexibility when lowering flowlines in harsh waters down to depths reaching 3,500 meters.
Paper applications span the supply chain from the raw pulp and paper mills to the downstream paper converters. Mills use steam joint and siphon systems and water unions for heating and cooling. Converters use rotary unions for heating and cooling rolls, as well as winders with air clutches and brakes.
The manufacturing of plastic materials encompasses a wide variety of applications including cast film, blown film, foam, flexible and rigid sheet extrusion, single and multi-layer co-extrusion, blow molding, thermoforming, pelletizing, wire and cable, injection molding, and winding. Rotary unions are used for heating or cooling the many processing rolls throughout a wide variety of applications. In addition, rotary unions for air and hydraulic service are used in winding and injection molding applications. Many of today's modern winding applications will also utilize electrical slip rings.
Printing on flexible rolls of paper or plastic films requires rotary unions for air or hydraulics, as well as chill rolls for temperature control. Web offset and sheet printing equipment use many rotary unions on the ink vibrator and chill rolls.
Rubber is compounded on big industrial mixers which use rotary unions for water-cooled rolls. Rubber extrusion is similar to plastic extrusion, with rotary unions used to cool the extruder screw.
The steel industry is one of the largest users of rotary unions primarily for continuous casting machines (CCM) which use rotary unions to cool the numerous rolls that support molten slabs as they move by gravity through various segments onto a run-out table to downstream annealing and heat treating. The slab is formed into a coil or sheet. The coil is further converted into processing centers that require hydraulic unions for the actuation of mandrels.
The textile industry is a large user of water, steam, and hot oil unions. Weaving, dyeing and finishing processes are the largest users of rotary unions.
Rubber tire plants use industrial mixers , extruders, calendar train cooling stacks, and rayon slashers to make tire cords. Rotary unions are used in every process for temperature control. [ 7 ] | https://en.wikipedia.org/wiki/Rotary_union |
A Rotary Vacuum Filter Drum consists of a cylindrical filter membrane that is partly sub-merged in a slurry to be filtered. The inside of the drum is held lower than the ambient pressure. As the drum rotates through the slurry, the liquid is sucked through the membrane, leaving solids to cake on the membrane surface while the drum is submerged. A knife or blade is positioned to scrape the product from the surface. [ 1 ]
The technique is well suited to slurries , flocculated suspensions , and liquids with a high solid content, which could clog other forms of filter. It is common to pre-coated with a filter aid, typically of diatomaceous earth (DE) or Perlite . In some implementations, the knife also cuts off a small portion of the filter media to reveal a fresh media surface that will enter the liquid as the drum rotates. Such systems advance the knife automatically as the surface is removed.
Rotary vacuum drum filter (RVDF), patented in 1872, [ 2 ] is one of the oldest filters used in the industrial liquid-solids separation. It offers a wide range of industrial processing flow sheets and provides a flexible application of dewatering, washing and/or clarification.
A rotary vacuum filter consists of a large rotating drum covered by a cloth. The drum is suspended on an axial over a trough containing liquid or solids slurry with approximately 50-80% of the screen area immersed in the slurry.
As the drum rotates into and out of the trough, the slurry is sucked on the surface of the cloth and rotated out of the liquid or solids suspension as a cake. When the cake is rotating out, it is dewatered in the drying zone. The cake is dry because the vacuum drum is continuously sucking the cake and taking the water out of it. At the final step of the separation, the cake is discharged as solids products and the drum rotates continuously to another separation cycle.
Applications:
The advantages and limitations of rotary vacuum drum filter compared to other separation methods are:
Basically there are five types of discharge that are used for the rotary vacuum drum filter such as belt, scraper, roll, string and pre coat discharge.
The filter cloth is washed on both sides with each drum rotation while discharging filter cakes. The products for this mechanism are usually sticky, wet and thin thus, requiring the aid of a discharge roll. Belt discharge is used if slurry with moderate solid concentration is used or if the slurry is easy to filter to produce cake formation or if a longer wear resistance is desired for the separation of the mentioned slurry. [ 3 ] [ 4 ] ....
This is the standard drum filter discharge. A scraper blade, which serves to redirect the filter cake into the discharge chute, removes the cake from the filter cloth just before re-entering the vat. Scraper discharge is used if the desired separation requires high filtration rate or if heavy solid slurry is used or if the slurry is easy to filter to produce cake formation or if a longer wear resistance is desired for the separation of the mentioned slurry. [ 3 ] [ 4 ]
It is a suitable discharge option for cakes that are thin and have the tendency to stick with one another. Filter cakes on the drum and discharged roll are pressed against one another to ensure that the thin filter cake is peeled or pulled from the drum. Removal of solids from the discharge roll is done via a knife blade. Roll discharged is used if the desired separation requires high filtration rate, if high solid content slurry is used or if the slurry is easy to filter to produce cake formation or if the discharged solid is sticky or mud-like cake. [ 3 ] [ 4 ]
The filtrate cakes that are thin and fragile are usually the end products of this discharge lie. The materials are capable of changing phases, from solid to liquid, due to instability and disturbance. Two rollers guide the strings back to drum surface and at the same time separation of the filtrate cake occurs as they pass the rollers. Application of the string discharge can be seen at the pharmaceutical and starch industries. String discharge is used if the high solid concentration slurry is used or if the slurry is easy to filter to produce cake formation or if the discharged solid is fibrous, stringy or pulpy or if a longer wear resistance is desired for the separation of the mentioned slurry.
Application of this discharge are usually seen where production of filter cakes that blind the filter media thoroughly and processes that have low solid concentration slurry. Pre coat discharge is used if slurry with very low solid concentration slurry is used that resulted in difficult cake formation or if the slurry is difficult to filter to produce cake . [ 3 ] [ 4 ]
Generally, the main process in a rotary vacuum drum filter is continuous filtration whereby solids are separated from liquids through a filter medium by a vacuum. The filter cloth is one of the most important components on a filter and is typically made of weaving polymer yarns. The best selection of cloth can increase the performance of filtration. Initially, slurry is pumped into the trough and as the drum rotates, it is partially submerged in the slurry. The vacuum draws liquid and air through the filter media and out the shaft hence forming a layer of cake. An agitator is used to regulate the slurry if the texture is coarse and it is settling rapidly. Solids that are trapped on the surface of the drum are washed and dried after 2/3 of revolution, removing all the free moisture. [ 5 ]
During the washing stage, the wash liquid can either be poured onto the drum or sprayed on the cake. Cake pressing is optional but its advantages are preventing cake cracking and removing more moisture. Cake discharge is when all the solids are removed from the surface of the cake by a scraper blade, leaving a clean surface as drum re-enters the slurry. [ 5 ] There are a few types of discharge which are scraper, roller, string, endless belt and pre coat. [ 4 ] The filtrate and air flow through internal pipes, valve and into the vacuum receiver where the separation of liquid and gas occurs producing a clear filtrate. [ 6 ] Pre coat filtration is an ideal method to produce a high clarity of filtrate. Basically, the drum surface is pre coated with a filter aid such as diatomaceous earth (DE) or perlite to improve filtration and increase cake permeability. It then undergoes the same process cycle as the conventional rotary vacuum drum filter however, pre coat filtration uses a higher precision blade to scrape off the cake. [ 5 ]
The filter is assessed by the size of the drum or filter area and its possible output. Typically, the output is in the units of pounds per hour of dry solids per square foot of filter area. The size of the auxiliary parts depends on the area of the filter and the type of usage. Rotary vacuum filters are flexible in handling variety of materials therefore the estimated solids yield from 5 to 200 pounds per hour per square foot. For pre coat discharge, the solid output is approximately 2 to 40 gallons per hour per square foot. [ 4 ] Filtration efficiencies can also be improved in terms dryness of filter cake by significantly preventing filtrate liquid from getting stuck in the filter drum during filtration phase. Usage of multiple filters for example, running 3 filter units instead of 2 units yields a thicker cake hence, producing a clearer filtrate. This becomes beneficial in terms of production cost and also quality. [ 5 ]
Vat level and drum speed are the two basic operating parameters for any rotary vacuum drum filter. These parameters are adjusted dependently to each other to optimize the filtration performance.
Valve level determines the proportion filter cycle in the filter. The filter cycle consist of the filter drum rotation, release of cake formation from slurry and the drying period for the cake formation shown in figure 1. By default, operate the vat at its maximum level to maximise the rate of filtration. Reduce vat level if discharged solid is in the form of thin and slimy cake or if the discharged solid is very thick. [ 4 ] [ 7 ] [ 8 ]
Decrease in the vat level eventually leads to a decrease in the portion of the drum being submerge under the slurry, more surface exposure for the cake dying surface hence, larger cake formation to dry time ratio. This result in less moisture content of formed solid and lessen the thickness of the form solid. In addition to operating at lower vat level, the flow rate per drum revolution decreases and ultimately thinner cake formation occurs. In the case of pre coat discharge the filter aid efficiency increases.
Drum speed is the driving factor for the filter output and its units is in the form of minutes per drum revolution. At steady operating conditions, adjusting the drum speed gives a proportional relationship with the filter throughput as shown as in figure 2.
Select filter cloth to obtain a good surface for cake formation. Use twill weave variation in the construction pattern of the fabric for better wear resistance. The belt tension, de-mooning bar height, wash water quantity and discharge roll speed are carefully tuned to maintain a good path for the cake formation to prevent excessive wear of the filter cloth.
Select filter cloth to obtain good wear and solid binding characteristics. Use moderate blowback pressure to avoid high wear. Adjust duration of blow back pressure short enough to remove the cake from the filter cloth. The tuning of valve body is important for the blow back to prevent the excess filtrated being force back out of the pipe to with the release cake solid as this minimises wear and filter media maintenance.
Select filter cloth to obtain solid binding resistance and good cake release. Use coated fabric for more effective cake release and have a longer-lasting cloth media due to solid binding filter cloth. Both the discharge roll speed and drum speed must be the same. Adjust the scraper knife to leave a significant heal on discharge roll to produce a continuous cake transfer.
Minimise the lateral pressure of the strings by adjusting the alignment tine bar to avoid the string being cut off. Have ceramic tube place over each aligning tine bar to act as bearing surface for the strings. [ 4 ]
Select filter cloth based on the type filter aid used (refer Filter aid selection), adjust the advancing knife to optimize the knife advance rate per drum revolution. (Detail explained in Advance blade section)
Filter aid selection: filter aid are recoat cake that act as the actual filter media and there two different types which are diatomaceous earth or perlite.
Important parameter to consider is the solid penetration into the pre coat cake and its limits 0.002 to 0.005 inch penetration thickness.
Large amount of filter aid is used i.e. “open”, more filter aid is aid removed which lead to higher disposal cost. If little amount of filter aid is used i.e. “tight” will lead to no flow rate into the drum. This comparison can be illustrated in figure 5 as below. [ 4 ]
The approximate knife advance rate can be determined for a set of operating conditions using table 6 below. The table indicates the number of hours that the filter can operate in a one-inch pre coat cake; the required condition is that the advance blade must be at a constant position. This method can be used to check for optimum operation range.
If the operating parameter is higher than the optimum range, the user can reduce the knife advance rate and use a tighter grade of filter aid. This will result in less filter aid used (lower capital cost) and less filter aid being removed (lower disposal cost). However, if the operating parameter is lower than the optimum range, the user can increase the knife advanced rate (more production) and decrease the drum speed for less filter air usage (reduced operating cost). [ 4 ]
Most commonly used post treatment, where chlorine is dissolved in water to form and hydrochloric acid hypochlorous acid. The latter act as a disinfectant that is able to eliminate pathogens such as bacteria, viruses and protozoa by penetrating the cell walls. [ 9 ]
The waste stream is irradiated with Ultraviolet radiation. The UV radiation disinfect by disrupting the pathogen cell to be mutated and prevent the cell from replicating. Eventually the mutated cell becomes extinct and this process eliminates odour. [ 9 ]
The stream is exposed to ozone and ozone is unstable at atmospheric condition. The ozone (O3) decomposes into oxygen (O2) and more oxygen is dissolved into the stream. The pathogen is oxidised to form carbon dioxide. This process eliminates the odour of the stream but result in slightly acidic product due to the effect of carbon dioxide present. [ 9 ]
The waste discharge can be used as land stabilizer as dry bio-solids that can be distributed to the market. The land stabilizer is used in reclaiming marginal land such as mining waste land. This process will help to restore the land to its initial appearance. [ 9 ]
The waste discharge can be sent into incineration plant, where the organic solid undergoes combustion process. The combustion process produces heat that can be used to generate electricity. [ 9 ]
The rotary vacuum drum filter designs available vary in physical aspects and their characteristics. The filtration area ranges from 0.5 m 2 to 125 m 2 . Disregarding the size of the design, filter cloth washing is a priority as it ensures efficiency of cake washing and acting vacuum. However, a smaller design would be more economical as the maintenance, energy usage and investment cost would be less than a bigger rotary vacuum drum filter.
Over the years, the technology drive has pushed development to further heights revolving around rotary vacuum drum filter in terms of design, performance, maintenance and cost. This has also led to the development of smaller rotary drum vacuum filters, ranging from laboratory scale to pilot scale, both of which can be used for smaller applications (such as at a lab in a university) [ 10 ] High performance capacity, optimised filtrate drainage with low flow resistance and minimal pressure loss are just a few of the benefits.
With advanced control systems prompting automation, this has reduced the operation of attention needed hence, reducing the operational cost. Advancements in technology also means that precoat can be cut to 1/20th the thickness of human hair, thus making the use of precoat more efficient [ 11 ] Lowered operational and capital cost can also be achieved nowadays due to easier maintenance and cleaning. Complete cell emptying can be done quickly with the installation of leading and trailing pipes.
Given that the filter cloth is usually one of the more expensive component in the rotary vacuum drum filter build up, priority on its maintenance must be kept quite high. A longer lifetime, protection from damage and consistent performance are the few criteria that must not be overlooked.
Besides considering production cost and quality, cake washing and cake thickness are essential issues that are important in the process. Methods have been performed to ensure a minimal amount of cake moisture while undergoing good cake washing with large cake dewatering angle. An even thickness of filter cake besides having a complete cake discharge is also possible. [ 5 ] [ 12 ]
Drum Filter Made in Viet Nam | https://en.wikipedia.org/wiki/Rotary_vacuum-drum_filter |
Rotary wheel blow molding systems are used for the high-output production of a wide variety of plastic extrusion blow molded articles. Containers may be produced from small, single serve bottles to large containers up to 20-30 liters in volume - but wheel machines are often sized for the volume and dimensional demands of a specific container, and are typically dedicated to a narrow range of bottle sizes once built. Multiple parison machines, with high numbers of molds are capable of producing over one million bottles per day in some configurations.
Rotary blow molding "wheels" are targeted to the high output production of containers. They are used to produce containers from one to seven layers. View stripe and In Mold Labeling (IML) options are available in some configurations. Rotary wheels, which may contain from six to thirty molds, feature continuously extruded parisons. Revolving sets of blow molds capture the parison or parisons as they pass over the extrusion head. The revolving sets of molds are located on clamp "stations".
Rotary wheels come in different variations, including both continuous motion and indexing wheels, and vertical or horizontal variations. [ 1 ] [ 2 ] Wheel machines are favored for their processing ease, due to having only single (or in some cases, two) parisons, and mechanical repeatability.
In some machinery configurations, the molds take on the shape of a "pie" sector. Thus, if two or more parisons are used, each blow molded "log" has a unique length, requiring special downstream handling and trimming requirements. In other machine configurations, the molds utilize "book style" opening mechanisms, allowing multiple parisons of equal length. However, machines of this style typically have lower clamp force, limiting the available applications.
The mold close and open actuation is typically carried out through a toggle mechanism linkage that is activated during the rotational process by stationary cams. This mechanical repeatability is considered an advantage by most processors.
The method of wheel rotation is typically conducted through an electric motor with a "pinion" gear or small gear to or in mesh with a rotating "bull" gear or large gear. All utilities for blowing containers and for mold cooling are carried through the main shaft or the axle from which the wheel rotates about. These utilities include compressed air and water. Sequencing functions necessary to inflate the parison, hold the container prior to discharge and discharge are completed by mechanical actuation to pneumatic valves – resulting in a high degree of repeatability.
Very tight weight and dimensional tolerances can be obtained on wheel equipment, as the parison is captured on both ends. It is pinched in the preceding mold on the leading end, and positioned by the stationary flowhead die on the other end. In shuttle machinery and reciprocating screw machinery multiple parisons are extruded and are free hanging. Because there is always some variation in the parison length on these machines, bottle weight and tolerance consistencies are not as tight as on rotary wheel machinery.
Other advantages of wheel equipment include:
Disadvantages:
The growth of wheel machinery in the United States was spurred by the conversion of motor oil containers from paperboard cans to plastic bottles , and the conversion of laundry detergent from powder to liquid form. Additional high volume applications have included single-serve juices and drinkable yogurt , condiments , and household cleaning supplies. | https://en.wikipedia.org/wiki/Rotary_wheel_blow_molding_systems |
Rotating-polarization coherent anti-Stokes Raman spectroscopy , ( RP-CARS ) is a particular implementation of the coherent anti-Stokes Raman spectroscopy (CARS). RP-CARS takes advantage of polarization-dependent selection rules in order to gain information about molecule orientation anisotropy and direction within the optical point spread function .
Coherent anti-Stokes Raman spectroscopy (CARS) is a non-
linear process in which the energy difference of a pair of incoming photons
matches the energy of the vibrational mode of a molecular bond of interest.
This phonon population is coherently probed by a third photon and anti-
Stokes radiation is emitted. [ 1 ]
In presence of molecular orientation anisotropy in the sample, CARS images often display artefacts due to polarization-dependent selection rules that affects the measured intensity with respect of the alignment between the polarization plane of the incident light and the main orientation plane of the molecular bonds. [ 2 ] This is due because the four-wave mixing process is more efficient when the polarization plane of the incident light is aligned with the main orientation plane of the molecular vibrations.
RP-CARS takes advantage of the polarization-dependent selection rules to detect the local microscopic orientation of the chemical bonds under investigation. By means of RP-CARS it is possible to visualize the
degree of orientation anisotropy of selected molecular bonds and to detect their average orientation direction. [ 3 ] It is possible by continuously rotating the orientation of the polarization plane of the incident light with a rotating waveplate and then, sequentially, for each image pixel, analysing the orientation dependence of the CARS signal intensity. This allows measuring for each pixel the average-orientation plane of the molecular bonds of interest and the degree of this spatial anisotropy in the point-spread-function volume. [ 4 ]
Possible biomedical-oriented applications of this technique are related to the study of the myelin and myelopathies . Myelin is a highly ordered structure, in which many lipid-
enriched, densely compacted phospholipid bilayers are spirally rolled up around the
cylindrical axons. The linear acyl chains of the phospholipid molecules present a
perpendicular orientation with respect to the myelin surface. Therefore, in a myelinated nerve
fiber, a large number of molecular bonds are ordered around a radial axis of symmetry. Such a
strong molecular anisotropy and azimuthal symmetry make RP-CARS a suitable tool to investigate
neural white matter. [ 4 ] | https://en.wikipedia.org/wiki/Rotating-polarization_coherent_anti-Stokes_Raman_spectroscopy |
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance . In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low. [ 1 ] Explicitly, terms in the Hamiltonians that oscillate with frequencies ω L + ω 0 {\displaystyle \omega _{L}+\omega _{0}} are neglected, while terms that oscillate with frequencies ω L − ω 0 {\displaystyle \omega _{L}-\omega _{0}} are kept, where ω L {\displaystyle \omega _{L}} is the light frequency, and ω 0 {\displaystyle \omega _{0}} is a transition frequency.
The name of the approximation stems from the form of the Hamiltonian in the interaction picture , as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system ket , leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.
The rotating-wave approximation is closely related to, but different from, the secular approximation . [ 2 ]
For simplicity consider a two-level atomic system with ground and excited states | g ⟩ {\displaystyle |{\text{g}}\rangle } and | e ⟩ {\displaystyle |{\text{e}}\rangle } , respectively (using the Dirac bracket notation ). Let the energy difference between the states be ℏ ω 0 {\displaystyle \hbar \omega _{0}} so that ω 0 {\displaystyle \omega _{0}} is the transition frequency of the system. Then the unperturbed Hamiltonian of the atom can be written as
Suppose the atom experiences an external classical electric field of frequency ω L {\displaystyle \omega _{L}} , given by E → ( t ) = E → 0 e − i ω L t + E → 0 ∗ e i ω L t {\displaystyle {\vec {E}}(t)={\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}} ; e.g., a plane wave propagating in space. Then under the dipole approximation the interaction Hamiltonian between the atom and the electric field can be expressed as
where d → {\displaystyle {\vec {d}}} is the dipole moment operator of the atom. The total Hamiltonian for the atom-light system is therefore H = H 0 + H 1 . {\displaystyle H=H_{0}+H_{1}.} The atom does not have a dipole moment when it is in an energy eigenstate , so ⟨ e | d → | e ⟩ = ⟨ g | d → | g ⟩ = 0. {\displaystyle \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{e}}\right\rangle =\left\langle {\text{g}}\left|{\vec {d}}\right|{\text{g}}\right\rangle =0.} This means that defining d → eg := ⟨ e | d → | g ⟩ {\displaystyle {\vec {d}}_{\text{eg}}\mathrel {:=} \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{g}}\right\rangle } allows the dipole operator to be written as
(with ∗ {\displaystyle ^{*}} denoting the complex conjugate ). The interaction Hamiltonian can then be shown to be
where Ω = ℏ − 1 d → eg ⋅ E → 0 {\displaystyle \Omega =\hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}} is the Rabi frequency and Ω ~ := ℏ − 1 d → eg ⋅ E → 0 ∗ {\displaystyle {\tilde {\Omega }}\mathrel {:=} \hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}} is the counter-rotating frequency. To see why the Ω ~ {\displaystyle {\tilde {\Omega }}} terms are called counter-rotating consider a unitary transformation to the interaction or Dirac picture where the transformed Hamiltonian H 1 , I {\displaystyle H_{1,I}} is given by
where Δ ω := ω L − ω 0 {\displaystyle \Delta \omega \mathrel {:=} \omega _{L}-\omega _{0}} is the detuning between the light field and the atom.
This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near resonance with the atomic transition. This means that Δ ω ≪ ω L + ω 0 {\displaystyle \Delta \omega \ll \omega _{L}+\omega _{0}} and the complex exponentials multiplying Ω ~ {\displaystyle {\tilde {\Omega }}} and Ω ~ ∗ {\displaystyle {\tilde {\Omega }}^{*}} can be considered to be rapidly oscillating. Hence on any appreciable time scale, the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms may be neglected and thus the Hamiltonian can be written in the interaction picture as
Finally, transforming back into the Schrödinger picture , the Hamiltonian is given by
Another criterion for rotating wave approximation is the weak coupling condition, that is, the Rabi frequency should be much less than the transition frequency. [ 1 ]
At this point the rotating wave approximation is complete. A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.
Given the above definitions the interaction Hamiltonian is
as stated. The next step is to find the Hamiltonian in the interaction picture , H 1 , I {\displaystyle H_{1,I}} . The required unitary transformation is
where the 3rd step can be proved by using a Taylor series expansion, and using the orthogonality of the states | g ⟩ {\displaystyle |{\text{g}}\rangle } and | e ⟩ {\displaystyle |{\text{e}}\rangle } . Note that a multiplication by an overall phase of e i ω 0 t / 2 {\displaystyle e^{i\omega _{0}t/2}} on a unitary operator does not affect the underlying physics, so in the further usages of U {\displaystyle U} we will neglect it. Applying U {\displaystyle U} gives:
Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section:
Finally, we transform the approximate Hamiltonian H 1 , I RWA {\displaystyle H_{1,I}^{\text{RWA}}} back to the Schrödinger picture:
The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is | https://en.wikipedia.org/wiki/Rotating-wave_approximation |
A rotating biological contactor or RBC is a biological fixed-film treatment process used in the secondary treatment of wastewater following primary treatment . [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] The primary treatment process involves removal of grit , sand and coarse suspended material through a screening process, followed by settling of suspended solids. The RBC process allows the wastewater to come in contact with a biological film in order to remove pollutants in the wastewater before discharge of the treated wastewater to the environment , usually a body of water (river, lake or ocean). A rotating biological contactor is a type of secondary (biological) treatment process. It consists of a series of closely spaced, parallel discs mounted on a rotating shaft which is supported just above the surface of the wastewater. Microorganisms grow on the surface of the discs where biological degradation of the wastewater pollutants takes place.
Rotating biological contactors (RBCs) are capable of withstanding surges in organic load. To be successful, micro-organisms need both oxygen to live and food to grow. Oxygen is obtained from the atmosphere as the disks rotate. As the micro-organisms grow, they build up on the media until they are sloughed off due to shear forces provided by the rotating discs in the sewage. Effluent from the RBC is then passed through a clarifier where the sloughed biological solids in suspension settle as a sludge. [ 6 ]
The rotating packs of disks (known as the media) are contained in a tank or trough and rotate at between 2 and 5 revolutions per minute. Commonly used plastics for the media are polyethylene , PVC and expanded polystyrene . The shaft is aligned with the flow of wastewater so that the discs rotate at right angles to the flow, with several packs usually combined to make up a treatment train. About 40% of the disc area is immersed in the wastewater. [ 8 ] : Ch 2
Biological growth is attached to the surface of the disc and forms a slime layer. The discs contact the wastewater with the atmospheric air for oxidation as it rotates. The rotation helps to slough off excess solids. The disc system can be staged in series to obtain nearly any detention time or degree of removal required. Since the systems are staged, the culture of the later stages can be acclimated to the slowly degraded materials. [ 8 ] : Ch 2
The discs consist of plastic sheets ranging from 2 to 4 m in diameter and are up to 10 mm thick. Several modules may be arranged in parallel and/or in series to meet the flow and treatment requirements. The discs are submerged in waste water to about 40% of their diameter. Approximately 95% of the surface area is thus alternately submerged in waste water and then exposed to the atmosphere above the liquid. Carbonaceous substrate is removed in the initial stage of RBC. Carbon conversion may be completed in the first stage of a series of modules, with nitrification being completed after the 5th stage. Most design of RBC systems will include a minimum of 4 or 5 modules in series to obtain nitrification of waste water. As the biofilm biomass changes from Carbon metabolizing to nitrifying, a visual colour change from grey/beige to brown can be seen which is illustrated by the adjacent photo.
Biofilms , which are biological growths that become attached to the discs, assimilate the organic materials (measured as BOD5) in the wastewater. Aeration is provided by the rotating action, which exposes the media to the air after contacting them with the wastewater, facilitating the degradation of the pollutants being removed. The degree of wastewater treatment is related to the amount of media surface area and the quality and volume of the inflowing wastewater.
RBC's regularly achieve the following effluent parameters for treated waste water: BOD5: 20 mg/L, Suspended Solids: 30 mg/L and Ammonia N: 20 mg/L. They consume very low power and make little noise due to the slow rotation of the rotor (2-5 RPM). They are generally considered very robust and low maintenance systems. Better discharge effluent parameters can be achieved by adding a tertiary polishing filter after the RBC to lower BOD5, SS and Ammonia Nitrogen. An additional UV or Chlorination step can achieve effluent parameters that make the water suitable for irrigation or toilet flushing.
Secondary clarifiers following RBCs are identical in design to conventional humus tanks, as used downstream of trickling filters . Sludge is generally removed daily, or pumped automatically to the primary settlement tank for co-settlement. Regular sludge removal reduces the risk of anaerobic conditions from developing within the sludge, with subsequent sludge flotation due to the release of gases. [ citation needed ]
The first RBC was installed in West Germany in 1959, later it was introduced in the United States and Canada. [ 8 ] : Ch 2:History In the United States, rotating biological contactors are used for industries producing wastewaters high in biochemical oxygen demand (BOD) (e.g., petroleum industry and dairy industry). In the UK, the first GRP RBC's - manufactured by KEE Process Ltd. originally known as KLARGESTER - go back to 1955.
A properly designed RBC produced a very high quality final effluent. However both the organic and hydraulic loading had to be addressed in the design phase.
In the 1980s problems were encountered in the USA prompting the Environmental Agency to commission a number of reports.
These reports identified a number of issues and criticized the RBC process. One author suggested that since manufacturers were aware of the problem, the problems would be resolved and suggested that design engineers should specify a long life. [ citation needed ]
Severn Trent Water Ltd, a large UK Water Company based in the Midlands, employed RBCs as the preferred process for their small works which amount to over 700 sites Consequently, long life was essential to compliance.
This issue was successfully addressed by Eric Findlay C Eng when he was employed by Severn Trent Water Ltd in the UK following a period of failure of a number of plants. As a result, the issue of short life failure became fully understood in the early 1990s when the correct process and hydraulic issues had been identified to produce a high quality nitrified effluent.
There are several other papers which address the whole issue of RBCs. Findlay also developed a system for repairing defective RBCs enabling shaft and frame life to be extended up to 30 years based on the Cranfield designed frame. Where additional capacity was required intermediate frames are used. [ 9 ] [ 10 ] | https://en.wikipedia.org/wiki/Rotating_biological_contactor |
A rotating detonation engine ( RDE ) uses a form of pressure gain combustion , where one or more detonations continuously travel around an annular channel. [ 1 ] Computational simulations and experimental results have shown that the RDE has potential in transport and other applications. [ 2 ] [ 3 ]
In detonative combustion, the flame front expands at supersonic speed . It is theoretically up to 25% more efficient than conventional deflagrative combustion , [ 4 ] offering potentially major fuel savings. [ 5 ] [ 6 ]
Disadvantages include instability and noise. [ citation needed ]
The basic concept of an RDE is a detonation wave that travels around a circular channel (annulus). Fuel and oxidizer are injected into the channel, normally through small holes or slits. A detonation is initiated in the fuel/oxidizer mixture by some form of igniter. After the engine is started, the detonation is self-sustaining. One detonation ignites the fuel/oxidizer mixture, which releases the energy necessary to sustain the detonation. The combustion products expand out of the channel and are pushed out of the channel by the incoming fuel and oxidizer. [ 3 ]
Although the RDE's design is similar to the pulse detonation engine (PDE), the RDE can function continuously because the waves cycle around the chamber, while the PDE requires the chambers to be purged after each pulse. [ 7 ]
Several organizations work on RDEs.
On May 14, 2025 Venus Aerospace successfully completed the first US flight test of a RDRE at Spaceport America in New Mexico. [ 8 ]
In 2023 GE Aerospace demonstrated a subscale laboratory turbine based combined cycle (TBCC) system that combined a Mach 2.5-class turbofan paired with a rotating detonation-dual-mode ramjet (RD-DMRJ). The test came 18 months after program launch. The company reported rotating detonations of a compressed fuel-air mixture in the presence of the supersonic airflow necessary for speeds above Mach 5. [ 9 ]
DARPA is working with RTX on Gambit, researching the application of rotating detonation engines for supersonic air-launched standoff missiles. [ 10 ] [ 11 ] DARPA is also working with Venus Aerospace which successfully tested its RDRE engine in March 2024. [ 12 ]
The US Navy has been pushing development. [ 13 ] Researchers at the Naval Research Laboratory (NRL) have a particular interest in the capability of detonation engines such as the RDE to reduce the fuel consumption of their ships. [ 14 ] [ 13 ] Several obstacles must still be overcome in order to use the RDE in the field. As of 2012, NRL researchers were focusing on better understanding how the RDE works. [ 15 ]
Since 2010, Aerojet Rocketdyne has conducted over 520 tests of multiple configurations. [ 16 ]
Daniel Paxson [ 17 ] at the Glenn Research Center used simulations in computational fluid dynamics (CFD) to assess the RDE's detonation frame of reference and compare performance with the PDE. [ 18 ] He found that an RDE can perform at least on the same level as a PDE. Furthermore, he found that RDE performance can be directly compared to the PDE as their performance was essentially the same.
On January 25, 2023, NASA reported successfully testing its first full-scale rotating detonation rocket engine (RDRE). This engine produced 4,000 lbf (18 kN) of thrust. NASA has stated their intention to create a 10,000-pound-force (44 kN) thrust unit as the next research step. [ 19 ] On December 20, 2023, a full-scale Rotating Detonation Rocket Engine combustor was reportedly fired for 251 seconds, achieving more than 5,800-pound-force (26 kN) of thrust. Test stand video captured at NASA's Marshall Space Flight Center in Huntsville, Alabama, US, demonstrated ignition. [ 20 ]
According to Russian Vice Prime Minister Dmitry Rogozin , in mid-January 2018 NPO Energomash company completed the initial test phase of a 2-ton class liquid propellant RDE and plans to develop larger models for use in space launch vehicles. [ 21 ]
In May 2016, a team of researchers affiliated with the US Air Force developed a rotating detonation rocket engine operating with liquid oxygen and natural gas as propellants. [ 22 ] Additional RDE testing was conducted at Purdue University , including a test article called "Detonation Rig for Optical, Non-intrusive Experimental measurements (DRONE)", an "unwrapped" semi-bounded, linear detonation channel experiment. [ 23 ] IN Space LLC, in a contract with the US Air Force , tested a
22,000 N (4,900 lbf) thrust rotating detonation rocket engine (RDRE) while testing with liquid oxygen and gaseous methane at Purdue University in 2021. [ 24 ]
In May 2020, a team of engineering researchers affiliated with the US Air Force claimed to have developed a highly experimental working model rotating detonation engine capable of producing 200 lbf (890 N) of thrust operating on a hydrogen/oxygen fuel mix. [ 25 ]
In 2021 the group demonstrated an oblique detonation wave engine with a ramp angle of 30 degrees. [ 26 ] [ 27 ]
On July 26, 2021 (UTC), Japan Aerospace Exploration Agency (JAXA) succeeded in testing the RDE in space for the first time in the world by launching the S-520-31 sounding rocket equipped with a 500 N class RDE in the second stage. [ 28 ] The engine used gaseous methane and oxygen as propellants, generating an average thrust of 518 N and delivering 290 seconds of specific impulse . Rotating combustion also created a torque of 0.26 N·m, so a S-shaped pulse detonation engine was used to reduce the spin of the stage. [ 29 ] [ 30 ]
S-520-34 launched on November 14, 2024, experimented successfully with a liquid ethanol / N 2 O propellant. [ 31 ]
On September 15, 2021, the Warsaw Institute of Aviation performed the first successful flight test of an experimental rocket powered by a rotating detonation rocket engine, powered by liquid propellants . The test took place on September 15, 2021, at the testing ground of the Military Institute of Armament Technology in Zielonka near Warsaw in Poland. The rocket engine, according to the plan, worked for 3.2 s, accelerating the rocket to a speed of about 90 m/s, which allowed the rocket to reach an altitude of 450 m. [ 32 ]
In 2023 researchers announced a demonstration unit of a hybrid air-breathing engine. It combines a continuous RDE for propulsion at below Mach 7 with an oblique detonation engine for use at speeds up to Mach 16. The oblique detonation waves are stationary and stabilized. BPMI is China's leading ramjet manufacturer. [ 33 ]
Earlier in 2023, China achieved the world's first RDE drone flight. The drone successfully flew at an undisclosed airfield in Gansu province . The FB-1 Rotating Detonation Engine was developed jointly by Chongqing University Industrial Technology Research Institute and private company Thrust-to-Weight Ratio Engine (TWR). [ 33 ]
On March 4, 2025, Pratt & Whitney reported they successfully completed tests on their RDE. It was tested in extreme conditions they are looking to operate in, with their eventual goal: to propel “Vehicles critical to future defence applications”. They believe the engine with no moving parts can increase efficiency and cost due to the lower complexity, allowing for more mass to be budgeted in other subsystems, like fuel and payloads. [ 34 ]
Other experiments have used numerical procedures to better understand the flow-field of the RDE. [ 35 ] In 2020, a study from the University of Washington explored an experimental device that allowed control of parameters such as the width of the annulus. Using a high-speed camera, researchers were able to view it operating in extreme slow motion. Based on that they developed a mathematical model to describe the process. [ 36 ]
In 2021, the Institute of Mechanics, Chinese Academy of Sciences , successfully tested the world's first hypersonic detonation wave engine powered by kerosene , which could propel a plane at Mach 9. [ 37 ] | https://en.wikipedia.org/wiki/Rotating_detonation_engine |
Rotating radio transients ( RRATs ) are sources of short, moderately bright, radio pulses, which were first discovered in 2006. [ 1 ] RRATs are thought to be pulsars , i.e. rotating magnetised neutron stars which emit more sporadically and/or with higher pulse-to-pulse variability than the bulk of the known pulsars. The working definition of what a RRAT is, is a pulsar which is more easily discoverable in a search for bright single pulses, as opposed to in Fourier domain searches so that 'RRAT' is little more than a label (of how they are discovered) and does not represent a distinct class of objects from pulsars. As of March 2015 [update] over 100 have been reported. [ 2 ]
Pulses from RRATs are short in duration, lasting from a few milliseconds . The pulses are comparable to the brightest single pulses observed from pulsars with flux densities of a few Jansky at 1.4 GHz . [ 1 ] Andrew Lyne , a radio astronomer involved in the discovery of RRATs, "guesses that there are only a few dozen brighter radio sources in the sky." [ 3 ] The time intervals between detected bursts range from seconds (one pulse period) to hours. Thus radio emission from RRATs is typically only detectable for less than one second per day. [ 1 ]
The sporadic emission from RRATs means that they are usually not detectable in standard periodicity searches which use Fourier techniques. Nevertheless, underlying periodicity in RRATs can be determined by finding the greatest common denominator of the intervals between pulses. This yields the maximum period but once many pulse arrival times have been determined the periods which are shorter (by an integer factor) can be deemed statistically unlikely. The periods thus determined for RRATs are on the order of 1 second or longer, implying that the pulses are likely to be coming from rotating neutron stars, and led to the name "Rotating Radio Transient" being given. The periods seen in some RRATs are longer than in most radio pulsars , somewhat expected for sources which are (by definition) discovered in searches for individual pulses. Monitoring of RRATs for the past few years has revealed that they are slowing down. For some of the known RRATs this slow-down rate, while small, is larger than that for typical pulsars, and which is again more in line with that of magnetars . [ 4 ]
The neutron star nature of RRATs was further confirmed when X-ray observations of the RRAT J1819-1458 were made using the space-based Chandra X-ray Observatory . [ 5 ] Cooling neutron stars have temperatures of order 1 million kelvins and so thermally emit at X-ray wavelengths. Measurement of an x-ray spectrum allows the temperature to be determined, assuming it is thermal emission from the surface of a neutron star. The resulting temperature for RRAT J1819-1458 is much cooler than that found on the surface of magnetars, and suggests that despite some shared properties between RRATs and magnetars, they belong to different populations of neutron stars. None of the other pulsars identified as RRATs has yet been detected in X-ray observation. This is in fact the only detection of these sources outside of the radio band.
After the discovery of pulsars in 1967, searches for more pulsars relied on two key characteristics of pulsar pulses in order to distinguish pulsars from noise caused by terrestrial radio signals. The first is the periodic nature of pulsars. By performing periodicity searches through data, "pulsars are detected with much higher signal-to-noise ratios" than when simply looking for individual pulses. [ 6 ] The second defining characteristic of pulsar signals is the dispersion in frequency of an individual pulse, due to the frequency dependence of the phase velocity of an electromagnetic wave that travels through an ionized medium. As the interstellar medium features an ionized component, waves traveling from a pulsar to Earth are dispersed, and thus pulsar surveys also focused on searching for dispersed waves. The importance of the combination of the two characteristics is such that in initial data processing from the Parkes Multibeam Pulsar Survey, which is the largest pulsar survey to date, "no search sensitive to single dispersed pulses was included." [ 6 ]
After the survey itself had finished, searches began for single dispersed pulses. About a quarter of the pulsars already detected by the survey were found by searching for single dispersed pulses, but there were 17 sources of single dispersed pulses which were not thought to be associated with a pulsar. [ 6 ] During follow-up observations, a few of these were found to be pulsars that had been missed in periodicity searches, but 11 sources were characterized by single dispersed pulses, with irregular intervals between pulses lasting from minutes to hours. [ 1 ]
As of March 2015 [update] over 100 have been reported, with dispersion measures up to 764 cm −3 pc. [ 2 ]
In order to explain the irregularity of RRAT pulses, we note that most of the pulsars which have been labelled as RRATs are entirely consistent with pulsars which have regular underlying emission which is simply undetectable due to the low intrinsic brightness or large distance of the sources. However, assuming that when we do not detect pulses from these pulsars that they are truly 'off', several authors have proposed mechanisms whereby such sporadic emission could be explained. For example, as pulsars gradually lose energy, they approach what is called the pulsar "death valley," a theoretical area in pulsar pulsar period—period derivative space, where the pulsar emission mechanism is thought to fail but may become sporadic as pulsars approach this region. However although this is consistent with some of the behavior of RRATs, [ 7 ] the RRATs with known periods and period derivatives do not lie near canonical death regions. [ 6 ] Another suggestion is that asteroids might form in the debris of the supernova that formed the neutron star, and infall of these debris in to the light cone of RRATs and some other types of pulsars might cause some of the irregular behavior observed. [ 8 ] Since most RRATs have large dispersion measures that indicate larger distances, combining with the similar emission properties, some RRATs could be due to the telescope detection threshold. Nevertheless, the possibility that RRATs share the similar emission mechanism with those pulsars with so called "giant pulses" can neither be excluded. [ 9 ] To fully understand the emission mechanisms of RRATs would require directly observing the debris surrounding a neutron star, which is not possible now, but may be possible in the future with the Square Kilometer Array . Nevertheless, as more RRATs are detected by observatories such as Arecibo , the Green Bank Telescope , and the Parkes Observatory at which RRATs were first discovered, some of the characteristics of RRATs may become clearer. | https://en.wikipedia.org/wiki/Rotating_radio_transient |
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame . An everyday example of a rotating reference frame is the surface of the Earth . (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles .)
All non-inertial reference frames exhibit fictitious forces ; rotating reference frames are characterized by three: [ 1 ]
and, for non-uniformly rotating reference frames,
Scientists in a rotating box can measure the rotation speed and axis of rotation by measuring these fictitious forces. For example, Léon Foucault was able to show the Coriolis force that results from Earth's rotation using the Foucault pendulum . If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning carousel .
In classical mechanics , centrifugal force is an outward force associated with rotation . Centrifugal force is one of several so-called pseudo-forces (also known as inertial forces ), so named because, unlike real forces , they do not originate in interactions with other bodies situated in the environment of the particle upon which they act. Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ]
The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis in connection with hydrodynamics , and also in the tidal equations of Pierre-Simon Laplace in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology .
Perhaps the most commonly encountered rotating reference frame is the Earth . Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the northern hemisphere , and to the left in the southern . Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator , and to the left of this direction south of the equator. This effect is responsible for the rotation of large cyclones (see Coriolis effects in meteorology ).
In classical mechanics , the Euler acceleration (named for Leonhard Euler ), also known as azimuthal acceleration [ 8 ] or transverse acceleration [ 9 ] is an acceleration that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame 's axis. This article is restricted to a frame of reference that rotates about a fixed axis.
The Euler force is a fictitious force on a body that is related to the Euler acceleration by F = m a , where a is the Euler acceleration and m is the mass of the body. [ 10 ] [ 11 ]
The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.
To derive these fictitious forces, it's helpful to be able to convert between the coordinates ( x ′ , y ′ , z ′ ) {\displaystyle \left(x',y',z'\right)} of the rotating reference frame and the coordinates ( x , y , z ) {\displaystyle (x,y,z)} of an inertial reference frame with the same origin. [ note 1 ] If the rotation is about the z {\displaystyle z} axis with a constant angular velocity Ω {\displaystyle \Omega } (so z ′ = z {\displaystyle z'=z} and d θ d t ≡ Ω , {\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\equiv \Omega ,} which implies θ ( t ) = Ω t + θ 0 {\displaystyle \theta (t)=\Omega t+\theta _{0}} for some constant θ 0 {\displaystyle \theta _{0}} where θ ( t ) {\displaystyle \theta (t)} denotes the angle in the x − y {\displaystyle x-y} -plane formed at time t {\displaystyle t} by ( x ′ , y ′ ) {\displaystyle \left(x',y'\right)} and the x {\displaystyle x} -axis),
and if the two reference frames coincide at time t = 0 {\displaystyle t=0} (meaning ( x ′ , y ′ , z ′ ) = ( x , y , z ) {\displaystyle \left(x',y',z'\right)=(x,y,z)} when t = 0 , {\displaystyle t=0,} so take θ 0 = 0 {\displaystyle \theta _{0}=0} or some other integer multiple of 2 π {\displaystyle 2\pi } ), the transformation from rotating coordinates to inertial coordinates can be written x = x ′ cos ( θ ( t ) ) − y ′ sin ( θ ( t ) ) {\displaystyle x=x'\cos(\theta (t))-y'\sin(\theta (t))} y = x ′ sin ( θ ( t ) ) + y ′ cos ( θ ( t ) ) {\displaystyle y=x'\sin(\theta (t))+y'\cos(\theta (t))} whereas the reverse transformation is x ′ = x cos ( − θ ( t ) ) − y sin ( − θ ( t ) ) {\displaystyle x'=x\cos(-\theta (t))-y\sin(-\theta (t))} y ′ = x sin ( − θ ( t ) ) + y cos ( − θ ( t ) ) . {\displaystyle y'=x\sin(-\theta (t))+y\cos(-\theta (t))\ .}
This result can be obtained from a rotation matrix .
Introduce the unit vectors ı ^ , ȷ ^ , k ^ {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}} representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose the frames are aligned at t = 0 {\displaystyle t=0} and the z {\displaystyle z} -axis is the axis of rotation. Then for a counterclockwise rotation through angle Ω t {\displaystyle \Omega t} : ı ^ ( t ) = ( cos θ ( t ) , sin θ ( t ) ) {\displaystyle {\hat {\boldsymbol {\imath }}}(t)=(\cos \theta (t),\ \sin \theta (t))} where the ( x , y ) {\displaystyle (x,y)} components are expressed in the stationary frame. Likewise, ȷ ^ ( t ) = ( − sin θ ( t ) , cos θ ( t ) ) . {\displaystyle {\hat {\boldsymbol {\jmath }}}(t)=(-\sin \theta (t),\ \cos \theta (t))\ .}
Thus the time derivative of these vectors, which rotate without changing magnitude, is d d t ı ^ ( t ) = Ω ( − sin θ ( t ) , cos θ ( t ) ) = Ω ȷ ^ ; {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}(t)=\Omega (-\sin \theta (t),\ \cos \theta (t))=\Omega {\hat {\boldsymbol {\jmath }}}\ ;} d d t ȷ ^ ( t ) = Ω ( − cos θ ( t ) , − sin θ ( t ) ) = − Ω ı ^ , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}(t)=\Omega (-\cos \theta (t),\ -\sin \theta (t))=-\Omega {\hat {\boldsymbol {\imath }}}\ ,} where Ω ≡ d d t θ ( t ) . {\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t).} This result is the same as found using a vector cross product with the rotation vector Ω {\displaystyle {\boldsymbol {\Omega }}} pointed along the z-axis of rotation Ω = ( 0 , 0 , Ω ) , {\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega ),} namely, d d t u ^ = Ω × u ^ , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega \times }}{\hat {\boldsymbol {u}}}\ ,} where u ^ {\displaystyle {\hat {\boldsymbol {u}}}} is either ı ^ {\displaystyle {\hat {\boldsymbol {\imath }}}} or ȷ ^ . {\displaystyle {\hat {\boldsymbol {\jmath }}}.}
Introduce unit vectors ı ^ , ȷ ^ , k ^ {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}} , now representing standard unit basis vectors in the general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of Ω ( t ) {\displaystyle \Omega (t)} about an axis along the rotation vector Ω ( t ) {\displaystyle {\boldsymbol {\Omega }}(t)} then each unit vector u ^ {\displaystyle {\hat {\boldsymbol {u}}}} of the rotating coordinate system (such as ı ^ , ȷ ^ , {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},} or k ^ {\displaystyle {\hat {\boldsymbol {k}}}} ) abides by the following equation: d d t u ^ = Ω × u ^ . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega }}\times {\boldsymbol {\hat {u}}}\ .} So if R ( t ) {\displaystyle R(t)} denotes the transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by Ω × = R ′ ( t ) ⋅ R ( t ) T {\displaystyle {\boldsymbol {\Omega }}\times =R'(t)\cdot R(t)^{T}} .
If f {\displaystyle {\boldsymbol {f}}} is a vector function that is written as [ note 2 ] f ( t ) = f 1 ( t ) ı ^ + f 2 ( t ) ȷ ^ + f 3 ( t ) k ^ , {\displaystyle {\boldsymbol {f}}(t)=f_{1}(t){\hat {\boldsymbol {\imath }}}+f_{2}(t){\hat {\boldsymbol {\jmath }}}+f_{3}(t){\hat {\boldsymbol {k}}}\ ,} and we want to examine its first derivative then (using the product rule of differentiation): [ 12 ] [ 13 ] d d t f = d f 1 d t ı ^ + d ı ^ d t f 1 + d f 2 d t ȷ ^ + d ȷ ^ d t f 2 + d f 3 d t k ^ + d k ^ d t f 3 = d f 1 d t ı ^ + d f 2 d t ȷ ^ + d f 3 d t k ^ + [ Ω × ( f 1 ı ^ + f 2 ȷ ^ + f 3 k ^ ) ] = ( d f d t ) r + Ω × f {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\imath }}}}{\mathrm {d} t}}f_{1}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\jmath }}}}{\mathrm {d} t}}f_{2}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+{\frac {\mathrm {d} {\hat {\boldsymbol {k}}}}{\mathrm {d} t}}f_{3}\\&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+\left[{\boldsymbol {\Omega }}\times \left(f_{1}{\hat {\boldsymbol {\imath }}}+f_{2}{\hat {\boldsymbol {\jmath }}}+f_{3}{\hat {\boldsymbol {k}}}\right)\right]\\&=\left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times {\boldsymbol {f}}\end{aligned}}} where ( d f d t ) r {\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }} denotes the rate of change of f {\displaystyle {\boldsymbol {f}}} as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as: d d t f = [ ( d d t ) r + Ω × ] f . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]{\boldsymbol {f}}\ .}
This result is also known as the transport theorem in analytical dynamics and is also sometimes referred to as the basic kinematic equation . [ 14 ]
A velocity of an object is the time-derivative of the object's position, so
The time derivative of a position r ( t ) {\displaystyle {\boldsymbol {r}}(t)} in a rotating reference frame has two components, one from the explicit time dependence due to motion of the object itself in the rotating reference frame, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement r ( t ) , {\displaystyle {\boldsymbol {r}}(t),} the velocities in the two reference frames are related by the equation
where subscript i {\displaystyle \mathrm {i} } means the inertial frame of reference, and r {\displaystyle \mathrm {r} } means the rotating frame of reference.
Acceleration is the second time derivative of position, or the first time derivative of velocity
where subscript i {\displaystyle \mathrm {i} } means the inertial frame of reference, r {\displaystyle \mathrm {r} } the rotating frame of reference, and where the expression, again, Ω × {\displaystyle {\boldsymbol {\Omega }}\times } in the bracketed expression on the left is to be interpreted as an operator working onto the bracketed expression on the right.
As Ω × Ω = 0 {\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}={\boldsymbol {0}}} , the first time derivatives of Ω {\displaystyle {\boldsymbol {\Omega }}} inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.
Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame, a r {\displaystyle \mathbf {a} _{\mathrm {r} }}
where a r = d e f ( d 2 r d t 2 ) r {\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }} is the apparent acceleration in the rotating reference frame, the term − Ω × ( Ω × r ) {\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )} represents centrifugal acceleration , and the term − 2 Ω × v r {\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }} is the Coriolis acceleration . The last term, − d Ω d t × r {\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} } , is the Euler acceleration and is zero in uniformly rotating frames.
When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in fictitious forces in the rotating reference frame, that is, apparent forces that result from being in a non-inertial reference frame , rather than from any physical interaction between bodies.
Using Newton's second law of motion F = m a , {\displaystyle \mathbf {F} =m\mathbf {a} ,} we obtain: [ 1 ] [ 12 ] [ 13 ] [ 15 ] [ 16 ]
where m {\displaystyle m} is the mass of the object being acted upon by these fictitious forces . Notice that all three forces vanish when the frame is not rotating, that is, when Ω = 0 . {\displaystyle {\boldsymbol {\Omega }}=0\ .}
For completeness, the inertial acceleration a i {\displaystyle \mathbf {a} _{\mathrm {i} }} due to impressed external forces F i m p {\displaystyle \mathbf {F} _{\mathrm {imp} }} can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as electromagnetic forces ) using Newton's second law in the inertial frame: F i m p = m a i {\displaystyle \mathbf {F} _{\mathrm {imp} }=m\mathbf {a} _{\mathrm {i} }} Newton's law in the rotating frame then becomes
In other words, to handle the laws of motion in a rotating reference frame: [ 16 ] [ 17 ] [ 18 ]
Treat the fictitious forces like real forces, and pretend you are in an inertial frame.
Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces.
This equation has exactly the form of Newton's second law, except that in addition to F , the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame provided we agree that in the noninertial frame we must add an extra force-like term, often called the inertial force .
It is convenient to consider magnetic resonance in a frame that rotates at the Larmor frequency of the spins. This is illustrated in the animation below. The rotating wave approximation may also be used. | https://en.wikipedia.org/wiki/Rotating_reference_frame |
Isaac Newton 's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres. The basis of the argument is that all observers make two observations: the tension in the string joining the bodies (which is the same for all observers) and the rate of rotation of the spheres (which is different for observers with differing rates of rotation). Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation. For all other observers a "correction" is required (a centrifugal force) that accounts for the tension calculated being different from the one expected using the observed rate of rotation. [ 1 ] It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space . Alternatively, these experiments provide an operational definition of what is meant by " absolute rotation ", and do not pretend to address the question of "rotation relative to what ?" [ 2 ] General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime . [ 3 ]
Newton was concerned to address the problem of how it is that we can experimentally determine the true motions of bodies in light of the fact that absolute space is not something that can be perceived. Such determination, he says, can be accomplished by observing the causes of motion (that is, forces ) and not simply the apparent motions of bodies relative to one another (as in the bucket argument ). As an example where causes can be observed, if two globes , floating in space , are connected by a cord, measuring the amount of tension in the cord, with no other clues to assess the situation, alone suffices to indicate how fast the two objects are revolving around the common center of mass. (This experiment involves observation of a force, the tension). Also, the sense of the rotation—whether it is in the clockwise or the counter-clockwise direction—can be discovered by applying forces to opposite faces of the globes and ascertaining whether this leads to an increase or a decrease in the tension of the cord (again involving a force). Alternatively, the sense of the rotation can be determined by measuring the apparent motion of the globes with respect to a background system of bodies that, according to the preceding methods, have been established already as not in a state of rotation, as an example from Newton's time, the fixed stars .
In the 1846 Andrew Motte translation of Newton's words: [ 4 ] [ 5 ]
We have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes kept at a given distance one from the other, by means of a cord that connects them, were revolved about their common center of gravity; we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion. ... And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared.
To summarize this proposal, here is a quote from Born: [ 6 ]
If the earth were at rest, and if, instead, the whole stellar system were to rotate in the opposite sense once around the earth in twenty-four hours, then, according to Newton, the centrifugal forces [presently attributed to the earth's rotation] would not occur.
Mach took some issue with the argument, pointing out that the rotating sphere experiment could never be done in an empty universe, where possibly Newton's laws do not apply, so the experiment really only shows what happens when the spheres rotate in our universe, and therefore, for example, may indicate only rotation relative to the entire mass of the universe. [ 2 ] [ 7 ]
For me, only relative motions exist...When a body rotates relatively to the fixed stars, centrifugal forces are produced; when it rotates relatively to some different body and not relative to the fixed stars, no centrifugal forces are produced.
An interpretation that avoids this conflict is to say that the rotating spheres experiment does not really define rotation relative to anything in particular (for example, absolute space or fixed stars); rather the experiment is an operational definition of what is meant by the motion called absolute rotation . [ 2 ]
This sphere example was used by Newton himself to discuss the detection of rotation relative to absolute space. [ 8 ] Checking the fictitious force needed to account for the tension in the string is one way for an observer to decide whether or not they are rotating – if the fictitious force is zero, they are not rotating. [ 9 ] (Of course, in an extreme case like the gravitron amusement ride, you do not need much convincing that you are rotating, but standing on the Earth's surface, the matter is more subtle.) Below, the mathematical details behind this observation are presented.
Figure 1 shows two identical spheres rotating about the center of the string joining them. The axis of rotation is shown as a vector Ω with direction given by the right-hand rule and magnitude equal to the rate of rotation: |Ω| = ω. The angular rate of rotation ω is assumed independent of time ( uniform circular motion ). Because of the rotation, the string is under tension. (See reactive centrifugal force .) The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference.
Adopt an inertial frame centered at the midpoint of the string. The balls move in a circle about the origin of our coordinate system. Look first at one of the two balls. To travel in a circular path, which is not uniform motion with constant velocity, but circular motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a centripetal force . The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 2. These two forces are provided by the string, putting the string under tension, also shown in Figure 2.
Adopt a rotating frame at the midpoint of the string. Suppose the frame rotates at the same angular rate as the balls, so the balls appear stationary in this rotating frame. Because the balls are not moving, observers say they are at rest. If they now apply Newton's law of inertia, they would say no force acts on the balls, so the string should be relaxed. However, they clearly see the string is under tension. (For example, they could split the string and put a spring in its center, which would stretch.) [ 10 ] To account for this tension, they propose that in their frame a centrifugal force acts on the two balls, pulling them apart. This force originates from nowhere – it is just a "fact of life" in this rotating world, and acts on everything they observe, not just these spheres. In resisting this ubiquitous centrifugal force, the string is placed under tension, accounting for their observation, despite the fact that the spheres are at rest. [ 11 ]
What if the spheres are not rotating in the inertial frame (string tension is zero)? Then string tension in the rotating frame also is zero. But how can that be? The spheres in the rotating frame now appear to be rotating and should require an inward force to do that. According to the analysis of uniform circular motion : [ 12 ] [ 13 ]
where u R is a unit vector pointing from the axis of rotation to one of the spheres, and Ω is a vector representing the angular rotation, with magnitude ω and direction normal to the plane of rotation given by the right-hand rule , m is the mass of the ball, and R is the distance from the axis of rotation to the spheres (the magnitude of the displacement vector, | x B | = R , locating one or the other of the spheres). According to the rotating observer, shouldn't the tension in the string be twice as big as before (the tension from the centrifugal force plus the extra tension needed to provide the centripetal force of rotation)? The reason the rotating observer sees zero tension is because of yet another fictitious force in the rotating world, the Coriolis force , which depends on the velocity of a moving object. In this zero-tension case, according to the rotating observer, the spheres now are moving, and the Coriolis force (which depends upon velocity) is activated. According to the article fictitious force , the Coriolis force is: [ 12 ]
where R is the distance to the object from the center of rotation, and v B is the velocity of the object subject to the Coriolis force, | v B | = ω R .
In the geometry of this example, this Coriolis force has twice the magnitude of the ubiquitous centrifugal force and is exactly opposite in direction. Therefore, it cancels out the ubiquitous centrifugal force found in the first example, and goes a step further to provide exactly the centripetal force demanded by uniform circular motion, so the rotating observer calculates there is no need for tension in the string − the Coriolis force looks after everything.
What happens if the spheres rotate at one angular rate, say ω I ( I = inertial), and the frame rotates at a different rate ω R ( R = rotational)? The inertial observers see circular motion and the tension in the string exerts a centripetal inward force on the spheres of:
This force also is the force due to tension seen by the rotating observers. The rotating observers see the spheres in circular motion with angular rate ω S = ω I − ω R ( S = spheres). That is, if the frame rotates more slowly than the spheres, ω S > 0 and the spheres advance counterclockwise around a circle, while for a more rapidly moving frame, ω S < 0, and the spheres appear to retreat clockwise around a circle. In either case, the rotating observers see circular motion and require a net inward centripetal force:
However, this force is not the tension in the string. So the rotational observers conclude that a force exists (which the inertial observers call a fictitious force) so that:
or,
The fictitious force changes sign depending upon which of ω I and ω S is greater. The reason for the sign change is that when ω I > ω S , the spheres actually are moving faster than the rotating observers measure, so they measure a tension in the string that actually is larger than they expect; hence, the fictitious force must increase the tension (point outward). When ω I < ω S , things are reversed so the fictitious force has to decrease the tension, and therefore has the opposite sign (points inward).
The introduction of F Fict allows the rotational observers and the inertial observers to agree on the tension in the string. However, we might ask: "Does this solution fit in with general experience with other situations, or is it simply a "cooked up" ad hoc solution?" That question is answered by seeing how this value for F Fict squares with the general result (derived in Fictitious force ): [ 14 ]
The subscript B refers to quantities referred to the non-inertial coordinate system. Full notational details are in Fictitious force . For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres:
and the velocity of this sphere as seen in the rotating frame:
where u θ is a unit vector perpendicular to u R pointing in the direction of motion.
The frame rotates at a rate ω R , so the vector of rotation is Ω = ω R u z ( u z a unit vector in the z -direction), and Ω × u R = ω R ( u z × u R ) = ω R u θ ; Ω × u θ = −ω R u R . The centrifugal force is then:
which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is
and has the ability to change sign, being outward when the spheres move faster than the frame ( ω S > 0 ) and being inward when the spheres move slower than the frame ( ω S < 0 ). [ 15 ] Combining the terms: [ 16 ]
Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ad hoc solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ω I , ω S is the greater, inasmuch as the centrifugal force contribution always is outward.
The isotropy of the cosmic background radiation is another indicator that the universe does not rotate. [ 17 ] | https://en.wikipedia.org/wiki/Rotating_spheres |
A rotating tank is a device used for fluid dynamics experiments. Typically cylinders filled with water on a rotating platform, the tanks can be used in various ways to simulate the atmosphere or ocean .
For example, a rotating tank with an ice bucket in the center can represent the Earth, with a cold pole simulated by the ice bucket. Just as in the atmosphere, eddies and a westerly jetstream form in the water.
This article about atmospheric science is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotating_tank |
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an axis of rotation . A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation . A solid figure has an infinite number of possible axes and angles of rotation , including chaotic rotation (between arbitrary orientations ), in contrast to rotation around a fixed axis .
The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or autorotation ). [ 1 ] In that case, the surface intersection of the internal spin axis can be called a pole ; for example, Earth's rotation defines the geographical poles .
A rotation around an axis completely external to the moving body is called a revolution (or orbit ), e.g. Earth's orbit around the Sun . The ends of the external axis of revolution can be called the orbital poles . [ 1 ]
Either type of rotation is involved in a corresponding type of angular velocity (spin angular velocity and orbital angular velocity) and angular momentum (spin angular momentum and orbital angular momentum).
Mathematically , a rotation is a rigid body movement which, unlike a translation , keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line).
All rigid body movements are rotations, translations, or combinations of the two.
A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion.
If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse ( inverse ) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group . However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation.
Rotations around the x , y and z axes are called principal rotations . Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.
The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time.
In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about a plane is not in general a rotation in a single plane.
2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity.
The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle θ {\displaystyle \theta } in counterclockwise direction can be quite simply represented by the following matrix :
A standard eigenvalue determination leads to the characteristic equation
which has
as its eigenvalues. Therefore, there is no real eigenvalue whenever cos θ ≠ ± 1 {\displaystyle \cos \theta \neq \pm 1} , meaning that no real vector in the plane is kept unchanged by A .
Knowing that the trace is an invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by
Using the principal arc-cosine, this formula gives a rotation angle satisfying 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} . The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis m {\displaystyle m} can always be written as a rotation having 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} if the axis is replaced with n = − m {\displaystyle n=-m} .)
Every proper rotation A {\displaystyle A} in 3D space has an axis of rotation, which is defined such that any vector v {\displaystyle v} that is aligned with the rotation axis will not be affected by rotation. Accordingly, A v = v {\displaystyle Av=v} , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle α {\displaystyle \alpha } is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial ). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In the degenerate case of a rotation angle α = 180 ∘ {\displaystyle \alpha =180^{\circ }} , the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary).
A spectral analysis is not required to find the rotation axis. If n {\displaystyle n} denotes the unit eigenvector aligned with the rotation axis, and if α {\displaystyle \alpha } denotes the rotation angle, then it can be shown that 2 sin ( α ) n = { A 32 − A 23 , A 13 − A 31 , A 21 − A 12 } {\displaystyle 2\sin(\alpha )n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}} . Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that sin ( α ) = 0 {\displaystyle \sin(\alpha )=0} . In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of A + I {\displaystyle A+I} that has a nonzero magnitude. [ 2 ]
This discussion applies to a proper rotation, and hence det A = 1 {\displaystyle \det A=1} . Any improper orthogonal 3x3 matrix B {\displaystyle B} may be written as B = − A {\displaystyle B=-A} , in which A {\displaystyle A} is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of A {\displaystyle A} is also the eigenvector of B {\displaystyle B} corresponding to an eigenvalue of −1.
As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has a plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation.
The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we do not have a proper rotation, but either the identity or the result of a sequence of reflections.
It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in the previous topic, v ¯ {\displaystyle {\bar {v}}} is also an eigenvector, and v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are such that their scalar product vanishes:
because, since v ¯ T v ¯ {\displaystyle {\bar {v}}^{\text{T}}{\bar {v}}} is real, it equals its complex conjugate v T v {\displaystyle v^{\text{T}}v} , and v ¯ T v {\displaystyle {\bar {v}}^{\text{T}}v} and v T v ¯ {\displaystyle v^{\text{T}}{\bar {v}}} are both representations of the same scalar product between v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} .
This means v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} , which is an invariant subspace under the application of A . Therefore, they span an invariant plane.
This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A , with eigenvalue 1, because of the orthogonality of the eigenvectors of A .
A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector A → {\displaystyle {\vec {A}}} which is parameterized by some variable t {\textstyle t} for which:
d | A → | 2 d t = d ( A → ⋅ A → ) d t ⇒ d | A → | d t = d A → d t ⋅ A ^ {\displaystyle {d|{\vec {A}}|^{2} \over dt}={d({\vec {A}}\cdot {\vec {A}}) \over dt}\Rightarrow {d|{\vec {A}}| \over dt}={d{\vec {A}} \over dt}\cdot {\hat {A}}}
Which also gives a relation of rate of change of unit vector by taking A → {\displaystyle {\vec {A}}} , to be such a vector: d A ^ d t ⋅ A ^ = 0 {\displaystyle {d{\hat {A}} \over dt}\cdot {\hat {A}}=0} showing that d A ^ d t {\textstyle {d{\hat {A}} \over dt}} vector is perpendicular to the vector, A → {\displaystyle {\vec {A}}} . [ 3 ]
From:
d A → d t = d ( | A → | A ^ ) d t = d | A → | d t A ^ + | A → | ( d A ^ d t ) {\displaystyle {d{\vec {A}} \over dt}={d(|{\vec {A}}|{\hat {A}}) \over dt}={d|{\vec {A}}| \over dt}{\hat {A}}+|{\vec {A}}|\left({d{\hat {A}} \over dt}\right)} ,
since the first term is parallel to A → {\displaystyle {\vec {A}}} and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against the vector itself.
As dimensions increase the number of rotation vectors increases. Along a four dimensional space (a hypervolume ), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various volumes , where each intersection is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, where a 3D object can be rotated perpendicular to the z axis. [ 4 ] [ 5 ]
The speed of rotation is given by the angular frequency (rad/s) or frequency ( turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s 2 ), caused by torque . The ratio of torque τ to the angular acceleration α is given by the moment of inertia : I = τ α . {\displaystyle I={\frac {\tau }{\alpha }}.}
The angular velocity vector (an axial vector ) also describes the direction of the axis of rotation. Similarly, the torque is an axial vector.
The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule , the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw .
It is possible for objects to have periodic circular trajectories without changing their orientation . These types of motion are treated under circular motion instead of rotation, more specifically as a curvilinear translation. Since translation involves displacement of rigid bodies while preserving the orientation of the body, in the case of curvilinear translation, all the points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation. [ 6 ]
In rotation, the orientation of the object changes and the change in orientation is independent of the observers whose frames of reference have constant relative orientation over time. By Euler's theorem , any change in orientation can be described by rotation about an axis through a chosen reference point. [ 6 ] Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through instantaneous center of circle and perpendicular to the plane of motion . In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast, a rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion. [ 7 ]
More generally, due to Chasles' theorem , any motion of rigid bodies can be treated as a composition of rotation and translation , called general plane motion. [ 6 ] A simple example of pure rotation is considered in rotation around a fixed axis .
The laws of physics are currently believed to be invariant under any fixed rotation . (Although they do appear to change when viewed from a rotating viewpoint: see rotating frame of reference .)
In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.
In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant . According to Noether's theorem , if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved .
Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z , the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.
These rotations are called precession , nutation , and intrinsic rotation .
In astronomy , rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution.
Stars , planets and similar bodies may spin around on their axes. The rotation rate of planets in the Solar System was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features. An example is sunspots , which rotate around the Sun at the same velocity as the outer gases that make up the Sun.
Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect is called tidal locking ; the Moon is tidal-locked to the Earth.
This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator . Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth is slightly deformed into an oblate spheroid ; a similar equatorial bulge develops for other planets.
Another consequence of the rotation of a planet are the phenomena of precession and nutation . Like a gyroscope , the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth 's axis to its orbital plane ( obliquity of the ecliptic ) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole Star .)
While revolution is often used as a synonym for rotation , in many fields, particularly astronomy and related fields, revolution , often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planets, planets revolve about their stars (such as the Earth around the Sun); and stars slowly revolve about their galaxial centers . The motion of the components of galaxies is complex, but it usually includes a rotation component.
Most planets in the Solar System , including Earth , spin in the same direction as they orbit the Sun . The exceptions are Venus and Uranus . Venus may be thought of as rotating slowly backward (or being "upside down"). Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. The dwarf planet Pluto (formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.
In flight dynamics , the principal rotations described with Euler angles above are known as pitch , roll and yaw . The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.
Principal rotations have the advantage of modelling a number of physical systems such as gimbals , and joysticks , so are easily visualised, and are a very compact way of storing a rotation. But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do, and suffer from a form of gimbal lock where the angles cannot be uniquely calculated for certain rotations.
Many amusement rides provide rotation. A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result, at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centripetal force . In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.
Rotation of a ball or other object, usually called spin , plays a role in many sports, including topspin and backspin in tennis , English , follow and draw in billiards and pool , curve balls in baseball , spin bowling in cricket , flying disc sports, etc. Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball.
Rotation of a player one or more times around a vertical axis may be called spin in figure skating , twirling (of the baton or the performer) in baton twirling , or 360 , 540 , 720 , etc. in snowboarding , etc. Rotation of a player or performer one or more times around a horizontal axis may be called a flip , roll , somersault , heli , etc. in gymnastics , waterskiing , or many other sports, or a one-and-a-half , two-and-a-half , gainer (starting facing away from the water), etc. in diving , etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a möbius in waterskiing freestyle jumping .
Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called a spin move and is used as a deceptive or avoidance manoeuvre, or in an attempt to play, pass, or receive a ball or puck, etc., or to afford a player a view of the goal or other players. It is often seen in hockey , basketball , football of various codes, tennis , etc. | https://en.wikipedia.org/wiki/Rotation |
The turn (symbol tr or pla ) is a unit of plane angle measurement that is the measure of a complete angle —the angle subtended by a complete circle at its center. One turn is equal to 2 π radians , 360 degrees or 400 gradians . As an angular unit , one turn also corresponds to one cycle (symbol cyc or c ) [ 1 ] or to one revolution (symbol rev or r ). [ 2 ] Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm). [ a ] The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers ), rotating objects, and the winding number of curves.
Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle , respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.
In the ISQ , an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation , defined as the ratio of a given angle and a full turn. It is represented by the symbol N . (See below for the formula.)
Because one turn is 2 π {\displaystyle 2\pi } radians, some have proposed representing 2 π {\displaystyle 2\pi } with the single letter tau ( τ {\displaystyle \tau } ).
There are several unit symbols for the turn.
The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns. [ 3 ] [ 4 ] Covered in DIN 1301-1 [ de ] (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit . However, it is a legal unit of measurement in the EU [ 5 ] [ 6 ] and Switzerland. [ 7 ]
The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+ , HP 49g+ , HP 39gs , and HP 40gs in 2017. [ 8 ] [ 9 ] An angular mode TURN was suggested for the WP 43S as well, [ 10 ] but the calculator instead implements "MUL π " ( multiples of π ) as mode and unit since 2019. [ 11 ] [ 12 ]
Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.
Using metric prefixes , the turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″ . [ 13 ] [ 14 ] A protractor divided in centiturns is normally called a " percentage protractor". While percentage protractors have existed since 1922, [ 15 ] the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. [ 13 ] [ 14 ] Some measurement devices for artillery and satellite watching carry milliturn scales. [ 16 ] [ 17 ]
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points , which implicitly have an angular separation of 1 / 32 turn. The binary degree , also known as the binary radian (or brad ), is 1 / 256 turn. [ 18 ] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte . Other measures of angle used in computing may be based on dividing one whole turn into 2 n equal parts for other values of n . [ 19 ]
One turn is equal to 2 π {\displaystyle 2\pi } = τ {\displaystyle \tau } ≈ 6.283 185 307 179 586 [ 20 ] radians , 360 degrees , or 400 gradians .
In the International System of Quantities (ISQ), rotation (symbol N ) is a physical quantity defined as number of revolutions : [ 21 ]
N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:
where 𝜑 denotes the measure of rotational displacement .
The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time), [ 21 ] and adopted in the International System of Units (SI). [ 22 ] [ 23 ]
Rotation count or number of revolutions is a quantity of dimension one , resulting from a ratio of angular displacement.
It can be negative and also greater than 1 in modulus.
The relationship between quantity rotation, N , and unit turns, tr, can be expressed as:
where {𝜑} tr is the numerical value of the angle 𝜑 in units of turns (see Physical quantity § Components ).
In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by n :
The SI unit of rotational frequency is the reciprocal second (s −1 ). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one", [ b ] which also received other special names, such as the radian. [ c ] Despite their dimensional homogeneity , these two specially named dimensionless units are applicable for non-comparable kinds of quantity : rotation and angle, respectively. [ 25 ] "Cycle" is also mentioned in ISO 80000-3, in the definition of period . [ d ] | https://en.wikipedia.org/wiki/Rotation_(physics) |
In geometry , there exist various rotation formalisms to express a rotation in three dimensions as a mathematical transformation . In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion . The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.
According to Euler's rotation theorem , the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin ) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom .
An example where rotation representation is used is in computer vision , where an automated observer needs to track a target. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system ). The basic problem is to specify the orientation of these three unit vectors , and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.
Rotation formalisms are focused on proper ( orientation-preserving ) motions of the Euclidean space with one fixed point , that a rotation refers to. Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame , or motions of a joint ), this approach creates a knowledge about all motions. Any proper motion of the Euclidean space decomposes to a rotation around the origin and a translation . Whichever the order of their composition will be, the "pure" rotation component wouldn't change, uniquely determined by the complete motion.
One can also understand "pure" rotations as linear maps in a vector space equipped with Euclidean structure, not as maps of points of a corresponding affine space . In other words, a rotation formalism captures only the rotational part of a motion, that contains three degrees of freedom, and ignores the translational part, that contains another three.
When representing a rotation as numbers in a computer, some people prefer the quaternion representation or the axis+angle representation, because they avoid the gimbal lock that can occur with Euler rotations. [ 1 ]
The above-mentioned triad of unit vectors is also called a basis . Specifying the coordinates ( components ) of vectors of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors, û , v̂ and ŵ , that form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters.
These parameters can be written as the elements of a 3 × 3 matrix A , called a rotation matrix . Typically, the coordinates of each of these vectors are arranged along a column of the matrix (however, beware that an alternative definition of rotation matrix exists and is widely used, where the vectors' coordinates defined above are arranged by rows [ 2 ] ) A = [ u ^ x v ^ x w ^ x u ^ y v ^ y w ^ y u ^ z v ^ z w ^ z ] {\displaystyle \mathbf {A} ={\begin{bmatrix}{\hat {\mathbf {u} }}_{x}&{\hat {\mathbf {v} }}_{x}&{\hat {\mathbf {w} }}_{x}\\{\hat {\mathbf {u} }}_{y}&{\hat {\mathbf {v} }}_{y}&{\hat {\mathbf {w} }}_{y}\\{\hat {\mathbf {u} }}_{z}&{\hat {\mathbf {v} }}_{z}&{\hat {\mathbf {w} }}_{z}\\\end{bmatrix}}}
The elements of the rotation matrix are not all independent—as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom.
The rotation matrix has the following properties:
The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding to the eigenvalue of 1 is the accompanying Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix.
The above properties are equivalent to | u ^ | = | v ^ | = | w ^ | = 1 u ^ ⋅ v ^ = 0 u ^ × v ^ = w ^ , {\displaystyle {\begin{aligned}|{\hat {\mathbf {u} }}|=|{\hat {\mathbf {v} }}|=|{\hat {\mathbf {w} }}|&=1\\{\hat {\mathbf {u} }}\cdot {\hat {\mathbf {v} }}&=0\\{\hat {\mathbf {u} }}\times {\hat {\mathbf {v} }}&={\hat {\mathbf {w} }}\,,\end{aligned}}} which is another way of stating that ( û , v̂ , ŵ ) form a 3D orthonormal basis . These statements comprise a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just 3 degrees of freedom, as required.
Two successive rotations represented by matrices A 1 and A 2 are easily combined as elements of a group, A total = A 2 A 1 {\displaystyle \mathbf {A} _{\text{total}}=\mathbf {A} _{2}\mathbf {A} _{1}} (Note the order, since the vector being rotated is multiplied from the right).
The ease by which vectors can be rotated using a rotation matrix, as well as the ease of combining successive rotations, make the rotation matrix a useful and popular way to represent rotations, even though it is less concise than other representations.
From Euler's rotation theorem we know that any rotation can be expressed as a single rotation about some axis. The axis is the unit vector (unique except for sign) which remains unchanged by the rotation. The magnitude of the angle is also unique, with its sign being determined by the sign of the rotation axis.
The axis can be represented as a three-dimensional unit vector e ^ = [ e x e y e z ] {\displaystyle {\hat {\mathbf {e} }}={\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}}} and the angle by a scalar θ .
Since the axis is normalized, it has only two degrees of freedom . The angle adds the third degree of freedom to this rotation representation.
One may wish to express rotation as a rotation vector , or Euler vector , an un-normalized three-dimensional vector the direction of which specifies the axis, and the length of which is θ , r = θ e ^ . {\displaystyle \mathbf {r} =\theta {\hat {\mathbf {e} }}\,.}
The rotation vector is useful in some contexts, as it represents a three-dimensional rotation with only three scalar values (its components), representing the three degrees of freedom. This is also true for representations based on sequences of three Euler angles (see below).
If the rotation angle θ is zero, the axis is not uniquely defined. Combining two successive rotations, each represented by an Euler axis and angle, is not straightforward, and in fact does not satisfy the law of vector addition, which shows that finite rotations are not really vectors at all. It is best to employ the rotation matrix or quaternion notation, calculate the product, and then convert back to Euler axis and angle.
The idea behind Euler rotations is to split the complete rotation of the coordinate system into three simpler constitutive rotations, called precession , nutation , and intrinsic rotation , being each one of them an increment on one of the Euler angles . Notice that the outer matrix will represent a rotation around one of the axes of the reference frame, and the inner matrix represents a rotation around one of the moving frame axes. The middle matrix represents a rotation around an intermediate axis called line of nodes .
However, the definition of Euler angles is not unique and in the literature many different conventions are used. These conventions depend on the axes about which the rotations are carried out, and their sequence (since rotations on a sphere are non-commutative ).
The convention being used is usually indicated by specifying the axes about which the consecutive rotations (before being composed) take place, referring to them by index (1, 2, 3) or letter (X, Y, Z) . The engineering and robotics communities typically use 3-1-3 Euler angles. Notice that after composing the independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are 3 × 3 × 3 = 27 possible combinations of three basic rotations but only 3 × 2 × 2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around the same axis (such as XXY) which would reduce the degrees of freedom that can be represented.
Therefore, Euler angles are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. Other conventions (e.g., rotation matrix or quaternions ) are used to avoid this problem.
In aviation orientation of the aircraft is usually expressed as intrinsic Tait-Bryan angles following the z - y ′- x ″ convention, which are called heading , elevation , and bank (or synonymously, yaw , pitch , and roll ).
Quaternions , which form a four-dimensional vector space , have proven very useful in representing rotations due to several advantages over the other representations mentioned in this article.
A quaternion representation of rotation is written as a versor (normalized quaternion): q ^ = q i i + q j j + q k k + q r = [ q i q j q k q r ] {\displaystyle {\hat {\mathbf {q} }}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}}
The above definition stores the quaternion as an array following the convention used in (Wertz 1980) and (Markley 2003). An alternative definition, used for example in (Coutsias 1999) and (Schmidt 2001), defines the "scalar" term as the first quaternion element, with the other elements shifted down one position.
In terms of the Euler axis e ^ = [ e x e y e z ] {\displaystyle {\hat {\mathbf {e} }}={\begin{bmatrix}e_{x}\\e_{y}\\e_{z}\end{bmatrix}}}
and angle θ this versor's components are expressed as follows: q i = e x sin θ 2 q j = e y sin θ 2 q k = e z sin θ 2 q r = cos θ 2 {\displaystyle {\begin{aligned}q_{i}&=e_{x}\sin {\frac {\theta }{2}}\\q_{j}&=e_{y}\sin {\frac {\theta }{2}}\\q_{k}&=e_{z}\sin {\frac {\theta }{2}}\\q_{r}&=\cos {\frac {\theta }{2}}\end{aligned}}}
Inspection shows that the quaternion parametrization obeys the following constraint: q i 2 + q j 2 + q k 2 + q r 2 = 1 {\displaystyle q_{i}^{2}+q_{j}^{2}+q_{k}^{2}+q_{r}^{2}=1}
The last term (in our definition) is often called the scalar term, which has its origin in quaternions when understood as the mathematical extension of the complex numbers, written as a + b i + c j + d k with a , b , c , d ∈ R {\displaystyle a+bi+cj+dk\qquad {\text{with }}a,b,c,d\in \mathbb {R} } and where { i , j , k } are the hypercomplex numbers satisfying i 2 = j 2 = k 2 = − 1 i j = − j i = k j k = − k j = i k i = − i k = j {\displaystyle {\begin{array}{ccccccc}i^{2}&=&j^{2}&=&k^{2}&=&-1\\ij&=&-ji&=&k&&\\jk&=&-kj&=&i&&\\ki&=&-ik&=&j&&\end{array}}}
Quaternion multiplication, which is used to specify a composite rotation, is performed in the same manner as multiplication of complex numbers , except that the order of the elements must be taken into account, since multiplication is not commutative. In matrix notation we can write quaternion multiplication as q ~ ⊗ q = [ q r q k − q j q i − q k q r q i q j q j − q i q r q k − q i − q j − q k q r ] [ q ~ i q ~ j q ~ k q ~ r ] = [ q ~ r − q ~ k q ~ j q ~ i q ~ k q ~ r − q ~ i q ~ j − q ~ j q ~ i q ~ r q ~ k − q ~ i − q ~ j − q ~ k q ~ r ] [ q i q j q k q r ] {\displaystyle {\tilde {\mathbf {q} }}\otimes \mathbf {q} ={\begin{bmatrix}\;\;\,q_{r}&\;\;\,q_{k}&-q_{j}&\;\;\,q_{i}\\-q_{k}&\;\;\,q_{r}&\;\;\,q_{i}&\;\;\,q_{j}\\\;\;\,q_{j}&-q_{i}&\;\;\,q_{r}&\;\;\,q_{k}\\-q_{i}&-q_{j}&-q_{k}&\;\;\,q_{r}\end{bmatrix}}{\begin{bmatrix}{\tilde {q}}_{i}\\{\tilde {q}}_{j}\\{\tilde {q}}_{k}\\{\tilde {q}}_{r}\end{bmatrix}}={\begin{bmatrix}\;\;\,{\tilde {q}}_{r}&-{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{j}&\;\;\,{\tilde {q}}_{i}\\\;\;\,{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{r}&-{\tilde {q}}_{i}&\;\;\,{\tilde {q}}_{j}\\-{\tilde {q}}_{j}&\;\;\,{\tilde {q}}_{i}&\;\;\,{\tilde {q}}_{r}&\;\;\,{\tilde {q}}_{k}\\-{\tilde {q}}_{i}&-{\tilde {q}}_{j}&-{\tilde {q}}_{k}&\;\;\,{\tilde {q}}_{r}\end{bmatrix}}{\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}}
Combining two consecutive quaternion rotations is therefore just as simple as using the rotation matrix. Just as two successive rotation matrices, A 1 followed by A 2 , are combined as A 3 = A 2 A 1 , {\displaystyle \mathbf {A} _{3}=\mathbf {A} _{2}\mathbf {A} _{1},} we can represent this with quaternion parameters in a similarly concise way: q 3 = q 2 ⊗ q 1 {\displaystyle \mathbf {q} _{3}=\mathbf {q} _{2}\otimes \mathbf {q} _{1}}
Quaternions are a very popular parametrization due to the following properties:
Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix.
Quaternions also capture the spinorial character of rotations in three dimensions. For a three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a rotation changes from a scalar +1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two full turns). This cycle repeats every 2 turns. After 2 n turns (integer n > 0 ), without any intermediate untangling attempts, the strings/bands can be partially untangled back to the 2( n − 1) turns state with each application of the same procedure used in untangling from 2 turns to 0 turns. Applying the same procedure n times will take a 2 n -tangled object back to the untangled or 0 turn state. The untangling process also removes any rotation-generated twisting about the strings/bands themselves. Simple 3D mechanical models can be used to demonstrate these facts.
The Rodrigues vector (sometimes called the Gibbs vector , with coordinates called Rodrigues parameters ) [ 3 ] [ 4 ] can be expressed in terms of the axis and angle of the rotation as follows: g = e ^ tan θ 2 {\displaystyle \mathbf {g} ={\hat {\mathbf {e} }}\tan {\frac {\theta }{2}}}
This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane.
It has a discontinuity at 180° ( π radians): as any rotation vector r tends to an angle of π radians, its tangent tends to infinity.
A rotation g followed by a rotation f in the Rodrigues representation has the simple rotation composition form
( g , f ) = g + f − f × g 1 − g ⋅ f . {\displaystyle (\mathbf {g} ,\mathbf {f} )={\frac {\mathbf {g} +\mathbf {f} -\mathbf {f} \times \mathbf {g} }{1-\mathbf {g} \cdot \mathbf {f} }}\,.}
Today, the most straightforward way to prove this formula is in the (faithful) doublet representation , where g = n̂ tan a , etc.
The combinatoric features of the Pauli matrix derivation just mentioned are also identical to the equivalent quaternion derivation below. Construct a quaternion associated with a spatial rotation R as, S = cos ϕ 2 + sin ϕ 2 S . {\displaystyle S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .} Then the composition of the rotation R B with R A is the rotation R C = R B R A , with rotation axis and angle defined by the product of the quaternions, A = cos α 2 + sin α 2 A and B = cos β 2 + sin β 2 B , {\displaystyle A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{and}}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,} that is C = cos γ 2 + sin γ 2 C = ( cos β 2 + sin β 2 B ) ( cos α 2 + sin α 2 A ) . {\displaystyle C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).}
Expand this quaternion product to cos γ 2 + sin γ 2 C = ( cos β 2 cos α 2 − sin β 2 sin α 2 B ⋅ A ) + ( sin β 2 cos α 2 B + sin α 2 cos β 2 A + sin β 2 sin α 2 B × A ) . {\displaystyle \cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).}
Divide both sides of this equation by the identity resulting from the previous one, cos γ 2 = cos β 2 cos α 2 − sin β 2 sin α 2 B ⋅ A , {\displaystyle \cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,} and evaluate
tan γ 2 C = tan β 2 B + tan α 2 A + tan β 2 tan α 2 B × A 1 − tan β 2 tan α 2 B ⋅ A . {\displaystyle \tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.}
This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two component rotations. He derived this formula in 1840 (see page 408). [ 3 ] The three rotation axes A , B , and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles.
Modified Rodrigues Parameters (MRPs)
Modified Rodrigues parameters (MRPs) can be expressed in terms of Euler axis and angle by p = e ^ tan θ 4 . {\displaystyle \mathbf {p} ={\hat {\mathbf {e} }}\tan {\frac {\theta }{4}}\,.} Its components can be expressed in terms of the components of a unit quaternion representing the same rotation as p x , y , z = q i , j , k 1 + q r . {\displaystyle p_{x,y,z}={\frac {q_{i,j,k}}{1+q_{r}}}\,.}
The modified Rodrigues vector is a stereographic projection mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane. The projection of the opposite quaternion − q results in a different modified Rodrigues vector p s than the projection of the original quaternion q . Comparing components one obtains that p x , y , z s = − q i , j , k 1 − q r = − p x , y , z p 2 . {\displaystyle p_{x,y,z}^{s}={\frac {-q_{i,j,k}}{1-q_{r}}}={\frac {-p_{x,y,z}}{\mathbf {p} ^{2}}}\,.} Notably, if one of these vectors lies inside the unit 3-sphere, the other will lie outside.
See definition at Wolfram Mathworld .
Active rotations of a 3D vector p in Euclidean space around an axis n over an angle η can be easily written in terms of dot and cross products as follows:
p ′ = p ∥ n + cos η p ⊥ + sin η p ∧ n {\displaystyle \mathbf {p} '=p_{\parallel }\mathbf {n} +\cos {\eta }\,\mathbf {p} _{\perp }+\sin {\eta }\,\mathbf {p} \wedge \mathbf {n} } wherein p ∥ = p ⋅ n {\displaystyle p_{\parallel }=\mathbf {p} \cdot \mathbf {n} } is the longitudinal component of p along n , given by the dot product , p ⊥ = p − ( p ⋅ n ) n {\displaystyle \mathbf {p} _{\perp }=\mathbf {p} -(\mathbf {p} \cdot \mathbf {n} )\mathbf {n} } is the transverse component of p with respect to n , and p ∧ n {\displaystyle \mathbf {p} \wedge \mathbf {n} }
is the cross product of p with n .
The above formula shows that the longitudinal component of p remains unchanged, whereas the transverse portion of p is rotated in the plane perpendicular to n . This plane is spanned by the transverse portion of p itself and a direction perpendicular to both p and n . The rotation is directly identifiable in the equation as a 2D rotation over an angle η .
Passive rotations can be described by the same formula, but with an inverse sign of either η or n .
The Euler angles ( φ , θ , ψ ) can be extracted from the rotation matrix A by inspecting the rotation matrix in analytical form.
Using the x -convention, the 3-1-3 extrinsic Euler angles φ , θ and ψ (around the z -axis, x -axis and again the Z {\displaystyle Z} -axis) can be obtained as follows: ϕ = atan2 ( A 31 , A 32 ) θ = arccos ( A 33 ) ψ = − atan2 ( A 13 , A 23 ) {\displaystyle {\begin{aligned}\phi &=\operatorname {atan2} \left(A_{31},A_{32}\right)\\\theta &=\arccos \left(A_{33}\right)\\\psi &=-\operatorname {atan2} \left(A_{13},A_{23}\right)\end{aligned}}}
Note that atan2( a , b ) is equivalent to arctan a / b where it also takes into account the quadrant that the point ( b , a ) is in; see atan2 .
When implementing the conversion, one has to take into account several situations: [ 5 ]
The rotation matrix A is generated from the 3-2-1 intrinsic Euler angles by multiplying the three matrices generated by rotations about the axes. A = A 3 A 2 A 1 = A Z A Y A X {\displaystyle \mathbf {A} =\mathbf {A} _{3}\mathbf {A} _{2}\mathbf {A} _{1}=\mathbf {A} _{Z}\mathbf {A} _{Y}\mathbf {A} _{X}}
The axes of the rotation depend on the specific convention being used. For the x -convention the rotations are about the x -, y - and z -axes with angles ϕ , θ and ψ , the individual matrices are as follows: A X = [ 1 0 0 0 cos ϕ − sin ϕ 0 sin ϕ cos ϕ ] A Y = [ cos θ 0 sin θ 0 1 0 − sin θ 0 cos θ ] A Z = [ cos ψ − sin ψ 0 sin ψ cos ψ 0 0 0 1 ] {\displaystyle {\begin{aligned}\mathbf {A} _{X}&={\begin{bmatrix}1&0&0\\0&\cos \phi &-\sin \phi \\0&\sin \phi &\cos \phi \end{bmatrix}}\\[5px]\mathbf {A} _{Y}&={\begin{bmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{bmatrix}}\\[5px]\mathbf {A} _{Z}&={\begin{bmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{bmatrix}}\end{aligned}}}
This yields A = [ cos θ cos ψ − cos ϕ sin ψ + sin ϕ sin θ cos ψ sin ϕ sin ψ + cos ϕ sin θ cos ψ cos θ sin ψ cos ϕ cos ψ + sin ϕ sin θ sin ψ − sin ϕ cos ψ + cos ϕ sin θ sin ψ − sin θ sin ϕ cos θ cos ϕ cos θ ] {\displaystyle \mathbf {A} ={\begin{bmatrix}\cos \theta \cos \psi &-\cos \phi \sin \psi +\sin \phi \sin \theta \cos \psi &\sin \phi \sin \psi +\cos \phi \sin \theta \cos \psi \\\cos \theta \sin \psi &\cos \phi \cos \psi +\sin \phi \sin \theta \sin \psi &-\sin \phi \cos \psi +\cos \phi \sin \theta \sin \psi \\-\sin \theta &\sin \phi \cos \theta &\cos \phi \cos \theta \\\end{bmatrix}}} Note: This is valid for a right-hand system, which is the convention used in almost all engineering and physics disciplines.
The interpretation of these right-handed rotation matrices is that they express coordinate transformations ( passive ) as opposed to point transformations ( active ). Because A expresses a rotation from the local frame 1 to the global frame 0 (i.e., A encodes the axes of frame 1 with respect to frame 0 ), the elementary rotation matrices are composed as above. Because the inverse rotation is just the rotation transposed, if we wanted the global-to-local rotation from frame 0 to frame 1 , we would write A T = ( A Z A Y A X ) T = A X T A Y T A Z T . {\displaystyle \mathbf {A} ^{\mathsf {T}}=(\mathbf {A} _{Z}\mathbf {A} _{Y}\mathbf {A} _{X})^{\mathsf {T}}=\mathbf {A} _{X}^{\mathsf {T}}\mathbf {A} _{Y}^{\mathsf {T}}\mathbf {A} _{Z}^{\mathsf {T}}\,.}
If the Euler angle θ is not a multiple of π , the Euler axis ê and angle θ can be computed from the elements of the rotation matrix A as follows: θ = arccos A 11 + A 22 + A 33 − 1 2 e 1 = A 32 − A 23 2 sin θ e 2 = A 13 − A 31 2 sin θ e 3 = A 21 − A 12 2 sin θ {\displaystyle {\begin{aligned}\theta &=\arccos {\frac {A_{11}+A_{22}+A_{33}-1}{2}}\\e_{1}&={\frac {A_{32}-A_{23}}{2\sin \theta }}\\e_{2}&={\frac {A_{13}-A_{31}}{2\sin \theta }}\\e_{3}&={\frac {A_{21}-A_{12}}{2\sin \theta }}\end{aligned}}}
Alternatively, the following method can be used:
Eigendecomposition of the rotation matrix yields the eigenvalues 1 and cos θ ± i sin θ . The Euler axis is the eigenvector corresponding to the eigenvalue of 1, and θ can be computed from the remaining eigenvalues.
The Euler axis can be also found using singular value decomposition since it is the normalized vector spanning the null-space of the matrix I − A .
To convert the other way the rotation matrix corresponding to an Euler axis ê and angle θ can be computed according to Rodrigues' rotation formula (with appropriate modification) as follows: A = I 3 cos θ + ( 1 − cos θ ) e ^ e ^ T + [ e ^ ] × sin θ {\displaystyle \mathbf {A} =\mathbf {I} _{3}\cos \theta +(1-\cos \theta ){\hat {\mathbf {e} }}{\hat {\mathbf {e} }}^{\mathsf {T}}+\left[{\hat {\mathbf {e} }}\right]_{\times }\sin \theta }
with I 3 the 3 × 3 identity matrix , and [ e ^ ] × = [ 0 − e 3 e 2 e 3 0 − e 1 − e 2 e 1 0 ] {\displaystyle \left[{\hat {\mathbf {e} }}\right]_{\times }={\begin{bmatrix}0&-e_{3}&e_{2}\\e_{3}&0&-e_{1}\\-e_{2}&e_{1}&0\end{bmatrix}}}
is the cross-product matrix .
This expands to: A 11 = ( 1 − cos θ ) e 1 2 + cos θ A 12 = ( 1 − cos θ ) e 1 e 2 − e 3 sin θ A 13 = ( 1 − cos θ ) e 1 e 3 + e 2 sin θ A 21 = ( 1 − cos θ ) e 2 e 1 + e 3 sin θ A 22 = ( 1 − cos θ ) e 2 2 + cos θ A 23 = ( 1 − cos θ ) e 2 e 3 − e 1 sin θ A 31 = ( 1 − cos θ ) e 3 e 1 − e 2 sin θ A 32 = ( 1 − cos θ ) e 3 e 2 + e 1 sin θ A 33 = ( 1 − cos θ ) e 3 2 + cos θ {\displaystyle {\begin{aligned}A_{11}&=(1-\cos \theta )e_{1}^{2}+\cos \theta \\A_{12}&=(1-\cos \theta )e_{1}e_{2}-e_{3}\sin \theta \\A_{13}&=(1-\cos \theta )e_{1}e_{3}+e_{2}\sin \theta \\A_{21}&=(1-\cos \theta )e_{2}e_{1}+e_{3}\sin \theta \\A_{22}&=(1-\cos \theta )e_{2}^{2}+\cos \theta \\A_{23}&=(1-\cos \theta )e_{2}e_{3}-e_{1}\sin \theta \\A_{31}&=(1-\cos \theta )e_{3}e_{1}-e_{2}\sin \theta \\A_{32}&=(1-\cos \theta )e_{3}e_{2}+e_{1}\sin \theta \\A_{33}&=(1-\cos \theta )e_{3}^{2}+\cos \theta \end{aligned}}}
When computing a quaternion from the rotation matrix there is a sign ambiguity, since q and − q represent the same rotation.
One way of computing the quaternion q = [ q i q j q k q r ] = q i i + q j j + q k k + q r {\displaystyle \mathbf {q} ={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}} from the rotation matrix A is as follows: q r = 1 2 1 + A 11 + A 22 + A 33 q i = 1 4 q r ( A 32 − A 23 ) q j = 1 4 q r ( A 13 − A 31 ) q k = 1 4 q r ( A 21 − A 12 ) {\displaystyle {\begin{aligned}q_{r}&={\frac {1}{2}}{\sqrt {1+A_{11}+A_{22}+A_{33}}}\\q_{i}&={\frac {1}{4q_{r}}}\left(A_{32}-A_{23}\right)\\q_{j}&={\frac {1}{4q_{r}}}\left(A_{13}-A_{31}\right)\\q_{k}&={\frac {1}{4q_{r}}}\left(A_{21}-A_{12}\right)\end{aligned}}}
There are three other mathematically equivalent ways to compute q . Numerical inaccuracy can be reduced by avoiding situations in which the denominator is close to zero. One of the other three methods looks as follows: [ 6 ] [ 7 ] q i = 1 2 1 + A 11 − A 22 − A 33 q j = 1 4 q i ( A 12 + A 21 ) q k = 1 4 q i ( A 13 + A 31 ) q r = 1 4 q i ( A 32 − A 23 ) {\displaystyle {\begin{aligned}q_{i}&={\frac {1}{2}}{\sqrt {1+A_{11}-A_{22}-A_{33}}}\\q_{j}&={\frac {1}{4q_{i}}}\left(A_{12}+A_{21}\right)\\q_{k}&={\frac {1}{4q_{i}}}\left(A_{13}+A_{31}\right)\\q_{r}&={\frac {1}{4q_{i}}}\left(A_{32}-A_{23}\right)\end{aligned}}}
The rotation matrix corresponding to the quaternion q can be computed as follows: A = ( q r 2 − q ˇ T q ˇ ) I 3 + 2 q ˇ q ˇ T + 2 q r Q {\displaystyle \mathbf {A} =\left(q_{r}^{2}-{\check {\mathbf {q} }}^{\mathsf {T}}{\check {\mathbf {q} }}\right)\mathbf {I} _{3}+2{\check {\mathbf {q} }}{\check {\mathbf {q} }}^{\mathsf {T}}+2q_{r}\mathbf {\mathcal {Q}} } where q ˇ = [ q i q j q k ] , Q = [ 0 − q k q j q k 0 − q i − q j q i 0 ] {\displaystyle {\check {\mathbf {q} }}={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\end{bmatrix}}\,,\quad \mathbf {\mathcal {Q}} ={\begin{bmatrix}0&-q_{k}&q_{j}\\q_{k}&0&-q_{i}\\-q_{j}&q_{i}&0\end{bmatrix}}} which gives A = [ 1 − 2 q j 2 − 2 q k 2 2 ( q i q j − q k q r ) 2 ( q i q k + q j q r ) 2 ( q i q j + q k q r ) 1 − 2 q i 2 − 2 q k 2 2 ( q j q k − q i q r ) 2 ( q i q k − q j q r ) 2 ( q j q k + q i q r ) 1 − 2 q i 2 − 2 q j 2 ] {\displaystyle \mathbf {A} ={\begin{bmatrix}1-2q_{j}^{2}-2q_{k}^{2}&2\left(q_{i}q_{j}-q_{k}q_{r}\right)&2\left(q_{i}q_{k}+q_{j}q_{r}\right)\\2\left(q_{i}q_{j}+q_{k}q_{r}\right)&1-2q_{i}^{2}-2q_{k}^{2}&2\left(q_{j}q_{k}-q_{i}q_{r}\right)\\2\left(q_{i}q_{k}-q_{j}q_{r}\right)&2\left(q_{j}q_{k}+q_{i}q_{r}\right)&1-2q_{i}^{2}-2q_{j}^{2}\end{bmatrix}}}
or equivalently A = [ − 1 + 2 q i 2 + 2 q r 2 2 ( q i q j − q k q r ) 2 ( q i q k + q j q r ) 2 ( q i q j + q k q r ) − 1 + 2 q j 2 + 2 q r 2 2 ( q j q k − q i q r ) 2 ( q i q k − q j q r ) 2 ( q j q k + q i q r ) − 1 + 2 q k 2 + 2 q r 2 ] {\displaystyle \mathbf {A} ={\begin{bmatrix}-1+2q_{i}^{2}+2q_{r}^{2}&2\left(q_{i}q_{j}-q_{k}q_{r}\right)&2\left(q_{i}q_{k}+q_{j}q_{r}\right)\\2\left(q_{i}q_{j}+q_{k}q_{r}\right)&-1+2q_{j}^{2}+2q_{r}^{2}&2\left(q_{j}q_{k}-q_{i}q_{r}\right)\\2\left(q_{i}q_{k}-q_{j}q_{r}\right)&2\left(q_{j}q_{k}+q_{i}q_{r}\right)&-1+2q_{k}^{2}+2q_{r}^{2}\end{bmatrix}}}
This is called the Euler–Rodrigues formula for the transformation matrix A {\displaystyle \mathbf {A} }
We will consider the x -convention 3-1-3 extrinsic Euler angles for the following algorithm. The terms of the algorithm depend on the convention used.
We can compute the quaternion q = [ q i q j q k q r ] = q i i + q j j + q k k + q r {\displaystyle \mathbf {q} ={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}} from the Euler angles ( ϕ , θ , ψ ) as follows:
q i = cos ϕ − ψ 2 sin θ 2 q j = sin ϕ − ψ 2 sin θ 2 q k = sin ϕ + ψ 2 cos θ 2 q r = cos ϕ + ψ 2 cos θ 2 {\displaystyle {\begin{aligned}q_{i}&=\cos {\frac {\phi -\psi }{2}}\sin {\frac {\theta }{2}}\\q_{j}&=\sin {\frac {\phi -\psi }{2}}\sin {\frac {\theta }{2}}\\q_{k}&=\sin {\frac {\phi +\psi }{2}}\cos {\frac {\theta }{2}}\\q_{r}&=\cos {\frac {\phi +\psi }{2}}\cos {\frac {\theta }{2}}\end{aligned}}}
A quaternion equivalent to yaw ( ψ ), pitch ( θ ) and roll ( ϕ ) angles. or intrinsic Tait–Bryan angles following the z - y ′- x ″ convention, can be computed by
q i = sin ϕ 2 cos θ 2 cos ψ 2 − cos ϕ 2 sin θ 2 sin ψ 2 q j = cos ϕ 2 sin θ 2 cos ψ 2 + sin ϕ 2 cos θ 2 sin ψ 2 q k = cos ϕ 2 cos θ 2 sin ψ 2 − sin ϕ 2 sin θ 2 cos ψ 2 q r = cos ϕ 2 cos θ 2 cos ψ 2 + sin ϕ 2 sin θ 2 sin ψ 2 {\displaystyle {\begin{aligned}q_{i}&=\sin {\frac {\phi }{2}}\cos {\frac {\theta }{2}}\cos {\frac {\psi }{2}}-\cos {\frac {\phi }{2}}\sin {\frac {\theta }{2}}\sin {\frac {\psi }{2}}\\q_{j}&=\cos {\frac {\phi }{2}}\sin {\frac {\theta }{2}}\cos {\frac {\psi }{2}}+\sin {\frac {\phi }{2}}\cos {\frac {\theta }{2}}\sin {\frac {\psi }{2}}\\q_{k}&=\cos {\frac {\phi }{2}}\cos {\frac {\theta }{2}}\sin {\frac {\psi }{2}}-\sin {\frac {\phi }{2}}\sin {\frac {\theta }{2}}\cos {\frac {\psi }{2}}\\q_{r}&=\cos {\frac {\phi }{2}}\cos {\frac {\theta }{2}}\cos {\frac {\psi }{2}}+\sin {\frac {\phi }{2}}\sin {\frac {\theta }{2}}\sin {\frac {\psi }{2}}\end{aligned}}}
Given the rotation quaternion q = [ q i q j q k q r ] = q i i + q j j + q k k + q r , {\displaystyle \mathbf {q} ={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}\,,} the x -convention 3-1-3 extrinsic Euler Angles ( φ , θ , ψ ) can be computed by
ϕ = atan2 ( ( q i q k + q j q r ) , − ( q j q k − q i q r ) ) θ = arccos ( − q i 2 − q j 2 + q k 2 + q r 2 ) ψ = atan2 ( ( q i q k − q j q r ) , ( q j q k + q i q r ) ) {\displaystyle {\begin{aligned}\phi &=\operatorname {atan2} \left(\left(q_{i}q_{k}+q_{j}q_{r}\right),-\left(q_{j}q_{k}-q_{i}q_{r}\right)\right)\\\theta &=\arccos \left(-q_{i}^{2}-q_{j}^{2}+q_{k}^{2}+q_{r}^{2}\right)\\\psi &=\operatorname {atan2} \left(\left(q_{i}q_{k}-q_{j}q_{r}\right),\left(q_{j}q_{k}+q_{i}q_{r}\right)\right)\end{aligned}}}
Given the rotation quaternion q = [ q i q j q k q r ] = q i i + q j j + q k k + q r , {\displaystyle \mathbf {q} ={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}\,,} yaw , pitch and roll angles, or intrinsic Tait–Bryan angles following the z - y ′- x ″ convention, can be computed by
roll = atan2 ( 2 ( q r q i + q j q k ) , 1 − 2 ( q i 2 + q j 2 ) ) pitch = arcsin ( 2 ( q r q j − q k q i ) ) yaw = atan2 ( 2 ( q r q k + q i q j ) , 1 − 2 ( q j 2 + q k 2 ) ) {\displaystyle {\begin{aligned}{\text{roll}}&=\operatorname {atan2} \left(2\left(q_{r}q_{i}+q_{j}q_{k}\right),1-2\left(q_{i}^{2}+q_{j}^{2}\right)\right)\\{\text{pitch}}&=\arcsin \left(2\left(q_{r}q_{j}-q_{k}q_{i}\right)\right)\\{\text{yaw}}&=\operatorname {atan2} \left(2\left(q_{r}q_{k}+q_{i}q_{j}\right),1-2\left(q_{j}^{2}+q_{k}^{2}\right)\right)\end{aligned}}}
Given the Euler axis ê and angle θ , the quaternion q = [ q i q j q k q r ] = q i i + q j j + q k k + q r , {\displaystyle \mathbf {q} ={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\\q_{r}\end{bmatrix}}=q_{i}\mathbf {i} +q_{j}\mathbf {j} +q_{k}\mathbf {k} +q_{r}\,,}
can be computed by q i = e ^ 1 sin θ 2 q j = e ^ 2 sin θ 2 q k = e ^ 3 sin θ 2 q r = cos θ 2 {\displaystyle {\begin{aligned}q_{i}&={\hat {e}}_{1}\sin {\frac {\theta }{2}}\\q_{j}&={\hat {e}}_{2}\sin {\frac {\theta }{2}}\\q_{k}&={\hat {e}}_{3}\sin {\frac {\theta }{2}}\\q_{r}&=\cos {\frac {\theta }{2}}\end{aligned}}}
Given the rotation quaternion q , define q ˇ = [ q i q j q k ] . {\displaystyle {\check {\mathbf {q} }}={\begin{bmatrix}q_{i}\\q_{j}\\q_{k}\end{bmatrix}}\,.} Then the Euler axis ê and angle θ can be computed by e ^ = q ˇ ‖ q ˇ ‖ θ = 2 arccos q r {\displaystyle {\begin{aligned}{\hat {\mathbf {e} }}&={\frac {\check {\mathbf {q} }}{\left\|{\check {\mathbf {q} }}\right\|}}\\\theta &=2\arccos q_{r}\end{aligned}}}
Since the definition of the Rodrigues vector can be related to rotation quaternions: { g i = q i q r = e x tan ( θ 2 ) g j = q j q r = e y tan ( θ 2 ) g k = q k q r = e z tan ( θ 2 ) {\displaystyle {\begin{cases}g_{i}={\dfrac {q_{i}}{q_{r}}}=e_{x}\tan \left({\dfrac {\theta }{2}}\right)\\g_{j}={\dfrac {q_{j}}{q_{r}}}=e_{y}\tan \left({\dfrac {\theta }{2}}\right)\\g_{k}={\dfrac {q_{k}}{q_{r}}}=e_{z}\tan \left({\dfrac {\theta }{2}}\right)\end{cases}}} By making use of the following property 1 = q r 2 + q i 2 + q j 2 + q k 2 = q r 2 ( 1 + q i 2 q r 2 + q j 2 q r 2 + q k 2 q r 2 ) = q r 2 ( 1 + g i 2 + g j 2 + g k 2 ) {\displaystyle 1=q_{r}^{2}+q_{i}^{2}+q_{j}^{2}+q_{k}^{2}=q_{r}^{2}\left(1+{\frac {q_{i}^{2}}{q_{r}^{2}}}+{\frac {q_{j}^{2}}{q_{r}^{2}}}+{\frac {q_{k}^{2}}{q_{r}^{2}}}\right)=q_{r}^{2}\left(1+g_{i}^{2}+g_{j}^{2}+g_{k}^{2}\right)} The formula can be obtained by factoring q 2 r from the final expression obtained for quaternions:
A = q r 2 [ 1 q r 2 − 2 q j 2 q r 2 − 2 q k 2 q r 2 2 ( q i q r q j q r − q k q r ) 2 ( q i q r q k q r + q j q r ) 2 ( q i q r q j q r + q k q r ) 1 q r 2 − 2 q i 2 q r 2 − 2 q k 2 q r 2 2 ( q j q r q k q r − q i q r ) 2 ( q i q r q k q r − q j q r ) 2 ( q j q r q k q r + q i q r ) 1 q r 2 − 2 q i 2 q r 2 − 2 q j 2 q r 2 ] {\displaystyle \mathbf {A} =q_{r}^{2}{\begin{bmatrix}{\frac {1}{q_{r}^{2}}}-2{\frac {q_{j}^{2}}{q_{r}^{2}}}-2{\frac {q_{k}^{2}}{q_{r}^{2}}}&2\left({\frac {q_{i}}{q_{r}}}{\frac {q_{j}}{q_{r}}}-{\frac {q_{k}}{q_{r}}}\right)&2\left({\frac {q_{i}}{q_{r}}}{\frac {q_{k}}{q_{r}}}+{\frac {q_{j}}{q_{r}}}\right)\\2\left({\frac {q_{i}}{q_{r}}}{\frac {q_{j}}{q_{r}}}+{\frac {q_{k}}{q_{r}}}\right)&{\frac {1}{q_{r}^{2}}}-2{\frac {q_{i}^{2}}{q_{r}^{2}}}-2{\frac {q_{k}^{2}}{q_{r}^{2}}}&2\left({\frac {q_{j}}{q_{r}}}{\frac {q_{k}}{q_{r}}}-{\frac {q_{i}}{q_{r}}}\right)\\2\left({\frac {q_{i}}{q_{r}}}{\frac {q_{k}}{q_{r}}}-{\frac {q_{j}}{q_{r}}}\right)&2\left({\frac {q_{j}}{q_{r}}}{\frac {q_{k}}{q_{r}}}+{\frac {q_{i}}{q_{r}}}\right)&{\frac {1}{q_{r}^{2}}}-2{\frac {q_{i}^{2}}{q_{r}^{2}}}-2{\frac {q_{j}^{2}}{q_{r}^{2}}}\end{bmatrix}}}
Leading to the final formula:
A = 1 1 + g i 2 + g j 2 + g k 2 [ 1 + g i 2 − g j 2 − g k 2 2 ( g i g j − g k ) 2 ( g i g k + g j ) 2 ( g i g j + g k ) 1 − g i 2 + g j 2 − g k 2 2 ( g j g k − g i ) 2 ( g i g k − g j ) 2 ( g j g k + g i ) 1 − g i 2 − g j 2 + g k 2 ] {\displaystyle \mathbf {A} ={\frac {1}{1+g_{i}^{2}+g_{j}^{2}+g_{k}^{2}}}{\begin{bmatrix}1+g_{i}^{2}-g_{j}^{2}-g_{k}^{2}&2\left(g_{i}g_{j}-g_{k}\right)&2\left(g_{i}g_{k}+g_{j}\right)\\2\left(g_{i}g_{j}+g_{k}\right)&1-g_{i}^{2}+g_{j}^{2}-g_{k}^{2}&2\left(g_{j}g_{k}-g_{i}\right)\\2\left(g_{i}g_{k}-g_{j}\right)&2\left(g_{j}g_{k}+g_{i}\right)&1-g_{i}^{2}-g_{j}^{2}+g_{k}^{2}\end{bmatrix}}}
The angular velocity vector ω = [ ω x ω y ω z ] {\displaystyle {\boldsymbol {\omega }}={\begin{bmatrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{bmatrix}}} can be extracted from the time derivative of the rotation matrix d A / d t by the following relation: [ ω ] × = [ 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 ] = d A d t A T {\displaystyle [{\boldsymbol {\omega }}]_{\times }={\begin{bmatrix}0&-\omega _{z}&\omega _{y}\\\omega _{z}&0&-\omega _{x}\\-\omega _{y}&\omega _{x}&0\end{bmatrix}}={\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{\mathsf {T}}}
The derivation is adapted from Ioffe [ 8 ] as follows:
For any vector r 0 , consider r ( t ) = A ( t ) r 0 and differentiate it: d r d t = d A d t r 0 = d A d t A T ( t ) r ( t ) {\displaystyle {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {r} _{0}={\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{\mathsf {T}}(t)\mathrm {r} (t)}
The derivative of a vector is the linear velocity of its tip. Since A is a rotation matrix, by definition the length of r ( t ) is always equal to the length of r 0 , and hence it does not change with time. Thus, when r ( t ) rotates, its tip moves along a circle, and the linear velocity of its tip is tangential to the circle; i.e., always perpendicular to r ( t ) . In this specific case, the relationship between the linear velocity vector and the angular velocity vector is d r d t = ω ( t ) × r ( t ) = [ ω ] × r ( t ) {\displaystyle {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\boldsymbol {\omega }}(t)\times \mathbf {r} (t)=[{\boldsymbol {\omega }}]_{\times }\mathbf {r} (t)} (see circular motion and cross product ).
By the transitivity of the abovementioned equations, d A d t A T ( t ) r ( t ) = [ ω ] × r ( t ) {\displaystyle {\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{\mathsf {T}}(t)\mathbf {r} (t)=[{\boldsymbol {\omega }}]_{\times }\mathbf {r} (t)}
which implies d A d t A T ( t ) = [ ω ] × {\displaystyle {\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{\mathsf {T}}(t)=[{\boldsymbol {\omega }}]_{\times }}
The angular velocity vector ω = [ ω x ω y ω z ] {\displaystyle {\boldsymbol {\omega }}={\begin{bmatrix}\omega _{x}\\\omega _{y}\\\omega _{z}\end{bmatrix}}} can be obtained from the derivative of the quaternion d q / d t as follows: [ 9 ] [ 0 ω x ω y ω z ] = 2 d q d t q ~ {\displaystyle {\begin{bmatrix}0\\\omega _{x}\\\omega _{y}\\\omega _{z}\end{bmatrix}}=2{\frac {\mathrm {d} \mathbf {q} }{\mathrm {d} t}}{\tilde {\mathbf {q} }}} where q̃ is the conjugate (inverse) of q .
Conversely, the derivative of the quaternion is d q d t = 1 2 [ 0 ω x ω y ω z ] q . {\displaystyle {\frac {\mathrm {d} \mathbf {q} }{\mathrm {d} t}}={\frac {1}{2}}{\begin{bmatrix}0\\\omega _{x}\\\omega _{y}\\\omega _{z}\end{bmatrix}}\mathbf {q} \,.}
The formalism of geometric algebra (GA) provides an extension and interpretation of the quaternion method. Central to GA is the geometric product of vectors, an extension of the traditional inner and cross products , given by a b = a ⋅ b + a ∧ b {\displaystyle \mathbf {ab} =\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \wedge \mathbf {b} }
where the symbol ∧ denotes the exterior product or wedge product . This product of vectors a , and b produces two terms: a scalar part from the inner product and a bivector part from the wedge product. This bivector describes the plane perpendicular to what the cross product of the vectors would return.
Bivectors in GA have some unusual properties compared to vectors. Under the geometric product, bivectors have a negative square: the bivector x̂ŷ describes the xy -plane. Its square is ( x̂ŷ ) 2 = x̂ŷx̂ŷ . Because the unit basis vectors are orthogonal to each other, the geometric product reduces to the antisymmetric outer product, so x̂ and ŷ can be swapped freely at the cost of a factor of −1. The square reduces to − x̂x̂ŷŷ = −1 since the basis vectors themselves square to +1.
This result holds generally for all bivectors, and as a result the bivector plays a role similar to the imaginary unit . Geometric algebra uses bivectors in its analogue to the quaternion, the rotor , given by R = exp ( − B ^ θ 2 ) = cos θ 2 − B ^ sin θ 2 , {\displaystyle \mathbf {R} =\exp \left({\frac {-{\hat {\mathbf {B} }}\theta }{2}}\right)=\cos {\frac {\theta }{2}}-{\hat {\mathbf {B} }}\sin {\frac {\theta }{2}}\,,} where B̂ is a unit bivector that describes the plane of rotation . Because B̂ squares to −1, the power series expansion of R generates the trigonometric functions . The rotation formula that maps a vector a to a rotated vector b is then b = R a R † {\displaystyle \mathbf {b} =\mathbf {RaR} ^{\dagger }} where R † = exp ( 1 2 B ^ θ ) = cos θ 2 + B ^ sin θ 2 {\displaystyle \mathbf {R} ^{\dagger }=\exp \left({\frac {1}{2}}{\hat {\mathbf {B} }}\theta \right)=\cos {\frac {\theta }{2}}+{\hat {\mathbf {B} }}\sin {\frac {\theta }{2}}} is the reverse of R {\displaystyle \scriptstyle R} (reversing the order of the vectors in B {\displaystyle B} is equivalent to changing its sign).
Example. A rotation about the axis v ^ = 1 3 ( x ^ + y ^ + z ^ ) {\displaystyle {\hat {\mathbf {v} }}={\frac {1}{\sqrt {3}}}\left({\hat {\mathbf {x} }}+{\hat {\mathbf {y} }}+{\hat {\mathbf {z} }}\right)} can be accomplished by converting v̂ to its dual bivector, B ^ = x ^ y ^ z ^ v ^ = i v ^ , {\displaystyle {\hat {\mathbf {B} }}={\hat {\mathbf {x} }}{\hat {\mathbf {y} }}{\hat {\mathbf {z} }}{\hat {\mathbf {v} }}=\mathbf {i} {\hat {\mathbf {v} }}\,,} where i = x̂ŷẑ is the unit volume element, the only trivector (pseudoscalar) in three-dimensional space. The result is B ^ = 1 3 ( y ^ z ^ + z ^ x ^ + x ^ y ^ ) . {\displaystyle {\hat {\mathbf {B} }}={\frac {1}{\sqrt {3}}}\left({\hat {\mathbf {y} }}{\hat {\mathbf {z} }}+{\hat {\mathbf {z} }}{\hat {\mathbf {x} }}+{\hat {\mathbf {x} }}{\hat {\mathbf {y} }}\right)\,.}
In three-dimensional space, however, it is often simpler to leave the expression for B̂ = iv̂ , using the fact that i commutes with all objects in 3D and also squares to −1. A rotation of the x̂ vector in this plane by an angle θ is then
x ^ ′ = R x ^ R † = e − i v ^ θ 2 x ^ e i v ^ θ 2 = x ^ cos 2 θ 2 + i ( x ^ v ^ − v ^ x ^ ) cos θ 2 sin θ 2 + v ^ x ^ v ^ sin 2 θ 2 {\displaystyle {\hat {\mathbf {x} }}'=\mathbf {R} {\hat {\mathbf {x} }}\mathbf {R} ^{\dagger }=e^{-i{\hat {\mathbf {v} }}{\frac {\theta }{2}}}{\hat {\mathbf {x} }}e^{i{\hat {\mathbf {v} }}{\frac {\theta }{2}}}={\hat {\mathbf {x} }}\cos ^{2}{\frac {\theta }{2}}+\mathbf {i} \left({\hat {\mathbf {x} }}{\hat {\mathbf {v} }}-{\hat {\mathbf {v} }}{\hat {\mathbf {x} }}\right)\cos {\frac {\theta }{2}}\sin {\frac {\theta }{2}}+{\hat {\mathbf {v} }}{\hat {\mathbf {x} }}{\hat {\mathbf {v} }}\sin ^{2}{\frac {\theta }{2}}}
Recognizing that i ( x ^ v ^ − v ^ x ^ ) = 2 i ( x ^ ∧ v ^ ) {\displaystyle \mathbf {i} ({\hat {\mathbf {x} }}{\hat {\mathbf {v} }}-{\hat {\mathbf {v} }}{\hat {\mathbf {x} }})=2\mathbf {i} ({\hat {\mathbf {x} }}\wedge {\hat {\mathbf {v} }})} and that − v̂x̂v̂ is the reflection of x̂ about the plane perpendicular to v̂ gives a geometric interpretation to the rotation operation: the rotation preserves the components that are parallel to v̂ and changes only those that are perpendicular. The terms are then computed: v ^ x ^ v ^ = 1 3 ( − x ^ + 2 y ^ + 2 z ^ ) 2 i x ^ ∧ v ^ = 2 i 1 3 ( x ^ y ^ + x ^ z ^ ) = 2 3 ( y ^ − z ^ ) {\displaystyle {\begin{aligned}{\hat {\mathbf {v} }}{\hat {\mathbf {x} }}{\hat {\mathbf {v} }}&={\frac {1}{3}}\left(-{\hat {\mathbf {x} }}+2{\hat {\mathbf {y} }}+2{\hat {\mathbf {z} }}\right)\\2\mathbf {i} {\hat {\mathbf {x} }}\wedge {\hat {\mathbf {v} }}&=2\mathbf {i} {\frac {1}{\sqrt {3}}}\left({\hat {\mathbf {x} }}{\hat {\mathbf {y} }}+{\hat {\mathbf {x} }}{\hat {\mathbf {z} }}\right)={\frac {2}{\sqrt {3}}}\left({\hat {\mathbf {y} }}-{\hat {\mathbf {z} }}\right)\end{aligned}}}
The result of the rotation is then x ^ ′ = x ^ ( cos 2 θ 2 − 1 3 sin 2 θ 2 ) + 2 3 y ^ sin θ 2 ( sin θ 2 + 3 cos θ 2 ) + 2 3 z ^ sin θ 2 ( sin θ 2 − 3 cos θ 2 ) {\displaystyle {\hat {\mathbf {x} }}'={\hat {\mathbf {x} }}\left(\cos ^{2}{\frac {\theta }{2}}-{\frac {1}{3}}\sin ^{2}{\frac {\theta }{2}}\right)+{\frac {2}{3}}{\hat {\mathbf {y} }}\sin {\frac {\theta }{2}}\left(\sin {\frac {\theta }{2}}+{\sqrt {3}}\cos {\frac {\theta }{2}}\right)+{\frac {2}{3}}{\hat {\mathbf {z} }}\sin {\frac {\theta }{2}}\left(\sin {\frac {\theta }{2}}-{\sqrt {3}}\cos {\frac {\theta }{2}}\right)}
A simple check on this result is the angle θ = 2 / 3 π . Such a rotation should map x̂ to ŷ . Indeed, the rotation reduces to x ^ ′ = x ^ ( 1 4 − 1 3 3 4 ) + 2 3 y ^ 3 2 ( 3 2 + 3 1 2 ) + 2 3 z ^ 3 2 ( 3 2 − 3 1 2 ) = 0 x ^ + y ^ + 0 z ^ = y ^ {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}'&={\hat {\mathbf {x} }}\left({\frac {1}{4}}-{\frac {1}{3}}{\frac {3}{4}}\right)+{\frac {2}{3}}{\hat {\mathbf {y} }}{\frac {\sqrt {3}}{2}}\left({\frac {\sqrt {3}}{2}}+{\sqrt {3}}{\frac {1}{2}}\right)+{\frac {2}{3}}{\hat {\mathbf {z} }}{\frac {\sqrt {3}}{2}}\left({\frac {\sqrt {3}}{2}}-{\sqrt {3}}{\frac {1}{2}}\right)\\&=0{\hat {\mathbf {x} }}+{\hat {\mathbf {y} }}+0{\hat {\mathbf {z} }}={\hat {\mathbf {y} }}\end{aligned}}}
exactly as expected. This rotation formula is valid not only for vectors but for any multivector . In addition, when Euler angles are used, the complexity of the operation is much reduced. Compounded rotations come from multiplying the rotors, so the total rotor from Euler angles is R = R γ ′ R β ′ R α = exp ( − i z ^ ′ γ 2 ) exp ( − i x ^ ′ β 2 ) exp ( − i z ^ α 2 ) {\displaystyle \mathbf {R} =\mathbf {R} _{\gamma '}\mathbf {R} _{\beta '}\mathbf {R} _{\alpha }=\exp \left({\frac {-\mathbf {i} {\hat {\mathbf {z} }}'\gamma }{2}}\right)\exp \left({\frac {-\mathbf {i} {\hat {\mathbf {x} }}'\beta }{2}}\right)\exp \left({\frac {-\mathbf {i} {\hat {\mathbf {z} }}\alpha }{2}}\right)} but x ^ ′ = R α x ^ R α † and z ^ ′ = R β ′ z ^ R β ′ † . {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}'&=\mathbf {R} _{\alpha }{\hat {\mathbf {x} }}\mathbf {R} _{\alpha }^{\dagger }\quad {\text{and}}\\{\hat {\mathbf {z} }}'&=\mathbf {R} _{\beta '}{\hat {\mathbf {z} }}\mathbf {R} _{\beta '}^{\dagger }\,.\end{aligned}}} These rotors come back out of the exponentials like so: R β ′ = cos β 2 − i R α x ^ R α † sin β 2 = R α R β R α † {\displaystyle \mathbf {R} _{\beta '}=\cos {\frac {\beta }{2}}-\mathbf {i} \mathbf {R} _{\alpha }{\hat {\mathbf {x} }}\mathbf {R} _{\alpha }^{\dagger }\sin {\frac {\beta }{2}}=\mathbf {R} _{\alpha }\mathbf {R} _{\beta }\mathbf {R} _{\alpha }^{\dagger }} where R β refers to rotation in the original coordinates. Similarly for the γ rotation, R γ ′ = R β ′ R γ R β ′ † = R α R β R α † R γ R α R β † R α † . {\displaystyle \mathbf {R} _{\gamma '}=\mathbf {R} _{\beta '}\mathbf {R} _{\gamma }\mathbf {R} _{\beta '}^{\dagger }=\mathbf {R} _{\alpha }\mathbf {R} _{\beta }\mathbf {R} _{\alpha }^{\dagger }\mathbf {R} _{\gamma }\mathbf {R} _{\alpha }\mathbf {R} _{\beta }^{\dagger }\mathbf {R} _{\alpha }^{\dagger }\,.} Noting that R γ and R α commute (rotations in the same plane must commute), and the total rotor becomes R = R α R β R γ {\displaystyle \mathbf {R} =\mathbf {R} _{\alpha }\mathbf {R} _{\beta }\mathbf {R} _{\gamma }}
Thus, the compounded rotations of Euler angles become a series of equivalent rotations in the original fixed frame.
While rotors in geometric algebra work almost identically to quaternions in three dimensions, the power of this formalism is its generality: this method is appropriate and valid in spaces with any number of dimensions. In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in. It has been known that pairs of quaternions can be used to generate rotations in 4D, yielding six degrees of freedom, and the geometric algebra approach verifies this result: in 4D, there are six linearly independent bivectors that can be used as the generators of rotations. | https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions |
In linear algebra , a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space . For example, using the convention below, the matrix
rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system . To perform the rotation on a plane point with standard coordinates v = ( x , y ) , it should be written as a column vector , and multiplied by the matrix R :
If x and y are the coordinates of the endpoint of a vector with the length r and the angle ϕ {\displaystyle \phi } with respect to the x -axis, so that x = r cos ϕ {\textstyle x=r\cos \phi } and y = r sin ϕ {\displaystyle y=r\sin \phi } , then the above equations become the trigonometric summation angle formulae : R v = r [ cos ϕ cos θ − sin ϕ sin θ cos ϕ sin θ + sin ϕ cos θ ] = r [ cos ( ϕ + θ ) sin ( ϕ + θ ) ] . {\displaystyle R\mathbf {v} =r{\begin{bmatrix}\cos \phi \cos \theta -\sin \phi \sin \theta \\\cos \phi \sin \theta +\sin \phi \cos \theta \end{bmatrix}}=r{\begin{bmatrix}\cos(\phi +\theta )\\\sin(\phi +\theta )\end{bmatrix}}.} Indeed, this is the trigonometric summation angle formulae in matrix form. One way to understand this is to say we have a vector at an angle 30° from the x -axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.
The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system ( y counterclockwise from x ) by pre-multiplication (the rotation matrix R applied on the left of the column vector v to be rotated). If any one of these is changed (such as rotating axes instead of vectors, a passive transformation ), then the inverse of the example matrix should be used, which coincides with its transpose .
Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry , physics , and computer graphics . In some literature, the term rotation is generalized to include improper rotations , characterized by orthogonal matrices with a determinant of −1 (instead of +1). An improper rotation combines a proper rotation with reflections (which invert orientation ). In other cases, where reflections are not being considered, the label proper may be dropped. The latter convention is followed in this article.
Rotation matrices are square matrices , with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1 . The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO( n ) , one example of which is the rotation group SO(3) . The set of all orthogonal matrices of size n with determinant +1 or −1 is a representation of the (general) orthogonal group O( n ) .
In two dimensions, the standard rotation matrix has the following form:
This rotates column vectors by means of the following matrix multiplication ,
Thus, the new coordinates ( x ′, y ′) of a point ( x , y ) after rotation are
For example, when the vector (initially aligned with the x -axis of the Cartesian coordinate system )
is rotated by an angle θ , its new coordinates are
and when the vector (initially aligned with the y -axis of the coordinate system)
is rotated by an angle θ , its new coordinates are
The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. −90°) for R ( θ ) {\displaystyle R(\theta )} . Thus the clockwise rotation matrix is found as (by replacing θ with -θ and using the trigonometric symmetry of sin ( − θ ) = − sin ( θ ) {\textstyle \sin(-\theta )=-\sin(\theta )} and cos ( − θ ) = cos ( θ ) {\textstyle \cos(-\theta )=\cos(\theta )} )
An alternative convention uses rotating axes (instead of rotating a vector), [ 1 ] and the above matrices also represent a rotation of the axes clockwise through an angle θ .
The two-dimensional case is the only non-trivial case where the rotation matrices group is commutative; it does not matter in which order rotations are multiply performed. For the 3-dimensional case, for example, a different order of multiple rotations gives a different result. (E.g., rotating a cell phone along z -axis then y -axis is not equal to rotations along the y -axis then z -axis.)
If a standard right-handed Cartesian coordinate system is used, with the x -axis to the right and the y -axis up, the rotation R ( θ ) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R ( θ ) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics , which often have the origin in the top left corner and the y -axis down the screen or page. [ 2 ]
See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.
Matrices
are 2D rotation matrices corresponding to counter-clockwise rotations of respective angles of 90°, 180°, and 270°.
The matrices of the shape [ x − y y x ] {\displaystyle {\begin{bmatrix}x&-y\\y&x\end{bmatrix}}} form a ring , since their set is closed under addition and multiplication.
Since [ 0 − 1 1 0 ] 2 = [ − 1 0 0 − 1 ] = − I {\displaystyle {\begin{bmatrix}0&-1\\1&0\end{bmatrix}}^{2}\ =\ {\begin{bmatrix}-1&0\\0&-1\end{bmatrix}}\ =-I} (where I {\textstyle I} is the identity matrix ), the map
(where i = [ 0 − 1 1 0 ] {\displaystyle i={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}} ) is a ring isomorphism from this ring to the field of the complex numbers C {\displaystyle \mathbb {C} } (incidentally, this shows that this ring is a field). Under this isomorphism, the rotation matrices correspond to the circle of the unit complex numbers , the complex numbers of modulus 1 .
If one identifies R 2 {\displaystyle \mathbb {R} ^{2}} with C {\displaystyle \mathbb {C} } through the linear isomorphism ( a , b ) ↦ a + i b {\displaystyle (a,b)\mapsto a+ib} , where ( a , b ) ∈ R 2 {\displaystyle (a,b)\in \mathbb {R} ^{2}} and a + i b ∈ C {\displaystyle a+ib\in \mathbb {C} } , the action of a matrix [ x − y y x ] {\displaystyle {\begin{bmatrix}x&-y\\y&x\end{bmatrix}}} on a vector ( a , b ) {\displaystyle (a,b)} corresponds to multiplication on the complex number a + i b {\displaystyle a+ib} by x + iy , and a rotation correspond to multiplication by a complex number of modulus 1 .
As every 2-dimensional rotation matrix can be written
the above correspondence associates such a matrix with the complex number
where the first equality is Euler's formula , the matrix I = [ 1 0 0 1 ] {\displaystyle I={\begin{bmatrix}1&0\\0&1\end{bmatrix}}} corresponds to 1, and the matrix [ 0 − 1 1 0 ] {\displaystyle {\begin{bmatrix}0&-1\\1&0\end{bmatrix}}} corresponds to the imaginary unit i {\textstyle i} .
A basic 3D rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate vectors by an angle θ about the x -, y -, or z -axis, in three dimensions, using the right-hand rule —which codifies their alternating signs. [ 3 ] Notice that the right-hand rule only works when multiplying R ⋅ x → {\displaystyle R\cdot {\vec {x}}} . (The same matrices can also represent a clockwise rotation of the axes. [ nb 1 ] )
For column vectors , each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle θ is positive. R z , for instance, would rotate toward the y -axis a vector aligned with the x -axis , as can easily be checked by operating with R z on the vector (1,0,0) :
This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices.
Other 3D rotation matrices can be obtained from these three using matrix multiplication . For example, the product
represents a rotation whose yaw, pitch, and roll angles are α , β and γ , respectively. More formally, it is an intrinsic rotation whose Tait–Bryan angles are α , β , γ , about axes z , y , x , respectively.
Similarly, the product
represents an extrinsic rotation whose (improper) Euler angles are α , β , γ , about axes x , y , z .
These matrices produce the desired effect only if they are used to premultiply column vectors , and (since in general matrix multiplication is not commutative ) only if they are applied in the specified order (see Ambiguities for more details). The order of rotation operations is from right to left; the matrix adjacent to the column vector is the first to be applied, and then the one to the left. [ 4 ]
Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis ( Euler rotation theorem ).
There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation ). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.
Given a 3 × 3 rotation matrix R , a vector u parallel to the rotation axis must satisfy
since the rotation of u around the rotation axis must result in u . The equation above may be solved for u which is unique up to a scalar factor unless R is the identity matrix I .
Further, the equation may be rewritten
which shows that u lies in the null space of R − I .
Viewed in another way, u is an eigenvector of R corresponding to the eigenvalue λ = 1 . Every rotation matrix must have this eigenvalue, the other two eigenvalues being complex conjugates of each other. It follows that a general rotation matrix in three dimensions has, up to a multiplicative constant, only one real eigenvector.
One way to determine the rotation axis is by showing that:
Since ( R − R T ) is a skew-symmetric matrix , we can choose u such that
The matrix–vector product becomes a cross product of a vector with itself, ensuring that the result is zero:
Therefore, if
then
The magnitude of u computed this way is ‖ u ‖ = 2 sin θ , where θ is the angle of rotation.
This does not work if R is symmetric. Above, if R − R T is zero, then all subsequent steps are invalid. In this case, the angle of rotation is 0° or 180° and any nonzero column of I + R is an eigenvector of R with eigenvalue 1 because R ( I + R ) = R + R 2 = R + RR T = I + R . [ 5 ]
To find the angle of a rotation, once the axis of the rotation is known, select a vector v perpendicular to the axis. Then the angle of the rotation is the angle between v and R v .
A more direct method, however, is to simply calculate the trace : the sum of the diagonal elements of the rotation matrix. Care should be taken to select the right sign for the angle θ to match the chosen axis:
from which follows that the angle's absolute value is
For the rotation axis n = ( n 1 , n 2 , n 3 ) {\displaystyle \mathbf {n} =(n_{1},n_{2},n_{3})} , you can get the correct angle [ 6 ] from
{ cos θ = tr ( R ) − 1 2 sin θ = − tr ( K n R ) 2 {\displaystyle \left\{{\begin{matrix}\cos \theta &=&{\dfrac {\operatorname {tr} (R)-1}{2}}\\\sin \theta &=&-{\dfrac {\operatorname {tr} (K_{n}R)}{2}}\end{matrix}}\right.}
where
K n = [ 0 − n 3 n 2 n 3 0 − n 1 − n 2 n 1 0 ] {\displaystyle K_{n}={\begin{bmatrix}0&-n_{3}&n_{2}\\n_{3}&0&-n_{1}\\-n_{2}&n_{1}&0\\\end{bmatrix}}}
The matrix of a proper rotation R by angle θ around the axis u = ( u x , u y , u z ) , a unit vector with u 2 x + u 2 y + u 2 z = 1 , is given by: [ 7 ] [ 8 ] [ 9 ] [ 10 ]
A derivation of this matrix from first principles can be found in section 9.2 here. [ 11 ] The basic idea to derive this matrix is dividing the problem into few known simple steps.
This can be written more concisely as [ 12 ]
where [ u ] × is the cross product matrix of u ; the expression u ⊗ u is the outer product , and I is the identity matrix . Alternatively, the matrix entries are:
where ε jkl is the Levi-Civita symbol with ε 123 = 1 . This is a matrix form of Rodrigues' rotation formula , (or the equivalent, differently parametrized Euler–Rodrigues formula ) with [ nb 2 ]
In R 3 {\displaystyle \mathbb {R} ^{3}} the rotation of a vector x around the axis u by an angle θ can be written as:
or equivalently:
This can also be written in tensor notation as: [ 13 ]
If the 3D space is right-handed and θ > 0 , this rotation will be counterclockwise when u points towards the observer ( Right-hand rule ). Explicitly, with ( α , β , u ) {\displaystyle ({\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathbf {u} )} a right-handed orthonormal basis,
Note the striking merely apparent differences to the equivalent Lie-algebraic formulation below .
For any n -dimensional rotation matrix R acting on R n , {\displaystyle \mathbb {R} ^{n},}
It follows that:
A rotation is termed proper if det R = 1 , and improper (or a roto-reflection) if det R = –1 . For even dimensions n = 2 k , the n eigenvalues λ of a proper rotation occur as pairs of complex conjugates which are roots of unity: λ = e ± iθ j for j = 1, ..., k , which is real only for λ = ±1 . Therefore, there may be no vectors fixed by the rotation ( λ = 1 ), and thus no axis of rotation. Any fixed eigenvectors occur in pairs, and the axis of rotation is an even-dimensional subspace.
For odd dimensions n = 2 k + 1 , a proper rotation R will have an odd number of eigenvalues, with at least one λ = 1 and the axis of rotation will be an odd dimensional subspace. Proof:
Here I is the identity matrix, and we use det( R T ) = det( R ) = 1 , as well as (−1) n = −1 since n is odd. Therefore, det( R – I ) = 0 , meaning there is a nonzero vector v with ( R – I ) v = 0 , that is R v = v , a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to v , so the total dimension of fixed eigenvectors is odd.
For example, in 2-space n = 2 , a rotation by angle θ has eigenvalues λ = e iθ and λ = e − iθ , so there is no axis of rotation except when θ = 0 , the case of the null rotation. In 3-space n = 3 , the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ , e − iθ . In 4-space n = 4 , the four eigenvalues are of the form e ± iθ , e ± iφ . The null rotation has θ = φ = 0 . The case of θ = 0, φ ≠ 0 is called a simple rotation , with two unit eigenvalues forming an axis plane , and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of θ = φ is called an isoclinic rotation , having eigenvalues e ± iθ repeated twice, so every vector is rotated through an angle θ .
The trace of a rotation matrix is equal to the sum of its eigenvalues. For n = 2 , a rotation by angle θ has trace 2 cos θ . For n = 3 , a rotation around any axis by angle θ has trace 1 + 2 cos θ . For n = 4 , and the trace is 2(cos θ + cos φ ) , which becomes 4 cos θ for an isoclinic rotation.
In Euclidean geometry , a rotation is an example of an isometry , a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave " handedness " unchanged. In contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, a glide reflection does both, and an improper rotation combines a change in handedness with a normal rotation.
If a fixed point is taken as the origin of a Cartesian coordinate system , then every point can be given coordinates as a displacement from the origin. Thus one may work with the vector space of displacements instead of the points themselves. Now suppose ( p 1 , ..., p n ) are the coordinates of the vector p from the origin O to point P . Choose an orthonormal basis for our coordinates; then the squared distance to P , by Pythagoras , is
which can be computed using the matrix multiplication
A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties it can be shown that a rotation is a linear transformation of the vectors, and thus can be written in matrix form, Q p . The fact that a rotation preserves, not just ratios, but distances themselves, is stated as
or
Because this equation holds for all vectors, p , one concludes that every rotation matrix, Q , satisfies the orthogonality condition ,
Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,
Equally important, it can be shown that any matrix satisfying these two conditions acts as a rotation.
The inverse of a rotation matrix is its transpose, which is also a rotation matrix:
The product of two rotation matrices is a rotation matrix:
For n > 2 , multiplication of n × n rotation matrices is generally not commutative .
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative , we may summarize all these properties by saying that the n × n rotation matrices form a group , which for n > 2 is non-abelian , called a special orthogonal group , and denoted by SO( n ) , SO( n , R ) , SO n , or SO n ( R ) , the group of n × n rotation matrices is isomorphic to the group of rotations in an n -dimensional space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.
The interpretation of a rotation matrix can be subject to many ambiguities.
In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix inversion (for these orthogonal matrices equivalently matrix transpose ).
Consider the 3 × 3 rotation matrix
If Q acts in a certain direction, v , purely as a scaling by a factor λ , then we have
so that
Thus λ is a root of the characteristic polynomial for Q ,
Two features are noteworthy. First, one of the roots (or eigenvalues ) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3 × 3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any n × n rotation matrix. If the dimension, n , is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree n and thus n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix , so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n / 2 of them.
The sum of the entries on the main diagonal of a matrix is called the trace ; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for 2 × 2 and 3 × 3 rotation matrices that the trace reveals the angle of rotation , θ , in the two-dimensional space (or subspace). For a 2 × 2 matrix the trace is 2 cos θ , and for a 3 × 3 matrix it is 1 + 2 cos θ . In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any 3 × 3 rotation matrix a rotation axis and an angle, and these completely determine the rotation.
The constraints on a 2 × 2 rotation matrix imply that it must have the form
with a 2 + b 2 = 1 . Therefore, we may set a = cos θ and b = sin θ , for some angle θ . To solve for θ it is not enough to look at a alone or b alone; we must consider both together to place the angle in the correct quadrant , using a two-argument arctangent function.
Now consider the first column of a 3 × 3 rotation matrix,
Although a 2 + b 2 will probably not equal 1, but some value r 2 < 1 , we can use a slight variation of the previous computation to find a so-called Givens rotation that transforms the column to
zeroing b . This acts on the subspace spanned by the x - and y -axes. We can then repeat the process for the xz -subspace to zero c . Acting on the full matrix, these two rotations produce the schematic form
Shifting attention to the second column, a Givens rotation of the yz -subspace can now zero the z value. This brings the full matrix to the form
which is an identity matrix. Thus we have decomposed Q as
An n × n rotation matrix will have ( n − 1) + ( n − 2) + ⋯ + 2 + 1 , or
entries below the diagonal to zero. We can zero them by extending the same idea of stepping through the columns with a series of rotations in a fixed sequence of planes. We conclude that the set of n × n rotation matrices, each of which has n 2 entries, can be parameterized by 1 / 2 n ( n − 1) angles.
In three dimensions this restates in matrix form an observation made by Euler , so mathematicians call the ordered sequence of three angles Euler angles . However, the situation is somewhat more complicated than we have so far indicated. Despite the small dimension, we actually have considerable freedom in the sequence of axis pairs we use; and we also have some freedom in the choice of angles. Thus we find many different conventions employed when three-dimensional rotations are parameterized for physics, or medicine, or chemistry, or other disciplines. When we include the option of world axes or body axes, 24 different sequences are possible. And while some disciplines call any sequence Euler angles, others give different names (Cardano, Tait–Bryan, roll-pitch-yaw ) to different sequences.
One reason for the large number of options is that, as noted previously, rotations in three dimensions (and higher) do not commute. If we reverse a given sequence of rotations, we get a different outcome. This also implies that we cannot compose two rotations by adding their corresponding angles. Thus Euler angles are not vectors , despite a similarity in appearance as a triplet of numbers.
A 3 × 3 rotation matrix such as
suggests a 2 × 2 rotation matrix,
is embedded in the upper left corner:
This is no illusion; not just one, but many, copies of n -dimensional rotations are found within ( n + 1) -dimensional rotations, as subgroups . Each embedding leaves one direction fixed, which in the case of 3 × 3 matrices is the rotation axis. For example, we have
fixing the x -axis, the y -axis, and the z -axis, respectively. The rotation axis need not be a coordinate axis; if u = ( x , y , z ) is a unit vector in the desired direction, then
where c θ = cos θ , s θ = sin θ , is a rotation by angle θ leaving axis u fixed.
A direction in ( n + 1) -dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, S n . Thus it is natural to describe the rotation group SO( n + 1) as combining SO( n ) and S n . A suitable formalism is the fiber bundle ,
where for every direction in the base space, S n , the fiber over it in the total space, SO( n + 1) , is a copy of the fiber space, SO( n ) , namely the rotations that keep that direction fixed.
Thus we can build an n × n rotation matrix by starting with a 2 × 2 matrix, aiming its fixed axis on S 2 (the ordinary sphere in three-dimensional space), aiming the resulting rotation on S 3 , and so on up through S n −1 . A point on S n can be selected using n numbers, so we again have 1 / 2 n ( n − 1) numbers to describe any n × n rotation matrix.
In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of n − 1 Givens rotations brings the first column (and row) to (1, 0, ..., 0) , so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave (1, 0, ..., 0) fixed.
When an n × n rotation matrix Q , does not include a −1 eigenvalue, thus none of the planar rotations which it comprises are 180° rotations, then Q + I is an invertible matrix . Most rotation matrices fit this description, and for them it can be shown that ( Q − I )( Q + I ) −1 is a skew-symmetric matrix , A . Thus A T = − A ; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, A contains 1 / 2 n ( n − 1) independent numbers.
Conveniently, I − A is invertible whenever A is skew-symmetric; thus we can recover the original matrix using the Cayley transform ,
which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180° rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions.
In three dimensions, for example, we have ( Cayley 1846 )
If we condense the skew entries into a vector, ( x , y , z ) , then we produce a 90° rotation around the x -axis for (1, 0, 0), around the y -axis for (0, 1, 0), and around the z -axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as x → ∞ , ( x , 0, 0) does approach a 180° rotation around the x axis, and similarly for other directions.
For the 2D case, a rotation matrix can be decomposed into three shear matrices ( Paeth 1986 ):
This is useful, for instance, in computer graphics, since shears can be implemented with fewer multiplication instructions than rotating a bitmap directly. On modern computers, this may not matter, but it can be relevant for very old or low-end microprocessors.
A rotation can also be written as two shears and a squeeze mapping (an area preserving scaling ) ( Daubechies & Sweldens 1998 ):
Below follow some basic facts about the role of the collection of all rotation matrices of a fixed dimension (here mostly 3) in mathematics and particularly in physics where rotational symmetry is a requirement of every truly fundamental law (due to the assumption of isotropy of space ), and where the same symmetry, when present, is a simplifying property of many problems of less fundamental nature. Examples abound in classical mechanics and quantum mechanics . Knowledge of the part of the solutions pertaining to this symmetry applies (with qualifications) to all such problems and it can be factored out of a specific problem at hand, thus reducing its complexity. A prime example – in mathematics and physics – would be the theory of spherical harmonics . Their role in the group theory of the rotation groups is that of being a representation space for the entire set of finite-dimensional irreducible representations of the rotation group SO(3). For this topic, see Rotation group SO(3) § Spherical harmonics .
The main articles listed in each subsection are referred to for more detail.
The n × n rotation matrices for each n form a group , the special orthogonal group , SO( n ) . This algebraic structure is coupled with a topological structure inherited from GL n ( R ) {\displaystyle \operatorname {GL} _{n}(\mathbb {R} )} in such a way that the operations of multiplication and taking the inverse are analytic functions of the matrix entries. Thus SO( n ) is for each n a Lie group . It is compact and connected , but not simply connected . It is also a semi-simple group , in fact a simple group with the exception SO(4). [ 14 ] The relevance of this is that all theorems and all machinery from the theory of analytic manifolds (analytic manifolds are in particular smooth manifolds ) apply and the well-developed representation theory of compact semi-simple groups is ready for use.
The Lie algebra so ( n ) of SO( n ) is given by
and is the space of skew-symmetric matrices of dimension n , see classical group , where o ( n ) is the Lie algebra of O( n ) , the orthogonal group . For reference, the most common basis for so (3) is
Connecting the Lie algebra to the Lie group is the exponential map , which is defined using the standard matrix exponential series for e A [ 15 ] For any skew-symmetric matrix A , exp( A ) is always a rotation matrix. [ nb 3 ]
An important practical example is the 3 × 3 case. In rotation group SO(3) , it is shown that one can identify every A ∈ so (3) with an Euler vector ω = θ u , where u = ( x , y , z ) is a unit magnitude vector.
By the properties of the identification s u ( 2 ) ≅ R 3 {\displaystyle \mathbf {su} (2)\cong \mathbb {R} ^{3}} , u is in the null space of A . Thus, u is left invariant by exp( A ) and is hence a rotation axis.
According to Rodrigues' rotation formula on matrix form , one obtains,
where
This is the matrix for a rotation around axis u by the angle θ . For full detail, see exponential map SO(3) .
The BCH formula provides an explicit expression for Z = log( e X e Y ) in terms of a series expansion of nested commutators of X and Y . [ 16 ] This general expansion unfolds as [ nb 4 ]
In the 3 × 3 case, the general infinite expansion has a compact form, [ 17 ]
for suitable trigonometric function coefficients, detailed in the Baker–Campbell–Hausdorff formula for SO(3) .
As a group identity, the above holds for all faithful representations , including the doublet (spinor representation), which is simpler. The same explicit formula thus follows straightforwardly through Pauli matrices; see the 2 × 2 derivation for SU(2) . For the general n × n case, one might use Ref. [ 18 ]
The Lie group of n × n rotation matrices, SO( n ) , is not simply connected , so Lie theory tells us it is a homomorphic image of a universal covering group . Often the covering group, which in this case is called the spin group denoted by Spin( n ) , is simpler and more natural to work with. [ 19 ]
In the case of planar rotations, SO(2) is topologically a circle , S 1 . Its universal covering group, Spin(2), is isomorphic to the real line , R , under addition. Whenever angles of arbitrary magnitude are used one is taking advantage of the convenience of the universal cover. Every 2 × 2 rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2 π . Correspondingly, the fundamental group of SO(2) is isomorphic to the integers, Z .
In the case of spatial rotations, SO(3) is topologically equivalent to three-dimensional real projective space , RP 3 . Its universal covering group, Spin(3), is isomorphic to the 3-sphere , S 3 . Every 3 × 3 rotation matrix is produced by two opposite points on the sphere. Correspondingly, the fundamental group of SO(3) is isomorphic to the two-element group, Z 2 .
We can also describe Spin(3) as isomorphic to quaternions of unit norm under multiplication, or to certain 4 × 4 real matrices, or to 2 × 2 complex special unitary matrices , namely SU(2). The covering maps for the first and the last case are given by
and
For a detailed account of the SU(2)-covering and the quaternionic covering, see spin group SO(3) .
Many features of these cases are the same for higher dimensions. The coverings are all two-to-one, with SO( n ) , n > 2 , having fundamental group Z 2 . The natural setting for these groups is within a Clifford algebra . One type of action of the rotations is produced by a kind of "sandwich", denoted by qvq ∗ . More importantly in applications to physics, the corresponding spin representation of the Lie algebra sits inside the Clifford algebra. It can be exponentiated in the usual way to give rise to a 2-valued representation, also known as projective representation of the rotation group. This is the case with SO(3) and SU(2), where the 2-valued representation can be viewed as an "inverse" of the covering map. By properties of covering maps, the inverse can be chosen ono-to-one as a local section, but not globally.
The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form
where dθ is vanishingly small and A ∈ so (n) , for instance with A = L x ,
The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. [ 20 ] It turns out that the order in which infinitesimal rotations are applied is irrelevant . To see this exemplified, consult infinitesimal rotations SO(3) .
We have seen the existence of several decompositions that apply in any dimension, namely independent planes, sequential angles, and nested dimensions. In all these cases we can either decompose a matrix or construct one. We have also given special attention to 3 × 3 rotation matrices, and these warrant further attention, in both directions ( Stuelpnagel 1964 ).
Given the unit quaternion q = w + x i + y j + z k , the equivalent pre-multiplied (to be used with column vectors) 3 × 3 rotation matrix is [ 21 ]
Now every quaternion component appears multiplied by two in a term of degree two, and if all such terms are zero what is left is an identity matrix. This leads to an efficient, robust conversion from any quaternion – whether unit or non-unit – to a 3 × 3 rotation matrix. Given:
we can calculate
Freed from the demand for a unit quaternion, we find that nonzero quaternions act as homogeneous coordinates for 3 × 3 rotation matrices. The Cayley transform, discussed earlier, is obtained by scaling the quaternion so that its w component is 1. For a 180° rotation around any axis, w will be zero, which explains the Cayley limitation.
The sum of the entries along the main diagonal (the trace ), plus one, equals 4 − 4( x 2 + y 2 + z 2 ) , which is 4 w 2 . Thus we can write the trace itself as 2 w 2 + 2 w 2 − 1 ; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: 2 x 2 + 2 w 2 − 1 , 2 y 2 + 2 w 2 − 1 , and 2 z 2 + 2 w 2 − 1 . So we can easily compare the magnitudes of all four quaternion components using the matrix diagonal. We can, in fact, obtain all four magnitudes using sums and square roots, and choose consistent signs using the skew-symmetric part of the off-diagonal entries:
Alternatively, use a single square root and division
This is numerically stable so long as the trace, t , is not negative; otherwise, we risk dividing by (nearly) zero. In that case, suppose Q xx is the largest diagonal entry, so x will have the largest magnitude (the other cases are derived by cyclic permutation); then the following is safe.
If the matrix contains significant error, such as accumulated numerical error, we may construct a symmetric 4 × 4 matrix,
and find the eigenvector , ( x , y , z , w ) , of its largest magnitude eigenvalue. (If Q is truly a rotation matrix, that value will be 1.) The quaternion so obtained will correspond to the rotation matrix closest to the given matrix ( Bar-Itzhack 2000 ) (Note: formulation of the cited article is post-multiplied, works with row vectors).
If the n × n matrix M is nonsingular, its columns are linearly independent vectors; thus the Gram–Schmidt process can adjust them to be an orthonormal basis. Stated in terms of numerical linear algebra , we convert M to an orthogonal matrix, Q , using QR decomposition . However, we often prefer a Q closest to M , which this method does not accomplish. For that, the tool we want is the polar decomposition ( Fan & Hoffman 1955 ; Higham 1989 ).
To measure closeness, we may use any matrix norm invariant under orthogonal transformations. A convenient choice is the Frobenius norm , ‖ Q − M ‖ F , squared, which is the sum of the squares of the element differences. Writing this in terms of the trace , Tr , our goal is,
Though written in matrix terms, the objective function is just a quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a 3 × 3 matrix, the orthogonality constraint implies six scalar equalities that the entries of Q must satisfy. To incorporate the constraint(s), we may employ a standard technique, Lagrange multipliers , assembled as a symmetric matrix, Y . Thus our method is:
Consider a 2 × 2 example. Including constraints, we seek to minimize
Taking the derivative with respect to Q xx , Q xy , Q yx , Q yy in turn, we assemble a matrix.
In general, we obtain the equation
so that
where Q is orthogonal and S is symmetric. To ensure a minimum, the Y matrix (and hence S ) must be positive definite. Linear algebra calls QS the polar decomposition of M , with S the positive square root of S 2 = M T M .
When M is non-singular , the Q and S factors of the polar decomposition are uniquely determined. However, the determinant of S is positive because S is positive definite, so Q inherits the sign of the determinant of M . That is, Q is only guaranteed to be orthogonal, not a rotation matrix. This is unavoidable; an M with negative determinant has no uniquely defined closest rotation matrix.
To efficiently construct a rotation matrix Q from an angle θ and a unit axis u , we can take advantage of symmetry and skew-symmetry within the entries. If x , y , and z are the components of the unit vector representing the axis, and
then
Determining an axis and angle, like determining a quaternion, is only possible up to the sign; that is, ( u , θ ) and (− u , − θ ) correspond to the same rotation matrix, just like q and − q . Additionally, axis–angle extraction presents additional difficulties. The angle can be restricted to be from 0° to 180°, but angles are formally ambiguous by multiples of 360°. When the angle is zero, the axis is undefined. When the angle is 180°, the matrix becomes symmetric, which has implications in extracting the axis. Near multiples of 180°, care is needed to avoid numerical problems: in extracting the angle, a two-argument arctangent with atan2 (sin θ , cos θ ) equal to θ avoids the insensitivity of arccos; and in computing the axis magnitude in order to force unit magnitude, a brute-force approach can lose accuracy through underflow ( Moler & Morrison 1983 ).
A partial approach is as follows:
The x -, y -, and z -components of the axis would then be divided by r . A fully robust approach will use a different algorithm when t , the trace of the matrix Q , is negative, as with quaternion extraction. When r is zero because the angle is zero, an axis must be provided from some source other than the matrix.
Complexity of conversion escalates with Euler angles (used here in the broad sense). The first difficulty is to establish which of the twenty-four variations of Cartesian axis order we will use. Suppose the three angles are θ 1 , θ 2 , θ 3 ; physics and chemistry may interpret these as
while aircraft dynamics may use
One systematic approach begins with choosing the rightmost axis. Among all permutations of ( x , y , z ) , only two place that axis first; one is an even permutation and the other odd. Choosing parity thus establishes the middle axis. That leaves two choices for the left-most axis, either duplicating the first or not. These three choices gives us 3 × 2 × 2 = 12 variations; we double that to 24 by choosing static or rotating axes.
This is enough to construct a matrix from angles, but triples differing in many ways can give the same rotation matrix. For example, suppose we use the zyz convention above; then we have the following equivalent pairs:
Angles for any order can be found using a concise common routine ( Herter & Lott 1993 ; Shoemake 1994 ).
The problem of singular alignment, the mathematical analog of physical gimbal lock , occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles.
The singularities are avoided when considering and manipulating the rotation matrix as orthonormal row vectors (in 3D applications often named the right-vector, up-vector and out-vector) instead of as angles. The singularities are also avoided when working with quaternions.
In some instances it is interesting to describe a rotation by specifying how a vector is mapped into another through the shortest path (smallest angle). In R 3 {\displaystyle \mathbb {R} ^{3}} this completely describes the associated rotation matrix. In general, given x , y ∈ S {\displaystyle \mathbb {S} } n , the matrix
belongs to SO( n + 1) and maps x to y . [ 22 ]
In materials science , the four-dimensional stiffness and compliance tensors are often simplified to a two-dimensional matrix using Voigt notation . When applying a rotational transform through angle θ {\displaystyle \theta } in this notation, the rotation matrix is given by [ 23 ]
This is particularly useful in composite laminate design, where plies are often rotated by a certain angle to bring the properties of the laminate closer to isotropic .
We sometimes need to generate a uniformly distributed random rotation matrix. It seems intuitively clear in two dimensions that this means the rotation angle is uniformly distributed between 0 and 2 π . That intuition is correct, but does not carry over to higher dimensions. For example, if we decompose 3 × 3 rotation matrices in axis–angle form, the angle should not be uniformly distributed; the probability that (the magnitude of) the angle is at most θ should be 1 / π ( θ − sin θ ) , for 0 ≤ θ ≤ π .
Since SO( n ) is a connected and locally compact Lie group, we have a simple standard criterion for uniformity, namely that the distribution be unchanged when composed with any arbitrary rotation (a Lie group "translation"). This definition corresponds to what is called Haar measure . León, Massé & Rivest (2006) show how to use the Cayley transform to generate and test matrices according to this criterion.
We can also generate a uniform distribution in any dimension using the subgroup algorithm of Diaconis & Shahshahani (1987) . This recursively exploits the nested dimensions group structure of SO( n ) , as follows. Generate a uniform angle and construct a 2 × 2 rotation matrix. To step from n to n + 1 , generate a vector v uniformly distributed on the n -sphere S n , embed the n × n matrix in the next larger size with last column (0, ..., 0, 1) , and rotate the larger matrix so the last column becomes v .
As usual, we have special alternatives for the 3 × 3 case. Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. Arvo (1992) takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation.
Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere. Since the homomorphism is a local isometry , we immediately conclude that to produce a uniform distribution on SO(3) we may use a uniform distribution on S 3 . In practice: create a four-element vector where each element is a sampling of a normal distribution. Normalize its length and you have a uniformly sampled random unit quaternion which represents a uniformly sampled random rotation. Note that the aforementioned only applies to rotations in dimension 3. For a generalised idea of quaternions, one should look into Rotors .
Euler angles can also be used, though not with each angle uniformly distributed ( Murnaghan 1962 ; Miles 1965 ).
For the axis–angle form, the axis is uniformly distributed over the unit sphere of directions, S 2 , while the angle has the nonuniform distribution over [0, π ] noted previously ( Miles 1965 ). | https://en.wikipedia.org/wiki/Rotation_matrix |
In mathematics , a rotation of axes in two dimensions is a mapping from an xy - Cartesian coordinate system to an x′y′ -Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle θ {\displaystyle \theta } . A point P has coordinates ( x , y ) with respect to the original system and coordinates ( x′ , y′ ) with respect to the new system. [ 1 ] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle θ {\displaystyle \theta } . A rotation of axes in more than two dimensions is defined similarly. [ 2 ] [ 3 ] A rotation of axes is a linear map [ 4 ] [ 5 ] and a rigid transformation .
Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry . To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas , the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola , ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates . [ 6 ]
The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.
The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle θ {\displaystyle \theta } into the x′y′ axes, are derived as follows.
In the xy system, let the point P have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} . Then, in the x′y′ system, P will have polar coordinates ( r , α − θ ) {\displaystyle (r,\alpha -\theta )} .
Using trigonometric functions , we have
and using the standard trigonometric formulae for differences , we have
Substituting equations ( 1 ) and ( 2 ) into equations ( 3 ) and ( 4 ), we obtain [ 7 ]
Equations ( 5 ) and ( 6 ) can be represented in matrix form as [ x ′ y ′ ] = [ cos θ sin θ − sin θ cos θ ] [ x y ] , {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}},}
which is the standard matrix equation of a rotation of axes in two dimensions. [ 8 ]
The inverse transformation is [ 9 ]
or [ x y ] = [ cos θ − sin θ sin θ cos θ ] [ x ′ y ′ ] . {\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x'\\y'\end{bmatrix}}.}
Find the coordinates of the point P 1 = ( x , y ) = ( 3 , 1 ) {\displaystyle P_{1}=(x,y)=({\sqrt {3}},1)} after the axes have been rotated through the angle θ 1 = π / 6 {\displaystyle \theta _{1}=\pi /6} , or 30°.
Solution: x ′ = 3 cos ( π / 6 ) + 1 sin ( π / 6 ) = ( 3 ) ( 3 / 2 ) + ( 1 ) ( 1 / 2 ) = 2 {\displaystyle x'={\sqrt {3}}\cos(\pi /6)+1\sin(\pi /6)=({\sqrt {3}})({\sqrt {3}}/2)+(1)(1/2)=2} y ′ = 1 cos ( π / 6 ) − 3 sin ( π / 6 ) = ( 1 ) ( 3 / 2 ) − ( 3 ) ( 1 / 2 ) = 0. {\displaystyle y'=1\cos(\pi /6)-{\sqrt {3}}\sin(\pi /6)=(1)({\sqrt {3}}/2)-({\sqrt {3}})(1/2)=0.}
The axes have been rotated counterclockwise through an angle of θ 1 = π / 6 {\displaystyle \theta _{1}=\pi /6} and the new coordinates are P 1 = ( x ′ , y ′ ) = ( 2 , 0 ) {\displaystyle P_{1}=(x',y')=(2,0)} . Note that the point appears to have been rotated clockwise through π / 6 {\displaystyle \pi /6} with respect to fixed axes so it now coincides with the (new) x′ axis.
Find the coordinates of the point P 2 = ( x , y ) = ( 7 , 7 ) {\displaystyle P_{2}=(x,y)=(7,7)} after the axes have been rotated clockwise 90°, that is, through the angle θ 2 = − π / 2 {\displaystyle \theta _{2}=-\pi /2} , or −90°.
Solution: [ x ′ y ′ ] = [ cos ( − π / 2 ) sin ( − π / 2 ) − sin ( − π / 2 ) cos ( − π / 2 ) ] [ 7 7 ] = [ 0 − 1 1 0 ] [ 7 7 ] = [ − 7 7 ] . {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos(-\pi /2)&\sin(-\pi /2)\\-\sin(-\pi /2)&\cos(-\pi /2)\end{bmatrix}}{\begin{bmatrix}7\\7\end{bmatrix}}={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}{\begin{bmatrix}7\\7\end{bmatrix}}={\begin{bmatrix}-7\\7\end{bmatrix}}.}
The axes have been rotated through an angle of θ 2 = − π / 2 {\displaystyle \theta _{2}=-\pi /2} , which is in the clockwise direction and the new coordinates are P 2 = ( x ′ , y ′ ) = ( − 7 , 7 ) {\displaystyle P_{2}=(x',y')=(-7,7)} . Again, note that the point appears to have been rotated counterclockwise through π / 2 {\displaystyle \pi /2} with respect to fixed axes.
The most general equation of the second degree has the form
Through a change of coordinates (a rotation of axes and a translation of axes ), equation ( 9 ) can be put into a standard form , which is usually easier to work with. It is always possible to rotate the coordinates at a specific angle so as to eliminate the x′y′ term. Substituting equations ( 7 ) and ( 8 ) into equation ( 9 ), we obtain
where
If θ {\displaystyle \theta } is selected so that cot 2 θ = ( A − C ) / B {\displaystyle \cot 2\theta =(A-C)/B} we will have B ′ = 0 {\displaystyle B'=0} and the x′y′ term in equation ( 10 ) will vanish. [ 11 ]
When a problem arises with B , D and E all different from zero, they can be eliminated by performing in succession a rotation (eliminating B ) and a translation (eliminating the D and E terms). [ 12 ]
A non-degenerate conic section given by equation ( 9 ) can be identified by evaluating B 2 − 4 A C {\displaystyle B^{2}-4AC} . The conic section is: [ 13 ]
Suppose a rectangular xyz -coordinate system is rotated around its z axis counterclockwise (looking down the positive z axis) through an angle θ {\displaystyle \theta } , that is, the positive x axis is rotated immediately into the positive y axis. The z coordinate of each point is unchanged and the x and y coordinates transform as above. The old coordinates ( x , y , z ) of a point Q are related to its new coordinates ( x′ , y′ , z′ ) by [ 14 ] [ x ′ y ′ z ′ ] = [ cos θ sin θ 0 − sin θ cos θ 0 0 0 1 ] [ x y z ] . {\displaystyle {\begin{bmatrix}x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}.}
Generalizing to any finite number of dimensions, a rotation matrix A {\displaystyle A} is an orthogonal matrix that differs from the identity matrix in at most four elements. These four elements are of the form
for some θ {\displaystyle \theta } and some i ≠ j . [ 15 ]
Find the coordinates of the point P 3 = ( w , x , y , z ) = ( 1 , 1 , 1 , 1 ) {\displaystyle P_{3}=(w,x,y,z)=(1,1,1,1)} after the positive w axis has been rotated through the angle θ 3 = π / 12 {\displaystyle \theta _{3}=\pi /12} , or 15°, into the positive z axis.
Solution: [ w ′ x ′ y ′ z ′ ] = [ cos ( π / 12 ) 0 0 sin ( π / 12 ) 0 1 0 0 0 0 1 0 − sin ( π / 12 ) 0 0 cos ( π / 12 ) ] [ w x y z ] ≈ [ 0.96593 0.0 0.0 0.25882 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 − 0.25882 0.0 0.0 0.96593 ] [ 1.0 1.0 1.0 1.0 ] = [ 1.22475 1.00000 1.00000 0.70711 ] . {\displaystyle {\begin{aligned}{\begin{bmatrix}w'\\x'\\y'\\z'\end{bmatrix}}&={\begin{bmatrix}\cos(\pi /12)&0&0&\sin(\pi /12)\\0&1&0&0\\0&0&1&0\\-\sin(\pi /12)&0&0&\cos(\pi /12)\end{bmatrix}}{\begin{bmatrix}w\\x\\y\\z\end{bmatrix}}\\[4pt]&\approx {\begin{bmatrix}0.96593&0.0&0.0&0.25882\\0.0&1.0&0.0&0.0\\0.0&0.0&1.0&0.0\\-0.25882&0.0&0.0&0.96593\end{bmatrix}}{\begin{bmatrix}1.0\\1.0\\1.0\\1.0\end{bmatrix}}={\begin{bmatrix}1.22475\\1.00000\\1.00000\\0.70711\end{bmatrix}}.\end{aligned}}} | https://en.wikipedia.org/wiki/Rotation_of_axes_in_two_dimensions |
In astronomy , the rotation period or spin period [ 1 ] of a celestial object (e.g., star, planet, moon, asteroid) has two definitions. The first one corresponds to the sidereal rotation period (or sidereal day ), i.e., the time that the object takes to complete a full rotation around its axis relative to the background stars ( inertial space ). The other type of commonly used "rotation period" is the object's synodic rotation period (or solar day ), which may differ, by a fraction of a rotation or more than one rotation, to accommodate the portion of the object's orbital period around a star or another body during one day.
For solid objects, such as rocky planets and asteroids , the rotation period is a single value. For gaseous or fluid bodies, such as stars and giant planets , the period of rotation varies from the object's equator to its pole due to a phenomenon called differential rotation . Typically, the stated rotation period for a giant planet (such as Jupiter, Saturn, Uranus, Neptune) is its internal rotation period, as determined from the rotation of the planet's magnetic field . For objects that are not spherically symmetrical , the rotation period is, in general, not fixed, even in the absence of gravitational or tidal forces. This is because, although the rotation axis is fixed in space (by the conservation of angular momentum ), it is not necessarily fixed in the body of the object itself. [ citation needed ] As a result of this, the moment of inertia of the object around the rotation axis can vary, and hence the rate of rotation can vary (because the product of the moment of inertia and the rate of rotation is equal to the angular momentum, which is fixed). For example, Hyperion , a moon of Saturn , exhibits this behaviour, and its rotation period is described as chaotic . | https://en.wikipedia.org/wiki/Rotation_period_(astronomy) |
In astronomy, rotational Brownian motion is the random walk in orientation of a binary star 's orbital plane, induced by gravitational perturbations from passing stars.
Consider a binary that consists of two massive objects (stars, black holes etc.) and that is embedded in a stellar system containing a large number of stars. Let M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} be the masses of the two components of the binary whose total mass is M 12 = M 1 + M 2 {\displaystyle M_{12}=M_{1}+M_{2}} . A field star that approaches the binary with impact parameter p {\displaystyle p} and velocity V {\displaystyle V} passes a distance r p {\displaystyle r_{p}} from the binary, where
p 2 = r p 2 ( 1 + 2 G M 12 / V 2 r p ) ≈ 2 G M 12 r p / V 2 ; {\displaystyle p^{2}=r_{p}^{2}\left(1+2GM_{12}/V^{2}r_{p}\right)\approx 2GM_{12}r_{p}/V^{2};}
the latter expression is valid in the limit that gravitational focusing dominates the encounter rate. The rate of encounters with stars that interact strongly with the binary, i.e. that satisfy r p < a {\displaystyle r_{p}<a} , is approximately n π p 2 σ = 2 π G M 12 n a / σ {\displaystyle n\pi p^{2}\sigma =2\pi GM_{12}na/\sigma } where n {\displaystyle n} and σ {\displaystyle \sigma } are the number density and velocity dispersion of the field stars and a {\displaystyle a} is the semi-major axis of the binary.
As it passes near the binary, the field star experiences a change in velocity of order
Δ V ≈ V b i n = G M 12 / a {\displaystyle \Delta V\approx V_{\rm {bin}}={\sqrt {GM_{12}/a}}} ,
where V b i n {\displaystyle V_{\rm {bin}}} is the relative velocity of the two stars in the binary.
The change in the field star's specific angular momentum with respect to the binary, l {\displaystyle l} , is then Δ l ≈ a V bin . Conservation of angular momentum implies that the binary's angular momentum changes by Δ l bin ≈ -(m/μ 12 )Δ l where m is the mass of a field star and μ 12 is the binary reduced mass . Changes in the magnitude of l bin correspond to changes in the binary's orbital eccentricity via the relation e = 1 - l b 2 / GM 12 μ 12 a . Changes in the direction of l bin correspond to changes in the orientation of the binary, leading to rotational diffusion. The rotational diffusion coefficient is
⟨ Δ ξ 2 ⟩ = ⟨ Δ l b i n 2 ⟩ / l b i n 2 ≈ ( m M 12 ) 2 ⟨ Δ l 2 ⟩ / G M 12 a ≈ m M 12 G ρ a σ {\displaystyle \langle \Delta \xi ^{2}\rangle =\langle \Delta l_{\rm {bin}}^{2}\rangle /l_{\rm {bin}}^{2}\approx \left({m \over M_{12}}\right)^{2}\langle \Delta l^{2}\rangle /GM_{12}a\approx {m \over M_{12}}{G\rho a \over \sigma }}
where ρ = mn is the mass density of field stars.
Let F (θ, t ) be the probability that the rotation axis of the binary is oriented at angle θ at time t . The evolution equation for F is [ 1 ]
∂ F ∂ t = 1 sin θ ∂ ∂ θ ( sin θ ⟨ Δ ξ 2 ⟩ 4 ∂ F ∂ θ ) . {\displaystyle {\partial F \over \partial t}={1 \over \sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\langle \Delta \xi ^{2}\rangle \over 4}{\partial F \over \partial \theta }\right).}
If <Δξ 2 >, a , ρ and σ are constant in time, this becomes
∂ F ∂ τ = 1 2 ∂ ∂ μ [ ( 1 − μ 2 ) ∂ F ∂ μ ] {\displaystyle {\partial F \over \partial \tau }={1 \over 2}{\partial \over \partial \mu }\left[(1-\mu ^{2}){\partial F \over \partial \mu }\right]}
where μ = cos θ and τ is the time in units of the relaxation time t rel , where
t r e l ≈ M 12 m σ G ρ a . {\displaystyle t_{\rm {rel}}\approx {M_{12} \over m}{\sigma \over G\rho a}.}
The solution to this equation states that the expectation value of μ decays with time as
μ ¯ = μ ¯ 0 e − τ . {\displaystyle {\overline {\mu }}={\overline {\mu }}_{0}e^{-\tau }.}
Hence, t rel is the time constant for the binary's orientation to be randomized by torques from field stars.
Rotational Brownian motion was first discussed in the context of binary supermassive black holes at the centers of galaxies. [ 2 ] Perturbations from passing stars can alter the orbital plane of such a binary, which in turn alters the direction of the spin axis of the single black hole that forms when the two coalesce.
Rotational Brownian motion is often observed in N-body simulations of galaxies containing binary black holes. [ 3 ] [ 4 ] The massive binary sinks to the center of the galaxy via dynamical friction where it interacts with passing stars. The same gravitational perturbations that induce a random walk in the orientation of the binary, also cause the binary to shrink, via the gravitational slingshot . It can be shown [ 2 ] that the rms change in the binary's orientation, from the time the binary forms until the two black holes collide, is roughly
δ θ ≈ 20 m / M 12 . {\displaystyle \delta \theta \approx {\sqrt {20m/M_{12}}}.}
In a real galaxy, the two black holes would eventually coalesce due to emission of gravitational waves . The spin axis of the coalesced hole will be aligned with the angular momentum axis of the orbit of the pre-existing binary. Hence, a mechanism like rotational Brownian motion that affects the orbits of binary black holes can also affect the distribution of black hole spins. This may explain in part why the spin axes of supermassive black holes appear to be randomly aligned with respect to their host galaxies. [ 5 ] | https://en.wikipedia.org/wiki/Rotational_Brownian_motion_(astronomy) |
Rotational correlation time ( τ c {\displaystyle \tau _{c}} ) is the average time it takes for a molecule to rotate one radian .
[ 1 ] In solution, rotational correlation times are in the order of picoseconds . For example, the τ c = {\displaystyle \tau _{c}=} 1.7 ps for water, [ 2 ] and 100 ps for a pyrroline nitroxyl radical in a DMSO-water mixture. [ 3 ] Rotational correlation times are employed in the measurement of microviscosity (viscosity at the molecular level) and in protein characterization.
Rotational correlation times may be measured by rotational (microwave) , dielectric , and nuclear magnetic resonance (NMR) spectroscopy. [ 4 ] Rotational correlation times of probe molecules in media have been measured by fluorescence lifetime or for radicals , from the linewidths of electron spin resonances . [ 3 ]
This nuclear magnetic resonance –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotational_correlation_time |
Rotational modulation collimators (or RMCs ) are a specialization of the modulation collimator , an imaging device invented by Minoru Oda [ fr ; de ] . Devices of this type create images of high energy X-rays (or other radiations that cast shadows). Since high energy X-rays are not easily focused, such optics have found applications in various instruments. RMCs selectively block and unblock X-rays in a way which depends on their incoming direction, converting image information into time variations. Various mathematical transformations can then reconstitute the image of the source.
The Small Astronomy Satellite 3 , launched in 1975, was one orbiting experiment that used RMCs. A more recent satellite that used RMCs was RHESSI .
This astronomy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotational_modulation_collimator |
In chemistry , the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy.
The total canonical partition function Z {\displaystyle Z} of a system of N {\displaystyle N} identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions ζ {\displaystyle \zeta } : [ 1 ] Z = ζ N N ! {\displaystyle Z={\frac {\zeta ^{N}}{N!}}} with: ζ = ∑ j g j e − E j / k B T , {\displaystyle \zeta =\sum _{j}g_{j}e^{-E_{j}/k_{\text{B}}T},} where g j {\displaystyle g_{j}} is the degeneracy of the j th quantum level of an individual particle, k B {\displaystyle k_{\text{B}}} is the Boltzmann constant , and T {\displaystyle T} is the absolute temperature of system.
For molecules, under the assumption that total energy levels E j {\displaystyle E_{j}} can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom) [ 2 ] E j = ∑ i E j i = E j trans + E j ns + E j rot + E j vib + E j e {\displaystyle E_{j}=\sum _{i}E_{j}^{i}=E_{j}^{\text{trans}}+E_{j}^{\text{ns}}+E_{j}^{\text{rot}}+E_{j}^{\text{vib}}+E_{j}^{\text{e}}} and the number of degenerate states are given as products of the single contributions g j = ∏ i g j i = g j trans g j ns g j rot g j vib g j e , {\displaystyle g_{j}=\prod _{i}g_{j}^{i}=g_{j}^{\text{trans}}g_{j}^{\text{ns}}g_{j}^{\text{rot}}g_{j}^{\text{vib}}g_{j}^{\text{e}},} where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions ζ = ∑ j g j e − E j / k B T {\displaystyle \zeta =\sum _{j}g_{j}e^{-E_{j}/k_{\text{B}}T}} can be written as a product itself ζ = ∏ i ζ i = ζ trans ζ ns ζ rot ζ vib ζ e . {\displaystyle \zeta =\prod _{i}\zeta ^{i}=\zeta ^{\text{trans}}\zeta ^{\text{ns}}\zeta ^{\text{rot}}\zeta ^{\text{vib}}\zeta ^{\text{e}}.}
Rotational energies are quantized. For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E J rot = J 2 2 I = J ( J + 1 ) ℏ 2 2 I = J ( J + 1 ) B . {\displaystyle E_{J}^{\text{rot}}={\frac {\mathbf {J} ^{2}}{2I}}={\frac {J(J+1)\hbar ^{2}}{2I}}=J(J+1)B.} J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0 , 1 , 2 , … {\displaystyle J=0,1,2,\ldots } , B = ℏ 2 2 I {\displaystyle B={\frac {\hbar ^{2}}{2I}}} is the rotational constant, and I {\displaystyle I} is the moment of inertia . Here we are using B in energy units. If it is expressed in frequency units, replace B by hB in all the expression that follow, where h is the Planck constant . If B is given in units of c m − 1 {\displaystyle \mathrm {cm^{-1}} } , then replace B by hcB where c is the speed of light in vacuum.
For each value of J, we have rotational degeneracy, g j {\displaystyle g_{j}} = (2J+1), so the rotational partition function is therefore ζ rot = ∑ J = 0 ∞ g j e − E J / k B T = ∑ J = 0 ∞ ( 2 J + 1 ) e − J ( J + 1 ) B / k B T . {\displaystyle \zeta ^{\text{rot}}=\sum _{J=0}^{\infty }g_{j}e^{-E_{J}/k_{\text{B}}T}=\sum _{J=0}^{\infty }(2J+1)e^{-J(J+1)B/k_{\text{B}}T}.}
For all but the lightest molecules or the very lowest temperatures we have B ≪ k B T {\displaystyle B\ll k_{\text{B}}T} . This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable. ζ rot ≈ ∫ 0 ∞ ( 2 J + 1 ) e − J ( J + 1 ) B / k B T d J = k B T B . {\displaystyle \zeta ^{\text{rot}}\approx \int _{0}^{\infty }(2J+1)e^{-J(J+1)B/k_{\text{B}}T}dJ={\frac {k_{\text{B}}T}{B}}.}
This approximation is known as the high temperature limit. It is also called the classical approximation as this is the result for the canonical partition function for a classical rigid rod.
Using the Euler–Maclaurin formula an improved estimate can be found [ 3 ] ζ rot = k B T B + 1 3 + 1 15 ( B k B T ) + 4 315 ( B k B T ) 2 + 1 315 ( B k B T ) 3 + ⋯ . {\displaystyle \zeta ^{\text{rot}}={\frac {k_{\text{B}}T}{B}}+{\frac {1}{3}}+{\frac {1}{15}}\left({\frac {B}{k_{\text{B}}T}}\right)+{\frac {4}{315}}\left({\frac {B}{k_{\text{B}}T}}\right)^{2}+{\frac {1}{315}}\left({\frac {B}{k_{\text{B}}T}}\right)^{3}+\cdots .}
For the CO molecule at T = 300 K {\displaystyle T=\mathrm {300~K} } , the (unit less) contribution ζ rot {\displaystyle \zeta ^{\text{rot}}} to ζ {\displaystyle \zeta } turns out to be in the range of 10 2 {\displaystyle 10^{2}} .
The mean thermal rotational energy per molecule can now be computed by taking the derivative of ζ rot {\displaystyle \zeta ^{\text{rot}}} with respect to temperature T {\displaystyle T} . In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k B T {\displaystyle k_{\text{B}}T} .
For a diatomic molecule with a center of symmetry, such as H 2 , N 2 , C O 2 , {\displaystyle {\rm {H_{2},N_{2},CO_{2},}}} or H 2 C 2 {\displaystyle \mathrm {H_{2}C_{2}} } (i.e. D ∞ h {\displaystyle D_{\infty h}} point group ), rotation of a molecule by π {\displaystyle \pi } radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The spin–statistics theorem of quantum mechanics requires that the total molecular wavefunction be either symmetric or antisymmetric with respect to this rotation depending upon whether an even or odd number of pairs of fermion nuclear pairs are exchanged. A given electronic & vibrational wavefunction will either be symmetric or antisymmetric with respect to this rotation. The rotational wavefunction with quantum number J will have a sign change of ( − 1 ) J {\displaystyle (-1)^{J}} . The nuclear spins states can be separated into those that are symmetric or antisymmetric with respect to the nuclear permutations produced by the rotation. For the case of a symmetric diatomic with nuclear spin quantum number I for each nucleus, there are ( I + 1 ) ( 2 I + 1 ) {\displaystyle (I+1)(2I+1)} symmetric spin functions and I ( 2 I + 1 ) {\displaystyle I(2I+1)} are antisymmetric functions for a total number of nuclear functions g ns = ( 2 I + 1 ) 2 {\displaystyle g^{\text{ns}}=(2I+1)^{2}} . Nuclei with an even nuclear mass number are bosons and have integer nuclear spin quantum number, I . Nuclei with odd mass number are fermions and had half integer I . For the case of H 2 , rotation exchanges a single pair of fermions and so the overall wavefunction must be antisymmetric under the half rotation. The vibration-electronic function is symmetric and so the rotation-vibration-electronic will be even or odd depending upon whether J is an even or odd integer. Since the total wavefunction must be odd, the even J levels can only use the antisymmetric functions (only one for I = 1/2) while the odd J levels can use the symmetric functions ( three for I = 1/2). For D2, I = 1 and thus there are six symmetric functions, which go with the even J levels to produce an overall symmetric wavefunction, and three antisymmetric functions that must go with odd J rotational levels to produce an overall even function. The number of nuclear spin functions that are compatible with a given rotation-vibration-electronic state is called the nuclear spin statistical weight of the level, often represented as g J {\displaystyle g_{J}} . Averaging over both even and odd J levels, the mean statistical weight is ( 1 / 2 ) ( 2 I + 1 ) 2 {\displaystyle (1/2)(2I+1)^{2}} , which is one half the value of g ns {\displaystyle g^{\text{ns}}} expected ignoring the quantum statistical restrictions. In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor σ = 2 {\displaystyle \sigma =2} with σ {\displaystyle \sigma } known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without.
A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A , B , {\displaystyle A,B,} and C {\displaystyle C} , which can often be determined by rotational spectroscopy . In terms of these constants, the rotational partition function can be written in the high temperature limit as [ 4 ] ζ rot ≈ π σ ( k B T ) 3 A B C {\displaystyle \zeta ^{\text{rot}}\approx {\frac {\sqrt {\pi }}{\sigma }}{\sqrt {\frac {(k_{\text{B}}T)^{3}}{ABC}}}} with σ {\displaystyle \sigma } again known as the rotational symmetry number [ 5 ] which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Like in the case of the diatomic treated explicitly above, this factor corrects for the fact that only a fraction of the nuclear spin functions can be used for any given molecular level to construct wavefunctions that overall obey the required exchange symmetries. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: ζ rot ≈ 5.34 × 10 6 σ T 3 A B C {\displaystyle \zeta ^{\text{rot}}\approx {\frac {5.34\times 10^{6}}{\sigma }}{\sqrt {\frac {T^{3}}{ABC}}}} where the prefactor comes from ( π k B ) 3 h 3 = 5.34 × 10 6 {\displaystyle {\sqrt {\frac {(\pi k_{\text{B}})^{3}}{h^{3}}}}=5.34\times 10^{6}} when A , B , and C are expressed in units of MHz. [ 6 ]
The expressions for ζ rot {\displaystyle \zeta ^{\text{rot}}} works for asymmetric, symmetric and spherical top rotors. | https://en.wikipedia.org/wiki/Rotational_partition_function |
Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase . The rotational spectrum ( power spectral density vs. rotational frequency ) of polar molecules can be measured in absorption or emission by microwave spectroscopy [ 1 ] or by far infrared spectroscopy. The rotational spectra of non-polar molecules cannot be observed by those methods, but can be observed and measured by Raman spectroscopy . Rotational spectroscopy is sometimes referred to as pure rotational spectroscopy to distinguish it from rotational-vibrational spectroscopy where changes in rotational energy occur together with changes in vibrational energy, and also from ro-vibronic spectroscopy (or just vibronic spectroscopy ) where rotational, vibrational and electronic energy changes occur simultaneously.
For rotational spectroscopy, molecules are classified according to symmetry into spherical tops, linear molecules, and symmetric tops; analytical expressions can be derived for the rotational energy terms of these molecules. Analytical expressions can be derived for the fourth category, asymmetric top, for rotational levels up to J=3, but higher energy levels need to be determined using numerical methods. The rotational energies are derived theoretically by considering the molecules to be rigid rotors and then applying extra terms to account for centrifugal distortion , fine structure , hyperfine structure and Coriolis coupling . Fitting the spectra to the theoretical expressions gives numerical values of the angular moments of inertia from which very precise values of molecular bond lengths and angles can be derived in favorable cases. In the presence of an electrostatic field there is Stark splitting which allows molecular electric dipole moments to be determined.
An important application of rotational spectroscopy is in exploration of the chemical composition of the interstellar medium using radio telescopes .
Rotational spectroscopy has primarily been used to investigate fundamental aspects of molecular physics. It is a uniquely precise tool for the determination of molecular structure in gas-phase molecules. It can be used to establish barriers to internal rotation such as that associated with the rotation of the CH 3 group relative to the C 6 H 4 Cl group in chlorotoluene ( C 7 H 7 Cl ). [ 2 ] When fine or hyperfine structure can be observed, the technique also provides information on the electronic structures of molecules. Much of current understanding of the nature of weak molecular interactions such as van der Waals , hydrogen and halogen bonds has been established through rotational spectroscopy. In connection with radio astronomy, the technique has a key role in exploration of the chemical composition of the interstellar medium. Microwave transitions are measured in the laboratory and matched
to emissions from the interstellar medium using a radio telescope . NH 3 was the first stable polyatomic molecule to be identified in the interstellar medium. [ 3 ] The measurement of chlorine monoxide [ 4 ] is important for atmospheric chemistry . Current projects in astrochemistry involve both laboratory microwave spectroscopy and observations made using modern radiotelescopes such as the Atacama Large Millimeter/submillimeter Array (ALMA). [ 5 ]
A molecule in the gas phase is free to rotate relative to a set of mutually orthogonal axes of fixed orientation in space, centered on the center of mass of the molecule. Free rotation is not possible for molecules in liquid or solid phases due to the presence of intermolecular forces . Rotation about each unique axis is associated with a set of quantized energy levels dependent on the moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, J {\displaystyle J} , which defines the magnitude of the rotational angular momentum.
For nonlinear molecules which are symmetric rotors (or symmetric tops - see next section), there are two moments of inertia and the energy also depends on a second rotational quantum number, K {\displaystyle K} , which defines the vector component of rotational angular momentum along the principal symmetry axis . [ 6 ] Analysis of spectroscopic data with the expressions detailed below results in quantitative determination of the value(s) of the moment(s) of inertia. From these precise values of the molecular structure and dimensions may be obtained.
For a linear molecule, analysis of the rotational spectrum provides values for the rotational constant [ notes 2 ] and the moment of inertia of the molecule, and, knowing the atomic masses, can be used to determine the bond length directly. For diatomic molecules this process is straightforward. For linear molecules with more than two atoms it is necessary to measure the spectra of two or more isotopologues , such as 16 O 12 C 32 S and 16 O 12 C 34 S. This allows a set of simultaneous equations to be set up and solved for the bond lengths ). [ notes 3 ] A bond length obtained in this way is slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, to which the rotational states refer, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by
where v is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated if the B values for two different vibrational states can be found. [ 7 ]
For other molecules, if the spectra can be resolved and individual transitions assigned both bond lengths and bond angles can be deduced. When this is not possible, as with most asymmetric tops, all that can be done is to fit the spectra to three moments of inertia calculated from an assumed molecular structure. By varying the molecular structure the fit can be improved, giving a qualitative estimate of the structure. Isotopic substitution is invaluable when using this approach to the determination of molecular structure.
In quantum mechanics the free rotation of a molecule is quantized , so that the rotational energy and the angular momentum can take only certain fixed values, which are related simply to the moment of inertia , I {\displaystyle I} , of the molecule. For any molecule, there are three moments of inertia: I A {\displaystyle I_{A}} , I B {\displaystyle I_{B}} and I C {\displaystyle I_{C}} about three mutually orthogonal axes A , B , and C with the origin at the center of mass of the system. The general convention, used in this article, is to define the axes such that I A ≤ I B ≤ I C {\displaystyle I_{A}\leq I_{B}\leq I_{C}} , with axis A {\displaystyle A} corresponding to the smallest moment of inertia. Some authors, however, define the A {\displaystyle A} axis as the molecular rotation axis of highest order.
The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes, based on the symmetry of their structure. These are
Transitions between rotational states can be observed in molecules with a permanent electric dipole moment . [ 9 ] [ notes 4 ] A consequence of this rule is that no microwave spectrum can be observed for centrosymmetric linear molecules such as N 2 ( dinitrogen ) or HCCH ( ethyne ), which are non-polar. Tetrahedral molecules such as CH 4 ( methane ), which have both a zero dipole moment and isotropic polarizability, would not have a pure rotation spectrum but for the effect of centrifugal distortion; when the molecule rotates about a 3-fold symmetry axis a small dipole moment is created, allowing a weak rotation spectrum to be observed by microwave spectroscopy. [ 10 ]
With symmetric tops, the selection rule for electric-dipole-allowed pure rotation transitions is Δ K = 0 , Δ J = ±1 . Since these transitions are due to absorption (or emission) of a single photon with a spin of one, conservation of angular momentum implies that the molecular angular momentum can change by at most one unit. [ 11 ] Moreover, the quantum number K is limited to have values between and including + J to - J . [ 12 ]
For Raman spectra the molecules undergo transitions in which an incident photon is absorbed and another scattered photon is emitted. The general selection rule for such a transition to be allowed is that the molecular polarizability must be anisotropic , which means that it is not the same in all directions. [ 13 ] Polarizability is a 3-dimensional tensor that can be represented as an ellipsoid. The polarizability ellipsoid of spherical top molecules is in fact spherical so those molecules show no rotational Raman spectrum. For all other molecules both Stokes and anti-Stokes lines [ notes 5 ] can be observed and they have similar intensities due to the fact that many rotational states are thermally populated. The selection rule for linear molecules is ΔJ = 0, ±2. The reason for the values ±2 is that the polarizability returns to the same value twice during a rotation. [ 14 ] The value ΔJ = 0 does not correspond to a molecular transition but rather to Rayleigh scattering in which the incident photon merely changes direction. [ 15 ]
The selection rule for symmetric top molecules is
Transitions with Δ J = +1 are said to belong to the R series, whereas transitions with Δ J = +2 belong to an S series. [ 15 ] Since Raman transitions involve two photons, it is possible for the molecular angular momentum to change by two units.
The units used for rotational constants depend on the type of measurement. With infrared spectra in the wavenumber scale ( ν ~ {\displaystyle {\tilde {\nu }}} ), the unit is usually the inverse centimeter , written as cm −1 , which is literally the number of waves in one centimeter, or the reciprocal of the wavelength in centimeters ( ν ~ = 1 / λ {\displaystyle {\tilde {\nu }}=1/\lambda } ). On the other hand, for microwave spectra in the frequency scale ( ν {\displaystyle \nu } ), the unit is usually the gigahertz . The relationship between these two units is derived from the expression
where ν is a frequency , λ is a wavelength and c is the velocity of light . It follows that
As 1 GHz = 10 9 Hz, the numerical conversion can be expressed as
The population of vibrationally excited states follows a Boltzmann distribution , so low-frequency vibrational states are appreciably populated even at room temperatures. As the moment of inertia is higher when a vibration is excited, the rotational constants ( B ) decrease. Consequently, the rotation frequencies in each vibration state are different from each other. This can give rise to "satellite" lines in the rotational spectrum. An example is provided by cyanodiacetylene , H−C≡C−C≡C−C≡N. [ 16 ]
Further, there is a fictitious force , Coriolis coupling , between the vibrational motion of the nuclei in the rotating (non-inertial) frame. However, as long as the vibrational quantum number does not change (i.e., the molecule is in only one state of vibration), the effect of vibration on rotation is not important, because the time for vibration is much shorter than the time required for rotation. The Coriolis coupling is often negligible, too, if one is interested in low vibrational and rotational quantum numbers only.
Historically, the theory of rotational energy levels was developed to account for observations of vibration-rotation spectra of gases in infrared spectroscopy , which was used before microwave spectroscopy had become practical. To a first approximation, the rotation and vibration can be treated as separable , so the energy of rotation is added to the energy of vibration. For example, the rotational energy levels for linear molecules (in the rigid-rotor approximation) are
In this approximation, the vibration-rotation wavenumbers of transitions are
where B ″ {\displaystyle B''} and B ′ {\displaystyle B'} are rotational constants for the upper and lower vibrational state respectively, while J ″ {\displaystyle J''} and J ′ {\displaystyle J'} are the rotational quantum numbers of the upper and lower levels. In reality, this expression has to be modified for the effects of anharmonicity of the vibrations, for centrifugal distortion and for Coriolis coupling. [ 17 ]
For the so-called R branch of the spectrum, J ′ = J ″ + 1 {\displaystyle J'=J''+1} so that there is simultaneous excitation of both vibration and rotation. For the P branch, J ′ = J ″ − 1 {\displaystyle J'=J''-1} so that a quantum of rotational energy is lost while a quantum of vibrational energy is gained. The purely vibrational transition, Δ J = 0 {\displaystyle \Delta J=0} , gives rise to the Q branch of the spectrum. Because of the thermal population of the rotational states the P branch is slightly less intense than the R branch.
Rotational constants obtained from infrared measurements are in good accord with those obtained by microwave spectroscopy, while the latter usually offers greater precision.
Spherical top molecules have no net dipole moment. A pure rotational spectrum cannot be observed by absorption or emission spectroscopy because there is no permanent dipole moment whose rotation can be accelerated by the electric field of an incident photon. Also the polarizability is isotropic, so that pure rotational transitions cannot be observed by Raman spectroscopy either. Nevertheless, rotational constants can be obtained by ro–vibrational spectroscopy . This occurs when a molecule is polar in the vibrationally excited state. For example, the molecule methane is a spherical top but the asymmetric C-H stretching band shows rotational fine structure in the infrared spectrum, illustrated in rovibrational coupling . This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band.
The rigid rotor is a good starting point from which to construct a model of a rotating molecule. It is assumed that component atoms are point masses connected by rigid bonds. A linear molecule lies on a single axis and each atom moves on the surface of a sphere around the centre of mass. The two degrees of rotational freedom correspond to the spherical coordinates θ and φ which describe the direction of the molecular axis, and the quantum state is determined by two quantum numbers J and M. J defines the magnitude of the rotational angular momentum, and M its component about an axis fixed in space, such as an external electric or magnetic field. In the absence of external fields, the energy depends only on J. Under the rigid rotor model, the rotational energy levels, F (J), of the molecule can be expressed as,
where B {\displaystyle B} is the rotational constant of the molecule and is related to the moment of inertia of the molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I B = I C , I A = 0 {\displaystyle I_{B}=I_{C},I_{A}=0} , so
For a diatomic molecule
where m 1 and m 2 are the masses of the atoms and d is the distance between them.
Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., Δ J = J ′ − J ′ ′ = ± 1 {\displaystyle \Delta J=J^{\prime }-J^{\prime \prime }=\pm 1} . Thus, the locations of the lines in a rotational spectrum will be given by
where J ′ ′ {\displaystyle J^{\prime \prime }} denotes the lower level and J ′ {\displaystyle J^{\prime }} denotes the upper level involved in the transition.
The diagram illustrates rotational transitions that obey the Δ J {\displaystyle \Delta J} =1 selection rule. The dashed lines show how these transitions map onto features that can be observed experimentally. Adjacent J ′ ′ ← J ′ {\displaystyle J^{\prime \prime }{\leftarrow }J^{\prime }} transitions are separated by 2 B in the observed spectrum. Frequency or wavenumber units can also be used for the x axis of this plot.
The probability of a transition taking place is the most important factor influencing the intensity of an observed rotational line. This probability is proportional to the population of the initial state involved in the transition. The population of a rotational state depends on two factors. The number of molecules in an excited state with quantum number J , relative to the number of molecules in the ground state, N J / N 0 is given by the Boltzmann distribution as
where k is the Boltzmann constant and T the absolute temperature . This factor decreases as J increases. The second factor is the degeneracy of the rotational state, which is equal to 2J + 1 . This factor increases as J increases. Combining the two factors [ 18 ]
The maximum relative intensity occurs at [ 19 ] [ notes 6 ]
The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it.
When a molecule rotates, the centrifugal force pulls the atoms apart. As a result, the moment of inertia of the molecule increases, thus decreasing the value of B {\displaystyle B} , when it is calculated using the expression for the rigid rotor. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. [ 20 ]
where D {\displaystyle D} is the centrifugal distortion constant.
Therefore, the line positions for the rotational mode change to
In consequence, the spacing between lines is not constant, as in the rigid rotor approximation, but decreases with increasing rotational quantum number.
An assumption underlying these expressions is that the molecular vibration follows simple harmonic motion . In the harmonic approximation the centrifugal constant D {\displaystyle D} can be derived as
where k is the vibrational force constant . The relationship between B {\displaystyle B} and D {\displaystyle D}
where ω ~ {\displaystyle {\tilde {\omega }}} is the harmonic vibration frequency, follows. If anharmonicity is to be taken into account, terms in higher powers of J should be added to the expressions for the energy levels and line positions. [ 20 ] A striking example concerns the rotational spectrum of hydrogen fluoride which was fitted to terms up to [J(J+1)] 5 . [ 21 ]
The electric dipole moment of the dioxygen molecule, O 2 is zero, but the molecule is paramagnetic with two unpaired electrons so that there are magnetic-dipole allowed transitions which can be observed by microwave spectroscopy. The unit electron spin has three spatial orientations with respect to the given molecular rotational angular momentum vector, K, so that each rotational level is split into three states, J = K + 1, K, and K - 1, each J state of this so-called p-type triplet arising from a different orientation of the spin with respect to the rotational motion of the molecule. The energy difference between successive J terms in any of these triplets is about 2 cm −1 (60 GHz), with the single exception of J = 1←0 difference which is about 4 cm −1 . Selection rules for magnetic dipole transitions allow transitions between successive members of the triplet (ΔJ = ±1) so that for each value of the rotational angular momentum quantum number K there are two allowed transitions. The 16 O nucleus has zero nuclear spin angular momentum, so that symmetry considerations demand that K have only odd values. [ 22 ] [ 23 ]
For symmetric rotors a quantum number J is associated with the total angular momentum of the molecule. For a given value of J, there is a 2 J +1- fold degeneracy with the quantum number, M taking the values + J ...0 ... - J . The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. In the absence of an external electrical field, the rotational energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the energy of each rotational state is given by
where B = h 8 π 2 c I B {\displaystyle B={h \over {8\pi ^{2}cI_{B}}}} and A = h 8 π 2 c I A {\displaystyle A={h \over {8\pi ^{2}cI_{A}}}} for a prolate symmetric top molecule or A = h 8 π 2 c I C {\displaystyle A={h \over {8\pi ^{2}cI_{C}}}} for an oblate molecule.
This gives the transition wavenumbers as
which is the same as in the case of a linear molecule. [ 24 ] With a first order correction for centrifugal distortion the transition wavenumbers become
The term in D JK has the effect of removing degeneracy present in the rigid rotor approximation, with different K values. [ 25 ]
The quantum number J refers to the total angular momentum, as before. Since there are three independent moments of inertia, there are two other independent quantum numbers to consider, but the term values for an asymmetric rotor cannot be derived in closed form. They are obtained by individual matrix diagonalization for each J value. Formulae are available for molecules whose shape approximates to that of a symmetric top. [ 26 ]
The water molecule is an important example of an asymmetric top. It has an intense pure rotation spectrum in the far infrared region, below about 200 cm −1 . For this reason far infrared spectrometers have to be freed of atmospheric water vapour either by purging with a dry gas or by evacuation. The spectrum has been analyzed in detail. [ 27 ]
When a nucleus has a spin quantum number , I , greater than 1/2 it has a quadrupole moment. In that case, coupling of nuclear spin angular momentum with rotational angular momentum causes splitting of the rotational energy levels. If the quantum number J of a rotational level is greater than I , 2 I + 1 levels are produced; but if J is less than I , 2 J + 1 levels result. The effect is one type of hyperfine splitting . For example, with 14 N ( I = 1 ) in HCN, all levels with J > 0 are split into 3. The energies of the sub-levels are proportional to the nuclear quadrupole moment and a function of F and J . where F = J + I , J + I − 1, …, | J − I | . Thus, observation of nuclear quadrupole splitting permits the magnitude of the nuclear quadrupole moment to be determined. [ 28 ] This is an alternative method to the use of nuclear quadrupole resonance spectroscopy. The selection rule for rotational transitions becomes [ 29 ]
In the presence of a static external electric field the 2 J + 1 degeneracy of each rotational state is partly removed, an instance of a Stark effect . For example, in linear molecules each energy level is split into J + 1 components. The extent of splitting depends on the square of the electric field strength and the square of the dipole moment of the molecule. [ 30 ] In principle this provides a means to determine the value of the molecular dipole moment with high precision. Examples include carbonyl sulfide , OCS, with μ = 0.71521 ± 0.00020 debye . However, because the splitting depends on μ 2 , the orientation of the dipole must be deduced from quantum mechanical considerations. [ 31 ]
A similar removal of degeneracy will occur when a paramagnetic molecule is placed in a magnetic field, an instance of the Zeeman effect . Most species which can be observed in the gaseous state are diamagnetic . Exceptions are odd-electron molecules such as nitric oxide , NO, nitrogen dioxide , NO 2 , some chlorine oxides and the hydroxyl radical . The Zeeman effect has been observed with dioxygen , O 2 [ 32 ]
Molecular rotational transitions can also be observed by Raman spectroscopy . Rotational transitions are Raman-allowed for any molecule with an anisotropic polarizability which includes all molecules except for spherical tops. This means that rotational transitions of molecules with no permanent dipole moment, which cannot be observed in absorption or emission, can be observed, by scattering, in Raman spectroscopy. Very high resolution Raman spectra can be obtained by adapting a Fourier Transform Infrared Spectrometer . An example is the spectrum of 15 N 2 . It shows the effect of nuclear spin, resulting in intensities variation of 3:1 in adjacent lines. A bond length of 109.9985 ± 0.0010 pm was deduced from the data. [ 33 ]
The great majority of contemporary spectrometers use a mixture of commercially available and bespoke components which users integrate according to their particular needs. Instruments can be broadly categorised according to their general operating principles. Although rotational transitions can be found across a very broad region of the electromagnetic spectrum , fundamental physical constraints exist on the operational bandwidth of instrument components. It is often impractical and costly to switch to measurements within an entirely different frequency region. The instruments and operating principals described below are generally appropriate to microwave spectroscopy experiments conducted at frequencies between 6 and 24 GHz.
A microwave spectrometer can be most simply constructed using a source of microwave radiation, an absorption cell into which sample gas can be introduced and a detector such as a superheterodyne receiver . A spectrum can be obtained by sweeping the frequency of the source while detecting the intensity of transmitted radiation. A simple section of waveguide can serve as an absorption cell. An important variation of the technique in which an alternating current is applied across electrodes within the absorption cell results in a modulation of the frequencies of rotational transitions. This is referred to as Stark modulation and allows the use of phase-sensitive detection methods offering improved sensitivity. Absorption spectroscopy allows the study of samples that are thermodynamically stable at room temperature. The first study of the microwave spectrum of a molecule ( NH 3 ) was performed by Cleeton & Williams in 1934. [ 34 ] Subsequent experiments exploited powerful sources of microwaves such as the klystron , many of which were developed for radar during the Second World War . The number of experiments in microwave spectroscopy surged immediately after the war. By 1948, Walter Gordy was able to prepare a review of the results contained in approximately 100 research papers. [ 35 ] Commercial versions [ 36 ] of microwave absorption spectrometer were developed by Hewlett-Packard in the 1970s and were once widely used for fundamental research. Most research laboratories now exploit either Balle- Flygare or chirped-pulse Fourier transform microwave (FTMW) spectrometers.
The theoretical framework [ 37 ] underpinning FTMW spectroscopy is analogous to that used to describe FT-NMR spectroscopy . The behaviour of the evolving system is described by optical Bloch equations . First, a short (typically 0-3 microsecond duration) microwave pulse is introduced on resonance with a rotational transition. Those molecules that absorb the energy from this pulse are induced to rotate coherently in phase with the incident radiation. De-activation of the polarisation pulse is followed by microwave emission that accompanies decoherence of the molecular ensemble. This free induction decay occurs on a timescale of 1-100 microseconds depending on instrument settings. Following pioneering work by Dicke and co-workers in the 1950s, [ 38 ] the first FTMW spectrometer was constructed by Ekkers and Flygare in 1975. [ 39 ]
Balle, Campbell, Keenan and Flygare demonstrated that the FTMW technique can be applied within a "free space cell" comprising an evacuated chamber containing a Fabry-Perot cavity . [ 40 ] This technique allows a sample to be probed only milliseconds after it undergoes rapid cooling to only a few kelvins in the throat of an expanding gas jet. This was a revolutionary development because (i) cooling molecules to low temperatures concentrates the available population in the lowest rotational energy levels. Coupled with benefits conferred by the use of a Fabry-Perot cavity, this brought a great enhancement in the sensitivity and resolution of spectrometers along with a reduction in the complexity of observed spectra; (ii) it became possible to isolate and study molecules that are very weakly bound because there is insufficient energy available for them to undergo fragmentation or chemical reaction at such low temperatures. William Klemperer was a pioneer in using this instrument for the exploration of weakly bound interactions. While the Fabry-Perot cavity of a Balle-Flygare FTMW spectrometer can typically be tuned into resonance at any frequency between 6 and 18 GHz, the bandwidth of individual measurements is restricted to about 1 MHz. An animation illustrates the operation of this instrument which is currently the most widely used tool for microwave spectroscopy. [ 41 ]
Noting that digitisers and related electronics technology had significantly progressed since the inception of FTMW spectroscopy, B.H. Pate at the University of Virginia [ 42 ] designed a spectrometer [ 43 ] which retains many advantages of the Balle-Flygare FT-MW spectrometer while innovating in (i) the use of a high speed (>4 GS/s) arbitrary waveform generator to generate a "chirped" microwave polarisation pulse that sweeps up to 12 GHz in frequency in less than a microsecond and (ii) the use of a high speed (>40 GS/s) oscilloscope to digitise and Fourier transform the molecular free induction decay. The result is an instrument that allows the study of weakly bound molecules but which is able to exploit a measurement bandwidth (12 GHz) that is greatly enhanced compared with the Balle-Flygare FTMW spectrometer. Modified versions of the original CP-FTMW spectrometer have been constructed by a number of groups in the United States, Canada and Europe. [ 44 ] [ 45 ] The instrument offers a broadband capability that is highly complementary to the high sensitivity and resolution offered by the Balle-Flygare design. | https://en.wikipedia.org/wiki/Rotational_spectroscopy |
Rotational symmetry , also known as radial symmetry in geometry , is the property a shape has when it looks the same after some rotation by a partial turn . An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.
Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids . [ 1 ] [ 2 ]
Formally the rotational symmetry is symmetry with respect to some or all rotations in m -dimensional Euclidean space . Rotations are direct isometries , i.e., isometries preserving orientation . Therefore, a symmetry group of rotational symmetry is a subgroup of E + ( m ) (see Euclidean group ).
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole E ( m ) . With the modified notion of symmetry for vector fields the symmetry group can also be E + ( m ) .
For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO( m ) , the group of m × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3) .
In another definition of the word, the rotation group of an object is the symmetry group within E + ( n ) , the group of direct isometries ; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem , the rotational symmetry of a physical system is equivalent to the angular momentum conservation law.
Rotational symmetry of order n , also called n -fold rotational symmetry , or discrete rotational symmetry of the n th order , with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360 ∘ n {\displaystyle {\tfrac {360^{\circ }}{n}}} (180°, 120°, 90°, 72°, 60°, 51 3 ⁄ 7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).
The notation for n -fold symmetry is C n or simply n . The actual symmetry group is specified by the point or axis of symmetry, together with the n . For each point or axis of symmetry, the abstract group type is cyclic group of order n , Z n . Although for the latter also the notation C n is used, the geometric and abstract C n should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D .
The fundamental domain is a sector of 360 ∘ n . {\displaystyle {\tfrac {360^{\circ }}{n}}.}
Examples without additional reflection symmetry :
C n is the rotation group of a regular n -sided polygon in 2D and of a regular n -sided pyramid in 3D.
If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.
A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller .
For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:
In the case of the Platonic solids , the 2-fold axes are through the midpoints of opposite edges, and the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry . The fundamental domain is a half-line .
In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates . The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric and axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut ( torus ). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms .
2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups . A rotocenter is the fixed, or invariant, point of a rotation. [ 3 ] There are two rotocenters per primitive cell .
Together with double translational symmetry the rotation groups are the following wallpaper groups , with axes per primitive cell:
Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore, the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.
3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is 2 3 {\displaystyle 2{\sqrt {3}}} times their distance. | https://en.wikipedia.org/wiki/Rotational_symmetry |
The characteristic rotational temperature ( θ R or θ rot ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. It has units of temperature and is defined as [ 1 ]
where B ¯ = B / h c {\displaystyle {\overline {B}}=B/hc} is the rotational constant , I is a molecular moment of inertia , h is the Planck constant , c is the speed of light , ħ = h /2 π is the reduced Planck constant and k B is the Boltzmann constant .
The physical meaning of θ R is as an estimate of the temperature at which thermal energy (of the order of k B T ) is comparable to the spacing between rotational energy levels (of the order of hcB ). At about this temperature the population of excited rotational levels becomes important. Some typical values are given in the table. In each case the value refers to the most common isotopic species.
This atomic, molecular, and optical physics –related article is a stub . You can help Wikipedia by expanding it .
This molecular physics –related article is a stub . You can help Wikipedia by expanding it .
This thermodynamics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rotational_temperature |
In quantum mechanics , a rotational transition is an abrupt change in angular momentum . Like all other properties of a quantum particle , angular momentum is quantized , meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.
Rotational transitions are important in physics due to the unique spectral lines that result. Because there is a net gain or loss of energy during a transition, electromagnetic radiation of a particular frequency must be absorbed or emitted. This forms spectral lines at that frequency which can be detected with a spectrometer , as in rotational spectroscopy or Raman spectroscopy .
Molecules have rotational energy owing to rotational motion of the nuclei about their center of mass . Due to quantization , these energies can take only certain discrete values. Rotational transition thus corresponds to transition of the molecule from one rotational energy level to the other through gain or loss of a photon . Analysis is simple in the case of diatomic molecules .
Quantum theoretical analysis of a molecule is simplified by use of Born–Oppenheimer approximation . Typically, rotational energies of molecules are smaller than electronic transition energies by a factor of m / M ≈ 10 −3 –10 −5 , where m is electronic mass and M is typical nuclear mass. [ 1 ] From uncertainty principle , period of motion is of the order of the Planck constant h divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the Schrödinger Equation for a nuclear wave function F s ( R ) , in an electronic state s , is written as (neglecting spin interactions) [ − ℏ 2 2 μ R 2 ∂ ∂ R ( R 2 ∂ ∂ R ) + ⟨ Φ s | N 2 | Φ s ⟩ 2 μ R 2 + E s ( R ) − E ] F s ( R ) = 0 {\displaystyle \left[-{\frac {\hbar ^{2}}{2\mu R^{2}}}{\frac {\partial }{\partial R}}\left(R^{2}{\frac {\partial }{\partial R}}\right)+{\frac {\langle \Phi _{s}|N^{2}|\Phi _{s}\rangle }{2\mu R^{2}}}+E_{s}(R)-E\right]F_{s}(\mathbf {R} )=0} where μ is reduced mass of two nuclei, R is vector joining the two nuclei, E s ( R ) is energy eigenvalue of electronic wave function Φ s representing electronic state s and N is orbital momentum operator for the relative motion of the two nuclei given by N 2 = − ℏ 2 [ 1 sin Θ ∂ ∂ Θ ( sin Θ ∂ ∂ Θ ) + 1 sin 2 Θ ∂ 2 ∂ Φ 2 ] {\displaystyle N^{2}=-\hbar ^{2}\left[{\frac {1}{\sin \Theta }}{\frac {\partial }{\partial \Theta }}\left(\sin \Theta {\frac {\partial }{\partial \Theta }}\right)+{\frac {1}{\sin ^{2}\Theta }}{\frac {\partial ^{2}}{\partial \Phi ^{2}}}\right]} The total wave function for the molecule is Ψ s = F s ( R ) Φ s ( R , r 1 , r 2 , … , r N ) {\displaystyle \Psi _{s}=F_{s}(\mathbf {R} )\Phi _{s}(\mathbf {R} ,\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})} where r i are position vectors from center of mass of molecule to i th electron.
As a consequence of the Born-Oppenheimer approximation, the electronic wave functions Φ s is considered to vary very slowly with R . Thus the Schrödinger equation for an electronic wave function is first solved to obtain E s ( R ) for different values of R . E s then plays role of a potential well in analysis of nuclear wave functions F s ( R ) .
The first term in the above nuclear wave function equation corresponds to kinetic energy of nuclei due to their radial motion. Term ⟨Φ s | N 2 |Φ s ⟩ / 2 μR 2 represents rotational kinetic energy of the two nuclei, about their center of mass, in a given electronic state Φ s . Possible values of the same are different rotational energy levels for the molecule.
Orbital angular momentum for the rotational motion of nuclei can be written as N = J − L {\displaystyle \mathbf {N} =\mathbf {J} -\mathbf {L} } where J is the total orbital angular momentum of the whole molecule and L is the orbital angular momentum of the electrons.
If internuclear vector R is taken along z-axis, component of N along z-axis – N z – becomes zero as N = R × P {\displaystyle \mathbf {N} =\mathbf {R} \times \mathbf {P} } Hence J z = L z {\displaystyle J_{z}=L_{z}} Since molecular wave function Ψ s is a simultaneous eigenfunction of J 2 and J z , J 2 Ψ s = J ( J + 1 ) ℏ 2 Ψ s {\displaystyle J^{2}\Psi _{s}=J(J+1)\hbar ^{2}\Psi _{s}} where J is called rotational quantum number and J can be a positive integer or zero. J z Ψ s = M j ℏ Ψ s {\displaystyle J_{z}\Psi _{s}=M_{j}\hbar \Psi _{s}} where − J ≤ M j ≤ J .
Also since electronic wave function Φ s is an eigenfunction of L z , L z Φ s = ± Λ ℏ Φ s {\displaystyle L_{z}\Phi _{s}=\pm \Lambda \hbar \Phi _{s}} Hence molecular wave function Ψ s is also an eigenfunction of L z with eigenvalue ±Λ ħ .
Since L z and J z are equal, Ψ s is an eigenfunction of J z with same eigenvalue ±Λ ħ . As | J | ≥ J z , we have J ≥ Λ . So possible values of rotational quantum number are J = Λ , Λ + 1 , Λ + 2 , … {\displaystyle J=\Lambda ,\Lambda +1,\Lambda +2,\dots } Thus molecular wave function Ψ s is simultaneous eigenfunction of J 2 , J z and L z .
Since molecule is in eigenstate of L z , expectation value of components perpendicular to the direction of z-axis (internuclear line) is zero. Hence ⟨ Ψ s | L x | Ψ s ⟩ = ⟨ L x ⟩ = 0 {\displaystyle \langle \Psi _{s}|L_{x}|\Psi _{s}\rangle =\langle L_{x}\rangle =0} and ⟨ Ψ s | L y | Ψ s ⟩ = ⟨ L y ⟩ = 0 {\displaystyle \langle \Psi _{s}|L_{y}|\Psi _{s}\rangle =\langle L_{y}\rangle =0} Thus ⟨ J . L ⟩ = ⟨ J z L z ⟩ = ⟨ L z 2 ⟩ {\displaystyle \langle \mathbf {J} .\mathbf {L} \rangle =\langle J_{z}L_{z}\rangle =\langle {L_{z}}^{2}\rangle }
Putting all these results together, ⟨ Φ s | N 2 | Φ s ⟩ F s ( R ) = ⟨ Φ s | ( J 2 + L 2 − 2 J ⋅ L ) | Φ s ⟩ F s ( R ) = ℏ 2 [ J ( J + 1 ) − Λ 2 ] F s ( R ) + ⟨ Φ s | ( L x 2 + L y 2 ) | Φ s ⟩ F s ( R ) {\displaystyle {\begin{aligned}\langle \Phi _{s}|N^{2}|\Phi _{s}\rangle F_{s}(\mathbf {R} )&=\langle \Phi _{s}|\left(J^{2}+L^{2}-2\mathbf {J} \cdot \mathbf {L} \right)|\Phi _{s}\rangle F_{s}(\mathbf {R} )\\&=\hbar ^{2}\left[J(J+1)-\Lambda ^{2}\right]F_{s}(\mathbf {R} )+\langle \Phi _{s}|\left({L_{x}}^{2}+{L_{y}}^{2}\right)|\Phi _{s}\rangle F_{s}(\mathbf {R} )\end{aligned}}}
The Schrödinger equation for the nuclear wave function can now be rewritten as − ℏ 2 2 μ R 2 [ ∂ ∂ R ( R 2 ∂ ∂ R ) − J ( J + 1 ) ] F s ( R ) + [ E ′ s ( R ) − E ] F s ( R ) = 0 {\displaystyle -{\frac {\hbar ^{2}}{2\mu R^{2}}}\left[{\frac {\partial }{\partial R}}\left(R^{2}{\frac {\partial }{\partial R}}\right)-J(J+1)\right]F_{s}(\mathbf {R} )+[{E'}_{s}(R)-E]F_{s}(\mathbf {R} )=0} where E ′ s ( R ) = E s ( R ) − Λ 2 ℏ 2 2 μ R 2 + 1 2 μ R 2 ⟨ Φ s | ( L x 2 + L y 2 ) | Φ s ⟩ {\displaystyle {E'}_{s}(R)=E_{s}(R)-{\frac {\Lambda ^{2}\hbar ^{2}}{2\mu R^{2}}}+{\frac {1}{2\mu R^{2}}}\langle \Phi _{s}|\left({L_{x}}^{2}+{L_{y}}^{2}\right)|\Phi _{s}\rangle } E′ s now serves as effective potential in radial nuclear wave function equation.
Molecular states in which the total orbital momentum of electrons is zero are called sigma states . In sigma states Λ = 0 . Thus E ′ s ( R ) = E s ( R ) . As nuclear motion for a stable molecule is generally confined to a small interval around R 0 where R 0 corresponds to internuclear distance for minimum value of potential E s ( R 0 ) , rotational energies are given by, E r = ℏ 2 2 μ R 0 2 J ( J + 1 ) = ℏ 2 2 I 0 J ( J + 1 ) = B J ( J + 1 ) {\displaystyle E_{r}={\frac {\hbar ^{2}}{2\mu {R_{0}}^{2}}}J(J+1)={\frac {\hbar ^{2}}{2I_{0}}}J(J+1)=BJ(J+1)} with J = Λ , Λ + 1 , Λ + 2 , … {\displaystyle J=\Lambda ,\Lambda +1,\Lambda +2,\dots } I 0 is moment of inertia of the molecule corresponding to equilibrium distance R 0 and B is called rotational constant for a given electronic state Φ s .
Since reduced mass μ is much greater than electronic mass, last two terms in the expression of E ′ s ( R ) are small compared to E s . Hence even for states other than sigma states, rotational energy is approximately given by above expression.
When a rotational transition occurs, there is a change in the value of rotational quantum number J . Selection rules for rotational transition are,
when Λ = 0 , Δ J = ±1 and
when Λ ≠ 0 , Δ J = 0, ±1 as absorbed or emitted photon can make equal and opposite change in total nuclear angular momentum and total electronic angular momentum without changing value of J .
The pure rotational spectrum of a diatomic molecule consists of lines in the far infrared or microwave region. The frequency of these lines is given by ℏ ω = E r ( J + 1 ) − E r ( J ) = 2 B ( J + 1 ) {\displaystyle \hbar \omega =E_{r}(J+1)-E_{r}(J)=2B(J+1)} Thus values of B , I 0 and R 0 of a substance can be determined from observed rotational spectrum. | https://en.wikipedia.org/wiki/Rotational_transition |
Viscosity is usually described as the property of a fluid which determines the rate at which local momentum differences are equilibrated. Rotational viscosity is a property of a fluid which determines the rate at which local angular momentum differences are equilibrated. In the classical case, by the equipartition theorem , at equilibrium, if particle collisions can transfer angular momentum as well as linear momentum, then these degrees of freedom will have the same average energy. If there is a lack of equilibrium between these degrees of freedom, then the rate of equilibration will be determined by the rotational viscosity coefficient. [ 1 ] : p.304
Rotational viscosity has traditionally been thought to require rotational degrees of freedom for the fluid particles, such as in liquid crystals . In these fluids, the rotational degrees of freedom allow angular momentum to become a dynamical quantity that can be locally relaxed, leading to rotational viscosity. However, recent theoretical work [ 2 ] has predicted that rotational viscosity ought to also be present in viscous electron fluids (see Gurzhi effect ) in anisotropic metals . In these cases, the ionic lattice explicitly breaks rotational symmetry and applies torques to the electron fluid, implying non-conservation of angular momentum and hence rotational viscosity.
The angular momentum density of a fluid element is written either as an antisymmetric tensor ( J i j {\displaystyle J_{ij}} ) or, equivalently, as a pseudovector . As a tensor,
the equation for the conservation of angular momentum for a simple fluid with no external forces is written:
where v i {\displaystyle v_{i}} is the fluid velocity and P i j {\displaystyle P_{ij}} is the total pressure tensor (or, equivalently, the negative of the total stress tensor ). Note that the Einstein summation convention is used, where summation is assumed over pairs of matched indices. The angular momentum of a fluid element can be separated into extrinsic angular momentum density due to the flow ( L i j {\displaystyle L_{ij}} ) and intrinsic angular momentum density due to the rotation of the fluid particles about their center of mass ( S i j {\displaystyle S_{ij}} ):
where the extrinsic angular momentum density is:
and ρ {\displaystyle \rho } is the mass density of the fluid element. The conservation of linear momentum equation is written:
and it can be shown that this implies that:
Subtracting this from the equation for the conservation of angular momentum yields:
Any situation in which this last term is zero will result in the total pressure tensor being symmetric, and the conservation of angular momentum equation will be redundant with the conservation of linear momentum. If, however, the internal rotational degrees of freedom of the particles are coupled to the flow (via the velocity term in the above equation), then the total pressure tensor will not be symmetric, with its antisymmetric component describing the rate of angular momentum exchange between the flow and the particle rotation.
In the linear approximation for this transport of angular momentum, the rate of flow is written: [ 1 ] : p.308
where ω i j {\displaystyle \omega _{ij}} is the average angular velocity of the rotating particles (as an antisymmetric tensor rather than a pseudovector) and η r {\displaystyle \eta _{r}} is the rotational viscosity coefficient. | https://en.wikipedia.org/wiki/Rotational_viscosity |
In physics , rotational–vibrational coupling [ 1 ] occurs when the rotation frequency of a system is close to or identical to a natural frequency of internal vibration . The animation on the right shows ideal motion, with the force exerted by the spring and the distance from the center of rotation increasing together linearly with no friction .
In rotational-vibrational coupling, angular velocity oscillates. By pulling the circling masses closer together, the spring transfers its stored strain energy into the kinetic energy of the circling masses, increasing their angular velocity. The spring cannot bring the circling masses together, since the spring's pull weakens as the circling masses approach. At some point, the increasing angular velocity of the circling masses overcomes the pull of the spring, causing the circling masses to increasingly distance themselves. This increasingly strains the spring, strengthening its pull and causing the circling masses to transfer their kinetic energy into the spring's strain energy, thereby decreasing the circling masses' angular velocity. At some point, the pull of the spring overcomes the angular velocity of the circling masses, restarting the cycle.
In helicopter design, helicopters must incorporate damping devices, because at specific angular velocities, the rotorblade vibrations can reinforce themselves by rotational-vibrational coupling, and build up catastrophically. Without damping, these vibrations would cause the rotorblades to break loose.
The animation on the right provides a clearer view on the oscillation of the angular velocity. There is a close analogy with harmonic oscillation .
When a harmonic oscillation is at its midpoint then all the energy of the system is kinetic energy . When the harmonic oscillation is at the points furthest away from the midpoint all the energy of the system is potential energy . The energy of the system is oscillating back and forth between kinetic energy and potential energy.
In the animation with the two circling masses there is a back and forth oscillation of kinetic energy and potential energy. When the spring is at its maximal extension then the potential energy is largest, when the angular velocity is at its maximum the kinetic energy is at largest.
With a real spring there is friction involved. With a real spring the vibration will be damped and the final situation will be that the masses circle each other at a constant distance, with a constant tension of the spring.
This discussion applies the following simplifications: the spring itself is taken as being weightless, and the spring is taken as being a perfect spring; the restoring force increases in a linear way as the spring is stretched out. That is, the restoring force is exactly proportional to the distance from the center of rotation. A restoring force with this characteristic is called a harmonic force .
The following parametric equation of the position as a function of time describes the motion of the circling masses:
where
The motion as a function of time can also be seen as a vector combination of two uniform circular motions. The parametric equations (1) and (2) can be rewritten as:
A transformation to a coordinate system that subtracts the overall circular motion leaves the eccentricity of the ellipse-shaped trajectory. the center of the eccentricity is located at a distance of ( a + b ) / 2 {\displaystyle (a+b)/2} from the main center:
That is in fact what is seen in the second animation, in which the motion is mapped to a coordinate system that is rotating at a constant angular velocity. The angular velocity of the motion with respect to the rotating coordinate system is 2ω, twice the angular velocity of the overall motion.
The spring is continuously doing work. More precisely, the spring is oscillating between doing positive work (increasing the weight's kinetic energy) and doing negative work (decreasing the weight's kinetic energy)
The centripetal force is a harmonic force.
The set of all solutions to the above equation of motion consists of both circular trajectories and ellipse-shaped trajectories. All the solutions have the same period of revolution. This is a distinctive feature of motion under the influence of a harmonic force; all trajectories take the same amount of time to complete a revolution.
When a rotating coordinate system is used the centrifugal term and the Coriolis term are added to the equation of motion. The following equation gives the acceleration with respect to a rotating system of an object in inertial motion.
Here, Ω is the angular velocity of the rotating coordinate system with respect to the inertial coordinate system. v is velocity of the moving object with respect to the rotating coordinate system. It is important to note that the centrifugal term is determined by the angular velocity of the rotating coordinate system; the centrifugal term does not relate to the motion of the object.
In all, this gives the following three terms in the equation of motion for motion with respect to a coordinate system rotating with angular velocity Ω .
Both the centripetal force and the centrifugal term in the equation of motion are proportional to r . The angular velocity of the rotating coordinate system is adjusted to have the same period of revolution as the object following an ellipse-shaped trajectory. Hence the vector of the centripetal force and the vector of the centrifugal term are at every distance from the center equal to each other in magnitude and opposite in direction, so those two terms drop away against each other. It is only in very special circumstances that the vector of the centripetal force and the centrifugal term drop away against each other at every distance from the center of rotation. This is the case if and only if the centripetal force is a harmonic force. In this case, only the Coriolis term remains in the equation of motion.
Since the vector of the Coriolis term always points perpendicular to the velocity with respect to the rotating coordinate system, it follows that in the case of a restoring force that is a harmonic force, the eccentricity in the trajectory will show up as a small circular motion with respect to the rotating coordinate system. The factor 2 of the Coriolis term corresponds to a period of revolution that is half the period of the overall motion.
As expected, the analysis using vector notation results in a straight confirmation of the previous analysis: The spring is continuously doing work. More precisely, the spring is oscillating between doing positive work (increasing the weight's kinetic energy) and doing negative work (decreasing the weight's kinetic energy).
In the earlier section titled 'Energy conversions', the dynamics is followed by keeping track of the energy conversions . The increase of angular velocity on contraction is in accordance with the principle of conservation of angular momentum . Since there is no torque acting on the circling weights, angular momentum is conserved. However, this disregards the causal mechanism, which is the force of the extended spring, and the work done during its contraction and extension.
Similarly, when a cannon is fired, the projectile will shoot out of the barrel toward the target, and the barrel will recoil, in accordance with the principle of conservation of momentum . This does not mean that the projectile leaves the barrel at high velocity because the barrel recoils. While recoil of the barrel must occur, as described by Newton's third law , it is not a causal agent.
The causal mechanism is in the energy conversions: the explosion of the gunpowder converts potential chemical energy to the potential energy of a highly compressed gas. As the gas expands, its high pressure exerts a force on both the projectile and the interior of the barrel. It is through the action of that force that potential energy is converted to kinetic energy of both projectile and barrel.
In the case of rotational-vibrational coupling, the causal agent is the force exerted by the spring. The spring is oscillating between doing work and doing negative work. (The work is taken to be negative when the direction of the force is opposite to the direction of the motion.) | https://en.wikipedia.org/wiki/Rotational–vibrational_coupling |
Rotational–vibrational spectroscopy is a branch of molecular spectroscopy that is concerned with infrared and Raman spectra of molecules in the gas phase . Transitions involving changes in both vibrational and rotational states can be abbreviated as rovibrational (or ro-vibrational ) transitions. When such transitions emit or absorb photons ( electromagnetic radiation ), the frequency is proportional to the difference in energy levels and can be detected by certain kinds of spectroscopy . Since changes in rotational energy levels are typically much smaller than changes in vibrational energy levels, changes in rotational state are said to give fine structure to the vibrational spectrum. For a given vibrational transition, the same theoretical treatment as for pure rotational spectroscopy gives the rotational quantum numbers , energy levels, and selection rules . In linear and spherical top molecules, rotational lines are found as simple progressions at both higher and lower frequencies relative to the pure vibration frequency. In symmetric top molecules the transitions are classified as parallel when the dipole moment change is parallel to the principal axis of rotation, and perpendicular when the change is perpendicular to that axis. The ro-vibrational spectrum of the asymmetric rotor water is important because of the presence of water vapor in the atmosphere.
Ro-vibrational spectroscopy concerns molecules in the gas phase . There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band . [ 1 ]
In the simplest cases the part of the infrared spectrum involving vibrational transitions with the same rotational quantum number (ΔJ = 0) in ground and excited states is called the Q-branch. On the high frequency side of the Q-branch the energy of rotational transitions is added to the energy of the vibrational transition. This is known as the R-branch of the spectrum for ΔJ = +1. The P-branch for ΔJ = −1 lies on the low wavenumber side of the Q branch. The appearance of the R-branch is very similar to the appearance of the pure rotation spectrum (but shifted to much higher wavenumbers ), and the P-branch appears as a nearly mirror image of the R-branch. [ note 1 ] The Q branch is sometimes missing because of transitions with no change in J being forbidden.
The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation .
The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number J {\displaystyle J} , and depends on temperature. Since there are actually 2 J + 1 {\displaystyle 2J+1} states with rotational quantum number J {\displaystyle J} , the population with value J {\displaystyle J} increases with J {\displaystyle J} initially, and then decays at higher J {\displaystyle J} . This gives the characteristic shape of the P and R branches.
A general convention is to label quantities that refer to the vibrational ground and excited states of a transition with double prime and single prime, respectively. For example, the rotational constant for the ground state is written as B ′ ′ , {\displaystyle B^{\prime \prime },} and that of the excited state as B ′ . {\displaystyle B^{\prime }.} Also, these constants are expressed in the molecular spectroscopist's units of cm −1 . so that B {\displaystyle B} in this article corresponds to B ¯ = B / h c {\displaystyle {\bar {B}}=B/hc} in the definition of rotational constant at Rigid rotor .
Numerical analysis of ro-vibrational spectral data would appear to be complicated by the fact that the wavenumber for each transition depends on two rotational constants, B ′ ′ {\displaystyle B^{\prime \prime }} and B ′ {\displaystyle B^{\prime }} . However combinations which depend on only one rotational constant are found by subtracting wavenumbers of pairs of lines (one in the P-branch and one in the R-branch) which have either the same lower level or the same upper level. [ 2 ] [ 3 ] For example, in a diatomic molecule the line denoted P ( J + 1) is due to the transition ( v = 0, J + 1) → ( v = 1, J ) (meaning a transition from the state with vibrational quantum number ν going from 0 to 1 and the rotational quantum number going from some value J + 1 to J , with J > 0), and the line R ( J − 1) is due to the transition ( v = 0, J − 1) → ( v = 1, J ). The difference between the two wavenumbers corresponds to the energy difference between the ( J + 1) and ( J − 1) levels of the lower vibrational state and is denoted by Δ 2 {\displaystyle \Delta _{2}} since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by [ 4 ]
where ν ¯ ( ) {\displaystyle {\bar {\nu }}()} means the frequency (or wavenumber ) of the given line. The main term, 2 B ″ ( 2 J + 1 ) , {\displaystyle 2B''(2J+1),} comes from the difference in the energy of the J + 1 {\displaystyle J+1} rotational state, B ″ ( ( J + 1 ) ( J + 2 ) ) , {\displaystyle B''((J+1)(J+2)),} and that of the J − 1 {\displaystyle J-1} state, B ″ ( ( J − 1 ) J ) . {\displaystyle B''((J-1)J).}
The rotational constant of the ground vibrational state B ′′ and centrifugal distortion constant, D ′′ can be found by least-squares fitting this difference as a function of J . The constant B ′′ is used to determine the internuclear distance in the ground state as in pure rotational spectroscopy . (See Appendix )
Similarly the difference R ( J ) − P ( J ) depends only on the constants B ′ and D ′ for the excited vibrational state ( v = 1), and B ′ can be used to determine the internuclear distance in that state (which is inaccessible to pure rotational spectroscopy).
Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values G ( v ) {\displaystyle G(v)} , [ note 3 ] for an anharmonic oscillator are given, to a first approximation, by
where v is a vibrational quantum number , ω e is the harmonic wavenumber and χ e is an anharmonicity constant.
When the molecule is in the gas phase, it can rotate about an axis, perpendicular to the molecular axis, passing through the centre of mass of the molecule. The rotational energy is also quantized, with term values to a first approximation given by
where J is a rotational quantum number and D is a centrifugal distortion constant . The rotational constant, B v depends on the moment of inertia of the molecule, I v , which varies with the vibrational quantum number, v
where m A and m B are the masses of the atoms A and B, and d represents the distance between the atoms. The term values of the ro-vibrational states are found (in the Born–Oppenheimer approximation ) by combining the expressions for vibration and rotation.
The first two terms in this expression correspond to a harmonic oscillator and a rigid rotor, the second pair of terms make a correction for anharmonicity and centrifugal distortion. A more general expression was given by Dunham .
The selection rule for electric dipole allowed ro-vibrational transitions, in the case of a diamagnetic diatomic molecule is
The transition with Δv=±1 is known as the fundamental transition. The selection rule has two consequences.
The calculation of the transition wavenumbers is more complicated than for pure rotation because the rotational constant B ν is different in the ground and excited vibrational states. A simplified expression for the wavenumbers is obtained when the centrifugal distortion constants D ′ {\displaystyle D^{\prime }} and D ′ ′ {\displaystyle D^{\prime \prime }} are approximately equal to each other. [ 5 ]
where positive m values refer to the R-branch and negative values refer to the P-branch. The term ω 0 gives the position of the (missing) Q-branch, the term ( B ′ + B ′ ′ ) m {\displaystyle (B^{\prime }+B^{\prime \prime })m} implies an progression of equally spaced lines in the P- and R- branches, but the third term, ( B ′ − B ′ ′ ) m 2 {\displaystyle (B^{\prime }-B^{\prime \prime })m^{2}} shows that the separation between adjacent lines changes with changing rotational quantum number. When B ′ ′ {\displaystyle B^{\prime \prime }} is greater than B ′ {\displaystyle B^{\prime }} , as is usually the case, as J increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of carbon monoxide , gives value of B ′ ′ {\displaystyle B^{\prime \prime }} of 1.915 cm −1 and B ′ {\displaystyle B^{\prime }} of 1.898 cm −1 . The bond lengths are easily obtained from these constants as r 0 = 113.3 pm, r 1 = 113.6 pm. [ 7 ] These bond lengths are slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by
where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide r eq = 113.0 pm. [ 8 ]
Nitric oxide , NO, is a special case as the molecule is paramagnetic , with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes lambda-doubling [ note 5 ] with calculated harmonic frequencies of 1904.03 and 1903.68 cm −1 . Rotational levels are also split. [ 9 ]
The quantum mechanics for homonuclear diatomic molecules such as dinitrogen , N 2 , and fluorine , F 2 , is qualitatively the same as for heteronuclear diatomic molecules, but the selection rules governing transitions are different. Since the electric dipole moment of the homonuclear diatomics is zero, the fundamental vibrational transition is electric-dipole-forbidden and the molecules are infrared inactive. [ 10 ] However, a weak quadrupole-allowed spectrum of N 2 can be observed when using long path-lengths both in the laboratory and in the atmosphere. [ 11 ] The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.
Dioxygen is a special case as the molecule is paramagnetic so magnetic-dipole-allowed transitions can be observed in the infrared. [ 11 ] The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N, [ note 6 ] so that each rotational level is split into three states with total angular momentum (molecular rotation plus electron spin) J ℏ {\displaystyle \mathrm {J\hbar } \,} , J = N + 1, N, and N - 1, each J state of this so-called p-type triplet arising from a different orientation of the spin with respect to the rotational motion of the molecule. [ 12 ] Selection rules for magnetic dipole transitions allow transitions between successive members of the triplet (ΔJ = ±1) so that for each value of the rotational angular momentum quantum number N there are two allowed transitions. The 16 O nucleus has zero nuclear spins angular momentum, so that symmetry considerations demand that N may only have odd values. [ 13 ] [ 14 ]
The selection rule is
so that the spectrum has an O-branch (∆ J = −2), a Q-branch (∆ J = 0) and an S-branch (∆ J =+2). In the approximation that B ′′ = B ′ = B the wavenumbers are given by
since the S-branch starts at J=0 and the O-branch at J=2. So, to a first approximation, the separation between S (0) and O (2) is 12 B and the separation between adjacent lines in both O- and S- branches is 4 B . The most obvious effect of the fact that B ′′ ≠ B ′ is that the Q-branch has a series of closely spaced side lines on the low-frequency side due to transitions in which Δ J =0 for J =1,2 etc. [ 15 ] Useful difference formulae, neglecting centrifugal distortion are as follows. [ 16 ]
Molecular oxygen is a special case as the molecule is paramagnetic, with two unpaired electrons. [ 17 ]
For homonuclear diatomics, nuclear spin statistical weights lead to alternating line intensities between even- J ′ ′ {\displaystyle J^{\prime \prime }} and odd- J ′ ′ {\displaystyle J^{\prime \prime }} levels. For nuclear spin I = 1/2 as in 1 H 2 and 19 F 2 the intensity alternation is 1:3. For 2 H 2 and 14 N 2 , I =1 and the statistical weights are 6 and 3 so that the even- J ′ ′ {\displaystyle J^{\prime \prime }} levels are twice as intense. For 16 O 2 ( I =0) all transitions with even values of J ′ ′ {\displaystyle J^{\prime \prime }} are forbidden. [ 16 ]
These molecules fall into two classes, according to symmetry : centrosymmetric molecules with point group D ∞h , such as carbon dioxide , CO 2 , and ethyne or acetylene, HCCH; and non-centrosymmetric molecules with point group C ∞v such as hydrogen cyanide , HCN, and nitrous oxide , NNO. Centrosymmetric linear molecules have a dipole moment of zero, so do not show a pure rotation spectrum in the infrared or microwave regions. On the other hand, in certain vibrational excited states the molecules do have a dipole moment so that a ro-vibrational spectrum can be observed in the infrared.
The spectra of these molecules are classified according to the direction of the dipole moment change vector. When the vibration induces a dipole moment change pointing along the molecular axis the term parallel is applied, with the symbol ∥ {\displaystyle \parallel } . When the vibration induces a dipole moment pointing perpendicular to the molecular axis the term perpendicular is applied, with the symbol ⊥ {\displaystyle \perp } . In both cases the P- and R- branch wavenumbers follow the same trend as in diatomic molecules. The two classes differ in the selection rules that apply to ro-vibrational transitions. [ 18 ] For parallel transitions the selection rule is the same as for diatomic molecules, namely, the transition corresponding to the Q-branch is forbidden. An example is the C-H stretching mode of hydrogen cyanide. [ 19 ]
For a perpendicular vibration the transition Δ J =0 is allowed. This means that the transition is allowed for the molecule with the same rotational quantum number in the ground and excited vibrational state, for all the populated rotational states. This makes for an intense, relatively broad, Q-branch consisting of overlapping lines due to each rotational state. The N-N-O bending mode of nitrous oxide , at ca. 590 cm −1 is an example. [ 6 ]
The spectra of centrosymmetric molecules exhibit alternating line intensities due to quantum state symmetry effects, since rotation of the molecule by 180° about a 2-fold rotation axis is equivalent to exchanging identical nuclei. In carbon dioxide, the oxygen atoms of the predominant isotopic species 12 C 16 O 2 have spin zero and are bosons , so that the total wavefunction must be symmetric when the two 16 O nuclei are exchanged. The nuclear spin factor is always symmetric for two spin-zero nuclei, so that the rotational factor must also be symmetric which is true only for even-J levels. The odd-J rotational levels cannot exist and the allowed vibrational bands consist of only absorption lines from even-J initial levels. The separation between adjacent lines in the P- and R- branches is close to 4B rather than 2B as alternate lines are missing. [ 20 ] For acetylene the hydrogens of 1 H 12 C 12 C 1 H have spin-1/2 and are fermions , so the total wavefunction is antisymmetric when two 1 H nuclei are exchanged. As is true for ortho and para hydrogen the nuclear spin function of the two hydrogens has three symmetric ortho states and one antisymmetric para states. For the three ortho states, the rotational wave function must be antisymmetric corresponding to odd J, and for the one para state it is symmetric corresponding to even J. The population of the odd J levels are therefore three times higher than the even J levels, and alternate line intensities are in the ratio 3:1. [ 21 ] [ 22 ]
These molecules have equal moments of inertia about any axis, and belong to the point groups T d (tetrahedral AX 4 ) and O h (octahedral AX 6 ). Molecules with these symmetries have a dipole moment of zero, so do not have a pure rotation spectrum in the infrared or microwave regions. [ 24 ]
Tetrahedral molecules such as methane , CH 4 , have infrared-active stretching and bending vibrations, belonging to the T 2 (sometimes written as F 2 ) representation. [ note 7 ] These vibrations are triply degenerate and the rotational energy levels have three components separated by the Coriolis interaction . [ 25 ] The rotational term values are given, to a first order approximation, by [ 26 ]
where ζ r {\displaystyle \zeta _{r}} is a constant for Coriolis coupling. The selection rule for a fundamental vibration is
Thus, the spectrum is very much like the spectrum from a perpendicular vibration of a linear molecule, with a strong Q-branch composed of many transitions in which the rotational quantum number is the same in the vibrational ground and excited states, J ′ = J ′ ′ = 1 , 2... {\displaystyle J^{\prime }=J^{\prime \prime }=1,2...} The effect of Coriolis coupling is clearly visible in the C-H stretching vibration of methane, though detailed study has shown that the first-order formula for Coriolis coupling, given above, is not adequate for methane. [ 27 ] [ 28 ] [ 29 ]
These molecules have a unique principal rotation axis of order 3 or higher. There are two distinct moments of inertia and therefore two rotational constants. For rotation about any axis perpendicular to the unique axis, the moment of inertia is I ⊥ {\displaystyle I_{\perp }} and the rotational constant is B = h 8 π 2 c I ⊥ {\displaystyle B={h \over {8\pi ^{2}cI_{\perp }}}} , as for linear molecules. For rotation about the unique axis, however, the moment of inertia is I ∥ {\displaystyle I_{\parallel }} and the rotational constant is A = h 8 π 2 c I ∥ {\displaystyle A={h \over {8\pi ^{2}cI_{\parallel }}}} . Examples include ammonia , NH 3 and methyl chloride , CH 3 Cl (both of molecular symmetry described by point group C 3v ), boron trifluoride , BF 3 and phosphorus pentachloride , PCl 5 (both of point group D 3h ), and benzene , C 6 H 6 (point group D 6h ).
For symmetric rotors a quantum number J is associated with the total angular momentum of the molecule. For a given value of J, there is a 2 J +1- fold degeneracy with the quantum number, M taking the values + J ...0 ... - J . The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. As with linear molecules, transitions are classified as parallel , ∥ {\displaystyle \parallel } or perpendicular , ⊥ {\displaystyle \perp } , in this case according to the direction of the dipole moment change with respect to the principal rotation axis. A third category involves certain overtones and combination bands which share the properties of both parallel and perpendicular transitions. The selection rules are
The fact that the selection rules are different is the justification for the classification and it means that the spectra have a different appearance which can often be immediately recognized.
An expression for the calculated wavenumbers of the P- and R- branches may be given as [ 30 ]
in which m = J +1 for the R-branch and - J for the P-branch. The three centrifugal distortion constants D J , D J K {\displaystyle D_{J},D_{JK}} , and D K {\displaystyle D_{K}} are needed to fit the term values of each level. [ 1 ] The wavenumbers of the sub-structure corresponding to each band are given by
ν ¯ s u b {\displaystyle {\bar {\nu }}_{sub}} represents the Q-branch of the sub-structure, whose position is given by
The C-Cl stretching vibration of methyl chloride , CH 3 Cl, gives a parallel band since the dipole moment change is aligned with the 3-fold rotation axis. The line spectrum shows the sub-structure of this band rather clearly; [ 6 ] in reality, very high resolution spectroscopy would be needed to resolve the fine structure fully. Allen and Cross show parts of the spectrum of CH 3 D and give a detailed description of the numerical analysis of the experimental data. [ 31 ] [ 32 ]
The selection rule for perpendicular bands give rise to more transitions than with parallel bands. A band can be viewed as a series of sub-structures, each with P, Q and R branches. The Q-branches are separated by approximately 2( A ′- B ′). The asymmetric HCH bending vibration of methyl chloride is typical. It shows a series of intense Q-branches with weak rotational fine structure. [ 6 ] Analysis of the spectra is made more complicated by the fact that the ground-state vibration is bound, by symmetry, to be a degenerate vibration, which means that Coriolis coupling also affects the spectrum. [ 33 ]
Overtones of a degenerate fundamental vibration have components of more than one symmetry type. For example, the first overtone of a vibration belonging to the E representation in a molecule like ammonia, NH 3 , will have components belonging to A 1 and E representations. A transition to the A 1 component will give a parallel band and a transition to the E component will give perpendicular bands; the result is a hybrid band. [ 34 ]
For ammonia, NH 3 , the symmetric bending vibration is observed as two branches near 930 cm −1 and 965 cm −1 . This so-called inversion doubling arises because the symmetric bending vibration is actually a large-amplitude motion known as inversion , in which the nitrogen atom passes through the plane of the three hydrogen atoms, similar to the inversion of an umbrella. The potential energy curve for such a vibration has a double minimum for the two pyramidal geometries, so that the vibrational energy levels occur in pairs which correspond to combinations of the vibrational states in the two potential minima. The two v = 1 states combine to form a symmetric state (1 + ) at 932.5 cm −1 above the ground (0 + ) state and an antisymmetric state (1 − ) at 968.3 cm −1 . [ 35 ]
The vibrational ground state (v = 0) is also doubled although the energy difference is much smaller, and the transition between the two levels can be measured directly in the microwave region, at ca. 24 GHz (0.8 cm −1 ). [ 36 ] [ 37 ] This transition is historically significant and was used in the ammonia maser , the fore-runner of the laser . [ 38 ]
Asymmetric top molecules have at most one or more 2-fold rotation axes. There are three unequal moments of inertia about three mutually perpendicular principal axes . The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.
The water molecule is an important example of this class of molecule, particularly because of the presence of water vapor in the atmosphere. The low-resolution spectrum shown in green illustrates the complexity of the spectrum. At wavelengths greater than 10 μm (or wavenumbers less than 1000 cm −1 ) the absorption is due to pure rotation. The band around 6.3 μm (1590 cm −1 ) is due to the HOH bending vibration; the considerable breadth of this band is due to the presence of extensive rotational fine structure. High-resolution spectra of this band are shown in Allen and Cross, p 221. [ 46 ] The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600 cm −1 and 3700 cm −1 are shown in Banwell and McCash, p91.
Ro-vibrational bands of asymmetric top molecules are classed as A-, B- or C- type for transitions in which the dipole moment change is along the axis of smallest moment of inertia to the highest. [ 47 ]
Ro-vibrational spectra are usually measured at high spectral resolution . In the past, this was achieved by using an echelle grating as the spectral dispersion element in a grating spectrometer . [ 9 ] This is a type of diffraction grating optimized to use higher diffraction orders. [ 48 ] Today at all resolutions the preferred method is FTIR . The primary reason for this is that infrared detectors are inherently noisy, and FTIR detects summed signals at multiple wavelengths simultaneously achieving a higher signal to noise by virtue of Fellgett's advantage for multiplexed methods. The resolving power of an FTIR spectrometer depends on the maximum retardation of the moving mirror. For example, to achieve a resolution of 0.1 cm −1 , the moving mirror must have a maximum displacement of 10 cm from its position at zero path difference. Connes measured the vibration-rotation spectrum of Venusian CO 2 at this resolution. [ 49 ] A spectrometer with 0.001 cm −1 resolution is now available commercially. The throughput advantage of FTIR is important for high-resolution spectroscopy as the monochromator in a dispersive instrument with the same resolution would have very narrow entrance and exit slits .
When measuring the spectra of gases it is relatively easy to obtain very long path-lengths by using a multiple reflection cell. [ 50 ] This is important because it allows the pressure to be reduced so as to minimize pressure broadening of the spectral lines, which may degrade resolution. Path lengths up to 20m are commercially available.
The method of combination differences uses differences of wavenumbers in the P- and R- branches to obtain data that depend only on rotational constants in the vibrational ground or excited state. For the excited state
This function can be fitted, using the method of least-squares to data for carbon monoxide, from Harris and Bertolucci. [ 51 ] The data calculated with the formula
in which centrifugal distortion is ignored, are shown in the columns labelled with (1). This formula implies that the data should lie on a straight line with slope 2B′′ and intercept zero. At first sight the data appear to conform to this model, with a root mean square residual of 0.21 cm −1 . However, when centrifugal distortion is included, using the formula
the least-squares fit is improved markedly, with ms residual decreasing to 0.000086 cm −1 . The calculated data are shown in the columns labelled with (2). | https://en.wikipedia.org/wiki/Rotational–vibrational_spectroscopy |
Rotavirus translation , the process of translating mRNA into proteins , occurs in a different way in Rotaviruses . Unlike the vast majority of cellular proteins in other organisms, in Rotaviruses the proteins are translated from capped but nonpolyadenylated mRNAs. The viral nonstructural protein NSP3 specifically binds the 3'-end consensus sequence of viral mRNAs and interacts with the eukaryotic translation initiation factor eIF4G . The Rotavirus replication cycle occurs entirely in the cytoplasm. Upon virus entry, the viral transcriptase synthesizes capped but nonpolyadenylated mRNA The viral mRNAs bear 5' and 3' untranslated regions ( UTR ) of variable length and are flanked by two different sequences common to all genes.
In the group A rotaviruses , the 3'-end consensus sequence UGACC is highly conserved among the 11 genes. Rotavirus NSP3 presents several similarities to PABP ; in rotavirus-infected cells, NSP3 can be cross-linked to the 3' end of rotavirus mRNAs and is coimmunoprecipitated with eIF4G . The binding of NSP3A to eIF4G and its specific interaction with the 3' end of viral mRNA brings the viral mRNA and the translation initiation machinery into contact, thus favoring efficient translation of the viral mRNA. NSP3 interacts with the same region of eIF4G as PABP does. As a consequence, during rotavirus infection PABP is evicted from eIF4G, probably impairing the translation of polyadenylated mRNA and leading to the shutoff of cellular mRNA translation observed during rotavirus infection. [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Rotavirus_translation |
A rotaxane (from Latin rota ' wheel ' and axis ' axle ' ) is a mechanically interlocked molecular architecture consisting of a dumbbell -shaped molecule which is threaded through a macrocycle (see graphical representation). The two components of a rotaxane are kinetically trapped since the ends of the dumbbell (often called stoppers ) are larger than the internal diameter of the ring and prevent dissociation (unthreading) of the components since this would require significant distortion of the covalent bonds .
Much of the research concerning rotaxanes and other mechanically interlocked molecular architectures, such as catenanes , has been focused on their efficient synthesis or their utilization as artificial molecular machines . However, examples of rotaxane substructure have been found in naturally occurring peptides , including: cystine knot peptides, cyclotides or lasso-peptides such as microcin J25.
The earliest reported synthesis of a rotaxane in 1967 relied on the statistical probability that if two halves of a dumbbell-shaped molecule were reacted in the presence of a macrocycle that some small percentage would connect through the ring. [ 2 ] To obtain a reasonable quantity of rotaxane, the macrocycle was attached to a solid-phase support and treated with both halves of the dumbbell 70 times and then severed from the support to give a 6% yield. However, the synthesis of rotaxanes has advanced significantly and efficient yields can be obtained by preorganizing the components utilizing hydrogen bonding , metal coordination, hydrophobic forces , covalent bonds , or coulombic interactions . The three most common strategies to synthesize rotaxane are "capping", "clipping", and "slipping", [ 3 ] though others do exist. [ 4 ] [ 5 ] Recently, Leigh and co-workers described a new pathway to mechanically interlocked architectures involving a transition-metal center that can catalyse a reaction through the cavity of a macrocycle. [ 6 ]
Synthesis via the capping method relies strongly upon a thermodynamically driven template effect; that is, the "thread" is held within the "macrocycle" by non-covalent interactions, for example rotaxinations with cyclodextrin macrocycles involve exploitation of the hydrophobic effect. This dynamic complex or pseudorotaxane is then converted to the rotaxane by reacting the ends of the threaded guest with large groups, preventing disassociation. [ 8 ]
The clipping method is similar to the capping reaction except that in this case the dumbbell shaped molecule is complete and is bound to a partial macrocycle. The partial macrocycle then undergoes a ring closing reaction around the dumbbell-shaped molecule, forming the rotaxane. [ 9 ]
The method of slipping is one which exploits the thermodynamic [ 10 ] stability of the rotaxane. If the end groups of the dumbbell are an appropriate size it will be able to reversibly thread through the macrocycle at higher temperatures. By cooling the dynamic complex, it becomes kinetically trapped as a rotaxane at the lower temperature.
Snapping involves two separate parts of the thread, both containing a bulky group. one part of the thread is then threaded to the macrocycle, forming a semi rotaxane, and end is closed of by the other part of the thread forming the rotaxane.
Leigh and co-workers recently began to explore a strategy in which template ions could also play an active role in promoting the crucial final covalent bond forming reaction that captures the interlocked structure (i.e., the metal has a dual function, acting as a template for entwining the precursors and catalyzing covalent bond formation between the reactants).
Rotaxane-based molecular machines have been of initial interest for their potential use in molecular electronics as logic molecular switching elements and as molecular shuttles . [ 12 ] [ 13 ] These molecular machines are usually based on the movement of the macrocycle on the dumbbell. The macrocycle can rotate around the axis of the dumbbell like a wheel and axle or it can slide along its axis from one site to another. Controlling the position of the macrocycle allows the rotaxane to function as a molecular switch, with each possible location of the macrocycle corresponding to a different state. These rotaxane machines can be manipulated both by chemical [ 14 ] and photochemical inputs. [ 15 ] Rotaxane based systems have also been shown to function as molecular muscles. [ 16 ] [ 17 ] In 2009, there was a report of a "domino effect" from one extremity to the other in a Glycorotaxane Molecular Machine. In this case, the 4 C 1 or 1 C 4 chair-like conformation of the manno pyranoside stopper can be controlled, depending on the localization of the macrocycle. [ 18 ] In 2012, unique pseudo-macrocycles consisting of double-lasso molecular machines (also called rotamacrocycles) were reported in Chem. Sci. These structures can be tightened or loosened depending on pH. A controllable jump rope movement was also observed in these new molecular machines. [ 19 ]
Potential application as long-lasting dyes is based on the enhanced stability of the inner portion of the dumbbell-shaped molecule. [ 20 ] [ 21 ] Studies with cyclodextrin -protected rotaxane azo dyes established this characteristic. More reactive squaraine dyes have also been shown to have enhanced stability by preventing nucleophilic attack of the inner squaraine moiety . [ 22 ] The enhanced stability of rotaxane dyes is attributed to the insulating effect of the macrocycle , which is able to block interactions with other molecules.
In a nanorecording application, [ 23 ] a certain rotaxane is deposited as a Langmuir–Blodgett film on ITO -coated glass. When a positive voltage is applied with the tip of a scanning tunneling microscope probe, the rotaxane rings in the tip area switch to a different part of the dumbbell and the resulting new conformation makes the molecules stick out 0.3 nanometer from the surface. This height difference is sufficient for a memory dot . It is not yet known how to erase such a nanorecording film.
Accepted nomenclature is to designate the number of components of the rotaxane in brackets as a prefix. [ 24 ] Therefore, the rotaxane consisting of a single dumbbell-shaped axial molecule with a single macrocycle around its shaft is called a [2]rotaxane, and two cyanostar molecules around the central phosphate group of dialkylphosphate is a [3]rotaxane. | https://en.wikipedia.org/wiki/Rotaxane |
In mathematics , a Rota–Baxter algebra is an associative algebra , together with a particular linear map R {\displaystyle R} which satisfies the Rota–Baxter identity . It appeared first in the work of the American mathematician Glen E. Baxter [ 1 ] in the realm of probability theory . Baxter's work was further explored from different angles by Gian-Carlo Rota , [ 2 ] [ 3 ] [ 4 ] Pierre Cartier , [ 5 ] and Frederic V. Atkinson , [ 6 ] among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory. [ 7 ] [ 8 ]
In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation , [ 9 ] named after the well-known physicists Chen-Ning Yang and Rodney Baxter .
The study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory , [ 10 ] dendriform algebras , associative analogue of the classical Yang–Baxter equation [ 11 ] and mixable shuffle product constructions. [ 12 ]
Let k {\displaystyle k} be a commutative ring and let λ {\displaystyle \lambda } be given. A linear operator R {\displaystyle R} on a k {\displaystyle k} -algebra A {\displaystyle A} is called a Rota–Baxter operator of weight λ {\displaystyle \lambda } if it satisfies the Rota–Baxter relation of weight λ {\displaystyle \lambda } :
for all x , y ∈ A {\displaystyle x,y\in A} . Then the pair ( A , R ) {\displaystyle (A,R)} or simply A {\displaystyle A} is called a Rota–Baxter algebra of weight λ {\displaystyle \lambda } . In some literature, θ = − λ {\displaystyle \theta =-\lambda } is used in which case the above equation becomes
called the Rota-Baxter equation of weight θ {\displaystyle \theta } . The terms Baxter operator algebra and Baxter algebra are also used.
Let R {\displaystyle R} be a Rota–Baxter of weight λ {\displaystyle \lambda } . Then − λ I d − R {\displaystyle -\lambda Id-R} is also a Rota–Baxter operator of weight λ {\displaystyle \lambda } . Further, for μ {\displaystyle \mu } in k {\displaystyle k} , μ R {\displaystyle \mu R} is a Rota-Baxter operator of weight μ λ {\displaystyle \mu \lambda } .
Integration by parts
Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let C ( R ) {\displaystyle C(R)} be the algebra of continuous functions from the real line to the real line. Let f ( x ) ∈ C ( R ) {\displaystyle f(x)\in C(R)} be a continuous function. Define integration as the Rota–Baxter operator
Let G ( x ) = I ( g ) ( x ) {\displaystyle G(x)=I(g)(x)} and F ( x ) = I ( f ) ( x ) {\displaystyle F(x)=I(f)(x)} . Then the formula for integration for parts can be written in terms of these variables as
In other words
which shows that I {\displaystyle I} is a Rota–Baxter algebra of weight 0.
The Spitzer identity appeared is named after the American mathematician Frank Spitzer . It is regarded as a remarkable
stepping stone in the theory of sums of independent random variables in fluctuation theory of probability. It can naturally be understood in terms of Rota–Baxter operators. | https://en.wikipedia.org/wiki/Rota–Baxter_algebra |
The Rote Liste , full name Rote Liste der bedrohten Nutztierrassen im Bundesgebiet , [ a ] is a red list of threatened breeds of domestic animal published annually by the Gesellschaft zur Erhaltung alter und gefährdeter Haustierrassen , the German national association for the conservation of historic and endangered domestic animal breeds. [ 1 ]
The GEH was founded in Witzenhausen , in Hesse , central Germany, in 1981. In 1987 it established the criteria on which the Rote Liste is based. The list is published annually, and attributes one of four categories of conservation risk to domestic breeds of cattle, dogs, goats, horses, pigs, rabbits and sheep, of chickens, ducks, geese and turkeys, and of bees; listing of domestic pigeon breeds is in preparation. [ 2 ] Some breeds from outside Germany are listed separately. The four levels of risk are:
The risk level is calculated using a formula that takes into account five criteria: the number of breeding animals or breeding females; the percentage of pure-bred matings; the five-year trend in breed numbers; the number of breeders or herds; and the interval between generations of the animal. [ 3 ]
The GEH also publishes, in conjunction with the Bund Deutscher Rassegeflügelzüchter [ de ] , the German national association of poultry breeders, a separate list of the historic poultry breeds and colour varieties that were raised in Germany before 1930. The same levels of conservation risk are assigned as in the main red list. [ 4 ]
In 2014 the breeds listed were: [ 2 ]
Murnau-Werdenfelser
Glan
Ansbach-Triesdorfer [ de ]
Angeln (old breeding standards)
Original Braunvieh
Rotes Höhenvieh
Hinterwälder
Pinzgauer
Deutsches Schwarzbuntes Niederungsrind
Vorderwälder
Augsburger
Bergische Kräher
Bergischer Schlotterkamm
German Langshan
Dominique
Krüper
Malines
Minorca
Sachsenhühn
Ramelsloher
Sundheimer [ de ]
Deutsches Reichshuhn [ de ]
Lakenfelder
Ostfriesische Möwen
Thüringer Barthuhn
Barnevelder
Hamburg
Italiener [ de ]
Orpington
Rheinländer
Vorwerk
Westfälische Totleger
Wyandotte
Großspitz
Altdeutscher Hütehund (South and Middle German types)
Pomeranian Duck
Orpington Duck
Rouen Duck
Indian Runner Duck
Muscovy Duck
Lippegans [ de ]
Bayerische Landgans [ de ]
Leinegans [ de ]
Emden Goose
Alt-Württemberger
Leutstettener
Dülmener
Senner
Arenberg-Nordkirchner
Lehmkuhlener
Ostfriesen and Alt-Oldenburger
Rhenish German Coldblood
Deutsches Sattelschwein [ de ]
Bentheim Black Pied
Rotbuntes Husumer Schwein
Angora
Deutsche Großsilber
Harlequin (Japaner)
Luxkaninchen [ de ]
American Sable (Marderkaninchen)
Brillenschaf
Leineschaf [ de ] (original type)
Weiße gehörnte Heidschnucke
German Karakul
Schwarzes Bergschaf
Weißes Bergschaf
Waldschaf [ de ]
Merinolangwollschaf [ de ]
Weiße Hornlose Heidschnucke [ de ] (Moorschnucke)
Skudde
Pomeranian Coarsewool
Weißköpfiges Fleischschaf [ de ]
Merinofleischschaf [ de ]
Rhönschaf [ de ]
Deutsche Puten | https://en.wikipedia.org/wiki/Rote_Liste |
In mathematics , Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers . It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue ( 1909 ), Carl Ludwig Siegel ( 1921 ), Freeman Dyson ( 1947 ), and Klaus Roth ( 1955 ).
Roth's theorem states that every irrational algebraic number α {\displaystyle \alpha } has approximation exponent equal to 2. This means that, for every ε > 0 {\displaystyle \varepsilon >0} , the inequality
can have only finitely many solutions in coprime integers p {\displaystyle p} and q {\displaystyle q} . Roth's proof of this fact resolved a conjecture by Siegel. It follows that every irrational algebraic number α satisfies
with C ( α , ε ) {\displaystyle C(\alpha ,\varepsilon )} a positive number depending only on ε > 0 {\displaystyle \varepsilon >0} and α {\displaystyle \alpha } .
The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of d for an algebraic number α of degree d ≥ 2. This is already enough to demonstrate the existence of transcendental numbers . Thue realised that an exponent less than d would have applications to the solution of Diophantine equations and in Thue's theorem from 1909 established an exponent d / 2 + 1 + ε {\displaystyle d/2+1+\varepsilon } which he applied to prove the finiteness of the solutions of Thue equation . Siegel's theorem improves this to an exponent about 2 √ d , and Dyson's theorem of 1947 has exponent about √ 2 d .
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting ε = 0 {\displaystyle \varepsilon =0} : by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case. However, there is a stronger conjecture of Serge Lang that
can have only finitely many solutions in integers p and q . If one lets α run over the whole of the set of real numbers , not just the algebraic reals, then both Roth's conclusion and Lang's hold
for almost all α {\displaystyle \alpha } . So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero . [ 1 ]
The theorem is not currently effective : that is, there is no bound known on the possible values of p , q given α {\displaystyle \alpha } . [ 2 ] Davenport & Roth (1955) showed that Roth's techniques could be used to give an effective bound for the number of p / q satisfying the inequality, using a "gap" principle. [ 2 ] The fact that we do not actually know C (ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.
The proof technique involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon ε {\displaystyle \varepsilon } , leading to a contradiction in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see effective results in number theory ); this is of particular interest since a major application of this type of result is to bound the number of solutions of some Diophantine equations.
There is a higher-dimensional version, Schmidt's subspace theorem , of the basic result. There are also numerous extensions, for example using the p -adic metric , [ 3 ] based on the Roth method.
William J. LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field . Define the height H (ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its minimal polynomial . Fix κ>2. For a given algebraic number α and algebraic number field K , the equation
has only finitely many solutions in elements ξ of K . [ 4 ] | https://en.wikipedia.org/wiki/Roth's_theorem |
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers . It was first proven by Klaus Roth in 1953. [ 1 ] Roth's theorem is a special case of Szemerédi's theorem for the case k = 3 {\displaystyle k=3} .
A subset A of the natural numbers is said to have positive upper density if
Roth's theorem on arithmetic progressions (infinite version) : A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression.
An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N]=\{1,\dots ,N\}} . Let r 3 ( [ N ] ) {\displaystyle r_{3}([N])} be the size of the largest subset of [ N ] {\displaystyle [N]} which contains no 3-term arithmetic progression.
Roth's theorem on arithmetic progressions (finitary version) : r 3 ( [ N ] ) = o ( N ) . {\displaystyle r_{3}([N])=o(N).}
Improving upper and lower bounds on r 3 ( [ N ] ) {\displaystyle r_{3}([N])} is still an open research problem.
The first result in this direction was Van der Waerden's theorem in 1927, which states that for sufficiently large N, coloring the integers 1 , … , n {\displaystyle 1,\dots ,n} with r {\displaystyle r} colors will result in a k {\displaystyle k} term arithmetic progression. [ 2 ]
Later on in 1936 Erdős and Turán conjectured a much stronger result that any subset of the integers with positive density contains arbitrarily long arithmetic progressions. In 1942, Raphaël Salem and Donald C. Spencer provided a construction of a 3-AP-free set (i.e. a set with no 3-term arithmetic progressions) of size N e O ( log N / log log N ) {\displaystyle {\frac {N}{e^{O(\log N/\log \log N)}}}} , [ 3 ] disproving an additional conjecture of Erdős and Turán that r 3 ( [ N ] ) = N 1 − δ {\displaystyle r_{3}([N])=N^{1-\delta }} for some δ > 0 {\displaystyle \delta >0} . [ 4 ]
In 1953, Roth partially resolved the initial conjecture by proving they must contain an arithmetic progression of length 3 using Fourier analytic methods. Eventually, in 1975, Szemerédi proved Szemerédi's theorem using combinatorial techniques, resolving the original conjecture in full.
The original proof given by Roth used Fourier analytic methods. Later on another proof was given using Szemerédi's regularity lemma .
In 1953, Roth used Fourier analysis to prove an upper bound of r 3 ( [ N ] ) = O ( N log log N ) {\displaystyle r_{3}([N])=O\left({\frac {N}{\log \log N}}\right)} . Below is a sketch of this proof.
Define the Fourier transform of a function f : Z → C {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {C} } to be the function f ^ : [ 0 , 1 ) → C {\displaystyle {\widehat {f}}:[0,1)\rightarrow \mathbb {C} } satisfying
where e ( t ) = e 2 π i t {\displaystyle e(t)=e^{2\pi it}} .
Let A {\displaystyle A} be a 3-AP-free subset of { 1 , … , N } {\displaystyle \{1,\dots ,N\}} . The proof proceeds in 3 steps.
For functions, f , g , h : Z → C , {\displaystyle f,g,h:\mathbb {Z} \rightarrow \mathbb {C} ,} define
Counting Lemma Let f , g : Z → C {\displaystyle f,g:\mathbb {Z} \rightarrow \mathbb {C} } satisfy ∑ n ∈ Z | f ( n ) | 2 , ∑ n ∈ Z | g ( n ) | 2 ≤ M {\displaystyle \sum _{n\in \mathbb {Z} }|f(n)|^{2},\sum _{n\in \mathbb {Z} }|g(n)|^{2}\leq M} . Define Λ 3 ( f ) = Λ ( f , f , f ) {\displaystyle \Lambda _{3}(f)=\Lambda (f,f,f)} . Then | Λ 3 ( f ) − Λ 3 ( g ) | ≤ 3 M ‖ f − g ^ ‖ ∞ {\displaystyle |\Lambda _{3}(f)-\Lambda _{3}(g)|\leq 3M\|{\widehat {f-g}}\|_{\infty }} .
The counting lemma tells us that if the Fourier Transforms of f {\displaystyle f} and g {\displaystyle g} are "close", then the number of 3-term arithmetic progressions between the two should also be "close." Let α = | A | / N {\displaystyle \alpha =|A|/N} be the density of A {\displaystyle A} . Define the functions f = 1 A {\displaystyle f=\mathbf {1} _{A}} (i.e the indicator function of A {\displaystyle A} ), and g = α ⋅ 1 [ N ] {\displaystyle g=\alpha \cdot \mathbf {1} _{[N]}} . Step 1 can then be deduced by applying the Counting Lemma to f {\displaystyle f} and g {\displaystyle g} , which tells us that there exists some θ {\displaystyle \theta } such that
Given the θ {\displaystyle \theta } from step 1, we first show that it's possible to split up [ N ] {\displaystyle [N]} into relatively large subprogressions such that the character x ↦ e ( x θ ) {\displaystyle x\mapsto e(x\theta )} is roughly constant on each subprogression.
Lemma 1: Let 0 < η < 1 , θ ∈ R {\displaystyle 0<\eta <1,\theta \in \mathbb {R} } . Assume that N > C η − 6 {\displaystyle N>C\eta ^{-6}} for a universal constant C {\displaystyle C} . Then it is possible to partition [ N ] {\displaystyle [N]} into arithmetic progressions P i {\displaystyle P_{i}} with length N 1 / 3 ≤ | P i | ≤ 2 N 1 / 3 {\displaystyle N^{1/3}\leq |P_{i}|\leq 2N^{1/3}} such that sup x , y ∈ P i | e ( x θ ) − e ( y θ ) | < η {\displaystyle \sup _{x,y\in P_{i}}|e(x\theta )-e(y\theta )|<\eta } for all i {\displaystyle i} .
Next, we apply Lemma 1 to obtain a partition into subprogressions. We then use the fact that θ {\displaystyle \theta } produced a large coefficient in step 1 to show that one of these subprogressions must have a density increment:
Lemma 2: Let A {\displaystyle A} be a 3-AP-free subset of [ N ] {\displaystyle [N]} , with | A | = α N {\displaystyle |A|=\alpha N} and N > C α − 12 {\displaystyle N>C\alpha ^{-12}} . Then, there exists a sub progression P ⊂ [ N ] {\displaystyle P\subset [N]} such that | P | ≥ N 1 / 3 {\displaystyle |P|\geq N^{1/3}} and | A ∩ P | ≥ ( α + α 2 / 40 ) | P | {\displaystyle |A\cap P|\geq (\alpha +\alpha ^{2}/40)|P|} .
We now iterate step 2. Let a t {\displaystyle a_{t}} be the density of A {\displaystyle A} after the t {\displaystyle t} th iteration. We have that α 0 = α , {\displaystyle \alpha _{0}=\alpha ,} and α t + 1 ≥ α + α 2 / 40. {\displaystyle \alpha _{t+1}\geq \alpha +\alpha ^{2}/40.} First, see that α {\displaystyle \alpha } doubles (i.e. reach T {\displaystyle T} such that α T ≥ 2 α 0 {\displaystyle \alpha _{T}\geq 2\alpha _{0}} ) after at most 40 / α + 1 {\displaystyle 40/\alpha +1} steps. We double α {\displaystyle \alpha } again (i.e reach α T ≥ 4 α 0 {\displaystyle \alpha _{T}\geq 4\alpha _{0}} ) after at most 20 / α + 1 {\displaystyle 20/\alpha +1} steps. Since α ≤ 1 {\displaystyle \alpha \leq 1} , this process must terminate after at most O ( 1 / α ) {\displaystyle O(1/\alpha )} steps.
Let N t {\displaystyle N_{t}} be the size of our current progression after t {\displaystyle t} iterations. By Lemma 2, we can always continue the process whenever N t ≥ C α t − 12 , {\displaystyle N_{t}\geq C\alpha _{t}^{-12},} and thus when the process terminates we have that N t ≤ C α t − 12 ≤ C α − 12 . {\displaystyle N_{t}\leq C\alpha _{t}^{-12}\leq C\alpha ^{-12}.} Also, note that when we pass to a subprogression, the size of our set decreases by a cube root. Therefore
Therefore α = O ( 1 / log log N ) , {\displaystyle \alpha =O(1/\log \log N),} so | A | = O ( N log log N ) , {\displaystyle |A|=O\left({\frac {N}{\log \log N}}\right),} as desired. ◼ {\displaystyle \blacksquare }
Unfortunately, this technique does not generalize directly to larger arithmetic progressions to prove Szemerédi's theorem. An extension of this proof eluded mathematicians for decades until 1998, when Timothy Gowers developed the field of higher-order Fourier analysis specifically to generalize the above proof to prove Szemerédi's theorem. [ 5 ]
Below is an outline of a proof using the Szemerédi regularity lemma .
Let G {\displaystyle G} be a graph and X , Y ⊆ V ( G ) {\displaystyle X,Y\subseteq V(G)} . We call ( X , Y ) {\displaystyle (X,Y)} an ϵ {\displaystyle \epsilon } -regular pair if for all A ⊂ X , B ⊂ Y {\displaystyle A\subset X,B\subset Y} with | A | ≥ ϵ | X | , | B | ≥ ϵ | Y | {\displaystyle |A|\geq \epsilon |X|,|B|\geq \epsilon |Y|} , one has | d ( A , B ) − d ( X , Y ) | ≤ ϵ {\displaystyle |d(A,B)-d(X,Y)|\leq \epsilon } .
A partition P = { V 1 , … , V k } {\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{k}\}} of V ( G ) {\displaystyle V(G)} is an ϵ {\displaystyle \epsilon } -regular partition if
Then the Szemerédi regularity lemma says that for every ϵ > 0 {\displaystyle \epsilon >0} , there exists a constant M {\displaystyle M} such that every graph has an ϵ {\displaystyle \epsilon } -regular partition into at most M {\displaystyle M} parts.
We can also prove that triangles between ϵ {\displaystyle \epsilon } -regular sets of vertices must come along with many other triangles. This is known as the triangle counting lemma.
Triangle Counting Lemma: Let G {\displaystyle G} be a graph and X , Y , Z {\displaystyle X,Y,Z} be subsets of the vertices of G {\displaystyle G} such that ( X , Y ) , ( Y , Z ) , ( Z , X ) {\displaystyle (X,Y),(Y,Z),(Z,X)} are all ϵ {\displaystyle \epsilon } -regular pairs for some ϵ > 0 {\displaystyle \epsilon >0} . Let d X Y , d X Z , d Y Z {\displaystyle d_{XY},d_{XZ},d_{YZ}} denote the edge densities d ( X , Y ) , d ( X , Z ) , d ( Y , Z ) {\displaystyle d(X,Y),d(X,Z),d(Y,Z)} respectively. If d X Y , d X Z , d Y Z ≥ 2 ϵ {\displaystyle d_{XY},d_{XZ},d_{YZ}\geq 2\epsilon } , then the number of triples ( x , y , z ) ∈ X × Y × Z {\displaystyle (x,y,z)\in X\times Y\times Z} such that x , y , z {\displaystyle x,y,z} form a triangle in G {\displaystyle G} is at least
Using the triangle counting lemma and the Szemerédi regularity lemma, we can prove the triangle removal lemma, a special case of the graph removal lemma . [ 6 ]
Triangle Removal Lemma: For all ϵ > 0 {\displaystyle \epsilon >0} , there exists δ > 0 {\displaystyle \delta >0} such that any graph on n {\displaystyle n} vertices with less than or equal to δ n 3 {\displaystyle \delta n^{3}} triangles can be made triangle-free by removing at most ϵ n 2 {\displaystyle \epsilon n^{2}} edges.
This has an interesting corollary pertaining to graphs G {\displaystyle G} on N {\displaystyle N} vertices where every edge of G {\displaystyle G} lies in a unique triangle. In specific, all of these graphs must have o ( N 2 ) {\displaystyle o(N^{2})} edges.
Take a set A {\displaystyle A} with no 3-term arithmetic progressions. Now, construct a tripartite graph G {\displaystyle G} whose parts X , Y , Z {\displaystyle X,Y,Z} are all copies of Z / ( 2 N + 1 ) Z {\displaystyle \mathbb {Z} /(2N+1)\mathbb {Z} } . Connect a vertex x ∈ X {\displaystyle x\in X} to a vertex y ∈ Y {\displaystyle y\in Y} if y − x ∈ A {\displaystyle y-x\in A} . Similarly, connect z ∈ Z {\displaystyle z\in Z} with y ∈ Y {\displaystyle y\in Y} if z − y ∈ A {\displaystyle z-y\in A} . Finally, connect x ∈ X {\displaystyle x\in X} with z ∈ Z {\displaystyle z\in Z} if ( z − x ) / 2 ∈ A {\displaystyle (z-x)/2\in A} .
This construction is set up so that if x , y , z {\displaystyle x,y,z} form a triangle, then we get elements y − x , z − x 2 , z − y {\displaystyle y-x,{\frac {z-x}{2}},z-y} that all belong to A {\displaystyle A} . These numbers form an arithmetic progression in the listed order. The assumption on A {\displaystyle A} then tells us this progression must be trivial: the elements listed above are all equal. But this condition is equivalent to the assertion that x , y , z {\displaystyle x,y,z} is an arithmetic progression in Z / ( 2 N + 1 ) Z {\displaystyle \mathbb {Z} /(2N+1)\mathbb {Z} } . Consequently, every edge of G {\displaystyle G} lies in exactly one triangle. The desired conclusion follows. ◼ {\displaystyle \blacksquare }
Szemerédi's theorem resolved the original conjecture and generalized Roth's theorem to arithmetic progressions of arbitrary length. Since then it has been extended in multiple fashions to create new and interesting results.
Furstenberg and Katznelson [ 7 ] used ergodic theory to prove a multidimensional version and Leibman and Bergelson [ 8 ] extended it to polynomial progressions as well. Most recently, Green and Tao proved the Green–Tao theorem which says that the prime numbers contain arbitrarily long arithmetic progressions. Since the prime numbers are a subset of density 0, they introduced a "relative" Szemerédi theorem which applies to subsets with density 0 that satisfy certain pseudorandomness conditions. Later on Conlon , Fox , and Zhao [ 9 ] [ 10 ] strengthened this theorem by weakening the necessary pseudorandomness condition. In 2020, Bloom and Sisask [ 11 ] proved that any set A {\displaystyle A} such that ∑ n ∈ A 1 n {\displaystyle \sum _{n\in A}{\frac {1}{n}}} diverges must contain arithmetic progressions of length 3; this is the first non-trivial case of another conjecture of Erdős postulating that any such set must in fact contain arbitrarily long arithmetic progressions.
There has also been work done on improving the bound in Roth's theorem. The bound from the original proof of Roth's theorem showed that
for some constant c {\displaystyle c} . Over the years this bound has been continually lowered by Szemerédi, [ 12 ] Heath-Brown , [ 13 ] Bourgain , [ 14 ] [ 15 ] and Sanders . [ 16 ] [ 17 ] The current (July 2020) best bound is due to Bloom and Sisask [ 11 ] who have shown the existence of an absolute constant c>0 such that
In February 2023 a preprint [ 18 ] [ 19 ] (later published [ 20 ] ) by Kelley and Meka gave a new bound of:
r 3 ( [ N ] ) ≤ 2 − Ω ( ( log N ) 1 / 12 ) ⋅ N {\displaystyle r_{3}([N])\leq 2^{-\Omega ((\log N)^{1/12})}\cdot N} .
Four days later, Bloom and Sisask published a preprint giving an exposition of the result [ 21 ] (later published [ 22 ] ), simplifying the argument and yielding some additional applications. Several months later, Bloom and Sisask obtained a further improvement to r 3 ( [ N ] ) ≤ exp ( − c ( log N ) 1 / 9 ) N {\displaystyle r_{3}([N])\leq \exp(-c(\log N)^{1/9})N} , and stated (without proof) that their techniques can be used to show r 3 ( [ N ] ) ≤ exp ( − c ( log N ) 5 / 41 ) N {\displaystyle r_{3}([N])\leq \exp(-c(\log N)^{5/41})N} . [ 23 ]
There has also been work done on the other end, constructing the largest set with no 3-term arithmetic progressions. The best construction has barely been improved since 1946 when Behrend [ 24 ] improved on the initial construction by Salem and Spencer and proved
Due to no improvements in over 70 years, it is conjectured that Behrend's set is asymptotically very close in size to the largest possible set with no 3-term progressions. [ 11 ] If correct, the Kelley-Meka bound will prove this conjecture.
As a variation, we can consider the analogous problem over finite fields . Consider the finite field F 3 n {\displaystyle \mathbb {F} _{3}^{n}} , and let r 3 ( F 3 n ) {\displaystyle r_{3}(\mathbb {F} _{3}^{n})} be the size of the largest subset of F 3 n {\displaystyle \mathbb {F} _{3}^{n}} which contains no 3-term arithmetic progression. This problem is actually equivalent to the cap set problem, which asks for the largest subset of F 3 n {\displaystyle \mathbb {F} _{3}^{n}} such that no 3 points lie on a line. The cap set problem can be seen as a generalization of the card game Set .
In 1982, Brown and Buhler [ 25 ] were the first to show that r 3 ( F 3 n ) = o ( 3 n ) . {\displaystyle r_{3}(\mathbb {F} _{3}^{n})=o(3^{n}).} In 1995, Roy Mesuhlam [ 26 ] used a similar technique to the Fourier-analytic proof of Roth's theorem to show that r 3 ( F 3 n ) = O ( 3 n n ) . {\displaystyle r_{3}(\mathbb {F} _{3}^{n})=O\left({\frac {3^{n}}{n}}\right).} This bound was improved to O ( 3 n / n 1 + ϵ ) {\displaystyle O(3^{n}/n^{1+\epsilon })} in 2012 by Bateman and Katz. [ 27 ]
In 2016, Ernie Croot , Vsevolod Lev, Péter Pál Pach, Jordan Ellenberg and Dion Gijswijt developed a new technique based on the polynomial method to prove that r 3 ( F 3 n ) = O ( 2.756 n ) {\displaystyle r_{3}(\mathbb {F} _{3}^{n})=O(2.756^{n})} . [ 28 ] [ 29 ] [ 30 ]
The best known lower bound is 2.2202 n {\displaystyle 2.2202^{n}} , discovered in December 2023 by Google DeepMind researchers using a large language model (LLM). [ 31 ]
Another generalization of Roth's theorem shows that for positive density subsets, there not only exists a 3-term arithmetic progression, but that there exist many 3-APs all with the same common difference.
Roth's theorem with popular differences: For all ϵ > 0 {\displaystyle \epsilon >0} , there exists some n 0 = n 0 ( ϵ ) {\displaystyle n_{0}=n_{0}(\epsilon )} such that for every n > n 0 {\displaystyle n>n_{0}} and A ⊂ F 3 n {\displaystyle A\subset \mathbb {F} _{3}^{n}} with | A | = α 3 n , {\displaystyle |A|=\alpha 3^{n},} there exists some y ≠ 0 {\displaystyle y\neq 0} such that | { x : x , x + y , x + 2 y ∈ A } | ≥ ( α 3 − ϵ ) 3 n . {\displaystyle |\{x:x,x+y,x+2y\in A\}|\geq (\alpha ^{3}-\epsilon )3^{n}.}
If A {\displaystyle A} is chosen randomly from F 3 n , {\displaystyle \mathbb {F} _{3}^{n},} then we would expect there to be α 3 3 n {\displaystyle \alpha ^{3}3^{n}} progressions for each value of y {\displaystyle y} . The popular differences theorem thus states that for each | A | {\displaystyle |A|} with positive density, there is some y {\displaystyle y} such that the number of 3-APs with common difference y {\displaystyle y} is close to what we would expect.
This theorem was first proven by Green in 2005, [ 32 ] who gave a bound of n 0 = tow ( ( 1 / ϵ ) O ( 1 ) ) , {\displaystyle n_{0}={\text{tow}}((1/\epsilon )^{O(1)}),} where tow {\displaystyle {\text{tow}}} is the tower function. In 2019, Fox and Pham recently improved the bound to n 0 = tow ( O ( log 1 ϵ ) ) . {\displaystyle n_{0}={\text{tow}}(O(\log {\frac {1}{\epsilon }})).} [ 33 ]
A corresponding statement is also true in Z {\displaystyle \mathbb {Z} } for both 3-APs and 4-APs. [ 34 ] However, the claim has been shown to be false for 5-APs. [ 35 ] | https://en.wikipedia.org/wiki/Roth's_theorem_on_arithmetic_progressions |
Rothalpy (or trothalpy) I {\displaystyle I} , a short name of rotational stagnation enthalpy , is a fluid mechanical property of importance in the study of flow within rotating systems. [ 1 ]
Consider we have an inertial frame of reference X Y Z {\displaystyle XYZ} and a rotating frame of reference x y z {\displaystyle xyz} which both are sharing common origin O {\displaystyle O} . Assume that frame x y z {\displaystyle xyz} is rotating around a fixed axis with angular velocity ω {\displaystyle \mathbf {\omega } } . Now assuming fluid velocity to be V {\displaystyle \mathbf {V} } and fluid velocity relative to rotating frame of reference to be w = V − u {\displaystyle \mathbf {w} =\mathbf {V} -\mathbf {u} } :
Rothalpy of a fluid point P {\displaystyle P} can be defined as
I = h 0 , r e l − u 2 2 {\displaystyle I=h_{0,rel}-{\frac {u^{2}}{2}}}
where u = ω × r {\displaystyle \mathbf {u} =\mathbf {\omega } \times \mathbf {r} } and r = O P → {\displaystyle \mathbf {r} ={\vec {OP}}} and h 0 , r e l {\displaystyle h_{0,rel}} is the stagnation enthalpy of fluid point P {\displaystyle P} relative to the rotating frame of reference x y z {\displaystyle xyz} , which is given by
h 0 , r e l = h + w 2 2 {\displaystyle h_{0,rel}=h+{\frac {w^{2}}{2}}}
and is known as relative stagnation enthalpy.
Rothalpy can also be defined in terms of absolute stagnation enthalpy:
I = h 0 − u V θ {\displaystyle I=h_{0}-uV_{\theta }}
where V θ {\displaystyle V_{\theta }} is tangential component of fluid velocity V {\displaystyle \mathbf {V} } . [ 1 ] [ 2 ] [ 3 ] [ 4 ]
Rothalpy has applications in turbomachinery and study of relative flows in rotating systems.
One such application is that for steady , adiabatic and irreversible flow in a turbomachine, the value of rothalpy across a blade remains constant along a flow streamline :
I = c o n s t . {\displaystyle I=const.} [ 1 ]
so Euler equation of turbomachinery can be written in terms of rothalpy.
This form of the Euler work equation shows that, for rotating blade rows, the relative stagnation enthalpy is constant through the blades provided the blade speed is constant. In other words, h 0 , r e l = c o n s t . {\displaystyle h_{0,rel}=const.} , if the radius of a streamline passing through the blades stays the same. This result is important for analyzing turbomachinery flows in the relative frame of reference. [ 2 ]
The function I {\displaystyle I} was first introduced by Wu (1952) and has acquired the widely used name rothalpy. [ 2 ]
This quantity is commonly called rothalpy, a compound word combining the terms rotation and enthalpy. However, its construction does not conform to the established rules for formation of new words in the English language, namely, that the roots of the new word originate from the same language. The word trothalpy satisfies this requirement as trohos is the Greek root for wheel and enthalpy is to put heat in, whereas rotation is derived from Latin rotare. [ 3 ] | https://en.wikipedia.org/wiki/Rothalpy |
In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle . A Rothberger space is a space in which for every sequence of open covers U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are sets U 1 ∈ U 1 , U 2 ∈ U 2 , … {\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\ldots } such that the family { U n : n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} covers the space.
In 1938, Fritz Rothberger introduced his property known as C ″ {\displaystyle C''} . [ 1 ]
For subsets of the real line, the Rothberger property can be characterized using continuous functions into the Baire space N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} . A subset A {\displaystyle A} of N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} is guessable if there is a function g ∈ A {\displaystyle g\in A} such that the sets { n : f ( n ) = g ( n ) } {\displaystyle \{n:f(n)=g(n)\}} are infinite for all functions f ∈ A {\displaystyle f\in A} . A subset of the real line is Rothberger iff every continuous image of that space into the Baire space is guessable. In particular, every subset of the real line of cardinality less than c o v ( M ) {\displaystyle \mathrm {cov} ({\mathcal {M}})} [ 2 ] is Rothberger.
Let X {\displaystyle X} be a topological space. The Rothberger game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} played on X {\displaystyle X} is a game with two players Alice and Bob.
1st round : Alice chooses an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} of X {\displaystyle X} . Bob chooses a set U 1 ∈ U 1 {\displaystyle U_{1}\in {\mathcal {U}}_{1}} .
2nd round : Alice chooses an open cover U 2 {\displaystyle {\mathcal {U}}_{2}} of X {\displaystyle X} . Bob chooses a set U 2 ∈ U 2 {\displaystyle U_{2}\in {\mathcal {U}}_{2}} .
etc.
If the family { U n : n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} is a cover of the space X {\displaystyle X} , then Bob wins the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} . Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game G 1 ( O , O ) {\displaystyle {\text{G}}_{1}(\mathbf {O} ,\mathbf {O} )} (formally, a winning strategy is a function). | https://en.wikipedia.org/wiki/Rothberger_space |
The Rothemund reaction is a condensation / oxidation process that converts four pyrroles and four aldehydes into a porphyrin . It is based on work by Paul Rothemund , who first reported it in 1936. [ 1 ] The method underpins more modern synthesis such as those described by Adler and Longo and by Lindsey. The Rothemund reactions is common in university teaching labs. [ 2 ]
The reaction employs an organic acidic medium such as acetic acid or propionic acid as typical reaction solvents. Alternatively p -toluenesulfonic acid or various Lewis acids can be used with chlorinated solvents. The aldehyde and pyrrole are heated in this medium to afford modest yields of the meso tetrasubstituted porphyrins [RCC 4 H 2 N] 4 H 2 . The reaction entails both condensation of the aldehydes with the 2,5-positions of the pyrrole but also oxidative dehydrogenation of the porphyrinogen [RCC 4 H 2 NH] 4 .
The multi-step syntheses of hemin and chlorophyll by Hans Fischer were awarded by a Nobel Prize in Chemistry . [ 3 ] [ 4 ] This has inspired the work of his student Paul Rothemund to develop a simple one pot synthesis of porphyrins. In 1935, Paul Rothemund reported the formation of porphyrin, from a simple reaction of pyrrole with gaseous acetaldehyde or formaldehyde in methanol followed by treatment with various concentrations of hydrochloric acid . [ 5 ] One year later Paul Rothemund announced the applicability of his reaction to other aldehydes , by which he was able to explore large number of porphyrins. [ 6 ] Here he detailed the synthesis of porphine , the fundamental ring system in all the porphyrins. He performed the porphin synthesis at a temperature of 90-95 °C and high pressure in sealed pyrex glass tubes, by reacting pyrrole, 2 % formaldehyde and pyridine in methanol for 30 hours. [ 7 ]
A simplified version of Rothemund porphyrin synthesis was described by Alan D. Adler and Frederick R. Longo in 1966. It utilizes mild organic acids as catalysts and reaction medium and is conducted in open air. Seventy aldehydes gave corresponding meso -substituted porphyrins. The reaction time was shortened to 30 minutes and yields improved to 20%. [ 8 ] The Alder-Logo reaction protocol was further modified by Lindsey et al. Using Lewis acid catalyst ( boron trifluoride ) or strong organic acids ( trifluoroacetic acid ) in chlorinated solvents , yields improved to 30-40%. [ 9 ]
Green chemistry variants have been developed in which the reaction is performed with microwave irradiation using reactants adsorbed on acidic silica gel [ 10 ] or at high temperature in the gas phase . [ 11 ] | https://en.wikipedia.org/wiki/Rothemund_reaction |
The Rotterdam Convention (formally, the Rotterdam Convention on the Prior Informed Consent Procedure for Certain Hazardous Chemicals and Pesticides in International Trade ) is a multilateral treaty to promote shared responsibilities in relation to importation of hazardous chemicals. The convention promotes open exchange of information and calls on exporters of hazardous chemicals to use proper labeling, include directions on safe handling, and inform purchasers of any known restrictions or bans. Signatory nations can decide whether to allow or ban the importation of chemicals listed in the treaty, and exporting countries are obliged to make sure that producers within their jurisdiction comply.
In 2012, the Secretariats of the Basel and Stockholm conventions, as well as the UNEP -part of the Rotterdam Convention Secretariat, merged to a single Secretariat with a matrix structure serving the three conventions. [ 1 ] The three conventions now hold back to back Conferences of the Parties as part of their joint synergies decisions.
The ninth meeting of the Rotterdam Conference [ 2 ] was held from 29 April to 10 May 2019 in Geneva, Switzerland.
The following chemicals are listed in Annex III to the convention: [ 3 ]
The Chemical Review Committee of the Rotterdam Convention decided to recommend to the conference of the parties meeting that it consider the listing of the following chemicals in Annex III to the convention: [ 4 ]
As of October 2018, the convention has 161 parties, which includes 158 UN member states , the Cook Islands , the State of Palestine , and the European Union . Non-member states include the United States .
At the 2011 meeting of the Rotterdam Convention in Geneva, the Canadian delegation surprised many with a refusal to allow the addition of chrysotile asbestos fibers to the Rotterdam Convention. [ 5 ] [ 6 ] [ 7 ] [ 8 ] Hearings are scheduled in the EU in the near future to evaluate the position of Canada and decide on the possibility of a punitive course of action. [ 9 ] [ 10 ] [ 11 ]
In continuing its objection, Canada is the only G8 country objecting to the listing. Kyrgyzstan , Kazakhstan and Ukraine also objected. Vietnam had also raised an objection, but missed a follow-up meeting on the issue. [ 12 ] In taking its position, the Canadian Government contrasted with India, which withdrew its long-standing objection to the addition of chrysotile to the list just prior to the 2011 conference. (India later reversed this position in 2013.) [ 13 ]
Numerous non-governmental organizations have publicly expressed criticism of Canada's decision to block this addition. [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ]
In September 2012, Canadian Industry minister Christian Paradis announced the Canadian government would no longer oppose inclusion of chrysotile in the convention. [ 19 ]
Eight of the largest chrysotile producing and exporting countries opposed such a move at the Rotterdam Conference of Parties in 2015: Russia, Kazakhstan, India, Kyrgyzstan, Pakistan, Cuba, and Zimbabwe. [ 13 ] | https://en.wikipedia.org/wiki/Rotterdam_Convention |
Rouché's theorem , named after Eugène Rouché , states that for any two complex -valued functions f and g holomorphic inside some region K {\displaystyle K} with closed contour ∂ K {\displaystyle \partial K} , if | g ( z )| < | f ( z )| on ∂ K {\displaystyle \partial K} , then f and f + g have the same number of zeros inside K {\displaystyle K} , where each zero is counted as many times as its multiplicity . This theorem assumes that the contour ∂ K {\displaystyle \partial K} is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.
The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part. For example, the polynomial z 5 + 3 z 3 + 7 {\displaystyle z^{5}+3z^{3}+7} has exactly 5 zeros in the disk | z | < 2 {\displaystyle |z|<2} since | 3 z 3 + 7 | ≤ 31 < 32 = | z 5 | {\displaystyle |3z^{3}+7|\leq 31<32=|z^{5}|} for every | z | = 2 {\displaystyle |z|=2} , and z 5 {\displaystyle z^{5}} , the dominating part, has five zeros in the disk.
It is possible to provide an informal explanation of Rouché's theorem.
Let C be a closed, simple curve (i.e., not self-intersecting). Let h ( z ) = f ( z ) + g ( z ). If f and g are both holomorphic on the interior of C , then h must also be holomorphic on the interior of C . Then, with the conditions imposed above, the Rouche's theorem in its original (and not symmetric) form says that
Notice that the condition | f ( z )| > | h ( z ) − f ( z )| means that for any z , the distance from f ( z ) to the origin is larger than the length of h ( z ) − f ( z ), which in the following picture means that for each point on the blue curve, the segment joining it to the origin is larger than the green segment associated with it. Informally we can say that the blue curve f ( z ) is always closer to the red curve h ( z ) than it is to the origin.
The previous paragraph shows that h ( z ) must wind around the origin exactly as many times as f ( z ). The index of both curves around zero is therefore the same, so by the argument principle , f ( z ) and h ( z ) must have the same number of zeros inside C .
One popular, informal way to summarize this argument is as follows: If a person were to walk a dog on a leash around and around a tree, such that the distance between the person and the tree is always greater than the length of the leash, then the person and the dog go around the tree the same number of times.
Consider the polynomial z 2 + 2 a z + b 2 {\displaystyle z^{2}+2az+b^{2}} with a > b > 0 {\displaystyle a>b>0} . By the quadratic formula it has two zeros at − a ± a 2 − b 2 {\displaystyle -a\pm {\sqrt {a^{2}-b^{2}}}} . Rouché's theorem can be used to obtain some hint about their positions. Since | z 2 + b 2 | ≤ 2 b 2 < 2 a | z | for all | z | = b , {\displaystyle |z^{2}+b^{2}|\leq 2b^{2}<2a|z|{\text{ for all }}|z|=b,}
Rouché's theorem says that the polynomial has exactly one zero inside the disk | z | < b {\displaystyle |z|<b} . Since − a − a 2 − b 2 {\displaystyle -a-{\sqrt {a^{2}-b^{2}}}} is clearly outside the disk, we conclude that the zero is − a + a 2 − b 2 {\displaystyle -a+{\sqrt {a^{2}-b^{2}}}} .
In general, a polynomial f ( z ) = a n z n + ⋯ + a 0 {\displaystyle f(z)=a_{n}z^{n}+\cdots +a_{0}} . If | a k | r k > ∑ j ≠ k | a j | r j {\displaystyle |a_{k}|r^{k}>\sum _{j\neq k}|a_{j}|r^{j}} for some r > 0 , k ∈ 0 : n {\displaystyle r>0,k\in 0:n} , then by Rouche's theorem, the polynomial has exactly k {\displaystyle k} roots inside B ( 0 , r ) {\displaystyle B(0,r)} .
This sort of argument can be useful in locating residues when one applies Cauchy's residue theorem .
Rouché's theorem can also be used to give a short proof of the fundamental theorem of algebra . Let p ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n , a n ≠ 0 {\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},\quad a_{n}\neq 0} and choose R > 0 {\displaystyle R>0} so large that: | a 0 + a 1 z + ⋯ + a n − 1 z n − 1 | ≤ ∑ j = 0 n − 1 | a j | R j < | a n | R n = | a n z n | for | z | = R . {\displaystyle |a_{0}+a_{1}z+\cdots +a_{n-1}z^{n-1}|\leq \sum _{j=0}^{n-1}|a_{j}|R^{j}<|a_{n}|R^{n}=|a_{n}z^{n}|{\text{ for }}|z|=R.} Since a n z n {\displaystyle a_{n}z^{n}} has n {\displaystyle n} zeros inside the disk | z | < R {\displaystyle |z|<R} (because R > 0 {\displaystyle R>0} ), it follows from Rouché's theorem that p {\displaystyle p} also has the same number of zeros inside the disk.
One advantage of this proof over the others is that it shows not only that a polynomial must have a zero but the number of its zeros is equal to its degree (counting, as usual, multiplicity).
Another use of Rouché's theorem is to prove the open mapping theorem for analytic functions. We refer to the article for the proof.
A stronger version of Rouché's theorem was published by Theodor Estermann in 1962. [ 1 ] It states: let K ⊂ G {\displaystyle K\subset G} be a bounded region with continuous boundary ∂ K {\displaystyle \partial K} . Two holomorphic functions f , g ∈ H ( G ) {\displaystyle f,\,g\in {\mathcal {H}}(G)} have the same number of roots (counting multiplicity) in K {\displaystyle K} , if the strict inequality | f ( z ) − g ( z ) | < | f ( z ) | + | g ( z ) | ( z ∈ ∂ K ) {\displaystyle |f(z)-g(z)|<|f(z)|+|g(z)|\qquad \left(z\in \partial K\right)} holds on the boundary ∂ K . {\displaystyle \partial K.}
THIS PART HAS TO BE REVISED, IT IS BASED ON HAVING + IN THE ABOVE FORMULA. THE PROOF BELOW IS CORRECT. The original version of Rouché's theorem then follows from this symmetric version applied to the functions f + g , f {\displaystyle f+g,f} together with the trivial inequality | f ( z ) + g ( z ) | ≥ 0 {\displaystyle |f(z)+g(z)|\geq 0} (in fact this inequality is strict since f ( z ) + g ( z ) = 0 {\displaystyle f(z)+g(z)=0} for some z ∈ ∂ K {\displaystyle z\in \partial K} would imply | g ( z ) | = | f ( z ) | {\displaystyle |g(z)|=|f(z)|} ).
The statement can be understood intuitively as follows.
By considering − g {\displaystyle -g} in place of g {\displaystyle g} , the condition can be rewritten as | f ( z ) + g ( z ) | < | f ( z ) | + | g ( z ) | {\displaystyle |f(z)+g(z)|<|f(z)|+|g(z)|} for z ∈ ∂ K {\displaystyle z\in \partial K} .
Since | f ( z ) + g ( z ) | ≤ | f ( z ) | + | g ( z ) | {\displaystyle |f(z)+g(z)|\leq |f(z)|+|g(z)|} always holds by the triangle inequality, this is equivalent to saying that | f ( z ) + g ( z ) | ≠ | f ( z ) | + | g ( z ) | {\displaystyle |f(z)+g(z)|\neq |f(z)|+|g(z)|} on ∂ K {\displaystyle \partial K} , which in turn means that for z ∈ ∂ K {\displaystyle z\in \partial K} the functions f ( z ) {\displaystyle f(z)} and g ( z ) {\displaystyle g(z)} are non-vanishing and arg f ( z ) ≠ arg g ( z ) {\displaystyle \arg {f(z)}\neq \arg {g(z)}} .
Intuitively, if the values of f {\displaystyle f} and g {\displaystyle g} never pass through the origin and never point in the same direction as z {\displaystyle z} circles along ∂ K {\displaystyle \partial K} , then f ( z ) {\displaystyle f(z)} and g ( z ) {\displaystyle g(z)} must wind around the origin the same number of times.
Let C : [ 0 , 1 ] → C {\displaystyle C\colon [0,1]\to \mathbb {C} } be a simple closed curve whose image is the boundary ∂ K {\displaystyle \partial K} . The hypothesis implies that f has no roots on ∂ K {\displaystyle \partial K} , hence by the argument principle , the number N f ( K ) of zeros of f in K is 1 2 π i ∮ C f ′ ( z ) f ( z ) d z = 1 2 π i ∮ f ∘ C d z z = I n d f ∘ C ( 0 ) , {\displaystyle {\frac {1}{2\pi i}}\oint _{C}{\frac {f'(z)}{f(z)}}\,dz={\frac {1}{2\pi i}}\oint _{f\circ C}{\frac {dz}{z}}=\mathrm {Ind} _{f\circ C}(0),} i.e., the winding number of the closed curve f ∘ C {\displaystyle f\circ C} around the origin; similarly for g . The hypothesis ensures that g ( z ) is not a negative real multiple of f ( z ) for any z = C ( x ), thus 0 does not lie on the line segment joining f ( C ( x )) to g ( C ( x )), and H t ( x ) = ( 1 − t ) f ( C ( x ) ) + t g ( C ( x ) ) {\displaystyle H_{t}(x)=(1-t)f(C(x))+tg(C(x))} is a homotopy between the curves f ∘ C {\displaystyle f\circ C} and g ∘ C {\displaystyle g\circ C} avoiding the origin. The winding number is homotopy-invariant: the function I ( t ) = I n d H t ( 0 ) = 1 2 π i ∮ H t d z z {\displaystyle I(t)=\mathrm {Ind} _{H_{t}}(0)={\frac {1}{2\pi i}}\oint _{H_{t}}{\frac {dz}{z}}} is continuous and integer-valued, hence constant. This shows N f ( K ) = I n d f ∘ C ( 0 ) = I n d g ∘ C ( 0 ) = N g ( K ) . {\displaystyle N_{f}(K)=\mathrm {Ind} _{f\circ C}(0)=\mathrm {Ind} _{g\circ C}(0)=N_{g}(K).} | https://en.wikipedia.org/wiki/Rouché's_theorem |
In stochastic analysis , a rough path is a generalization of the classical notion of a smooth path. It extends calculus and differential equation theory to handle irregular signals—paths that are too rough for traditional analysis, such as a Wiener process . This makes it possible to define and solve controlled differential equations of the form d y t = f ( y t ) , d x t , y 0 = a {\displaystyle \mathrm {d} y_{t}=f(y_{t}),\mathrm {d} x_{t},\quad y_{0}=a} even when the driving path x t {\displaystyle x_{t}} lacks classical differentiability . The theory was introduced in the 1990s by Terry Lyons . [ 1 ] [ 2 ] [ 3 ]
Rough path theory captures how nonlinear systems interact with highly oscillatory or noisy input. It builds on the integration theory of L. C. Young , the geometric algebra of Kuo-Tsai Chen , and the Lipschitz function theory of Hassler Whitney , while remaining compatible with key ideas in stochastic calculus . The theory also extends Itô's theory of stochastic differential equations far beyond the semimartingale setting. Its definitions and uniform estimates form a robust framework that can recover classical results—such as the Wong–Zakai theorem , the Stroock–Varadhan support theorem , and the construction of stochastic flows—without relying on probabilistic properties like martingales or predictability.
A central concept in the theory is the Signature of a path: a noncommutative transform that encodes the path as a sequence of iterated integrals . Formally, it is a homomorphism from the monoid of paths (under concatenation) into the group-like elements of a tensor algebra . The Signature is faithful—it uniquely characterizes paths up to certain negligible modifications—making it a powerful tool for representing and comparing paths. These iterated integrals play a role similar to monomials in a Taylor expansion : they provide a coordinate system that captures the essential features of a path. Just as Taylor’s theorem allows a smooth function to be approximated locally by polynomials , the terms of the Signature offer a structured, hierarchical summary of a path’s behavior. This enriched representation forms the basis for defining a rough path and enables analysis without directly examining its fine-scale structure.
The theory has widespread applications across mathematics and applied fields. Notably, Martin Hairer used rough path techniques to help construct a solution theory for the KPZ equation , [ 4 ] and later developed the more general theory of regularity structures , [ 5 ] for which he was awarded the Fields Medal in 2014.
Rough path theory aims to make sense of the controlled differential equation
where the control, the continuous path X t {\displaystyle X_{t}} taking values in a Banach space , need not be differentiable nor of bounded variation. A prevalent example of the controlled path X t {\displaystyle X_{t}} is the sample path of a Wiener process . In this case, the aforementioned controlled differential equation can be interpreted as a stochastic differential equation and integration against " d X t j {\displaystyle \mathrm {d} X_{t}^{j}} " can be defined in the sense of Itô . However, Itô's calculus is defined in the sense of L 2 {\displaystyle L^{2}} and is in particular not a pathwise definition. Rough paths give an almost sure pathwise definition of stochastic differential equations. The rough path notion of solution is well-posed in the sense that if X ( n ) t {\displaystyle X(n)_{t}} is a sequence of smooth paths converging to X t {\displaystyle X_{t}} in the p {\displaystyle p} -variation metric (described below), and
then Y ( n ) {\displaystyle Y(n)} converges to Y {\displaystyle Y} in the p {\displaystyle p} -variation metric.
This continuity property and the deterministic nature of solutions makes it possible to simplify and strengthen many results in Stochastic Analysis, such as the Freidlin-Wentzell's Large Deviation theory [ 6 ] as well as results about stochastic flows.
In fact, rough path theory can go far beyond the scope of Itô and Stratonovich calculus and allows to make sense of differential equations driven by non- semimartingale paths, such as Gaussian processes and Markov processes . [ 7 ]
Rough paths are paths taking values in the truncated free tensor algebra (more precisely: in the free nilpotent group embedded in the free tensor algebra), which this section now briefly recalls. The tensor powers of R d {\displaystyle \mathbb {R} ^{d}} , denoted ( R d ) ⊗ n {\displaystyle {\big (}\mathbb {R} ^{d}{\big )}^{\otimes n}} , are equipped with the projective norm ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert } (see Topological tensor product , note that rough path theory in fact works for a more general class of norms).
Let T ( n ) ( R d ) {\displaystyle T^{(n)}(\mathbb {R} ^{d})} be the truncated tensor algebra
Let △ 0 , 1 {\displaystyle \triangle _{0,1}} be the simplex { ( s , t ) : 0 ≤ s ≤ t ≤ 1 } {\displaystyle \{(s,t):0\leq s\leq t\leq 1\}} .
Let p ≥ 1 {\displaystyle p\geq 1} . Let X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } be continuous maps △ 0 , 1 → T ( ⌊ p ⌋ ) ( R d ) {\displaystyle \triangle _{0,1}\to T^{(\lfloor p\rfloor )}(\mathbb {R} ^{d})} .
Let X j {\displaystyle \mathbf {X} ^{j}} denote the projection of X {\displaystyle \mathbf {X} } onto j {\displaystyle j} -tensors and likewise for Y j {\displaystyle \mathbf {Y} ^{j}} . The p {\displaystyle p} -variation metric is defined as
where the supremum is taken over all finite partitions { 0 = t 0 < t 1 < ⋯ < t n = 1 } {\displaystyle \{0=t_{0}<t_{1}<\cdots <t_{n}=1\}} of [ 0 , 1 ] {\displaystyle [0,1]} .
A continuous function X : △ 0 , 1 → T ( ⌊ p ⌋ ) ( R d ) {\displaystyle \mathbf {X} :\triangle _{0,1}\rightarrow T^{(\lfloor p\rfloor )}(\mathbb {R} ^{d})} is a p {\displaystyle p} -geometric rough path if there exists a sequence of paths with finite 1-variation (or, equivalently, of bounded variation) X ( 1 ) , X ( 2 ) , … {\displaystyle X(1),X(2),\ldots } such that
converges in the p {\displaystyle p} -variation metric to X {\displaystyle \mathbf {X} } as n → ∞ {\displaystyle n\rightarrow \infty } . [ 8 ]
A central result in rough path theory is Lyons ' Universal Limit theorem. [ 1 ] One (weak) version of the result is the following:
Let X ( n ) {\displaystyle X(n)} be a sequence of paths with finite total variation and let
Suppose that X ( n ) {\displaystyle \mathbf {X} (n)} converges in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path X {\displaystyle \mathbf {X} } as n → ∞ {\displaystyle n\to \infty } . Let ( V j i ) j = 1 , … , d i = 1 , … , n {\displaystyle (V_{j}^{i})_{j=1,\ldots ,d}^{i=1,\ldots ,n}} be functions that have at least ⌊ p ⌋ {\displaystyle \lfloor p\rfloor } bounded derivatives and the ⌊ p ⌋ {\displaystyle \lfloor p\rfloor } -th derivatives are α {\displaystyle \alpha } -Hölder continuous for some α > p − ⌊ p ⌋ {\displaystyle \alpha >p-\lfloor p\rfloor } . Let Y ( n ) {\displaystyle Y(n)} be the solution to the differential equation
and let Y ( n ) {\displaystyle \mathbf {Y} (n)} be defined as
Then Y ( n ) {\displaystyle \mathbf {Y} (n)} converges in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path Y {\displaystyle \mathbf {Y} } .
Moreover, Y {\displaystyle \mathbf {Y} } is the solution to the differential equation
driven by the geometric rough path X {\displaystyle \mathbf {X} } .
The theorem can be interpreted as saying that the solution map (aka the Itô-Lyons map) Φ : G Ω p ( R d ) → G Ω p ( R e ) {\displaystyle \Phi :G\Omega _{p}(\mathbb {R} ^{d})\to G\Omega _{p}(\mathbb {R} ^{e})} of the RDE ( ⋆ ) {\displaystyle (\star )} is continuous (and in fact locally lipschitz) in the p {\displaystyle p} -variation topology. Hence rough paths theory demonstrates that by viewing driving signals as rough paths, one has a robust solution theory for classical stochastic differential equations and beyond.
Let ( B t ) t ≥ 0 {\displaystyle (B_{t})_{t\geq 0}} be a multidimensional standard Brownian motion. Let ∘ {\displaystyle \circ } denote the Stratonovich integration . Then
is a p {\displaystyle p} -geometric rough path for any 2 < p < 3 {\displaystyle 2<p<3} . This geometric rough path is called the Stratonovich Brownian rough path .
More generally, let B H ( t ) {\displaystyle B_{H}(t)} be a multidimensional fractional Brownian motion (a process whose coordinate components are independent fractional Brownian motions) with H > 1 4 {\displaystyle H>{\frac {1}{4}}} . If B H m ( t ) {\displaystyle B_{H}^{m}(t)} is the m {\displaystyle m} -th dyadic piecewise linear interpolation of B H ( t ) {\displaystyle B_{H}(t)} , then
converges almost surely in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path for 1 H < p {\displaystyle {\frac {1}{H}}<p} . [ 9 ] This limiting geometric rough path can be used to make sense of differential equations driven by fractional Brownian motion with Hurst parameter H > 1 4 {\displaystyle H>{\frac {1}{4}}} . When 0 < H ≤ 1 4 {\displaystyle 0<H\leq {\frac {1}{4}}} , it turns out that the above limit along dyadic approximations does not converge in p {\displaystyle p} -variation. However, one can of course still make sense of differential equations provided one exhibits a rough path lift, existence of such a (non-unique) lift is a consequence of the Lyons–Victoir extension theorem .
In general, let ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} be a R d {\displaystyle \mathbb {R} ^{d}} -valued stochastic process. If one can construct, almost surely, functions ( s , t ) → X s , t j ∈ ( R d ) ⊗ j {\displaystyle (s,t)\rightarrow \mathbf {X} _{s,t}^{j}\in {\big (}\mathbb {R} ^{d}{\big )}^{\otimes j}} so that
is a p {\displaystyle p} -geometric rough path, then X s , t {\displaystyle \mathbf {X} _{s,t}} is an enhancement of the process X {\displaystyle X} . Once an enhancement has been chosen, the machinery of rough path theory will allow one to make sense of the controlled differential equation
for sufficiently regular vector fields V j i . {\displaystyle V_{j}^{i}.}
Note that every stochastic process (even if it is a deterministic path) can have more than one (in fact, uncountably many) possible enhancements. [ 10 ] Different enhancements will give rise to different solutions to the controlled differential equations. In particular, it is possible to enhance Brownian motion to a geometric rough path in a way other than the Brownian rough path. [ 11 ] This implies that the Stratonovich calculus is not the only theory of stochastic calculus that satisfies the classical product rule
In fact any enhancement of Brownian motion as a geometric rough path will give rise a calculus that satisfies this classical product rule. Itô calculus does not come directly from enhancing Brownian motion as a geometric rough path, but rather as a branched rough path.
Rough path theory allows to give a pathwise notion of solution to (stochastic) differential equations of the form
provided that we can construct a rough path which is almost surely a rough path lift of the multidimensional stochastic process X t {\displaystyle X_{t}} and that the drift b {\displaystyle b} and the volatility σ {\displaystyle \sigma } are sufficiently smooth (see the section on the Universal Limit Theorem).
There are many examples of Markov processes, Gaussian processes, and other processes that can be enhanced as rough paths. [ 12 ]
There are, in particular, many results on the solution to differential equation driven by fractional Brownian motion that have been proved using a combination of Malliavin calculus and rough path theory. In fact, it has been proved recently that the solution to controlled differential equation driven by a class of Gaussian processes, which includes fractional Brownian motion with Hurst parameter H > 1 4 {\displaystyle H>{\frac {1}{4}}} , has a smooth density under the Hörmander's condition on the vector fields. [ 13 ] [ 14 ]
Let L ( V , W ) {\displaystyle L(V,W)} denote the space of bounded linear maps from a Banach space V {\displaystyle V} to another Banach space W {\displaystyle W} .
Let B t {\displaystyle B_{t}} be a d {\displaystyle d} -dimensional standard Brownian motion. Let b : R n → R d {\displaystyle b:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{d}} and σ : R n → L ( R d , R n ) {\displaystyle \sigma :\mathbb {R} ^{n}\rightarrow L(\mathbb {R} ^{d},\mathbb {R} ^{n})} be twice-differentiable functions and whose second derivatives are α {\displaystyle \alpha } -Hölder for some α > 0 {\displaystyle \alpha >0} .
Let X ε {\displaystyle X^{\varepsilon }} be the unique solution to the stochastic differential equation
where ∘ {\displaystyle \circ } denotes Stratonovich integration.
The Freidlin Wentzell's large deviation theory aims to study the asymptotic behavior, as ϵ → 0 {\displaystyle \epsilon \rightarrow 0} , of P [ X ε ∈ F ] {\displaystyle \mathbb {P} [X^{\varepsilon }\in F]} for closed or open sets F {\displaystyle F} with respect to the uniform topology.
The Universal Limit Theorem guarantees that the Itô map sending the control path ( t , ε B t ) {\displaystyle (t,{\sqrt {\varepsilon }}B_{t})} to the solution X ε {\displaystyle X^{\varepsilon }} is a continuous map from the p {\displaystyle p} -variation topology to the p {\displaystyle p} -variation topology (and hence the uniform topology). Therefore, the Contraction principle in large deviations theory reduces Freidlin–Wentzell's problem to demonstrating the large deviation principle for ( t , ε B t ) {\displaystyle (t,{\sqrt {\varepsilon }}B_{t})} in the p {\displaystyle p} -variation topology. [ 6 ]
This strategy can be applied to not just differential equations driven by the Brownian motion but also to the differential equations driven any stochastic processes which can be enhanced as rough paths, such as fractional Brownian motion.
Once again, let B t {\displaystyle B_{t}} be a d {\displaystyle d} -dimensional Brownian motion. Assume that the drift term b {\displaystyle b} and the volatility term σ {\displaystyle \sigma } has sufficient regularity so that the stochastic differential equation
has a unique solution in the sense of rough path. A basic question in the theory of stochastic flow is whether the flow map ϕ s , t ( x ) {\displaystyle \phi _{s,t}(x)} exists and satisfy the cocyclic property that for all s ≤ u ≤ t {\displaystyle s\leq u\leq t} ,
outside a null set independent of s , u , t {\displaystyle s,u,t} .
The Universal Limit Theorem once again reduces this problem to whether the Brownian rough path B s , t {\displaystyle \mathbf {B_{s,t}} } exists and satisfies the multiplicative property that for all s ≤ u ≤ t {\displaystyle s\leq u\leq t} ,
outside a null set independent of s {\displaystyle s} , u {\displaystyle u} and t {\displaystyle t} .
In fact, rough path theory gives the existence and uniqueness of ϕ s , t ( x ) {\displaystyle \phi _{s,t}(x)} not only outside a null set independent of s {\displaystyle s} , t {\displaystyle t} and x {\displaystyle x} but also of the drift b {\displaystyle b} and the volatility σ {\displaystyle \sigma } .
As in the case of Freidlin–Wentzell theory, this strategy holds not just for differential equations driven by the Brownian motion but to any stochastic processes that can be enhanced as rough paths.
Controlled rough paths, introduced by M. Gubinelli, [ 15 ] are paths Y {\displaystyle \mathbf {Y} } for which the rough integral
can be defined for a given geometric rough path X {\displaystyle X} .
More precisely, let L ( V , W ) {\displaystyle L(V,W)} denote the space of bounded linear maps from a Banach space V {\displaystyle V} to another Banach space W {\displaystyle W} .
Given a p {\displaystyle p} -geometric rough path
on R d {\displaystyle \mathbb {R} ^{d}} , a γ {\displaystyle \gamma } - controlled path is a function Y s = ( Y s 0 , Y s 1 , … , Y s ⌊ γ ⌋ ) {\displaystyle \mathbf {Y} _{s}=(\mathbf {Y} _{s}^{0},\mathbf {Y} _{s}^{1},\ldots ,\mathbf {Y} _{s}^{\lfloor \gamma \rfloor })} such that Y j : [ 0 , 1 ] → L ( ( R d ) ⊗ j + 1 , R n ) {\displaystyle \mathbf {Y} ^{j}:[0,1]\rightarrow L((\mathbb {R} ^{d})^{\otimes j+1},\mathbb {R} ^{n})} and that there exists M > 0 {\displaystyle M>0} such that for all 0 ≤ s ≤ t ≤ 1 {\displaystyle 0\leq s\leq t\leq 1} and j = 0 , 1 , … , ⌊ γ ⌋ {\displaystyle j=0,1,\ldots ,\lfloor \gamma \rfloor } ,
and
Let X = ( 1 , X 1 , … , X ⌊ p ⌋ ) {\displaystyle \mathbf {X} =(1,\mathbf {X} ^{1},\ldots ,\mathbf {X} ^{\lfloor p\rfloor })} be a p {\displaystyle p} -geometric rough path satisfying the Hölder condition that there exists M > 0 {\displaystyle M>0} , for all 0 ≤ s ≤ t ≤ 1 {\displaystyle 0\leq s\leq t\leq 1} and all j = 1 , , 2 , … , ⌊ p ⌋ {\displaystyle j=1,,2,\ldots ,\lfloor p\rfloor } ,
where X j {\displaystyle \mathbf {X} ^{j}} denotes the j {\displaystyle j} -th tensor component of X {\displaystyle \mathbf {X} } .
Let γ ≥ 1 {\displaystyle \gamma \geq 1} . Let f : R d → R n {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} ^{n}} be an ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -times differentiable function and the ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -th derivative is γ − ⌊ γ ⌋ {\displaystyle \gamma -\lfloor \gamma \rfloor } Hölder, then
is a γ {\displaystyle \gamma } -controlled path.
If Y {\displaystyle \mathbf {Y} } is a γ {\displaystyle \gamma } -controlled path where γ > p − 1 {\displaystyle \gamma >p-1} , then
is defined and the path
is a γ {\displaystyle \gamma } -controlled path.
Let V : R n → L ( R d , R n ) {\displaystyle V:\mathbb {R} ^{n}\rightarrow L(\mathbb {R} ^{d},\mathbb {R} ^{n})} be functions that has at least ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } derivatives and the ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -th derivatives are γ − ⌊ γ ⌋ {\displaystyle \gamma -\lfloor \gamma \rfloor } -Hölder continuous for some γ > p {\displaystyle \gamma >p} . Let Y {\displaystyle Y} be the solution to the differential equation
Define
where D {\displaystyle D} denotes the derivative operator, then
is a γ {\displaystyle \gamma } -controlled path.
Let X : [ 0 , 1 ] → R d {\displaystyle X:[0,1]\rightarrow \mathbb {R} ^{d}} be a continuous function with finite total variation. Define
The signature of a path is defined to be S ( X ) 0 , 1 {\displaystyle S(X)_{0,1}} .
The signature can also be defined for geometric rough paths. Let X {\displaystyle \mathbf {X} } be a geometric rough path and let X ( n ) {\displaystyle \mathbf {X} (n)} be a sequence of paths with finite total variation such that
converges in the p {\displaystyle p} -variation metric to X {\displaystyle \mathbf {X} } . Then
converges as n → ∞ {\displaystyle n\rightarrow \infty } for each N {\displaystyle N} . The signature of the geometric rough path X {\displaystyle \mathbf {X} } can be defined as the limit of S ( X ( n ) ) s , t {\displaystyle S(X(n))_{s,t}} as n → ∞ {\displaystyle n\rightarrow \infty } .
The signature satisfies Chen's identity, [ 16 ] that
for all s ≤ u ≤ t {\displaystyle s\leq u\leq t} .
The set of paths whose signature is the trivial sequence, or more precisely,
can be completely characterized using the idea of tree-like path.
A p {\displaystyle p} -geometric rough path is tree-like if there exists a continuous function h : [ 0 , 1 ] → [ 0 , ∞ ) {\displaystyle h:[0,1]\rightarrow [0,\infty )} such that h ( 0 ) = h ( 1 ) = 0 {\displaystyle h(0)=h(1)=0} and for all j = 1 , … , ⌊ p ⌋ {\displaystyle j=1,\ldots ,\lfloor p\rfloor } and all 0 ≤ s ≤ t ≤ 1 {\displaystyle 0\leq s\leq t\leq 1} ,
where X j {\displaystyle \mathbf {X} ^{j}} denotes the j {\displaystyle j} -th tensor component of X {\displaystyle \mathbf {X} } .
A geometric rough path X {\displaystyle \mathbf {X} } satisfies S ( X ) 0 , 1 = ( 1 , 0 , … ) {\displaystyle S(\mathbf {X} )_{0,1}=(1,0,\ldots )} if and only if X {\displaystyle \mathbf {X} } is tree-like. [ 17 ] [ 18 ]
Given the signature of a path, it is possible to reconstruct the unique path that has no tree-like pieces. [ 19 ] [ 20 ]
It is also possible to extend the core results in rough path theory to infinite dimensions, providing that the norm on the tensor algebra satisfies certain admissibility condition. [ 21 ] | https://en.wikipedia.org/wiki/Rough_path |
Roughness length ( z 0 {\displaystyle z_{0}} ) is a parameter used in modeling the horizontal mean wind speed near the ground. In wind vertical profile such the log wind profile , the roughness length (with dimension of length and SI unit of metres) is equivalent to the height at which the wind speed theoretically becomes zero in the absence of wind-slowing obstacles and under neutral conditions. In reality, the wind at this height no longer follows a logarithm. It is so named because it is typically related to the height of terrain roughness elements (i.e. protrusions from and/or depressions into the surface). For instance, forests tend to have much larger roughness lengths than tundra. The roughness length does not exactly correspond to any physical length; however, it can be considered as a length-scale representation of the roughness of the surface. [ 1 ]
The roughness length z 0 {\displaystyle z_{0}} appears in the expression for the mean wind speed u z {\displaystyle u_{z}} near the ground derived using the Monin–Obukhov similarity theory : [ 2 ]
u z = u ∗ κ [ ln ( z − d z 0 ) + ψ ( z − d − z 0 L ) ] , {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)+\psi \left({\frac {z-d-z_{0}}{L}}\right)\right],}
where
In the simplest possible case (statically neutral conditions and no wind-slowing obstacles), the mean wind speed simplifies to:
u z = u ∗ κ ln ( z z 0 ) . {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\ln \left({\frac {z}{z_{0}}}\right).}
This provides a method to calculate the roughness length by measuring the friction velocity and the mean wind velocity (at known elevation) in a given, relatively flat location (under neutral conditions) using an anemometer . [ 4 ] Of note is that, in this simplified form, the log wind profile is identical in form to the dimensional law of the wall .
If we don't know the friction velocity, one can calculate the surface roughness as follow
z 0 = exp ( u ( z 2 ) ln ( z 1 ) − u ( z 1 ) ln ( z 2 ) u ( z 2 ) − u ( z 1 ) ) {\displaystyle z_{0}=\exp \left({\frac {u(z_{2})\ln(z_{1})-u(z_{1})\ln(z_{2})}{u(z_{2})-u(z_{1})}}\right)}
Due to the limitation of observation instruments and the theory of mean values, the levels (z) should be chosen where there is enough difference between the measurement readings. If one has more than two readings, the measurements can be fit to the above equation to find the roughness length.
When calculating the surface roughness, the displacement height can be neglected.
As an approximation, the roughness length is approximately one-tenth of the height of the surface roughness elements. For example, short grass of height 0.01 meters has a roughness length of approximately 0.001 meters. Surfaces are rougher if they have more protrusions. Forests have much larger roughness lengths than tundra, for example. Roughness length is an important concept in urban meteorology as the building of tall structures, such as skyscrapers, has an effect on roughness length and wind patterns.
[ 5 ]
For urban areas, the roughness length changes with the wind direction [ 6 ]
The roughness length is one of many possible measures of the roughness of a surface. For example, in classical mechanics the coefficient of friction is commonly used to measure the roughness of a surface as it relates to the force exerted on another contacted object. And, in fluid dynamics, hydraulic roughness is a measure of the resistance water experiences when flowing over land or through a channel. All of these measures ultimately derive from frictional forces, which result from irregularities on the surfaces of relevance. [ citation needed ] | https://en.wikipedia.org/wiki/Roughness_length |
Rouging is a form of corrosion found in stainless steel . [ 1 ] It can be due to iron contamination of the stainless steel surface due to welding of non-stainless steel for support columns, or other temporary means, which when welded off leaves a low chromium area. [ 2 ]
There are three classes of rouging: Class I, Class II, and Class III. [ 3 ]
Class I – stainless steel surface and the Cr/Fe ratio [ clarification needed ] of the metal surface beneath such deposits usually remain unaltered. Class II – Iron particles originating in-situ on unpassivated or improperly passivated stainless steel surfaces. By their formation the Cr/Fe ratio of the metal surface is altered. Class III – Iron oxide (or scale) which forms on surfaces in high temperature steam systems. The Cr/Fe ratio of the protective film is usually altered.
This corrosion -related article is a stub . You can help Wikipedia by expanding it .
This article about a mechanical engineering topic is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rouging |
Round-trip engineering ( RTE ) in the context of model-driven architecture is a functionality of software development tools that synchronizes two or more related software artifacts, such as, source code, models , configuration files, documentation, etc. between each other. [ 1 ] The need for round-trip engineering arises when the same information is present in multiple artifacts and when an inconsistency may arise in case some artifacts are updated. For example, some piece of information was added to/changed in only one artifact (source code) and, as a result, it became missing in/inconsistent with the other artifacts (in models).
Round-trip engineering is closely related to traditional software engineering disciplines: forward engineering (creating software from specifications), reverse engineering (creating specifications from existing software), and reengineering (understanding existing software and modifying it). Round-trip engineering is often wrongly defined as simply supporting both forward and reverse engineering. In fact, the key characteristic of round-trip engineering that distinguishes it from forward and reverse engineering is the ability to synchronize existing artifacts that evolved concurrently by incrementally updating each artifact to reflect changes made to the other artifacts. Furthermore, forward engineering can be seen as a special instance of RTE in which only the specification is present and reverse engineering can be seen as a special instance of RTE in which only the software is present. Many reengineering activities can also be understood as RTE when the software is updated to reflect changes made to the previously reverse engineered specification.
Various books describe two types of RTE: [ 2 ] : 459
Another characteristic of round-trip engineering is automatic update of the artifacts in response to automatically detected inconsistencies. In that sense, it is different from forward- and reverse engineering which can be both manual (traditionally) and automatic (via automatic generation or analysis of the artifacts). The automatic update can be either instantaneous or on-demand . In instantaneous RTE, all related artifacts are immediately updated after each change made to one of them. In on-demand RTE, authors of the artifacts may concurrently update the artifacts (even in a distributed setting) and at some point choose to execute matching to identify inconsistencies and choose to propagate some of them and reconcile potential conflicts.
Round trip engineering may involve an iterative development process. After you have synchronized your model with revised code, you are still free to choose the best way to work – make further modifications to the code or make changes to your model. You can synchronize in either direction at any time and you can repeat the cycle as many times as necessary.
Many commercial tools and research prototypes support this form of RTE; a 2007 book lists Rational Rose , Micro Focus Together , ESS-Model , BlueJ , and Fujaba among those capable, with Fujaba said to be capable to also identify design patterns . [ 3 ]
A 2005 book on Visual Studio notes for instance that a common problem in RTE tools is that the model reversed is not the same as the original one, unless the tools are aided by leaving laborious annotations in the source code. [ 4 ] The behavioral parts of UML impose even more challenges for RTE.
Usually, UML class diagrams are supported to some degree; however, certain UML concepts, such as associations and containment do not have straightforward representations in many programming languages which limits the usability of the created code and accuracy of code analysis/reverse engineering (e.g., containment is hard to recognize in the code).
A more tractable form of round-trip engineering is implemented in the context of framework application programming interfaces (APIs), whereby a model describing the usage of a framework API by an application is synchronized with that application's code. In this setting, the API prescribes all correct ways the framework can be used in applications, which allows precise and complete detection of API usages in the code as well as creation of useful code implementing correct API usages. Two prominent RTE implementations in this category are framework-specific modeling languages and Spring Roo (Java).
Round-trip engineering is critical for maintaining consistency among multiple models and between the models and the code in Object Management Group 's (OMG) Model-driven architecture . OMG proposed the QVT (query/view/transformation) standard to handle model transformations required for MDA. To date [ when? ] , a few implementations of the standard have been created. (Need to present practical experiences with MDA in relation to RTE).
Code generation (forward-engineering) from models means that the user abstractly models solutions, which are connoted by some model data, and then an automated tool derives from the models parts or all of the source code for the software system. In some tools, the user can provide a skeleton of the program source code, in the form of a source code template where predefined tokens are then replaced with program source code parts during the code generation process.
UML (if used for MDA) diagrams specification was criticized for lack the detail which is needed to contain the same information as is covered with the program source. Some developers even claim that "the Code is the design". [ 5 ] [ 6 ]
There is a serious risk that the generated code will rapidly differ from the model or that the reverse-engineered model will lose its reflection on the code or a mix of these two problems as result of cycled reengineering efforts. [ 7 ]
Regarding behavioral/dynamic part of UML for features like statechart diagram there is no equivalents in programming languages. Their translation during code-generation will result in common programming statement (.e.g if,switch,enum ) being either missing or misinterpreted. If edited and imported back may result in different or incomplete model. [ 8 ] [ 9 ] The same goes for code snippets used for code generation stage for the pattern-implementation and user-specific logic: intermixed they may not be easily reverse-engineered back. [ 8 ] [ 9 ]
There is also general lack of advanced tooling for modelling that are comparable to that of modern IDEs (for testing, debugging, navigation, etc.) for general-purpose programming languages and domain-specific languages . [ 9 ]
Perhaps the most common form of round-trip engineering is synchronization between UML ( Unified Modeling Language ) models and the corresponding source code and entity–relationship diagrams in data modelling and database modelling .
Round-trip engineering based on Unified Modeling Language (UML) needs three basic tools for software development: [ citation needed ] | https://en.wikipedia.org/wiki/Round-trip_engineering |
A round number is an integer that ends with one or more " 0 "s (zero-digit) in a given base . [ 1 ] So, 590 is rounder than 592, but 590 is less round than 600. In both technical and informal language, a round number is often interpreted to stand for a value or values near to the nominal value expressed. For instance, a round number such as 600 might be used to refer to a value whose magnitude is actually 592, because the actual value is more cumbersome to express exactly. Likewise, a round number may refer to a range of values near the nominal value that expresses imprecision about a quantity. [ 2 ] Thus, a value reported as 600 might actually represent any value near 600, possibly as low as 550 or as high as 649, all of which would round to 600.
In decimal notation , a number ending in the digit "5" is also considered more round than one ending in another non-zero digit (but less round than any which ends with "0"). [ 2 ] [ 3 ] For example, the number 25 tends to be seen as more round than 24. Thus someone might say, upon turning 45, that their age is more round than when they turn 44 or 46. These notions of roundness are also often applied to non-integer numbers ; so, in any given base, 2.3 is rounder than 2.297, because 2.3 can be written as 2.300. Thus, a number with fewer digits which are not trailing "0"s is considered to be rounder than others of the same or greater precision.
Numbers can also be considered "round" in numbering systems other than decimal (base 10). For example, the number 1024 would not be considered round in decimal, but the same number ends with a zero in several other numbering systems including binary (base 2: 10000000000), octal (base 8: 2000), and hexadecimal (base 16: 400). The previous discussion about the digit "5" generalizes to the digit representing b /2 for base- b notation, if b is even .
Psychologically, round numbers form waypoints in pricing and negotiation. So, starting salaries are usually round numbers. Prices are often pitched just below round numbers to avoid breaking the psychological barrier of paying the price of the round number.
Round-number anniversaries are often especially celebrated. For example, a fiftieth birthday, the centenary of an event, or the millionth visitor or customer to a location or business.
On January 1, 2000, the round-number year 2000 was widely celebrated . Technically, the 3rd millennium did not begin until January 1, 2001, a year later, as there is no year zero in the Gregorian calendar .
Round number bias is the psychological tendency to prefer round numbers over others, [ 4 ] [ 5 ] which is passed onto a person through socialization. [ 6 ] Round numbers are also easier for a person to remember, process, and perform mathematical operations on. [ 5 ]
Round number bias has been observed mainly in retail and grocery , where prices are often just slightly less than a rounded number (ex. $9.99 or $9.95), in investments, including crowdfunding , in the real estate market through mortgages , and number milestones . [ 7 ] [ 8 ] [ 9 ] [ 10 ]
Round numbers are often used when estimating the time taken to complete a task. [ 11 ]
A round number is mathematically defined as an integer which is the product of a considerable number of comparatively small factors [ 12 ] [ 13 ] as compared to its neighboring numbers, such as 24 = 2 × 2 × 2 × 3 (4 factors, as opposed to 3 factors for 27; 2 factors for 21, 22, 25, and 26; and 1 factor for 23). | https://en.wikipedia.org/wiki/Round_number |
The Roundabout ( Robo ) family of proteins are single-pass transmembrane receptors that are highly conserved across many branches of the animal kingdom, from C. elegans to humans. [ 1 ] They were first discovered in Drosophila , through a mutant screen for genes involved in axon guidance . The Drosophila roundabout mutant was named after its phenotype, which resembled the circular traffic junctions (see roundabout ). [ 2 ] The Robo receptors are most well known for their role in the development of the nervous system, where they have been shown to respond to secreted Slit ligands. [ 2 ] [ 3 ] [ 4 ] One well-studied example is the requirement for Slit-Robo signaling in regulation of axonal midline crossing. Slit-Robo signaling is also critical for many neurodevelopmental processes including formation of the olfactory tract , the optic nerve , and motor axon fasciculation . [ 5 ] [ 6 ] In addition, Slit-Robo signaling contributes to cell migration and the development of other tissues such as the lung, kidney, liver, muscle and breast. [ 7 ] [ 8 ] Mutations in Robo genes have been linked to multiple neurodevelopmental disorders in humans.
A large-scale screen of the Drosophila genome for mutants that exhibited axon guidance defects led to the discovery of the roundabout (robo) mutation. [ 9 ] In robo mutants, axons were observed to inappropriately cross and recross the midline. It was subsequently found that the secreted protein Slit was the ligand for the Roundabout receptor. [ 10 ] Vertebrate Slit proteins were identified shortly after, and were shown to bind both vertebrate and Drosophila Robo receptors and to mediate axonal repulsion of spinal cord explants. [ 4 ] It was several more years before a functional analysis of the vertebrate Slit and Robo mutants was performed; this analysis demonstrated that Slit-Robo signaling regulates commissural axon guidance in vertebrates as well. [ 11 ] While the vertebrate receptors Robo1 and Robo2 signal repulsion in response to Slit to prevent inappropriate midline crossing, a novel function for Robo3/Rig1 was discovered; unlike the other Robo receptors, it is required to promote midline crossing. [ 12 ]
Phylogenetic analysis reveals that all Robo receptors have evolved from a common ancestral protein, with many subsequent diversification events occurring independently in different lineages. [ 1 ] The Robo gene was initially identified in Drosophila and has since been cloned in various species including mice and humans. [ 13 ] Drosophila have three Robo receptors: Robo1, Robo2, and Robo3. [ 14 ] [ 15 ] In vertebrates, four Robo receptors have been identified: Robo1 , Robo2 , Robo3 /Rig-1, and Robo4 /Magic Roundabout. [ 16 ]
In humans, Robo1 and Robo2 are located on chromosome 3p 12.3, while Robo3 and Robo4 are found on chromosome 11p24.2. In mice, the corresponding Robo genes 1 and 2 are found on chromosome 16 and Robo genes 3 and 4 are located on chromosome 9.
In vertebrates, Robo1 undergoes complex alternative splicing , generating several isoforms including DUTT1, a variant that has been identified as a tumor suppressor gene . [ 17 ] Vertebrate Robo3/Rig1 is also alternatively spliced; its two splice products are expressed at different times during commissural axon guidance, and have opposing activities. [ 18 ]
In humans, Robo1 is expressed generally throughout the central nervous system. [ 17 ] Robo2 is enriched in most regions of the adult and fetal brain, as well as in the adult ovary. Intermediate expression of Robo2 is seen in the fetal liver and adult lung, kidney, spleen, testes, and spinal cord. [ 19 ] Robo3/Rig1 is found in the hindbrain and spinal cord. [ 20 ] Robo4 is expressed in the heart, liver, lungs, kidney, muscle, small intestine, endothelial cells, and largely in the placenta. [ 21 ]
Each member of the Robo family has a similar structure, consisting of five immunoglobulin-like domains , three fibronectin type III (FN3) repeats, a transmembrane domain, and a cytoplasmic domain with up to four conserved motifs (CC0-3). In all identified Robo receptors except for vertebrate Robo4, the Ig1 and Ig2 domains have been evolutionarily conserved and are crucial for binding to Slit ligands. Robo4 is unusual as it only contains two Ig and FN3 domains. However, recent research proposes that the vertebrate Slit2 protein can in fact bind to Robo4. [ 22 ]
In bilaterian animals, including insects and mammals, most axons in the CNS cross the midline during nervous system development. The Robo proteins are critical regulators of midline crossing across species. In Drosophila embryos, Robo1 and Robo2 are required to keep ipsilaterally projecting axons from inappropriately crossing the midline, and to prevent contralateral axons from remaining stuck at the midline. Robo3, while it also binds Slit, does not appear to play a major role in regulating midline crossing. Instead, it is required for the lateral pathway selection of axons after crossing. [ 14 ] Robo2 also contributes to lateral pathway formation.
In the vertebrate spinal cord, Robo1 and Robo2 are expressed on commissural axons and act as repulsive receptors for the Slit ligands expressed by floor plate cells located at the midline. [ 11 ] In contrast, Robo3/Rig1 is required for midline crossing, and acts in part by antagonizing Slit-mediated repulsion by Robo1 and Robo2. [ 23 ]
Robo receptors have also been shown to be crucial regulators of many other axon pathfinding decisions during development, including the projection of axons in the optic tract and the olfactory epithelium. [ 5 ] [ 6 ]
The Robo gene family contributes to the guidance and migration of non-neural cells, including neuronal precursor cells, muscle cells, tracheal cells, Langerhans cells , and vascular smooth muscle cells.
Robo1 is thought to play a role in the inhibition of glioma invasion and migration. Glioblastoma cells grow away from areas that contain high concentrations of Slit2 and its receptor Robo1, suggesting that the Robo1/Slit2 complex can serve as a chemorepellent for glioma cells, inhibiting the invasion and migration of the tumor cells.
The binding of Slit to Robo receptors leads to reorganization of the actin cytoskeleton . Actin polymerization is regulated by several adaptor proteins that can bind to the cytoplasmic motifs of the Robo receptors. In Drosophila , several signaling proteins downstream of Robo1 have been identified, including Enabled, Son of Sevenless (SOS), Rac , and Dock . [ 24 ] [ 25 ] [ 26 ] It is thought that activation of Robo1 by Slit leads to increased depolymerization of actin, resulting in growth cone collapse. It remains unclear how Drosophila Robo2 and Robo3 signal, although multiple studies suggest that they have distinct signaling capabilities that cannot be recapitulated by Robo1. [ 27 ] [ 28 ]
The vertebrate Robo3/Rig1 homolog is a more distant relative of the Robo gene family, and is thought to play a distinct role in axonal guidance. [ 16 ] [ 29 ] Robo3/Rig1 is alternatively spliced to generate a protein that inhibits Robo1/2-mediated repulsion, effectively leading to the promotion of midline crossing. [ 23 ] The exact mechanism by which Robo3 achieves this anti-repulsive activity is unknown. [ 29 ]
The Robo4 receptor has been linked to angiogenesis in both mice and zebrafish . It is also present in human microvascular endothelial cells (HMVEC) and human umbilical vein endothelial cells ( HUVEC ). Exposure of Robo4 to Slit2 inhibits angiogenesis. However, exposure to a protein that inhibits Slit2 also inhibits angiogenesis. [ 30 ] Due to these inconclusive results, the role of Robo4 in blood vessel growth is not completely understood.
Robo1 has been linked to cancerous tumor growth and suppression. The Slit2/Robo1 pathway has been associated with tumor angiogenesis, leading to subsequent tumor growth. Slit2 proteins have been identified in several varieties of tumors, including melanoma , breast cancer , small cell lung cancer , and bladder cancer. Furthermore, inhibition of the Slit2/Robo1 pathway via R5 and RoboN reduced tumor mass and volume, while also reducing microvessel density. [ 31 ] However, Slit2 proteins have not been identified in all kinds of tumors, and other research suggests that Slit-2 expression may suppress tumors in small cell lung cancer and breast cancer. [ 30 ]
The Robo1 protein is thought to be associated with dyslexia , possibly through chromosomal translocation . [ 32 ] The role of Robo1 in regards to dyslexia is not fully understood at this time.
Recently, a genome-wide linkage study by Viding and colleagues (2010)reported that the Robo2 gene could be involved in developmental disorders such as psychopathy.
A defect in the Robo3/Rig1 protein results in horizontal gaze palsy with progressive scoliosis (HGPPS), a rare genetic disorder. HGPPS is characterized by a lack of horizontal eye movement within the socket (although vertical movement remains unaffected) and the gradual curvature of the spine throughout development. [ 33 ] [ 34 ] The disorder is caused by a genetic mutation on chromosome 11, and is autosomal recessive . [ 35 ] During normal brain development, Robo3/Rig1 decreases sensitivity of Robo1 to Slit proteins, allowing the axon to grow past the midline . [ 34 ] This process allows axons to cross to the other side of the brain, which is crucial for motor function as well as sensory processing. In patients with HGPPS, the absence of Robo3/Rig1 prevents axons in the corticospinal tract and the trochlear nerve [ 33 ] from growing past the midline. This abnormal growth of the hindbrain and spinal cord manifests itself as the symptoms associated with HGPPS.
[ 36 ] | https://en.wikipedia.org/wiki/Roundabout_family |
A roundel is a circular disc used as a symbol. The term is used in heraldry , but also commonly used to refer to a type of national insignia used on military aircraft , generally circular in shape and usually comprising concentric rings of different colours. Other symbols also often use round shapes.
In heraldry , a roundel is a circular charge . Roundels are among the oldest charges used in coats of arms , dating from at least the twelfth century. Roundels in British heraldry have different names depending on their tincture . [ 2 ] Thus, while a roundel may be blazoned by its tincture, e.g., a roundel vert (literally "a roundel green"), it is more often described by a single word, in this case pomme (literally "apple", from the French) or, from the same origins, pomeis —as in "Vert; on a cross Or five pomeis" (a green field with a golden/yellow cross on which are drawn five green roundels/circles). [ 3 ]
One special example of a named roundel is the fountain , depicted as a roundel barry wavy argent and azure , that is, containing alternating horizontal wavy bands of blue and silver (or white).
The French Air Service originated the use of roundels on military aircraft during the First World War . [ 1 ] The chosen design was the French national cockade , whose colours are the blue-white-red of the flag of France . Similar national cockades, with different ordering of colours, were designed and adopted as aircraft roundels by their allies, including the British Royal Flying Corps and Royal Naval Air Service , and (in the last few months of the war) the United States Army Air Service . After the First World War, many other air forces adopted roundel insignia, distinguished by different colours or numbers of concentric rings.
The term "roundel" is often used even for those military aircraft insignia that are not round, like the Iron Cross - Balkenkreuz symbol of the Luftwaffe or the red star of the Russian Air Force . [ citation needed ]
Among national flags which display a roundel are the flags of Bangladesh , Belize , Brazil , Burundi , Dominica , Ethiopia , Grenada , India , Japan , Kazakhstan , Kyrgyzstan , Laos , Mongolia , Namibia , Niger , North Korea , North Macedonia , Palau , Paraguay , Rwanda , South Korea , Republic of China (Taiwan) , Tunisia , and Uganda .
Flags for British Overseas Territories used a British Blue Ensign defaced with a roundel displaying the arms or badge of the dependency until 1999. The same pattern is still used for all the states of Australia except Victoria .
Some of the design elements that appear in logos that utilize roundels include variables such as harmony, balance, symmetry, proportion, and circularity, as established by Pamela W. Henderson & Joseph A. Cote [ 4 ] However, for a simple logo, such as the Target Logo , to become associated with the brand, the brand needs to be well known and have unique branding. [ 5 ]
Some corporations and organizations make use of roundels in their branding. | https://en.wikipedia.org/wiki/Roundel |
Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle . Roundness applies in two dimensions , such as the cross sectional circles along a cylindrical object such as a shaft or a cylindrical roller for a bearing . In geometric dimensioning and tolerancing , control of a cylinder can also include its fidelity to the longitudinal axis, yielding cylindricity . The analogue of roundness in three dimensions (that is, for spheres ) is sphericity .
Roundness is dominated by the shape's gross features rather than the definition of its edges and corners, or the surface roughness of a manufactured object. A smooth ellipse can have low roundness, if its eccentricity is large. Regular polygons increase their roundness with increasing numbers of sides, even though they are still sharp-edged.
In geology and the study of sediments (where three-dimensional particles are most important), roundness is considered to be the measurement of surface roughness and the overall shape is described by sphericity.
The ISO definition of roundness is the ratio of the radii of inscribed to circumscribed circles , i.e. the maximum and minimum sizes for circles that are just sufficient to fit inside and to enclose the shape. [ 1 ] [ 2 ]
Having a constant diameter , measured at varying angles around the shape, is often considered to be a simple measurement of roundness. This is misleading. [ 3 ]
Although constant diameter is a necessary condition for roundness, it is not a sufficient condition for roundness: shapes exist that have constant diameter but are far from round. Mathematical shapes such as the Reuleaux triangle and, an everyday example, the British 50p coin demonstrate this.
Roundness does not describe radial displacements of a shape from some notional centre point, [ note 1 ] merely the overall shape.
This is important in manufacturing, such as for crankshafts and similar objects, where not only the roundness of a number of bearing journals must be measured, but also their alignment on an axis. A bent crankshaft may have perfectly round bearings, yet if one is displaced sideways, the shaft is useless. Such measurements are often performed by the same techniques as for roundness, but also considering the centre position and its relative position along an additional axial direction.
A single trace covering the full rotation is made and at each equally spaced angle, θ i {\displaystyle \theta _{i}} , a measurement, R i {\displaystyle R_{i}} , of the radius or distance between the center of rotation and the surface point. A least-squares fit to the data gives the following estimators of the parameters of the circle: [ 4 ]
The deviation, sometimes referred to as noncircularity , is then measured as:
Roundness measurement is very important in metrology . It includes measurement of a collection of points.
For this two fundamental methods exist: intrinsic or extrinsic.
The intrinsic method is limited to small deformations only. For large deformations extrinsic method has to be followed. In this case the datum is not a point or set of points on the object, but is a separate precision bearing usually on the measuring instrument. The axis of the object or part of the object to be measured is aligned with the axis of the bearing. Then a stylus from the instrument is just made to touch the part to be measured. A touch sensor connected to the tip of the stylus makes sure that the stylus just touches the object. A minimum of three readings are taken and an amplified polar plot is drawn to get the required error.
A common definition used in digital image processing (image analysis) for characterizing 2-D shapes is:
This ratio will be 1 for a circle and greater than 1 for non-circular shapes. Another definition is the inverse of that:
which is 1 for a perfect circle and goes down as far as 0 for highly non-circular shapes. | https://en.wikipedia.org/wiki/Roundness |
Roundup is a brand name of herbicide originally produced by Monsanto , which Bayer acquired in 2018. Prior to the late-2010s formulations, it used broad-spectrum glyphosate-based herbicides . [ 1 ] As of 2009, sales of Roundup herbicides still represented about 10 percent of Monsanto's revenue despite competition from Chinese producers of other glyphosate-based herbicides. [ 2 ] The overall Roundup line of products represented about half of Monsanto's yearly revenue in 2009. [ 3 ] The product is marketed to consumers by Scotts Miracle-Gro Company . [ 4 ] In the late-2010s other non-glyphosate containing herbicides were also sold under the Roundup brand. [ 5 ] [ 6 ]
Monsanto patented the herbicidal use of glyphosate and derivatives in 1971. [ 7 ] Commercial sale and usage in significant quantities started in 1974. [ 8 ] It retained exclusive rights to glyphosate in the US until its US patent expired in September 2000; in other countries the patent expired earlier. The Roundup trademark is registered with the US Patent and Trademark Office and still extant. However, glyphosate is no longer under patent, so similar products use it as an active ingredient. [ 9 ]
The main active ingredient of Roundup is the isopropylamine salt of glyphosate. Another ingredient of Roundup is the surfactant POEA ( polyethoxylated tallow amine ).
Monsanto also produced seeds which grow into plants genetically engineered to be tolerant to glyphosate, which are known as Roundup Ready crops. The genes contained in these seeds are patented. Such crops allow farmers to use glyphosate as a post-emergence herbicide against most broadleaf and cereal weeds.
The health impacts of the product as well as its effects on the environment have been at the center of substantial legal and scientific controversies. In June 2020, Bayer agreed to pay $9.6 billion to settle tens of thousands of claims, mostly alleging that glyphosate-based Roundup had caused cancer. [ 10 ] [ 11 ]
Glyphosate-based formulations may contain a number of adjuvants , the identities of which may be proprietary. [ 12 ] Surfactants are used in herbicide formulations as wetting agents, to maximize coverage and aid penetration of the herbicide(s) through plant leaves. As agricultural spray adjuvants, surfactants may be pre-mixed into commercial formulations or they may be purchased separately and mixed on-site. [ 13 ]
Polyethoxylated tallow amine (POEA) is a surfactant used in the original Roundup formulation and was commonly used in 2015. [ 14 ] Different versions of Roundup have included different percentages of POEA. A 1997 US government report said that Roundup is 15% POEA while Roundup Pro is 14.5%. [ 15 ] Since POEA is more toxic to fish and amphibians than glyphosate alone, POEA is not allowed in aquatic formulations. [ 16 ] [ 15 ] [ 17 ]
Non-glyphosate formulations of Roundup are typically used for lawns that glyphosate would otherwise kill. Both type of products being sold under the Roundup brand name can be a source of confusion for consumers. [ 6 ] Active ingredients for non-glyphosate formulations of Roundup can include MCPA , quinclorac , dicamba , and sulfentrazone , penoxsulam , and 2,4-D [ 5 ] [ 6 ]
The lethal dose of different glyphosate-based formulations varies, especially with respect to the surfactants used. Formulations intended for terrestrial use that include the surfactant POEA can be more toxic than other formulations for aquatic species. [ 18 ] [ 19 ] Due to the variety in available formulations, including five different glyphosate salts and different combinations of inert ingredients, it is difficult to determine how much surfactants contribute to the overall toxicity of each formulation. [ 20 ] [ 21 ] Independent scientific reviews and regulatory agencies have repeatedly concluded that glyphosate-based herbicides do not lead to a significant risk for human or environmental health when the product label is properly followed. [ 22 ]
The acute oral toxicity for mammals is low, [ 18 ] but death has been reported after deliberate overdose of concentrated Roundup. [ 23 ] The surfactants in glyphosate formulations can increase the relative acute toxicity of the formulation. [ 21 ] Surfactants generally do not, however, cause synergistic effects (as opposed to additive effects) that increase the acute toxicity of glyphosate within a formulation. [ 21 ] The surfactant POEA is not considered an acute toxicity hazard, and has an oral toxicity similar to vitamin A and less toxic than aspirin . [ 24 ] Deliberate ingestion of Roundup ranging from 85 to 200 ml (of 41% solution) has resulted in death within hours of ingestion, although it has also been ingested in quantities as large as 500 ml with only mild or moderate symptoms. [ 25 ] Consumption of over 85 ml of concentrated product is likely to cause serious symptoms in adults, including burns due to corrosive effects as well as kidney and liver damage. More severe cases lead to "respiratory distress, impaired consciousness, pulmonary edema , infiltration on chest X-ray, shock, arrhythmias, kidney failure requiring haemodialysis, metabolic acidosis, and hyperkalaemia" and death is often preceded by bradycardia and ventricular arrhythmias . [ 21 ]
Skin exposure can cause irritation, and photocontact dermatitis has been occasionally reported. Severe skin burns are very rare. [ 21 ] In a 2017 risk assessment, the European Chemicals Agency (ECHA) wrote: "There is very limited information on skin irritation in humans. Where skin irritation has been reported, it is unclear whether it is related to glyphosate or co-formulants in glyphosate-containing herbicide formulations." The ECHA concluded that available human data was insufficient to support classification for skin corrosion or irritation. [ 26 ]
Inhalation is a minor route of exposure, but spray mist may cause oral or nasal discomfort, an unpleasant taste in the mouth, or tingling and irritation in the throat. Eye exposure may lead to mild conjunctivitis. Superficial corneal injury is possible if irrigation is delayed or inadequate. [ 21 ]
Glyphosate formulations with POEA, such as Roundup, are not approved for aquatic use due to aquatic organism toxicity. [ 16 ] Due to the presence of POEA, glyphosate formulations only allowed for terrestrial use are more toxic for amphibians and fish than glyphosate alone. [ 16 ] [ 15 ] [ 17 ] Terrestrial glyphosate formulations that include the surfactants POEA and MON 0818 (75% POEA) may have negative impacts on various aquatic organisms like protozoa , mussels , crustaceans , frogs and fish . [ 18 ] Aquatic organism exposure risk to terrestrial formulations with POEA is limited to drift or temporary water pockets. [ 16 ] While laboratory studies can show effects of glyphosate formulations on aquatic organisms, similar observations rarely occur in the field when instructions on the herbicide label are followed. [ 22 ]
Studies in a variety of amphibians have shown the toxicity of products containing POEA to amphibian larvae. These effects include interference with gill morphology and mortality from either the loss of osmotic stability or asphyxiation. At sub-lethal concentrations, exposure to POEA or glyphosate/POEA formulations have been associated with delayed development, accelerated development, reduced size at metamorphosis , developmental malformations of the tail, mouth, eye and head, histological indications of intersex and symptoms of oxidative stress. [ 17 ] Glyphosate-based formulations can cause oxidative stress in bullfrog tadpoles. [ 27 ] The use of glyphosate-based pesticides are not considered the major cause of amphibian decline, the bulk of which occurred prior to widespread use of glyphosate or in pristine tropical areas with minimal glyphosate exposure. [ 28 ]
A 2000 review of the toxicological data on Roundup concluded that "for terrestrial uses of Roundup minimal acute and chronic risk was predicted for potentially exposed nontarget organisms". It also concluded that there were some risks to aquatic organisms exposed to Roundup in shallow water. [ 29 ]
Roundup Ready‐To‐Use, Roundup No Glyphosate, and Roundup ProActive have all been found to cause significant mortality in bumblebees when sprayed directly on them. It has been hypothesized that this is due to surfactants in the formulations blocking the tracheal system of the bees. [ 30 ]
There is limited evidence that human cancer risk might increase as a result of occupational exposure to large amounts of glyphosate, such as agricultural work, but no good evidence of such a risk from home use, such as in domestic gardening. [ 31 ] The consensus among national pesticide regulatory agencies and scientific organizations is that labeled uses of glyphosate have demonstrated no evidence of human carcinogenicity . [ 32 ] Organizations such as the Joint FAO / WHO Meeting on Pesticide Residues and the European Commission , Canadian Pest Management Regulatory Agency , and the German Federal Institute for Risk Assessment [ 33 ] have concluded that there is no evidence that glyphosate poses a carcinogenic or genotoxic risk to humans. The final assessment of the Australian Pesticides and Veterinary Medicines Authority in 2017 was that "glyphosate does not pose a carcinogenic risk to humans". [ 34 ] The EPA has evaluated the carcinogenic potential of glyphosate multiple times since 1986. In 1986, glyphosate was initially classified as Group C: "Possible Human Carcinogen", but later recommended as Group D: "Not Classifiable as to Human Carcinogenicity" due to lack of statistical significance in previously examined rat tumor studies. In 1991, it was classified as Group E: "Evidence of Non-Carcinogenicity for Humans", and in 2015 and 2017, "Not Likely to be Carcinogenic to Humans". [ 35 ] [ 36 ]
One international scientific organization, the International Agency for Research on Cancer ( IARC ), classified glyphosate in Group 2A , "probably carcinogenic to humans" in 2015. [ 27 ] The variation in classification between this agency and others has been attributed to "use of different data sets" and "methodological differences in the evaluation of the available evidence". [ 32 ] In 2017, California environmental regulators listed glyphosate as “known to the state to cause cancer.” The state's Office of Environmental Health Hazard Assessment made the decision based in part on the report from the IARC . State Proposition 65 requires the state office to add substances the international agency deems carcinogenic in humans or laboratory animals to a state list of cancer-causing items. [ 37 ]
In the ten months following Bayer's June 2018 acquisition of Monsanto, its stock lost 46% of its value because of investor apprehension concerning the 11,200 lawsuits filed against its subsidiary. [ 38 ] As of 2023, around 165,000 claims have been made against Bayer, mostly alleging that Roundup had caused cancer. [ 11 ] Bayer has settled tens of thousands of those claims and has agreed to pay billions in damages, but, as of 2023, more than 50,000 similar claims were still pending. [ 11 ] In December 2023, Bayer won a case against a claim that Roundup had caused a man's cancer. In a statement they said the outcome was "consistent with the evidence in this case that Roundup does not cause cancer and is not responsible for the plaintiff's illness". At that time, Bayer had previously won 10 of 15 such cases. [ 11 ]
Most cases claiming injury from Roundup are based on a failure-to-warn theory of liability, meaning Monsanto is liable for a plaintiff's injury because it failed to warn the plaintiff that Roundup can cause cancer. The United States Court of Appeals for the Ninth Circuit in 2021, and the United States Court of Appeals for the Eleventh Circuit in early 2024, held that such state-law failure to warn claims were not preempted by the Federal Insecticide, Fungicide, and Rodenticide Act ("FIFRA"). In August 2024, however, the United States Court of Appeals for the Third Circuit held that FIFRA does preempt state-law failure to warn claims involving Roundup, expressly recognizing that its holding conflicts with that of the Ninth and Eleventh Circuits. [ 39 ] [ 40 ] This conflict among the Third, Ninth, and Eleventh Circuit creates a heightened potential that the United States Supreme Court will review the Third Circuit's decision so that the Supreme Court can resolve the conflict among the Courts of Appeals. [ 41 ] [ 42 ]
As of October 30, 2019, there were more than 42,000 plaintiffs who said that glyphosate herbicides caused their cancer. [ 43 ] After the IARC classified glyphosate as "probably carcinogenic to humans" in March 2015, [ 27 ] [ 44 ] many state and federal lawsuits were filed in the United States. Early on, over 300 of them were consolidated into a multidistrict litigation called In re: RoundUp Products Liability . [ 45 ]
On August 10, 2018, Dewayne Johnson , who has non-Hodgkin's lymphoma , was awarded $289 million in damages (later cut to $78 million on appeal [ 46 ] then reduced to $21 million after another appeal [ 47 ] ) after a jury in San Francisco found that Monsanto had failed to adequately warn consumers of cancer risks posed by the herbicide. [ 48 ] [ 49 ] Johnson had routinely used two different glyphosate formulations in his work as a groundskeeper, RoundUp and another Monsanto product called Ranger Pro. [ 50 ] [ 51 ] The jury's verdict addressed the question of whether Monsanto knowingly failed to warn consumers that RoundUp could be harmful, but not whether RoundUp causes cancer. [ 52 ] Court documents from the case alleged the company's efforts to influence scientific research via ghostwriting . [ 53 ]
In January 2019, Costco decided to stop carrying Roundup or other glyphosate-based herbicides. The decision was reportedly influenced in part by the public court cases. [ 54 ]
In March 2019, a man was awarded $80 million (later cut to $26 million on appeal [ 55 ] ) in a lawsuit claiming Roundup was a substantial factor in his cancer. [ 56 ] [ 57 ] U.S. District Judge Vince Chhabria stated that a punitive award was appropriate because the evidence "easily supported a conclusion that Monsanto was more concerned with tamping down safety inquiries and manipulating public opinion than it was with ensuring its product is safe." Chhabria stated that there was evidence on both sides as to whether glyphosate causes cancer, and that the behavior of Monsanto showed "a lack of concern about the risk that its product might be carcinogenic." [ 55 ]
On May 13, 2019, a jury in California ordered Bayer to pay a couple $2 billion in damages (later cut to $87 million on appeal [ 58 ] ) after finding that the company had failed to adequately inform consumers of the possible carcinogenicity of Roundup. [ 59 ] On December 19, 2019, it was announced that Timothy Litzenburg, the lawyer for the RoundUp Virginia plaintiffs had been charged with extortion after offering to stop searching for more plaintiffs if he was paid a $200 million consulting fee by a manufacturer of glyphosate. [ 60 ] [ 61 ] [ 62 ] Litzenburg and his partner Daniel Kincheloe pleaded guilty to the charges and they were sentenced to two and one years in prison respectively. [ 63 ]
In June 2020, Bayer agreed to settle more than a hundred thousand Roundup lawsuits, agreeing to pay $8.8 to $9.6 billion to settle those claims, and $1.5 billion for any future claims. The settlement does not include three cases that have already gone to jury trials and are being appealed. [ 64 ] However the settlement was not allowed to cover future cases. [ 11 ]
In the 2020s, facing billions of dollars in more claims, Bayer lobbied the U.S. Congress and state legislatures to change legal standards for pesticide labeling in an attempt to reduce its liability. [ 65 ] As of 2024 [update] , the US EPA planned to reevaluate regulations in 2026. [ 65 ]
In May 2025, Bayer announced that it was making another push to settle the pending lawsuits and that it would consider a Chapter 11 bankruptcy for its Monsanto division if the settlement plan was not successful. [ 66 ] The company engaged law firms Latham & Watkins and AlixPartners to review its options. [ 66 ]
In 1996, Monsanto was accused of false and misleading advertising of glyphosate products, prompting a lawsuit by the New York State attorney general. [ 67 ] Monsanto had made claims that its spray-on glyphosate based herbicides, including Roundup, were safer than table salt and "practically non-toxic" to mammals, birds, and fish, "environmentally friendly", and "biodegradable". [ 68 ] Citing avoidance of costly litigation, Monsanto settled the case, admitting no wrongdoing, and agreeing to remove the offending advertising claims in New York State. [ 68 ]
Environmental and consumer rights campaigners brought a case in France in 2001 accusing Monsanto of presenting Roundup as "biodegradable" and claiming that it "left the soil clean" after use; glyphosate, Roundup's main ingredient, was classed by the European Union as "dangerous for the environment" and "toxic for aquatic organisms". In January 2007, Monsanto was convicted of false advertising and fined 15,000 euros. The result was confirmed in 2009. [ 69 ] [ 70 ]
On 27 March 2020 Bayer settled claims in a proposed class action alleging that it falsely advertised that the active ingredient in Roundup Weed & Grass Killer only affects plants with a $39.5 million deal that included changing the labels on its products. [ 71 ]
In June 2023, Bayer reached a $6.9 million settlement agreement with the New York attorney general, settling false advertising allegations concerning the safety of Roundup. [ 72 ]
Some tests originally conducted on glyphosate by contractors were later found to have been fraudulent, along with tests conducted on other pesticides. Concerns were raised about toxicology tests conducted by Industrial Bio-Test Laboratories in the 1970s [ 73 ] and Craven Laboratories was found to have fraudulently analysed samples for residues of glyphosate in 1991. [ 74 ] Monsanto has stated that the studies have since been repeated. [ 75 ]
In January 2019, Roundup Pro 360 was banned in France following a Lyon court ruling that regulator ANSES had not given due weight to safety concerns when they approved the product in March 2017. The ban went into effect immediately. The court's decision cited research by the IARC , based in Lyon. [ 76 ] [ 77 ]
Monsanto first developed Roundup in the 1970s. End-users initially used it in a similar way to paraquat and diquat – as a non-selective herbicide. Application of glyphosate-based herbicides to row crops resulted in problems with crop damage and kept them from being widely used for this purpose. In the United States, use of Roundup experienced rapid growth following the commercial introduction of a glyphosate-resistant soybean in 1996. [ 78 ] "Roundup Ready" became Monsanto's trademark for its patented line of crop seeds that are resistant to Roundup. Between 1990 and 1996 sales of Roundup increased around 20% per year. [ 79 ] As of 2015 [update] the product was used in over 160 countries. [ 80 ] Roundup is used most heavily on corn, soy, and cotton crops that have been genetically modified to withstand the chemical, but as of 2012 [update] glyphosate treated approximately 5 million acres in California for crops like almond , peach , cantaloupe , onion , cherry , sweet corn , and citrus , [ 81 ] although the product is only applied directly to certain varieties of sweet corn. | https://en.wikipedia.org/wiki/Roundup_(herbicide) |
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